complete intersections The Crepant Transformation Conjecture for toric Advances in Mathematics

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Advances in Mathematics

www.elsevier.com/locate/aim

The Crepant Transformation Conjecture for toric complete intersections

Tom Coates^a,∗, Hiroshi Iritani^b, Yunfeng Jiang^c

a DepartmentofMathematics,ImperialCollegeLondon,UnitedKingdom

b DepartmentofMathematics,KyotoUniversity,Japan

cDepartmentofMathematics,UniversityofKansas,UnitedStates

a r t i c l e i n f o a bs t r a c t

Article history:

Received14November2017 Accepted15November2017 Availableonline23March2018 CommunicatedbytheManaging Editors

MSC:

primary14N35

secondary14A20,14E16,14F05, 53D45

Keywords:

ToricDeligne–Mumfordstacks CrepantResolutionConjecture Mirrorsymmetry

Quantumcohomology Fourier–Mukaitransformation Mellin–Barnesmethod

LetX andY beK-equivalenttoricDeligne–Mumfordstacks relatedbyasingletoricwall-crossing.WeprovetheCrepant Transformation Conjecture in this case, fully-equivariantly andingenuszero.Thatis,weshowthattheequivariantquan- tumconnectionsforX andY becomegauge-equivalentafter analytic continuation in quantum parameters. Furthermore weidentifythegaugetransformationinvolved,whichcanbe thoughtofasalinearsymplectomorphismbetweentheGiven- talspacesforXandY,withaFourier–Mukaitransformation betweentheK-groupsofXandY,viaanequivariantversion oftheGamma-integralstructureonquantumcohomology.We provesimilarresultsfortoriccompleteintersections.Weim- poseonlyveryweakgeometrichypothesesonX andY:they canbenon-compact,forexample,andneednotbeweakFano or haveGorensteincoarsemodulispace. Ourmaintoolsare theMirror TheoremsfortoricDeligne–Mumfordstacks and toriccompleteintersections,andtheMellin–Barnes method foranalyticcontinuationofhypergeometricfunctions.

* Correspondingauthor.

E-mailaddresses:t.coates@imperial.ac.uk(T. Coates),iritani@math.kyoto-u.ac.jp(H. Iritani), y.jiang@ku.edu(Y. Jiang).

https://doi.org/10.1016/j.aim.2017.11.017

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1. Introduction

A birational map ϕ: X+ X₋ between smooth varieties, orbifolds, or Deligne–

Mumford stacks is called a K-equivalence ifthere exists asmooth variety, orbifold,or Deligne–MumfordstackX andprojectivebirationalmorphismsf_±:X →X_± suchthat f₋=ϕ◦f+ andf₊KX+=f₋KX₋:

X

f₋ f+

X+

ϕ X₋

(1.1)

In this case, the celebrated Crepant Transformation Conjecture of Y. Ruan predicts that the quantum (orbifold) cohomology algebras of X₊ and X₋ should be related by analytic continuation in the quantum parameters. This conjecture has stimulated agreat deal ofinterest inthe connectionsbetween quantum cohomology (or Gromov–

Wittentheory)andbirational geometry:see, forexample,[9,10,17,18,20,22,23,27,40,44, 52,55–58,61,67,70,74,75]. Ruan’s original conjecture was subsequently reﬁned, revised, and extended to higher genus Gromov–Witten invariants, ﬁrst by Bryan–Graber [19]

undersomeadditionalhypotheses,andthenbyCoates–Iritani–Tseng,Iritani,andRuan ingeneral[33,34,49].Recall thatatoricDeligne–Mumford stack X canbe constructed as aGITquotient

C^m//_ωK

ofC^m byanaction of acomplextorus K,where ω is an appropriatestabilitycondition,andthatwall-crossinginthespaceofstabilityconditions induces birational transformationsbetween GITquotients[36,71]. Ourmain resultim- pliestheCIT/RuanversionoftheCrepantTransformationConjectureingenus zero,in thecasewhere X+ andX₋ arecompleteintersections intoricDeligne–Mumford stacks and ϕ: X₊ X₋ arises from a toric wall-crossing. We concentrate initially on the casewhereX+andX₋ aretoric,deferringthediscussionoftoriccompleteintersections to §1.3.

1.1. The toric case

We consider toric Deligne–Mumford stacks X_± of the form

C^m//ωK

, where K is a complex torus,and consider a K-equivalence ϕ:X₊ X₋ determined by a wall- crossinginthespaceofstabilityconditionsω.TheactionofT = (C^×)^monC^mdescends togive(ineffective)actionsofT onX_±,andweconsider theT-equivariantChen–Ruan cohomologygroupsH_CR,T^• (X_±) [25]. Thereis aT-equivariantbigquantum product _τ onH_CR,T^• (X_±),parametrizedbyτ∈H_CR,T^• (X_±) anddefinedintermsofT-equivariant Gromov–WitteninvariantsofX_±.TheT-equivariantquantumconnection isapencilof flatconnections:

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∇=d+z⁻¹ N i=0

(φiτ)dτⁱ (1.2)

on the trivialH_CR,T^• (X_±)-bundle over anopen set inH_CR,T^• (X_±); here z ∈ C^× is the pencilvariable,τ∈H_CR,T^• (X_±) istheco-ordinateonthebaseofthebundle,φ₀,. . . ,φ_N are a basis for H_CR,T^• (X_±), and τ⁰,. . . ,τ^N are the corresponding co-ordinates of τ ∈ H_CR,T^• (X_±),sothatτ=N

i=0τⁱφ_i.

