Tài liệu Đề tài " Gromov-Witten theory, Hurwitz theory, and completed cycles " ppt

Gromov-Witten theory, Hurwitz theory,and completed cycles By A.. We define the stationary sector of the Gromov-Witten theory X to be the integrals volving only the descendents of ω.. We p

Gromov-Witten theory, Hurwitz theory,

and completed cycles

By A Okounkov and R Pandharipande

1 The geometry of descendents

1.1 Motivation: nondegenerate maps

1.2 Relative Gromov-Witten theory

1.3 Degeneration

1.4 The abstract GW/H correspondence

1.5 The leading term

1.6 The full GW/H correspondence

1.7 Completion coefficients

2 The operator formalism

2.1 The finite wedge

2.2 OperatorsE

3 The Gromov-Witten theory of P1

3.1 The operator formula

3.2 The 1-point series

3.3 The n-point series

4 The Toda equation

4.1 The τ -function

4.2 The string equation

4.3 The Toda hierarchy

5 The Gromov-Witten theory of an elliptic curve

0 Introduction

0.1 Overview.

0.1.1 There are two enumerative theories of maps from curves to curves.Our goal here is to study their relationship All curves in the paper will beprojective overC

The first theory, introduced in the 19th century by Hurwitz, concerns the

enumeration of degree d covers,

π : C → X,

of nonsingular curves X with specified ramification data In 1902, Hurwitz

published a closed formula for the number of covers,

π : P1→ P1,

with specified simple ramification over A1 ⊂ P1 and arbitrary ramificationover∞ (see [17] and also [10], [36]).

Cover enumeration is easily expressed in the class algebra of the symmetric

group S(d) The formulas involve the characters of S(d) Though great strides have been taken in the past century, the characters of S(d) remain objects of

substantial combinatorial complexity While any particular Hurwitz numbermay be calculated, very few explicit formulas are available

The second theory, the Gromov-Witten theory of target curves X, is ern It is defined via intersection in the moduli space M g,n (X, d) of degree d

mod-stable maps,

π : C → X, from genus g, n-pointed curves A sequence of descendents,

τ0 (γ), τ1(γ), τ2(γ), ,

is determined by each cohomology class γ ∈ H ∗ (X, Q) The descendents τ k (γ) correspond to classes in the cohomology of M g,n (X, d) Full definitions are given in Section 0.2 below The Gromov-Witten invariants of X are defined

as integrals of products of descendent classes against the virtual fundamental

class of M g,n (X, d).

Let ω ∈ H2(X,Q) denote the (Poincar´e dual) class of a point We define

the stationary sector of the Gromov-Witten theory X to be the integrals volving only the descendents of ω The stationary sector is the most basic and fundamental part of the Gromov-Witten theory of X.

in-Since Gromov-Witten theory and Hurwitz theory are both enumerativetheories of maps, we may ask whether there is any precise relationship between

the two We prove the stationary sector of Gromov-Witten is in fact equivalent

to Hurwitz theory

0.1.2 Let X be a nonsingular target curve The main result of the paper is a correspondence, termed here the GW/H correspondence, between

the stationary sector of Gromov-Witten theory and Hurwitz theory

Each descendent τ k (ω) corresponds to an explicit linear combination of

ramification conditions in Hurwitz theory A stationary Gromov-Witten

in-variant of X is equal to the sum of the Hurwitz numbers obtained by replacing

τ k (ω) by the associated ramification conditions The ramification conditions associated to τ k (ω) are universal — independent of all factors including the target X.

0.1.3 The GW/H correspondence may be alternatively expressed as

associating to each descendent τ k (ω) an explicit element of the class algebra

of the symmetric group The associated elements, the completed cycles, have

been considered previously in Hurwitz theory — the term completed cycle first

appears in [12] following unnamed appearances of the associated elements in[1], [11] In fact, completed cycles, implicitly, are ubiquitous in the theory ofshifted symmetric functions

The completed k-cycle is the ordinary k-cycle corrected by a nonnegative

linear combination of permutations with smaller support (except, possibly, forthe constant term corresponding to the empty permutation, which may be ofeither sign) The corrections are viewed as completing the cycle In [12], the

corrections to the ordinary k-cycle were understood as counting degenerations

of Hurwitz coverings with appropriate combinatorial weights Similarly, inGromov-Witten theory, the correction terms will be seen to arise from the

boundary strata of M g,n (X, d).

