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\section*{TCFT}

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

\hypertarget{functorial_quantum_field_theory}{}\paragraph*{{Functorial quantum field theory}}\label{functorial_quantum_field_theory}

[[!include functorial quantum field theory - contents]]

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{Classification}{Classification}\dotfill \pageref*{Classification} \linebreak
\noindent\hyperlink{with_coefficients_in_algebras_in_chain_complexes}{With coefficients in (algebras in) chain complexes}\dotfill \pageref*{with_coefficients_in_algebras_in_chain_complexes} \linebreak
\noindent\hyperlink{general_version}{General version}\dotfill \pageref*{general_version} \linebreak
\noindent\hyperlink{ActionFunctionals}{Worldsheet and effective background theories}\dotfill \pageref*{ActionFunctionals} \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}

The term \emph{topological conformal field theory} (TCFT) is used for a linearization or [[stabilization]] of something that is like a [[conformal field theory]] (CFT) up to [[homotopy]]. It is a notion somewhere half-way between a (2-dimensional) [[TQFT]] and a [[CFT]].

\begin{quote}%
(Actually, the remnant of conformal structure here should be just an artefact of the way to parameterize the moduli space of surfaces. As the classification result by Lurie discussed below shows, TCFTs are really $(\infty,2)$-TFTs.)


\end{quote}
This formalizes the physics notion of ``the [[topological string]]'', a \emph{topologically twisted} superconformal field theory, such as, notably, the [[A-model]] and the [[B-model]]. TCFTs are therefore a tool for formalizing [[homological mirror symmetry]].

Recall that an ordinary [[conformal field theory]] (CFT) is, in [[FQFT]]-language, a [[symmetric monoidal functor]] on a [[category]] $Bord_2^{conf}$ whose objects are disjoint unions of intervals and circles, and whose [[morphism]]s are [[Riemann surface]]s with these 1d manifolds as incoming and outgoing punctures.

Since Riemann surfaces form a well-understood [[moduli space]], one can turn this also into a [[Top]]-[[enriched category]], i.e. an [[(∞,1)-category]], $Bord_{2}^{conf,top}$ whose [[hom-space]]s are these [[moduli space]]s of [[Riemann surface]]s with given 1d manifolds as incoming and outgoing punctures.

A ``truly topological conformal field theory'' would be an [[(∞,1)-functor]] of the form

\begin{displaymath}
Bord_2^{conf,top} \to \infty Grpd
\end{displaymath}
or similar. But what is actually called a ``topological conformal field theory'' is the linearization or [[stabilization]] of this:

in a TCFT, this [[(∞,1)-category]] of conformal [[cobordism]]s is replaced by a [[stable (∞,1)-category]] whose [[hom-object]]s (when modeled by a [[dg-category]]) are just the [[homology]] [[chain complex]]s of the original [[hom-space]]s.

Write $Bord_2^{conf,dg}$ for the resulting [[symmetric monoidal category|symmetric monoidal]] [[dg-category]] of Riemann cobordisms. Then a TCFT is a an homotopy-symmetric monoidal [[chain complex]]-[[enriched functor]]

\begin{displaymath}
F : Bord_2^{conf,dg} \to Ch_\bullet
\end{displaymath}
to the symmetric monoidal [[dg-category]] of chain complexes.

This means in particular that when two [[Riemann surface]]s $\Sigma_1$ and $\Sigma_2$ are homologous as chains in the [[moduli space]] of Riemann surfaces, then the TCFT will send them to two equivalent morphisms $f_{\Sigma_1}$ and $f_{\Sigma_2}$ of chain complexes between the in- and the output states. The equivalence between $f_{\Sigma_1}$ and $f_{\Sigma_2}$, however, is not unique neither up to equivalence. Rather, it funtorially depends on the 1-chain realizing the homology equivalence between $\Sigma_1$ and $\Sigma_2$ as 0-chains in the moduli space. In particular, two non-homologous 1-chains between $\Sigma_1$ and $\Sigma_2$ will in general lead to non-equivalent equivalences between $f_{\Sigma_1}$ and $f_{\Sigma_2}$.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

According to [[On the Classification of Topological Field Theories|ClassTFT]] the original definition of the domain for TCFTs can be formulated as follows (without reference to any conformal or Riemann structure).

