2.1 Descendant-ancestor correspondence

To elucidate step (i), we first recall the descendant/ancestor correspondence [Reference Coates and Givental6, Appendix 2]:

$$\begin{align*}\mathcal{D} = e^{F(\tau)} \widehat{S(\tau)^{-1}}\ {\mathcal A} (\tau).\end{align*}$$

Here, $\mathcal {D}$ is the total descendant, and ${\mathcal A}$ the total ancestor potential of E. The latter is defined [Reference Coates and Givental6, Reference Givental13] by replacing the $\psi $ -classes in the definition of Gromov–Witten invariants with their counterparts pulled back from Deligne–Mumford spaces by the contraction maps $\operatorname {ct} \colon \overline {\mathcal {M}}_{g,n+m}(E,d)\to \overline {\mathcal {M}}_{g,n}$ and also inserting a primary class $\tau \in H$ at each of the m free marked points (which results in the dependence of ${\mathcal A}$ on the parameter $\tau $ ). The function F in the exponent is the potential for primary (no descendants) genus- $1$ Gromov–Witten invariants. The operator S is lower-triangular (i.e., represented by a series in $z^{-1}$ ) and is uniquely determined by the overruled Lagrangian cone ${\mathcal L}_E$ , as follows. Each tangent space to ${\mathcal L}_E$ is tangent to ${\mathcal L}_E$ at a point of the form $-1z+\tau +\operatorname {\mathrm {O}}(1/z)$ , and thus the tangent spaces depend on a parameter $\tau $ . For a tangent space $T_{\tau }$ , the $\operatorname {\mathrm {\Xi }}[z]$ -linear projection ${\mathcal H}_{+}\to T_{\tau }$ along ${\mathcal H}_{-}$ determines (and is determined by) the lower-triangular loop group element $S(\tau )^{-1} \colon H \subset {\mathcal H}_{+} \to T_{\tau } \subset {\mathcal H}$ .

The operator S can be expressed in terms of Gromov–Witten invariants and thus it is subject to dimensional constraints. This guarantees that conjugation by S respects the grading operators, that is, transforms the grading operator $l_0$ for Gromov–Witten theory into the grading operator for the ancestor theory. Thus, it remains to find another grading-preserving loop group transformation relating $e^{F(\tau )}{\mathcal A} (\tau )$ with $\mathcal {D}_{E^T}$ .

2.2 Fixed-point localization

In the torus-equivariant version of Gromov–Witten theory, there exists an ‘upper-triangular’ element $R (\tau )$ of the loop group which provides the following relationship between suitable ancestor potentials of the target space E and its fixed point locus $E^T$ :

(3) $$ \begin{align} {\mathcal A}^{{{eq}}}(\tau) = \widehat{R(\tau)} \ \prod_{\alpha \in F} {\mathcal A}^{\alpha,{{tw}}}_{B}(u_\alpha(\tau)). \end{align} $$

Here, ${\mathcal A}^{{{eq}}}$ refers to the total ancestor potential of E in T-equivariant Gromov–Witten theory (of which we are reminded by the superscript $eq$ ), and ${\mathcal A}^{\alpha ,{ {tw}}}_{B}$ is a similar ancestor potential of one component $E^\alpha = B$ of the fixed point locus $E^T$ . The superscript $tw$ indicates that we are dealing here not with the Gromov–Witten theory of $E^{\alpha }=B$ per se, but with the Gromov–Witten theory of the normal bundle of $E^{\alpha }$ in E; this is the local theoryFootnote ¹ of $E^\alpha $ , or in other words the twisted theory of B. The product over the fixed point set $F=X^T$ means that each function depends on its own group of variables according to the decomposition $H_T:=H_T^*(E;\operatorname {\mathrm {\Xi }}) = \oplus _{\alpha \in F} H_T^*(E^{\alpha };\operatorname {\mathrm {\Xi }})$ of equivariant cohomology induced by the embedding $E^T\subset E$ and localization. Let us write $\tau = \oplus _{\alpha } \tau _{\alpha }$ using the same decomposition. The quantities $u_\alpha (\tau )$ in Equation (3) are certain block-canonical coordinates on $H_T$ , defined in §3.4 below, which have the property that $u_\alpha (\tau ) \equiv \tau _\alpha $ modulo Novikov variables. The operator $R(\tau )$ here is a power series in z, and it does not have a nonequivariant limit. The existence of the operator $R(\tau )$ and the validity of formula (3) are established in §3 below.

Next, the ancestor potential of the normal bundle of $E^\alpha $ in E is related to the corresponding descendant potential by the equivariant version of the descendant/ancestor correspondence:

$$ \begin{align*} {\mathcal A}^{\alpha,{{tw}}}_{B} (u_\alpha(\tau)) = e^{-F^{\alpha,{{tw}}}_{B}(u_\alpha(\tau))} \widehat{S_{B}^{\alpha,{{tw}}}(u_\alpha(\tau))} \mathcal{D}_{B}^{\alpha,{{tw}}} && \text{for each } \alpha \in F. \end{align*} $$

Finally, according to the Quantum Riemann–Roch theoremFootnote ² [Reference Coates and Givental6]:

(4) $$ \begin{align} \mathcal{D}_{B}^{\alpha,{{tw}}} = \widehat{\Gamma_{\alpha}^{-1}} \mathcal{D}_{B}^{{{eq}}}. \end{align} $$

That is, the twisted equivariant descendant potential is obtained from the untwisted equivariant descendant potential of the fixed point manifold (i.e., $|F|$ copies of the base B of the toric bundle in our case) by quantized loop group transformations. The operators involved have the form

$$\begin{align*}\Gamma_{\alpha}^{-1} =\exp \left(\sum_{m\geq 0} \rho_m^{\alpha} z^{2m-1}\right), \end{align*}$$

where $\rho _m^{\alpha }$ are certain operators of multiplication in the classical equivariant cohomology of $E^{\alpha }$ ; see [Reference Coates and Givental6] for the precise definition.

Composing the above transformations, we obtain

$$ \begin{align*} e^{F^{{{eq}}}(\tau)}{\mathcal A}^{{{eq}}}(\tau) =\widehat{M(\tau)} \prod_{\alpha \in F} \mathcal{D}^{{{eq}}}_B && \text{where} && \widehat{M(\tau)}=e^{F^{{{eq}}}(\tau)-\sum_{\alpha} F^{\alpha,{{tw}}}_{B}(u_\alpha(\tau))} \widehat{R}(\tau) \left(\bigoplus_{\alpha\in F} \widehat{S_{B}^{\alpha,{{tw}}}}\big(u_{\alpha}(\tau)\big) \widehat{\Gamma_{\alpha}^{-1}}\right). \end{align*} $$

Both $e^{F^{{{eq}}}(\tau )}{\mathcal A}^{{{eq}}}(\tau )$ and $\prod _{\alpha \in F} \mathcal {D}_{B}^{{{eq}}}$ have nonequivariant limits, but some ingredients of the operator $M(\tau )$ do not. Nonetheless, we prove:

This implies Theorem 1.4.

2.3 Why does the limit exist?

To understand why the nonequivariant limit of $M(\tau )$ exists, consider the equivariant descendant/ancestor relation

$$\begin{align*}e^{F^{{{eq}}}(\tau)}{\mathcal A}^{{{eq}}}(\tau) = \widehat{S^{{{eq}}}}(\tau) \mathcal{D}^{{{eq}}}. \end{align*}$$

The descendant potential here does not depend on the parameter $\tau \in H_T$ . The upper-triangular loop group element $S^{{{eq}}}(\tau )$ is a fundamental solution to the (equivariant) quantum differential equationsFootnote ³ for E:

$$ \begin{align*} z \partial_v S (\tau) & = v \bullet_{\tau} S (\tau) & v \in H_T, \end{align*} $$

where $\bullet _{\tau }$ is the equivariant big quantum product with parameter $\tau \in H_T$ . Therefore, the function $e^{F^{{{eq}}}}{\mathcal A}^{{{eq}}}$ depends on $\tau $ in the same way, that is, satisfies:

$$ \begin{align*} \partial_v \Big(e^{F^{{{eq}}}(\tau)}{\mathcal A}^{{{eq}}}(\tau)\Big) &= \widehat{\left(\frac{v\bullet_{\tau}}{z}\right)} \Big(e^{F^{{{eq}}}(\tau)}{\mathcal A}^{{{eq}}}(\tau)\Big) & v \in H_T. \end{align*} $$

On the other hand, $e^{F^{ {{eq}}}}{\mathcal A}^{{{eq}}}=\widehat {M(\tau )}\prod _{\alpha \in F} \mathcal {D}_{B}^{{{eq}}}$ and $\prod _{\alpha \in F} \mathcal {D}_{B}^{{{eq}}}$ does not depend on $\tau $ . Therefore, the dependence of M on $\tau $ is governed by the same connection:

$$ \begin{align*} z\partial_v M (\tau) & = v\bullet_{\tau} M(\tau) & v \in H_T. \end{align*} $$

The main result of the paper [Reference Brown4] about toric bundles provides, informally speaking, a fundamental solution to this connection for the total space E of a toric bundle assuming that the fundamental solution for the base B is known. The solution is given in the form of an oscillating integral – the ‘equivariant mirror’ to the toric fiber X. The proof of the above claim is based on the identification of M with another form of this solution, obtained by replacing the oscillating integrals with their stationary phase asymptotics. In the nonequivariant limit of the oscillating integrals, the stationary phase asymptotics tend to a well-defined limit because the critical points of the phase function remain well defined and nondegenerate in the limit – this is equivalent to the semisimplicity of the nonequivariant quantum cohomology algebra of the toric fiber [Reference Iritani18]. We explain this in detail in §4 below. In the next section, we discuss localization in T-equivariant Gromov–Witten theory of toric bundles.

3.1 The T-action on E

Let $\pi \colon E \to B$ be a toric bundle with fiber X, constructed as in §1.6. Let $\mathfrak {k}$ and $\mathfrak {t}$ denote the Lie algebras of $K = (S^1)^k$ and $T = (S^1)^N$ , respectively. Our assumption that $\operatorname {rk} H^2(X)=k$ impliesFootnote ⁴ that there is a canonical isomorphism $\mathfrak {k}^\vee \cong H^2(X;\mathbb {R})$ , and so the symplectic form on the toric fiber X determines a point $\omega \in \mathfrak {k}^\vee $ . The embedding $K \to T$ determines and is determined by a linear map $\mathfrak {k} \to \mathfrak {t}$ ; this map is given by a $k \times N$ matrix with integer entries $(m_{ij})_{1 \leq i \leq k,1 \leq j \leq N}$ . The columns of $(m_{ij})$ determine elements $D_1,\ldots ,D_N \in H^2(X;\mathbb {R})$ , the toric divisors on X.

As discussed above, the total space E of the toric bundle carries a fiberwise left action of the big torus T. The T-fixed set $E^T$ consists of $n = \operatorname {rk} H^*(X)$ copies of B, which are the images of sections of $\pi $ . The T-fixed points on X, and also the T-fixed strata on E, are indexed by subsets $\alpha \subset \{1,2,\ldots ,N\}$ of size k such that $\omega $ lies in the cone spanned by $\{D_i : i \in \alpha \}$ . Denote the set of all such subsets $\alpha $ by F. Given such a subset $\alpha \in F$ , we write $x^\alpha $ for the corresponding T-fixed point in X and $E^\alpha $ for the corresponding T-fixed stratum in E. One-dimensional T-orbits on E are nonisolated (unless B is a point), and components of this space are indexed by one-dimensional T-orbits on X. A one-dimensional T-orbit in X that connects $x^\alpha $ to $x^\beta $ corresponds to the component of the space of one-dimensional orbits in E consisting of orbits that connect $E^\alpha $ to $E^\beta $ ; this component is again a copy of the base B. We write $\beta \to \alpha $ if there is a one-dimensional T-orbit in X from $x^\alpha $ to $x^\beta $ . For each $\beta \in F$ such that $\beta \to \alpha $ there is a line bundle over $E^\alpha \cong B$ formed by tangent lines (at $E^\alpha )$ to closures of the one-dimensional orbits connecting $E^\alpha $ to $E^\beta $ ; we denote the first Chern class of this line bundle by $\chi _{\alpha \beta }$ .

