Theorem 4.2. The F -invariance for descendent invariants of f -special type is equivalent to theF -invariance of big quantum rings.
Proof. We only need to prove “⇐”:
Consider the generating series hτk1a1,· · · , τknaniβS,d2 of f -special type with (βS, d2) 6= (0, 0). Let k = ∑iki be the total descendent degree. We will prove the theorem by induction on k.
If k = 0, we may assume that n ≥3 by adding divisors ξ or D ∈ H2(S) into the insertions. Since (ξ.`) = 0 = (D.`), this only affects the series by a nonzero constant, hence the F -invariance reduces to the case of big quantum ring.
Now let k > 0. Without loss of generality we assume that k1 ≥ 1. By induction the results holds for strictly smaller descendent degree and for any n≥1.
We first treat the case n≥3. By the topological recursion relation ψ1 = [D1|2,3]virt,
we get
hτk1a1,· · · , τknaniβS,d2
=
∑
µ
hτk1−1a1,· · · , Tµiβ0
S,d02hTµ, τk2a2, τk3a3,· · · iβ00 S,d002,
where the sum is over all splitting of curve classes such that (β0S, d20) + (β00S, d002) = (βS, d2).
Notice that on the RHS, the case(β0S, d02) = (0, 0)is excluded since ξ|a1 and it will lead to trivial invariants. The(β0S, d02)series is thenF -invariant since it has strictly smaller descendent order k1−1<k. (Recall that on the X0 side we may chooseF Tµ andF Tµ for the splitting sinceF preserves the Poincar´e pairing.)
The(β00S, d002)series is alsoF -invariant: It has strictly smaller descendent degree and it has at least 3 insertions. So even if(β00S, d200) = (0, 0)we still get theF -invariance.
The case n=1 can be reduced to the case n =2 by the divisor equation for descendant invariants. Namely let b be a divisor coming from the base S or ξ such that b.(βS+d2γ) 6=0. Then(b.β) 6=0 is independent of d and
hb, τkaiβS,d2 = (b.β)hτkaiβS,d2+ hτk−1abiβS,d2.
The case n=2 can be similarly reduced to the case n=3. If there is only one descendent insertion, sayha1, τka2iβS,d2, then
hb, a1, τka2iβS,d2 = (b.β)ha1, τka2iβS,d2+ ha1, τk−1a2biβS,d2. If there are two descendent insertions, sayhτla1, τk−la2iβS,d2, then
hb, τla1, τk−la2iβS,d2 = (b.β)hτla1, τk−la2iβS,d2
+ hτl−1a1b, τk−la2iβS,d2+ hτla1, τk−l−1a2biβS,d2. All the other series are either 3-point functions or have descendent degree drops by one. Thus by induction the proof is complete. 4.2. Divisorial reconstruction and quasi-linearity. Theorem 4.2 reduces the analytic continuation problem to the local models completely. How-ever, in the actual determination of GW invariants (as will see in later sec-tions), another natural set of initial GW invariants are those with at most one descendent insertion. This suggests another reconstruction procedure.
Definition 4.3 (Quasi-linearity). We say that the flop f is quasi-linear if for every special insertion α∈ H(X) ∪τ•H(E), ¯ti ∈H(S)and(βS, d2) 6= (0, 0), we have
Fh¯t1,· · · , ¯tn−1, αiXβ
S,d2
∼= h¯t1,· · · , ¯tn−1,F αiXβ0
S,d2.
We call invariants of the above type (with only one insertion not from the base) elementary. Quasi-linearity is the F -invariance for elementary
f -special invariants.
Notice that the similar statement for descendent invariants, even for sim-ple flops, is generally wrong if α= τka with k>0 but a 6∈H(E)(c.f. [11]).
Theorem 4.4. Suppose that f is quasi-linear. Then all descendent invariants of f -special type are F -invariant. Namely for α = (α1, . . . , αn)(n ≥ 1) with αi ∈ H(X) ∪τ•H(E)and for(βS, d2) 6= (0, 0), we have
FhαiXβ
S,d2
∼= hF αiXβ0
S,d2.
More precisely, any series of f -special type can be reconstructed, in an F -compatible manner, from the extremal functions with n≥3 points and elementary
f -special series.
