Overview of Mirror Symmetry

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

Historical Background

Mirror symmetry is not a field that arose naturally within the mathematical framework, but is grounded in string theory. According to string theory, the world we live in rests on a single assumption: its fundamental degrees of freedom are not point particles but one-dimensional strings. Then the trajectory in spacetime of a particle moving along the time axis is no longer a \(1\)-dimensional worldline but a \(2\)-dimensional worldsheet, and its equation of motion is determined as a specific action-minimizing solution, just as in §Classical Mechanics, §§Principle of Least Action. To reconcile this description with the framework of existing quantum mechanics, spacetime is forced to be \(10\)-dimensional; thus physicists regard this 10-dimensional spacetime as the product of \(4\)-dimensional Minkowski spacetime and a compact manifold \(M\) that accounts for the remaining \(6\) dimensions. Writing out the conditions that this space \(X\) must satisfy physically, one finds that \(X\) must be a Calabi-Yau threefold.

Meanwhile, 10-dimensional superstring theory is divided into five types according to the choice of boundary conditions and quantum conditions that the worldsheet must satisfy. Among these, the direct stage for mirror symmetry is Type IIA and Type IIB superstring theory, which, as their names suggest, are closely related to each other. Type IIA string theory furnishes a Kähler structure and a complex structure on a Calabi-Yau threefold \(X\), while Type IIB string theory interchanges these two structures and defines a new Calabi-Yau threefold \(\check{X}\).

Therefore, if two distinct Calabi-Yau threefolds \(X\) and \(\check{X}\) appear as the Type IIA and Type IIB manifestations of a single theory, they yield a relation between the Kähler structure of \(X\) and the complex structure of \(\check{X}\). We call such a pair \((X, \check{X})\) a mirror pair, and the symmetry between them mirror symmetry.

This relationship was supported almost entirely by physicists’ intuition and had not been formulated in mathematical language, so in the early days it was not a particularly interesting problem for mathematicians outside mathematical physics. The situation changed at a mirror symmetry workshop held at MSRI in May 1991, where Candelas, de la Ossa, Green, and Parkes computed the number of degree \(d\) rational curves on a quintic Calabi-Yau threefold by transferring the calculation to \(\check{X}\) via the mirror symmetry assumption. There is an interesting anecdote here: initially, the values predicted by algebraic geometers through intersection theory differed from those predicted by physicists using this method. Later, a bug was found in the algebraic geometers’ code, and after correcting it and recalculating, the physicists’ computation turned out to be correct, causing mirror symmetry to emerge as a core research area in mathematics as well.

However, because physicists’ intuition fundamentally derived from quantum mechanical results, it was impossible to formalize it mathematically as such, and bringing it into mathematics required an appropriate formalism. The canonical framework that mathematicians generally accept is the Givental formalism; briefly put, it states that if one packages the Gromov–Witten invariants—the invariants of the A-model—into data called the \(J\)-function, and similarly packages the oscillating integrals—the invariants of the B-model—into the \(I\)-function, then these are equal to each other through an appropriate change of variables.

In the posts in this category we will explain these A-model and B-model separately, and based on this we will explore topics in mirror symmetry. In the remainder of this post, as motivation, we examine duality in toric varieties.

Hori-Vafa Mirror Construction

In the case of toric varieties (§Definition of Toric Varieties, ⁋Definition 3), mirror symmetry takes a very concrete form, so before starting the main discussion we examine how mirror symmetry works in this setting.

Let \(\Sigma\) be the fan of a smooth projective toric variety \(X=X_\Sigma\), and let \(v_1, \ldots, v_N \in N \cong \mathbb{Z}^n\) be the primitive generators of its one-dimensional cones. If \(\Sigma\) is a complete fan, the \(v_i\) span \(N_\mathbb{R}\). However, since \(N>n\), they are \(\mathbb{Z}\)-linearly dependent, and thus there exist \(r=N-n\) integral relations among them.

Definition 1 The charge matrix of \(X_\Sigma\) is the integer matrix

\[Q = (Q_{ji}) \in \Mat_{r \times N}(\mathbb{Z})\]

consisting of the coefficients of the integral relations among the above rays

\[\sum_{i=1}^N Q_{ji}\, v_i = 0,\qquad j = 1, \ldots, r.\]

Although the charge matrix is simply the matrix collecting the coefficients of the ray relations, if one writes \(X_\Sigma\) as the GIT quotient via the Cox construction

\[X_\Sigma \;=\; \big(\mathbb{C}^N \setminus Z\big) \,\big/\!/\, (\mathbb{C}^\ast)^r\]

then the \(j\)-th \((\mathbb{C}^\ast)\) factor acts on the Cox ring variables \(\x_i\) with weight \(Q_{ji}\), and these become important numbers determining the geometry of the toric variety.

