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String theory
Lecture notes – Fall 2018
Niko JokelaDecember 12, 2018
Contents
1 Introduction 3
2 The relativistic point particle 42.1 Polyakov action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 The relativistic string 63.1 Polyakov action for a string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Symmetries of the Polyakov action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Gauge fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Noether currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.6 Example: rotating string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.7 Light-cone coordinates on the worldsheet . . . . . . . . . . . . . . . . . . . . . . . . 143.8 Closed strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.9 Hamiltonian analysis of the closed string . . . . . . . . . . . . . . . . . . . . . . . . . 163.10 Open strings with Neumann-Neumann boundary conditions . . . . . . . . . . . . . . 183.11 SL(2,R) subalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 Quantization of the bosonic string 204.1 Light-cone quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Spectrum of open strings in the light-cone gauge . . . . . . . . . . . . . . . . . . . . 244.3 Comments on tachyons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.4 Comments on critical dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.5 Comments of Lorentz symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.6 Covariant quantization of open string . . . . . . . . . . . . . . . . . . . . . . . . . . 294.7 Closed string spectrum (in the light-cone gauge) . . . . . . . . . . . . . . . . . . . . 32
5 CFT in 2D 345.1 Scalar field in d dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2 Conformal symmetry in d = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.3 Primary fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.4 Ward identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6 Operator formalism 416.1 Equal-time commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.2 Mode expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.2.1 State-operator correspondence for primaries recap . . . . . . . . . . . . . . . 50
7 Closed strings in complex variables 507.1 Scalar primary state created by the vertex operator . . . . . . . . . . . . . . . . . . . 507.2 Next level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
8 String scattering 518.1 Tree level closed string amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538.2 4-point tachyon amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538.3 Regge behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548.4 Soft high energy behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548.5 Open string scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
9 Strings in background fields 579.1 Sigma model expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579.2 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589.3 S-matrix picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
1
9.4 Comments: two expansions: gs and α′ . . . . . . . . . . . . . . . . . . . . . . . . . . 60
10 Superstring theory 6110.1 Classical RNS action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6210.2 Boundary conditions: Ramond vs. Neveu-Schwarz . . . . . . . . . . . . . . . . . . . 6310.3 Open strings and doubling trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6410.4 Light-cone gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6510.5 Superstring spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6510.6 GSO projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6710.7 Modular invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6710.8 Twisted partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6910.9 Super-spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
11 Supergravity 7411.1 D=11 SUGRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7511.2 Down to D=10 SUGRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
11.2.1 Coupling RR fields to extended objects . . . . . . . . . . . . . . . . . . . . . 7611.2.2 Electric-magnetic duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7911.2.3 Type IIA SUGRA action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7911.2.4 Type IIB SUGRA action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8011.2.5 Black p-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
12 Compactification and dualities 8112.1 KK compactification in field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8112.2 KK compactification of bosonic closed string . . . . . . . . . . . . . . . . . . . . . . 8312.3 (Microscopic) T-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8512.4 Type IIA/B with T-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8612.5 T-duality for open strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8612.6 S-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8812.7 String dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2
Editorial note This is a course given on a black board. The following text is the write-up (thanksto Miika Sarkkinen!) but does not include the many figures from the black board. The wholecourse is 14×2 hours and it has weekly exercises. While most of the contents are identical to thosein the notes of Keski-Vakkuri & Hassan, I have decided to cover some parts in different order,maintain slightly different convention (e.g. α′ = l2s always), skip some material and replace themwith something else. Also, many ideas and discussions from various lecture notes/books have beenrecycled in these notes.
As this note is being updated during the course, the numbering of the equations and the sectioningmost likely will undergo changes, so please do not go and print this out if not necessary.
1 Introduction
We give a quick introduction to string theory below. If something is not understood, that is ok, asmany of the statements will become clear as we progress during the course. The bullet point listhere is meant to luring in interested readers to study through the course.
• String theory (ST) is a quantum theory of 1D objects: (fig of closed and open strings).
• It can be defined as a (1+1)d QFT with SWS =´d2σLstring. Since there are many such
QFTs, there are also many STs.
• Fundamental strings are infinitely thin filaments of energy.
• Spin-2 massless mode is an excitation of a closed string. For couplings to be consistent thisexcitation mode must be a graviton.
• Open strings can join their endpoints, so open STs always contain closed strings. Thus ST isa theory of quantum gravity.
• It is also finite.
• In ST spacetime is treated as a target space of quantum fields. Consistency requires thatD = 26 (bosonic) or D = 10 (superstrings). Some other D > 3 + 1 are also possible, but wewill not discuss them in this course.
• Extra dimensions do not necessarily have the same meaning as the ordinary ones in D = 3 + 1Minkowski space. Instead, extra dimensions should be regarded as a configuration space (likein classical mechanics).
• Metric on the target space obeys Einstein equations!
• Open strings contain non-Abelian gauge fields and chiral fermions. ST is naturally supersym-metrized, so one might hope that it could be the UV completion of the Standard Model or itsextensions.
Recall a toy model of scalars:
L =1
2(∂φ)2 − 1
2m2φ2 − λ4φ
4 − λ6
M2φ6 + ... (1.1)
interactions of λk with
• + VE mass dimension are renormalizable,
• 0 mass dimension are renormalizable,
• + VE mass dimension are non-renormalizable.
Recall that [GN ] = −2 implies that gravity is non-renormalizable, i.e. at high energies higher ordercorrections become important. GR is bad at the UV region. Consider a scattering process in GR
3
and QFT. In such a process we get micro BH at high energies at intermediate state. As a resultthe process is highly non-linear. Consider then a string scattering. Higher energy strings have moreoscillations than strings with a low energy. We just produce another ball of yarn larger than theSchwarzschild radius. This results in a good UV behavior!
ST has an extremely rich structure and has led to important insight into a range of topics:
• non-perturbative dualities
• gauge theories (especially at strong coupling)
• mathematics (algebraic geometry)
• black holes (micro-counting of entropy)
• holography (AdS/CFT)
• theories on branes (inflation, brane engineering).
In this course we are not going to talk about ST as a fundamental theory. Instead, we are going tofocus on background to arriving to the bonuses listed above.
2 The relativistic point particle
Consider a relativistic point particle in Minkowski space where the metric is ηµν = diag(−1, 1, . . . , 1).The position of the particle is denoted by Xµ(τ) where τ parametrizes the path of the particle. Hereτ = worldline coordinate of the particle. Thus the particle parametrizes the space in which it ismoving. We call it the target space.
The Lorentz invariant line element in Minkowski space is
ds =√−ηµνdXµdXν = dτ
√−ηµνdXµdXν , (2.1)
where Xµ ≡ dxµ/dτ .
Action
S =
ˆLdt = −m
ˆds = −m
ˆ τ1
τ0
dτ
√−ηµνdXµdXν (2.2)
=⇒ L = −m√−X2 , (2.3)
where we have written X2 ≡ XµXµ = ηµνdX
µdXν .
Momentum conjugate
pµ =∂L∂Xµ
=mXµ√−X2
=⇒ pµpµ =
mXµ√−X2
mXµ√−X2
= −m2 (2.4)
=⇒ pµpµ +m2 = 0. (2.5)
Equation of motion:
∂τ
[∂L∂Xµ
]= 0 =⇒ ∂τ
[mXµ√−X2
]= 0 (2.6)
The Hamiltonian:
H =∂L∂Xµ
Xµ − L = 0 (2.7)
4
Thus the canonical Hamiltonian vanishes and the system is completely governed by the constraints.The reason behind this is the invariance under arbitrary reparametrizations of τ . This is the analogueof general coordinate invariance of GR in 0 + 1 dimensions. Let us check this: make a transformationτ → τ ′(τ) with dτ ′
dτ > 0. As dτ ′ = dτ ′
dτ dτ , we have that
Xµ(τ ′) =dXµ
dτ ′=Xµ(τ)dτ ′
dτ
=⇒√−X2(τ ′) =
√−X2(τ)
|dτ ′/dτ |(2.8)
=⇒ dτ ′√−X2(τ ′) = dτ
√−X2(τ), (2.9)
just like it should be. This invariance can be fixed by choosing a “gauge”. We could, for example,fix the static gauge in which X0 = t ≡ τ . In this gauge Xµ = (1, ~v) =⇒ X2 = −1 + ~v2 =⇒L = −m
√1− ~v2, which is a standard form of the Lagrangian density of a point particle in special
relativity: ~p = ∂L
∂~v = m~v√1−~v2
E = ~v · ∂L~v − L = m~v√1−~v2
+m√
1− ~v2 = m√1−~v2
.(2.10)
These are just the standard dispersion relations of special relativity.
Let us now return to general gauge. We introduce a Lagrange multiplier N to incorporate theconstraint
pµpµ +m2 = 0. (2.11)
Thus we find that the Hamiltonian is
H =N
2m(p2 +m2). (2.12)
Impose then canonical Poisson bracket Xµ, pν = δµν . From the equation of motion it follows nowthat
Xµ = Xµ, H =N
2mXµ, pνp
ν =N
mpµ =
NXµ√−X2
(2.13)
=⇒ X2 = −N2, (2.14)
where N is arbitrary. Choosing N amounts to fixing a gauge symmetry. For N ∈ R, we have timeliketrajectories. If we choose N = 1 by choosing a scale for the parameter τ , then τ acts as the propertime in this gauge. Under this choice of N the equation of motion is
0 = ∂τ
[mXµ√−X2
]=⇒ Xµ = 0 =⇒ Xµ(τ) = x0 +
pµ
mτ, (2.15)
and the momentum reads as
pµ =mXµ√−X2
= mXµ =⇒ Xµ = pµ/m, (2.16)
which is the same relation as in non-relativistic theory.
There are some problems with the above action:
• Square root is undesirable (hard to quantize).
• As m→ 0, the action vanishes.
For these reasons we will introduce Polyakov action, which avoids these problems and can be usedto generalize to strings.
5
2.1 Polyakov action
Define a metric on 1D worldline as
ds2 = gττdτ2 = e2(τ)dτ2, (2.17)
where we have introduced an einbein e =√gττ , gττ = e−2(τ). With this metric we get an action
S =1
2
ˆdτe(τ)
[e−2X2 −m2
]=
1
2
ˆdτ√
detg[gττ∂τX · ∂τX −m2
]. (2.18)
Then vary this action with respect to e:
δeS =1
2
ˆdτ[δ(e−1)X2 −m2δe
]= −1
2
ˆdτ
[X2
e2+m2
]δe, (2.19)
which yields the equation of motion:
e−2x2 +m2 = 0 =⇒ e =1
m
√−X2 (2.20)
Vary next with respect to Xµ:
δXS = −1
2
ˆdτe−1 · 2Xµ∂τδXµ =
ˆdτ∂τ
[e−1Xµ
]δXµ, (2.21)
where the last equality comes from integration by parts. The equation of motion is now
∂τ
[e−1Xµ
]= 0 =⇒ ∂τ
[mXµ√−X2
]= 0. (2.22)
This is the same equation of motion as with the original action with the undesirable square root. Italso follows that
S =1
2
ˆdτ[e−1X2 −m2
]= ... = −
ˆdτm
√−X2, (2.23)
which is the same as before, too. In other words,
• Polyakov action reduces to the original action.
• For m 6= 0 both actions are classically equivalent.
• m 6= 0 Polyakov action makes sense.
Polyakov action is invariant under reparametrizations of the worldline of the type
τ → τ + ξ(τ). (exercise) (2.24)
3 The relativistic string
In analogy to point particles, we write Nambu-Goto action:
SNG = −TˆdA, (3.1)
where dA is an area element and T is a constant with dimension [T ] = [length]−2
= [mass]2, the
string tension. Conventionally, we write T = 12πα′ , where α′ is Regge slope. It has a dimension
6
[α′] = [length]2, which implies that
√α′ is a length scale, referred to as the string length. Its inverse
Ms is the string mass scale:
ls =√α′, Ms =
1√α′
=1
ls. (3.2)
Parametrize the worldsheet with two coordinates ξα, α = 0, 1: (ξ0, ξ1) = (τ, σ). Evolution isdescribed by a function Xµ(ξα) (embedding).
Suppose the metric of spacetime is Gµν . Let’s compute the induced metric on M, i.e. the pullbackmetric for the embedding map:
Σ→M (3.3)
ξα → Xµ(ξα) (3.4)
We get
ds2 = GµνdXµdXν = Gµν
∂Xµ
∂ξα∂Xν
∂ξβ︸ ︷︷ ︸≡ Gαβ
dξαdξβ = Gαβdξαdξβ (3.5)
Area element on M:
dA =
√−detGαβd
2ξ =⇒ SNG = −Tˆd2ξ
√−detGαβ (3.6)
Consider flat spacetime Gµν = ηµν . Now the induced metric is
Gαβ = ηµν∂αXµ∂βX
ν = −∂αX0∂βX0 + ∂α ~X∂β ~X ≡ ∂αX · ∂βX (3.7)
Recall that
X ·X ′ = ηµνXµX ′ν , X2 = ηµνX
µXν , X ′2 = ηµνX′µX ′ν (3.8)
which give the components:
G00 = X2, G01 = G10 = X ·X ′, G11 = X ′2 (3.9)
=⇒ det Gαβ = G00G11 − G201 = X2X ′2 − (X ·X ′)2 (3.10)
=⇒ SNG = −Tˆ √
(X ·X ′)2 − X2X ′2d2ξ ≡ˆd2ξL (3.11)
The equation of motion is
∂τ
(∂L∂Xµ
)+ ∂σ
(∂L∂X ′µ
)= 0 . (3.12)
Note that this is a non-linear partial differential equation. The canonical momentum reads
Πµ =∂L∂Xµ
= −T (X ·X ′)Xµ −X ′2Xµ√(X ·X ′)2 − X2X ′2
. (3.13)
Πµ satisfies the following constraints (exercise):
Π ·X ′ = 0 , Π2 + T 2X ′2 = 0 (3.14)
The Hamiltonian again vanishes: ˆdσ[X ·Π− L
]= . . . = 0 . (3.15)
Going through the calculation is left for an exercise. As in the particle case the dynamics of thesystem is governed by the constraints.
7
3.1 Polyakov action for a string
Quantizing Nambu-Goto action is very involved. Therefore we consider another (classically equiva-lent) action instead. This is the so-called Polyakov action or the string sigma model action. This newaction contains an auxiliary worldsheet metric gαβ(τ, σ), which plays the same role as the auxiliaryfield e(τ) in the point particle case.
SP =T
2
ˆd2ξ√−det ggαβ∂αX
ν∂βXνηµν , (3.16)
where gαβ is the inverse of of gαβ . We also assumed that Gµν = ηµν .
The metric can be eliminated by using its equation of motion, and the resulting action coincideswith NG action. Let us write the equation of motion for gαβ :
Tαβ ≡ −2
T
1√det g
δS
δgαβ= 0. (3.17)
(N.b. one can also use the Noether procedure for defining Tαβ , which of course matches this.) Sinceδ(det g) = (exercise) = −det g gαβδg
αβ , we have that δ(√−det g) = − 1
2
√−det ggαβδg
αβ . We getthe energy-momentum tensor :
Tαβ = ∂αX · ∂βX −1
2gαβg
γδ∂γX · ∂δX (3.18)
Note that Tαα = 0 (scale invariance). Using Tαβ = 0 yields
∂αX · ∂βX =
[1
2gγδ∂γX · ∂δX
]gαβ (3.19)
=⇒ −det ∂αX · ∂βX =
[1
2gγδ∂γX · ∂δX
]2
(−det gαβ) (3.20)
=⇒ −T√−det ∂αX · ∂βX︸ ︷︷ ︸
Nambu-Goto Lagrangian
= −T2gγδ∂γX · ∂δX (−det gαβ)︸ ︷︷ ︸
Polyakov Lagrangian
(3.21)
Thus NG Lagrangian and Polyakov Lagrangian coincide when g is eliminated by using the equationof motion.
Let us now derive the equation of motion for Xµ. The variation of the action with respect to Xµ is
δSP = −Tˆd2ξ√−det ggαβ∂αδX
µ∂βXνηµν (3.22)
= T
ˆd2ξ∂α
[√−det ggαβ∂βX
νηµν
]δXµ , (3.23)
where the second line comes from neglecting the total derivatives. Since this is true for all variationsof Xµ, we have that
∂α
[√−det ggαβ∂βX
µ]
= 0 (3.24)
We can rewrite this as
∆Xµ ≡ − 1√−det g
∂α
[√−det ggαβ∂βX
µ]
= 0 wave equation (3.25)
This is the Laplace equation in 2D space with metric gαβ . Thus spacetime coordinates Xµ are scalarfields in the two-dimensional string worldsheet!
********************************************
End of lecture 1
8
3.2 Symmetries of the Polyakov action
Let’s study symmetries of SP = −T2´d2ξ√−det ggαβ∂αX
µ∂βXνηµν for a string moving in Minkowski
spacetime.
1. Poincare transformationsThese are global, internal symmetries. In infinitesimal form, the Poincare transformations are:
δXµ = ωµνXν + aµ, δgαβ = 0, (3.26)
where ωµν are constants satisfying ωµν = −ωνµ and aµ correspond to translations.
2. 2D reparametrizationsThis is a local symmetry that is induced by the diffeomorphisms of the worldsheet coordinates:
ξα → fα(ξ) = ξ′α, (3.27)
under which
gαβ(ξ) =∂fγ
∂ξα∂fδ
∂ξβgγδ. (3.28)
Thus gαβ is a second-order tensor. This is a symmetry of GR in 2D telling us that we canchoose coordinates however we like.
3. Weyl transformations (rescaling)The SP is invariant under the local rescaling (clearly related to CFT):
gαβ → eφ(τ,σ)gαβ , δXµ = 0 . (3.29)
As√− det g → eφ
√−det g and gαβ → e−φgαβ , the eφ factors drop out and SP is invariant.
Notice that this only happens for 2D worldsheets. N.B. this is not a symmetry of the Nambu-Goto action. This means that local dilatations are an additional redundancy of Polyakovactions. Notice that φ(τ, σ) is not a physical field and has no degrees of freedom associatedwith it.
The symmetry 3) is the one responsible for Tαα = 0. Let’s perform an infinitesimal transformation:δgαβ = φgαβ
δgαβ = −φgαβ
δXµ = 0
=⇒ δSP =δSPδgαβ
δgαβ = −T2
√− det gTαβ(−gαβ)φ (3.30)
=T
2
√−det gTαβg
αβφ (3.31)
Thus (infinitesimal Weyl transformations)
δSP = 0 =⇒ Tαα = 0 . (3.32)
3.3 Gauge fixing
In principle, the worldsheet metric gαβ has 3 independent components:
gαβ =
(g00 g01
g10 g11
), g01 = g10. (3.33)
By using reparametrization invariance, we can make the metric conformally flat:
gαβ = eΛ ηαβ , ηαβ = diag(−1, 1). (3.34)
9
In 2D (with trivial topology) this is always possible. Moreover, we can use Weyl invariance to reducegαβ to ηαβ , i.e.
gαβ →diff.
eΛ ηαβ →Weyl
ηαβ . (3.35)
From now on we will take gαβ = ηαβ = diag(−1, 1). This is called unit or covariant gauge. Thegauge-fixed Polyakov action is:
SP = −T2
ˆ τ2
τ1
dτ
ˆ σ
0
dσηαβ∂αXµ∂βX
νηµν (3.36)
The equation of motion: ∂α∂αXµ = 0 [
∂2
∂τ2− ∂2
∂σ2
]Xµ = 0 , (3.37)
which are the Kaluza-Klein equations in 2D! However, we shouldn’t forget about the constraintstαβ = 0, which originate from the equation of motion of auxiliary gαβ . In more detail (gαβ = ηαβ),first
gγδ∂γx · ∂δx = −x2 + x′2. (3.38)
This implies that T00 = X2 −
(− 1
2
) (−X2 +X ′2
)= 1
2
(X2 +X ′2
)T11 = X ′2 −
(− 1
2
) (−X2 +X ′2
)= 1
2
(X2 +X ′2
)= T00
T01 = X ·X ′ = T10.
(3.39)
Check trace: Tαα = ηαβTαβ = −T00 + T11 = 0.
Therefore the constraints are: X2 +X ′2 = 0
X ·X ′ = 0.(3.40)
=⇒ 0 = X2 +X ′2 ± 2 · X ·X ′ =(X ±X ′
)2
(3.41)
=⇒(X ±X ′
)2
= 0 Virasoro constraints (3.42)
In summary, we can simplify our lives by sending the metric to a flat one, gαβ → ηαβ , but we have
to additionally impose(X ±X ′
)2
= 0.
Thinking ahead: how to proceed upon quantizing the string? We have three options:
1. (Old) covariant quantization
We first quantize and then impose Virasoro constraints. After quantizing the Hilbert space istoo big and includes negative norm states (it is not manifestly unitary). Virasoro constraintsproject onto physical Hilbert subspace. The good side in this is that no gauge choice is neededand the result is manifestly covariant.
2. Ligh-cone quantization
Before quantizing we use symmetries to pick a particular gauge that solves Virasoro constraints.This is manifestly unitary, but not covariant and Lorentz invariance is obscured. Good side:quick (and dirty)
10
3. BRST (or new covariant) quantization
This method is too advanced for this course, see Polchinski Ch. 4 or Green, Schwarz & WittenCh. 4 for details.
We will highlight some steps in 1. but will then focus on 2.
3.4 Noether currents
Let’s now find the conserved currents associated with the Poincare symmetries of SP . Considerglobal transformation: φ → φ + δεφ, with φ a field of the theory and ε << 1. If we allow ε = ε(ξ),then
L → L+ ε ∂αJα, (3.43)
where Jα is the conserved current; i.e. ∂αJα = 0 when the equation of motion is imposed. Let’s
first focus on spacetime translations:
Xµ → Xµ + aµ and aµ = aµ(ξ) (3.44)
(3.45)
=⇒ δS = δ
[−T
2
ˆd2ξηαβ∂αX
µ∂βXµ
]= −T
ˆd2ξηαβ∂αX
µ∂βaµ (3.46)
=int.byparts
T
ˆdξaµ∂
α [∂αXµ] ≡
ˆd2ξaµ∂
αP µα , (3.47)
where the conserved current associated with with Poincare invariance is:
P µα = T∂αX
µ . (3.48)
The associated charge is:
pµ =
ˆ σ
0
P µτ dσ = T
ˆ σ
0
Xµdσ (3.49)
which should be identified with the momentum of the string. Notice that since ∂α∂αXµ = 0 (the
equation of motion), ∂αP µα = T∂α∂αX
µ = 0 is conserved.
Let us then consider Lorentz transformations of the form
δXµ = ωµνXν , δXµ = ωµνX
ν , ωµν = −ωνµ. (3.50)
Then δS = (exercise) leads to a conserved current:
Jµνα ≡ T (Xµ∂αXν −Xν∂αX
µ) . (3.51)
Again ∂αJµνα = 0 upon the equation of motion.
The generator of Lorentz transformations, i.e. the angular momentum of the string is
Jµν =
ˆ σ
0
dσJµντ = T
ˆ σ
0
dσ(XµXν −XνXµ
). (3.52)
11
3.5 Boundary conditions
Usually in QFT one derives equations of motion and discards the boundary terms since one assumesthe spacetime is infinite and that all relevant quantities die off sufficiently quickly. For strings, theboundaries matter. Consider thus a general variation of SP :
δSP = −Tˆdτdσηαβ∂α(δXµ)∂βXµ (3.53)
= −Tˆdτdσ
[δXµηαβ∂α∂βXµ − ηαβ∂α (δXµ∂βXµ)
]. (3.54)
Assume that σ ∈ [0, σ] and τ ∈ [τ1, τ2]. Then
δSP = −Tˆ τ2
τ1
dτ
ˆ σ
0
dσ∂α∂αXµδXµ + T
ˆ τ2
τ1
dτ
ˆ σ
0
dσ∂α [∂αXµδXµ] (3.55)
= −Tˆ τ2
τ1
dτ
ˆ σ
0
dσ∂α∂αXµδXµ − T
ˆ σ
0
dσ[XµδX
µ]τ2τ1
+ T
ˆ τ2
τ1
dτ[X ′µδX
µ]σ=σ
σ=0. (3.56)
The vanishing of the first term implies the equations of motion. The vanishing of the second one isautomatic because δXµ = 0 at time boundaries in the variational principle, i.e. δXµ
∣∣τ=τ1,τ2
= 0.
The vanishing of the third term is non-trivial and implies
X ′µδXµ
∣∣σ=σ
σ=0= 0. (3.57)
Let’s consider the consequences of this separately for open and closed strings.
Closed strings
Here σ = 2π and we impose that the embedding functions are periodic:
Xµ(τ, σ + 2π) = Xµ(τ, σ) and X ′µ(τ, σ + 2π) = X ′µ(τ, σ). (3.58)
This solves the boundary condition. N.b. Xµ = −Xµ is also possible but this breaks the spacetimeLorentz symmetry. For example, Xµ → −Xµ does not commute with Xµ → Xµ + aµ.
Open strings
Now we take σ = π. There are two possibilities:
X ′(τ, σ)∣∣σ=∂
= 0 Neumann (3.59)
δXµ∣∣σ=∂
= 0 Dirichlet (3.60)
Notice that for the Dirichlet boundary condition the string is fixed at σ = 0, π, i.e.
Xµ∣∣σ=0
= const. = Xµ0 (3.61)
Xµ∣∣σ=π
= const. = Xµπ (3.62)
One of the unexpected things is that string theory is not only a theory of string but also of extendedobjects called D-branes. We shall denote the dimensionality of a brane by Dp-brane as an objecthaving p space dimensions. Mixed options are also possible.
Let’s say this in another way. Consider a momentum flow, i.e. the variation of the total momentumwith time. Since
pµ =
ˆ σ
0
P µτ dσ = T
ˆ σ
0
Xµdσ (3.63)
=⇒ dpµ
dτ=
ˆ σ
0
dσdP µ
τ
dτ= T
ˆ σ
0
dσ∂2Xµ
dτ2=
EoMT
ˆ σ
0
dσ∂2Xµ
dσ2(3.64)
= T [X ′µ(τ, σ = σ)−X ′µ(τ, σ = 0)] (3.65)
12
The expression vanishes in two cases:Closed strings
Neumann boundary conditions for open strings(3.66)
=⇒ No momentum flow off the ends of the string (3.67)
But note that momentum flow does not vanish for Dirichlet boundary conditions for open strings.There thus has to be an object in which they end and absorbs the momentum. These objects areprecisely the D-branes.
