Givental J-function and Mirror Theorem

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.

In §Dubrovin Connection, §§D-module we showed that the quantum differential equation (QDE) on the quantum cohomology side admits a flat section of dimension $\dim_\mathbb{C} H^\ast(X)$ in its space of solutions, and we announced that the explicit fundamental solution is Givental’s $J$-function. Meanwhile, in §Gauss-Manin Connection, ⁋Proposition 7 we examined how the period matrix $\mathcal{I}^a_p$ on the B-side forms the fundamental solution matrix of the B-model connection $\nabla^z_B$. The mirror theorem we foreshadowed will be concretized as the statement that the A-side $J$-function and the B-side period matrix are fundamental solutions of the same $D$-module, and this post will fill in the missing A-side slot in that claim.

Descendant Gromov-Witten Invariant

Before beginning the definition of the $J$-function, let us briefly summarize the descendant invariant used throughout this post, adapted to our notation from ([Symplectic Geometry] §Gromov-Witten Invariants, ⁋Definition 2). On the moduli space of genus $0$, $(n+1)$-marked, class $\beta$ stable maps

\[\overline{\mathcal{M}}_{0, n+1}(X, \beta)\]

there are evaluation maps $\ev_i: \overline{\mathcal{M}}_{0, n+1}(X, \beta) \to X$ at each marked point $i$, and the universal cotangent line bundle $\mathbb{L}_i$. Intuitively, $\mathbb{L}_i$ is the cotangent line at the $i$-th marked point, assembled over the moduli space; the first Chern class of this universal cotangent line bundle

\[\psi_i := c_1(\mathbb{L}_i) \in H^2(\overline{\mathcal{M}}_{0, n+1}(X, \beta))\]

is called the $\psi$-class.

Definition 1 (Descendant Gromov-Witten Invariant) For arbitrary cohomology classes $\gamma_i \in H^\ast(X)$ and $k_i \geq 0$, the descendant Gromov-Witten invariant is defined by

\[\left\langle \tau_{k_1}(\gamma_1), \ldots, \tau_{k_{n+1}}(\gamma_{n+1})\right\rangle_{0, n+1, \beta} := \int_{[\overline{\mathcal{M}}_{0, n+1}(X, \beta)]^{\mathrm{vir}}} \prod_{i=1}^{n+1} \psi_i^{k_i} \smile \ev_i^\ast \gamma_i\]

When all $k_i = 0$, it is called a primary invariant; if at least one $k_i \geq 1$, it is called a gravitational descendant.

Although dealing with moduli spaces requires a slight upgrade in language, the Chern class of a stack is essentially the same intuitive quantity as the Chern class of an ordinary line bundle. That is, $\mathbb{L}_i$ measures how twisted the bundle is over the moduli space. More concretely, given a $2$-cycle $\Sigma$ in the moduli base, the quantity obtained by pairing with $\psi_i$

\[\int_\Sigma \psi_i\in \mathbb{Z}\]

is the same as looking at the monodromy action: when we traverse $\mathbb{L}_i$ once around $\Sigma$, it tells us how twisted the bundle is before returning to the original fiber. In the simplest case, when this value is $1$, it means the degree of $\mathbb{L}_i$ restricted to $\Sigma$ is $1$; more intuitively, we can interpret this as a phase factor, i.e., monodromy, being concentrated at a virtual point on $\Sigma$. Of course there is freedom in choosing this point, but this comes exactly from choosing a representative of the cohomology class of $\psi_i$, and thus more generally $\int_\Sigma\psi_i = n$ corresponds to arranging $n$ such points.

A well-known fact is that these $n$ points can be thought of as the intersections of $\Sigma$ with the boundary divisors of the moduli space, i.e., the loci where the source curve degenerates; thus $\psi_i$ can be thought of as carrying information about the degeneration of the source curve.

Then it is obvious that in Definition 1 (Descendant Gromov-Witten Invariant), the $\psi_i^{k_i}$ components determine the degeneration of the source curve, while $\ev_i^\ast\gamma_i$ determines the incidence condition on the target. In particular, the latter condition is that the image of the $i$-th marked point $p_i$ passes through $\gamma_i$. However, the first condition is somewhat less intuitive, so let us unpack its meaning.

We have seen above that $\psi_i$ can essentially be represented as a sum of degenerations of the source curve. More concretely, suppose the source curve degenerates by splitting the marked points $1,\ldots, n$ into two components $S$ and $S^c$, and let $D_S$ be the corresponding boundary divisor. Here we assume $i\in S$, and call the component containing $i$ the tail. To make $\mathbb{P}^1$ stable, three points are needed, so both $S$ and $S^c$ must contain at least two points.

Intuitively, $\psi_i$ can be thought of as the sum $\sum_{i\in S} D_S$ of boundary divisors arising in this manner, but if left as is, the same cohomology class would be counted multiple times, so a reference choice is needed to remove redundancy. That is, fixing $j,k$, we must kill the automorphism of this $\mathbb{P}^1$ by taking the component corresponding to $S^c$ together with $j,k$ and the nodal point.

