abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
To every complex 3-dimensional Calabi-Yau variety $X$ are associated two similar but differing typesof 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 $SCFT_B(\hat X)$.
This is called mirror symmetry. At least in some cases this can be understood as a special case of T-duality.
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 TCFTs, $A(X)$ 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 exchanges the A-model with the $B$-model and conversely:
By a version of the cobordism hypothesis-theorem, these TCFTs (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 Lagrangean intersection theory on the underlying symplectic manifold; and the B-model by an 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 $X\leftrightarrow \hat{X}$:
This categorical formulation was introduced by Maxim Kontsevich in 1994 under the name 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 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 (what can 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 (Strominger-Yau-Zaslow conjecture).
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 Hori-Vafa; 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, …
duality in physics, duality in string theory
The original statement of the homological mirror symmetry conjecture is in
A review and status report is in
M. Ballard, Meet homological mirror symmetry (arxiv:0801.2014)
A. Port, An Introduction to Homological Mirror Symmetry and the Case of Elliptic Curves (arXiv:1501.00730)
Other reviews include
Paul Aspinwall, Tom Bridgeland, Alastair Craw, Michael Douglas, Mark Gross, Anton Kapustin, Gregory Moore, Graeme Segal, Balázs Szendrői, P. Wilson,
Dirichlet branes and mirror symmetry,
Clay Mathematics Monogrph Volume 4, Amer. Math. Soc. Clay Math. Institute 2009
(pdf) (very readable!)
The relation to T-duality was established in
Further references include
Cumrun Vafa, S-T. Yau editors, Winter school on mirror symmetry, vector bundles, and Lagrangian submanifolds, Harvard 1999, AMS, Intern. Press (includes A. Strominger, S-T. Yau, E. Zaslow, Mirror symmetry is $T$-duality as pages 333–347; ).
K. Hori, S. Katz, A. Klemm et al. Mirror symmetry I, AMS, Clay Math. Institute 2003.
Paul Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zürich, 2008. viii+326 pp
Mark Gross, Bernd Siebert, Mirror symmetry via logarithmic degeneration data I, math.AG/0309070, From real affine geometry to complex geometry, math.AG/0703822, Mirror symmetry via logarithmic degeneration data II, arxiv/0709.2290
Anton Kapustin, Dmitri Orlov, Lectures on mirror symmetry, derived categories, and D-branes, Uspehi Mat. Nauk 59 (2004), no. 5(359), 101–134; translation in Russian Math. Surveys 59 (2004), no. 5, 907–940, math.AG/0308173
Maxim Kontsevich, Yan Soibelman, Homological mirror symmetry and torus fibrations, math.SG/0011041
Yong-Geun Oh, Kenji Fukaya, Floer homology in symplectic geometry and mirror symmetry, Proc. ICM 2006, pdf
wikipedia: mirror symmetry (string theory), homological mirror symmetry
partial notes from Miami 08 workshop: miami08-notes and abstracts from miami09, miami10
Mark Gross, Tropical geometry and mirror symmetry, CBMS regional conf. ser. 114 (2011), based on the CBMS course in Kansas, AMS book page, pdf
Here is a list with references that give complete proofs of homological mirror symmetry on certain (types of) spaces.
M. Abouzaid, I. Smith, Homological mirror symmetry for the four-torus, Duke Math. J. 152 (2010), 373–440, arXiv:0903.3065
A. Polishchuk and E. Zaslow, Categorical mirror symmetry: the elliptic curve, Adv. Theor. Math. Phys. 2:443470, 1998.
V. Golyshev, V. Lunts, D. Orlov, Mirror symmetry for abelian varieties, J. Alg. Geom. 10 (2001), no. 3, 433–496, math.AG/9812003
P. Seidel, Homological mirror symmetry for the quartic surface, arXiv:0310414
Alexander I. Efimov, Homological mirror symmetry for curves of higher genus, Inventiones Math. 166 (2006), 537–582, arXiv:0907.3903
D. Auroux, L. Katzarkov, D. Orlov, Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves, ; Mirror symmetry for weighted projective planes and their noncommutative deformations, Ann. Math. 167 (2008), 867–943, math.AG/0404281
Mohammed Abouzaid, Denis Auroux, Alexander I. Efimov, Ludmil Katzarkov, Dmitri Orlov, Homological mirror symmetry for punctured spheres, arxiv/1103.4322
Paul Seidel, Homological mirror symmetry for the genus two curve, J. Algebraic Geometry, to appear, arXiv:0812.1171