A Review of Topological Strings ICTA 2011

8/3/2019 A Review of Topological Strings ICTA 2011

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A Review of Topological Strings

Imran Parvez Khan

Department of Physics,CIIT, Islamabad.

Email:[email protected].

Submitted on September 30, 2011

Abstract

We have reviewed a series of papers ([6] - [12]) by Edward Witten1,which laid the foundations of topological strings and topological fieldtheory. We have concisely presented the hallmarks of this theory, es-pecially focussing on the topological aspects. We extracted invariants,and discussed the cohomological field theory. Then we studied the re-

lationship of Calabi-Yau spaces with the cohomological field theories.After studying supersymmetry and R-symmetry, we partially provedthe topological field theory to be a cohomolgical field theory. We twistthe N= 2 supersymmetric sigma model to obtain the A/B-models ofthe topological field theory. Finally coupling of these models to gravityis sketched.

KEYWORDS: supersymmetry, spinor, holomorphic, correlator, chern class.

1School of Natural Sciences, Institute for Advanced Study, Princeton NJ 08540.


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1 EMERGENCE 1

1 Emergence

The stregnth of topology lies in its quality of being metric-independent. Be-cause this helps to find invariants of any theory.

1.1 Chern-Simons Theory

The Chern-Simons action, with a gauge group G on a three-manifold M isgiven by

S =k

4

M

Tr(A dA +2

3A A A),

where k is the coupling constant, and A is a G-gauge connection on the trivial

bundle over M. Notice that the Chern-Simons action does not involve themetric, which implies that the Chern-Simons theory is topological, and wecan look for invariants. One invariant is found to be the partition function,[2]

Z(M) =

[DA]eiS.(1)

One more invariant can be constructed from the Chern-Simons theory asfollows[1]. Given a Lie algebra element A, we have a Lie group element gsuch that

g = eA

Now consider a path (t) inside M. Given the instantaneous tangent vectors

on , d

dtt

For the entire closed curve ,

g = limt0

exp

A(

d

dt(0)t

exp

A(

d

dt(t)t

exp

A(

d

dt(2t)t

From the Baker-Campbell-Housdorff theorem, eUeV = eU+V if and only if [U, V] =0 But here we may face anti-commutativity, therefore we express,

g = Pexp

A

where P is the path ordering. The element g is defined as the holonomy ofA. The trace of holonomy ofA is an invariant, called the Wilson loop W(A):

W(A) = Tr Pexp

A(2)


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1 EMERGENCE 2

Equations (1) and (2) are very important invariants, which boosted the study

of this subject in its early days.

1.2 Cohomological Field Theories

If the action S has no boundary2, i.e., S = 0, then notice that this propertycould serve as an invariant of the theory. Which by Noethers thoerem impliesthe existence of an invariance operator, say Q, such that for an operator O

O =

i[Q,O] = i(QO OQ) if at least one operator is bosonic,i{Q,O} = i(QO + OQ) if both operators are fermionic.

But this theory is not cohomological yet, to make it so, we impose the fol-lowing:

Nilpotency of the invariance operator Q: Q2 = 0,

Closure of all Ois under the action of Q3: [Q,Oi]=0

If the invariant states |j satisfy Q|j=0, then in particular for anunbroken symmetry, the vacuum satisfies

Q|0 = 0.

These facts enable us to deduce that any correlator containing an oper-ator of the form O = [Q,L], i.e. a Q exact operator, is automaticallyzero

0|,O1 . . . [Q,L]pm . . .On|0 = 0.

This, infact implies that the physical operators of this theory are givenby the cohomology ring of Q, viz.,

O O + [Q,L].

2We like to study topological strings over Riemann surfaces without boundary.3[,] represents both commutator and anti-commutator


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1 EMERGENCE 3

The final requirement for a cohomological field theory is that the energy-

momentum tensor must be Q-exact:

T S

h= [Q, G],(3)

for some operator G. Last requirement shows the metric independence ofthe theory. So, in order to ensure (3), we use such Langrangean which itselfis Q-exact:

L = [Q, V] hence S = [Q,

V],

for some operator V.

