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\section*{string topology}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{the_string_operations}{The string operations}\dotfill \pageref*{the_string_operations} \linebreak
\noindent\hyperlink{the_string_product}{The string product}\dotfill \pageref*{the_string_product} \linebreak
\noindent\hyperlink{the_bvoperator}{The BV-operator}\dotfill \pageref*{the_bvoperator} \linebreak
\noindent\hyperlink{InTermsOfTQFTs}{String operations as operators in a topological quantum field theory}\dotfill \pageref*{InTermsOfTQFTs} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In \textbf{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 operad|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 \emph{string topology} was initated by [[Moira Chas]] and [[Dennis Sullivan]].

\hypertarget{the_string_operations}{}\subsection*{{The string operations}}\label{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 [[integer]]s $\mathbb{Z}$).

\hypertarget{the_string_product}{}\subsubsection*{{The string product}}\label{the_string_product}

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{string product} is a morphism of [[abelian group]]s

\begin{displaymath}
(-)\cdot(-) : 
  H_\bullet(L X) \otimes H_\bullet(L X)
  \to 
  H_{\bullet - dim X}(L X)
  \,,
\end{displaymath}
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 [[transversal map|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 \emph{string product} is then defined using such representatives by

\begin{displaymath}
[\alpha] \cdot [\beta] := [\alpha \cdot \beta]
  \,.
\end{displaymath}
\end{defn}
\begin{theorem}
\label{}\hypertarget{}{}
The string product is [[associativity|associative]] and graded-commutative.

\end{theorem}
This is due to (\hyperlink{ChasSullivan}{ChasSullivan}). There is is a more elegant way to capture this, due to (\hyperlink{CohenJones}{CohenJones}):

Let

\begin{displaymath}
S^1 \coprod S^1 \to 8 \leftarrow S^1
\end{displaymath}
be the [[cospan]] that exhibts the inner and the outer circle of the figure ``8'' topological space. By forming [[hom space]]s this induces the [[span]]

\begin{displaymath}
\itexarray{
      && X^8
      \\
      & {}^{\mathllap{in}}\swarrow && \searrow^{\mathrlap{out}}
     \\
     L X \times L X &&&& L X
  }
  \,.
\end{displaymath}
Write $in^!$ for the ``pullback'' in [[ordinary homology]] along $in$ (the dual [[fiber integration]]) and $out_*$ for the ordinary pushforward.

\begin{theorem}
\label{}\hypertarget{}{}
The string product is the pull-push operation

\begin{displaymath}
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)
  \,.
\end{displaymath}
\end{theorem}
This is due to (\hyperlink{CohenJones}{CohenJones}).

\hypertarget{the_bvoperator}{}\subsubsection*{{The BV-operator}}\label{the_bvoperator}

\begin{defn}
\label{}\hypertarget{}{}
Define a morphism of [[abelian group]]s

\begin{displaymath}
\Delta : H_\bullet(L X) \to H_{\bullet + 1}(L X)
\end{displaymath}
as follows. Consider first the rotation map

\begin{displaymath}
\rho : S^1 \times L X \to L X
\end{displaymath}
that sends $(\theta, \gamma) \mapsto (t \mapsto \gamma(\theta + t))$. Then take

\begin{displaymath}
\Delta : a \mapsto \rho_* ([S^1] \times a)
  \,,
\end{displaymath}
where $[S^1] \in H_1(S^1)$ is the [[fundamental class]] of the [[circle]].

\end{defn}
This is called the \textbf{BV-operator} for string topology.

\begin{prop}
\label{}\hypertarget{}{}
The [[Goldman bracket]] on $H_0(L X)$ is equivalent to the string product applied to the image of the BV-operator

\begin{displaymath}
\{[\gamma_1], [\gamma_2]\}
  = 
  \Delta[\Gamma_1] \cdot \Delta[\Gamma_2]
  \,.
\end{displaymath}
\end{prop}
This is due to (\hyperlink{ChasSullivan}{ChasSullivan}).

