symmetric monoidal (∞,1)-category of spectra
In string topology one studies the BV-algebra-structure on the ordinary homology of the free loop space of an oriented manifold , or more generally the framed little 2-disk algebra-structure on the singular chain complex. This is a special case of the general algebraic structure on higher order Hochschild cohomology, as discussed there.
The string product is a morphism of abelian groups
where is the dimension of , defined as follows:
Write for the evaluation map at the basepoint of the loops.
For and we can find representatives and such that and intersect transversally. There is then an -chain such that is the chain given by that intersection: above this is the loop obtained by concatenating and at their common basepoint. The string product is then defined using such representatives by
The string product is associative and graded-commutative.
The string product is the pull-push operation
This is due to (CohenJones).
Define a morphism of abelian groups
as follows. Consider first the rotation map
that sends . Then take
This is called the BV-operator for string topology.
The Goldman bracket on is equivalent to the string product applied to the image of the BV-operator
This is due to (ChasSullivan).
The structures studied in the string topology of a smooth manifold may be understood as being essentially the data of a 2-dimensional topological field theory sigma model with target space , or rather its linearization to an HQFT (with due care on some technical subtleties).
The idea is that the configuration space of a closed or open string-sigma-model propagating on is the loop space or path space of , respectively. The space of states of the string is some space of sections over this configuration space, to which the (co)homology is an approximation. The string topology operations are then the cobordism-representation with coefficients in the category of chain complexes
given by the FQFT corresponding to the -modelon these state spaces, acting on these state spaces.
For a collection of oriented compact submanifolds write for the path space of paths in that start in and end in .
For closed strings this is discussed in (Cohen-Godin 03, Tamanoi 07). For open strings on a single brane this was shown in (Godin 07), where the general statement for arbitrary branes is conjectured. A detailed proof of this general statement is in (Kupers 11).
These constructions work by regarding the mapping spaces from 2-dimensional cobordisms with maps to the base space as correspondences and then applying pull-push (pullback followed by push-forward in cohomology/Umkehr maps) to these. Hence these quantum field theory realizations of string topology may be thought of as arising from a quantization process of the form path integral as a pull-push transform/motivic quantization.
The original references include the following:
Ralph Cohen, Alexander Voronov, Notes on string topology, math.GT/05036259, 95 pp. published as a part of R. Cohen, K. Hess, A. Voronov, String topology and cyclic homology, CRM Barcelona courseware, Springer, description, doi, pdf
Dennis Sullivan, Open and closed string field theory interpreted in classical algebraic topology, Topology, geometry, and quantum field theory, 344–357. London Math. Soc. Lec. Notes 308, Cambridge Univ. Press. 2004.
The interpretation of closed string topology as an HQFT is discussed in
The generalization to multiple D-branes is discussed in
(see example 4.2.16, remark 4.2.17).
For the string product and the BV-operator this extension has been known early on, it yields a homotopy BV algebra considered around page 101 of
Evidence for the existence of the TCFT version by exhibiting a dg-category that looks like it ought to be the dg-category of string-topology branes (hence ought to correspond to the TCFT under the suitable version of the TCFT-version of the cobordism hypothesis) is discussed in
A generalization of string topology with target manifolds generalized to orbifolds is discussed in
Kai Behrend, Gregory Ginot, Behrang Noohi, Ping Xu, String topology for stacks, (89 pages) arxiv/0712.3857; String topology for loop stacks, C. R. Math. Acad. Sci. Paris, 344 (2007), no. 4, 247–252, (6 pages, pdf)
The relation between string topology and Hochschild cohomology in dimenion is discussed in
More developments are in