×

zbMATH — the first resource for mathematics

Moduli space actions on the Hochschild co-chains of a Frobenius algebra. I: Cell operads. (English) Zbl 1145.55008
This is the first of two papers to prove that the cell model of the moduli space of curves with marked points and tangent vectors acts on the Hochschild cochains of a Frobenius algebra. There are several combinatorially defined objects that underlie operads and PROPs which act on chain and cochain modules. These include Cacti, Arcs, chord diagrams, little discs, etc. Interest is strong for these structures because they induce operations on homology under the right circumstances. The second part of this series promises the definitions of the actions.
In this paper, the constructions and relations between constructions are given of the various operads and PROPs, sometimes defined weakly to gain flexibility. The starring role is played by versions of the Arc operad, studied by R. M. Kaufmann, M. Livernet and R. C. Penner [Geom. Topol. 7, 511–568 (2003; Zbl 1034.18009)] which is then related to moduli spaces via a dual graph construction taking us through marked metric ribbon graphs.

MSC:
55P48 Loop space machines and operads in algebraic topology
18D50 Operads (MSC2010)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
17A99 General nonassociative rings
32G81 Applications of deformations of analytic structures to the sciences
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[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. Preprint 2004, · Zbl 1171.14038
[9] J. L. Harer, Stability of the homology of the mapping class groups of orientable surfaces. 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.
[13] A. Kapustin andY. Li, Topological correlators in Landau-Ginzburg models with bound- aries. Adv. Theor. Math. Phys. 7 (2003), 727-749. · Zbl 1058.81061
[14] A. Kapustin and L. Rozansky, On the relation between open and closed topological strings. Comm. Math. Phys. 252 (2004), 393-414. · Zbl 1102.81064
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.