Mirror symmetry got its start in 1989 with work of Greene and Plesser [17] and Candelas, Lynker and Schimmrigk [9]. These two works first observed the existence of pairs of Calabi-Yau manifolds exhibiting an exchange of Hodge numbers. Recall that by Yau’s proof of the Calabi conjecture [76], a Calabi-Yau manifold is an $n$-dimensional complex manifold $X$ with a nowhere vanishing holomorphic $n$-form $\Omega$ and a Ricci-flat Kähler metric with Kähler form $\omega$. Ricci-flatness is equivalent to $\omega^{n}=C\Omega\wedge\bar{\Omega}$ for a constant $C$.

The most famous example of a Calabi-Yau manifold is a smooth quintic three-fold $X\subseteq\mathbb{P}^{4}$. The Hodge numbers of $X$ are $h^{1,1}(X)=1$ and $h^{1,2}(X)=101$, with topological Euler characteristic $-200$. The original construction of Greene and Plesser gave a mirror to $X$, as follows. Let $Y\subseteq\mathbb{P}^{4}$ be given by the equation

$$x_{0}^{5}+\cdots+x_{4}^{5}=0,$$

and let

$$G=\{(a_{0},\ldots,a_{4})\in\mathbb{Z}_{5}^{5}\,|\,\sum_{i}a_{i}=0\}.$$

An element $(a_{0},\ldots,a_{4})\in G$ acts on $Y$ by

$$(x_{0},\ldots,x_{4})\mapsto(\xi^{a_{0}}x_{0},\ldots,\xi^{a_{4}}x_{4})$$

for $\xi$ a primitive fifth root of unity. The quotient $Y/G$ is highly singular, but there is a resolution of singularities $\check{X}\rightarrow Y/G$ such that $\check{X}$ is also Calabi-Yau, and one finds $h^{1,1}(\check{X})=101$ and $h^{1,2}(\check{X})=1$, so that $\check{X}$ has topological Euler characteristic $200$.

The relationship between these two Calabi-Yau manifolds proved to be much deeper than just this exchange of Hodge numbers. Pioneering work of Candelas, de la Ossa, Greene and Parkes [10] performed an amazing calculation, following string-theoretic predictions which suggested that certain enumerative calculations on $X$ should give the same answer as certain period calculations on $\check{X}$. The calculations on $\check{X}$, though subtle, could be carried out: these involved integrals of the holomorphic form on $\check{X}$ over three-cycles as the complex structure on $\check{X}$ is varied. On the other hand, the corresponding calculations on $X$ involved numbers of rational curves on $X$ of each degree. For example, the number of lines on a generic quintic threefold is $2875$ and the number of conics is $609250$. String theory thus gave predictions for these numbers for every degree, an astonishing feat given that most of these numbers seemed far beyond the reach of algebraic geometry at the time.

More generally, string theory introduced the concepts of the $A$-model and $B$-model. The $A$-model involves the symplectic geometry of Calabi-Yau manifolds. Properly defined, the counts of rational curves are in fact symplectic invariants, now known as Gromov-Witten invariants. The $B$-model, on the other hand, involves the complex geometry of Calabi-Yau manifolds. Holomorphic forms of course depend on the complex structure, so the period calculations mentioned above can be thought of as $B$-model calculations. Ultimately, string theory predicts an isomorphism between the $A$-model of a Calabi-Yau manifold $X$ and the $B$-model of its mirror, $\check{X}$. The equality of numerical invariants is then a consequence of this isomorphism.

Proofs of these string-theoretic predictions of curve-counting invariants were given by Givental [15] and Lian, Liu and Yau [58], with successively simpler proofs by many other researchers. However, all the proofs relied on the geometry of the ambient space $\mathbb{P}^{4}$ in which the quintic is contained. Roughly speaking, one considers all rational curves in $\mathbb{P}^{4}$, and tries to understand how to compute how many of these are contained in a given quintic hypersurface.

This raised the question: is there some underlying intrinsic geometry to mirror symmetry?

Historically the first approach to an intrinsic formulation of mirror symmetry is Kontsevich’s Homological Mirror Symmetry conjecture, stated in 1994 in [51]. This made mathematically precise the notion of an isomorphism between the $A$- and $B$-models. The homological mirror symmetry conjecture posits an isomorphism between two categories, the Fukaya category of Lagrangian submanifolds of $X$ (the $A$-model) and the derived category of coherent sheaves on the mirror $\check{X}$ (the $B$-model). Morally, this states that the symplectic geometry of $X$ is the same as the complex geometry of $\check{X}$. At the time this conjecture was made, however, there was no clear idea as to how such an isomorphism might be realised, nor did this conjecture state how to construct mirror pairs.