3.6 Example: rotating string
Consider the following configuration:X0 = Bτ
X1 = B cos τ cosσ
X2 = B sin τ cosσ
X3 = X4 = ... = const.
(3.68)
where B = const. This represents a string rotating on a (X1, X2)-plane. It obviously satisfies theequation of motion:
[∂2σ − ∂2
τ
]Xµ = 0. Let’s check that constraints are satisfied:
Xµ = (B,−B sin τ cosσ,B cos τ cosσ, 0, . . .)
X ′µ = (0,−B cos τ sinσ,−B sin τ sinσ, 0, . . .)(3.69)
=⇒
Xµ +X ′µ = B(1,− sin(τ + σ), cos(τ + σ), 0, . . .)
Xµ −X ′µ = B(1,− sin(τ − σ), cos(τ − σ), 0, . . .)(3.70)
=⇒ Xµ ±X ′µ = B(1,− sin(τ ± σ), cos(τ ± σ), 0, . . .) (3.71)
=⇒ (Xµ ±X ′µ)2 = B2(−1 + sin2(τ ± σ) + cos2(τ ± σ)) = 0. (3.72)
Let’s compute the momentum:
P µτ = TXµ = TB(1,− sin τ cosσ, cos τ cosσ, 0, . . .) (3.73)
=⇒ pµ =
ˆ π
0
P µτ dσ = πTB(1, 0, 0, 0, . . .). (3.74)
So the energy of the string is E = p0 = πTB. Angular momentum is:
J12 = T
ˆ π
0
dσ[X1X2 −X2X1
]= ... =
πTB2
2= J. (3.75)
Notice that
E2
J= 2πT =
1
α′=⇒ J = α′E2 (3.76)
This is the Regge trajectory, where α′ is the Regge slope. Notice also that the rotating string satisfiesthe Neumann boundary condition:
X ′µ∣∣σ=0,π
= (0,−B cos τ sinσ,−B sin τ sinσ, 0, . . .)∣∣σ=0,π
= 0. (3.77)
Let’s finally compute the speed of different parts of the string:
~v = (v1, v2) =
(dX1
dX0,dX2
dX0
)=
1
B
(dX1
dτ,dX2
dτ
)= (− sin τ, cos τ) cosσ (3.78)
|~v| = |cosσ|, (3.79)
and as |~v|σ=0,π = 1 the two ends of the string move at the speed of light.
13
********************************************
End of lecture 2
3.7 Light-cone coordinates on the worldsheet
Define ξ± as ξ+ = τ + σ
ξ− = τ − σ(3.80)
=⇒ dξ+dξ− = dτ2 − dσ2. (3.81)
These are similar to holomorphic and antiholomorphic variables in C. Now the worldsheet metric is
ds2 = −dξ+dξ−,
g++ = g−− = 0
g+− = g−+ = −1/2.(3.82)
Define further ∂± = 12 (∂τ ± ∂σ). Thus we have that
∂±ξ± = 1
∂±ξ∓ = 0.
(3.83)
(3.84)
Polyakov action becomes:
∂+Xµ∂−Xµ = ... =
1
4ηαβ∂αX
µ∂βXµ (3.85)
=⇒ SP = 2T
ˆd2ξ∂+X
µ∂−Xνηµν , (3.86)
and the equation of motion is
∂+∂−Xµ = 0. (3.87)
Energy-momentum tensor,
gγδ∂γX∂δX = 2 g+−︸︷︷︸−2
∂+X · ∂−X = −4∂+X · ∂−X (3.88)
=⇒
T++ = ∂+X · ∂+X − 1
2g++(gγδ∂γX∂δX) = ∂+X · ∂+X
T−− = (similar) = ∂−X · ∂−XT+− = ∂+X · ∂−X − 1
2g+−(−4∂+X · ∂−X) = 0 = T−+.
(3.89)
The equation of motion can be solved as:
Xµ(τ, σ) = XµL(τ + σ)︸ ︷︷ ︸
left movers
+XµR(τ − σ)︸ ︷︷ ︸
right movers
. (3.90)
We can expand XL and XR in modes as
XµR(τ − σ) =
xµ
2+l2s2pµ(τ − σ) +
ils√2
∑n6=0
αµnNe−in(τ−σ) (3.91)
XµL(τ + σ) =
xµ
2+l2s2pµ(τ + σ) +
ils√2
∑n 6=0
αµnne−in(τ+σ) (3.92)
As XL, XR ∈ R, the zero modes are real xµ, pµ, pµ ∈ R and and the oscillators must satisfy
(αµn)∗ = αµ−n and (αµn)∗ = αµ−n . (3.93)
14
3.8 Closed strings
From periodicity condition Xµ(τ, σ + 2π) = Xµ(τ, σ) we get n ∈ Z \ 0 , pµ = pµ,
=⇒
XµR(τ − σ) = xµ
2 +l2s2 p
µ(τ − σ) + ils√2
∑n∈Z\0
αµnn e−in(τ−σ)
XµL(τ + σ) = xµ
2 +l2s2 p
µ(τ + σ) + ils√2
∑n∈Z\0
αµnn e−in(τ+σ)
(3.94)
Let’s define αµ0 ≡ ls√2pµ ≡ αµ0 . Compute
∂+XµL =
l2spµ
2+
ls√2
∑n∈Z\0
αµne−in(τ+σ) (3.95)
=⇒
∂+X
µL = ls√
2
∑n∈Z α
µne−in(τ+σ)
∂−XµR = ls√
2
∑n∈Z α
µne−in(τ−σ).
(3.96)
Let’s compute the center-of-mass position of the string:
XµCM ≡
1
2π
ˆ 2π
0
dσXµ(τ, σ) =1
2π
ˆ 2π
0
dσ[xµ + l2sp
µτ]
= xµ + l2spµτ , (3.97)
where the second equality follows from the fact that only the zero modes make a contribution. Inother words, xµ is the center-of-mass position of the string at τ = 0, which moves as a free particle.The momentum of the center-of-mass is
pµCM = T
ˆ 2π
0
dσXµ = T l2spµ
ˆ 2π
0
dσ = pµ. (3.98)
Thus pµ is the total momentum of the string.
Recall that the Virasoro constraints can be written as
T±± = ∂±Xµ∂±Xµ = 0 . (3.99)
Define then the Fourier transforms asLm ≡ T
´ 2π
0dσT−−e
im(τ−σ)
Lm ≡ T´ 2π
0dσT++e
im(τ+σ),(3.100)
where Lm, Lm are Virasoro generators.
Let’s next obtain the mode expansion of the Virasoro generators:
T−− = ∂−Xµ∂−Xµ = ∂−X
µR∂−X
Rµ =
l2s2
∑n,k∈Z
αµnαkµe−i(n+k)(τ−σ) (3.101)
=⇒ Lm =T l2s2︸︷︷︸
1/4π
∑n,k∈Z
αµnαkµe−i(n+k−m)τ
ˆ 2π
0
dσei(n+k−m)σ︸ ︷︷ ︸2πδk,m−n
(3.102)
=1
2
∑n∈Z
αm−n · αn (3.103)
Lm = (similarly) =1
2
∑n∈Z
αm−n · αn (3.104)
15
The inverse Fourier relations are1T−− = l2s
∑∞m=−∞ Lme
−im(τ−σ)
T++ = l2s∑∞m=−∞ Lme
−im(τ+σ)(3.105)
Clearly, we must have the following reality conditions:
L∗m = L−m , L∗m = L−m . (3.106)
Notice also that the constraints in terms of modes reads:
Lm = 0 = Lm, m = 0,±1,±2, ... (3.107)
Let’s focus on L0 = 0 = L0:
L0 =1
2
∞∑n=−∞
αm−n ·αn∣∣m=0
=1
2
[(n = 0) +
−1∑n=−∞
α−n · αn +
∞∑n=1
α−n · αn
]=
1
2α2
0 +
∞∑n=1
α−n ·αn .
(3.108)
Since, α20 =
l2s2 p
µpµ = − l2s
2 M2, where M is the rest mass of string in target space, yields:
M2 =4
α′
∞∑n=1
α−n · αn . (3.109)
Similarly for L0 leads to M2 = 4α′
∑∞n=1 α−n · αn. For consistency we must satisfy the following
level matching condition:∞∑n=1
α−n · αn =
∞∑n=1
α−n · αn. (3.110)
So we can write the mass formula symmetrically:
M2 =2
α′
∞∑n=1
(α−n · αn + α−n · αn). (3.111)
Classically this is a continuous quantity and the actual value of M depends on αn, αn.
3.9 Hamiltonian analysis of the closed string
Canonical momentum conjugate to Xµ is:
Πµ(τ, σ) =δSP
δXµ(τ, σ), L =
T
2(X2 −X ′2) =⇒ Πµ = TXµ (3.112)
yielding the Hamiltonian
H =
ˆ 2π
0
dσ(XµΠµ − L) =T
2
ˆ 2π
0
dσ(X2 +X ′2) . (3.113)
1Proof:∑∞m=−∞ Lme−im(τ−σ) = 1
2
∑n,m∈Z αm−n · αne−im(τ−σ) =
m−n=k
12
∑k,n∈Z
αk · αne−i(k+n)(τ−σ)
︸ ︷︷ ︸2l2sT−−
.
16
In terms of the EM-tensor:
T++ + T−− =∂±= 1
2 (∂τ±∂σ)
1
4(X ′ + X)2 +
1
4(X −X ′)2 =
1
2(X2 +X ′2) (3.114)
=⇒ H = T
ˆ 2π
0
dσ [T++ + T−−] = L0 + L0 (3.115)
The canonical equal-time Poisson brackets (PB) are:[Πµ(τ, σ),Πν(τ, σ′)]PB = 0
[Xµ(τ, σ), Xν(τ, σ′)]PB = 0
[Πµ(τ, σ), Xν(τ, σ′)]PB = T[Xµ(τ, σ), Xν(τ, σ′)
]PB
= ηµνδ(σ − σ′).(3.116)
In terms of modes, these are (exercise):
[xµ, xν ]PB = [pµ, pν ]PB = 0
[pµ, xν ]PB = ηµν
[αµm, ανn]PB = imδm+nη
µν
[αµm, ανn]PB = imδm+nη
µν
[αµn, ανm]PB = 0.
(3.117)
Now we can establish the Poisson algebra of two Virasoro generators. One can show (exercise) thatLm’s close the classical Virasoro algebra:
[Lm, Ln] = i(m− n)Lm+n. (3.118)
The Virasoro is an infinite-dimensional algebra whose origin is the residual symmetry left by thefixing of worldsheet gαβ . We can easily check this as follows. Let’s make a change of coordinates inwhich light-cone coordinates ξ+, ξ− do not mix:
ξ+ → ξ+′ = f+(ξ+)
ξ− → ξ−′
= f−(ξ−).(3.119)
The ± derivatives of Xµ transform as
∂′±Xµ =
∂Xµ
∂ξ±∂ξ±
∂ξ±′=
1
∂±f±∂±X
µ . (3.120)
Thus
∂′+X · ∂′−X =1
∂+f+∂−f−∂+X · ∂−X. (3.121)
and since d2ξ′ = ∂+f+∂−f
−d2ξ we have d2ξ−∂′+X · ∂′−X = d2ξ∂+X · ∂−X and so SP with trivialmetric is invariant.
Now consider operators V ± = f±(ξ±) ∂∂ξ± , where f± is infinitesimal. They generate the infinitesimal
transformation: δξ± = f±. Consider the following basis of functions f±:
f±n (ξ±) = einξ±, n ∈ Z. (3.122)
The corresponding generators are: V ± = einξ± ∂∂ξ± . The commutator acting on an arbitrary function
reads [V +n , V
+m
]F (ξ+) = einξ
+ ∂
∂ξ+(eimξ
+
∂+F )− eimξ+ ∂
∂ξ+(einξ
+
∂+F ) (3.123)
= i(m− n)ei(m+n)ξ+
∂+F = i(m− n)V +m+nF . (3.124)
17
Similarly for V −n . Therefore the V ±n satisfy the algebra[V ±n , V
±m
]= i(n−m)V ±m+n. (3.125)
These are two copies of the Virasoro algebra we have found. The Ln and Ln are the realizations ofthese diffeomorphisms in the space of the string oscillators.
3.10 Open strings with Neumann-Neumann boundary conditions
Let’s take σ = π and impose Neumann boundary conditions at both ends of the string:
Xµ′(τ, σ)∣∣σ=0,π
= 0. (3.126)
Since the mode expansion of Xµ is:
Xµ = xµ +l2s2
(pµ + pµ)τ +l2s2
(pµ − pµ)σ +ils√
2
∑k 6=0
1
k
[αµke
−ik(τ−σ) + αµke−ik(τ+σ)
](3.127)
(3.128)
we have
Xµ′(τ, σ) =l2s2
(pµ − pµ) +ls√2
∑k 6=0
[αµke
−ik(τ+σ) − αµke−ik(τ−σ)
](3.129)
−→σ→0
l2s2
(pµ − pµ) +ls√2
∑k 6=0
e−ikτ [αµk − αµk ] (3.130)
=⇒ pµ = pµ and αµk = αµk (3.131)
The left and right movers get identified.
=⇒ Xµ′(τ, σ) = ... = −i√
2ls∑k 6=0
αµk sin kσe−ikτ (3.132)
Notice further that Xµ′(τ, σ = π) = 0 implies k ∈ Z \ 0:
=⇒ Xµ(τ, σ) = xµ + l2spµτ + i
√2ls
∑k∈Z\0
1
kαµk cos kσe−ikτ . (3.133)
For convenience we rescale pµ → 2pµ:
Xµ = xµ + 2l2spµτ + i
√2ls
∑k∈Z\0
αµkk
cos kσe−ikτ (3.134)
Further,
∂±Xµ = l2sp
µ +ls√2
∑k∈Z\0
αµke−ik(τ±σ) =
ls√2
∑k∈Z
αµke−ik(τ±σ), (3.135)
where we have defined αµ0 ≡√
2pµls.
Let’s compute center-of-mass position of the open string:
xµCM =1
π
ˆ π
0
dσXµ(τ, σ) = xµ + 2l2spµτ , (3.136)
18
since´ π
0cosnσdσ = 0,whenn 6= 0. Thus xµ is the center of mass position of the string, which is
moving as a free particle. Compute next the center-of-mass momentum:
pµCM = T
ˆ π
0
dσXµ = T
ˆ π
0
dσ[2l2sp
µ]
= pµ . (3.137)
This justifies the rescaling we did earlier.
We now study the Hamiltonian:
H =T
2
ˆ π
0
dσ[X2 +X ′2
]= T
ˆdσ[(∂+X)2 + (∂−X)2
]. (3.138)
Plugging in mode expansions
(∂±Xµ)2 =
l2s2
∑n,m∈Z
αnαme−i(n+m)τe∓i(n+m)σ (3.139)
=⇒ (∂+X)2 + (∂−X)2 = l2s∑n,m∈Z
αnαme−i(n+m)τ cos [(n+m)σ] (3.140)
and using´ π
0dσ cos [(n+m)σ] = πδn+m, we have that
HNN =1
2
∑n∈Z
α−n · αn = l2sp2 +
∞∑n=1
α−n · αn. (3.141)
Let us define the classical Virasoro generator:
Lm = T
ˆ π
0
dσ[T++e
im(τ+σ) + T−−eim(τ−σ)
]. (3.142)
As T±± = (∂±X)2 we can write the integrals as
ˆ π
0
dσT±±eim(τ±σ) = ... =
πl2s2
∑n∈Z
αm−n · αn (3.143)
Lm = Tπl2s∑n∈Z
αm−n · αn =1
2
∑n∈Z
αm−n · αn, open, Neumann-Neumann (3.144)
Notice that we get the same expression as in the closed string, but we now have only one set ofoscillators. The classical Virasoro constraints are
Lm = 0, m = 0,±1,±2, ... (3.145)
Notice that, in particular
HNN = L0. (3.146)
Then, the constraint L0 = 0 implies
l2s p2︸︷︷︸=−M2
+
∞∑n=1
α−n · αn = 0 (3.147)
=⇒ M2 =1
α′
∞∑n=1
α−n · αn (3.148)
This is classical continuous mass spectrum of open string with Neumann-Neumann boundary con-ditions.
19
3.11 SL(2,R) subalgebra
The Virasoro algebra contains an SL(2,R) subalgebra that is generated by L0, L±1. This is anoncompact form of the familiar SU(2) algebra. Just as SU(2) and SO(3) have the same Liealgebra, so do SL(2,R) and SO(2, 1). Thus, in the case of closed strings, the complete Virasoroalgebra of both left-movers and right-movers contains the subalgebra SL(2,R)×SL(2,R) = SO(2, 2).This is a noncompact version of the Lie algebra identity SU(2)× SU(2) = SO(4).
********************************************
End of lecture 3
4 Quantization of the bosonic string
Let us now quantize the bosonic string. We follow the standard rule:
[. . . , . . .]PB → i [. . . , . . .] , (4.1)
where ~ = 1. This means that the classical Poisson bracket
[A,B]PB = C (4.2)
is substituted in the quantum theory by
i[A, B
]= C =⇒
[A, B
]= −iC, (4.3)
where hats denote operators. We will suppress them from now on.
Let’s apply this to canonical Poisson bracket between Xµ(τ, σ) and Πµ(τ, σ):[Xµ(τ, σ), Xν(τ, σ′)] = 0
[Πµ(τ, σ),Πν(τ, σ′)] = 0
[Xµ(τ, σ),Πν(τ, σ′)] = iηµνδ(σ − σ′).(4.4)
Let’s write these in terms of modes. We only consider closed strings:[xµ, pν ] = iηµν
[αµm, ανn] = [αµm, α
νn] = mηµνδm+n
[αµm, ανn] = 0.
(4.5)
Notice that the commutator between xµ and pµ is realized by taking pµ = −i ∂∂xµ . Let’s continue by
first redefining the oscillators:
aµm ≡1√mαµm, a
µ†m ≡
1√mαµ−m, a
µm ≡
1√mαµm, a
µ†m ≡
1√mαµ−m, m > 0. (4.6)
Thus, we are defining αµ−m and αµ−m as Hermitian conjugates of αµm and αµm for m > 0. Notice thatclassically αµ−m = (αµm)∗ and αµ−m = (αµm)∗, which is consistent with our definition.
The commutator algebra is [aµm, a
ν†m
]=[aµm, a
ν†m
]= ηµνδmn. (4.7)
This is exactly the algebra of creation and annihilation operators of an infinite set of harmonicoscillators. As every physicist anticipates, this algebra will determine the mass levels of the string.
20
The Hilbert space is obtained by applying raising operators on the ground state. The ground state|0; k〉 is defined as the state that is annihilated by all lowering operators:
αµm |0; k〉 = αµm |0; k〉 = 0,m > 0. (4.8)
In addition it is an eigenstate of pµ:
pµ |0; k〉 = kµ |0; k〉 . (4.9)
A general state of Hilbert space generated in this way is a linear combination of states of the form:
αµ1
−m1αµ2
−m2· · · αν1
−n1αν2−n2· · · |0; k〉 , m1,m2, ..., n1, n2, ... > 0. (4.10)
Notice that the general state is also an eigenstate of pµ. These states are one-particle states, i.e. weare in a first quantization formalism. Indeed, we will verify that pµp
µ is determined by the occupationnumbers. Different states carry different representations of the Lorentz group with different spinand different number of spacetime Lorentz indices.
An important fact is that the Hilbert space is not positive definite since there are states with negativenorm. Indeed, since (η00 = −1)
[a0m, a
0†m
]= −1 we have
‖a0†m |0〉‖2 = 〈0| a0
ma0†m |0〉 = 〈0|
[a0m, a
0†m
]|0〉 = −〈0|0〉 = −1. (4.11)
Thus the state a0†m |0〉 has negative norm and the Hilbert space is not unitary. This is similar problem
that shows up in the quantization of the gauge field in QED. In order to bypass it we need to imposesubsidiary conditions and construct a smaller Hilbert space in which unitarity is recovered. Thenatural subsidiary conditions in this case are the Virasoro constraints
Lm = Lm = 0. (4.12)
Let’s explore the realization of these constraints in the quantum theory. As in QED, it is notconsistent to require the vanishing of Lm and Lm as operators. We should require that they vanishin the weak sense, as it is done in the Gupta-Bleurer formalism of QED. Accordingly, let us demandthat the expectation value of Lm and Lm vanish on any physical state |ψ〉:
〈ψ|Lm |ψ〉 = 〈ψ| Lm |ψ〉 = 0. (4.13)
It is important to notice that, to satisfy this condition, it is enough to require:
Lm |ψ〉 = Lm |ψ〉 = 0, m ≥ 0. (4.14)
i.e. that Lm and Lm vanish weakly for m ≥ 0. Indeed, since L†n = L−n (due to Hermiticitycondition) we have for n > 0:
〈ψ|L−n |ψ〉 = 〈ψ|L†n |ψ〉 = 〈ψ|Ln |ψ〉∗ = 0, (4.15)
since Ln |ψ〉 = 0, n > 0. Similarly for Ln.
Note that Ln is a quadratic expression in oscillators. Therefore, as in ordinary QFT, in orderto write down the quantum version of the Virasoro operators we have to introduce the normalordering of creation and annihilation operators. Recall that normal ordering is such that all lowering(annihilation) operators are to the right of all raising (creation) operators. Thus, the quantum Lmoperator is defined as:
Lm =1
2
∞∑n=−∞
: αm−n · αn : (4.16)
21
(Similarly for Lm) This definition is unambiguous except for L0 and L0. In this case a C-number isgenerated in the reordering. In order to avoid this ambiguity, we define L0 and L0 as:
L0 = 12α
20 +
∑∞n=1 α−n · αn
L0 = 12 α
20 +
∑∞n=1 α−n · αn
(4.17)
and write the physical state condition as:
(Lm − aδm) |ψ〉 = 0, (Lm − aδm) |ψ〉 = 0, m ≥ 0 (4.18)
where a is a C-number to be determined, which represents the zero-point energy. Note that we havetaken the same constant (they turn out to match actually, similarly to level matching) for the leftmovers and right movers. For open strings we would only have the condition for L0.
4.1 Light-cone quantization
As in gauge theories, it is possible to find a Fock space that is manifestly free of negative norm statesat the expense of not having explicit Lorentz covariance in the target space. Indeed, as above, evenafter choosing the gauge gαβ = ηαβ for the worldsheet metric we can perform change of variables inthe worldsheet coordinates of the type (see section 3.9)
ξ± → f±(ξ±), (4.19)
which keep the form of worldsheet metric invariant. As ξ± = τ ± σ, this corresponds to:τ → τ = 1
2 [f+(ξ+) + f−(ξ−)]
σ → σ = 12 [f+(ξ+)− f−(ξ−)]
(4.20)
It thus follows that τ can be an arbitrary solution of the free massless wave equation (exercise):(∂2
∂τ2− ∂2
∂σ2
)τ = 0. (4.21)
Notice also that once τ is determined then σ is specified up to a constant. Recall that Xµ solves alsothe wave-equation for any µ. So, τ can be expressed as a linear combination of the Xµ. Supposethe string moves in D-dimensional Minkowski spacetime. We define the light-cone coordinates inspacetime as:
X± =1√2
(X0 ±XD−1
). (4.22)
In these coordinates Xµ = (X+, X−, Xi), i = 1, ..., D− 2. The inner product of any two vectors V µ
and Wµ is (exercise):
V ·W = −V +W− − V −W+ +
D−2∑i=1
V iW i (4.23)
= V +W+ + V −W− +
D−2∑i=1
V iWi. (4.24)
In the light-cone gauge we choose τ to be proportional to X+:
τ ∝ X+ (light-cone gauge) (4.25)
22
The proportionality constant one chooses differs in closed and open strings. Let’s first focus onclosed strings:
τ =X+
l2sp+
+ const. (4.26)
In other words,
X+ = x+ + l2sp+τ (closed string) (4.27)
This is manifestly independent of σ and corresponds to setting all the oscillators α+n = 0:
α+n = 0, n 6= 0 (closed strings) (4.28)
In this non-covariant gauge the oscillators α−n can be written in terms of transverse ones. Let’s startwith Virasoro constraints: (X ±X ′)2 = 0. In light-cone
−2(X− ±X−′)(X+ ±X+′) + (Xi ±Xi′)2 = 0 (4.29)
but for closed strings
X+ = l2sp+, X+′ = 0 =⇒ −2l2sp
+(X− ± (X−)′) + (Xi ± (Xi)′)2 = 0 (4.30)
=⇒ X− ± (X−)′ =1
2p+l2s(Xi ± (Xi)′)2 (4.31)
∂± =1
2(∂τ ± ∂σ) =⇒ 2∂±X
− =1
2p+l2s· 4(∂±X
i)2 (4.32)
=⇒ ∂±X− =
1
p+l2s(∂±X
i)2 (4.33)
But from (3.95): ∂−X
− = ls√2
∑n∈Z α
−n e−in(τ−σ)
∂+X+ = ls√
2
∑n∈Z α
−n e−in(τ+σ)
(4.34)
From 4.30 and 4.34 we get
ls√2
∑n∈Z
α−n e−in(τ−σ) =
1
p+l2s
l2s2
∑p,k
αikαipe−i(k+p)(τ−σ) (4.35)
=k+p=n
1
2p+
∑n∈Z
∑p∈Z
αin−pαip
e−in(τ−σ) (4.36)
=⇒ α−n =1√
2lsp+
∑m∈Z
αin−mαim. (4.37)
This is the classical relation. Quantum mechanically, this relation gets modified to (similarly for ∂+
one gets α−n ):
α−n =1√
2lsp+
[ ∞∑m=−∞
: αin−mαim : −2aδn
]
α−n =1√
2lsp+
[ ∞∑m=−∞
: αin−mαim : −2aδn
] (4.38)
(4.39)
for closed strings.
