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# Advances in Theoretical and Mathematical Physics

## Volume 10 (2006)

### Number 2

### Inherited duality and quiver gauge theory

Pages: 159 – 179

DOI: http://dx.doi.org/10.4310/ATMP.2006.v10.n2.a1

#### Authors

#### Abstract

We study the duality group of $\widehat{A}_{n-1}$ quiver gauge theories, primarily using their M5-brane construction. For $\CN=2$ supersymmetry, this duality group was first noted by Witten to be the mapping class group of a torus with $n$ punctures. We find that it is a certain quotient of this group that acts faithfully on gauge couplings. This quotient group contains the affine Weyl group of $\widehat{A}_{n-1}$, $\ZZ_n$ and $SL(2,\ZZ)$. In fact there are $n$ non-commuting $SL(2,\ZZ)$ subgroups, related to each other by conjugation using the $\ZZ_n$. When supersymmetry is broken to $\CN=1$ by masses for the adjoint chiral superfields, a renormalization group (RG) flow ensues which is believed to terminate at a conformal field theory (CFT) in the infrared. We find the explicit action of this duality group for small values of the adjoint masses, paying special attention to when the sum of the masses is non-zero. In the $\CN=1$ CFT, Seiberg duality acts non-trivially on both gauge couplings and superpotential couplings and we interpret this duality as inherited from the $\CN=2$ parent theory. We conjecture the action of $S$-duality in the CFT based on our results for small mass deformations. We also consider non-conformal deformations of these $\CN=1$ theories. The cascading RG flows that ensue are a one-parameter generalization of those found by Klebanov and Strassler and by Cachazo et al. The universality exhibited by these flows is shown to be a simple consequence of paths generated by the action of the affine Weyl group.