3.2 The T-action on the moduli space of stable maps

In this section and the next, we describe the technique of fixed point localization on moduli spaces of stable maps. This material, which is well known, is included for completeness, but we will need very little of it in what follows. In what comes afterwards, torus-invariant stable maps will be chopped into ‘macroscopic’ pieces. What one needs to digest from the ‘microscopic’ description given here is that the moduli spaces of torus-invariant stable maps factor according to the pieces, that integrals over the factors are naturally assembled into appropriate Gromov–Witten invariants, and that the only integrands which do not behave multiplicatively with respect to the pieces are the ‘smoothing factors’ defined below. The impatient reader should therefore skip straight to §3.4, pausing only to examine the explicit forms of the smoothing factors, which are given just after Equation (6).

Let $E_{g,n,d}$ denote the moduli space of degree-d stable maps to E from curves of genus g with n marked points; here, g and n are nonnegative integers and $d \in H_2(E;{\mathbb {Z}})$ . The action of T on E induces an action of T on $E_{g,n,d}$ . A stable map $f \colon C \to E_{g,n,d}$ representing a T-fixed point in $E_{g,n,d}$ necessarily has $T_{\mathbb {C}}$ -invariant image, which lies therefore in the union of zero- and one-dimensional $T_{\mathbb {C}}$ -orbits in E. More precisely, some rational irreducible components of C are mapped onto one-dimensional $T_{\mathbb {C}}$ -orbit closures as multiple covers $z \mapsto z^k$ (in obvious coordinates). We call them legs of multiplicity k. After removing the legs, the domain curve C falls into connected components $C_v$ , and the restrictions $f |_{C_v}$ are stableFootnote ⁵ maps to $E^T$ which we call stable pieces. Note that each leg maps $z=0$ and $z=\infty $ to the T-fixed locus $E^T \subset E$ , and each of $z=0$ and $z=\infty $ is one of:

1. 1. a node, connecting to a stable piece;

2. 2. a node, connecting to another leg;

3. 3. a marked point;

4. 4. an unmarked smooth point.

We associate to possibilities (1)–(4) vertices of $\Gamma $ , of types 1–4, respectively, thereby ensuring that each leg connects precisely two vertices. The combinatorial structure of a T-fixed stable map can be represented by a decorated graph $\Gamma $ , with vertices as above and an edge for each leg. The edge e of $\Gamma $ is decorated by the multiplicity $k_e$ of the corresponding leg. The vertex v of $\Gamma $ is decorated with $(\alpha _v,g_v,n_v,d_v)$ , where $\alpha _v \in F$ is the component of the T-fixed set determined by v, and $(g_v,n_v,d_v)$ record the genus, number of marked points and degree of the stable map $f|_{C_v}$ for vertices of type $1$ , and:

$$\begin{align*}(g_v,n_v,d_v) = \begin{cases} (0,2,0) & \text{for type 2} \\ (0,1,0) & \text{for type 3} \\ (0,0,0) & \text{for type 4.} \end{cases} \end{align*}$$

As a T-fixed stable map varies continuously, the combinatorial type $\Gamma $ does not change. Each connected componentFootnote ⁶ is isomorphic to a substack of:

(5) $$ \begin{align} \left(\prod_{\text{vertices } v \text{ of } \Gamma} B_{g_v,n_v,d_v} \right) \Big/ \left(\operatorname{\mathrm{Aut}}(\Gamma) \times \prod_{\text{edges } e \text{ of } \Gamma} {\mathbb{Z}}/k_e{\mathbb{Z}} \right). \end{align} $$

Here, $B_{g_v,n_v,d_v}$ is the moduli space of stable maps to B, and the ‘missing’ moduli spaces $B_{0,2,0}$ , $B_{0,1,0}$ and $B_{0,0,0}$ are taken to be copies of B. The substack is defined by insisting that, for each edge e between vertices v, w of $\Gamma $ , the evaluation maps

$$ \begin{align*} \operatorname{\mathrm{ev}}_e \colon B_{g_v,n_v,d_v} \to B && \text{and} && \operatorname{\mathrm{ev}}_e \colon B_{g_{w},n_{w},d_{w}} \to B \end{align*} $$

determined by the edge e have the same image. These constraints reflect the fact that the one-dimensional orbit in E determined by the edge e runs along a fiber of the toric bundle $E \to B$ .

3.3 Virtual localization

We will compute Gromov–Witten invariants of E by virtual localization. The localization theorem in equivariant cohomology states that, given a holomorphic action of a complex torus $T_{\mathbb {C}}$ on a compact complex manifold M and $\omega \in H^\bullet _T(M)$ , we have

$$\begin{align*}\int_{[M]} \omega = \int_{[M^T]} \frac{i^* \omega}{\operatorname{\mathrm{Euler}}(N_{M^T})}, \end{align*}$$

where $i \colon M^T \to M$ is the inclusion of the T-fixed submanifold, $N_{M^T}$ is the normal bundle to $M^T$ in M, $\operatorname {\mathrm {Euler}}$ denotes the T-equivariant Euler class, and the integrals denote the evaluation of a T-equivariant cohomology class against the T-equivariant fundamental cycle. According to Graber–Pandharipande [Reference Graber and Pandharipande17], the same formula holds when M is the moduli space of stable maps to a smooth projective T-variety; $[M]$ and $[M^T]$ denote T-equivariant virtual fundamental classes [Reference Li and Tian24, Reference Behrend and Fantechi2]; and $N_{M^T}$ denotes the virtual normal bundle to $M^T$ , that is, the moving part of the virtual tangent bundle to M restricted to $M^T$ . To apply this virtual localization formula, we need to describe $[M^T]$ and $\operatorname {\mathrm {Euler}}(N_{M^T})$ . The T-fixed components (5) come equipped with virtual fundamental classes from the Gromov–Witten theory of B, and these give the virtual fundamental class $[M^T]$ . We next describe $\operatorname {\mathrm {Euler}}(N_{M^T})$ .

Consider a connected component (5) consisting of stable maps with combinatorial type $\Gamma $ . The analysis of the virtual tangent bundle to the moduli space of stable maps in [Reference Graber and Pandharipande17, Reference Kontsevich23] shows that $\operatorname {\mathrm {Euler}}(N_{M^T})$ takes the form:

(6) $$ \begin{align} \operatorname{\mathrm{Euler}}(N_{M^T}) = C_{\text{smoothing}} \, C_{\text{vertices}} \, C_{\text{edges}}. \end{align} $$

Here, the factor $C_{\text {smoothing}}$ records the contribution from deformations which smooth nodes in the T-fixed stable maps of type $\Gamma $ . To each type- $1$ flag, that is, each pair $(v,e)$ , where v is a type- $1$ vertex and e is an edge of $\Gamma $ incident to v, there corresponds a one-dimensional smoothing mode which contributes

$$\begin{align*}\frac{\operatorname{\mathrm{ev}}_e^* \chi_{\alpha,\beta}}{k_e} - \psi_e \end{align*}$$

to $C_{\text {smoothing}}$ . Here, $\psi _e$ is the cotangent line class (on $B_{g_v,n_v, d_v}$ ) at the marked point determined by e. To each type- $2$ flag, there corresponds a one-dimensional smoothing mode which contributes

$$\begin{align*}\frac{\chi_{\alpha,\beta_1}}{k_1} + \frac{\chi_{\alpha,\beta_2}}{k_2} \end{align*}$$

to $C_{\text {smoothing}}$ , where the legs incident to the type-2 vertex connect $E^\alpha $ to $E^{\beta _1}$ and $E^{\beta _2}$ , with multiplicities $k_1$ and $k_2$ , respectively. Flags of types 3 and 4 do not contribute to $C_{\text {smoothing}}$ .

The factor $C_{\text {vertices}}$ in Equation (6) is a product over the vertices of $\Gamma $ . A type- $1$ vertex contributes the T-equivariant Euler class of the virtual vector bundle $N^{\alpha _v}_{g_v,n_v,d_v}$ over $B_{g_v,n_v,d_v}$ with fiber at a stable map $f \colon C \to B$ given by:Footnote ⁷

$$\begin{align*}H^0(C,f^* N^{\alpha_v}) \ominus H^1(C,f^* N^{\alpha_v}). \end{align*}$$

Here, $N^{\alpha _v} \to B$ is the normal bundle to $E^{\alpha _v} \cong B$ in E. Vertices of type $2$ , $3$ , and $4$ contribute, respectively:

$$ \begin{align*} \operatorname{\mathrm{Euler}}_T(N^{\alpha_v}), && \operatorname{\mathrm{Euler}}_T(N^{\alpha_v}), && \text{and} && \frac{\operatorname{\mathrm{Euler}}_T(N^{\alpha_v})}{\chi_{\alpha_v,\beta}/k_e}, \end{align*} $$

where the leg terminating at a type-4 vertex has multiplicity $k_e$ and connects $E^{\alpha _v}$ to $E^\beta $ .

The factor $C_{\text {edges}}$ in Equation (6) contains all other contributions from the moving part of the virtual tangent bundle. We will not need the explicit formula in what follows, but include it here for completeness. The factor $C_{\text {edges}}$ is a product over edges of $\Gamma $ . An edge of multiplicity k connecting fixed points indexed by $\alpha $ , $\beta \in F$ contributes

$$\begin{align*}\prod_{j=1}^N \frac{ \prod_{m=-\infty}^0 \big( U_j(\alpha) + m \frac{\chi_{\alpha, \beta}}{k} \big) }{ \prod_{m=-\infty}^{D_j \cdot d - 1} \big( U_j(\alpha) + m \frac{\chi_{\alpha, \beta}}{k} \big), } \end{align*}$$

where $U_j(\alpha ) \in H^2_T(E)$ is the T-equivariant class of the jth toric divisor, restricted to the fixed locus $E^\alpha $ and $D_j \cdot d$ is the intersection index of that divisor with the degree of the corresponding multiply-covered one-dimensional T-orbit. See [Reference Givental12] and [Reference Brown4] for details.

3.4 Virtual localization in genus zero

Consider the following generating functions for genus-zero Gromov–Witten invariants:

• • the J-function $J(\tau ,z) \in H^\bullet _T(E)$ defined by $J(\tau ,z) = \sum _\mu J(\tau ,z)^\mu \phi _\mu $ , where

  (7) $$ \begin{align} J(\tau,z)^\mu = (1,\phi^\mu) z + \tau^\mu + \sum_{d \in H_2(E)} \sum_{n=0}^\infty \frac{Q^d}{n!} \left \langle \tau,\tau,\ldots,\tau,\frac{\phi^\mu}{z-\psi_{n+1}} \right \rangle_{0,n+1,d}^E \end{align} $$

• • the fundamental solution $S(\tau ,z) \colon H^\bullet _T(E) \to H^\bullet _T(E)$ defined by

  (8) $$ \begin{align} {S(\tau,z)^\mu}_\nu = (\phi^\mu,\phi_\nu) + \sum_{d \in H_2(E)} \sum_{n=0}^\infty \frac{Q^d}{n!} \left \langle \phi^\mu,\tau,\ldots,\tau,\frac{\phi_\nu}{z-\psi_{n+2}} \right \rangle_{0,n+2,d}^E \end{align} $$

• • the bilinear form $V(\tau ,w,z)$ on $H^\bullet _T(E)$ defined by

  (9) $$ \begin{align} V(\tau,w,z)_{\mu \nu} = \frac{(\phi_\mu,\phi_\nu)}{w+z} + \sum_{d \in H_2(E)} \sum_{n=0}^\infty \frac{Q^d}{n!} \left \langle \frac{\phi_\mu}{w-\psi_1},\tau,\ldots,\tau,\frac{\phi_\nu}{z-\psi_{n+2}} \right \rangle_{0,n+2,d}^E. \end{align} $$

Here, $\phi _1,\ldots ,\phi _{\operatorname {rk} H_T^\bullet (E)}$ and $\phi ^1,\ldots ,\phi ^{\operatorname {rk} H_T^\bullet (E)}$ are bases for $H^\bullet _T(E)$ that are dual with respect to the T-equivariant Poincaré pairing on E, endomorphisms M of $H^\bullet _T(E)$ have matrix coefficients such that $M(\phi _\nu ) = \sum _\mu {M^\mu }_\nu \phi _\mu $ , and bilinear forms V on $H^\bullet _T(E)$ have matrix coefficients such that $V(\phi _\mu ,\phi _\nu ) = V_{\mu \nu }$ .