\textbf{Definition} The $(\infty,2)$-category $Bord^{nc}_2$ of non-compact 2-dimensional cobordism is defined as follows:

\begin{itemize}%
\item The objects of $Bord^{nc}_2$ are oriented 0-manifolds.


\item Given a pair of objects $X, Y \in Bord^{nc}_2$ , a 1-morphism from $X$ to $Y$ is an oriented bordism $B : X \to Y$.


\item Given a pair of 1-morphsims $B,B' : X \to Y$ in $Bord^{nc}_2$, a 2-morphism from $B$ to $B'$ in $Bord^{nc}_2$ is an oriented bordism $\Sigma: B \to B'$ (which is trivial along $X$ and $Y$) with the following property: every connected component of $\Sigma$ has nonempty intersection with $B'$.


\item Higher morphisms in $Bord^{nc}_2$ are given by (orientation preserving) diffeomorphisms, isotopies between diffeomorphisms, and so forth.



\end{itemize}
Then, the [[cobordism hypothesis]]-theorem for $Bord^{nc}_2$ becomes

\begin{theorem}
\label{}\hypertarget{}{}
Let $C$ be a [[symmetric monoidal (∞,2)-category]]. Then symmetric monoidal [[(∞,2)-functor]]s

\begin{displaymath}
Z : Bord^{nc}_2 \to C
\end{displaymath}
are equivalent to [[Calabi-Yau objects]] $A$ in $C$: the functor $Z$ sends the point to $A$.

\end{theorem}
This is [[On the Classification of Topological Field Theories|ClassTFT, theorem 4.2.11]]. One can ``unfold'' $Bord^{nc}_2$ and the theorem above, obtaining a statement in terms of [[symmetric monoidal (∞,1)-category|symmetric monoidal (∞,1)-categories]]. Actually it was the unfolded version to be proven first, (\hyperlink{Costello04}{Costello 04}).

in the particular case $C=Ch_\bullet$. We state it below in the general version given by [[Jacob Lurie]] in [[On the Classification of Topological Field Theories|ClassTFT]].

\begin{defn}
\label{}\hypertarget{}{}
Let $\mathcal{OC}$ be the [[(infinity,1)-category]] of open-closed strings, described as follows:

\begin{enumerate}%
\item objects are oriented 1-manifolds with boundary;


\item morphisms are oriented bordisms between 1-manifolds such that each connected component has non-vanishing intersection with the codomain 1-manifold;


\item the higher morphisms are given by orientation preserving [[diffeomorphism]]s, isotopies between these, and so forth.



\end{enumerate}
Write $\mathcal{O}$ for the full [[sub-(∞,1)-category]] on disjoint unions of intervals (open strings sector).

\end{defn}
This is [[On the Classification of Topological Field Theories|ClassTFT, above theorem 4.2.13]].

\hypertarget{Classification}{}\subsection*{{Classification}}\label{Classification}

\hypertarget{with_coefficients_in_algebras_in_chain_complexes}{}\subsubsection*{{With coefficients in (algebras in) chain complexes}}\label{with_coefficients_in_algebras_in_chain_complexes}

The original statement of the classification result for TCFTs concerned symmetric homotopy-monoidal functors $Bord_2^{conf,dg} \to Ch_\bullet$:

\begin{defn}
\label{}\hypertarget{}{}
([[Kevin Costello|Costello]], following [[Maxim Kontsevich|Kontsevich]])

\begin{enumerate}%
\item The category of open TCFTs with set $\Lambda$ of D-branes is equivalent to that of [[Calabi-Yau categories]] with set $\Lambda$ of objects.


\item The homology of the chain complex of closed states of the universal extension of an open TCFT to an open-closed TCFT is the [[Hochschild homology]] of the corresponding [[Calabi-Yau category]].



\end{enumerate}
\end{defn}
In (\hyperlink{Costello04}{Costello 04}) this is proven using information about cell decompositions of the moduli space of punctured Riemann surfaces, thus effectively presenting $Bord_2^{conf,dg}$ by generators-and-relations, The then theorem amounts to noticing that representations of these generators and relations define the operations in an $A_\infty$-category with pairing operation.