Proposition 2 (Genus 0 Topological Recursion Relation) For fixed three indices $i,j,k$ of $\{1, \ldots, n\}$ ($n\geq 4$), the $\psi$-class $\psi_i$ is represented as the sum of the following boundary divisors:

\[\psi_i=\sum_{\substack{S \subset \{1, \ldots, n\} \\ i \in S, j, k \notin S, \lvert S\rvert \geq 2}} D_S \in H^2(\overline{\mathcal{M}}_{0, n})\]

Proof

When $n = 4$, we have $\overline{\mathcal{M}}_{0,4} \cong \mathbb{P}^1$ and $\psi_i$ is the point class of degree $1$, which is linearly equivalent to the unique boundary divisor $D_{\{i, l\}}$ separating $i$ from $j, k$ (where $l$ is the fourth index), so it matches the right-hand side for $n = 4$. For general $n$, comparing cotangent line classes under the forgetful morphism $\pi: \overline{\mathcal{M}}_{0,n} \to \overline{\mathcal{M}}_{0,4}$ that forgets all but $i, j, k$ and one point, we obtain that $\psi_i$ differs from $\pi^\ast \psi_i$ by the boundary where $i$ is separated from $j, k$; iterating this yields the boundary sum on the right-hand side. This is a standard fact for $\overline{\mathcal{M}}_{0,n}$ ([CK, §10]).

Then for the moduli space of stable maps $\overline{\mathcal{M}}_{0, n}(X, \beta)$ with target $X$ given, we can pull back the above formula via the forgetful morphism $\overline{\mathcal{M}}_{0, n}(X, \beta) \to \overline{\mathcal{M}}_{0, n}$.

Now we can understand $\psi_i^{k_i}$ more clearly. From this perspective, $\psi_i^{k_i}$ means that the part corresponding to marked point $p_i$ degenerates $k_i$ times, giving a degenerate cycle belonging to the tail; putting this all together, the descendant GW invariant can be thought of as the virtual counting of stable maps simultaneously satisfying both the target incidence condition and the source depth-$k_i$ tail degeneration condition.

To consider all these gravitational invariants at once, we introduce the formal geometric series

\[\frac{1}{z - \psi} = \frac{1}{z}\cdot\frac{1}{1 - \psi/z} = \sum_{k \geq 0} z^{-k-1}\psi^k\]

For example, inserting the $\psi$ class only in the last factor,

\[\begin{aligned}\left\langle \gamma_1, \ldots, \gamma_n, \frac{\gamma_{n+1}}{z - \psi}\right\rangle_{0, n+1, \beta}&=\left\langle \gamma_1, \ldots, \gamma_n, \sum_{k \geq 0} z^{-k-1}\psi^k\gamma_{n+1}\right\rangle_{0, n+1, \beta}=\sum_{k\geq 0}z^{-k-1}\left\langle \gamma_1, \ldots, \gamma_n, \psi^k\gamma_{n+1}\right\rangle_{0, n+1, \beta}\\&=\sum_{k \geq 0} z^{-k-1} \left\langle \gamma_1, \ldots, \gamma_n, \tau_k(\gamma_{n+1})\right\rangle_{0, n+1, \beta}\end{aligned}\]

so the descendant invariant at each depth $k$ is separated as the coefficient of $z^{-k-1}$ and captured all at once. At first glance this is just an infinite sum on the right-hand side, but in the next section, when we introduce the $J$-function, we will see that even the $z$ on the right-hand side carries a specific meaning.

Introduction of the $J$-Function

Now we introduce the $J$-function using the gravitational invariants examined above. Consider a smooth projective variety $X$, a homogeneous basis $\{ T_a \}_{a=0,\ldots,s}$ of $H^\ast(X, \mathbb{C})$ ($T_0 = 1$), and their Poincaré dual basis $\{ T^a \}$. Taking the $H^2$ components as $\{ T_a \}_{a=1,\ldots,r}$ (for notational simplicity), the flat coordinates $t^a$ and Novikov variables $q_a := e^{t^a}$ ($a = 1, \ldots, r$) introduced in §Dubrovin Connection, §§D-module are defined. For convenience, let

\[t_{(2)} := \sum_{b=1}^r t^b T_b\]

then $q^\beta = e^{t_{(2)} \cdot \beta}$. Meanwhile, the Dubrovin connection was defined by the formula

\[\nabla^z = \partial + z^{-1}\mathcal{C}\]

but in this post, to match signs with the oscillating integral side, we consider its dual Dubrovin connection

\[\nabla^{z, \vee} := -\nabla^{-z} = \partial - z^{-1}\mathcal{C}\]

and the horizontal section equation of this connection

\[z q_a\partial_{q_a} s = T_a \qtimes s \qquad (a = 1, \ldots, r)\tag{$\ast$}\]

To find the solution to this equation, first consider the situation as $z\rightarrow \infty$. Then as $z\rightarrow\infty$, the $z^{-1}\mathcal{C}$ term vanishes and the equation degenerates to the standard differential $\partial$, so the leading-order horizontal section of ($\ast$) becomes a constant section independent of $q$. More generally, the solution vanishing in the $z\rightarrow \infty$ limit should be of the form

\[s=s_0+\frac{s_1}{z}+\frac{s_2}{z^2}+\cdots\]

and from the above calculation we saw that the $0$-th order term $s_0$ can be taken as a constant section, so substituting the form

\[s=1+\frac{s_1}{z}+\frac{s_2}{z^2}+\cdots\]

directly into ($\ast$) allows us to determine the $s_i$ recursively. Specifically, organizing both sides of ($\ast$) by powers of $z$, since $s_0$ is constant the left-hand side is

\[z \cdot \sum_k z^{-k} q_a\partial_{q_a} s_k = \sum_k z^{-k} q_a\partial_{q_a} s_{k+1}\]

and the right-hand side is $\sum_k z^{-k} T_a\qtimes s_k$, so comparing coefficients of each $z^{-k}$ yields the general recursion

\[q_a\partial_{q_a} s_{k+1} = T_a \qtimes s_k \qquad (a = 1, \ldots, r, k \geq 0)\]

