MIXED-SPIN-P FIELDS OF FERMAT POLYNOMIALS HUAI-LIANG CHANG

HUAI-LIANG CHANG¹, JUN LI², WEI-PING LI³, AND CHIU-CHU MELISSA LIU⁴

Abstract. This is the first part of the project toward an e↵ective algorithm to evaluate all genus Gromov-Witten invariants of quintic Calabi-Yau threefolds.

In this paper, we introduce the notion of Mixed-Spin-P fields, construct their moduli spaces, and construct the virtual cycles of these moduli spaces.

1. Introduction

Explicitly solving all genus Gromov-Witten invariants (in short GW invariants) of Calabi-Yau threefolds is one of the major goals in the subject of Mirror Symmetry.

For quintic Calabi-Yau threefolds, the mirror formula of genus-zero GW invariants was conjectured in [CdGP] and proved in [Gi, LLY]. The mirror formula of genus- one GW invariants was conjectured in [BCOV] and proved in [LZ, Zi]. A complete determination of all genus GW invariants based on degeneration is provided in [MP] and plays a crucial role in the proof of the GW/Pairs correspondence [PP].

However, the mirror prediction on genusg GW invariants for 2g51 in [HKQ]

is still open, even in theg= 2 case.

The mirror prediction in [HKQ] includes both GW invariants of quintic threefolds and FJRW invariants of the Fermat quintic. In this paper, we introduce the notion of Mixed-Spin-P fields (in short MSP fields) of the Fermat quintic polynomial, construct their moduli spaces, and establish basic properties of these moduli spaces.

This class of moduli spaces will be employed in the sequel of this paper [CLLL]

toward developing an e↵ective theory evaluating all genus GW invariants of quintic threefolds and all genus FJRW invariants of the Fermat quintic.

The theory of MSP fields, for the Calabi-Yau quintic polynomial F5,5(x) =x⁵₁+· · ·+x⁵₅,

provides a transition between FJRW invariants [FJR] and GW invariants of stable maps with p-fields [CL]. It is known that the FJRW invariants of the Fermat quintic is the LG theory taking values in [C⁵/µ₅] (via spin fields), and the GW invariants of stable maps with p-fields is the LG theory taking values in the canonical line bundleK_P⁴ (via P-fields). Our idea is to use the master space technique to study the two GIT quotients [C⁵/µ₅] andK_P⁴ of [C⁶/G^m], which led to the notion of

1Partially supported by Hong Kong GRF grant 600711.

2Partially supported by NSF grant DMS-1104553 and DMS-1159156.

3Partially supported by by Hong Kong GRF grant 602512 and HKUST grant FSGRF12SC10.

4Partially supported by NSF grant DMS-1206667 and DMS-1159416.

1

MSP fields, providing a geometric transition between the LG theories of the two GIT quotients of [C⁶/G^m].

In this paper, we will introduce the notion of MSP field for the Fermat polynomial (1.1) Fn,r(x) :=x^r₁+· · ·+x^r_n.

WithFn,r understood, an MSP field is a collection (1.2) ⇠= (⌃^C⇢C,L,N,',⇢,⌫),

consisting of a pointed twisted curve ⌃^C ⇢ C, fields ' 2 H⁰(C,L ⁿ) and ⇢ 2 H⁰(C,L^_⌦r⌦!^log_C ), and a gauge field ⌫ = (⌫1,⌫2) 2 H⁰(C,L⌦N N). The numerical invariants of⇠are the genus ofC, the monodromy i ofLat the marking

⌃^C_i (of⌃^C), and the bi-degreesd0= deg(L⌦N) andd₁= degN.

For a choice of g, = ( 1,· · ·, `) and d = (d0, d₁), we form the moduli W^{g, ,d}of equivalence classes of stable MSP fields of numerical data (g, ,d). It is a separated DM stack, locally of finite type, though usually not proper. The stack W^{g, ,d}admits aT =C^⇤ action, via

t·(⌃^C⇢C,L,N,',⇢,⌫1,⌫2) = (⌃^C⇢C,L,N,',⇢, t⌫1,⌫2).

It comes with a perfect (relative) obstruction theory, of virtual dimension (in case

=;)

vir.dimWg, =;,d= (1 +n r)d0+ (1 n+r)d₁+ (4 n)(g 1).

Theorem 1.1. The moduli stackW^{g, ,d}(of stable MSP fields of the Fermat polyno- mialFn,r) is a separated DM stack, locally of finite type. It has a cosection localized T-equivariant virtual cycle[W^{g, ,d}]^vir_loc, lying in a proper substackWg, ,d⇢W^{g, ,d}:

[W^{g, ,d}]^vir_loc2A^T_⇤(Wg, ,d)^T.

In the sequel [CLLL], for the quintic Fermat polynomialF5,5we apply the virtual localization formula to derive a doubly indexed polynomial relations among the GW invariants of quintic threefolds and the FJRW invariants of the Ferman polynomial F5,5. These relations provide an e↵ective algorithm in evaluating all genus GW invariants of quintics in terms of FJRW invariants ofF5,5(with the insertion 2/5), and provide a collection of relations among all genus FJRW invariants ofF5,5(with the insertion 2/5).

This work is inspired by Witten’s vision that the “Landau-Ginzburg looks like the analytic continuation of Calabi-Yau to negative Kahler class.” (See [Wi, 3.1].) One interpretation of his proposed transition of theories is that the LG theory of [C⁵/Z⁵] and that ofK_P⁴ di↵er by a fields version of “wall-crossing”. The MSP fields introduced can be viewed as a geometric construction to realize this “wall-crossing”.

Around the time of the completion of the first draft of this paper, there have been other approaches for high genus LG/CY correspondence [CK, FJR2]. Since then, there are a few notable further developments based on the theory of MSP developed.

In [CLLL], the virtual localization formula of the C^⇤ equivariant moduli of MSP fields were developed, and the mentioned recursion relations among GW and FJRW

of the quintic Fermat polynomial is derived. In [GR], Guo-Ross proved the genus- one Landau-Ginzburg/Calabi-Yau conjecture of Chiodo and Ruan. In [CGLZ], the explicit formula of the genus one GW invariants of the quintic CY threefold has been recovered using the recursion relations. Recently, it is shown that the localization of the genus two MSP theory recovers the genus two BCOV’s Feyman rule of the GW invariants of the quintic Calabi-Yau threefolds [CGL]. Along the way, the finite generation conjecture of Yamaguchi-Yau and the holomorphic anomaly equation for g = 2 GW invariants of the quintic Calabi-Yau threefolds is establised. (cf.

