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\section*{mirror symmetry}

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

\hypertarget{duality_in_string_theory}{}\paragraph*{{Duality in string theory}}\label{duality_in_string_theory}

[[!include duality in string 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_tropical_geometry}{Relation to tropical geometry.}\dotfill \pageref*{relation_to_tropical_geometry} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{CompleteProofs}{Complete proofs}\dotfill \pageref*{CompleteProofs} \linebreak
\noindent\hyperlink{computation_via_topological_recursion}{Computation via topological recursion}\dotfill \pageref*{computation_via_topological_recursion} \linebreak


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

To every complex 3-dimensional [[Calabi-Yau variety]] $X$ are associated two similar but differing types of [[sigma-model]] $N=2$-supersymmetric [[2d CFTs]]. There is at least for some CY $X$ a map $X \mapsto \hat X$ which exchanges the [[Hodge numbers]] $h^{1,1}$ and $h^{1,2}$ such that $SCFT_A(X)$ is expected to be equivalent to $SCFT_B(\hat X)$.

\begin{displaymath}
SCFT_A(X) \simeq SCFT_B(\hat X)
  \,.
\end{displaymath}
This is called \emph{mirror symmetry}. At least in some cases this can be understood as a special case of [[T-duality]] (\hyperlink{StromingerYauZaslow96}{Strominger-Yau-Zaslow 96}).

In this form mirror symmetry remains a conjecture, not the least because for the moment there is no complete construction of these SCFTs. But to every such $SCFT(X)$ one can associate two [[TCFT]]s, $A(X)$ and $B(X)$, the [[A-model]] and the [[B-model]]. These $N=1$ supersymmetric field theories were obtained by [[Edward Witten]] using a ``topological twist''. The topological A-model can be expressed in terms of [[symplectic geometry]] of a variety and the topological B-model can be expressed in terms of the [[algebraic geometry]] of a variety.

These topological theories are easier to understand and do retain a little bit of the information encoded in the full SCFTs. In terms of these the statement of mirror symmetry says that passing to mirror CYs \emph{exchanges} the A-model with the $B$-model and conversely:

\begin{displaymath}
A(X) \simeq B(\hat X),\,\,\,\,\,\,\,B(X)\simeq A(\hat X)
  \,.
\end{displaymath}
By a version of the [[cobordism hypothesis]]-theorem, these [[TCFT]]s (see there) are encoded by [[A-∞ categories]] that are [[Calabi-Yau categories]]: the [[A-model]] by the [[Fukaya category]] $Fuk(X)$ of $X$ which can be understood as a [[stable (∞,1)-category]] representing the Lagrangian intersection theory on the underlying [[symplectic manifold]]; and the [[B-model]] by an [[enhanced triangulated category|enhancement]] of the [[derived category]] of [[coherent sheaves]] $D^b_\infty(\hat X)$ on $\hat X$.

In terms of this data, mirror symmetry is the assertion that these [[A-∞ categories]] are equivalent and simultaneously the same under exchange $X\leftrightarrow \hat{X}$:

\begin{displaymath}
Fuk(X) \simeq D^b_\infty(\hat{X}),
\,\,\,\, and \,\,\,\,
Fuk(\hat{X}) \simeq D^b_\infty(X).
\end{displaymath}
This categorical formulation was introduced by [[Maxim Kontsevich]] in 1994 under the name \textbf{homological mirror symmetry}. The equivalence of the categorical expression of mirror symmetry to the SCFT formulation has been proven by [[Maxim Kontsevich]] and independently by [[Kevin Costello]], who showed how the datum of a topological conformal field theory is equivalent to the datum of a [[Calabi-Yau category|Calabi-Yau A-∞-category]](see [[TCFT]]).

The mirror symmetry conjecture roughly claims that every Calabi-Yau 3-fold has a mirror. In fact one considers (mirror symmetry for) degenerating families for Calabi-Yau 3-folds in large volume limit (which may be expressed precisely via the Gromov-Hausdorff metric). The appropriate definition of (an appropriate version of) the [[Fukaya category]] of a symplectic manifold is difficult to achieve in desired generality. Invariants/tools of Fukaya category include symplectic [[Floer homology]] and Gromov-Witten invariants (building up the [[quantum cohomology]]).

