Mirror Symmetry: An Overview

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 did not arise naturally from within mathematics itself; rather, it is rooted in superstring theory. In superstring theory, the fundamental degrees of freedom are not point particles but one-dimensional strings. Consequently, when a particle moves along the time axis, its trajectory in spacetime is no longer a \(1\)-dimensional worldline but a \(2\)-dimensional worldsheet, and its equation of motion is determined by a specific action-minimizing solution, just as in [Symplectic Geometry] §Classical Mechanics, §§Principle of Least Action. To reconcile this interpretation with the existing framework of quantum mechanics, spacetime is forced to be \(10\)-dimensional; thus physicists regard this \(10\)-dimensional spacetime as the product of a \(4\)-dimensional Minkowski spacetime and a compact manifold \(X\) that accounts for the remaining \(6\) dimensions. Writing out the physical conditions that this space \(X\) must satisfy, one finds that \(X\) must be a Calabi-Yau threefold.

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

Therefore, if two different Calabi-Yau threefolds \(X\) and \(\check{X}\) arise as the Type IIA and Type IIB manifestations of a single theory, they should exhibit a relationship 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 its 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, when Candelas, de la Ossa, Green, and Parkes used the mirror symmetry assumption to transfer the computation of the number of degree \(d\) rational curves on a quintic Calabi-Yau threefold to a calculation on \(\check{X}\). An interesting anecdote surrounds this: initially, the values predicted by algebraic geometers via intersection theory differed from those predicted by the physicists. Subsequently, 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 central research area in mathematics as well.

However, since physicists’ intuition fundamentally derives from results in quantum mechanics, it was impossible to formalize this directly in mathematics, and an appropriate formalism was needed to import it. The canonical framework universally accepted by mathematicians is the Givental formalism. Briefly put, this packages the Gromov–Witten invariants (the A-model invariants) into data called the \(J\)-function, and similarly packages the oscillating integrals (the B-model invariants) into the \(I\)-function; these are then identified via an appropriate change of variables.

In the posts of this category we will explain the A-model and B-model in turn, 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 ([Toric Geometry] §Definition of Toric Varieties, ⁋Definition 3), mirror symmetry takes a very concrete form, so before embarking on the full story we examine how mirror symmetry works in this setting.

Let the fan of a smooth projective toric variety \(X=X_\Sigma\) be \(\Sigma\), and let the primitive generators of its \(1\)-dimensional cones be \(v_1, \ldots, v_m \in \mathbb{Z}^n\). If \(\Sigma\) is a complete fan, then the \(v_i\) span \(\mathbb{R}^n\). However, since \(m>n\), they are \(\mathbb{Z}\)-linearly dependent, and hence there exist \(r=m-n\) integral relations among them.

Definition 1 The charge matrix of \(X_\Sigma\) is the integer matrix formed from the coefficients of the integral relations

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

among the rays above:

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

Here the rows of \(Q\) are chosen to form a \(\mathbb{Z}\)-basis of the kernel (that is, the relation lattice) of the morphism \(\mathbb{Z}^m \rightarrow \mathbb{Z}^n\) defined by the \(v_i\); consequently \(Q\) is uniquely determined up to left multiplication by \(\GL_r(\mathbb{Z})\).

Although the charge matrix is simply the matrix collecting the coefficients of the ray relations, when we write \(X_\Sigma\) via the Cox construction as a GIT quotient

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

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

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

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

Among these there is a unique relation \(v_0 + v_1 + \cdots + v_n = 0\), 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 \((0,1)+(0,-1)=0\), two in total; hence the charge matrix becomes

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

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

From the perspective of mirror symmetry, the charge matrix carries the data of the \(B\)-model. One must be careful, however, that the situation we are currently treating is more general than the Calabi-Yau manifolds explained in the introduction. A smooth projective toric variety \(X_\Sigma\) can never be Calabi-Yau, and in this post we treat the Fano case among these. In this case the mirror dual of \(X_\Sigma\) is represented not by a Calabi-Yau but by a Landau-Ginzburg model.

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

The purpose of this post is to examine this phenomenon through light computations before defining the concepts of mirror symmetry in earnest. Therefore, instead of explaining the data on both sides precisely, we substitute 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 above. 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 \(\partial_i\) are the partial derivatives with respect to these. Geometrically \(\Jac(W)\) is the coordinate ring of the critical scheme \(\Crit(W) = \{\dd{W} = 0\} \subseteq \check{X}\) 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 \(q=(q_1,\ldots, q_r)\in (\mathbb{C}^\ast)^r\), the Hori-Vafa mirror defined by this data is the following Landau-Ginzburg model.

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

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

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

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

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 \(q\). That is, for each value of \(q\) a unique LG model \((\check{X}, W_q)\) is determined, and it is more accurate to say that the entire family \(\{(\check{X}, W_q)\}_q\) appears as the mirror of \(X_\Sigma\). Here the complex structure \(q\) appears as the Novikov parameter \(q\) in the A-model.

