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\section*{Gromov-Witten invariants}

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

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory}

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

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{relation_to_tqft}{Relation to TQFT}\dotfill \pageref*{relation_to_tqft} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{expositions}{Expositions}\dotfill \pageref*{expositions} \linebreak
\noindent\hyperlink{via_geometric_quantization}{Via geometric quantization}\dotfill \pageref*{via_geometric_quantization} \linebreak
\noindent\hyperlink{as_a_tcft}{As a TCFT}\dotfill \pageref*{as_a_tcft} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{in_higher_differential_geometry__on_orbifolds}{In higher differential geometry / on orbifolds}\dotfill \pageref*{in_higher_differential_geometry__on_orbifolds} \linebreak
\noindent\hyperlink{in_terms_of_motives}{In terms of motives}\dotfill \pageref*{in_terms_of_motives} \linebreak


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

\ldots{}

\hypertarget{relation_to_tqft}{}\subsection*{{Relation to TQFT}}\label{relation_to_tqft}

Gromov-Witten invariants may be understood (and have originally been found) as arising from a particular [[TQFT]], or actually a [[TCFT]], called the [[A-model]].

For a useful exposition of this see (\hyperlink{Tolland}{Tolland}).

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

\begin{itemize}%
\item [[stable map]]


\item [[orbifold cohomology]]


\item [[quantum sheaf cohomology]]


\item [[Donaldson theory]]


\item [[A-model]]



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

\hypertarget{expositions}{}\subsubsection*{{Expositions}}\label{expositions}

here are some seminar notes:

\begin{itemize}%
\item [[basic ideas of moduli stacks of curves and Gromov-Witten theory]]

\end{itemize}
Some introductory notes:

\begin{itemize}%
\item Simon Rose, \emph{Introduction to Gromov-Witten theory}, \href{http://arxiv.org/abs/1407.1260v1}{arXiv}.

\end{itemize}
And this introductory bit on the [[moduli stack]] of [[elliptic curve]]s:

\begin{itemize}%
\item [[A Survey of Elliptic Cohomology - elliptic curves]].

\end{itemize}
An exposition of GW theory as a [[TCFT]] is at

\begin{itemize}%
\item [[AJ Tolland]], \emph{Gromov-Witten Invariants and Topological Field Theory} (\href{http://sbseminar.wordpress.com/2008/12/11/gromov-witten-invariants-and-topological-field-theory}{blog})

\end{itemize}
The origin of Gromov-Witten theory in and relation to [[string theory]] and other [[physics]] motivation is recalled and surveyed in

\begin{itemize}%
\item Daniel Grunberg, \emph{Gromov-Witten Theory and Threshold Corrections} (\href{http://arxiv.org/abs/hep-th/0605087}{arXiv:hep-th/0605087})

\end{itemize}
\hypertarget{via_geometric_quantization}{}\subsubsection*{{Via geometric quantization}}\label{via_geometric_quantization}

Discussion in the context of [[geometric quantization]] is in

\begin{itemize}%
\item Emily Clader, Nathan Priddis, Mark Shoemaker, \emph{Geometric Quantization with Applications to Gromov-Witten Theory} (\href{http://arxiv.org/abs/1309.1150}{arXiv:1309.1150})

\end{itemize}
\hypertarget{as_a_tcft}{}\subsubsection*{{As a TCFT}}\label{as_a_tcft}

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

\end{itemize}
See also the references at [[A-model]].

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

A discussion by quantization of quadratic Hamiltonians is in

\begin{itemize}%
\item [[Alexander Givental]], \emph{Gromov-Witten invariants and quantization of quadratic Hamiltonians} (\href{http://math.berkeley.edu/~giventh/papers/gwi.pdf}{pdf})


\item M. Kontsevich, Yu. Manin, \emph{Gromov-Witten classes, quantum cohomology, and enumerative geometry}, Comm. Math. Phys. 164 (1994), no. 3, 525--562 (\href{http://projecteuclid.org/euclid.cmp/1104270948}{euclid}).


\item [[Yuri Manin]], \emph{Frobenius manifolds, quantum cohomology and moduli spaces}, Amer. Math. Soc., Providence, RI, 1999,


\item W. Fulton, R. Pandharipande, \emph{Notes on stable maps and quantum cohomology}, in: Algebraic Geometry- Santa uz 1995 ed. Kollar, Lazersfeld, Morrison. Proc. Symp. Pure Math. 62, 45--96 (1997)


\item J Robbin, D A Salamon, \emph{A construction of the Deligne-Mumford orbifold}, J. Eur. Math. Soc. (JEMS) 8 (2006), no. 4, 611--699 (\href{http://arxiv.org/abs/math/0407090}{arxiv}; \href{http://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=8&iss=4&rank=3}{pdf at JEMS}); corrigendum J. Eur. Math. Soc. (JEMS) 9 (2007), no. 4, 901--905 (\href{http://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=9&iss=4&rank=11}{pdf at JEMS}).


\item J Robbin, Y Ruan, D A Salamon, \emph{The moduli space of regular stable maps}, Math. Z. 259 (2008), no. 3, 525--574 (\href{http://dx.doi.org/10.1007/s00209-007-0237-x}{doi}).


