The D-module structure on S 1 -equivariant Floer cohomology 165

In this section we explain why theS¹-equivariant Floer cohomologyHF_S1(LP^w) should carry a naturalD-module structure. Recall that Ω is the K¨ahler form onLP^w induced by the K¨ahler structure onP^w, and that we consider the covering spacep:LP]^w!LP^w. We have [Ω]=ev^?₁P. The formp^?Ω is not equivariantly closed, so it does not define anS¹ -equivariant cohomology class onLP]^w, butp^?Ω+zµis equivariantly closed—this follows from the fact thatm is a moment map. Let ℘ be the class ofp^?Ω+zµin H_S²1(LP]^w).

Consider the mapP:HF_S1(LP^w)!HF_S1(LP^w) given by multiplication by℘, and the mapQ:HF_S1(LP^w)!HF_S1(LP^w) given by pull-back by the deck transformationQ⁻¹. Since

(Q⁻¹)^?℘=℘−z

we have [P,Q]=zQ. In other words, if we defineDto be the Heisenberg algebra D=C[z][[Q]][Q⁻¹]hPi such that [P,Q] =zQ,

thenHF_S1(LP^w) should carry the structure of a D-module whereQ acts by pull-back byQ⁻¹andPacts by multiplication by℘.

In the non-equivariant limit (z!0) this structure degenerates to aC((Q))[P]-module structure onHF(LP^w), wherePacts via (23). Thus we can recover the part of the small orbifold quantum cohomology algebra generated byP—which, as we will see below, is the whole thing—from theD-module structure onHF_S1(LP^w). It is clear that HF(LP^w) should be generated as aC((Q))[P]-module by{Q^f1f}, so we expectHF_S1(LP^w) to be finitely generated as aD-module. Our analysis below will show that HF_S1(LP^w) is of rank 1, generated by 10Q⁰. This generator is Givental’s “fundamental Floer cycle”—it represents the semi-infinite cycle in LP]^w swept out by upward gradient flow from the component of the critical set at level 0. The projection toLP^wof the fundamental Floer cycle consists of all loops which bound holomorphic discs with a possibly-stacky point at the origin.

The link between Floer cohomology and Gromov–Witten theory appears here as a conjectural D-module isomorphism between HF_S1(LP^w) and the D-module generated by the smallJ-function. We have seen howDacts onHF_S1(LP^w). Another realization ofDis by differential operators

P:f7−!z∂f

∂t and Q:f7−!Qe^tf

acting on the space of analytic functionsf:C!H_orb (P^w; Λ[Q⁻¹])⊗C((z⁻¹)). The small J-function is such a function (see§2.3.1) and so it generates aD-module; relations in this

D-module are differential equations satisfied byJP^w(t) (see equations (14), (19) and the discussions thereafter). We will make use of this conjectural D-module isomorphism in the next section, where we write down a concrete model forHF_S1(LP^w) as aD-module and then identify the fundamental Floer cycle in this model with the small J-function JP^w(t). This will give a conjectural formula for the smallJ-function.

3.7. Computing the D-module structure

As we lack a concrete model forLP]^w, we consider instead the space ofpolynomial loops Lpoly={(f⁰, ..., fⁿ) :fⁱ∈C[t, t⁻¹] and not all the fⁱ are zero}/C^×,

whereα∈C^× acts on a vector-valued Laurent polynomial as (f⁰, ..., fⁿ)7−!(α^−w0f⁰, ..., α^−wnfⁿ).

The spaceLpolyis quite different fromLP]^w—it is, for example, certainly not a covering space(¹¹) of LP^w. But Lpoly is in some ways a good analog for LP]^w. We will see below that there is an S¹-action on Lpoly such that the S¹-fixed subset is a covering space of the inertia stack IP^w with deck transformation groupZ. So for computations involving quantities which localize to the S¹-fixed set—such as S¹-equivariant semi-infinite cohomology—L_polyis a good substitute for LP]^w. Working by analogy with the discussion in the previous section, we now construct an action ofDon the “S¹-equivariant semi-infinite cohomology” ofL_poly. This will be our concrete model forHF_S1(LP^w).

The space Lpoly is an infinite-dimensional weighted projective space. It carries an S¹-action coming from loop rotation, which is Hamiltonian with respect to the Fubini–

Study form Ω⁰∈Ω²(L_poly). The moment map for this action is µ⁰:

X

k∈Z

a⁰_kt^k, ...,X

k∈Z

aⁿ_kt^k

7−!− Pn

l=0

P

k∈Zk|a^l_k|² Pn

l=0

P

k∈Zw_l|a^l_k|². A polynomial loop

[(f⁰(t), ..., fⁿ(t))]∈Lpoly

is fixed by loop rotation if and only if

(f⁰(λt), ..., fⁿ(λt)) = (α(λ)^−w0f⁰(t), ..., α(λ)^−wnfⁿ(t))

(¹¹) The “obvious map”L_poly!LP^w, given by restricting a polynomial map f(t) to the circle {t∈C:|t|=1}and filling in where necessary using continuity, is not even continuous.

for allλ∈S¹ and some possibly multi-valued functionα(λ). We needα(λ)=λ^−k/wi for some integerk, so components of theS¹-fixed set are indexed by

Fe={k/wi:k∈Zand 06i6n}.

