Abstract

Motivated by studies on 4d black holes and q-deformed 2d Yang–Mills theory, and borrowing ideas from compact geometry of the blowing up of affine ADE singularities, we build a class of local Calabi–Yau threefolds (CY 3) extending the local 2-torus model O ( m ) ⊕ O ( − m ) → T 2 considered in [C. Gomez, S. Montanez, A comment on quantum distribution functions and the OSV conjecture, hep-th/0608162] to test OSV conjecture. We first study toric realizations of T 2 and then build a toric representation of X 3 using intersections of local Calabi–Yau threefolds O ( m ) ⊕ O ( − m − 2 ) → P 1 . We develop the 2d N = 2 linear σ-model for this class of toric CY 3s. Then we use these local backgrounds to study partition function of 4d black holes in type IIA string theory and the underlying q-deformed 2d quiver gauge theories. We also make comments on 4d black holes obtained from D-branes wrapping cycles in O ( m ) ⊕ O ( − m − 2 ) → B k with m = ( m 1 , … , m k ) a k-dim integer vector and B k a compact complex one dimension base consisting of the intersection of k 2-spheres S i 2 with generic intersection matrix I i j . We give as well the explicit expression of the q-deformed path integral measure of the partition function of the 2d quiver gauge theory in terms of I i j . A comment on the link between our analysis and the construction of [N. Caporaso, M. Cirafici, L. Griguolo, S. Pasquetti, D. Seminara, R.J. Szabo, Topological strings, two-dimensional Yang–Mills theory and Chern–Simons theory on torus bundles, hep-th/0609129] is also given.

Highlights

  • Few years ago, Ooguri, Strominger and Vafa (OSV) have made a conjecture [1] relating the microstates counting of 4d BPS black holes in type II string theory on Calabi-Yau threefoldsX3 to the topological string partition function Ztop on the same manifold

  • Motivated by studies on 4d black holes and q-deformed 2d Yang Mills theory, and borrowing ideas from compact geometry of the blowing up of affine ADE singularities, we build a class of local Calabi-Yau threefolds (CY3) extending the local 2-torus model O(m) ⊕ O(−m) → T 2 considered in hep-th/0406058 to test OSV conjecture

  • It has been first tested in [2] by considering configurations of D-branes wrapped cycles O(m) ⊕ O(−m) → T 2 for m a positive definite integer. It has been checked in [18] by using wrapped D-branes in a vector bundle of rank 2 non trivial fibration over a genus g-Riemann surface O(2g + m − 2) ⊕ O(−m) → Σg. It has been shown in these studies that the BPS black hole partition function localizes onto field configuration which are invariant under U (1) actions of the fibers

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Summary

Introduction

Ooguri, Strominger and Vafa (OSV) have made a conjecture [1] relating the microstates counting of 4d BPS black holes in type II string theory on Calabi-Yau threefolds. We mention that 2d quiver gauge theories, associated with BPS black holes in type IIA string on the local CY3s we have considered, are classified by the ”sign” of the intersection matrix Iik of the real 2-spheres Si2 forming the compact base. We shall use our construction to complete partial results in literature on field theoretic realization using 2d N = (2, 2) linear sigma model for affine ADE geometries with special focus on the An case To our knowledge, this supersymmetric 2d field realization of local Calabi-Yau threefolds has not been considered before.

Toric realization of local T 2
Toric realization
General on toric graphs
Supersymmetric field model
Example
Local 2-torus
Brane theory and 4d black holes in type II string
Derivation of 2d quiver gauge field action
Path integral measure in 2d q-deformed quiver gauge theory
Conclusion
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