23
For open strings with NN boundary conditions we gauge fix:
X+ = x+ + 2l2sp+. (4.40)
Then, X+ = 2l2sp+, (X+)′ = 0 =⇒ ∂±X
− = 12l2sp
+ (∂±Xi)2. But for open strings with NN boundary
conditions (see (3.135)):
∂±Xµ =
ls√2
∑n∈Z
αµne−in(τ±σ). (4.41)
This is the same relation as for closed strings with αn = αn. So we are readily get, by dividing by 2:
α−n =1
2√
2lsp+
[ ∞∑m=−∞
: αin−mαim : −2aδm
](open NN) (4.42)
In the light-cone gauge the Hilbert space is generated with the D− 2 transverse oscillators αin. Thisis because α+
n = 0 and α−n can be written in terms of αin. In this non-covariant gauge there areno negative norm states and unitarity is manifest. However, as we will explicitly show, Lorentzinvariance is not guaranteed unless certain conditions are met.
4.2 Spectrum of open strings in the light-cone gauge
Let |ψ〉 be a physical state. It must satisfy:
(L0 − a) |ψ〉 = 0 , (4.43)
where a is a constant generated by the normal ordering ambiguities (vacuum energy). Here,
L0 =1
2
∞∑n=−∞
: α−n · αn :=1
2α2
0 +
∞∑n=1
α−n · αn . (4.44)
But αµ0 =√
2lspµ → α2
0 = 2l2sp2 = −2l2sm
2 =⇒ L0 = −l2sm2 +∑∞n=1 α−n · αn. In light-cone
α−n · αn = −α+−nα
−n − α−−nα+
n +
D−2∑i=1
αi−nαin (4.45)
=⇒ L0 = −l2sm2 +
D−2∑i=1
∞∑n=1
αi−nαin . (4.46)
Let’s define the number operator N as:
N ≡D−2∑i=1
∞∑n=1
αi−nαin . (4.47)
So, in terms of αi−n =√nai†n , α
in =√nain we get
N =
D−2∑i=1
∞∑n=1
nai†n ain (4.48)
=⇒ L0 = −α′m2 +N , (4.49)
and the state |ψ〉 must satisfy:
(−α′m2 +N − a) |ψ〉 = 0 , (4.50)
24
and if we work in a basis where N is diagonal, we have:
m2 =1
α′(N − a) , (4.51)
where now N should be understood as the eigenvalue of the number operator.
Let’s study the first excited states. We begin with the vacuum |0〉, which is the state withoutoscillators and has N = 0. Clearly this state is a scalar and its mass is
m2 = − a
α′vacuum scalar (4.52)
Notice that this scalar is a tachyonic particle with m2 < 0 if vacuum energy a > 0. More on theinterpretation later.
Next, we consider the first excited state with N = 1, which is a state αi−1 |0〉. This state is a vectorin SO(D − 2) with D − 2 components. In a relativistic theory it should correspond to a masslessparticle with spin 1. Its mass, however, is:
m2 =1
α′(1− a) vector (4.53)
The condition that this state is massless will fix the vacuum energy constant a = 1. Let’s computeit.
L0 = −α′m2 +1
2
D−2∑i=1
∞∑n=−∞,n6=0
αi−nαin (4.54)
But∞∑
n=−∞,n6=0
αi−nαin =
∞∑n=1
(αi−nαin + αinα
i−n︸ ︷︷ ︸
αi−nαin+[αin,αi−n]=αi−nαin+n
) (4.55)
= 2
∞∑n=1
αi−nαin︸ ︷︷ ︸
:αi−nαin:
+
∞∑n=1
n︸ ︷︷ ︸∞-const. indep. of i
(4.56)
=⇒ L0 = −α′m2 +
∞∑n=1
D−2∑i=1
: αi−nαin : +
D − 2
2
∞∑n=1
n︸ ︷︷ ︸=−a
(4.57)
=⇒ a = −D − 2
2
∞∑n=1
n =exercise
D − 2
24(4.58)
Therefore, the first two excited states in the open string spectrum are
scalar =⇒ |0〉 =⇒ m2 = −D − 2
24tachyon if D > 2 (4.59)
vector =⇒ αi−1 |0〉 =⇒ m2 =26−D
24α′(4.60)
Let us look in more detail these vector states. A vector particle in a D-dimensional spacetime hasonly D− 2 components if it is massless (and has D− 1 components if it is massive). This is dictatedby Lorentz symmetry. If we want to preserve this symmetry in the target spacetime we shouldrequire that the states αi−1 |0〉 are massless
=⇒ D = 26 =⇒ a = 1 (4.61)
25
D = 26 is called the critical dimension of the bosonic string.
Let us now study N = 2. We have:
αi−2 |0〉 , αi−1αj−1 |0〉 (4.62)
The squared masses of these states are m2 = + 1α′ . In D dimensions the number of states of these
are:
D − 2 +1
2(D − 2)(D − 1) =
1
2(D − 2)(D + 1) =
D(D − 1)
2− 1. (4.63)
This is the number of components of a symmetric traceless second-rank tensor representation ofSO(D − 1), which corresponds to a particle of spin 2 and mass2 = 1/α′.
In general, the spectrum is a tower of states with increasing mass and spin.
(FIGURE IN CLASS)
Notice that the massive states become infinitely heavy when α′ → 0 (or ls → 0) and they decouplein this limit. This is the field theory limit.
4.3 Comments on tachyons
As E2 − p2 = m2, a particle with m2 < 0 has |p| > |E|, which means that it is traveling faster than
light (recall |p|E = |~v|). In QFT, however, having particles with m2 < 0 is not so exotic. Recall thatif T (x) is a field describing the fluctuations around some configuration (say T = 0) in a potentialV (T ), then the mass of the associated particle is
m2 =∂2V (T )
∂T 2
∣∣T=0
(4.64)
Thus if m2 < 0, then we have a maximum of a potential. This is exactly what happens when thereis a spontaneously broken symmetry. In this case the Higgs field H at H = 0 is also a tachyon.
(FIGURE IN CLASS)
In principle the tachyon signals an instability and its presence indicates that we are not quantizingthe field around a good vacuum. As in the Higgs mechanism, it could be that the potential V (T )has a minimum at T 6= 0:
(FIGURE IN CLASS)
In this case the tachyon will roll to the minimum. For bosonic string it is not clear what happensactually, the presence of the tachyon is one of the main motivations to consider superstrings sincein this case there is no tachyon. Since we are heading towards superstrings, we shall ignore thetachyon. If you are interested more on (open string) tachyons in string theory, you can take a lookat myPhD thesis: https://helda.helsinki.fi/handle/10138/23286 orMaster’s thesis: http://ethesis.helsinki.fi/julkaisut/mat/fysik/pg/jokela/.
4.4 Comments on critical dimension
The critical D was found to be D = 26 for a bosonic string. In the SUSY case D = 10. It is quiteremarkable that the mere presence of an object moving in a spacetime determines D by imposingthe consistency of the quantum theory! One important remark is that the extra dimensions arenot necessarily the spacetime dimensions in the usual sense. They could be compactified, as in
26
Kaluza-Klein theories, and they could be internal directions. Indeed, in the formulation of KK, therotations of these compactified dimensions are associated to the gauge symmetries of the theoriesdescribing the strong, weak, and electromagnetic interactions. This is actually a first hint of thefact that string theory unifies (i.e. the dynamics of spacetime) with the other interactions. In themodern holographic formulations the energy scale is identified with one of these extra dimensions.
********************************************
End of lecture 4
4.5 Comments of Lorentz symmetry
An important issue in the light-cone gauge is whether or not the Lorentz symmetry is realizedwithout anomalies in the quantum theory. Recall the generator of Lorentz transformation in stringtheory (3.52):
Jµν =
ˆdσJµντ = T
ˆdσ [Xµxν −Xν xµ] (4.65)
and the generator of translations pµ (3.49):
pµ =
ˆdσPµτ = T
ˆXµdσ . (4.66)
They should satisfy the Poincare algebra:[pµ, pν ] = 0
[pµ, Jνρ] = −iηµνpρ + iηµρpν[Jµν , Jρλ
]= −iηνρJµλ + iηµρJνλ + iηνλJµρ − iηλµJνρ.
(4.67)
Let’s find the expression of Jµν in modes. Let’s start with closed string. Since:
Xµ = xµ + l2spµτ +
ils√2
∑n∈Z\0
1
n
[αµne
−in(τ−σ) + αµne−in(τ+σ)
](4.68)
=⇒ Xµ = l2spµ +
ls√2
∑n∈Z\0
[αµne
−in(τ−σ) + αµne−in(τ+σ)
]. (4.69)
The zero-mode contribution to Jµν is:
T
ˆ 2π
0
dσ[(xµ + l2sp
µτ)(l2spν)− (µ↔ ν)
]= T
ˆ 2π
0
dσl2s (xµpν − xνpµ) (4.70)
= xµpν − xνpµ . (4.71)
Let’s next compute the contribution from the oscillators:
Eµν =il2sT
2
∑n,m
1
nαµnα
νm
ˆ 2π
0
dσ e−i(n+m)τei(n+m)σ︸ ︷︷ ︸2πδm+n
−(µ↔ ν) (4.72)
=i
2T l2s2π
∑n
1
n(αµnα
ν−n − ανnα
µ−n) (4.73)
=n→−n
− i2
∞∑n=−∞n 6=0
1
n(αµ−nα
νn − αν−nαµn). (4.74)
27
Let’s consider the first term in this sum:
∞∑n=−∞n6=0
1
nαµ−nα
νn =
∞∑n=1
1
nαµ−nα
νn +
−∞∑n=−1
1
nαµ−nα
νn (4.75)
=
∞∑n=1
1
nαµ−nα
νn︸ ︷︷ ︸
A
−∞∑n=1
1
nαµnα
ν−n︸ ︷︷ ︸
B
(4.76)
Similarly the second term:
−∞∑
n∈Z\0
1
nαν−nα
µn = −
∞∑n=1
1
nαν−nα
µn︸ ︷︷ ︸
C
+
∞∑n=1
1
nανnα
µ−n︸ ︷︷ ︸
D
. (4.77)
Then reorder B as follows:
B = −∞∑n=1
1
nαµnα
ν−n = −
∞∑n=1
1
nαν−nα
µn −
∞∑n=1
1
n
[αµn, α
ν−n]︸ ︷︷ ︸
nηµν
(4.78)
= −∞∑n=1
1
nαν−nα
µn︸ ︷︷ ︸
C
−∞∑n=1
ηµν (4.79)
Similarly for D:
D =
∞∑n=1
1
nανnα
µ−n =
∞∑n=1
1
nαµ−nα
νn +
∞∑n=1
[ανn, α
µ−n]︸ ︷︷ ︸
nηµν
(4.80)
=
∞∑n=1
1
nαµ−nα
νn︸ ︷︷ ︸
A
+
∞∑n=1
ηµν . (4.81)
The C-number terms in Eµν cancel and A + B + C + D = 2(A + C):
Eµν = −i∞∑n=1
1
n(αµ−nα
νn − αν−nαµn) (4.82)
Notice that the RHS is normal-ordered. Similarly for α:
Eµν = −i∞∑n=1
1
n(αµ−nα
νn − αν−nαµn) (4.83)
The total angular momentum for closed strings is
Jµν = xµpν − xνpµ + Eµν + Eµν closed (4.84)
For open strings we have the same expression but with only one type of oscillators:
Jµν = xµpν − xνpµ + Eµν open string (4.85)
One can check that Jµν satisfies the Poincare algebra (exercise) in the covariant gauge for any valueof D and a. However, in the light-cone gauge this is not so. Indeed, the commutator
[J i−, Jj−
]28
should be zero. As in the light-cone gauge α− ∼∑αiαi it is not obvious that this commutator is
zero. Actually, one gets (exercise):
[J i−, Jj−
]∼∞∑m=1
∆m
[αi−mα
jm − α
j−mα
im
], (4.86)
where
∆m = m
[26−D
12
]+
1
m
[D − 26
12+ 2(1− a)
]. (4.87)
∆m should vanish ∀m
=⇒ D = 26 and a = 1 (4.88)
in agreement with our previous analysis of the massless spectrum of the open string. The conclusionis that only for these values of D and a Lorentz invariance is unbroken in the light-cone gauge.
4.6 Covariant quantization of open string
Let’s start by analysing the representation theory of the Virasoro algebra. This algebra is modifiedwhen passing from the classical to the quantum theory. In the latter case, the generators Ln aredefined using the normal-ordering:
Lm =1
2
∞∑−∞
: αm−n · αn : (4.89)
By using the commutators for the modes αµm, one can show that the quantum Virasoro operatorssatisfy the relation:
[Lm, Ln] = (m− n)Lm+n +c
12m(m2 − 1)δm+n (4.90)
In the above formula c = D is the spafetime dimension. The term proportional to c is a quantumeffect not present in the classical theory (an anomaly). This term is called a central extension of theVirasoro algebra and the number c is called the central charge. Many interacting gapless quantumsystems do not possess simple particle-like excitations, making it difficult to quantify their effectivedegrees of freedom at low energy. CFTs constitute an important example. For 2D CFTs, theVirasoro central charge is a good measure of degrees of freedom.
The states which are annihilated by all positively moded Virasoro operators and are eigenstatesof L0 with eigenvalue a are called Virasoro primaries. The number a is called the weight of theVirasoro primary. Therefore, if |φ〉 is one of such primaries, it satisfies:
(L0 − a) |φ〉 = 0, Ln |φ〉 = 0, n > 0 (4.91)
The Virasoro algebra is graded by the number operator N , defined as:
N =
∞∑n=1
αµ−nαnµ. (4.92)
N.b. this is the covariant version of the operator defined in (4.47) for the light-cone. This operatorsatisfies (exercise): [
N,αµ−n]
= nαµ−n. (4.93)
29
Thus the oscillator α−n has ”charge” n. It follows that the Virasoro generator L−n has grade +n.
A Virasoro descendant of a given primary state |φ〉 is a state that can be written as a finite lin-ear combination of products of negatively moded Virasoro operators acting on the primary. Let’sconsider some examples of descendants. If |φ〉 is a primary, L−1 |φ〉 is a descendant at level one:
L−1 |φ〉 → descendant at level one (4.94)
The level is the eigenvalue of the number operator relative to the eigenvalue of the primary state.In general, as
[N,L−n] = nL−n (4.95)
one can prove that, if N |φ〉 = Nφ |φ〉 one has:
N (L−1 |φ〉) = L−1N |φ〉+ [N,L−1] |φ〉 = (Nφ + 1)L−1 |φ〉 (4.96)
Thus, L−1 |φ〉 is an eigenstate of N with eigenvalue Nφ+1. Similarly, the descendants at level 2 are:
L−2 |φ〉 , L−1L−1 |φ〉 . (4.97)
The descendants at level 3 are:
L−3 |φ〉 , L−2L−1 |φ〉 , L−1L−2 |φ〉 , L3−1 |φ〉 (4.98)
Notice that not all the states written above are independent because
L−1L−2 = [L−1, L−2] + L−2L−1 = L−3 + L−2L−1. (4.99)
In general, one can take the following basis of descendants at level n:
L−n1L−n2 · · ·L−nk |φ〉 , n1 ≥ n2 ≥ ... ≥ nk, n =∑i
ni, (4.100)
where on the left we have the conventional ordering of Virasoro operators. Therefore, the descendantshave the structure of the Verma module
(FIGURE IN CLASS)
Claim: The descendants are orthogonal to any primary.
Proof. Any descendant can be written as
|des〉 =∑i
ciL−ni |χi〉 (4.101)
with ci constants and |χi〉 states and ni > 0. Then, since L†−n = Ln we have:
〈des| =∑i
c∗i 〈χi|Lni (4.102)
and
〈des|primary〉 =∑i
c∗i 〈χi|Lni |primary〉 = 0. (4.103)
30
Definition:
A null state is a state which is both primary and descendant.
=⇒ Null state ⊥ any primary state
Moreover, a null state has zero norm. This is so because a null state is also primary and it thusmust be orthogonal to itself. Therefore, if |φ〉 is a primary and |null〉 is a null state:
|φ〉 and |φ〉+ |null〉have the same inner products with all primary states. (4.104)
Thus, adding null states to primaries cannot change any physical amplitude and two primary staesthat differ by a null state should be thought as physically indistinguishable. Therefore, the trans-formation:
|φ〉 → |φ〉+ |null〉 (4.105)
is a gauge symmetry. Recall that the states of ST must be primary states with weight a = 1.
Definition
A physical state of ST is an equivalence class of a primary state of weight a = 1 modulo the nullstates:
|phys〉 ∼ |phys〉+ |null〉 (4.106)
Thus, a physical state is a state that satisfies the Virasoro constraints with weight one and it is nota descendant.
Example
Let’s now illustrate these abstract ideas with a concrete example. Consider vector states of openstring, e.g.
|ξ; k〉 ≡ ξµαµ−1 |0; k〉 (vector state in covariant form) (4.107)
where ξµ is a constant polarization vector. Recall that
L0 =1
2α2
0 +N = −α′M2 +N, (4.108)
where N =∑∞n=1 α
µ−nαnµ is the number operator for all the oscillators.
=⇒ (L0 − 1)ξµαµ−1 |0; k〉 =
(−α′M2 +N − 1
)ξµα
µ−1 |0; k〉 (4.109)
Since Nαµ−1 |0〉 =[N,αµ−1
]︸ ︷︷ ︸αµ−1
|0〉+ αµ−1N |0〉︸ ︷︷ ︸=0
= αµ−1 |0〉
=⇒ (L0 − 1) |ξ; k〉 = −α′M2ξµαµ−1 |0; k〉 = −α′M2 |ξ; k〉 (4.110)
=⇒ (L0 − 1) |ξ; k〉 = 0 =⇒ M2 = 0 (4.111)
Let’s next study physical state conditions Ln |ξ; k〉 = 0, n > 0. (Lm = 12
∑∞n=−∞ : αm−n · αn :)
n = 1 (4.112)
L1 |ξ; k〉 = (α0α1 + α−1α2 + ...)ξµαµ−1 |0; k〉 = α0ν
[αν1 , α
µ−1
]ξµ |0; k〉 (4.113)
= ξµ αµ0︸︷︷︸√2lspµ
|0; k〉 =√
2lsξµ pµ |0; k〉︸ ︷︷ ︸kµ|0;k〉
=√
2ls(ξ · k) |0; k〉 (4.114)
=⇒ L1 |ξ; k〉 =√
2ls(ξ · k) |0; k〉 (4.115)
31
Thus, the condition that |ξ; k〉 must be physical is
L1 |ξ; k〉 = 0 =⇒ ξ · k = 0 (4.116)
Therefore the polarization and the momentum must be orthogonal for a physical state of spin 1 (avector particle). Notice that this is the same condition as the Lorentz gauge condition in QED.
It can be easily verified that the Lm,m > 0 automatically annihilate the state |ξ; k〉.
Let’s consider the following descendant state
|des〉 =λ√2ls
L−1 |0; k〉 , (4.117)
where λ = const. and we require k2 = 0. Taking into account that L−1 = α−1α0 + α−2α1 + ... werewrite:
|des〉 =λ√2ls
α−1α0 + α−2α1︸ ︷︷ ︸→0
+...
|0; k〉 =λ√2ls
αµ−1 α0µ |0; k〉︸ ︷︷ ︸=√
2lspµ|0;k〉
= λαµ−1kµ |0; k〉 (4.118)
=⇒ |des〉 = λkµαµ−1 |0; k〉 . (4.119)
Thus, |des〉 is a vector state with polarization ξµ = λkµ. Notice, that this state is physical because
kµξµ = λk2 = 0, (4.120)
Thus, for k2 = 0, the state |des〉 is both physical and descendant and therefore |des〉 is null. We candirectly compute the norm:
〈des|des〉 = |λ|2kµkν 〈0; k| αν1αµ−1︸ ︷︷ ︸
[αν1 ,αµ−1]=ηνµ
|0〉 = |λ|2kµkµ = 0 (4.121)
Since |des〉 is null, the states |ξ; k〉 and |ξ; k〉+ |des〉 are equivalent. This translates into the equiva-lence between polarizations ξµ and ξµ + λkµ : ξµ ≡ ξµ + λkµ. Notice that, for the associated vectorfield Aµ, the equivalence is the invariance under the transformation:
Aµ → Aµ + ∂µΛ, (4.122)
where Λ is an arbitrary function. (The polarization vector of ∂µΛ is just of the form λkµ.) Then,the gauge symmetry |phys〉 → |phys〉 + |null〉 for the Virasoro algebra implies the ordinary gaugesymmetry for an Abelian vector field. This is remarkable since the gauge symmetry is not imposeda priori (as in QFT). This has origins in the conformal invariance of the string.
4.7 Closed string spectrum (in the light-cone gauge)
Let’s study the closed string spectrum in the light-cone gauge. To determine this spectrum we’llanalyze the physical state conditions:
(L0 − a) |ψ〉 = 0
(L0 − a) |ψ〉 = 0,(4.123)
where we have used the same zero-point energy in both sectors, i.e. a = D−224 = 1. By subtracting
these two equations we get the so-called level-matching condition,
(L0 − L0) |ψ〉 = 0 (4.124)
32
Recall: L0 = − l
2s
4 m2 +
∑∞n=1 α−n · αn
L0 = − l2s
4 m2 +
∑∞n=1 α−n · αn
(4.125)
In the light-cone α+n = α+
n = 0
=⇒
L0 = −α
′
4 m2 +N
L0 = −α′
4 m2 + N ,
N =
∑D−2i=1
∑∞n=1 α
i−n · αin
N =∑D−2i=1
∑∞n=1 α
in · αin
(4.126)
=⇒ (L0 + L0 − 2) |ψ〉 = 0 (4.127)
In a basis where N, N are diagonal:
−α′
2m2 +N + N − 2 = 0 (4.128)
=⇒ m2 =2
α′(N + N − 2) and N = N (level matcing) (4.129)
Ground state
N = N = 0 =⇒ |0〉 scalar
From mass formula: m2 = − 4
α′tachyon
This is the so-called closed string tachyon. As in the open string case this state is removed in thesuperstring and we will not consider it anymore.
First excited state
N = N = 1 =⇒ Ωij = αi−1αj−1 |0〉
For these states: m2 = 0 massless
The state Ωij has the structure of a second order tensor (two indices). Let us decompose Ωij inthree parts:
• Symmetric traceless hij
• Antisymmetric Bij
• Trace φ
More concretely,
Ωij = hij +Bij +φ
24δij , φ = δijΩij (4.130)
Symmetric tensor can be obtained as follows:
hij =1
2
[Ωij + Ωji
]− φ
24δij (4.131)
One can check that hij = hji and δijhij = 0, i.e. it is symmetric and traceless. The antisymmetriccompoment of Ωij is defined as:
Bij =1
2
[Ωij − Ωji
], Bij = −Bji (4.132)
Let us now study the different particles represent by hij , Bij and φ.
33
• hij this represents a spin 2 particle with m = 0. In a covariant gauge it can be represented bya second order symmetric tensor hµν , which can be naturally identified with a graviton, i.e.with the fluctuations of the metric around the flat geometry gµν ≈ ηµν + hµν . We concludethat closed string theory contains quantum gravity.
• Bij is a two-form gauge potential which, in a covariant gauge, can be represented as theantisymmetric tensor Bµν = −Bνµ. It is massless.
• φ is a massless scalar field called the dilaton.
Thus, the massless spectrum of the closed string is
(gµν , Bµν , φ) (4.133)
In the covariant approach one can check that the null vectors of the Virasoro algebra generate generalcoordinate transformations and the gauge symmetry of the antisymmetric field.
********************************************
End of lecture 5
5 CFT in 2D
5.1 Scalar field in d dimensions
Before we jump into the conformal field theory in 2d dimensions, let us work out a toy example ofa scalar field φ coupled to gravity. The Euclidean action we have in mind is as follows
S =
ˆddx√gL(φ) , (5.1)
where the precise form of the Lagrangian density L(φ) is irrelevant to the discussion in hand.
Recall, that the energy-momentum tensor is defined as the variation of the action with respect to themetric gµν , i.e., the variation of an action under infinitesimal change of coordinates (diffeomorphism)gives
δS =1
2
ˆddx√gTµνδgµν (N.B. Euclidean) (5.2)
from which follows
Tµν =2√g
δS
δgµν. (5.3)
Consider dilatations (scale transformations)
xµ → xµ + εxµ, ε infinitesimal, i.e. xµ → xµ + εµ, εµ = εxµ (5.4)
Now gµν = δµν ,
=⇒ δS = −1
2
ˆddxTµν (∂µεν + ∂νεµ) = −ε
ˆddxTµµ = 0 (5.5)
=⇒ Tµµ = 0 (5.6)
Thus invariance under scale transformations implies tracelessness of Tµν .
What is the current whose conservation implies Tµµ = 0? Simply
jµd = Tµν xν (5.7)
34
(check ∂µjµd = (∂µT
µν )xν + Tµν δ
νµ = Tµµ = 0) This current is called the dilatation current.
Claim
If a theory with a local conserved EM tensor is scale-invariant, then it is invariant under a largergroup of symmetries.
Proof. Let’s construct a current:
Jµf (x) = Tµν(x)fν(x) (5.8)
The divergence reads
∂µJµf (x) = (∂µT
µν)︸ ︷︷ ︸=0
fν + Tµν∂µfν (5.9)
=1
2Tµν(∂µfν + ∂νfµ), (5.10)
assuming that Tµν = Tνµ. Suppose fν satisfies the so-called conformal Killing equation:
∂µfν + ∂νfµ = gµνF (x) , gµν metric (5.11)
This implies that
∂µJµf =
1
2TµνgµνF (x) =
1
2TµµF (x) = 0. (5.12)
The conformal transformations are the ones that multiply the metric by an overall factor
gµν(x)→ Ω(x)gµν(x) (5.13)
Let’s assume that the change of metric is induced by a change of coordinates:
xµ → xµ + εfµ =⇒ δgµν = −ε(∂µfν + ∂νfµ), (5.14)
where we assume fµ satisfies the conformal Killing equation (5.11) and so
gµν → [1− εF (x)] gµν . (5.15)
Take a trace of (5.11): 2∂µfµ = dF (x)
=⇒ ∂µfν + ∂νfµ =2
d(∂ · f)gµν , ∂ · f = ∂µfµ , (5.16)
which is the conformal Killing equation in d dimensions. The solutions to this equation whengµν = δµν i.e. Rd for d > 2 and d = 2 are very different. When d > 2 one can show that fµ is atmost quadratic in the coordinates xµ. There exist 1
2 (d+ 2)(d+ 1) solutions of the conformal Killingequation which correspond to
1. Translations
fµ = aµ , aµ = const. (d transformations) (5.17)
2. Rotations
fµ = bµνxν , bµν = −bνµ ,
(d(d− 2)
2transformations
)(5.18)
35
3. Dilatations
fµ = αxµ , (1 transformation) (5.19)
4. Special conformal transformations
fµ = 2(~x ·~b)xµ − bµ~x2, bµ = const. (d transformations) (5.20)
It can be shown that these transformations generate the group SO(d+ 1, 1).
