Contents

Idea

In string topology one studies the BV-algebra-structure on the ordinary homology of the free loop space $X^{S^1}$ of an oriented manifold $X$, 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 study of string topology was initated by Moira Chas and Dennis Sullivan.

The string operations

Let $X$ be a smooth manifold, write $L X$ for its free loop space (for $X$ regarded as a topological space) and $H_\bullet(L X)$ for the ordinary homology of this space (with coefficients in the integers $\mathbb{Z}$).

The string product

Definition

The string product is a morphism of abelian groups

$(-)\cdot(-) : H_\bullet(L X) \otimes H_\bullet(L X) \to H_{\bullet - dim X}(L X) \,,$

where $dim X$ is the dimension of $X$, defined as follows:

Write $ev_* : L X \to X$ for the evaluation map at the basepoint of the loops.

For $[\alpha] \in H_i(L X)$ and $[\beta] \in H_j(L X)$ we can find representatives $\alpha$ and $\beta$ such that $ev(\alpha)$ and $ev(\beta)$ intersect transversally. There is then an $((i+j)-dim X)$-chain $\alpha \cdot \beta$ such that $ev(\alpha \cdot \beta)$ is the chain given by that intersection: above $x \in ev(\alpha \cdot \beta)$ this is the loop obtained by concatenating $\alpha_x$ and $\beta_x$ at their common basepoint. The string product is then defined using such representatives by

$[\alpha] \cdot [\beta] := [\alpha \cdot \beta] \,.$
Theorem

The string product is associative and graded-commutative.

This is due to (ChasSullivan). There is is a more elegant way to capture this, due to (CohenJones):

Let

$S^1 \coprod S^1 \to 8 \leftarrow S^1$

be the cospan that exhibts the inner and the outer circle of the figure “8” topological space. By forming hom spaces this induces the span

$\array{ && X^8 \\ & {}^{\mathllap{in}}\swarrow && \searrow^{\mathrlap{out}} \\ L X \times L X &&&& L X } \,.$

Write $in^!$ for the “pullback” in ordinary homology along $in$ (the dual fiber integration) and $out_*$ for the ordinary pushforward.

Theorem

The string product is the pull-push operation

$out_* \circ in^! : H_\bullet(L X \times L X) \simeq H_\bullet(L X) \otimes H_\bullet(L X) \to H_{\bullet - dim X}(L X) \,.$

This is due to (CohenJones).

The BV-operator

Definition

Define a morphism of abelian groups

$\Delta : H_\bullet(L X) \to H_{\bullet + 1}(L X)$

as follows. Consider first the rotation map

$\rho : S^1 \times L X \to L X$

that sends $(\theta, \gamma) \mapsto (t \mapsto \gamma(\theta + t))$. Then take

$\Delta : a \mapsto \rho_* ([S^1] \times a) \,,$

where $[S^1] \in H_1(S^1)$ is the fundamental class of the circle.

This is called the BV-operator for string topology.

Proposition

The Goldman bracket on $H_0(L X)$ is equivalent to the string product applied to the image of the BV-operator

$\{[\gamma_1], [\gamma_2]\} = \Delta[\Gamma_1] \cdot \Delta[\Gamma_2] \,.$

This is due to (ChasSullivan).

String operations as operators in a topological quantum field theory

The structures studied in the string topology of a smooth manifold $X$ may be understood as being essentially the data of a 2-dimensional topological field theory sigma model with target space $X$, 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 $X$ is the loop space or path space of $X$, respectively. The space of states of the string is some space of sections over this configuration space, to which the (co)homology $H_\bullet(L X)$ is an approximation. The string topology operations are then the cobordism-representation with coefficients in the category of chain complexes

$H_\bullet(Bord_2) \to Ch_\bullet$

given by the FQFT corresponding to the $\sigma$-modelon these state spaces, acting on these state spaces.

$\,,$

Let $X$ be an oriented compact manifold of dimension $d$.

For $\mathcal{B} = \{A, B , \cdots\}$ a collection of oriented compact submanifolds write $P_X(A,B)$ for the path space of paths in $X$ that start in $A \subset X$ and end in $B \subset X$.

Theorem

The tuple $(H_\bullet(L M, \mathbb{Q}), \{H_\bullet(P_X(A,B), \mathbb{Q})\}_{A,B \in \mathcal{B}})$ carries the structure of a $d$-dimensional HCFT with positive boundary and set of branes $\mathcal{B}$, such that the correlators in the closed sector are the standard string topology operation.

For closed strings this is discussed in (Cohen-Godin 03, Tamanoi 07). For open strings on a single brane $\mathcal{B} = \{*\}$ 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).

Remark

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.

• Ralph Cohen, John R. Klein, Dennis Sullivan, The homotopy invariance of the string topology loop product and string bracket, J. of Topology 2008 1(2):391-408; doi

• Ralph Cohen, Homotopy and geometric perspectives on string topology, pdf

In

• Ralph Cohen and J.D.S. Jones, A homotopy theoretic realization of string topology , Math. Ann. 324

(2002), no. 4, (arXiv:0107187)

the string product was realized as genuine pull-push (in terms of dual fiber integration via Thom isomorphism).

The interpretation of closed string topology as an HQFT is discussed in

• Hirotaka Tamanoi, Loop coproducts in string topology and triviality of higher genus TQFT operations (2007) (arXiv)

A detailed discussion and generalization to the open-closed HQFT in the presence of a single space-filling brane is in

The generalization to multiple D-branes is discussed in

For target space a classifying space of a finite group or compact Lie group this is discussed in

• David Chataur, Luc Menichi, String topology of classifying spaces (pdf)

Arguments that this string-topology HQFT should refine to a chain-level theory – a TCFT – were given in

• Kevin Costello, Topological conformal field theories and Calabi-Yau $A_\infty$-categories (2004) , (arXiv:0412149)

and

(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

• Scott Wilson, On the Algebra and Geometry of a Manifold’s

Chains and Cochains_ (2005) (pdf)

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

Refinements of string topology from homology groups to the full ordinary homology-spectra is discussed in (Blumberg-Cohen-Teleman 09) and in

A generalization of string topology with target manifolds generalized to orbifolds is discussed in

• Alejandro Adem, Johanna Leida, Yongbin Ruan, Orbifolds and string topology, Cambridge Tracts in Mathematics 171, 2007 (pdf)

Further generalization to target spaces that are more generally differentiable stacks/Lie groupoids is discussed in

The relation between string topology and Hochschild cohomology in dimenion $\gt 1$ is discussed in

More developments are in