0.1.4 The GW/H correspondence is important from several points ofview From the geometric perspective, the correspondence provides a combi-

natorial approach to the stationary Gromov-Witten invariants of X, leading

to very concrete and efficient formulas From the perspective of symmetricfunctions, a geometrization of the theory of completed cycles is obtained.Hurwitz theory with completed cycles is combinatorially much more acces-sible than standard Hurwitz theory — a major motivation for the introduction

of completed cycles Completed cycles calculations may be naturally ated in the operator formalism of the infinite wedge representation, Λ∞2V In

evalu-particular, closed formulas for the completed cycle correction terms are

ob-tained If the target X is either genus 0 or 1, closed form evaluations of all

corresponding generating functions may be found; see Sections 3 and 5 Infact, the completed cycle corrections appear in the theory with target genus 0.Hurwitz theory, while elementary to define, leads to substantial combi-natorial difficulties Gromov-Witten theory, with much more sophisticatedfoundations, provides a simplifying completion of Hurwitz theory

0.1.5 The present paper is the first of a series devoted to the

Gromov-Witten theory of target curves X In subsequent papers, we will consider the

equivariant theory for P1, the descendents of the other cohomology classes

of X, and the connections to integrable hierarchies The equivariant

Gromov-Witten theory of P1 and the associated 2-Toda hierarchy will be the subject

of [32]

The introduction is organized as follows We review the definitions ofGromov-Witten and Hurwitz theory in Sections 0.2 and 0.3 Shifted symmetricfunctions and completed cycles are discussed in Section 0.4 The basic GW/Hcorrespondence is stated in Section 0.5

0.2 Gromov-Witten theory The Gromov-Witten theory of a nonsingular target X concerns integration over the moduli space M g,n (X, d) of stable degree

d maps from genus g, n-pointed curves to X Two types of cohomology classes are integrated The primary classes are:

ev∗ i (γ) ∈ H2(M g,n (X, d), Q),

where evi is the morphism defined by evaluation at the ith marked point,

evi : M g,n (X) → X, and γ ∈ H ∗ (X, Q) The descendent classes are:

ψ k iev∗ i (γ), where ψ i ∈ H2(M g,n (X, d),Q) is the first Chern class of the cotangent line

bundle L i on the moduli space of maps

Let ω ∈ H2(X,Q) denote the Poincar´e dual of the point class We will be

interested here exclusively in the integrals of the descendent classes of ω:

The theory is defined for all d ≥ 0.

Let g(X) denote the genus of the target The integral (0.1) is defined to

vanish unless the dimension constraint,

If the resulting genus is not an integer, the integral is defined as vanishing.Unless emphasis is required, the genus subscript will be omitted

The integrals (0.1) constitute the stationary sector of the Gromov-Witten theory of X since the images in X of the marked points are pinned by the

integrand The total Gromov-Witten theory involves also the descendants of

the identity and odd classes of H ∗ (X,Q)

The moduli space M g,n (X, d) parametrizes stable maps with connected

domain curves However, Gromov-Witten theory may also be defined with

disconnected domains If C =l

i=1 C i is a disconnected curve with connected

components C i , the arithmetic genus of C is defined by:

g(C) =

i

g(C i)− l + 1,

where g(C i ) is the arithmetic genus of C i In the disconnected theory, the genus

may be negative Let M • g,n (X, d) denote the moduli space of stable maps with

possibly disconnected domains

We will use the brackets  ◦ as above in (0.1) for integration in connectedGromov-Witten theory The brackets   • will be used for the disconnected

theory obtained by integration against [M • g,d (X, d)]vir The brackets   will

be used when it is not necessary to distinguish between the connected anddisconnected theories

