Convolution Integrals and a Mirror Theorem from Toric Fiber Geometry

Let E be a toric fibration arising from symplectic reduction of a direct sum of complex line bundles over (almost) Kähler base B. Then each torus-fixed point of the toric manifold fiber defines a section of the fibration. Let a L be convex line bundles over B, a A smooth divisors of B arising as the zero loci of generic sections of a L , and : B E α → a particular fixed-point section of E. Further assume the { } a A to be mutually disjoint. The manifold ( ) E A α  is a new manifold with tautological line bundles over new projective spaces in the geometry, where previously there was a simpler vector bundle in the given local geometry (Section 1.5). Thus, we compute genus-0 Gromov-Witten invariants of ( ) a a E A α   in terms of genus-0 Gromov-Witten invariants of B and of { } a A , the matrix used for the symplectic reduction description of the fiber of the toric fibration E B → , and the restriction maps ( ) ( ) * * * : a A a i H B H A → . The proofs utilize the fixed-point localization technique describing the geometry of ( ) E A α  and its genus-0 Gromov-Witten theory, as well as the Quantum Lefschetz theorem relating the genus-0 Gromov-Witten theory of A with that of B.


Toric Fibrations
by rigid translation in the direction of the dilaton shift. Thus  is also a Lagrangian submanifold. Henceforth we consider  as a submanifold of  . The work of Coates-Givental [1], establishes that  is a (Lagrangian) cone as a formal Lagrangian section of * T +  near q z = − ; that is, In particular, each tangent space is preserved by multiplication by z.
The shift of the formal variable ( ) t z in the z-(or ψ -) direction appears to be well-understood, so perhaps formality of the geometry (to guarantee convergence of ( ) q  ) in the z-direction need not be assumed. This existence (via convergence) of the "vertex" or the "limiting vertex" of the cone gives an intuitive way to think about the introductory material; however, the author has not studied this convergence sufficiently. In our main theorem, the domain variable ( )
We will call the resulting manifold the projective-space (surgery, gluing,  , for simplicity of notation.

Simplification: Toric Manifold
Let X be a compact symplectic toric manifold and let ( ) be the maximal unitary torus, and let Y be a T-invariant submanifold of X. Then X Y  is again a toric manifold. As explained in Section 1.5, though not in the generality needed here, the action of T on X induces an action of T on X Y  .
Thus, we may study T-equivariant genus-0 Gromov-Witten invariants of X Y  , the *  -quotient of X along (or normal to) Y directly, using fixed-point localization. All faces of the moment polytope of Y are faces of the moment polytope of X. The moment polytope of X Y  admits a canonical inclusion into the moment polytope of X, for which all faces of the moment polytope of . By a mirror theorem of Givental [6] and its extensions [7], a particular family of points on the Lagrangian cone of the genus-0 Gromov-Witten theory of a toric manifold is given by an explicit formula 4 in terms of 1

∑ ∏
This project has its roots in the following instructive example. Let E be the total space of the projective bundle described by symplectic reduction with respect to the matrix  . These latter are the weights of the T-action on the normal bundle to γ in the toric manifold.
Apply this first to the original projective bundle X E = . Then compute the weights of the T-action on the normal bundles in X Y  to the T-fixed points of the exceptional divisor. Finally, compute the normals to the codimension one faces of the momentum polyhedron of X Y  . A basis of linear relations among them is given by the rows of the matrix.
However, in fact, our main theorem arises as a generalization of this example. 4 The product formula convention is given in Section 4. Advances in Pure Mathematics Here we are using the toric mirror theorems [6] [7] [8] as a guide to the structure of genus-0 Gromov-Witten invariants more generally (following the initial proposals of A. Elezi and A. Givental). Elezi's work focused on projective bundles [13]. In [14], Givental proposed a toric bundles generalization of Elezi's approach using toric mirror integral representations [6] [7]. This is an ingredient in [8] and in the present work.

