# 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 singular 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 $LX$ for its free loop space (for $X$ regarded as a topological space) and ${H}_{•}\left(LX\right)$ for the homology of this space (with coefficients in the integers $ℤ$).

### The string product

###### Definition

The string product is a morphism of abelian groups

$\left(-\right)\cdot \left(-\right):{H}_{•}\left(LX\right)\otimes {H}_{•}\left(LX\right)\to {H}_{•-\mathrm{dim}X}\left(LX\right)\phantom{\rule{thinmathspace}{0ex}},$(-)\cdot(-) : H_\bullet(L X) \otimes H_\bullet(L X) \to H_{\bullet - dim X}(L X) \,,

where $\mathrm{dim}X$ is the dimension of $X$, defined as follows:

Write ${\mathrm{ev}}_{*}:LX\to X$ for the evaluation map at the basepoint of the loops.

For $\left[\alpha \right]\in {H}_{i}\left(LX\right)$ and $\left[\beta \right]\in {H}_{j}\left(LX\right)$ we can find representatives $\alpha$ and $\beta$ such that $\mathrm{ev}\left(\alpha \right)$ and $\mathrm{ev}\left(\beta \right)$ intersect transversally. There is then an $\left(\left(i+j\right)-\mathrm{dim}X\right)$-chain $\alpha \cdot \beta$ such that $\mathrm{ev}\left(\alpha \cdot \beta \right)$ is the chain given by that intersection: above $x\in \mathrm{ev}\left(\alpha \cdot \beta \right)$ 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

$\left[\alpha \right]\cdot \left[\beta \right]:=\left[\alpha \cdot \beta \right]\phantom{\rule{thinmathspace}{0ex}}.$[\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

${S}^{1}\coprod {S}^{1}\to 8←{S}^{1}$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

$\begin{array}{ccc}& & {X}^{8}\\ & {}^{\mathrm{in}}↙& & {↘}^{\mathrm{out}}\\ LX×LX& & & & LX\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ && X^8 \\ & {}^{\mathllap{in}}\swarrow && \searrow^{\mathrlap{out}} \\ L X \times L X &&&& L X } \,.

Write ${\mathrm{in}}^{!}$ for the “pullback” in homology along $\mathrm{in}$ (the dual fiber integration) and ${\mathrm{out}}_{*}$ for the ordinary pushforward.

###### Theorem

The string product is the pull-push operation

${\mathrm{out}}_{*}\circ {\mathrm{in}}^{!}:{H}_{•}\left(LX×LX\right)\simeq {H}_{•}\left(LX\right)\otimes {H}_{•}\left(LX\right)\to {H}_{•-\mathrm{dim}X}\left(LX\right)\phantom{\rule{thinmathspace}{0ex}}.$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

$\Delta :{H}_{•}\left(LX\right)\to {H}_{•+1}\left(LX\right)$\Delta : H_\bullet(L X) \to H_{\bullet + 1}(L X)

as follows. Consider first the rotation map

$\rho :{S}^{1}×LX\to LX$\rho : S^1 \times L X \to L X

that sends $\left(\theta ,\gamma \right)↦\left(t↦\gamma \left(\theta +t\right)\right)$. Then take

$\Delta :a↦{\rho }_{*}\left(\left[{S}^{1}\right]×a\right)\phantom{\rule{thinmathspace}{0ex}},$\Delta : a \mapsto \rho_* ([S^1] \times a) \,,

where $\left[{S}^{1}\right]\in {H}_{1}\left({S}^{1}\right)$ is the fundamental class of the circle.

This is called the BV-operator for string topology.

###### Proposition

The Goldman bracket on ${H}_{0}\left(LX\right)$ is equivalent to the string product applied to the image of the BV-operator

$\left\{\left[{\gamma }_{1}\right],\left[{\gamma }_{2}\right]\right\}=\Delta \left[{\Gamma }_{1}\right]\cdot \Delta \left[{\Gamma }_{2}\right]\phantom{\rule{thinmathspace}{0ex}}.$\{[\gamma_1], [\gamma_2]\} = \Delta[\Gamma_1] \cdot \Delta[\Gamma_2] \,.

This is due to (ChasSullivan).

## In terms of TQFTs

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 ${H}_{•}\left(LX\right)$ is an approximation. The string topology operations are then the cobordism-representation

${H}_{•}\left({\mathrm{Bord}}_{2}\right)\to {\mathrm{Ch}}_{•}$H_\bullet(Bord_2) \to Ch_\bullet

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

(…)

### Open closed string TQFT

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

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

###### Theorem

The tuple $\left({H}_{•}\left(LM,ℚ\right),\left\{{H}_{•}\left({P}_{X}\left(A,B\right),ℚ\right){\right\}}_{A,B\in ℬ}\right)$ carries the structure of a $d$-dimensional HCFT with positive boundary and set of branes $ℬ$, such that the correlators in the closed sector are the standard string topology operation.

For a single brane $ℬ=\left\{*\right\}$ this was shown in (Godin), where the general statement is conjectured. The detailed proof for the general statement is in (Kupers).

## 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, 773798.

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

A generalization of string topology with target manifolds generalized to target differentiable stacks/Lie groupoids is discussed in

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

• Dmitry Vaintrob?, The String topology BV algebra, Hochschild cohomology and the Goldman bracket on surfces (arXiv:0702859)
Revised on July 24, 2011 18:35:36 by Urs Schreiber (82.113.99.30)