射影直線上的Gromov-Witten理論

國立臺灣大學理學院數學系 碩士論文

Department of Mathematics College of Science

National Taiwan University Master Thesis

射影直線上的 Gromov-Witten 理論

The Gromov-Witten theory of

賴冠文 Kuan-Wen Lai

指導教授：林惠雯 教授 Advisor: Professor Hui-Wen Lin

中華民國 100 年 7 月

謝辭

這篇論文雖然沒有足以為道的貢獻，卻已經讓我學到了很多。欠缺基 礎的我研讀文獻的過程滿是胡亂摸索，常常以為觸及了什麼，過一陣子 才驚覺自己正身陷於錯誤的理解中。跌跌撞撞的最後總算能夠約略掌握 目的與動機的全貌，而這時我早已結束了論文口詴、學期的課程以及法 國的暑期學校兼研討會──那是一趟對我影響深遠的旅程。撰寫論文的 過程短短半年，其中有太多恩人在各個層面伴演了關鍵的角色。

知識不存在於特定的個人、文章或是任何的討論、演說，它即是這一 切動靜本身，獲得它的方法就是讓自己融入這一切。令我深刻體會到這 點的，是兩年來不厭其煩地引導我、督促我並改變我的林惠雯教授，感 謝您無論在什麼方面總是提醒我我應該重視什麼。

感謝王金龍教授，您所豎立的目標與營造的學術氛圍讓來自物理系的 我能毫無顧忌地投入其中──雖然我這個人總是顯得異常緊繃。這裡我 要同時感謝賀培銘教授，因為您讓我知道進入數學系以後應該去敲哪些 教授的門。

李元斌教授與莊武諺教授，謝謝您們擔任我的口詴委員。另外李元斌 教授在法國的指導與照顧讓我獲益良多。

特別感謝翁秉仁老師，您的拓樸學課程讓我見識了一場又一場對學問 細緻的組織與呈現。

感謝王賜聖學長時常給予有用的建議。廖宏仁，每次問你問題總是能 獲得令人滿意的解答。另外，蔡政江…你果真是個數學奇才！

謝謝雅真助教，沒有妳的「 」這份論文無以完成。

最後我要感謝父母一直以來讓我走自己的夢。還有文窈，謝謝妳無時 無刻的陪伴，以及，透過妳那人類學的視野，讓知識之於我有著多樣的 樣貌與想像，謝謝妳讓我的世界充滿色彩。

兩年間，我在物理與數學、古典與當代之間擺盪，這是激烈的拉扯，

亦是令人興奮的奔走。碩士論文的完成一方面宣告它的完結，另一方面 也敲開了一扇門。我要延續這份熱忱，在學問的宇宙中不停蹄地拓自己 見識的荒。

2011.07.27

摘要

在這篇文章中，我概述了關於 Gromov-Witten 不變量與 Hurwitz 數之 間如何建立對應的工作，以及詳細探討 Toda 階序的 Hirota 方程。該階 序能夠提供相當程度的遞迴關係以計算射影直線上的 Gromov-Witten 不 變量。主要的參考文獻為 A. Okounkov 與 R. Pandharipande 的一系列論 文[11, 12]。

關鍵辭：ELSV 方程，Gromov-Witten 理論，Hurwitz 數， -函數，Toda 階序，完備輪換，偏移對稱函數，無限維楔表示論。

Abstract

In this article, I would like to outline the work about the correspondence between Gromov-Witten invariants and Hurwitz numbers, and concentrate mainly on the detailed study of Hirota equations for the Toda hierarchy which provides certain recurrence relations for relative Gromov-Witten invariants of . The papers of A. Okounkov and R. Pandharipande [11, 12]

are the main sources of my study.

Keywords: completed cycle, ELSV formula, Gromov-Witten theory, Hurwitz number, infinite wedge representation, shifted symmetric function, -function, Toda hierarchy.

Contents

Acknowledgments i

Abstract ii

Table of contents iii

Introduction 1

1. Gromov-Witten invariants and Hurwitz numbers 2 1.1. Gromov-Witten invariants . . . 2 1.2. Hurwitz numbers . . . 4 1.3. Completed cycles . . . 7

2. The operator formalism 9

2.1. The Fock representations. . . 9 2.2. Boson-Fermion correspondence . . . 12 2.3. The operatorsE . . . 17 3. The Gromov-Witten/Hurwitz correspondence 18 3.1. Hodge integrals and equivariant n+ m-point functions . . 18 3.2. The operator formula for Hodge integrals. . . 23 3.3. Special GW/H correspondence. . . 25 3.4. Full GW/H correspondence . . . 27

4. Main results 31

4.1. Hirota equations for the Toda hierarchy . . . 32 4.2. The Toda hierarchy in the GW-theory . . . 33

References 40

Introduction

In the absolute Gromov-Witten theory of ⁿ, the localization technique is applied to compute the 1-point invariants with descendents. For multi- point cases, the divisor relation [7] was used to reduce them to the 1-point case [6]. In the relative theory, however, the computation becomes rather difficult even for target space ¹.

The main purpose of this article is to study some recurrence relations [11] for the stationary Gromov-Witten invariants of ¹ relative to 0 and∞, which is a conclusion of a general correspondence between the Gromov- Witten invariants and the Hurwitz numbers proved in the same paper.