0.3 Hurwitz theory.

0.3.1 The Hurwitz theory of a nonsingular curve X concerns the meration of covers of X with specified ramification The ramifications are

enu-determined by the profile of the cover over the branch points

For Hurwitz theory, we will only consider covers,

π : C → X, where C is nonsingular and π is dominant on each component of C Let d > 0

be the degree of π The profile of π over a point q ∈ X is the partition η of d obtained from multiplicities of π −1 (q).

By definition, a partition η of d is a sequence of integers,

η = (η1 ≥ η2 ≥ · · · ≥ 0),

where |η| = η i = d Let (η) denote the length of the partition η, and let m i (η) denote the multiplicity of the part i The profile of π over q is the

partition (1d ) if and only if π is unramified over q.

Let d > 0, and let η1, , η n be partitions of d assigned to n distinct points q1, , q n of X A Hurwitz cover of X of genus g, degree d, and monodromy

(ii) π has profile η i over q i,

(iii) π is unramified over X \ {q1, , q n }.

Hurwitz covers may exist with connected or disconnected domains TheRiemann-Hurwitz formula,

is valid for both connected and disconnected Hurwitz covers In disconnected

theory, the domain genus may be negative Since g(C) is uniquely determined

by the remaining data, the domain genus will be omitted in the notation below

Two covers π : C → X, π  : C  → X are isomorphic if there exists an isomorphism of curves φ : C → C  satisfying π  ◦ φ = π Up to isomorphism, there are only finitely many Hurwitz covers of X of genus g, degree d, and monodromy η i at q i Each cover π has a finite group of automorphisms Aut(π).

The Hurwitz number,

HX d (η1, , η n ),

is defined to be the weighted count of the distinct, possibly disconnected Hurwitz covers π with the prescribed data Each such cover is weighted by 1/ |Aut(π)|.

The GW/H correspondence is most naturally expressed as a relationshipbetween the disconnected theories, hence the disconnected theories will be ofprimary interest to us

0.3.2 We will require an extended definition of Hurwitz numbers valid

in the degree 0 case and in case the ramification conditions η satisfy |η| = d.

The Hurwitz numbers HX d are defined for all degrees d ≥ 0 and all partitions

η i by the following rules:

(i) HX0 (∅, , ∅) = 1, where ∅ denotes the empty partition.

(ii) If|η i | > d for some i then the Hurwitz number vanishes.

(iii) If|η i | ≤ d for all i then

In other words, the monodromy condition η at q ∈ X with |η| < d

corre-sponds to counting Hurwitz covers with monodromy η at q together with the

data of a subdivisor of π −1 (q) of profile η.

0.3.3 The enumeration of Hurwitz covers of P1 is classically known to

be equivalent to multiplication in the class algebra of the symmetric group

We review the theory here

Let S(d) be the symmetric group Let QS(d) be the group algebra The

class algebra,

Z(d) ⊂ QS(d),

is the center of the group algebra

Hurwitz covers with profile η i over q i ∈ P1 canonically yield n-tuples of permutations (s1, , sn) defined up to conjugation satisfying:

(i) s i has cycle type η i,

(ii) s1s2· · · s n = 1.

The elements s i are determined by the monodromies of π around the points q i.Therefore, HP1

d (η1, , η n ) equals the number of n-tuples satisfying

con-ditions (ii) and (ii) divided by |S(d)| The factor |S(d)| accounts for over

counting and automorphisms

Let C η ∈ Z(d) be the conjugacy class corresponding to η We have shown:

dim λ , |λ| = |η| ,

(0.7)

where χ λ

η is the character of any element of C η in the representation λ The

trace in equation (0.6) may be evaluated to yield the basic character formulafor Hurwitz numbers:

HPd1(η1, , η n) = 

|λ|=d

dim λ d!

