An Introduction to T-Duality in String Theory

TL;DR

The paper surveys T-duality in string theory, focusing on abelian duality via Buscher and Roček-Verlinde formalisms and on non-abelian duality via Quevedo, analyzing how duality transforms background fields, dilaton, and topology. It shows that conformal invariance requires a dilaton shift in abelian duality and reveals potential anomalies in non-semisimple non-abelian cases, highlighting conditions for consistency. A canonical transformation perspective is developed for the abelian case, deriving Buscher's rules in arbitrary coordinates and clarifying the duality’s action on currents and topology, while noting the ongoing challenges for non-abelian canonical formulations and global properties. The discussion also addresses the duality-induced changes to the cosmological constant and to the notion of distance in string theory, emphasizing conceptual and practical implications for interpreting geometry in the stringy regime and outlining open problems for future work.

Abstract

In these lectures a general introduction to T-duality is given. In the abelian case the approaches of Buscher, and Roucek and Verlinde are reviewed. Buscher's prescription for the dilaton transformation is recovered from a careful definition of the gauge integration measure. It is also shown how duality can be understood as a quite simple canonical transformation. Some aspects of non-abelian duality are also discussed, in particular what is known on relation to canonical transformations. Some implications of the existence of duality on the cosmological constant and the definition of distance in String Theory are also suggested.