HIGHER GENUS GROMOV-WITTEN THEORY OF Hilb

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ANDCohFTs ASSOCIATED TO LOCAL CURVES

RAHUL PANDHARIPANDE AND HSIAN-HUA TSENG

Dedicated to A. Givental on the occasion of his 60th birthday

ABSTRACT. We study the higher genus equivariant Gromov-Witten theory of the Hilbert scheme of npoints of C². Since the equivariant quantum cohomology, computed in [29], is semisimple, the higher genus theory is determined by an R-matrix via the Givental-Teleman classification of Cohomological Field Theories (CohFTs). We uniquely specify the requiredR-matrix by explicit data in degree0. As a consequence, we lift the basic triangle of equivalences relating the equivariant quantum cohomology of the Hilbert schemeHilbⁿ(C²)and the Gromov-Witten/Donaldson-Thomas correspondence for 3-fold theories of local curves to a triangle of equivalences in all higher genera.

The proof uses the analytic continuation of the fundamental solution of the QDE of the Hilbert scheme of points determined in [30]. The GW/DT edge of the triangle in higher genus concerns new CohFTs defined by varying the 3-fold local curve in the moduli space of stable curves.

The equivariant orbifold Gromov-Witten theory of the symmetric productSymⁿ(C²)is also shown to be equivalent to the theories of the triangle in all genera. The result establishes a complete case of the crepant resolution conjecture [1, 5, 6].

CONTENTS

0. Introduction 2

1. Cohomological field theory 10

2. TheR-matrix forHilbⁿ(C²) 14

3. TheR-matrix forSymⁿ(C²) 20

4. Local theories of curves: stable maps 26

5. Local theories of curves: stable pairs 29

6. The restricted CohFTs 32

7. Quantum differential equations 35

8. Solutions of the QDE 36

9. Analytic continuation 45

References 51

Date: May 2019.

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0. INTRODUCTION

0.1. Quantum cohomology. TheHilbert schemeHilbⁿ(C²)ofnpoints in the planeC²parameterizes idealsI ⊂C[x, y]of colengthn,

dim_CC[x, y]/I =n .

The Hilbert scheme Hilbⁿ(C²) is a nonsingular, irreducible, quasi-projective variety of dimension 2n, see [14, 28] for an introduction. An open dense set ofHilbⁿ(C²)parameterizes ideals associated to configurations ofndistinct points.

The symmetries ofC²lift to the Hilbert scheme. The algebraic torus T= (C^∗)²

acts diagonally onC²by scaling coordinates,

(z₁, z₂)·(x, y) = (z₁x, z₂y). The inducedT-action onHilbⁿ(C²)will play basic role here.

The Hilbert scheme carries a tautological ranknvector bundle,

(0.1) O/I →Hilbⁿ(C²),

with fiberC[x, y]/Iover[I]∈ Hilbⁿ(C²), see [22]. TheT-action onHilbⁿ(C²)lifts canonically to the tautological bundle (0.1). Let

D=c₁(O/I)∈H_T²(Hilbⁿ(C²),Q) be theT-equivariant first Chern class.

TheT-equivariant quantum cohomology ofHilbⁿ(C²)has been determined in [29]. Thematrix elementsof theT-equivariant quantum product count¹rational curves meeting three given subvari- eties ofHilbⁿ(C²). The (non-negative)degreeof an effective²curve class

β ∈H₂(Hilbⁿ(C²),Z) is defined by pairing withD,

d= Z

β

D.

Curves of degree d are counted with weightq^d, where q is the quantum parameter. The ordinary multiplication inT-equivariant cohomology is recovered by settingq= 0.

LetM_D denote the operator ofT-equivariant quantum multiplication by the divisorD. A central result of [29] is an explicit formula forM_D an as operator on Fock space.

1The count is virtual.

2Theβ= 0is considered here effective.

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0.2. Fock space formalism. We review the Fock space description of theT-equivariant cohomology of the Hilbert scheme of points of C² following the notation of [29, Section 2.1], see also [15, 28].

By definition, theFock spaceF is freely generated overQby commuting creation operatorsα−k, k ∈Z>0, acting on the vacuum vectorv∅. The annihilation operatorsα_k,k∈Z>0, kill the vacuum

α_k·v_∅ = 0, k >0, and satisfy the commutation relations

[α_k, α_l] =k δ_k+l. A natural basis ofF is given by the vectors

(0.2) |µi= 1

z(µ) Y

i

α−µ_iv∅

indexed by partitionsµ. Here,

z(µ) =|Aut(µ)| Y

i

µ_i

is the usual normalization factor. Let the length`(µ)denote the number of parts of the partitionµ.

TheNakajima basisdefines a canonical isomorphism, (0.3) F ⊗_QQ[t₁, t₂]=^∼ M

n≥0

H_T^∗(Hilbⁿ(C²),Q).

The Nakajima basis element corresponding to|µiis 1 Π_iµ_i[V_µ]

where [V_µ] is (the cohomological dual of) the class of the subvariety of Hilb^|µ|(C²) with generic element given by a union of schemes of lengths

µ₁, . . . , µ_`(µ)

supported at`(µ)distinct points³ofC². The vacuum vectorv_∅ corresponds to the unit in 1∈H_T^∗(Hilb⁰(C²),Q).

The variablest₁andt₂ are the equivariant parameters corresponding to the weights of theT-action on the tangent space Tan₀(C²)at the origin ofC².

The subspace of F ⊗_Q Q[t1, t2] corresponding to H_T^∗(Hilbⁿ(C²),Q) is spanned by the vectors (0.2) with|µ|=n. The subspace can also be described as then-eigenspace of theenergy operator:

| · |=X

k>0

α−kαk.

