The Worldsheet Perspective of T-duality Symmetry in String Theory

TL;DR

This paper provides a pedagogical review of T-duality in string theory, emphasizing its covariance properties, vertex operators, and applications to superstring backgrounds, highlighting the role of worldsheet formulations and specific compactification schemes.

Contribution

It demonstrates that vertex operators at all massive levels can be expressed in T-duality invariant form using the Hassan-Sen compactification scheme.

Findings

Equations of motion for compact coordinates are $O(d,d)$ covariant.

Vertex operators for massive levels can be T-duality invariant.

Worldsheet superfield equations are $O(d,d)$ covariant under certain conditions.

Abstract

The purpose of this article is to present a pedagogical review of T-dualityin in string theory. The evolution of the closed string is envisaged on the worldsheet in the presence of its massless excitations. The duality symmetry is studied when some of the spacial coordinates are compactified on d-dimensional torus, $T^d$. The known results are reviewed to elucidate that equations of motion for the compact coordinates are $O(d,d)$ covariant, $d$ being the number of compact directions. Next, the vertex operators of excited massive levels are considered in a simple compactification scheme. It is shown that the vertex operators for each massive level can be cast in a T-duality invariant form in such a case. Subsequently, the duality properties of superstring is investigated in the NSR formulation for the massless backgrounds such as graviton and antisymmetric tensor. The worldsheet superfield formulation is found to be very suitable for our purpose. The Hassan-Sen compactification is adopted and it is shown that the worldsheet equations of motion for compact superfields to be very suitable for our purpose. The Hassan-Sen compactification is adopted and it is shown that the worldsheet equations of motion for compact superfields are $O(d,d)$ covariant when the backgrounds are independent of superfields along compact directions. The vertex operators for excited levels are presented in the NS-NS sector and it is shown that they can be cast in T-duality invariant form for the case of Hassan-Sen compactification scheme. An illustrative example is presented to realize our proposal.