nLab string topology

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

$$(-)\cdot(-) \;\colon\; H_\bullet(L X) \otimes H_\bullet(L X) \longrightarrow H_{\bullet - dim X}(L X)$$

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

Write $ev_* \colon L X \to X$ 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 $\big((i+j)-dim X\big)$-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

$$[\alpha] \cdot [\beta] \coloneqq [\alpha \cdot \beta] \,.$$

Theorem

The string product is associative and graded-commutative.

This is due to Chas & Sullivan. There is is a more elegant way to capture this, due to Cohen & Jones:

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 Cohen & Jones.

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

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

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 Chas & Sullivan.

Properties

As a TQFT

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 by Cohen & Godin 2003, Tamanoi 2007. For open strings on a single space-filling brane, $\mathcal{B} = \{ X \}$ this was shown by Godin 2007, where the general statement for arbitrary branes is conjectured. A detailed proof of this general statement is given by Kupers 2011.

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.

Example

(open string product on 0-brane is Pontrjagin product)
For the case of a single brane being a 0-brane, hence a point $\{x\} \hookrightarrow X$, then:

1. the open string configuration spaces is the based loop space $P(x,x) = \Omega_x X$,

2. whence the homology is the homology of loop spaces $H_\bullet(\Omega_x X)$, and

3. the open string product coincides with the Pontrjagin product,

   (1)$$(conc_x)_\ast \colon H_\bullet(\Omega X) \otimes H_\bullet(\Omega X) \longrightarrow H_\bullet(\Omega X) \mathrlap{\,,}$$

   given simply by pushforward in homology along the loop concatenation map

   (2)$$conc_x \colon \Omega_x X \times \Omega_x X \longrightarrpw \Omega_x X \mathrlap{\,.}$$

(Sullivan 2005 SFT, Ex. 3)

Proof

The first two statements are immediate from the definitions. The third statement is a special case of the explicit formula for the open string product $\mu_{a,b,c}$ (here: $\mu_{x,x,x}$) given in Kupers 2011 p. 137:

In our case of a 0-brane, the map denoted “$M^i$” there (on the previous p. 136), is an isomorphism, and the map “$M^j$” there (beware the typo in the orientation of the arrow) becomes the loop concatenation operation $conc_x$ (2), whence the formula for the string product reduces to the Pontrjagin product (1):

$$\begin{aligned} \mu_{x,x,x} & \coloneqq (M^j)_\ast \circ (M^i)^! \\ & \equiv (conc_x)_\ast \circ (id)^! \\ & \simeq (conc_x)_\ast \mathrlap{\,.} \end{aligned}$$

Related concepts

• Goldman bracket

• Sullivan chord diagram

• path integral as a pull-push transform

References

General

Original references:

• Moira Chas, Dennis Sullivan: String topology [math.GT/9911159]

• Ralph Cohen, John R. Klein, Dennis Sullivan: The homotopy invariance of the string topology loop product and string bracket, J. of Topology 1 2 (2008) 391-408 [doi:10.1112/jtopol/jtn001]

Exposition:

• Ralph Cohen: Homotopy and geometric perspectives on string topology (2005) [pdf, pdf]

Realizing the string product as a pull-push (in terms of dual fiber integration via Thom isomorphism):

• Ralph Cohen, John David Stuart Jones, A homotopy theoretic realization of string topology , Math. Ann. 324 4 (2002) [arXiv:0107187]

More on relation to homology of the based loop space and its Pontrjagin product:

• Eric J. Malm: String Topology and the Based Loop Space, talk at UC Riverside (2009) [slides: pdf, pdf]

• Eric J. Malm: String Topology and the Based Loop Space, PhD thesis, Stanford (2010) [proquest:2010. 28168917]

• Eric J. Malm: String topology and the based loop space [arXiv:1103.6198]

On string topology operations in the generality of (the homology of loop spaces of) Poincaré duality spaces:

• David Chataur: A bordism approach to string topology, International Mathematics Research Notices, 2005 46 (2005) 2829–2875 [arXiv:math/0306080, doi:10.1155/IMRN.2005.2829]

• Alastair Hamilton, Andrey Lazarev: Symplectic $A_\infty$-algebras and string topology operations, Amer. Math. Soc. Transl. 224 2 (2008) 147–157 [arXiv:0707.4003]

Refinements of string topology from homology groups to the full ordinary homology-spectra:

• Blumberg, Cohen & Teleman 09

• Ralph Cohen, John Jones, A homotopy theoretic realization of string topology, Mathematische Annalen (arXiv:math/0107187)

• Ralph Cohen, John Jones, Gauge theory and string topology (arXiv:1304.0613)