The vector|1ⁿicorresponds to the unit

1∈H_T^∗(Hilbⁿ(C²),Q). A straightforward calculation shows

(0.4) D =−

2,1ⁿ⁻² .

3The points and parts ofµare considered here to be unordered.

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The standard inner product on theT-equivariant cohomology ofHilbⁿ(C²)induces the following nonstandardinner product on Fock space after an extension of scalars:

(0.5) hµ|νi= (−1)^|µ|−`(µ)

(t₁t₂)^`(µ) δµν

z(µ). With respect to the inner product,

(0.6) (α_k)^∗ = (−1)^k−1(t₁t₂)^sgn(k)α−k.

0.3. Quantum multiplication byD. The formula of [29] for the operatorMD of quantum multiplication byDis:

M_D(q, t₁, t₂) = (t₁+t₂)X

k>0

k 2

(−q)^k+ 1

(−q)^k−1α−kα_k − t₁+t₂ 2

(−q) + 1 (−q)−1| · | +1

2 X

k,l>0

h

t₁t₂α_k+lα−kα−l−α−k−lα_kα_li .

The q-dependence of M_D occurs only in the first two terms (which acts diagonally in the basis (0.2)). The two parts of the last sum are known respectively as the splitting and joining terms.

Letµ¹ andµ² be partitions of n. The T-equivariant Gromov-Witten invariants of Hilbⁿ(C²)in genus0with 3 cohomology insertions given (in the Nakajima basis) byµ¹,D, andµ²are determined byM_D:

∞

X

d=0

µ¹, D, µ²Hilbⁿ(C²)

0,d q^d =

µ¹ M_D

µ² .

Equivalently, denoting the 2-cycle(2,1ⁿ)by(2), we have (0.7)

∞

X

d=0

µ¹,(2), µ²Hilbⁿ(C²)

0,d q^d =

µ¹ −MD

µ²

.

Letµ¹, . . . , µ^r ∈Part(n). TheT-equivariant Gromov-Witten series in genusg, µ¹, µ², . . . , µ^rHilbⁿ(C²)

g =

∞

X

d=0

µ¹, µ², . . . , µ^rHilbⁿ(C²)

g,d q^d ∈Q[[q]],

is a sum over the degreedwith variableq. TheT-equivariant Gromov-Witten series in genus 0, µ¹, µ², . . . , µ^rHilbⁿ(C²)

0 =

∞

X

d=0

µ¹, µ², . . . , µ^rHilbⁿ(C²) 0,d q^d,

can be calculated from the special 3-point invariants (0.7), see [29, Section 4.2].

0.4. Higher genus. Our first result here is a determination of the T-equivariant Gromov-Witten theory of Hilb(C², d) in all higher genera g. We will use the Givental-Teleman classification of semisimple Cohomological Field Theories (CohFTs). The Frobenius structure determined by the T-equivariant genus0 theory of Hilb(C², d) is semisimple, but not conformal. Therefore, the R- matrix isnotdetermined by theT-equivariant genus 0theory alone. Fortunately, together with the divisor equation, an evaluation of theT-equivariant higher genus theory in degree 0is enough to uniquely determine theR-matrix.

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Let Part(n)be the set of of partitions ofncorresponding to theT-fixed points ofHilbⁿ(C²). For eachη ∈ Part(n), let Tan_η(Hilbⁿ(C²))be theT-representation on the tangent space at theT-fixed point corresponding toη. Let

Eg → M_g

be the Hodge bundle of differential forms over the moduli space of stable curves of genusg.

Theorem 1. TheR-matrix of theT-equivariant Gromov-Witten theory ofHilbⁿ(C²)is uniquely determined from theT-equivariant genus 0 theory by the divisor equation and the degree 0 invariants

µHilbⁿ(C²)

1,0 = X

η∈Part(n)

µ|_η Z

M1,1

e(E^∗1⊗Tan_η(Hilbⁿ(C²))) e(Tan_η(Hilbⁿ(C²))) ,

Hilbⁿ(C²)

g≥2,0 = X

η∈Part(n)

Z

Mg

e E^∗g⊗Tanη(Hilbⁿ(C²)) e(Tanη(Hilbⁿ(C²))) .

The insertionµin the genus 1 invariant is in the Nakajima basis, andµ|_ηdenotes the restriction to theT-fixed point corresponding toη— which can be calculated from the Jack polynomialJ^η. The integral of the Euler classein the formula may be explicitly expressed in terms of Hodge integrals and the tangent weights of theT-representation Tan_η(Hilbⁿ(C²)).

Apart from

µHilbⁿ(C²)

1,0 and ^Hilb

n(C²)

g≥2,0 , all other degree 0 invariants of Hilbⁿ(C²) in positive genus vanish. Theorem 1 can be equivalently stated in the following form: the R-matrix of the T-equivariant Gromov-Witten theory ofHilbⁿ(C²) is uniquely determined from theT-equivariant genus 0 theory by the divisor equation and the degree 0 invariants in positive genus.

0.5. Lifting the triangle of correspondences. The calculation of theT-equivariant quantum cohomology ofHilbⁿ(C²)is a basic step in the proof [2, 29, 31] of the following triangle of equivalences:

Gromov-Witten theory ofC²×P¹

Quantum cohomology ofHilbⁿ(C²)

Donaldson-Thomas theory ofC²×P¹

Our second result is a lifting of the above triangle to all higher genera. The top vertex is replaced by theT-equivariant Gromov-Witten theory ofHilbⁿ(C²)in genusg withrinsertions. The bottom vertices of the triangle arenewtheories which are constructed here.

LetM_g,r be the moduli space of Deligne-Mumford stable curves of genusg withr markings.⁴ Let

C → M_g,r be the universal curve with sections

p₁, . . . ,p_r :M_g,r → C

4We will always assumegandrsatisfy thestabilitycondition2g−2 +r >0.

