Instanton Counting and Chern-Simons Theory

The instanton partition function of N=2, D=4 SU(2) gauge theory is obtained by taking the field theory limit of the topological open string partition function, given by a Chern-Simons theory, of a CY3-fold. The CY3-fold on the open string side is obtained by geometric transition from local F_0 which is used in the geometric engineering of the SU(2) theory. The partition function obtained from the Chern-Simons theory agrees with the closed topological string partition function of local F_0 proposed recently by Nekrasov. We also obtain the partition functions for local F_1 and F_2 CY3-folds and show that the topological string amplitudes of all local Hirzebruch surfaces give rise to the same field theory limit. It is shown that a generalization of the topological closed string partition function whose field theory limit is the generalization of the instanton partition function, proposed by Nekrasov, can be determined easily from the Chern-Simons theory.Comment: 39 pages, references added, typos corrected, cosmetic change

The Acceleration of the Universe, a Challenge for String Theory

Recent astronomical observations indicate that the universe is accelerating. We argue that generic quintessence models that accommodate the present day acceleration tend to accelerate eternally. As a consequence the resulting spacetimes exhibit event horizons. Hence, quintessence poses the same problems for string theory as asymptotic de Sitter spaces.Comment: JHEP, LaTeX, 12 pages, 4 figures. Added a reference, corrected typo

Moduli Potentials in Type IIA Compactifications with RR and NS Flux

We describe a simple class of type IIA string compactifications on Calabi-Yau manifolds where background fluxes generate a potential for the complex structure moduli, the dilaton, and the K\"ahler moduli. This class of models corresponds to gauged N=2 supergravities, and the potential is completely determined by a choice of gauging and by data of the N=2 Calabi-Yau model - the prepotential for vector multiplets and the quaternionic metric on the hypermultiplet moduli space. Using mirror symmetry, one can determine many (though not all) of the quantum corrections which are relevant in these models.Comment: 30 pages, 2 figures; v2: minor change

The toroidal block and the genus expansion

We study the correspondence between four-dimensional supersymmetric gauge theories and two-dimensional conformal field theories in the case of N=2* gauge theory. We emphasize the genus expansion on the gauge theory side, as obtained via geometric engineering from the topological string. This point of view uncovers modular properties of the one-point conformal block on a torus with complexified intermediate momenta: in the large intermediate weight limit, it is a power series whose coefficients are quasi-modular forms. The all-genus viewpoint that the conformal field theory approach lends to the topological string yields insight into the analytic structure of the topological string partition function in the field theory limit.Comment: 36 page

Transformations of Spherical Blocks

We further explore the correspondence between N=2 supersymmetric SU(2) gauge theory with four flavors on epsilon-deformed backgrounds and conformal field theory, with an emphasis on the epsilon-expansion of the partition function natural from a topological string theory point of view. Solving an appropriate null vector decoupling equation in the semi-classical limit allows us to express the instanton partition function as a series in quasi-modular forms of the group Gamma(2), with the expected symmetry Weyl group of SO(8) semi-direct S_3. In the presence of an elementary surface operator, this symmetry is enhanced to an action of the affine Weyl group of SO(8) semi-direct S_4 on the instanton partition function, as we demonstrate via the link between the null vector decoupling equation and the quantum Painlev\'e VI equation.Comment: 31 pages, 1 figure; v2: typos corrected, references adde

The Vertex on a Strip

We demonstrate that for a broad class of local Calabi-Yau geometries built around a string of IP^1's - those whose toric diagrams are given by triangulations of a strip - we can derive simple rules, based on the topological vertex, for obtaining expressions for the topological string partition function in which the sums over Young tableaux have been performed. By allowing non-trivial tableaux on the external legs of the corresponding web diagrams, these strips can be used as building blocks for more general geometries. As applications of our result, we study the behavior of topological string amplitudes under flops, as well as check Nekrasov's conjecture in its most general form.Comment: 26 pages, 12 figures; v2: minor corrections, version to appear in ATM