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Abstract. We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to Gelfand-Dickey hierarchies. That is, we show that intersection numbers on the moduli space of stable r-spin curves assemble into a generating function which yields a solution of the semiclassical limit of the KdVr equations. We formulate axioms for a cohomology class on this moduli space which allow one to construct a cohomological field theory of rank r −1 in all genera. In genus zero it produces a Frobenius manifold which is isomorphic to the Frobenius manifold structure on the base of the versal deformation of the singularity Ar−1. We prove analogs of the puncture, dilaton, and topological recursion relations by drawing an analogy with the construction of Gromov-Witten invariants and quantum cohomology. The moduli space of stable curves of genus g with n marked points Mg,n is a fascinating object. Mumford [37] introduced tautological cohomology classes associated to the universal curve Cg,n

Abstract. We prove that the BRST complex of a topological conformal field theory is a homotopy Gerstenhaber algebra, as conjectured by Lian and Zuckerman in 1992. We also suggest a refinement of the original conjecture for topological vertex operator algebras. We illustrate the usefulness of our main tools, operads and “string vertices ” by obtaining new results on Vassiliev invariants of knots and double loop spaces. Two-dimensional topological quantum field theory (TQFT) at its most elementary level is the theory of Z-graded commutative associative algebras (with some additional structure) [34]. Thus, it came as something of a surprise when several groups of mathematicians realized that the physical state space of a 2D TQFT has the structure of a Z-graded Lie algebra, relative to a new grading equal to the old grading minus one. Moreover, the commutative and Lie products fit together nicely to give the structure of a Gerstenhaber algebra (Galgebra), a Z-graded Poisson algebra for which the Poisson bracket has degree −1 (see Section 1). This G-algebra structure is best understood in the framework of 2D topological conformal field theories (TCFTs) (see Section 5.2) wherein operads of moduli spaces of Riemann surfaces play a fundamental role. G-algebras arose explicitly in M. Gerstenhaber’s work on the Hochschild cohomology theory for associative algebras (see Section 1 for this and several other contexts for the theory of G-algebras). Operads arose in the work of J. Stasheff, Gerstenhaber and later work of P. May on the recognition problem for iterated loop spaces. Eventually, F. Cohen discovered that the homology of a double loop space is naturally a G-algebra, see Section 1; in fact, a double loop space is naturally

Contents 3. The NC-affine space and Feynman-Maslov operator calculus. 4. Detailed study of algebraic NC-manifolds. 5. Examples of NC-manifolds. The term “noncommutative geometry ” has come to signify a vast framework of ideas

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We discuss general properties of A∞-algebras and their applications to the theory of open strings. The properties of cyclicity for A∞-algebras are examined in detail. We prove the decomposition theorem, which is a stronger version of the minimal model theorem, for A∞algebras and cyclic A∞-algebras and discuss various consequences of it. In particular it is applied to classical open string field theories and it is shown that all classical open string field theories on a fixed conformal background are cyclic A∞-isomorphic to each other. The same results hold for classical closed string field theories, whose algebraic structure is governed by cyclic L∞-algebras. Contents 1 Introduction and Summary 2 1.1 A∞-space and A∞-algebras.............................. 2 1.2 A∞-structure and classical open string field theory................. 6 1.3 Dual description; formal noncommutative supermanifold.............. 13

Abstract. We show that the generating function for the higher Weil–Petersson volumes of the moduli spaces of stable curves with marked points can be obtained from Witten’s free energy by a change of variables given by Schur polynomials. Since this generating function has a natural extension to the moduli space of invertible Cohomological Field Theories, this suggests the existence of a “very large phase space”, correlation functions on which include Hodge integrals studied by C. Faber and R. Pandharipande. From this formula we derive an asymptotical expression for the Weil–Petersson volume as conjectured by C. Itzykson. We also discuss a topological interpretation of the genus expansion formula of Itzykson–Zuber, as well as a related bialgebra acting upon quantum cohomology as a complex version of the classical path groupoid. 0. Introduction and summary. The aim of this paper is to record some progress in understanding intersection numbers on moduli spaces of stable pointed curves and their generating functions. Continuing the study started in [KoM], [KoMK] and pursued further in [KaMZ], [KabKi], we work with Cohomological

We introduce and study wheeled PROPs, an extension of the theory of PROPs which can treat traces and, in particular, solutions to the master equations which involve divergence operators. We construct a dg free wheeled PROP whose representations are in one-to-one correspondence with formal germs of SP-manifolds, key geometric objects in the theory of Batalin-Vilkovisky quantization. We also construct minimal wheeled resolutions of classical operads Com and As s as rather non-obvious extensions of Com ∞ and As s∞, involving, e.g., a mysterious mixture of associahedra with cyclohedra. Finally, we apply the above results to a computation of cohomology of a directed version of Kontsevich’s complex of ribbon graphs.

Dedicated to Dennis Sullivan on the occasion of his sixtieth birthday. Abstract. These are notes of a mini-course given at Dennisfest in June 2001. The goal of these notes is to give a self-contained survey of deformation quantization, operad theory, and graph homology. Some new results related to “String Topology ” and cacti are announced in Section 2.7.

Abstract. We obtain a simple, recursive presentation of the tautological (κ, ψ, and λ) classes on the moduli space of curves in genus zero and one in terms of boundary strata (graphs). We derive differential equations for the generating functions for their intersection numbers which allow us to prove a simple relationship between the genus zero and genus one potentials. As an application, we describe the moduli space of normalized, even, rank one cohomological field theories in genus one in coordinates which are additive under taking tensor products. Our results simplify and generalize those of Kaufmann, Manin, and Zagier. Recently, there has been a great deal of interest in the topology of the moduli space of curves. Much of this interest has been due to the important role that these spaces (and their cousins, the moduli space of stable maps) play in the theory of Gromov-Witten invariants and quantum cohomology [18, 27, 25] whose origins in the physical literature are called a topological gravity [27]. They furnish nontrivial examples of cohomological field theories (CohFTs) [18, 21], in genus zero (and conjecturally for higher genera). Often, this structure is enough to completely determine the Gromov-Witten invariants themselves. The moduli spaces of curves are endowed with tautological classes whose generating functions for their associated intersection numbers obey a system of differential equations which often possess remarkable properties [27, 17]. In this paper, we apply a mixture of algebraic geometry and combinatorics to such find a simple presentation of these classes in genus zero and one to obtain a generalization of some equations due to Witten and Dijkgraaf [27, 4]. These generating functions parametrize the potentials associated to the space of all normalized, even, rank one cohomological field theories in genus one and endow this space with coordinates which are additive with respect to tensor product in the category of CohFTs, a nontrivial operation. Our results were motivated by the work of Kaufmann, Manin, and Zagier [15]. The moduli space of genus g curves with n marked points, Mg,n:= { [C; x1, x2,..., xn]}, is the moduli space of configurations of n marked points, on a smooth, complex curve (Riemann surface) C of genus g. If