Virasoro Constraints under Projectivization

1 Introduction

Virasoro constraints organize the descendant Gromov–Witten invariants of a smooth projective variety into differential equations for a single generating function. A basic structural question is whether these equations pass from a base to the total space of a geometric fibration. We prove that they pass through projectivization of an arbitrary algebraic vector bundle.

For a smooth connected projective complex variety Y, let Z_Y be its total ordinary descendant potential: the exponential of the connected stable-map potentials, with genus weight \hbar^{g-1}, ordinary cotangent-line classes at the markings, and monomials Q^d indexed by the actual effective classes d\in H_2(Y;\Z). We use all of H^*(Y;\C) with its cohomological parity and Poincaré pairing. The Virasoro operators are the normally ordered operators associated to the first Hodge grading

Their zero-mode constant is C_Y=\chi(Y)/16-\str(\mu_Y^2)/4, and the other modes have no scalar correction. Section 2 specifies the loop-space and quantization conventions completely. Write \V(Y) for the coefficientwise identities

Thus \V(Y) includes every genus, individual integral curve class, and finite list of descendants, including primitive and odd insertions.

The bundle is arbitrary: it need not split or admit a filtration by line bundles, and it is subject to no positivity condition. The base is allowed to have nonsemisimple quantum cohomology and arbitrary Hodge types. In particular the assertion applies successively to towers of projective bundles once the constraints hold on the initial base. The conclusion concerns the ordinary theory of each total space.

1.1 Historical context and related results

The Virasoro conjecture in Gromov–Witten theory extends the differential constraints for intersection numbers on moduli spaces of stable curves arising in Witten’s conjecture and Kontsevich’s theorem [21, 15]. Eguchi–Hori–Xiong proposed the corresponding constraints for quantum cohomology [4]. The extension to general Hodge types and odd cohomology, including Katz’s correction of the grading, is described by Eguchi–Jinzenji–Xiong [5]; see also Getzler’s account [7]. Liu–Tian prove the constraints in genus zero [18]. In all genera, Givental’s quantization formalism and Teleman’s reconstruction theorem establish them for targets with generically semisimple quantum cohomology [9, 10, 20]. Okounkov–Pandharipande prove them for smooth target curves, including odd insertions [19]. Together with Theorem 1.1, this gives the full constraints for every projective bundle over a smooth projective curve, and for towers of such bundles.

The closest transfer theorem is due to Coates–Givental–Tseng [3]. They prove that the Virasoro constraints hold for the base of a toric bundle if and only if they hold for its total space, for toric bundles constructed from sums of line bundles. Their proof combines ancestor localization with Brown’s mirror theorem [1]; split projective bundles are among its examples. For arbitrary vector bundles, Fan [6] proves all-genus reconstruction and Chern-class dependence of projective-bundle invariants. His argument uses the projective completion \mathbb P_B(E\oplus\mathcal O), which is also the auxiliary space used below. Reconstruction of invariants by itself does not establish compatibility with the Virasoro differential operators.

Iritani–Koto [14] construct a mirror theorem and a decomposition of the quantum D-module for arbitrary projective bundles. Koto [17] proves a mirror theorem for nonsplit toric bundles. These results concern genus zero. In particular, Iritani–Koto’s equivalence of generic semisimplicity for a projective bundle and its base, combined with Givental–Teleman, already gives the conclusion of Theorem 1.1 when the base has generically semisimple quantum cohomology. The transfer proved here also covers bases whose quantum cohomology is not semisimple.

As in Fan’s reconstruction, we use a projective completion of the bundle as an auxiliary space. The localization calculation adapts the ancestor organization of Coates–Givental–Tseng, including their use of the genus-zero cone and ancestor identities. We give the additional gluing argument required when the orbit lines are parametrized by a positive-dimensional variety. The mechanism for passing constraints between the two fixed components uses Iritani’s equivariant shift operator [13]. A local logarithm of that shift cancels equivariant differentiation in the grading. The resulting spectral modes can be quantized and their equations continued between the components. Quantum Riemann–Roch [2] identifies the local equations with the ordinary constraints. The argument uses virtual localization [12] and the ancestor–descendant formalism [16, 8, 9]; the necessary forms of these results are recalled at their points of use.

1.2 The proof mechanism

Twisting E by a sufficiently negative line bundle leaves X unchanged and makes E^* globally generated. Set

Let \C^* scale the trivial summand, and let \lambda be its equivariant parameter. Its fixed components are B and X. Write y for the variable measuring tautological degree and retain separate variables Q^\beta for the full integral base classes. The proof has five stages.

1. 1.

   Express the auxiliary ancestors using the two fixed theories. Localization factors the ancestor potential of W through the product of the inverse-Euler-twisted ancestor potentials of B and X. The transformation is the quantization of an upper symplectic series R(z). The same R factors the genus-zero fundamental solution. Section 4 proves both statements together, keeping the families of orbit lines as evaluation correspondences. Localization provides a relation involving both fixed theories; a second structure is needed to transfer constraints from one to the other.

2. 2.

   Separate the shift operator into spectral blocks. The genus-zero shift operator translates \lambda by the loop variable z. Its coefficients, at each base degree, are polynomial in y. After reducing modulo z and positive base degree, its distinct spectral values are \lambda+h, where

   h^r(h+\lambda)=y.

   There are r+1 branches at a generic value of y. Near y=0, one tends to the eigenvalue zero and corresponds to B; the other r tend to \lambda and together correspond to X. Section 5 establishes this description without diagonalizing the quantum cohomology of B.

3. 3.

   Construct quantizable modes on the blocks. The equivariant grading contains \lambda\partial_\lambda. On each block the logarithm of the shift contains z\partial_\lambda. Subtracting \lambda/z times this logarithm removes coefficient differentiation. The remaining grading defines Virasoro-type loop operators. Section 6 constructs their projections and logarithms coefficientwise and compares them at y=0 with the fixed-component operators. Section 7 uses quantum Riemann–Roch to identify the latter, up to a triangular change of modes, with the ordinary modes on B or X.

4. 4.

   Continue equations between branches. For fixed genus, base class, and insertions, the connected ancestor invariants of W are polynomial in y. Moreover, ancestor powers are bounded independently of fiber degree. Every coefficient of a quantized equation therefore involves finitely many coefficients of the spectral modes and is an actual identity of germs. The covering h\mapsto h^r(h+\lambda) is connected off its branch values. Its monodromy transports the equations from the B branch to every branch. Adding the r branches of the X cluster and returning to y=0 gives the ordinary equations for X up to scalar.

