Complex Structures, T-duality and Worldsheet Instantons in Born Sigma Models

TL;DR

This paper explores the mathematical structures underlying Born geometries, revealing how various complex structures are realized and classified, and examines the T-duality properties of worldsheet instantons in these models.

Contribution

It demonstrates that complex structures in Born geometries can be represented as subalgebras of bi-quaternions and classifies parts of these structures using Clifford algebras, also analyzing T-duality of instantons.

Findings

Born structures and generalized Kähler structures are subalgebras of bi-quaternions.

Parts of these structures are classified by Clifford algebras.

Instantons in Kähler geometries relate non-trivially to those in bi-hermitian geometries.

Abstract

We investigate doubled (generalized) complex structures in $2D$-dimensional Born geometries where T-duality symmetry is manifestly realized. We show that K\"{a}hler, hyperk\"{a}hler, bi-hermitian and bi-hypercomplex structures of spacetime are implemented in Born geometries as doubled structures. We find that the Born structures and the generalized K\"{a}hler (hyperk\"{a}hler) structures appear as subalgebras of bi-quaternions and split-tetra-quaternions. We find parts of these structures are classified by Clifford algebras. We then study the T-duality nature of the worldsheet instantons in Born sigma models. We show that the instantons in K\"{a}hler geometries are related to those in bi-hermitian geometries in a non-trivial way.