Born reciprocity

Born Reciprocityand the Nature of Spacetime in String Theory

Rob Leigh

University of Illinois

Based on 1307.7080 +...with Laurent Freidel [Perimeter] and Djordje Minic [Virginia Tech]

Rob Leigh (UIUC) Born Reciprocity Wits: March 2014 1 / 25

Quotes

Some Inspiration

Our mistake is not that we take our theories too seriously, but that wedo not take them seriously enough.S. Weinberg (The First Three Minutes: A Modern View of the Origin of the Universe)

Pretty smart, them strings.J. Polchinski (Combinatorics of Boundaries in String Theory, hep-th/9407031)


Quotes

Some Inspiration

Our mistake is not that we take our theories too seriously, but that wedo not take them seriously enough.S. Weinberg (The First Three Minutes: A Modern View of the Origin of the Universe)

Pretty smart, them strings.J. Polchinski (Combinatorics of Boundaries in String Theory, hep-th/9407031)


Born Reciprocity

Born Reciprocity

it is a familiar feature of quantum mechanics that a choice of basisfor Hilbert space is immaterial

I e.g., for particle states, {|q〉} is just as good as {|p〉}I one choice may be preferred given a choice of observablesI e.g., for atomic systems:

F interaction with light → use energy basisF for material properties → use position basis

a change of basis is accomplished by Fourier transformfast forward to Quantum Gravity: Max Born in the 1930’s pointedout that this should be a feature of any QG theory too→ there should be no invariant significance to space-time!!this seemingly incontravertible fact went largely ignored, and Bornhimself was unable to make sense of it


Born Reciprocity

Born Reciprocity

it is a familiar feature of quantum mechanics that a choice of basisfor Hilbert space is immaterial

I e.g., for particle states, {|q〉} is just as good as {|p〉}I one choice may be preferred given a choice of observablesI e.g., for atomic systems:

F interaction with light → use energy basisF for material properties → use position basis

a change of basis is accomplished by Fourier transformfast forward to Quantum Gravity: Max Born in the 1930’s pointedout that this should be a feature of any QG theory too→ there should be no invariant significance to space-time!!this seemingly incontravertible fact went largely ignored, and Bornhimself was unable to make sense of it


Born Reciprocity

Born Reciprocity in String Theory

in string theory, we have a supposedly consistent quantum theory,so we ask, what of Born reciprocity (BR)?open any book on string theory, and you will find that Bornreciprocity is broken explicitly within the first few pagesWhy? because we force string theory to provide us with a locallow energy theory on spacetimeIndeed, as we will see, the basic formulation of string theory as aquantum theory is itself BR-symmetric.

I It is the truncation to specific boundary conditions (which we willrelate to space-time locality) and the subsequent reduction to a lowenergy sector that is non-generic.


Born Reciprocity

Born Reciprocity in String Theory

Does this picture ever break down?! String theory is consistentquantum mechanically!Do we know much about the nature of space-time at shortdistances? Are we making unwarranted assumptions?indeed, there are classic tests, such as

I hard scattering (Gross-Mende,...)I finite temperature (Atick-Witten)

each of these rode off into the sunset with profound questionsunansweredClear thinking along these lines must have something profound tosay about “the short-distance nature of spacetime".


Born Reciprocity

Born Reciprocity and T-duality

of course, what often comes to mind in thinking about shortdistances is T-dualitycompactification gives us a way to define a short distance – i.e.,small radiiseems to indicate that space-time looks the same at shortdistances as at long distancesmore precisely, short distance is governed by the long distanceproperties of a dual space-timedoes this apply generically? what about curvature? Are thereother questions/probes where it doesn’t apply?if a dual space-time is involved in such a fundamental way, why dowe say that string theory is defined by maps into a space-time?in fact, T-duality is closely related to Born reciprocity


Born Reciprocity

Closed strings

we usually define the string path integral as a summation of mapsX : Σ→ M ∑

g

Z [gαβ] =∑

g

∫[DX ]e

iλ2

∫Σ ηµν(∗dXµ∧dXν)

specifically, there is an assumption of single-valuedness of thesemaps – the string is closed – X is periodicis this the right prescription? Is it required for consistency, orwell-definedness?the action would be well-defined if the integrand is single-valued,that is, for periodic dX .this does not mean that X (σ, τ) has to be periodic, even if M isnon-compact. Instead, it means that X must be quasi-periodic.


Born Reciprocity

Quasi-periodicity

quasi-periodicity means

Xµ(σ + 2π, τ) = Xµ(σ, τ) + δµ.

If δµ is not zero, there is no a priori geometrical interpretation of aclosed string propagating in a flat spacetime — periodicity goeshand-in-hand with a space-time interpretation.Of course, if M were compact and space-like, then δµ 6= 0 cancorrespond to periodic X . This is interpreted as winding, and it isnot in general zero.more generally, δµ 6= 0 corresponds to a tear in the embedding ofthe worldsheet in the target space.

δ =

∮C

dX

are there operators that would induce such a tear?