5. 5.

   Fix the scalar normalization. Quantization is projective, so its covariance has only determined the equations up to insertion-independent scalars. The commutators [L_{-1},L_1]=-2L_0 and [L_0,L_k]=-kL_k remove those scalars and give exactly the prescribed constant in L_0. Section 8 proves this calculation and completes the transfer.

Two features of the argument are useful beyond the localization formula itself. Subtracting the block logarithm of a difference operator can turn a grading that differentiates parameters into a loop operator over the coefficient ring. Ancestor bounds then let identities for such operators continue coefficientwise, even when continuation of an entire quantized transformation has not been defined. In the present setting these two steps supply the passage from the ordinary constraints on one fixed component to those on the other.

2 Descendant potentials and quantization

We fix the ordinary theory and its operator normalization before introducing the auxiliary equivariant target. Quantized changes of frame are naturally defined up to a scalar. We therefore record both the exact Virasoro constraints and the weaker, projective form that can be transported between frames. The scalar ambiguity will be removed at the end of the proof.

2.1 Cohomology, curve classes, and descendants

Let Y be a smooth connected projective complex variety of dimension d. Its state space is the full super vector space H_Y=H^*(Y;\C), with parity given by cohomological degree modulo two and pairing

Every tensor product, permutation, contraction, and derivative below uses the Koszul convention. In particular, the coevaluation tensor is the categorical inverse of this pairing, without an additional parity involution. Define even endomorphisms

Thus \mu_Y uses the first Hodge index. Poincaré duality and the Hodge type of c_1(TY) give \mu_Y^*=-\mu_Y, R_Y^*=R_Y, and [\mu_Y,R_Y]=R_Y.

Choose a homogeneous Hodge basis (\phi_a) with \phi_0=1_Y. For j\geq0, let t_j^a have the parity of \phi_a, and put

These are formal supercommuting variables; the aggregate insertion t_j is even. The connected, ordinary, unreduced descendant potentials are

Only stable maps contribute. Here \psi_i is the cotangent class at the marked point of the stable map. It is not an ancestor class. The exponential includes disconnected domains and the empty domain.

The Novikov symbols retain the actual effective integral classes \beta\in H_2(Y;\Z), including \beta=0; multiplication is Q^{\beta_1}Q^{\beta_2}=Q^{\beta_1+\beta_2}. Fixing an ample integral class gives the Novikov completion: only finitely many labels occur below each fixed ample degree. All identities are read coefficientwise in this completion, at finite order in the descendant variables, and with the coefficientwise Laurent interpretation in \hbar. In particular, classes are not identified merely because their divisor degrees agree. The finiteness of effective homology labels of bounded degree follows, for example, from the finite-type Chow spaces of cycles of bounded degree in a projective embedding.

2.2 The ordinary Virasoro operators

On the loop space and its standard polarization use

The infinitesimal symplectic operators are

Their infinitesimal symplectic property follows from the adjoint identities after (2.1), including the sign of z in the residue pairing.

For an even infinitesimal symplectic operator A on a paired loop space with pairing \eta, write

Use homogeneous Darboux coordinates for the displayed polarization, with positive coordinates q_j^a and dual negative coordinates p_{j,a}. Normal ordering quantizes a pp monomial as \hbar times the corresponding second derivative, a pq monomial as the first-order operator with multiplication before differentiation, and a qq monomial as multiplication by that monomial divided by \hbar. Derivatives are left derivatives and all reorderings have their graded signs. We denote the resulting operator, after the dilaton translation

by \op_\eta(A). The negative coordinates p_{j,a} in this paragraph are unrelated to the tautological divisor p introduced below.

With this sign convention the string operator is

For all other modes set

The supertrace uses even minus odd cohomological parity. There are no scalar corrections in positive modes. We write \V(Y) for the full set of identities

These equations use every genus, every effective integral class, and all insertions in H_Y, including odd and primitive classes.

2.3 Projective constraints and changes of frame

This formulation discards exactly the scalar ambiguity of quadratic quantization. It is useful for comparing different paired state spaces. For a symplectic map M between them, denote its quantized action, when defined in the completions used here, by U_M. Our group-action convention is the one in the following covariance formula. With the Hamiltonian sign above, its infinitesimal form uses -\op_\eta(A) for the action of \exp(A).

2.4 Coefficientwise finiteness near zero curve variables

Some later mode matrices have infinitely many positive powers of z, and can also contain finite powers of z\partial_z. We interpret their residue identities entrywise in the polarized mode coordinates. The arrays need not act on every uncompleted loop-space input; only the coefficientwise actions specified here are used. The following elementary rule specifies the formal setting for their local quantizations; the continuation argument will establish its separate finiteness assertion at generic fiber parameter.

Ordinary descendant potentials have this support property: for fixed genus, class, and number of marks, sufficiently high powers of the cotangent classes vanish on the finite-dimensional stable-map space. It also holds for the twisted descendants used below, where the torus acts trivially on the target and the cotangent classes remain ordinary classes. Ancestor potentials have the stronger bound supplied by the dimension of the moduli space of stable curves. Stability ensures the stated Laurent property after exponentiation: degree-zero genus-zero components consume marked points, whereas positive-degree components consume positive Novikov degree.

The standard splitting and cotangent-class identities used in the paper are tensor identities with the diagonal given by the categorical coevaluation. Their marked-point proofs therefore apply to full cohomology with precisely these signs. Characteristic-class multiplications are even; quantum Riemann–Roch and the two-mark shift identities likewise use arbitrary evaluation classes and this diagonal. Throughout, a splitting adds the actual integral curve classes. These conventions permit the standard formulas to be used without discarding odd insertions or merging Novikov labels.

3 The master space and its fixed theories

We place the base and its projectivization inside a single smooth projective variety with a torus action. Its fixed components will carry inverse-Euler twists of their ordinary theories. This section specifies their curve labels and pairings, and recalls the ancestor and fundamental solution conventions needed for the localization comparison.