Theorem 1.1.LetX₊ andX₋ betoric Deligne–Mumfordstacks, andletϕ:X₊X₋ be aK-equivalencethat arisesfromawall-crossing of GITstabilityconditions. Then:

(1) the equivariant quantum connections of X_± become gauge-equivalent afteranalytic continuation in τ, via a gauge transformation Θ(τ,z) :H_CR,T^• (X₋)→H_CR,T^• (X₊) whichishomogeneousof degreezero,regularatz= 0,andpreserves theequivariant orbifold Poincarépairing;

(2) there exists acommon toric blowup X of X_± as in (1.1)such that gauge transfor- mationΘcoincides withtheFourier–Mukaitransformation

FM:K_T⁰(X₋)→K_T⁰(X₊) E→(f₊)(f₋)(E) via theequivariantGamma-integral structureintroducedin §3below.

Here:

• TheGamma-integralstructureonequivariantquantumcohomologyisanassignment, to eachclassE ∈K_T⁰(X_±) ofT-equivariantvector bundlesonX_±,of aflatsection s(E) fortheequivariantquantumconnectiononX_±.Thisgivesalatticeinthespace offlatsectionswhichisisomorphictotheintegralequivariantK-groupK_T⁰(X_±).The flatsections(E) is,roughlyspeaking,givenbytheCherncharacterofE multiplied by a characteristic class of X_±, called the Γ-class, that is defined in terms of the Γ-function. Part (2) of Theorem 1.1 asserts thatthe flat section s(E) analytically continuestos(FM(E)).

• The gauge transformation Θ(τ,z) will in general be non-constant: it depends on the parameter τ for the equivariant quantum product, and also on the parameter z appearing inthe equivariantquantum connection. When writteninterms of the integralstructure,however,itbecomesaconstant,integrallineartransformation.

Remark1.2. Throughoutthis paper,when weconsider K-equivalence (1.1)ofDeligne–

MumfordstacksX_±,K_X_±meansthecanonicalclassasastack;ingeneralthisisdiﬀerent from the(Q-Cartier)canonical divisorK_|X_±| of the coarsemoduli space|X_±|. In par- ticular,wedonotrequirethecoarsemoduli spaces|X_±| tobeGorenstein.

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Remark 1.3. Gonzalez and Woodward [44] have proved a very general wall-crossing formula for Gromov–Witten invariants under variation of GIT quotient, using gauged Gromov–Witten theory. Their result, which is a quantum version of Kalkman’s wall- crossing formula, gives a complete description of how non-equivariant genus-zero Gromov–Witten invariants change under wall-crossing. Thus their theorem must imply the non-equivariant version of the first part of Theorem 1.1, and the first part of Theorem 1.4.Ourmethodsaresignificantlylessgeneral—theyapplyonlytotoricstacks andtoric completeintersections — butgive amuch moreexplicitrelationshipbetween thegenus-zeroGromov–Wittentheories.

Theorem 1.1isslightlyimpreciselystated:wegiveprecisestatements,oncetheneces- sarynotationanddeﬁnitionsareinplace,asTheorems 5.14, 6.1,and 6.3below.Wenow explainhowTheorem 1.1impliestheCIT/RuanversionoftheCrepantTransformation Conjecture.

TheCIT/Ruanversion oftheCrepantTransformationConjecture isstatedinterms ofGivental’ssymplecticformalismforGromov–Wittentheory[43].Inourcontext,this associates to X_± the vector spaces H(X_±):= H_CR,T^• (X_±)((z⁻¹)) equipped with a cer- tainsymplecticform,and encodesT-equivariant genus-zeroGromov–Witten invariants viaaLagrangiancone L± ⊂ H(X_±).The Giventalcone L± for X_± determines thebig quantum product τ on H_CR,T^• (X_±), and vice versa. The CIT/Ruan Crepant Trans- formationConjecture, made inthe context of non-equivariant Gromov–Witten theory, asserts that there exists a C((z⁻¹))-linear grading-preserving symplectic isomorphism U: H(X₋)→ H(X₊),suchthatafteranalyticcontinuationofL±wehaveU(L−)=L+. See[33,34]formoredetails.

Therearevarioussubtlepointsinthenotionofanalyticcontinuationofthe(inﬁnite- dimensional) cones L±, especially under the weak convergence hypotheses that we impose,and some necessary foundational material is missing.Thus we chooseto state Theorem 1.1 in terms of the equivariant quantum connections for X_± rather than in termsoftheGiventalconesL±.Thetwo formulationsareverycloselyrelated,however, aswenowexplain.LetL_±(τ,z) denoteafundamentalsolutionfortheequivariantquan- tumconnection∇,thatis,amatrixwithcolumnsthatgiveabasisofﬂatsectionsfor∇. Theassignment

τ→L_±(τ, z)⁻¹H+ τ∈H_CR,T^• (X_±) where H+:=H_CR,T^• (X_±)⊗C[z]

givesthefamilyoftangentspacesto theGiventalcone L±.As emphasizedin[33],this defines avariationof semi-infiniteHodge structureinthe sense ofBarannikov[5]. The Giventalcone L± canbe reconstructedfrom thesemi-infinitevariationas:

L± =

τ

zL_±(τ, z)⁻¹H+

Thuspart (1)ofTheorem 1.1impliestheCIT/Ruan-styleCrepantTransformationCon- jecturewheneveritmakessense,withthesymplectictransformation Udeﬁnedinterms

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of the gauge transformation Θ by U = L⁻₊¹ΘL₋. The fact that U is independent of τ follows from the fact that Θ is a gauge equivalence. The fact that U is symplectic (orequivalently,thefactthatΘ ispairing-preserving)followsfromtheidentiﬁcation,in part (2)ofTheorem 1.1,ofΘ withtheFourier–MukaitransformationFM.TheFourier–

MukaitransformationisaderivedequivalenceandthuspreservestheMukaipairingson K_T⁰(X_±); thisimplies, viathe equivariantHirzebruch–Riemann–Roch theorem,thatΘ ispairing-preserving.TheidentificationofΘ withFMalsomakesclearthatthesymplec- tic transformation Uhas awell-defined non-equivariant limit,since theFourier–Mukai transformation itselfcanbedefinednon-equivariantly.

IntermsofthesymplectictransformationU,part(2)ofTheorem 1.1canberephrased as thecommutativityofthediagram

K_T⁰(X₋) ^FM

Ψ₋

K_T⁰(X₊)

Ψ+

H(X₋) ^U H(X+)

where H(X_±) is avariant of Givental’s symplectic space and Ψ_± are certain ‘framing maps’ builtfromtheGamma-integral structure:see Theorem 6.1.This identiﬁcationof U with a Fourier–Mukai transformation was proposed in [49]. Our results also imply Ruan’s original conjecture thatthe quantum cohomology rings of X_± are (abstractly) isomorphic, andthattheassociatedF-manifoldstructuresareisomorphic.Wereferthe reader to [27,28,33,34,50] for discussions on the consequence of these conjectures and several concreteexamples.

1.2. TheMellin–Barnes methodand theworkofBorisov–Horja

The main ingredients inthe proofof Theorem 1.1 are theMirror Theorem fortoric stacks [26,29], which determines the equivariant quantum connection ∇ (or, equivalently, the Giventalcone L±) interms of acertain cohomology-valued hypergeometric function called the I-function, and the Mellin–Barnes method [6,21], which allows us to analyticallycontinue the I-functions for X_±. From this point of view, thesymplec- tic transformation U arises as the matrix which intertwines the two I-functions (see Theorem 6.1):

UI₋=I+.

On the otherhand, componentsof theI-functiongive hypergeometricsolutionsto the Gelfand–Kapranov–Zelevinsky(GKZ)systemofdiﬀerentialequations.Theanalyticcon- tinuationofsolutionsto theGKZsystemhasbeenstudiedbyBorisov–Horja[12].They showedthat,underanappropriateidentiﬁcationofthespacesofGKZsolutionswiththe

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K-groupsofthecorrespondingtoricDeligne–Mumfordstacks,theanalyticcontinuation of solutions to a GKZsystem is induced by a Fourier–Mukai transformation between the K-groups.Our computation maybe viewed as astraightforward generalization of theirs.Thediﬀerencesfromtheirsituationare:

(a) we work with a fully equivariant version, that is, the parameters β_j in the GKZ systemarearbitrary and weuse theequivariant K-groups(here βj corresponds to theequivariantparameter);

(b) wecomputeanalyticcontinuationoftheI-functioncorrespondingtothebigquantum cohomology;intermsoftheGKZsystem, wedonotassumethatlatticevectorsin theset¹A lieonahyperplaneofheight one.

Sinceweworkequivariantly,wecanusetheﬁxedpointbasisinlocalizedequivariant cohomology tocalculate theanalytic continuationof theI-functions.It turns outthat analyticcontinuation viatheMellin–Barnesmethod becomesmucheasier to handlein thefullyequivariantsetting,becauseweonlyneedtoevaluateresiduesatsimplepoles.² Itisalso straightforwardto computetheFourier–Mukaitransformationintermsofthe ﬁxed point basis inthe localized equivariant K-group, and hence to see that analytic continuationcoincideswithFourier–Mukai.

Regardingpart(b)above,wechooseAto betheset{b1,. . . ,bm}⊂Nofrayvectors of anextended stacky fan [11,53]. Since we do notrestrict ourselvesto the weakFano case,andsinceweworkwithJiang’sextendedstackyfans,thegenericrankoftheGKZ systemcanbebiggerthantherankofH_CR,T^• (X_±).Toremedythis,wetreatonespecial variableanalyticallyandworkformallyintheothervariables.Infact,thebigI-functions are notnecessarily convergent in allof the variables, and we analytically continue the I-functionwith respecttoonespeciﬁcvariable yr.This amountsto consideringanadic completion of the Borisov–Horja better-behaved GKZsystem [14] with respect to the othervariables.TheanalyticcontinuationinTheorem 1.1occursacrossa“globalKähler modulispace”M^◦ whichistreatedasananalyticspaceinonedirectionandasaformal schemeintheotherdirections.

1.3. The toric completeintersection case

Letϕ:X₊ X₋ be aK-equivalence betweentoric Deligne–Mumford stacks that arisesfrom atoric wall-crossing,as in§1.1.LetX bethecommontoric blow-upofX_± and letX₀ denote thecommon blow-down; X₀ hereis a(singular)toric variety,nota stack.