The fundamental solution $S(\tau ,z)$ satisfies the T-equivariant quantum differential equations:

(10) $$ \begin{align} z \partial_v S (\tau,z) = v \bullet_{\tau} S (\tau,z) && v \in H_T^\bullet(E) \end{align} $$

together with the normalization condition $S(\tau ,z) = \operatorname {\mathrm {Id}} + \operatorname {\mathrm {O}}(z^{-1})$ . Standard results in Gromov–Witten theory imply that

(11) $$ \begin{align} J(\tau,z) = z S(\tau,z)^* 1 && \text{and} && V(\tau,w,z) = \frac{S(\tau,w)^* S(\tau,z)}{w+z}, \end{align} $$

where $S(\tau ,z)^*$ denotes the adjoint of $S(\tau ,z)$ with respect to the equivariant Poincaré pairing on $H_T^\bullet (E)$ , and we identify $H^\bullet _T(E)$ with its dual space via the equivariant Poincaré pairing (thus equating the bilinear form V with an operator). The analogous statements hold in the Gromov–Witten theory of B twisted by the normal bundle $N^\alpha $ to the T-fixed locus $E^\alpha \cong B$ .

We begin by processing $S(\tau ,z)$ by fixed-point localization. Henceforth, we work over the field of fractions of the coefficient ring $H^\bullet _T(pt) = H^\bullet (BT)$ of T-equivariant cohomology theory and insist that our basis $\phi _1,\ldots ,\phi _{\operatorname {rk} H_T^\bullet (E)}$ for $H^\bullet _T(E)$ is compatible with the fixed-point localization isomorphism

$$ \begin{align*} H^{\bullet}_{T}(E) \longrightarrow \bigoplus_{\alpha \in F} H^{\bullet}_{T}(E^{\alpha}) \end{align*} $$

in the sense that each $\phi _i$ restricts to zero on all except one of the T-fixed loci, which we denote by $E^{\alpha _i} \subset E$ .

Proposition 3.1. The fundamental solution $S(\tau ,z)$ can be factorized as the product

(12) $$ \begin{align} S(\tau,z) = R(\tau,z) \, S_{{\textit{block}}}(\tau,z), \end{align} $$

where R has no pole at $z=0$ , and $S_{{\textit {block}}}(\tau ,z)$ is the block-diagonal transformation

$$\begin{align*}S_{{\textit{block}}}(\tau,z) = \bigoplus_{\alpha \in F} S^{\alpha, {\textit{tw}}}\big(u_\alpha(\tau),z\big). \end{align*}$$

Here, $S^{\alpha , {{tw}}}(u,z)$ is the fundamental solution in the Gromov–Witten theory of B twisted by the normal bundle $N^\alpha $ to the T-fixed locus $E^\alpha \cong B$ , and $\tau \mapsto \oplus _\alpha u_\alpha (\tau )$ is a certain nonlinear change of coordinates with $u_\alpha (\tau ) \in H^\bullet (E^\alpha ) \subset H^\bullet (E)$ .

Proof of Proposition 3.1.

The z-dependence in ${S(\tau ,z)^\mu }_\nu $ arises only from the input $\frac {\phi _\nu }{z-\psi _{n+2}}$ in Equation (8) at the last marked point. Let $E^\alpha $ denote the fixed-point component on which $\phi _\nu $ is supported. By fixed-point localization, we see that ${S(\tau ,z)^\mu }_\nu $ is a sum of contributions from fixed-point components $E^\Gamma _{0,n+2,d}$ where the graphs $\Gamma $ can be described as follows. A typical $\Gamma $ has a distinguished vertex, called the head vertex, that carries the $(n+2)$ nd marked point with insertion $\frac {\phi _\nu }{z-\psi _{n+2}}$ . The head vertex is a stable map to $E^\alpha $ ; it is incident to m trees (the ends) that do not carry the first marked point and also to one distinguished tree (the tail) that carries the first marked point. Thus, ${S(\tau ,z)^\mu }_\nu $ has the form

(13) $$ \begin{align} \begin{aligned} {\delta^\mu}_{\nu} + \sum_{\beta: \beta \to \alpha} \sum_{k=1}^\infty \frac{\Big(T^{k}_{\beta,\alpha}(\phi^\mu),\phi_\nu\Big)}{\frac{\chi_{\alpha \beta}}{k} + z} + \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \\ \qquad \qquad \sum_d \sum_{m=0}^\infty \frac{Q^d}{m!} \left \langle \phi^\mu + \sum_{\beta: \beta \to \alpha} \sum_{k=1}^\infty \frac{T^{k}_{\beta,\alpha}(\phi^\mu)}{\frac{\chi_{\alpha \beta}}{k} - \psi_1}, \epsilon_\alpha(\psi_2),\ldots,\epsilon_\alpha(\psi_{m+1}),\frac{\phi_\nu}{z-\psi_{m+2}} \right \rangle^{B,{\textit{tw}},\alpha}_{0,m+2,d}, \end{aligned} \end{align} $$

where the correlators represent integration over the moduli space of stable maps given by the head vertex, $\epsilon _\alpha $ represents the contribution of all possible ends, the linear map $T^{k}_{\beta ,\alpha }\colon H_T^\bullet (E) \to H_T^\bullet (E^\alpha ;\operatorname {\mathrm {\Xi }})$ records the contribution of all possible tails that approach $E^\alpha $ along an edge from the T-fixed component $E^\beta \subset E$ with multiplicity k and $\chi _{\alpha \beta }$ is the cohomology class on $E^\alpha \cong B$ defined in Section 3.1. The sum is over degrees $d \in H_2(B;{\mathbb {Z}})$ , and $Q^d$ represents, in the Novikov ring of E, the degree of these curves in $E^\alpha \cong B$ . Here, $\epsilon _\alpha = \tau $ modulo Novikov variables, as we include in our definition of $\epsilon _\alpha $ the degenerate case where the end consists just of a single marked point attached to the head vertex. Except ${\delta ^\mu }_{\nu }$ , the terms on the first line of Equation (13) arise from those exceptional graphs $\Gamma $ where the head vertex is unstable as a map to $E^\alpha \cong B$ , that is, where the head vertex is a type-3 vertex in the sense of §3.2.

In fact, Equation (13) can be written as

(14) $$ \begin{align} {\delta^\mu}_{\nu} + \sum_{\beta: \beta \to \alpha} \sum_{k=1}^\infty \frac{1}{z + \frac{\chi_{\alpha \beta}}{k}} \Bigg( S^{\alpha,{{tw}}}\big(u_\alpha(\tau),{\textstyle \frac{\chi_{\alpha \beta}}{k}}\big) T^k_{\beta,\alpha}(\phi^\mu), \, S^{\alpha,{{tw}}}\big(u_\alpha(\tau),z\big) \phi_\nu \Bigg)^{\alpha,{{tw}}}, \end{align} $$

where $(\cdot ,\cdot )^{\alpha , {{tw}}}$ is the twisted Poincaré pairing on $E^\alpha $ . This is a consequence of a general result about the structure of genus-zero Gromov–Witten invariants, applied to the Gromov–Witten theoryFootnote ⁸ of $E^\alpha $ . Recall that the tangent space to the Lagrangian cone $\mathcal {L}^{{{tw}}}_{E^\alpha }$ at the point $\mathbf {t} \in {\mathcal H}_+$ is the graph of the differential of the quadratic form on $\mathcal {H}_+$ given by

$$\begin{align*}\epsilon \longmapsto \operatorname{Res}_{z=0} \operatorname{Res}_{w=0} \mathbb{V}(\mathbf{t},{-w},{-z})\big(\epsilon(z),\epsilon(w)\big) \, dz \, dw, \end{align*}$$

where

$$\begin{align*}\mathbb{V}(\mathbf{t},w,z) \big(\epsilon_1,\epsilon_2 \big) = \frac{\big(\epsilon_1(z),\epsilon_2(w)\big)^{\alpha,{\textit{tw}}}}{z+w} + \sum_d \sum_{n=0}^\infty \frac{Q^d}{n!} \left \langle \frac{\epsilon_1(z)}{z-\psi_1},\mathbf{t}(\psi_2),\mathbf{t}(\psi_3),\ldots,\mathbf{t}(\psi_{n+1}), \frac{\epsilon_2(w)}{w-\psi_{n+2}} \right \rangle^{B,\alpha, {\textit{tw}}}_{0,n+2,d}. \end{align*}$$

But this tangent space has the form $S^{\alpha , {{tw}}}(u_\alpha ,z)^{-1} \mathcal {H}_+$ for some point $u_\alpha \in H^\bullet _T(E^\alpha )$ that is determined by $\mathbf {t}$ . In fact,

(15) $$ \begin{align} u_\alpha = \sum_{\epsilon} \sum_d \sum_{n=0}^\infty \frac{Q^d}{n!} \left \langle 1,\phi^\epsilon,\mathbf{t},\ldots,\mathbf{t} \right \rangle^{\alpha,{{tw}}}_{0,n+2,d} \phi_\epsilon, \end{align} $$

where the sum is over $\epsilon $ such that $\phi _\epsilon $ is supported on $E^\alpha $ ; this is the Dijkgraaf–Witten formula [Reference Dijkgraaf and Witten7, Equation (2.2)] (in the context of Gromov-Witten theory, see also the proof of [Reference Givental15, Theorem 1] and [Reference Getzler11, Section 4.4]). As a result,

$$\begin{align*}\mathbb{V}(\mathbf{t},w,z) \big(\epsilon_1,\epsilon_2 \big) =\mathbb{V}(u_\alpha,w,z) \big(\epsilon_1,\epsilon_2 \big) = \frac{1}{z+w} \Bigg( S^{\alpha,{{tw}}}(u_\alpha,z) \epsilon_1(z),\, S^{\alpha,{{tw}}}(u_\alpha,w) \epsilon_1(w) \Bigg)^{\alpha,{{tw}}}, \end{align*}$$

where we used Equation (11). Applying this to Equation (13) yields Equation (14), where $u_\alpha (\tau )$ is given by Equation (15) with $\mathbf {t}$ replaced by the contribution $\epsilon _\alpha $ from all possible ends.

Setting

(16) $$ \begin{align} R(\tau,z) = \operatorname{\mathrm{Id}} + \sum_{\alpha \in F} \sum_{\beta: \beta \to \alpha} \sum_{k=1}^\infty \frac{1}{z + \frac{\chi_{\alpha \beta}}{k}} \Big(T^k_{\beta,\alpha}\Big)^* S^{\alpha,{{tw}}}\big(u_\alpha(\tau),{\textstyle \frac{\chi_{\alpha \beta}}{k}}\big)^* \end{align} $$

yields the result. This expression has no pole at $z=0$ .

Remark 3.4. The end contribution $\epsilon _\alpha $ occurring in the proof of Proposition 3.1 can be identified in terms of fixed-point localization for the J-function:

$$\begin{align*}J(\tau,{-z})\big|_{E^\alpha} = {-1z} + \epsilon_\alpha(z) + \sum_d \sum_{m=0}^\infty \sum_\nu \frac{Q^d}{m!} \left \langle \epsilon_\alpha(\psi_1),\ldots,\epsilon_\alpha(\psi_m),\frac{\phi^\nu}{z-\psi_{m+1}} \right \rangle^{B,\alpha, {{tw}}}_{0,m+1,d} \phi_\nu \big|_{E^\alpha}, \end{align*}$$

where we processed the virtual localization formulas exactly as in the proof of Proposition 3.1. Since $\psi _{n+1}$ is nilpotent, the correlator terms have poles (and no regular part) at $z=0$ . At the same time, the summand $\epsilon _\alpha (z)$ has no pole at $z=0$ , as it is equal to $\tau |_{E^\alpha }$ plus a sum of terms of the form $\frac {c}{-z + \chi _{\alpha \beta }/k}$ , where c is independent of z. Thus,

(17) $$ \begin{align} \epsilon_\alpha(z) = z + \Big[ J(\tau,{-z})\big|_{E^\alpha} \Big]_+ = z - \Big[ z S(\tau,{-z})^* 1\big|_{E^\alpha} \Big]_+, \end{align} $$

where $[ \cdot ]_+$ denotes taking the power series part of the Laurent expansion at $z=0$ . For the last equality here, we used Equation (11).

