# Examples in mirror symmetry that can be understood.

It seems to me, that a typical science often has simple and important examples whose formulation can be understood (or at least there are some outcomes that can be understood). So if we consider mirror symmetry as science, what are some examples there, that can be understood?

I would like to explain a bit this question. If we consider the article "Meet homological mirror symmetry" http://arxiv.org/abs/0801.2014 it turns out, that in order to understand something we need to know huge amount of material, including $A_{\infty}$ algebras, Floer cohomology, ect.

Here, on the contrary, is an example, that "can be understood" (for my taste): According to Arnold, the first instance of symplectic geometry was "last geometric theorem of Poincare".

This is the following statement: Let $F: C\to C$ be any area-preserving self map of a cylinder $A$ to itself, that rotates the boundaries of $A$ in opposite directions. Then the map has at least two fixed points. (this was proven by Birkhoff http://en.wikipedia.org/wiki/George_David_Birkhoff)

So, I would like to ask if there are some phenomena related to mirror symmetry that can be formulated in simple words.

Added. I would like to thank everyone for the given answers! I decided to give a bit of bounty for this question, to encourage people share phenomena related to mirror symmetry that can be simply formulated (or at least look exciting). Since there are lot of people in this area I am sure there must be more examples.

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Another interesting explanation of Mirror symmetry in certain cases related to combinatorics is in terms of typical shapes for certain classes of partitions. You start with a class of partitions related to some variety, consider the typical shape and this gives you the dual variety. –  Gil Kalai Mar 10 '11 at 22:31
Gil, I would love to see such examples that illustrate what you say! –  aglearner Mar 10 '11 at 22:58
I heard about this typical-shape-of-partition approach to mirror symmetry in a lecture by Okounkov. I dont know the precise papers (some probably with Pandharipande,and Nekrasov). Maybe one can start by reading Okounkov's paper the use of random partitions arxiv.org/PS_cache/math-ph/pdf/0309/0309015v1.pdf and then maybe look at Okounkov-Reshetikhin-Wafa arxiv.org/PS_cache/hep-th/pdf/0309/0309208v2.pdf But explicit mention of Mirror symmetry there is sparse. –  Gil Kalai Mar 11 '11 at 7:22
I think that research in mirror symmetry goes in the opposite direction to what happened with Poincare'-Birkhoff discovery. In that case a simple statement lead to a beautiful rich theory. In mirror symmetry a very complicated statement (such as the counting formula for curves on the quintic), which no-one understood at first, lead to a theory which is slowly becoming clearer and enriched with simpler examples. –  Diego Matessi Mar 11 '11 at 13:31
Section 2.3 of the paper by O-R-W linked above is called "Mirror symmetry and the limit shape". –  Gil Kalai Mar 12 '11 at 11:33

## 9 Answers

Here is my biased view of a simple example: the two-torus. Everything I know about homological mirror symmetry stems from this example.

Because the example is one-dimensional, a symplectic form is just an area form, and Lagrangians are simply curves, and the holomorphic maps which are part of the Fukaya category are simply topological disks. (By uniformization of Riemann surfaces, there is one holomorphic map for each topological disk satisfying the appropriate boundary conditions.) Even better, you can go to the universal cover, which is $R^2,$ and just draw Lagrangians as straight lines with rational slope. The holomorphic disks which determine compositions in the category are simply triangles.

On the mirror side, we're talking about a complex two-torus, or elliptic curve. A typical object would be a line bundle on the elliptic curve, such as the theta line bundle, whose sections are theta functions, once we lift them up to the complex plane.

The two-torus is circle-fibered over a base circle, and the elliptic curve is circle-fibered by the dual circle (i.e., $U(1)$ local systems on the original circle). This is called T-duality, and it explains how to construct the mirror equivalence going from Lagrangians to line bundles, or vice versa. For example, the Lagrangian $\{y=0\}$ represents a family of trivial $U(1)$ local systems, corresponding to the trivial holomorphic line bundle whose sections are just holomorphic functions. The Lagrangian $\{y=nx\}$ corresponds to a line bundle of degree $n$. After making these definitions, one checks that compositions match up.