Forr∈Fe, the correspondingS¹-fixed component

Fixr={[(b0t^w0r, ..., bnt^wnr)]∈Lpoly:bi= 0 unlesswir∈Z}

is a copy of the componentP(V^hri) of the inertia stack, and the value ofµ⁰ on this fixed component is−r. The normal bundle to Fixr is

n

M

i=0

M

j∈Z j6=wir

O(wiP+(j−wir)z),

where O(aP+bz) denotes the bundle O(a) on Fixr=P(V^hri) which has weight b with respect to loop rotation.

Let℘⁰ be the class of Ω⁰+zµ⁰ in H_S²1(Lpoly), so that H_S1(Lpoly) =C[z, ℘⁰],

and introduce an action ofZonLpoly by “deck transformations”:

Q^m:

X

k∈Z

a⁰_kt^k, ...,X

k∈Z

aⁿ_kt^k

7−! X

k∈Z

a⁰_kt^k−mw0, ...,X

k∈Z

aⁿ_kt^k−mwn

, m∈Z.

The deck transformationQ^m changes the value of µ⁰ by m, and sends Fixr to Fix_r−m. We let Q act on H_S1(Lpoly) by pull-back by Q⁻¹, and P act on H_S1(Lpoly) by cup product with℘⁰. As

(Q⁻¹)^?℘⁰=℘⁰−z, so that [P,Q]=zQ, this gives an action ofDonH_S1(Lpoly).

We now consider the “S¹-equivariant semi-infinite cohomology” of Lpoly. We will work formally, representing semi-infinite cohomology classes by infinite products in

H_S1(L_poly).

These products, interpreted na¨ıvely, definitely diverge, but one can make rigorous sense of them by considering them as the limits of finite products and at the same time considering L_polyas the limit of spaces of Laurent polynomials of bounded degree. This is explained in [19] and [29]. Recall that the fundamental Floer cycle inLP]^wconsists (roughly speaking)

of loops which bound holomorphic discs. The analog of the fundamental Floer cycle in Lpoly is the cycle of Laurent polynomials which are regular att=∞. We represent this by the infinite product

∆ =

n

Y

i=0

Y

k>0

(wi℘⁰+kz).

To interpret this, observe that the Fourier coefficientaⁱ_k of the loop

X

k∈Z

a⁰_kt^k, ...,X

k∈Z

aⁿ_kt^k

∈L_poly

gives a section of the bundleO(wi) over

Lpoly∼=P(..., wn, w0, w1, ..., wn, w0, w1, ..., wn, w0, ...),

which has weight k with respect to loop rotation. Our candidate for the Floer fun-damental cycle is cut out by the vanishing of the aⁱ_k, k>0, and so ∆ is a candidate for the S¹-equivariant Thom class of its normal bundle—that is, for its S¹-equivariant Poincar´e-dual. We have

n

Y

i=0 w_i−1

Y

j=0

(wiP−jz)∆ =Q∆. (24)

This is an equation in theS¹-equivariant semi-infinite cohomology ofLpoly, regarded as a D-module via the actions ofPandQdefined above. As aD-module, theS¹-equivariant semi-infinite cohomology ofLpolyis generated by ∆.

We cannot directly identify ∆ with the smallJ-function, as theD-module generated by ∆ involves shift operators

P:g(℘⁰)7−!℘⁰g(℘⁰) and Q:g(℘⁰)7−!g(℘⁰−z),

whereas that generated by the smallJ-function involves differential operators P:f(t)7−!z∂f

∂t and Q:f(t)7−!Qe^tf(t).

We move between the two via a sort of Fourier transform. We expect, by analogy with the Atiyah–Bott localization theorem [6], that there should be a localization map Loc from localizedS¹-equivariant semi-infinite cohomology of Lpoly to the cohomology H_S

1(L^S_poly¹ )⊗C(z) of theS¹-fixed set. We consider

Loc(e^℘0t/z∆) (25)

as this should satisfy PLoc(e^℘0t/z∆) =z∂

∂tLoc(e^℘0t/z∆) = Loc(e^℘0t/z℘⁰∆) = Loc(e^℘0t/zP∆),

QLoc(e^℘0t/z∆) =Qe^tLoc(e^℘0t/z∆) =e^tLoc((Q⁻¹)^?(e^℘0t/z∆)) = Loc(e^℘0t/zQ∆).

The class ℘⁰∈H_S²1(L_poly) restricts to the class c₁(O(1))−zr∈H²(Fix_r)⊗C(z), and we can write this as the Chen–Ruan orbifold cup product

(P−zr)1_h−ri. Thus Loc(e^℘0t/z∆) should be something like

X where the numerator records the restriction of ∆ to Fixrand the denominator stands for theS¹-equivariant Euler class of the normal bundle to Fixr. We need to make sense of this expression.

Note first that if r>0, the numerator in (26) is divisible by P^dimhri+1 and hence vanishes for dimensional reasons. So our expression is

X This expression still does not make sense due to the divergent infinite product on the right. We “regularize” it by simply dropping these factors—which depend on r only throughhri—and multiplying by z, obtaining theI-function

I(t) =ze^{P t/z}X

so the D-modules generated by ∆ and by I are isomorphic (see (24)). We conjecture that this D-module is isomorphic to theD-module generated by the smallJ-function, and that

J_Pw(t) =I(t).

4. Calculation of the smallJ-function