The Cohomology of Algebras over Moduli Spaces

The purpose of this paper is to introduce the cohomology of various algebras over an operad of moduli spaces including the cohomology of conformal field theories (CFT’s) and vertex operator algebras (VOA’s). This cohomology theory produces a number of invariants of CFT’s and VOA’s, one of which is the space of their infinitesimal deformations. The paper is inspired by the ideas of Drinfeld [5], Kontsevich [16] and Ginzburg and Kapranov [10] on Koszul duality for operads. An operad is a gadget which parameterizes algebraic operations on a vector space. They were originally invented [21] in order to study the homotopy type of iterated loop spaces. Recently, operads have turned out to be an effective tool in describing various algebraic structures that arise in mathematical physics in terms of the geometry of moduli spaces particularly in conformal field theory (see [8], [12], [14], [19], [26], [27]) and topological gravity (see [15], [18]). In fact, a (tree level c = 0) conformal field theory is nothing more than a representation of the operad, P, consisting of moduli spaces of configurations of holomorphically embedded unit disks in the Riemann sphere. Alternately, a conformal field theory is said to be a P-algebra or an algebra over P. In the first part of this paper, we use the idea of Ginzburg-Kapranov [10] of homology of an algebra over a quadratic operad to define the cohomology of an algebra over a quadratic operad with values in an arbitrary module. We prove that P is a quadratic operad and construct its Koszul dual operad. We then construct the cohomology theory associated to a conformal field theory. We demonstrate that the second cohomology group, as it should, parameterizes deformations of conformal field theories. Our approach to deformations is morally the same as the one developed by Dijkgraaf and E. and H. Verlinde [3, 4] and Ranganathan, Sonoda and Zwiebach [22] using (1,1)-fields, except that we fix the action of the Virasoro algebra. It would be very interesting to find an explicit connection between the two approaches. By analogy with the case of associative algebras, one expects the third cohomology group to contain obstructions to extending an infinitesimal deformation to a formal neighborhood. Another plausible interpretation of higher dimensional cohomology is that