[BCOV, YY, ASYZ, LP].)

This paper is organized as follows. In Section one, we will introduce the notion of Mixed-Spin-P fields of the Fermat quintic polynomial; construct the moduli spaces of stable Mixed-Spin-P fields, and construct the cosection localized virtual cycles of these moduli spaces. These cycles lie in the degeneracy loci of the cosection mentioned. In Section two and three, we will prove that these degeneration loci are proper, separated and of finite type.

Acknowledgement. The third author thanks the Stanford University for several months visit there in the spring of 2011 where the project started. The second and the third author thank the Shanghai Center for Mathematical Sciences at Fudan University for many visits. The authors thank Y.B. Ruan for stimulating discussions on the FJRW invariants.

2. The moduli of Mixed-Spin-P fields

In this section, we introduce the notion of MSP (Mixed-Spin-P) fields, construct their moduli stacks, and form their cosection localized virtual cycles. We introduce theT-structure on it. The proof of the localization formula of cosection localized virtual cycles will appear in [CKL].

2.1. Twisted curves and invertible sheaves. We recall the basic notions and properties of twisted curves with representable invertible sheaves on them. The materials are drawn from [ACV, AJ, AF, AGV, Cad].

A prestable twisted curve with`-markings is a one-dimensional proper, separated connected DM stackC, with at most nodal singularities, together with a collection of disjoint closed substacks⌃1,· · · ,⌃` of smooth locus ofC such thatC^sm [ⁱ⌃i

is a scheme, and node are balanced.

Here an indexrbalanced node looks like the following model Vr:=⇥

Spec C[u, v]/(uv) µ_r⇤

, ⇣·(u, v) = (⇣u,⇣ ¹v).

Similarly, an indexr marking of a twisted curve looks like the model U_r:=⇥

SpecC[u] µ_r⇤

, ⇣·u=⇣u.

Denote by

(2.1) ⇡r:Vr!Vr:= Spec C[x, y]/(xy) and ⇡r:Ur!Ur:=A¹

defined byx7!u^rand y7!v^r, the maps to their coarse moduli spaces. Note that Vr contains two subtwisted curves Vr,1 and Vr,2, each isomorphic to Ur in (2.1).

This process Vr 7! Ur,u`

Ur,v is called the decomposition of Vr along its node.

The reverse process is called the gluing, which can be defined via a push out.

We comment on out convention on invertible sheaves on a twisted curveCnear a stacky point. In the model caseC=Vr, an invertible sheaf onCis aµ_r-module Mm, for 0< m < r, so that

M_m:=u ^{(r m)}C[u] [0]v ^mC[v] := ker{u ^{(r m)}C[u] v ^mC[v]!C^m}, where the arrow is a homomorphism of µ_r-modules, and µ_r leaves 1 2C[u] and 12C[v] fixed and acts on 12C^m⇠=Cvia⇣·1 =⇣^m1, and bothu ^{(r m)}C[u]!C^m andv ^mC[v]!C^m are surjective. Whenm= 0,

M₀:=C[u] [0]C[v] := ker{C[u] C[v]!C},

where maps are defined similarly as the case ofm6= 0. Note that the isomorphism classes are indexed bym2{0,· · · , r 1}. Further, in the convention (2.1)

(2.2) ⇡r⇤Mm=C[x] C[y].

Similarly, invertible sheaves on C near an index r marking in the model case C=Urlooks like the µ_r-module

(2.3) Mm:=u ^mC[u].

Let⇣r= exp(2⇡i/r)2µ_r. Underu7!⇣ru, the generatoru ^m1u7!⇣_r^mu ^{(r m)}1u, we call (the inverse of⇣_r^m)⇣_r^mthe monodromy ofMmat the marking, and callm the monodromy index at the marking. Note that forMmoverVr,Mmrestricted to Vr,1and to Vr,2 have monodromies⇣_r^mand⇣_r^m, at their respective stacky points.

Definition 2.1. We call M_mrepresentable if(m, r) = 1.

Example 2.2. Let C = A¹/µ_r be the obvious global quotient twisted curve; let p 2 A¹ be its origin. Then we use OC(^m_rp) to denote the sheaf (2.3), having monodromy index m.

2.2. Definition of MSP fields. We fix a Fermat polynomial (1.1). We denote by µ_dC^⇤ the subgroup of thed-th roots of unity. We let

˜

µ⁺_r =µ_r[{(1,⇢),(1,')}, and µ˜_r= ˜µ⁺_r {1}.

For ↵ 2 µ˜⁺_r, we let h↵i  G^m be the subgroup generated by ↵; for the two exceptional element (1,⇢) and (1,'), we agree thath(1,⇢)i=h(1,')i=h1i.

We fix the Fermat polynomialFn,r; we let

g 0, = ( 1,· · ·, `)2( ˜µ<sub>r</sub>)<sup>⇥`</sup>, and d= (d0, d₁)2Q^⇥2,

and call the triple (g, ,d) a numerical data (for MSP fields), and call (g, ,d) a broad numerical data if 2( ˜µ⁺_r)^⇥` instead.

For an `-pointed twisted nodal curve⌃<sup>C</sup> ⇢C, denote!<sup>log</sup><sub>C/S</sub>: =!C/S(⌃<sup>C</sup>), and for↵2µ˜<sup>+</sup><sub>r</sub>, let⌃<sup>C</sup><sub>↵</sub>=`

i=↵⌃^C_i.

Definition 2.3. LetS be a scheme,(g, ,d)be a numerical data. An S-family of MSP-fields (of Fermat typeFn,r) of type(g, ,d)is a datum

(C,⌃^C,L,N,',⇢,⌫) such that

(1) [<sup>`</sup>i=1⌃<sup>C</sup><sub>i</sub> =⌃<sup>C</sup>⇢Cis an `-pointed, genusg, twisted curve overS such that the i-th marking⌃^C_i is banded by the grouph ⁱi G^m;

(2) LandNare representable invertible sheaves on C, andL⌦NandNhave fiberwise degreesd0 andd₁ respectively. The monodromy ofLalong⌃^C_i is

i whenh ⁱi 6=h1i;

(3) ⌫= (⌫1,⌫2)2H⁰(L⌦N) H⁰(N)such that(⌫1,⌫2)is nowhere vanishing;

(4) '= ('1, . . . ,'n)2H⁰(L) ⁿ so that(',⌫1)is nowhere zero, and'|⌃^C_(1,')= 0.