2 n i=1

fηi (λ)

(0.8)

The character formula is easily generalized to include the extended

Hur-witz numbers (of Section 0.3.2) of target curves X of arbitrary genus g The

character formula can be traced to Burnside (exercise 7 in§238 of [2]); see also

[4], [19]

Define fη (λ) for arbitrary partitions η and irreducible representations λ of S(d) by:

dim λ .

(0.9)

If η = ∅, the formula is interpreted as:

f (λ) = 1

For|η| < |λ|, the function χ λ

η is defined via the natural inclusion of symmetric

groups S( |η|) ⊂ S(d) If |η| > |λ|, the binomial in (0.9) vanishes.

The character formula for extended Hurwitz numbers of genus g targets

determines an isomorphism betweenZ(d) and the algebra of functions on P(d).

Formula (0.8) may be alternatively derived as a consequence of the Fouriertransform isomorphism

Let P denote the set of all partitions (including the empty partition ∅).

We may extend the Fourier transform (0.11) to define a map,

φ :

∞ d=0

Z(d) C µ µ ∈ Q P ,

(0.12)

via definition (0.9) The extended Fourier transform φ is no longer an phism of algebras However, φ is linear and injective.

isomor-We will see the image of φ in QP is the algebra of shifted symmetric

functions defined below (see [23] and also [31]).

0.4.2 The shifted action of the symmetric group S(n) on the algebra Q[λ1, , λn ] is defined by permutation of the variables λ i − i Let

Q[λ1, , λn]∗S(n)denote the invariants of the shifted action The algebraQ[λ1, , λn]∗S(n) has

a natural filtration by degree

Define the algebra of shifted symmetric functions Λ∗ in an infinite number

of variables by

Λ∗= lim←− Q[λ1, , λn]∗S(n) ,

(0.13)

where the projective limit is taken in the category of filtered algebras with

respect to the homomorphisms which send the last variable λ n to 0

Concretely, an element f ∈ Λ ∗is a sequence (usually presented as a series),

(i) the polynomials f(n) are of uniformly bounded degree,

(ii) the polynomials f(n) are stable under restriction,

f(n+1)

λ n+1=0 = f(n)

The elements of Λ∗ will be denoted by boldface letters

The algebra Λ∗ is filtered by degree The associated graded algebra gr Λ∗

is canonically isomorphic to the usual algebra Λ of symmetric functions asdefined, for example, in [27]

A point (x1, x2, x3, )∈ Q ∞ is finite if all but finitely many coordinates

vanish By construction, any element f ∈ Λ ∗ has a well-defined evaluation at

any finite point In particular, f can be evaluated at any point

λ = (λ1, λ2, , 0, 0, ) , corresponding to a partition λ. An elementary argument shows functions

f ∈ Λ ∗ are uniquely determined by their values f (λ) Hence, Λ ∗ is

canoni-cally a subalgebra of QP.

0.4.3 The shifted symmetric power sum pk will play a central role in our

study Define pk ∈ Λ ∗ by:

are of degree k and are stable under restriction Hence, p k is well-defined.The shifts by 12 in the definition of pk appear arbitrary — their signifi-

cance will be clear later The peculiar ζ-function constant term in p k will beexplained below

The image of pkin gr Λ∗ ∼ = Λ is the usual kth power-sum functions Sincethe power-sums are well known to be free commutative generators of Λ, weconclude that

Λ∗=Q[p1, p2, p3, ] The explanation of the constant term in (0.14) is the following Ideally,

we would like to define pk by

However, the above formula violates stability and diverges when evaluated at

any partition λ In particular, evaluation at the empty partition ∅ yields:

Definition (0.15) can be repaired by subtracting the infinite constant (0.16)

inside the sum in (0.14) and compensating by adding the ζ-regularized value

outside the sum

The same regularization can be obtained in a more elementary fashion bysumming the following generating series:

It is convenient to arrange the polynomials pk into a generating function:

where [z k ] denotes the coefficient of z k in the expansion of the meromorphic

function e(λ, z) in Laurent series about z = 0.

