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$K$-theoretic wall-crossing formulas and multiple basic hypergeometric series

Ryo Ohkawa, Jun'ichi Shiraishi

TL;DR

The work develops a comprehensive $K$-theoretic wall-crossing framework for framed quiver moduli, highlighting handsaw and chainsaw quiver varieties and two primary matter classes. By constructing enhanced master spaces and applying localization, the authors derive explicit recursions and functional equations that connect partition functions across stability chambers, and they geometrically interpret known transformation formulas for multiple basic hypergeometric series (Kajihara, HLNR, Noumi). The resulting results yield explicit combinatorial descriptions of fixed points, closed-form product formulas, and dualities across AL spaces and their duals, illuminating deep links between quiver moduli, $K$-theory, and hypergeometric identities with potential applications to Nekrasov-type invariants and non-stationary Ruijsenaars systems. The methods unify wall-crossing techniques with $q$-deformed special functions, providing a framework that generalizes to various quiver settings and supports conjectural adjoint/fundamental dualities. The findings offer new geometric interpretations of classical hypergeometric transformation formulas and contribute to the broader understanding of algebraic and geometric structures underlying quantum field-theoretic partition functions.

Abstract

We study $K$-theoretic integrals over famed quiver moduli via wall-crossing phenomena. We study the chainsaw quiver varieties, and consider generating functions defined by two types of $K$-theoretic classes. In particular, we focus on integrals over the handsaw quiver varieties of type $A_{1}$, and get functional equations for each of them. We also give explicit formula for these partition functions. In particular, we obtain geometric interpretation of transformation formulas for multiple basic hypergeometric series including the Kajihara transformation formula, and the one studied by Langer-Schlosser-Warnaar and Hallnäs-Langman-Noumi-Rosengren.

$K$-theoretic wall-crossing formulas and multiple basic hypergeometric series

TL;DR

The work develops a comprehensive -theoretic wall-crossing framework for framed quiver moduli, highlighting handsaw and chainsaw quiver varieties and two primary matter classes. By constructing enhanced master spaces and applying localization, the authors derive explicit recursions and functional equations that connect partition functions across stability chambers, and they geometrically interpret known transformation formulas for multiple basic hypergeometric series (Kajihara, HLNR, Noumi). The resulting results yield explicit combinatorial descriptions of fixed points, closed-form product formulas, and dualities across AL spaces and their duals, illuminating deep links between quiver moduli, -theory, and hypergeometric identities with potential applications to Nekrasov-type invariants and non-stationary Ruijsenaars systems. The methods unify wall-crossing techniques with -deformed special functions, providing a framework that generalizes to various quiver settings and supports conjectural adjoint/fundamental dualities. The findings offer new geometric interpretations of classical hypergeometric transformation formulas and contribute to the broader understanding of algebraic and geometric structures underlying quantum field-theoretic partition functions.

Abstract

We study -theoretic integrals over famed quiver moduli via wall-crossing phenomena. We study the chainsaw quiver varieties, and consider generating functions defined by two types of -theoretic classes. In particular, we focus on integrals over the handsaw quiver varieties of type , and get functional equations for each of them. We also give explicit formula for these partition functions. In particular, we obtain geometric interpretation of transformation formulas for multiple basic hypergeometric series including the Kajihara transformation formula, and the one studied by Langer-Schlosser-Warnaar and Hallnäs-Langman-Noumi-Rosengren.
Paper Structure (38 sections, 50 theorems, 237 equations)

This paper contains 38 sections, 50 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. Framed quiver moduli
  3. Chainsaw quiver variety
  4. Handsaw quiver variety of type $A_{1}$
  5. Framed quiver moduli and enhanced master space
  6. Setting
  7. Enhancement of quiver
  8. Enhanced master space
  9. Moduli stack of destabilizing representations $\widetilde{B}^{\sharp}$ on $\widetilde{V}^{\sharp}$
  10. Decomposition of $\mathcal{M}^{\mathbb{C}^{\ast}_{\hbar}}$
  11. Relative tangent bundles for flags
  12. Wall-crossing formula
  13. Equivariant framed quiver
  14. Equivariant Rieman-Roch
  15. Virtual fundamental cycle
  16. ...and 23 more sections

Key Result

Theorem 1.1

For $\Lambda=\Lambda(\mathcal{V})$ on $M_{Q}^{\zeta^{\pm}}(\alpha)$, we have where $\lfloor \alpha_{\ast} / \beta_{\ast} \rfloor$ is a round down of the rational number $\alpha_{\ast} / \beta_{\ast}$, and $\widetilde{C}_{\boldsymbol{\mathfrak I}} (\mathcal{V})$ is a localized $\mathbb T \times \prod_{k=1}^{j} \mathbb{C}^{\ast}_{\hbar_{k}}$-equivariant cohomology class define

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Theorem 1.6
  • Theorem 1.7: Langer-Schlosser-Warnaar LSW, Hallnäs-Langmann-Noumi-Rosengren HLNR2
  • Remark 1.8
  • Theorem 1.9: Noumi
  • Theorem 1.10
  • ...and 51 more