At the first step of the recursion, $k = 0$, we have $T_a \qtimes 1 = T_a$, so without any quantum correction, $q_a\partial_{q_a} s_1 = T_a$ holds. Integrating both sides with respect to $q_a$ gives

\[s_1=t_{(2)} + C_1,\qquad C_1\in H^\ast(X)\]

The interesting part is where the quantum correction first appears, at $k=1$. Considering the recursion formula

\[q_a\partial_{q_a} s_2 = T_a \qtimes t_{(2)}=T_a\qtimes \left(\sum_{b=1}^r t^b T_b\right)\]

the right-hand side now has, in addition to the classical cup product $T_a\smile t_{(2)}$, the quantum correction

\[\sum_{\beta \neq 0} q^\beta \sum_c \left(\sum_{b=1}^r t^b \langle T_a, T_b, T^c\rangle_{0, 3, \beta}\right) T_c\]

Since $T_b \in H^2$, applying the result of [Symplectic Geometry] §Gromov-Witten Invariants, ⁋Proposition 4

\[\langle T_a, T_b, T^c\rangle_{0, 3, \beta} = (T_b \cdot \beta)\langle T_a, T^c\rangle_{0, 2, \beta}\]

we obtain the formula

\[T_a \qtimes t_{(2)} = T_a \smile t_{(2)} + \sum_{\beta \neq 0} q^\beta (t_{(2)} \cdot \beta) \sum_c \langle T_a, T^c\rangle_{0, 2, \beta} T_c\]

Now let us integrate this with respect to $q_a$. The antiderivative of the classical part $T_a \smile t_{(2)}$ is $(t_{(2)})^2/2$, which can be directly verified using $q_a\partial_{q_a} = \partial_{t^a}$:

\[\partial_{t^a}\bigl((t_{(2)})^2/2\bigr) = \partial_{t^a}\!\left(\frac{1}{2}\sum_{b, c} t^b t^c T_b \smile T_c\right) = \sum_c t^c T_a \smile T_c = T_a \smile t_{(2)}\]

The antiderivative of the quantum part is solved $\beta$-by-$\beta$ using the relation $q_a\partial_{q_a} q^\beta = (T_a \cdot \beta) q^\beta$ coming from $q^\beta = e^{t_{(2)}\cdot \beta}$, and for each $\beta$ it is organized as an $H^\ast(X)$-valued correction determined by the $q^\beta$ factor and the primary GW invariant (descendant invariant without $\psi$-class insertion) $\langle T_a, T^c\rangle_{0, 2, \beta}$. Thus $s_2$ becomes the sum of the classical $(t_{(2)})^2/2$ and this quantum correction. At higher orders $z^{-k}$ ($k \geq 2$), the same recursion accumulates gravitational descendants of the form $\tau_{k-1}(T_a)$ in sequence, and the $J$-function is ultimately the explicit one-line formula for the fundamental solution forced in this manner.

Definition 3 The (small) Givental $J$-function $J_X: (\mathbb{C}^\ast)^r \times \mathbb{C}^\ast \to H^\ast(X)$ of $X$ is defined by

$J_X(q, z) := e^{t_{(2)}/z}\left( 1 + \sum_{\substack{\beta \in H_2(X, \mathbb{Z})_{\mathrm{eff}} \\ \beta \neq 0}} \sum_{a=0}^s q^\beta \left\langle \frac{T_a}{z(z - \psi)} \right\rangle_{0, 1, \beta} T^a \right)$

Here $H_2(X, \mathbb{Z})_{\mathrm{eff}}$ is the set of effective curve classes (defined in [Symplectic Geometry] §Quantum Cohomology, §§Novikov Ring), and as $\beta$ ranges over this set, each $\beta \neq 0$ contributes to the instanton correction of order $q^\beta$.

Since we have already verified the formula

\[\left\langle \gamma_1, \ldots, \gamma_n, \frac{\gamma_{n+1}}{z - \psi}\right\rangle_{0, n+1, \beta}=\sum_{k \geq 0} z^{-k-1} \left\langle \gamma_1, \ldots, \gamma_n, \tau_k(\gamma_{n+1})\right\rangle_{0, n+1, \beta}\]

the above $J$-function can be expanded as

\[J_X(q, z) = e^{t_{(2)}/z}\left(\mathbf{1} + \sum_{\substack{\beta \in H_2(X, \mathbb{Z})_{\mathrm{eff}} \\ \beta \neq 0}}\sum_{a = 0}^s \sum_{k \geq 0} q^\beta z^{-k-2} \langle\tau_k(T_a)\rangle_{0, 1, \beta} T^a\right)\]

Thus, this function can be thought of as a generating function obtained by attaching the single marked point descendant invariants

\[\bigl\{\langle\tau_k(T_a)\rangle_{0, 1, \beta}\bigr\}_{\beta \in H_2(X, \mathbb{Z})_{\mathrm{eff}} \setminus \{0\}, a = 0, \ldots, s, k \geq 0}\]

to the Novikov parameter $q^\beta$, the spectral parameter $z^{-k-2}$, and the cohomology basis elements $T^a$.

Note that while the descendant invariants we defined above are generally of the multi-marked point form

\[\left\langle \tau_{k_1}(\gamma_1), \ldots, \tau_{k_{n+1}}(\gamma_{n+1})\right\rangle_{0, n+1, \beta}\]

in the above we only put single marked points into the formula. This is possible because $\beta\neq 0$, so the target is sufficiently large that stability is guaranteed regardless of the source’s stability, and the reason it suffices to look at only these is that we are primarily thinking of the $H^2$ deformation, i.e., small quantum cohomology.