(5) ⇢2H⁰(L ^r⌦!_C/S^log )such that(⇢,⌫2)is nowhere vanishing, and⇢|⌃^C_(1,⇢) = 0;

In the future, we call ' (resp. ⇢) the '-field (resp. ⇢-field) of the MSP-field.¹ (Here we abbreviateL ^r=L^_⌦r.)

Definition 2.4. In case (g, ,d) is a broad numerical data, a similarly defined (C,⌃^C,L,N,',⇢,⌫)as in Definition 2.3 is called an S-family of broad MSP-fields.

We remark that in this paper we will only be concerned with MSP fields.

Definition 2.5. An arrow

(C⁰,⌃^C0,L⁰,N⁰,'⁰,⇢⁰,⌫⁰) !(C,⌃^C,L,N,',⇢,⌫)

from an S⁰-MSP-field to an S-MSP-field consists of a morphism S⁰ ! S and a 3-tuple (a, b, c)such that

(1) a: (⌃^C0 ⇢C⁰)!(⌃^C⇢C)⇥^SS⁰ is an S⁰-isomorphism of pointed twisted curves;

(2) b:a^⇤L!L⁰ andc:a^⇤N!N⁰ are isomorphisms of invertible sheaves such that the pullbacks of 'k,⇢and⌫i are identical to '⁰_k,⇢⁰ and ⌫_i⁰, where the pullbacks and the isomorphisms are induced by a,bandc.

We define Wg, ,d^pre to be the category fibered in groupoids over the category of schemes, such that the objects inWg, ,d^pre over S areS-families of MSP-fields, and morphisms are given by Definition 2.5.

Definition 2.6. ⇠2Wg, ,d^pre (C)isstableifAut(⇠)is finite. ⇠2Wg, ,d^pre (S)is stable if ⇠|^s is stable for every closed points2S.

LetW^{g, ,d}⇢Wg, ,d^pre be the open substack of families of stable objects inWg, ,d^pre . We introduce aT =Gmaction onW^{g, ,d}by

(2.4) t·(⌃^C,C,L,N,',⇢,(⌫1,⌫2)) = (⌃^C,C,L,N,',⇢,(t⌫1,⌫2)), t2T.

Theorem 2.7. The stack W^{g, ,d}is a DMT-stack, locally of finite type.

1We call (C,⌃^C,L,N,⌫) satisfying (1)-(3), and⌃^C_(1,') ⇢(⌫1 6= 0) and⌃^C_(1,⇢) ⇢(⌫2 6= 0), a gauged twistedS-curve.

Proof. The theorem follows immediately from that the stackM^twg,`of stable twisted

`-pointed curves is a DM stack, and each of its connected components is proper and

of finite type (see [AJ, Ol]). ⇤

In this paper, we will reserve the symbolT =G^m for this action onW^{g, ,d}. Example 2.8 (Stable maps withp-fields). A stable MSP-field ⇠ 2W^{g, ,d} having

⌫1= 0will haveN⇠=O_C,⌫2= 1. Then⇠= (⌃^C,C,· · ·)reduces to a stable mapf = ['] :⌃^C⇢C!P^{n 1} together with ap-field ⇢2H⁰(f^⇤O_P^{n 1}( r)⌦!^log_C ). Moduli of genusg `-pointed stable maps withp-fields will be denoted byMg,`(P^{n 1}, d)^p. Example 2.9 (r-Spin curves withp-fields). A stable MSP-field ⇠2W^{g, ,d} having

⌫2 = 0 will have N ⇠= L^_, ⌫1 = 1. Then ⇠ reduces to a pair of a r-spin curve (⌃^C,C,⇢:L^r⇠=!_C^log)andn p-fields'i2H⁰(L). Moduli ofr-spin curves with fixed monodromy andn p-fields will be denoted byM^1/r,np_g, .

2.3. Cosection localized virtual cycle. The DM stack W^{g, ,d} admits a tauto- logical T-equivariant perfect obstruction theory.

LetD^g, be the stack of triples (C,⌃^C,L,N), where⌃^C⇢Care`-pointed genus gconnected twisted curves (i.e. objects inM<sup>tw</sup>g,`),LandNare invertible sheaves on Csuch that the monodromy ofLalong the marked points are given by . Because M^twg,` is a smooth DM stack, D<sup>g,</sup> is a smooth Artin stack, locally of finite type and of dimension (3g 3) +`+ 2(g 1) = 5g 5 +`, where the automorphisms of (⌃^C⇢C,L,N) are triples (a, b, c) as in Definition 2.5.

Define

(2.5) q:W^{g, ,d} !D^g,

to be the forgetful morphism, forgetting (',⇢,⌫) from points ⇠ 2 W^{g, ,d}. The morphismqisT-equivariant withT acting onD^g, trivially. Let

(2.6) ⇡:⌃^C ⇢C!W^{g, ,d} with (L,N,',⇢,⌫)

being the universal family over W^{g, ,d}. Let 0mi r 1 be so that i =⇣_r^mi. For convenience, we let `' = #{i| <sup>i</sup> = (1,')}, and let`o = #{i| ⁱ 2 µ_r}. We abbreviate

P=L ^r⌦!_C/W^log

g, ,d.

Proposition 2.10. The pairq:W^{g, ,d}!D^g, admits a tautologicalT-equivariant relative perfect obstruction theory taking the form

⇣R⇡_⇤ L( ⌃^C_(1,')) ⁿ P( ⌃^C_(1,⇢)) (L⌦N) N ⌘_

!L^•_{Wg, ,d/Dg,} . The virtual dimension (g, ,d) := vir.dimW^{g, ,d}is

(1 +n r)d0+ (1 n+r)d₁+ (4 n)(g 1) +`+ (1 n) `'+

`o

X

i=1

mi

r .

