Large-Small Equivalence in String Theory

TL;DR

This paper demonstrates a universal equivalence between large and small geometries in all toroidal string compactifications, extending the known R to 1/R duality to more complex backgrounds.

Contribution

It generalizes the large-small duality in string theory to include all toroidal compactifications with background fields, showing an iterative transformation method.

Findings

Large-small geometry equivalence holds universally for toroidal compactifications.

An iterative mode transformation can increase the volume beyond a finite lower bound.

The duality extends beyond simple circle compactification to complex lattice backgrounds.

Abstract

The simplest toroidally compactified string theories exhibit a duality between large and small radii: compactification on a circle, for example, is invariant under R goes to 1/R. Compactification on more general Lorentzian lattices (i.e. toroidal compactification in the presence of background metric, antisymmetric tensor, and gauge fields) yields theories for which large-small invariance is not so simple. Here an equivalence is demonstrated between large and small geometries for all toroidal compactifications. By repeatedly transforming the momentum mode corresponding to the smallest winding length to another mode on the lattice, it is possible to increase the volume to exceed a finite lower bound.