Duality, Self-Duality, Sources and Charge Quantization in Abelian N-Form Theories

TL;DR

This paper explores the duality and charge quantization in Abelian N-form theories across different dimensions, highlighting symmetry properties, source coupling, and implications for self-dual forms and dimensional reduction.

Contribution

It provides a symmetric coupling framework for N-form fields and sources, clarifies duality properties in various dimensions, and connects self-dual forms to higher-dimensional invariances.

Findings

Duality is an SO(2) rotation in D=4k dimensions, invariant under Chern-Simons form.

In D=4k+2 dimensions, the duality rotation is not an invariance and lacks a generator.

Charges obey Dirac quantization, with symmetric treatment of electric and magnetic sources.

Abstract

We investigate duality properties of N-form fields, provide a symmetric way of coupling them to electric/magnetic sources, and check that these charges obey the appropriate quantization requirements. First, we contrast the D=4k case, in which duality is a well-defined SO(2) rotation generated by a Chern-Simons form leaving the action invariant, and D=4k+2 where the corresponding ostensibly SO(1,1) rotation is not only not an invariance but does not even have a generator. When charged sources are included we show explicitly in the Maxwell case how the usual Dirac quantization arises in a fully symmetric approach attaching strings to both types of charges. Finally, for D=4k+2 systems, we show how charges can be introduced for self-dual (2k)-forms, and obtain the D=4k models with sources by dimensional reduction, tracing their duality invariance to a partial invariance in the higher dimensions.