Topological Field Theory

Notes on Topological Field Theory

Xi Yin Harvard University

Introduction

The notes give a survey of the basics of the following topological ﬁeld theories:

• The Chern-Simons gauge theory on 3-manifolds, its renormalization, geomet- ric quantization, computation of partition functions by surgery, and relation with Jones Polynomials

• Twisted N = 2 supersymmetric Yang-Mills theory on 4-manifolds, its reduction to N = 1 theory with a mass gap, and computation of Donaldson’s polynomials

• Topological sigma model, A-model open string interpretation of Chern-Simons gauge theory, and the duality between topological closed string and Chern-Simons theory.

The Chern-Simons functional

Consider a G bundle E over compact oriented three manifold M. We put on E a connection 1-form A which takes value in the Lie algebra. The Chern-Simons action is given by Z k 2 S = Tr(A ∧ dA + A ∧ A ∧ A) 4π M 3 Including the Wilson line Z WR(C) = TrRP exp A C

Let L be a collection of oriented and non-intersecting knots Ci, we have the partition function Z Yr Z(M; L) = DA exp(iS) WRi (Ci) i=1 Under gauge transformation the Chern-Simons functional is well-deﬁned modulo some integer. This is because any compact, oriented three-manifold M can be arranged as the boundary of some four-manifold B. Extend E, A over B, we have Z k CS(A) = TrF ∧ F 4π B

1 Consider a gauge transformation A → A0 = g−1Ag + g−1dg. Take another four- manifold B0 with ∂B0 = −M (reversed orientation) and extend A0 to B0. Glue two pieces together, we have Z k CS(A) − CS(A0) = TrF ∧ F 4π X=B∪B0 known as the instanton number. The classical solutions to Chern-Simons action are flat connections, i.e. gauge fields with vanishing curvature. Gauge equivalent classes of flat connections are completely decided by their holonomy up to conjugation. Therefore the moduli ∼ space of classical solutions to Chern-Simons action is M = Hom(π1(M),G)/G, where G acts by conjugation (the gauge transformation acts as conjugation on the holonomy). It has dimension (2g − 2) dim G for genus g > 1.

A look at the weak coupling limit

(α) At weak coupling limit, expand gauge field Ai = Ai + Bi, the first terms of the action are Z k S = k · CS(A(α)) + Tr(B ∧ DB) 4π M where D is the covariant derivative with respect to A(α). To fix the gauge we have i to pick a metric: DiB = 0. The ghost action is Z i i SGF = Tr(φDiB +cD ¯ iD c) M Integrating out B, φ, c, c¯, the contribution to the partition function from expanding around A(α) is exp(iπη(A)/2) · Tα (α) 1 P where η(A ) is the “eta invariant” defined by 2 signλi, where λi’s are eigenvalues of operator L−, the restriction of ∗DA + DA∗ on odd forms; Tα is the torsion invariant of flat connection A(α). Let M = ∂X, extend A(α) to X, Atiyah-Patodi- Singer index theorem says Z 1 index L− = Aˆ(X) · c(E) − η(L−|M ) X 2 The integral of Chern class c(E) on X is just the Chern-Simons functional on the 1 (α) 1 (α) boundary M, we obtain the relation 2 η(A ) − 2 η(0) = Qg/2π · CS(A ). The partition function can be written as

π X (α) η(0) i(k+Qg/2)·CS(A ) Z = e 2 e · Tα α

2 We see that k + Qg/2 is the renormalized coupling constant, for SU(N) this is k + N. η(0) is the only term that depends on the metric, we need to introduce d counter term 24 I(g) with the gravitational Chern-Simons term being Z 1 2 I(g) = Tr(ω ∧ dω + ω ∧ ω ∧ ω) 4π M 3 ω is the affine connection on the spin bundle over M, d = dim G. By APS theorem 1 d 2 η(0) + 24π I(g) is independent of the metric, therefore the renormalized gauge- fixed partition function is a topological invariant. There is a subtlety in that, for I(g) to be well-defined one needs to choose a trivialization (“framing”) of the tangent bundle TM, but there is no canonical way of doing this. Under shifting of framing the partition function transform as Z → Z · exp(2πis · d/24), s ∈ Z. The following concrete example is interesting. For U(1) gauge group, the action is simply Z k ijk S = ² Ai∂jAk 8π M To compute the partition function in the weak coupling limit, consider some circles Ca that don’t intersect each other (but may link nontrivially). Include the Wilson line Yr Z W = exp(ina A) a=1 Ca The correlation function is   Z Z 1 X hW i = exp − n n dxi dyjhA (x)A (y)i 2 a b i j a,b Ca Cb   Z Z i X xk − yk = exp  n n dxi dyj²  2k a b ijk |x − y|3 a,b Ca Cb

The integral Z Z k k 1 i j x − y Φ(Ca,Cb) = dx dy ²ijk 3 4π Ca Cb |x − y| is some integer, known as the Gauss linking number, the most classical knot invariant. Again, the regularization of the case a = b requires a choice of framing.

