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 $ADE$ classification of tt*-structures. Under natural structural assumptions, a tt*-structure over ${\mathbb{C}}^{\ast }$ 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 ${\widetilde{Br}}_n$, 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 ${\widetilde{Br}}_n$-action, and that the ${\widetilde{Br}}_n$-orbit of a Stokes matrix determines a tt*-structure over ${\mathbb{C}}^{\ast }$. 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 ${\mathbb{C}}^{\ast }$. This provides a direct analytic realization of the $ADE$ classification and clarifies the interplay between Stokes phenomena, ${\widetilde{Br}}_n$-symmetry, and positivity of Cartan-type matrices.

24 pages, 2 figures