3.1 Geometry and integral curve labels

Tensoring E by a line bundle does not change the projective bundle of one-dimensional subspaces. Choose such a twist so that E^* is globally generated, and continue to denote the resulting bundle by E. Set

Let T=\C^* act with weight zero on E and weight one on \mathcal O_B, and let \lambda\in H_T^2(\mathrm{pt}) be the class of the weight-one representation. The fixed locus is W^T=X\sqcup B, where the second component is the section corresponding to the summand \mathcal O_B. Write \pi for either projective-bundle projection and put p=c_1^T(\mathcal O_W(1)). The ordinary normal bundles and their torus weights are

The restrictions of the divisor are

Here w_F is the common torus weight on the normal fibers; the integers j_F are recorded for the fixed-section curve factors in the shift operator below. The normal-bundle descriptions follow from the relative tangent space \Hom(L,V/L) at a line L\subset V.

We write Q^\beta y^l for the actual class corresponding to (\beta,l). Under the inclusion X\hookrightarrow W these two coordinates are unchanged, and under B\hookrightarrow W a class \beta becomes (\beta,0). Thus the fixed theories and the master-space theory have compatible full integral labels. We may allow all pairs of an effective base class and an integer l\geq0 as formal labels, assigning coefficient zero when a pair is not effective for the target in question.

Choose an ample integral class H on B. For a sufficiently large integer M, the class c_1(\mathcal O_P(1))+M\pi^*H is ample for both projective bundles. We complete in the degree

There are finitely many effective labels of bounded degree, as in Section 2. Unless specified otherwise, positive Novikov filtration means positive degree for this completion, so it includes positive fiber degree even when the base class is zero. The separate completion by positive base degree will be used only when the fiber parameter y is treated as a variable on a complex domain.

3.2 Equivariant pairings and inverse-Euler twists

The equivariant projective-bundle formula makes H_T^*(W) free over \C[\lambda], with basis

for a homogeneous Hodge basis (\phi_a) of H_B. Its equivariant integration pairing is perfect over \C[\lambda]. Indeed, integration over the projective fibers gives an antitriangular matrix in the powers of p, whose antidiagonal blocks are the ordinary Poincaré pairing of B.

After extending scalars to \C(\lambda), restriction to the fixed locus is an isomorphism of paired spaces

The unit restricts to the sum of the two fixed units. Every normal weight is nonzero, so the Euler classes in this formula are invertible over \C(\lambda).

The corresponding twisted Gromov–Witten theory on F inserts

in each stable-map integral. Here \rho and f are the universal curve and universal map, and the inverse Euler class is extended multiplicatively to the indicated K-theory class. The torus acts trivially on F and with weight w_F on N_F. We use subscripts F,\mathrm{tw} for its potentials and fundamental solutions. Its degree-zero three-point pairing is exactly \eta_F^t.

Give p and \lambda Hodge bidegree (1,1). In a homogeneous polynomial basis the first-Hodge grading acts on basis vectors, while differentiation in \lambda records the degree of equivariant coefficients when needed. The virtual dimension identities can be expressed in this first degree: algebraicity forces the sum of the Hodge-degree differences of insertions in a nonzero invariant to be zero. This remains true equivariantly, by finite-dimensional approximations or by fixed-locus integration. It is distinct from the ordinary cohomological-degree count used to bound fiber degree later.

3.3 Ancestors and the fundamental solution

We give the same definitions for an ordinary theory, the equivariant theory of W, or one of the twisted fixed theories. Denote its pairing by \eta, its Novikov monomial for a class d by \mathsf q^d, and its stable-map integrals by brackets. Let u be a formal even primary background. For distinguished insertions a_1,\ldots,a_m, set

where d runs over effective classes and only stable-map terms are included. Descendant insertions in these brackets use the map cotangent classes.

For 2g-2+m>0, there is a stabilization map

which forgets the map and the n background marks and stabilizes the remaining curve. Define \bar\psi_i=\operatorname{st}_{g,m}^*\psi_i for 1\leq i\leq m. The ancestor potential at background u is

The bracket in this display denotes integration of the displayed product, with the twist when appropriate. All terms with curve-unstable distinguished data are omitted, even when the stable-map space itself exists. The genus-one term with no distinguished marks is therefore omitted as well. Since the ancestors come from \overline{\mathcal M}_{g,m}, their total power exceeds the dimension 3g-3+m only when the corresponding product vanishes.

The genus-zero splitting and cotangent-class identities give

where \star_u is the quantum product and \partial_v differentiates the primary background. For clarity, the genus-zero cone \mathcal L is the formal Lagrangian graph of dF_0 in the Darboux coordinates (q,p), with q=t-z1. Its point with descendant input \epsilon has positive coordinate \epsilon-z1, and the tangent space there is the graph of the Hessian of F_0 at \epsilon. The genus-zero string, dilaton, and topological recursion identities say that each tangent space T is tangent along zT\subset\mathcal L. At primary coordinate u that tangent space is S(u,z)^{-1}\cH_+; see [11]. The ancestor–descendant correspondence in our group-action convention is

where c(u) is independent of ancestor variables. These formulas follow from the splitting identity for \psi_i-\bar\psi_i and its genus-zero specializations; see [16, 8, 9]. They hold with the super contractions of Section 2 and with the inverse-Euler twists above.

We use (3.8) only on the fixed theories, with u of positive total Novikov filtration. There S(u,z)-\id has positive filtration, and every Novikov coefficient has only finitely many negative powers of z: only finitely many background marks occur at that coefficient, and the map cotangent classes on the fixed targets are nilpotent. Thus the quantized action is covered by Lemma 2.3. For the master space we abbreviate

Its rational loop-variable expansions and the quantized action used to compare it with the fixed theories will be established directly by localization in the next section.

4 Ancestor localization for the master space

The torus action on W=\mathbb P_B(E\oplus\mathcal O) separates its Gromov–Witten theory into contributions from B and X. We need this separation at the level of ancestor potentials, together with the corresponding factorization of the genus-zero fundamental solution. We abbreviate the fixed paired space of (3.3) by

Restriction identifies this paired space with H_T^*(W) after inverting \lambda. All series below use the full curve labels Q^\beta y^l.

The master space and its fixed-graph geometry also occur in Fan’s reconstruction of projective-bundle invariants [6, §§3–4]. For the ancestor factorization we adapt the localization argument of Coates–Givental–Tseng [3, §3]. Their toric-bundle proof has a finite set of orbit directions in each fiber. Here the orbit lines form the projective bundle X, so their contributions are cohomology correspondences. We first establish the geometry and gluing formula for these families. After this step, the genus-zero identities determine R, and stabilization of the localization graphs gives its quantized action. This supplies, for the action at hand, the leg-moduli and virtual-class details left open in [3, Remark 3.7].