3.5 Virtual Localization for the ancestor potential

We will use virtual localization to express generating functions for Gromov–Witten invariants of E in terms of generating functions for twisted Gromov–Witten invariants of $E^T$ . The T-equivariant total ancestor potential of E is

(18) $$ \begin{align} \mathcal{A}(\tau;\mathbf{t}) = \exp\left(\sum_{g=0}^\infty \hbar^{g-1} \bar{\mathcal{F}}_g(\tau;\mathbf{t}) \right), \end{align} $$

where $\bar {\mathcal {F}}_g$ is the T-equivariant genus-g ancestor potential:Footnote ⁹

$$\begin{align*}\bar{\mathcal{F}}_g(\tau;\mathbf{t}) = \sum_{d \in H_2(E)} \sum_{m=0}^\infty \sum_{n=0}^\infty \frac{Q^d}{m!n!} \int_{[E_{g,m+n,d}]^{\text{vir}}} \prod_{i=1}^m \left( \sum_{k=0}^\infty \operatorname{\mathrm{ev}}_i^* (t_k) \bar{\psi}_i^k \right) \prod_{i=m+1}^{m+n} \operatorname{\mathrm{ev}}_i^* \tau. \end{align*}$$

Here, $\tau \in H_T^\bullet (E)$ , $\mathbf {t} = t_0 + t_1 z + t_2 z^2 + \cdots \in H_T^\bullet (E)[z]$ , $Q^d$ is the representative of $d \in H_2(E;{\mathbb {Z}})$ in the Novikov ring, $\operatorname {\mathrm {ev}}_i \colon E_{g,n,d} \to E$ is the evaluation map at the ith marked point, and the ‘ith ancestor class’ $\bar {\psi }_i \in H_T^2(E_{g,m+n,d};\mathbb {Q})$ is the pullback of the cotangent line class $\psi _i \in H^2(\overline {\mathcal {M}}_{g,m};\mathbb {Q})$ along the contraction morphism $\operatorname {ct} \colon E_{g,m+n,d} \to \overline {\mathcal {M}}_{g,m}$ that forgets the map and the last n marked points and then stabilizes the resulting m-pointed curve. We will express the total ancestor potential $\mathcal {A}$ , via virtual localization in terms of the total ancestor potentials $\mathcal {A}_B^{\alpha , { {tw}}}$ of the base manifold B twisted [Reference Coates and Givental6] by the T-equivariant inverse Euler class of the normal bundle $N^\alpha $ to the T-fixed locus $E^\alpha \cong B$ in E. These are defined exactly as above, but replacing E by B and replacing the virtual class $[E_{g,m+n,d}]^{\text {vir}}$ by $[B_{g,m+n,d}]^{\text {vir}} \cap e_T^{-1} \big (N^{\alpha }_{g,m+n,d}\big )$ for an appropriate twisting class $N^{\alpha }_{g,m+n,d} \in K^0_T(B_{g,m+n,d})$ .

Theorem 3.5.

$$\begin{align*}\mathcal{A}(\tau;\mathbf{t}) = \widehat{R(\tau)} \prod_{\alpha \in F} \mathcal{A}^{\alpha,{{tw}}}_B(u_\alpha(\tau);\mathbf{t}_\alpha), \end{align*}$$

where $\mathbf {t} = \oplus _{\alpha \in F} \mathbf {t}_\alpha $ with $\mathbf {t}_\alpha \in H^*_T(E^\alpha )[z]$ and $\widehat {R(\tau )}$ is the quantization of the linear symplectomorphism $f \mapsto R(\tau ,z) f$ , and the map $\tau \mapsto \oplus _{\alpha \in F} u_{\alpha }(\tau )$ is defined by Equation (15).

The rest of this Section contains a proof of Theorem 3.5. According to [Reference Givental13, Proposition 7.3], the action of $\widehat {R(\tau )}$ is given by a Wick-type formula

(19) $$ \begin{align} \widehat{R(\tau)} \, \left(\prod_{\alpha \in F} \mathcal{A}^{\alpha, {{tw}}}_B\big(u_\alpha(\tau);\mathbf{t}_\alpha\big) \right) = \left( \exp(\Delta) \prod_{\alpha \in F} \mathcal{A}_B^{\alpha, {{tw}}} \big(u_\alpha(\tau), \mathbf{t}_\alpha\big)\middle) \right|{}_{\mathbf{t} \mapsto R(\tau,z)^{-1} \mathbf{t} + z( \operatorname{\mathrm{Id}} - R(\tau,z)^{-1}) 1}, \end{align} $$

where the propagator $\Delta $ , which depends on $\tau $ , is defined by

$$ \begin{align*} \Delta = \frac{\hbar}{2} \sum_{i,j} \sum_{\lambda,\mu} \Delta^{ij}_{\lambda,\mu} \frac{\partial}{\partial t^\lambda_i} \frac{\partial}{\partial t^\mu_j} && \text{and} && \sum_{i,j} \sum_{\lambda} \Delta^{ij}_{\lambda,\mu} (-1)^{i+j} w^i z^j \phi^\lambda = \left(\frac{R(\tau,w)^* R(\tau,z) - \operatorname{\mathrm{Id}}}{w+z}\right)\phi_\mu. \end{align*} $$

We will compute the T-equivariant total ancestor potential $\mathcal {A}(\tau ;\mathbf {t})$ using virtual localization, obtaining a Wick-type formula which matches precisely with Equation (19).

We begin by factoring the fixed point loci $E^T_{g,m+n,d}$ into ‘macroscopic’ pieces called stable vertices, stable edges, tails and ends, which are defined somewhat informally as follows. Given a T-fixed stable map $C \to E$ with $m+n$ marked points, forgetting the last n marked points yields a stable curve $C'$ with m marked points, and a stabilization morphism $C\to C'$ given by contracting the unstable components. We label points of C according to their fate under the stabilization morphism: a tree of rational components of C which contracts to a node, a marked point or a regular point of $C'$ is called, respectively, a stable edge, a tail or an end, while each maximal connected component of C which remains intact in $C'$ is called a stable vertex. Under virtual localization, the contribution of $E^T_{g,m+n,d}$ into $\mathcal {A}_E$ can be assembled from these pieces. We will see that the contributions from stable edges, tails and ends together give rise to the operator R, while the contribution of stable vertices gives $\prod _{\alpha \in F} \mathcal {A}^{\alpha , {{tw}}}_B$ .

More formally, virtual localization expresses $\mathcal {A}(\tau ;\mathbf {t})$ as a sum over T-fixed strata $E_{g,n,d}^\Gamma $ in $E_{g,m+n,d}$ which are indexed by decorated graphs $\Gamma $ as in §3.2. Consider a prestable curve C with combinatorial structure $\Gamma $ and the curve $C'$ obtained from C by forgetting the last n marked points and contracting unstable components. The curve C can be partitioned into pieces according to the fate of points of C under this process. Those components of C that survive as components of $C'$ are called stable vertices of C. Maximal connected subsets which contract to nodes of $C'$ are called stable edges of C. Maximal connected subsets which contract to a marked point of $C'$ are called tails. Maximal connected subsets which contract to smooth unmarked points of $C'$ are called ends. We denote by $\operatorname {ct}(\Gamma )$ the combinatorial structure of $C'$ , that is, the graph $\gamma $ with vertices and edges given respectively by the stable vertices and stable edges and with each vertex decorated by its genus and number of tails. We arrange the sum over $\Gamma $ from virtual localization according to the stable graphs $\operatorname {ct}(\Gamma )$ :

$$\begin{align*}\mathcal{A}(\tau;\mathbf{t}) = \sum_\Gamma c_\Gamma = \sum_\gamma \sum_{\Gamma : \operatorname{ct}(\Gamma) = \gamma} c_{\Gamma}. \end{align*}$$

We begin by analysing certain integrals over stable vertices which occur in the virtual localization formulas. These take the form

(20) $$ \begin{align} \left \langle \frac{T_1(\bar{\psi}_1)}{\chi_1-\psi_1}, \ldots, \frac{T_m(\bar{\psi}_m)}{\chi_m-\psi_m}, \epsilon_\alpha(\psi_{m+1}), \ldots, \epsilon_\alpha(\psi_{m+n}) \right \rangle^{\alpha,{{tw}}}_{g,m+n,d}, \end{align} $$

where $T_1,\ldots ,T_m$ arise from tails and/or stable edges, $\epsilon _\alpha $ arises from ends, and each $\chi _i$ is equal to $\frac {\chi _{\alpha \beta }}{k}$ for some $\beta \to \alpha $ and some $k \in \mathbb {N}$ . Note the presence of descendant classes $\psi _i$ . Our first task is to express these vertex integrals in terms of ancestor potentials, where no descendant classes occur. Consider the sum

(21) $$ \begin{align} \sum_{d,n} \frac{Q^d}{n!} \left \langle \frac{T_1(\bar{\psi}_1)}{\chi_1-\psi_1}, \ldots, \frac{T_m(\bar{\psi}_m)}{\chi_m-\psi_m}, \epsilon_\alpha(\psi_{m+1}), \ldots, \epsilon_\alpha(\psi_{m+n}) \right \rangle^{\alpha,{{tw}}}_{g,m+n,d} \end{align} $$

and note the identity

(22) $$ \begin{align} \frac{T_i(y)}{\chi_i - x} = \frac{T_i(y)}{\chi_i - y} + \frac{(x - y) T_i (y)}{(\chi_i - x)(\chi_i - y)} \end{align} $$

in $H_T^\bullet (E)[\![x,y]\!]$ . We can replace the insertion $\frac {T_i(\bar {\psi }_i)}{\chi _i - \psi _i}$ in Equation (21) first by the right-hand side of Equation (22) with $x=\psi _i$ and $y=\bar {\psi }_i$ , and then by

(23) $$ \begin{align} \frac{T_i(\bar{\psi}_i)}{\chi_i - \bar{\psi}_i} + \sum_{n,d} \frac{Q^d}{n!} \sum_{\mu,\nu} \phi_\mu \left \langle \frac{\phi^\mu}{\chi_i-\psi_1}, \epsilon_\alpha(\psi_2),\ldots,\epsilon_\alpha(\psi_{n+1}),\phi_\nu \right \rangle^{\alpha,{{tw}}}_{0,n+2,d} \left( \phi^\nu,\frac{T_i(\bar{\psi}_i)}{\chi_i-\bar{\psi}_i} \right)^{\alpha,{{tw}}}. \end{align} $$

Here, we used the fact, first exploited by Getzler [Reference Getzler11] and Kontsevich–Manin [Reference Kontsevich and Manin21], that $\psi _i - \bar {\psi }_i$ is Poincaré-dual to the virtual divisor which is total range of the gluing map

$$\begin{align*}\bigsqcup_{\substack{n_1+n_2=n\\d_1+d_2=d}} E_{0,n_1+2,d_1} \times_E E_{g,m-1+n_2+1,d_2} \longrightarrow E_{g,m+n,d} \end{align*}$$

that attaches a genus-zero stable map carrying the ith marked point and $n_1$ marked points with insertions in $\{m+1,\ldots ,m+n\}$ to a genus-g stable map carrying the marked points $1,\ldots ,m$ omitting i, and $n_2$ marked points with insertions in $\{m+1,\ldots ,m+n\}$ . The insertion (23) simplifies to

(24) $$ \begin{align} \frac{S^{\alpha,{{tw}}}(u_\alpha(\tau),\chi_i) T(\bar{\psi}_i)}{\chi_i-\bar{\psi}_i} \end{align} $$

using [Reference Givental15, equation 2] and the fact that the contribution $\epsilon _\alpha $ from ends here coincides with the contribution $\epsilon _\alpha $ from ends in genus zero which occurred in Equation (15). In what follows, we will see that whenever contributions from tails and stable edges are expressed in terms of $\bar {\psi }_i$ -classes, the transformations $S^{\alpha ,{{tw}}}(u_\alpha (\tau ),\chi _i)$ will occur. We call these transformations the dressing factors.

