nLab mirror symmetry

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 $SCFT_B(\hat X)$.

$$SCFT_A(X) \simeq SCFT_B(\hat X) \,.$$

This is called mirror symmetry. At least in some cases this can be understood as a special case of T-duality (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 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:

$$A(X) \simeq B(\hat X),\,\,\,\,\,\,\,B(X)\simeq A(\hat X) \,.$$

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 Lagrangian 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 $X\leftrightarrow \hat{X}$:

$$Fuk(X) \simeq D^b_\infty(\hat{X}), \,\,\,\, and \,\,\,\, Fuk(\hat{X}) \simeq D^b_\infty(X).$$

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 (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 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 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, …

Properties

Relation to tropical geometry.

Close relation to tropical geometry, see e.g. Gross 11.

Related concepts

• mirror map

• moduli space of Calabi-Yau spaces

• 3d mirror symmetry

• duality in physics, duality in string theory

• geometric Langlands duality

• Landau-Ginzburg model

References

General

The original statement of the homological mirror symmetry conjecture is in

• Maxim Kontsevich, Homological algebra of mirror symmetry, Proc. ICM Zürich 1994, alg-geom/9411018

See also:

• Paul Aspinwall, Brian Greene, David Morrison, Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory, Nucl.Phys. B416 (1994) 414-480 (arXiv:hep-th/9309097)

• Philip Candelas, Xenia C. de la Ossa, Paul S. Green, Linda Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1):21–74, 1991

Review:

• Kentaro Hori, Cumrun Vafa, Mirror Symmetry (arXiv:hep-th/0002222)

• 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)

• Luis Ibáñez, Angel Uranga, section 10.1.2 of String Theory and Particle Physics – An Introduction to String Phenomenology, Cambridge University Press 2012

History:

• Maxim Kontsevich, On the History of quantum cohomology and homological mirror symmetry (2021) [video:YT]

Discussion amplifying the role of category theory, and higher geometry:

• Eric Sharpe, Categorical equivalence and the renormalization group, Proceedings of LMS/EPSRC Symposium Higher Structures in M-Theory, Fortschritte der Physik 2019 (arXiv:1903.02880)

Further review:

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

  (pdf, pdf) (very readable!)

• Felipe Espreafico G. Ramos, Mirror Symmetry and Fukaya Categories (2020) [pdf, pdf]

The relation to T-duality was established in

• Andrew Strominger, Shing-Tung Yau, Eric Zaslow, Mirror Symmetry is T-Duality, Nucl.Phys.B479:243-259,1996 (DOI 10.1016/0550-3213(96)00434-8) hep-th/9606040

Further references:

• Cumrun Vafa, Shing-Tung Yau (eds.), 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

Discussion in the context of derived Morita equivalence includes

• So Okada, Homological mirror symmetry of Fermat polynomials (arXiv:0910.2014)

Complete proofs

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

Computation via topological recursion

Computation via topological recursion in matrix models and all-genus proofs of mirror symmetry is due to

• Vincent Bouchard, Albrecht Klemm, Marcos Marino, Sara Pasquetti, Remodeling the B-model, Commun.Math.Phys.287:117-178, 2009 (arXiv:0709.1453)

• Bertrand Eynard, Amir-Kian Kashani-Poor, Olivier Marchal, A matrix model for the topological string I: Deriving the matrix model (arXiv:1003.1737)

• Bertrand Eynard, Amir-Kian Kashani-Poor, Olivier Marchal, A matrix model for the topological string II: The spectral curve and mirror geometry (arXiv:1007.2194)

• Bertrand Eynard, Nicolas Orantin, Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture (arXiv:1205.1103)

• Bohan Fang, Chiu-Chu Melissa Liu, Zhengyu Zong, All Genus Open-Closed Mirror Symmetry for Affine Toric Calabi-Yau 3-Orbifolds (arXiv:1310.4818)