The Hurwitz numbers enumerates the numbers of covers with assigned ramification conditions over a smooth target curve X. It has a character- theoretic expression, more precisely, an expression in the space of shifted symmetric functions. The definition of the Hurwitz numbers could be ex- tended over the space of such functions, which provides a necessary plat- form for establishing the GW/H correspondence.

The construction of the GW/H correspondence starts with the degenera- tion formula with some reinterpretation of the 0-point factors by the Hur- witz numbers:

⟨∏n

i=1

τki(ω), η¹, . . . , η^m

⟩•X

d

= ∑

|µ¹|=···=|µⁿ|=d

H_d^X(µ¹, . . . , µⁿ, η¹, . . . , η^m)

∏n i=1

z(µⁱ)⟨

µⁱ, τki(ω)⟩_•X .

Then the remaining work is to show that the right hand side can be inte- grated to form the required (extended) Hurwitz numbers. The whole pro- cess is taken within the function space mentioned above.

However, in the above strategy, a special case is needed and is refered to [12]. The special correspondence requires the method about the equivariant technique as well as the operator formalism, in which the ELSV formula [1] plays an essential role. Actually, the ELSV formula equates the simple Hurwitz numbers with the Hodge integrals up to a multiple of constants, thus can be seen as a toefold of the connection between the Gromov-Witten theory and the Hurwitz numbers.

After the construction of the GW/H correspondence, we then get into the main part: the 2-Toda hierarchy as well as the recurrence relations for the GW-invariants of ¹ relative to two points. As a consequence of the com- mutativity of specific operators, the τ-function introduced in the operator formalism satisfies a series of equations called 2-Toda hierarchy. On the other hand, a generating funciton for the relative GW-invariants will arises

as a special case of theτ-function, and the recurrence relations are given in fact by the Toda hierarchy.

Organization of the paper. Section 1 starts with the definition of Gromov- Witten invariants, including a word on the disconnected theory. Next the definition of Hurwitz numbers is recalled, which will be translated into the representation-theoretical setting to lead in the notion of completed cycles.

Some motivations about introducing the completed cycles are organized at the end.

Section 2, as a set-up of the later sections, is a brief summary of the oper- ator formalism. Infinite wedge space, Murnaghan-Nakayama rule together with the notions and relations of some important operators, say, bosons, fermions and the operatorsE, are the main contents of this section.

Section 3 is devoted to the GW/H correspondence, whose proof is split- ted into two stages: the special correspondence and the full correspon- dence. The special correspondence, derived from the localization formula, is worked out essentially through the operator formalism. Here some com- putational details, especially on how the localization formula works, will be given; The full correspondence is a conclusion of the degeneration formula, where the special case provides as a sufficient constraint to confirm that the degeneration formula actually gives the exact answer.

Section 4 is an appication of the GW/H correspondence. First a general τ-function is introduced and is proved to satisfy the Hirota equations for the Toda hierarchy. Then we show that a generating function for the relative GW-invariants over two points of ¹ is a special case of theτ-function, so that the Toda hierarchy would provides a recursive recipe for computing those invariants. I this part I’d try to work out the computation details that were omitted in the source paper.

1. Gromov-Witten invariants and Hurwitz numbers

1.1. Gromov-Witten invariants. Fix a smooth projective curve X. Let Mg,n(X, d) be the moduli space of genus g, degree d stable maps with n marked points and target space X. Also let ev_i : M_g,n(X, d) −→ X be the evaluation map at the i-th marked point.

We take into account two kinds of classes in A¹(M_g,n(X, d)). One is ev^∗_i(ω) withω ∈ A¹(X) being the Poincar´e dual of the point class, the other is the first Chern classψiof the i-th cotangent bundle on Mg,n(X, d). The stationary Gromov-Witten invariants with descendents (”GW-invariants” for short) are

defined to be

⟨∏n

i=1

τki(ω)

⟩◦X

g,d

=

∫

[Mg,n(X,d)]^vir

∏n i=1

ψ^kiev^∗_i(ω).

Here the upperright circle is used to emphasize that we’re considering con- nected domain curves.

1.1.1. The disconnected theory. The moduli space of stable maps with pos- sibly disconnected domain, roughly speaking, can be thought of as a Deligne- Mumford stack in the following way:

M^•_g,[n](X, d) = ⨿

{(gi,[ni],di)}∈Part[g,[n],d]



∏

i

M_gi,[ni](X, di)



 /

Aut{(gi, [ni], di)}, where [n]= {1, . . . , n}, and

∑

i

2g_i− 2 = 2g − 2, ⨿

i

[n_i]= [n], ∑

i

d_i = d.

The quotient by Aut{(gi, [ni], di)} is necessary since nontrivial stabilizers and repetition of moduli points occurs whenever there’re repeated copies in the direct product. Principally, the generating function of disconnected GW- invariants over all possible data is the exponential function with variables being the connected ones.

1.1.2. Relative Gromov-Witten invariants. Consider a branched covering p : C −→ X of degree d. We define a pro f ile over a point q ∈ X to be a partitionη of d obtained from the multipilicities of p⁻¹(q).

Fix m points q1, . . . , qm ∈ X and partitions η¹, . . . , η^m of degree d. The moduli space M_g,n(X, η¹, . . . , η^m) parameterizes all genus g, n-pointed rela- tive stable maps toward X with profilesηⁱ at q_i, i = 1, . . . , m. The relative GW-invariants are defined by

⟨∏n

i=1

τki(ω), η¹, . . . , η^m

⟩◦X

g,d

=

∫

[Mg,n(X,η¹,...,η^m)]^vir

∏n i=1

ψ^k_iev^∗_i(ω).