5.2 Conformal symmetry in d = 2
Let’s solve the conformal Killing equation in d = 2 : ∂µfν + ∂νfµ = (∂ · f)δµν
µ = 0, ν = 1 =⇒ ∂0f1 = −∂1f0 (5.21)
µ = 0, ν = 0 =⇒ ∂0f0 = ∂1f1 (5.22)
µ = 1, ν = 1 =⇒ ∂1f1 = ∂0f0 (5.23)
=⇒ ∂0f0 = ∂1f1, ∂0f1 = −∂1f0 (5.24)
Introduce complex coordinatesz = x0 + ix1
z = x0 − ix1=⇒
x0 = (z + z)/2
x1 = (z − z)/(2i)(5.25)
and f = f0 + if1
f = f0 − if1
=⇒
f0 = (f + f)/2
f1 = (f − f)/(2i)(5.26)
We thus find Cauchy-Riemann equations. Define∂∂z = 1
2
(∂∂x0 − i ∂
∂x1
)∂∂z = 1
2
(∂∂x0 + i ∂
∂x1
) =⇒
∂∂x0 = ∂
∂z + ∂∂z
∂∂x1 = i
(∂∂z −
∂∂z
) (5.27)
which satisfy
∂
∂zz = 1 =
∂
∂zz,
∂
∂zz = 0 =
∂
∂zz (5.28)
so,
∂
∂zf = 0
holomorphic f=f(z)
or equivalently∂
∂zf = 0
anti−holomorphic f=f(z)
. (5.29)
Otherwise f, f are arbitrary. This means that 2d conformal Killing equation has infinitely manysolutions.
Notice that the transformations arex0 = x0 + εf0
x1 = x1 + εf1⇐⇒
z → z + εf(z)
z → z + εf(z)(5.30)
In finite form
z = g(z) , z = g(z) (5.31)
36
which means that holomorphic and anti-holomorphic coordinates transform into themselves. Theseare nothing but the standard conformal transformations of the C-plane. The only true conformalsymmetries in d = 2 are the ones corresponding to the conformal mapping the (extended) C-planeinto itself. These are the so-called Mobius (or bilinear) transformations of the form
f(z) =az + b
cz + d, a, b, c, d ∈ C, ad− bc = 1. (5.32)
The Mobius transformations are isomorphic to the group SL(2,C). This fact can be proved byassociating to the function f(z) the matrix
A =
(a bc d
). (5.33)
It is easy to verify that the composition of two maps corresponds to the product of two matrices.The condition ad− bc = 1 ⇐⇒ detA = 1, which is required to have an invertible mapping. Noticethat SL(2,C) is isomorphic to SO(3, 1), which is the Lorentz group in 4d and has 6 real parameters.We will refer also to these transformations as global conformal transformations.
Let us study the infinitesimal form of these mappings. As the identity is obtained for a = 1 = d, wetake
a = 1 + α
b = β
c = γ
d = 1 + δ
, at first order ad− bc ≈ 1 + α+ δ = 1 (5.34)
=⇒ δ = −α =⇒ f(z) =(1 + α)z + β
γz + 1− α≈ [(1 + α)z + β] [1 + α− γz] ≈ z + ε(z), (5.35)
ε(z) = β + 2αz − γz2 (5.36)
We can now identify different types of global conformal transformations.
ε(z) ∝ const. =⇒ translations (5.37)
ε(z) ∝ (const.)z =⇒ dilatations + rotations (5.38)
ε(z) ∝ (const.)z2 =⇒ special conformal transformations (5.39)
The Euclidean metric in z-coordinates:
ds2 = dzdz, gαβ =
(0 1/2
1/2 0
), gαβ =
(0 22 0
)(5.40)
Since gαβ is not diagonal, one should be careful wih the sub- and superscripts. For a vector Vµ:
Vz = gzzVz =
1
2V z, Vz = gzzV
z =1
2V z. (5.41)
EM-tensor becomes
T00 =
(∂z
∂x0
)2
︸ ︷︷ ︸=1
Tzz + 2∂z
∂x0︸︷︷︸=1
∂z
∂x0︸︷︷︸=1
Tzz +
(∂z
∂x0
)2
︸ ︷︷ ︸=1
Tzz (5.42)
=⇒
T00 = Tzz + 2Tzz + Tzz
T11 = −Tzz + 2Tzz − TzzT01 = i (Tzz − Tzz)
=⇒
Tzz = 1
4 (T00 − T11 − 2iT01)
Tzz = 14 (T00 − T11 + 2iT01)
Tzz = Tzz = 14 (T00 + T11)
(5.43)
37
Notice that Tzz = 14T
µµ , the EM tensor conservation can be written as
gµα∂αTµν = 0 (5.44)
ν = z: gµα∂αTµz = gzz∂zTzz + gzz∂zTzz = 0 =⇒ ∂zTzz + ∂zTzz = 0 (5.45)
ν = z: gµα∂αTµz = 0 =⇒ ∂zTzz + ∂zTzz = 0 (5.46)
If we have scale invariance (and thus conformal invariance), then Tzz = 0. Thus,
∂zTzz = 0 =⇒ Tzz holomorphic (5.47)
∂zTzz = 0 =⇒ Tzz anti-holomorphic (5.48)
The 2d problem has effectively reduced to two 1d systems.
5.3 Primary fields
Consider a covariant tensor φµ1...µn(x) of order n. Under change of coordinates x→ x′:
φµ1...µn(x)→ φ′µ1...µn(x) =∂x′ν1
∂xµ1· · · ∂x
′νn
∂xµnφν1...νn(x′). (5.49)
In complex coordinates z, z, conformal transformations are of the type:
z → f(z), z → f(z). (5.50)
A tensor φ then transforms as:
φzz...︸︷︷︸m
zz...︸︷︷︸m
(z, z)→ φ′z...zz...z(z, z) =
(∂f
∂z
)m(∂f
∂z
)mφz...z︸︷︷︸
m
z...z︸︷︷︸m
(f(z), f(z)) (5.51)
In general a primary field of conformal weight (h, h) is defined as a field which under an analyticcoordinate transformation behaves as:
φ(z, z)→ φ′(z, z) =
(∂f
∂z
)h(∂f
∂z
)hφ(f(z), f(z)), (5.52)
where h, h ∈ R. When h, h are integers we recover the transformation law for a covariant tensor asabove.
Notice that for a dilatation
f(z) = λz, f(z) = λz =⇒ φ(z, z)→ φ′(z, z) = λh+hφ(λz, λz). (5.53)
Changing λ→ λ−1, z → λz we get φ′(λz, λz) = λ−h−hφ(z, z), i.e. the scaling dimension of φ is
∆ = h+ h (5.54)
Under rotations z → eiαz, z → e−iαz:
φ(z, z)→ ei(h−h)αφ(eiαz, e−iαz) (5.55)
The difference h− h is called the conformal spin:
s = h− h (5.56)
38
Let’s consider infinitesimal transformations of the type:f(z) = z + ε(z)
f(z) = z + ε(z)=⇒
(∂f∂z
)h= (1 + ∂ε)
h ≈ 1 + h∂ε(∂f∂z
)h=(1 + ∂ε
)h ≈ 1 + h∂ε(5.57)
=⇒(∂f
∂z
)h(∂f
∂z
)hφ(f(z), f(z) ≈ (1 + h∂ε)(1 + h∂ε)
[φ(z, z) + ε∂φ+ ε∂φ
](5.58)
≈ φ+ h∂εφ+ ε∂φ+ h∂εφ+ ε∂φ (5.59)
=⇒ δε,εφ(z, z) =[(h∂ε+ ε∂) + (h∂ε+ ε∂)
]φ(z, z) (5.60)
We already argued that the only conformal transformations which are symmetries are the onescorresponding to the SL(2,C) subgroup, corresponding to ε ∝ 1, z, z2. Let U be the operator thatimplements any of these SL(2,C) transformations on the fields:
φ→ U−1φU, U−1 = U† (5.61)
We will assume that the vacuum of the theory is invariant under these transformations, i.e.
U |0〉 = |0〉 , 〈0|U† = 〈0|U−1 = 〈0| (5.62)
Let us then find the constraints induced on the correlators of primary fields by this invariance. Ifφ1, ..., φn are primary fields, then
〈0|φ1...φn |0〉 = 〈0|U−1φ1...φnU |0〉 = 〈0|U−1φUU−1φ2U...U−1φnU |0〉 (5.63)
i.e. the correlator is invariant under substitution φi → U−1φiU . On the other hand,
U−1φi(z, z)U = (∂f)h(∂f)hφ(f(z), f(z)) (5.64)
and infinitesimally U−1φiU = φi + δε,εφi. Thus
n∑i=1
〈φ1(z1, z1) · · · δε,εφi(zi, zi) · · ·φn(zn, zn)〉 = 0 (5.65)
Let us concentrate on the holomorphic part asε ∼ 1 =⇒ δ1φ = ε∂φ (translations)
ε ∼ z =⇒ δzφ = (ε∂ + h)φ (dilatations)
ε ∼ z2 =⇒ δz2φ = (ε∂ + 2zh)φ (special conformal transformations)
(5.66)
We get the following differential equations
n∑i=1
∂i 〈φ1(z1, z1) · · ·φn(zn, zn)〉 = 0 invariant under translations (5.67)
n∑i=1
(zi∂i + hi) 〈φ1(z1, z1) · · ·φn(zn, zn)〉 = 0 invariant under dilatations (5.68)
n∑i=1
(z2i ∂i + 2zihi) 〈φ1(z1, z1) · · ·φn(zn, zn)〉 = 0 inv. under special conf. transf. (5.69)
Similar conditions exist involving anti-holomorphic derivative.
The SL(2,C) conditions are enough to fix two- and three-point correlators of primary fields. Forexample (exercise)
G(z1, z2) = 〈0|φ1(z1)φ2(z2) |0〉 =δh1,h2
(z1 − z2)h1+h2(5.70)
39
5.4 Ward identities
The full set of conformal symmetries, although they are not a symmetry of the theory, can be usedto generate Ward identities for the correlators. In order to obtain these identities let us considerfirst a more general situation in which we tranform the fields as follows:
φ(x)→ φ′(x) = φ+ δωφ. (5.71)
Let X be a product of such fields
X = φ1(xi) · · ·φn(xn) (5.72)
The vacuum expectation value of X is
〈X〉 =1
Z
ˆ[dφ]Xe−S[φ] (5.73)
Let us then perform a change of variables in the functional integral of the above type φ → φ′.Obviously 〈X〉 does not change. However, X and S change as follows
X → X + δωX
S[φ]→ S[φ′] = S[φ]−´ddx jµa ∂µωa
(5.74)
where jµa follows from Noether’s theorem and the ωa are a set of infinitesimal parameters such thatδXµ = ωa
δXµ
δωa. Thus
〈X〉 =1
Z
ˆ[dφ′] (X + δωX) e−S[φ]+
´ddx jµa ∂µωa (5.75)
Let us assume that the integration measure does not change [dφ′] = [dφ] and let’s expand
(X + δωX) e−S+´jµa ∂µωa = Xe−S +
(δωX +X
ˆjµa ∂µωa
)e−S + ... (5.76)
Plugging back in:
〈X〉 =1
Z
ˆ[dφ]
[Xe−S +
(δωX +X
ˆjµa ∂µωa
)e−S
](5.77)
= 〈X〉+ 〈δωX〉+
ˆ〈jµa (x)X〉 ∂µωa(x) (5.78)
Therefore we get
〈δωX〉+
ˆ〈jµa (x)X〉 ∂µωa(x) = 0 (5.79)
or more explicitly, which is called the Ward identity :
n∑i=1
〈φ1(x1) · · · δωφi(xi) · · ·φn(xn)〉+
ˆddx ∂µωa(x) 〈jµa (x)φ1(x1)...φn(xn)〉 = 0 (5.80)
In particular, for infinitesimal coordinate transformation xµ → xµ + εµ we must substitute ωa → ενand jµa → Tµν :
n∑i=1
〈φ1(x1) · · · δεφi(xi) · · ·φn(xn)〉+
ˆddx ∂µεν(x) 〈Tµν(x)φ1(x1)...φn(xn)〉 = 0 (5.81)
40
Let us now apply this to 2d CFTs and suppose φ’s are primaries. In complex coordinates:
∂µενTµν = ∂µενTµν = ∂zεzTzz + ∂zεzTzz (5.82)
= 2∂zεzTzz + 2∂zε
zTzz = 2∂εTzz + 2∂εTzz, (5.83)
where we used Tzz = Tzz = 0.
=⇒n∑i=1
〈φ1(z1, z1) · · · δε,εφi(zi, zi) · · ·φn(zn, zn)〉 = −2
ˆd2x 〈
[∂εTzz + ∂εTzz
]φ1...φn〉 (5.84)
As ε and ε can be chosen independently, we shall concentrate on the holomorphic part. Let usconsider the particular function
ε(z) =1
w − z, (5.85)
which has a pole at z = w. We have
∂ε = ∂z1
w − z= −πδ(2)(z − w) (by Cauchy theorem) (5.86)
=⇒ δεφi(zi, zi) =
[hi
(∂zi
1
w − zi
)+
1
w − zi∂zi
]φi =
[hi
(w − zi)2+
1
w − zi∂zi
]φi (5.87)
The (holomorphic part of the) Ward identity becomes
n∑i=1
[hi
(w − zi)2+
1
w − zi∂zi
]〈φ1...φn〉 = −2
ˆd2x(−πδ(2)(z − w)) 〈Tzz(z)φ1...φn〉 (5.88)
Defining T (z) ≡ 2πTzz(z) we finally arrive at
〈T (w)φ1(z1, z1)...φn(zn, zn)〉 =
n∑i=1
[hi
(w − zi)2+
1
w − zi∂zi
]〈φ1(z1, z1)...φn(zn, zn)〉 (5.89)
Similarly one can get an expression for the insertion of the anti-holomorphic component of theenergy-momentum tensor.
********************************************
End of lecture 6
6 Operator formalism
We want to construct a formalism which incorporates at an operator level some of the features wehave obtained for the correlation fuctions in the path integral approach. As is standard in QFT,for this purpose we need to define a direction for time. In Euclidean theory the choice for this timedirection is rather arbitrary. Actually, it would be useful to work with a time axis along radialdirection of the plane. Let us motivate this choice. Take a theory in the (x0, x1)-plane, where x0 isthe Euclidean time coordinate. The complex coordinates:
ξ = x0 + ix1
ξ = x0 − ix1.(6.1)
Suppose that our system has finite spatial extension L and that we identify x1 = x1 + L. This isequivalent to say that (x0, x1) coordinates define a cylinder.
41
(FIGURE IN CLASS)
We now map the strip 0 ≤ Im ξ ≤ L to the plane by:
ξ → z = e2πL ξ, i.e. z = exp
(2π
L(x0 + ix1)
)(6.2)
(6.3)
(FIGURE IN CLASS)
Thus x0 flows in the radial direction as desired. The asymptotic time x0 = −∞ is the origin z = 0and x0 = ∞ corresponds to |z| = ∞. With this choice of time, the operator formalism is calledradial quantization.
Let’s define the following “in” state:
|φin〉 = limz,z→0
φ(z, z) |0〉 (6.4)
The ”out” states can be obtained from the “in” states by a conformal map
w =1
z(6.5)
which maps z = 0 to z =∞. If φ′(w, w) is the field after this transformation, it is natural to define
〈φout| ∼ limw,w→0
〈0|φ′(w, w) (6.6)
But under w → f(w):
φ′(w, w) = (∂f)h(∂f)hφ(f(w), f(w)) (6.7)
With f = 1w = z we have ∂f = − 1
w2 , ∂f = − 1w2 . Thus
φ′(w, w) = (−w−2)h(−w−2)hφ
(1
w,
1
w
)(6.8)
Omitting signs, we define (relabel z → w):
〈φout| = limz,z→0
〈0|φ(
1
z,
1
z
)1
z2h,
1
z2h(6.9)
We next wish to introduce a notion of Hermitean conjugation such that 〈φout| = |φin〉†. Let’sfirst motivate this choice. Recall that in these Euclidean cylindrical coordinates (x0, x1) the timeevolution is given by
φ(x0, x1) = eHx0
φ(0, x1)e−Hx0
(6.10)
If we want this time evolution to be compatible with the Hermitean conjugation we should takex0 → −x0 when taking the adjoint (if x0 = it with t Minkowski time, this corresponds to t → tunder the adjoint). After mapping from cylinder to plane, the effect of x0 → −x0 is equivalent to
z = exp
[2π
L(x0 + ix1)
]→ exp
[2π
L(−x0 + ix1)
]= exp
[−2π
L(x0 − ix1)
]=
1
z. (6.11)
Therefore, we should include in the definition of the adjoint the substitutions z → 1z , z →
1z . Taking
into account the powers of z and z induced by the mapping w = 1z , we define
[φ(z, z)]†
= φ
(1
z,
1
z
)1
z2h
1
z2h(6.12)
Let’s check that this works:
|φin〉† =
[limz,z→0
φ(z, z) |0〉]†
=
[limz,z→0
φ(z, z) |0〉]†
= limz,z→0
〈0| [φ(z, z)]†
(6.13)
which coincides with 〈φout|.
42
6.1 Equal-time commutators
Another important advantage of the radial quantization formalism is that it allows a systematiccalculation of equal-time commutators in CFT. To illustrate this point let us consider first a 2DQFT in Minkowski and let (t, x) be the coordinates. Let’s consider an operator Q(t) which can berepresented as the space integral of some local operator O(t, x):
Q(t) =
ˆdxO(t, x). (6.14)
Let’s for simplicity assume that O is bosonic. The equal-time commutator with another bosonicfield is
[Q(t), φ(t, x′)] =
ˆdx [O(t, x), φ(t, x′)] (6.15)
Generally the commutator of two local operators on the RHS is ill-defined and one needs some kindof regularization. We do this by time-splitting as follows:
[O(t, x), φ(t, x′)] = limε→0+
[O(t+ ε, x)φ(t, x′)− φ(t+ ε, x′)O(t, x)] (6.16)
Notice that the products of operators are time-ordered. This definition is quite natural if we wantto make contact with the path integral formalism, since in the latter the products of fields in acorrelator correspond to time-ordered products of operators in a vacuum expectation value. Bymaking use of the T-product the previous equation can be written as
[O(t, x), φ(t, x′)] = limε→0+
[T (O(t+ ε, x)φ(t, x′))− T (O(t, x)φ(t+ ε, x′))] (6.17)
so we get
[Q(t), φ(t, x′)] = limt→t′
[(ˆt>t′
dx−ˆt<t′
dx
)T [O(t, x)φ(t′, x′)]
]. (6.18)
In the radial quantization approach the time-ordering corresponds to radial ordering, which for twobosonic fields A(w) and B(w) is defined as:
R [A(z)B(w)] =
A(z)B(w), |z| > |w|B(w)A(z), |w| > |z|.
(6.19)
Let Q be defined as
Q =1
2πi
‰|z|∼const.
dzO(z) (6.20)
where the integration is performed anti-clockwise along a contour with fixed value of |z|. The equal-time (ET) commutator of Q with another field φ(w) is defined as:
[Q,φ(w)]ET = lim|z|→|w|
[[‰|z|>|w|
−‰|z|<|w|
]R [O(z)φ(w)]
dz
2πi
](6.21)
Let us manipulate the contour as follows:
(FIGURE IN CLASS)
i.e.|z|>|w|−
|z|<|w| =
w∼ integral along a small circle centered at z = w0.
=⇒[‰
dz
2πiO(z), φ(w)
]ET
=
‰w
dz
2πiR [O(z)φ(w)] (6.22)
43
The RHS is a contour integral which can be evaluated if the singularities of the integrand, whenz → w, are known. These singularities can be obtained by Wick theorem. Consider e.g. two scalarfields A(z) and B(w). According to Wick theorem we can write
R (A(z)B(w)) = A(z)B(w)+ : A(z)B(w) : (6.23)
where the normal-ordered expression is regular when z → w, and all singularities are contained in
contraction AB:
A(z)B(w) = 〈A(z)B(w)〉 (6.24)
Thus we can write the following operator product expansion (OPE):2
R (A(z)B(w)) ∼ 〈A(z)B(w)〉 (6.25)
(R symbol is typically omitted.) As an example take the scalar field ϕ. In the exercises you showthat
〈ϕ(x)ϕ(y)〉 = − 1
4πglog(x− y)2 + const. (6.26)
From this result we will infer the OPE:
ϕ(x)ϕ(y) ∼ − 1
4πglog(x− y)2 (6.27)
(We are not including the IR and UV regulators.) In complex coordinates, as (x−y)2 = (z−w)(z−w),we have
(ϕL(z) + ϕR(z))(ϕL(w) + ϕR(w)) ∼ − 1
4πg[log(z − w) + log(z − w)] (6.28)
which means that the non-vanishing OPEs are
ϕL(z)ϕL(w) ∼ − 1
4πglog(z − w) (6.29)
ϕR(z)ϕR(w) ∼ − 1
4πglog(z − w) (6.30)
Let us now consider the operator
Qε =1
2πi
˛dzε(z)T (z)− 1
2πi
˛dzε(z)T (z) . (6.31)
Let X = φ1(z1z1) · · ·φn(zn, zn) be a product of n primaries. Then, according to our prescription
[Qε, X]ET =1
2πi
˛C
dzε(z)R(T (z)X)− 1
2πi
˛C
dzε(z)R(T (z)X), (6.32)
where C is a contour which encircles the positions of the fields in X and can be chosen in such a waythat ε(z) is analytic (ε(z) antianalytic) inside C. We now need to know the singularities as z → ziof the OPE T (z)X. These singularities can be extracted from the value of the correlator 〈T (z)X〉,
2Notice that we are implicitly assuming that the operators are always sitting inside time-ordered correlators.Formally OPE is the asymptotic expansion: A(z)B(w) ∼
∑i Ci(z − w)O(w), where Oi are a complete set of local
operators and Ci are singular numerical coeffs. By dimensional analysis Ci ∼ 1/|z − w|∆A+∆B−∆Oi , where ∆ arethe scaling dimensions (5.54) of the operators.
44
which is determined by the conformal Ward identity. It follows that the singular behavior of theproduct of T (z) with a primary φ(w) of conformal weight h is given by (5.89):
R(T (z)φ(w)) =h
(z − w)2φ(w) +
∂φ(w)
z − w+ regular terms . (6.33)
Actually, the operator Qε should be the generator of coordinate transformations z → z+ ε, z → z+ εin the space of fields. If this is true the equal-time commutator [Qε, X]ET must be equal to δε,εX(exercise).
Example
Consider again a scalar field ϕ. Its classical EM tensor is:
Tµν = g
[−∂µϕ∂νϕ+
1
2ηµν∂λϕ∂
λϕ
](6.34)
As ηzz = ηzz = 0, ηzz = 12 , we haveTzz = g(−∂zϕ∂zϕ) = −g∂ϕL∂ϕLTzz = g(−∂zϕ∂zϕ) = −g∂ϕR∂ϕRTzz = g
[−∂zϕ∂zϕ+ 1
2ηzz(2ηzz∂zϕ∂zϕ)
]= 0.
(6.35)
Thus, since T (z) = 2πTzz, T (z) = 2πTzz, we get
T (z) = −2πg∂ϕL∂ϕL (6.36)
T (z) = −2πg∂ϕR∂ϕR (6.37)
These are classical expressions. As these are operators we need to be careful about normal ordering.
Let us concentrate on the holomorphic piece and drop L label subsequently. We define
T = −2πg : ∂ϕ∂ϕ :≡ −2πg limz→w
[∂ϕ(z)∂ϕ(w)− 〈∂ϕ(z)∂ϕ(w)〉] (6.38)
As
〈∂ϕ(z)∂ϕ(w)〉 = ∂z∂w 〈ϕ(z)ϕ(w)〉 = ∂z∂w
[− 1
4πglog(z − w)
](6.39)
= − 1
4πg∂w
1
z − w= − 1
4πg
1
(z − w)2(6.40)
we have
T (z) = −2π limz→w
[∂ϕ(z)∂ϕ(w) +
1
4πg
1
(z − w)2
](6.41)
Notice that, as the correlator 〈ϕ(z)ϕ(w)〉 has a logarithm log(z −w), the scalar ϕ is not a primary.However, from the correlator 〈∂ϕ(z)∂ϕ(w)〉 it seems that ∂ϕ is a primary with dimension 1. Let’sverify this point by computing the OPE T∂ϕ:
T (z)∂ϕ(w) = −2πg : ∂zϕ∂zϕ : ∂wϕ(w) ∼ −2πg · 2 : ∂zϕ∂zϕ∂wϕ(w) : (6.42)
= −4πg
[− 1
4πg
1
(z − w)2
]∂zϕ(z)
∣∣∣Taylor expand ∂ϕ(z) around z = w (6.43)
=1
(z − w)2
[∂wϕ(w) + (z − w)∂2ϕ(w) + ...
](6.44)
=∂ϕ(w)
(z − w)2+∂2ϕ(w)
z − w+ regular terms (6.45)
45
which indeed corresponds to a primary with weight 1.
It is important to notice that EM tensor itself is not a primary. To verify this, one can computethe OPE T (z)T (w) for the scalar field:
T (z)T (w) = (2πg)2 : ∂zϕ∂zϕ :: ∂wϕ∂wϕ : (6.46)
∼ (exercise) ∼ 1
2
1
(z − w)4+
2T (w)
(z − w)2+∂T (w)
z − w(6.47)
This OPE does not correspond to that of a primary field with h = 2 because of the presence of theanomalous term 1/(z − w)4. In general the OPE T (z)T (w) has the form
T (z)T (w) ∼ 1
2
c
(z − w)4+
2T (w)
(z − w)2+∂T (w)
z − w(6.48)
where c is the central charge. Only if c = 0 would T be a primary of dimension 2.3 For a scalarfields we have already seen that c = 1.