Consider the following generating function which we call the mixed potential:

$$\begin{align*}\mathcal{A}(\boldsymbol{\epsilon};\mathbf{t}) = \exp \left(\sum_{g=0}^\infty \sum_{m,n=0}^\infty \sum_d \frac{\hbar^{g-1} Q^d}{m! n!} \left \langle \mathbf{t}(\bar{\psi}_1),\ldots,\mathbf{t}(\bar{\psi}_m);\boldsymbol{\epsilon}(\psi_{m+1}),\ldots,\boldsymbol{\epsilon}(\psi_{m+n}) \right \rangle_{g,m+n,d}\right). \end{align*}$$

Note that specializing $\boldsymbol {\epsilon } \in \mathcal {H}_+$ to $\tau \in H^\bullet _T(E) \subset \mathcal {H}_+$ recovers the ancestor potential $\mathcal {A}(\tau ;\mathbf {t})$ . The notion of mixed potential makes sense for a general Gromov–Witten-type theory, including the Gromov–Witten theory of arbitrary target spaces (equivariant or not), twisted Gromov–Witten theory, and so on and Proposition 3.6 below expresses the mixed potential of such a theory in terms of the ancestor potential for that theory. The argument just given showed that the sum (21) can be expressed by including appropriate dressing factors, in terms of the mixed potential associated to the twisted Gromov–Witten theory of the T-fixed locus $E^\alpha $ .

Proposition 3.6.

$$\begin{align*}\mathcal{A}(\boldsymbol{\epsilon};\mathbf{t}) = \mathcal{A}\Big(u(\boldsymbol{\epsilon});\mathbf{t}+\big[S(u(\boldsymbol{\epsilon}),z)(\boldsymbol{\epsilon}(z)-u(\boldsymbol{\epsilon}))\big]_+\Big), \end{align*}$$

where $u(\boldsymbol {\epsilon })\in H$ is characterized by

$$\begin{align*}\big[S(u(\boldsymbol{\epsilon}),z) \big(\boldsymbol{\epsilon}(z) - u(\boldsymbol{\epsilon})\big)\big]_+ \in z\mathcal{H}_+. \end{align*}$$

Proof. Set $\mathbf {y}(z) = \boldsymbol {\epsilon }(z) - u(\boldsymbol {\epsilon })$ so that

$$\begin{align*}\mathcal{A}(\boldsymbol{\epsilon};\mathbf{t}) = \exp \left(\sum_{g=0}^\infty \sum_{k,l,m=0}^\infty \sum_d \frac{\hbar^{g-1} Q^d}{k! l! m!} \left \langle \mathbf{t}(\bar{\psi}_1),\ldots,\mathbf{t}(\bar{\psi}_m);\mathbf{y}(\psi_{m+1}),\ldots,\mathbf{y}(\psi_{m+k}),u(\boldsymbol{\epsilon}),\ldots,u(\boldsymbol{\epsilon}) \right \rangle_{g,k+l+m,d}\right). \end{align*}$$

Then consider the morphism $X_{g,m+k+l,d} \to \overline {\mathcal {M}}_{g,m+k}$ forgetting the map and the last l marked points and then stabilizing; the Getzler/Kontsevich–Manin ancestor-to-descendant argument discussed above then gives

$$\begin{align*}\mathcal{A}(\boldsymbol{\epsilon};\mathbf{t}) = \exp \left(\sum_{g=0}^\infty \sum_{k,l,m=0}^\infty \sum_d \frac{\hbar^{g-1} Q^d}{k! l! m!} \left \langle \mathbf{t}(\bar{\bar{\psi}}_1),\ldots,\mathbf{t}(\bar{\bar{\psi}}_m),\mathbf{x}(\bar{\psi}_{m+1}),\ldots,\mathbf{x}(\bar{\psi}_{m+k}); u(\boldsymbol{\epsilon}),\ldots,u(\boldsymbol{\epsilon}) \right \rangle_{g,k+l+m,d}\right), \end{align*}$$

where $\mathbf {x}(z) = \big [S({u(\boldsymbol {\epsilon })},z) \mathbf {y}(z)\big ]_+$ and the classes $\bar {\bar {\psi }}_i$ differ from the ancestor classes $\bar {\psi }_i$ by being lifts of $\psi $ -classes from $\overline {\mathcal {M}}_{g,m}$ rather than from $\overline {\mathcal {M}}_{g,m+k}$ . For $\psi $ -classes on Deligne–Mumford spaces, we have

$$ \begin{align*} \left(\prod_{r=m+1}^{m+k} \psi_r\right) \cup (\psi_i - \pi^* \psi_i) = 0 && \text{for } i = 1,2,\ldots,m, \end{align*} $$

where $\pi \colon \overline {\mathcal {M}}_{g,m+k} \to \overline {\mathcal {M}}_{g,m}$ is the map that forgets the last k marked points and then stabilizes. This is because $\psi _i - \pi ^* \psi _i$ is the divisor in $\overline {\mathcal {M}}_{g,m+k}$ given by the image of the gluing map

$$\begin{align*}\bigsqcup_{k_1+k_2=k} \overline{\mathcal{M}}_{g,m-1+k_2+1} \times \overline{\mathcal{M}}_{0,k_1+1} \to \overline{\mathcal{M}}_{g,m+k}, \end{align*}$$

where the second factor carries $k_1$ marked points with indices in $\{m+1,\ldots ,m+k\}$ , and on this divisor the product $\prod _{r=m+1}^{m+k} \psi _r$ vanishes for dimensional reasons. Thus,

$$ \begin{align*} \left(\prod_{r=m+1}^{m+k} \bar{\psi}_r\right) \cup (\bar{\psi}_i - \bar{\bar{\psi}}_i) = 0 && \text{for } i = 1,2,\ldots,m \end{align*} $$

and, since $\mathbf {x}(z)$ is divisible by z – this is exactly how we chose $u(\boldsymbol {\epsilon })$ – we may replace $\bar {\bar {\psi }}_i$ s by $\bar {\psi }_i$ s in our expression for $\mathcal {A}(\boldsymbol {\epsilon },\mathbf {t})$ above. Polylinearity then gives the proposition.

Now, we apply Proposition 3.6 to the twisted Gromov–Witten theory of the T-fixed locus $E^\alpha $ so that S there is $S^{\alpha ,{{tw}}}$ . We will show that if we set $\boldsymbol {\epsilon }$ equal to the end contribution $\epsilon _\alpha $ in the vertex integrals (21), then $u(\boldsymbol {\epsilon })$ becomes the block-canonical coordinate $u_\alpha (\tau )$ defined in Proposition 3.1 and that

(25) $$ \begin{align} \big[S^{\alpha,{{tw}}}(u_\alpha(\tau),z)(\epsilon_\alpha(z)-u_\alpha(\tau))\big]_+ = z \big(\operatorname{\mathrm{Id}} - R(\tau,z)^{-1}\big) 1. \end{align} $$

Note that this precisely matches the contribution to Wick’s formula in Equation (19). Since $S^{\alpha , {{tw}}}(u_\alpha (\tau ),z)$ is a power series in $z^{-1}$ , we have

$$\begin{align*}\Big[ S^{\alpha,{{tw}}}(u_\alpha(\tau),z) \big[ A \big]_+ \Big]_+ = \Big[ S^{\alpha,{{tw}}}(u_\alpha(\tau),z) \, A \Big]_+ \end{align*}$$

for any A. From Equation (17), we have

$$\begin{align*}\epsilon_\alpha(z) = z - \Big[ z S(\tau,{-z})^* 1\big|_{E^\alpha} \Big]_+ \end{align*}$$

and Proposition 3.1 gives

$$\begin{align*}S(\tau,-z)^* 1\big|_{E^\alpha} = S^{\alpha, {{tw}}}(u_\alpha(\tau),{-z})^* \Big( R(\tau,{-z})^* 1 \big|_{E^\alpha}\Big). \end{align*}$$

We compute

$$\begin{align*}\big[S^{\alpha,{{tw}}}(u_\alpha(\tau),z)(\epsilon_\alpha(z)-u_\alpha(\tau))\big]_+ = z + u_\alpha(\tau) - z R(\tau,{-z})^* 1 \big|_{E^\alpha} - u_\alpha(\tau), \end{align*}$$

where the first two terms are $\big [S^{\alpha , {{tw}}}(u_\alpha (\tau ),z) z\big ]_+$ , the last term is $\big [S^{\alpha ,{{tw}}}(u_\alpha (\tau ),z) u_\alpha (\tau )\big ]_+$ and we used $S^{\alpha , {{tw}}}(u_\alpha (\tau ),z) S^{\alpha , {{tw}}}(u_\alpha (\tau ),{-z})^* = \operatorname {\mathrm {Id}}$ . The right-hand side here lies in $z \mathcal {H}_+$ , and therefore $u(\epsilon _\alpha ) = u_\alpha (\tau )$ , as claimed; furthermore, since $R(\tau ,-z)^* R(\tau ,z) = \operatorname {\mathrm {Id}}$ we obtain Equation (25).

We return now to the integrals (20) over stable vertices. Here, tails – which carry one nonforgotten marked point each – give rise to insertions of

$$\begin{align*}\mathbf{t}(\bar{\psi}) + \sum_{\alpha \in F} \sum_{\beta: \beta \to \alpha} \sum_{k=1}^\infty \frac{1}{\frac{\chi_{\alpha \beta}}{k} - \psi} T^k_{\beta,\alpha} \mathbf{t}(\bar{\psi}), \end{align*}$$

where the linear maps $T^k_{\beta ,\alpha }$ were defined in the proof of Proposition 3.1 and $\mathbf {t}$ is the argument of the ancestor potential (18). Applying Equations (22–24), we can write this in terms of $\bar {\psi }$ only by including dressing factors:

$$\begin{align*}\mathbf{t}(\bar{\psi}) + \sum_{\alpha \in F} \sum_{\beta: \beta \to \alpha} \sum_{k=1}^\infty \frac{1}{\frac{\chi_{\alpha \beta}}{k} - \bar{\psi}} S^{\alpha,{{tw}}}\big(u_\alpha(\tau),{\textstyle \frac{\chi_{\alpha \beta}}{k}}\big) T^k_{\beta,\alpha} \mathbf{t}(\bar{\psi}). \end{align*}$$

From Equation (16) and $R(\tau ,-z)^* R(\tau ,z) = \operatorname {\mathrm {Id}}$ , we see that this is $\big (R(\tau ,z)^{-1} \mathbf {t}(z) \Big )\big |_{E^\alpha }$ with $z = \bar {\psi }$ . Again, this precisely matches the contribution to Wick’s formula in Equation (19).

In the localization formula for the ancestor potential, the edge contributions occur in the form

$$\begin{align*}\sum_{\beta \to \alpha} \sum_{\beta' \to \alpha'} \sum_{k=1}^\infty \sum_{k'=1}^\infty \frac{E^{k,k'}_{\beta\to\alpha,\beta'\to\alpha'}} { \Big(\frac{\chi_{\alpha\beta}}{k} - \psi\Big) \Big(\frac{\chi_{\alpha'\beta'}}{k'} - \psi'\Big), }, \end{align*}$$

each functioning as two vertex insertions (which may be to the same vertex or different vertices). Here, $\psi $ and $\psi '$ are the $\psi $ -classes on the vertex moduli spaces at the corresponding marked points, and the class $E^{k,k'}_{\beta \to \alpha ,\beta '\to \alpha '} \in H^\bullet _T(B) \otimes H^\bullet _T(B)$ is pulled back to these moduli spaces by the product $\operatorname {\mathrm {ev}} \times \operatorname {\mathrm {ev}}'$ of the corresponding evaluation maps. This expression participates, in the same role, in localization formulas for the genus-zero quantity $V(\tau ,w,z)$ . As before, we can replace $\psi $ -classes by $\bar {\psi }$ -classes provided that we introduce dressing factors:

$$\begin{align*}\sum_{\beta \to \alpha} \sum_{\beta' \to \alpha'} \sum_{k=1}^\infty \sum_{k'=1}^\infty \frac{ S^{\alpha,{{tw}}}\big(u_\alpha(\tau),{\textstyle \frac{\chi_{\alpha \beta}}{k}}\big) \otimes S^{\alpha',{{tw}}}\big(u_{\alpha'}(\tau),{\textstyle \frac{\chi_{\alpha' \beta'}}{k'}}\big) \Big(E^{k,k'}_{\beta\to\alpha,\beta'\to\alpha'}\Big)} { \Big(\frac{\chi_{\alpha\beta}}{k} - \bar{\psi}\Big) \Big(\frac{\chi_{\alpha'\beta'}}{k'} - \bar{\psi}'\Big) } =: E_{\alpha,\alpha'}(\bar{\psi},\bar{\psi}'). \end{align*}$$

