zbMATH — the first resource for mathematics

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.

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
Full Text: DOI Link arXiv
[1] C. Berger and B. Fresse, Une décomposition prismatique de l’opérade de Barratt- Eccles. C. R. Acad. Sci. Paris Sér. I Math. 335 (2002), 365-370. · Zbl 1016.18005
[2] M. Chas and D. Sullivan, String topology. Ann. of Math. , to appear; preprint 1999, · Zbl 1185.55013
[3] D. Chataur, A bordism approach to string topology. Internat. Math. Res. Notices 2005 (2005), 2829-2875. · Zbl 1086.55004
[4] R. L. Cohen, Multiplicative properties of Atiyah duality. Homology Homotopy Appl. 6 (2004), 269-281. · Zbl 1072.55004
[5] R. L. Cohen and V. Godin, A polarized view of string topology. In Topology, geometry and quantum field theory (Oxford, 2002), London Math. Soc. Lec- ture Note Ser. 308, Cambridge University Press, Cambridge, 2004, 127-154. · Zbl 1095.55006
[6] R. L. Cohen and J. D. S. Jones, A homotopy theoretic realization of string topology. Math. Ann. 324 (2002), 773-798. · Zbl 1025.55005
[7] J. Conant and K. Vogtmann, On a theorem of Kontsevich. Algebr. Geom. Topol. 3 (2003), 1167-1224. · Zbl 1063.18007
[8] K. J. Costello, Topological conformal field theories and Calabi-Yau categories. Adv. Math. 210 (2007), 165-214 Geom. Topol. 11 (2007), 1637-1652. · Zbl 1171.14038
[9] J. L. Harer, Stability of the homology of the mapping class groups of orientable sur- faces. Ann. of Math. (2) 121 (1985), 215-249. · Zbl 0579.57005
[10] J. Hubbard and H. Masur, Quadratic differentials and foliations. Acta Math. 142 (1979), 221-274. · Zbl 0415.30038
[11] J. D. S. Jones, Cyclic homology and equivariant homology. Invent. Math. 87 (1987), 403-423. · Zbl 0644.55005
[12] A. Kapustin and Y. Li, D-branes in Landau-Ginzburg models and algebraic geometry. J. High Energy Phys. 2003 , no. 12, 005.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.