$tt^{*}$ Geometry of Modular Curves

TL;DR

This paper explores the mathematical structure of a class of $tt^{*}$ geometries related to modular curves, revealing connections between physics, elliptic curves, and number theory, with implications for understanding vacua in certain quantum models.

Contribution

It introduces a novel class of models parametrized by modular curves, linking $tt^{*}$ geometry with elliptic curves and modular forms, and analyzes their spectral and symmetry properties.

Findings

Models are parametrized by modular curves $Y_{1}(N)$ and $Y(N)$.

The $tt^{*}$ equations reduce to $ abla$-diagonalized Toda equations.

Modular properties classify critical limits and reveal geometric-number theoretic links.

Abstract

Motivated by Vafa's model, we study the $tt^{*}$ geometry of a degenerate class of FQHE models with an abelian group of symmetry acting transitively on the classical vacua. Despite it is not relevant for the phenomenology of the FQHE, this class of theories has interesting mathematical properties. We find that these models are parametrized by the family of modular curves $Y_{1}(N)= \mathbb{H}/\Gamma_{1}(N)$, labelled by an integer $N\geq 2$. Each point of the space of level $N$ is in correspondence with a one dimensional $\mathcal{N}=4$ Landau-Ginzburg theory, which is defined on an elliptic curve with $N$ vacua and $N$ poles in the fundamental cell. The modular curve $Y(N)= \mathbb{H}/\Gamma(N)$ is a cover of degree $N$ of $Y_{1}(N)$ and plays the role of spectral cover for the space of models. The presence of an abelian symmetry allows to diagonalize the Berry's connection of the vacuum bundle and the $tt^{*}$ equations turn out to be the well known $\hat{A}_{N-1}$ Toda equations. The underlying structure of the modular curves and the connection between geometry and number theory emerge clearly when we study the modular properties and classify the critical limits of these models.