On duality symmetry in perturbative quantum theory

TL;DR

This paper explores how duality symmetries in quantum theories, especially in 2d models, can be made manifest and preserved at the quantum level using a doubled formulation, with implications for supergravity.

Contribution

It demonstrates that duality symmetries can be realized as manifest symmetries in a doubled formulation, preserving Lorentz invariance at the quantum level in 2d models and discusses implications for 4d supergravities.

Findings

Duality symmetry can be made manifest in a doubled formulation.

Quantum effective actions and S-matrices preserve duality symmetry.

Discrete Z_2 duality corresponds to specific transformations in 2d and 4d models.

Abstract

Non-compact symmetries of extended 4d supergravities involve duality rotations of vectors and thus are not manifest off-shell invariances in standard "second-order" formulation. To study how such symmetries are realised in the quantum theory we consider examples in 2 dimensions where vector-vector duality is replaced by scalar-scalar one. Using a "doubled" formulation, where fields and their momenta are treated on an equal footing and the duality becomes a manifest symmetry of the action (at the expense of Lorentz symmetry), we argue that the corresponding on-shell quantum effective action or S-matrix are duality symmetric as well as Lorentz invariant. The simplest case of discrete Z_2 duality corresponds to a symmetry of the S-matrix under flipping the sign of the negative-chirality scalars in 2 dimensions or phase rotations of chiral (definite-helicity) parts of vectors in 4 dimensions. We also briefly discuss some 4d models and comment on implications of our analysis for extended supergravities.