Notes on Topological Field Theory Xi Yin Harvard University Introduction The notes give a survey of the basics of the following topological field theories: ² The Chern-Simons gauge theory on 3-manifolds, its renormalization, geomet- ric quantization, computation of partition functions by surgery, and relation with Jones Polynomials ² 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 ² 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) 4¼ 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 par- tition function Z Yr Z(M; L) = DA exp(iS) WRi (Ci) i=1 Under gauge transformation the Chern-Simons functional is well-defined mod- ulo 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 4¼ 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 4¼ 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(¼1(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 (®) At weak coupling limit, expand gauge field Ai = Ai + Bi, the first terms of the action are Z k S = k ¢ CS(A(®)) + Tr(B ^ DB) 4¼ M where D is the covariant derivative with respect to A(®). To fix the gauge we have i to pick a metric: DiB = 0. The ghost action is Z i i SGF = Tr(ÁDiB +cD ¯ iD c) M Integrating out B; Á; c; c¯, the contribution to the partition function from expanding around A(®) is exp(i¼´(A)=2) ¢ T® (®) 1 P where ´(A ) is the “eta invariant” defined by 2 sign¸i, where ¸i’s are eigenval- ues of operator L¡, the restriction of ¤DA + DA¤ on odd forms; T® is the torsion invariant of flat connection A(®). Let M = @X, extend A(®) to X, Atiyah-Patodi- Singer index theorem says Z 1 index L¡ = Aˆ(X) ¢ c(E) ¡ ´(L¡jM ) X 2 The integral of Chern class c(E) on X is just the Chern-Simons functional on the 1 (®) 1 (®) boundary M, we obtain the relation 2 ´(A ) ¡ 2 ´(0) = Qg=2¼ ¢ CS(A ). The partition function can be written as ¼ X (®) ´(0) i(k+Qg=2)¢CS(A ) Z = e 2 e ¢ T® ® 2 We see that k + Qg=2 is the renormalized coupling constant, for SU(N) this is k + N. ´(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(! ^ d! + ! ^ ! ^ !) 4¼ M 3 ! is the affine connection on the spin bundle over M, d = dim G. By APS theorem 1 d 2 ´(0) + 24¼ 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(2¼is ¢ d=24); s 2 Z. The following concrete example is interesting. For U(1) gauge group, the action is simply Z k ijk S = ² Ai@jAk 8¼ 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 0 1 Z Z 1 X hW i = exp @¡ n n dxi dyjhA (x)A (y)iA 2 a b i j a;b Ca Cb 0 1 Z Z i X xk ¡ yk = exp @ n n dxi dyj² A 2k a b ijk jx ¡ yj3 a;b Ca Cb The integral Z Z k k 1 i j x ¡ y Φ(Ca;Cb) = dx dy ²ijk 3 4¼ Ca Cb jx ¡ yj is some integer, known as the Gauss linking number, the most classical knot in- variant. 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 8¼ Σ dt 3 The Poisson bracket is 4¼ fAa(x);Ab(y)g = ² ±ab±2(x ¡ y) i j k ij together with the constraint ±S k ij a = ² Fij = 0 ±A0 4¼ 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 ! = ® ^ ¯ (1) 4¼ Σ where ®; ¯ 2 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 L­k. 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. ² 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. ² r = 1, the Hilbert space has dimension 1 if R is the trivial representation and 0 otherwise. ² r = 2, HΣ = Inv(R1 ­ R2) has dimesion 1 if R2 is the conjugate representation of R1 and 0 otherwise. ² 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) 2 HΣ. It is certainly a smooth invariant because the metric doesn’t appear in the path integral.