The moduli spaces M^•_g,n(X, η¹, . . . , η^m) and invariants⟨∏n

i=1τki(ω), η¹, . . . , η^m⟩_•X

g,d

in the disconnected notion are similarly defined.

Finally, I would like to mention that the expected dimension of the rela- tive moduli space, whether disconnected or not, is

2g− 2 + n + d(2 − 2g(X)) −

∑n i=1

(d− l(ηⁱ)), where l(η) stands for the length of the partition η.

1.2. Hurwitz numbers. The Hurwitz number H_d^X(η¹, . . . , η^m) is defined to be the number of isomorphism classes of coverings, possibly disconnected, over X which are unbranched except over q₁, . . . , qmwith profilesη¹, . . . , η^m, respectively. The countings are weighted by 1/|Aut(−)|.

In fact, for the dimension being zero, the moduli space M^•_g,0(X, η¹, . . . , η^m) with no marked point parametrizes the coverings satisfying the Riemann- Hurwitz formula

(1.1) 2g− 2 + d(2 − 2g(X)) =

∑m i=1

(d− l(ηⁱ)).

Hence the Hurwitz number is exactly (1.2) H^X_d(η¹, . . . , η^m)= ⟨

η¹, . . . , η^m⟩_•X

d .

1.2.1. Description by permutations. For convenience let X = ¹ at this stage.

Let S_d be the d-th symmetric group. For any partitionη of degree d, we can associate it with the conjugacy class C_ηin S_dof the corresponding cycle type. The Hurwitz number has the following equivalent description

H^_d¹(η¹, . . . , η^m)= |{(s1, . . . , sm)∈∏m

i=1C_ηi : s₁· · · sm= 1}|

|Sd| = d! .

This follows from the one-to-one correspondence between the isomorphism classes of Hurwitz coverings and the conjugate classes of the above m- tuples (si) as in the following:

Given a Hurwitz covering C → ^{p 1} with profilesη¹, . . . , η^m, let C^◦ → U be the associated cover of U := ¹\{q1, . . . , qm}. We’ve known that π1(U)=

⟨γ1, . . . , γm :γ1· · · γm = 1⟩, where each γi is a generator around the punc- tured point qi. For each i, consider a neighborhood V around qi which is small enough such that p maps each connected sheet of p⁻¹(V) isomorphi- cally toward V, then the liftings ofγi will associate a permutation si of the sheets over V. Thus the cover C → ¹ will give us an m-tuple (s₁, . . . , sm) up to conjugation, and γ1· · · γm = 1 imples s1· · · sm = 1. It’s straightfor- ward to see that their cycle types coincide with the original profile data.

Note that an isomorphism f between two coverings C → ¹and C^′ → ¹ induces a conjugation between the m-tuples (s_i) and f⁻¹(s^′_i) f thus a conju- gation between (s_i) and (s^′_i).

Conversely, assume an m-tuple (s₁, . . . , sm) ∈ ∏

iC_ηi is given. Let ˜U be the universal covering of U and recall thatπ1(U)  Deck( ˜U/U). Also let

[d]= {1, . . . , d} be a set equipped with π1(U)-action given by the homomor- phism

π1(U)−→ Sd

γi 7−→ si.

A Hurwitz cover of ¹ with profiles η¹, . . . , η^m can be constructed by ex- tending the following cover of U to ¹:

U˜ ×_π1(U)[d]−→ U

under the equivalence relation (xγ⁻¹, k) ∼ (x, γk). The isomorphism class of the resulting covering is independent of the conjugation of (s1, . . . , sm).

The maps constructed above are converse to each other.

Consider the S_d-action on∏

iC_ηi by conjugation: g· (si)= g⁻¹(s_i)g. Let Orb(s_i) be the orbit and Stab(s_i) be the stablizer of (s_i) under the action.

By the above correspondence, the countings for both sides regardless of the weights can be explicitly written down as

∑

Isom. class of C→¹

1= ∑

(si)∈∏

iC_ηi

∏

isi=1

1

|Orb(si)|.

Given a covering C → ¹ and a corresponding (s_i), observe that, using the above construction, there’s a one-to-one correspondence between the sets Aut(C/¹) and Stab(s_i). It follows that

∑

Isom. class of C→¹

1

|Aut(C/¹)| = ∑

(si)∈∏

iC_ηi

∏

isi=1

1

|Orb(si)| · |Stab(si)| = ∑

(si)∈∏

iC_ηi

∏

isi=1

1 d!,

which is exactly the required identity.

1.2.2. Burnside formula. Counting the number of the elements as above in the symmetric group allows another viewpoint from the representation theory. Recall the following standard correspondences:

‘A partitionλ of d ’

←→ ‘A conjugacy class Cλ ⊂ Sd’

←→ ‘An irreducible representation ρλ : S_d → End(Vλ) ’.

We’ll usually abuse notations to denote by dimλ the dimension of Vλ and byλ the irreducible representation ρλ.

Let c_η =∑

s∈Cη s. It is easy to get that (1.3) H^_d¹(η¹, . . . , η^m)= 1

d![1_Sd]∏

c_ηⁱ = 1

(d!)²Tr_QSd(∏

c_ηⁱ) ,

where [1_Sd] stands for the operation to capture the coefficient of the identity element and Tr_QSd means the trace for the regular representation.

