Algebraic structure of $tt^*$ equations for Calabi-Yau sigma models

TL;DR

This paper explores the algebraic structure of $tt^*$ equations in Calabi-Yau sigma models, revealing a finite set of generators that form a Lie algebra and encode the non-holomorphic content of the equations across different dimensions.

Contribution

It introduces a finite generator framework for the non-holomorphic $tt^*$ equations in Calabi-Yau sigma models, connecting them to special functions, Lie algebras, and modular forms.

Findings

Generators form a Lie algebra structure.

Differential rings are constructed for lattice polarized K3 manifolds.

In the quartic mirror case, the ring coincides with quasi-modular forms.

Abstract

The $tt^*$ equations define a flat connection on the moduli spaces of $2d, \mathcal{N}=2$ quantum field theories. For conformal theories with $c=3d$, which can be realized as nonlinear sigma models into Calabi-Yau d-folds, this flat connection is equivalent to special geometry for threefolds and to its analogs in other dimensions. We show that the non-holomorphic content of the $tt^*$ equations in the cases $d=1,2,3$ is captured in terms of finitely many generators of special functions, which close under derivatives. The generators are understood as coordinates on a larger moduli space. This space parameterizes a freedom in choosing representatives of the chiral ring while preserving a constant topological metric. Geometrically, the freedom corresponds to a choice of forms on the target space respecting the Hodge filtration and having a constant pairing. Linear combinations of vector fields on that space are identified with generators of a Lie algebra. This Lie algebra replaces the non-holomorphic derivatives of $tt^*$ and provides these with a finer and algebraic meaning. For sigma models into lattice polarized $K3$ manifolds, the differential ring of special functions on the moduli space is constructed, extending known structures for $d=1$ and 3. The generators of the differential rings of special functions are given by quasi-modular forms for $d=1$ and their generalizations in $d=2,3$. Some explicit examples are worked out including the case of the mirror of the quartic in $CP^3$, where due to further algebraic constraints, the differential ring coincides with quasi modular forms.