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On the geometry of quiver gauge theories

Christopher P. Herzog, Robert L. Karp

2009
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Advances in Theoretical and Mathematical Physics
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In this paper we advance the program of using exceptional collections to understand the gauge theory description of a D-brane probing a Calabi-Yau singularity. To this end, we strengthen the connection between strong exceptional collections and fractional branes. To demonstrate our ideas, we derive a strong exceptional collection for every Y p,q singularity, and also prove that this collection is simple. Contents 1 Introduction 600 2 Preliminary ideas 602 e-print archive:
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... bs/hep-th/0605177 600 CHRISTOPHER P. HERZOG AND ROBERT L. KARP 2.1 Physical motivation 602 2.2 Quiver representations and path algebras 605 2.3 D(S ) versus D(K S) 608 2.4 The physical quiver 610 2.5 The stacky version 612 3 Vanishing results -absence of tachyons 613 3.1 Absence of Ext 0 's 614 3.2 Absence of Ext d+1 's 616 3.3 Invariance of the quiver 618 4 The geometry of the Y p,q spaces 618 4.1 An exceptional collection on F n 622 4.2 The exceptional collection 623 5 The toric approach 628 5.1 Toric Kawamata-Viehweg vanishing 629 5.2 Specializing to a toric surface 629 5.3 The Y p,q case 631 Acknowledgments 633 References 633 1. An object A ∈ D(S) is called exceptional if Ext q (A, A) = 0 for q = 0 and Ext 0 (A, A) = C. 2. An exceptional collection A = (A 1 , A 2 , . . . , A n ) in D(S) is an ordered collection of exceptional objects such that Ext q (A i , A j ) = 0, for all q, whenever i > j . A strong exceptional collection A is an exceptional collection which in addition satisfies: Ext q (A i , A j ) = 0 for q = 0. 4. An exceptional collection is complete or full if it generates D(S). 2 It is common practice to mention only the compact dimensions of the brane. Accordingly, a D0-brane in this topological setting could refer to a D3-brane which fills the three non-compact dimensions of the full string theory. 3 Given a local Calabi-Yau X and a del Pezzo S, by shrinking down to a point, what we really mean is that there is a partial crepant resolution of the singularity π : X → X * where X * is singular at a point p and π * (p) = S.

doi:10.4310/atmp.2009.v13.n3.a1
fatcat:uwjprswk6rboleatoiecmm2lta