Now given an irreducible representationλ, for each partition η, let fη(λ) be the eigenvalue ofρλ(c_η). By the Wedderburn’s structure theoremQSd ≃

⊕

λV_λ^{⊕ dim λ}, whereλ runs through all non-equivalent irreducible represen- tations of S_d.

Tr_QSd(

∏m i=1

c_ηⁱ)= ∑

λ

(dimλ) TrV_λ





∏m i=1

c_ηⁱ



 =∑

λ

(dimλ)²

∏m i=1

f_ηⁱ(λ).

We plug this into (1.3) to yield

H^_d¹(η¹, ..., η^m)= ∑

|λ|=d

(dimλ d!

)2∏m

i=1

f_ηⁱ(λ).

So the Hurwitz numbers can be expressed in terms of functions of partitions under the Fourier transform:

Z_d −→ Q^P(d) : c_η7→ f_η,

where Z_dis the center ofQSdandP(d) stands for the set of partitions of d.

What we’ve proved is a special case of the Burnside formula:

(1.4) H_d^X(η¹, ..., η^m)= ∑

|λ|=d

(dimλ d!

)_{2−2g(X) m}∏

i=1

f_ηⁱ(λ).

Having this in hand, we can generalize the definition of the Hurwitz num- bers. It is done by first generalizing f_ηto be a function onP =⊕

d≥0P(d).

Note that

f_η(λ) = TrV_λ(c_η) dimλ =

∑

s∈CηTr_Vλ(s)

dimλ = |Cη| χ^λ_η dimλ

whereχ^λ_ηis the value of the character ofλ at the class Cη. Thus an extended function f_η ∈ Q^P can be defined by

f_∅ ≡ 1; f_η(λ) = 0 if |η| > |λ|

(1.5)

f_η(λ) = (|λ|

|η|

)

|C_η| χ^λ_η

dimλ if 0< |η| ≤ |λ|.

Hence the Fourier transform can be extended to

(1.6) ⊕

d≥0

Z_d −→ Q^P : c_η 7→ f_η.

And the Hurwitz number can be generalized to profiles with arbitrary de- grees keeping the form (1.4).

Remark. Given any partition µ, we denote by mi(µ) the number of i’s ap- pearing inµ. Suppose there’re partitions η¹, . . . , η^m with degree ≤ d. Let η¹, . . . , η^m be the partitions obtained fromη¹, . . . , η^m by adding 1’s until all the degrees are equal to d. Then by using (1.5) as well as a direct computa- tion, we can get

H^X_d(η¹, . . . , η^m)=





∏m i=1

(m₁(ηⁱ) m1(ηⁱ)

)H^Xd(η¹, . . . , η^m).

1.3. Completed cycles. The shifted action of Sd on Q[x1, . . . , xd] is by permuting the shifted variables x_i− i, i = 1, . . . , d. Let

Λ^∗(d)= Q[x1, . . . , xd]^∗Sd

be the invariant subalgebra of the shifted action. There’s a natural map Λ^∗(d) → Λ^∗(d− 1) by setting the last variable xdto be zero. The algebra of shifted symmetric functionsΛ^∗is obtained by taking the projective limit

Λ^∗= lim←−−_d Λ^∗(d). It’s easy to see thatΛ^∗⊂ Q^P.

Define the completed cycle for each k∈ N ∪ {0} by (1.7) p_k(x)= k![z^k]e(x, z), e(x, z) =∑^∞

i=1

e^z(xi−i+12),

where [z^k] means we only take the coefficient of z^k. However, buy directly expanding it one might get

p_k(x) “= ”∑^∞

i=1

(x_i− i + 1 2)^k,

which doesn’t belong to the algebra Λ^∗! The correct functions must be obtained by taking the Riemannζ-function regularization:

p_k(x)=∑^∞

i=1

[(x_i− i +1

2)^k− (−i + 1

2)^k]+ (1 − 2^−k)ζ(−k).

Such shifted symmetric power sums canonically form a set of generators forΛ^∗:

Λ^∗= Q[p1, p2, . . .].

By [5], the image of the Fourier transform (1.6) lies in Λ^∗. Moreover there’s a transformation between those f’s and p’s:

(1.8) f_µ = 1

∏µi

p_µ+(lower degree terms), p_µ :=

l(µ)

∏

i=1

p_µi µi

.

In particular f₍₂₎ = ^p_2!². Hence the Fourier transform arises as an isomor- phism between⊕

d≥0Z_dandΛ^∗. Furthermore, using the expression (1.4)the Hurwitz number can be extended as a function onΛ^∗, especially bringing into the completed cycles would be the backbone of the following discus- sions.

1.3.1. Why completed cycles? Consider the smooth locus M_g^•_,n(X, d) ,→^ι M^•_g,n(X, d). A correspondence was found by a quite direct manner in Propo- sition 1.1 [11]:

∫

[Mg^•,n(X,d)]

∏n i=1

τki(ω) = 1

∏k_i!H_d^X((k₁+ 1), . . . , (kn+ 1)).

Note that [M^•_g,n(X, d)] = ι^∗[M^•_g,n(X, d)]^vir (see Proposition 5.2 [10]). There- fore

⟨∏n i=1

τki(ω)

⟩•X

d

= 1

∏ki!H_d^X((k₁+ 1), . . . , (kn+ 1)) + △

= ∑

|λ|=d

(dimλ d!