0.4.4 The function fµ (λ), arising in the character formulas for Hurwitz

numbers, is shifted symmetric,

pµi and the dots stand for terms of degree lower than |µ|.

The combinatorial interplay between the two mutually triangular linearbases {p µ } and {f µ } of Λ ∗ is a fundamental aspect of the algebra Λ∗ In fact,

these two bases will define the GW/H correspondence

Following [12], we define the completed conjugacy classes by

cycles are:

(1) =(1)− 1

24· () , (2) =(2) ,

is our shorthand notation for conjugacy classes

Since fµ(∅) = 0 for any µ = ∅, the coefficient of the empty partition,

() = C ∅ ,

in (k) equals the constant term of 1kpk

The completion coefficients ρ k,µdetermine the expansions of the completedcycles,

The term completed cycle is appropriate as (k) is obtained from (k) by

adding nonnegative multiples of conjugacy classes of strictly smaller size (withthe possible exception of the constant term, which may be of either sign) The

nonnegativity of ρ k,µ for µ = ∅ is clear from formula (0.22) Also, the coefficient

ρ k,µ vanishes unless the integer k + 1 − |µ| − (µ) is even and nonnegative.

We note the transposition (2) is the unique cycle with no correctionsrequired for completion

0.4.5 The term completed cycle was suggested in [12] when the functions

pk in [1], [11] were understood to count degenerations of Hurwitz coverings.The GW/H correspondence explains the geometric meaning of the completedcycles and, in particular, identifies the degenerate terms as contributions fromthe boundary of the moduli space of stable maps

In fact, completed cycles implicitly penetrate much of the theory of shiftedsymmetric functions While the algebra Λ∗ has a very natural analog of the

Schur functions (namely, the shifted Schur functions, studied in [31] and many

subsequent papers), there are several competing candidates for the analog ofthe power-sum symmetric functions The bases {f µ } and {p µ } are arguably

the two finalists in this contest The relationship between these two linearbases can be studied using various techniques; in particular, the methods of[31], [33], [24] can be applied

Let X be a nonsingular target curve The GW/H correspondence is the

following relation between the disconnected Gromov-Witten and disconnectedHurwitz theories:

2−2g(X) n

i=1

pki+1(λ) (k i+ 1)! .(0.25)

For g(X) = 0 and 1, the right side can be expressed in the operator formalism

of the infinite wedge Λ∞2V and explicitly evaluated, see Sections 3 and 5.

The GW/H correspondence naturally extends to relative Gromov-Wittentheory; see Theorem 1 In the relative context, the GW/H correspondence

provides an invertible rule for exchanging descendent insertions τ k (ω) for

ram-ification conditions

The coefficients ρ k,µ are identified as connected 1-point Gromov-Witten

invariants of P1 relative to 0∈ P1 The explicit formula (0.22) for the cients is a particular case of the formula for 1-point connected GW invariants

coeffi-of P1 relative to 0, ∞ ∈ P1; see Theorem 2

0.5.2 Let us illustrate the GW/H correspondence in the special case ofmaps of degree 0 In particular, we will see the role played by the constant

terms in the definition of pk

In the degree 0 case, the only partition λ in the sum (0.25) is the empty partition λ = ∅ Since, by definition,

together with the following evaluation of the connected degree 0, 1-point tion,

And, indeed, the result is correct; see [13], [34]

0.5.3 A useful convention is to formally set the contributionτ −2 (ω)  •X 0,0

of the unstable moduli space M 0,1 (X, 0) to equal 1,

τ −2 (ω)  •X 0,0 = 1

(0.27)

This convention simplifies the form of the generating function (0.26) and eral others functions in the paper In the disconnected theory, the unstablecontribution (0.27) is allowed to appear in any degree and genus Hence, inthe disconnected theory, the convention is equivalent to setting

sev-τ −2 (ω) = 1

(0.28)