We will prove the reconstruction by induction on(βS, d2) ∈W, and then on m which is the number of insertions not coming from base classes. This is based on the following observations:
(1) Under divisorial reconstruction: ψi+ψj= [Di|j]virt, and for L∈Pic(X), (4.1) e∗iL=e∗jL+ (β.L)ψj−
∑
β1+β2=β
(β1.L)[Diβ1|jβ2]virt
([13], c.f. also [11]), the degree β is either preserved or split into effective classes β= β1+β2.
(2) When summing over β ∈ (d2γ+ψ¯∗βS.Hr+Z`) ∩NE(X), the split-ting terms can usually be written as the product of two generasplit-ting series with no more marked points in a manner which will be clear in each con-text during the proof.
We also need to comment on the excluded cases(βS, d2) = (0, 0):
(3) Let αi = τkiai. If k = ∑ ki 6=0, say ξ|a1, then the extremal invariants survive only for the case β=0. Since M0,n(X, 0) ∼= M0,n×X, we have (4.2) hτk1a1,· · · , τknanin,β=0 =
Z
M0,n
ψ1k×
Z
Xa1· · ·an. It is non-trivial only if k=dim M0,n=n−3, and then
Z
Xa1· · ·an =
Z
X0F a1· · ·F an since the flop f restricts to an isomorphism on E.
(4) For extremal invariants with k = 0, since ξ|Z = 0 and the extremal curves will always stay in Z, we get trivial invariant if one of the insertions involves ξ. Hence by Theorem 2.9 the statement in the theorem still holds in this initial case except for the 2-point invariantsh¯t1hr, ¯t2hri. By the divisor axiom
δhh¯t1hr, ¯t2hri = hh, ¯t1hr, ¯t2hri+,
the 2-point invariants will satisfy theF -invariance functional equation up to analytic continuation only after incorporated with classical defect. Thus we may base our induction on(βS, d2) = (0, 0)with special care taken to handle this case.
Proof. Let(βS, d2) 6= (0, 0). If m=1 then we are done, so let m≥2 .
Step 1. First we handle the type I case, i.e. with the appearance of ξ in some αi.
By reordering we may assume that αn =τsξ a, s≥0. Write α1 = ¯t1τkhlξj.
We will reduce m by moving divisors in α1into αnin the order of ψ, h and ξ. This process is compatible withF since F a.F ξ=F(a.ξ).
For ψ, we use the equation
ψ1 = −ψn+ [D1|n]virt. If k≥1 then j6=0 and we get
h¯t1τkhlξj,· · · , τsξ aiβS,d2 = −h¯t1τk−1hlξj,· · · , τs+1ξ aiβS,d2 +
∑
µ
h¯t1τk−1hlξj,· · · , Tµiβ0
S,d02hTµ,· · · , τsξ aiβ00
S,d002. For each i, if one of (β0S, d02) and (β00S, d002) is (0, 0) then since both terms contain ξ the splitting term must vanish. So we may assume that
(β0S, d02) < (βS, d2) and (β00S, d002) < (βS, d2)
and these terms are done by the induction hypothesis. (By performing this procedure to α1, . . . , αn−1 we may assume that the only descendent inser-tion is αn.)
For h, if l ≥1 we use the divisor relation (4.1) for L=h to get h¯t1hlξj,· · · , τsξ aiβS,d2
= h¯t1hl−1ξj,· · · , τsξ ahiβS,d2+δhh¯t1hl−1ξj,· · · , τs+1ξ aiβS,d2
−
∑
µ
δhh¯t1hl−1ξj,· · · , Tµiβ0
S,d02hTµ,· · · , τsξ aiβ00 S,d002.
The only cases for the splitting term to have one factor with the same (βS, d2) and m are of the form (denote by ¯t∗ some set of insertions αj ∈ H(S))
δhh¯t1hl−1ξj, ¯t∗, Tµi0,0hTµ,· · · , τsξ aiβS,d2, where the LHS has n0points, or
δhh¯t1hl−1ξj,· · · , TµiβS,d2hTµ, ¯t∗, τsξ ai0,0.
But l−1 < r forces the former LHS invariants to vanish: For j 6=0 this is trivial. For j=0, the codimension (c.f. §2)
(4.3) µ= |h| − (2r+1+n0−3) <2r−2r=0.