More concretely, we already saw from [Symplectic Geometry] §Gromov-Witten Invariants, ⁋Proposition 4 that inserting a divisor $D\in H^2(X)$ into a Gromov-Witten invariant has the effect of multiplying by $D\cdot \beta$, and expanding this as a power series, looking at the $H^2$ direction is nothing more than multiplying by $e^{t_{(2)}\cdot \beta}$ in front, i.e., reparametrizing the Novikov variable as $q^\beta \mapsto e^{t_{(2)}\cdot \beta}q^\beta$. In Definition 3 we further multiplied by an additional $z^{-1}$ in the $T_a/(z-\psi)$ term,¹ so reflecting this $z$-shift as well, the overall factor attached in front of the $J$-function becomes $e^{t_{(2)}/z}$. Once all the $H^2$ dependence is put into $e^{t_{(2)}/z}$ and the Novikov variable, the Gromov-Witten information is condensed into the remaining single marked point, so it suffices to look at only single marked point invariants.

Meanwhile, let us recall that our motivation for introducing the $J$-function was to find the fundamental solution of the QDE ($\ast$). In §Dubrovin Connection, §§D-module we saw that the A-model $D$-module is a bundle with fiber $H^\ast(X)$ at each point $(q, z)$ of the base $M_A = (\mathbb{C}^\ast)^r \times \mathbb{C}^\ast_z$, and the connection $1$-form $\mathcal{C}_a = T_a \qtimes -$ is an endomorphism on this fiber, whose solutions are precisely the horizontal sections of ($\ast$). These solutions are $\dim_\mathbb{C}H^\ast(X)$-dimensional, and if we solve this equation with the $T_a$ as initial values, we obtain all the solutions. Through this process we obtain the following fundamental solution matrix.

Proposition 4 (A-side Fundamental Solution Matrix) Define the endomorphism $S(q, z) \in \End(H^\ast(X))$ by the matrix element

\[\eta\bigl(S(q,z)T_a, T_b\bigr) := \eta(T_a, T_b) + \sum_{\beta \neq 0} q^\beta \left\langle \frac{T_a}{z - \psi}, T_b \right\rangle_{0, 2, \beta}\]

where $\eta$ is the Poincaré pairing. Then $S$ satisfies the following.

(Flat section property) Each column $S(q, z) T_b$ of $S(q, z)$ is a horizontal section of the dual small Dubrovin connection $\nabla^{z, \vee}$. That is,
\[z q_a\partial_{q_a} \bigl(S(q,z)T_b\bigr) = T_a \qtimes \bigl(S(q,z)T_b\bigr)\qquad (a = 1, \ldots, r)\]
($J$ = $T_0$ column) $J_X(q, z) = e^{t_{(2)}/z} S(q, z) T_0$ holds. In particular, $J$ itself is also a horizontal section of ($\ast$), and
\[z q_a\partial_{q_a} J_X = T_a \qtimes J_X \qquad (a = 1, \ldots, r)\]

Proof

First, let us show that substituting $T_b = T_0 = 1$ in the second position of the defining equation of $S$ yields the $J = T_0$ column. In this post we consistently follow the convention $1/(z - \psi) = \sum_{k \geq 0} z^{-k-1}\psi^k$, and accordingly we keep the divisor equation correction term with a $+$ sign (in some literature $1/(z + \psi)$ is used with the opposite sign). Now the 2-point descendant in the definition is of the form with $T_0 = 1$ inserted, and expanding this as a geometric series,

\[\left\langle \frac{T_a}{z - \psi}, 1\right\rangle_{0, 2, \beta} = \sum_{k \geq 0} z^{-k-1}\langle \tau_k(T_a), 1\rangle_{0, 2, \beta}\]

Applying [Symplectic Geometry] §Gromov-Witten Invariants, ⁋Proposition 3 to each term,

\[\langle \tau_k(T_a), 1\rangle_{0, 2, \beta} = \langle \tau_{k-1}(T_a)\rangle_{0, 1, \beta}\]

holds for all $k\geq 1$. The $k = 0$ term vanishes as $\langle T_a, 1\rangle_{0, 2, \beta} = 0$ ($\beta \neq 0$) because there is no $\psi$ to lower in the string equation right-hand side (this is precisely the fact that a marked point evaluating the fundamental class does not contribute when $\beta \neq 0$). Thus the power series shifts down by one degree, giving

\[\sum_{k \geq 1} z^{-k-1}\langle \tau_{k-1}(T_a)\rangle_{0, 1, \beta} = \sum_{m \geq 0} z^{-m-2}\langle \tau_m(T_a)\rangle_{0, 1, \beta} = \left\langle \frac{T_a}{z(z - \psi)}\right\rangle_{0, 1, \beta}\]

This is exactly the term appearing in the small $J$ formula of Definition 3, and multiplying by the prefactor $e^{t_{(2)}/z}$ gives $J_X = e^{t_{(2)}/z} S(q, z) T_0$. This is the origin of the additional $z^{-1}$ that may have appeared somewhat awkward in Definition 3.

Now let us show that each column $S(q,z)T_b$ of $S$ is a horizontal section. For this, we directly apply $z q_a\partial_{q_a}$ to $S(q, z) T_b$. First, $S(q, z) T_b$ is expressed as a sum of terms with $q^\beta$ as a factor for each $\beta$; using the recursion relation examined before introducing the $J$-function and [Symplectic Geometry] §Gromov-Witten Invariants, ⁋Proposition 4, $z q_a\partial_{q_a}$ ultimately inserts $T_a$ at a point while, through the multiplied $z$, lowering the degree of $\psi$ in $1/(z - \psi)$ at the distinguished marked point by one.