1.2.1 Semi-classical limit

If we explicitly include the Plancks constant in our description, then thequantum measure in (1) reads

expi

h

Q,

M

V

.

One can show thatd

dhOi1 Oin = 0.

That is, the correlators are independent of h, and therefore we can calculate

them exactly in the classical limit.

1.3 Calabi-Yau Manifolds

1.3.1 Kahler Metric

A (local) Kahler metric on a certain patch is defined to be

gij =K(z, z)

z izj(4)

where K is known as the Kahler potential. The Kahler metric belongs to the

category of Hermitean metrics, such thatgij = 0 = g ij .

If the patch is an m-dimensional ball, then notice that the Kahler metricis nothing but second derivative of the Kahler potential, and hence we can


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1 EMERGENCE 4

invoke the Poincare lemma, showing the equivalence of (4) to the integrability

condition4

: gijzk

=gkjz i

.(5)

It is not difficult to show that in notation of n forms, (5) becomes

d = 0,(6)

where 2iK

implies ij = ij = igij, while all other components of vanish. It followsthat a globalKahler manifold is a complex manifold with a Hermitean metricsatisfying (6).

1.3.2 Ricci-flat Kahler manifold

From differential geometry, we know that given a metric we can evaluate theRicci(contracted curvature) tensor, R on it. For the Kahler metric, we findRij and R ij to vanish. In particular,

Rij = 0.(7)

Now a Calabi-Yau manifold is defined to be a Ricci-flat Kahler manifold, i.e.,one for which (7) holds.

1.3.3 Moduli Spaces

Theorem 1 Ricci-flatness implies that the first Chern class, c1(M) of themanifoldM vanishes.

In [3] and [4], the converse of Theorem 1 was established for Kahler metrics,stated as under:

Theorem 2 Given a complex manifold M with c1(M) equal to zero, thenthere exists at most one Ricci-flat Kahler metric in each Kahler class.

From the above theorems, we can say that given a certain topological space,a complex structure and a Kahler class on it, we can obtain a unique Calabi-Yau metric. Hence the Calabi-Yau moduli space is defined to be the spaceof all possible Kahler classes and complex structures on M.

4Similarly for the anti-holomorphic derivatives.


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1 EMERGENCE 5

Figure 1: A 6-dim Calabi-Yau manifold. A 2-dim Calabi-Yau manifold is atorus.


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2 TOPOLOGICAL STRINGS 6

2 Topological Strings

2.1 Supersymmetry

Supersymmetry a.k.a SUSY, generators transform as fermions under theLorentz group. AnN= n supersymmetric theory has n such spinor of super-charges. In two-dimensions, the Lorentz group is SO(2) = U(1), so we usetheN= (p,q) when there are p irreducible spinor supercharges with positiveU(1)-charge, and q irreduible spinor supercharges with negative U(1)-charge.

2.1.1 N=2 supersymmetric sigma model

We begin with a Riemann surface M, the manifold on which our theory isdefined. Then consider the maps such that

: M X,(8)

where X is the target space5. We define a set of fields i on M by thecompositon

i = xi,

where xi are the local coordinates for X. The action, in this case is so definedas to minimize the area of the Riemann surface, M. For the case ofN = 2supersymmetry,

S =M

dzdz

1

2gijz

izj +

i

2gij

iz

j +

i

2gij

i+z

j+ +

1

4Rijkl

i+

j+

k

l

.