Born Reciprocity

Quasi-periodicity and locality

in fact, eliminating such operators is central to the consistency ofthe usual string – operators are required to be mutually local.this is closely related to locality in space-time as well, to theinterpretation of the string path integral as giving rise to local QFT.classically, one can motivate this through the local constraints

Constraints

H :12

p2 +12δ2 = N + N − 2

D : p · δ = N − N

if the spectrum is level-matched, then it is consistent to take δ = 0


Phase Space Formulation

a hint that δ 6= 0 might be consistent: given a boundary ∂Σparameterized by σ, a string state |Ψ〉 may be represented by aPolyakov path integral

Ψ[x(σ)] =

∫X |∂Σ=x

[DX Dg] eiSP [X ]/λ2(1)

if X is periodic, then∫

C dX = 0 and α′p =∫

C ∗dXWe define a Fourier transform of this state by

Ψ[y(σ)] ≡∫

[Dx(σ)] ei∫∂Σ xµdyµΨ[x(σ)].

In fact, this state can also be represented as a string stateassociated to a dual Polyakov action: by extending y(σ) to the bulkof the worldsheet, and integrating out X then gives

Ψ[y(σ)] =

∫Y |∂Σ=y

[DY Dg] e−iλ2SP [Y ]. (2)



The momentum may now be expressed as p =∮

C dY , and so wewill refer to Y as coordinates in momentum space.note also that δ =

∮dX = λ2 ∮ ∗dY (by EOM)

so from the point of view of the Y space-time, momentum is zero,but quasi-period is non-zero.in the compact case, Y is the usual dual coordinate to X, and thusthe Fourier transform reproduces T-dualitythis is an indication that we might generalize to arbitrary (p, δ),even in the non-compact case (or the curved case)



Dyons

if we allow quasi-periodicity, then we can consider generic vertices∼ eip·X+iδ·Y

there is a close analogy here with 2d electro-magneto-staticsI p → elec chg, δ → mag chg

if we quantize in the usual form, then the diff constraint looks like

p · δ = N − N

(usually this is trivially satisfied by δ = 0 and N = N)so here, we need to enforce the diff constraint, but not necessarilyin the usual way – in fact p · δ need only be an integerthis condition should be thought of as a Dirac quantizationconditionin E& M, one way to solve the Dirac quantization is to forbidmonopoles. But it’s not the only way.



Phase Space

it is convenient to go to a first-order formalism, in which weintegrate in worldsheet 1-forms Pµ = Pµdτ + Qµdσ

S1 =

∫Σ

(Pµ ∧ dXµ +

λ

2εηµν(∗Pµ ∧ Pν)

).

I integrate out P: back to SP [X ]/λ2

I integrate out X : find dP = 0, so locally P = dY , and get λ2SP [Y ]I (however, Y is quasi-periodic)

obtain a ‘phase space’ formalism if we partially integrate Pindeed, integrating out Q, and introducing P = ∂σY , we find theTseytlin action (c.f. Floreani-Jackiw)

1~

SPS =

∫ [1λε∂σY · ∂τX − 1

2ε2∂σY · ∂σY − 12λ2∂σX · ∂σX

]Rob Leigh (UIUC) Born Reciprocity Wits: March 2014 13 / 25


Born Geometry

It is convenient, as suggested by the double field formalism tointroduce a coordinate X on phase space, together with a neutral1

metric η, and a metric H.

XA ≡(

Xµ/λYµ/ε

), ηAB =

(0 δ

δ−1 0

), HAB ≡

(η 00 η−1

)we then obtain

1~

SPS =12

∫ (∂τXA∂σXBηAB − ∂σXA∂σXBHAB

).

note that the data (η,H) are not independent: J ≡ η−1H is aninvolution that preserves η, i.e., JTηJ = η or

J2 = 1 (chiral structure)

1Here neutral means that η is of signature (d, d), while H is of signature (2, 2(d − 2)).



Dyons

if we allow quasi-periodicity, then we can consider generic vertices∼ eiP·X where P = (p, δ)

if we quantize in the usual form, then the diff constraint looks like

p · δ = N − N

(usually this is trivially satisfied by δ = 0 and N = N)so here, we need to enforce the diff constraint, but not necessarilyin the usual way – in fact p · δ need only be an integer



Hamiltonian form

Note that the momentum conjugate to XA is

ΠA =12ηAB∂σXB

and the Hamiltonian is

H = 2∫

dσ Π.JΠ

the canonical bracket implies{ΠA(σ),ΠB(σ′)

}= πηAB∂σ′δ(σ − σ′)

and one then finds

14π

{H,XA(σ)

}= (J∂σX)A = ∂τXA − SA



Classical EOM

so the classical EOM imply

S ≡ ∂τX− J∂σX = 0

the classical constraints from linear coordinate transformations are

W = 0, H = 12∂σX · J(∂σX),

L = 12S · S, D = ∂σX · ∂σX,



Lorentz!

we note the free Tseytlin action is not worldsheet Lorentz invariant

1~

SPS =12

∫ (∂τXA∂σXBηAB − ∂σXA∂σXBHAB

).