Canonical quantization of Chern-Simons action

Consider the special three manifold of the form Σ × R1, where Σ is a Riemann surface. Fix the gauge A0 = 0, the action is reduced to Z Z k ij d S = dt ² TrAi Aj 8π Σ dt

3 The Poisson bracket is 4π {Aa(x),Ab(y)} = ² δabδ2(x − y) i j k ij together with the constraint

δS k ij a = ² Fij = 0 δA0 4π The idea is to impose constraints first and then quantize the moduli space of flat connections M. There is a natural symplectic structure on M: Z k ω = α ∧ β (1) 4π Σ where α, β ∈ T M = H1 (Σ,E ⊗ g), because the tagent space of M at A is dA precisely the first cohomology group w.r.t. DA. If the symplectic form lies in the integral cohomology, which is satisfied in our case, we can find a holomorphic line bundle L over M such that the symplectic structure is represented by the curvature form. In fact the determinant bundle L of the Cauchy-Riemann operator ∂¯A with the Quillen metric has this property. As Quillen showed, its first Chern class is precisely given by (1) with k = 1. For general k, we simply take the tensor product L⊗k. In the usual canonical quantization, we pick canonical pairs of coordinates q and p together with a symplectic structure. In our case we pick z ∼ q+ip, z¯ ∼ q−ip to be the ”canonical pairs”, and the Hilbert space is understood as a suitable space of ”functions” in z. More precisely, the Hilbert space associated with Σ is the space ∼ ⊗k of holomorphic sections HΣ = Γ(M, g ⊗ L ). However there is no canonical way to pick a complex structure on M, usually a difficult step is to prove that the quantization is independent of the complex structure. The key observation is, the quantum Hilbert space of the Chern-Simons gauge theory is identified with the space of conformal blocks. As a simple example consider Σ of genus one. The moduli space of flat connections modulo gauge ∼ transformation is MΣ = T ×T/W , where T is the maximal torus and W the Weyl group. For G = SU(2) this is CP1. The line bundle over CP1 of degree k is given by gluing together the trivialization over two coordinate patch U1, U2 with transition function f(z) = zk. The Hilbert space, i.e. the space of holomorphic ⊗k 2 k sections HΣ = Γ(M, L ) is spanned by 1, z, z , ··· , z in local coordinates. This d is identified with the first k + 1 characters of SU(2)k. Now consider Σ as a Riemann sphere with marked points Pi associated to representation Ri. This arises when we pick a sphere in M pierced by Wilson lines. In the weak coupling limit, the physical Hilbert space would be as if the “static charges” are not coupled to the gauge field, under the constraint that the total charge being zero. Therefore the Hilbert space under the large k limit is the singlet in the product of Ri’s:

HΣ = Inv(R1 ⊗ R2 · · · ⊗ Rr) (2)

4 For finite k the correct Hilbert space is a subspace of (2). The following are well known from conformal field theory. • r = 0, for the Riemann sphere with no marked points, the moduli space of flat connections M is simply one point. The Hilbert space is 1 dimensional. • r = 1, the Hilbert space has dimension 1 if R is the trivial representation and 0 otherwise. • r = 2, HΣ = Inv(R1 ⊗ R2) has dimesion 1 if R2 is the conjugate representation of R1 and 0 otherwise. • r = 3, dim HΣ = Nijk ≤ dim Inv(⊗Ri). Nijk is the coefficient that appears in the fusion rules. For Riemann surface of higher genus, we pick a number of simple loops (genera- tors of H1(M; Z)) and collapse them to get copies of 3-punctured Riemann spheres, glued together along the marked points. The total Hilbert space is obtained by G fusing the subspaces (Ri ⊗ Rj ⊗ Rk) (of dimension Nijk) together.