4.1 The moving components and their gluing

Call an irreducible component of a torus-fixed stable map a moving leg if its image is not contained in W^T=B\sqcup X. A connected part mapping into a fixed component will instead be kept as a stable-map factor for that component, including its internal boundary strata.

The geometry is illustrated in Figure 1. Nonsplitting of E affects the global family of lines and its cohomology classes, but does not change the description of an individual leg or the nonzero characters in the lemma.

We now apply virtual localization [12]. To make its use with these families explicit, first distinguish all branches at gluing nodes and divide by graph automorphisms afterward. A torus-fixed deformation cannot smooth any of the nodes cut in Lemma 4.2: the smoothing parameter has nonzero character. The fixed loci are therefore obtained by matching leg endpoints and stable maps to B or X along their evaluation maps. This description holds in families. Indeed, degree zero over B forces the projection of a rational leg family to factor through its base, and the local monomial description then applies over a trivialization of E. On a stable-map piece with image in F, the lifted torus acts trivially on the domain: its automorphism group as a pointed stable map is finite. Fixed deformations of that piece consequently remain maps into F.

For completeness, the compatibility of obstruction theories can be read directly from normalization. Denote the normalized pieces by C_\alpha and the cut nodes by q. The map deformation complexes, relative to their prestable domains, fit into the triangle

Cut branches are marked on the pieces. Passing to absolute map deformations adds their pointed-domain deformation and automorphism complexes; releasing a cut node adds its smoothing line T_{q,+}\otimes T_{q,-}. The ambient virtual tangent complex, restricted to this fixed gluing locus, consequently has class

Here each piece carries its pointed stable-map obstruction theory; for a leg this is the ambient theory restricted to \mathfrak L_m. The identity comes from the displayed compatible triangle, so its fixed part gives the obstruction theory of diagonal matching, not only an equality of virtual ranks. On fixed-map pieces the fixed part is their theory in F; on legs it is T_{\mathfrak L_m}. The matching terms are T_F, and every smoothing line is moving. The fixed virtual class is therefore the diagonal virtual pullback of these classes, including when two legs meet or a graph has a cycle. The moving part contains R\pi_*f^*N_F on each fixed-map piece, the moving leg complexes, the matching terms -N_F, and the smoothing lines. These are the factors required by virtual localization.

In particular, cutting a leg off a fixed-map vertex contributes the normal matching numerator e_T(N_F) and the smoothing denominator

Here \psi belongs to the fixed-map vertex and a to the leg. The matching numerator is exactly what makes the contraction use the inverse of \eta_F^t. All remaining leg data are cohomology correspondences independent of the descendants on the vertex. There is also an endpoint version of this rule: if an external marked point replaces the attachment to a fixed-map vertex, there is no smoothing denominator, its cotangent class is -a, and contraction with \eta_F^t cancels the normal matching numerator. This convention includes endpoints with no fixed-map component.

The ordinary class in a need not descend along X\to B. We therefore expand it on the leg stack before any pushforward. If a=a_0+\nu, with a_0=w_F\lambda/m\ne0, then

The sum is finite. More explicitly, let C be a correspondence coefficient on a cut piece, with endpoint map e to F, and put h_j=e_*(\nu^jC), with its virtual and Euler factors included. The label h will retain this entire finite family of moments, together with the scalar a_0. For any function K at this flag define

The symbol \alpha is a dummy scalar: the derivative acts on K alone, holding C and all h_j fixed, even if they depend on \lambda. For example the later expression S_f(a)h/(a+z) means (4.6) with K(\alpha)=S_f(\alpha)/(\alpha+z). At several attachment flags use the joint moments and independent dummy scalars. A scalar tangent class and an ordinary cohomology vector are the special case in which only h_0 is present. This convention makes the correspondence formulas precise even when the tangent class does not descend to F.

All graph sums in this section are coefficientwise finite. Each moving leg has positive tautological degree, so its multiplicity and the number of legs are bounded at fixed ample degree. There are finitely many splittings into effective full curve labels, and stability bounds the remaining degree-zero pieces once genus and the distinguished markings are fixed. The fixed-map moduli spaces bound the powers of their cotangent classes. These observations also justify the finite nilpotent expansions and all termwise pushforwards used below.

4.2 The genus-zero factor

We first construct R from two-point invariants. Fix F=B or X. An end at F is a genus-zero unmarked tree attached to a fixed-map vertex by a moving leg, with the vertex itself omitted. Let \epsilon_F(s) be the sum of its contributions, including the denominator (a-s)^{-1} at the attachment. It is a cohomology-valued power series in s, of positive Novikov filtration. Likewise a tail has one external primary insertion b\in H_T^*(W); write its contribution as h_b/(a-s), with summation over tails understood. The moment data h_b include the curve monomial, the stack and symmetry factors, and the linear dependence on b. These definitions include an external insertion directly at a leg endpoint.

End denominators have infinite Taylor expansions. Accordingly, in identities involving these backgrounds we use the completion \widehat{\cH}_+=H_F[[z]] of the positive space, with localized Novikov coefficients, and [\,\cdot\,]_+ denotes the nonnegative part. The usual cone identities extend to these backgrounds coefficientwise: positive filtration bounds their number of insertions, and nilpotence of the map cotangent classes bounds the indices which can contribute.

In the twisted theory of F, let u_F be the primary parameter of the tangent space to the genus-zero cone at descendant background \epsilon_F. Recall the particular genus-zero facts that we use. The string, dilaton, and topological recursion relations make the cone overruled, and its tangent spaces are S_{F,\mathrm{tw}}(u,z)^{-1}\widehat{\cH}_+ in this completion [11]. The parameter u_F is equivalently determined by

Indeed, applying S_f to the cone point makes it belong to z\widehat{\cH}_+, and [S_f(z)(-z+u_F)]_+=-z. The vanishing of the constant term in (4.7) has linearization -\id with respect to u_F at zero Novikov degree. It therefore determines u_F uniquely by the formal implicit equation, and u_F has positive filtration.

The tangent space is the graph of the Hessian of the genus-zero potential. Thus its two-point function depends on \epsilon_F only through u_F. In the following formula the bracket sums the stable genus-zero terms with any number of additional \epsilon_F(\psi) insertions:

At primary background this is the standard two-point fundamental solution identity; the Hessian description gives it at the present background. It follows alternatively by genus-zero topological recursion. The metric term records the unstable two-point convention. Every occurrence of this identity is rational in w,z at a fixed curve coefficient.