\hypertarget{general_version}{}\subsubsection*{{General version}}\label{general_version}

\begin{theorem}
\label{}\hypertarget{}{}
Let $C$ be a [[symmetric monoidal (∞,1)-category]]. Then symmetric monoidal [[(∞,1)-functor]]s

\begin{displaymath}
Z : \mathcal{O} \to C
\end{displaymath}
are equivalent to [[Calabi-Yau algebra]] [[Calabi-Yau object|objects]] $A$ in $C$: the functor $Z$ sends the interval $[0,1]$ to $A$.

\end{theorem}
This is the result of spring \href{http://arxiv.org/abs/math/0412149}{Cos04} reformulated and generalized according to [[On the Classification of Topological Field Theories|ClassTFT, theorem 4.2.14]].

This is a special case of the general [[cobordism hypothesis]]-theorem.

The idea of the proof is that a topological open string theory, i.e., a symmetric monoidal [[(∞,1)-functor]] $Z : \mathcal{O} \to C$ has a [[Kan extension]] to an open-closed topological string theory, i.e., to a symmetric monoidal [[(∞,1)-functor]] $Z : \mathcal{OC} \to C$, which is the unfolded version of a symmetric monoidal [[(∞,2)-functor]] from $Bord^{nc}_2$ to a symmetric monoidal $(\infty,2)$-category $C'$.

\hypertarget{ActionFunctionals}{}\subsection*{{Worldsheet and effective background theories}}\label{ActionFunctionals}

One imagines generally that one obtains TCFTs, in their formal definition given above, from worldsheet [[action functional]]s as familiar from the physics literature (such as on the [[A-model]] and the [[B-model]]) by performing the [[path integral]] and finding from it a collection of [[differential form]]s on [[moduli space]] of bosonic field configurations.

It seems there is at this point no literature giving a direct construction along these lines, but there is the following:

In \href{http://arxiv.org/abs/math/0605647}{Cos06} is constructed from the geometric input datum of a generalized [[Calabi-Yau space]] $(X,Q)$ and it is shown that

\begin{enumerate}%
\item there is a collection of differential forms $K_{g,h}(\cdots)$on the [[moduli space]] $\mathcal{M}_{g}^{h,n}$ of [[Riemann surface]]s such that these define a 2d TCFT;

(In the discussion leading up to Lemma 4.5.1 there. The proof that this yields a TCFT is theorem 4.5.4.)


\item the partition function of the string perturbation series for the above TCFT is

\begin{displaymath}
\sum_{{g,n \geq 0, h \gt 0} \atop {2g-2+h+\frac{n}{2}}}
  \lambda^{2g-2+h}N^h
  \frac{1}{n!}
  \int_{\mathcal{M}_g^{h,n}}
  K_{g,h}(a^\otimes n)
\end{displaymath}
which is shown to be the partition function of a background [[Chern-Simons theory]] coming from the [[action functional]]

\begin{displaymath}
a \mapsto S(a) = \int_X \frac{1}{2} a Q a + \frac{1}{3}a^3
  \,.
\end{displaymath}


\end{enumerate}
So this constructs a 2d TCFT and shows that its [[effective field theory|effective]] background quantum field theory is a [[Chern-Simons theory]]. While the action functionl on the worldsheet itself, whose [[path integral]] should give the [[differential forms]] on [[moduli space]] considered above, is not explicitly considered here, this does formalizes at least some aspects of an observation that was earlier made in (\hyperlink{Witten92}{Witten 92}) where it was observed that Chern-Simons theory is the effective background string theory of 2d TFTs obtained from action functionals of the A-model and the B-model.

Similarly the effective background QFT of the [[B-model]] topological string can be identified. This is known as \emph{[[Kodeira-Spencer gravity]]} or as \emph{[[BCOV theory]]}.

(See also at \emph{[[world sheets for world sheets]]} for a similar mechanism and see at \emph{[[super 1-brane in 3d]]} for related ``physical'' strings.)

So via the detour over the effective background field theory, this sort of shows that the physicist's A-model and B-model are indeed captured by the abstract [[FQFT]] definition of TCFT as given above.