Surgery on three-manifolds

Given a three manifold M with boundary Σ a Riemann surface, the path integral on M gives Z(M) ∈ HΣ. It is certainly a smooth invariant because the metric doesn’t appear in the path integral. However, to ﬁx the gauge we need to pick a metric, and regularization is required as discuss above. Include Wilson lines (“link” L) gives Z(M; L). We can further consider ∂M = Σ as a Riemann surface with marked points, and Wilson lines connecting the marked points, associated with representation Ri(i = 1, ..., r) of the gauge group G. The path integral satisﬁes the gluing axiom: Suppose we cut a three manifold M into two pieces ML, MR along a Riemann surface Σ, then

Z(M) = (Z(ML),Z(MR))

∗ where Z(ML) ∈ HΣ and Z(MR) ∈ HΣ. This can be generalized to include Wilson lines: Z(M1 + M2; L1 + L2) = (Z(M1; L1),Z(M2; L2)) where L1 + L2 is the connected sum of links connected to the marked points on Σ = ∂M1 = −∂M2. We can perform a diﬀeomorphism K on the boundary of M2, accordingly the path integral on M2 transform as Z(M2) → K · Z(M2), and glue M2 back to M1. Denote the new three-manifold by M˜ , the path integral on M˜ is simply:

Z(M˜ ) = (Z(M1),K · Z(M2))

Let vi = Z(M; Ri) be the path integral with a Wilson line in the representation Ri. K acts on the Verlinde basis as: X j K · Z(M; Ri) = Ki Z(M; Rj) (3) j

5 In the case of genus 1, Kj for modular transformations can be determined i P ¯ l explicitly. Define gij = 1 if Ri = Rj and zero otherwise, Kij = gljKi. For 1 modular transformation S : τ → − τ , G = SU(2) we have s µ ¶ 2 (m + 1)(n + 1)π S = sin mn k + 2 k + 2 This can be determined by the fusion rules or from the modular transformation d property of SU(2)k characters. It is known that any compact, oriented three manifold can be made into S3 by repeated surgery of cutting out a solid torus along a knot, make a diffeomorphism on the surface of the solid torus and glue it back. The simplest example is X = S2 × S1. We can think of X as two copies of solid tori D2 × S1 with boundaries 1 identified. Perform modular transformation S : τ → − τ on the second solid torus and glue it back, we get S3. Suppose the partition function with Wilson lines on S3 (or S2 × S1) is known, by repeated surgery one can obtain the partition function on any compact, oriented three-manifold. The transformation property of the partition function under surgery follows easily from (3): X ˜ j Z(M; Ri) = Ki Z(M; Rj) j i.e. the partition function on the new manifold is obtained from the partition function on the old manifold with Wilson lines sitting in the place of surgery. For three-manifolds of the form X ×S1, we can use the Hamiltonian formalism. Fixing the gauge A0 = 0, Chern-Simons action involves only the first order derivative of the gauge field, therefore has vanishing Hamiltonian. We have

1 Z(X × S ) = TrHX 1 = dim HX In particular Z(S2 × S1) = 1 For S2 × S1 with Wilson lines, the Hilbert space is associated to the Riemann sphere with punctures where the Wilson lines pierce through

2 1 Z(S × S ; R) = dim HS2;R By surgery operation S we obtain the partition function on S3 (G = SU(2)) s µ ¶ X 2 π Z(S3) = SjZ(S2 × S1; R ) = S = sin 0 j 0,0 k + 2 k + 2 j The general formula for G = SU(N) is given by s NY−1 µ ¶N−j iπN(N−1)/8 1 N + k jπ Z 3 (SU(N), k) = e 2 sin (4) S N/2 N N + k (N + k) j=1

6 The Jones Polynomials

Let M be the connected sum of two compact oriented three manifold ML and 2 ∨ ∨ 3 MR joined along S , and ML ,MR denote ML,MR with a three ball D cut out. 3 Similarly let S = DL ∪ DR. We can write the partition functions as Z(M) = (χ, ψ) and Z(S3) = (u, v) ∨ where χ, u ∈ HS2 are the path integrals on ML and DL, while ψ, v the path ∨ ∗ integrals on MR and DR, living in the dual Hilbert space HS2 . From dim HS2 = 1, (χ, ψ) · (u, v) = (χ, u) · (ψ, v) in other words 3 Z(M) · Z(S ) = Z(ML) · Z(MR) This allows us to solve Chern-Simons theory exactly, by attaching pieces of three manifolds to form more complicated manifolds. To analyze a general link L ⊂ M, pick a small sphere Σ around a crossing. Let the gauge group be SU(N), and the Wilson lines are associated to the deﬁning representation R. The part of the link inside the sphere consists two Wilson lines connecting four marked points on Σ, associated to representation R,R, R,¯ R¯; dim HΣ = 2. M is therefore cut into two pieces: MR containing the small ball bounded by Σ and ML containing the rest of the link. The path integral on ML,MR ∗ with Wilson lines are vectors χ ∈ HΣ and ψ ∈ HΣ. The partition function on M is Z(M; L) = (χ, ψ). Consider the braiding operation (half-monodromy) B on Σ, i.e. interchange the two marked points associated with representation R¯ by rotating a half-twist around each other. This operation allows one to unknot the link. We have

