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Moduli space actions on the Hochschild co-chains of a Frobenius algebra. II: Correlators. (English) Zbl 1169.55005
In the first paper of this series [J. Noncommut. Geom. 1, No. 3, 333–384 (2007; Zbl 1145.55008)], certain topological and cellular operads were introduced and interrelated on which actions on the Hochschild co-chains of a Frobenius algebra are developed in this second paper. The motivation for the existence of such actions lies in the structure of $$D$$-branes, string topology, and the solution of the Deligne conjecture. The main idea is given by operadic correlation functions which are defined for a vector space $$A$$ and a choice of sum $$C\in A\otimes A$$, $$C = \sum c^{(1)}\otimes c^{(2)}$$ using the operations
$\circ_i: \text{Hom}(A^{\otimes n+1}, k)\otimes \text{Hom}(A^{\otimes m+1},k) \to \text{Hom}(A^{\otimes n+m}, k)$
and given on $$\varphi \in \text{Hom}(A^{\otimes n+1}, k)$$ and $$\psi \in \text{Hom}(A^{\otimes m+1}, k)$$ by
$\begin{split} \varphi\circ_i\psi (a_1\otimes \cdots \otimes a_{n+m})\\ = \sum \varphi(a_1\otimes \cdots \otimes a_{i-1} \otimes c^{(1)}\otimes a_{i+m} \otimes \cdots \otimes a_{n+m})\cdot \psi(c^{(2)}\otimes a_i \otimes \cdots \otimes a_{i+m-1}).\end{split}$
The operadic correlation functions for an operad $${\mathcal O}$$ are given by $$Y_{n+1}: {\mathcal O}(n) \to \text{Hom}(A^{\otimes n+1}, k)$$ with $$Y_{n+1}$$ $$\Sigma_{n+1}$$-equivariant and for $$u\in {\mathcal O}(n)$$ and $$v\in {\mathcal O}(m)$$, $$Y_{n+m}(u\circ_i v) = Y_{n+1}(u)\circ_i Y_{m+1}(v)$$. Examples abound in the author’s menagerie of operads. The correlators for Hochschild co-chains of a Frobenius algebra are given in §3 and they are shown to unify and extend the motivating examples.

##### MSC:
 55P48 Loop space machines and operads in algebraic topology 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) 17A99 General nonassociative rings 18D50 Operads (MSC2010) 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 32G81 Applications of deformations of analytic structures to the sciences 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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