The parallel convention for the completed cycles

p0 = 0 , (−1)!1 p−1= 1fits well with the formula (0.18)

0.6 Plan of the paper.

0.6.1 A geometric study of descendent integrals concluding with a proof

of the GW/H correspondence in the context of relative Gromov-Witten theory

is presented in Section 1 The GW/H correspondence is Theorem 1 A specialcase of GW/H correspondence is assumed in the proof The special case, the

GW/H correspondence for the absolute Gromov-Witten theory of P1, will beestablished by equivariant computations in [32]

Relative Gromov-Witten theory is discussed in Section 1.2 The tion coefficients (0.21) are identified in Section 1.7 as 1-point Gromov-Witten

comple-invariants of P1 relative to 0∈ P1

0.6.2 The remainder of the paper deals with applications of the GW/Hcorrespondence In particular, generating functions for the stationary Gromov-Witten invariants of targets of genus 0 and 1 are evaluated These computa-tions are most naturally executed in the infinite wedge formalism We reviewthe infinite representation Λ∞2V in Section 2 The formalism also provides a

convenient and powerful approach to the study of integrable hierarchies; seefor example [20], [28], [35]

The stationary GW theory of P1 relative to 0, ∞ ∈ P1 is considered inSection 3 We obtain a closed formula for the corresponding 1-point function

in Theorem 2 The formula (0.22) for the completion coefficients is obtained

as a special case A generalization of Theorem 2 for the n-point function is

given in Theorem 3

0.6.3 The 2-Toda hierarchy governing the Gromov-Witten theory of P1

relative to{0, ∞} ⊂ P1is discussed in Section 4 The main result is Theorem 4which states that the natural generating function for relative GW-invariants

is a τ -function of the 2-Toda hierarchy of Ueno and Takasaki [38] Theorem 4

generalizes a result of [30]

The flows of the Toda hierarchy are associated to the ramification

con-ditions µ and ν imposed at {0, ∞} The equations of the Toda hierarchy are

equivalent to certain recurrence relations for relative Gromov-Witten ants, the simplest of which is made explicit in Proposition 4.3

invari-0.6.4 The Gromov-Witten theory of P1 was conjectured to be governed

by the Toda equation by Eguchi and Yang [8], and also by Dubrovin [5] TheToda conjecture was further studied in in [6], [7], [16], [30], [34]

The Toda conjecture naturally extends to the C×-equivariant

Gromov-Witten theory of P1 We will prove in [32] that the equivariant theory of P1

is governed by an integrable hierarchy which can also be identified with the2-Toda of [38] The flows in the equivariant 2-Toda correspond to the insertions

of τ k ([0]) and τ k([∞]), where

[0], [ ∞] ∈ H ∗

C×(P1, Q) ,

are the classes of the torus fixed points

The equivariant 2-Toda hierarchy is different from the relative 2-Toda

studied here However, the lowest equations of both hierarchies agree on theircommon domain of applicability

0.6.5 In Section 5, we discuss the stationary Gromov-Witten theory of

an elliptic curve E The GW/H correspondence identifies the n-point function

of Gromov-Witten invariants of E with the character of the infinite wedge

representation of gl(∞) This character has been previously computed in [1],

see also [29], [11] We quote the results of [1] here and briefly discuss some oftheir implications, in particular, the appearance of quasimodular forms.While the GW/H correspondence is valid for all nonsingular target curves

X, we do not know closed form evaluations for targets of genus g(X) ≥ 2.

The targets P1 and E yield very beautiful theories Perhaps the study of the

Gromov-Witten theory of higher genus targets will lead to the discovery of newstructures

0.7 Acknowledgments An important impulse for this work came from

the results of [12] and, more generally, from the line of research pursued in

[11], [12] Our interaction with S Bloch, A Eskin, and A Zorich played avery significant role in the development of the ideas presented here.