2 ψ1 = Bψ, ψ2 = B ψ

Since the Hilbert space has dimension two, there is a linear relation between ψ, ψ1 and ψ2. In fact B is a linear transformation on HΣ, satisﬁes characteristic equation: B2 − yB + z = 0 and the skein relation can be written as

z · ψ − y · ψ1 + ψ2 = 0 This algorithm uniquely determine the invariants of all knots. For example, apply 3 the relation z · Z(L) − y · Z(L1) + Z(L2) = 0 to a “twisted” circle in S , we ﬁnd Z(S3; C) z + 1 (z + 1)Z(S3; C) − yZ(S3; C2) = 0 =⇒ hCi = = Z(S3) y

7 Let hR be the conformal weight of a primary field transforming in representation R, and Ei be the irreducible representation appearing in R ⊗ R. The key point is to find the eigenvalues of B. In conformal field theory we have duality identity

B = F −1(1 ⊗ Ω)F Ã ! Ã ! i i where F is the fusing operation, Ω = eiπ(hi+hj −hk) . It comes from jk kj the following commutative diagram of fusing operation

i p i p Vjp ⊗ Vkl → Vkp ⊗ Vjl ↓ ↑ i p i p Vpl ⊗ Vjk → Vpl ⊗ Vkj with the horizontal maps being B and 1 ⊗ Ω, vertial maps F and F −1. V i is the Ã ! jk i vector space of chiral vertex operators of type . The eigenvalues of B are jk thus given by

λi = ± exp [iπ(2hR − hEi )] , i = 1, 2

The ± sign depends on the symmetry of Ei appearing in R ⊗ R = E1 ⊕ E2. From the Sugawara construction of the energy-stress tensor in current algebra, we obtain the conformal weight QR hR = 2 kψ + Qg where QR is the second Casimir of representation R and Qg of the adjoint representation. In our case R is the deﬁning representation of SU(N), the eigenvalues of B are µ ¶ µ ¶ iπ(−N + 1) iπ(N + 1) λ = exp , λ = − exp 1 N(N + k) 2 N(N + k) Eventually the skein relation is determined to be

N/2 1/2 −1/2 −N/2 −q Z(L) + (q − q )Z(L1) + q Z(L2) = 0 ³ ´ 2πi where q = exp N+k .

Twisted N=2 supersymmetric Yang-Mills theory on four-manifolds

Starting from the usual N = 2 supersymmetric gauge theory in Euclidean space 4 R , the Lagrangian has global internal symmetry SU(2)I × U(1)U . Replace the 0 0 rotation symmetry group SU(2)L ⊗ SU(2)R by SU(2)L ⊗ SU(2)R, where SU(2)R is the diagonal of SU(2)R × SU(2)I . Under this exotic twisting the usual right

8 fermion ψR is replaced by self-dual two form χαβ and singlet η, ψL becomes a 1-form ψα (they are of wrong spin-statistics). The twisted action is Z · √ 1 1 S = d4x gTr F F αβ + φD Dαλ − iηD ψα + iD ψ χαβ 4 αβ 2 α α α β M ¸ Z i αβ i α i 1 2 1 − φ[χαβ, χ ] − λ[ψα, ψ ] − φ[η, η] − [φ, λ] + TrF ∧ F 8 2 2 8 8 M The supersymmetry transformation of operator O takes the form δO = −i² · {Q, O}. We are going to think of Q as the BRST operator of the theory. The energy-stress tensor satisﬁes Tαβ = {Q, λαβ} R This implies that the partition function Z = DX exp(−S0/e2) is a topological (smooth) invariant. In fact, under variation of the metric, ½ Z ¾ 1 √ δZ = − h Q, gδgαβλ i e2 αβ Z M ½ Z ¾ 1 0 2 √ αβ = − 2 DX Q· exp(−S /e ) gδg λαβ = 0 e M because the measure is invariant under supersymmetry. We are assuming the measure DX is independent of the metric g even at quantum level. Moreover, the action satisﬁes S = {Q,V } for some operator V . For the same reason the partition 2 function is independent of the coupling constant: δeZ = δ(−1/e )h{Q,V }i = 0. Go to small limit of e2, the classical equation of motion gives:

α F = − ∗ F,Dαψ = 0,DAψ + ∗DAψ = 0

Solutions to the ﬁrst equation are anti-instantons. If we try to make small variation in the instanton moduli space A → A + δA, δA has to satisfy

α DαδA = 0,DAδA + ∗DAδA = 0

Compare with (), we see that the dimension of the instanton moduli space M is the same as the number of ψα zero modes, which must be absorbed in the path integral, and leads to a violation of U number (the charge of R-symmetry). Actually our discussion above is only valid generically. More precisely

dim M = ∆U = (# of ψ zero modes) − (# of (η, χ) zero modes)