1 RecallthatGelfand–Kapranov–ZelevinskydeﬁnedtheGKZsystemintermsofaﬁnitesetA⊂Z^d.They calledittheA-hypergeometricsystem.

2 For anexample of the complexities causedby non-simple poles,see the orbifoldﬂop calculation in [27, §7].

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X

f−

f+

X+

ϕ

g+

X₋

g₋

X₀

(1.3)

Consider adirectsum of semiample line bundles E₀ →X₀, and pull this back to give vector bundles E₊ → X₊, E → X, and E₋ → X₋. Lets₊, ˜s, and s₋ be sections of, respectively,E+, E, andE₋ thatarecompatibleviaf+ andf₋ (sof₊s+=s=f₋s₋) such thatthe zeroloci of s_± intersectthe ﬂoppinglocus of ϕtransversely. Let Y+, Y, and Y₋ denote thesubstacksdeﬁnedbythezerolociof, respectively,s+, ˜s, ands₋. In this situationthere isacommutativediagram:

Y

F− F+

˜ι

Y₋

ι−

X

f− f+

Y₊

ι+

X₋ X₊

(1.4)

where thevertical mapsareinclusions,thebottomtriangleis(1.1),andthesquaresare Cartesian. The K-equivalence ϕ:X+ X₋ induces aK-equivalence ϕ:Y+ Y₋. Wenow considertheCrepantTransformationConjectureforϕ:Y+ Y₋.

Since the complete intersections Y_± will not in general be T-invariant we consider non-equivariant Gromov–Witten invariantsand the non-equivariant quantum product.

(Our assumptionsonX_± ensure thatthe non-equivarianttheory makessense.) Denote byH_amb^• (Y_±) theimageimι_±⊂H_CR^• (Y_±),whereι_±:Y_± →X_± istheinclusionmap.If τ ∈H_amb^• (Y_±) thenthebigquantumproductτpreservestheambientpartH_amb^• (Y_±)⊂ H_CR^• (Y_±).Wecanthereforedeﬁne aquantumconnectionontheambientpart:

∇=d+z⁻¹ N i=0

(φ_i_τ)dτⁱ

This is a pencilof ﬂatconnectionsonthe trivialH_amb^• (Y_±)-bundleover anopen set in H_amb^• (Y_±) where,as in (1.2), z ∈ C^× is the pencil variable, τ ∈ H_amb^• (Y_±) is the co- ordinateonthebaseofthebundle,φ0,. . . ,φN areabasisforH_amb^• (Y_±),andτ⁰,. . . ,τ^N are thecorresponding co-ordinatesofτ.

In§7.1belowweconstructanambientversionoftheGamma-integralstructure,which is anassignment toeachclassE intheambientpartofK-theory

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K_amb⁰ (Y_±) = imι_±⊂K⁰(Y_±)

ofaﬂatsections(E) forthequantum connectiononthe ambientpartH_amb^• (Y_±).This gives alattice inthe space of ﬂat sections which is isomorphic to the ambientpart of (integral)K-theoryK_amb⁰ (Y_±).

Theorem1.4.Letϕ: Y₊Y₋ beaK-equivalencebetweentoriccomplete intersections asabove.Then:

(1) thequantumconnectionsontheambientpartsH_amb^• (Y_±)⊂H_CR^• (Y_±)becomegauge- equivalent afteranalytic continuationinτ,via agauge transformation

ΘY(τ, z) :H_amb^• (Y₋)→H_amb^• (Y+)

which ishomogeneousofdegree zero andregularatz= 0.If Y iscompactthen Θ_Y preserves theorbifold Poincarépairing;

(2) when expressedintermsoftheambientintegralstructure,thegauge transformation ΘY coincideswith theFourier–Mukaitransformation

FM:K_amb⁰ (Y₋)→K_amb⁰ (Y+) E→(F+)(F₋)(E) given bythetop trianglein(1.4).

Asbefore,Theorem 1.4isslightlyimpreciselystated:precisestatementscanbefound asTheorems 7.2, 7.9,and 7.11below.Arguingasin §1.1showsthatTheorem 1.4implies the CIT/Ruan version of the Crepant Transformation Conjecture for ϕ: Y+ Y₋ wheneveritmakessense,with thecorresponding map

UY:Hamb(Y₋)→ Hamb(Y+)

betweentheambientparts oftheGiventalspacesforY_± beinggivenby:

UY = (L^amb₊ )⁻¹Θ_YL^amb₋

whereL^amb_± arethefundamentalsolutionsforthequantum connectionsontheambient partsH_amb^• (Y_±).

The proof of Theorem 1.4 relies on the Mirror Theorem for toric complete intersections [30], and on non-linear Serre duality [31,41,42,73],which relates thequantum cohomology of Y_± to the quantum cohomology of the total space of the dual bundles E_±^∨.SinceE_±^∨ istoric,itcanbe analyzedusingTheorem 1.1.

Remark 1.5. The idea of using non-linear Serre duality to analyze wall-crossing has been developed independently by Lee–Priddis–Shoemaker [59], in the context of the Landau–Ginzburg/Calabi–Yaucorrespondence.