The latter RHS invariants also vanish since they contain ξ.
If j = 0, the case (β0S, d02) = (0, 0) may still support nontrivial invari-ants with 3 or more points. In that case m decreases in the RHS. For the
other terms, the only possible appearance of type II invariants (i.e. without ξ insertion) is
(4.4) δhh¯t1hl−1,· · · , Tµiβ0
S,d02 = hh, ¯t1hl−1,· · · , Tµiβ0 S,d02,
where j=0, which has at least 3 points and(0, 0) < (β0S, d02) < (βS, d2). For ξ, the argument is entirely similar. For j≥1, the divisor relation says that
h¯t1ξj,· · · , τsξ aiβS,d2
= h¯t1ξj−1,· · · , τsξ2aiβS,d2+δξh¯t1ξj−1,· · · , τs+1ξ aiβS,d2
−
∑
µ
δξh¯t1ξj−1,· · · , Tµiβ0
S,d20hTµ,· · · , τsξ aiβ00 S,d002.
We then have(β0S, d02) < (βS, d2)and(β00S, d002) < (βS, d2)as before. Notice that only type I invariants appear in the reduction.
Step 2. Next we deal with the type II case: αi = ¯tihli, 1 ≤ i ≤ n. In case βS = 0, we can add one ξ into the insertions and then go back to Step 1. From (4.4),(βS, d2)will be getting smaller when the possible type II invariants appear again, so it is done by induction. Thus we can allow βS 6= 0 here. By adding base divisors into the insertions we may always assume that n≥3.
We can not apply (4.1) to move divisors since it will produce non f -special invariants. Instead, since n≥3 we may apply (2.1), the descendent-free form of the divisor relation, as we have used in the proof of Theorem 2.9.
Suppose that l1 > 0 and l2 > 0 and we move h from α1 to α2. We run induction on l1. Namely we assume the F -invariant reduction holds for α1 = ¯t1hj with j ≤ l1−1. The initial case j = 0 holds since m drops by 1.
Then
h¯t1hl1, ¯t2hl2, α3,· · · iβS,d2
= h¯t1hl1−1, ¯t2hl2+1, α3,· · · iβS,d2 +
∑
µ
h¯t1hl1−1, α3,· · · , Tµiβ0
S,d02δhh¯t2hl2,· · · , Tµiβ00
S,d002
−δhh¯t1hl1−1,· · · , Tµiβ0
S,d02h¯t2hl2, α3,· · · , Tµiβ00
S,d002.
If l2≤r−1, the processes on X and X0are clearlyF -compatible and the splitting terms are all handled by induction. Indeed, if(β0S, d02) = (βS, d2) and m0 = m then(β00S, d002) = (0, 0)which gives an extremal function with m00 ≤2. The analogous codimension condition as in (4.3) forces the term to vanish. Similar consideration applies to the case(β00S, d002) = (βS, d2)as well.
If l2 = r, the first term is no longerF -compatible. The topological de-fect of the second insertion is given by Lemma 2.8: F(hr+1) − (F h)r+1 = (−1)r+1F Θr+1, where Θr+1 is the dual class of pt.hrξ0. Meanwhile, the splitting terms also contain one term not of lower order in(βS, d2)and m.
By the codimension consideration as in (4.3), we have Tµ = ˇ¯t2hr and the term is given by
h¯t1hl1−1, α3,· · · , αn, ¯t2Θr+1iβS,d2δhh¯t2hr, ˇ¯t2hri0,0. Comparing with its corresponding term on X0
h¯t1F hl1−1,F α3,· · · ,F αn, ¯t2F Θr+1iβS,d2δF hh¯t2F hr, ˇ¯t2F hri0,0 and using the induction, we get the difference to be
− h¯t1F hl1−1,F α3,· · · ,F αn, ¯t2F Θr+1iβS,d2× (−1)r+1
= −h¯t1F hl1−1, ¯t2F(hr+1),· · · iβS,d2 + h¯t1F hl1−1, ¯t2(F h)r+1,· · · iβS,d2. This cancels the defect of the nonF -compatible terms.
Thus the whole reduction isF -invariant and the proof is complete. 4.3. WDVV equations. We may strengthen Theorem 4.4 to
Theorem 4.5. If the quasi-linearity holds for elementary type I series h¯t1,· · · , ¯tn−1, τkaξi,
then theF -invariance holds for all series of f -special type.