The path integral Z Z(O) = DX exp(−S/e2) ·O vanishes unless the U number of O is equal to dim M. In the case G = SU(2), 3 dim M = 8k − (χ + σ) 2

9 where k is the instanton number, χ the Euler characteristic of M and σ the signature. The factor 8 is essentially why the U(1) R-symmetry is broken to Z8 by instantons. In order for Z(O) to be a topological invariant, we need O to be BRST 2 2 closed. W0(x) = Trφ (x)/8π is such an operator. Since hW0(x)i is independent of the metric, it shouldn’t depend on the choice of x, i.e. dW0 is BRST exact. Starting from W0 we can choose a set of forms Wk on M, with the property dWk = i{Q,Wk+1}. On a homology k-cycle, the integral Z I(γ) = Wk γ R is BRST invariant because i{Q,I} = γ dWk−1 = 0, and depends only the homology class of γ. In particular hW (x ) ··· W (x )i is independent of the position of Q0 R1 0 n insertions x , ··· , x . Let O = W , Z(O) is a topological invariant. Gener- 1 n i γi ki alize to four manifolds with boundary ∂M = B, the physical Hilbert space on B is the cohomology group of Q, with states being functional Ψ(X|B) annihilated by Q. The path integral Z 2 Z(O, Ψ) = DX exp(−S/e ) ·O· Ψ(X|B) determines a topological invariant if QΨ = 0, and depends only on the cohomology class of Ψ. This corresponds to Donaldson polynomials taking values in Floer homology group HF∗(Y ).

Reduction to N = 1 case and computations of Donaldson’s polynomials

Consider simply connected compact, oriented four manifold M and gauge group G = SU(2). Because of Poincare duality nontrivial homology k-cycle Σ exists only for k = 0, 2, 4. The case k = 4 is rather trivial, Σ is just M, hI(Σ)i turns out to be R 2 the instanton number M TrF ∧ F/8π . A general correlation function takes the form hO(x1) ···O(xr) · I(Σ1) ··· I(Σs)i Rescale the metric by g → tg and let t → ∞. If the theory has a mass gap, in large t limit the correlation function can be expressed by a sum of local operators. Unfortunately this is not true for N = 2 theory. Assuming M is Kahler, the idea is to reduce to N = 1 supersymmetry by adding a mass term to chiral supermultiplet Φ = (φ, ψ): Z ∆L = − ω ∧ d2zd¯ 2θTrΦ2 + h.c. where ω is a (non-vanishing) holomorphic 2-form. This is a sum of I(Σi) terms up to a BRST commutator, the correlation functions are still topological invariant. The existence of nonzero holomorphic 2-form requires H2,0(M) 6= 0, which is also

10 Donaldson’s consideration. The general partition function as a response to the background metric is h1i = exp(−Leff ) where the eﬀective action Leff can be expanded as a sum of integral of local R 4 √ 2 operators. The only topological invariant of this form (e.g. M d x gR ) is a linear combination of the Euler characteristic χ and the signature σ:

h1i = exp(aχ + bσ)

For insertions of local operator O(xi), we can consider the limit that xi’s are far apart by scaling up the metric. Assuming there is only one vacuum we simply get

hexp(λO)i = h1i · exp(λhOiΩ)

Its generalization to several vacua is straightforward. For the 2-form observ- R mn ables write hI(Σ)i = Σ dσ hZmni. By similar arguments of scaling dimension hZmn(x)i = 0, and hZmn(x)Zpq(y)i = 0 unless x = y. By perturbing Σ1 and Σ2, the contribution to correlation function Z hI(Σ1)I(Σ2)i = hZ(x) ∧ Z(y)i Σ1×Σ2 is localized at the intersection points of Σ1 and Σ2. The only such topological invariant is the relative orientation at the intersection point, which is counted in the algebraic intersection number. Therefore we obtain the important relation

hI(Σ1)I(Σ2)i = η · #(Σ1 ∩ Σ2) · h1i

For the general case of more than two observables, by perturbing Σi’s we can assume there is only pairwise intersections. Including the possibility of several vacua, we arrive at the formula µ ¶ ³X ´ X η X hexp αiI(Σi) + λO i = h1iρ · exp αiαj#(Σi ∩ Σj) + λhOiΩρ (5) vacua 2 where h1i = exp(aρχ + bρσ). 0 In the case G = SU(2), the theory has symmetry Z4 × Z2 generated by α : λ → iλ, λ¯ → −iλ,¯ φ → −φ and β :Φ → −Φ. The chiral symmetry Z4 is believed spontaneously broken to Z2. In this case there are two vacua |+i and |−i, the correlation functions are related by the broken symmetry as η+ = −η− and hOi+ = −hOi−. h1i+ and h1i− are related by the action of α on fermion zero modes: ∆ h1i− = i h1i+, where ∆ is the diﬀerence of λ zero modes and λ¯ zero modes, i.e. the index of Dirac operator, which is by the index theorem 1 ∆ = 4k − 3(1 − h1,0 + h2,0) = dim M 2