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associated to the markings. Let

π :C²× C → M_g,r

be the universallocal curveoverM_g,r. The torusTacts on theC² factor. The Gromov-Witten and Donaldson-Thomas theories of the morphismπ are defined by theπ-relativeT-equivariant virtual class of the universalπ-relative moduli spaces of stable maps and stable pairs.⁵

Theorem 2. For all generag ≥0, there is a triangle of equivalences ofT-equivariant theories:

Gromov-Witten theory of π :C²× C → M_g,r

Gromov-Witten theory ofHilbⁿ(C²) in genusg withrinsertions

Donaldson-Thomas theory of π :C²× C → M_g,r

The triangle of Theorem 2 may be viewed from different perspectives. First, all three vertices define CohFTs. Theorem 2 may be stated as simply an isomorphism of the three CohFTs. A second point of view of the bottom side of the triangle of Theorem 2 is as a GW/DT correspondence in families of 3-folds as the complex structure of the local curve varies. While a general GW/DT correspondence for families of 3-folds can be naturally formulated, there are very few interesting⁶ cases studied.

For a pointed curve of fixed complex structure (C, p₁, . . . , p_r), the triangle of Theorem 2 is a basic result of the papers [2, 29, 31] since C can be degenerated to a curve with all irreducible components of genus 0.

0.6. Crepant resolution. The Hilbert scheme of points ofC² is well-known to be a crepant resolution of the symmetric product,

:Hilbⁿ(C²) → Symⁿ(C²) = (C²)ⁿ/S_n, .

Viewed as anorbifold, the symmetric productSymⁿ(C²)has aT-equivariant Gromov-Witten theory with insertions indexed by partitions ofn. TheT-equivariant Gromov-Witten generating series⁷,

µ¹, µ², . . . , µ^rSymⁿ(C²)

g =

∞

X

b=0

µ¹, µ², . . . , µ^rSymⁿ(C²)

g,b u^b ∈Q[[u]], is a sum over the number of free ramification pointsbwith variableu.

The Frobenius structure determined by theT-equivariant genus0theory ofSymⁿ(C²)is semisimple, butnotconformal. Again, theR-matrix isnotdetermined by theT-equivariant genus 0theory

5While the triangle of equivalences originally included the Donaldson-Thomas theory of ideal sheaves, the theory of stable pairs [36] is much better behaved, see [26, Section 5] for a discussion valid forC²×P¹. We will use the theory of stable pairs here.

6The equivariant GW/DT correspondence is a special case of the GW/DT correspondence for families (and is well studied).

7A full discussion of the definition appears in Section 3.2.

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alone. The determination of theR-matrix ofSymⁿ(C²)is given by the following result parallel to Theorem 1.

Theorem 3. The R-matrix of the T-equivariant Gromov-Witten theory of Symⁿ(C²) is uniquely determined from theT-equivariant genus 0 theory by the divisor equation and all the unramified invariants,

µ¹, . . . , µ^rSymⁿ(C²)

g,0 ,

in positive genus.

For eachη∈Part(n), LetH^η_g be the moduli space ´etale covers with`(η)connected components⁸ of degrees

η₁, . . . , η_`(η) of a nonsingular genusg ≥2curve. The degree of the map

H_g^η → M_g is an unramified Hurwitz number. Let

H^η_g → M_g be the compactification by admissible covers.

Since the genus of the component of the cover corresponding to the partη_iisη_i(g−1) + 1, there is a Hodge bundle

E^∗ηi(g−1)+1→ H^η_g obtained via the corresponding component.

The simplest unramified invariants required in Theorem 3 are

Symⁿ(C²)

g≥2,0 = X

η∈Part(n)

1

|Aut(η)|

Z

H^η_g

`(η)

Y

i=1

e E^∗ηi(g−1)+1⊗Tan₀(C²) e(Tan₀(C²)) .

However, there are many further unramified invariants in positive genus obtained by including insertions. Unlike the case ofHilbⁿ(C²), the unramified invariants with insertions forSymⁿ(C²)do not all vanish.

In genus 0, the equivalence of theT-equivariant Gromov-Witten theories of Hilbⁿ(C²) and the orbifoldSymⁿ(C²)was proven⁹in [1]. Our fourth result is a proof of the equivalence for all genera.

Theorem 4. For all generag ≥0andµ¹, µ², . . . , µ^r ∈Part(n), we have µ¹, µ², . . . , µ^rHilbⁿ(C²)

g = (−i)^P^rⁱ⁼¹^`(µⁱ^)−|µⁱ^|

µ¹, µ², . . . , µ^rSymⁿ(C²) g

after the variable change−q=e^iu.

The variable change of Theorem 4 is well-defined by the following result (which was previously proven in genus 0 in [29]).

8For the definition ofH^η_g, we consider the parts ofηto beorderedand in bijection with the components of the cover.

9The prefactor(−i)^P^rⁱ⁼¹^`(µⁱ^)−|µⁱ^|was treated incorrectly in [1] because of an arthimetical error. The prefactor here is correct.

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Theorem 5. For all generag ≥0andµ¹, µ², . . . , µ^r ∈Part(n), the series¹⁰ µ¹, µ², . . . , µ^rHilbⁿ(C²)

g ∈Q(t₁, t₂)[[q]]

is the Taylor expansion inqof a rational function inQ(t₁, t₁, q).

Calculations in closed form in higher genus are not easily obtained. The first nontrivial example occurs in genus 1 for the Hilbert scheme of 2 points:

(0.8)

(2)Hilb²(C²)

1 =− 1

24

(t₁ +t₂)²

t₁t₂ · 1 +q 1−q.

While the formula is simple, our virtual localization [16] calculation of the integral is rather long.

Since (0.8) captures all degrees, the full graph sum must be controlled – we use the valuation at (t₁+t₂)as in [29].

