On the geometry of quiver gauge theories
Christopher P. Herzog, Robert L. Karp
Advances in Theoretical and Mathematical Physics
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:
... 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.