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Quantum TBA for refined BPS indices

Sergei Alexandrov, Khalil Bendriss

TL;DR

The paper addresses how refined BPS indices $\Omega(\gamma,y)$ induce a quantum Riemann-Hilbert problem, i.e., a non-commutative deformation of the classical RH framework for BPS data on moduli spaces.It develops a reformulation in terms of a non-commutative, Moyal-like star-product version of TBA equations, and constructs a formal perturbative solution in the refinement parameter via a two-step construction involving refined Darboux coordinates and adjoint actions.A key outcome is the adjoint representation of the refined solution and the extraction of generating functions: an all-orders generating function for the refined Darboux coordinates $\mathcal{X}_\gamma$ and a well-defined unrefined limit $y\to 1$ that matches the classical RH problem, with explicit perturbative expansions and tree-based representations.The work checks consistency in the uncoupled case against known results and discusses Setup-specific issues (notably Setup 2 lacking a smooth unrefined limit) while pointing toward broader implications for quaternionic/HK/QK moduli spaces, twistor methods, and potential modular properties in string theory contexts.

Abstract

Refined BPS indices give rise to a quantum Riemann-Hilbert problem that is inherently related to a non-commutative deformation of moduli spaces arising in gauge and string theory compactifications. We reformulate this problem in terms of a non-commutative deformation of a TBA-like equation and obtain its formal solution as an expansion in refined indices. As an application of this construction, we derive a generating function of solutions of the TBA equation in the unrefined case.

Quantum TBA for refined BPS indices

TL;DR

The paper addresses how refined BPS indices $\Omega(\gamma,y)$ induce a quantum Riemann-Hilbert problem, i.e., a non-commutative deformation of the classical RH framework for BPS data on moduli spaces.It develops a reformulation in terms of a non-commutative, Moyal-like star-product version of TBA equations, and constructs a formal perturbative solution in the refinement parameter via a two-step construction involving refined Darboux coordinates and adjoint actions.A key outcome is the adjoint representation of the refined solution and the extraction of generating functions: an all-orders generating function for the refined Darboux coordinates $\mathcal{X}_\gamma$ and a well-defined unrefined limit $y\to 1$ that matches the classical RH problem, with explicit perturbative expansions and tree-based representations.The work checks consistency in the uncoupled case against known results and discusses Setup-specific issues (notably Setup 2 lacking a smooth unrefined limit) while pointing toward broader implications for quaternionic/HK/QK moduli spaces, twistor methods, and potential modular properties in string theory contexts.

Abstract

Refined BPS indices give rise to a quantum Riemann-Hilbert problem that is inherently related to a non-commutative deformation of moduli spaces arising in gauge and string theory compactifications. We reformulate this problem in terms of a non-commutative deformation of a TBA-like equation and obtain its formal solution as an expansion in refined indices. As an application of this construction, we derive a generating function of solutions of the TBA equation in the unrefined case.

Paper Structure

This paper contains 25 sections, 1 theorem, 121 equations, 6 figures.

Table of Contents

  1. Introduction
  2. BPS indices and TBA
  3. Setup 1.
  4. Setup 2.
  5. Setup 3.
  6. Refinement and non-commutativity
  7. Refined BPS indices and quantum RH problem
  8. Star product
  9. Refined variables
  10. Solution
  11. Perturbative expansion
  12. Reconstructing integral equations
  13. Other Setups
  14. Setup 2.
  15. Setup 3.
  16. ...and 10 more sections

Key Result

Lemma 1

For a labeled rooted tree $T$ with $n$ vertices, one has where $\mathds{T}_{n}^{\rm lin}(T)$ is the set of linear labeled rooted trees with $n$ vertices such that the labeling preserves the partial ordering of the original tree $T$ in the sense that $v<v'\Rightarrow \ell_{T'}^{-1}(\ell_T(v))< \ell_{T'}^{-1}(\ell_T(v'))$.

Figures (6)

  • Figure 1: A representation of the identity \ref{['cyc3']} in terms of rooted trees. Here $T_k$'s are any rooted subtrees.
  • Figure 2: The almost linear tree appearing in the proof of Lemma \ref{['lemma-tree']} and the result of application of the basic identity.
  • Figure 3: The rooted trees corresponding to the expansions of $\mathsf{X}_0$ in $\mathsf{X}^{\rm ref}_0$ and $\mathsf{X}^{\rm sf}_0$, respectively.
  • Figure 4: Cancellation between trees after application of Lemma \ref{['lemma-tree']}.
  • Figure 5: Contributions remaining after cancellations.
  • ...and 1 more figures

Theorems & Definitions (3)

  • Conjecture 1
  • Lemma 1
  • proof