11 ³ ´ 3iπ Therefore h1i− = exp − 8 (χ + σ) h1i+. For hyper-Kahler manifolds there exists non-vanishing holomorphic 2-form ω, our previous formula can be applied directly. The only compact hyper-Kahler four 4 4 manifolds are T and K3 surface. T has χ = σ = 0, therefore h1i+ = h1i− = 1; 1,0 2,0 K3 has h = 0, h = 1, h1i+ = −h1i−, their Donaldson polynomials are directly obtained from (5).

The A-model

By twisting the (2,2) supersymmetric sigma model one gets topological theory. Consider maps from a Riemann surface Σ to the target space, Kahler manifold X. The bosonic ﬁeld is a map Φ : Σ → X. The fermi ﬁelds are χI , a section of Φ∗(TX) and a 1-form ψ = ψ¯i dz ⊗ ∂ + ψi dz¯ ⊗ ∂ taking value in Φ∗(TX) with z ∂φ¯i z¯ ∂φi the self-duality condition. The BRST transformation is

δφI = iαχI , δχI = 0 ¯i ¯i ¯j ¯i m¯ δψz = −α∂φ − iαχ Γ¯jm¯ ψz i ¯ i j i m δψz¯ = −α∂φ − iαχ Γjmψz¯

The action is Z µ ¶ 2 1 I J ¯i i i ¯i i ¯i i ¯j S = 2t dz gIJ ∂zφ ∂z¯φ + iψzDz¯χ g¯ii + iψz¯Dzχ g¯ii − Ri¯ij¯jψz¯ψzχ χ Σ 2

I I In the usual (2,2) sigma model, fermionic fields ψ+ and ψ− have superconformal weight 1/2, they are sections of K1/2 ⊗ Φ∗(TX) and K¯1/2 ⊗ Φ∗(TX) respectively, where K = T (1,0)Σ(K¯ = T (0,1)Σ) is the canonical (anit-canonical) line bundle of Riemann surface Σ. The parameter α of fermionic symmetry is a section of the line bundle K1/2. The point of the twisted model is that the fermionic fields χ and ψ are taken to be sections of Φ∗(T 1,0X) and K⊗Φ∗(T 0,1X), i.e. of superconformal weight 0 and 1. Half of α’s are globally defined as functions on Σ, the rest are set to zero. Our construction above is known as the A-model, an alternative twist would give rise to the B-model. It follows that the action can be put in the form Z Z S = it d2z{Q,V } + t Φ∗K Σ Σ

i ¯j where K = −igijdz ∧ dz is the Kahler form. The last term only depends on the cohomology class of K, for simplicity we can choose it to be in 2πZ. Except for a factor of exp(2πnt), the partition function is independent of the coupling, or scale t. It follows that the stress tensor is a BRST commutator: Tαβ = {Q, bαβ}. The partition function is independent of the both target space metric and “world- sheet” metric. In particular the graviton vertex operator is a BRST commutator, this implies that the metric ﬂuctuation in the target space is not observable.

12 For Σ of non-empty boundary we require Φ map ∂Σ into Langragian submanifold M ⊂ X, the normal derivative at ∂Σ lies in Φ∗(NM) (pullback of normal bundle of M); χ and ψ take values in Φ∗(TM) on ∂Σ. Suppose ∂Σ consists of circles Ci mapped into Mi ⊂ X. We can introduce Chan-Paton factors and couple the theory to gauge ﬁelds in the target space by Z Y Z ∗ DΨi exp(−S(Ψi)) · TrP exp Φ A i Ci We require the Wilson lines to preserve the fermionic symmetry, i.e. the variation

Z Z J Z ∗ I dφ ∗ δ Tr exp Φ A = Tr δφ FIJ (τ)dτ · P exp Φ A C C dτ Cτ should vanish. Cτ is the loop C starting and ending at τ, coming from taking the variation of the path-ordered integral. δφI = iαχI =⇒ F = 0. We will show that A-model in the target space is precise Chern-Simons gauge theory on Mi’s, with classical solutions being ﬂat connections. In other words, string theory can only be coupled to background ﬁelds as classical solutions of space-time theory!