Mirror symmetry is related to the [[T-duality]] on each fiber of an associated Lagrangian fibration \hyperlink{StromingerYauZaslow96}{Strominger-Yau-Zaslow 96}.

Although the non-Calabi-Yau case may be of lesser interest to physics, one can still formulate some mirror symmetry statements for, for instance, Fano manifolds. The mirror to a Fano manifold is a Landau-Ginzburg model (see \hyperlink{HoriVafa00}{Hori-Vafa 00}; see also work of Auroux for an explanation via the Strominger-Yau-Zaslow T-duality philosophy). Then the statements are: the A-model of the Fano (given by the Fukaya category) is equivalent to the B-model of the Landau-Ginzburg model (given by the category of matrix factorizations); and the B-model of the Fano (given by the derived category of sheaves) is equivalent to the A-model of the Landau-Ginzburg model (given by the Fukaya-Seidel category). A few of the relevant names: Kontsevich, Hori-Vafa, Auroux, Katzarkov, Orlov, Seidel, \ldots{}

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_tropical_geometry}{}\subsubsection*{{Relation to tropical geometry.}}\label{relation_to_tropical_geometry}

Close relation to [[tropical geometry]], see e.g. \hyperlink{Gross11}{Gross 11}.

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

\begin{itemize}%
\item [[moduli space of Calabi-Yau spaces]]


\item [[3d mirror symmetry]]


\item [[duality in physics]], [[duality in string theory]]


\item [[geometric Langlands duality]]



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

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

The original statement of the homological mirror symmetry conjecture is in

\begin{itemize}%
\item [[Maxim Kontsevich]], \emph{Homological algebra of mirror symmetry}, Proc. ICM Z\"u{}rich 1994, \href{http://arxiv.org/abs/alg-geom/9411018}{alg-geom/9411018}

\end{itemize}
See also

\begin{itemize}%
\item [[Paul Aspinwall]], [[Brian Greene]], [[David Morrison]], \emph{Calabi-Yau Moduli Space, Mirror Manifolds and Spacetime Topology Change in String Theory}, Nucl.Phys. B416 (1994) 414-480 (\href{http://arxiv.org/abs/hep-th/9309097}{arXiv:hep-th/9309097})

\end{itemize}
Review contains

\begin{itemize}%
\item [[Kentaro Hori]], [[Cumrun Vafa]], \emph{Mirror Symmetry} (\href{https://arxiv.org/abs/hep-th/0002222}{arXiv:hep-th/0002222})


\item M. Ballard, \emph{Meet homological mirror symmetry} (\href{http://arxiv.org/abs/0801.2014}{arxiv:0801.2014})


\item A. Port, \emph{An Introduction to Homological Mirror Symmetry and the Case of Elliptic Curves} (\href{http://arxiv.org/abs/1501.00730}{arXiv:1501.00730})


\item [[Luis Ibáñez]], [[Angel Uranga]], section 10.1.2 of \emph{[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]}, Cambridge University Press 2012



\end{itemize}
Discussion amplifying the role of [[category theory]], and [[higher geometry]] is in

\begin{itemize}%
\item [[Eric Sharpe]], \emph{Categorical Equivalence and the Renormalization Group}, Proceedings of LMS/EPSRC Symposium \emph{\href{http://www.maths.dur.ac.uk/lms/109/index.html}{Higher Structures in M-Theory}}, Fortschritte der Physik 2019 (\href{https://arxiv.org/abs/1903.02880}{arXiv:1903.02880})

\end{itemize}
Other reviews include

\begin{itemize}%
\item [[Paul Aspinwall]], [[Tom Bridgeland]], [[Alastair Craw]], [[Michael Douglas]], Mark Gross, [[Anton Kapustin]], [[Gregory Moore]], [[Graeme Segal]], [[Balázs Szendrői]], P. Wilson,

\emph{Dirichlet branes and mirror symmetry},

Clay Mathematics Monograph Volume 4, Amer. Math. Soc. Clay Math. Institute 2009

(\href{http://www.claymath.org/library/monographs/cmim04c.pdf}{pdf}, \href{http://www2.maths.ox.ac.uk/cmi/library/monographs/cmim04.pdf}{pdf}) (very readable!)