We explained mirror symmetry earlier as a symmetry between complex structure and symplectic structure, and the Novikov parameter described above is precisely what carries 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, it is somewhat cumbersome to consider a moduli space directly; thus we choose \(B\in H^2(X, \mathbb{R})\) and form the complexified Kähler class

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

Intuitively this fills out the Kähler form \(\omega\) in the complex direction to perform a complexification, and physically it corresponds to the \(B\)-field appearing in superstring theory. Now the Novikov parameter is exactly the exponential of this \(t\), given for a curve class \(\beta_0 \in H_2(X)\) 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 \(q^{\beta_0}\) carries the symplectic volume \(\int_{\beta_0} \omega\) of the curve class \(\beta_0\), and the phase \(\arg q^{\beta_0} = 2\pi \int_{\beta_0} B\) carries the \(B\)-field. Hence when the symplectic volume goes to \(0\), the magnitude of \(q\) goes to \(1\) so that quantum effects appear in full, while conversely when the symplectic volume goes to infinity, the magnitude of \(q\) goes to \(0\) so that quantum effects disappear.

Now in the computation above, fixing a single \(q\) is the same as determining the complexified Kähler class \(t\), i.e. fixing the \(B\)-field and \(\omega\) respectively. Then from the formula above \(B\) has period \(1\), and \(\omega\) determines the radius in the direction fixed by \(B\), so the moduli space of \(q\) (or of \(t\)) becomes the algebraic torus \((\mathbb{C}^\ast)^r\) with \(r=\dim_\mathbb{R} H^2(X, \mathbb{R})\). 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 space is 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.

On the B-side, \(q\) appeared as the coefficient of the superpotential, as we saw above. Thinking intuitively of the case where the critical point equation obtained from this is of the form \(\x^k=q\), the solutions of \(\x^k=q\) (that is, the critical points) degenerate to a single point as \(q\) goes to \(0\), while for nonzero \(q\) one gets singularities that are appropriately separated.

To state the mirror symmetry correspondence properly, we now need to 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, we can define the quantum cup product on the cohomology \(H^\ast(X, \mathbb{C})\) of \(X\) by the formula

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

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

Now the mirror symmetry statement asserts that

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

This statement matches the picture we already knew in several respects; for example, in the classical limit where \(q\rightarrow 0\), the quantum cohomology ring returns to the classical cohomology ring, which from the viewpoint of \(\Jac(W_q)\) corresponds to the singularities clumping together to produce a degenerate non-reduced singularity. Conversely, 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 regarded as a relatively complex and difficult task, but mirror symmetry reduces this to a simple ring computation. Let us verify that this actually holds in the two simple cases \(\mathbb{P}^1\) and \(\mathbb{P}^2\).

Example 5 (\(\mathbb{P}^1\) case) In Example 2 we checked 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

\[\x_0 \x_1 = q.\]

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 exactly one \(\mathbb{P}^1\) passing through three points, we have \(\langle H, H, H \rangle_{0,3,1}^{\mathbb{P}^1} = 1\), and the classical cup product \(H\smile H\) is \(0\) for dimensional reasons. Hence the quantum cup product becomes \(H \star_q H = q\), and from this the quantum cohomology is the graded \(\mathbb{C}[q]\)-polynomial algebra

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

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

As a slightly more complicated example, consider \(\mathbb{P}^2\).

Example 6 (\(\mathbb{P}^2\) case) The mirror dual of \(\mathbb{P}^2\) satisfies

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

and the superpotential is given by

\[W_q(\x_1, \x_2) = \x_1 + \x_2 + \frac{q}{\x_1 \x_2}.\]

Now the critical points are obtained by solving

\[\partial_{\x_1} W_q = 1 - \frac{q}{\x_1^2 \x_2} = 0, \qquad \partial_{\x_2} W_q = 1 - \frac{q}{\x_1 \x_2^2} = 0,\]

and their solutions are given by the three points satisfying \(\x_1=\x_2\), \(\x_1^3=q\). Now computing the Jacobi ring explicitly gives

\[\Jac(W_q) = \mathbb{C}[\x_1^\pm, \x_2^\pm, q^\pm] \big/ (\partial_{\x_1} W_q, \partial_{\x_2} W_q) \cong \mathbb{C}[H, q^\pm]/(H^3 - q).\]

Meanwhile, to compute the quantum cohomology in the A-model it suffices to use 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 exists a unique line \(L \cong \mathbb{P}^1 \subseteq \mathbb{P}^2\) passing through two generic points \(P_1, P_2 \in \mathbb{P}^2\), (ii) this line meets a generic line \(H_1 \subseteq \mathbb{P}^2\) at exactly one point, and (iii) the three points thus obtained uniquely determine \(f : \mathbb{P}^1 \xrightarrow{\sim} L\). From this one knows that the quantum cohomology is determined as the graded \(\mathbb{C}[q]\)-polynomial algebra

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

In this case as well, one can check that the expected isomorphism holds.

More generally, the two examples above hold for an arbitrary smooth projective toric Fano variety. In the next post we will examine 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.