A geometric unification of dualities

Abstract

We study the dynamics of a large class of N = 1 quiver theories, geometrically realized by type IIB D-brane probes wrapping cycles of local Calabi-Yau threefolds. These include N = 2 (affine) A-D-E quiver theories deformed by superpotential terms, as well as chiral N = 1 quiver theories obtained in the presence of vanishing 4-cycles inside a Calabi-Yau.We consider the various possible geometric transitions of the 3-fold and show that they correspond to Seiberg-like dualities (represented by Weyl reflections in the A-D-E case or 'mutations' of bundles in the case of vanishing 4-cycles) or large N dualities involving gaugino condensates (generalized conifold transitions). Also duality cascades are naturally realized in these classes of theories, and are related to the affine Weyl group symmetry in the A-D-E case.October 2001 then consider wrapping general classes of D3,D5 and D7 branes. In this case, it is more convenient to use the mirror IIA picture of the manifold and branes, as it does not suffer from quantum corrections. Using the appropriate mirror symmetry in the context of branes [19], we write down the corresponding quiver theory, as well as the corresponding Seiberg-like dualities. The dualities involve changes of the classical parameters in the type IIA mirror. We specialize to the Calabi-Yau threefolds involving delPezzo and their transitions. Certain aspects of this case have been noted recently in [20,21].The organization of this paper is as follows: In section 2 we give an overview of the N = 1 A-D-E quiver theories and the results we will find for them in this paper. In section 3 we give the description of classical aspects of the A-D-E quiver gauge theories under consideration. In section 4 we discuss some aspects of the quantum dynamics of the gauge couplings and their running. In section 5 we discuss gaugino condensation in the nonaffine A-D-E N = 1 quiver theories. In section 6 we consider the geometric engineering of these theories and their large N dual, involving the leading quantum corrections and the geometric realization of gaugino condensates. In section 7 we discuss Seiberg-like dualities for the A-D-E quiver theories anticipated from geometry. In section 8 we discuss the gauge theoretic interpretation of these dualities. In section 9 we consider the gauge theory dynamics of the A 2 quiver in more detail, as a typical situation where the Seiberg-like duality is relevant. In section 10 we discuss dynamical aspects of the affine quiver theory and its relation to the non-affine case. We also note the connection of RG cascades in this class of theories with affine Weyl reflection. In section 11 we discuss examples of N = 1 superconformal A-D-E quiver theories. In section 12 we setup the geometric engineering of the type (ii) local threefolds, as well as dualities predicted by geometry. In section 13 we specialize to a class of examples and illustrate how the gaugino condensation takes place in these chiral theories and what geometric transition they correspond to.2. Basic structure of the ty...

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