\hypertarget{InTermsOfTQFTs}{}\subsection*{{String operations as operators in a topological quantum field theory}}\label{InTermsOfTQFTs}

The structures studied in the \emph{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 [[state]]s 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]]

\begin{displaymath}
H_\bullet(Bord_2) \to Ch_\bullet
\end{displaymath}
given by the [[FQFT]] corresponding to the $\sigma$-modelon these state spaces, acting on these state spaces.

$\,,$

Let $X$ be an [[orientation|oriented]] [[compact space|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 $B \subset X$.

\begin{theorem}
\label{}\hypertarget{}{}
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 \emph{positive boundary} and set of [[branes]] $\mathcal{B}$, such that the correlators in the closed sector are the standard string topology operation.

\end{theorem}
For [[closed strings]] this is discussed in (\hyperlink{CohenGodin03}{Cohen-Godin 03}, \hyperlink{Tamanoi07}{Tamanoi 07}). For [[open strings]] on a single [[brane]] $\mathcal{B} = \{*\}$ this was shown in (\hyperlink{Godin}{Godin 07}), where the general statement for arbitrary branes is conjectured. A detailed proof of this general statement is in (\hyperlink{Kupers}{Kupers 11}).

\begin{remark}
\label{}\hypertarget{}{}
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 generalized cohomology|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 \emph{[[path integral as a pull-push transform]]/[[motivic quantization]]}.

\end{remark}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Goldman bracket]]


\item [[path integral as a pull-push transform]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The original references include the following:

\begin{itemize}%
\item [[Moira Chas]], [[Dennis Sullivan]], \emph{String topology}, Ann. Math. \href{http://arxiv.org/abs/math/9911159}{math.GT/9911159}

\end{itemize}
\begin{itemize}%
\item [[Ralph Cohen]], [[Alexander Voronov]], \emph{Notes on string topology}, \href{http://arxiv.org/abs/math/0503625}{math.GT/05036259}, 95 pp. published as a part of R. Cohen, [[Kathryn Hess|K. Hess]], A. Voronov, \emph{String topology and cyclic homology}, CRM Barcelona courseware, Springer, \href{http://www.springer.com/birkhauser/mathematics/book/978-3-7643-2182-6}{description}, \href{http://dx.doi.org/10.1007/3-7643-7388-1}{doi}, \href{http://gen.lib.rus.ec/get?md5=adde9464705ede0fea6b435edb58fbe7}{pdf}


\item [[Dennis Sullivan]], \emph{Open and closed string field theory interpreted in classical algebraic topology}, Topology, geometry, and quantum field theory, 344--357. London Math. Soc. Lec. Notes \textbf{308}, Cambridge Univ. Press. 2004.


\item [[Ralph Cohen]], John R. Klein, Dennis Sullivan, \emph{The homotopy invariance of the string topology loop product and string bracket}, J. of Topology 2008 \textbf{1}(2):391-408; \href{http://dx.doi.org/10.1112/jtopol/jtn001}{doi}


\item [[Ralph Cohen]], \emph{Homotopy and geometric perspectives on string topology}, \href{http://math.stanford.edu/~ralph/skyesummary.pdf}{pdf}



\end{itemize}
In

\begin{itemize}%
\item [[Ralph Cohen]] and J.D.S. Jones, \emph{A homotopy theoretic realization of string topology} , Math. Ann. 324 (2002), no. 4, (\href{http://arxiv.org/abs/math/0107187}{arXiv:0107187})

\end{itemize}
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

\begin{itemize}%
\item [[Ralph Cohen]], [[Veronique Godin]], \emph{[[A Polarized View of String Topology]]} (\href{http://arxiv.org/abs/math/0303003}{arXiv:math/0303003})

\end{itemize}
\begin{itemize}%
\item Hirotaka Tamanoi, \emph{Loop coproducts in string topology and triviality of higher genus TQFT operations} (2007) (\href{http://arxiv.org/abs/0706.1276}{arXiv})

\end{itemize}
A detailed discussion and generalization to the open-closed [[HQFT]] in the presence of a single space-filling [[brane]] is in

\begin{itemize}%
\item [[Veronique Godin]], \emph{Higher string topology operations} (2007)(\href{http://arxiv.org/abs/0711.4859}{arXiv:0711.4859})