\end{itemize}
The relation to [[T-duality]] was established in

\begin{itemize}%
\item [[Andrew Strominger]], [[Shing-Tung Yau]], [[Eric Zaslow]], \emph{Mirror Symmetry is T-Duality}, Nucl.Phys.B479:243-259,1996 (DOI 10.1016/0550-3213(96)00434-8) \href{http://arxiv.org/abs/hep-th/9606040}{hep-th/9606040}

\end{itemize}
Further references include

\begin{itemize}%
\item [[Cumrun Vafa]], [[Shing-Tung Yau]] (eds.), \emph{Winter school on mirror symmetry, vector bundles, and Lagrangian submanifolds}, Harvard 1999, AMS, Intern. Press (includes A. Strominger, S-T. Yau, E. Zaslow, \emph{Mirror symmetry is $T$-duality} as pages 333--347; ).


\item K. Hori, S. Katz, A. Klemm et al. \emph{Mirror symmetry I}, AMS, Clay Math. Institute 2003.


\item Paul Seidel, \emph{Fukaya categories and Picard-Lefschetz theory}, Zurich Lectures in Advanced Mathematics. European Mathematical Society, Z\"u{}rich, 2008. viii+326 pp


\item Mark Gross, Bernd Siebert, \emph{Mirror symmetry via logarithmic degeneration data I}, \href{http://arxiv.org/abs/math/0309070}{math.AG/0309070}, \emph{From real affine geometry to complex geometry}, \href{http://arxiv.org/abs/math/0703822}{math.AG/0703822}, \emph{Mirror symmetry via logarithmic degeneration data II}, \href{http://arxiv.org/abs/0709.2290}{arxiv/0709.2290}


\item [[Anton Kapustin]], [[Dmitri Orlov]], \emph{Lectures on mirror symmetry, derived categories, and D-branes}, Uspehi Mat. Nauk \textbf{59} (2004), no. 5(359), 101--134; translation in Russian Math. Surveys \textbf{59} (2004), no. 5, 907--940, \href{http://arxiv.org/abs/math/0308173}{math.AG/0308173}


\item [[Maxim Kontsevich]], [[Yan Soibelman]], \emph{Homological mirror symmetry and torus fibrations}, \href{http://arxiv.org/abs/math/0011041}{math.SG/0011041}


\item Yong-Geun Oh, Kenji Fukaya, \emph{Floer homology in symplectic geometry and mirror symmetry}, Proc. ICM 2006, \href{http://www.math.wisc.edu/~oh/Oh-icm2006.pdf}{pdf}


\item wikipedia: \href{http://en.wikipedia.org/wiki/Mirror_symmetry_%28string_theory%29}{mirror symmetry (string theory)}, \href{http://en.wikipedia.org/wiki/Homological_mirror_symmetry}{homological mirror symmetry}


\item partial notes from Miami 08 workshop: \href{http://www-math.mit.edu/~auroux/frg/miami08-notes}{miami08-notes} and abstracts from \href{http://www-math.mit.edu/~auroux/frg/miami09-abstracts.html}{miami09}, \href{http://www-math.mit.edu/~auroux/frg/miami10-abstracts.html}{miami10}


\item [[Mark Gross]], \emph{Tropical geometry and mirror symmetry}, CBMS regional conf. ser. 114 (2011), based on the CBMS course in Kansas, \href{http://www.ams.org/bookstore-getitem/item=CBMS-114}{AMS book page}, \href{http://www.math.ucsd.edu/~mgross/kansas.pdf}{pdf}



\end{itemize}
Discussion in the context of [[derived Morita equivalence]] includes

\begin{itemize}%
\item So Okada, \emph{Homological mirror symmetry of Fermat polynomials} (\href{http://arxiv.org/abs/0910.2014}{arXiv:0910.2014})