\item Martin A. Guest, \emph{From quantum cohomology to integrable systems}, Oxford Graduate Texts in Mathematics, 15. Oxford University Press, Oxford, 2008. xxx+305 pp.


\item Joachim Kock, Israel Vainsencher, \emph{An invitation to quantum cohomology. Kontsevich's formula for rational plane curves}, Progress in Mathematics, 249. Birkh\"a{}user Boston, Inc., Boston, MA, 2007. xiv+159 pp.


\item Dusa McDuff, Dietmar Salamon, \emph{Introduction to symplectic topology}, 2 ed. Oxford Mathematical Monographs 1998. x+486 pp.


\item Sheldon Katz, \emph{Enumerative geometry and string theory}, Student Math. Library \textbf{32}. IAS/Park City AMS \& IAS 2006. xiv+206 pp.


\item Eleny-Nicoleta Ionel, Thomas H. Parker, \emph{Relative Gromov-Witten invariants}, Ann. of Math. (2) 157 (2003), no. 1, 45--96 (\href{http://dx.doi.org/10.4007/annals.2003.157.45}{doi}).


\item [[Edward Frenkel]], [[Constantin Teleman]], [[AJ Tolland]], \emph{Gromov-Witten Gauge Theory I} (\href{http://arxiv.org/abs/0904.4834}{arXiv:0904.4834})


\item [[Constantin Teleman]], \emph{The structure of 2D semi-simple field theories} (\href{http://arxiv.org/abs/0712.0160}{arXiv:0712.0160})


\item Oliver Fabert, \emph{Floer theory, Frobenius manifolds and integrable systems}, (\href{http://arxiv.org/abs/1206.1564}{arxiv/1206.1564})



\end{itemize}
A generalization is discussed in

\begin{itemize}%
\item [[Edward Frenkel]], A. Losev, [[Nikita Nekrasov]], \emph{Instantons beyond topological theory I} (\href{http://arxiv.org/abs/hep-th/0610149}{arXiv:hep-th/0610149})


\item [[Edward Frenkel]], A. Losev, [[Nikita Nekrasov]], \emph{Instantons beyond topological theory II} (\href{http://arxiv.org/abs/0803.3302}{arXiv:hep-th/0610149})



\end{itemize}
Expositions and summaries of this are in

\begin{itemize}%
\item [[Edward Frenkel]], A. Losev, [[Nikita Nekrasov]], \emph{Notes on instantons in topological field theory and beyond} (\href{http://arxiv.org/abs/hep-th/0702137}{arXiv:hep-th/0702137})


\item [[Jacques Distler]], \emph{Localized} (2006) (\href{Localized}{blog})



\end{itemize}
\hypertarget{in_higher_differential_geometry__on_orbifolds}{}\subsubsection*{{In higher differential geometry / on orbifolds}}\label{in_higher_differential_geometry__on_orbifolds}

GW theory of [[orbifolds]] (hence in [[higher differential geometry]]) has been introduced in

\begin{itemize}%
\item Weimin Chen, [[Yongbin Ruan]], \emph{Orbifold Gromov-Witten Theory}, in \emph{Orbifolds in mathematics and physics} (Madison, WI, 2001), 25--85, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002 (\href{http://arxiv.org/abs/math/0103156}{arXiv:math/0103156})

\end{itemize}
A review with further pointers is in

\begin{itemize}%
\item Dan Abramovich, \emph{Lectures on Gromov-Witten invariants of orbifolds} (\href{http://arxiv.org/abs/math/0512372}{arXiv:math/0512372})

\end{itemize}
\hypertarget{in_terms_of_motives}{}\subsubsection*{{In terms of motives}}\label{in_terms_of_motives}

That the [[path integral as a pull-push transform|pull-push quantization]] of Gromov-Witten theory is naturally understood as a ``[[motivic quantization]]'' in terms of [[Chow motives]] of [[Deligne-Mumford stacks]] was suggested in

\begin{itemize}%
\item [[Kai Behrend]], [[Yuri Manin]], \emph{Stacks of Stable Maps and Gromov-Witten Invariants} (\href{http://arxiv.org/abs/alg-geom/9506023}{arXiv:alg-geom/9506023})

\end{itemize}
Further investigation of these stacky Chow motives then appears in

\begin{itemize}%
\item [[Bertrand Toën]], \emph{On motives for Deligne-Mumford stacks}, International Mathematics Research Notices 2000, 17 (2000) 909-928 (\href{http://arxiv.org/abs/math/0006160}{arXiv:math/0006160}, \href{http://hal.archives-ouvertes.fr/hal-00773027}{web}, \href{http://hal.archives-ouvertes.fr/docs/00/77/30/27/PDF/motdm.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item Utsav Choudhury, \emph{Motives of Deligne-Mumford Stacks} (\href{http://arxiv.org/abs/1109.5288}{arXiv:1109.5288})

\end{itemize}
[[!redirects Gromov-Witten invariant]] [[!redirects Gromov-Witten invariants]] [[!redirects Gromov–Witten invariant]] [[!redirects Gromov–Witten invariants]] [[!redirects Gromov--Witten invariant]] [[!redirects Gromov--Witten invariants]]

[[!redirects Gromov-Witten theory]] [[!redirects Gromov–Witten theory]] [[!redirects Gromov--Witten theory]]



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