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Vi! Vj] = lCwi3: [V;, Vj] = 0, [V;, V,] = 0. It is importmit to note thal: 1) 6% preserves holornorphicity. This can be verified by showing that it. commute with complex derivatives A;. 2) = central. This means that it is a projectively flat connection. Next we consider more interesting case of an afirie symplectic space wilh a Lie group G action which preserves the symplectic form. We then have symplcctic reduction and the symplectic quotient is M = p-'(Oj/G, where p is the moment map. In the c a x of a finite dimensional affine space we can push down objects like symplectic farm, complex structure, natural line bundle, connection etc.
It can be quantized by the usual methods as described in chapter 1. ,btgl = Hnm(xl(C),S’) = R2”/ZZg= Jac(C). It is the Jacobi variety. Given a complex structure J an C, it induces a complex structure on Jac(C). >ug and Jac(C) = Cg/Z’g is represented by divided by its periods where cul! ,figis a basis of HI@,2). To quantize Jac(E:),we first const,ruct a line bundle L -+ Jac(C) with a natural connection A on C whose curvature is --cJ, i . e . R),the tangent space of the moduli space I Jam! Note also that w E H’l’(Jac, Z ) -+ H2(Jac,R) is the first Chern class of C.
Hefore doing that we shall explain how to describe deforrriations of a complex structure in our case. The tangent space of the moduli space is T,M = H i ( C ;E @ g) = T"' @ To)* An infinitesimal deformation of the complex structure I is I : T O J+ T110,I I + I j = 0. The deformation is given by G: 3 Tor1 -f, T1'0, G(cu,cu)= J,Tr(cuA * a ) , T r a 2 E H o ( C ;K 2 ) is a holomorphic quadratic form. In other words, 6 J = Gijwji& @ dzk. Finally the quantum connection is: xi,j,k 1 6* = 6 + -(ViSJijVj 2t + 6 J i j ( V i l o g H ) V j + -21-k *k*+ h J l o g H ) , 44 Quantization of Chern-Simons Action where 6 is the derivation of complex structures, H = log A and Witten et a1 proved that S* is a projective flat connection.