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Coulomb branch localization, quasimaps, and surface counting in Calabi--Yau fourfolds

Duiliu-Emanuel Diaconescu, Nicolo Piazzalunga

Abstract

We present a string theoretic approach to surface counting in local Calabi--Yau fourfolds via supersymmetric localization in topologically twisted four-dimensional gauge theories. This approach is based on a spectral correspondence between PT1-stable pairs on local fourfolds and twisted quasimaps with fixed two-dimensional domain associated to the ADHM quiver, or, equivalently, ADHM sheaves. For local toric fourfolds, we derive a conjectural residue formula for the K-theoretic quasimap partition function via Coulomb branch localization. As a result, in this case, we obtain a conjectural prescription fixing all usual sign ambiguities in the equivariant computation of such invariants. We present some explicit computations for local P2, extending the results available in the literature, and describe the formalism in general. This is the first instance of Coulomb branch localization for a quasimap theory in the context of four-dimensional gauge theories.

Coulomb branch localization, quasimaps, and surface counting in Calabi--Yau fourfolds

Abstract

We present a string theoretic approach to surface counting in local Calabi--Yau fourfolds via supersymmetric localization in topologically twisted four-dimensional gauge theories. This approach is based on a spectral correspondence between PT1-stable pairs on local fourfolds and twisted quasimaps with fixed two-dimensional domain associated to the ADHM quiver, or, equivalently, ADHM sheaves. For local toric fourfolds, we derive a conjectural residue formula for the K-theoretic quasimap partition function via Coulomb branch localization. As a result, in this case, we obtain a conjectural prescription fixing all usual sign ambiguities in the equivariant computation of such invariants. We present some explicit computations for local P2, extending the results available in the literature, and describe the formalism in general. This is the first instance of Coulomb branch localization for a quasimap theory in the context of four-dimensional gauge theories.

Paper Structure

This paper contains 26 sections, 27 theorems, 154 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The setup
  3. ADHM sheaves and spectral correspondence
  4. PT1/Coulomb correspondence
  5. Open questions
  6. Organization
  7. Acknowledgments
  8. Stable pairs on fourfolds as ADHM sheaves
  9. ADHM sheaves
  10. Beilinson Theorem
  11. Stable pair moduli spaces
  12. From stable pairs to ADHM sheaves
  13. Topological invariants
  14. Compactness
  15. Gauge theory
  16. ...and 11 more sections

Key Result

Theorem 1.2

The derived algebraic space parametrizing $\mathrm{PT}_1$ stable pairs on $X$ is naturally identified with a derived moduli space of framed ADHM sheaves on $S$ via spectral correspondence.

Figures (2)

  • Figure 1: $\mathrm{PT}_1$/Coulomb correspondence.
  • Figure 2: An example of renormalized $\tilde{K}$. We have ${\tilde{n}}_1 = 1 + q_2$, ${\tilde{n}}_2=1$ and $\lambda = 1 + q_1 + q_1^2 + q_1^3 + q_2 + q_1 q_2 + q_2^2$. One can check that eq:regK is satisfied, since $\tilde{K} = \frac{1+q_2}{1-q_1} + \frac{q_2^2}{1-q_2} + \lambda q_1 q_2^2$. The case $d=2$ is special, as the renormalized partition is still a partition (this is not true for $d>2$).

Theorems & Definitions (61)

  • Remark 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Remark 1.4
  • Theorem 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Corollary 2.4
  • Lemma 2.5
  • proof
  • ...and 51 more