Proof. The construction of the obstruction theory is parallel to that in [CL, Prop 2.5], and will be omitted. We compute its virtual dimension. Let⇠= (⌃^C,C,L,· · ·) be a closed point inW^{g, ,d}. Observe that whenh ⁱi 6={1},'|⌃^C_i = 0. Thus by that (',⌫1) is nowhere vanishing, we see that⌫1|⌃^C_i 6= 0, and the monodromy ofL⌦N along⌃^C_i is trivial. Therefore, let⇡:C!C be the coarse moduli morphism, the degrees of⇡_⇤(L⌦N),⇡_⇤Land⇡_⇤Nared0,d0 d₁ P`o

i=1 mi

r andd₁ `o+P`o

i=1 mi

r , respectively. Since the relative virtual dimension ofW^{g, ,d}!D^g, is

(L( ⌃^C_(1,')) ⁵ L^_⌦5( ⌃^C_(1,⇢))⌦!_C^log L⌦N N).

Here we insert ⌃^C_(1,') and ⌃^C_(1,⇢) because of (4) and (5) in Definition 2.3. Using that dimD^g, = 5g 5 +`, applying Riemann-Roch theorem to (L) = (⇡_⇤L), we obtain the formula of (g, ,d), as stated in the proposition. ⇤

The relative obstruction sheaf ofW^{g, ,d}!D^g, is

Ob_{Wg, ,d/D}g, :=R¹⇡_⇤ L( ⌃^C_(1,')) ⁿ P( ⌃^C_(1,⇢)) (L⌦N) N , and the absolute obstruction sheafOb_{Wg, ,d} is the cokernel of the tautological map q^⇤T_Dg, !Ob_{Wg, ,d}/Dg, , fitting into the exact sequence

(2.7) q^⇤T_Dg, !Ob_{Wg, ,d}/Dg, !Ob_{Wg, ,d} !0.

We define a cosection

(2.8) :Ob_{Wg, ,d}/Dg, !O_{Wg, ,d}

by the rule that at an S-point⇠2W^{g, ,d}(S), (in the notation ⇠= (C,⌃^C,· · ·) as in (1.2)),

(2.9) (⇠)( ˙',⇢,˙ ⌫˙1,⌫˙2) =r⇢X

'^r_i ¹'˙i+ ˙⇢X

'^r_i 2H¹(!C/S)⌘H⁰(OC)^_, where

( ˙',⇢,˙ ⌫˙1,⌫˙2)2H¹ L( ⌃^C_(1,')) ⁿ P( ⌃^C_(1,⇢)) L⌦N N . (Here P=L^_⌦5⌦!_C/^log_W

g, ,d.) Note that the term r⇢P'^r_i ¹'˙i+ ˙⇢P'^r_i a priori lies in H¹ C,!^log_C/S( ⌃^C_(1,⇢)) . However, whenh ⁱi 6= (1,⇢),'j|⌃^C_i = 0. Thus it lies inH¹(C,!C/S).

Lemma 2.11. The rule (2.9)defines aT-equivariant homomorphism as in (2.8).

Via (2.7)the homomorphism lifts to aT-equivariant cosection ofOb_{Wg, ,d}. Proof. The proof that the cosection lifts is exactly the same as in [CLL], and will be omitted. That the homomorphism isT-equivariant is becauseT acts on

W^{g, ,d}via scaling⌫1 and is independent of⌫1. ⇤

As in [KL], we define the degeneracy locus of to be Wg, ,d(C) ={⇠2W^{g, ,d}(C)| |^⇠ = 0}, (2.10)

endowed with the reduced structure. It is a closed substack ofW^{g, ,d}.

Lemma 2.12. The closed points ofWg, ,d(C)are ⇠2W^{g, ,d}(C)such that (2.11) ('= 0)[('^r₁+· · ·+'^r_n=⇢= 0) =C.

Proof. We consider individual terms in (2.9). Taking the term ⇢'^r_i ¹'˙i, by the vanishing along⌃^C_i recalled before the statement of Lemma 2.11, we conclude that

⇢'^r_i ¹2H⁰ L^_⌦!_C^log( ⌃^C) =H⁰ L^_⌦!C .

By Serre duality, when⇢'^r_i ¹6= 0, there is a ˙'i2H¹(L) so that⇢'^r_i ¹·'˙i6= 02 H¹(!C).

Repeating this argument, we conclude that |^⇠ = 0 if and only if

⇢'^r₁ ¹=· · ·=⇢'^r_n ¹='^r₁+· · ·+'^r_n = 0.

This is equivalent to that ('= 0)[('^r₁+· · ·+'^r₅=⇢= 0) =C. ⇤ Note that (2.10) makes sense for ⇠ 2 Wg, ,d^pre (C) as well. For convenience, we denote

Wg, ,d^pre (C) = ⇠2Wg, ,d^pre (C)|(2.10) holds for⇠ ; Applying [KL, CKL], we obtain the cosection localized virtual cycle

[W^{g, ,d}]^vir_loc2A^TWg, ,d, = (g, ,d).

2.4. MSP invariants. Using the universal family (2.6) we define the evaluation maps (associated to the marked sections⌃^C_i):

evi:W^{g, ,d}!X :=Pⁿ[(µ_r)

as follows. In case h ⁱi 6= 1, we define evi to be the constant map to h ⁱi 2 µ_r; in case i = (1,'), we define evi( i) = 1 2 µ_r. In case i = (1,⇢), for si : W^{g, ,d} ! C^{g, ,d} the i-th marked section of the universal curve, by (2) of Definition 2.3 we haves^⇤_i⇢= 0. Thuss^⇤_i⌫2is nowhere vanishing, ands^⇤_iN⇠=OWg, ,d. Therefore, s^⇤_i(',⌫1) is a nonwhere vanishing section of s^⇤_iL ⁽ⁿ⁺¹⁾, defining the desired evaluation morphism

(2.12) evi= [s^⇤_i'1,· · · , s^⇤_i'n, s^⇤_i⌫1] :W^{g, ,d}!Pⁿ such that ev^⇤_iOPⁿ(1) =s^⇤_iL.

LetT act onPⁿ by

t·['1, . . . ,'n,⌫1] = ['1, . . . ,'n, t⌫1], and letT act trivially on µ_r. It makes evi T-equivariant.