The significance of this reduction will become clear after we introduce the practical method to calculate GW invariants. The proof is based on Proposition 4.6. Any type II series over (βS, d2) can be transformed into sum of products of (1) type I series over (β0S, d02) ≤ (βS, d2), (2) type II series over β0S < βS, and (3) extremal functions. Also, the processes can be done in a F -compatible manner.
Indeed, with Proposition 4.6, Theorem 4.5 then follows from the proof of Theorem 4.4: Simply replace Step 2 by the proposition and run the in-duction. All type II special series eventually disappear. (Degenerate type II series with(βS, d2) = (0, 0)are simply extremal functions.)
The remaining of this subsection is devoted to the proof of Proposition 4.6. Notice that if d26=0 then this is trivial: By the divisor axiom,
ha1,· · · , aniβS,d2 = ha1,· · · , an, ξiβS,d2/d2.
Thus we considerha1,· · · , an−1, ¯tihjiβS,0with a1, . . . , an−1∈ H(Z). Let{T¯i}be a basis for H(S)and{ˇ¯Ti}be its dual basis. We start with the case of three-point functions ha, b, ¯TihjiβS,0 for any a, b ∈ H(Z). This cer-tainly includes also the one-point and two-point cases by picking suitable a, b∈ H2(S).
For any c, d∈ H(X), the WDVV equations
m,n
∑
∂ijmF0gmn∂nklF0=
∑
m,n
∂ikmF0gmn∂njlF0
lead to the diagram
[a∨b7→ ξc∨ξd] = [a∨ξc7→b∨ξd].
We apply it to split the curve classes over (βS, d2 = 1) and get a linear equation
(4.5)
∑
i,j
ha, b, ¯TihjiβS,0hˇ¯TiHr−jΘr+1, ξc, ξdi0,d2 = Ic,d,
where all terms in the LHS of WDVV with either (1) β0S < βS, (2) d20 6= 0, or (3) with basis class insertion Tµ = T¯ihjξk (k > 0) from the diagonal splitting, have been moved into the RHS. Since the original RHS of WDVV are all type I series, any series in Ic,d over(β0S, d02)must satisfy β0S < βS or (β0S, d02) = (βS, 0).
Let m = ∑ihi(S). We intend to form an N×N invertible system with N=m(r+1). The virtual dimension of the second series is
d2(r+2) +2r+1+s.
Thus for d2 = 1, we should require |c| + |d| = r+ |T¯i| +j to match the dimension.
Natural choices of{(c, d)}are
(4.6) c=ck,l :=T¯kξl, d= hr. The set{ck,l}is partially ordered by|T¯k|and then by l.
We claim that the resulting system is upper triangular with non-zero di-agonal. Indeed,
hˇ¯TiHr−jΘr+1, ¯Tkξl+1, ξhri0,1 6=0 only if|T¯k| +l= |T¯i| +j.
The key point is to use the fiber bundle structure M0,n(X, β) → S for β= d` +d2γas in the extremal case (where d2 = 0). The fiber is given by M0,nof the toric local model for the simple flop case.
Thus if |T¯k| > |T¯i| then|ˇ¯Ti| + |T¯k| > s and the invariant is zero. Even in the case|T¯k| = |T¯i|, and so l = j, we must have ¯Tk = T¯i to avoid trivial invariants. The other cases|T¯k| < |T¯i|belong to the strict upper triangular region which do not affect our concern.
It remains to calculate the diagonal fiber series (sum in d≥0)
∑
ihˇ¯TiHr−jΘr+1, ¯Tiξj+1, ξhri0,1= hhr−j(ξ−h)r+1, ξj+1, ξhrisimpled
2=1 . We had done a similar calculation before for the extremal case in [11], Proposition 3.8. In the current case we have
Lemma 4.7. For simple flops, the fiber series in d with d2=1 are given by hhr−j(ξ−h)r+1, ξj+1, ξhrid2=1 =
((−1)jq`qγ, 0≤j≤r−1;
(1− (−1)r+1q`)qγ, j=r.