The second approach is due to Strominger, Yau and Zaslow in their 1996 paper [74]. They made a remarkable proposal, based on new ideas in string theory, which gave a very concrete geometric interpretation for mirror symmetry.

Let me summarize, very roughly, the physical argument they used here. Developments in string theory in the mid-1990s introduced the notion of Dirichlet branes, or $D$-branes. These are submanifolds of space-time, with some additional data, which serve as boundary conditions for open strings, i.e., we allow open strings to propagate with their endpoints constrained to lie on a $D$-brane. Remembering that space-time, according to string theory, looks like $\mathbb{R}^{1,3}\times X$, where $\mathbb{R}^{1,3}$ is ordinary space-time and $X$ is a Calabi-Yau three-fold, we can split a $D$-brane into a product of a submanifold of $\mathbb{R}^{1,3}$ and one on $X$. It turned out, simplifying a great deal, that there are two particular types of submanifolds on $X$ of interest: holomorphic $D$-branes, i.e., holomorphic submanifolds with a holomorphic line bundle, and special Lagrangian $D$-branes, which are special Lagrangian submanifolds with flat $U(1)$-bundle:

Holomorphic $D$-branes can be viewed as $B$-model objects, and special Lagrangian $D$-branes as $A$-model objects. The isomorphism between the $B$-model on $X$ and the $A$-model on $\check{X}$ then suggests that the moduli space of holomorphic $D$-branes on $X$ should be isomorphic to the moduli space of special Lagrangian $D$-branes on $\check{X}$. (This is now seen as a physical manifestation of the homological mirror symmetry conjecture). Now $X$ itself is the moduli space of points on $X$. So each point on $X$ should correspond to a pair $(M,\nabla)$, where $M\subseteq\check{X}$ is a special Lagrangian submanifold and $\nabla$ is a flat $U(1)$-connection on $M$.

A theorem of McLean [60] tells us that the tangent space to the moduli space of special Lagrangian deformations of a special Lagrangian submanifold $M\subseteq\check{X}$ is $H^{1}(M,\mathbb{R})$. Of course, the moduli space of flat $U(1)$-connections modulo gauge equivalence on $M$ is the torus $H^{1}(M,\mathbb{R})/H^{1}(M,\mathbb{Z})$. In order for this moduli space to be of the correct dimension, we need $\dim H^{1}(M,\mathbb{R})=n$, the complex dimension of $X$. This suggests that $X$ consists of a family of tori which are dual to a family of special Lagrangian tori on $\check{X}$. An elaboration of this argument yields the following conjecture:

This conjecture motivated a great deal of work in the five years following its introduction in 1996, some of which will be summarized in the following sections. There was a certain amount of success, as we shall see, with the conjecture proved for some cases, including the quintic three-fold, at the topological level. Further, the conjecture gave a solid framework for thinking about mirror symmetry at an intuitive level. However, work of Dominic Joyce demonstrated that the conjecture was unlikely to be literally true. Nevertheless, it is possible that weaker limiting forms of the conjecture still hold.

In the first several sections of this survey, I will clarify the conjecture, review what is known about it, and state a weaker form which seems accessible. Most importantly, I will explain how the SYZ conjecture leads to the study of affine manifolds (manifolds with transition functions being affine linear) and hence to an algebro-geometric interpretation of the conjecture, developed by me and Bernd Siebert. This removes the hard analysis, and gives a powerful framework for understanding mirror symmetry at a conceptual level.

The bulk of the paper is devoted to outlining this framework as developed over the last ten years. I explain how affine manifolds are related to degenerations of Calabi-Yau manifolds. Once one begins to consider degenerations, log geometry of K. Kato and Fontaine–Illusie comes into the picture. Conjecturally, the base of the SYZ fibration incorporates key combinatorial information about log structures on degenerations of Calabi-Yau manifolds. Log geometry then gives a connection with tropical geometry and log Gromov-Witten theory, which theoretically allows a description of $A$-model curve counting using tropical geometry. On the mirror side, we explain how again tropical geometry is used to describe complex structures. This identifies tropical geometry as the geometry underlying both sides of mirror symmetry, and guides us towards a conceptual understanding of mirror symmetry. We end with a description of recent work with Pandharipande and Siebert [27] which provides a snapshot of the relationship between the two sides of mirror symmetry.

I would like to thank the organizers of Current Developments in Mathematics 2012 for inviting me to take part in the conference, and Bernd Siebert, my collaborator on much of the work described here. Some of the material appearing in this article was first published in my article “The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations,” in Algebraic Geometry: Seattle 2005, edited by D. Abramovich, et al., Proceedings of Symposia in Pure Mathematics Vol. 80, part 1, 149-192, published by the American Mathematical Society. (c) 2009 by the American Mathematical Society. Finally, I would like to thank Lori Lejeune and the Clay Institute for Figure 3.