Instantons beyond topological theory II
E. Frenkel, A. Losev, N. Nekrasov
TL;DR
This work extends the Part I program by treating two- and four-dimensional instanton theories in the infinite radius limit, where the path integral localizes to finite-dimensional instanton moduli and couplings are complexified into ${\tau}$ and ${\overline{\tau}}$. The authors demonstrate that, beyond the BPS sector, the Hamiltonian is non-diagonalizable due to instanton-induced extensions, yielding a logarithmic CFT with central charge $c=0$ and logarithmic mixing of jet-observation operators. They develop jet-evaluation observables tied to the jet space ${\mathcal J}X$ and show that their correlators require regularization, producing logarithmic partners and nontrivial OPE corrections mediated by holomortex operators and GC-type maps within a chiral-anti-chiral de Rham framework. The analysis carries over to gauged sigma models and to four-dimensional ${\mathcal N}=2$ Yang–Mills via a gauged Morse-theory approach, revealing similar logarithmic structures and connecting to Donaldson invariants and geometric Langlands through the appearance of chiral de Rham-type cohomology. Overall, the paper uncovers a rich LCFT structure in instanton theories at infinite radius, with precise operator extensions and OPE corrections governed by moduli-space singularities and holomorphic factors. $${\cal N}=2$$-theoretic and geometric Langlands-inspired aspects emerge from the gauged constructions, suggesting deep links between instanton counting, jet data, and chiral algebra structures in nonperturbative QFT.
Abstract
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 (arXiv:hep-th/0610149) 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.