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

\begin{itemize}%
\item [[A-model]], [[B-model]], [[Landau-Ginzburg model]], [[homological mirror symmetry]]


\item [[HCFT]]



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

The concept is essentially a formalization of what used to be called [[cohomological field theory]] in

\begin{itemize}%
\item [[Edward Witten]], \emph{Introduction to cohomological field theory}, InternationalJournal of Modern Physics A, Vol. 6,No 6 (1991) 2775-2792 ([[WittenCQFT.pdf:file]])

\end{itemize}
The definition was given independently by

\begin{itemize}%
\item [[Ezra Getzler]], \emph{Batalin-Vilkovisky algebras and two-dimensional topological field theories} , Comm. Math. Phys. 159(2), 265--285 (1994) (\href{http://arxiv.org/abs/hep-th/9212043}{arXiv:hep-th/9212043})

\end{itemize}
and

\begin{itemize}%
\item [[Graeme Segal]], \emph{Topological field theory} , (1999), Notes of lectures at Stanford university. (\href{http://www.cgtp.duke.edu/ITP99/segal/}{web}). See in particular \href{http://www.cgtp.duke.edu/ITP99/segal/stanford/lect5.pdf}{lecture 5} (``topological field theory with cochain values'').

\end{itemize}
The classification of TCFTs by [[Calabi-Yau categories]] was discussed in

\begin{itemize}%
\item [[Kevin Costello]], \emph{Topological conformal field theories and Calabi-Yau categories} Advances in Mathematics, Volume 210, Issue 1, (2007), (\href{http://arxiv.org/abs/math/0412149}{arXiv:math/0412149})


\item [[Kevin Costello]], \emph{The Gromov-Witten potential associated to a TCFT} (\href{http://arxiv.org/abs/math/0509264}{arXiv:math/0509264})



\end{itemize}
following conjectures by [[Maxim Kontsevich]], e.g.

\begin{itemize}%
\item [[Maxim Kontsevich]], \emph{Homological algebra of mirror symmetry} , in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z\"u{}rich, 1994), pages 120--139, Basel, 1995, Birkh\"a{}user.

\end{itemize}
This classification is a precursor of the full [[cobordism hypothesis]]-theorem. This, and the reformulation of the original TCFT constructions in full generality is in

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

\end{itemize}
Here are notes from a seminar on these definitions and results:

\begin{itemize}%
\item [[Peter Teichner]] and [[Kevin Costello]] \emph{TCFT seminar} (\href{http://math.berkeley.edu/~cpries/Hot-Topics-07.pdf}{pdf notes})

\end{itemize}
Discussion of the construction of TCFTs from differential forms on moduli space and the way this induces by ``[[second quantization]]'' effective background Chern-Simons theories is in

\begin{itemize}%
\item [[Kevin Costello]], \emph{Topological conformal field theories and gauge theories} (\href{http://arxiv.org/abs/math/0605647}{arXiv:math/0605647})

\end{itemize}
formalizing at least aspects of the observations in

\begin{itemize}%
\item [[Edward Witten]], \emph{Chern-Simons Gauge Theory As A String Theory} (\href{http://arxiv.org/abs/hep-th/9207094}{arXiv:hep-th/9207094})


\item P.A. Grassi, [[Giuseppe Policastro]], \emph{Super-Chern-Simons Theory as Superstring Theory} (\href{http://arxiv.org/abs/hep-th/0412272}{arXiv:hep-th/0412272})



\end{itemize}
Discussion of how the [[second quantization]] of the [[B-model]] yields [[Kodeira-Spencer gravity]]/[[BCOV theory]] is in

\begin{itemize}%
\item M. Bershadsky, S. Cecotti, [[Hirosi Ooguri]], [[Cumrun Vafa]], \emph{Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes}, Commun.Math.Phys.165:311-428,1994 (\href{http://arxiv.org/abs/hep-th/9309140}{arXiv:hep-th/9309140})


\item [[Kevin Costello]], Si Li, \emph{Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model} (\href{http://arxiv.org/abs/1201.4501}{arXiv:1201.4501})


\item Si Li, \emph{BCOV theory on the elliptic curve and higher genus mirror symmetry} (\href{http://arxiv.org/abs/1112.4063}{arXiv:1112.4063})


\item Si Li, \emph{Variation of Hodge structures, Frobenius manifolds and Gauge theory} (\href{http://arxiv.org/abs/1303.2782}{arXiv:1303.2782})



\end{itemize}
[[!redirects TCFTs]]

[[!redirects topological conformal field theory]] [[!redirects topological conformal field theories]]



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