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Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
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Moduli spaces of higher spin curves and integrable hierarchies
 Compositio Math
"... Abstract. We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to GelfandDickey hierarchies. That is, we show that intersection numbers on the moduli space of stable rspin curves assemble into a generating function which yields a solution of the s ..."
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Cited by 44 (8 self)
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Abstract. We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to GelfandDickey hierarchies. That is, we show that intersection numbers on the moduli space of stable rspin 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 GromovWitten 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
Noncommutative geometry based on commutator expansions
 J. Reine Angew. Math
, 1998
"... Contents 3. The NCaffine space and FeynmanMaslov operator calculus. 4. Detailed study of algebraic NCmanifolds. 5. Examples of NCmanifolds. The term “noncommutative geometry ” has come to signify a vast framework of ideas ..."
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Cited by 27 (0 self)
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Contents 3. The NCaffine space and FeynmanMaslov operator calculus. 4. Detailed study of algebraic NCmanifolds. 5. Examples of NCmanifolds. The term “noncommutative geometry ” has come to signify a vast framework of ideas
Homotopy Gerstenhaber algebras and topological field theory, Operads
 Proceedings of Renaissance Conferences
, 1996
"... 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 mai ..."
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Cited by 26 (3 self)
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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. Twodimensional topological quantum field theory (TQFT) at its most elementary level is the theory of Zgraded 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 Zgraded 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 Zgraded Poisson algebra for which the Poisson bracket has degree −1 (see Section 1). This Galgebra 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. Galgebras 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 Galgebras). 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 Galgebra, see Section 1; in fact, a double loop space is naturally
A New Point of View in the Theory of Knot and Link Invariants
, 2001
"... Recent progress in string theory has led to a reformulation of quantumgroup polynomial invariants for knots and links into new polynomial invariants whose coefficients can be understood in topological terms. We describe in detail how to construct the new polynomials and we conjecture their general ..."
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Cited by 18 (5 self)
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Recent progress in string theory has led to a reformulation of quantumgroup polynomial invariants for knots and links into new polynomial invariants whose coefficients can be understood in topological terms. We describe in detail how to construct the new polynomials and we conjecture their general structure. This leads to new conjectures on the algebraic structure of the quantumgroup polynomial invariants. We also describe the geometrical meaning of the coefficients in terms of the enumerative geometry of Riemann surfaces with boundaries in a certain CalabiYau threefold.
Noncommutative homotopy algebras associated with open strings
 Rev. Math. Phys
"... 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 a ..."
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Cited by 18 (5 self)
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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
Invertible cohomological field theories and WeilPetersson volumes
"... 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 m ..."
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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
Notes on universal algebra
 Graphs and Patterns in Mathematics and Theoretical Physics (M. Lyubich and L. Takhtajan, eds.), Proc. Sympos. Pure Math
, 2005
"... Dedicated to Dennis Sullivan on the occasion of his sixtieth birthday. Abstract. These are notes of a minicourse given at Dennisfest in June 2001. The goal of these notes is to give a selfcontained survey of deformation quantization, operad theory, and graph homology. Some new results related to “ ..."
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Cited by 15 (1 self)
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Dedicated to Dennis Sullivan on the occasion of his sixtieth birthday. Abstract. These are notes of a minicourse given at Dennisfest in June 2001. The goal of these notes is to give a selfcontained survey of deformation quantization, operad theory, and graph homology. Some new results related to “String Topology ” and cacti are announced in Section 2.7.
WHEELED PROPS, GRAPH COMPLEXES AND THE MASTER EQUATION
, 2007
"... 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 onetoone correspondence with formal germs of ..."
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Cited by 12 (6 self)
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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 onetoone correspondence with formal germs of SPmanifolds, key geometric objects in the theory of BatalinVilkovisky quantization. We also construct minimal wheeled resolutions of classical operads Com and As s as rather nonobvious 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.