The above calculation (0.8) for the Hilbert scheme of 2 points yields new Hodge integral calculations for the bielliptic locusH₁((2)²ⁿ), the moduli space of double covers of elliptic curves with 2nordered branch points,

H₁ (2)²ⁿ

→ M_1,2n.

Since the domain curves parameterized byH₁((2)²ⁿ)have genusn+ 1, there is a Hodge bundle E^∗n+1 → H1 (2)²ⁿ

.

By a direct application¹¹of Theorem 4,

∞

X

n=1

u²ⁿ⁻¹ (2n−1)!

Z

H1((2)²ⁿ)

λ_n+1λn−1 = i

24 ·1−e^iu 1 +e^iu

= 1

48u+ 1

576u³+ 1

5760u⁵+. . . .

Theuandu³ coefficients can be checked geometrically using the bielliptic calculations of [9]. The u⁵ coefficient has been checked in [39, Remark 5.14].

The corresponding series for highern,

(2,1ⁿ⁻²)Hilbⁿ(C²)

1 ∈Q(t₁, t₂, q),

very likely has a simple closed formula. We will return to these questions in a future paper.

0.7. Plan of proof. Theorems 2 and 4 are proven together by studying the R-matrices of all four theories. The R-matrix of the CohFT associated to the local Donaldson-Thomas theories of curves is easily proven to coincide with theR-matrix of theT-equivariant Gromov-Witten theory of Hilbⁿ(C²) determined in Theorem 1. Similarly, the R-matrices of the CohFTs associated to Symⁿ(C²)and the local Gromov-Witten theories of curves are straightforward to match (with determination by Theorem 3). The above results require a detailed study of the divisor equation in the four cases.

10As always,gandrare required to be in the stable range2g−2 +r >0.

11We follow the standard conventionλi=ci(En+1).

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The main step of the joint proof of Theorems 2 and 4 is to match theR-matrices of Theorems 1 and 3. The matter is non-trivial since the former is a function ofqand the latter is a function of u.

The matching after

−q=e^iu

requires an analytic continuation. Our method here is to express the R-matrices in terms of the solution of the QDE associated toHilbⁿ(C²). Fortunately, the analytic continuation of the solution of the QDE proven in [30] is exactly what is needed, after a careful study of asymptotic expansions, to match the twoR-matrices.

The addition of Symⁿ(C²)via Theorem 4 to the triangle of Theorem 2 yields a tetrahedron of equivalences ofT-equivariant theories (as first formulated in genus 0 in [1]).

@

Gromov-Witten theory ofHilbⁿ(C²) in genusgwithrinsertions

Orbifold Gromov-Witten theory ofSymⁿ(C²) in genusgwithrinsertions

Donaldson-Thomas theory of π:C²× C → Mg,r

Gromov-Witten theory of π:C²× C → Mg,r

Before the development of the orbifold Gromov-Witten theory ofSymⁿ(C²), a theory of Hurwitz- Hodge integrals was proposed by Cavalieri [3]. While the orbifold Gromov-Witten theory is formulated in terms of principalS_n-bundles over curves, Cavalieri’s theory is formulated in terms of the associated Hurwitz covers of curves. In fact, the virtual class of the orbifold theory ofSymⁿ(C²)exactly coincides with the Hodge integrand proposed by Cavalieri, so the two theories areequal. The orbifold vertex of the above tetrahedron may therefore also be viewed via Cavalieri’s definition.¹² 0.8. Acknowledgments. We are grateful to J. Bryan, A. Buryak, R. Cavalieri, T. Graber, J. Gu´er´e, F. Janda, Y.-P. Lee, H. Lho, D. Maulik, A. Oblomkov, A. Okounkov, D. Oprea, A. Pixton, Y. Ruan, R. Thomas, and D. Zvonkine for many conversations over the years related to the Gromov-Witten theory of Hilbⁿ(C²), 3-fold theories of local curves, and CohFTs. We thank A. Givental and C.

Teleman for discussions about semisimple CohFTs and O. Costin and S. Gautam for discussions about asymptotic expansions.

R. P. was partially supported by SNF-200020-162928, SNF-200020-182181, ERC-2012-AdG- 320368-MCSK, ERC-2017-AdG-786580-MACI, SwissMAP, and the Einstein Stiftung. H.-H. T.

12Our discussion concerns Cavalieri’s level(0,0)theory extended naturally over the moduli space of genusgcurves.

In genus0, Cavalieri [3] noted the equivalence of his theory to the Gromov-Witten vertex. He also defined theories of other levels(a, b)which do not exactly agree with the corresponding(a, b)-theories of the Gromov-Witten vertex (even in genus0). We do not explore here the interesting geometry of the(a, b)-level structure over the moduli space of genus gcurves.

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was partially supported by a Simons foundation collaboration grant and NSF grant DMS-1506551.

The research presented here was furthered during visits of H.-H. T. to ETH Z¨urich in March 2016 and March 2017. The project has received funding from the European Research Council (ERC) under the European Union Horizon 2020 Research and Innovation Program (grant No. 786580).

1. COHOMOLOGICAL FIELD THEORY

1.1. Definitions. The notion of a cohomological field theory (CohFT) was introduced in [20], see also [24]. We follow closely here the treatment of [35, Section 0.5], and we refer the reader also to the survey [33].

Letkbe an algebraically closed field of characteristic0. LetAbe a commutativek-algebra. Let Vbe a freeA-module of finite rank, let

η :V⊗V→A

be an even, symmetric, non-degenerate¹³ pairing, and let 1 ∈ V be a distinguished element. The data(V, η,1)is the starting point for defining a cohomological field theory. Given a basis{e_i}of V, we write the symmetric form as a matrix

η_jk =η(e_j, e_k). The inverse matrix is denoted byη^jk as usual.