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Example 1.6.AmirrorY to thequintic3-foldarises[7,21,46]asacrepantresolutionof an anticanonicalhypersurface inX =

P⁴/(Z/5Z)³

.A mirrortheorem forY hasbeen provedbyLee–Shoemaker[60].ThevarietyY isaCalabi–Yau3-foldwithh^1,1(Y)= 101.

TherearemanybirationalmodelsofY astorichypersurfaces,correspondingtothemany diﬀerent latticetriangulationsof theboundaryofthefan polytopefor X. Theorem 1.4 implies that the quantum connections (and quantum cohomology algebras) of all of these birational models become isomorphicafter analytic continuationover theKähler modulispace(whichis101-dimensional),andthattheisomorphismsinvolvedarisefrom Fourier–Mukai transformations.

1.4. Anote onhypotheses

Since we work with T-equivariant Gromov–Witten invariants of the toric Deligne–

Mumford stacks X_±, we donot needto assume thatthecoarse moduli spaces |X_±| of X_± are projective. Weinsist insteadthat|X_±| issemi-projective,i.e. that |X_±|is projective overthe aﬃnizationSpec(H⁰(|X_±|,O)),andalso thatX_± containsatleast one torus ﬁxedpoint.These conditionsareequivalent todemandingthatX_± isobtainedas theGITquotient

C^m//_ωK

ofavectorspacebythelinearactionofacomplextorusK;

they ensurethatthe equivariantquantum cohomologyofX_± admitsanon-equivariant limit.Inparticular,therefore,thenon-equivariantversionoftheCrepantTransformation Conjecture followsautomaticallyfromTheorem 1.1.

We do not assume, either, that the stacks X_± or Y_± satisfy any sort of positivity or weakFano condition; put diﬀerently, we do not impose any additional convergence hypotheses ontheI-functionsforX_± andY_±. Thisextrageneralityispossible because of ourhybrid formal/analytic approach, where we single out onevariable yr and analytically continue inthatvariable alone. The sametechniqueallows us to describe the analytic continuation ofbig quantum cohomology (or its ambientpart),as opposed to small quantum cohomology. Ingeneral, obtainingconvergence results forbig quantum cohomology ishard.

1.5. Thehemisphere partition function

Recentlythere wassomeprogressinphysicsintheexactcomputationofhemisphere partition functionsforgauged linearsigmamodels. Hori–Romo[48] explainedwhy the Mellin–Barnes analytic continuation of hemisphere partitionfunctions shouldbe com- patible with branetransportation[47] inthe B-brane category. Inthelanguage ofthis paper,thehemispherepartitionfunctioncorrespondstoacomponentoftheK-theoretic flatsection s(E), andbranetransportationcorresponds to theFourier–Mukaitransfor- mation. Theorem 1.1 thus confirms the result of Hori–Romo. Note that the relevant equivalence betweenB-brane categoriesshoulddependonachoiceofapathofanalytic continuation, and thatthe Fourier–Mukai transformation in Theorem 1.1 corresponds to aspecificchoiceofpath(seeFig. 1).

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1.6. Plan ofthepaper

Weﬁxnotation forequivariantGromov–Witten invariantsandequivariant quantum cohomology in §2, and introduce the equivariant Gamma-integral structure in §3. We establishnotationfortoricDeligne–Mumfordstacksin §4.In§5westudyK-equivalences ϕ:X+X₋ oftoricDeligne–Mumfordstacksarisingfromwall-crossing,constructing global versionsof the equivariantquantum connections forX_±. Weprovethe Crepant TransformationConjecture fortoricDeligne–Mumford stacks(Theorem 1.1)in §6, and theCrepant Transformation Conjecture fortoric complete intersections (Theorem 1.4) in §7.

1.7. Notation

Weusethefollowing notationthroughoutthepaper.

• X denotes a general smooth Deligne–Mumford stack in §2 and §3; it denotes a smoothtoricDeligne–Mumford stackin §4and later.

• T = (C^×)^m.

• RT =H_T^•(pt,C).

• λj ∈H_T²(pt,C)= Lie(T) isthecharacterofT = (C^×)^mgiven byprojectiontothe jthfactor, sothatR_T =C[λ₁,. . . ,λ_m].

• ST isthelocalizationofRT withrespecttothesetofnon-zerohomogeneouselements.

• Z[T]=K_T^•(pt),sothatZ[T]=Z[e^±λ¹,. . . ,e^±λ^m].

• μ_l={z∈C^×:z^l= 1}isacyclicgroupoforderl.

2. Equivariantquantumcohomology

InthissectionweestablishnotationforvariousobjectsinequivariantGromov–Witten theory.WeintroduceequivariantChen–Ruancohomologyin §2.2,equivariantGromov–

Witteninvariantsin §2.3,equivariantquantumcohomologyin §2.4,Givental’ssymplectic formalismin §2.5, andtheequivariantquantumconnectionin §2.6.

