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Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence

Sergei Alexandrov, Daniel Persson, Boris Pioline

TL;DR

The work establishes a general QK/HK correspondence that pairs quaternion-Kähler spaces with a quaternionic circle isometry to hyperkähler spaces with a rotational isometry, augmented by a canonical hyperholomorphic circle bundle. In the physical setting of $\mathcal{N}=2$ theories and string vacua, D-instanton corrections to the hypermultiplet moduli space are captured by this dual HK data, with the hyperholomorphic bundle encoding BPS invariants and central charges. A key technical advance is expressing the wall-crossing consistency conditions through the Rogers dilogarithm $L(z)$, whose functional identities (pentagon, hexagon, octagon, and beyond) follow from the motivic KS wall-crossing formula; this ensures a globally well-defined twistor construction for the corrected moduli spaces. The paper further connects these geometric constructions to local/rigid $c$-maps, cluster algebras (rank-2 Dynkin types $A_2,B_2,G_2$), and outlines potential quantization extensions, revealing deep links between wall-crossing, dilogarithm identities, and cluster structures in $\mathcal{N}=2$ theories. Together, these results provide a rigorous geometric framework for D-instanton corrections and illuminate the shared mathematical structure between gauge theory and string theory moduli spaces.

Abstract

When formulated in twistor space, the D-instanton corrected hypermultiplet moduli space in N=2 string vacua and the Coulomb branch of rigid N=2 gauge theories on $R^3 \times S^1$ are strikingly similar and, to a large extent, dictated by consistency with wall-crossing. We elucidate this similarity by showing that these two spaces are related under a general duality between, on one hand, quaternion-Kahler manifolds with a quaternionic isometry and, on the other hand, hyperkahler manifolds with a rotational isometry, further equipped with a hyperholomorphic circle bundle with a connection. We show that the transition functions of the hyperholomorphic circle bundle relevant for the hypermultiplet moduli space are given by the Rogers dilogarithm function, and that consistency across walls of marginal stability is ensured by the motivic wall-crossing formula of Kontsevich and Soibelman. We illustrate the construction on some simple examples of wall-crossing related to cluster algebras for rank 2 Dynkin quivers. In an appendix we also provide a detailed discussion on the general relation between wall-crossing and the theory of cluster algebras.

Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence

TL;DR

The work establishes a general QK/HK correspondence that pairs quaternion-Kähler spaces with a quaternionic circle isometry to hyperkähler spaces with a rotational isometry, augmented by a canonical hyperholomorphic circle bundle. In the physical setting of theories and string vacua, D-instanton corrections to the hypermultiplet moduli space are captured by this dual HK data, with the hyperholomorphic bundle encoding BPS invariants and central charges. A key technical advance is expressing the wall-crossing consistency conditions through the Rogers dilogarithm , whose functional identities (pentagon, hexagon, octagon, and beyond) follow from the motivic KS wall-crossing formula; this ensures a globally well-defined twistor construction for the corrected moduli spaces. The paper further connects these geometric constructions to local/rigid -maps, cluster algebras (rank-2 Dynkin types ), and outlines potential quantization extensions, revealing deep links between wall-crossing, dilogarithm identities, and cluster structures in theories. Together, these results provide a rigorous geometric framework for D-instanton corrections and illuminate the shared mathematical structure between gauge theory and string theory moduli spaces.

Abstract

When formulated in twistor space, the D-instanton corrected hypermultiplet moduli space in N=2 string vacua and the Coulomb branch of rigid N=2 gauge theories on are strikingly similar and, to a large extent, dictated by consistency with wall-crossing. We elucidate this similarity by showing that these two spaces are related under a general duality between, on one hand, quaternion-Kahler manifolds with a quaternionic isometry and, on the other hand, hyperkahler manifolds with a rotational isometry, further equipped with a hyperholomorphic circle bundle with a connection. We show that the transition functions of the hyperholomorphic circle bundle relevant for the hypermultiplet moduli space are given by the Rogers dilogarithm function, and that consistency across walls of marginal stability is ensured by the motivic wall-crossing formula of Kontsevich and Soibelman. We illustrate the construction on some simple examples of wall-crossing related to cluster algebras for rank 2 Dynkin quivers. In an appendix we also provide a detailed discussion on the general relation between wall-crossing and the theory of cluster algebras.

Paper Structure

This paper contains 42 sections, 291 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. The QK/HK correspondence
  3. Twistorial descriptions of HK and QK manifolds: a reminder
  4. Hyperkähler manifolds and symplectic geometry
  5. Quaternion-Kähler manifolds and contact geometry
  6. The QK/HK correspondence
  7. From QK to HK
  8. From HK to QK
  9. Darboux coordinates and transition functions
  10. The QK/HK correspondence in one quaternionic dimension
  11. Tod Ansatz for one-dimensional QK manifolds with one isometry
  12. Toda Ansatz for one-dimensional HK manifolds with one rotational isometry
  13. Hyperholomorphic connection and QK/HK correspondence
  14. Examples
  15. Local c-map vs. rigid c-map
  16. ...and 27 more sections

Figures (1)

  • Figure 1: Overview of the QK/HK correspondence.