Journal of Noncommutative Geometry
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Moduli space actions on the Hochschild co-chains of a Frobenius algebra I: cell operads
Ralph M. Kaufmann (1)
(1) Department of Mathematics, Purdue University, 150 N. University St., IN 47907-2067, WEST LAFAYETTE, UNITED STATES
This is the first of two papers in which we prove that a cell
model of the moduli space of curves with marked points and tangent
vectors at the marked points acts on the Hochschild co-chains of
a Frobenius algebra. We also prove that a there is dg-PROP action
of a version of Sullivan chord diagrams which acts on the
normalized Hochschild co-chains of a Frobenius algebra. These
actions lift to operadic correlation functions on the co-cycles.
In particular,
the PROP action gives an action on the homology of a loop space of
a compact simply-connected manifold.
In this first part, we set up the topological operads/PROPs and
their cell models. The main theorems of this part are: There
is a cell model operad for the moduli space of genus g curves
with n punctures and a tangent vector at each of these punctures,
there exists a CW complex whose chains are isomorphic to
a certain type of Sullivan chord diagrams and they form a
PROP. Furthermore there exist weak versions of these structures on
the topological level which all lie inside an all encompassing
cyclic (rational) operad.
Keywords: Moduli spaces, operads, Hochschild cohomology, foliations, cell models, string topology, Sullivan chord diagrams, PROPs, conformal field theory