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Thank you! Nice to hear that the elliptic curve example is so imporant for HMS. Do you think, there is a resonable math reference for this, at least, could you advise something? –  aglearner Mar 14 '11 at 16:58
Well, I can offer you my own article with Polishchuk: arxiv.org/abs/math/9801119 Below, AByer mentions that HMS implies that we get an isomorphism of our category for each loop in moduli space. These isomorphisms ("autoequivalences") alone can lead to the mirror map, as demonstracted by a calculation for this example in arxiv.org/abs/math/0506359. The T-duality aspect can be pushed in many other directions and examples, too. (Sorry for the lazy self-promotion! There are, of course, many other articles by other authors, some of which are included in the answers below.) –  Eric Zaslow Mar 14 '11 at 17:31
I decided to accept this answer, since it looks like this answer go in the direction of I wanted. Namely it says that there is a very important example: elliptic curves, and moreover it gives references to articles that one might try to understand (I just had a quick look at them but will try to do this more seriously). On the other had, Eric, if you want to add anything to your answer (like references, or whatsoever), you are more than welcome. –  aglearner Mar 16 '11 at 9:07
The paper with Polishchuk works through the example in detail, as opposed to just the sketch above. You seem to be still somewhat dissatisfied. Why don't you say precisely which aspect of mirror symmetry you are looking to uncover in your example? (Three references for torus fibrations are Arinkin-Polishchuk, Leung-Yau-Z, and Mark Gross's "Topological Mirror Symmetry.") –  Eric Zaslow Mar 16 '11 at 12:27
This is a nice answer, and I am happy with it, and will try to study the references that you proposed. I just wanted to hear a bit more... For example, since you are a physicist, I was curious, which side of mirror in this example is "closer" to you -- Fukaya, or derived categories. Or maybe this example is pure math (and does not require any physics intuition)...? But, again, I am happy with the answer (just will need time to read the articles) –  aglearner Mar 16 '11 at 22:17

Mirror symmetry gives some remarkable connections between certain varieties. The first step in this connection is that certain homology groups have the same rank. An explicit case for mirror symmetry duals is the case coming from toric varieties. In this case, the dual objects comes from duality of polytopes. So duality of polytopes: associating the octahedron to the cube and the icosahederon to the dodecahederon is related to Mirror symmetry.

Perhaps the very first facts about polytopes which demonstrates unexpected equalities for certain homologies can be described as follows:

For 2-dimensional polytopes this is the following numerical fact: A polygon has the same number of edges as the dual. (Well, this is not so unexpected.)

For 4 dimensional polytope P it is the following numerical fact. Start with a 4-polytope with n vertices and e edges. Triangulate every 2-face by non crossing diagonals. Let $e^+$ be the number of edges including the added diagonals. Consider the quantity

$$\gamma (P) = e^+ - 4n .$$

It is true that for every dual pair of 4-polytopes $P$ and $P^*$,

$$\gamma (P^*)=\gamma(P).$$

This is more surprising.

For example, let P be the 4-dimensional cross polytope and Q be the 4-dimensional cube. P has 8 bertices 24 edges and all the 2-faces are triangles so $\gamma (P)=24~-~4\cdot 8~=~-8$. The 4-cube Q has 16 vertices, and 32 edges and it has also 24 2-faces which are squares, so $e^+(Q)=56$. $\gamma (Q)=56-64 = -8$. Voila!

This reflects some properties of toric varieties (unexpected equalities between Hodge numbers) which express (sort of the 0-th step of) mirror symmetry.

Related papers: V. Batyrev and L. Borisov, Mirror duality and string-theoretic Hodge numbers; V. Batyrev and B. Nill, Combinatorial aspect of mirror symmetry. Here is a lecture by B. Nill.

Another manifestation of mirror symmetry of combinatorial nature, that can be formulated in simple words, is in terms of typical shape of various classes of partitions. I mentioned it in a remark above and let me quote a description taken from my adventure book.

A partition is just a way to write a number as a sum of other numbers. Like 9=4+2+1+1+1. Partitions have attracted mathematicians for centuries. Among others, the famous Indian mathematician Ramanujan was well known for his identities regarding partitions. And now enters another idea, baring the names of Ulam, Vershik, Kerov, Shepp and others who studied partitions as stochastic objects. In particular, it was discovered that "most" partitions, say of a number n, come in a "typical shape".

The emergent picture drawn by Okounkov and his coauthors goes very roughly like this: an "algebraic variety" (a manifold of some sort) that takes part in a certain string theory is related to a class of partitions, and when we consider the typical shape of a partition in the class this gives us another algebraic variety, and - lo and behold - the typical shape IS the mirror image of the original one. The mirror relations translate to asymptotic results on the number of partitions, somewhat in the spirit of the famous asymptotic formulas of the mathematicians Hardy and Ramanujan for p(n)- the total number of partitions for the number n.

