My preprint arXiv:1106.5352v1 is mostly inspired by the perfect explanation of Damien Calaque of the Kevin Costello’s project. Given a -algebra and a compact parallelized -manifold without borders a morphism from homology of the Lie algebra associated with to some manifoldic homology is constructed in the preprint. I guess that for a proper this morphism gives perturbative Chern-Simons invariants of and that this construction is closely connected with the Costello’s approach. Some comments are under the cut. (more…)
The universal enveloping -algebra is a functor from the category of homotopy Lie algebras over to the category of algebras over the operad of rational chains of the operad of little discs. For one get (the derived functor of) the usual universal enveloping algebra. (more…)
Below there is a construction that from any augmented dg algebra over the operad of chains on framed little -discs produces a coalgebra over the same operad.
This construction generalizes the Koszul duality, for this is the usual Koszul duality for algebras.
Another case of this duality is from topology. Take a topological space . Then is a module over framed little -discs. The union of symmetric powers of is a comodule over the (trivial) operad and thus over framed discs (I am grateful to Victor Turchin for this observation). (Co)chains of this (co)modules over operads are dual to each other in our sence. This explains why one may calculate homology of mapping spaces from a -manifold to either by means of the generalization of the higher Hochschild homology or the spherical homology.
It seems that it is connected with work in progress of M. Ching and P. Salvatore “Geometric self-duality for the little discs operads” (I found it on the homepage of Michael Ching).
UPD: This Koszul duality is mentioned in the draft of J. Lurie’s “Higher algebra”(see e.g. example 126.96.36.199). Unfortunately, details are postponed for future work. See also “Moduli Problems for Ring Spectra”.
Nowadays the theory of quantum cohomology is treated as a part of symplectic geometry. But it is not clear (for me) how much information about the symplectic structure does quantum cohomology contain. Perhaps, the situation is analagous to the Morse theory, where the definition is given in differential geometric terms (integral curves and so on) but the result is purely topological.
If one take as a starting point the paper of Witten and two papers of Givental (this and this), one may develop a theory of a different flavour sketched below. Conjecturally it coincides with the usual one, at least in some cases.
I already made some unclear hints about connections between the perturbative Chern-Simons theory and spherical homology. Below I would like to clarify it. It seems that nearly everything I am going to say is known as “BV formalism”. But until I learn this language let me continue with my own.
Given a 3-manifold and a (say) connection on it with zero cohomology, the perturbative Chern-Simons theory gives an invariant of it, that is a power series of the quantization parameter. It would be plausible to make this invariant two-step: on get from the initial data some structure (like a homotopy algebra over a operad) and then take an invariant of it.
The structure involved is a 1-parametric family of homotopy BV algebras that is curvy trivial and trivialized . The spherical homology gives invariants of this structure.
Let me explain and comment last two phrases.
UPD: Things described here and in some previous posts are particular cases of the topological chiral homology introduced in J. Lurie, “Derived Algebraic Geometry VI: E_k Algebras”. The only difference is that I need framed discs. I thank Damien for this comment.
Speculating about the question posed in the end of this confused post I recognised, that the generalization involved makes some sense. Spherical homology is generalised in two directions. Firstly 2-sphere is replaced with any compact oriented 2-manifold. Secondly a BV (f2-)algebra is replaced with a cyclic 2-category. As a result one get a structure that strongly reminds TQFT up to decorations. By decorations I mean metric, counit and so on. This conjectural structure plays for cyclic 2-categories the same role as Hochschild homology plays for (1-)categories.
In the paper “What do DG categories form?” D. Tamarkin introduced the notion of a homotopy 2-category. Its important feature is that for any object on the operad of chains of little discs acts. Below a notion of homotopy cyclic 2-category is introduced. For it the complex is acted by the operad of chains of framed little discs.