Robbert Dijkgraaf

Matrix string theory

Via compactification on a circle, the matrix mode] of M-theory proposed by Banks et a]. suggests a concrete identification between the large N limit of two-dimensional N = 8 supersymmetric Yang-Mills theory and type IIA string theory. In this paper we collect evidence that supports this identification. We explicitly identify the perturbative string states and their interactions, and describe the appearance of D-particle and D-membrane states.

Topological strings and integrable hierarchies

We consider the topological B-model on local Calabi-Yau geometries. We show how one can solve for the amplitudes by using -algebra symmetries which encode the symmetries of holomorphic diffeomorphisms of the Calabi-Yau. In the highly effective fermionic/brane formulation this leads to a free fermion description of the amplitudes. Furthermore we argue that topological strings on Calabi-Yau geometries provide a unifying picture connecting non-critical (super)strings, integrable hierarchies, and various matrix models. In particular we show how the ordinary matrix model, the double scaling limit of matrix models, and Kontsevich-like matrix model are all related and arise from studying branes in specific local Calabi-Yau three-folds. We also show how an A-model topological string on P1 and local toric threefolds (and in particular the topological vertex) can be realized and solved as B-model topological string amplitudes on a Calabi-Yau manifold.

Elliptic genera of symmetric products and second quantized strings

Abstract: In this note we prove an identity that equates the elliptic genus partition function of a supersymmetric sigma model on the N-fold symmetric product of a manifold M to the partition function of a second quantized string theory on the space . The generating function of these elliptic genera is shown to be (almost) an automorphic form for . In the context of D-brane dynamics, this result gives a precise computation of the free energy of a gas of D-strings inside a higher-dimensional brane.

Counting Dyons in N =4 String Theory

Abstract We present a microscopic index formula for the degeneracy of dyons in four-dimensional N = 4 string theory. This counting formula is manifestly symmetric under the duality group, and its asymptotic growth reproduces the macroscopic Bekenstein-Hawking entropy. We give a derivation of this result in terms of the type 11 five-brane compactified on K3, by assuming that its fluctuations are described by a closed string theory on its world-volume. We find that the degeneracies are given in terms of the denominator of a generalized super Kac-Moody algebra. We also discuss the correspondence of this result with the counting of D-brane states.

C=1 conformal field theories on Riemann surfaces

We study the theory ofc=1 torus and ℤ2-orbifold models on general Riemann surfaces. The operator content and occurrence of multi-critical points in this class of theories is discussed. The partition functions and correlation functions of vertex operators and twist fields are calculated using the theory of double covered Riemann surfaces. It is shown that orbifold partition functions are sensitive to the Torelli group. We give an algebraic construction of the operator formulation of these nonchiral theories on higher genus surfaces. Modular transformations are naturally incorporated as canonical transformations in the Hilbert space.

Instanton strings and hyper-Kähler geometry

Abstract We discuss two-dimensional sigma models on moduli spaces of instantons on K 3 surfaces. These N = (4, 4) superconformal field theories describe the near-horizon dynamics of the D1-D5-brane system and are dual to string theory on AdS 3 . We derive a precise map relating the moduli of the K 3 type 1113 string compactification to the moduli of these conformal field theories and the corresponding classical hyper-Kahler geometry. We conclude that in the absence of background gauge fields, the metric on the instanton moduli spaces degenerates exactly to the orbifold symmetric product of K 3. Turning on a self-dual NS B -field deforms this symmetric product to a manifold that is diffeomorphic to the Hilbert scheme. We also comment on the mathematical applications of string duality to the global issues of deformations of hyper-Kahler manifolds.

Quantum Geometry of Refined Topological Strings

A bstractWe consider branes in refined topological strings. We argue that their wavefunctions satisfy a Schrödinger equation depending on multiple times and prove this in the case where the topological string has a dual matrix model description. Furthermore, in the limit where one of the equivariant rotations approaches zero, the brane partition function satisfies a time-independent Schrödinger equation. We use this observation, as well as the back reaction of the brane on the closed string geometry, to offer an explanation of the connection between integrable systems and

Intersection Theory, Integrable Hierarchies and Topological Field Theory

\mathcal{N}=2

Mirror Symmetry and Elliptic Curves

gauge systems in four dimensions observed by Nekrasov and Shatashvili.

BPS spectrum of the five-brane and black hole entropy

The last two years have seen the emergence of a beautiful new subject in mathematical physics. It manages to combine a most exotic range of disciplines: two-dimensional quantum field theory, intersection theory on the moduli space of Riemann surfaces, integrable hierarchies, matrix integrals, random surfaces, and many more. The common denominator of all these fields is two-dimensional quantum gravity or, more general, low-dimensional string theory. Here the application of large-N techniques in matrix models, that are used to simulate fluctuating triangulated surfaces [1]–[3], has led to complete solvability [4]–[8]. (See e.g. the review papers [9], and also the lectures of S. Shenker in this volume.) Shortly after the onset of the remarkable developments in matrix models, Edward Witten presented compelling evidence for a relationship between random surfaces and the algebraic topology of moduli space [10, 11]. This proposal involved a particular quantum field theory, known as topological gravity [12], whose properties were further established in [13, 14] and generalized to the so-called multi-matrix models in [15]–[19]. This subject can, among many other things, be considered as a fruitful application of quantum field theory techniques to a particular problem in pure mathematics, and as such is a prime example of a much bigger program, also largely due to Witten, that has been taking shape in recent years. It is impossible to do fully justice to this subject within the confines of these lecture notes. I will however make an effort to indicate some of the more startling interconnections.