(9)Here gij is the metric ofX, Rijkl is the Riemann tensor, and is the covariantderivative (cf. equation 11). The s transform as spinors w.r.t manfoldM rotations, and as vectors w.r.t Lorentz transformations on X. The +and - subscripts denote the positive U(1)-charge and negative U(1)-chargerespectively. Equation (9) gives the action for our nonlinear sigma model.We now introduce the notion of canonical bundle, B. Given an n-dimensionalmanifold, the (n, 0)-forms6

= i1,...indz1 . . . dzn(10)

5physically speaking, X is space-time.6(n,m)-forms are F = Fi1...in,j1...jm(z, z)dz

i1 . . . dzin dzj1 . . . dzjm .


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are called canonical bundle7. Such structure is also called a complex line

bundle. On a Riemann surface, the (complex) line bundle is the space of(1,0) forms, which is dual to the bundle of holomorphic vectors (vectors ofthe form v = vzz) T

1,0M Hence, we make the identification

B = T1,0M = T0,1M,

where the last expression employs the idea of anti-canonical bundle, intro-duced in footnote [7] . Formally, K as the bundle of (0, n)-forms is an anti-canonical bundle. Further we make the following identifications,

K 1,0 = T1,0 = T0,1,

K 0,1

= T1,0

= T0,1.A (1, 0)-form transforms as follows, under a U(1) gauge transformation:

z z = zei

z z = zei

If ei/2, then (8) implies that the positive U(1)-charges transform as

square roots of (1, 0)-forms, and negative U(1)-charges transform as squareroots of (0, 1)-forms. Thus we may define line bundles K1/2 and K1/2, whichcan be thought of as square roots of the canonical and anti-canonical bun-dles, respectively. In terms of global data, this means that the transitionfunctions defining the spinors are elements of U(1) that square to the tran-sition functions defining K and K. K1/2 and K1/2, are the line bundles inwhich these spinors live. [5] In terms of the index of the local coordinates xi

of X, spinors +, can be defined as:

i+ (M, K1/2 (T X)),

i (M, K1/2 (T X)),

where (M, E) means space of sections of the bundle E, and is the pull-back map of the tangent bundle T X onto the worldsheet M. The covariant

derivative on the spinor is defined as

i+ = i+ +

iijkk+.(11)

7It wouldnt be inapt to introduce anti-canonical bundle here, which is simply any(0, n)-form.


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Similarly,

i

= i

+ i

i

jkk

,(12)where we have used the Christoffel symbol on X. Now the theory exhibitsN= 2 supersymmetry under the following transformations:

i = ii+ + i+

i,

i+ = zi i+

k

ikm

m+ ,

i = +zi i

k+

ikm

m .

Here the transformation parameter + is a holomorphic section of K1/2 and

is an anti-holomorphic section ofK1/2. holomorphism/anti-holomorphismis a necessary condition for the +/ to be pulled through the derivativesupon varying the Lagrangean.

2.1.2 N=2 SUSY and the Kahler Metric

Now consider if the target space X in (8) is a complex manifold. Analogically,we know that a real 2n-dimensional manifold can be described by a family ofcharts {U, zi, z

i}, where i runs from 1 through n, such that the transition

functions do not mix the holomorphic and anti-holomorphic coordinates, i.e.,given an intersection U U,

zi = zi(z

1, . . . , z

n), and z

i = z

i(z

1, . . . , z

n).

This implies that we may write the nonlinear sigma model in (9) be decom-posing the fields as

i = {i, [i, i}, i = {i,

i}, g

ij = {gij, gij}.

Now this decomposition, would be consistent with the complex structure ofX, if and only if the metric on X is such that the parallel transport8 ofvectors preserves the decomposition

T X = T1,0X T0,1X.

The metric that satisfy this condition is the Kahler metric given by (4),because for

gij =K(z, z)

z izj,

8The Christoffel symbol terms in Covariant derivative expressions (11) and (12).