\end{itemize}
\hypertarget{CompleteProofs}{}\subsubsection*{{Complete proofs}}\label{CompleteProofs}

Here is a list with references that give complete proofs of \emph{homological} mirror symmetry on certain (types of) spaces.

\begin{itemize}%
\item M. Abouzaid, I. Smith, \emph{Homological mirror symmetry for the four-torus}, Duke Math. J. 152 (2010), 373--440, \href{http://arxiv.org/abs/0903.3065}{arXiv:0903.3065}


\item A. Polishchuk and E. Zaslow, \emph{Categorical mirror symmetry: the elliptic curve}, Adv. Theor. Math. Phys. 2:443470, 1998.


\item V. Golyshev, V. Lunts, D. Orlov, \emph{Mirror symmetry for abelian varieties}, J. Alg. Geom. \textbf{10} (2001), no. 3, 433--496, \href{http://arxiv.org/abs/math/9812003}{math.AG/9812003}


\item P. Seidel, \emph{Homological mirror symmetry for the quartic surface}, \href{http://arxiv.org/abs/math/0310414}{arXiv:0310414}


\item Alexander I. Efimov, \emph{Homological mirror symmetry for curves of higher genus}, Inventiones Math. \textbf{166} (2006), 537--582, \href{http://arxiv.org/abs/0907.3903}{arXiv:0907.3903}


\item [[D. Auroux]], [[L. Katzarkov]], [[D. Orlov]], \emph{Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves}, \href{}{}; \emph{Mirror symmetry for weighted projective planes and their noncommutative deformations}, Ann. Math. \textbf{167} (2008), 867--943, \href{http://arxiv.org/abs/math/0404281}{math.AG/0404281}


\item [[Mohammed Abouzaid]], Denis Auroux, Alexander I. Efimov, Ludmil Katzarkov, Dmitri Orlov, \emph{Homological mirror symmetry for punctured spheres}, \href{http://arxiv.org/abs/1103.4322}{arxiv/1103.4322}


\item [[Paul Seidel]], \emph{Homological mirror symmetry for the genus two curve}, J. Algebraic Geometry, to appear, \href{http://arxiv.org/abs/0812.1171}{arXiv:0812.1171}



\end{itemize}
\hypertarget{computation_via_topological_recursion}{}\subsubsection*{{Computation via topological recursion}}\label{computation_via_topological_recursion}

Computation via [[topological recursion]] in [[matrix models]] and all-[[genus of a surface|genus]] proofs of mirror symmetry is due to

\begin{itemize}%
\item [[Vincent Bouchard]], [[Albrecht Klemm]], [[Marcos Marino]], [[Sara Pasquetti]], \emph{Remodeling the B-model}, Commun.Math.Phys.287:117-178, 2009 (\href{https://arxiv.org/abs/0709.1453}{arXiv:0709.1453})


\item [[Bertrand Eynard]], [[Amir-Kian Kashani-Poor]], Olivier Marchal, \emph{A matrix model for the topological string I: Deriving the matrix model} (\href{https://arxiv.org/abs/1003.1737}{arXiv:1003.1737})


\item [[Bertrand Eynard]], [[Amir-Kian Kashani-Poor]], Olivier Marchal, \emph{A matrix model for the topological string II: The spectral curve and mirror geometry} (\href{https://arxiv.org/abs/1007.2194}{arXiv:1007.2194})


\item [[Bertrand Eynard]], [[Nicolas Orantin]], \emph{Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture} (\href{https://arxiv.org/abs/1205.1103}{arXiv:1205.1103})


\item Bohan Fang, Chiu-Chu Melissa Liu, Zhengyu Zong, \emph{All Genus Open-Closed Mirror Symmetry for Affine Toric Calabi-Yau 3-Orbifolds} (\href{https://arxiv.org/abs/1310.4818}{arXiv:1310.4818})



\end{itemize}
[[!redirects homological mirror symmetry]]



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