Historically, mirror symmetry started as a remarkable set of observations made by physicists working in the field of string theory. In classical physical models one represents particles as 0-dimensional objects propagating in four dimensional space-time. String theory postulates that at “string scale” particles should be treated as 1-dimensional objects (and hence the name “strings”) rather than 0-dimensional. This is motivated by the fact that the study of interactions between strings leads to a formal framework which elegantly explains various phenomena observed in particle physics. Interestingly, for its consistency, the (super-string) theory also requires the extension of classical four dimensional space-time by six additional dimensions, which are also noticeable only at “string scale”. From a mathematical point of view this means that one considers space to be of the form ℝ^{4}XM^{6} where M^{6 }is a compact manifold of “tiny” size. It was further discovered by Candelas, Horowitz, Strominger and Witten [CHSW] that if one requires the theory to preserve the condition of super-symmetry the manifold must admit a complex structure and a Ricci flat metric. In other words, the manifold M^{6 }must be a complex Calabi-Yau threefold (in fact, the term Calabi-Yau was coined in [CHSW]). On the other hand, any Calabi-Yau manifold M^{6 }can be used to define a (non-linear sigma model) of string theory. Mirror symmetry started from the observation that there exists pairs of Calabi-Yau threefolds M and W which lead to equivalent theories, but, with the role of certain features intertwined (A-model and B-model). For instance, one of the first examples of such pairs was given by Greene and Plesser in [GP]:

In the original physical literature, two Calabi-Yau threefolds were considered mirror to one another if “their sigma models induce isomorphic super-conformal field theories whose N=2 super-conformal representations are intertwined via sign change”. Even-though, from a mathematical point-of-view this implies h^{p,q}^{}(M)=h^{n-p,q}(W) it was not clear, at first, what are the mathematical foundations of such a duality, as well as possible applications. However, great interest began to arise when mirror symmetry was used by Candelas-de la Ossa-Green and Parkes [CdlOGP] to answer very hard questions in enumerative geometry.

At this point, various questions required answer. For instance: (a) Is mirror symmetry a phenomena restricted only to Calabi-Yau threefolds or is it a more general duality principle (b) Given a manifold M is there an “algorithm” to find its mirror(s) W (for instance generalizing the Greene-Plesser construction described above)? Is there a geometric mechanism relating the geometry of M to that of W? and finally (c) what does it mean for two manifolds to be mirror of one another from a mathematical viewpoint. It is fair to say that the study of this circle of questions led to some of the most fascinating mathematical developments of the last twenty five years.

Let us make a digression and make a small remark regarding the classical notion of projective duality, see [GKZ]. Recall that to any projective variety X⊂ℙ^{N }corresponds another variety X^{V}⊂ℙ^{N} called its dual. On the one hand, one has a geometric definition of the dual variety X^{V}, which explains the relation between X and X^{V} in terms of tangents. One then has an algebraic “algorithm” (which covertly incorporates the geometric definition), that enables to compute the dual variety, in terms of discriminants. This algebraic recipe works especially well for spacial classes of varieties, for instance, toric varieties. The reason we make this remark is that a somewhat similar situation occurs in the setting of mirror symmetry duality.

First, one has the Storminger-Yau and Zaslow viewpoint (known shortly as SYZ) which gives a deep and general geometric interpretation of mirror symmetry in terms of the T-duality principle, see [SYZ]. However, it is rather hard to apply the T-duality principle directly in order to find the mirror of a given specific manifold.