As mentioned in the comments I am not sure about good references to this connection between mirror symmetry and limit shapes of classes of partitions. The 2003 paper Quantum Calabi-Yau and Classical Crystals by Andrei Okounkov, Nikolai Reshetikhin, and Cumrun Vafa describes this connection in Section 2.3 called "mirror symmetry and the limit shape".

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Here is the simplest example that I can think of...

The ordinary cohomology ring of $\mathbb{CP}^n$ is given by $\mathbb{C}[a]/(a^{n+1})$. The structure of this ring can be thought of as describing the intersection theory of subvarieties / submanifolds / linear subspaces of $\mathbb{CP}^n$. For example, the relation $a^3 \cdot a^3 = 0$ in the cohomology ring of $\mathbb{CP}^5$ reflects the fact that the intersection of two generic dimension 2 subspaces of $\mathbb{CP}^5$ is empty.

Now the quantum cohomology ring of $\mathbb{CP}^n$ is $\mathbb{C}[a]/(a^{n+1} - q)$, where we can think of $q$ as being a nonzero constant, or a formal parameter if you like. The quantum cohomology ring is a deformation (in a suitable sense) of the ordinary cohomology ring. The structure of the deformed ring now encodes "enumerative geometry" information. For example, it is a fact that given generic linear subspaces $A,B,C$ of $\mathbb{CP}^n$ of total dimension $n-1$, there is a unique degree 1 map $\mathbb{CP}^1 \to \mathbb{CP}^n$ sending the points $0,1,\infty$ to $A,B,C$ respectively. Writing $q$ as $1 \cdot q^1$, the coefficient $1$ corresponds to the uniqueness of the map, and the exponent $1$ corresponds to the degree of the map. I like to think of this as a generalization of the fact that there is a unique line passing through any two distinct points in the plane, which has been known since at least Euclid... :-)

But so far I haven't said anything about "mirror symmetry"...

Mirror symmetry says that the story I've described above is echoed by certain properties of the function $W = x_1 + \cdots + x_n + \frac{q}{x_1\cdots x_n}$ on $(\mathbb{C}^\ast)^n$. For example, the Jacobian ring of $W$, which is by definition the ring $\mathbb{C}[x_i^{\pm 1}]/(\partial_i W)$, is isomorphic to $\mathbb{C}[a]/(a^{n+1} - q)$.

EDIT: The relation between $\mathbb{CP}^n$ and $W$ goes much deeper. For another elementary(-ish) mirror symmetry statement, there is Seidel(I think?)'s proof that the derived category of $\mathbb{CP}^n$ is equivalent to the Fukaya-Seidel category of $W$. In this case these categories can be described fairly easily, without too much fancy language, via the "Beilinson quiver", which on the derived category side corresponds to the line bundles $\mathcal{O}, \mathcal{O}(1), \cdots , \mathcal{O}(n)$ and the fact that there is an $(n+1)$-dimensional set of morphisms from $\mathcal{O}(i)$ to $\mathcal{O}(i+1)$. For example, consider the morphisms from $\mathcal{O}$ to $\mathcal{O}(1)$; these are just the sections of $\mathcal{O}(1)$, which are the homogeneous degree 1 polynomials in $n+1$ variables.

On the other side, one can see the Beilinson quiver via the "vanishing cycles" $L_0, L_1, \dots, L_n$ of $W$, and the $n+1$-many morphisms above correspond to the $n+1$ intersection points between $L_i$ and $L_{i+1}$. For more on this, see the notes from Bohan Fang's talk here and this paper of Seidel.

This kind of correspondence between vector bundles and cycles, and between morphisms of vector bundles and intersections points of cycles, is a first approximation of homological mirror symmetry, or "categorical" mirror symmetry. For a better approximation, the statement is that compositions of morphisms of vector bundles correspond to "compositions" of intersection points, where these "compositions" are defined via $J$-holomorphic discs. But for the elliptic curve / symplectic torus, things are still pretty simple, and one can avoid saying the word "$J$-holomorphic disc" if one wishes. In this situation, the correspondence between compositions reduces to a correspondence between some classical facts about theta functions on elliptic curves and some very elementary observations about lines and triangles on a torus.

And finally, here is the most trivial example of mirror symmetry. Let $X$ be a point $\operatorname{Spec} \mathbb{C}$. Then the mirror of $X$, call it $Y$, is also a point. Notice that the point is a Lagrangian submanifold of $Y$. Notice that the point intersect the point is the point. On the other hand, take $\mathbb{C}$ as a $\mathbb{C}$-module. Then there is a 1-dimensional set of $\mathbb{C}$-module morphisms from $\mathbb{C}$ to $\mathbb{C}$.