2.1. SmoothDeligne–Mumford stackswithtorus action

Let X be a smooth Deligne–Mumford stack of ﬁnite type over C equipped with an actionof an algebraic torus T ∼= (C^×)^m.Let |X|denote the coarse moduli spaceof X and letIX denote theinertia stack X×|X|X ofX: apoint on IX is given by apair (x,g) withx∈X andg∈Aut(x).Wewrite

IX =_v∈B

^X^v

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for thedecompositionof IX into connectedcomponents.Weassumethefollowingcon- ditions:

(1) thecoarsemodulispace|X|issemi-projective,i.e. isprojectiveovertheaﬃnization SpecH⁰(|X|,O)= SpecH⁰(X,O);

(2) all the T-weights appearing in the T-representation H⁰(X,O) are contained in a strictly convexconeinLie(T)^∗,andtheT-invariantsubspaceH⁰(X,O)^T isC; (3) theinertiastack IX isequivariantly formal,thatis, theT-equivariantcohomology

H_T^•(IX;C) is afree module over R_T := H_T^•(pt;C) and onehas a (non-canonical) isomorphismofRT-modulesH_T^•(IX;C)∼=H^•(IX;C)⊗CRT.

TheseconditionsallowustodefineGromov–WitteninvariantsofXandalsotheequivari- ant(Dolbeault)indexofcoherentsheavesonX.Thefirstandsecondconditionstogether implythatthefixed setX^T iscompact.Thethirdconditionseemstobe closelyrelated to thefirst two,butit implies forexample thelocalization of equivariantcohomology:

therestrictionH_T^•(IX;C)→H_T^•(IX^T;C) to theT-ﬁxed locusisinjectiveand becomes anisomorphismafter localization(see[45]).Later weshallrestricttothecasewhereX is atoricDeligne–Mumfordstack,whereconditions(1)–(3) automatically hold,butthe deﬁnitionsinthissectionmakesense forgeneralX satisfyingtheseconditions.

2.2. EquivariantChen–Ruan cohomology

Let H_CR,T^• (X) denote theevenpartof theT-equivariantorbifoldcohomologygroup ofChenandRuan.ItisdeﬁnedastheevendegreepartoftheT-equivariantcohomology

H_CR,T^k (X) =

v∈B:k−2ιv∈2Z

H_T^k⁻^2ι^v(Xv;C)

oftheinertiastackIX.ThegradingofH_CR,T^• (X) isshiftedfromthatofH_T^•(IX) bythe so-calledage ordegree shiftingnumber ι_v∈Q[24];notethatweconsideronlytheeven degree classes inH_T^•(IX). (For toric stacks,all cohomology classeson IX are of even degree.)EquivariantformalityofIX givesthatH_CR,T^• (X) isafreemoduleoverRT.We write

(α, β) =

IX

α∪inv^∗β, α, β∈H_CR,T^• (X)

fortheequivariantorbifoldPoincarépairing:hereinv :IX →IX denotestheinvolution on the inertiastack IX thatsends a point (x,g) with x∈ X, g ∈Aut(x) to(x,g⁻¹).

SinceX isnotnecessarilyproper,theequivariantintegralontheright-handsidehereis deﬁnedviatheAtiyah–Bottlocalizationformula[3]andtakes valuesinthelocalization S_T ofR_T withrespecttothemultiplicativesetofnon-zerohomogeneouselements³inR_T.

3 NotethatRT ST Frac(RT);weuseST insteadofFrac(RT) sinceweneedagradingonST later.

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2.3. Equivariant Gromov–Witteninvariants

LetXg,n,d denote themoduli spaceof degree-dstable mapsto X from genus g orb- ifoldcurves with nmarked points [1,2]; hered∈ H₂(|X|;Z). Themoduli space carries a T-action and a virtual fundamental cycle [X_g,n,d]^vir ∈ A_•,T(X_g,n,d;Q). There are T-equivariant evaluation maps evi: Xg,n,d → IX, 1 ≤ i ≤ n, to the rigidified iner- tia stack IX (see [2]). Let ψi ∈ H_T²(Xg,n,d) denote the psi-class at the ith marked point, i.e. the equivariant first Chern class of the ith universal cotangent line bundle Li → Xg,n,d. For α1,. . . ,αn ∈ H_CR,T^• (X) and non-negative integers k1,. . . ,kn, the T-equivariantGromov–Witteninvariant isdefinedtobe:

α1ψ^k¹, . . . , αnψ^kⁿX g,n,d=

[Xg,n,d]^vir

n i=1

(ev^∗_i αi)ψ^k_iⁱ (2.1)

where we regard α_i as a class in H_T^•(IX) via the canonical isomorphism H_T^•(IX) ∼= H_T^•(IX). Themoduli space hereisnot necessarilyproper: theright-hand sideis again definedviathe Atiyah–Bottlocalizationformula and so belongs to ST. Conditions (1) and (2) in §2.1ensurethattheT-fixedlocusX_g,n,d^T inthemodulispaceiscompact,and thusthattheright-handsideof(2.1)iswell-defined.