• Kate Gruher, Paolo Salvatore: Generalized string topology operations, Proc. London Math. Soc. 96 1 (2008) 78–106 [doi:10.1112/plms/pdm030, math.AT/0602210]

  (and generalizing from $L M \to M$ to any homotopy monoid bundle over $M$)

Further generalization to target spaces that need not be compact and moreover may be differentiable stacks/Lie groupoids:

• Kai Behrend, Gregory Ginot, Behrang Noohi, Ping Xu: String topology for stacks, Astérisque 343 (2012) [arxiv/0712.3857, numdam:AST_2012__343__R1_0]

• String topology for loop stacks, C. R. Math. Acad. Sci. Paris, 344 4 (2007) 247–252

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

On therelation between string topology and Hochschild cohomology:

• Dmitry Vaintrob: The String topology BV algebra, Hochschild cohomology and the Goldman bracket on surfaces [arXiv:0702859]

• Manuel Rivera, Zhengfang Wang: Singular Hochschild cohomology and algebraic string operations, Journal of Noncommutative Geometry 13 1 (2017) 297–361 [doi;10.4171/jncg/325]

Specifically on string topology of n-spheres (in particular on the ordinary homology of free loop spaces of n-spheres):

• Ralph L. Cohen, John D. S. Jones, Jun Yan: The loop homology algebra of spheres and projective spaces, in: Categorical Decomposition Techniques in Algebraic Topology, Progress in Mathematics 215 Birkhäuser (2004) [arXiv:math/0210353, doi:10.1007/978-3-0348-7863-0_5]

• Luc Menichi, Gerald Gaudens: String topology for spheres, Comment. Math. Helv. 84 (2009) 135–157 [arXiv:math/0609304, ems:43186]

• Ricardo Cabral, Basto Rodriguez, A Geometric Approach to the String Topology of Spheres, extended abstract [pdf]

As a TQFT

The interpretation of closed string topology as an HQFT:

• Ralph Cohen, Veronique Godin, A Polarized View of String Topology (arXiv:math/0303003)

• Alberto S. Cattaneo, Juerg Froehlich, Bill Pedrini: Topological Field Theory Interpretation of String Topology, Commun. Math. Phys. 240 (2003) 397–421 [doi:10.1007/s00220-003-0917-2, arXiv:math/0202176]

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

The open-closed HQFT in the presence of a single space-filling brane:

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

The generalization to multiple branes:

• Sander Kupers: String topology operations, MS thesis, Utrecht (2011) [hdl:20.500.12932/7283, pdf]

Exposition of the perspective of regarding string topology-operations as the TQFT of a topological string sigma model:

• Ralph Cohen, Alexander Voronov: Notes on String Topology, Part I in: Ralph Cohen, Kathryn Hess, Alexander Voronov (eds.): String topology and cyclic homology, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser (2006) [math.GT/05036259, doi:10.1007/3-7643-7388-1, pdf]

• Dennis Sullivan: Open and closed string field theory interpreted in classical algebraic topology, chapter 11 in: Ulrike Tillmann (ed.): Topology, Geometry and Quantum Field Theory, Cambridge University Press (2005) [doi:10.1017/CBO9780511526398.014, pdf]

• Dennis Sullivan: Sigma models and string topology, in: Mikhail Lyubich, Leon Takhtajan (eds.): Graphs and Patterns in Mathematics and Theoretical Physics, Proc. Symp. Pure Math. 73 (2005) [doi:10.1090/pspum/073, spire:1697823, pdf]

For target space the classifying space $B G$ of a compact Lie group $G$ (such as a finite group):

• Kate Gruher: String Topology of Classifying Spaces, PhD thesis, Stanford (2007) [pdf]

• David Chataur, Luc Menichi: String topology of classifying spaces [arXiv:0801.0174, pdf]

• Richard Hepworth, Anssi Lahtinen: On string topology of classifying spaces, Advances in Mathematics

  281 (2015) 394-507 [arXiv:1308.6169, doi:10.1016/j.aim.2015.03.022]

• Katsuhiko Kuribayashi, Luc Menichi: The Batalin–Vilkovisky Algebra in the String Topology of Classifying Spaces, Canadian Journal of Mathematics 71 4 (2019) 843-889 [doi:10.4153/CJM-2018-021-9]

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

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

• Jacob Lurie, ex. 4.2.16, rem. 4.2.17 in: On the Classification of Topological Field Theories (2009)

For the string product and the BV-operator this extension has been known early on, it yields a homotopy BV algebra considered in

• Scott Wilson, around page 101 of: 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, in: Alpine Perspectives on Algebraic Topology, Contemp. Math. 504, Amer. Math. Soc. (2009) 53-56 [arXiv:0906.5198]