A cohomological field theory consists of a systemΩ = (Ω_g,r)2g−2+r>0of elements Ω_g,r∈H^∗(M_g,r,A)⊗(V^∗)^⊗r.

We viewΩ_g,ras associating a cohomology class onM_g,rto elements ofVassigned to thermark- ings. The CohFT axioms imposed onΩare:

(i) Each Ωg,r isSr-invariant, where the action of the symmetric group Sr permutes both the marked points ofM_g,rand the copies ofV^∗.

(ii) Denote the basic gluing maps by

q :Mg−1,r+2 → M_g,r, qe:M_g₁_,r₁₊₁× M_g₂_,r₂₊₁ → M_g,r .

The pull-backsq^∗(Ω_g,r)andqe^∗(Ω_g,r)are equal to the contractions of Ωg−1,r+2 and Ω_g₁_,r₁₊₁⊗Ω_g₂_,r₂₊₁

by the bi-vector

X

j,k

η^jke_j⊗e_k inserted at the two identified points.

(iii) Letv₁, . . . , v_r ∈Vand letp:M_g,r+1 → M_g,rbe the forgetful map. We require Ω_g,r+1(v₁⊗ · · · ⊗v_r⊗1) =p^∗Ω_g,r(v₁⊗ · · · ⊗v_r),

Ω_0,3(v₁⊗v₂⊗1) =η(v₁, v₂).

13The pairingηinduces an isomorphism betweenVand the dual moduleV^∗=Hom(V,A).

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Definition 1.1. A systemΩ = (Ω_g,r)2g−2+r>0of elements

Ω_g,r ∈H^∗(M_g,r,A)⊗(V^∗)^⊗r

satisfying properties (i) and (ii) is acohomological field theory(CohFT). If (iii) is also satisfied,Ω is aCohFT with unit.

A CohFTΩyields aquantum product?onVvia

(1.1) η(v₁? v₂, v₃) = Ω_0,3(v₁⊗v₂⊗v₃).

Associativity of?follows from (ii). The element1∈Vis the identity for?by (iii).

A CohFTωcomposed only of degree 0 classes,

ω_g,r∈H⁰(M_g,r,A)⊗(V^∗)^⊗r ,

is called atopological field theory. Via property (ii),ω_g,r(v₁, . . . , v_r)is determined by considering stable curves with a maximal number of nodes. Every irreducible component of such a curve is of genus 0 with 3 special points. The value ofω_g,r(v₁ ⊗ · · · ⊗v_r) is thus uniquely specified by the values of ω_0,3 and by the pairing η. In other words, givenV and η, a topological field theory is uniquely determined by the associated quantum product.

1.2. Gromov-Witten theory. Let X be a nonsingular projective variety over C. The stack of stable maps¹⁴Mg,r(X, β)of genusg curves toX representing the classβ ∈ H2(X,Z)admits an evaluation

ev :M_g,r(X, β)→X^r and a forgetful map

ρ:M_g,r(X, β)→ M_g,r,

when 2g −2 +r > 0. TheGromov-Witten CohFT is constructed from the virtual classes of the moduli of stable maps ofX,

Ω^GW_g,r(v₁ ⊗...⊗v_r) = X

β∈H₂(X,Z)

q^βρ∗

ev^∗(v₁⊗...⊗v_r)∩[M_g,r(X, β)]^vir .

The ground ringAis theNovikov ring, a suitable completion of the group ring¹⁵associated to the semi-group of effective curve classes inX,

V=H^∗(X,Q)⊗_QA=H^∗(X,A),

the pairingηis the extension of the Poincar´e pairing, and1∈Vis the unit in cohomology.

1.3. Semisimplicity.

14See [10] for an introduction to stable maps.

15We take the Novikov ring withQ-coefficients.

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1.3.1. Classification. LetΩbe a CohFT with respect to(V, η,1). We are concerned here with theories for which the algebraVwith respect to the quantum product?defined by (1.1) is semisimple.

Such theories are classified in [40]. Specifically,Ωis obtained from the algebraVvia the action of anR-matrix

R ∈1+z·End(V)[[z]], satisfying the symplectic condition

R(z)R^?(−z) = 1.

Here, R^? denotes the adjoint with respect to η and 1 is the identity matrix. The explicit reconstruction of the semisimple CohFT from the R-matrix action will be explained below, following [35].

1.3.2. Actions on CohFTs. LetΩ = (Ω_g,r)be a CohFT with respect to(V, η,1). Fix a symplectic matrix

R∈1+z·End(V)[[z]]

as above. A new CohFT with respect to(V, η,1)is obtained via the cohomology elements RΩ = (RΩ)_g,r,

defined as sums over stable graphsΓof genusg withrlegs, with contributions coming from vertices, legs, and edges. Specifically,

(1.2) (RΩ)_g,r =X

Γ

1

|Aut(Γ)|(ι_Γ)_? Y

v

Cont(v)Y

l

Cont(l)Y

e

Cont(e)

!

where:

(i) the vertex contribution is

Cont(v) = Ω_g(v),r(v),

withg(v)andr(v)denoting the genus and number of half-edges and legs of the vertex, (ii) the leg contribution is

Cont(l) = R(ψl)

whereψ_lis the cotangent class at the marking corresponding to the leg, (iii) the edge contribution is

Cont(e) = η⁻¹−R(ψ_e⁰)η⁻¹R(ψ⁰⁰_e)^>

ψ⁰_e+ψ_e⁰⁰ .

Here ψ_e⁰ and ψ⁰⁰_e are the cotangent classes at the node which represents the edge e. The symplectic condition guarantees that the edge contribution is well-defined.

A second action on CohFTs is given by translations. As before, letΩbe a CohFT with respect to (V,1, η)and consider a power seriesT∈V[[z]]with no terms of degrees0and1:

T(z) =T2z²+T3z³+. . . , Tk ∈V.

