Finite in All Directions

TL;DR

This paper explores the unique features of toroidal string compactifications, especially with timelike dimensions, revealing ergodic duality actions, infinite symmetries, and a novel flat connection on moduli space.

Contribution

It introduces a new perspective on moduli space structure, identifies unbroken infinite symmetries, and formulates Ward identities via a flat connection in toroidal string compactifications.

Findings

Narain moduli space is not a manifold due to ergodic duality action.

Certain compactifications preserve infinite dimensional symmetries.

A flat connection on moduli space enables formulation of broken symmetry Ward identities.

Abstract

We study toroidal compactifications of string theories which include compactification of a timelike coordinate. Some new features in the theory of toroidal compactifications arise. Most notably, Narain moduli space does not exist as a manifold since the action of duality on background data is ergodic. For special compactifications certain infinite dimensional symmetries, analogous to the infinite dimensional symmetries of the $2D$ string are unbroken. We investigate the consequences of these symmetries and search for a universal symmetry which contains all unbroken gauge groups. We define a flat connection on the moduli space of toroidally compactified theories. Parallel transport by this connection leads to a formulation of broken symmetry Ward identities. In an appendix this parallel transport is related to a definition of conformal perturbation theory.