For the case of toric Fano manifolds one has a remarkably elegant description of mirror symmetry in terms of the polar polytope, discovered by V. Batyrev, see for instance [Ba2]. Let us give a short description: Let Δ be an integral polytope. To such a polytope one associates the embedding i_{Δ}:(ℂ∖0)^{n}→ℙ^{N} where N=#ℤ^{n}∩Δ which is given by i_{Δ}(z)=[z:n∈Δ∩ℤ^{n}]. The compactification of i_{Δ}((ℂ\0)^{n}) in ℙ^{N }is called the *(polarized) toric variety* defined by and we denote it by X_{Δ}. The variety X_{Δ }is smooth if for any vertex of Δ the normal of all facets meeting at the vertex form a generating set of ℤ^{n}. To a polytope Δ one can associate another polytope Δ^{0}. The polar polytope Δ^{0 }satisfies the following duality property Δ^{00}^{}=Δ. Typically, the polar polytope of an integral polytope is not nessecerally an integral polytope. We refer to a polytope Δ as *reflexive *if 0∈Δ and Δ^{0 }is integral. One has:

In particular, a toric Fano variety X is associated with its polar X^{0}. Moreover, a hyperplane section M’_{Δ}=X∩H of a toric Fano variety X⊂ℙ^{N} (a section of the anticanonical bundle) is always a Calabi-Yau variety. Batyrev shows that the singularities of M' can be resolved in such a way that keeps the Calabi-Yau property. The following example shows how Batyrev's construction is related to mirror symmetry:

In fact, many of the examples of mirror symmetric Calabi-Yau threefolds, first discovered by physicists, can be realized in terms of this construction. A study of the T-duality properties of this construction (and more general settings) can be found in [A].

As mentioned, at first, physicists considered two Calabi-Yau threefolds as mirror to one another if they give rise to equivalent super-conformal field theories. In particular, such a physical definition applies only in (complex) dimension 3. A formal mathematical definition of what it means for two manifolds to be mirror to each others is therefore needed. It should be noted that the search for a “formal definitions” is not “just a formality”. Indeed, many of the most fundamental objects of mathematics arouse out of the attempt to “formalize” concepts first studied in physical context (derivative as a formalization of velocity, differential equations from equations of motion, vector fields from gravitational and electro-magnetic fields, etc..). Kontsevich's homological symmetry conjectures give a deep formal framework to what it means for two manifolds to be mirror to each other, from a mathematical point of view. The conjectures were first described for Calabi-Yau manifolds in [K1] and later extended to a larger class in [K2].

It was noted already in the first works on mirror symmetry done by physicists that the duality between field theories interchanges the so-called A-model and B-model where the A-model depends strictly on the Kahler (symplectic) geometry of the manifold while the B-model depends strictly on the complex geometry of the manifold.

From a mathematical point of view, both the complex and the symplectic geometry of a given manifold M are studied by studying the properties of additional objects associated to M. In the case of complex algebraic geometry the objects one studies are coherent sheaves (sheaves of sections of bundles, structure sheaves of subvarieties, etc..,) while in symplectic topology one typically studies Lagrangian submanifolds (graphs of Hamiltonian flows, zero sections of cotangent bundles, etc..). Moreover, the *objects* one studies have morphisms between them. Indeed, in complex gometry, to two sheaves F,G one classicaly associates the space Hom(F,G). In symplectic topology, two Lagrangian submanifolds L_{1},L_{2} which are regular enough can also be associated with a morphism space, given in terms of Floer homology HF(L_{1},L_{2}). In the language of homological algebra such collections of objects are called a “category”. In the complex side coherent sheaves form the Abelian category Coh(M). While, in the symplectic side, the category of Lagrangian submanifolds (satisfying a few natural regularity conditions) is the Fukaya category Fuk(M), which actually has the structure of an A_{∞}-category. For both types of categories, a fundamental construction in homological algebra associates a new category, called the derived category, denoted by D^{b}(Coh(M)) and D^{b}(Fuk(M)), respectively. The original formulation of the homological mirror symmetry conjecture for Calabi-Yau manifolds states:

In particular, what the conjecture means is that whenever one has a pair of Calabi-Yau manifolds which are supposed to be mirror to one another via a physical consideration\geometrical construction, the equivalence of categories mentioned above should hold.