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only ijk and ijk are non-zero, while all other components vanish. This

enables us to rewrite our model (Equation. (9)) as

S =M

dzdz

1

2gi

jziz

j +1

2gijz

izj + igij

iDz

j + igij

i+Dz

j+ +

1

4Rijk l

i+

j+

k

The corresponding supersymmetry rules are

i = ii+ + i+

i,

i = ii+ + i+

i,

i+ = zi i+

k

ikm

m+ ,

i+ = zbari i+kikmm+ ,

i = +zi i+

k+

ikm

m ,

i = +zbari i+

k

ikm

m+ ,

As mentioned in the introduction, this model is known as N = 2 super-symmetry, because there two spinors of supercharges, or equivalently twoholomorphic and two anti-holomorphic SUSY parameters. The spinors andparameters are sections of the following bundles:

i+ (M, K1/2 (T1,0X)), i+ (M, K

1/2 (T0,1X)),(13)

i (M, K1/2 (T1,0X)), i (M, K

1/2 (T0,1X)),

+, + (M, K1/2),

, (M, K1/2).

2.2 R-symmetry

Besides supersymmetry, the sigma model remains conditionally invariantunder two more symmetries. They are known as the vector R-symmetryand the axial R-symmetry, their generators being denoted by FV and FA,

respectively. These two symmetries act on fermions only, and are defined asfollows:

eiFV{i, i} {e

ii, eii},

eiFA{i, i} {e

ii, eii}.


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The total supersymmetery variation is expressed as follows:

= iQ+ + i+Q + iQ+ + i+Q,

where Q and Q are supersymmetry generators. These generators obeythe following Lie-algebra:

{Q,Q} = P H,(14)

where P and H are the Euclidean versions of the generators of space andtime translations. Let M, be the generator of Euclidean Lorentz boosts, i.e.,SO(2) rotations. Then it satisfies the following Lie-algebra:

[M,Q] = Q and [M,Q] = Q.(15)

So in the same analogy, we have a Lie-algebra corresponding to the R-symmetry generators:

[FV,Q] = Q,(16)

[FV,Q] = Q,

[FA,Q] = Q,

[FA,Q] = Q.

We conclude this section with the following important result:

Result 1 The vectorR-symmetry is conserved for any Kahler target spaces.But the axialR-symmetry is conserved if and only if the target space is Calabi-Yau.

2.3 Twisted N=2 theories

Dont forget that our N = 2 supersymmetric field theory is not yet a com-pletely cohomological field theory. From (13), we note that,

{Q+ + Q,Q+ Q} = 2H,

{Q+ + Q,Q+ +Q} = 2P,

{Q+ + Q,Q+ Q} = 2H,

{Q+ + Q,Q+ +Q} = 2P.


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Defining QA = Q++Q and QB = Q++Q, we can find two operators which

square to zero, such that the Hamiltonian and the momentum are QA/B -exact. Now we try to make our theory a cohomological field theory. Noticefrom (14) and (16) that the operators M, FA and FV have exactly same typesof commutators with the other operators, and have vanishing commutatorsamong themselves. Therefore, we may define a new operator,

MA = M FV or MB = M FA

relating the Lorentz transformation properties of the new fields with the oldones.Now consider the following commutators:

[MA,Q+] = 2Q+, [MB,Q+] = 2Q+,

[MA,Q] = 0, [MB,Q] = 2Q,

[MA,Q+] = 0, [MB,Q] = 0,

[MA,Q] = 2Q, [MB,Q] = 0.

Notice that for MA, the operator QA has become a scalar, similarly for MB,the operator QB serves as a scalar. And therefore the corresponding symme-try operations can now be defined on any arbitrary curved worldsheet. Thisconstruction is called twisting, and we may conclude that these twisted

theories can be viewed as cohomological field theories.

2.4 The A-model

In the last section, the transformation MA = MFV corresponds to what isknown as the A-model9 in the literature. After making this transformation,spinors become

i+ iz,

i+

i

i i, i

i.z

The A-model action now reads

SA = 2tM

(gijziz

j+gijziz

j+igijizz

j+igijjzz

i+1

2Rijk l

iz

jz

kl),

9So named, because of its relation to Type IIA strings.