We thank E Getzler and A Givental for discussions of the Gromov-Witten

theory of P1, and T Graber and Y Ruan for discussions of the relative theory.A.O was partially supported by DMS-0096246 and fellowships from theSloan and Packard foundations R.P was partially supported by DMS-0071473and fellowships from the Sloan and Packard foundations

1 The geometry of descendents

1.1 Motivation: nondegenerate maps We begin by examining the relation

between Gromov-Witten and Hurwitz theory in the context of nondegeneratemaps with nonsingular domains

Let M g,n • (X, d) ⊂ M • g,n (X, d) be the open locus of maps,

π : (C, p1, , p n)→ X, where each connected component C i ⊂ C is nonsingular and dominates X Let q1, , qn ∈ X be distinct points Define the closed substack V by:

V = ev −11 (q1)∩ · · · ∩ ev −1 n (q n)⊂ M g,n • (X, d) The stacks M g,n • (X, d) and V are nonsingular Deligne-Mumford stacks of the

expected dimensions — see [14] for proofs

The Hurwitz number HX

d ((k1+ 1), , (k n+ 1)) may be defined by theenumeration of pointed Hurwitz covers

π : (C, p1, , p n)→ (X, q1, , q n ) ,

where

(i) π(p i ) = q i,

(ii) π has ramification order k i at p i

Here, π has ramification order k at p if π takes the local form z → z k+1 at p i

The count of pointed Hurwitz covers is weighted by 1/ |Aut(π)| where Aut(π)

is the automorphism group of the pointed cover.

The above enumeration of pointed covers coincides with the definition of

is represented by the locus of covers enumerated by H X

d ((k1+ 1), , (k n + 1)).

Proof Since V represents n

i=1ev∗ i (ω) in the Chow theory of M g,n • (X, d),

we may prove that the locus of Hurwitz covers represents

n



i=1

k i ! c1(L i)k i ∩ [V ]

in the Chow theory of V

First, consider the marked point p1 There exists a canonical section

s ∈ H0(V, L1) obtained from π by the following construction Let π ∗ denotethe pull-back map on functions:

π ∗ : m q1/m2q1 → m p1/m2p1,

(1.1)

where m q1, m p1 are the maximal ideals of the points q1 ∈ X and p1 ∈ C

respectively Via the canonical isomorphisms,

m q1/m2q1 = T ∼ q ∗1(X), m p1/m2p1 = T ∼ p ∗1(C), the map (1.1) is the dual of the differential of π Since q1 is fixed, the identifi-

cation m q1/m2q1 =∼ C yields a section s of L1 by (1.1)

The scheme theoretic zero locus Z(s) ⊂ V is easily seen to be the (reduced) substack of maps where p1 has ramification order at least 1 over q1 The cycle

Z(s) represents c1 (L1)∩ [V ] in the Chow theory of V

When restricted to Z(s), the pull-back of functions yields a map:

a canonical section s  ∈ H0(Z(s), L ⊗21 ) is obtained A direct scheme theoretic

verification shows that Z(s ) ⊂ Z(s) is the (reduced) substack where p1 has

ramification order at least 2 over q Hence the cycle Z(s ) represents the cycle

class 2c1(L1)2

After iterating the above construction, we find that k1! c1(L1)k1 is

repre-sented by the substack where p1 has ramification order at least k1 At eachstage, the reducedness of the zero locus is obtained by a check in the versaldeformation space of the ramified map (the issue of reducedness is local).Since the cycles determined by ramification conditions at distinct mark-

ings p i are transverse, we conclude that n

where ∆ is a correction term obtained from the boundary,

M • g,n (X, d) \ M •

g,n (X, d).

The GW/H correspondence gives a description of this correction term ∆

For example, consider the case where k i = 1 for all i Then, since 2-cycles

are already complete (see Section 0.4), the basic GW/H correspondence (0.24)yields an exact equality,

1.2 Relative Gromov-Witten theory We will study the GW/H dence in the richer context of the Gromov-Witten theory of X relative to a finite set of distinct points q1, , q m ∈ X Let η1, , η m be partitions of d.

correspon-The moduli space

M g,n (X, η1, , η m)

parametrizes genus g, n-pointed relative stable maps with monodromy η i at q i.Foundational developments of relative Gromov-Witten theory in symplecticand algebraic geometry can be found in [9], [18], [25], [26] The stationarysector of the relative Gromov-Witten theory is:

the integrals of descendents of ω relative to q1, , qm ∈ X.

