Ivancevic_Applied-Diff-Geom

1110 Applied Differential Geometry: A Modern Introductiondimension of the space of solutions of these equations is the virtual dimensionof the moduli space M. Thus, within the context of our quantum fieldtheoretical model, the virtual dimension of M is identified with the numberof the zero modes of the quantum fields ψ and N.For simplicity we assume that there are no zero modes of ψ and N, i.e.,the moduli space is zero–dimensional. Then no zero modes exist for theother two fermionic fields χ and µ. To compute the partition function inthis case, we first observe that the quadratic action S q(p) is invariant underthe supersymmetry obtained by expanding Q to first order in the quantumfields around the monopole solution A o , M o (equations of motion for thenonpropagating fields H and ν should also be used.). This supersymmetrytransforms the set of 8 real bosonic fields (each complex field is counted astwo real ones; the a i contribute 2 upon gauge fixing.) and the set of 16fermionic fields to each other. Thus at a given monopole background weget [Zhang et. al. (1995)]∫DF ′ exp(−S (p)q ) = Pfaff(∇ F)|Pfaff(∇ F )| = ɛ(p) ,where ɛ (p) is +1 or -1. In the above equation, ∇ F is the skew symmetricfirst order differential operator defining the fermionic ( part ) of the actionS q(p) , which can be read off from S q (p)0 Tto be ∇ F =−T ∗ . Therefore,0ɛ (p) is the sign of the determinant of the elliptic operator T at the monopolebackground A o , M o , and the partition function Z = ∑ p ɛ(p) coincides withthe SW invariant of the 4–manifold X.When the dimension of the moduli space M is greater than zero, thepartition function Z vanishes identically, due to integration over zero modesof the fermionic fields. In order to get any non trivial topological invariantsfor the underlying manifold X, we need to examine correlations functions ofoperators satisfying equations (6.173) and (6.174). A class of such operatorscan be constructed following the standard procedure [Witten (1994)]. Wedefine the following set of operators