)2−2g(X) n∏

i=1

f(ki+1)(λ) + △, where△ stands for the contribution from the boundary divisors.

The defect can be modified by considering the completed cycles{p_k}. For instance, when X = ¹the following special GW/H correspondence will be introduced in Section 3:

(1.9)

⟨∏n i=1

τki(ω)

⟩•¹

d

= ∑

|λ|=d

(dimλ d!

)2∏n i=1

p_ki+1(λ) (ki+ 1)!.

We can see that completing the cycles corresponds to including the bound- ary strata of the moduli space.

If we denote by (k) the preimage of p_k/k under the Fourier transform (1.6), the right hand side of (1.9) can be rewritten as

∏1

k_i!H^_d¹((k₁+ 1), . . . , (kn+ 1))

under the extended notion of Hurwitz numbers. This is a “completion” of the classical Hurwitz theory. In this modern fashion the Hurwitz theory would become more accessible in combinatorial sense. One known reason is that the completed cycles could be naturally manipulated in the operator formalism of the infinite wedge representation.

Remark. From (1.8) we could only know that the transformation matrix between the two basis{fµ} and {p_µ} is triangular. The explicit formulae for computing the completion coefficients for expressing the completed cycles in {fµ} are given by Proposition 1.6 and 3.2, [11], where Proposition 1.6, as a conclusion of the GW/H correspondence, expresses the coefficients in some 1-point relative invariants that would reveal some geometric meanings about the coefficients, while Proposition 3.2 is obtained from Proposition 1.6 by the operator formalism and serves as a down-to-number solution.

A further geometric interpretation of the completed cycles could be found in [13].

2. The operator formalism

In this section a brief summary of the operator formalism is given.

The first part is about Fock representations, in which we’ll take a glance at the bosonic Fock space, and then focus on the infinite wedge space (fermionic Fock space). The later one will be our operating platform for deriving the special GW/H correspondence and the main results. Some fun- damental operators, for example, bosons and fermions, will be introduced in this part.

Next is a discussion about the equivalence of the above two Fock repre- sentations. The equivalence would allow us to write the fermions in terms of bosons, which is essential to the Hirota equations for the Toda hierarchy in Section 4. A brief introduction of the Murnaghan-Nakayama rule is also included, which plays an important role in translating the operator language into the language of characters and vice versa.

The final part is an introduction to the operators E, which can be seen as a generalization of the bosons and will be frequently used in the later sections.

Besides the source paper [11], the main reference for this section is [8].

The part about fermions in bosons is refered to Chapter 14, [4].

2.1. The Fock representations.

2.1.1. Bosonic Fock space. Let R_B be a C-algebra generated by symbols {αn: n∈ Z\{0}}, called bosons, with relations given by the commutators:

(2.1) [αl, αr]= lδl+r,0.

We define the bosonic Fock space to be B = C[x1, x2, . . . ; q, q⁻¹]. The following representation

ρB : RB −→ End(B) αn 7−→ ∂

∂xn

, n > 0 α−n 7−→ nxn, n > 0 is called the bosonic Fock representation.

2.1.2. Infinite wedge space (Fermionic Fock space). Given fermionsψk,ψ^∗_k indexed by the half-integers, i.e. k∈ Z+¹₂, let R_F be theC-algebra generated by them with the following anti-commutation relations

{ψi, ψj} = {ψ^∗_i, ψ^∗_j} = 0, {ψi, ψ^∗_j} = δi j.

In what follows, I’ll introduce the notion of the Infinite wedge space (or Fermionic Fock space), which serves as a representation space for the fermionic algebra.

Consider subsets S = {s1 > s2 > s3 > · · · } ⊂ Z + ¹₂ satisfying:

(i) S₊:= S \(Z + ¹₂)_<0is finite, (ii) S₋:= (Z + ¹₂)_<0\S is finite.

Denote by|S ⟩ the following infinite wedge product:

|S ⟩ = s1∧ s2∧ s3∧ . . . .

then the infinite wedge space is defined to be the following infinite dimen- sional vector space

F =⊕

S

C |S ⟩ .

In this fermionic case, we also consider the dual space F^∗with dual elements denoted by⟨S |. The inner product on F is defined by the conditions ⟨S |S^′⟩ = δS,S^′ for all S, S^′.

The fermionic Fock representation is defined by ρF : RF −→ End(F)

ψk 7−→ k∧

whileρF(ψ^∗_k) is defined to be the adjoint ofρF(ψk) with respect to the inner product. Such operation is the same as contracting k from the left. For simplicity, the symbol ρF( ) will be omitted later if there’s no confusion caused.

2.1.3. Charge and energy. Define the normal ordering :· : by :ψiψ^∗j :=

{ ψiψ^∗_j, j > 0

−ψ^∗_jψi, j < 0

and let E_{i, j} =: ψiψ^∗_j : . The charge and the Hamiltonian operators are defined to be

C = ∑

k∈Z+¹2

E_k,k and H = ∑

k∈Z+¹2

kE_k,k

respectively. An eigenvalue of any eigenstate for C would be called charge of this state; Similarly an eigenvalue for H would be called energy. For instance, by acting them on arbitrary state|S ⟩ of the basis for F, we can get

C|S ⟩ = (|S₊| − |S₋|) |S ⟩ and H |S ⟩ =



∑

s∈S+

s− ∑

s∈S−

s^′



 |S ⟩ . Thus the elements in the basis have definite charges and energies. It follows that the infinite wedge space F can be decomposed into subspaces F_l con- sisting of states with charge l ∈ Z, and each Fl can be further decomposed with respect to the energies, say,

F= ⊕

charge l∈Z

F_l = ⊕

charge l∈Z energy d≥^l2₂

F^d_l.

