Back to Heterotic Strings on ALE Spaces: Part II -- Geometry of T-dual Little Strings

TL;DR

This paper explores the geometric aspects of T-duality in heterotic string compactifications on ALE spaces, using F-theory and M-theory dualities to analyze moduli spaces and classify T-dualities, especially in extremal K3-based Little String Theories.

Contribution

It introduces a geometric framework for understanding T-dualities in heterotic ALE instantonic Little String Theories via F-theory, revealing a T-hexality structure from elliptic K3 fibrations.

Findings

Identified T-hexality as a 6-fold family of T-dualities.

Demonstrated the role of elliptic K3 fibrations in classifying T-dualities.

Analyzed extremal K3 surfaces with flavor groups of maximal rank 18.

Abstract

This work is the second of a series of papers devoted to revisiting the properties of Heterotic string compactifications on ALE spaces. In this project we study the geometric counterpart in F-theory of the T-dualities between Heterotic ALE instantonic Little String Theories (LSTs) extending and generalising previous results on the subject by Aspinwall and Morrison. Since the T-dualities arise from a circle reduction one can exploit the duality between F-theory and M-theory to explore a larger moduli space, where T-dualities are realised as inequivalent elliptic fibrations of the same geometry. As expected from the Heterotic/F-theory duality the elliptic F-theory Calabi-Yau we consider admit a nested elliptic K3 fibration structure. This is central for our construction: the K3 fibrations determine the flavor groups and their global forms, and are the key to identify various T-dualities. We remark that this method works also more generally for LSTs arising from non-geometric Heterotic backgrounds. We study a first example in detail: a particularly exotic class of LSTs which are built from extremal K3 surfaces that admit flavor groups with maximal rank 18. We find all models are related by a so-called T-hexality (i.e. a 6-fold family of T-dualities) which we predict from the inequivalent elliptic fibrations of the extremal K3.