The genus and the degree may be omitted in the notation (1.4) as long

as m > 0 Again, the corresponding disconnected theory is denoted by the

brackets  •

The stationary theory relative to q1, , qm specializes to the stationary

theory relative to q1, , qm −1 when η m is the trivial partition (1d) In

par-ticular, when all the partitions η i are trivial, the standard stationary theory

of X is recovered A proof of this specialization property is obtained from the

degeneration formula discussed in Section 1.3 below

The stationary Gromov-Witten theory of P1 relative to 0, ∞ ∈ P1 will

play a special role Let µ, ν be partitions of d prescribing the profiles over

0, ∞ ∈ P1 respectively We will use the notation,

to denote integrals in the stationary theory of P1 relative to 0, ∞ ∈ P1

1.3 Degeneration The degeneration formula for relative Gromov-Witten

theory provides a formal approach to the descendent integrals



τ k1(ω) · · · τ k n (ω), η1, , η m•X

d Let x1, , xn ∈ X be distinct fixed points Consider a family of curves with

n sections over the affine line,

π : (X , s1, , s n)→ A1,

defined by the following properties:

(i) (X t , s1 (t), , s n (t)) is isomorphic to the fixed data (X, x1, , xn) for all

t = 0.

(ii) (X0, s1 (0), , s n (0)) is a comb consisting of n + 1 components (1

back-bone isomorphic to X and n teeth isomorphic to P1) The teeth are

attached to the points x1, , xn of the backbone The section s i(0) lies

on the ith tooth

The degeneration π can be easily constructed by blowing-up the n points (x i , 0)

of the trivial family X × A1

The following result is obtained by viewing the family π as a degeneration

of the target in relative Gromov-Witten theory

Proposition 1.2 ([9], [18], [25], [26]) A degeneration formula holds for relative Gromov-Witten invariants:

Here, the factor z(µ) is defined by:

i ≥1 S(m i (µ)) is the symmetry group permuting equal parts

of µ The factor z(µ) will occur often.

The right side of the degeneration formula (1.6) involves the Hurwitz

num-bers and 1-point stationary Gromov-Witten invariants of P1relative to 0∈ P1.The degeneration formula together with the definition of the Hurwitz numbersimplies the specialization property of relative Gromov-Witten invariants when

where the sum is again over partitions µ i of d.

1.4 The abstract GW/H correspondence Formula (1.6) can be restated

as a substitution rule valid in degree d:

ram-A degree independent substitution rule is obtained by studying the nected relative invariants Disconnected invariants may be expressed as sums

con-of products con-of connected invariants obtained by all possible decompositions con-ofthe domain and distributions of the integrand As the invariant µ, τ k (ω)  •P1

has a single term in the integrand and

m1 (µ)

i z(µ − 1 i ) ,

we may rewrite (1.9) as:

where the summation is over all partitions ν.

Proposition 1.3 is a degree independent, abstract form of the GW/H

corre-spondence Clearly, only partitions ν of size at most d contribute to the degree

d invariants What remains is the explicit identification of the coefficients in

|ν| ≤ k + 1 Moreover, ν = (k + 1) is the only partition of size k + 1 which actually appears

in (1.10) All other partitions ν appearing in (1.10) have a strictly smaller size.

We will now determine the coefficient of ν = (k + 1) in (1.10) by the

method of Proposition 1.1 The corresponding relative invariant is computed

in the following lemma

Lemma 1.4 For d > 0,

(d), τ d −1 (ω) P1

= 1

d! . Proof We first note that the connected and disconnected invariants coin-

The completion coefficients ρ k,µdetermine the expansions of the completedcycles,

The...