\end{itemize}
The generalization to multiple [[D-brane]]s is discussed in

\begin{itemize}%
\item [[Sander Kupers]], \emph{String topology operations} MS thesis (2011) (\href{http://math.stanford.edu/~kupers/thesis7thjune2011.pdf}{pdf})

\end{itemize}
For target space a [[classifying space]] of a [[finite group]] or [[compact space|compact]] [[Lie group]] this is discussed in

\begin{itemize}%
\item David Chataur, [[Luc Menichi]], \emph{String topology of classifying spaces} (\href{http://math.univ-angers.fr/perso/lmenichi/String_Classifiant09.pdf}{pdf})

\end{itemize}
Arguments that this string-topology [[HQFT]] should refine to a chain-level theory -- a [[TCFT]] -- were given in

\begin{itemize}%
\item [[Kevin Costello]], \emph{Topological conformal field theories and Calabi-Yau $A_\infty$-categories} (2004) , (\href{http://arxiv.org/abs/0412149}{arXiv:0412149})

\end{itemize}
and

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[On the Classification of Topological Field Theories]]}

\end{itemize}
(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

\begin{itemize}%
\item [[Scott Wilson]], \emph{On the Algebra and Geometry of a Manifold's Chains and Cochains} (2005) (\href{http://qcpages.qc.cuny.edu/~swilson/main.pdf}{pdf})

\end{itemize}
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

\begin{itemize}%
\item [[Andrew Blumberg]], [[Ralph Cohen]], [[Constantin Teleman]], \emph{Open-closed field theories, string topology, and Hochschild homology} (\href{http://arxiv.org/abs/0906.5198}{arXiv:0906.5198})

\end{itemize}
Refinements of string topology from [[homology groups]] to the full [[ordinary homology]]-[[spectra]] is discussed in (\hyperlink{BlumbergCoheneTeleman09}{Blumberg-Cohen-Teleman 09}) and in

\begin{itemize}%
\item [[Ralph Cohen]], [[John Jones]], \emph{A homotopy theoretic realization of string topology}, Mathematische Annalen (\href{http://arxiv.org/abs/math/0107187}{arXiv:math/0107187})


\item [[Ralph Cohen]], [[John Jones]], \emph{Gauge theory and string topology} (\href{http://arxiv.org/abs/1304.0613}{arXiv:1304.0613})



\end{itemize}
A generalization of string topology with target manifolds generalized to [[orbifolds]] is discussed in

\begin{itemize}%
\item Alejandro Adem, Johanna Leida, Yongbin Ruan, \emph{Orbifolds and string topology}, Cambridge Tracts in Mathematics 171, 2007 (\href{http://www.math.colostate.edu/~renzo/teaching/Orbifolds/Ruan.pdf}{pdf})

\end{itemize}
Further generalization to target spaces that are more generally [[differentiable stacks]]/[[Lie groupoids]] is discussed in

\begin{itemize}%
\item [[Kai Behrend]], [[Gregory Ginot]], [[Behrang Noohi]], [[Ping Xu]], \emph{String topology for stacks}, (89 pages) \href{http://arxiv.org/abs/0712.3857}{arxiv/0712.3857}; \emph{String topology for loop stacks}, C. R. Math. Acad. Sci. Paris, \textbf{344} (2007), no. 4, 247--252, (6 pages, \href{}{pdf})


\item Po Hu, \emph{Higher string topology on general spaces}, Proc. London Math. Soc. \textbf{93} (2006) 515-544, \href{http://dx.doi.org/10.1112/S0024611506015838}{doi}, \href{http://www.math.wayne.edu/~po/koszul04.ps}{ps}



\end{itemize}
The relation between string topology and [[Hochschild cohomology]] in dimenion $\gt 1$ is discussed in

\begin{itemize}%
\item [[Dmitry Vaintrob]], \emph{The String topology BV algebra, Hochschild cohomology and the Goldman bracket on surfces} (\href{http://arxiv.org/abs/math/0702859}{arXiv:0702859})

\end{itemize}
More developments are in

\begin{itemize}%
\item Eric Malm, \emph{String topology and the based loop space}, 2009 (\href{http://arxiv.org/abs/1103.6198}{arXiv:1103.6198}, \href{http://math.ucr.edu/~jbergner/ucr-st-present.pdf}{slides})

\end{itemize}


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