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I won't claim this means ''understanding mirror symmetry'', but if you are familiar with the derived category of coherent sheaves, then there is a consequence of Kontsevich's homological mirror symmetry that can be understood, without knowing anything about the Fukaya category:

For every symplectic diffeomorphism of the mirror $\hat X$, there is a autoequivalence of $\mathrm D^b(X)$.

Examples:

1. If $X$ is a an elliptic curve, then $\mathrm{SL}_2(\mathbb Z)$ acts as a group of symplectic diffeomorphisms on the mirror $\hat X = \mathbb{R}^2/\mathbb{Z}^2$. There is a corresponding action of (a central extension of) $\mathrm{SL}_2(\mathbb Z)$ generated by Fourier-Mukai transform induced by the Poincare line bundle on $X \times X$, and by tensoring by a line bundle of degree one (and shifts).

2. A Dehn twist corresponds to the ''spherical twist'' $\mathrm{ST}_E$ at an spherical object $E$ (see arXiv:math.AG/0001043).

3. More examples have been studied by Horja, see arXiv:0103.5231.

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This is a nice point. I would find even more exiting if it were possible to go in the opposite direction: we start with an auto-equivalence of $D^b(X)$ and get a symplectomorphism of the "mirror". I wonder if this direction ever appeared in the literature? Also, do I understand correctly, that we don't really know what is the symplectiomorphism group of symplectic manifolds of dimesnion $4$ an higher? We don't even know what is the group of connected components of the symplectomorphism group in any single example? –  aglearner Mar 6 '11 at 11:15
@aglearner: There doesn't seem to be any a priori reason for an arbitrary derived autoequivalence of Fukaya to come from a symplectomorphism. –  S. Carnahan Mar 6 '11 at 14:58
@Scott. Indeed, they don't all come from symplectomorphisms. For instance, the group of real line bundles acts on the spin-structures that decorate the Lagrangian submanifolds. @aglearner. Some 4-dimensional symplectomorphism groups are well understood. In higher dimensions, a main difficulty is that there are possibly non-trivial things which Floer cohomology doesn't distinguish from the identity: does $\pi_0 \mathrm{Diff}(S^6)=\mathbb{Z}/28$ inject into $\pi_0 \mathrm{Symp}(T^\ast S^6)$?. HMS doesn't help with this. –  Tim Perutz Mar 6 '11 at 15:49

A toy model for mirror symmetry is the following. Consider a real manifold (not necessarily compact) $B$ with an atlas of affine coordinates, i.e. such that the change of coordinate maps are of the type $x \mapsto Mx + b$ where $M$ and $b$ are constant. Then the tangent bundle $TB$ has natural complex coordinates given by $z = x +i y$, where $y$ are coordinates on the fibre. If further one assumes that $\det M = 1$ then $TM$ also has a nowhere vanishing holomorphic $n$-form. On the other hand $T^*B$ has it's usual simplectic structure. So $TB$ and $T^*B$ can be thought of being mirror. One can also twist the complex structure on $TB$ with a B-field. One can go a bit further too, in fact suppose $\gamma$ in $B$ is a affine curve (i.e. a straight line in affine coordinates). Then one can lift $\gamma$ to a Lagrangian submanifold $L_{\gamma}$ of $T^*B$ by adding the anhilator of $\gamma'(t)$ in the fibre at $\gamma(t)$. On the other hand the same curve lifts to a complex object in $TB$ by adding the line generated by $\gamma'$ inside the fibre $T_{\gamma} B$. If the affine structure is also integral (i.e. $M$ and $b$ have integral coefficients), then one can also partially compactify by taking latices $\Lambda \subset TB$ and its dual $\Lambda^'$ and then form torus bundles $X = TB / \Lambda$ and $X^* = T^*B / \Lambda^'$. This picture is too simple to work in the compact case, but it is expected that actual mirror symmetry is a perturbation of this. What I just described is the SYZ approach to mirror symmetry.