We noticed previously that ∂ϕ is a primary with h = 1. We can define the following operator:
j(z) = i√
4πg∂ϕ(z) (6.49)
whose 2-point function is
〈j(z)j(w)〉 =1
(z − w)2. (6.50)
j(z) is called the U(1) current. Associated to this current we can define the following charge:
Q =
˛j(z)
dz
2πi(6.51)
It can be easily checked thatQ commutes with T and in particularQ commutes with the Hamiltonian.It is therefore conserved:
[Q,T (w)]ET =
˛dz
2πij(z)T (w). (6.52)
But
j(z)T (w) = T (w)j(z) ∼ j(z)
(z − w)2+∂j(z)
z − w∼ j(w)
(z − w)2+
∂j(w)
z − w−∂j(z)
z − w(6.53)
=⇒ [Q,T (w)]ET = j(w)
˛dz
2πi
1
(z − w)2= 0. (6.54)
Notice that the energy-momentum tensor can be written as a quadratic expression in the currents:
T (z) =1
2: j(z)j(z) : (6.55)
This is the so-called Sugawara construction of the EM tensor.
For simplicity, let’s now pick g such that 4πg = 1:
T = −1
2∂ϕ∂ϕ, ϕ(z)ϕ(w) ∼ − log(z − w), j(z) = i∂ϕ(z). (6.56)
3Notice that c determines the correlator of two EM tensors 〈T (z)T (w)〉 =c/2
(z−w)4as the VEV of T (z) vanishes,
see below in (6.84).
46
We are going to consider OPEs involving exponential operators : eiαϕ :. For this purpose it is useful
to present a general result for OPEs of the form : Am :: Bn :. We’ll assume AB 6= 0 and that A,B
have bosonic statistics. Note that if we perform L contractions AB there are(mL
)ways to choose L
A’s which are going to be contracted. Once the A’s are chosen we have n(n− 1) · · · (n−L+ 1) waysof choosing B’s. Thus, the combinatorial factor for L contractions is:(
m
L
)n(n− 1) · · · (n− L+ 1) =
m!n!
(m− L)!L!(n− L)!(6.57)
Therefore:
: Am :: Bn :=
min(m,n)∑L=0
m!n!
(m− L)!L!(n− L)!(AB)L : Am−LBn−L : (6.58)
Let’s now apply this to computing the OPE j(z)Vα(w), where Vα(w) =: eiαϕ(w) :
j(z)Vα(w) = j(z) : eiαϕ(w) := i∂ϕ(z)
∞∑n=0
: (iαϕ(w))n :
n!· 2πg (6.59)
∼ −α∂ϕ(z)ϕ(w)︸ ︷︷ ︸− 1z−w
∞∑n=1
: (iαϕ(w))(n−1) :
(n− 1)!︸ ︷︷ ︸Vα(w)
·2πg (6.60)
=α
z − wVα(w) · 2πg (6.61)
It then follows that
[Q,Vα] = 2πgα
˛dz
2πi
1
z − w︸ ︷︷ ︸=1
Vα(w) = 2πgαVα(w) (6.62)
i.e. Vα(w) is also a primary of weight h = α2
8πg and scaling dimension ∆ = α2
4πg (exercise). Theoperators Vα are usually called vertex operators.
6.2 Mode expansions
Let φ(z, z) be a field with conformal weights (h, h). We shall expand it in modes as follows:
φ(z, z) =∑m,n∈Z
z−m−hz−n−hφm,n (6.63)
Notice that in this expansion there are positive and negative powers and therefore it is actually aLaurent expansion around z = z = 0. The mode operators φm,n can be determined as contourintegrals of φ(z, z) as follows:
φm,n =
˛dz
2πizm+h−1
˛dz
2πizn+h−1φ(z, z) (6.64)
where the integration contours enclose the origin.4
4 Proof: ˛dz
2πizm+h−1
˛dz
2πizn+h−1
∑p,q∈Z
z−p−hz−q−hφp,q
∣∣∣∣ ˛ dz
2πizm =
˛dz
2πizm = δm,−1
=∑p,q∈Z
δm−1−p,1δn−1−q,−1φp,q = φm,n
47
The Hermitean conjugate, according to our definition is
φ(z, z)† = z−2hz−2hφ
(1
z,
1
z
)= z−2hz−2h
∑m,n∈Z
zm+hzn+hφm,n
∣∣∣∣ (m,n)→ (−m,−n) (6.65)
=∑m,n∈Z
z−m−hz−n−hφ−m,−n (6.66)
If we take the Hermitean conjugate with the rules z† = z and z† = z we get instead from (6.63):
φ(z, z)† =∑m,n∈Z
z−m−hz−n−hφ†m,n (6.67)
If we want these expressions to be compatible we require
φ†m,n = φ−m,−n (6.68)
On the other hand,
φ(z, z) |0〉 =∑m,n∈Z
z−m−hz−n−hφm,n |0〉 (6.69)
The “in” state |φIN 〉 = limz,z→0
φ(z, z) |0〉 is well-defined if we require that
φm,n |0〉 = 0 for m > −h, n > −h (6.70)
From now on we shall drop the dependence on the anti-holomorphic coordinate and we’ll write themode expansion as:
φ(z) =∑m∈Z
z−m−hφm, φm =1
2πi
˛dz zm+h−1φ(z) (6.71)
Let us apply this formalism to the EM tensor. As T (z) is a field with (h, h) = (2, 0), we can write
T (z) =∑m∈Z
z−m−2Lm
T (z) =∑m∈Z
z−m−2Lm
(6.72)
(6.73)
These can be inverted:
Lm =
˛dz
2πizn+1T (z) Lm =
˛dz
2πizn+1T (z) (6.74)
where L†m = L−m and L†m = L−m.
The generator of conformal transformations is (its holomorphic part):
Qε =1
2πi
˛dzε(z)T (z). (6.75)
Expanding ε(z) in Laurent series ε(z) =∑m∈Z z
m+1εm we get
˛dz
2πiε(z)T (z) =
˛dz
2πi
∑m,n∈Z
zn+1z−m−2εnLm =∑m,n∈Z
δn−m−1,−1εnLn =∑n
εnLn (6.76)
48
Thus
Qε =∑n∈Z
εnLn (6.77)
From this expression we conclude that Ln is the generator of the conformal transformations of theform z → z + εzn+1. Thus:
Ln generates z → z + εzn+1
Ln generates z → z + εzn+1
(6.78)
(6.79)
Notice that the global conformal transformations correspond to n = −1 (translations), n = 0(dilatations), and n = 1 (special conformal transformations). In particular L0 + L0 generates thetransformation (z, z) → λ(z, z), which corresponds to radial translations. Therefore, in the radialquantization framework one can identify L0 + L0 with the Hamiltonian of the theory.
Requiring that the vacuum |0〉 is invariant under global conformal transformations is equivalent tothe conditions:
Ln |0〉 = 0, n = −1, 0, 1 (6.80)
and similarly for the anti-holomorphic modes. A generalization of this condition is obtained byrequiring that T (z) |0〉 be regular at z = 0,
T (z) |0〉 =∑m∈Z
z−m−2Lm |0〉 (6.81)
If we want that limz→0
T (z) |0〉 exists, only terms with −m− 2 ≥ 0 are allowed . We thus impose
Lm |0〉 = 0, m ≥ −1 (6.82)
As Lm = L†−m, L†−m |0〉 = 0,m ≥ −1 or L†m |0〉 = 0,m ≤ 1. Taking Hermitean conjugate, this gives
〈0|Lm = 0, m ≤ 1 (6.83)
Notice that the only operators which annhilate both |0〉 and 〈0| are L0, L±1. This generalizedcondition ensures that the VEV 〈0|T (z) |0〉 vanishes:
〈0|T (z) |0〉 =∑m∈Z
z−m−2 〈0|Lm |0〉 =
0,m ≥ −1, since Lm |0〉 = 0,
0,m ≤ −2 since 〈0|Lm = 0(6.84)
= 0 (6.85)
One can easily check that Ln’s satisfy the Virasoro algebra (exercise):
[Ln, Lm] = (n−m)Ln+m +c
12n(n2 − 1)δm+n (6.86)
If φ(z) is a field of conformal weight h, then (exercise)
[Ln, φ(w)] = h(n+ 1)wnφ(w) + wn+1∂φ(w) (6.87)
[Ln, φm] = [n(h− 1)−m]φm+n . (6.88)
49
6.2.1 State-operator correspondence for primaries recap
Let φ(w) be a primary with weight h. Let’s define the state |h〉 = φ(0) |0〉. As [Ln, φ(0)] =h(n+ 1)wnφ(w) + wn+1∂φ(w)
∣∣w=0
= 0, n > 0, for n = 0: [L0, φ(0)] = hφ(0)
=⇒ ... (6.89)
=⇒ Ln |h〉 = 0, n > 0 and L0 |h〉 = h |h〉 (6.90)
In string theory physical states are primaries with weight h = 1.
********************************************
End of lecture 7
7 Closed strings in complex variables
Now let’s return to string theory and resum what we learned, very quickly.
Wick rotate first τ → −iτ and definez = eτ−iσ
z = eτ+iσ, (7.1)
then collect from before
Xµ(z, z) = XµR(z) +Xµ
L(z), (7.2)
XµR(z) =
xµ
2− iα
′
2pµ log z + i
√α′
2
∑n 6=0
αµnnz−n, (7.3)
∂Xµ = −i√α′
2
∑n∈Z
αµnz−n−1, (7.4)
αµm = i
√2
α′
‰dz
2πizm∂Xµ(z), (7.5)
αµ0 =
√α′
2pµ (7.6)
OPEs of coordinates:
Xµ(z)Xν(w) ∼ −α′
2ηµν log(z − w), g =
1
2πα′(7.7)
This is a CFT with c = D.
7.1 Scalar primary state created by the vertex operator
|k〉 =: eikµXµ(0) : |0〉 (7.8)
50
Now act on |k〉 with αµ0 : Since
pµ |k〉 =
√2
α′αµ0
(limz→0
: eik·X(z) : |0〉)
(7.9)
= limz→0
√2
α′
(i
√2
α′
)˛dw
2πi∂Xµ(w) : eik·X(z) : |0〉 (7.10)
= i limz→0
2
α′
˛dw
2πi
[−iα′kµ
2
: eik·X(z) :
w − z+ ...
]|0〉 (7.11)
= kµ limz→0
: eik·X(z) : |0〉 = kµ |k〉 , (7.12)
we have that
αµ0 |k〉 =
√α′
2kµ |k〉 =⇒ pµ |k〉 = kµ |k〉 (7.13)
Thus |k〉 has momentum kµ. The conformal weight is
h =α′
4k2 = 1 ⇐⇒ k2 =
4
α′= −m2 (closed string tachyon) (7.14)
7.2 Next level
Consider next the tensor state
|ξ, k〉 = limz,z→0
ξµν : ∂Xµ(z)∂Xν(z)eikρXρ(z,z) : |0〉 (7.15)
And compute the OPE of T with the vertex operator:
T (z)ξµν : ∂Xµ(w)∂Xν(w)eikρXρ(w,w) : (7.16)
∼(exercise) (7.17)
∼− iα′
2
kµξµν(z − w)3
: ∂Xν(w)eikρXρ(w,w) : (7.18)
+
[α′k2/4 + 1
(z − w)2+
∂wz − w
]ξµν : ∂Xµ(w)∂Xν(w)eikρX
ρ(w,w) : (7.19)
This corresponds to a primary with h = 1 if:
m = −k2 = 0 (massless particles) (7.20)
and
kµξµν = 0 (polarization condition) (7.21)
8 String scattering
Recall the Polyakov action (Euclidean)
SEP =T
2
ˆΣ
d2σ√ggab∂aX · ∂bX, (8.1)
where Σ is the 2D surface of the worldsheet. At face value, this is a free theory. The string inter-actions are implicit and incorporated in string splittings and joinings, corresponding to worldsheets
51
of different topology. We learn that we need to sum over these worldsheet Riemann surfaces, byweighing them differently. The prescription is (Euclidean)
Sstring = SEP + λχ = SEP +λ
4π
ˆd2σ√gR, (8.2)
where χ = 2 − 2g for a genus g (can add crosscaps and boundaries...). The added Einstein termdoes not make gravity dynamical in 2D as it can be locally gauged away. The path integral becomes(λ = eφ) ∑
toposmetrics
∼∑topos
(eφ)2g−2
ˆDXDg
vole−SP [g,X] (8.3)
For small string coupling gs = eφ the genus expansion and the tree level approximation is givenby the sphere (closed string) or disk (open string). Notice that eφ plays the role of 1/N in gaugetheories; we have replaced Feynman diagrams by Riemann surfaces. The factor vol reminds the hugegauge redundancy, which is discussed at length using FP trick in Esko’s notes.
Now the upshot is that all string interactions are already included: one only focuses on one stringand considers different topologies! Think, e.g., closed string splitting into two
(FIGURE IN CLASS)
This can be mapped conformally to a sphere with 3 punctures:
(FIGURE IN CLASS)
Each branch of the cylinder, or at each puncture, we have an on-shell state string state. On-shellstrings states are created by vertex operators, so we insert, say, a tachyon Vα(w) =: eikρX
ρ(w,w) : ifthat’s what we are scattering. So, we wish to compute amplitudes of the form:
A(m)(Λi, pi) =∑topos
g−χs
ˆDgvol
ˆDXe−Sp
m∏i=1
V (Λi, pi), (8.4)
where V (Λi, pi) are the operators that insert the relevant asymptotic states at the punctures of theRiemann surface. These are the vertex operators, which must be such that:
1. Create states |Λ, p〉 of well-defined momentum, i.e. that they transform with phase eipiΛi undershifts Xµ → Xµ + aµ
2. Have the correct Lorentz group structure
3. Do not depend on local coordinates on the worldsheet
4. Be quantum-conformally invariant
A good initial guess is V (Λi, pi) =´d2zV (z, z; Λ, p), where
V (z, z; Λ, p) =: Λ(∂X(z, z))eip·X(z,z) : (8.5)
and where Λ(∂X) is some differential polynomial of X. Recall e.g. that for a graviton
V (Λ, p) =:
ˆd2zξµν∂X
µ∂XνeikρXρ
:, (8.6)
so we map correlation functions of 2D CFT into S-matrix elements of a string theory.
52
8.1 Tree level closed string amplitude
At three level we would write
A(m)(Λi, pi) =1
g2s
1
vol(SL(2,C))
ˆDXe− 1
2πα′´d2z∂X·∂X
m∏i=1
ˆd2ziVi(zi, zi) (8.7)
=1
g2s
∏mi=1
´d2zi
vol(SL(2,C))〈V1(z1, z1), ..., Vm(zm, zm)〉CFT (8.8)
Hence, we are essentially instructed to compute correlation functions of a scalar field on the sphere.The residual gauge symmetry is give by the global analytic transformations of C which are regularwhen z →∞. One can show that the group of such transformations is indeed SL(2,C). This meansthat global conformal transformations act on the coordinates zi and therefore relate many portionsof the multidimensional integral. Dividing by the volume of this redundancy is tantamount to fixingthe residual gauge symmetry. SL(2,C) has 3 independent complex parameters. This means thatwe can fix the residual gauge symmetry by placing 3 punctures to desired positions. Conventionalchoice is
z1 = 0, z2 = 1, z4 =∞ (8.9)
and z3 = z is the true relevant modulus (for 4 points) which parametrizes inequivalent contributions.
8.2 4-point tachyon amplitude
In the bosonic string the simplest vertex operator is the one for the tachyon state N = 0, i.e.m2 = −4/α′:
V (0; p) = gs
ˆd2z : eip·X := gs
ˆd2zV (z, z; p), (8.10)
where we have included a normalization factor gs. The theory is free, so Wick’s theorem is all wehave to do. First, we separate out the zero mode X(z, z, p) = xµ0 + (Xµ)′(z, z).
=⇒ 〈m∏i=1
V (zj , pj)〉CFT = gms
ˆdDxµ0e
ipµxµ0
ˆD(Xµ)′e−SP
ˆ m∏i=1
d2zi : eip·X′(zi,zi) : (8.11)
= gms δ
(∑i
pi
) ˆ m∏i=1
d2zi∏
1≤i<j≤m
|zi − zj |α′pi·pj (8.12)
=⇒ A(m)(Λi, pi) = δ
(∑i
pi
)gm−2s
vol(SL(2,C))
ˆ m∏i=1
d2zi∏i<j
|zi − zj |α′pi·pj (8.13)
Now we focus on m = 4 and fix SL(2,C) invariance by putting the insertions at 0, 1, z, and ∞:
A(4) ∼ g2sδ
(∑i
pi
) ˆd2z|z|α
′p1p3 |1− z|α′p2p3 (8.14)
The factor |z4|α′p4
∑3i=1 pi = |z4|−α
′p24 = |z4|−4 is absorbed into the overall normalization.5 By usingˆ
d2z|z|2a−2|1− z|2b−2 =2πΓ(a)Γ(b)Γ(c)
Γ(1− a)Γ(1− b)Γ(1− c), a+ b+ c = 1 (8.16)
5More honestly, one needs to properly compute the Jacobian of the transformation of the Riemann sphere:
z → z + ε(α+ βz + γz2) +O(ε2).
The transformation
z′ =(z − z1)(z2 − z4)
(z − z4)(z2 − z1)
53
one finds
A(4) ∼ g2sδ
(∑i
pi
)Γ(−1− α′s
4 )Γ(−1− α′t4 )Γ(−1− α′u
4 )
Γ(2 + α′s4 )Γ(2 + α′t
4 )Γ(2 + α′u4 )
(8.17)
(FIGURE IN CLASS) In terms of Mandelstam variabless = −(p1 + p2)2
t = −(p1 + p3)2
u = −(p1 + p4)2
(8.18)
which satisfy on-shell
s+ t+ u = −4∑i=1
p2i =
∑m2i = −16
α′. (8.19)
The amplitude we found is called the Virasoro-Shapiro amplitude.
8.3 Regge behavior
Consider the large s, fixed t limit. The A(4) has poles at s = 4(N − 1)/α′. These are exactly thevalues given by the states in the closed string mass spectrum. Moreover, the residues of these poleshave a leading behavior (in p) that goes as t2n i.e. p4n. Therefore the single string amplitude isequivalent to a sum over field theory amplitudes, where all the spectrum contributes, and at level nwith particles with maximum spin J = 2n:
A(4) ∼∞∑n=0
t2n
s−m2n
+O(t2n−1). (8.20)
We could also fix s, and take t → ∞. We would find poles in the t channel, corresponding to thesame resonances. In QFT these 2 channels should be added up. In string theory this is automatic.Pictorially:
(FIGURE IN CLASS)
In the early days of strings this dual way of writing the amplitude was called the dual resonancemodel.
8.4 Soft high energy behavior
Let’s now focus on a limit of colliding highly energetic strings. This can be done in many differentways. Let’s focus on s, t → ∞, s/t fixed. This corresponds to a limit, where we keep θ, the anglebetween p1 and p4 fixed. Take
p1 =√s
2 (1, 1, 0, ...)
p2 =√s
2 (1,−1, 0, ...)
p3 =√s
2 (1, cos θ, sin θ, 0, ...)
p4 =√s
2 (1,− cos θ,− sin θ, 0, ...)
(8.21)
brings any 3 points z1, z2 and z4 to 0, 1,∞. So the Jacobian∣∣∣∣∂(z1, z2, z3)
∂(α, β, γ)
∣∣∣∣2 = |(z1 − z2)(z1 − z4)(z2 − z4)|2 z4→∞−−−−−→ |z4|4 (8.15)
will cancel the |z4|−4 we got.
54
Using Γ(x) ∼ ex log x, and avoiding poles, as s, t→∞ we find
A(4) ∼ g2sδ
(∑i
pi
)eα′2 (s log s+t log t+u log u) (8.22)
which does not diverge. Had we truncated the sum to finite terms it would have! Conclusions:
1. Mathematically, infinite sums behave differently from finite ones.
2. Physically, the string length ls =√α′ prevents the string from probing distances smaller than
ls.
8.5 Open string scattering
The 2D Einstein-Hilbert action does not lead to well-defined equations of motion when boundariesare present. One needs to add a Gibbons-Hawking term:
χ =1
4π
ˆΣ
d2σ√gR+
1
2π
ˆ∂Σ
dsK, (8.23)
where K is the extrinsic curvature of the boundary
K = −tαnβ∇αtβ , (8.24)
where tα(nβ) is tangential (normal) to the boundary. Again χ is an integer (the Euler characteristic)for 2D manifolds with boundary:
χ = 2− 2g − b, (8.25)
where g is the number of handles (or genus), and b is the number of boundaries. Hence gχs weighsthe disc with g−1
s , the annulus with g0s etc. In the same spirit as for the closed string, the infinite
cylinder is now an infinite strip.
(FIGURE IN CLASS)
Notice that the ordering is fixed. mapping the operator insertions onto real line one needs to becareful to integrate over it to make the amplitude diffeomorphism invariant:
V (Λi, pi) =√gs
ˆ ∞+i0
−∞+i0
dz : Λ(∂x(z))eip·X(z) : (8.26)
where we normalized the open string vertex by gopen =√gclosed =
√gs. Let’s then compute the
same 4-point tachyon amplitude as for closed strings. However, there is a subtlety as X(z, z) is onlydefined in the upper half-plane. Suppose we wish to implement Neumann boundary conditions. Thepropagator is defined as
〈X(z, z)X(z′, z′)〉 = G(z, z; z′, z′), (8.27)
which obeys G = −2πα′δ(z − z′, z − z′), and is subject to Neumann boundary condition
∂σG(z, z; z′, z′)∣∣σ=0,π
= 0. (8.28)
Notice that
∂σ∣∣σ=0,π
= (−iz∂z + iz∂z)Im z=0 = −iRe z(∂z − ∂z)∣∣Im z=0
(8.29)
So the Neumann boundary condition reads
∂X(z) = ∂X(z) at Im (z) = 0. (8.30)
55
This means that there are fewer states propagating (than for closed strings) along the worldsheet asthe operators ∂X and ∂X give rise to the same state (for closed strings ∂X and ∂X are independent).
We solve this problem as in electrostatics. We let X(z, z) vary over the whole complex plane andadd an image charge in the lower half-plane. In other words, we define X over the whole complexplane with dynamics given by the propagator
G(z, z; z′, z′) = −α′
2log|z − z′|2 − α′
2log|z − z′|2 (8.31)
So, in complex coordinates z = eτ−iσ:
∂σG = (−iz∂z + iz∂z)G =iα′
2
(z
z − z′+
z
z − z′− z
z − z′− z
z − z′
)(8.32)
=iα′
2
((z − z)z′
(z − z′)(z − z′)+
(z − z)z′
(z − z′)(z − z′)
)(8.33)
=iα′
2
(z′
(z − z′)(z − z′)+
z′
(z − z′)(z − z′)
)· Im (z) (8.34)
which vanishes for Im(z) = 0. The amplitude is now computed with operator insertions along theboundary condition of the disk which maps onto the real axis of the complex plane. There is thusa fixed ordering implicitly in A. The Wick contraction is the same as for closed string with anadditional factor of 2 from the doubling trick. We also have the residual SL(2,R) (which keeps theordering) that maps the real line onto itself. We get
A(4) =gs
vol(SL(2,R))
ˆ 4∏i=1
dxi 〈eip1X(x1) · · · eip4X(x4)〉 (8.35)
∼ gsvol(SL(2,R)
δ
(∑i
pi
) ˆ 4∏i=1
dxi∏i<j
|xi − xj |2α′pipj (8.36)
For a given ordering, the residual symmetry can be used to fix 3 (real) points:
x1 = 0, x2 = 1, x3 = x, x4 =∞. (8.37)
The resulting expression contains a single integration for x ∈ [0, 1]:
A(4) ∼ gsˆ 1
0
dx|x|2α′p1p2 |1− x|2α
′p3p4
∣∣∣∣Euler beta; usem2 = − 1
α′(8.38)
∼ gsΓ(−1− α′s)Γ(−1− α′t)
Γ(−1− α′(s+ t))(8.39)
This is called the Veneziano amplitude.
The other channels come from considering all possible orderings of the vertex operators along thereal line and one thus obtains the full symmetry
s↔ t↔ u, (8.40)
A(4) ∼ gs [B(−1− α′s,−1− α′t) +B(−1− α′s,−1− α′u) +B(−1− α′t,−1− α′u)] (8.41)
From the pole analysis of this formula α(s) = α0 +α′s = N with an intercept α0 = 1 the resonancesoccur at
s = m2 =N − 1
α′(8.42)
which is precisely the spectrum of the open string. The level N states have the highest angularmomentum in the maximally symmetrixed situation αµ1
−1 · · ·αµN−1 |0〉, hence J = N . The Regge
behavior is:
J = 1 + α′m2. (8.43)
56
********************************************
End of lecture 8
9 Strings in background fields
Let’s write down an action for light-modes of string (Gµν , Bµν , φ) at low energies. Recall, e.g., forQFT with spontaneously broken global symmetry, the light modes are (pseudo)-Nambu-Goldstonebosons; pions in chiral perturbation theory for QCD. Such effective theories are called non-linearsigma models. Since low-lying modes of strings are gravitons etc., we need to study strings on ageneral target space metric Gµν .
Let’s start with the Polyakov action (Euclidean):
SP =1
4πα′
ˆd2σ√ggαβ∂αX
µ∂βXνGµν(X). (9.1)
Let’s then expand this action about flat spacetime Gµν(X) = ηµν + χµν(X):
e−S = e−SP(
1− 1
4πα′
ˆd2σ√ggαβ∂αX
µ∂βXνχµν(X) + . . .
). (9.2)
Recall from previous lectures the vertex operator for gravitons etc.:
V (ξ, k) =gsα′
ˆd2σ√ggαβ : ∂αX
µ∂βXνξµνe
ik·X : (9.3)
where ξµν was the polarization tensor (ξµνkµ = 0).
Let’s now compare this expansion in terms of flat space:
〈· · ·〉G = 〈· · ·〉G=η +
∞∑j=1
⟨· · ·
j∏i=1
Vi(ξiµν , ki)
⟩∣∣∣∣∣G=η
(9.4)
i.e.