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where we have introduced a coupling constant t. With respect to the old

supersymmetry transformations (13), and + are now scalars, and + and are sections of the canonical and anti-canonical line buncle, respectively.Now we can have a globally supersymmetric system, by doing away withthe + and , and keeping the scalar SUSY parameters only. With theparameters denoted by and , the transformations are:

i = i,

i = ii,

iz = zi ikikm

m,z

i

z = zi

ik

i

kmm,

z

i = i = 0.

2.5 The B-model

As in the previous case, the transformation MB = MFA corresponds to theB-model10 in the literature. After making this transformation, the spinorsbecome:

i+ iz, gij(

j+

j) i,

i iz,

i+

j i,

The action for the B-model then reads,

S = tM

(gijziz

i+igiji(z

iz+z

iz)+ii(z

izz

iz)+

1

2Rlijk

iz

kz

jl).

The action is invariant uder the following supersymmetry transformations11:

i = 0, i = ii,

iz = , iz = d

i,

i = i = 0.

10So named because of its link with the Type IIB strings.11We have set = for simplicitys sake.


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3 COUPLING TO GRAVITY 13

3 Coupling to Gravity

Hitherto, we have assumed a fixed worldsheet geometry, viz., a fixed Riemannsurface. What if we allow to integrate over different worldsheet geometries? Infact this integration over metrics, is what is referred tp as quantumgravity/geometry. The idea is to couple the sigma models to worldsheetgeometry.[13]

4 Conclusion

We have reviewed the story of topological field theory, beginning from Chern-Simons theory upto the A and B models. The question that which of themany Calabi-Yaus is the string theory Calabi-Yau is still open. We developedthe A and B models of topological strings which correspond to the type IIAand type IIB strings respectively. In the end, we discussed how to couple thesigma models to gravity, an issue which is the principle motivation behindall string theories.


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REFERENCES 14

References

[1] Vonk, Marcel. A mini-course on topological strings, (2005),[arXiv:hep-th/0504147v1].

[2] Marino, Marcos. Chern-Simons Theory and Topological Strings,(2005), arXiv:hep-th/0406005v4

[3] Calabi, E. On Kahler manifolds with vanishing canonical class, inAlgebraic geometry and topology: a symposium in honor of S. Lef-schetz, eds. R. H. Fox, D. C. Spencer and A. W. Tucker, PrincetonUniversity Press, (1957).

[4] Yau, S.-T. On the Ricci curvature of a compact Kahler manifold andthe complex Monge-Ampere equation, I, Comm. Pure Appl. Math.31, 339-411 (1978).

[5] Collinucci, Andres and Wyder, Thomas. Introduction to topologicalstring theory, (2007).

[6] Witten, Edward. Topological Gravity, Phys. Lett. B 206, 601 (1988).

[7] Witten, Edward. Topological Sigma Models, Commun. Math. Phys.118, 411 (1988).

[8] Witten, Edward. Topological Quantum Field Theory, Commun.Math. Phys. 117, 353 (1988).

[9] Witten, Edward. On The Structure Of The Topological Phase Of Two-

Dimensional Gravity, Nucl.Phys.B 340, 281 (1990).[10] Witten, Edward. Introduction To Cohomological Field Theories, Int.

J. Mod. Phys. A 6, 2775 (1991).

[11] Witten, Edward. Mirror manifolds and topological eld theory, in Es-says on mirror manifolds, ed. S.-T. Yau, International Press 1992,120-158 [arXiv:hep-th/9112056].

[12] Witten, Edward. Chern-Simons gauge theory as a string theory, Prog.Math. 133, 637 (1995) [arXiv:hep-th/9207094].

[13] Hori, Vafa, Klemm, et al., Mirror Symmetry, Clay Mathematics In-stitute (2003)

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A Review of Topological Strings ICTA 2011