Deformed WZW Models and Hodge Theory -- Part I

TL;DR

This paper explores a novel connection between two-dimensional integrable sigma-models and Hodge theory, showing that certain solutions correspond to variations of Hodge structures via the Weil operator.

Contribution

It establishes a link between the classical dynamics of the $ ext{lambda}$-deformed $G/G$ model and variations of Hodge structures, introducing a new perspective in integrable models and algebraic geometry.

Findings

Special solutions correspond to one-parameter variations of Hodge structures.

Identification of the group-valued field with the Weil operator.

Suggests a connection between string theory on Hodge classifying spaces and period mappings.

Abstract

We investigate a relationship between a particular class of two-dimensional integrable non-linear $\sigma$-models and variations of Hodge structures. Concretely, our aim is to study the classical dynamics of the $\lambda$-deformed $G/G$ model and show that a special class of solutions to its equations of motion precisely describes a one-parameter variation of Hodge structures. We find that this special class is obtained by identifying the group-valued field of the $\sigma$-model with the Weil operator of the Hodge structure. In this way, the study of strings on classifying spaces of Hodge structures suggests an interesting connection between the broad field of integrable models and the mathematical study of period mappings.