Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Geometry of the tt*-Toda equations I: universal centralizer and symplectic groupoids

Martin A. Guest, Nan-Kuo Ho

Abstract

We investigate the geometry of a certain space of meromorphic connections with irregular singularities, and prove in particular that it is a (real) symplectic Lie groupoid. The connections have a physical meaning: they correspond to certain solutions of the topological-antitopological fusion (tt*) equations of Cecotti and Vafa, and hence to deformations of supersymmetric quantum field theories. The groupoid structure arises because we restrict ourselves to the tt* equations of Toda type, whose monodromy data has a Lie theoretic description. To obtain these results, we show first that the universal centralizer of a Lie group is a holomorphic symplectic groupoid over the Steinberg cross section.

Geometry of the tt*-Toda equations I: universal centralizer and symplectic groupoids

Abstract

We investigate the geometry of a certain space of meromorphic connections with irregular singularities, and prove in particular that it is a (real) symplectic Lie groupoid. The connections have a physical meaning: they correspond to certain solutions of the topological-antitopological fusion (tt*) equations of Cecotti and Vafa, and hence to deformations of supersymmetric quantum field theories. The groupoid structure arises because we restrict ourselves to the tt* equations of Toda type, whose monodromy data has a Lie theoretic description. To obtain these results, we show first that the universal centralizer of a Lie group is a holomorphic symplectic groupoid over the Steinberg cross section.

Paper Structure

This paper contains 13 sections, 27 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Monodromy data
  3. The connection form $\hat{\alpha}$.
  4. The Stokes matrices.
  5. The Steinberg cross section.
  6. The connection matrices.
  7. The space of monodromy data.
  8. The space $\mathcal{S}_{n+1}^{\text{local}}$.
  9. The universal centralizer.
  10. Main Results
  11. Groupoids.
  12. The universal centralizer.
  13. Anti-symmetry and Reality.

Key Result

Proposition 2.2

(cf. section 2 of GIL2) $Q^{(0)} _{k+\frac{2}{n+1}} = \Pi\ Q^{(0)} _k \ \Pi^{-1}$. ∎ $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (64)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Remark 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Definition 2.10
  • ...and 54 more