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S-duality and the universal isometries of instanton corrected q-map spaces

Vicente Cortés, Iván Tulli

Abstract

Given a conical affine special Kähler (CASK) manifold together with a compatible mutually local variation of BPS structures, one can construct a quaternionic-Kähler (QK) manifold. We call the resulting QK manifold an instanton corrected c-map space. Our main aim is to study the isometries of a subclass of instanton corrected c-map spaces associated to projective special real (PSR) manifolds with a compatible mutually local variation of BPS structures. We call the latter subclass instanton corrected q-map spaces. In the setting of Calabi-Yau compactifications of type IIB string theory, instanton corrected q-map spaces are related to the hypermultiplet moduli space metric with perturbative corrections, together with worldsheet, D(-1) and D1 instanton corrections. In the physics literature, it has been shown that the hypermultiplet metric with such corrections must have an $\mathrm{SL}(2,\mathbb{Z})$ acting by isometries, related to S-duality. We give a mathematical treatment of this result, specifying under which conditions instanton corrected q-map spaces carry an action by isometries by $\mathrm{SL}(2,\mathbb{Z})$ or some of its subgroups. We further study the universal isometries of instanton corrected q-map spaces, and compare them to the universal isometries of tree-level q-map spaces. Finally, we give an explicit example of a non-trivial instanton corrected q-map space with full $\mathrm{SL}(2,\mathbb{Z})$ acting by isometries and admitting a quotient of finite volume by a discrete group of isometries.

S-duality and the universal isometries of instanton corrected q-map spaces

Abstract

Given a conical affine special Kähler (CASK) manifold together with a compatible mutually local variation of BPS structures, one can construct a quaternionic-Kähler (QK) manifold. We call the resulting QK manifold an instanton corrected c-map space. Our main aim is to study the isometries of a subclass of instanton corrected c-map spaces associated to projective special real (PSR) manifolds with a compatible mutually local variation of BPS structures. We call the latter subclass instanton corrected q-map spaces. In the setting of Calabi-Yau compactifications of type IIB string theory, instanton corrected q-map spaces are related to the hypermultiplet moduli space metric with perturbative corrections, together with worldsheet, D(-1) and D1 instanton corrections. In the physics literature, it has been shown that the hypermultiplet metric with such corrections must have an acting by isometries, related to S-duality. We give a mathematical treatment of this result, specifying under which conditions instanton corrected q-map spaces carry an action by isometries by or some of its subgroups. We further study the universal isometries of instanton corrected q-map spaces, and compare them to the universal isometries of tree-level q-map spaces. Finally, we give an explicit example of a non-trivial instanton corrected q-map space with full acting by isometries and admitting a quotient of finite volume by a discrete group of isometries.
Paper Structure (23 sections, 29 theorems, 212 equations)

This paper contains 23 sections, 29 theorems, 212 equations.

Table of Contents

  1. Introduction
  2. Summary of main results
  3. Organization of topics
  4. Instanton corrected QK metrics and their twistor description
  5. QK metrics associated to CASK manifolds with mutually local variations of BPS structures
  6. Associated instanton corrected HK manifold
  7. Associated instanton corrected QK manifold via HK/QK correspondence
  8. The case of a CASK domain
  9. Twistor space description and Darboux coordinates
  10. Darboux coordinates for c-map spaces associated to CASK domains
  11. The case with instanton corrections
  12. S-duality on instanton corrected q-map spaces
  13. Setting
  14. The quantum corrected mirror map and the S-duality action
  15. Type IIB Darboux coordinates
  16. ...and 8 more sections

Key Result

Theorem 2.4

CT Let $(M,g_M,\omega_M,\nabla,\xi,\Gamma)$ and $(M,\Gamma,Z,\Omega)$ be as before. Furthermore, let $\omega_i\in \Omega^2(N)$ for $i=1,2,3$ be defined by Then the triple of real $2$-forms $(\omega_1,\omega_2, \omega_3)$ corresponds to the Kähler forms of a pseudo-HK structureOur terminology is such that the signature of the metric is not assumed to be constant, in case $N$ has several components

Theorems & Definitions (88)

  • Remark 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Remark 2.6
  • Theorem 2.7
  • Remark 2.8
  • Definition 2.9
  • ...and 78 more