Proof. By applying the divisor relation to move one ξ class with respect to (i, j, k) = (2, 1, 3), we get (notice that ξ(ξ−h)r+1 =0)
hhr−j(ξ−h)r+1, ξj+1, ξhrid2=1
=
∑
µ
hξj, ξhr, Tµi0δξhTµ, hr−j(ξ−h)r+1i1−δξhξj, Tµi1hTµ, hr−j(ξ−h)r+1, ξhri0
= hhr−j(ξ−h)r+1, ξj+1hri1.
By another divisor relation (4.1), we can keep track on the 2-point invari-ants as follows:
hhr−j(ξ−h)r+1, ξj+1hri1
= hψhr−j(ξ−h)r+1, ξjhri1−
∑
µ
δξhξjhr, Tµi1hTµ, hr−j(ξ−h)r+1i0
= hψhr−j(ξ−h)r+1, ξjhri1= · · ·
= hψj+1hr−j(ξ−h)r+1, hri1.
Here we use the fact that there is no extremal invariants with any insertion involving ξ (notice that(ξ−h)r+1 =ξ(· · · )since hr+1 =0).
Next we move the divisor class h in hrto the left one by one:
hψj+1hr−j(ξ−h)r+1, hri1
= hψj+1hr−j+1(ξ−h)r+1, hr−1i1+δhhψj+2hr−j(ξ−h)r+1, hr−1i1
−
∑
µ
δhhhr−1, Tµi0hTµ, ψj+1hr−j(ξ−h)r+1i1
= hψj+1(h+dψ)hr−j(ξ−h)r+1, hr−1i1= · · ·
= hψj+1(h+dψ)r−1hr−j(ξ−h)r+1, hi1.
Note thathhr−1, Tµi0=0 since the power of h is less than r.
Finally, the divisor axiom helps us to obtain the result:
hψj+1(h+dψ)r−1hr−j(ξ−h)r+1, hi1
=dhψj+1(h+dψ)r−1hr−j(ξ−h)r+1i1+ hhψj(h+dψ)r−1hr−j(ξ−h)r+1i1
= hψj(h+dψ)rhr−j(ξ−h)r+1i1,
which is the constant term in the z expansion in D
∑
k≥0
ψk
zkzj(h+dz)rhr−j(ξ−h)r+1E
1
=zj+2e1∗ 1 z(z−ψ)e
∗
1(h+dz)rhr−j(ξ−h)r+1.
According to the same discussion of quasi-linearity in [11], if d2−d< 0 then Pβvanishes after multiplication by ξ. Here hr−j(ξ−h)r+1does contain at least one ξ. Hence we only need to consider d2 ≥ d. Now d2 = 1, thus d=0 or 1.
If d =0, then hrhr−j(ξ−h)r+1 is nontrivial only if j =r and in this case we get hr(ξ−h)r+1 =hrξr+1=pt. It is clear that the constant term of z in
zr+2Jβ.pt= zr+2 1
(ξ−h+z)r+1(ξ+z).pt is equal to 1.
If d=1, then Jβ =1/(h+z)r+1(ξ+z). Thus zj+2(h+z)rhr−j(ξ−h)r+1
(h+z)r+1(ξ+z)
= z
j+2
z2
hr−j(ξ−h)r+1 (1+h/z)(1+ξ/z)
=zjhr−j(ξ−h)r+11− h z +h
2
z2 − · · · (−1)jh
j
zj + · · ·1− ξ
z + · · ·. Since ξ(ξ−h)r+1=0, the constant term is given by
(−1)jhr(ξ−h)r+1= (−1)jhrξr+1 = (−1)j.
The proof is complete.
Now we consider n-point functions with n ≥3. The WDVV equation is for triple derivatives of the g = 0 potential function. Let t ∈ H>2(X)be a general insertion without the fundamental class and divisors. Then we have
(4.7)
∑
i,j
ha, b, ¯TihjiβS,0(t)hˇ¯TiHr−jΘr+1, ¯Tkξl+1, ξhri0,1(t) = Ik,l(t)
where any series in Ic,d over (β0S, d02) must satisfy β0S < βS or (β0S, d02) = (βS, 0).
By dimension counting, one more marked point increases one virtual dimension while t has Chow degree more than one, so we find that
hˇ¯TiHr−jΘr+1, ¯Tkξl+1, ξhri0,1(t) = hˇ¯TiHr−jΘr+1, ¯Tkξl+1, ξhri0,1
is in fact independent of t when|T¯i| +j= |T¯k| +l. The linear system (4.7) is thusF -compatible by the quantum invariance of simple flop case [11].
