Noncommutative Geometry and String Duality

TL;DR

This paper reviews how noncommutative geometry provides a systematic framework for understanding duality symmetries in string theory, linking algebraic structures to geometric transformations.

Contribution

It introduces spectral triples related to lattice vertex operator algebras and demonstrates their role in encoding target space and worldsheet dualities as geometric isometries.

Findings

Target space duality acts as a gauge transformation.

Constructs spectral triples for vertex operator algebras.

Connects noncommutative geometry with string dualities.

Abstract

A review of the applications of noncommutative geometry to a systematic formulation of duality symmetries in string theory is presented. The spectral triples associated with a lattice vertex operator algebra and the corresponding Dirac-Ramond operators are constructed and shown to naturally incorporate target space and discrete worldsheet dualities as isometries of the noncommutative space. The target space duality and diffeomorphism symmetries are shown to act as gauge transformations of the geometry. The connections with the noncommutative torus and Matrix Theory compactifications are also discussed.