Level/rank duality and Chern-Simons-matter theories

Po-Shen Hsin,Nathan Seiberg

doi:10.1007/jhep09(2016)095

Po-Shen Hsin, Nathan Seiberg

Open Access

https://doi.org/10.1007/jhep09(2016)095

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Abstract

We discuss in detail level/rank duality in three-dimensional Chern-Simons theories and various related dualities in three-dimensional Chern-Simons-matter theories. We couple the dual Lagrangians to appropriate background fields (including gauge fields, spin$_c$ connections and the metric). The non-trivial maps between the currents and the line operators in the dual theories is accounted for by mixing of these fields. In order for the duality to be valid we must add finite counterterms depending on these background fields. This analysis allows us to resolve a number of puzzles with these dualities, to provide derivations of some of them, and to find new consistency conditions and relations between them. In addition, we find new level/rank dualities of topological Chern-Simons theories and new dualities of Chern-Simons-matter theories, including new boson/boson and fermion/fermion dualities.

Highlights

We discuss in detail level/rank duality in three-dimensional Chern-Simons theories and various related dualities in three-dimensional Chern-Simons-matter theories
In order for the duality to be valid we must add finite counterterms depending on these background fields
Where the notation GL represents the chiral algebra of G with level L and in the right hand side of (1.1) we have a GKO coset [6]

Summary

A brief summary of spinc for Chern-Simons theory

We will be mostly interested in U(N )K,K (K′ = 0), which we will abbreviate as U(N )K and U(N )K,K±N (K′ = ±1) We couple these theories to background fields in such a way that they do not depend on the spin structure. U(1) gauge field C and the spin theories to a background spinc-connection A. We will couple these Chern-Simons theories to matter fields and the monopole operators will be less trivial. This is a slight generalization of appendix B in [14], where more details can be found. We will use L0[A] as a Lagrangian to denote the fact that the theory includes a transparent line with spin one-half

A useful fact

Self-duality of QED with two fermions

A Often used equations