(FIGURE IN CLASS)
We could had picked our reference metric any way we wished, but for the time-being, you only knowhow to compute amplitudes in Minkowski metric. Recall that the general background of masslessclosed string states can be decomposed into a graviton Gµν(X), an antisymmetric tensor Bµν(X),and a dilaton φ(X). We may thus write our non-linear σ-model action as:
Sσ =1
4πα′
ˆd2σ√g[(gαβGµν(X) + iεαβBµν(X)
)∂αX
µ∂βXν + α′R(2)φ(X)
]. (9.5)
We have written R(2) which stands for 2D Ricci scalar to distinguish it from the geometric quanti-ties of target spacetime. This non-linear σ-model corresponds to coherent states of massless fieldsgenerated by exponenting the massless vertex operators V.6
9.1 Sigma model expansion
Let’s now keep the background fields general and expand the worldsheet-to-target space map Xµ
about a point in the target space xµ0 :
Xµ(τ, σ) = xµ0︸︷︷︸classical config.
+ Y µ(τ, σ)︸ ︷︷ ︸N.b. not a vector
(9.6)
6An operator V creates a single quantum of a vacuum, whereas eV will create a coherent state.
57
Then the sigma model action expands as follows
Sσ =1
4πα′
ˆd2σ√g[gαβ∂αY
µ∂βYν (Gµν(x0) +Gµν,ω(x0)Y ω + . . .) (9.7)
iεαβ∂αYµ∂βY
ν (Bµν(x0) +Bµν,ω(x0)Y ω + . . .) (9.8)
+α′R(2) (φ(x0) + φ,ωYω + . . .)
]. (9.9)
In principle there should also be a term for the tachyon (which is a very interesting story), but sinceit will be projected out for superstrings we will not write it down. We interpret this action as a 2Dinteracting QFT. Notice the usual kinetic term for Y µ and that the couplings are given by spacetimederivatives of the metric G etc. There are infinitely many couplings: interactions depend on wherewe are in the field space. To make the expansion more transparent, we can introduce dimensionlesscoordinates:
φµ =Y µ√α′. (9.10)
It is then clear that the derivative expansion is in powers of
α′
L2, (9.11)
where L is the characteristic curvature scale of the target space. Recall also T = 12πα′ . So, the
expansion is valid when the string is too small to probe the curvature of spacetime.
9.2 Consistency
Recall from before that we were very careful about the consistency conditions when quantizingthe string theory. Let’s now check how to implement these consistency conditions in a generalbackground.
We ask if the path integral is independent of choices of gαβ related by Weyl transformations:
Z(Λ(σ)gαβ)?= Z(gαβ). (9.12)
We should also care about vacuum expectation values:
〈· · ·〉g =
ˆDXe−SP (· · · ). (9.13)
Then Weyl invariance would require
〈· · ·〉Λg = 〈· · ·〉g . (9.14)
Recall δSδgαβ
∼ Tαβ :
δ 〈· · ·〉g = − 1
4πα′
ˆd2σ√−gδgαβ 〈Tαβ〉g . (9.15)
One might require that the Weyl anomaly vanishes:
〈Tαα 〉 = 0. (9.16)
Weyl rescaling is just a dilatation so that in particular we have to require (at least) invariance withrespect to rigid dilatations. The vanishing of the Weyl anomaly is stronger than this, but we can
58
live with this weaker condition. Invariance under rigid dilatations tells us that the theory must bescale-invariant, i.e., the couplings must have vanishing β functions:
βλ(M0) = M∂
∂Mλ(M)
∣∣M=M0
= 0 (9.17)
In our 2D QFT we may compute 〈Tαα 〉 to any order in the sigma model expansion parameter α′/L2.
In unit gauge on a flat worldsheet embedding, we already know that energy-momentum tensor istraceless. This was part of our Virasoro constraints. For a more general embedding we know thatthe trace of EM tensor must be propto curvature, in order to ensure correct flat space limit:
Tαα = aR(2). (9.18)
After a very hard work one finds
a =26−D
12. (9.19)
There are corrections at higher orders of α′/L2:
Tαα = − 1
2α′βGµνg
αβ∂αXµ∂βX
ν − 1
2α′βBµνε
αβ∂αXµ∂βX
ν − 1
2βφR(2) (9.20)
where (after even harder work) the beta functions are:βGµν = α′Rµν + 2α′∇µ∇νφ− α′
4 HµλωHλων +O(α′2)
βBµν = α′
2 ∇ωHωµν + α′∇ωφHωµν +O(α′2)
βφ = D−266 − α′
2 ∇2φ+ α′∇ωφ∇ωφ− α′
24HµνλHµνλ +O(α′2) .
(9.21)
(For a detailed derivation, see e.g. the Tasi lectures by Callan & Thorlacius ’88.) and the 3-formHµνλ is the exterior derivative of the Bµν tensor, H = dB:
Hµνλ = ∂µBνρ + (permutations) . (9.22)
We can think of these as loop corrections (in the sigma model) to the beta functions. The conditionof Weyl invariance requires
βG(X) = βB(X) = βφ(X) = 0. (9.23)
It turns out that these equations can be derived from the 26D low-energy effective action, in thestring frame,
S26 =1
2κ2
ˆd26x√−Ge−2φ
[R− 1
12HµνλH
µνλ + 4∂µφ∂µφ+O(α′)
], (9.24)
where R = Rii is the scalar curvature. Here κ is yet-undetermined constant and all geometricquantities (
√−G,R etc.) are evaluated on the target space. Up to φ-rescaling this action is just the
Einstein-Hilbert + matter. The H2 is a 3-form generalization of the usual field strength with theusual kinetic term. We can now perform a metric transformation to get to Einstein frame:
GEµν = e−φ/6Gµν (9.25)
yielding
SE =1
2κ2
ˆd26x
√−GE
[RE − 1
12e−φ/3HµνλH
µνλ − 1
6∂µφ∂
µφ+ · · ·]. (9.26)
This is the familiar Einstein-Hilbert action with the canonical kinetic term for a scalar in 26D. Wethus make an identification of the overall factor
κ = (8πGN )(1/2) =(8π)(1/2)
MPl. (9.27)
59
Going to higher order in α′ one finds (very hard!)
βGµν = α′Rµν +α′2
2RµγλτR
γλτν +O(α′3), (9.28)
where the 2nd term is the stringy correction to GR.
It might seem discomforting that we need to impose these extra spacetime equations of motion ontop of Xµ equation of motion. One might be worried that only special subset of theories, such asS26, are relevant. This means that only a subset of string theories are consistent.
9.3 S-matrix picture
There is a different perspective to understand where these “additional” constraints are coming from.Let’s think of computing S-matrix elements of massless states (Gµν , Bµν , φ):
ˆDXV1 · · ·Vne−SP−λχ ∼ (FIGURE IN CLASS) (9.29)
We would need to do this for all number of insertions and for all Riemann surfaces. This is extremelyhard beyond torus. Let’s simplify and look at S-matrices for things like:
(FIGURE IN CLASS)
So, the question we pose: Is there an effective field theory in 26D, coupled to gravity, that hasS-matrix elements that match string theory up to O(α′)?:
AEFT?= Astring up toO(α′) (9.30)
that would hold by computing normal Feynman diagrams:
(FIGURE IN CLASS)
The answer is yes! The EFT that reproduces precisely the string S-matrix is S26!
Summary: We found out that (target space) Einstein equations appear both from Weyl consistencyin string theory and the scattering of gravitons etc. in the low-energy effective theory. Thus, justby starting with the Polyakov action SP we find that we force the background of our target spaceto obey Einstein equations!
9.4 Comments: two expansions: gs and α′
There are two expansion schemes in string theory. We just discussed the sigma model expansion,an expansion in α′. Earlier we have also discussed the string loop (genus) expansion, that is, anexpansion in gs.
If we want to use string theory to study, e.g., 4-graviton scattering, we can write our amplitude withrespect to each of these expansions. Let’s denote Aij , where i is the order of sigma model expansionand j in the loop expansion:
A(4) =A00 +A0
1gs +A02g
2s + . . . (9.31)
+ (9.32)
A10
(α′
L2
)+A1
1
(α′
L2
)gs +A1
2
(α′
L2
)g2s + . . . (9.33)
+ (9.34)
... (9.35)
60
By brute force one never gets further than this. However, the use of symmetries and non-renormalizationtheorems allows to get results to all orders in gs and α′.
Recall the stringy correction (9.28). Now we could ask where the α′2 term came from? It appearsto be an expansion in α′ × (curvature).
The true quantum gravity effects are gs effects since quantum gravity in target spacetime requires~ 6= 0. However, string theory provides other high-scale effects from “stringy” α′ physics. Forexample, we know that Einstein-Hilbert action for GR is L =
√gR. The correction to this at string
tree-level (i.e. O(g2s)↔ S2) is:
∆L =√gα′3ζ(3)
(RαβγδR
αβγδ + . . .). (9.36)
Note that we are at O(α′3) but still at string tree-level. Including string loop would introduce O(gs)terms which come from corrections like
〈g...g〉T 2 . (9.37)
In sum:
1. gs terms represent genuine loop expansion capturing effects from string theory as a quantumgravity.
2. α′ represents “stringy” corrections coming from string theory as a theory of extended objects.
Comment on S26: This is a long-wavelength effective field theory. It is not finite; we knew verywell that combining QFT with GR does not give a finite theory. What repairs this are α′ and gscorrections. However, beware we cannot compute the α′ corrections within an effective field theory.
********************************************
End of lecture 9
10 Superstring theory
Thus far we have focused on bosonic strings, which suffered from two shortcomings:
1. The tachyonic ground state, which implies vacuum instability.
2. String spectrum was bosonic. The lack of fermions makes it unrealistic.
Both of these issues are fixed in superstring theory. Regarding 1., the idea of including fermionswould somehow cancel the negative contribution to the vacuum energy from the bosons.
We will learn that both problems will be solved when we replace the bosonic string theory in 26Dwith the superstring theory in 10D. Supersymmetry (SUSY) exchanges bosons and fermions. Thesuperstring worldsheet theory contains both bosons and fermions. The bosonic sector is identical tothe worldsheet theory of the bosonic string.
There exist two major formulations of superstring theory. Both have SUSY on the worldsheet, butdiffer:
1. in the Ramond-Neveu-Schwarz (RNS) formulation, SUSY is manifest on the worldsheet, butnot on spacetime.
2. in the Green-Schwarz (GS) formulations, SUSY is manifest in spacetime, but not on theworldsheet.
More recently, the pure-spinor or Berkovits formulation has been developed as a yet another approachto superstrings. We will only consider RNS formalism.
61
10.1 Classical RNS action
Recall Polyakov action in unit gauge:
SP = − 1
4πα′
ˆd2σ∂αX
µ∂αXµ, (10.1)
where Xµ is a worldsheet scalar. Now we wish to add 2D worldsheet spinors. Consider thus:
S = − 1
4πα′
ˆd2σ
(∂αX
µ∂βXνηαβηµν + iα′Ψµρα∂αΨµ
), (10.2)
where we work in flat metric both in worldsheet and spacetime. It is important to have Minkowskisignature on worldsheet, since we want Minkowski signature Clifford algebra. New incredients areworldsheet fermions Ψµ obeying Ψµ,Ψν = 0, where µ is a target space Lorentz index and we havesuppressed spinor indices. The ρα are 2D Dirac matrices which obey 2D Clifford algebra
ρα, ρβ
= 2ηαβ . (10.3)
This admits a representation of 2× 2 real matrices
ρ0 =
(0 1−1 0
), ρ1 =
(0 11 0
). (10.4)
With ρ0 we can define the conjugate spinor
Ψ = Ψ†ρ0. (10.5)
Now the fact that ρ matrices are real, we can take Ψµ to be real, i.e. Majorana, two-componentspinors. We shall label them as Ψ = ψ− ⊕ ψ+, i.e. ψ∗+ = ψ+ and ψ∗− = ψ−.
We define a chirality operator
Γ = ρ0ρ1 =
(1 00 −1
). (10.6)
This is a 2D version of the 4D −iγ5 = diag (I2×2,−I2×2). These obey the usual relations
Γ2 = 1, Γ, ρα = 0. (10.7)
The existence of this operator which commutes with the Hamiltonian tells us that Γ eigenstatesdo not mix and we can decompose the reducible Ψ representation into irreducible representationsaccoding to chirality: Ψ = ψ− ⊕ ψ+.
Recap: We started with 2-complex-D Dirac spinors. The projection operator allowed us to constructchiral irreducible representations ψ± which are Weyl spinors, since Weyl spinor is the one that isprojected onto a chirality. But we also learned that our Dirac spinors could in fact be written as2-real-D Majorana spinors. So we actually have 2D Majorana-Weyl spinors.
This can be slightly counter-intuitive to those who are used to working with spinors in 4D, wherethere are no Majorana-Weyl representations. One can only simultaneously impose Majorana andWeyl conditions in spaces with dimension 2 mod 8. Now is a good time to recall that superstringslive in 10D.
The Dirac equation in 2D is
ρα∂αΨµ = 0 i.e.
(0 −∂0 + ∂1
∂0 + ∂1 0
)(ψµ−ψµ+
)= 0. (10.8)
62
In terms of ∂± = 12 (∂0 ± ∂1):
∂∓ψµ± = 0. (10.9)
Now we can work with the irreducible, single real-dimensional Majorana-Weyl spinors. This meansthat we can choose our fermions to be a single chirality and decompose into left- and right-movers.(Choice of ψ+ or ψ− is irrelevant.) We will write the left-movers as ψµ and right-movers as ψµ sothat the Lagrangian takes the form
LF ∼ ψµ∂ψµ + ψµ∂ψµ. (10.10)
Now what property does our original action have? Supersymmetry of course, take the SUSY trans-formation parameter to be
ε =
(ε−ε+
), (10.11)
a Majorana spinor, then the original action obeys a symmetry (exercise):δXµ = iεΨµ√
α′
2
δΨµ = 12ρα∂αX
µε√
2α′
(10.12)
Our action has N = (1, 1) superconformal symmetry. We have two N = 1 sectors in our theory: theindependent Qα and Qα Majorana-Weyl generators of SUSY (α, α = 1).
10.2 Boundary conditions: Ramond vs. Neveu-Schwarz
Let’s consider boundary conditions for closed superstring. We will use complex coordinate w = σ+iτ.Recall the action
Sψ =1
4π
ˆd2wψµ∂ψµ + ψµ∂ψµ. (10.13)
Note the invariance under w → w + 2π. The variation of the fermion action allows more choices,each mode can independently be periodic or anti-periodic:
ψµ(w + 2π) = ±ψµ(w)
ψµ(w + 2π) = ±ψµ(w)(10.14)
These boundary conditions have names Ramond (R) for periodic and Neveu-Schwarz (NS) for anti-periodic. Thus there are four distinct sectors, corresponding to the choice of boundary conditions:R-R, R-NS, NS-R, NS-NS.
A handy notation in the w-coordinates is to writeψµ(w + 2π) = e2πiνψµ(w)
ψµ(w + 2π) = e2πiνψµ(w),(10.15)
where ν, ν = 0, 1/2 according to R or NS boundary conditions, respectively. We can expand inFourier modes
ψµ(w) = i−1/2∑r∈Z+ν ψ
µr eirw
ψµ(w) = i−1/2∑r∈Z+ν ψ
µr eirw
(10.16)
63
In terms of z = e−iw coordinate we need to transform the fields:
ψµ(z) = (∂zw)1/2ψµ(w) = i1/2z−1/2ψµ(w). (10.17)
So, the mode expansions are ψµ(z) =
∑r∈Z+ν
ψrzr+1/2
ψµ(z) =∑r∈Z+ν
ψrzr+1/2
(10.18)
The extra 1/2 is the sign that ψµ(z) is a primary of weight h = 1/2. Therefore, on the sphere inRamond sector, the fields have a branch cut. This feature carries over to the vertex operators forphysical states. In particular, one can show that the vertex operator associated with the vacuumstate in the R sector introduces a branch cut on the sphere.
• Two such R vertex operators encircling will then result in a monodromy. These monodromiesmust be absent in a consistent theory so that OPEs are single-valued. This imposes severeconstraints on which sectors can be combined with each other into a consistent theory.
• If one computes one-loop amplitudes, i.e., using CFT on the torus, one finds that the result ismodular invariant if there is at least one left- and right-moving R sector in the theory.
10.3 Open strings and doubling trick
We have mentioned several times that the closed string is essentially two copies of the open string.Let’s describe this a bit more in the case of superstrings by what is called a doubling trick.
The boundary condition terms in the variation of the action for open strings (in light-cone gauge)are
δSψ =1
2π
ˆdτψi+δψ
i+ − ψi−δψi−
∣∣∣σ=0,π
. (10.19)
In order to have a non-trivial solution (imposing ψ±(τ, 0) = 0 would force the entire field to vanishby equation of motion) one has to relate the ψ+, ψ− chiralities so that
ψi−(τ, σ = 0, π) = ±ψi+(τ, σ = 0, π). (10.20)
Since we can choose the overall sign of ψ− and ψ+ arbitrarily (separately), we can choose – withoutloss of generality – to set
ψi−(τ, 0) = +ψi+(τ, 0). (10.21)
On the other end of the string we have a choice:
ψi−(τ, π) = ±ψi+(τ, π), (10.22)
+ for Ramond, - for Neveu-Schwarz. Now the trick: we can combine these boundary conditions intoa closed string over the interval σ ∈ [−π, π] (one can then shift to the usual interval σ ∈ [0, 2π]):
ψi(τ, σ) =
ψi−(τ, σ), σ ∈ [0, π]
ψi+(τ,−σ), σ ∈ [−π, 0] .(10.23)
This closed string has R or NS boundary conditions according to (10.22).
64
10.4 Light-cone gauge
We are not going to discuss superconformal field theory on the worldsheet in detail, but we will juststate what would happen upon quantizing. We could be naive and start with the canonical (anti-)commutation relations:
[αµm, ανn] = mδm+nη
µν
ψµm, ψνn = δm+nηµν .
(10.24)
Both of these are bad. We know from experience that these relations will give negative norm statesthat would invalidate our theory.
Let’s recall how we dealt with this in the bosonic case. We introduced the Polyakov action whichhad extra degrees of freedom gαβ , whose equation of motion gave the Virasoro constraints Tαβ = 0.We learned that these constraints were an avatar of conformal invariance of SP , associated with theleft-over diffeomorphism and Weyl gauge freedom and furnished a constraint (Virasoro) algebra toidentify physically allowed states.
For superstrings we could follow the same story to formulate a theory with a local worldsheet SUSYand derive the resulting constraint algebra. The equation of motion for ψµ will give us a conditionfor the worldsheet fermionic EM tensor TR = 0. This will be our superconformal symmetry, whosealgebra will act as our generalization of the Virasoro algebra acting on states. Anomaly cancellationwill give us the critical dimension: the SCFT calculations give D = 10.
In the bosonic string, conformal symmetry (diffeomorphisms ∩ Weyl) allowed us to set X+(τ, σ) =x+ + p+τ and thus remove the + oscillators α+ = 0, i.e. to go to light-cone gauge. In this gauge wewere able to solve for X−(τ, σ) in terms of transverse modes with[
αim, αjn
]= mδm+nδ
ij . (10.25)
Similarly, superconformal symmetry allows us to set ψ+(τ, σ) = 0 and solve for ψ−(τ, σ) in terms oftransverse ψi(τ, σ) with
ψir, ψjs
= δr+sδ
ij . (10.26)
10.5 Superstring spectrum
Let us go through the spectrum very quickly. The relevant details leading to that can be found e.g.in Polchinski. Let’s start with NS vacuum |0〉NS and force it to obey
ψµr>0 |0〉NS = 0, r =1
2,
3
2, ... (10.27)
The excited states are built in the usual way
ψµ−r |0〉NS , r =1
2,
3
2, ... (10.28)
However, because this is a fermionic raising operator, we can add one at a time. Now a bit confusingpoint: This is a worldsheet fermion but a spacetime boson.
Following the same arguments as for the bosonic string, we find a general mass formula:
α′M2 =
∞∑n=1
αi−nαni︸ ︷︷ ︸bosonic contribution
+
∞∑r=1/2
rψi−rψri︸ ︷︷ ︸fermionic
− 1
2︸︷︷︸zero-point energy
. (10.29)
65
The − 12 comes from the same argument as before: we count degrees of freedom and argue that they
are only consistent with a massless state.
Let’s now move to the Ramond sector and look into the low-lying, zero-mode, anti-commutationrelation
ψµ0 , ψν0 = ηµν (10.30)
and rescale Γµ ≡√
2ψµ0 :
Γµ,Γν = 2ηµν (10.31)
which is nothing but the D-dimensional Clifford algebra, where D = 10. We choose, as usual, that
ψµr |0〉R = 0, r = 1, 2, 3, ... (10.32)
But what about ψµ0 ? Imposing ψµ0 |0〉R = 0 turns out to be inconsistent, since from the anti-commutation relations we get
ψµ0ψν0 |0〉R + ψν0ψ
µ0 |0〉R = ηµν |0〉R 6= 0. (10.33)
ψµ0 takes ground states to ground states, which is to say that it is degenerate and furnishes arepresentation of 10D Clifford algebra. By the spin-statistics theorem in spacetime |0〉R is a spacetimefermion. These fermions turn out to be massless; showing this in detail would take too long.
To identify physical states, it is immediate in the light-cone gauge. The relevant anti-commutationrelation is √
2ψi0,√
2ψj0
= 2δij . (10.34)
This is the 8-dimensional Euclidean Clifford algebra, which is represented by eight 16× 16(= 28/2×28/2) Γ-matrices Γ1, ...,Γ8. One can again construct a chirality operator with usual properties7,which means that we can decompose 16 → 8+ ⊕ 8−; 16 refers to SO(9, 1) which goes to its littlegroup SO(8) in the light-cone gauge. To follow Polchinski’s notation we write as 8⊕ 8′. SO(8) hasthree eight-dimensional representations:
• two spinors (8, 8′)
• one vector 8v.
These are related by triality of the SO(8) Dynkin diagram D4:
(FIGURE IN CLASS)
Notice interesting analog to Standard Model: (FIGURE IN CLASS)
We know that R vacuum has to be a spinor representation of SO(8). Which one? We are free tochoose either 8 (|0〉+R) or 8′ (|0〉−R) since these are related by parity. The mass formula for the Rsector is
α′M2 =
∞∑n=1
αi−nαni +
∞∑r=1
rψi−rψri + 0, (10.35)
where we have explicitly written aR = 0. We now just claim that the ground states |0〉±R aremassless. Unlike with bosonic target spacetime representations, we are now dealing with targetspacetime fermions and it is no longer a trivial exercise to count degrees of freedom.
7Γ9 = Γ1 · · ·Γ8, (Γ9)2 = 1,
Γ9,Γi
= 0.
66
Summary: Our low-lying spectrum looks like
|0〉NS , α′M2 = −1
2tachyon (10.36)
ψi−1/2 |0〉NS , α′M2 = 0 8v (10.37)
|0〉+R , α′M2 = 0 8 (10.38)
|0〉−R , α′M2 = 0 8′ (10.39)
We have only written the left-movers; for right-movers one just adds tildes and bars. A full stateis written as a tensor product of left-moving and right-moving state, e.g. |0〉+R ⊗ |0〉
−R. But our
spectrum still has a tachyon, which is a negative-norm state; the situation is even worse as it breaksspacetime SUSY.
N.b. this naive spectrum contains a massless spin 3/2 field, a gravitino. Such a theory is onlyconsistent if it is SUSY.
10.6 GSO projection
So, we have a tachyon, which we wish to project out. We propose (which will be ”derived” later) asolution: perhaps there is a parity operator O (i.e. O2 = 1) which can be used to project out thetachyon in the same way that we use a chirality operator to project a Dirac spinor representation toa Weyl representation. Since our tachyon is just the NS vacuum, we would require
O |0〉NS = − |0〉NS . (10.40)
Of course, it is not a priori clear that such an operator exists. We argue for this later on. Gliozzi,Sherk, and Olive (GSO) proposed one such operator (−1)F , by
(−1)F =
(−1)
∑∞r=1/2 ψ
i−rψri+1 (NS)
Γ9(−1)∑∞r=1 ψ
i−rψri (R)
(10.41)
F is an operator which counts the worldsheet fermion number. The Γ9 turns out to be importantfor projecting out extra fermions to get a supersymmetric spectrum of spacetime fields. With thisoperator we find following parities (n.b. only left-movers):
(−1)F |0〉NS = − |0〉NS , (−1)Fψi−1/2 |0〉NS = +ψi−1/2 |0〉NS , (−1)F |0〉±R = ± |0〉±R . (10.42)
So it projects out the tachyon together with one of the R vacua. Great! But where did this operatorcome from?
********************************************
End of lecture 10
10.7 Modular invariance
To understand better where the GSO-projection came from, we need to take a peek at the quantumconsistency of our theory. Recall, the chiral anomaly cancellation condition in QCD by Fujikawa:the path integral measure has to be diffeomorphism invariant. For infinitesimal transformations thisis clear, but there is a special subset of diffeomorphic transformations where life is subtle: modulartransformations.8
Consider computing the vacuum amplitude on T 2, the analog of bubble diagrams on QFT:
8See Ginsparg CFT lectures hep-th/9108028.
67
(FIGURE IN CLASS)
T 2 is periodic in two directions, e.g., 0 ≤ σ1, σ2 ≤ 2π. Alternately, we can define T 2 with a complexstructure τ and a flat metric via
w = σ1 + τσ2 (10.43)
w = σ1 + τσ2 (10.44)
with the identifications w ∼ w + 2πn
w ∼ w + 2πτm, m, n ∈ Z (10.45)
and the metric ds2 = dwdw. The complex number τ determines the shape of T 2.