Now this lowered $\psi$ can be decomposed into the sum of boundary divisors $\sum_S D_S$ using Proposition 2 (Genus 0 Topological Recursion Relation) (more precisely, the version of Proposition 2 (Genus 0 Topological Recursion Relation) with the target lifted). On each $D_S$, the source curve splits into two components with effective class $\beta = \beta_1 + \beta_2$, and at the node between them the diagonal class $\sum_c T_c \otimes T^c$ is inserted. The component where the newly inserted $T_a$ lands gives a 3-point invariant $\langle T_a, T_c, T_d\rangle_{0, 3, \beta_1}$, i.e., the structure constant expressing the quantum product $T_a \qtimes$ in the basis, while the other component reconstructs the (lower degree) $\beta_2$ part of $S(q, z) T_b$ having $T^c$ at the node. Thus summing over all $c$ and splittings $\beta = \beta_1 + \beta_2$ reproduces exactly $T_a \qtimes$ applied to the vector $S(q, z) T_b$, i.e., the right-hand side $T_a \qtimes \bigl(S(q, z) T_b\bigr)$.

Therefore, as $q\rightarrow 0$, all $q^\beta$ terms vanish, so the classical limit of $S$ is $\id$. On the A-side, this can be thought of as a consequence of the fact that the classical limit of the quantum cup product is the ordinary cup product.

Proposition 4 (A-side Fundamental Solution Matrix) shows that the $J$-function is more than a mere bundle of enumerative data: it forms a column of the A-side fundamental solution matrix $S$ (specifically, the column corresponding to the normalization element $T_0 = 1$). This is precisely the counterpart to the fact that in §Gauss-Manin Connection, ⁋Proposition 7, the period matrix $\mathcal{I}$ formed the fundamental solution matrix of $\nabla^z_B$ trivialized by the frame $\{e_a\}$.

In particular, when $H^\ast(X)$ is generated by $H^2(X)$, for example when $X = \mathbb{P}^n$ or most toric Fano varieties, this single column $J$ in fact determines all of $S$. Indeed, repeatedly applying the flat section equation $z q_a\partial_{q_a} J = T_a \qtimes J$ from Proposition 4 (A-side Fundamental Solution Matrix) generates the quantum products $T_{a_1} \qtimes \cdots \qtimes T_{a_k} \qtimes J$ of $H^2$ classes in sequence; since $H^\ast(X)$ is generated by $H^2$, these quantum products sweep through all $T_b$ as a basis of cohomology, so the remaining columns $S(q, z) T_b$ are also all reconstructed from differentials of $J$. This justifies in Conjecture 5 (Mirror theorem, $J$-function form) why our “Mirror theorem” only makes a claim about the first column.

Meanwhile, this calculation is also related to the handling of integration constants obtained by repeatedly integrating from ($\ast$): namely, Proposition 4 (A-side Fundamental Solution Matrix)’s $S$ is the solution where these integration constants are set to $0$ at all orders. That is, after stripping off the components coming from $t_{(2)}$ as the prefactor $e^{t_{(2)}/z}$, all remaining corrections have only instanton orders $\beta \neq 0$, so that the classical limit as $q \to 0$ emerges as above.

Mirror Theorem

We have independently constructed the A-side fundamental solution matrix $S$ (Proposition 4 (A-side Fundamental Solution Matrix)) and the B-side period matrix $\mathcal{I}$ (§Gauss-Manin Connection, ⁋Proposition 7). The insight of the mirror theorem is that these two matrices actually coincide.

Conjecture 5 (Mirror theorem, $J$-function form) For a mirror pair $(X, \check{X})$, the $J$-function of $X$ is written in terms of the cohomology basis $\{T^a\}$ components of the oscillating integral over a specific Lefschetz thimble $\Gamma_0$ (the distinguished thimble determined in the neighborhood of the large radius limit), corresponding to the $T_0 = 1$ normalization. That is,

\[J_X(q, z) = \sum_a J^a(q, z) T^a,\qquad J^a(q, z) \;\propto\; \mathcal{I}^a_{\Gamma_0}(q, z) = \int_{\Gamma_0} T_a\, e^{W_q/z}\,\omega\]

holds up to normalization. Here the right-hand side is the $p = \Gamma_0$ column of the period matrix $\mathcal{I}^a_p$ introduced in §Gauss-Manin Connection, ⁋Proposition 7.

The above Conjecture 5 (Mirror theorem, $J$-function form) is one of the strongest expressions of mirror symmetry: all A-side descendant Gromov-Witten invariants are reconstructed from B-side period integrals. Then in particular, the classical version of ring-level mirror symmetry $QH^\ast(X) \cong \Jac(W_q)$ is recovered as the leading order as $z\rightarrow 0$. Specifically, the $z \to 0$ stationary phase asymptotic expands into a sum over critical points of $W_q$ (§Gauss-Manin Connection, ⁋Proposition 3), and the leading order critical values $\{W_q(p)\}$ recover the canonical coordinates of quantum cohomology.

This claim is not merely a repetition of §Dubrovin Connection, ⁋Conjecture 4 (Mirror theorem, $D$-module form). To show this, we must first demonstrate that the QDE satisfied by the $J$-function (Proposition 4 (A-side Fundamental Solution Matrix)) and the Gauss-Manin system satisfied by the period matrix (§Gauss-Manin Connection, ⁋Proposition 7) define the same $D$-module; this is the content of §Dubrovin Connection, ⁋Conjecture 4 (Mirror theorem, $D$-module form). After that, we must verify that the $J$-function, i.e., the first column of the matrix $S$, is mapped not to an arbitrary linear combination of thimble periods under the isomorphism of §Dubrovin Connection, ⁋Conjecture 4 (Mirror theorem, $D$-module form), but specifically to the oscillating integral over the distinguished thimble $\Gamma_0$. For this, an additional claim that the integral structures on both sides match is needed; on the A-side, the lattice defined by $K$-theory and the $\hat{\Gamma}$-class plays this role, and on the B-side, the lattice generated by Lefschetz thimbles does.