We introduce the MSP state space. As aC-vector space, the MSP state space and theT-equivariant MSP state space are the cohomology group and theT-equivariant cohomology group of the evaluation spaceX=Pⁿ[(µ_r):

H^MSP=H^⇤(X;C), and H^MSP,T =H_T^⇤(X;C).

In terms of generators, we have

H_T^⇤(Pⁿ;C) =C[H,t]/hHⁿ(H+t)i, and H_T^⇤(µ_r;C) = Mr j=1

C[t]1^j

r,

and the (non-equivariant) MSP state space is by settingt= 0, while the grading is given by

(2.13) degH = 2, degt= 2 and deg1^j

r =2(n 1)

r j.

We formulate the gravitational descendants. Given

a1, . . . , a`2Z <sup>0</sup>, 1, . . . , `2H^MSP =H^⇤(X;C), we define the MSP-invariants

(2.14) h⌧a1 1· · ·⌧a` `i^MSPg,`,d:=

ˆ

[Wg,`,d]^vir_loc

Y` k=1

ak

k ev^⇤_{k k}2C. where

[W^g,`,d]^vir_loc= X

2( ˜µ_r)^`

[W^{g, ,d}]^vir_loc. Similarly, we defineT-equivariant genusg MSP-invariants to be (2.15) h⌧a1 1· · ·⌧a` `i^MSP,Tg,`,d :=

ˆ

[Wg,`,d]^vir_loc

Y` k=1

ak

k ev^⇤_{k k}2H^⇤(BT;Q) =Q[t], where i2H^MSP,T, and

[W^g,`,d]^vir_loc= X

2( ˜µ_r)^`

[W^{g, ,d}]^vir_loc.

(Here we use the same [·]^vir_loc to mean theT-equivariant class.) Suppose 1, . . . , `

are homogeneous, and let e(a_·, _·) := P`

k=1

⇣ak+^deg₂ ^k⌘

⇣(1 +n r)d0+ (1 n+r)d`+ (4 n)(g 1) +`⌘ . By the formula of the virtual dimension ofW^g,`,d, we see that

h⌧a1 1· · ·⌧a` `i^MSP,Tg,`,d 2Ct^{e(a·, ·)}. In casee(a_·, _·)<0, we have vanishing

(2.16) h

t ^{e(a·, ·)}·h⌧a1 1· · ·⌧a` `i^MSP,Tg,`,d

i

0= 0, where [·]0is the dimension 0 part of the pushforward toH0(pt).

By virtual localization, we will express all genus full descendant MSP invariants in terms of (1): GW invariants of the quintic threefold Q ⇢ P⁴; (2): FJRW in- variants of the Fermat quintic; and (3): the descendant integrals on M^g,n. The invariants in item (1) has been solved in genus zero [Gi, LLY] and genus one for all degrees [LZ, Zi], and in all genus for degree zero, those in item (2) has been solved in genus zero [CR], and those in item (3) have been solved in all genera. One of our goals to introduce MSP invariants is to use vanishing (2.16) to obtain recursive relations to determine (1) and (2) in all genus. This will be addressed in details in the sequel [CLLL].

3. Properness of the degeneracy loci In this section, we will prove thatWg, ,dis proper overC.

3.1. The conventions. In this section, we denote by⌘0 2S a closed point in an affine smooth curve, and denoteS_⇤=S ⌘0its complement.

In using valuative criterion to prove properness, we need to take a finite base changeS⁰!S ramified over⌘0. By shrinkingSif necessary, we assume there is an

´etaleS!A¹ so that⌘0 is the only point lying over 02A¹. This way, we can take S⁰ to be of the formS⁰ =S⇥A¹A¹, whereA¹ !A¹ is viat 7!t^k for some integer k 2. This way,⌘₀⁰ 2S⁰ lying over⌘02S is also the only point lying over 02A¹. One particular choice ofS⁰ is the degreerbase change: Sr=S⇥A¹A¹!S, where A¹!A¹ is viat7!t^r.

To keep notations easy to follow, for a property P that holds after a finite base change S⁰ !S of a family ⇠ over S, we will say “after a finite base change, the family⇠ has the property P”, meaning that we have already done the finite base changeS⁰ !S and then replaceS⁰ byS for abbreviation of notations.

In this and the next section, for⇠2Wg, ,d^pre (C) or Wg, ,d^pre (S), we understand (3.1) ⇠= ⌃^C,C,L,N,',⇢,⌫ 2Wg, ,d^pre .

Similarly, we will use subscript “⇤” to denote families overS_⇤. Hence⇠_⇤2Wg, ,d^pre (S_⇤) will be of the form

(3.2) ⇠_⇤= ⌃^C⇤,C_⇤,L_⇤,N_⇤,'_⇤,⇢_⇤,⌫_⇤ .

We first prove a simple version of the extension result we need.

Proposition 3.1. Let ⇠_⇤ 2 Wg, ,d(S_⇤) be such that ⇢_⇤ = 0. Then after a finite base change,⇠_⇤ extends to a⇠2Wg, ,d(S).

Proof. Since ⇢_⇤ = 0, ⌫2⇤ is nowhere vanishing and N_⇤ ⇠= O_⇤. Thus ('_⇤,⌫1⇤) is a nowhere vanishing section of H⁰(L_⇤ⁿ L_⇤), and induces a morphismf_⇤ from (⌃^C⇤,C_⇤) to Pⁿ, such that (f_⇤)^⇤O_Pⁿ(1) ⇠= L_⇤. By the stability assumption of ⇠_⇤, this morphism is an S_⇤-family of stable maps. By the properness of moduli stack of stable maps, after a finite base change we can extend (⌃^C⇤,C_⇤) to (⌃^C,C) and extendf_⇤ to anS-family of stable maps f from (⌃^C,C) toPⁿ. LetL=f^⇤O_Pⁿ(1), which is an extension ofL_⇤. Thenf is provided by a section (',⌫1)2H⁰(L ⁿ L), extending ('_⇤,⌫1⇤). Since [f,⌃^C,C] is stable, the central fiber C₀ is a connected curve with at worst nodal singularities. Define N⇠=O_C and ⌫2 to be the isomor- phismN⇠=O_Cextending⌫_2⇤. Define⇢= 0. Then⇠= ⌃^C,C,L,',⇢,⌫ is a desired

extension. ⇤

The case involving'_⇤= 0 over some irreducible components is technically more involved. We will treat this case by first studying the case C_⇤ is smooth. For this, we characterize stable objects inWg, ,d^pre (C). We say that an irreducible component E⇢Cis a rational curve if it is smooth and its coarse moduli is isomorphic toP¹.