Computing $V(\tau ,w,z)$ by virtual localization, as in the proof of Proposition 3.1, and applying the identities

$$\begin{align*}V^{\alpha,{{tw}}}\Big(u_\alpha,z,{\textstyle\frac{\chi_{\alpha\beta}}{k}}\Big) = \frac{S^{\alpha,{{tw}}}\Big(u_\alpha,z\Big)^* S^{\alpha,{{tw}}}\Big(u_\alpha,\frac{\chi_{\alpha\beta}}{k}\Big)}{z+\frac{\chi_{\alpha\beta}}{k}} \end{align*}$$

for each $\alpha \in F$ , $\beta \to \alpha $ , and $k \in \{1,2,\ldots \}$ yields

$$\begin{align*}V(\tau,z,z') = S_{{{block}}}(\tau,z)^* \circ \left( \frac{\operatorname{\mathrm{Id}}}{z+z'} + \bigoplus_{\alpha,\alpha' \in F} E_{\alpha,\alpha'}(z,z') \right) \circ S_{{{block}}}(\tau,z'), \end{align*}$$

where we regard the bivector $E_{\alpha ,\alpha '}$ as an operator via the twisted Poincaré pairing. But

$$\begin{align*}V(\tau,z,z') = \frac{S(\tau,z)^* S(\tau,z')}{z+z'} = \frac{S_{{{block}}}(\tau,z)^* R(\tau,z)^* R(\tau,z') S_{{{block}}}(\tau,z')}{z+z'} \end{align*}$$

from Equation (11) and Proposition 3.1, and we conclude that

$$\begin{align*}\bigoplus_{\alpha,\alpha' \in F} E_{\alpha,\alpha'}(z,z') = \frac{R(\tau,z)^* R(\tau,z') - \operatorname{\mathrm{Id}}}{z+z'}. \end{align*}$$

In other words, the edge contributions $E_{\alpha ,\alpha '}$ (which include dressing factors) are precisely what is inserted by the propagator in Wick’s formula (19). This completes the proof of Theorem 3.5.

Remark 3.7. Consider a Kähler manifold E equipped with the action of a torus T, with no further assumptions about the structure of the fixed point manifold $E^T$ or the one-dimensional orbits. Theorem 3.5 can be extended to this general situation. First, virtual localization in genus zero shows, as in §3.4, that the fundamental solution matrix $S_E(\tau ,z)$ can be factored as

$$\begin{align*}S_E(\tau,z)=R(\tau,z) \, S_{E^T}^{{{tw}}}\big(u(\tau),z\big), \end{align*}$$

where R is a power series in z, and $\tau \mapsto u(\tau )$ is a certain nonlinear diffeomorphism between the parameter spaces $H^\bullet _T(E)$ and $H^\bullet _T(E^T)$ which is defined over the field of fractions of the coefficient ring $H^\bullet _T(\{ \text {point} \} )$ tensored with suitable Novikov ring (and can be specified in terms of genus-zero Gromov–Witten invariants). Then, virtual localization in all genera shows that

$$\begin{align*}\mathcal{A}_E(\tau; {\mathbf t}) = \widehat{R (\tau)} \, \mathcal{A}_{E^T}^{{{tw}}}\big(u(\tau); {\mathbf t}|_{E^T}\big). \end{align*}$$

However, we refrained from phrasing our proof of Theorem 3.5 in this abstract setting. In the context of general torus actions, one-dimensional $T_{\mathbb C}$ -orbits – called legs in §3.2 – may depend on parameters. Some foundational work is needed here – a systematic description of leg moduli spaces and their virtual fundamental cycles – and establishing these details, which are unimportant to the essence of our argument, would carry us too far away from the current aim. This is the only reason why we limit the proof of Theorem 3.5 to the case of toric bundles.

4.1 The I-function of E

Recall that our toric bundle $E \to B$ is obtained from the total space of a direct sum of line bundles $L_1 \oplus \cdots \oplus L_N \to B$ by fiberwise symplectic reduction for the action of a subtorus $K = (S^1)^k$ of $T = (S^1)^N$ . Let $\mu $ denote the moment map for the action of K on the total space of $L_1 \oplus \cdots \oplus L_N \to B$ so that $E = \mu ^{-1}(\omega )/K$ . The quotient map $\mu ^{-1}(\omega ) \to E$ exhibits $\mu ^{-1}(\omega )$ as a principal K-bundle over E. Since $K = (S^1)^k$ , this defines k tautological $S^1$ -bundles over E, each of which carries an action of T. Let $P_1,\ldots ,P_k \in H^2_T(E;{\mathbb {Z}})$ be the T-equivariant first Chern classes of the corresponding antitautological bundles, and let $p_1,\ldots ,p_k$ be the restrictions of $P_1,\ldots ,P_k$ to the fiber X. Without loss of generality, by changing the identification of K with $(S^1)^k$ if necessary, we may assume that the classes $p_1,\ldots ,p_k$ are ample. The classes $p_1,\ldots ,p_k$ generate the T-equivariant cohomology algebra $H^\bullet _T(X)$ ; let

(28) $$ \begin{align} \Delta_\beta(p_1,\ldots,p_k) \end{align} $$

be monomials in $p_1,\ldots ,p_k$ , indexed by $\beta $ , that together form a basis for $H^\bullet _T(X)$ .

Let $\lambda _1,\ldots ,\lambda _N$ denote the first Chern classes of the N antitautological bundles on $BT = (\mathbb {C} P^\infty )^N$ so that $R_T := H^\bullet _T(\{\text {point}\};\mathbb {Q}) = \mathbb {Q}[\lambda _1,\ldots ,\lambda _N]$ . The T-equivariant cohomology algebra $H^\bullet _T(E)$ has basis $\Delta _\beta (P_1,\ldots ,P_k)$ over $H^\bullet (B) \otimes R_T$ (cf. [Reference Sankaran and Uma27]). Let $\Lambda _j$ denote the first Chern class of the dual bundle $L_j^\vee $ so that the T-equivariant first Chern class of $L_j$ is $-\Lambda _j - \lambda _j$ , and define

$$ \begin{align*} u_j = \sum_{i=1}^{k} m_{ij} p_i - \lambda_j && U_j=\sum_{i=1}^{k} m_{ij} P_i -\Lambda_j - \lambda_j && 1 \leq j \leq N. \end{align*} $$

Then $u_j$ is the T-equivariant cohomology class Poincaré dual to the jth toric divisor in X, and $U_j$ is the T-equivariant cohomology class Poincaré dual to the jth toric divisor in E.

Let $J^B(\tau _B,z)$ denote the J-function of B, and write

$$\begin{align*}J^B(\tau_B,z) =\sum_{\beta\in \operatorname{\mathrm{Eff}}(B)} J^B_\beta(\tau_B,z) Q_B^\beta. \end{align*}$$

Here, $\tau _B\in H^\bullet (B)$ , which can be written as $\tau _B=\sum _b \tau _b\phi _b$ after choosing a basis $\phi _1,\ldots ,\phi _{\operatorname {rk} H^\bullet (B)}$ for $H^\bullet (B)$ .

The I-function of E is

(29) $$ \begin{align} I_E(t,\tau_B,z):=e^{Pt/z} \sum_{d\in {\mathbb{Z}}^k} \sum_{\beta\in \operatorname{\mathrm{Eff}}(B)} J^B_\beta(\tau_B,z) Q_B^\beta q^d e^{dt} \prod_{j=1}^N \frac{\prod_{m=-\infty}^{0} (U_j+mz)}{\prod_{m=-\infty}^{U_j(d, \beta)} (U_j+mz)}, \end{align} $$

where

$$ \begin{align*} &t:=(t_1,\dots, t_k) & &d = (d_1,\ldots,d_k) & &Pt:=\sum_{i=1}^k P_it_i \\ &dt:=\sum_{i=1}^k d_it_i & &q^d := q_1^{d_1} \cdots q_k^{d_k} & &U_j(d, \beta) :=\sum_{i=1}^k d_im_{ij}-\int_{\beta}\Lambda_j. \end{align*} $$

We have that

(30) $$ \begin{align} I_E(t,\tau_B,z) = z e^{Pt/z} e^{\tau_B/z} + \operatorname{\mathrm{O}}(Q). \end{align} $$

Brown proves [Reference Brown4, Theorem 1] that $I_E(t,\tau _B,{-z})$ lies on the Lagrangian cone $\mathcal {L}_E$ defined by the T-equivariant Gromov–Witten theory of E.

Recall the monomials $\Delta _\beta $ defined in Equation (28). The classes

(31) $$ \begin{align} \Delta_\beta(P_1,\ldots,P_k) \phi_b, \end{align} $$

where $\{\phi _b\}$ is a basis for $H^\bullet (B)$ , form a basis for $H^\bullet _T(E)$ , and this basis has a well-defined nonequivariant limit. Elements of this basis are indexed by pairs $(\beta , b)$ . Denote by T the following matrix whose column vectors are also indexed by pairs $(\beta ,b)$ :

(32) $$ \begin{align} T(z) = \begin{bmatrix} \Delta_\beta\Big(z \frac{\partial}{\partial {t_1}},\ldots,z \frac{\partial}{\partial {t_k}}\Big) \partial_{\phi_b} I_E(t,\tau_B,z) \end{bmatrix}. \end{align} $$

More precisely, the column of $T(z)$ corresponding to the pair $(\beta ,b)$ is $\Delta _\beta \Big (z \frac {\partial }{\partial {t_1}},\ldots ,z \frac {\partial }{\partial {t_k}}\Big ) \partial _{\phi _b} I_E(t,\tau _B,z)$ .

The columns of $T(-z)$ form a basis for $T_{I(t,\tau _B,{-z})} \mathcal {L}_E$ over the ring $\operatorname {\mathrm {\Xi }}\{z\}$ of power series in $q_1,\ldots ,q_k$ and Novikov variables of B with coefficients which are polynomials of z: See Equation (30). For an appropriate value of $\tau $ , determined by t and $\tau _B$ , the columns of $S_E(\tau ,{-z})^*$ form another such basis, which consists of series in $z^{-1}$ . Expressing the columns of T in terms of columns of S yields

$$\begin{align*}T({-z}) = S_E(\tau,{-z})^* L(\tau, z), \end{align*}$$

where L is a matrix with entries in $\operatorname {\mathrm {\Xi }}\{z\}$ ; this is the Birkhoff factorization of T. Since all other ingredients in this identity admit a nonequivariant limit, $L(\tau , z)$ does too.

The products

$$\begin{align*}\frac{\prod_{m=-\infty}^{0} (U_j+mz)}{\prod_{m=-\infty}^{U_j(d, \beta)} (U_j+mz)} \end{align*}$$

that occur in Equation (29), and thus in Equation (32), are rational functions of z. When $\lambda _j \ne 0$ for each j, we expand these as Laurent series near $z=0$ . We saw in §3.4 that the same operation applied to $S_E(\tau ,z)$ yields

$$\begin{align*}S_E(\tau,z) = R(\tau,z) S_{{{block}}}(\tau,z). \end{align*}$$

Thus,

$$\begin{align*}T(-z) \sim S_{{{block}}}(\tau,z)^{-1} R(\tau,z)^{-1} L(\tau, z), \end{align*}$$

where $\sim $ denotes the expansion near $z=0$ . In the next two subsections, we will show that the expansion of $\Gamma _{{{block}}} (z) \, T({-z})$ near $z=0$ has a nonequivariant limit by identifying this expansion with the stationary phase expansion of certain oscillating integrals.

4.2 Oscillating integrals

Consider $W = \sum _{j=1}^{N} (x_j + \lambda _j \log x_j)$ , and oscillating integrals

$$\begin{align*}\int_{\gamma} e^{W/z} \, \frac{ \prod_{j=1}^{N} d \log x_j }{ \prod_{i=1}^{k} d \log (q_ie^{t_i}), } \end{align*}$$

where $\gamma $ is a cycle in the subvariety of $(\mathbb {C}^\times )^N$ defined by

(33) $$ \begin{align} \prod_{j=1}^{N} x_j^{m_{ij}} = q_i e^{t_i} && 1 \leq i \leq k \end{align} $$

given by downward gradient flow of $\Re (W/z)$ from a critical point of W. Such oscillating integrals, over an appropriate set of cycles $\gamma $ , together form the mirror to the T-equivariant quantum cohomology of the toric manifold X [Reference Givental12]. We now relate these integrals to the q-series $I_E(t,\tau _B,z)$ by expanding the integrand as a q-series, following [Reference Brown4].