In the case of Calabi-Yau hypersurfaces in toric varieties, this was proven by Givental and Lian-Liu-Yau, and for our main object of interest, toric Fano varieties, the $I = J$ theorem in the next section provides an explicit and computable form of this claim.

Givental’s Mirror Theorem

In the case of toric Fano varieties, the B-side oscillating integral is computed in an explicit hypergeometric form. The object defined through this is the $I$-function, which serves as the toric mirror counterpart of the $J$-function.

Definition 5 (Givental’s $I$-function) Let $X$ be a smooth projective toric Fano variety, $D_1, \ldots, D_m$ the toric divisors, and $\beta \in H_2(X, \mathbb{Z})$ an effective curve class. The $I$-function of $X$ is defined by

$I_X(q, z) := e^{t_{(2)}/z} \sum_{\beta \in H_2(X, \mathbb{Z})_{\mathrm{eff}}} q^\beta \prod_{i=1}^m \frac{\prod_{k=-\infty}^{0} (D_i + kz)}{\prod_{k=-\infty}^{D_i \cdot \beta}(D_i + kz)}$

Here the formally appearing infinite product to $-\infty$ precisely cancels between numerator and denominator, so it is actually a well-defined expression reducing to a finite product (or its inverse) depending on the sign of $D_i \cdot \beta$. The data determining this product is only the toric divisor $D_i$ and its intersection number $D_i \cdot \beta$, and the information contained in these two is the same as the information contained in the charge matrix $Q = (Q_{ji}) \in \Mat_{r \times m}(\mathbb{Z})$ introduced in §Overview of Mirror Symmetry, ⁋Definition 1.

Let $X$ be a smooth projective toric variety. Then we have the following short exact sequence

\[0 \longrightarrow H_2(X, \mathbb{Z}) \xrightarrow{\ \iota\ } \mathbb{Z}^m \xrightarrow{\ e_i \mapsto v_i\ } \mathbb{Z}^n \longrightarrow 0\]

Here $m$ is the number of rays, and $\mathbb{Z}^n$ is the cocharacter lattice. That is, $H_2$ is identified with the set of relations among rays

\[\{(a_i)_i \in \mathbb{Z}^m : \sum_i a_i v_i = 0\}\]

By definition, the charge matrix is the matrix whose rows are these $r$ relations, so it exactly encodes the information about the basis of $H_2(X, \mathbb{Z})$. That is, the preimage $\beta_j=\iota^{-1}(Q_{j\bullet})$ of each row $(Q_{j\bullet})$ of $\mathbb{Z}^m$ forms a basis of $H_2(X, \mathbb{Z})$, and in this setting the intersection with the toric divisor $D_i$ is merely reading the $i$-th coordinate of $\beta=(a_i)_i$, so $D_i\cdots\beta_j=Q_{ji}$.

The origin of this formula is the oscillating integral on the Hori-Vafa mirror $\check{X}$ of $X$, or more precisely, expanding that integral over the distinguished thimble $\Gamma_0$ of Conjecture 5 (Mirror theorem, $J$-function form) in terms of charge data gives exactly $I_X$. That is, $I_X$ is nothing more than the explicit formula for the distinguished column of the period matrix $\mathcal{I}$ of §Gauss-Manin Connection, ⁋Proposition 7. We defer the detailed expansion to Example 7 ($X = \mathbb{P}^n$), and first translate this into a claim about the $J$-function.

Proposition 6 (Givental’s mirror theorem) When $X$ is a smooth projective toric Fano variety, the $I$-function and $J$-function of $X$ satisfy the relation

\[J_X(\tau(q), z) = I_X(q, z)\]

where $\tau(q)$ is the mirror map defined from the asymptotic expansion of the $I$-function

\[I_X(q, z) = 1 + \tau(q)/z + O(z^{-2})\]

Looking at each side for the proof, $J_X$ is, as seen in Proposition 4 (A-side Fundamental Solution Matrix), the fundamental solution of the small QDE ($\ast$), and $I_X$ is the hypergeometric function explicitly given by charge data in Definition 5 (Givental’s $I$-function). The key of the proof is to directly verify that this explicit $I_X$ is also a solution of the same $D$-module as $J_X$, i.e., the same QDE ($\ast$). If $X$ is toric Fano, then $H^\ast(X)$ is generated by $H^2(X)$, so ($\ast$) reduces to a differential equation for a single component of $I_X$, and applying $z q_a\partial_{q_a}$ to the hypergeometric product of $I_X$ directly shows that each factor of the product satisfies this equation term by term. Meanwhile, both $I_X$ and $J_X$ have the normalization $1 + O(z^{-1})$ as $z \to \infty$, and since the solution of the QDE is uniquely determined from this leading asymptotic in a recurrence form once $a_0$ is fixed, we obtain $J_X(\tau(q), z) = I_X(q, z)$. We will verify this concretely for $X = \mathbb{P}^n$. (Example 7 ($X = \mathbb{P}^n$))

Note that the mirror map being well-defined is thanks to the Fano property of $X$. If $X$ is Fano, then for all non-zero effective curve classes $\beta$ we have $-K_X \cdot \beta > 0$, so all $q^\beta$ corrections appear only at orders $z^{-1}$ and below, and thus the $z^0$ term of $I_X$ is exactly $1$ ($I_X = 1 + O(z^{-1})$), allowing us to read off $\tau(q)$ from the $z^{-1}$ coefficient. More generally, in the semi-positive case where $-K_X$ is merely nef, there are directions with $-K_X \cdot \beta = 0$ producing a correction $I_0(q) \neq 1$ in the $z^0$ order, and the relation is modified to $J_X(\tau(q), z) = I_X(q, z)/I_0(q)$.