(FIGURE IN CLASS)
Now we consider a special set of transformations(σ′1σ′2
)=
(a bc d
)(σ1
σ2
),
ad− bc = 1
a, b, c, d ∈ Z
PSL(2,Z) (10.46)
These transformations are those of SL(2,Z), where P refers to reflections (a, b, c, d)→ −(a, b, c, d).This implies that a torus with modular parameter τ is conformally equivalent to one with
τ ′ =aτ + b
cτ + d, (10.47)
and we can restrict τ to upper half-plane H without loss of generality. Accordingly, the moduli spaceof conformally inequivalent Riemann surfaces of genus one is
Mg=1 = H/PSL(2,Z) . (10.48)
PSL(2Z) is generated by modular transformations
T : τ → τ + 1 ↔(
1 10 1
)and S : τ → −1/τ ↔
(0 −11 0
). (10.49)
A natural choice for the fundamental domain F is:
|Re τ | ≤ 1
2, Im τ > 0, |τ | ≥ 1. (10.50)
(FIGURE IN CLASS)
The moduli space has three cusps and singularities, where there is a deficit angle, located at τ =i,∞, eiπ/3. Therefore it is not a smooth manifold. Now let’s discuss the boundary conditions in σ1
and σ2, where we work in Euclidean time σ2. Recall from statistical mechanics that the path integralfor fermions has anti-periodic boundary conditions in time, i.e., under σ2 → σ2 + 2π in the torus.(Bosons have the usual periodic boundary condition.) For σ1 we know that there is a choice that wecan pick, + for Ramond and - for Neveu-Schwarz. We will denote these boundary conditions as
(σ1, σ2) =
(+,−), R
(−,−), NS(10.51)
Now the critical question: how do modular transformations affect the choice of boundary conditions?
Under S (τ → −1/τ): (σ′1σ′2
)=
(0 −11 0
)(σ1
σ2
)=
(−σ2
σ1
). (10.52)
68
Notice that we have swapped σ1 ↔ σ2, i.e. which circle is which. For boundary condition (α, β) thisproduces boundary condition (β, α). For example, the R boundary condition (+,−) gets mapped to(−,+). But what is (−,+)? This is a boundary condition that is anti-periodic in σ1 but periodic inσ2 (i.e. bosonic). This is a new type of a boundary condition! This was the goal that we wanted toachieve.
We started with emphasizing that diffeomorphism invariance should be anomaly free. We focusedon a special subset of diffeomorphisms (modular transformations). Starting with a theory of R & NSboundary conditions we found out that in order to be consistent, we must consider other boundaryconditions as well.
Similar thing happens with T -transformation (τ → τ + 1):(σ′1σ′2
)=
(1 10 1
)(σ1
σ2
). (10.53)
One can check that boundary condition (α, β) maps to (αβ, β). In particular, we map the NSboundary condition
(−,−)→ (+,−) (10.54)
which is nothing but a R boundary condition. The modular invariance then dictates that we musthave a Ramond state. Modular invariance thus relates
(−,−)↔ (+,−)↔ (−,+), (10.55)
but not (+,+). If one continues to two loops, one finds that (+,+) also joins in. But beforecontinuing, we need to understand + boundary condition in the Euclidean time for these 2D fermions.
10.8 Twisted partition function
Recall that computing the vacuum expectation value of an unit operator corresponds to the partitionfunction:
〈1〉T 2(τ) = Z(τ) (10.56)
(FIGURE IN CLASS)
Recall then from QM how we compute this:
Z = Tr e−βH =∑α
〈α| e−βH |α〉 . (10.57)
Now we read e−βH |α〉 as “|α〉 at tE = 2πτ2”:
e−βH |φ, tE = 0〉 = |φ, tE = 2πτ2〉 . (10.58)
In other words e−βH generates a time translation by an amount of β = 2πτ2. In order to understand(±,+) boundary condition, we want a similar transformation but witha sign flip. How do weintroduce such sign flip? Recall the operator (−1)F and let’s compute a twisted partition function:
ZF = Tr(−1)F e−βH =∑α
〈α| (−1)F e−βH |α〉 . (10.59)
This gives precisely the wanted sign flip upon time translation, at least for states with odd numberof worldsheet fermions. This “twisted partition function” is properly an index, it gives
Tr(−1)F e−βH |φ, tE = 0〉 = dim(+)− dim(−) |eigenstate〉 (10.60)
69
where dim(+) is the number of + chirality zero modes and dim(−) is the number of - chirality zeromodes. The reason that one does not get any contributions from excited states is that, since theSUSY generator Qα commutes with Pµ, any excited, non-zero enery state is paired with anotherstate of same energy and momentum but opposite statistics. As each non-zero energy state is pairedwith another with opposite statistics, they can only make the transition to/from zero energy in pairs,so the number of zero-energy bosonic states minus the zero-energy fermionic states does not changeunder dialing the parameters of the theory. The object capturing this difference is called the Wittenindex.
The last step is to combine this twisted partition function with the untwisted partition function,so that we account for all of our boundary conditions. The obvious thing is to just sum the twopartition functions, since the modular transformation just maps single states to other single states.For example, the map (−,−) → (−,+) also maps TrNS e
−βH → TrNS e−βH(−1)F . We are thus
lead to consider
Z =1
2Tre−βH + (−1)F e−βH
= Tr
(1
2(1 + (−1)F )e−βH
)(10.61)
The operator 12 (1+(−1)F ) is nothing but a projection onto the + eigenspace of the (−1)F operator.
This is precisely the GSO projection acting on the partition function! The GSO projection is thusrequired by the quantum consistency of the theory. We have a freedom to choose the overall chargeof (−1)F which just corresponds to between projecting out NS tachyon, 2nd, 4th excited states or1st, 3rd excited states. Obviously we choose the former.
We learned that if we start with NS sector, the modular invariance implies the existence of bothtwisted (−,−) and untwisted (−,+) NS sectors, as well as untwisted (+,−) R sector. We foundout that this meant that we needed to include GSO projection 1
2 (1 + (−1)F ) on the NS sector.Well, what would happen if we started from the R sector? We already know from 1-loop modularinvariance that the untwisted R sector (+,−) is related to the others in the NS sector.
The twisted (+,+) R sector has its own orbit under SL(2,Z) and it is not related to them. Onewould then be tempted to just take it and think it as decoupled from the rest. However, it turnsout that the twisted R sector is needed to be included for OPEs to close – for this one needs to lookinto two-loops. Thus, one cannot have a theory with only R sector and even in the R sector onemust also implement the GSO projection 1
2 (1 + (−1)F ). The correct prescription then is to includeall boundary conditions and project with the GSO in each sector.
10.9 Super-spectrum
Let us now work out the low-lying spectrum for closed superstrings. We introduce a notation NS±for ± eigenstates of (−1)F in the NS sector (R± similarly). Recall first that we have a choice to pickfor an overall sign of (−1)F , we choose +, so that our definition is
(−1)F ≡
+(−1)
∑∞r=1/2 ψ
i−rψri+1, NS
+Γ9(−1)∑∞r=1 ψ
i−rψri , R.
(10.62)
This then explicitly nails down, that for a generic NS state[D−1∏i=2
∞∏n=1
(αi−n)Ni,n
]m∏j=1
ψij−rj |0〉NS , (10.63)
(10.64)
are
NS+ ↔ m odd
NS− ↔ m even
(10.65)
(10.66)
70
For the R sector similarly: [D−1∏i=2
∞∏n=1
(αi−n)Ni,n
]l∏
j=1
ψij−rj |0〉
±R (10.67)
We have
R+ ↔
l even for |0〉+Rl odd for |0〉−R
R− ↔
l odd for |0〉+Rl even for |0〉−R
(10.68)
(10.69)
To remind, the lowest states are
NS+ : ψi−1/2 |0〉NS (10.70)
NS− : |0〉NS (10.71)
R± : |0〉±R (10.72)
We have only dealt with left-movers thus far. Now we need to combine them to right-movers bylevel-matching. We thus wish to construct the Fock space building upon |0〉left ⊗ |0〉right. Because
the modular transformations act exactly the same way for ψ(z) and ψ(z) boundary conditions, theresulting story for right-movers readily follows.
When combining left- and right-moving sector, we have some freedom. We thus far have chosen(−1)F |0〉NS = − |0〉NS , i.e. that the tachyon is projected out both for left- and right-movers. Westill have choices:
Left RightNS+ NS+
IIB R+ R+
IIB’ R− R−IIA R+ R−IIA’ R− R+
The symbols of the left-most column are the names of different string theories. The primed andunprimed ST’s are physically equivalent as they are related by chirality. This is because if we saythat our theory has fixed chirality, we cannot tell the difference between + and - (unless there ismore structure). For example, the vacuum states |0〉+R ⊗ |0〉
+R of type IIB are equivalent to vacuum
states |0〉−R ⊗ |0〉−R of type IIB’, up to chirality.
Digression: Other string theories
• Type IIA and IIB are not the only string theories. There are other which are consistent and wewill only shortly mention. One possibility is to take right-moving superstrings and left-movingbosonic strings. Such a combination leads to heterotic string theories – it turns out that thereare two possible ways to make such a merging with the corresponding gauge groups SO(32)or E8 × E8.
• Other possibilities include Type 0A and Type 0B theories. To understand them, let’s introducea notation of boundary conditions (left, right), for which our TypeIIA and IIB read as:
IIA = (NS+, NS+), (NS+, R−), (R+, NS+), (R+, R−) (10.73)
IIB = (NS+, NS+), (NS+, R+), (R+, NS+), (R+, R+) (10.74)
Notice that Type IIA and IIB are related by
(..., R−)→ (..., R+). (10.75)
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These boundary conditions were chosen so as to project out the tachyon. Had we chosendifferently we would have landed on (see Polchinski-II p. 27)
0A = (NS+, NS+), (NS−, NS−), (R+, R−), (R−, R+) (10.76)
0B = (NS+, NS+), (NS−, NS−), (R+, R+), (R−, R−). (10.77)
These theories are modular-invariant but they have a tachyon in their spectra. They also donot possess target spacetime SUSY. (Thus name 0; Type IIA/B have N = (1, 1)/(2, 0) SUSY.)
• There is yet another possibility: Type I theory. This is the only consistent 10D superstringtheory with open strings. Type IIA/B are oriented closed superstring theories. One can definea new unoriented theory by taking the quotient
(Type I)closed ≡Type IIB
Ω, (10.78)
where Ω is the worldsheet parity operation
Ω : σ → 2π − σ (10.79)
under which the chiral Type IIB is invariant. It turns out this quotient theory preserves 1/2of the SUSY of Type IIB and thus realizes N = 1 SUSY in 10D.
However, the theory defined as above is not fully consistent as such. Computation of one-loop amplitude requires summing over all topologies. Now one needs to consider also theunoriented genus 1 surface: the Klein bottle. The Klein bottle amplitude happens to be IR-divergent and so the appearance of the tadpole renders (Type I)closed inconsistent. The wayout is to add space-filling D9-branes. This means that we add open strings with Neumannboundary condition in every direction. The tadpole is exactly cancelled iff we add 32 D9-branes. The gauge group for 32 coincident D9-branes, subject to Ω, is SO(32) or more preciselySpin(32)/Z2.
• Until 1995 all these 5 (TypeIIA/B, I, Het SO(32) &E8×E8) seemed independent. It was thenrealized that all of them were related by dualities. They are all just different manifestations ofone underlying theory.
Let us now return to constructing the low-lying spectra of Type IIA/B ST’s. For Type IIA theorywe have ψi−1/2 |0〉NS
or
|0〉+R
⊗ψi−1/2 |0〉NS
or
|0〉−R
(10.80)
and for Type IIB we have ψi−1/2 |0〉NSor
|0〉+R
⊗ψi−1/2 |0〉NS
or
|0〉+R
. (10.81)
We can work out the SO(8) representations of our fields by again recalling that SO(8) came fromSO(9, 1) upon going to the light-cone gauge. Recall again the three 8D representations:
8v : ψi−1/2 |0〉NS (10.82)
8 : |0〉+R (10.83)
8′ : |0〉−R (10.84)
Now we just tensor them together and find
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8v ⊗ 8v NS-NS IIA & IIB8⊗ 8 R-R IIB8⊗ 8′ R-R IIA8v ⊗ 8 NS-R IIA
R-NS IIB8v ⊗ 8′ NS-R IIA
Tensor product decompositions are very straightforward.
8v ⊗ 8v
This is a product of two vector representations. We decompose the 2-index tensor cij :
cij → (sij − 1
8sii)︸ ︷︷ ︸
symm. traceless
+ aij︸︷︷︸antisymm.
+1
8sii︸︷︷︸
trace
(10.85)
c has 8× 8 = 64 degrees of freedom:
• symmetric traceless: 8·92 − 1 = 35
• antisymmetric: 8·(8−1)2 = 28
• trace: 1
So we get
8v ⊗ 8v = 35⊕ 28⊕ 1 (10.86)
another way of writing this is
8v ⊗ 8v = (2) + [2] + [0] (10.87)
where (n) means n symmetric indices and [n] means n antisymmetric indices. The correspondingfields that we found are the old familiar graviton Gij , antisymmetric tensor Bij and the dilaton φ.However, notice that for the superstrings these came from NS-NS sector!
Comment: We never consider states coming fromXµ oscillators. The masslessXµ fields are projectedout by GSO so that the lightest Xµ excitations live at the string scale and do not participate in thelow-energy action.
8v ⊗ 8
Let us write the generic state as |ξ〉αi , where i is the vector index and α is the spinor index. We canconstruct irreducible representations by
|ξ〉αi Γiαβ′ , (10.88)
where we recall that Γ converts indices of one chirality (8, α) to the other (8′, β′). Thus the stateabove is in the 8’ representation (dilatino). It turns out that we cannot decompose any further. Ourdecomposition is
8v ⊗ 8 : 64→ 8′ ⊕ 56 , (10.89)
where 56 is the irreducible vector-spinor representation ψαi (gravitino).
8v ⊗ 8′ By parity, 8v ⊗ 8′ has the same structure as above
8v ⊗ 8′ : 64→ 8⊕ 56′ , (10.90)
where 56′ → ψβ′
i .
8⊗ 8
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From QFT we know how to tensor spinors together. For example in 4D QFT we can write fermionbilinears as linear combinations of gamma matrices in between fermions. The relations were calledFierz identities. Thus for two spinors ζ and χ we would write:
ζ ⊗ χ ∼∑
ζ Γ[µ1...µm]χ, (10.91)
where sum is restricted over m in such a way that each term is a m-form. The decomposition is asfollows
8⊗ 8 = [0] + [2]︸︷︷︸a[ij]
+ [4]+︸︷︷︸c[ijkl]
= 1⊕ 28⊕ 35+ (10.92)
The subscript + for the 4-index tensor refers to the self-dual representation. Explicitly 35+ is a fieldC4 such that ∗dC4 = dC4. Recall that dC4 is a five-form which can be self-dual in 10D.
8′ ⊗ 8′
In the same way we can write
8′ ⊗ 8′ = [0] + [2] + [4]− = 1⊕ 28⊕ 35−, (10.93)
where - now refers to the anti-self dual ∗dC4 = −dC4.
8⊗ 8′
The tensor product of different chirality representations gives
8⊗ 8′ = [1] + [3] = 8v ⊕ 56T (10.94)
where the subscript T reminds that this is a tensor and not a vector representation.
Let us recap what we did. We started with 8v, 8, and 8′ representations and decomposed the relevanttensor products into the irreducible representations. We can write this into a table to compare thelow-lying spectra of Type IIA/B:
NS-NS bosons R-NS & NS-R fermions R-R bosonsIIA: (1⊕ 28⊕ 35) ⊕ (8 + 56′)⊕ (8′ ⊕ 56) ⊕ (8v ⊕ 56T )
φ,Bij , Gij λβ , ψβ′
i λβ′ , ψβi C1, C3
IIB: (1⊕ 28⊕ 35) ⊕ (8′ ⊕ 56)⊕ (8′ ⊕ 56) ⊕ (1⊕ 28⊕ 35+)
φ,Bij , Gij λβ′ , ψβi λβ′ , ψ
βi C0, C2, C4
Notice that the NS-NS sector is the same for both. This is of course expected since the onlydifference between IIA and IIB is the right-moving ground state: |0〉−R vs. |0〉+R, respectively. Theλ’s are dilatinos and ψ’s gravitinos and they always come in pairs.
Note also that IIA has fermion pairs of opposite chiralities while IIB has fermion pairs with samechirality. The Type IIB is chiral.
Finally, note that IIA has odd R-R (gauge potential) forms C1, C3, while IIB has even-formsC0, C2, C4. This will correlate with the occurrence of having (stable) Dp-branes in Type IIA forp = even and in Type IIB for p = odd.
********************************************
End of lecture 11
11 Supergravity
We are now familiar with Type IIA/B superstrings. However, we have thus produced the low-energyspectra and we would desire to do something. In particular , we need a prescription which describes
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the interactions between the fields. Recall the situation in 4D with rigid SUSY. The SUSY spectrumfollows almost trivially from the SUSY algebra. The interactions is the tricky part, for that one needsan action.
So, we need an action for superstrings, at least for low-lying states. More precisely, we need asupersymmetric action that is the low-energy effective theory associated with our string theory.Since the metric itself is one of the low-energy fields, we really want a supergravity (SUGRA) action.
Counting supercharges Let us first understand theories which have more than 1 supercharge.A classic example in 4D is the case with the largest amount of supercharges, whose SUSY algebrais coined N = 8. So why is there an upper bound on the possible SUSY? This can be understoodas follows. Imagine starting with a state with spin s. Acting with N supercharges would then giveus a state with spin |s − N2 |. If we restrict our theory to fields of spin no more than 2, then weimmediately find that N ≤ 8; as there are 8 half-steps between -2 and 2.
Now we would question, why do we restrict to fields with s ≤ 2? A hand-waving argument is thatsuch fields do not have a conserved current to which they can couple. For example, the gravitoncan couple to Tµν , but no higher-index current naturally occurs in our theories. We will now justassume that no interacting theory of spin s > 2 exists, which then restricts the possible theories, sothat when reduced to 4D cannot have more than N = 8 SUSY.
This 4D N = 8 SUSY algebra contains 8 × 4 = 32 real supercharges (the supercharges Qiα ∈ 2and Qα,i ∈ 2′ complex plus h.c., i = 1, ..., 8 so that R symmetry is R = U(8)). Thus any higherdimensional SUSY theory which is supposed to give a realistic 4D theory must live in a dimensionwhich admits spinors with at most 32 elements (not the same as the number of spinors). For example,we could consider a SUSY theory in D=12. The minimal Dirac algebra in 12D is a 64-componentspinor. This means that SUSY generators must have at least 64 components so that there are atleast 64 supercharges. A reduction down to 4D would give us more than N = 8, so D ≥ 12 cannotgive consistent 4D theories.
Comment: There are 3 notable exceptions that we are excluding:
1. The number of supercharges inside a spinor also depends on the signature of spacetime. So,one can have a 12D two-time theory; see the works by Itzhak Bars.
2. One can have an extra dimension to 11D as a book-keeping device, but not regarded as a newphysical dimension; see Cumrun Vafa’s F-theory.
3. If one considers an infinite tower of massless fields (see Vasiliev’s higher spin theory). This isan active field of study. One can understand some aspects in the context of string field theoryin the tensionless limit.
11.1 D=11 SUGRA
We are thus lead to theories with D ≤ 11. These have been studied a lot in the past. Only somewhatrecently it was noticed that the 11D SUGRA is the low-energy limit of M-theory.
The 11D spinor has 32 components and it has a unique action, called Cremmer-Julia-Scherk action.For simplicity, we will always only write the bosonic part of the action – the fermionic part followsfrom SUSY. In fact, fermions do not give contributions to the low-energy theory since they areinherently quantum and do not acquire vacuum expectation values in the classical limit. The actionreads
S11D =1
16πGN11
ˆ [d11x√−G(R− 1
48F 2
4 ) +1
6A3 ∧ F4 ∧ F4
]+ fermions. (11.1)
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Here GN11 is the 11D gravitational constant which sets the overall length scale (i.e. it does notrepresent an extra parameter). Since there are no parameters, this action really is unique. F4 = dA3,so that |F4|2 is the usual kinetic term. A3 ∧ F4 ∧ F4 is a Wess-Zumino term, which is necessary forSUSY to hold. This action does not have a cosmological term.
11.2 Down to D=10 SUGRA
Let’s just start with the previous 11D SUGRA action, and make a dimensional reduction to get to10D. In 10D, however, the action is not unique anymore, but there are 2 possibilities for SUGRAactions which possess the necessary 32 supercharges which resulted from 11D. These two theoriesthat we get are called Type IIA/B SUGRAs and they are low-energy limits of Type IIA/B stringtheories. These SUGRA actions were known before they were realized to describe the limits of TypeIIA/B STs.
We start with the 11D Majorana generator Qα, a real 32-component spinor. We can reduce this toIIA via
(IIA) Qα → Q(1)α +Q
(2)α′ . (11.2)
In terms of representations Q(1) is a 16 and Q(2) is a 16’ of SO(9, 1); these go to 8 and 8’ of SO(8)in the light-cone gauge. Both are 10D Majorana-Weyl spinors. Notice that only one index has aprime which signals that they denote opposite chirality.
Alternatively, we could reduce to 10D via
(IIB) Qα → Q(1)α +Q(2)
α , (11.3)
where now Q(2) is also 16 and the model is chiral. Both of the indices are unprimed, this correspondsto IIB SUGRA.
Instead of simply dimensionally reducing to get the IIA SUGRA action (exercise), one can find theaction as in the bosonic case. Recall, that the vanishing of the Weyl anomaly demands:
βµν(G) = βµν(B) = βµν(φ) = 0, (11.4)
where these equations are covariant complicated expressions of the massless fields. In Type IIsuperstrings, in addition, we need to include the RR-forms in a way compatible with SUSY. Thus,these equations coincide with those arising in 10D theories of SUGRA. Again, higher order α′
corrections would lead to higher powers of curvature, as in the bosonic string. But before we willpresent the Type IIA/B SUGRA actions, we should understand RR-forms a bit better.
11.2.1 Coupling RR fields to extended objects
Recall that the antisymmetric Bµν in the NS-NS sector couples directly to the string worldsheet,the string carries (electric) charge with respect to Bµν :
Sstring = − 1
4πα′
ˆd2σεαβBµν(X)∂αX
µ∂βXν . (11.5)
The Lagrangian changes by a total derivative under the gauge symmetry:
Bµν → Bµν + ∂µΛν − ∂νΛµ. (11.6)
In electromagnetism, the gauge invariant degrees of freedom are contained in the field strengthF = dA. Similarly,
H = dB, Hµνγ = ∂µBνγ + ∂νBγµ + ∂γBµν . (11.7)
76
However, the situation for RR-potentials C(n) is very different, because the vertex operators for theRR-states involve only F (n+1). Thus, only field strengths, not the potentials, would naively be ableto couple to strings
=⇒ Thus, elementary, perturbative string states cannot carry any charge with respect to RR gaugefields C(n).
We are therefore forced to look for non-perturbative degrees of freedom which couple to these po-tentials. Clearly, they must be extended objects that sweep out a (p+ 1)-dimensional world-volume∑p+1 as they propagate in time, generalizing the notion of a string:
q
ˆ∑p+1
dp+1ξεa0a1...ap∂Xµ1
∂ξa0
∂Xµ2
∂ξa1· · · ∂X
µp+1
∂ξapC(p+1)µ1...µp+1
→ q
ˆ∑p+1
C(p+1) (11.8)
in complete analogy with EM and B-field minimal couplings.
Now let’s consider F (p+2), a (p + 2)-form field strength representing an antisymmetric tensor fieldwith p + 2 indices in D dimensions. It is the field strength of a potential F (p+2) = dC(p+1), thatelectrically couples to a (p+ 1)-dimensional object:
µp
ˆ∑p+1
C(p+1). (11.9)
The Hodge dual of F (p+2) is
F (D−p−2) = ∗F (p+2) = dC(D−p−3). (11.10)
Its potential couples magnetically to the extended object
µD−p−4
ˆ∑D−p−3
C(D−p−3). (11.11)
In superstring theory then, in D = 10, we thus have two possible couplings:
F (p+2) couples to
electric p-branes
magnetic (6-p)-branes.(11.12)
Recall that F (n) forms result from tensor product of two Majorana-Weyl spinor representations in10D (notice that we now switch back to SO(9, 1) representation instead of SO(8)):
F (n)µ1...µn = ψleftΓ[µ1...µn]ψright. (11.13)
Thus given that
Γ11Γ[µ1...µn] ∝ εµ1···µnν1···ν10−n
Γ[ν1...ν10−n], (11.14)
there is an isomorphism, electric-magnetic duality,
F (n)µ1...µn ∼ ε
ν1···ν10−nµ1···µn F (10−n)
ν1...ν10−n. (11.15)
This identifies the representations [n]↔ [10− n], in particular [5]+ is self-dual.
Recall that in IIA: 16⊗ 16′ = [0]⊕ [2]⊕ [4] (again in terms of SO(9, 1) representations). Thus thereare even branes.
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IIA:
NS −NS (11.16)
H3 couples to
electric 1-branes ↔ F1-string
magnetic 5-branes ↔ NS5-brane(11.17)
R−R (11.18)
F (2) couples to
electric 0-branes ↔ D0-brane
magnetic 6-branes ↔ D6-brane(11.19)
F (4) couples to
electric 2-branes ↔ D2-brane
magnetic 4-branes ↔ D4-brane(11.20)
In Type IIB: 16⊗ 16 = [1]⊕ [3]⊕ [5]+. Thus there are odd branes.
IIB:
NS −NS (11.21)
H3 couples to
electric 1-branes ↔ F1-string
magnetic 5-branes ↔ NS5-brane(11.22)
R−R (11.23)
F (1) couples to
electric -1-branes ↔ D(-1)-brane
magnetic 7-branes ↔ D7-brane(11.24)
F (3) couples to
electric 1-branes ↔ D1-brane
magnetic 5-branes ↔ D5-brane(11.25)
F (5) couples to
electric 3-branes ↔ D3-brane
magnetic 3-branes ↔ D3-brane(11.26)
Since F (5) is self-dual, the EM duality states that D3-brane is both electric and magnetic.
Comments:
• Type IIA/B do have Dp-branes for odd/even. It just that they are not charged under the R-Rsector, they are non-BPS states, and are subject to decay.
• D(-1) brane is an instanton, which has Dirichlet boundary conditions on time and all thespatial directions.
• ∃D8-brane, but this is a special case and represents a domain wall as there is only one spatialorthogonal direction to its worldvolume. The sugra term generated would be S ∝ d10x
√−gF 2
10,but F10 turns out not to be dynamical but a constant. This term thus corresponds to acosmological constant term (Romans mass) and leads to what is called massive Type IIAsugra. It is rather tricky to have single D8-branes residing in otherwise flat and empty spaceas the dilaton diverges at a finite distance away from them on either side.