Given a T-fixed point $\alpha $ on X, one can solve the Equations (33) for $x_i$ , $i \in \alpha $ , in terms of $x_j$ , $j \not \in \alpha $ :

(34) $$ \begin{align} x_i = \prod_{l=1}^{k} (q_l e^{t_l})^{(m_\alpha^{-1})_{il}} \prod_{j \not \in \alpha} x_j^{-(m_\alpha^{-1} m)_{ij}} && i \in \alpha, \end{align} $$

where $m_\alpha $ is the $k \times k$ submatrix of $(m_{ij})$ given by taking the columns in $\alpha $ ; this defines a chart on the toric mirror (33). In this chart, the integrand $e^{W/z}$ becomes

$$\begin{align*}\Phi_\alpha := \left(\prod_{i=1}^k (q_i e^{t_i})^{\alpha^*(p_i)/z}\right) e^{\sum_{j \not \in \alpha} (x_j - \alpha^*(u_j) \log x_j )/z} \sum_{\substack{ d \in {\mathbb{Z}}^k : \\ u_j(d)\, \geq\, 0 \text{ for } j\, \in\, \alpha }} (q_1 e^{t_1}) ^{d_1} \cdots (q_k e^{t_k})^{d_k} \frac{\prod_{j \not \in \alpha} x_j^{-u_j(d)}} {\prod_{j \in \alpha} u_j(d)!z^{u_j(d)} }, \end{align*}$$

where $u_j(d) = \sum _{i=1}^k d_i m_{ij}$ ; cf. the proof of [Reference Brown4, Theorem 3]. Note that $u_j(d)$ is the value of the cohomology class $u_j \in H^2(X)$ on the element $d \in H_2(X)$ such that $p_i(d) = d_i$ , and that our ampleness assumption guarantees that all $d_i$ are nonnegative. We thus consider

$$\begin{align*}\mathcal{I}_\alpha(q e^t,z,\lambda) = \int_{(\mathbb{R}_+)_{j \not \in \alpha}} \Phi_\alpha \bigwedge_{j \not \in \alpha} d \log x_j \end{align*}$$

as a q-series of oscillating integrals with phase function ${\sum _{j \not \in \alpha } (x_j - \alpha ^*(u_j) \log x_j )}$ and monomial amplitudes. Replacing the variables $\lambda _j$ by the differential operator $\lambda _j + z \partial _{\Lambda _j}$ , we can consider $\mathcal {I}_\alpha $ as an operator to be applied to the J-function of the base B. According to the computation in [Reference Brown4, Section 5], this gives

(35) $$ \begin{align} \left(\prod_{i=1}^k q_i^{-\alpha^*(P_i)/z}\right) \mathcal{I}_\alpha(q e^t,z,\lambda + z \partial_\Lambda) J_B(\tau_B,z) = \alpha^* I_E (t,\tau_B,z) \prod_{j \not \in \alpha} \int_0^\infty e^{(x - \alpha^*(U_j) \log x)/z} d \log x. \end{align} $$

Applying the differential operators in Equation (32), we get expressions in terms of oscillating integrals for each entry $\Delta _\beta \Big (z \frac {\partial }{\partial {t_1}},\ldots ,z \frac {\partial }{\partial {t_k}}\Big ) \partial _{\phi _b} I_E$ of $T(z)$ . The stationary phase asymptotics of the oscillating integrals on the right (at the unique critical points $x = \alpha ^*(U_j)$ of the phase functions) combine to give $(2\pi z)^{\dim _{\mathbb {C}} X/2}\Gamma _\alpha (z)$ ; see [Reference Brown4, Section 5.2]. Consequently,

(36) $$ \begin{align} \Big(\Gamma_{{{block}}}({-z}) \, T(z) \Big)^{\beta,b}_\alpha \sim \left(\prod_{i=1}^k q_i^{-\alpha^*(P_i)/z}\right) \frac{\Delta_\beta\Big({z \textstyle \frac{\partial}{\partial {t_1}},\ldots,z \frac{\partial}{\partial {t_k}}}\Big)}{(2 \pi z)^{\frac{1}{2} \dim_{\mathbb{C}} X}} \mathcal{I}_\alpha(q e^t,z,\lambda + z \partial_\Lambda)\partial_{\phi_b}J_B(\tau_B,z), \end{align} $$

that is, the expansions near $z=0$ of the entries of $\Gamma _{{{block}}}(-z) \, T(z)$ coincide with stationary phase asymptotics of the oscillating integrals on the right-hand side of Equation (36). We now show that these stationary phase asymptotics admit a nonequivariant limit.

4.3 Stationary phase asymptotics

The expression

(37) $$ \begin{align} \Delta_\beta\Big({z \textstyle \frac{\partial}{\partial {t_1}},\ldots,z \frac{\partial}{\partial {t_k}}}\Big) \int_{\gamma} e^{W/z} \, \frac{ \prod_{j=1}^{N} d \log x_j }{ \prod_{i=1}^{k} d \log (q_ie^{t_i}) } \end{align} $$

is an oscillating integral with the phase function $\sum _{j=1}^N (x_j + \lambda _j \log x_j)$ over a Lefschetz thimble $\gamma $ in the complex torus with equations

(38) $$ \begin{align} \prod_{j=1}^{N} x_j^{m_{ij}} = q_i e^{t_i} && 1 \leq i \leq k. \end{align} $$

In the classical limit $q \to 0$ , this torus degenerates into a union of coordinate subspaces , where $\mathbb {C}^{n-k}_\alpha $ is the subspace of $\mathbb {C}^n$ given by the equations $x_j = 0$ , $j \in \alpha $ . The equations for critical points of W under the constraints (38) in the coordinate chart $\{x_j : j \not \in \alpha \}$ take the form

$$ \begin{align*} 0 = x_j - \alpha^*(u_j) + \text{terms involving positive powers of } q && j \not \in \alpha. \end{align*} $$

In the classical limit $q \to 0$ , exactly one of these critical points approaches the critical point

(39) $$ \begin{align} x_j = \alpha^*(u_j) && j \not \in \alpha \end{align} $$

of the phase function $\sum _{j \not \in \alpha } (x_j - \alpha ^*(u_j) \log x_j)$ on $\mathbb {C}^{n-k}_\alpha $ . Call this the $\alpha $ th critical point of W. On the right-hand side of Equation (36), we first expanded the oscillating integral (37) as a q-series and then took termwise stationary phase asymptotics at the critical point (39). General properties of oscillating integrals [Reference Brown4, Corollary 8] guarantee that this coincides with the q-series expansion of the stationary phase asymptotics of Equation (37) at the $\alpha $ th critical point of W. The key point here is that, for a generic value of q, if we let $\lambda _j \to 0$ for all j along a generic path, then the critical points of W corresponding to $\alpha \in F$ remain nondegenerate. The stationary phase expansions at these critical points depend continuously (indeed analytically) on $\lambda $ , and at $\lambda = 0$ remain well defined.

Arranging the integrals (37) into an $|F| \times |F|$ matrix and taking stationary phase asymptotics gives

$$\begin{align*}\left(\sum_{k=0}^\infty z^k \Psi_k \right) \begin{bmatrix} \ddots & & 0 \\ &e^{w_\alpha/z} & \\ 0 & & \ddots, \end{bmatrix} \end{align*}$$

where the factor on the right is a diagonal matrix, $w_\alpha $ is the value of W at the $\alpha $ th critical point and $\Psi _k$ is an $|F| \times |F|$ matrix. Here, $\Psi _k$ and $w_\alpha $ depend analytically on $(q,t,\lambda )$ and are well defined in the limit $\lambda =0$ ; also $\Psi _0$ is invertible, as a consequence of $\Delta _\beta (p_1,\ldots ,p_k)$ forming a basis in $H^\bullet _T(X)$ . The right-hand side of Equation (36) is obtained by replacing $\lambda _j$ here by $\lambda _j + z \partial _{\Lambda _j}$ , and applying the resulting differential operator to $\partial _{\phi _b} J_B(\tau _B,z)$ .

For a function $\lambda \mapsto w(\lambda )$ on $H_2(B)$ that depends on parameters (such as $q_i$ and $t_i$ ), consider first the action of $e^{w(z \partial _\Lambda )/z}$ on $J_B(\tau _B,z)$ . By the divisor equation, the action of $z \partial _\Lambda $ on the J-function $J_B(\tau _B,z)$ coincides with the action of $\Lambda + z Q \partial _Q$ where $Q \partial _Q$ is the derivation of Novikov variables (for B) corresponding to $\Lambda \in H^2(B)$ . For each $D \in H_2(B)$ , we have

$$\begin{align*}e^{w(\Lambda + z Q \partial_Q)/z} Q^D = e^{w(\Lambda + zD)/z} Q^D \end{align*}$$

and so $e^{w(\Lambda + z Q \partial _Q)/z}$ gives a well-defined operation on the space of cohomology-valued Laurent series in z with coefficients that converge Q-adically, provided that $w(0) = 0$ . Due to the string equation,

$$\begin{align*}e^{w(0)/z} J_B(\tau_B,z) = J_B\big(\tau_B + w(0)1,z\big). \end{align*}$$

On the other hand, $e^{w(\Lambda + z Q \partial _Q)/z - w(0)/z} J_B(\tau _B,z)$ , after flipping the sign of z, lies in $z T_{J_B(\tau ^*,{-z})}\mathcal {L}_B \subset \mathcal {L}_B$ for some point $\tau ^* \in H^\bullet (B)$ that depends on w and $\tau _B$ . This result was first used in [Reference Coates and Givental6] in the proof of the quantum Lefschetz theorem but was proved incorrectly there; an accurate proof is given in [Reference Coates, Corti, Iritani and Tseng5] and [Reference Givental16, Theorem 1].

Applying these arguments with $w = w_\alpha $ for each $\alpha \in F$ , we obtain

$$\begin{align*}e^{-w_\alpha(q,t,\lambda - z \partial_\Lambda)/z} J_B(\tau_B,{-z}) \in z T_{J_B(\tau_\alpha^*,{-z})}\mathcal{L}_B \subset \mathcal{L}_B \end{align*}$$

for certain $\tau ^*_\alpha \in H^\bullet (B)$ depending on $(q,t,\lambda )$ and $\tau _B$ . Differentiating with $\partial _{\phi _b}$ yields a basis, indexed by b, for $T_{J_B(\tau _\alpha ^*,{-z})} \mathcal {L}_B$ as a module over power series in z. These bases together give a basis for the direct sum $\oplus _{\alpha \in F} T_{J_B(\tau _\alpha ^*,{-z})}\mathcal {L}_B$ . Note that applying $z \partial _\Lambda $ to a family of tangent vectors $v(\tau _\alpha ^*) \in T_{J_B(\tau _\alpha ^*,{-z})} \mathcal {L}_B$ – here the family depends on $\tau _B$ via $\tau _\alpha ^*$ – yields another family of tangent vectors in $T_{J_B(\tau _\alpha ^*,{-z})} \mathcal {L}_B$ . Therefore, applying the z-series of matrix-valued differential operators $\sum _{k=0}^\infty (-z)^k \Psi _k(q,t,\lambda - z \partial _\Lambda )$ to our basis for $\oplus _{\alpha \in F} T_{J_B(\tau _\alpha ^*,{-z})}\mathcal {L}_B$ yields another basis for this direct sum. This space, however, has a standard basis, formed by the columns of $S_B(\tau _\alpha ^*,z)^{-1}$ , $\alpha \in F$ . Expressing our basis in terms of the standard one, we obtain

(40) $$ \begin{align} \left(\sum_{k=0}^\infty z^k \Psi_k(q,t,\lambda - z \partial_\Lambda) \right) \begin{bmatrix} \ddots & & 0 \\ &e^{-w_\alpha(q,t,\lambda - z \partial_\Lambda)/z} & \\ 0 & & \ddots \end{bmatrix} S_B(\tau_B,z)^{-1} = \begin{bmatrix} \ddots & & 0 \\ &S_B(\tau_\alpha^*,z)^{-1} & \\ 0 & & \ddots \end{bmatrix} R"(z)^{-1} \end{align} $$

for some invertible matrix-valued z-series $R"(z)$ with entries in the Novikov ring of B that depend analytically on $(q,t,\lambda )$ . Here, we used the fact that the columns of $S_B(\tau _B,z)^{-1}$ are $\partial _{\phi _b} J_B(\tau _B,{-z})$ . The left-hand side of Equation (40) is the expansion near $z=0$ of $\Gamma _{{{block}}}(z) \, T(-z)$ , and the right-hand side provides its analytic extension to the nonequivariant limit $\lambda =0$ . Thus, the expansion near $z=0$ of $\Gamma _{{{block}}}({-z}) \, T(z)$ has a well-defined nonequivariant limit, as claimed.