The condition d ≥ ^l₂² comes from the fact that for each F_l, there’s a unique ground state

|l⟩ = l − 1

2 ∧ l − 3

2∧ l − 5 2 ∧ . . . , which has energy l²/2. Especially,

|0⟩ = −1 2 ∧ −3

2 ∧ −5 2 ∧ . . . ,

called the vacuum state, is the ground state of F₀. For any operator X on F, we’ll call⟨X⟩ := ⟨0 |X| 0⟩ the vacuum expectation for X.

It’s easy to check that F₀, as the kernal of C, is spanned by the states

|λ⟩ = λ1− 1

2 ∧ λ2− 3

2 ∧ λ3− 5

2∧ . . . =∧^∞

i=1

λi− i + 1 2 indexed by all partitionsλ = (λ1 ≥ λ2≥ . . . ≥ 0 = . . .), and

H |λ⟩ = |λ| |λ⟩ .

Generally the subspace F_l is generated by the states

|λ, l⟩ = λ1+ l −1

2 ∧ λ2+ l − 3

2∧ λ3+ l − 5 2 ∧ . . . with

H |λ, l⟩ = (

|λ| +l² 2 )

|λ, l⟩ . 2.2. Boson-Fermion correspondence.

2.2.1. Bosons in Fermions. The bosons can be realized in the infinite wedge space. Define

B_n= ∑

k∈Z+¹2

E_k−n,k, n ∈ Z\{0}.

By the fundamental identities among commutators and anti-commutators [AB, C] = A{B, C} − {A, C}B

(2.2)

= A[B, C] + [A, C]B, (2.3)

one can check from (2.2) that

[B_n, ψk]= ψk−n, [Bn, ψ^∗_k]= −ψ^∗_k+n and then from (2.3) that

[B_n, Bm]= nδn+m,0, hence there’s a well-defined homomorphism

ρF : RB −→ End(F) αn 7−→ Bn.

It’s also straitforward to check in the same way that [H , B−n]= nB−n, n ∈ Z,

which means the operation of B_−nwould increase the energy of a state by n.

2.2.2. Equivalence of Fock representations. The Fock representations ρB

andρF are actually isomorphic to each other. An isomorphism can be con- structed explicitly as the following:

First form the formal sum of Bnin indeterminants{x1, x2, . . .}

B(x)=∑^∞

n=1

x_nB_n, then defineΦ to be a linear map from F =⊕

S C |S ⟩ to B = C[x1, x2, . . . ; q, q⁻¹] by

Φ(|S ⟩) =∑

l∈Z

q^l⟨l| e^B(x)|S ⟩ .

The following theorem shows thatΦ is an isomorphism of representations.

The isomorphicity would allows us to abuse the symbolαn, rather than Bn, to denote a boson in the infinite wedge space F.

Theorem 2.1. The morphismΦ is an isomorphism between the Fock repre- sentationsρBandρF. In particular, for n> 0,

ΦBnΦ⁻¹= ∂

∂xn

, ΦB_−nΦ⁻¹= nxn. Proof. We first show that

Φ(Bn|S ⟩) =

{ _∂

∂xnΦ(|S ⟩) if n> 0,

−nx_−nΦ(|S ⟩) if n < 0.

Since B_ncommutes with each other for n> 0, we have _∂x^∂_ne^B(x) = e^B(x)B_n. Then the first equality is obtained from

Φ(Bn|S ⟩) =∑

l∈Z

q^l⟨l| e^B(x)B_n|S ⟩ = ∑

l∈Z

q^l⟨l| ∂

∂xn

e^B(x)|S ⟩ = ∂

∂xn

Φ(|S ⟩).

For n< 0, we need the following useful formula (2.4) e^ABe^−A = B + [A, B] + 1

2![A, [A, B]] + 1

3![A, [A, [A, B]]] + · · · . The commutation relation [B(x), Bn]= −nxntogether with (2.4) yields

e^B(x)Bne^−B(x) = Bn− nxn.

Since Bm with m > 0 is an annihilator for the ground states, ⟨l| Bn = 0.

Therefore we have Φ(Bn|S ⟩) =∑

l∈Z

q^l⟨l| e^B(x)B_n|S ⟩ =∑

l∈Z

q^l⟨l| e^B(x)B_ne^−B(x)e^B(x)|S ⟩

= ∑

l∈Z

q^l⟨l| (Bn− nxn) e^B(x)|S ⟩ = −nx−nΦ(|S ⟩), as required.

Now we turn to prove thatΦ is an isomorphism of vector spaces, which will complete the proof of the theorem by combining with the above result.

Because B(x) annihilates all the ground states, we haveΦ(|l⟩) = q^l. Using the previous result, any monomials of B has a preimage in F, say,

Φ(∏

i

B_−ni

n_i |l⟩) = q^l∏

i

x_ni, so the morphismΦ is surjective.

The bijectivity is proved by counting dimensions: The bosonic Fock space can be rewritten as B = ⊕

l∈Zq^lC[x1, x2, . . .], in which we assign a q^l-homogeneous element a charge l. By definition those Bn’s leave the

charge unchanged under action, so the mapΦ would preserve charges un- der the assignments.