2.4. Equivariant quantumcohomology

Considerthe cone NE(X) ⊂H₂(|X|,R) generated byclasses ofeffective curves and setNE(X)_Z:={d∈H2(|X|,Z):d∈NE(X)}.ForaringR,define RJQKtobe thering offormalpowerserieswithcoefficientsinR:

RJQK=

⎧⎨

⎩

d∈NE(X)Z

adQ^d:ad∈R

⎫⎬

⎭

so thatQ is aso-called Novikov variable [62, III 5.2.1]. Let φ0,φ1,. . . ,φN be ahomo- geneousbasisforH_CR,T^• (X) over RT andletτ⁰,τ¹,. . . ,τ^N bethecorresponding linear co-ordinates.Weassumethatφ₀= 1 andφ₁,. . . ,φ_r∈H_T²(X) aredegree-twountwisted classesthatinduceaC-basisofH²(X;C)∼=H_T²(X)/H_T²(pt).Wewriteτ=N

i=0τⁱφifor ageneralelement ofH_CR,T^• (X).Theequivariant quantum product τ atτ ∈H_CR,T^• (X) isdeﬁnedbytheformula

(φ_i_τφ_j, φ_k) =

d∈NE(X)Z

∞ n=0

Q^d

n! φ_i, φ_j, φ_k, τ, . . . , τ^X0,n+3,d

or,equivalently, by

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φ_i_τφ_j =

d∈NE(X)_Z

∞ n=0

Q^d

n! inv^∗ev_3,∗

ev^∗₁(φ_i) ev^∗₂(φ_j)

n+3

l=4

ev^∗_l(τ)∩[X_0,n+3,d]^vir

.

(2.2) Conditions (1) and (2) in §2.1ensurethatev3:X0,n+3,d→IX isproper,andthusthat the push-forwardalong ev₃ is well-deﬁnedwithoutinverting equivariantparameters.It follows that:

φiτφj∈H_CR,T^• (X)⊗RT RTJτ, QK

where R_TJτ,QK=R_TJτ⁰,. . . ,τ^NKJQK. Theproduct _τ deﬁnes anassociativeand com- mutative ring structure on H_CR,T^• (X)⊗RT R_TJτ,QK. The non-equivariant limit of _τ exists, and this limit deﬁnes the non-equivariant quantum cohomology

H_CR^• (X)⊗C

CJτ,QK,τ

.

Remark 2.1. The divisor equation [2, Theorem 8.3.1] implies that exponentiated H²- variables andtheNovikovvariableQplaythesamerole:onehas

(φiτφj, φk) =

d∈NE(X)_Z

∞ n=0

Q^de^σ,d

n! φi, φj, φk, τ, . . . , τ^X_0,n+3,d where τ =σ+τ with σ=r

i=1τⁱφi andτ =τ0φ0+N

i=r+1τⁱφi. TheStringEqua- tion[2,Theorem8.3.1]impliesthattheright-handsidehereisinfactindependentofτ0. 2.5. Givental’sLagrangian cone

LetS_T((z⁻¹)) denote theringof formalLaurentseries inz⁻¹ withcoeﬃcientsinS_T. Givental’ssymplecticvectorspaceisthespace

H=H_CR,T^• (X)⊗RT ST((z⁻¹))JQK equipped withthenon-degenerateSTJQK-bilinear alternatingform:

Ω(f, g) =−Resz=∞(f(−z), g(z))dz with f,g∈ H.Thespaceisequipped withastandardpolarization

H=H+⊕ H−

where

H+:=H_CR,T^• (X)⊗RT ST[z]JQK and H−:=z⁻¹H_CR,T^• (X)⊗RT STJz⁻¹KJQK

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areisotropicsubspacesforΩ.ThestandardpolarizationidentiﬁesHwiththecotangent bundleofH+.Thegenus-zerodescendantGromov–Wittenpotential isaformalfunction FX⁰ : (H+,−z1) → STJQK deﬁned on the formal neighbourhood of −z·1 in H+ and takingvaluesinS_TJQK:

FX⁰(−z1 +t(z)) =

d∈NE(X)Z

∞ n=0

Q^d

n! t(ψ), . . . ,t(ψ)^X_0,n,d Heret(z)=_∞

n=0tnzⁿ withtn∈H_CR,T^• (X)⊗RTSTJQK.Let{φⁱ}⊂H_CR,T^• (X)⊗RT ST

denotethebasisPoincarédual to{φi},sothat(φi,φ^j)=δ_i^j.

Deﬁnition2.2 ([29,43]).Givental’s Lagrangian cone LX ⊂(H,−z1) isthegraphofthe diﬀerentialdFX⁰ :H+→T^∗H+∼=H.Itconsistsof pointsofHoftheform:

−z1 +t(z) +

d∈NE(X)_Z

∞ n=0

N i=0

Q^d n!

φ_i

−z−ψ,t(ψ), . . . ,t(ψ)

0,n+1,d

φⁱ (2.3)

where 1/(−z − ψ) in the correlator should be expanded as the power series _∞

k=0ψ^k(−z)⁻^k⁻¹ inz⁻¹. Inamoreformal language,we deﬁne thenotion ofa‘point onLX’asfollows.Letx= (x1,. . . ,xn) beformalparameters.AnSTJQ,xK-valuedpoint onLX isanelement ofHJxKoftheform (2.3)with t(z)∈ H+JxKsatisfying

t(z)|Q=x=0= 0.

Itshouldbethoughtofas aformal familyofelements onLX parametrizedbyx.

Thesubmanifold LX encodesallgenus-zero Gromov–Witten invariants (2.1). It has thefollowingspecialgeometricproperties[43]:itisacone,andatangentspaceT ofLX

istangenttoLX exactlyalongzT.KnowingGivental’sLagrangianconeLXisequivalent to knowingthe dataof thequantum product τ,i.e. LX canbe reconstructedfrom τ

andviceversa.SeeRemark 2.5.

