Instantons beyond Topological Theory Ii

The present paper is the second part of our project in which we describe quantum field theories with instantons in a novel way by using the " infinite radius limit " (rather than the limit of free field theory) as the starting point. The theory dramatically simplifies in this limit, because the correlation functions of all, not only topological (or BPS), observables may be computed explicitly in terms of integrals over finite-dimensional moduli spaces of instanton configurations. In Part I we discussed in detail the one-dimensional (that is, quantum mechanical) models of this type. Here we analyze the supersymmetric two-dimensional sigma models and four-dimensional Yang–Mills theory, using the one-dimensional models as a prototype. We go beyond the topological (or BPS) sectors of these models and consider them as full-fledged quantum field theories. We study in detail the space of states and find that the Hamiltonian is not diagonalizable, but has Jordan blocks. This leads to the appearance of logarithms in the correlation functions. We find that our theories are in fact logarithmic conformal field theories (theories of this type are of interest in condensed matter physics). We define jet-evaluation observables and consider in detail their correlation functions. They are given by integrals over the moduli spaces of holomorphic maps, which generalize the Gromov–Witten invariants. These integrals generally diverge and require regularization, leading to an intricate logarithmic mixing of the operators of the sigma model. A similar structure arises in the four-dimensional Yang–Mills theory as well. Contents 1. Introduction 3 1.1. Weak coupling limit 3 1.2. Topological sector 4 1.3. Sigma models 5 1.4. Logarithmic structure of the correlation functions 6 1.5. Yang–Mills theory 7 1.6. Contents 8 1.7. Acknowledgments 10 2. Two-dimensional sigma models 10 2.1. Definition of the supersymmetric sigma model 10 2.2. Infinite radius limit 12 2.3. Sigma model in the infinite radius limit as a CFT 13 2.4. Sigma model as quantum mechanics on the loop space 13 3. Quantum mechanics with Morse–Novikov functions 16 3.1. Path integral analysis on non-simply connected manifolds 16 3.2. Maximal abelian cover 18 3.3. Ground states 19