Seiberg Duality for Quiver Gauge Theories

By D. Berenstein, Michael R Douglas. 2002

164 citations recorded by the source corpus.

A popular way to study N = 1 supersymmetric gauge theories is to realize them geometrically in string theory, as suspended brane constructions, D-branes wrapping cycles in Calabi-Yau manifolds, orbifolds, and otherwise. Among the applications of this idea are simple derivations and generalizations of Seiberg duality for the theories which can be so realized. We abstract from these arguments the idea that Seiberg duality arises because a configu- ration of gauge theory can be realized as a bound state of a collection of branes in more than one way, and we show that different brane world-volume theories obtained this way have matching moduli spaces, the primary test of Seiberg duality. Furthermore, we do this by defining "brane" and all the other ingredients of such arguments purely algebraically, for a very large class of N = 1 quiver supersymmetric gauge theo- ries, making physical intuitions about brane-antibrane systems and tachyon condensation precise using the tools of homological algebra. These techniques allow us to compute the spectrum and superpotential of the dual theory from first principles, and to make contact with geometry and topological string theory when this is appropriate, but in general provide a more abstract notion of "noncommutative geometry" which is better suited to these problems. This makes contact with mathematical results in the representation theory of algebras; in this language, Seiberg duality is a tilting equivalence between the derived categories of the quiver algebras of the dual theories.

Subfields: Geometry and Topology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/80563e853035e519e53bd6e3e575d4a5df78e9b6

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Citations in the corpus

Three-dimensional N=6 SCFT's and their membrane dynamics (2008, background): “We can guarantee then that Z only has upper triangular form (these configurations should be though of as bound states of branes [68]).”
Quivers, tilings, branes and rhombi (2005, background): “These “phases” of the theory are believed to be Seiberg–dual to each other [55, 56, 57, 58].”
Triangulated Categories: Cluster algebras, quiver representations and triangulated categories (2008, unknown): “rature, cf. for example equation (12.2) on page 70 in [27] (I thank to S. Fomin for this reference). In the physics context, a crucial role is played by the so-called superpotentials, cf. for example [18]. This lead Derksen-Weyman-Zelevinsky to their systematic study of quivers with potentials and t”
Derived equivalences from mutations of quivers with potential (2009, methodology): “es between derived categories. Our approach is based on recent work by Derksen-Weyman-Zelevinsky [11] and by Ginzburg [18]. The idea of such an interpretation goes back at least to Berenstein-Douglas [6] and was further developed by Mukhopadhyay-Ray [33] in physics and by Bridgeland [9], Iyama-Reite”
Dirichlet Branes and Mirror Symmetry (2009, background): “Following [40], Seiberg duality can be understood as following from the fact that the dual theory describes the same configurations of Dirichlet branes, but as the bound states of a different “basis” of elementary branes.”
Quivers with potentials and their representations I: Mutations (2007, background): “ZELEVINSKY ideal,and tothequotientofthepath algebramodulo J(S)asthe Jacobian algebra. They appeared in physicists’work on superpotentials in thecontextoftheSeiberg duality in mirrorsymmetry (see e.g.,[17,2,6]). Since in some oftheirwork the superpotentialsarerequiredtosatisfysomeform ofSerreduality,”