For c supported on F, localize the matrix element \eta(b,S_W(z)c). If the input lies on a fixed-map vertex, all unmarked outgoing trees provide its background \epsilon_F. The insertion b is either on that same vertex, as b|_F, or in a tail with attachment h_b/(a-\psi). If the input itself is a marked leg endpoint, the endpoint rule supplies the metric term in (4.8), with denominator a+z. Taking also the coefficient at w^{-1} in that identity to treat a primary slot gives

This proves (4.1): explicitly,

expanded at z=0. The denominators are invertible there, so R is upper triangular. The fixed-theory S_f is finite in negative powers of z at each curve coefficient, and the tails give rational functions. This also proves the claimed rationality of S_W. Since S_W and S_{\mathrm{bl}} are symplectic, their rational identities imply

Every nonidentity term in (4.10) has a moving leg, so R-\id has positive Novikov filtration.

Two further genus-zero identities identify the translation and the edge contraction for the quantized action. They will allow us to recognize the higher-genus localization formula without computing the individual leg Euler factors.

First use the one-point function J_W(-z)=-zS_W(z)^{-1}1. The string equation identifies it with

where only stable one-marked terms are included. In the expansion at zero, its regular part on F is -z+\epsilon_F(z). Indeed, a marked leg endpoint supplies an end, whereas a marking on a fixed-map vertex contributes only strictly negative powers of z. Hence

Apply S_f and take the regular part. Because S_f has only nonpositive powers, inserting or omitting an inner regular-part projection does not alter the final regular part. The string equation gives [S_f(z)z]_+=z+u_F; using the factorization already proved yields

Next a bridge is an unmarked genus-zero tree with two attachments to fixed-map vertices, omitted from the tree, and moving legs at both attachments; a single leg is allowed. Its contribution has denominators (a-s)^{-1}(a'-s')^{-1}. Apply S_f(a) and S_{f'}(a') to its two correspondence slots, and denote the sum of these dressed contributions by D(s,s'). A tensor is here regarded as an operator by the twisted pairing; the first argument belongs to the output slot.

Localize the two-point function of W, including its metric term. If no moving leg separates the marked points, the contribution is the block identity (4.8). Otherwise the path between them determines a unique bridge, and (4.8) at its two ends includes the cases in which either head is an unstable marked endpoint. Consequently

Together with (4.1), this gives

The numerator vanishes at w=-z by (4.11). Thus the right side is a power series at (w,z)=(0,0), as is the dressed bridge expression. We may first establish these equalities with independent rational variables and then take these Taylor expansions.

4.3 From descendants at the flags to ancestors

We have constructed the upper factor and identified its translation and bridge kernel. To obtain (4.2), it remains to express every surviving vertex of a localization graph by its twisted ancestor theory. We record the two ancestor identities needed for that conversion, including their dependence on the background.

For a fixed target with twisted pairing \eta^t, let \mathcal A(\epsilon;t) denote the mixed potential with distinguished insertions t(\bar\psi) and additional insertions \epsilon(\psi). Here the ancestors forget the additional marks. The parameter \epsilon is of positive filtration, and u and x(z)=[S(u,z)(\epsilon(z)-u)]_+ are specified by (4.7).

4.4 Assembly of the stable graphs

Consider a fixed stable map contributing to an ancestor invariant of W with stable distinguished curve data. Forget the map and stabilize its distinguished marked curve. Every moving leg disappears, because it has at most two special points. On the graph whose vertices are entire fixed-map factors, prune unmarked rational trees and suppress rational chains with two remaining flags, keeping all distinguished markings. A fixed-map factor which survives is retained with its own stabilization at those flags and markings; its internal stable-map boundary remains part of its moduli space. The pieces removed between surviving vertices are bridges, those ending in a distinguished marking are tails, and all other removed pieces are ends.

This decomposition identifies the restriction of the global ancestor classes. At a surviving vertex they are its ancestors for the remaining distinguished flags and marks, with the end marks forgotten. For a marking on a tail, its global ancestor is the ancestor at the tail’s attachment after contraction. These statements follow from the composition of the stabilization maps with gluing of the surviving pointed curves.

Ends therefore supply the background \epsilon_F at each vertex. Lemma 4.3 converts its bridge and tail denominators to ancestor denominators and inserts the factors S_f(a). Bridges become exactly D(\bar\psi,\bar\psi'). The direct external markings together with all tails become, by (4.10),

Finally (4.16) replaces the background by u_F and adds x_F. In view of (4.12), the complete vertex substitution is consequently

The upper-triangular quantization formula [9, Proposition 7.3] is now applicable. If

its contraction operator is \hbar/2 times the second derivatives with coefficient tensors D_{ij}, using the categorical inverse of the twisted pairing. Exponentiating this operator on the product of the fixed ancestor potentials and then making (4.17) is U_R. Indeed its kernel is (4.14); the arguments -w,-z are the signs required by the negative Darboux modes (-z)^{-i-1}. This is also [3, Equation (19)] in the present notation.

The localization graph sum has precisely this form. Labeling the flags first and subsequently dividing by permutations gives its factorials and graph automorphism factors. A bridge joining two vertices, or two flags at the same vertex, contributes one factor of \hbar under the same contraction rule; cycles therefore have the required genus power. All pairings and permutations are those in super vector spaces. The tensor gluing proof consequently includes odd cohomology without a change of pairing convention. Components with no distinguished markings can change the overall scalar, which is invertible as a formal exponential. We have proved (4.2).

It remains to check the invertibility asserted in the proposition. The series R-\id has positive filtration, so its inverse and logarithm are defined successively in Novikov degree. The same holds for the actions obtained by exponentiating the corresponding quadratic operators. At a fixed curve coefficient, genus power, and number of variables, only finitely many filtration-positive factors can contribute. The remaining sums over ancestor indices are finite because an ancestor on a stable n-pointed genus-g curve has total degree at most 3g-3+n; in the fixed descendant identities the corresponding bound is the finite dimension of the fixed-target stable-map space. The inverse upper action has the same properties. Thus all transformations used here and their inverses act in the stated coefficientwise completion. This completes the proof of Proposition 4.1.