4.4 Grading

To complete the proof of Theorem 1.4, it remains to show that the loop group operator just defined respects the gradings. In the nonequivariant limit $\lambda =0$ , all the functions of q, t, Q and $\tau _B$ involved satisfy homogeneity conditions that reflect the natural grading in cohomology theory. To describe these conditions explicitly, introduce the Euler vector field

$$\begin{align*}\mathcal{E} := \sum_{i=1}^k c_i q_i\frac{\partial}{\partial q_i} + \sum_{a=1}^r \delta_a Q_a \frac{\partial}{\partial Q_a} + \sum_{b=1}^{\text{rk}\, H^\bullet (B)} \left(1-\frac{\deg (\phi_b)}{2}\right) \tau_b \frac{\partial}{\partial \tau_b}. \end{align*}$$

Here, $c_i$ and $\delta _a$ are the coefficients of the first Chern class of E with respect to an appropriate basis:

$$\begin{align*}c_1(E)=\sum_{j=1}^N U_j+\pi^*c_1(B) = \sum_{i=1}^k\left(\sum_{j=1}^N m_{ij}\right) P_i + \pi^*\left(c_1(B)-\sum_{j=1}^N\Lambda_j\right) = \sum_{i=1}^k c_iP_i+\sum_{a=1}^r \delta_a \pi^*\phi_a, \end{align*}$$

where we have chosen our basis $\phi _1,\ldots ,\phi _{\operatorname {rk} H^\bullet (B)}$ for $H^\bullet (B)$ such that $\phi _1,\ldots ,\phi _r$ is a basis for $H^2(B)$ .

In what follows, we abuse notations and use the same notation to denote both an equivariant object and its nonequivariant limit. Let $\mu _E$ denote the Hodge grading operator for E. That is, in a homogeneous basis of $H^\bullet (E)$ , $\mu _E$ is the diagonal matrix whose diagonal entries are half-integers expressing the degrees of the basis elements measured relative to the middle (complex) dimension of the target

$$\begin{align*}\mu_E\big(\Delta_\beta(P)\otimes \pi^*\phi_b\big) = \frac{\text{deg}\Delta_\beta+\text{deg}\phi_b-\text{dim}_{\mathbb{C}} E}{2} \big(\Delta_\beta(P) \otimes \pi^*\phi_b\big). \end{align*}$$

Lemma 4.1. In its nonequivariant limit the matrix T, defined in Equation (32), satisfies the grading condition

(41) $$ \begin{align} \left(z\frac{d}{dz}+\mathcal{E}\right) T(-z)= T(-z) \mu_E-\mu_E T(-z), \end{align} $$

where $\mu _E$ is the Hodge grading operator for E.

We can rewrite the grading condition (41) using the divisor equations

$$ \begin{align*} q_i\frac{\partial}{\partial q_i} T(-z) = \left(\frac{\partial}{\partial t_i} +\frac{P_i}{z}\right) T(-z) && \text{and} && Q_a\frac{\partial}{\partial Q_a} T(-z) = \left(\frac{\partial}{\partial \tau_a} +\frac{\phi_a}{z}\right) T(-z) \end{align*} $$

to replace $\mathcal {E}$ with $\partial _{\mathcal {E}} +c_1(E)/z$ , where

$$\begin{align*}\partial_{\mathcal{E}} = \sum_{i=1}^k c_i\frac{\partial}{\partial t_i} + \sum_{a=1}^r \delta_a \frac{\partial}{\partial \tau_a} +\sum_{b=1}^{\operatorname{rk} H^\bullet(B)} \left(1-\frac{\deg (\phi_b)}{2}\right) \tau_b \frac{\partial}{\partial \tau_b}. \end{align*}$$

From the nonequivariant version of the quantum differential equation (10) for $S_E$ , we see that $T(-z)=S_E(z)^{-1}L(z)$ satisfies the ordinary differential equation (ODE)

$$ \begin{align*} -z\partial_{\mathcal{E}} T(-z) = T(-z) \mathcal{E} (z), && \text{where} && \mathcal{E}(z) = L^{-1}(z)(\partial_{\mathcal{E}}\bullet) L(z) -z L^{-1}(z)\partial_{\mathcal{E}} L(z) \end{align*} $$

is regarded as a power series in z. Thus, considering T as an operator, we have

(42) $$ \begin{align} \left( z\frac{d}{dz}+\mu_E+\frac{c_1(E)}{z}\right) T(-z) = T(-z)\left( z\frac{d}{dz} + \mu_E+\frac{\mathcal{E}(z)}{z}\right). \end{align} $$

On the other hand, the matrix $\mathcal {I}$ of oscillating integrals, whose stationary phase asymptotics yield Equation (40), in the nonequivariant limit assumes the form

$$\begin{align*}\mathcal{I} (-z):= \bigoplus_{\alpha \in F} \left[ \frac{\Delta_{\beta}\left(\textstyle -z \frac{\partial}{\partial t_1,} \dots, -z \frac{\partial}{\partial t_k}\right)}{(-2\pi z)^{\frac{1}{2}\dim_{\mathbb{C}} X}} \int_{\gamma_\alpha} e^{-\sum_j x_j/z} \prod_j x_j^{\partial_{\Lambda_j}} \frac{\prod_j d\log x_j}{\prod_i d\log (q_ie^{t_i})} \right] S_B(\tau_B, z)^{-1}. \end{align*}$$

Here, columns of $\mathcal {I}$ are indexed by pairs $(\beta , b)$ that correspond to elements of the basis $\{ \Delta _\beta (P) \otimes \pi ^*\phi _b \}$ for $H^\bullet (E)$ , while the rows are indexed by pairs $(\alpha , a)$ where $\alpha $ determines the integration cycle $\gamma _\alpha $ and $\phi _a$ is an element of the basis for $H^\bullet (B)$ . Note that a and b are row and column indices in the matrix $S_B^{-1}$ .

Lemma 4.2. In its nonequivariant limit, the matrix $\mathcal {I}$ satisfies the grading condition

(43) $$ \begin{align} \left(z\frac{d}{dz}+\partial_{\mathcal{E}}+\frac{c_1(B)}{z}\right) \mathcal{I}(-z) = \mathcal{I} (-z) \mu_E-\mu_B \mathcal{I} (-z). \end{align} $$

Proof. The Hodge grading operator $\mu _E$ on the right is decomposed as $\mu _X+\mu _B$ (or, more precisely, as $\mu _X \otimes 1+1\otimes \mu _B$ in the basis $\Delta _\beta (P)\otimes \pi ^*\phi _b$ of $H^{\bullet }(E)$ indexing columns of $\mathcal {I}$ ), and the part $\mathcal {I}(-z) \mu _X$ arises here from the degrees of the operators $\Delta _\beta (-z\partial /\partial t)/(-2\pi z)^{\frac {1}{2}\dim _{\mathbb {C}} X}$ with respect to $z\frac {d}{dz}$ .

We examine now how $z\frac {d}{dz}+\partial _{\mathcal {E}}+\frac {c_1(B)}{z}$ acts on the integral in the definition of $\mathcal {I}(-z)$ . Since the Lie derivative of a closed form is exact, to differentiate an integral along a vector field it suffices to differentiate the integrand along a lift of the vector field to the domain of integration. The vector field $\sum _i c_i \partial /\partial t_i$ lifts along the projection (33) to the Euler operator $Eu:=\sum _j x_j \partial /\partial x_j$ . The latter acts trivially on $S_B^{-1}$ but satisfies $Eu \prod _jx_j^{\partial _{\Lambda _j}}=\prod _j x_j^{\partial _{\Lambda _j}}(Eu +\sum _j \partial _{\Lambda _j})$ . Since $\sum _a \delta _a \partial /\partial \tau _a + \sum _j \partial _{\Lambda _j} = \partial _{c_1(B)}$ , the action of $z \frac {d}{dz}+\partial _{\mathcal {E}}+\frac {c_1(B)}{z}$ on the integral in $\mathcal {I}(-z)$ amounts to applying $z\frac {d}{dz}+\partial _{c_1(B)}+\frac {c_1(B)}{z}+\sum _b \left (1-\frac {\deg (\phi _b)}{2}\right )\tau _b \frac {\partial }{\partial \tau _b}$ to $S_B^{-1}$ instead.

It remains to refer to the grading condition for $S_B^{-1}$ which, taking into account the divisor equations $Q_a\partial /\partial Q_a S_B^{-1}= (\partial /\partial \tau _a + \phi _a /z) S_B^{-1}$ , $a=1,\dots , \operatorname {rk} H^2(B)$ , assumes the form

$$\begin{align*}\left(z\frac{d}{dz}+\partial_{c_1(B)}+\frac{c_1(B)}{z}+\sum_b \left(1-\frac{\deg (\phi_b)}{2}\right)\tau_b \frac{\partial}{\partial\tau_b}\right) S_B^{-1}= S_B^{-1} \mu_B - \mu_B S_B^{-1}. \end{align*}$$

See [Reference Givental13, Section 8]. Altogether, we obtain

$$\begin{align*}\left(z\frac{d}{dz}+\partial_{\mathcal{E}}+\frac{c_1(B)}{z}\right) \mathcal{I}(-z) = \mathcal{I}(-z)\mu_X+\mathcal{I}(-z)\mu_B-\mu_B\mathcal{I}(-z)\end{align*}$$

as promised.

As a function of $(t, \tau _B)$ , $\mathcal {I}$ satisfies the same differential equations as T, since even before taking the nonequivariant limit $\mathcal {I}$ and T differ only by $\Gamma $ -function factors that are independent of t and $\tau _B$ . In particular,

$$\begin{align*}-z \partial_{\mathcal{E}}\mathcal{I}(-z)= \mathcal{I}(-z) \mathcal{E}(z).\end{align*}$$

Considering $\mathcal {I}(-z)$ as an operator, we thus arrive at the following commutation relation:

(44) $$ \begin{align} \mathcal{I}(-z)^{-1}\left( z\frac{d}{dz}+\mu_B +\frac{c_1(B)}{z}\right) = \left( z\frac{d}{dz}+\mu_E+\frac{{\mathcal{E}}(z)}{z}\right) \mathcal{I} (-z)^{-1}. \end{align} $$

Our previous results equate $\mathcal {D}_E$ , up to a constant factor, with

$$\begin{align*}\widehat{S_E^{-1}}\, \widehat{L} \, \widehat{R'}\, \left(\widehat{\oplus_\alpha S_B(\tau^*_{\alpha})}\right)\, \mathcal{D}_B^{\otimes |F|}.\end{align*}$$

Here, $S_E^{-1}(z) L(z)$ coincides with $T(-z)$ , and $R'(z)\left (\oplus _{\alpha }S_B(\tau ^*_{\alpha },z)\right )$ is the matrix inverse to the stationary phase asymptotics of $\mathcal {I} (-z)$ . Note that the values of the arguments $\tau $ and $\tau ^*_{\alpha }$ in $S_E^{-1}$ and $S_B$ are determined by certain mirror maps and cannot be described here explicitly. Nevertheless the complicated relationship between them holds automatically, because all ingredients of the formula were constructed from the same function $I_E$ . Now, the commutation relations (42) and (44) show that commuting the Virasoro grading operator $l_0(E)=zd/dz +1/2+\mu _E+c_1(E)/z$ first across $T(-z)$ and then across the matrix inverse of the stationary phase asymptotics of $\mathcal {I}(-z)$ yields $l_0(B)=zd/dz+1/2+\mu _B+c_1(B)/z$ , the Virasoro grading operator for B. Thus, the loop group transformation that relates $\mathcal {D}_B^{\otimes |F|}$ and $\mathcal {D}_E$ is grading preserving, as claimed.