Lemma 3.2. Let p16=p22P¹ be two distinct closed points, GAut(P¹) be the subgroup fixing p1 andp2, and L be a G-linearized line bundle on P¹ such thatG acts trivially onL|^p1. Then the following holds:

(1) any invariants2H⁰(L)^G with s(p1) = 0must be the zero section;

(2) suppose Gacts trivially onL|^p2, thenL⇠=O_P¹.

Proof. Both are well-known. ⇤

Lemma 3.3. Let ⇠2Wg, ,d^pre (C). It is unstable if and only if one of the following holds:

(1) Ccontains a rational curveEsuch thatE\(⌃^C[C_sing)contains two points, andL^⌦r|^E⇠=O_E;

(2) Ccontains a rational curveEsuch thatE\(⌃^C[C_sing)contains one point, and either L|^E⇠=N|^E⇠=O_E or⇢|^E is nowhere vanishing;

(3) Cis a smooth rational curve with⌃^C=;,d0=d₁= 0.

(4) Cis irreducible, g= 1,⌃^C=;, andL^⌦r⇠=O_C andL^_⇠=N.

Proof. We first prove the necessary part. Let ⇠ 2 Wg, ,d^pre (C) be unstable. For each irreducible E ⇢ C, let AutE(⇠) be the subgroup of Aut(⇠) mapping E to itself. There exists an E such that AutE(⇠) is of infinite order. If the image of AutE(⇠) ! Aut(E) is finite, then for a finite index subgroup G⁰  AutE(⇠), G⁰ leavesCfixed, thusG⁰ acts on⇠by acting on the line bundlesLandNvia scaling.

However, by Definition 2.3, thatG⁰ leaves (',⇢,⌫) invariant implies that the image ofG⁰!Aut(L)⇥Aut(N) is finite. Since this arrow is injective, it contradicts to the fact thatG⁰ is infinite. Thus the group G= im(AutE(⇠)!Aut(E)) is of infinite order.

We now consider the case whereEhas arithmetic genus zero (thus smooth). We divide it into several cases. The first case (when ga(E) = 0) is when E\(⌃^C[ C_sing) contains one point, say p 2 E. Suppose ⇢|^E = 0, then ⌫2|^E is nowhere vanishing, implying N|^E ⇠= O_E. Thus (',⌫1)|^E is a nowhere vanishing section of H⁰(L ⁽ⁿ⁺¹⁾|^E). SinceG is infinity and (',⌫1)|^E is G-equivariant, this is possible only if L|^E⇠=OE and (',⌫1)|^E is a constant section. This is Case (2).

The other case is when ⇢|^E 6= 0. We argue that ⇢|^E is nowhere vanishing.

Otherwise,⌫2|^E6= 0, and then degN|^E 0. Since ⇠2Wg, ,d^pre (C), we have'|^E= 0, thus⌫1|^Eis nowhere vanishing andL^_|^E⇠=N|^E. Because⇢|^E6= 0 and deg!^log_C |^E= 1, we must have degL|^E <0. Thus ⌫2 2 H⁰(N|^E) = H⁰(L^_|^E) must vanish at some point. Let p1 and p2 2 Ebe such that ⇢(p1) = 0 = ⌫2(p2). Since (⇢,⌫2) is nowhere vanishing, we havep16=p2. Furthermore, sinceGfixesp, p1 andp2, and is infinite,p=p1or p2.

Supposep=p1, a similar argument shows thatL ^r⌦!_C^log|^E⇠=O_E, contradicting to⇢6= 0 and vanishing somewhere. Supposep=p2, we conclude that degL|E= 0, contradicting to degL|^E < 0. Combined, we proved that if ⇢|^E 6= 0, then ⇢|^E is nowhere vanishing. This is Case (2).

The second case is whenE\(⌃^C[Csing) contains two points, sayp1 andp22E.

Then Gfixes bothp1 and p2. A parallel argument shows that Gacts trivially on L^r|^p1 and L^r|^p2. Applying Lemma 3.2, we conclude that L^r ⇠=OE. This is Case

(1). The third case is whenE\(⌃^C[Csing) =;. A parallel argument shows that in this case we must haveL⇠=N⇠=OC. This conclude the study of the casega(E) = 0.

The remaining case is whenga(E) = 1, thenE\⌃^C=;, and a similar argument shows that it must belong to Case (4). Combined, this proves that if⇠is unstable, then one of (1)-(4) holds.

We now prove the other direction that whenever there is anE⇢Cthat satisfies one of (1)-(4), then ⇠ is unstable. Most of the cases can be argued easily, except a sub-case of (2) when ⇢|^E is nowhere vanishing, which we now prove. Since E\ (⌃C[Csing) contains one point, say p 2 C, we have deg!^log_C |^E = 1. Since ⇢|^E is nowhere vanishing, we have degL^r|E = 1. Thus p must be a stacky point.

Hence E⇠=P^1,r as stacks. Let P^1,r= Proj(k[x, y]) where degx= 1 and degy=r.

Then p corresponds to the point [0,1]. Let G =C (the additive group) acts on P^1,r viax!x, y! x^r+y for 2G. TheG-action onP^1,r lifts to an action of

!_C^log|E as well asL^_|E⇠=O_P1,r(1). One can check via local calculations that Gacts trivially on !^log_C |^E |^p as well as L ¹|^E |^p, thus trivially on L ^r⌦!_C^log |^p. Since L ^r⌦!_C^log|^E⇠=O_E, and sinceG^a =Chas no non-trivial characters, by Prop.1.4 in [FMK],G acts on L ^r⌦!_C^log |^E trivially as well. Hence G acts trivially on⇢|^E. Therefore the groupGis a subgroup of the automorphism group of⇠|^E. ⇤ Corollary 3.4. Let ⇠ 2 Wg, ,d^pre (C). Let ⇡ : ˜C ! C be the normalization of C, let ⌃^C˜ = ⇡ ¹(⌃^C[Csing), and let ( ˜L,N,˜ ',˜ ⇢,˜ ⌫)˜ be the pullback of (L,N,',⇢,⌫) via ⇡^⇤. Write ˜C = `

aC˜a the connected component decomposition, and let ⇠˜a be (⌃^C˜\C˜a,C˜a)paired with( ˜L,N,˜ ',˜ ⇢,˜ ⌫˜)|C^˜a. Then⇠is stable if and only if all⇠˜a are stable.

