Gerbes and Duality

TL;DR

This paper develops a global geometric framework using gerbes to analyze duality transformations between antisymmetric fields, establishing quantum equivalence and refining duality proofs in supermembrane theories.

Contribution

It introduces a gerbe-based approach to duality, addressing potentials directly and proving quantum equivalence with a generalized Dirac quantization condition.

Findings

Gerbes provide a natural geometric setting for duality transformations.

Dual theories are shown to be quantum equivalent, satisfying quantization conditions.

Refined proof of duality between 11D supermembrane and 10D IIA supermembrane.

Abstract

We describe a global approach to the study of duality transformations between antisymmetric fields with transitions and argue that the natural geometrical setting for the approach is that of gerbes, these objects are mathematical constructions generalizing U(1) bundles and are similarly classified by quantized charges. We address the duality maps in terms of the potentials rather than on their field strengths and show the quantum equivalence between dual theories which in turn allows a rigorous proof of a generalized Dirac quantization condition on the couplings. Our approach needs the introduction of an auxiliary form satisfying a global constraint which in the case of 1-form potentials coincides with the quantization of the magnetic flux. We apply our global approach to refine the proof of the duality equivalence between d=11 supermembrane and d=10 IIA Dirichlet supermembrane.