Black holes in type IIA string on Calabi-Yau threefolds with affine ADE geometries and q-deformed 2d quiver gauge theories

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)->T2 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 T2 and then build a toric representation of X3 using intersections of local Calabi-Yau threefolds O(m)+O(-m-2)->P1. We develop the 2d N=2 linear σ-model for this class of toric CY3s. 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)->Bk with m=(m1,...,mk) a k-dim integer vector and Bk a compact complex one dimension base consisting of the intersection of k 2-spheres Si2 with generic intersection matrix Iij. 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 Iij. 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