A new CohFT wit respect to(V,1, η), denotedTΩ, is defined by setting (1.3) (TΩ)_g,r(v₁⊗. . .⊗v_r) =

∞

X

m=0

1

m!(p_m)_?Ω_g,r+m(v₁⊗. . .⊗v_r⊗T(ψ_r+1)⊗. . .⊗T(ψ_r+m))

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where

p_m :M_g,r+m → M_g,r is the forgetful morphism.

1.3.3. Reconstruction. With the above terminology understood, we can state the Givental-Teleman classification theorem [40]. FixΩa semisimple CohFT with respect to(V,1, η), and writeωfor the degree0topological part of the theory. Given any symplectic matrix

R∈1+z·End(V)[[z]]

as above, we form a power seriesTby plugging1 ∈ VintoR, removing the free term, and multi- plying byz:

T(z) =z(1−R(1))∈V[[z]].

Givental-Teleman classification. There exists a unique symplectic matrixRfor which Ω = RTω .

A proof of uniqueness can be found, for example, in [25, Lemma 2.2].

1.4. Targets Hilbⁿ(C²) and Symⁿ(C²). The CohFTs determined by the T-equivariant Gromov- Witten theories ofHilbⁿ(C²)andSymⁿ(C²)are based on the algebras

(1.4) A=Q(t₁, t₂)[[q]], Ae =Q(t₁, t₂)[[u]], and the corresponding free modules

(1.5) V=Fⁿ⊗_QA, Ve =Fⁿ⊗_QAe,

whereFⁿis the Fock space with basis indexed by Part(n). While the inner product for the CohFT determined byHilbⁿ(C²)is the inner product defined in (0.5),

η(µ, ν) = hµ|νi= (−1)^|µ|−`(µ) (t1t2)^`(µ)

δ_µν z(µ),

the inner product for the CohFT determined bySymⁿ(C²)differs by a sign, eη(µ, ν) = (−1)^|µ|−`(µ)hµ|νi= 1

(t₁t₂)^`(µ) δ_µν z(µ).

Since the CohFTs are both semisimple, we obtain unique R-matrices from the Givental-Teleman classification,

R^Hilb and R^Sym.

After rescaling the insertionµin the T-equivariant Gromov-Witten theory ofSymⁿ(C²) by the factor(−i)^|µ|−`(µ), the inner products match. Let

(1.6) |eµi= (−i)^`(µ)−|µ||µi ∈V.e

The linear transformationV→Ve defined by

µ7→µe

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respects the inner products η and η. Moreover, the correspondence claimed by Theorem 4 thene simplifies to

µ¹, µ², . . . , µ^rHilbⁿ(C²)

g =

eµ¹,µe², . . . ,µe^rSymⁿ(C²) g

after the variable change−q=e^iu.

To prove Theorems 2 and 4, we will construct an operator series Rdefined over an open set of values ofq ∈Ccontainingq = 0andq=−1with the two following properties

• R =R^Hilb nearq = 0,

• R =R^Sym nearq=−1after the variable change−q=e^iu.

The operatorR is obtained from the asymptotic solution to the QDE of Hilbⁿ(C²). The existence of R is guaranteed by results of [7, 11]. The asymptotic solution is shown to be the asymptotic expansion of an actual solution to the QDE. The behavior ofRnearq=−1is then studied by using the solution to the connection problem in [30].

2. THER-MATRIX FORHilbⁿ(C²)

2.1. The formal Frobenius manifold. We follow the notation of Section 1.4 for the CohFT determined by theT-equivariant Gromov-Witten theory ofHilbⁿ(C²). The genus 0 Gromov-Witten potential,

F^Hilb₀ ⁿ^(C²⁾(γ) =

∞

X

d=0

q^d

∞

X

r=0

1 r!

γ, . . . , γ

| {z }

r

Hilbⁿ(C²)

0,d , γ ∈V

is a formal series in the ringA[[V^∗]]where

A=Q(t₁, t₂)[[q]].

TheT-equivariant genus 0 potentialF^Hilb₀ ⁿ^(C²⁾defines a formal Frobenius manifold (V, ?, η)

at the origin ofV.

A basic result of [29] is therationalityof the dependence onq. Let Q=Q(t₁, t₂, q)

be the field of rational functions. By [29, Section 4.2], F^Hilb₀ ⁿ⁽^C²⁾(γ)∈Q[[V^∗]].

We will often view the formal Frobenius manifold (V, ?, η) as defined over the fieldQ instead of the ringA.

2.2. Semisimplicity. The formal Frobenius manifold (V, ?, η) is semisimple at the origin after extendingQto the algebraic closureQ. The semisimple algebra

(Tan₀V, ?₀, η)

is the small quantum cohomology of Hilbⁿ(C²). The matrices of quantum multiplication of the algebra Tan₀(V)have coefficients in Q. The restriction to q = 0is well-defined and semisimple with idempotents proportional to the classes of theT-fixed points of Hilbⁿ(C²). The idempotents of the algebra Tan₀(V) may be written in the Nakajima basis after extending scalars to Q. The

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algebraic closure¹⁶is required since the eigenvalues of the matrices of quantum multiplication lie in finite extensions ofQ.

Since the formal Frobenius manifold(V, ?, η)is semisimple at the origin, the full algebra

(2.1) (Tan0(V)⊗QQ[[V^∗]], ?, η)

is also semisimple (after extending scalars toQ). The idempotents of the algebra Tan₀(V)can be lifted to all orders to obtain idempotents_µ of the full algebra (2.1) parameterized by partitionsµ corresponding to theT-fixed points ofHilbⁿ(C²).