Organization of the Text
For each T-fixed section γ α ≠ of E, the strata ( ) gives rise to a T-fixed section over A in the exceptional divisor.
In the case ( ) .
From now on let the symbol γ stand for the T-fixed strata denoted γ above, or for the "substrata" Ker c L α ⋅ ⊂  of α . Let us denote the situation of a torus fixed point β connected to α by a 1-dimensional edge of the momentum polyhedron of a fiber of E, by β α . In this case : Similarly, we have the notation ( )

Geometric Preliminaries and Decomposition of Cohomology
The action of T on E decomposes ( ) ( ) Proposition. The action of T on In the following, we extend the definition of P  to the entire With this interpretation of P  , recall the isomorphism of vector spaces [15] ( ) ( ) Let us assume that This gives the inclusion. Finally, taking the quotients of ( ) Let us assume the map on the LHS is an isomorphism (an equality). This is also assumed as hypotheses for the main theorem (Section 4) and Theorem 2.
The RHS is used in the comparison of projection maps. Then,

Fixed-Point Localization
For each

The htA Function
Let 0 τ be the coordinate along  , by the dilaton shift.
Let us assume that ( ) has the property (Div + Str primary) that its , In the case that ( ) ( ) ( ) is only proper, our goal is to prove well-definedness of the least positive integer function ( ) The difficulty with this is that the Mori cone (resp Novikov ring) of A is only a subcone (resp. subring) of the Mori cone (resp. Novikov ring) of B. The natural algebraic tool for working with Langragian cones in genus-0 Gromov-Witten theory is the Birkhoff factorization technique. We will do this using the divisor equations. Thus, assume Namely, expand the RHS (right-hand side) at order D, be the unique 5 family of points of a A  whose truncation to order D′ ≤ on both of +  and −  in the Novikov's variables of the base The A series has been reindexed relative to the original A series.
This is a polynomial in powers of the nilpotent of maximal non-vanishing i P in the n B series; i.e., 3 n d = . The preceding discussion allows us to deduce the following.
Proof. Group each numerator factor with a denominator factor and expand analogously to the above. Each factor in the denominator that is not grouped with a factor in the numerator gives a power of 1 z − beyond those that come with powers of

The I-Function
Upon extension of scalars ⊂   of homology groups, the Mori cone of and for each smooth family with the property Div + Str primary, the z z → − version of the series , and the preceding series lies in the preceding cone without any truncation condition on either, while still assuming the property Div + Str primary for the smooth family Since the genus-0 generating functions of Gromov-Witten theory of E and 6 By an extension of the Novikov ring of ( ) Remark. When the fibers are copies of the point then we omit the sum over d  and we set P  to zero, since the projective fibers are also copies of the point.
Keeping these interpretations in mind, the theorem remains true when the fiber of the toric fibration is the point. The theorem reduces to the statement Remark. The natural generalization of the Main Theorem to the case of several T-fixed sections of E coincides, at the first level of analysis, with the natural generalization of the mirror theory of Section 7.
Remark. The analogue of the proof of Theorem 2 in [8] indicates the dependence of points of ( ) Conjecture. The dependence on domain variables The latter shift of the argument of A G by u is free, and then the shift of τ is  2) Let B be ( ) lies in the Lagrangian cone associated to the genus-0 Gromov-Witten theory of ( )

Graded Homogeneity
for all Novikov's variables Q  . This determines the degrees of Novikov's variables l Q as follows: , l a ρ denote the coefficent of a along the basis vector l ρ . Thus, The latter restricts correctly to the exceptional divisor and to the standard locus.
Let us now check the degree of the total series Thus, this need only be compared to the hypergeometric factors of the series. In view of the above remarks, we compare the Novikov variables degrees to the upper limit indices on the product series. The Novikov variable degrees and the product series degrees should be equal, so that they cancel out to 0. The comparison is immediately verified.

Localization of Stable Maps
The work of Graber-Pandharipande [18] justifies the fixed-point localization technique for computing integrals of T-equivariant cohomology classes over virtual fundamental cycles in the moduli spaces of stable maps to ( ) Here the T-equivariant normal "bundle" to a T-fixed stable map is actually a virtual (orbi-) virtual bundle in T-equivariant K-theory. The description of T-fixed stable maps is then analogous to the description in [8]. Namely, the connected components of the T-fixed loci in the moduli spaces of genus-0 stable maps are fiber products of moduli spaces of genus-0 stable maps into the T-fixed strata of ( ) . Let C be a leg of Σ ; i.e., an irreducible component of Σ that maps surjectively to a T-invariant edge of (a fiber of π of) ( )  , ii) C maps to a T-invariant connecting T-fixed strata ε and ε ′ . Let us assume that, in the normalization of Σ , C is a , bundle. Let us use the same notation for the bundle and for the fiber. This is along the same lines as in [8], with the only new subtlety coming from the case when