On the other hand, each xn could be assigned a weight n, hence there’s a decomposition

C[x1, x2, . . .] =⊕

n

⊕

|λ|=n

⟨x_λ1· · · xλn

⟩

C, where theλ runs through all partitions with prescript degree.

Now let’s focus on the charge-l level of both representation spaces. In this case the mapΦ has a more simple form

Φ_F

l : F_l −→ Bl = q^lC[x1, x2, . . .]

|S ⟩ 7−→ q^l⟨l| e^B(x)|S ⟩ ,

from which and the fact that Bn with n > 0 would lower down the energy by n, it can be observed that, to an energy-d state, the survived monomials under the map must be of weight d−^l₂². Therefore the mapΦ can be further decomposed with respect to the energy

Φ_F^d

l : F^d_l −→ B^d_l := q^l ⊕

|λ|=d−^l2₂

⟨x_λ1· · · x_λn

⟩

C.

The end of Section 2.1.3 indicates that the basis of F^d_l is parametrized by partitions with degree d−^l₂², hence the dimension of F^d_l is exactly the same as B^d_l. SinceΦ is surjective, it must be an isomorphism.  2.2.3. Murnaghan-Nakayama rule. Here’s a useful formula connecting the theory of symmetric functions and the operator formalism.

Given partitionsλ and ν of the same degree, starting with the state |λ⟩, one may wonder what will be left after a sequence of operations by∏_l(ν)

i=1α_νi. By the energy constraint the result must be some multiple of the vacuum state

|0⟩. But what’s the number?

Recall that any partition, thus any zero-charge state, corresponds uniquely to a Young diagram. Rearrange theανi’s in the product ∏l(ν)

i=1ανi such that the integersνi range from small to large from left to right, then apply them one-by-one to the state|λ⟩. It can be found that the manipulation on the cor- responding Young diagrams is exactly the recursive process for calculating the characterχ^λ_ν by the Murnaghan-Nakayama rule! Hence

l(ν)

∏

i=1

ανi|λ⟩ = χ^λ_ν|0⟩ . For more detail see [3].

What will be frequently used later is the dual form (2.5)

∏l(ν) i=1

α_−νi|0⟩ = ∑

|λ|=d

χ^λ_ν|λ⟩ .

2.2.4. Fermions in Bosons. The problem of realizing fermions in bosons is more subtle. Let z = (z, z², z³, . . .), the realization is attained by realizing the following generating functions for fermions

ψ(z) = ∑

k∈Z+¹₂

z^k+12ψk, ψ^∗(z)= ∑

k∈Z+¹₂

z^−k−12ψ^∗k

in terms of two kinds of operators:

The first is the translation operator T

T s₁∧ s2∧ s3∧ . . . = s1+ 1 ∧ s2+ 1 ∧ s3+ 1 ∧ . . . . It’s trivial that

TψkT⁻¹ = ψk+1, Tψ^∗_kT⁻¹= ψ^∗_k+1,

so that [C, Tⁿ] = nTⁿ, therefore the charge decomposition can be rewritten as

F=⊕

l∈Z

T^lF₀.

The same phenomenon can be revealed in the bosonic Fock space by di- rectly checking thatΦTΦ⁻¹ = q.

The second is the vertex operators Γ_±(t)= exp



∑

n>0

t_nα_±n n





with{t1, t2, . . .} a sequence of indeterminants. Since the bosonic operators αnwith n> 0 annihilate the ground states, we have

(2.6) Γ+(t)|l⟩ = |l⟩ .

Also observe thatΓ^∗_±= Γ_∓and

(2.7) Γ+(t)Γ−(s)= e^{∑tn snn} Γ−(s)Γ+(t). By definition we have

(2.8) [αn, ψ(z)] = zⁿψ(z), [αn, ψ^∗(z)]= −zⁿψ^∗(z), which implies that

Γ±(t)ψ(z) = γ(z^±1, t)ψ(z)Γ±(t) (2.9)

Γ_±(t)ψ^∗(z)= γ(z^±1, −t)ψ^∗(z)Γ_±(t), γ(z, t) = exp



∑

n≥1

t_nzⁿ n



 .

Proposition 2.2. ψ(z) and ψ^∗(z) can be formulated as ψ(z) = z^CTΓ−(z)Γ+(−z⁻¹) (2.10)

ψ^∗(z)= T⁻¹z^−CΓ−(−z)Γ+(z⁻¹),

where the operator z^Cacts on the charge-l subspace by multiplying z^l. The proof is proceeded in bosonic Fock space. Let

Ψ(z) = Φψ(z)Φ⁻¹, Ψ^∗(z)= Φψ^∗(z)Φ⁻¹

be the corresponding elements of the generating function for fermions. The commutation relation (2.8) can be rewritten as

[x_n, Ψ(z)] = z⁻ⁿ

n Ψ(z) and [ ∂

∂xn

, Ψ(z)] = zⁿΨ(z).

Lemma 2.3. Let D : C[x1, x2, . . .] → C[[x1, x2, . . .]] be a differential oper- ator, namely,

D=∑

r≥0

∑

1≤i1≤...≤ir

P_ii...ir ∂

∂xi1

· · · ∂

∂xir

, Pii...ir ∈ C[[x1, x2, . . .]].