Similarly, consider the case where the Fano index of $X$ is at least $2$. Here the Fano index $r_X$ is the largest positive integer such that there exists $H \in \Pic(X)$ with $-K_X = r_X H$. In this case $H = -K_X / r_X$ is also ample, so for non-zero effective curve classes $\beta$ we have $H \cdot \beta \geq 1$, and thus

\[-K_X \cdot \beta = r_X (H \cdot \beta) \geq 2\]

so all $q$-dependent corrections are pushed to orders $z^{-2}$ and below. As a result, no $q$-correction remains in the mirror map $\tau(q)$, and in this case $J_X(q, z) = I_X(q, z)$ holds without any coordinate change. The $\mathbb{P}^n$ in the following Example 7 ($X = \mathbb{P}^n$) ($-K = (n+1)H$, Fano index $n+1 \geq 2$) is such a case.

Example 7 ($X = \mathbb{P}^n$) The fan of $\mathbb{P}^n$ is the normal fan of the standard simplex, with $n+1$ rays

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

(§Definition of Toric Varieties, ⁋Example 10). The toric divisors are $D_0, \ldots, D_n$, each corresponding to a coordinate hyperplane, and since they are all linearly equivalent they define a single hyperplane class $H\in H^2(\mathbb{P}^n)$. Moreover, the cohomology calculation of $\mathbb{P}^n$ showed that this hyperplane class generates the entire cohomology of $\mathbb{P}^n$.

Now let us unpack the data entering the $I$-function formula of Definition 5 (Givental’s $I$-function) for $\mathbb{P}^n$. First, since $H^2(\mathbb{P}^n)$ is generated by $H$ alone, we have $t_{(2)} = tH$, and the Novikov variable is given by $q = e^t$. That is, since $t = \ln q$, we can write $e^{t_{(2)}/z} = e^{H \ln q / z}$.

Next, effective curve classes are parameterized by non-negative multiples of the line class $H^\vee$, so $\beta = d H^\vee$ ($d \geq 0$) and $q^\beta = q^d$, and the $n+1$ toric divisors $D_0, \ldots, D_n$ of $\mathbb{P}^n$ satisfy (§Torus-Invariant Divisors and Line Bundles, ⁋Example 11) $D_i \cdot \beta = H \cdot d H^\vee = d$ for all $i$, so substituting this, the infinite product to $-\infty$ cancels for each factor, giving

\[\frac{\prod_{k=-\infty}^{0}(D_i + kz)}{\prod_{k=-\infty}^{D_i \cdot \beta}(D_i + kz)} = \frac{\prod_{k=-\infty}^{0}(H + kz)}{\prod_{k=-\infty}^{d}(H + kz)} = \frac{1}{\prod_{j=1}^{d}(H + jz)}\]

and since there are $n+1$ such factors for each toric divisor, raising to the $(n+1)$-th power gives the $I$-function

\[I_{\mathbb{P}^n}(q, z) = e^{H \ln q /z} \sum_{d \geq 0} \frac{q^d}{\prod_{j=1}^d (H + jz)^{n+1}}\]

Since $H^{n+1} = 0$, expanding $(H + jz)^{-(n+1)}$ in the denominator as a power series in $z^{-1}$ and expanding $e^{H\ln q/z}$ as a Taylor series, then grouping by degree, we obtain the following expansion.

\[I_{\mathbb{P}^n}(q, z) = 1 + \frac{H \ln q}{z} + \frac{(H \ln q)^2}{2 z^2} + \cdots + q \frac{1}{(H+z)^{n+1}} + \cdots\]

Now the Fano index of $\mathbb{P}^n$ is $n+1 \geq 2$ from $-K_{\mathbb{P}^n} = (n+1) H$. In the above expansion, the $q$-dependent corrections (the $q^d$ terms for $d \geq 1$) appear only at orders $z^{-(n+1)}$ and below, so the $z^{-1}$ coefficient of $I_{\mathbb{P}^n}$ is only $H \ln q$ (i.e., $t_{(2)}$) from the prefactor. Therefore, as discussed immediately after Proposition 6 (Givental’s mirror theorem), the mirror map is the identity and

\[J_{\mathbb{P}^n}(q, z) = I_{\mathbb{P}^n}(q, z)\]

must hold. Let us verify this directly.

The B-side calculation is largely finished in §Gauss-Manin Connection, ⁋Example 8. According to it, for the fundamental solution matrix $\mathcal{I}_p^a(q,z)$ of $\mathbb{P}^n$, the first component $\mathcal{I}_p^0(q,z)$ of the vector corresponding to the Lefschetz thimble base $\Gamma_p$ must satisfy

\[(z\partial_q)(qz\partial_q)^n\mathcal{I}_p^0=\mathcal{I}_p^0 \tag{$\ast\ast$}\]

Now we show that the $I$-function with $H=0$ satisfies this equation. According to ($\ast$), applying $qz\partial_q$ is the same as taking the quantum product with $H$, and on the B-side this is exactly the same as computing the $\mathcal{I}_p^a$ inductively from $\mathcal{I}_p^0$, so showing that these are solutions of the same ODE when $H=0$ completes the verification of Proposition 6 (Givental’s mirror theorem).

