Comments on Holographic Level/Rank Dualities

TL;DR

This paper explores a holographic model in Type-IIB string theory that accurately reproduces various level/rank dualities of pure Chern-Simons theories, highlighting the role of boundary conditions and $SL(2, extbf{Z})$ duality.

Contribution

It provides a string theory holographic realization of level/rank dualities, clarifying the role of boundary conditions and duality transformations in these theories.

Findings

Holographically reproduces multiple level/rank dual pairs.

Shows $SL(2, extbf{Z})$ duality acts via RR potentials.

Establishes a precise boundary theory correspondence.

Abstract

We investigate a holographic realization in Type-IIB string theory of pure Chern-Simons theories, and focus on the level/rank dualities that they enjoy. The level/rank duality is established between the boundary theory, engineered utilizing D3-branes compactified on a $S^1$, and the theory on probe D7-branes located at a specific bulk location. Paying attention to the boundary conditions imposed on the Ramond-Ramond (RR) two- and six-forms, we show that holography precisely reproduces several different level/rank dual pairs. The $SL(2,\mathbf{Z})$ action relating all these theories is realized through the $SL(2,\mathbf{Z})$ electric/magnetic duality involving the RR potentials.