Notes on universal algebra

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.Comment: 22 pages, 10 figures; In the revised version, a few minor errors are corrected, including some signs for the A_infty operad and algebra

Homotopy Gerstenhaber algebras

The goal of this paper is to complete Getzler-Jones' proof of Deligne's Conjecture, thereby establishing an explicit relationship between the geometry of configurations of points in the plane and the Hochschild complex of an associative algebra. More concretely, it is shown that the $B_\infty$-operad, which is generated by multilinear operations known to act on the Hochschild complex, is a quotient of a certain operad associated to the compactified configuration spaces. Different notions of homotopy Gerstenhaber algebras are discussed: one of them is a $B_\infty$-algebra, another, called a homotopy G-algebra, is a particular case of a $B_\infty$-algebra, the others, a $G_\infty$-algebra, an $\bar E^1$-algebra, and a weak $G_\infty$-algebra, arise from the geometry of configuration spaces. Corrections to the paper arXiv:q-alg/9602009 of Kimura, Zuckerman, and the author related to the use of a nonextant notion of a homotopy Gerstenhaber algebra are made.Comment: 22 pages, 7 figures. A minor error in the description of Tamarkin's counterexample is corrected, Conference Moshe Flato 1999 (G. Dito and D. Sternheimer, eds.), vol. 2. Kluwer Academic Publishers, the Netherlands, 2000, pp. 307-33

Stability of the Rational Homotopy Type of Moduli Spaces

We show that for g > 2k+2 the k-rational homotopy type of the moduli space M_{g,n} of algebraic curves of genus g with n punctures is independent of g, and the space M_{g,n} is k-formal. This implies the existence of a limiting rational homotopy type of M_{g,n} as g goes to infinity and the formality of it.Comment: 7 pages, 1 figur

Quantizing deformation theory II

A quantization of classical deformation theory, based on the Maurer-Cartan Equation $dS + \frac{1}{2}[S,S] = 0$ in dg-Lie algebras, a theory based on the Quantum Master Equation $dS + \hbar \Delta S + \frac{1}{2} \{S,S\} = 0$ in dg-BV-algebras, is proposed. Representability theorems for solutions of the Quantum Master Equation are proven. Examples of "quantum" deformations are presented.Comment: 20 pages, for the collection dedicated to Yuri I. Manin on the occasion of his 80th birthday; the new version adds a few references and expands on example

Topological field theories, string backgrounds and homotopy algebras

String backgrounds are described as purely geometric objects related to moduli spaces of Riemann surfaces, in the spirit of Segal's definition of a conformal field theory. Relations with conformal field theory, topological field theory and topological gravity are studied. For each field theory, an algebraic counterpart, the (homotopy) algebra satisfied by the tree level correlators, is constructed.Comment: 12 page

The Swiss-Cheese Operad

We introduce a new operad, which we call the Swiss-cheese operad. It mixes naturally the little disks and the little intervals operads. The Swiss-cheese operad is related to the configuration spaces of points on the upper half-plane and points on the real line, considered by Kontsevich for the sake of deformation quantization. This relation is similar to the relation between the little disks operad and the configuration spaces of points on the plane. The Swiss-cheese operad may also be regarded as a finite-dimensional model of the moduli space of genus-zero Riemann surfaces appearing in the open-closed string theory studied recently by Zwiebach. We describe algebras over the homology of the Swiss-cheese operad.Comment: 9 pages, 2 figure

The MV formalism for IBL∞{\rm IBL}_\infty- and BV∞{\rm BV}_\infty-algebras

We develop a new formalism for the Quantum Master Equation $\Delta e^{S/\hbar} = 0$ and the category of ${\rm IBL}_\infty$-algebras and simplify some homotopical algebra arising in the context of oriented surfaces with boundary. We introduce and study a category of MV-algebras, which, on the one hand, contains such important categories as those of ${\rm IBL}_\infty$-algebras and ${\rm L}_\infty$-algebras, and on the other hand, is homotopically trivial, in particular allowing for a simple solution of the quantum master equation. We also present geometric interpretation of our results.Comment: 26 pages; this is a version accepted for publication in Lett. Math. Phys.; a new section on algebras of types other than commutative is added, a few minor changes mad

Categorification of Dijkgraaf-Witten Theory

The goal of the paper is to categorify Dijkgraaf-Witten theory, aiming at providing foundation for a direct construction of Dijkgraaf-Witten theory as an Extended Topological Quantum Field Theory. The main tool is cohomology with coefficients in a Picard groupoid, namely the Picard groupoid of hermitian lines.Comment: 28 pages; submitted version containing minor modification

The BV formalism for L∞_\infty-algebras

Functorial properties of the correspondence between commutative BV$_\infty$-algebras and L$_\infty$-algebras are investigated. The category of L$_\infty$-algebras with L$_\infty$-morphisms is characterized as a certain category of pure BV$_\infty$-algebras with pure BV$_\infty$-morphisms. The functor assigning to a commutative BV$_\infty$-algebra the L$_\infty$-algebra given by higher derived brackets is also shown to have a left adjoint. Cieliebak-Fukaya-Latschev's machinery of IBL$_\infty$- and BV$_\infty$-morphisms is further developed with introducing the logarithm of a map.Comment: 20 pages; a version, containing minor changes, to be published in Journal of Homotopy and Related Structure

PROPped up graph cohomology

We consider graph complexes with a flow and compute their cohomology. More specifically, we prove that for a PROP generated by a Koszul dioperad, the corresponding graph complex gives a minimal model of the PROP. We also give another proof of the existence of a minimal model of the bialgebra PROP from math.AT/0209007. These results are based on the useful notion of a 1/2 PROP introduced by Kontsevich in an e-mail message to the first author.Comment: 33 pages; this is a final version to appear in Maninfest; some corrections made, including adding a condition of connectedness of the differentia