In any case, if|T¯k| > |T¯i|then the invariants are still zero. In particular the N×N system is still upper triangular. Moreover the diagonal entries are still given by the original 3 point (finite) series. Thus the series
ha, b, ¯TihjiβS,0(t) are solvable in terms of the expected terms.
5. BIRKHOFF FACTORIZATION
In this section, a general framework for calculating the J function for a split toric bundle is discussed. It relies on a given (partial) section I of the Lagrangian cone generated by J. The process to go from I to J is introduced in a constructive manner, and Theorem 0.4 will be proved (=Proposition 5.6 + Theorem 5.10).
5.1. Lagrangian cone and the J function. We start with Givental’s sym-plectic space reformulation of Gromov–Witten theory arising from Witten’s basic dilaton, string, and topological recursion relation in two-dimensional grav-ity [25]. The main references for this section are [6, 2], with supplements and clarification from [15, 10]. In the following, the underlying ground ring is the Novikov ring
R=C[\NE(X)].
All the complicated issues on completion are deferred to [15].
Let H := H(X), H := H[z, z−1]], H+ := H[z]and H− := z−1H[[z−1]]. Let 1 ∈ H be the identity. One can identify H as T∗H+ and this gives a canonical symplectic structure and a vector bundle structure onH.
Let
be a general point, where{Tµ}form a basis of H. In the Gromov–Witten context, the natural coordinates onH+are t(z) =q(z) +1z (dilaton shift), with t(ψ) = ∑µ,ktµkTµψk serving as the general descendent insertion. Let F0(t)be the generating function of genus zero descendent Gromov–Witten invariants on X. Since F0 is a function on H+, the one form dF0 gives a section of π :H → H+.
Givental’s Lagrangian coneLis defined as the graph of dF0, which is con-sidered as a section of π. By construction it is a Lagrangian subspace. The existence ofC∗ action onLis due to the dilaton equation∑ qµk∂/∂qµkF0 = follows from the divisor equation for descendent invariants. Furthermore, the string equation for J says that we can take out the fundamental class 1 from the variable τ to get an overall factor eτ0/zin front of (5.1).
The J function can be considered as a map from H to zH−. Let Lf =TfL be the tangent space ofLat f∈ L. Let τ ∈H be embedded intoH+via
H∼= −1z+H⊂ H+.
Denote by Lτ = L(τ,dF0(τ)). Here we list the basic structural results from [6]:
(i) zL⊂ L and so L/zL∼= H+/zH+∼= H has rank N :=dim H.
(ii) L∩ L =zL, considered as subspaces insideH.
(iii) The subspace L of H is the tangent space at every f ∈ zL ⊂ L. Moreover, Tf = L implies that f∈ zL. zL is considered as the ruling of the cone.
(iv) The intersection ofLand the affine space−1z+zH−is parameter-ized by its image−1z+H∼= H3τvia the projection by π.
−zJ(τ,−z−1) = −1z+τ+O(1/z) is the function of τ whose graph is the intersection.
(v) The set of all directional derivatives z∂µJ = Tµ+O(1/z)spans an N dimensional subspace of L, namely L∩zH−, such that its projection to L/zL is an isomorphism.
Note that we have used the convention of the J function which differs from that of some more recent papers [6, 2] by a factor z.
Lemma 5.1. z∇J = (z∂µJν)forms a matrix whose column vectors z∂µJ(τ) gen-erates the tangent space Lτ of the Lagrangian cone Las an R{z}-module. Here a =∑ qβaβ(z) ∈R{z}if aβ(z) ∈C[z].
Proof. Apply (v) to L/zL and multiply zk to get zkL/zk+1L. We see that the germ ofLis determined by an N-dimensional submani-fold. In this sense, zJ generatesL. Indeed, all discussions are applicable to the Gromov–Witten context only as formal germs around the neighborhood of q= −1z.
5.2. Generalized mirror transform for toric bundles. Let ¯p : X → S be a smooth fiber bundle such that H(X)is generated by H(S)and fiber divisors Di’s as an algebra, such that there is no linear relation among Di’s and H2(S). An example of X is a toric bundle over S. Assume that H(X)is a free module over H(S)with finite generators{De:=∏iDeii}e∈Λ.