Large t limit

A simpler case is when the target space X is the cotangent bundle of the Lagrangian submanifold T ∗M. In this case there is no nonconstant instantons proved as follows. The bosonic part of sigma model action Z Z Z I ¯ J ¯i ¯ j ∗ I = i gIJ ∂φ ∧ ∂φ = 2i g¯ij∂φ ∧ ∂φ + Φ ω Σ Σ Σ ω is the Kahler form on X. The second term depends only the homotopy class of map Φ : Σ → X. Instantons minimize the classical solutions of a certain homotopy type, they are given by ∂φ¯ i = 0. If X = T ∗M, take ω to be the natural P a symplectic form dpa ∧ dq = dρ where ρ vanishes on M. Since Φ(∂Σ) ⊂ M, R ∗ R ∗ Σ Φ ω = ∂Σ Φ ρ = 0. I vanishes for instantons ⇒ φ = constant map. For general Calabi-Yau manifold X there will be instanton corrections to the space-time gauge theory action. Consider open string ﬁeld theory Z µ ¶ 1 2 S = Ψ ∗ QΨ + Ψ ∗ Ψ ∗ Ψ 2 3 with gauge transformation

δΨ = Q² − ² ∗ Ψ + Ψ ∗ ² R The BRST operator “derivation” Q has ghost number 1 and has ghost number −3. In the context of string theory we demand Ψ has ghost number 1, hence it

13 a must be of the form Ψ = χ Aa(q). We don’t consider closed strings because they 1 don’t appear at first order in t , and the theory is independent of t, the large t limit is exact. It is argued that the open and closed strings decouple in the topological string theory. In the large t limit the path integral reduces to integral over zero modes. The string field action reduces as Z Z Z 1 1 a b 1 Ψ ∗ QΨ → dqdχ Trχ Aa(q)Q· χ Ab(q) = TrA ∧ dA 2 2 2 M R R Similarly the interaction term Ψ ∗ Ψ ∗ Ψ reduces to M TrA ∧ A ∧ A. In the case X = T ∗M there is no instanton correction, the open string field action reduces exactly to the space-time Chern-Simons action on M.

Tree level S-matrix

The tree level computation provides a simple but interesting check of the equiva- lence between topological string theory and Chern-Simons gauge theory. Back to Chern-Simons perturbation theory, expand gauge field A = A(0) + B and fix the gauge d ∗ B = 0. The propagator is given by the “Hodge theory ∗ 1 ∗ ∗ inverse” d ∆ , where ∆ = dd +d d is the Laplacian. “Physical states” are elements of H1(Σ, End(E)). One would expect the four point (cyclic order) amplitude of topological string theory give rise to the amplitude of Chern-Simons gauge field theory Z µ ¶ d∗ d∗ I(αi) = Tr α1 ∧ α2 α3 ∧ α4 + α2 ∧ α3 α4 ∧ α1 (6) M ∆ ∆ corresponding to the degenerated graph of the disk. a Vertex operators Vi = (αi)aχ are inserted at the boundary of the disk. Taking the 1 modulus to be the position of V2, choose V1,V3,V4 to be at 0, 1, ∞ respectively. The antighost insertion effectively replaces V2 by dqa W = [b ,V ] = i(α ) + ψI χa∂ (α ) 2 0 2 2 a dτ τ I 2 a The amplitude is Z 1 dσhV1(0) · W2(σ) · V3(1) · V4(∞)i 0 at large t limit it is dominated by divergent contributions near σ = 0 (and σ = 1), taking the form Z ² π 1 ∗ dσ hχ(d (α1 ∧ α2))(0) · V3(1) · V4(∞)i 2t 0 σ ∗ To fix the divergence, notice that χ(d (α1 ∧ α2)) is not annihilated by L0, we need to include anomalous dimension L0 = π∆/2t. Integrating out the zero modes, we have Z ² Z ³ ´ π 1 ∗ π∆/2t dσ Tr d (α1 ∧ α2)σ ∧ α3 ∧ α4 2t 0 σ M

14 At the limit t → ∞ this (together with the contribution from σ → 1) gives precisely (6). t’Hooft expansion and closed string interpretation

The 1/N expansion of the free energy of Chern-Simons theory takes the form X X h 2g−2+h 2−2g 2g−2+h F = Cg,hN κ = Cg,hN λ g,h g,h

2π where κ = k+N is the renormalized gauge coupling, λ = κN is ﬁxed at large N limit. Interpreted as A-model topological string theory on T ∗M, N h comes from trace of Chan-Paton factors, κ is the string coupling, Cg,h = Zg,h is the partition function at genus g with h boundaries. In the case M = S3, the exact expression of the partition function is given by (4). Tedious but straight forward computation gives Cg,h (g ≥ 1): ζ(2g − 2 + 2p) C = (−1)g−1+pχ ,C = 0 g,2p g,2p (2π)2g−2+2p g,2p+1 where χg,h is the Euler characteristic of the moduli space of Riemann surfaces of genus g and h holes. Consider Cg,2p as an analytic function in h = 2p, going to h = 0 limit we have ζ(2g − 2) C = (−1)g−1χ g,h→0 g (2π)2g−2 One would expect this agree with the result obtained from topological closed string theory. We sketch the idea of computing genus g (g > 2) partition function of A- model closed string theory as follows. As we known the instantons of A-model are holomorphic maps from Σ to M, taking t → ∞, roughly speaking the partition function counts the “number” of holomorphic curves in M. The currents of the twisted theory can be identiﬁed with those of the bosonic string theory as follows