For the formal Frobenius manifold (V, ?, η), there are two bases of vector fields which play a fundamental role in the theory:

• theflatvector fields∂_µcorresponding the basis elements|µi ∈V,

• thenormalized idempotentse_µof the quantum product?.

The normalized idempotents are proportional to the idempotents and satisfy η(e_µ,e_ν) = δ_µν.

LetΨbe change of basis matrix,

Ψ^ν_µ=η(e_ν, ∂_µ) ∈ Q[[V^∗]]

between the two frames. Uniquecanonical coordinatesat the origin, u_µ∈Q[[V^∗]] _µ∈Part(n)

can be found satisfying

u_µ(0) = 0, ∂

∂u_µ =_µ. By [11, Prop. 1.1], the quantum differential equation¹⁷

(2.2) ∇zeS= 0

has a formal fundamental solution¹⁸in the basis of normalized idempotents of the form

(2.3) eS=R(z)e^u/z

Here,∇_zis the Dubrovin connection,

(2.4) R(z) =1+R₁z+R₂z²+R₃z³+. . .

is a series of|Part(n)| × |Part(n)|matrices starting with the identity matrix1, anduis a diagonal matrix with the diagonal entries given by the canonical coordinatesu_µ.

16More precisely, coefficients lie in the ringQ∩Asince the eigenvalues lie inA.

17We follow the notation of the exposition in [21]. See also [7].

18Often the solution is written in the basis of flat vector fields as S= Ψ⁻¹R(z)e^u/z.

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We viewR(z) as an End(V)-valued formal power series in z written in the basis of normalized idempotents. The seriesR(z)satisfies the symplectic condition

(2.5) R^†(−z)R(z) =1,

whereR^†(z)is the adjoint ofR(z)with respect to the inner productη. A detailed treatment of the construction and properties ofR(z)can found in [21, Section 4.6 of Chapter 1].

AnR-matrixassociated to the formal Frobenius manifold(V, ?, η)is a matrix series of the form (2.4) which determines a solution (2.3) of the quantum differential equation (2.2)andsatisfies the symplectic condition (2.5). The two basic properties ofR-matrices which we will use are:

(i) There exists a uniqueR-matrixR^Hilbassociated to(V, ?, η)with coefficients inA[[V^∗]]which generates theT-equivariant Gromov-Witten theory ofHilbⁿ(C²)in all higher genus.

(ii) TwoR-matrices associated to(V, ?, η)with coefficients inA[[V^∗]]must differ by right multiplication by

exp

∞

X

j=1

a2j−1z^2j−1

!

where eacha2j−1 is adiagonalmatrix with coefficients inA.

Property (i) is a statement of the Givental-Teleman classification of semisimple CohFTs applied to theT-equivariant Gromov-Witten theory ofHilbⁿ(C²). For properties (i) and (ii), we have left the field Q and returned to the ringA because the CohFT associated to Hilbⁿ(C²) is defined over A.

In Section 2.5, the unique R-matrix of property (i) will be shown to actually have coefficients in Q[[V^∗]].

2.3. Divisor equation. The formal Frobenius manifold(V, ?, η)is actually well-defined away from the origin along the line with coordinatetdetermined by the vector

|2,1ⁿ⁻²i ∈V.

At the point−t|2,1ⁿ⁻²i ∈V, the potential of the Frobenius manifold is F^Hilb₀ ⁿ⁽^C²⁾

−t|2,1ⁿ⁻²i+γ

=

∞

X

d=0

q^d

∞

X

r=0

∞

X

m=0

t^m r!m!

γ, . . . , γ

| {z }

r

, D, . . . , D

| {z }

m

Hilbⁿ(C²) 0,d

=

∞

X

d=0

q^d

∞

X

r=0

∞

X

m=0

(dt)^m r!m!

γ, . . . , γ

| {z }

r

Hilbⁿ(C²)

0,d ,

=

∞

X

d=0

q^de^dt

∞

X

r=0

1 r!

γ, . . . , γ

| {z }

r

i^Hilb_0,dⁿ⁽^C²⁾

forγ ∈ Vand D = − |2,1ⁿ⁻²ias in (0.4). We have used thedivisor equationof Gromov-Witten theory in the second equality. The potential near the point −t|2,1ⁿ⁻²i ∈ Vis obtained from the potential at0∈Vby the substitution

F^Hilb₀ ⁿ^(C²⁾

−t|2,1ⁿ⁻²i+γ

=F^Hilb₀ ⁿ^(C²⁾(γ) _q7→qe_t. The Frobenius manifold is semisimple at all the points−t|2,1ⁿ⁻²i ∈V.

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Let Ωbe the CohFT associated to the T-equivariant Gromov-Witten theory of Hilbⁿ(C²). The genus 0 data ofΩis exactly given by the formal Frobenius manifold (V, ?, η)at the origin. Define the−t|2,1ⁿ⁻²i-shifted CohFT by

Ω^−t|2,1_g,r ⁿ⁻²ⁱ(γ⊗ · · · ⊗γ) = X

m≥0

t^m

m!ρ^r+m_r∗ Ω_g,r+m(γ⊗ · · · ⊗γ⊗D^⊗m)

= Ω_g,r(γ⊗ · · · ⊗γ) _q7→qe_t. Here,ρ^r+m_r is the forgetful map which drops the lastmmarkings,

(2.6) ρ^r+m_r :M_g,r+m → M_g,r.

We have again used the divisor equation of Gromov-Witten theory in the second equality.

LetR^Hilbbe the uniqueR-matrix associated to theT-equivariant Gromov-Witten theory ofHilbⁿ(C²).

The shifted CohFTΩ^−t|2,1g,r ⁿ⁻²ⁱis obtained from the semisimple genus 0 data F^Hilb₀ ⁿ⁽^C²⁾

−t|2,1ⁿ⁻²i+γ by the uniqueR-matrix

R^Hilb

−t|2,1ⁿ⁻²i+γ .