of course, the full path integral is Lorentz invariantI this shows up on-shell, because J2 = 1 and SA = 0

(to do somewhat better, we can write the original Polyakov theoryas an integral over frames rather than metrics, and divide by Diff ×Weyl × Lorentz)

I then there are Liouville modes (θ, ρ) for both Lorentz and Weyl

dω → dω + d ∗ dρ, d ∗ ω → d ∗ ω + d ∗ dθ

(ω is worldsheet spin connection, dω its usual curvature)



The Quantum Theory and Chiral Structure

this classical structure depends crucially on J2 = 1. Defining

P± =12

(1± J)

one finds that worldsheet chirality is paired with J-chirality(“soldering")one can construct the algebra of components of the stress tensor,and one finds that T+− decouples2 and cL = cR = d iff J2 = 1this is the quantum version of the effect of the classical constraintsand EOM which imply S = ∂τX− J.∂σX = 0

2Recall T+− + T−+ = 0 is the Weyl anomaly, while T+− − T−+ is the Lorentz anomaly.


Phase Space Backgrounds


conjecture: there are consistent string theories (CFTs) for whichη = η(X) and H = H(X) (etc.)however, the phase space geometry is not arbitrary: we haveseen the importance of J2 = 1 and presumably this is to be kept inthe curved caseas well we have two notions of ‘metric’, η and Hin such a background, the equations of motion become

∇σSA = −12

(∇AHBC)∂σXB∂σXC

where ∇ has been assumed to be η-compatiblethis should be supplemented by the constraintswhat are solutions – what is the geometry of phase space?



Born Geometry

ingredient #1: ∃ an (almost) chiral (or para-complex) structure(η, J)

this allows for a bi-Lagrangian structure, a choice ofdecomposition TP = L⊕ L

I L, L are null wrt η, and J(L) = LI bi-Lagrangian also characterized by K

∣∣∣L

= Id , K∣∣∣L

= −Id , with

K 2 = 1, JK + KJ = 0, K TηK = −η.I ω = ηK is a symplectic structure on P 3

I I = KJ is an almost Kähler structure (I2 = −1, ITωI = ω)

we believe these are the primary features of the phase spacegeometry

3It is not obvious however that ω must be closed.Rob Leigh (UIUC) Born Reciprocity Wits: March 2014 21 / 25


Born Geometry

in summary, Born geometry4 is characterized by (η, I, J,K ) with

I2 = −1, J2 = +1, K 2 = +1IJ + JI = 0, IK + KI = 0, JK + KJ = 0

and possesses

η : neutral metricω = ηK : symplectic structure (3)

H = η−1J = ωI : generalized metric

we’ve got something for everyone: complex, real, symplecticgeometry

4This is a simpler name than hyper-para-Kähler, or some such.



Born Geometry

in summary, Born geometry4 is characterized by (η, I, J,K ) with

I2 = −1, J2 = +1, K 2 = +1IJ + JI = 0, IK + KI = 0, JK + KJ = 0

and possesses

η : neutral metricω = ηK : symplectic structure (3)

H = η−1J = ωI : generalized metric

we’ve got something for everyone: complex, real, symplecticgeometry

4This is a simpler name than hyper-para-Kähler, or some such.



Space-time?

there are a number of questions!the nature of the Born geometry should be determined byquantum consistency – we have to insist on both Weyl andLorentz symmetriesgiven such a geometry, how does a space-time emerge?what are the observables of this theory? What is the interpretationof the path integral in general?does locality emerge along with the space-time?

I we have a symplectic form so it makes sense to introduceLagrangian distributions – I’ll show you some evidence thatspace-time should be thought of in those terms

I another closely related idea is relative locality – each free statecarries its own notion of a spacetime (Lagrangian submfld)!

I when strings interact, they have to first agree on their space-times!perhaps this structure is more or less invisible in a suitable limit(e.g., 〈x〉 >> λ, or E << ε)Rob Leigh (UIUC) Born Reciprocity Wits: March 2014 23 / 25


Born Geometry and Space-time

the classical constraints following from the Tseytlin action are

W = 0, H = 12∂σX · J(∂σX),

L = 12S · S, D = ∂σX · ∂σX,

the diff constraint implies that ∂σX is null wrt ηthus ∂σX defines a Lagrangian Lin the flat case, S = 0 and thus ∂τX = J∂σX ∈ L, where L = J(L)

in the curved case, recall

∇σSA = −12

(∇AHBC)∂σXB∂σXC

so S is no longer zero, but the Lorentz constraint implies that it isnull with respect to η, as is ∂σXone argues that S ∈ L and ∂τX ∈ L, and that the induced metric onL is g = H

∣∣∣L



Summary

string theory can be reformulated without putting in assumptionsof space-time properties by hand. This necessarily involves givingup on a priori notions of locality.claim/hope/expectation: the theory can be consistently quantizedand non-trivial backgrounds exist (these claims rely onyet-to-be-finished anomaly calculations)expect that supersymmetry will play an interesting role, as will theusual non-perturbative aspects (D-branes, etc.)


Technology

Born reciprocity