+ − G ←→ 2jB,G ←→ b, J ←→ bc where G± are superconformal currents and J is the U(1) current of N = 2 SCFT. This can be confirmed by showing they satisfy the same commutation relations. Consider the moduli space of genus g Riemann surfaces Mg. The Beltrami differ- a ¯b entials µ = µa dm andµ ¯ =µ ¯¯b dm¯ are defined as z zz¯ z¯ zz¯ ¯ µz¯ = g ∂mgz¯z¯, µ¯z = g ∂mgzz As the bosonic string case, the genus g partition function (g > 1) is obtained by folding G− with the Beltrami differentials, and integrate over the moduli space: * + Z Z ^ Z 1 a α ¯b ¯ ¯ β¯ Fg = 3g−3 dm µaψα∂X dm¯ µ¯¯bψβ¯∂X Mg (3g − 3)!(2πi) Σg(m) Σg(m)

15 Go to large t limit, we expand around instantons. The leading contribution comes from constant maps, although for general Calabi-Yau M there would be instantons other than the constant map. The moduli space of such maps is obviously given by Mg × M, M being the Calabi-Yau target space. χ is a section of the pullback ∗ of TM, it has 3 zero modes spanning the ﬁber of π2TM. Similarly ψ has 3g zero modes spanning the ﬁber of vector bundle

∗ ∗ ∗ V = π1H ⊗ π2T M where H is the Hodge bundle over M , with ﬁbers being holomorphic 1-forms over g R Σg(m). H has a natural metric from NAB¯ = Σ ωA ∧ ω¯B¯ . (In analogous to the previous discussion on the moduli space of ﬂat connections.) The curvature can be computed as µZ ¶ ^ µZ ¶ CD¯ a ¯b PAB¯ = −N dm µaωA ∧ ωC dm¯ µ¯¯bω¯D¯ ∧ ω¯B¯

Integrate over the zero modes and use the free contraction of X ﬁelds

a ¯ β¯ αβ¯ AB¯ h∂X (z)∂X (w ¯)ig = G ωA(z)N ω¯B¯ (w ¯)

The only way to absorb χ zero modes in the path integral is by pulling down i ¯i j ¯j Ri¯ij¯jψz¯ψzχ χ from the exponential of the action. The result in the weak coupling limit is Z Z g R Fg|t→∞ = (−1) det = c(V) Mg×M 2πi Mg×M where R is the curvature on Mg × M. Let H be the de Rham bundle over Mg 1 ∗ with fiber being H (Σg, C), by Hodge decomposition H = H ⊕ H . H admits a natural flat connection, since the metric on H1(Σ, Z) is given by the cup product in the singular cohomology ring, one can make a local trivialization in which the metric is independent of the moduli. Therefore c(H)c(H∗) = c(H) = 1, and c(H∗) ∗ is related to c(H) by flipping the signs of odd degree terms. Write C3g(H ⊗ T M) in terms of the Chern classes of H and T ∗M, using the fact that M has vanishing first Chern class, eventually one gets Z 1 3 Fg|t→∞ = χ(M) cg−1 2 Mg In fact, a direct algebraic geometry computation by Faber and Pandharipande showed Z 3 g−1 ζ(2g − 2) cg−1 = (−1) χg 2g−2 Mg (2π) This would be the same as the free energy of Chern-Simons gauge theory if χ(M) = 2. A possible choice of M is the S2 resolution of T ∗S3, which at infinity looks like S2 × S3 with the S2 of finite size. This is analogous to the S5 in AdS/CFT. In

16 fact Gopakumar and Vafa showed the exact matching between the free energy of Chern-Simons theory on S3 " # X∞ 2g−2 g−1 2ζ(2g − 2) χg 2g−3 −nt Fg = gs (−1) χg 2g−2 − n e (2π) (2g − 3)! n=1 and closed topological string on S2 resolution of T ∗S3 " # Z X 2g−2 χ 3 −nt Fg = gs cg−1 + αne 2 Mg where the sum comes from the contribution from genus g curves wrapping around the S2 n times, t = iλ. They further conjectured that the two theories are exactly dual to each other. Further evidence was found in that the Wilson lines observable in the topological string theory gives the same knot invariants.

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