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Comments on Worldsheet Description of the Omega Background

Yu Nakayama, Hirosi Ooguri

TL;DR

This work addresses extending Nekrasov's Omega background beyond the self-dual limit by classifying four-dimensional field-strength configurations with graviphoton and vector multiplet components and Fayet-Iliopoulos terms. It proposes a refined topological string framework capable of encoding the ε_+ ≠ 0 background via a specific mixture of self-dual and anti-self-dual insertions and FI twists, and tests the approach in a heterotic setup where the zero-slope limit reproduces the Nekrasov partition function. The paper further provides a Type II interpretation through string dualities and outlines how the refined amplitudes can be computed in non-compact Calabi–Yau geometries, offering a concrete path to calculate refined BPS spectra. Overall, it establishes a string-theoretic realization of refined topological string theory aligned with the refined Omega background and clarifies the role of R-twists and FI terms in preserving supersymmetry.

Abstract

Nekrasov's partition function is defined on a flat bundle of R^4 over S^1 called the Omega background. When the fibration is self-dual, the partition function is known to be equal to the topological string partition function, which computes scattering amplitudes of self-dual gravitons and graviphotons in type II superstring compactified on a Calabi-Yau manifold. We propose a generalization of this correspondence when the fibration is not necessarily self-dual.

Comments on Worldsheet Description of the Omega Background

TL;DR

This work addresses extending Nekrasov's Omega background beyond the self-dual limit by classifying four-dimensional field-strength configurations with graviphoton and vector multiplet components and Fayet-Iliopoulos terms. It proposes a refined topological string framework capable of encoding the ε_+ ≠ 0 background via a specific mixture of self-dual and anti-self-dual insertions and FI twists, and tests the approach in a heterotic setup where the zero-slope limit reproduces the Nekrasov partition function. The paper further provides a Type II interpretation through string dualities and outlines how the refined amplitudes can be computed in non-compact Calabi–Yau geometries, offering a concrete path to calculate refined BPS spectra. Overall, it establishes a string-theoretic realization of refined topological string theory aligned with the refined Omega background and clarifies the role of R-twists and FI terms in preserving supersymmetry.

Abstract

Nekrasov's partition function is defined on a flat bundle of R^4 over S^1 called the Omega background. When the fibration is self-dual, the partition function is known to be equal to the topological string partition function, which computes scattering amplitudes of self-dual gravitons and graviphotons in type II superstring compactified on a Calabi-Yau manifold. We propose a generalization of this correspondence when the fibration is not necessarily self-dual.

Paper Structure

This paper contains 18 sections, 55 equations.

Table of Contents

  1. 1. Introduction
  2. 2. Review of the Omega background
  3. 2.1 Melvin-type geometry
  4. 2.2 Relation to the topological string
  5. 3. Supersymmetric configurations with field strengths in four dimensions
  6. 3.1. Configurations without FI terms
  7. 3.1.1 $F_{\alpha\beta}^G \neq 0$ and all other $=0$
  8. 3.1.2 $F_{\alpha\beta}^G, F_{\dot \alpha \dot \beta}^{\bar{A}} \neq 0$ and all other $=0$
  9. 3.1.3 $F_{\alpha\beta}^G, F_{\alpha \beta}^{A}, \partial t^A \neq 0$ and all other $=0$
  10. 3.2. Configurations with FI terms
  11. 3.2.1. $F_{\alpha\beta}^G, F_{\alpha \beta}^{A}, P^A \neq 0$ and all other $=0$
  12. 3.3. Summary
  13. 4. Omega Background and Refined Topological String Theory
  14. 4.1. Omega Background
  15. 4.2. R twist and FI terms
  16. ...and 3 more sections