• ∃D9-brane, which is also rather special and represents a spacetime filling brane. Notice thatC10 does not have a field strength, so there is no sugra solution corresponding to D9-brane.But C10 can still be included in the SUSY algebra and its gauge and SUSY trafos formallydetermine the worldvolume action of the D9-brane.
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11.2.2 Electric-magnetic duality
Let us recall how the story goes in Maxwell theory, where this was found out by Montonen andOlive. In the absence of charges and currents, the equations of motion are
dF = 0, d ∗ F = 0 , (11.27)
where F is the 2-form field strength describing electric and magnetic fields. The equations of motionare symmetric under interchanging F ↔ ∗F . Assuming that sources can be added symmetrically,
dF = ∗Jm, d ∗ F = ∗Je , (11.28)
we face the Dirac quantization condition: the wave function of an electrically charged particle movingin a field of a monopole is uniquely defined if
e ·m = 2πn, n ∈ Z. (11.29)
We saw that our D-branes can both couple electrically and magnetically. Their charges are measuredusing Gauss’ law. The Dirac quantization condition straightforwardly generalizes:
µp · µ6−p = 2πn, n ∈ Z (11.30)
11.2.3 Type IIA SUGRA action
The action of Type IIA SUGRA in the string frame reads:
SIIA =1
2κ2
ˆe−2φ
[ˆd10x√−gR+ 4dφ ∧ ∗dφ− 1
2 · 3!H(3) ∧ ∗H(3) (11.31)
−1
2
(1
2!F (2) ∧ ∗F (2) +
1
4!F (4) ∧ ∗F (4) +B(2) ∧ F (4) ∧ F (4)
)], (11.32)
where
F (2) = dC(1), F (4) = dC(3) − C(1) ∧H(3), H(3) = dB(2). (11.33)
We can also go to the Einstein frame by
(gµν)string = g−1/2s eφ/2(gµν)Einstein, (11.34)
where gs = eφ(r→∞) is the string coupling. Then,√|gstring| = (g−1/2
s eφ/2)5√|gEinstein| = g−5/2
s e5φ/2√|gEinstein|. (11.35)
In general, for any p-form
(F (p) ∧ ∗F (p))string = gp/2s e−pφ/2(F (p) ∧ ∗F (p))Einstein. (11.36)
The resulting action in the Einstein frame reads:
SEinsteinIIA =1
2κ2
ˆ [d10x√−gERE −
1
2dφ ∧ ∗dφ− 1
2 · 3!e−φH(3) ∧ ∗H(3) (11.37)
−1
2
(1
2!e3φ/2F (2) ∧ ∗F (2) +
1
4!eφ/2F (4) ∧ ∗F (4) +B(2) ∧ F (4) ∧ F (4)
)]. (11.38)
Gravity is now canonically normalized, as well as the dilaton kinetic term, but the coupling with theRR-forms is more involved.
Recall that one can land on this action either by demanding Weyl consistency or dimensionallyreducing down to 10D from 11D.
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11.2.4 Type IIB SUGRA action
The story in IIB is similar to IIA, there is one subtle point. Up until recently (see 1511.08220),there was no way of writing a Lorentz invariant action from which the equations of motion could bederived. Thus far what one did was to write down an action and supplement with it the constraintof self-duality of the 5-form field strength. This constraint needed to be imposed after derivingequations of motion from the action. However, since the construction 1511.08220 is rather involved,we will follow now the old method as it also leads to the same correct equations of motion. Thus,the action of Type IIB SUGRA in string frame reads:
SIIB =1
2κ2
ˆe−2φ
[ˆd10x√−gR+ 4dφ ∧ ∗dφ− 1
2 · 3!H(3) ∧ ∗H(3)
](11.39)
− 1
4κ2
ˆ [F (1) ∧ ∗F (1) +
1
3!F (3) ∧ ∗F (3) +
1
2 · 5!F (5) ∧ ∗F (5)− C(4) ∧ F (3) ∧H(3)
],(11.40)
where
F (1) = dC(0), F (3) = dC(2) − C(0) ∧H(3), H(3) = dB(2), (11.41)
F (5) = dC(4) − 1
2C(2) ∧H(3) +
1
2B(2) ∧ F (3), (11.42)
supplemented by the additional on-shell constraint
F (5) = ∗F (5). (11.43)
This can also be cast in the Einstein frame form
SEIIB =1
2κ2
ˆ [d10x√−gERE −
1
2dφ ∧ ∗dφ− 1
2 · 3!e−φH(3) ∧ ∗H(3)
](11.44)
−1
2
[e2φF (1) ∧ ∗F (1) +
1
3!eφF (3) ∧ ∗F (3) (11.45)
+1
2 · 5!F (5) ∧ ∗F (5)− C(4) ∧ F (3) ∧H(3)
]. (11.46)
11.2.5 Black p-branes
One is typically interested to look for specific types of solutions to the SUGRA actions. A simple caseis obtained when only one of the anti-symmetric tensors, say the (p + 1)-form potential is around.In that case our action generically looks like
Sn =1
2κ2
ˆd10x√−g[R− 1
2∂µφ∂
µφ− 1
2eanφF 2
n
]. (11.47)
We have a3 = −1 for the NS-NS 3-form H3 and an = (5−n)/2 for any RR n−form F (n). Notice thatfor n = 5, i.e., the self-dual 3-brane, the dilaton decouples. The equations of motion are (exercise):
Rµν −1
2gµνR =
1
2(∂µφ∂νφ−
1
2gµν∂λφ∂
λφ) +1
2(n− 1)!eanφTµν (11.48)
1√−g
∂µ(√−ggµν∂νφ) =
1
2
ann!eanφF 2
n (11.49)
1
(n− 1)!
1√−g
∂µ(√−geanφFµν2...νn) = 0 , (11.50)
where Tµν = Fµλ2...λnFλ2...λnν − 1
2ngµνF2n .
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The most general metric that incorporates all the symmetries is:
ds2 = −B2dt2 + C2δijdyidyj + F 2dr2 +G2r2dΩ2
d−1, (11.51)
with all the functions depending only on r. There is an extremal solution
ds2 = H−2(7−p)
∆ (−dt2 + δijdyidyj) +H
2(p+1)∆ (dr2 + r2dΩ2
d−1), (11.52)
with ∆ = (p+ 1)(7− p) + 4a2n and
H = 1 +
√∆
4(7− p)Q
r7−p (11.53)
The electric solutions are those with p = n− 2 while p = 8− n for magnetic. The dilaton
eφ =
H
8ap+2∆ , electric
H−8a8−p
∆ , magnetic(11.54)
whereas the RR form Fty1...ypr = d
drH−1
Fθ1...θ8−p = Qω8−p ∝ volume element of S8−p.(11.55)
The solutions look simpler in the string frame,
ds2 = H−1/2(−dt2 + δijdyidyj) +H1/2(dr2 + r2dΩ2
8−p) (11.56)
for any p. The mass of these solutions can be computed
M = |q|, q =LpΩ8−p
2κ2Q. (11.57)
They all saturate the BPS bound.
As mentioned earlier, the n = 5 (p = 3) is special. Plugging in a5 = 0,∆ = 16 we find
ds2p=3 = (1 +
1
4
Q
r4)−1/2(−dt2 + δijdy
idyj) + (1 +1
4
Q
r4)1/2(dr2 + r2dΩ2
5). (11.58)
If one focuses in the region close to the throat r → 0, the metric behaves as
ds2 ∼ r2(−dt2 +
3∑i=1
(dyi)2) + r−2(dr2 + r2dΩ25). (11.59)
This is AdS5 × S5 with equal radii of curvature. This is also the beginning of the beautiful story ofAdS/CFT or gauge/gravity duality.
********************************************
End of lecture 12
12 Compactification and dualities
12.1 KK compactification in field theory
Consider a field theory in D = 1 + d dimensions and let us imagine that the direction xd is circular:
xd = xd + 2πR. (12.1)
81
This corresponds to
R1,D−1 → R1,D−2 × S1. (12.2)
Terminology is such that S1 is called internal space. We expect three things to happen:
1. We get a Kaluza-Klein (KK) tower of massive states in (D − 1) dimensions.
2. We find an extra symmetry U(1) in (D − 1).
3. We get massless scalar fields (=moduli) in (D − 1).
Let us add more detail in all these cases.
1. Let M,N = 0, 1, ..., D − 1 and µ, ν = 0, 1, ..., d − 1 = D − 2, and for simplicity consider amassless scalar in D dimensions, Φ = Φ(xµ):
∂µ∂µΦ(xµ) = 0 . (12.3)
Φ must be periodic in xd. The most general ansatz is
Φ(xµ) =
∞∑−∞
φn(xµ)einRx
d
, (12.4)
which corresponds to a complete set of periodic functions in xd. Plugging this into the equationof motion yields
∂µ∂µφn(xµ) =
n2
R2φn(xµ) ∀n . (12.5)
Thus, the nth Fourier mode φn(xµ) appears as a scalar field of mass m2n = n2/R2 from the
perspective of (D − 1)-dimensional theory. The collection of these states is called the KKtower. Note also that the zero-mode φ0 is both massless and independent of xd.
As R → 0, the mass of the lowest-lying state m21 → ∞ and the whole tower disappears from
the low-energy spectrum. At energies E 1/R the theory looks (D−1)-dimensional. In otherwords, we landed on low-energy effective field theory.
2. The extra U(1) gauge potential arises from the components G(D)µd of the D-dimensional metric.
The most general metric ansatz is
ds2 = G(D)MNdx
MdxN = g(D−1)µν dxµdxν + gdd(dx
d +Aµdxµ)2 . (12.6)
This metric is invariant under general coordinate trafos xM → x′M (xN ). From (D − 1)-dimensional point of view, i.e., restricting to xµ → x′µ(xν) splits the original metric into
• a scalar field Gdd = gdd = e2φ
• a vector field Gµd = gddAµ = e2φAµ
• a metric Gµν = gµν + e2φAµAν .
Consider for simplicity the zero modes of Gµν , Gdd, and Aµ, i.e. let all components dependonly on xµ. The subgroup of diffeomorphisms in D dimensions compatible with this metricansatz transforms as follows:
• Diffeomorphism invariance in (D − 1) dimensions implies xµ → x′µ(xν).
• Diffeomorphism invariance along the circle S1 implies xd → x′d = xd + λ(xµ), whichyields
Aµ → A′µ = Aµ − ∂µλ, (12.7)
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which is the statement that Aµ is the gauge potential in D − 1. The group of diffeomor-phisms decomposes as
GL(D,R)→ GL(D − 1,R)× U(1). (12.8)
3. The metric component Gdd is a scalar field from the (D − 1)-dimensional perspective. It setsthe volume of the internal space
V ol(S1) =
ˆ 2πR
0
dxd√Gdd =
√Gdd · 2πR, (12.9)
i.e. the vacuum expectation value of the scalar field Gdd determines a geometric propertyof the internal space, in this case the volume of S1. The appearance of the scalar Gdd in(D − 1)-dimensional effective field theory follows by dimensional reduction, i.e. expansion ofD-dimensional Einstein action. In the exercises you will show that
√Gdd is not constrained
by a potential: the scalar field is flat (and massless). Such flat scalar fields which determinegeometric properties of internal space are called moduli.
12.2 KK compactification of bosonic closed string
Let us now apply this simple KK idea in ST. New stringy phenomena will appear!
Recall the mode expansion
Xµ(τ, σ) =xµ
2+xµ
2+
√α′
2(αµ0 + αµ0 )τ +
√α′
2(αµ0 − α
µ0 )σ + . . . , (12.10)
where αµ0 =
√α′
2 pµ
αµ0 =√
α′
2 pµ
, µ = 0, 1, ..., 25 = d. (12.11)
Under σ → σ + 2π : Xµ(τ, σ)→ Xµ(τ, σ) + 2π√
α′
2 (αµ0 − αµ0 ) yields αµ0 = αµ0 =
√α′
2 pµ.
Now consider KK compactification of Xd=25 along S1 by identifying Xd = Xd + 2πR. In stringtheory this has 2 consequences:
1. The momentum in xd-direction is quantized as before pd = n/R. Therefore:
αd0 + αd0 = 2
√α′
2
n
R↔ Xd(τ, σ + 2π) = Xd(τ, σ) (12.12)
This is as in field theory.
2. In addition, something new happens: the string can wind w times around S1 before closing onto itself Xd(τ, σ + 2π) = Xd(τ, σ) + 2πRw:
αd0 − αd0 =
√2
α′wR. (12.13)
(FIGURE IN CLASS)
Thus for winding strings the left- and right-moving momenta are independent :αd0 =(nR + wR
α′
)√α′
2 = pdL
√α′
2
αd0 =(nR −
wRα′
)√α′
2 = pdR
√α′
2 .(12.14)
83
Note that the center-of-mass momentum is still
pdL + pdR = 2n
R. (12.15)
The mass-shell condition follows as always from the Virasoro constraints
(L0 − 1) |phys〉 = 0 = (L0 − 1) |phys〉 . (12.16)
With L0 = 12α
20 +N and L0 = 1
2 α20 + N this implies
−d−1=24∑µ=0
pµpµ = (pdL)2 +
4
α′(N − 1) = (pdR)2 +
4
α′(N − 1). (12.17)
This leads to an effective mass in (D − 1 = 25) dimensions of the form
m2 = −pµpµ =n2
R2+w2R2
α′2+
2
α′(N + N − 2). (12.18)
The level-matching condition now relates left- and right-moving oscillation numbers as
(L0 − L0) |phys〉 = 0 =⇒ N −N = nw . (12.19)
We observe the following:
1. The sector n = w = 0 gives rise to familiar states also present for R→∞.
2. The sector w = 0, n 6= 0 contains the infinite tower of KK excitations present as in pointparticle theory on S1.
3. The sector w 6= 0 contains winding states of mass w2R2/α′2. Note that winding costs energydue to string tension. The winding sector is a truly stringy effect not present for point particles.
Let’s now consider the limit R→ 0:
1. The KK tower as in the point particle case.
2. However, the winding states become light because their mass scales as m2w ∼ 1
α′2w2R2.
We conclude that, unlike in the point particle case, ST compactified on S1 remains effectively(1 + d) = D-dimensional!
For generic values of R, the massless spectrum corresponds to the sectors:
m2 = 0 ⇐⇒ n = w = 0, N = N = 1. (12.20)
We have the following states:
1. αµ−1αν−1 |0, k〉 gives rise to G(µν), B[µν], φ in the non-compact 1 + (d− 1) = 1 + 24 dimensions.
2. (αµ−1αd−1 + αd−1α
µ−1) |0, k〉 corresponds to a vector from the perspective of the non-compact 1
+ 24 dimensions. This is the U(1) potential from G(D)µd encountered also for a point particle
theory.
3. (αµ−1αd−1 − αd−1α
µ−1) |0, k〉 gives another vector, corresponds to the component Bµd.
4. αd−1αd−1 |0, k〉 gives a scalar, corresponding to G
(D)dd .
Thus we find a U(1)× U(1) gauge symmetry and a modulus in the non-compact dimensions.
At a special value of R, called a self-dual radius, extra states appear in the massless spectrum. ForR =
√α′:
pdL,R =1√α′
(n± w). (12.21)
84
Massless states now require
(n+ w)2 + 4N = (n− w)2 + 4N = 4, N −N = nw . (12.22)
This gives rise to new states:
• The sector n = w = ±1, N = 0, N = 1 gives 2 more vectors.
• The sector n = −w = ±1, N = 1, N = 0 gives another 2 more vectors.
These are in addition to the two gauge bosons present for generic values of R. This suggests thatat R =
√α′ the symmetry
U(1)× U(1) is enhanced to SU(2)× SU(2) (12.23)
where each SU(2) factor accounts for three gauge bosons. This can be checked at the level ofinteractions.
Comments:
1. This non-abelian enhancement at R =√α′ is a stringy effect not obtainable in field theory.
2. The compactification is trivial to generalize to several dimensions on a torus T d = S1×· · ·×S1.One can further generalize to many other internal spaces.
3. Recall that the bosonic sector of the heterotic string is compactified on T 16. It turns out thatmodular invariance at 1-loop forces the radii of all circles to coincide and be critical. Thisenhances the symmetry:
U(1)16 →
SO(32)
E8 × E8.(12.24)
12.3 (Microscopic) T-duality
Consider closed superstrings of Type IIA/B.
Bosonic sector: From the mass formula
m2 =n2
R2+R2w2
α′2+
2
α′(N + N − 2) (12.25)
for closed superstrings compactified on S1 with radius R we observe that the spectrum is invariantunder the operation:
n↔ w, R↔ R′ =α′
R, (12.26)
which exchanges KK momentum and winding momentum. This is truly stringy feature that relieson the extended, non-local nature of a string. This transformation is called T-duality. It actuallyextends to an exact symmetry of the closed CFT, including interactions. To understand this, noticethat exchanging n↔ w corresponds to
pdL → pdL, pdR → −pdR. (12.27)
More generally, T-duality is defined by extending this to a full-fledged parity transformation on thewhole string field (including oscillators):
XdL(z)→ Xd
L(z), XdR(z)→ −Xd
R(z), (12.28)
85
i.e. we map Xd(z, z) = XdL(z) +Xd
R(z)→ X ′d(z, z) = XdL(z)−Xd
R(z). One can check that replacingXd → X ′d is indeed a symmetry of X-CFT.
Since the spectrum and all interactions are left invariant by the transformation n ↔ w,R ↔ R′ =α′/R, we have established that in string theory
physics at R <√α′ = physics at R >
√α′ (12.29)
The consequence of this is that there is a minimal distance R =√α′, corresponding to the self-dual
radius. It does not make sense to define distances smaller than this minimal radius, to the extent thatwe can always map all processes at such distances back to radii bigger than
√α′ = ls. N.b. precisely
at this self-dual radius R = ls we have the gauge enhancement U(1)× U(1)→ SU(2)× SU(2).
12.4 Type IIA/B with T-duality
Consider T-dualizing along Xd=9.
• The bosonic fields transform as: X9R(z)→ −X9
R(z)
• By worldsheet SCFT also: ψ9R(z)→ −ψ9
R(z).
In the R sector this implies for zero modes:
ψ9±(τ, σ) =
1√2
∑r=0,1,...
ψ9re−ir(τ±σ) : ψ9
−,0 → −ψ9−,0 (12.30)
Hence T : Γ11 → −Γ11 only on the right-handed sector.
We conclude that T-duality flips the chirality for right-moving spinors and therefore transforms thevarious superstring sectors as
(R+, R±)→ (R+, R∓) (12.31)
(NS+, R±)→ (NS+, R∓) (12.32)
This exchanges Type IIB and Type IIA. More precisely:
Type IIB on S1 with radius R = Type IIA on S1 with radius R =α′
R. (12.33)
Indeed, one can check that T-duality acts on RR-states such that it adds or removes an index:
IIA IIBCµ1µ2µ3 → Cµ1µ2µ39
Cµ1µ29 → Cµ1µ2
Cµ → Cµ9
C9 → C
In other words, T-duality along a direction spanned by a Dp-brane will render it a D(p − 1)-braneand vice versa. Pictorially for a D1-brane:
(FIGURE IN CLASS)
12.5 T-duality for open strings
Recall the mode expansion for open strings with Neumann boundary condition:
Xµ(τ, σ) = xµ0 + 2α′pµτ + i√
2α′∑n6=0
αµnne−inτ cosnσ, (12.34)
86
which satisfies
∂σXµ(τ, 0) = ∂σX
µ(τ, π) = 0. (12.35)
We can express this asXµ(τ, σ) = Xµ
L(τ + σ) +XµR(τ − σ)
XµL,R(τ ± σ) =
xµ02 ±
(x′0)µ
2 + α′pµ(τ ± σ) + i√
α′
2
∑n 6=0
αµnn e−in(τ±σ).
(12.36)
Now compactify X25 which quantizes the momentum as:
X25(τ, σ + 2π) ∼ X25(τ, σ) + 2πR ⇐⇒ p25 =n
R, (12.37)
as there are no winding states for open strings. Since the interior of an open string is indistinguishablefrom that of closed, we can implement the same T-duality rule and define the dual coordinate:
X ′25(τ, σ) = X25L (τ + σ)−X25
R (τ − σ) (12.38)
= x′250 + 2α′
n
Rσ + i
√2α′
∑n6=0
α25n
ne−inτ sinnσ. (12.39)
This dual coordinate has fixed endpoints:
∂τX′25(τ, 0) = ∂τX
′25(τ, π) = 0. (12.40)
T-duality thus mapped Neumann to Dirichlet boundary condition.
In fact we see that one of the end-points is fixed to (x′0)25. The other one is fixed too, but possiblyshifted by integer multiples:
X ′25(τ, 0) = x′250
X ′25(τ, π) = x′250 + 2πα′n
R .(12.41)
(FIGURE IN CLASS)
In the covering space we thus have an open string stretching between two Dp-branes, separated bya distance.
Now let us think about he situation, where to distinguish to which brane does each endpoint ter-minate. Imagine there being N coincident parallel D-branes, which means that each endpoint hasN possible places on which to end, giving N2 possibilities in total. Each open string has its ownmass-spectrum meaning that there are N2 different particles around.
It is natural to arrange the associated fields to reside inside N × N Hermitean matrices. We thenhave the open string tachyon Tmn and the massless fields
(φµ)mn, (Aa)mn, m, n = 1, ..., N. (12.42)
Diagonal components are from those strings whose endpoints live on the same brane. Off-diagonalcorrespond to strings who end on different branes.
The gauge field is particularly interesting. It turns out it transforms under U(N). If one separatesM branes one gets
U(N)→ U(N −M)× U(M). (12.43)
This is the Higgs mechanism, as the excitations between the two sets of branes acquire masses.Notice that the scalars φµ transform under adjoint and the tachyons under bifundamental of thissymmetry.
87
When we compactify X25 we can turn on a constant vacuum expectation value for the gauge fieldA25(X25) as follows:
A25 =1
2πRdiag (θ1, ..., θN ), (12.44)
which is a pure gauge field A25 = −iΛ−1∂25Λ with
Λ(X25) = diag (eiθ1
2πRX25
, ..., eiθn
2πRX25
). (12.45)
In a state |k; ij〉 the endpoints couple to the vector potential like a point particle. Hence, upon goingaround the circle, we only require quasi-periodicity of states
|k; ij〉 → ei(θi−θj) |k; ij〉 . (12.46)
For Neumann boundary condition this quantizes momentum as follows:
p25 =n
R+θi − θj2πR
, (12.47)
and thus the dual Dirichlet boundary conditions become (R′ = α′/R):
X ′25(τ, σ)−X ′25(τ, 0) = (2πn+ θi − θj)R′. (12.48)
T-duality thus maps Wilson lines to positions of dual D-branes.
12.6 S-duality
Recall the action of Type IIB SUGRA
SIIB =1
16πGN10
ˆd10x√−ge−2φ
[R+ 4∂µφ∂
µφ− 1
12H(3)2
− 1
2∂µχ∂
µχ (12.49)
− 1
12F 2
(3) −1
240F 2
(5)
]+
1
16πGN10
ˆC(4) ∧ F(3) ∧H(3), (12.50)
supplemented by the additional on-shell constraint F(5) = ∗F(5) = dC(4)− 12C(2)∧H(3) + 1
2B(2)∧F(3)
and we have denoted the RR scalar C(0) = χ0.
The scalars and 2-form potentials come in pairs. The equations of motion of Type IIB SUGRA areinvariant under SL(2,R). This can be seen as follows. Arrange the RR-scalar and the dilaton in acomplex scalar
λ = χ+ ie−φ =⇒ λ′ =aλ+ b
cλ+ d, ad− bc = 1, a, b, c, d ∈ R. (12.51)
A similar thing happens for NS-NS B(2) and RR C(2) 2-form potentials:(B(2)
C(2)
)→(dB(2) − cC(2)
aC(2) − bB(2)
), (12.52)
with an element of SL(2,R).
Since B(2) couples to F1 and the corresponding charge is quantized
SL(2,R)→ SL(2,Z). (12.53)
Consider for simplicity χ = 0 and the SL(2,Z) transformation:
gs → 1/gs, B(2) → C(2), C(2) → −B(2). (12.54)
88
This transformation is often referred to as S-duality. It is a non-perturbative duality, since it ex-changes weak and strong coupling. However, instead of exchanging at the same time electric andmagnetic degrees of freedom, it exchanges NS-NS and RR fields, both electric. We earlier arguedthat there are no RR-charged states in perturbative string spectrum. We thus again observe thenecessity of having non-perturbative objects carrying RR charge. Notice that SL(2,Z) is a dualityrelating different regimes of the same theory.
A general transformation maps the fundamental F1-string into a general (p, q)-string. It should bepossible to quantize it and reproduce the Type IIB theory. The (p, q)-string is solitonic, its tension
T(p,q) ∼1
gs. (12.55)
This gives the string scale of the dual theory. Thus, α′ must not be invariant under S-duality. SinceGN10 ∼ g2
sα′4 is invariant, we find α′ → gsα
′. Given the fact that Type IIB SUGRA is SL(2,Z)invariant, there should be a way of writing it down in a manifestly SL(2,Z) covariant way. Indeed,in Einstein frame:
SEIIB =1
16πG10N
ˆd10x√−g[R− ∂µλ∂
µλ
2(Imλ)2− Mij
2F i(3) · F
j(3) −
1
240F 2
(5)
](12.56)
+εij
32πGN10
ˆC(4) ∧ F i(3) ∧ F
j(3), (12.57)
where
F i(3) =
(H(3)
F(3)
), Mij =
1
Imλ
(|λ|2 −Reλ−Reλ 1
). (12.58)
This is invariant under
λ′ =aλ+ b
cλ+ d, (F i(3))
′ = ΛijFj(3), F ′(5) = F(5), (gEµν)′ = gEµν , (12.59)
where
Λij =
(d cb a
), M ′ij = (Λ−1)TMijΛ
−1. (12.60)
12.7 String dualities
We have encountered 5 perturbative consistent string theories
Type IIA, IIB, Het SO(32)&E8 × E8, Type I.
All of them are dual to one another using T- and S-dualities (and dimensionally reducing on appro-priate internal manifolds). We summarize these relations in a picture. More discussion can be foundin Esko’s lecture notes.
(FIGURE IN CLASS)
********************************************
End of lecture 13 (last)
89
