Issues in topological gauge theory

Abstract

We discuss topological theories, arising from the general N = 2 twisted gauge theories.We initiate a program of their study in the Gromov-Witten paradigm. We re-examine the low-energy effective abelian theory in the presence of sources and study the mixing between the various p-observables. We present the twisted superfield formalism which makes duality transformations transparent. We propose a scheme which uniquely fixes all the contact terms. We derive a formula for the correlation functions of p-observables on the manifolds of generalized simple type for 0 ≤ p ≤ 4 and on some manifolds with b + 2 = 1. We study the theories with matter and explore the properties of universal instanton. We also discuss the compactifications of higher dimensional theories. Some relations to sigma models of type A and B are pointed out and exploited.

Testing Seiberg-Witten Solution

“…The important constraint on the function C(P, Q) is that it is independent of p and q. This property was called in [17] the universality of the contact terms. It follows from the simple fact that the relative geometry of a pair of points on a four-fold X is independent of the dimensions of the cycles the points are confined to.…”

Superlocalization formulas and supersymmetric Yang–Mills theories

“…The computation for arbitrary winding number in the case of eight supersymmetries with matter in the fundamental and adjoint representation was presented in [31]. In turn this work stemmed from a line of research focused on computing topological invariants with the aid of topological field theories [26,29,28]. …”

Remarks on twisted theories with matter

“…As originally noticed in [4], an explicit expression for the contact terms can be derived from the blowup function by requiring invariance under Sp(2g, Z Z) duality transformations, and taking into account that they must vanish semiclassically [3]. We shall perform a similar analysis in what follows.…”

“…We first identify the fixed characteristic of the Θ-function as a singular one. This fact affects significantly the behaviour of the blowup function and, consequently, modifies the generic expression of the contact terms with respect to that of the pure gauge case, the latter being given by [4] T k,l = − 1 2πi . We further show how the methods of [9] can be extended to actually compute the polynomials…”

“…This interplay appears in a very natural way in the context of the so-called u-plane integral introduced by Moore and Witten [3], and has been explored in some detail in [4][5] [6][7] [8]. In a recent paper [9], we have extended and clarified the role of integrable systems in twisted N = 2 theories by using the theory of hyperelliptic Kleinian functions.…”