Example 2 Write the rays of \(\mathbb{P}^n\) as

\[v_0=-e_1-\cdots-e_n,\quad v_i=e_i\qquad (i=1,\ldots, n).\]

There is a unique relation \(v_0 + v_1 + \cdots + v_n = 0\) among them, and hence the charge matrix is the \(1\times(n+1)\) matrix

\[Q = (1,\, 1,\, \ldots,\, 1) \in \Mat_{1 \times (n+1)}(\mathbb{Z}).\]

According to the explanation above, this encodes the standard scaling action of the torus on \(\mathbb{P}^n\),

\[t\cdot(\x_0,\ldots, \x_n)=(t \x_0, \ldots, t \x_n).\]

As a slightly nontrivial example, consider \(\mathbb{P}^1\times \mathbb{P}^1\). Its rays are given by \((\pm 1, 0)\), \((0, \pm 1)\), and the relations are \((1,0)+(-1,0)=0\) and \((0,1)+(0,-1)=0\), so the charge matrix is

\[Q = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix}.\]

This encodes the information that the torus acts by the standard scaling action on each of the first and second \(\mathbb{P}^1\) factors.

From the viewpoint of mirror symmetry, the charge matrix encodes the data of the \(B\)-model. One should be somewhat careful that the situation we are currently dealing with is more general than the Calabi-Yau manifold explained in the introduction. In general, a smooth projective toric variety \(X_\Sigma\) is given as a Fano variety rather than a Calabi-Yau, and in this case the mirror dual of \(X_\Sigma\) is expressed not as a Calabi-Yau but as a Landau-Ginzburg model.

Definition 3 A Landau-Ginzburg model is a pair \((\check{X}, W)\) of a complex manifold \(\check{X}\) and a holomorphic function \(W : \check{X} \to \mathbb{C}\) defined on it. We call \(W\) the superpotential.

The purpose of this post is to examine this phenomenon through light calculations before formally defining the concepts of mirror symmetry. Therefore, instead of explaining the data on both sides precisely, we replace this with brief ideas and intuition. First, from the \(B\)-model side, the charge matrix defines the Jacobi ring \(\Jac(W_q)\), which can be viewed as the classical limit of the oscillating integral mentioned earlier. For a given Landau-Ginzburg model \((\check{X}, W)\), its Jacobi ring is given by definition as

\[\Jac(W) = \frac{\mathcal{O}(\check{X})}{(\partial_1 W, \ldots, \partial_n W)}.\]

Here \(\x_1, \ldots, \x_n\) are local coordinates on \(\check{X}\), and the \(\partial_i\) are the partial derivatives with respect to them. Geometrically, \(\Jac(W)\) is the coordinate ring of the critical scheme \(\Crit(W) = \{dW = 0\} \subset \check{X}\) of \(W\). Then the mirror symmetry statement is that the Jacobi ring of the Hori-Vafa mirror in Definition 4 recovers the data of the original A-side model.

Definition 4 For a smooth projective toric Fano variety \(X_\Sigma\) and additional data \(q=(q_1,\ldots, q_r)\in \mathbb{C}^r\), the Hori-Vafa mirror defined by this is the following Landau-Ginzburg model.

1. The mirror domain \(\check{X}\) is the submanifold of the algebraic torus \((\mathbb{C}^\ast)^N\) defined as the set of points satisfying the \(r\) restrictions imposed by the charge matrix \(Q\):

   \[\x_1^{Q_{j1}} \cdots \x_N^{Q_{jN}} = q_j\in \mathbb{C}^\ast \qquad (j = 1, \ldots, r).\]

2. The superpotential on \(\check{X}\) is defined as the sum of local coordinates

   \[W_q : \check{X} \to \mathbb{C}, \qquad W_q(\x_1, \ldots, \x_N) = \x_1 + \x_2 + \cdots + \x_N.\]

Here \(q = (q_1, \ldots, q_r) \in (\mathbb{C}^\ast)^r\) is the variable carrying the complex structure of the mirror LG model. The complex structure of the mirror domain \(\check{X}\) itself is always the same affine torus \((\mathbb{C}^\ast)^n\), but the superpotential \(W_q\) placed on it is determined by \(q\). In other words, for each value of \(q\) a unique LG model \((\check{X}, W_q)\) is determined, and it is more accurate to say that the whole family \(\{(\check{X}, W_q)\}_q\) appears as the mirror of \(X_\Sigma\). At this point, the complex structure \(q\) appears as the Novikov parameter \(q\) in the A-model.

