nLab
string topology

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 $H_{•}(LX)$ for the homology of this space (with coefficients in the integers $\mathbb{Z}$).

The string product

Definition

The string product is a morphism of abelian groups

$$(-)\cdot (-):H_{•}(LX)\otimes H_{•}(LX)\to H_{•-\dim X}(LX),$$(-)\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 ${\mathrm{ev}}_{*}:LX\to X$ for the evaluation map at the basepoint of the loops.

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

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

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

$$\begin{array}{ccc}& & X^8\\ & {}^{\mathrm{in}}\swarrow & & {\searrow }^{\mathrm{out}}\\ LX\times LX& & & & LX\end{array}.$$\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

$${\mathrm{out}}_{*}\circ {\mathrm{in}}^{!}:H_{•}(LX\times LX)\simeq H_{•}(LX)\otimes H_{•}(LX)\to H_{•-\dim X}(LX).$$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

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

as follows. Consider first the rotation map

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

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

$$\Delta :a\mapsto {\rho }_{*}([S^1]\times a),$$\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 $H_0(LX)$ is equivalent to the string product applied to the image of the BV-operator

$$\{[{\gamma }_1],[{\gamma }_2]\}=\Delta [{\Gamma }_1]\cdot \Delta [{\Gamma }_2].$$\{[\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 $H_{•}(LX)$ is an approximation. The string topology operations are then the cobordism-representation

$$H_{•}({\mathrm{Bord}}_2)\to {\mathrm{Ch}}_{•}$$H_\bullet(Bord_2) \to Ch_\bullet

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

Closed string TQFT

(…)

Open closed string TQFT

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

For $ℬ=\{A,B,\cdots \}$ 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 $(H_{•}(LM,\mathbb{Q}),\{H_{•}(P_X(A,B),\mathbb{Q}){\}}_{A,B\in ℬ})$ 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 $ℬ=\{*\}$ this was shown in (Godin), where the general statement is conjectured. The detailed proof for the general statement is in (Kupers).

Related concepts

• Goldman bracket

• path integral as a pull-push transform

References

The original references include the following:

• Moira Chas, Dennis Sullivan, String topology, Ann. Math. math.GT/9911159

• 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

• Ralph Cohen, Veronique Godin, A Polarized View of String Topology

• 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

• Veronique Godin, Higher string topology operations (2007)(arXiv:0711.4859)

The generalization to multiple D-branes is discussed in

• Sander Kupers, String topology operations MS thesis (2011)

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 $A_{\mathrm{\infty }}$-categories (2004) , (arXiv:0412149)

and

• Jacob Lurie, On the Classification of Topological Field Theories

(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

• Andrew Blumberg, Ralph Cohen, Constantin Teleman, Open-closed field theories, string topology, and Hochschild homology (arXiv:0906.5198)

A generalization of string topology with target manifolds generalized to target differentiable stacks/Lie groupoids 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)

• Po Hu, Higher string topology on general spaces, Proc. London Math. Soc. 93 (2006) 515-544, doi, ps

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)