Albert S. Schwarz

Noncommutative Geometry and Matrix Theory: Compactification on Tori

We study toroidal compactification of Matrix theory, using ideas and results of noncommutative geometry. We generalize this to compactification on the noncommutative torus, explain the classification of these backgrounds, and argue that they correspond in supergravity to tori with constant background three-form tensor field. The paper includes an introduction for mathematicians to the IKKT formulation of Matrix theory and its relation to the BFSS Matrix theory.

Instantons on noncommutative R**4 and (2,0) superconformal six-dimensional theory

Abstract:We show that the resolution of moduli space of ideal instantons parameterizes the instantons on noncommutative ℝ4. This moduli space appears to be the Higgs branch of the theory of kD0-branes bound to ND4-branes by the expectation value of the B field. It also appears as a regularized version of the target space of supersymmetric quantum mechanics arising in the light cone description of (2,0) superconformal theories in six dimensions.

Geometry of Batalin-Vilkovisky quantization

The geometry ofP-manifolds (odd symplectic manifolds) andSP-manifolds (P-manifolds provided with a volume element) is studied. A complete classification of these manifolds is given. This classification is used to prove some results about Batalin-Vilkovisky procedure of quantization, in particular to obtain a very general result about gauge independence of this procedure.

The Partition Function of Degenerate Quadratic Functional and Ray-Singer Invariants

The partition function of degenerate quadratic functional is defined and studied. It is shown that analytic torsion and similar invariants can be interpreted as partition functions of quadratic functionals.

Introduction to M(atrix) theory and noncommutative geometry

Abstract Noncommutative geometry is based on an idea that an associative algebra can be regarded as “an algebra of functions on a noncommutative space”. The major contribution to noncommutative geometry was made by A. Connes, who, in particular, analyzed Yang–Mills theories on noncommutative spaces, using important notions that were introduced in his papers (connection, Chern character, etc). It was found recently that Yang–Mills theories on noncommutative spaces appear naturally in string/M-theory; the notions and results of noncommutative geometry were applied very successfully to the problems of physics. In this paper we give a mostly self-contained review of some aspects of M(atrix) theory, of Connes’ noncommutative geometry and of applications of noncommutative geometry to M(atrix) theory. The topics include introduction to BFSS and IKKT matrix models, compactifications on noncommutative tori, a review of basic notions of noncommutative geometry with a detailed discussion of noncommutative tori, Morita equivalence and SO (d,d| Z ) -duality, an elementary discussion of noncommutative orbifolds, noncommutative solitons and instantons. The review is primarily intended for physicists who would like to learn some basic techniques of noncommutative geometry and how they can be applied in string theory and to mathematicians who would like to learn about some new problems arising in theoretical physics. The second part of the review (Sections 10–12) devoted to solitons and instantons on noncommutative Euclidean space is almost independent of the first part.

Khovanov-Rozansky Homology and Topological Strings

We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot homology groups and suggests that they can be interpreted directly in topological string theory. We use this approach in various examples to predict the sl(N) knot homology groups for all values of N. We verify that our predictions pass some non-trivial checks

Morita equivalence and duality

It was shown by Connes, Douglas, Schwarz [hep-th/9711162] that one can compactify M(atrix) theory on a non-commutative torus To. We prove that compactifications on Morita equivalent tori are in some sense physically equivalent. This statement can be considered as a generalization of non-classical SL(2,Z)N duality conjectured by Connes, Douglas and Schwarz for compactifications on two-dimensional non-commutative tori.

The partition function of a degenerate functional

The partition function of a degenerate quadratic functional is defined and studied. It is shown that Ray-Singer invariants can be interpreted as partition functions of quadratic functionals. In the case of a degenerate non-quadratic functional the semiclassical approximation to the partition function is considered.

Geometric interpretation of the partition function of 2D gravity

We construct explicity the subspace in the infinite dimensional grassmannian corresponding to the τ-function of the 2D topological gravity. This allows us to give a simple proof of some conjectures on the equations defining this function.

On regular solutions of Euclidean Yang-Mills equations

Abstract The number of instantons and the number of zero fermion modes in the field of instanton are calculated.