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On tt*-structures from $ADE$-type Stokes data

Tadashi Udagawa

Abstract

Cecotti and Vafa introduced the topological anti-topological fusion (tt*)-equation, whose solutions describe massive deformations of supersymmetric conformal field theories. We provide a rigorous analytic formulation of the $ADE$ classification of tt*-structures. Under natural structural assumptions, a tt*-structure over $\mathbb{C}^*$ can be described via isomonodromic deformations with upper unitriangular real Stokes matrices. Two fundamental issues arise: the ambiguities of Stokes matrices, governed by an action of a group $\tilde{Br}_n$, which is generated by reordering operations, and the solvability of the associated Riemann-Hilbert problem. Our first main result shows that the classification reduces to admissible Stokes matrices modulo $\tilde{Br}_n$-action, and that the $\tilde{Br}_n$-orbit of a Stokes matrix determines a tt*-structure over $\mathbb{C}^*$. Our second main result establishes that upper unitriangular matrices whose symmetrizations coincide with Cartan matrices of type $A_n, D_n, E_6, E_7,$ or $E_8$ give rise to tt*-structures over $\mathbb{C}^*$. This provides a direct analytic realization of the $ADE$ classification and clarifies the interplay between Stokes phenomena, $\tilde{Br}_n$-symmetry, and positivity of Cartan-type matrices.

On tt*-structures from $ADE$-type Stokes data

Abstract

Cecotti and Vafa introduced the topological anti-topological fusion (tt*)-equation, whose solutions describe massive deformations of supersymmetric conformal field theories. We provide a rigorous analytic formulation of the classification of tt*-structures. Under natural structural assumptions, a tt*-structure over can be described via isomonodromic deformations with upper unitriangular real Stokes matrices. Two fundamental issues arise: the ambiguities of Stokes matrices, governed by an action of a group , which is generated by reordering operations, and the solvability of the associated Riemann-Hilbert problem. Our first main result shows that the classification reduces to admissible Stokes matrices modulo -action, and that the -orbit of a Stokes matrix determines a tt*-structure over . Our second main result establishes that upper unitriangular matrices whose symmetrizations coincide with Cartan matrices of type or give rise to tt*-structures over . This provides a direct analytic realization of the classification and clarifies the interplay between Stokes phenomena, -symmetry, and positivity of Cartan-type matrices.
Paper Structure (14 sections, 22 theorems, 96 equations, 2 figures)

This paper contains 14 sections, 22 theorems, 96 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. tt*-structures
  4. Isomonodromy theory of tt*-equations
  5. The Stokes matrices and the Stokes factors
  6. From tt*-structures to the Riemann-Hilbert problem
  7. From the Riemann-Hilbert problem to tt*-structures
  8. Ambiguity of Stokes matrices
  9. Ambiguity (i)
  10. Ambiguity (ii)
  11. The equivalence relation on tt*-structures over $\mathbb{C}^*$
  12. The tt*-structures constructed from the Cartan matrices
  13. $\tilde{Br}_n$-orbits of the Stokes matrices
  14. The Cartan matrices of $A_n,D_n,E_6,E_7$ and $E_8$

Key Result

Lemma 2.1

There exists a frame $\tau = (\tau_1,\dots,\tau_n)$ of $E$ such that The frame $\tau$ is uniquely determined up to right multiplication by ${\rm diag}(\varepsilon_1,\cdots,\varepsilon_n)$, where $\varepsilon_j^2 = 1\ (j=1,\dots,n)$.

Figures (2)

  • Figure 1: The Stokes rays
  • Figure 2: The sectors $\Omega_j$ and $\Omega_j \cap \Omega_{j+1}$

Theorems & Definitions (53)

  • Definition 2.1: C. Hertling H2003, H. Fan, T. Yang, Z. Lan FLY2021
  • Example 1: The Tzitzeica equation
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.1
  • Example 2
  • Definition 2.2
  • Definition 2.3
  • ...and 43 more