Let ¯t := ∑s¯tsT¯s be a general cohomology class in H(S), which is iden-tified with ¯p∗H(S). Similarly denote D = ∑ tiDi the general fiber divisor.
Elements in H(X)can be written as linear combinations of{T(s,e) =T¯sDe}. Denote the ¯Ts directional derivative on H(S)by ∂T¯s ≡ ∂¯ts, and denote the multiple derivative
∂(s,e) :=∂¯ts
∏
i
∂etii.
Note, however, most of the time z will appear with derivative. For the notational convenience, denote the index(s, e)by e. We then denote (5.2) ∂ze≡∂z(s,e) :=z∂¯ts
∏
i
z∂etii =z|e|+1∂(s,e).
As usual, the Tedirectional derivative on H(X)is denoted by ∂e = ∂Te. This is a special choice of basis Tµ(and ∂µ) of H(X), which is denoted by
Te ≡T(s,e) ≡T¯sDe; e∈ Λ+.
The two operators ∂zeand z∂eare by definition very different, nevertheless they are closely related in the study of quantum cohomology as we will see below.
Assuming that ¯p : X → S is a toric bundle of the split type, i.e. toric quotient of a split vector bundle over S. Let JS(¯t, z−1)be the J function on S. The hypergeometric modification of JSby the ¯p-fibration takes the form (5.3) IX(¯t, D, z, z−1):=
∑
β∈NE(X)
qβeDz+(D.β)IβX/S(z, z−1)JβSS(¯t, z−1) with the relative factor IβX/S, whose explicit form for X = E˜ → S will be given in Section 6.2.
The major difficulty which makes IXbeing deviated from JX lies in the fact that in general positive z powers may occur in IX. Nevertheless for each β ∈ NE(X), the power of z in IβX/S(z, z−1) is bounded above by a constant depending only on β. Thus we may study IX in the space H := H[z, z−1]]over R.
Notice that the I function is defined only in the subspace (5.4) ˆt := ¯t+D∈ H(S) ⊕M
i
CDi ⊂ H(X). J. Brown recently established the following result:
Theorem 5.2([1] Theorem 1). (−z)IX(ˆt,−z)lies in the Lagrangian coneLof X.
Definition 5.3 (GMT). For each ˆt, zI(ˆt)lies in Lτ ofL. The correspondence ˆt7→τ(ˆt) ∈ H(X) ⊗R
is called the generalized mirror transformation (c.f. [2, 6]).
Remark 5.4. In general τ(ˆt)may be outside the submodule of the Novikov ring R generated by H(S) ⊕LiCDi. This is in contrast to the (classical) mir-ror transformation where τ is a transformation within(H0(X) ⊕H2(X))R (small parameter space).
To make use of Theorem 5.2, we start by outlining the idea behind the following discussions. By the properties ofL, Theorem 5.2 implies that I can be obtained from J by applying certain differential operator in z∂e’s
to it, with coefficients being series in z. However, what we need is the reverse direction, namely to obtain J from I, which amounts to removing the positive z powers from I. Note that, the I function has variables only in the subspace H(S) ⊕LiCDi. Thus a priori the reverse direction does not seem to be possible.
The key idea below is to replace derivatives in the missing directions by higher order differentiations in the fiber divisor variables ti’s, a pro-cess similar to transforming a first order ODE system to higher order scaler equation. This is possible since H(X) is generated by Di’s as an algebra over H(S).
Lemma 5.5. z∂1J = J and z∂1I = I.
Proof. The first one is the string equation. For the second one, by definition I =∑βqβeD/z+(D.β)IβX/SJβS
S(¯t), where IβX/Sdepends only on z. The differen-tiation with respect to t0(dual coordinate of 1) only applies to JβS
S(¯t). Hence the string equation on JβSS(¯t)concludes the proof. Proposition 5.6. (1) The GMT: τ =τ(ˆt)satisfies τ(ˆt, q=0) =ˆt.
(2) Under the basis{Te}e∈Λ+, there exists an invertible N×N matrix-valued formal series B(τ, z), which is free from cohomology classes, such that
(5.5)
∂zeI(ˆt, z, z−1)= z∇J(τ, z−1)B(τ, z), where(∂zeI)is the N×N matrix with ∂zeI as column vectors.