5 A grading and a shift in genus zero

We now construct two commuting operators on the equivariant quantum cohomology of W. One is the grading operator. The other translates \lambda by the loop variable z and is defined by genus-zero invariants of a bundle over \mathbb P^1. Its fixed-locus expression will connect the ordinary theories of B and X. Its spectrum, computed below using only fiber curves, will provide the branches along which we transport the constraints.

Throughout this section, \star means the small equivariant quantum product of W, at primary background zero. Put

The equivariant projective-bundle formula gives

Choose a homogeneous Hodge basis of H^*(B) and the resulting polynomial basis \pi^*b\,p^j, 0\leq j\leq r, of H_T^*(W) over \C[\lambda]. In this basis, \mu acts by the first Hodge degree minus \dim_\C(W)/2 and does not differentiate the coefficient \lambda.

5.1 The calibrated grading

Define

The subscripts distinguish the constant, or descendant, frame from the quantum, or ancestor, frame. Both operators act on coefficients in \lambda, whereas the Novikov variables are held fixed.

5.2 The shift operator and its fixed-locus formula

Here is the genus-zero input from Iritani that we use. For a smooth projective variety Y with a \C^*-action, form its associated bundle \widehat Y\to\mathbb P^1. Let \sigma_{\min} be the section class of the fixed component with positive normal weights. The operator defined by two-fiber section invariants, with monomials Q^{\widehat d-\sigma_{\min}}, is an unlocalized equivariant cohomology operator. After reversing the sign of Iritani’s loop variable, it translates \lambda to \lambda+z and satisfies

Here x runs through the ordinary Chern roots of the weight-m normal subbundle of F; each quotient has only finitely many remaining factors. This is the projective specialization of [13, Proposition 2.2, Remark 3.4, Definition 3.9, Remark 3.10, Definition 3.13, and Theorem 3.14]. The source’s seminegativity condition is automatic for a complete target, since its global functions are constant. Its calibration M is related to ours by S_Y(z)=M(-z)^{-1}.

We describe the associated geometry in this application to fix both the section classes and the signs. For the action that has weight one on the trivial summand of E\oplus\mathcal O, it is

Equivalently it is W\times(\C^2\setminus\{0\})/\C^*, with s\cdot(w,v_1,v_2)=(s\cdot w,s^{-1}v_1,s^{-1}v_2). An additional torus scales v_2, and its equivariant parameter is Iritani’s loop variable before sign reversal. The fibers over [1:0] and [0:1] carry the original action and the action composed with this additional torus. Identifying their equivariant cohomologies by the induced change of torus variables accounts for the translation in (5.6).

More explicitly, write \zeta for that parameter before sign reversal, and let \rho_0(t,u)w=t\cdot w and \rho_1(t,u)w=(tu)\cdot w be the actions on the two fibers. The change of torus variables induces \Phi_1:H^*_{T\times\C^*,\rho_0}(W)\to H^*_{T\times\C^*,\rho_1}(W) with

If \iota_0,\iota_\infty denote the fiber inclusions, the section correspondence is defined by

Only effective section classes are included. The inputs belong to the equivariant cohomologies of their respective fibers, and the pairing on the left is the one on the second fiber. The operator used here is T_a=(\Phi_1^{-1}\widetilde T)|_{\zeta=-z}. Its semilinearity translates \lambda to \lambda+z, as asserted.

In this construction the normal weight at X is positive, so X gives \sigma_{\min}. A section obtained from a point of X has tautological degree zero in (5.7); a section obtained from the trivial-summand fixed component B is the line \mathcal O(-1) and has tautological degree one. Both have base class zero. Thus \sigma_B-\sigma_{\min} is precisely the fiber-line class of W.

This also checks the curve labels in the shift theorem. The integral projective-bundle splitting identifies H_2(W,\Z) by (\beta,l). The same splitting on (5.7) identifies a section class by (\beta,1,l). Subtraction of \sigma_{\min}=(0,1,0) leaves (\beta,0,l). The inclusions of the two fibers preserve \beta and tautological degree, hence induce the same identification on integral homology. At every splitting of a section stable map the labels therefore add in actual integral homology. In particular the use of (5.6) does not replace classes by their numerical equivalence classes.

We have now obtained operators whose coefficients can be studied both as Novikov series at y=0 and as functions of a nonzero complex variable y. Before making that change of viewpoint, we verify that the shift respects the grading and determine its limiting spectrum.

5.3 The spectrum of the fiber theory

Let Q^{>0} denote the ideal of positive base degree in the Novikov ring. Since an effective nonzero base class has positive ample degree, reduction modulo this ideal retains exactly the maps with constant projection to B. The variable y is retained. We next identify T_a modulo z in this fiber theory.

Let I=H^{>0}(B;\C). This is a nilpotent ideal, also when H^*(B) has odd classes. Reducing the fiber quantum algebra modulo I gives

Indeed the classical projective-bundle relation reduces to p^r(p+\lambda)=0. In degree r+1 the only possible positive curve correction is a constant multiple of y. Its coefficient is the degree-one line invariant in a fiber \mathbb P^r, hence one. More explicitly, one may pair the relevant structure constant with a top-degree base class; the degree-zero base projection reduces the computation to the fiber. The base directions contribute no obstruction because H^1(C,\mathcal O_C)=0 for a genus-zero domain. The coefficient has degree zero, so is independent of \lambda and may equally be computed at \lambda=0. It is then

Here H is the ordinary hyperplane class of the fiber. Indeed two general point conditions determine a unique line, and the remaining marked point is its unique intersection with a general hyperplane. Convexity of projective space makes this the virtual count as well. This gives the coefficient of y in p\star p^r, hence the coefficient in (5.14).

The polynomial dependence of the two operators and the finite set of eigenvalue branches are the ingredients needed for the spectral construction in the next section. The calibration identities retain their full curve labels, so the later return to y=0 will still compare the individual descendant theories of the two fixed manifolds.

6 Spectral projections and block gradings

The shift operator T_a distinguishes r+1 spectral blocks at generic fiber parameter, whereas the localization formula distinguishes the two fixed manifolds B and X. We construct operators on the individual spectral blocks and compare their sum near y=0 with the fixed-manifold blocks. The construction must also remove differentiation in the equivariant parameter: only after that removal can the resulting loop operators be quantized over the coefficient ring. All assertions in this section are classical statements about operators.