Proof. If ⇠ is unstable, then it contains an irreducible E satisfying one of (1)-(4) in Lemma 3.3. This E (or its normalization) will appear in one of ˜⇠a, making it unstable. The other direction is the same. This proves the Corollary. ⇤ 3.2. The baskets. We will first studying a special case.

Special type: Let ⇠_⇤ 2Wg, ,d(S_⇤)be of the form (3.2)such that C_⇤ is smooth (and connected),'_⇤= 0,⌫_2⇤6= 0and⇢_⇤6= 0.

Proposition 3.5. Let⇠_⇤overS_⇤be of special type. Then after a finite base change, (a1) (⌃^C⇤,C_⇤)extends to a pointed twisted curve(⌃^C,C)overSsuch thatC ⌃^C is a scheme,C is smooth, and the central fiber C0 is reduced with at worst nodal singularities and smooth irreducible components;

(a2) L_⇤ and N_⇤ extend to invertible sheaves L andN respectively onC so that

⌫_⇤ extends to a surjective⌫ = (⌫1,⌫2) :L^_ OC!N;

(a3) ⇢_⇤ extends to a ⇢2 (L^_⌦5⌦!^log_C/S(D)) for a divisor D⇢C contained in the central fiberC₀such that⇢restricting to every irreducible component of C₀ is non-trivial;

(a4) (⇢_⇤= 0)\(⌫2⇤= 0) =;,(⇢_⇤ = 0)and(⌫2⇤= 0)intersectC0 transversally.

Proof. First, possibly after a finite base change, we can assume that (⇢_⇤= 0)[ (⌫_2⇤= 0) is a union of disjoint sections of C_⇤ !S_⇤, and that if we let ⌃^ex_⇤ be the union of those sections of (⇢_⇤= 0)[(⌫2⇤= 0) that are not contained in⌃^C⇤, then

⌃^ex_⇤ is disjoint from ⌃^C⇤. If (⌃^C⇤ [⌃^ex_⇤ ,C_⇤) is a stable pointed curve, let⌃^au_⇤ =;. Otherwise, let⌃^au_⇤ be some extra sections ofC_⇤!S_⇤, disjoint from⌃^C⇤ [⌃^ex_⇤ , so that after letting⌃^comb_⇤ =⌃^C⇤ [⌃^ex_⇤ [⌃^au_⇤ , the pair (⌃^comb_⇤ ,C_⇤) is stable.

Since (⌃^comb_⇤ ,C_⇤) is stable, possibly after a finite base change, it extends to an S-family of stable twisted curves (⌃^comb,C⁰) such that all singular points of its central fiber C⁰₀ are non-stacky. Thus after blowing up C⁰ along the singular points of C⁰₀ if necessary, taking a finite base change, and followed by a minimal desingularization, we can assume that the resulting family (⌃^C,C) is a family of pointed twisted curves with smoothCsatisfying Condition (a1). Condition (a4) is satisfied due to the construction.

Since '_⇤ is identically zero, ⌫_1⇤ is an isomorphism. We can extend N_⇤ to an invertible sheafNonCso that⌫2⇤ extends to a section⌫2 ofN. LetL⇠=N^_. We extend⌫1⇤to an isomorphism⌫1:L^_!N, and extend ⇢_⇤ to a section⇢satisfying (a3), for a choice ofD. This proves the proposition. ⇤

We will work with the coarse moduliC ofC.

Definition 3.6. AnS-family of pre-stacky pointed nodal curves is a flat S-family (⌃, C)of pointed nodal curves (i.e. not twisted curves) so that each marked-section

⌃i (of⌃) is either assigned pre-stacky or assigned regular. We call it a good family if in additionC is smooth, and all irreducible components of the central fiberC0= C⇥^S⌘0 are smooth.

Note that if we letCbe the coarse moduli of theCin Proposition 3.5, let⌃i⇢C be the image of ⌃^C_i ⇢Cunder C!C, and call ⌃i pre-stacky if⌃^C_i is stacky, and call it regular otherwise, then (⌃, C) with this assignment is a good S-family of pre-stacky pointed nodal curves.

Given anS-family of pointed twisted curve (⌃^C,C) so that the only non-scheme points ofCare possibly along⌃^C, applying the procedure described, we obtain a pre- stacky pointed nodal curve (⌃, C). We call this procedure un-stacking. Conversely, applying the root construction (cf. [AGV, Cad]) to the S-family of pre-stacky pointed nodal curves (⌃, C) obtained, we recover the original family (⌃^C,C); we call the later the stacking of (⌃, C).

Definition 3.7. Let(⌃, C) be a goodS-family of pre-stacky pointed nodal curves, and letDi,i2⇤, be irreducible components ofC0. A pre-basket of (⌃, C)is a data

(3.3) B= (B+X

i2⇤liDi, A+X

i2⇤miDi), where

(1) A = Pk1

i=1aiAi, where A1,· · ·, Ak1 are disjoint sections of C ! S such that for any pair (i, j), either Ai\⌃j =; or Ai =⌃j, ai 2 ¹rZ^>0 when Ai=⌃j for some pre-stacky⌃j, otherwiseai2Z^>0;

(2) B=Pk2

i=1biBi, wherebi2Z^>0,B1,· · ·, Bk2 are disjoint sections ofC!S such that for any pair (i, j), either Bi\⌃j = ; or Bi = ⌃j, and when Bi=⌃j,⌃j must be assigned regular;

(3) A1,· · ·, Ak1, B1,· · ·, Bk2 are mutually disjoint and intersect C0 transver- sally;

(4) rmi2Z andli2Z; such that

(3.4) OC(B+X

liDi)⇠=OC(rA+X

rmiDi)⌦!_C/S^log .

We call Ba basket if in addition it satisfiesli 0,mi 0 andlimi= 0for all i.

Definition 3.8. We say a basket Bfinal if it satisfies

(i). for every i2⇤,B\Di =; if mi6= 0, andA\Di=; if li6= 0;

(ii). for distincti6=j2⇤such that limj 6= 0,Di\Dj =;.

