No abstract

In Batalin-Vilkovisky formalism a classical mechanical system is specified by means of a solution to the {\sl classical master equation}. Geometrically such a solution can be considered as a $QP$-manifold, i.e. a super\m equipped with an odd vector field $Q$ obeying $\{Q,Q\}=0$ and with $Q$-invariant odd symplectic structure. We study geometry of $QP$-manifolds. In particular, we describe some construction of $QP$-manifolds and prove a classification theorem (under certain conditions). We apply these geometric constructions to obtain in natural way the action functionals of two-dimensional topological sigma-models and to show that the Chern-Simons theory in BV-formalism arises as a sigma-model with target space $\Pi {\cal G}$. (Here ${\cal G}$ stands for a Lie algebra and $\Pi$ denotes parity inversion.)Comment: 29 pages, Plain TeX, minor modifications in English are made by Jim Stasheff, some misprints are corrected, acknowledgements and references adde

We show that the resolution of moduli space of ideal instantons parameterizes the instantons on non-commutative IR 4 . This moduli space appears as a Higgs branch of the theory of k D0-branes bound to N D4-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. 02/98It seems natural to study all possible solutions U i to the consistency equations for the compactification of the matrix fields 1 [8].Recently, the non-commutative torus emerged as one of the solutions to (1.1) [9]. It has been argued that the parameter of non-commutativity is related to the flux of the B-field through the torus. It has been further shown in [10] that the compactification on a non-commutative torus can be thought of as a T -dual to a limit of the conventional compactification on a commutative torus. See [11] for further developments in the studies of compactifications on low-dimensional tori.On the other hand, the modified self-duality equations on the matrices in the Matrix description of fivebrane theory has been used in [12] in the study of quantum mechanics on the instanton moduli space. The modification is most easily described in the framework of ADHM equations. It makes the moduli space smooth and allows to define a six dimensional theory decoupled from the eleven-dimensional supergravity and all others M -theoretic degrees of freedom. The heuristric reason for the possibility of such decoupling is the fact that the Higgs branch of the theory is smooth and there is no place for the Coulomb branch to touch it.In this paper we propose an explanation of the latter construction in terms of noncommutative geometry. We show, that the solutions to modified ADHM equations parameterize (anti-)self-dual gauge fields on non-commutative IR 4 .1 The conjecture of [8] is that the non-abelian tensor fields in six dimensions would also appear as such solutions

The present paper is devoted to the study of geometry of Batalin-Vilkovisky quantization procedure. The main mathematical objects under consideration are P-manifolds and SP-manifolds (supermanifolds provided with an odd symplectic structure and, in the case of SP-manifolds, with a volume element). The Batalin-Vilkovisky procedure leads to consideration of integrals of the superharmonic functions over Lagrangian submanifolds. The choice of Lagrangian submanifold can be interpreted as a choice of gauge condition; Batalin and Vilkovisky proved that in some sense their procedure is gauge independent. We prove much more general theorem of the same kind. This theorem leads to a conjecture that one can modify the quantization procedure in such a way as to avoid the use of the notion of Lagrangian submanifold. In the next paper we will show that this is really so at least in the semiclassical approximation. Namely the physical quantities can be expressed as integrals over some set of critical points of solution S to the master equation with the integrand expressed in terms of Reidemeister torsion. This leads to a simplification of quantization procedure and to the possibility to get rigorous results also in the infinite-dimensional case. The present paper contains also a compete classification of P-manifolds and SP-manifolds. The classification is interesting by itself, but in this paper it plays also a role of an important tool in the proof of other results.Comment: 13 page

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.