On the other hand, theR-matrix

R^Hilb _q7→qe_t

also generates Ω^−t|2,1g,r ⁿ⁻²ⁱ from the same semisimple genus 0 data. By the uniqueness of the R- matrix in the Givental-Teleman classification,

R^Hilb

−t|2,1ⁿ⁻²i+γ

=R^Hilb q7→qe^t. We find the following differential equation:

(2.7) − ∂

∂tR^Hilb=q ∂

∂qR^Hilb.

The differential equation is anextra condition satisfied by R^Hilbⁿ⁽^C²⁾ in additionto determining a solution (2.3) of the quantum differential equation (2.2) and satisfying the symplectic condition (2.5).

Proposition 6. TwoR-matrices associated to(V, ?, η)with coefficients inA[[V^∗]]whichbothsatisfy the differential equation

−∂

∂tR=q ∂

∂qR must differ by right multiplication by

exp

∞

X

i=1

a2j−1z^2j−1

!

where eacha2j−1 is adiagonalmatrix with coefficients inQ(t₁, t₂).

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Proof. Let eR and Rb be two R-matrices associated to (V, ?, η) which both satisfy the additional differential equation. By property (ii) ofR-matrices associated to(V, ?, η),

Re=Rb·exp

∞

X

i=1

a_2j−1z^2j−1

!

where eacha2j−1 is a diagonal matrix with coefficients in A=Q(t₁, t₂)[[q]]. Differentiation yields

−∂

∂teR = −∂

∂tRb·exp

∞

X

i=1

a_2j−1z^2j−1

!

−Rb· ∂

∂texp

∞

X

i=1

a_2j−1z^2j−1

!

= −∂

∂tRb·exp

∞

X

i=1

a2j−1z^2j−1

!

= q ∂

∂qbR·exp

∞

X

i=1

a2j−1z^2j−1

! .

The second equality uses the independence of a2j−1 with respect to the coordinate t. The third equality uses the differential equation forR. Application of the operatorb q_∂q^∂ toReyields

q ∂

∂qRe =q ∂

∂qRb·exp

∞

X

i=1

a2j−1z^2j−1

!

+bR·q ∂

∂q exp

∞

X

i=1

a2j−1z^2j−1

! .

Finally, using the differential equation foreR, we find bR·q ∂

∂qexp

∞

X

i=1

a2j−1z^2j−1

!

= 0. By the invertibility ofR,b

q ∂

∂qexp

∞

X

i=1

a2j−1z^2j−1

!

= 0.

Hence, the matricesa2j−1are independent ofq.

2.4. Proof of Theorem 1. Since the CohFT Ωg,r associated to the T-equivariant Gromov-Witten theory ofHilbⁿ(C²)is defined over the ring

A=Q(t₁, t₂)[[q]], we can consider the CohFTΩ_g,r

_q=0defined over the ringQ(t₁, t₂). The CohFTΩ_g,r

_q=0is semisimple withR-matrix given by theq= 0restriction,

R^Hilb q=0,

of theR-matrix ofΩg,r. As a corollary of Proposition 6, we obtain the following characterization of R^Hilb.

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Proposition 7. R^Hilb is the uniqueR-matrix associated to(V, ?, η)which satisfies

−∂

∂tR^Hilb=q ∂

∂qR^Hilb

andhasq= 0restriction which equals theR-matrix of the restrictedCohFTΩ_g,r q=0. SinceΩg,r

q=0concerns constant maps of genusgcurves toHilbⁿ(C²), the CohFT can be written explicitly in terms of Hodge integrals. The moduli space of maps in degree 0 is

M_g,r(Hilbⁿ(C²),0) =M_g,r×Hilbⁿ(C²) with virtual class

E^∗g⊗Tan(Hilbⁿ(C²)).

Since the Hodge bundle is pulled back fromM_1,1in genus 1 andM_gin higher genera, all invariants in positive genus vanish other than

µHilbⁿ(C²)

1,0 = X

η∈Part(n)

µ|_η Z

M1,1

e(E^∗1⊗Tan_η(Hilbⁿ(C²))) e(Tan_η(Hilbⁿ(C²))) , (2.8)

Hilbⁿ(C²)

g≥2,0 = X

η∈Part(n)

Z

Mg

e E^∗g⊗Tan_η(Hilbⁿ(C²)) e(Tan_η(Hilbⁿ(C²))) .

Theorem 1 follows from Proposition 7 together with the fact that (2.8) is the complete list of degree

0 invariants.

The uniqueR-matrixR^Hilb

q=0 of the CohFT Ω_g,r

q=0 can be explicitly written after the coordinates onVare set to 0. The formula is presented in Section 6.1.

2.5. Proof of Theorem 5. As we have seen in Section 2.3 using the divisor equation, the dependence of the potential of the formal Frobenius manifold(V, ?, η)at the origin,

F^Hilb₀ ⁿ⁽^C²⁾∈Q[[V^∗]], along−t|2,1ⁿ⁻²ican be expressed as

F^Hilb₀ ⁿ⁽^C²⁾=

F^Hilb₀ ⁿ⁽^C²⁾ t=0

q7→qe^t.

The same dependence ontthen also holds for the matrices of quantum multiplication for(V, ?, η) and their common eigenvalues.

In the procedure¹⁹for constructing anR-matrix associated to(V, ?, η), we can take all the unde- termined diagonal constants for R2j−1 equal to 0 for all j. The resulting associated R-matrixR^# will satisfy

−∂

∂tR^#=q ∂

∂qR^#

since the sameq7→qe^tdependence ontholds for all terms in the procedure. By Proposition 6,

(2.9) R^Hilb=R^#·exp

∞

X

i=1

a2j−1z^2j−1

! ,

19See [21, Section 4.6 of Chapter 1]