This bundle is not quite what is needed for geometrical deformation theory.
For that, we might take the bundle of sections of However, that will not suffice for reasons that follow. In any case, we need some bundle that contains ( ) to use in the role of the third non-zero term in the short exact sequences defining the gluing maps of the deformation theory.
There is the complication here that we don't want the * C f TB ′′ bundle to contribute, via the Quantum Riemann Roch theorem, to the twisted cone α  .
Thus, the present solution to the deformation problem would not be consistent with the twisted cones conditions. Let  be an index value for local cooordinates with non-trivial T-action, as in Section 6. Let X be either the fiber of the toric bundle E or of the total space of the normal line bundle to the projective space bundle are defined by subsets, as the direct sum over l  ∀ in such subsets. In the case of toric bundles, the ambient set is { } is for the gluing construction defined by short exact sequences that glue the separate deformation bundles from the fiber product of stable map moduli spaces. Namely, the direct summand L provides constant deformations (within the T-fixed stratum α ) of C f ′′ i), to coincide with the given deformation of is deduced as a result in [8]; it is not a definition. By analogy with the derivation there, in the case that ( ) : In the case that ε α ′ =  ,

A Key Ingredient of Theorem 2
Let C′ have the same meaning as in Section 5.1 case i), and reserve the notation C for case ii) except that the first marked point will also be allowed the role of nodal point of Σ attaching C to C′ . As in 5.1 the connected component of ( ) Ker c L π ⋅ may be interpreted as constraining

Recursion
Finally, apply the discussion in 2.2 and 5.1 combined with the general computational details given in [8] , Let us note the orthogonal 10 direct sum decomposition is understood by observing that In the case ( ) Aside from the many cases to consider, the proof is identical to the proof of the corresponding Theorem 2 in [8].

Recursion
To prove the equivariant version of the Main Theorem, it suffices to show that , The hypergeometric modifications -series whose coefficients have simple poles at  is induced by that on E. We would like to take a lift of the projection π , * D π   , which we may assume [8] to be preserved by the action of T on E; i.e., that * D π   is represented by such a curve. Apriori, there may be any number of toric edge component curves among the curves representing * D π   . These may intersect with a curve component of * D π   in ( ) ,   In the following recursion verification, let 1, 0 , jj jj jj For the ( )   Proof. For 1, 0 jj +   =    , the inequalities describing the support of the series are , whose solution set is "a subset of The pullbacks * P γ  vanish. In particular Proof. The support of the series .
Proof. The theory of quantum  -modules can be adapted to the case of ( ) , z τ F in such a way that the exponent is processed as is a first-order differential operator that is weighted by a net positive-integer power of z. These may be processed as a shift of the τ variable and symmetries of the cone as in [8].   ( A  respectively). We will outline a proof (as in [8]) in the last check of Section 7.3, for the mirror integrals and their asymptotics.
The Propositions will be applied in Section 7.3 where a phase function Φ will be given. A further role of stationary phase asymptotics appears in 7.2.
There is the caveat that, when , given that the asymptotics is well-defined.

The Quantum Riemann-Roch Theorem
, and e 1, , When the base is the point and 1 l = , these are the defining equations of the fiber of ( ) is the non-compact cycle ( ) The differential operators may be processed at each order of the series By applying the Lemma, we deduce the differential operator version By applying the Lemma, we deduce the differential operator version  constructions will involve an extra step, defined in terms of additional parameters , v w graded-homogeneous of degree 0, for the asymptotics. Thus, the phase function ϕ will not quite be the lift of either separate phase function.
The main role of the stationary phase asymptotics is to provide a common 13 The latter is interpreted as 1, 0 v w = = , adding the two exponential functions. From there, we will arrive at the sum of pullback series, as required. Thus, we need to reduce the exponential to the sum of exponentials, while still giving a point on the cone B α  . This suggest starting with a variety 1 vw = and deforming it to 0 vw = on the " ϕ asymptotics". Thus, we will arrive at the "direct sum of asymptotics", by working with the " ϕ asymptotics". Thus, consider the curve vw ε = . Our goal is to arrive at the "direct sum of asymptotics", by taking the limit 0 ε → in the   The numerator and denominator from the Lemma' or Lemma" in the ratio of asymptotics, each produce the same net shift in the domain variable. Thus, the domain variable stays the same in either case, as required.
In summary, the series α  F is contained in the cone B α  , and is obtained from the section of the cone B α  depending on parameters , v w , obtained from the stationary phase asymptotics from the mirror integral operator with phase function ϕ applied to ( ) , B z τ F as above.