Notes on instantons in topological field theory and beyond

TL;DR

This paper investigates quantum field theories with instantons in a special limit, revealing exact correlation functions, non-diagonalizable Hamiltonians, and logarithmic conformal field theory behavior on Kahler manifolds.

Contribution

It provides a detailed analysis of models with instantons, showing explicit computation of correlation functions, spectrum, and uncovering logarithmic structures beyond topological sectors.

Findings

Correlation functions can be explicitly studied in these models.

The spectrum may include Jordan blocks, leading to logarithms.

Models on Kahler manifolds exhibit holomorphic factorization and are logarithmic CFTs.

Abstract

This is a brief summary of our studies of quantum field theories in a special limit in which the instantons are present, the anti-instantons are absent, and the perturbative corrections are reduced to one-loop. We analyze the corresponding models as full-fledged quantum field theories, beyond their topological sector. We show that the correlation functions of all, not only topological (or BPS), observables may be studied explicitly in these models, and the spectrum may be computed exactly. An interesting feature is that the Hamiltonian is not always diagonalizable, but may have Jordan blocks, which leads to the appearance of logarithms in the correlation functions. We also find that in the models defined on Kahler manifolds the space of states exhibits holomorphic factorization. In particular, in dimensions two and four our theories are logarithmic conformal field theories.