I like this example. Consider a surface $V$ in $(\mathbb{C}^{\ast})²$ given by some Laurent polynomial $p(z)$, then the hyperfurface $X$ defined by $xy = p(z)$ is Calabi-Yau. Its mirror $\check X$ can be constructed by taking the Newton polygon $\Delta$ of $p$ and then consider the toric variety defined by the cone over $\{ 1 \} \times \Delta$ in $\mathbb{R}³$. $\check X$ is a resolution of this toric variety obtained from some subdivision of $\Delta$. Now, the surface $V$ (the one we started from) has a "tropical amoeba". This can be thought of a graph in $\mathbb{R}²$ which is the limit (in some sense) of the image of $V$ under the standard torus fibration $(\mathbb{C}^{\ast})² \rightarrow \mathbb{R}²$. The interesting thing is that this graph gives a subdivision of $\mathbb{R}²$ which is dual to the subdivision of $\Delta$ (this is related Gil Kalai's answer). More over this graph is also the locus of singular fibres of a Lagrangian torus fibration defined on $X$. Such a Lagrangian fibration induces on the base an affine structure as I said previously. A construction such as the one I described above can be used to construct many Lagrangian $S^3$'s in $X$ over the bounded regions defined by the graph. The mirror of these objects are the divisors $\check X$ corresponding to interior integral points of $\Delta$, or better, line bundles supported on such divisors. There are some interesting correspondences between intersection points of these spheres and cohomology of the line bundles, even without getting into $\mathcal{A}^{\infty}$ constructions.

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Diego, thanks! This is simple indeed. This generalised the idea that $T\mathbb R^n$ is naturally complex, while $T^*\mathbb R^n$ is naturally symplectic. But what would be the first non-trivial statement that one could try to understand? –  aglearner Mar 10 '11 at 23:14
Diego, thanks for adding the example! It is nice. –  aglearner Mar 12 '11 at 18:16
General do we have a natural map from a manifold to its its mirror? For example, in the $TB$ and $T^{*}B$ case, seems we need a legendre transform to do this, but that required extre data.. Also about K3 case, do we have such map between manifolds? –  Jay Nov 15 '12 at 17:30

I'm not sure if this is what you're looking for, but the paper "Mirror symmetry and Elliptic curves" by R. Dijkgraaf might be provide a good example.

The example in that paper concerns the mirror of an elliptic curve. They have two moduli parameters, their complex moduli and their Kahler moduli. Mirror symmetry in this case simply states that the mirror of some $E_{\tau, \omega}$ is $E_{\omega,\tau}$ i.e. you simply switch the two moduli parameters.

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You need the machinery of triangulated categories and homological algebra to understand the mirror symmetry as it stand today, Homological mirror symmetry. But one can get an idea of mirror symmetry without delving into these concepts,I am talking about the classical picture of mirror symmetry as noticed by Physicists. i.e. Mirror symmetry as an isomorphism between the complex and Kahler moduli spaces of Calabi-Yau 3-folds.(Beware : this was first definition of mirror symmetry and even here I am overlooking the subtleties involving the large complex structure limit.) e.g. Elliptic curve (Dijgraaf), Quintic (Greene, Plesser and Candelas et al).This may give an idea of Mirror symmetry but to understand this picture properly one need an understanding of the geometry of Calabi-Yau manifolds, Variations of mixed Hodge structures, quantum cohomology and GW invariants.

Also there is a modern picture of mirror symmetry called SYZ conjecture which is more geometric and doesn't involve homological algebra and triangulated categories. But again you need the knowledge of geometry of special Lagrangian submanifolds of CY manifolds.

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Since you are asking for examples, you might want to take a look at the lecture notes of Mark Gross published in Calabi-Yau Manifolds and Related Geometries. In the chapter "Mirror Symmetry in Practice" the case of the quintic is worked out in some detail.

Although one does not need to know what an $A_\infty$ algebra is, one has instead to be familiar with variations of Hodge structure and some symplectic geometry. However, as far as I know, homological mirror symmetry is actually weaker than what is believed to be true, so it does probably not hurt to see, what one can show in specific examples. You might also want to look at this book, it is written for an audience of physicists and mathematicians, but probably does not represent the most recent view on mirror symmetry.

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Gernot, thanks for taking time to give these references. In fact I know about the existence of these books surely, and have opened them several times. But you see, this statement -- the last geometric theorem of Poincare can be explained to any calculus student, and this statement, led to important development in math (Arnold's conjectures), it is simple and deep. So, my hope is that people who studied the books that you mention, understand them to some extent and work professionally in the area could be able to say something at least relatively similar (if this is possible at all)-I can't :( –  aglearner Mar 9 '11 at 20:32

My master's thesis, An Introduction to Homological Mirror Symmetry and the Case of Elliptic Curves, might provide a piece of what you're looking for. It mostly concerns itself with the symplectic side of HMS (cause I have only a very superficial knowledge of algebraic geometry), but it includes a good amount background and some history. I tried to make it a useful document for other beginners to read... though then I let it rot on JSTOR for two years before posting it on arXiv :) Oh well, it's up there now. I hope it helps!

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