Proof. By Theorem 5.2, zI ∈ L, hence z∂I∈ TL =L. Then z(z∂)I ∈ zL⊂ L and so z∂(z∂)I lies again in L. Inductively, ∂zeI lies in L. The factoriza-tion(∂zeI) = (z∇J)B(z)then follows from Lemma 5.1. Also Lemma 5.5 says that the I (resp. J) function appears as the first column vector of(∂zeI) (resp.(z∇J)). By the R{z}module structure it is clear that B does not in-volve any cohomology classes.
By the definitions of J, I and ∂ze(c.f. (5.1), (5.3), (5.2)), it is clear that (5.6) ∂zeeˆt/z=Teeˆt/z, z∂eet/z =Teet/z
(t∈ H(X)). Hence modulo Novikov variables ∂zeI(ˆt) ≡Teeˆt/zand z∂eJ(τ) ≡ Teeτ/z
To prove (1), modulo all qβ’s we have eˆt/z≡
∑
e∈Λ+
Be,1(z)Teeτ(ˆt)/z. Thus
e(ˆt−τ(ˆt))/z≡
∑
e
Be,1(z)Te, which forces that τ(ˆt) ≡ ˆt (and Be,1(z) ≡δTe,1).
To prove (2), notice that by (1) and (5.6), B(τ, z) ≡ IN×N when modulo Novikov variables, so in particular B is invertible. Notice that in getting
(5.5) we do not need to worry about the sign on “−z” since it appears in
both I and J.
Definition 5.7 (BF). The left-hand side of (5.5) involves z and z−1, while the right-hand side is the product of a function of z and a function of z−1. Such a matrix factorization process is termed the Birkhoff factorization.
Besides its existence and uniqueness, for actual computations it will be important to know how to calculate τ(ˆt)directly or inductively.
Proposition 5.8. There are scalar-valued formal series Ce(ˆt, z)such that (5.7) J(τ, z−1) =
∑
e∈Λ+
Ce(ˆt, z)∂zeI(ˆt, z, z−1), where Ce ≡δTe,1modulo Novikov variables.
In particular τ(ˆt) =ˆt+ · · · is determined by the 1/z coefficients of the RHS.
Proof. Proposition 5.6 implies that
z∇J = (∂zeI)B−1.
Take the first column vector in the LHS, which is z∇1J = J by Lemma 5.5, one gets expression (5.7) by defining Ce to be the corresponding e-th entry of the first column vector of B−1. Modulo qβ’s, B−1 ≡ IN×N, hence
Ce ≡δTe,1.
Definition 5.9. A differential operator P is of degreeΛ+if P=∑e∈Λ+Ce∂ze for some Ce. Namely, its components are multi-derivatives indexed byΛ+. Theorem 5.10 (BF/GMT). There is a unique, recursively determined, scalar-valued degreeΛ+differential operator
P(z) =1+
∑
β∈NE(X)\{0}
qβPβ(ti, ¯ts, z; z∂ti, z∂¯ts),
with each Pβ being polynomial in z, such that P(z)I(ˆt, z, z−1) =1+O(1/z). Moreover,
J(τ(ˆt), z−1) =P(z)I(ˆt, z, z−1),
with τ(ˆt)being determined by the 1/z coefficient of the right-hand side.
Proof. The operator P(z)is constructed by induction on β ∈ NE(X). We set Pβ = 1 for β = 0. Suppose that Pβ0 has been constructed for all β0 < βin NE(X). We set P<β(z) =∑β0<βqβ0Pβ0. Let
(5.8) A1 =zk1qβ
∑
e∈Λ+
fe(ti, ¯ts)Te
be the top z-power term in P<β(z)I. If k1 < 0 then we are done. Other-wise we will remove it by introducing “certain Pβ”. Consider the “naive quantization”
(5.9) Aˆ1:=zk1qβ
∑
e∈Λ+
fe(ti, ¯ts)∂ze.
In the expression
(P<β(z) −Aˆ1)I =P<β(z)I−Aˆ1I, the target term A1is removed since
(P<β(z) −Aˆ1)I =P<β(z)I−Aˆ1I, the target term A1is removed since