(i) If [x_i, D] = λiD for i= 1, 2, . . ., then D = D(1) exp(

−∑

iλi ∂

∂xi

). (ii) If [_∂x^∂

i, D] = µiD for i= 1, 2, . . ., then D(1) = c exp (∑

iµix_i), c∈ C.

Proof. Note that, for any f ∈ C[x1, x2, . . .], the operator exp(

−∑

iλi ∂

∂xi

) =:

T_λis the parallel translation:

(T_λf )(x1, x2, . . .) = f (x1+ λ1, x2+ λ2, . . .).

For (i), we replace D by DT_λ, then the statement is equivalent to that [x_i, D] = 0 implies D = D(1), i = 1, 2, . . ., which is obvious.

For (ii), we replace D by exp (−∑

iµix_i) D, then the statement is equiva- lent to the one: [_∂x^∂

i, D] = 0 implies D(1) ≡ const., i = 1, 2, . . ., which is

also obvious. 

According to the fact that any linear operator onC[x1, x2, . . .] is a differ- ential operator, the above lemma would implies that

Ψ(z) = ˆc exp



∑

n

zⁿx_n



 exp



−∑

n

z⁻ⁿ n

∂

∂xn



 ,

where ˆc is an operator independent of those x_n’s. By definitionψ(z) lifts the charge by 1, so ˆc = c(z)q with c(z) some operator depending only on the variable z.

Translating it back into the fermionic side F we have ψ(z) = c(z)TΓ−(z)Γ+(−z⁻¹).

To figure out what c(z) is we directly act ψ(z) on |l⟩ by the above form as well as by definition and then compare the lowest energy terms

ψ(z) |l⟩ = c(z)TΓ₋(z)Γ₊(−z⁻¹)|l⟩ = c(z) |l + 1⟩ +(higher energy terms)

de f= ∑

k∈Z+¹2 z^k+12ψk|l⟩ = z^l+1|l + 1⟩ +(higher energy terms). Hence c(z)= z^C andψ(z) has the required expression.

The proof forψ^∗(z) is similar and omitted here.

2.3. The operatorsE. Let ς(z) = e^z/2− e^−z/2. For l∈ Z the operator El(z) is defined by

El(z)= ∑

k∈Z+¹₂

e^z(k−l2)Ek−l,k+ δl,0

ς(z).

These operators generalize the classical bosons. Indeed, for l, 0, El(0)= ∑

k∈Z+¹2

E_k−l,k = αl.

Note thatEl(z)^∗= E_−l(z). Similar to the bosons, [H , E−l(z)]= lE−l(z), The operatorsE satisfies the commutation relation (2.11) [El(z), Er(w)]= ς

( det

[ l z r w

])

El+r(z+ w).

In particular,

(2.12) [αl, Er(w)]= ς(lw)El+r(w).

If l+ r , 0, then El+r(w) is regular and (2.12) vanishes when w= 0. Other- wise the singuler term ofE0(w) contributes a constant factor as w→ 0

ς(lw)

ς(w) = e^lw/2− e^−lw/2 e^w/2− e^−w/2

w→0

−→ l.

This recovers the commutation relation for the bosons [αl, αr]= lδl+r.

By treating z as a formal symbol, we can rewriteE0(z)= ∑

k∈Z+¹2 e^zkψkψ^∗_k. From this expression it’s easy to see that

(2.13) E0(z)|λ⟩ = e(λ, z) |λ⟩ .

By extracting the z-coefficients of E0(z) we get the operatorsPkfor k > 0 (2.14) Pk = k![z^k]E0(z),

thenPk|λ⟩ = pk(λ) |λ⟩ by the definition of pk. The operator F2 = P2

2!

will play a special role in deriving the special GW/H correspondence.

3. The Gromov-Witten/Hurwitz correspondence

The first three subsections in this section are devoted to the proof of the special GW/H correspondence (1.9). In Section 3.1 the equivariant n + m- point functions over ¹and their generating function are introduced, which will be expressed in some vacuum expectation in Section 3.3, then the spe- cial correspondence is captured via the nonequivariant limit of this operator formula.

The operator formula is obtained by two steps: In Section 3.1 the Hodge integrals are introduced in both connected and disconnected fashions, then the generating function for the n+m-point functions is expressed in terms of the Hodge integrals using the localization formula — the proof is all about manipulations on the bipartite graghs and the switch between the connected and the disconnected theories. Next, in Section 3.2, we’ll show that the gen- erating function for the Hodge interals is an operator formula as a result of the ELSV formula [1] and the Murnaghan-Nakayama rule (2.5). Therefore the n+ m-point functions and the operator formalism can be bridged by the Hodge integrals through the above two steps.

The last part, Section 3.4, will deal with the full GW/H correspondence:

Theorem 3.1. For any smooth projective curve X, we have

⟨∏n

i=1

τki(ω), η¹, . . . , η^m

⟩•X

d

= 1

∏k_i!H^X_d((k₁+ 1), . . . , (kn+ 1), η¹, . . . , η^m).

As a conclusion of the degeneration formula the left hand side will be written in the algebra of shifted symmetric functions. The expression is the same as the right hand side except that it’s in some basis which we don’t know whether it’s the same as the one generated by the completed cycles.

The identification of the two bases will finally be confirmed by the special correspondence.

3.1. Hodge integrals and equivariant n+ m-point functions. This sub- section would introduce the formula (3.2), which is an alternating expres- sion of the localization formula in terms of the Hodge integrals.