Homological algebra related to surfaces with boundary. (English) Zbl 1468.18022
The authors describe an algebraic framework which can be used for three related but different purposes: (equivariant) string topology [M. Chas and D. Sullivan, in: The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3–8, 2002. Berlin: Springer. 771–784 (2004; Zbl 1068.55009)], symplectic field theory [Y. Eliashberg et al., in: GAFA 2000. Visions in mathematics—Towards 2000. Proceedings of a meeting, Tel Aviv, Israel, August 25–September 3, 1999. Part II. Basel: Birkhäuser. 560–673 (2000; Zbl 0989.81114)], and Lagrangian Floer theory [A. Floer, J. Differ. Geom. 28, No. 3, 513–547 (1988; Zbl 0674.57027)] of higher genus.
It turns out that the relevant algebraic structure for all three contexts is a homotopy version of involutive bi-Lie algebras, called \(IBL_\infty\)-algebras. First, the authors define \(IBL_\infty\)-algebras and their morphisms, and discuss the defining relations from various points of view. Next, for various applications and the discussion of Maurer-Cartan elements the more general notion of a filtered \(IBL_\infty\)-algebra is needed. The remaining part of the paper gives some ideas how \(IBL_\infty\)-structures arise in algebraic and symplectic topology.
It turns out that the relevant algebraic structure for all three contexts is a homotopy version of involutive bi-Lie algebras, called \(IBL_\infty\)-algebras. First, the authors define \(IBL_\infty\)-algebras and their morphisms, and discuss the defining relations from various points of view. Next, for various applications and the discussion of Maurer-Cartan elements the more general notion of a filtered \(IBL_\infty\)-algebra is needed. The remaining part of the paper gives some ideas how \(IBL_\infty\)-structures arise in algebraic and symplectic topology.
Reviewer: Marek Golasiński (Olsztyn)
MSC:
| 18M85 | Polycategories/dioperads, properads, PROPs, cyclic operads, modular operads |
| 53D42 | Symplectic field theory; contact homology |
| 19D55 | \(K\)-theory and homology; cyclic homology and cohomology |
| 55P50 | String topology |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
| 58J28 | Eta-invariants, Chern-Simons invariants |
Keywords:
cyclic homology; involutive Lie bialgebra; loop space; perturbative Chern-Simons theory; ribbon graph;string topology; symplectic field theoryReferences:
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