We previously described mirror symmetry as a symmetry between complex structure and symplectic structure, and the Novikov parameter explained above carries precisely the symplectic structure. Specifically, given a compact Kähler manifold \(X\), we define its Kähler form \(\omega\in H^2(X, \mathbb{R})\) as the symplectic form of \(X\). Since this is a real form and considering the moduli space is somewhat cumbersome, we choose \(B\in H^2(X, \mathbb{R})\) to form the complexified Kähler class

\[t = B + i\omega \in H^2(X, \mathbb{C}).\]

Intuitively, this fills in the Kähler form \(\omega\) in the complex direction to achieve complexification, and physically it means the \(B\)-field appearing in string theory. Now the Novikov parameter is precisely the exponential of this \(t\), and for a curve class \(\beta_0 \in H_2(X)\) it is given by

\[q^{\beta_0} = e^{2\pi i \int_{\beta_0} t} = e^{2\pi i \int_{\beta_0} B}\, e^{-2\pi \int_{\beta_0} \omega}.\]

Then the magnitude \(\lvert q^{\beta_0}\rvert = e^{-2\pi \int_{\beta_0} \omega}\) of \(q^{\beta_0}\) encodes the symplectic volume \(\int_{\beta_0} \omega\) of the curve class \(\beta_0\), and the phase \(\arg q^{\beta_0} = 2\pi \int_{\beta_0} B\) encodes the \(B\)-field. Therefore, when the symplectic volume goes to \(0\), the magnitude of \(q\) goes to \(1\) and quantum effects appear in full; conversely, when the symplectic volume goes to infinity, the magnitude of \(q\) goes to \(0\) and quantum effects disappear.

Now, from the above calculation, specifying a single \(q\) amounts to determining the complexified Kähler class \(t\), that is, specifying the \(B\)-field and \(\omega\) respectively. Then from the above formula, \(B\) has period \(2\pi\), and \(\omega\) determines the radius in the direction fixed by \(B\), so the moduli space of \(q\) (or of \(t\)) is the algebraic torus \((\mathbb{C}^\ast)^r\) of dimension \(r=\dim_\mathbb{R} H^2(X, \mathbb{R})\). However, since \(\omega\) is a Kähler form it must lie inside the Kähler cone (for an effective curve class \(\beta_0\) we have \(\int_{\beta_0} \omega > 0\), i.e. \(\lvert q^{\beta_0}\rvert < 1\)), so strictly speaking the moduli are not the whole torus but an open region near the large volume limit where \(q = 0\), and \((\mathbb{C}^\ast)^r\) is the ambient algebraic torus containing it.

As we saw above, \(q\) on the B-side appears as the coefficients of the superpotential. Intuitively, thinking of the case where the critical point equation obtained from this is of the form \(\x^k=q\), the solutions of \(\x^k=q\), i.e. the critical points, degenerate to a single point as \(q\) goes to \(0\), and in other cases one obtains singularities that are appropriately separated.

To write the mirror symmetry statement properly, we must define the (small) quantum cohomology of \(X\). Specifically, among the tools needed to examine the symplectic and complex structures of \(X\) are \(J\)-holomorphic curves. Using these, one can define the quantum cup product on the cohomology \(H^\ast(X, \mathbb{C})\) of \(X\) by the following formula

\[\alpha \star_q \beta \;=\; \alpha \smile \beta + \sum_{\beta_0} q^{\beta_0} \sum_\gamma \langle \alpha, \beta, \gamma^\vee \rangle_{0, 3, \beta_0}\, \gamma\]

and this structure gives the (small) quantum cohomology \(QH^\ast(X)\) of \(X\). Intuitively, if \(\alpha\smile \beta\) in the above formula encodes information about the intersection of the two classes \(\alpha, \beta\), then the remaining terms take into account “quantum” intersections that do not actually occur but meet via the curve class \(\beta_0\).

The mirror symmetry statement now claims that

\[\Jac(W_q) \cong QH^\ast(X_\Sigma)\]

holds. This statement fits the picture we already knew in many respects; for example, in the classical limit \(q\rightarrow 0\) the quantum cohomology ring returns to the classical cohomology ring, which corresponds in \(\Jac(W_q)\) to singularities clumping together to create a non-reduced singularity. Or, put the other way around, introducing quantum effects can be thought of on the A-side as resolving the classical cohomology using the Novikov variable \(q\), and on the B-side as smoothing out the clumped singularities.