6.1 Functional calculus for differential series

Let V be the finite-dimensional vector space underlying a chosen equivariant basis of H_T^*(W). On a small open set U in the (y,\lambda)-plane, write \mathcal O_U for holomorphic functions and

The multiplication is composition, so \partial_\lambda a=a\partial_\lambda+\partial_\lambda(a). There is no differentiation in y. Denote by \Lambda_B the base Novikov completion already fixed, and by \Lambda_{B,+} its positive degree ideal. We use the completed algebra

Here and below the notation \beta\geq0 means that \beta ranges over the effective base labels, including zero. The completion is taken with respect to z and ample base degree. In particular each A_{\beta,m} has finite differential order, without a bound uniform in \beta,m. Products at a fixed coefficient are finite by Novikov finiteness. The element z is central in this algebra. We may equally work with germs, and then all holomorphic assertions are coefficientwise. When a loop derivative occurs, we adjoin z\partial_z with

all expressions involving this extra derivative have finite order in it.

Suppose T\in\mathscr A_U has order-zero reduction

After shrinking U, choose contours in the complex \xi-plane that avoid the spectrum of T^{(0)} and enclose specified groups of its eigenvalues. The contours are held fixed while y and \lambda vary in U. Set R_0(\xi)=(\xi-T^{(0)})^{-1} and K=T-T^{(0)}. The formal resolvent is

Each factor K increases the joint z/base filtration. Consequently each coefficient in this expression is a finite sum of differential operators, and is meromorphic in \xi, with poles only at the eigenvalues of T^{(0)}.

We will compare this calculus in frames related by matrices with negative powers of z. The comparison requires a larger algebra, but does not require convergence in the loop variable. Let \mathcal O be a ring of holomorphic parameter germs stable under \partial_\lambda, and let \mathcal M be either the base Novikov monoid or the full monoid of labels (\beta,l), l\geq0. Write

The notation q^d means Q^\beta or Q^\beta y^l, respectively. The Novikov support condition is imposed as before. The lower z-bound can depend on d. Multiplication is well-defined: a fixed label has finitely many decompositions, and for each decomposition the two Laurent lower bounds leave only finitely many summands at a fixed power of z.

Two forms of M will occur. The first is a matrix equal to the identity in degree zero whose coefficient at each positive Novikov label is the Laurent expansion at z=0 of a rational function of z. Its Novikov inverse has the same property. The second is a matrix of the form \exp(N/z)M_+(z), where N is a nilpotent cup-product operator and M_+(z) and its inverse are regular at z=0. The exponential and its inverse are then Laurent polynomials in z^{-1}. Thus both forms satisfy the Laurent requirement of Lemma 6.2.

6.2 Block operators without equivariant differentiation

We now apply the calculus to T=T_a. By Lemma 5.3, its coefficients are polynomial in y,\lambda,z at fixed base degree before the shift \exp(z\partial_\lambda) is expanded. In particular it belongs to (6.1). By (5.15), the spectrum of its reduction modulo z and positive base degree consists of

Take \lambda\neq0, y\neq0, and exclude the values where these roots coincide. On a small open set in this locus, choose their labels and a logarithm near each a_i. A label refers to a whole generalized eigenspace: no diagonalizability within that space is assumed. Define

Here \gamma_i encloses the one spectral value a_i with its full multiplicity. For a group of these values we use the sum of the contours; its logarithm is specified on each component.

The subtraction in (6.9) has accomplished the first goal: it gives loop operators over the coefficient ring. We record their precise bounds before making the comparison at y=0.

In particular, changing a logarithm by 2\pi\mathrm i n changes D_i by -2\pi\mathrm i n\lambda z^{-1}P_i. All the modes for one logarithm therefore span the same collection as those for any other logarithm. Infinitesimal symplecticity will follow from the fixed-manifold calculation and continuation; it is not needed for the constructions above.

6.3 The two fixed-manifold blocks at y=0

Fix a germ of \lambda\neq0. At y=0, one eigenvalue a_B=\lambda+h_B tends to zero, while the other r tend to \lambda. We call the first the B branch and the second group the X cluster. Choose disjoint small contours about 0 and \lambda that enclose these groups for y sufficiently small. The projector calculus applies through y=0 on both contours. The logarithm applies through y=0 on the X contour, since that contour bounds a neighborhood not containing zero. Denote the resulting operators by P_B,P_X,\mathscr L_X.

For this comparison express both frames in the fixed-cohomology coordinates furnished by restriction to B\sqcup X. The change from the polynomial equivariant basis depends only on \lambda and is invertible at \lambda\neq0; it preserves the completed algebras and all assertions about holomorphic dependence on y. Let \Pi_B,\Pi_X be the constant projections onto these two summands. In these coordinates write

where the product is over the ordinary Chern roots of N_F. Thus (5.10) reads T_d|_F=y^{j_F}K_F E_z.

We have thus constructed the same block operators in two useful forms. At generic y, their individual coefficients continue with the roots in (6.6). Near y=0, they are conjugates of operators on the two fixed manifolds. The coefficients of the B-branch modes are finite polynomials in \log y with holomorphic coefficients. Indeed D_B=D_B^{\mathrm{reg}}-\lambda(\log y)z^{-1}P_B, where D_B^{\mathrm{reg}}=P_BG_a-(\lambda/z)\mathscr K_B has holomorphic coefficients; apply (6.12) with b=-\lambda\log y. For A_{k,B} the logarithmic degree is therefore at most k+1. The X-cluster modes are holomorphic through zero by Proposition 6.5. These statements require neither convergence in z nor an analytic interpretation of the Novikov series.

7 The ordinary theory on a fixed component

The operators of Section 6 were constructed from the equivariant theory of W. We now identify what their equations mean near y=0. On a fixed component, quantum Riemann–Roch removes the inverse Euler twist. The same transformation turns the logarithm of the shift operator into a translation generator in \lambda. Its derivative term cancels the derivative term of the equivariant grading, leaving the ordinary grading and a scalar multiple of z^{-1}.

We first prove the identity of classical operators. We then explain its quantization; this second step requires separating the rank factor of the Euler class before applying quantum Riemann–Roch.

7.1 Removing the normal bundle from the grading

Fix F and abbreviate n=n_F, w=w_F, and j=j_F. Put

After restriction to the descendant space of F, the grading and shift from Section 5 are

Here x runs over the ordinary Chern roots of N_F, and multiplication by a class is understood in the second formula. Symmetric expressions in the roots are interpreted by the splitting principle. All expansions in x are finite after evaluation in the cohomology of F.

