Quiver-Invariant Dualities between Brane Tilings

TL;DR

The paper discovers a new correspondence between pairs of 4d $\mathcal{N}=1$ quiver gauge theories realized by brane tilings on $T^2$ that share the same mesonic moduli space $\mathcal{M}^{mes}$ and the same quiver, yet differ in their superpotentials. This correspondence is implemented by a local tilting mutation along hexagon diagonals, equivalent to a specific sequence of Seiberg dualities that leaves the quiver unchanged while altering the periodic quiver and superpotential. An explicit example with Models A and B shows identical toric data $G_t$ up to $GL(2,\mathbb{Z})$ and identical $\mathcal{M}^{mes}$ and Hilbert series, but different realizations of generators in terms of chiral fields. The results imply an infinite family of brane tilings connected by tilting mutations and point to deeper algebro-geometric structures in brane tilings, with possible links to zig-zag path data and higher-dimensional gauge theories.

Abstract

We study pairs of 4d N=1 supersymmetric gauge theories that share the same vacuum moduli space and the same chiral field content, encoded by a common quiver, but differ in their superpotentials. These theories arise as worldvolume theories on a D3-brane probing a toric Calabi-Yau 3-fold and admit a description in terms of bipartite graphs on a 2-torus, known as brane tilings. Using an explicit example, we show that the correspondence is realized by a single `tilting' mutation along the diagonals of hexagonal faces in the brane tiling, which is equivalent to a specific sequence of Seiberg dualities performed at distinct gauge nodes in the quiver.

Figures (6)

• Figure 1: (a) The local 'tilting' mutation acts along the diagonals of hexagonal faces (blue) in the brane tiling. (b) The mutation modifies the periodic quiver on the 2-torus $T^2$ and hence the superpotential, while leaving the underlying quiver and the mesonic moduli space of the associated $4d$$\mathcal{N}=1$ theories invariant. The 'tilting’ mutation realizes a particular sequence of Seiberg dualities.
• Figure 2: Seiberg duality acts on (a) quadrilateral faces (red) of a brane tiling and (b) corresponding gauge nodes (red) in the periodic quiver of the associated $4d$$\mathcal{N}=1$ theory. The local mutation on the brane tiling and the periodic quiver is also referred to as a spider move or urban renewal 2011arXiv1107.5588GCIUCU1998341999math......3025K.
• Figure 3: (a) Quiver and (b) toric diagram of the toric Calabi-Yau 3-fold shared by Models A and B.
• Figure 4: (a) The brane tiling and (b) the labelled toric diagram for Model A, where vertices in the toric diagram correspond to GLSM fields $p_a$. The dotted quadrilateral in the brane tiling indicates the fundamental domain on the 2-torus $T^2$.
• Figure 5: (a) The brane tiling and (b) the labelled toric diagram for Model B, where vertices in the toric diagram correspond to GLSM fields $p_a$. The dotted quadrilateral in the brane tiling indicates the fundamental domain on the 2-torus $T^2$.