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The purpose of this paper is to describe an analogue of a construction of Costello in the context of finite-dimensional differential graded Frobenius algebras which produces closed forms on the decorated moduli space of Riemann surfaces. We show that this construction extends to a certain natural compactification of the moduli space which is associated to the modular closure of the associative operad, due to the absence of ultra-violet divergences in the finite-dimensional case. We demonstrate that this construction is equivalent to the dual construction of Kontsevich.
In this paper we show that the homology of a certain natural compactification of the moduli space, introduced by Kontsevich in his study of Wittens conjectures, can be described completely algebraically as the homology of a certain differential grade
The moduli space of flat SU(2) connections on a punctured surface, having prescribed holonomy around the punctures, is a compact smooth manifold if the prescription is generic. This paper gives a direct, elementary proof that the trace of the holonom
We give two proofs that appropriately defined congruence subgroups of the mapping class group of a surface with punctures/boundary have enormous amounts of rational cohomology in their virtual cohomological dimension. In particular we give bounds tha
Let ${mathcal I}(n)$ denote the moduli space of rank $2$ instanton bundles of charge $n$ on ${mathbb P}^3$. We know from several authors that ${mathcal I}(n)$ is an irreducible, nonsingular and affine variety of dimension $8n-3$. Since every rank $2$
Let $Gamma$ be a finite-index subgroup of the mapping class group of a closed genus $g$ surface that contains the Torelli group. For instance, $Gamma$ can be the level $L$ subgroup or the spin mapping class group. We show that $H_2(Gamma;Q) cong Q$ f