2.6. The equivariantquantum connection anditsfundamental solution Letv∈H_CR,T^• (X).Theequivariantquantumconnection

∇v:H_CR,T^• (X)⊗RT RT[z]Jτ, QK→z⁻¹H_CR,T^• (X)⊗RT RT[z]Jτ, QK isdeﬁnedby

∇vf(τ) =∂vf(τ) +z⁻¹v τf(τ)

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where ∂vf(τ)= _ds^df(τ+sv)|s=0 isthedirectional derivative.We write∇i for ∇φi and

∇f for N

i=0(∇if)dτⁱ. The associativity of τ implies that the connection ∇ is ﬂat, that is, [∇i,∇j] = 0 forall i, j. Let ρ denote the equivariant ﬁrstChern class (in the untwisted sector):

ρ:=c^T₁(T X)∈H_T²(X)⊂H_CR,T² (X)

For homogeneousφ∈ H_CR,T^• (X),we write degφ forthe age-shifted(real)degree ofφ, so thatφ ∈H_CR,T^deg^φ(X). The equivariant Euler vector ﬁeldE and the grading operator μ∈End_C(H_CR,T^• (X)) aredeﬁnedby

E:=

m i=1

λi

∂

∂λi

+ N i=0

1−degφ_i 2

τⁱ ∂

∂τⁱ +∂ρ

μ(φ) :=

degφ

2 −dim_CX 2

φ

(2.4)

where λ1,. . . ,λm ∈ H_T²(pt) are generators of RT (see §1.7). The grading operator on H_CR,T^• (X)⊗RT RT[z]Jτ,QKisdeﬁnedby

Gr(f(τ, z)φ) =

z_∂z^∂ +E f(τ, z)

φ+f(λ, τ, z)μ(φ)

where φ∈H_CR,T^• (X) and f(λ,τ,z)∈RT[z]Jτ,QK.Thequantum connectioniscompat- ible with thegrading operatorinthesense that[Gr,∇i]=∇[E,∂_{τ i}] = (¹₂degφi−1)∇i, i = 0,. . . ,N. This follows from the virtual dimensionformula for the moduli space of stable maps.

Notation 2.3. Letv∈H_T²(X) be adegree-two classintheuntwisted sector.The action of v on H_CR,T^• (X) is deﬁned by v·α = q^∗(v)∪α, where q:IX → X is the natural projection. (Thiscoincideswiththeactionofv viatheChen–Ruancupproduct.)

Considertheﬂatsectionequationsfor∇,and afundamentalsolution L(τ, z)∈End_R_T(H_CR,T^• (X))⊗RT R_T((z⁻¹))Jτ, QK determined bythefollowingconditions:

∇iL(τ, z)φ= 0 fori= 0, . . . , N (ﬂatness) (2.5)

vQ ∂

∂Q−∂v

L(τ, z)φ=L(τ, z)v

zφ forv∈H_T²(X) (divisor equation) (2.6)

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L(τ, z)|τ=Q=0 = id (initial condition) (2.7) Here φ ∈ H_CR,T^• (X) and vQ_∂Q^∂ with v ∈ H_T²(X) acts on Novikov variables as Q^d → v, dQ^d(itactsbyzerowhenv∈H_T²(pt)⊂H_T²(X)).TheflatnessequationfixesL(τ,z) uptorightmultiplication byanendomorphism-valuedfunctiong(z;Q) inz andQ;the divisorequation impliesthattheambiguity g(z;Q) isindependent ofQ andcommutes withv∪,v∈H_T²(X);finallytheinitialconditionfixesL(τ,z) uniquely.Thefundamental solution satisfying these conditions can be written explicitly interms of (descendant) Gromov–Witteninvariants:

L(τ, z)φi =φi+ N j=0

d∈NE(X)Z

∞ (n≥1 ifn=0d=0)

Q^d n!

φi

−z−ψ, τ, . . . , τ, φj

X 0,n+2,d

φ^j (2.8)

This is deﬁned over R_T (without inverting equivariant parameters) becauseit canbe rewrittenintermsofthepush-forwardalongthelast evaluationmapevn+2 as in(2.2).

A straightforward equivariant generalizationof [41, Corollary 6.3], [66, Proposition 2], [49,Proposition2.4]gives:

Proposition 2.4. The fundamental solution L(τ,z) in (2.8) satisﬁes the conditions (2.5)–(2.7).Furthermore itsatisﬁes:

L(τ, z) = id +O(z⁻¹) (regularity atz=∞) GrL(τ, z)φ=L(τ, z)

μ−ρ z

φ (homogeneity) (α, β) = (L(τ,−z)α, L(τ, z)β) (unitarity) whereφ,α,β ∈H_CR,T^• (X).

Remark 2.5 ([43]). The fundamental solution L(τ,z) is determined by the quantum product_τ viadiﬀerentialequations (2.5)–(2.7).Thenτ →T_τ =L(τ,−z)⁻¹H+ givesa versalfamilyoftangentspacestoGivental’sconeLX.TheconeLX isreconstructedas LX =

τzTτ.

We now study ∇-ﬂat sections s(τ,z) that are homogeneous of degree zero:

Gr(s(τ,z)) = 0. By Proposition 2.4, if a ﬂat section L(τ,z)f(z) is homogeneous of degreezero,then:

z ∂

∂z+μ−ρ z

f(z) = 0

Thisdiﬀerentialequationhasthefundamentalsolution: