Duality Symmetries and Noncommutative Geometry of String Spacetime

TL;DR

This paper explores the algebraic structure of string spacetime using noncommutative geometry, revealing how dualities and symmetries emerge from Dirac operators and connecting these to classical spacetime and larger symmetry groups.

Contribution

It provides a detailed algebraic framework for understanding string spacetime symmetries, dualities, and their relation to noncommutative geometry, including the role of Dirac operators and vertex operator algebras.

Findings

Spacetime duality arises from two independent Dirac operators.

Classical spacetime emerges as a low-energy limit of the string algebra.

Spacetime duality and general covariance are gauge symmetries in this formalism.

Abstract

We examine the structure of spacetime symmetries of toroidally compactified string theory within the framework of noncommutative geometry. Following a proposal of Frohlich and Gawedzki, we describe the noncommutative string spacetime using a detailed algebraic construction of the vertex operator algebra. We show that the spacetime duality and discrete worldsheet symmetries of the string theory are a consequence of the existence of two independent Dirac operators, arising from the chiral structure of the conformal field theory. We demonstrate that these Dirac operators are also responsible for the emergence of ordinary classical spacetime as a low-energy limit of the string spacetime, and from this we establish a relationship between T-duality and changes of spin structure of the target space manifold. We study the automorphism group of the vertex operator algebra and show that spacetime duality is naturally a gauge symmetry in this formalism. We show that classical general covariance also becomes a gauge symmetry of the string spacetime. We explore some larger symmetries of the algebra in the context of a universal gauge group for string theory, and connect these symmetry groups with some of the algebraic structures which arise in the mathematical theory of vertex operator algebras, such as the Monster group. We also briefly describe how the classical topology of spacetime is modified by the string theory, and calculate the cohomology groups of the noncommutative spacetime. A self-contained, pedagogical introduction to the techniques of noncommmutative geometry is also included.