The derived category of coherent sheaves D^{b}^{}(X)=D^{b}^{}(Coh(X)) encodes vast information regarding the manifold X. For instance, in the Fano case, it was proven that one can completely “reconstruct” the manifold from its derived category by Bondal and Orlov [BO]. In the sense that D^{b}^{}(X_{1})=D^{b}^{}(X_{2}_{}) implies X_{1}=X_{2} (this is not the case in general). On the other hand, there are various instances in which the derived category is known to be “generated as a triangulated category” by a certain collection {F_{i}_{}}_{i∈I}⊂D^{b}^{}(X) of well-described objects (vector bundles on elliptic curves, Pic(X) of toric manifolds, full strongly exceptional collections, etc..). In fact, most of the existing proofs of homological mirror symmetry, invovle finding analog collections {L_{i}}_{i∈I} of lagrangian submanifolds in the appropriate Fukaya category of the mirror, and modeling the mirror functor over the association HMS(F_{i}_{})=L_{i}. In order to explain homological mirror symmetry for toric Fano manifolds we need to mention a few properties of Landau-Ginzburg systems.

[A] D. Auroux. Mirror symmetry and T-duality in the complement of anti-canonical divisor, J. Gökova Geom. Topol. 1 (2007), 51-91.

[Ab1] M. Abouzaid. Homogeneous coordinate rings and mirror symmetry for toric varieties. Geometry and Topology 10 (2006) 1097-1156.

[Ab2] M. Abouzaid. Morse homology, tropical geometry, and homological mirror symmetry for toric varieties. Selecta Mathematica August 2009, Volume 15, Issue 2, pp 189-270.

[AKO] D. Auroux, L. Katzarkov, D. Orlov. Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves. Invent. Math. 166 (2006), no. 3, 537-582.

[Ba2] V. Batyrev. Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Algebraic Geom. 3 (1994), no. 3, 493-535.

[Ba3] V. Batyrev. On the classification of toric Fano 4 -folds. Algebraic geometry, 9. J. Math. Sci. (New York) 94 (1999), no. 1, 1021-1050.

[BO] A. Bondal, D. Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Mathematica 125 (03), 327-344.

[CHSW] P. Candeles, GT. Horowitz, A. Strominger, E. Witten, Vacuum configurations for superstrings, Nuclear Physics B 258, 46-74.

[CdlOGP] P. Candeles, X. C. De La Ossa, P. S. Green, L. Parkes. A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys, B359 (1991) 1.

[CK] D. A. Cox, S. Katz, Mirror symmetry and algebraic geometry, Mathematical surveys and monographs, 68, 1999.

[GP] B. R. Greene, R. Plesser, Duality in Calabi-Yau Moduli Space, Nucl. Phys. B338 (1990) 15.

[GM] S. Gelfand, Y. Manin. Methods of homological algebra. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003.

[GKZ] I. Gelfand, M. Kapranov, A. Zelevinsky. Discriminants, resultants, and multidimensional determinants. Mathematics: Theory and Applications. Birkhauser Boston, Inc., Boston, MA, 1994.

[HV] K. Hori, C. Vafa. Mirror Symmetry. arXiv:hep-th/0002222

[K] M. Kontsevich. Homological algebra of mirror symmetry. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zurich, 1994), 120-139, Birkhauser, Basel, 1995.

[K2] M. Kontsevich. Course at ENS. 1998, http://arxiv.org/abs/hep-th/0002222

[S] P. Seidel. More about vanishing cycles and mutation. Symplectic geometry and mirror symmetry (Seoul, 2000), 429-465, World Sci. Publ., River Edge, NJ, 2001.

[SYZ] A. Strominger, S. T. Yau, E. Zaslow, Mirror symmetry is T-duality, Nuclear Physics B, 1996, 479 (1–2): 243–259.

[T] R. Thomas. Derived categories for the working mathematician. Providence, R.I, Mirror symmetry; Winter school on mirror symmetry, vector bundles and Lagrangian submanifolds, Cambridge, MA, January 1999, Publisher: American Mathematical Society