Anti-de Sitter Space and the Center of the Gauge Group

TL;DR

This paper explores the emergence of a $Z_N$ topological symmetry in SU(N) gauge theories and superconformal theories via compactification, providing insights into the AdS/CFT correspondence and magnetic confinement phenomena.

Contribution

It demonstrates how a $Z_N$ symmetry arises in compactified gauge theories and superconformal theories, linking topological features to domain walls and flux tubes within the AdS/CFT framework.

Findings

$Z_N$ symmetry appears in compactified SU(N) gauge theories.

Domain walls are D-strings or M-theory membranes.

Flux tubes can end on topological domain walls.

Abstract

Upon compactification on a circle, SU(N) gauge theory with all fields in the adjoint representation acquires a $Z_N$ global symmetry because the center of the gauge group is $Z_N$. For N=4 super Yang-Mills theory, we show how this $Z_N$ "topological symmetry" arises in the context of the AdS/CFT correspondence, and why the symmetry group is $Z_N$ rather than U(1). This provides a test of the AdS/CFT correspondence for finite N. If the theory is formulated on $R^3 \times S^1$ with anti-periodic boundary conditions for fermions around the $S^1$, the topological symmetry is spontaneously broken; we show that the domain walls are D-strings, and hence that flux tubes associated with magnetic confinement can end on the domain walls associated with the topological symmetry. For the (0,2) $A_{N-1}$ superconformal field theory in six dimensions, we demonstrate an analogous phenomenon: a $Z_N$ global symmetry group arises if this theory is compactified on a Riemann surface. In this case, the domain walls are M-theory membranes.