Choose a germ of \log a at a nonzero value of \lambda. Define a multiplier \Delta_F by

The B_{2m} are the Bernoulli numbers. The coefficient of z^{-1} has positive ordinary cohomological degree, hence is nilpotent as a multiplication operator. Both \Delta_F and its inverse consequently have a finite lower bound on their powers of z. Their positive tails are interpreted as formal series at z=0.

The multiplier maps the ordinary paired loop space of F to the space with pairing \eta_F^t. Indeed, all terms in (7.3) except the half logarithm are odd in z, so that

The inverse Euler factor in \eta_F^t therefore cancels this product. This also fixes the direction of the square-root factor in (7.3).

For the scalar part of the shift calculation, put

Subtracting \lambda/z times (7.7) from (7.5) now removes \lambda\partial_\lambda. Thus the operator D_F on its fixed block, after the descendant change of space and then \Delta_F, is

up to a scalar multiple of \lambda/z from a different logarithm choice. In particular, the remaining operator differentiates none of \lambda, y, or the Novikov variables.

For a scalar b independent of z, write

The binomial identity

is triangular with diagonal entries one. Hence these operators are infinitesimal symplectic, and their simultaneous projective equations are equivalent to those of the ordinary modes. We have proved this identification at the classical level. To obtain the corresponding statement for the potentials, we next implement it by quantum Riemann–Roch.

7.2 Quantizing the comparison

The logarithm in (7.3) contains -\log(w\lambda)c_N/z. Although this term defines a perfectly good classical multiplication operator, directly exponentiating its quantization would require interpreting a translation that is not small in \lambda^{-1}. We avoid that interpretation by first using the normalized characteristic class

Let Z_{F,\mathrm{ntw}} and \eta_F^{\mathrm{ntw}} denote its descendant potential and pairing. The subscript distinguishes this theory from the inverse Euler twist used in localization.

Define \Delta_{F,0} by the formula (7.3), replacing -\log(a+x) by -\log(1+x/a) and replacing the half logarithm by \tfrac12\log(1+x/a). It maps the ordinary paired space to (\mathcal H_F,\eta_F^{\mathrm{ntw}}) and is the identity modulo \lambda^{-1}.

The relation between the two classical multipliers is particularly simple:

If C is an ordinary divisor class, then

Here R_F commutes with C, and the two displayed contributions to the first commutator are -C/z and C/z. Consequently C/z commutes with all the ordinary modes and with their scalar translates (7.11). After the coefficient derivatives have been removed in (7.10), the scalar a^{n/2} also cancels in conjugation. Thus \Delta_F and \Delta_{F,0} produce the same conjugated mode matrices. By Lemma 7.3 and the covariance of Lemma 2.2, the equations for those matrices on Z_{F,\mathrm{ntw}} are equivalent to the ordinary projective equations on Z_F.

It remains to restore the rank factor omitted in (7.12). This changes the paired space as well as the invariants, and both changes must be made together.

The preceding argument used quantum Riemann–Roch only as a formal identity in \lambda^{-1}. We also need its consequence as an identity of germs at nonzero \lambda. To make that passage, first remove the scalar b_F by the invertible binomial change (7.11). Each coefficient of the conjugated matrices \Delta_{F,0}\ell_{k,F}\Delta_{F,0}^{-1} is then rational in \lambda. These matrices have finite lower bounds on their loop powers. The coefficients of Z_{F,\mathrm{tw}} are likewise rational in \lambda: the torus acts trivially on F, and the inverse Euler class on each fixed stable-map space is expanded only to its finite cohomological dimension. For fixed genus, class, and number of marks, the descendant indices which can occur are bounded for the same reason.

It follows from the finite-support quantization convention that each nonconstant coefficient of

is a finite sum of rational functions of \lambda. Vanishing of its formal Laurent expansion at infinity is equivalent to vanishing as a rational function, and hence as a germ. At a fixed genus and curve coefficient, (7.15) only shifts finitely many powers of \lambda, so the same argument applies in both directions. We may now restore b_F and its logarithms using (7.11). In particular, no analytic value of the quantized infinite multiplier is being taken in this passage.

8 Continuation and the ordinary constraints

The fixed-component comparison has converted the hypothesis on B into equations on one spectral branch of the ancestor theory of W. We now continue those equations to every branch. The essential finiteness comes from ancestors: their cotangent powers are bounded by the dimension of a moduli space of curves, independently of the fiber degree. We then add the branches belonging to X and use the same fixed-component comparison in reverse.

8.1 Why the equations are identities of germs

Write the connected ancestor potential at background zero as

All coefficients below are taken in the polynomial equivariant basis of Section 3. In particular we have not localized the invariants themselves in order to define them.

The modes on the B branch are infinitesimal symplectic formally at y=0 by Proposition 7.1. By Proposition 6.5, their entries are finite polynomials in \log y with coefficients holomorphic at zero. Taylor expansion is injective on this ring, so the symplectic identities hold as germs on a punctured sector. This property continues entrywise along any path on which the modes continue: it is a linear identity between their matrix entries and their adjoints. For a branch and logarithm obtained by such continuation, put

This notation means that the differential operator acts on the potential, followed by multiplication by its inverse. For the entire X cluster, the notation \mathcal E_{k,X} means the same expression with A_{k,X}=\sum_{i\in X}A_{k,i}, using a single logarithm throughout the cluster. Projective satisfaction means that every coefficient of positive insertion degree in (8.2) vanishes.

The last statement is what allows formal identities to start analytic continuation. A finite sum \sum_{j=0}^J f_j(y)(\log y)^j, with f_j holomorphic at zero, whose formal series in y and \log y vanishes is zero on a punctured sector: every Taylor coefficient of every f_j vanishes. Thus we use convergent germs of individual coefficient tests, without requiring convergence of a total potential.

8.2 Transport from one branch to the other fixed component

Fix \lambda\neq0. The branch equation is

Its critical values are 0 and (-r)^r\lambda^{r+1}/(r+1)^{r+1}. Removing these values from the y-plane gives an unramified covering of degree r+1.

8.3 The prescribed scalar normalization

The use of projective equations has absorbed the scalar ambiguity of quantization. For the ordinary Virasoro operators that ambiguity is removed by two commutators. We give the scalar calculation because the constant in L_0 is part of the assertion.