In general, examining \(QH^\ast(X_\Sigma)\) on the right-hand side of the above isomorphism amounts to counting curves passing through given classes simultaneously, which is considered a relatively complex and difficult task, but mirror symmetry reduces this to a simple ring calculation. Let us verify that this actually holds in the two simple cases of \(\mathbb{P}^1\) and \(\mathbb{P}^2\).

Example 5 (\(\mathbb{P}^1\) case) We checked in Example 2 that the charge matrix of \(\mathbb{P}^1\) is \(Q = (1, 1)\). Hence the domain \(\check{X}\) of the Hori-Vafa mirror is the submanifold of \((\mathbb{C}^\ast)^2\) satisfying the equation

\[\x_0 \x_1 = q.\]

On this we have \(\x_0 = q/\x_1\), so the superpotential can be written as

\[W_q(\x_1) = \x_1 + \frac{q}{\x_1}\]

and its critical points are the solutions of

\[\partial_{\x_1} W_q = 1 - \frac{q}{\x_1^2} = 0,\]

namely the two points \(\x_1 = \pm\sqrt{q}\). From this one can check that the Jacobi ring is given by

\[\Jac(W_q) = \mathbb{C}[\x_1^\pm, q^\pm] / (\partial_{\x_1} W_q) \;\cong\; \mathbb{C}[H, q^\pm]/(H^2 - q),\qquad H := \x_1.\]

Meanwhile, the small quantum cohomology on the A-side is simple: since there is only one \(\mathbb{P}^1\) passing through three points, \(\langle H, H, H \rangle_{0,3,1}^{\mathbb{P}^1} = 1\), and the classical cup product \(H\smile H\) is \(0\) for dimensional reasons. That is, the quantum cup product becomes \(H \star_q H = q\), and from this the quantum cohomology is the graded \(\mathbb{C}[q]\)-polynomial algebra

\[QH^\ast(\mathbb{P}^1) = \mathbb{C}[H, q] \big/ (H^2 - q), \qquad \deg H = 2,\;\; \deg q = 4.\]

Now forgetting the grading and making \(q\) invertible, one can verify that this becomes exactly the same \(\mathbb{C}\)-algebra as the Jacobi ring above.

Let us look at \(\mathbb{P}^2\) as a slightly more complicated example.

Example 6 (\(\mathbb{P}^2\) case) The mirror dual of \(\mathbb{P}^2\) is the variety satisfying the equation

\[\x_0 \x_1 \x_2 = q\]

and the superpotential is given by

\[W_q(\z_1, \z_2) = \z_1 + \z_2 + \frac{q}{\z_1 \z_2}.\]

The critical points are now obtained by solving

\[\partial_{\z_1} W_q = 1 - \frac{q}{\z_1^2 \z_2} = 0, \qquad \partial_{\z_2} W_q = 1 - \frac{q}{\z_1 \z_2^2} = 0,\]

and their solutions are given by the three points satisfying \(\z_1=\z_2\) and \(\z_1^3=q\). Computing the Jacobi ring explicitly now gives

\[\Jac(W_q) = \mathbb{C}[\z_1^\pm, \z_2^\pm, q^\pm] \big/ (\partial_{\z_1} W_q, \partial_{\z_2} W_q) \;\cong\; \mathbb{C}[H, q^\pm]/(H^3 - q).\]

Meanwhile, to compute the quantum cohomology in the A-model one uses the following Gromov–Witten invariant:

\[\langle H, H^2, H^2 \rangle_{0,3,1}^{\mathbb{P}^2} = 1.\]

Geometrically this reflects the facts that (i) there is a unique \(\mathbb{P}^1 \subset \mathbb{P}^2\) passing through two generic points \(P_1, P_2 \in \mathbb{P}^2\), (ii) this line meets a generic line \(H_1 \subset \mathbb{P}^2\) at exactly one point, and (iii) the three points thus obtained uniquely determine \(f : \mathbb{P}^1 \xrightarrow{\sim} L\). Through this one knows that the quantum cohomology is the graded \(\mathbb{C}[q]\)-polynomial algebra

\[QH^\ast(\mathbb{P}^2) = \mathbb{C}[H, q] \big/ (H^3 - q), \qquad \deg H = 2,\;\; \deg q = 6.\]

In this case too, one can verify that the expected isomorphism works well.

More generally, the above two examples hold for an arbitrary smooth projective toric Fano variety. In the next post we will look at the Batyrev mirror, which extends this to Calabi-Yau hypersurfaces inside toric varieties.

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References

[CK] D. A. Cox, S. Katz, Mirror Symmetry and Algebraic Geometry, Mathematical Surveys and Monographs 68, AMS, 1999.

[MS] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, E. Zaslow, Mirror Symmetry, Clay Mathematics Monographs 1, AMS, 2003.