Algebra and Twisted Algebra in Toroidal Target Space

TL;DR

This paper explores the quantization of string theory in a toroidal space, revealing multiple inequivalent quantizations and a duality condition, along with a twisted algebraic structure involving topological features.

Contribution

It introduces a new algebraic framework for toroidal string quantization, showing the existence of infinite inequivalent quantizations and a twisted relation between quantum numbers and topology.

Findings

Existence of infinite inequivalent quantizations parametrized by s and t.

Duality in the spectrum occurs only when s = t or -t (mod 1).

Deformation by a central extension introduces a twist relating quantum numbers and winding.

Abstract

Target space duality is reconsidered from the viewpoint of quantization in a space with nontrivial topology. An algebra of operators for the toroidal bosonic string is defined and its representations are constructed. It is shown that there exist an infinite number of inequivalent quantizations, which are parametrized by two parameters $ 0 \le s, t < 1 $. The spectrum exhibits the duality only when $ s = t $ or $ -t $ (mod 1). A deformation of the algebra by a central extension is also introduced. It leads to a kind of twisted relation between the zero mode quantum number and the topological winding number.