# nLab string topology

### Context

#### Topology

topology

algebraic topology

## Examples

#### Higher algebra

higher algebra

universal algebra

# 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.

## References

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

Revised on February 6, 2014 03:08:00 by Urs Schreiber (89.204.137.107)