Superselection, Boundary Algebras and Duality in Gauge Theories

TL;DR

This paper explores the algebra of boundary and dual operators in gauge theories, revealing implications for superselection sectors, duality invariance, and mathematical structures like Drinfel'd doubles.

Contribution

It introduces dual boundary operators in gauge theories and analyzes their algebra, highlighting duality invariance and connections to advanced mathematical frameworks.

Findings

Identification of dual boundary operators and their algebraic relations.

Demonstration of $SL(2, \\mathbb{Z})$ duality transformations.

Insights into superselection sectors and confinement mechanisms.

Abstract

We consider the generators of gauge transformations with test functions which do not vanish on the boundary of a spacelike region of interest. These are known to generate the edge degrees of freedom in a gauge theory. In this paper, we augment these by introducing the dual or magnetic analogue of such operators. We then study the algebra of these operators, focusing on implications for the superselection sectors of the gauge theory. A manifestly duality-invariant action is also considered, from which alternate descriptions which are $SL(2, \mathbb{Z})$ transforms of each other can be obtained. We also comment on a number of issues related to local charges, definition of confinement and the appearance of interesting mathematical structures such as the Drinfel'd double and the Manin triple.