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Instantons beyond topological theory I

E. Frenkel, A. Losev, N. Nekrasov

TL;DR

This work analyzes instanton-rich quantum field theories in one, two, and four dimensions under a limit that preserves instantons while suppressing anti-instantons, enabling explicit solutions and revealing deep structural properties beyond the topological sector. It develops a Hamiltonian framework where the space of states becomes a holomorphic factorization into delta-forms supported on Morse strata, with the Hamiltonian acquiring Grothendieck-Cousin (GC) corrections that render it non-diagonalizable and produce logarithmic terms in correlation functions. In Kahler settings the theory exhibits holomorphic factorization and, in 2D and 4D, falls into the class of logarithmic conformal field theories in this limit. The paper also connects these quantum-mechanical constructions to two- and four-dimensional theories (2D sigma models and 4D Yang–Mills) via delta-form localization on instanton moduli spaces, and develops a general framework for perturbations around the infinite-coupling limit, perturbative expansions of correlators, and Morse–Novikov generalizations, with GC complexes clarifying cohomological structures (de Rham and Dolbeault) and their link to chiral algebras.

Abstract

Many quantum field theories in one, two and four dimensions possess remarkable limits 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 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. We conclude that in dimensions two and four our theories are logarithmic conformal field theories.

Instantons beyond topological theory I

TL;DR

This work analyzes instanton-rich quantum field theories in one, two, and four dimensions under a limit that preserves instantons while suppressing anti-instantons, enabling explicit solutions and revealing deep structural properties beyond the topological sector. It develops a Hamiltonian framework where the space of states becomes a holomorphic factorization into delta-forms supported on Morse strata, with the Hamiltonian acquiring Grothendieck-Cousin (GC) corrections that render it non-diagonalizable and produce logarithmic terms in correlation functions. In Kahler settings the theory exhibits holomorphic factorization and, in 2D and 4D, falls into the class of logarithmic conformal field theories in this limit. The paper also connects these quantum-mechanical constructions to two- and four-dimensional theories (2D sigma models and 4D Yang–Mills) via delta-form localization on instanton moduli spaces, and develops a general framework for perturbations around the infinite-coupling limit, perturbative expansions of correlators, and Morse–Novikov generalizations, with GC complexes clarifying cohomological structures (de Rham and Dolbeault) and their link to chiral algebras.

Abstract

Many quantum field theories in one, two and four dimensions possess remarkable limits 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 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. We conclude that in dimensions two and four our theories are logarithmic conformal field theories.

Paper Structure

This paper contains 48 sections, 361 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Description of the models
  3. Some puzzles
  4. Summary of the results
  5. Two-dimensional sigma models
  6. Four-dimensional Yang-Mills theory
  7. Plan of the paper
  8. Acknowledgments
  9. Supersymmetric quantum mechanics
  10. Recollections on Morse theory
  11. Important special case
  12. Path integral and gradient trajectories
  13. Correlation functions as integrals over moduli spaces of instantons
  14. Topological sector
  15. Analogy with the Gromov-Witten theory
  16. ...and 33 more sections

Figures (1)

  • Figure :