Then from the product formula of Definition 5 (Givental’s $I$-function), the prefactor becomes $e^0 = 1$, and each factor becomes $(H + jz) \mapsto jz$, so we obtain the formula

\[\Phi_0(q, z) := I_{\mathbb{P}^n}(q, z)\big\vert_{H=0} = \sum_{d \geq 0} \frac{q^d}{(d!)^{n+1}z^{(n+1)d}}\]

Let us directly verify that this $\Phi_0$ satisfies ($\ast\ast$). Direct computation shows that the action of $qz\partial_q$ pulls out $d$ term by term and attaches a $z^{-1}$ factor, so

\[(qz\partial_q)^n\Phi_0 = \sum_{d\geq 0}\frac{d^n q^d}{(d!)^{n+1}z^{(n+1)d - n}},\]

and applying $z\partial_q$ again pulls out $d$ once more and shifts $d$ to $d - 1$, giving

\[(z\partial_q)(qz\partial_q)^n\Phi_0 = \sum_{d \geq 1}\frac{d^{n+1}q^{d-1}}{(d!)^{n+1}z^{(n+1)d - n - 1}} = \sum_{d' \geq 0}\frac{(d'+1)^{n+1}q^{d'}}{((d'+1)!)^{n+1}z^{(n+1)d'}} = \Phi_0\]

which is exactly verified.

The stationary phase aspect of Conjecture 5 (Mirror theorem, $J$-function form) was also already computed in §Gauss-Manin Connection, ⁋Example 8, but this is slightly outside our central concerns in this post, so we omit it.

This example shows that mirror symmetry is realized not as an abstract isomorphism but as the concrete coincidence of two hypergeometric series. For general toric Fano varieties, similar explicit calculations proceed from the charge matrix, and in all such cases the $I = J$ theorem of Proposition 6 (Givental’s mirror theorem) provides a practical verification of mirror symmetry.

Extension to Big Quantum Cohomology

The small $J$-function we have dealt with so far was the fundamental solution for the small quantum $D$-module with base given by the $H^2$ direction deformation of §Dubrovin Connection, §§D-module. As seen in the earlier part of the same post, the Dubrovin connection itself was defined on big quantum cohomology with base given by the full $H^\ast(X)$ deformation, and the small version looking only at the $H^2$ direction was a specialization of this. The same elevation is possible for the $J$-function, and it is the big $J$-function.

To go to the big version setup, we must introduce the general deformation parameter $t = \sum_{a=0}^s t^a T_a$ of $H^\ast(X)$. For arbitrary $t \in H^\ast(X)$, the big quantum product $\star_t$ determines the connection $1$-form of the Dubrovin connection as $\mathcal{C}_\alpha = T_\alpha \star_t -$, as seen in §Dubrovin Connection, §§D-module. Its dual horizontal section equation is

\[z\partial_{t^a} s = T_a \star_t s\qquad (a = 0, 1, \ldots, \dim_\mathbb{C} H^\ast(X) - 1)\]

which is the big QDE extending ($\ast$) to the full $H^\ast$ direction.

Definition 8 (Big $J$-function) The big Givental $J$-function $J_X^{\mathrm{big}}: H^\ast(X) \times \mathbb{C}^\ast \to H^\ast(X)$ of $X$ is defined by

\[J_X^{\mathrm{big}}(t, z) := e^{t_{(2)}/z}\left( 1 + \sum_{\substack{\beta \in H_2(X, \mathbb{Z})_{\mathrm{eff}}, n \geq 0 \\ (\beta, n) \neq (0, 0)}} \sum_{a=0}^s \frac{q^\beta}{n!} \left\langle \frac{T_a}{z - \psi}, t, \ldots, t \right\rangle_{0, n+1, \beta} T^a \right)\]

Here $T_a/(z-\psi)$ is inserted at the first marked point (i.e., the pullback of $T_a$ with all orders of $\psi^k$ in generating function form), and $t \in H^\ast(X)$ is inserted at each of the remaining $n$ marked points, making a total of $n+1$ marked points.

The fact that the big $J$-function forms a horizontal section of the big QDE follows from the same argument as the small version in Proposition 4 (A-side Fundamental Solution Matrix). The small $J$-function is recovered as the specialization of the big $J$-function to $t = t_{(2)} \in H^2(X)$: applying the divisor equation (with descendant correction term) $n$ times to $H^2$ insertions, the $t_{(2)}$ insertions come out as the factor $(t_{(2)} \cdot \beta)^n$ with $\psi$-shift corrections, and these sum to $\sum_n (t_{(2)}\cdot \beta)^n/n! = q^\beta$ while the $\psi$-shift corrections produce an additional $z^{-1}$ factor, so the small $J$-function form of Definition 3 with a single marked point is directly recovered.

The additional information contained in the big $J$-function is all descendant invariants determined by arbitrary cohomology classes. Based on this, Conjecture 5 (Mirror theorem, $J$-function form) is also elevated to the big version, becoming a stronger statement claiming the coincidence of the full $S$-matrix and the full period matrix, and $I = J$ (Proposition 6 (Givental’s mirror theorem)) also has richer content in the big version where the mirror map $\tau(q)$ is generally non-trivial.

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.

Its origin is examined in the proof of Proposition 4 (A-side Fundamental Solution Matrix). ↩

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Givental J-function and Mirror Theorem

Descendant Gromov-Witten Invariant

Introduction of the \(J\)-Function

Mirror Theorem

Givental’s Mirror Theorem

Extension to Big Quantum Cohomology

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