Cumrun Vafa

Microscopic origin of the Bekenstein-Hawking entropy

Abstract The Bekenstein-Hawking area-entropy relation S BH = A 4 is derived for a class of five-dimensional extremal black holes in string theory by counting the degeneracy of BPS solition bound states.

Strings on orbifolds

String propagation on the quotient of a flat torus by a discrete group is considered. We obtain an exactly soluble and more or less realistic method of string compactification.

Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes

We develop techniques to compute higher loop string amplitudes for twistedN=2 theories withĉ=3 (i.e. the critical case). An important ingredient is the discovery of an anomaly at every genus in decoupling of BRST trivial states, captured to all orders by a master anomaly equation. In a particular realization of theN=2 theories, the resulting string field theory is equivalent to a topological theory in six dimensions, the Kodaira-Spencer theory, which may be viewed as the closed string analog of the Chern-Simons theory. Using the mirror map this leads to computation of the ‘number’ of holomorphic curves of higher genus curves in Calabi-Yau manifolds. It is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the correspondingN=2 theory. Relations withc=1 strings are also pointed out.

CFT's from Calabi–Yau four-folds

We consider F/M/Type IIA theory compactified to four, three, or two dimensions on a Calabi–Yau four-fold, and study the behavior near an isolated singularity in the presence of appropriate fluxes and branes. We analyze the vacuum and soliton structure of these models, and show that near an isolated singularity, one often generates massless chiral superfields and a superpotential, and in many instances in two or three dimensions one obtains nontrivial superconformal field theories. In the case of two dimensions, we identify some of these theories with certain Kazama–Suzuki coset models, such as the N=2 minimal models.

A Strong coupling test of S duality

Abstract By studying the partition function of N =4 topologically twisted supersymmetric Yang-Mills on four-manifolds, we make an exact strong coupling test of the Montonen-Olive strong-weak duality conjecture. Unexpected and exciting links are found with two-dimensional rational conformal field theory.

Chiral rings in N = 2 superconformal theories

We investigate the properties of chiral operators in N = 2 superconformal theories. In particular, we study the spectral flow of such models under a one-parameter family of twists generated by the U(1) current, and use this to deduce various properties of the ring of chiral primary fields. We furthermore investigate under what conditions a given superconformal theory can be represented as the fixed point of an N = 2 Landau-Ginzburg theory and show how to determine the superpotential. We also investigate the coset models of Kazama and Suzuki and find a simple cohomological characterization for the elements of the chiral primary ring. Moreover we show how some of them can be represented as LG models.

Matrix models, topological strings, and supersymmetric gauge theories

We show that B-model topological strings on local Calabi–Yau threefolds are large-N duals of matrix models, which in the planar limit naturally give rise to special geometry. These matrix models directly compute F-terms in an associated N=1 supersymmetric gauge theory, obtained by deforming N=2 theories by a superpotential term that can be directly identified with the potential of the matrix model. Moreover by tuning some of the parameters of the geometry in a double scaling limit we recover (p,q) conformal minimal models coupled to 2d gravity, thereby relating non-critical string theories to type II superstrings on Calabi–Yau backgrounds.

Compactifications of F-theory on Calabi-Yau threefolds (II)

Abstract We continue our study of compactifications of F-theory on Calabi-Yau threefolds. We gain more insight into F-theory duals of heterotic strings and provide a recipe for building F-theory duals for arbitrary heterotic compactifications on elliptically fibered manifolds. As a byproduct we find that string/string duality in six dimensions gets mapped to fiber/base exchange in F-theory. We also construct a number of new N = 1, d = 6 examples of F-theory vacua and study transitions among them. We find that some of these transition points correspond upon further compactification to 4 dimensions to transitions through analogues of Argyres-Douglas points of N = 2 moduli. A key idea in these transitions is the notion of classifying (0,4) fivebranes of heterotic strings.

The Topological Vertex

We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kähler classes of the threefold. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the quantum Kodaira-Spencer theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann surface as fermions related to the chiral boson by bosonization.

Black hole attractors and the topological string

A simple relationship of the form ZBH = |Ztop|2 is conjectured, where ZBH is a supersymmetric partition function for a four-dimensional BPS black hole in a Calabi-Yau compactification of Type II superstring theory and Ztop is a second-quantized topological string partition function evaluated at the attractor point in moduli space associated to the black hole charges. Evidence for the conjecture in a perturbation expansion about large graviphoton charge is given. The microcanonical ensemble of BPS black holes can be viewed as the Wigner function associated to the wave function defined by the topological string partition function.