Twisted Einstein Tensors and Orbifold Geometry

TL;DR

This paper explores the geometric structures of various orbifolds in string theory, focusing on twisted Einstein tensors and proposing twisted Einstein equations for conformal sigma models.

Contribution

It introduces a comprehensive framework for analyzing twisted Einstein tensors and formulates conjectured equations governing conformal orbifold sigma models.

Findings

Development of phase-space geometry for WZW orbifolds

Construction of sigma model orbifold actions including known cases

Evidence supporting twisted Einstein equations for conformal invariance

Abstract

Following recent advances in the local theory of current-algebraic orbifolds, we study various geometric properties of the general WZW orbifold, the general coset orbifold and a large class of (non-linear) sigma model orbifolds. Phase-space geometry is emphasized for the WZW orbifolds - while for the sigma model orbifolds we construct the corresponding {\it sigma model orbifold action}, which includes the previously-known general WZW orbifold action and general coset orbifold action as special cases. We focus throughout on the {\it twisted Einstein tensors} with diagonal monodromy, including the twisted Einstein metric, the twisted B field and the twisted torsion field of each orbifold sector. Finally, we present strong evidence for a conjectured set of {\it twisted Einstein equations} which should describe those sigma model orbifolds in this class which are also 1-loop conformal.