Integrating over the Coulomb branch in N=2 gauge theory.pdfVIP

a r X i v : h e p - t h / 9 7 1 2 0 6 2 v 1 5 D e c 1 9 9 7 1 Integrating over the Coulomb branch in N = 2 gauge theory Marcos Marin?o and Gregory Moore a? aDepartment of Physics, Yale University, Box 208120, New Haven, CT 06520, USA We review the relation of certain integrals over the Coulomb phase of d = 4, N = 2 SO(3) supersymmetric Yang-Mills theory with Donaldson-Witten theory. We describe a new way to write an important contact term in the theory and show how the integrals generalize to higher rank gauge groups. 1. Introduction In the past 15 years there has been a fruit- ful, sometimes dramatic, math/physics dialogue in the arena of 4-manifold theory and SYM. In the early 1980’s Donaldson defined diffeomorphism invariants of compact, oriented 4-folds X using Yang-Mills instantons. These invariants are best organized in terms of a single function (or for- mal series), on the homology of X called the the Donaldson-Witten function. Let P ∈ H0(X ;Z), S ∈ H2(X ;Z), then the Donaldson-Witten func- tion can be written as: ZDW (pP + vS) = ∑ d?,r p? ?! vr r! (1) In 1988 E. Witten [28] interpreted ZDW as the generating function of correlation functions in topological SO(3) SYM: ZDW (v·S+pP ) = ? exp[v·I(S)+pO(P )] ? SO(3),N=2 (2) where I(S) and O(P ) are certain operators de- fined below. This lead to an interesting refor- mulation of the problem of computing Donald- son invariants and, in 1994, when the structure of the vacuum became apparent due to the work of Seiberg and Witten [24], Witten [31] gave a beautiful and simple expression for Donaldson in- variants for X of b+2 1 and “of simple type,” reproducing and extending the results of Kron- heimer and Mrowka [17]. All known 4-manifolds of b+2 1 are of simple type. ?Supported by DOE grant DE-FG02-92ER40704. With the publication of [31] the program of de- riving 4-manifold invariants from supersymmetric Yang-Mills theory became a resounding success. Nevertheless, a few loose ends remained to be tied up. These