Let (C,L,N,⇢,⌫) andDbe given by Proposition 3.5. Let{D_i|i2⇤}be the set of irreducible components ofC0. We form

(3.5) A= (⌫2⇤= 0), B= (⇢_⇤= 0), (⌫2= 0) =A+X

miDi, D= X liDi, where the summations run over all i 2 ⇤. By the construction, ⌫2 and ⇢ induce isomorphismsN⇠=OC(A+P

miDi) andOC⇠=L ^r⌦!^log_C/S(D B). UsingL^_⇠=N, we obtain an isomorphism

(3.6) OC(B+X

liDi)⇠=OC(rA+X

rmiDi)⌦!_C/S^log .

Let (⌃, C) be the good S-family of pre-stacky pointed nodal curves that is the un-stacking of (⌃^C,C) as explained before Definition 3.7. LetDi⇢C0be the image ofD_i. SinceCaway from (⌫2= 0) is a scheme, and by the construction carried out in the proof of Proposition 3.5, we haveB =Pk2

i=1biB_i, whereB_i are sections of C!S andbi2Z^>0, and for any (i, j) eitherBi\⌃^C_j =; orBi =⌃^C_j, and in the later case⌃^C_j is a scheme. We letBi⇢C be the image ofB_i. ForA, it can also be written as A=Pk1

i=1aiA_i, whereA_i are sections ofC!S. Let Ai be the image ofA_i. We form

A= X

Ai62{pre-stacky⌃j}

aiAi+ X

Ai2{pre-stacky⌃j}

ai

rAi, and B=

k2

X

i=1

biBi. Lemma 3.9. Let (⌃, C)be as before, letB in (3.3) be such that the coefficients li

andmi are given in (3.5), and letA andB be given in the identities above. Then B is a pre-basket.

Proof. ThatBsatisfies (1)-(4) in Definition 3.7 follows from the proof of Proposition 3.5. For the isomorphism (3.4), we notice that by our choice ofAandB, we have

OC(B+X

liDi)⇠=OC(B+X

liDi)⌦^OC OC

and

O_C(rA+X

rmiD_i)⌦!_C/S^log ⇠= O_C(rA+X

rmiDi)⌦!^log_C/S ⌦^OC O_C.

Therefore, (3.4) follows from (3.6). This proves the Lemma. ⇤ 3.3. Restacking. In this subsection, we fix an S-family of pre-stacky n-pointed nodal curves (⌃, C) and a final basket B on it in the notations of Definition 3.7 and 3.8. Let t be a uniformizing parameter ofR, whereS = SpecR, that is the pullback of the standard coordinate variable ofA¹via the mapS !A¹ specified at the beginning of§3.1. LetRa=R[z]/(z^a t), and letSa= SpecRa.

Lemma 3.10. LetCbe a flatS-family of nodal curves, letN be the singular points of the central fiberC0, and letM be an (integral) e↵ective Cartier divisor onCsuch that M =rMh+M0, where M0 (resp. Mh) is an integral Weil divisor contained inC0 (resp. none of its irreducible components lie in C0). Then there is a unique Sr-family of twisted curvesC˜ such that

(1) letN˜ ⇢˜C0 be the singular points ofC˜0, thenC˜ N˜ ⇠= (C N)⇥^SSr; (2) let ˜ : ˜C N˜ !C be the morphism induced by (1), then ˜^⇤(M)is divisible

byr, and ¹_r˜⇤(M)extends to a Cartier divisor onC, denoted by˜ M˜¹

r; (3) each ⇣2N˜ is either a scheme point or a µ_a-stacky point ofC, where˜ a|r,

such that the tautological mapAut(⇣)!Aut(OC˜( ˜M¹

r)|^⇣)is injective.

Proof. Let p 2 N be a singular point of C0. Pick an ´etale open neighborhood q:V !C ofp2C so thatV is an open subscheme of (xy=t^k)⇢Spec(R[x, y]), as S-schemes. LetD1 and D2 in V be (x=t= 0) and (y =t= 0), respectively (Example 6.5.2 in [Har]). Whenk6= 1,D1andD2are Weil but not Cartier divisors.

Writeq ¹M =rA+n1D1+n2D2, whereA =q ¹Mh is an integral Weil divisor with no irreducible components contained in D1[D2. Let CL(V) (resp. Car(V)) be the Weil (resp. Cartier) divisor class groups ofV respectively. It is known that CL(V)/Car(V) = Z/kZ, generated by D1 (or D2). Thus A is linearly equivalent to lD1+B for an integerl and a Cartier divisorB. SinceM is a Cartier divisor, we have rl+n1 n2 ⌘ 0(k) (i.e. ⌘ 0 mod (k)). Here we used the fact that D1+D2= 02CL(V)/Car(V).

Consider the base change ˜C :=C⇥^SSr!C. Let ˜pbe the node in the central fiber of ˜Ccorresponding to p. Let ˜V =V ⇥^SSr and ˜q: ˜V !V be the projection.

It is an ´etale neighborhood of ˜p2 C, and is an open subsheme of (xy˜ = z^rk)⇢ Spec(Rr[x, y]).

Let ˜A be ˜q ¹(A), ˜D1 = (x=z = 0), and ˜D2 = (y =z = 0). Since ˜q ¹(Di) = rD˜i, the pullback ˜q ¹(M) =rA˜+rn1D˜1+rn2D˜2, and ¹_rq˜ ¹(M) away from ˜pis Cartier. SinceA=lD12CL(V)/Car(V), ˜A=l(rD˜1)2CL( ˜V)/Car( ˜V). Thus

˜

q ¹(M) =rA˜+rn1D˜1+rn2D˜2⌘(2rl+rn1 rn2) ˜D12CL( ˜V)/Car( ˜V).

Leta⁰ be defined by

(3.7) (r`+n1 n2, rk) =a⁰k, a⁰ 2[1, r],

and leta=r/a⁰. Sincek|(r`+n1 n2),a2Zanda|r. Thus ¹_rq˜ ¹(M) is Cartier only whena= 1.

To make it Cartier whena >1, we introduceµ_a-stacky structure at ˜pas follows.

Consider (uv =z^k)⇢Spec(Ra[u, v]) and the morphism (uv =z^k)!(xy =z^ak)