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

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520 A. OKOUNKOV AND R. PANDHARIPANDE
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 P
1
, the descendents of the other cohomology classes
of X, and the connections to integrable hierarchies. The equivariant Gromov-
Witten theory of P
1
and the associated 2-Toda hierarchy will be the subject
of [32].
The introduction is organized as follows. We review the definitions of
Gromov-Witten and Hurwitz theory in Sections 0.2 and 0.3. Shifted symmetric
functions and completed cycles are discussed in Section 0.4. The basic GW/H
correspondence 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) ∈ H
2
(M
g,n
(X, d), Q),
where ev
i
is the morphism defined by evaluation at the i
th
marked point,
ev
i
: M
g,n
(X) → X,
and γ ∈ H
∗
(X, Q). The descendent classes are:
ψ
k
i
ev
∗
i
(γ),
where ψ
i
∈ H
2
(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 ∈ H
2
(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:

n

i=1
τ
k
i
(ω)

◦X
g,d
=

[M
g,n
(X,d)]
vir
n

i=1
ψ
k
i
i
ev
∗
i
(ω).(0.1)
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,
2g − 2+d(2 − 2g(X)) =
n

i=1
k
i
,(0.2)
is satisfied. If the subscript g is omitted in the bracket notation 

i
τ
k
i
(ω)
X
d
,
the genus is specified by the dimension constraint from the remaining data.
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
GROMOV-WITTEN THEORY, HURWITZ THEORY, AND COMPLETED CYCLES
521
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 connected
Gromov-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 and
disconnected theories.
0.3. Hurwitz theory.
0.3.1. The Hurwitz theory of a nonsingular curve X concerns the enu-
meration of covers of X with specified ramification. The ramifications are
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 =(η
1
≥ η
2
≥···≥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 (1
d
) if and only if π is unramified over q.
Let d>0, and let η
1
, ,η
n
be partitions of d assigned to n distinct points
q
1
, ,q
n
of X. A Hurwitz cover of X of genus g, degree d, and monodromy
η
i
at q
i
is a morphism
π : C → X(0.3)
satisfying:
(i) C is a nonsingular curve of genus g,
522 A. OKOUNKOV AND R. PANDHARIPANDE
(ii) π has profile η
i
over q
i
,
(iii) π is unramified over X \{q
1
, ,q
n
}.
Hurwitz covers may exist with connected or disconnected domains. The
Riemann-Hurwitz formula,
2g(C) − 2+d(2 − 2g(X)) =
n

i=1
(d − (η
i
)) ,(0.4)
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,
H
X
d
(η
1
, ,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 relationship
between the disconnected theories, hence the disconnected theories will be of
primary 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 H
X
d
are defined for all degrees d ≥ 0 and all partitions
η
i
by the following rules:
(i) H
X
0
(∅, ,∅) = 1, where ∅ denotes the empty partition.
(ii) If |η
i
| >dfor some i then the Hurwitz number vanishes.
(iii) If |η
i
|≤d for all i then
H
X
d
(η
1
, ,η
n
)=
n

i=1

m
1
(η
i
)
m
1
(η
i
)

· H
X
d
(η
1
, ,η
n
) ,(0.5)
where η
i
is the partition of size d obtained from η
i
by adding d −|η
i
|
parts of size 1.
In other words, the monodromy condition η at q ∈ X with |η| <dcorre-
sponds to counting Hurwitz covers with monodromy η at q together with the
data of a subdivisor of π
−1
(q) of profile η.
GROMOV-WITTEN THEORY, HURWITZ THEORY, AND COMPLETED CYCLES
523
0.3.3. The enumeration of Hurwitz covers of P
1
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
∈ P
1
canonically yield n-tuples of
permutations (s
1
, ,s
n
) defined up to conjugation satisfying:
(i) s
i
has cycle type η
i
,
(ii) s
1
s
2
···s
n
=1.
The elements s
i
are determined by the monodromies of π around the points q
i
.
Therefore, H
P
1
d
(η
1
, ,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:
H
P
1
d
(η
1
, ,η
n
)=
1
d!

C
(1
d
)


C
η
i
(0.6)
=
1
(d!)
2
tr
Q
S(d)

C
η
i
where

C
(1
d
)

stands for the coefficient of the identity class and tr
Q
S(d)
denotes
the trace in the adjoint representation.
Let λ be an irreducible representation λ of S(d) of dimension dim λ. The
conjugacy class C
η
acts as a scalar operator with eigenvalue
f
η
(λ)=|C
η
|
χ
λ
η
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 formula
for Hurwitz numbers:
H
P
1
d
(η
1
, ,η
n
)=

|λ|=d

dim λ
d!

2
n

i=1
f
η
i
(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].
524 A. OKOUNKOV AND R. PANDHARIPANDE
Define f
η
(λ) for arbitrary partitions η and irreducible representations λ of
S(d)by:
f
η
(λ)=

|λ|
|η|

|C
η
|
χ
λ
η
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
X is:
H
X
d
(η
1
, ,η
n
)=

|λ|=d

dim λ
d!

2−2g(X)
n

i=1
f
η
i
(λ) .(0.10)
0.4. Completed cycles.
0.4.1. Let P(d) denote the set of partitions of d indexing the irreducible
representations of S(d). The Fourier transform,
Z(d)  C
µ
→ f
µ
∈ Q
P(d)
, |µ| = d,(0.11)
determines an isomorphism between Z(d) and the algebra of functions on P(d).
Formula (0.8) may be alternatively derived as a consequence of the Fourier
transform 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
→ f
µ
∈ Q
P
,(0.12)
via definition (0.9). The extended Fourier transform φ is no longer an isomor-
phism of algebras. However, φ is linear and injective.
We will see the image of φ in Q
P
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[λ
1
, ,λ
n
] is defined by permutation of the variables λ
i
− i. Let
Q[λ
1
, ,λ
n
]
∗S(n)
denote the invariants of the shifted action. The algebra Q[λ
1
, ,λ
n
]
∗S(n)
has
a natural filtration by degree.
GROMOV-WITTEN THEORY, HURWITZ THEORY, AND COMPLETED CYCLES
525
Define the algebra of shifted symmetric functions Λ
∗
in an infinite number
of variables by
Λ
∗
= lim
←−
Q[λ
1
, ,λ
n
]
∗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),
f =

f
(n)

, f
(n)
∈ Q[λ
1
, ,λ
n
]
∗S(n)
,
satisfying:
(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 as
defined, for example, in [27].
A point (x
1
,x
2
,x
3
, ) ∈ 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 =(λ
1
,λ
2
, ,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 Q
P
.
0.4.3. The shifted symmetric power sum p
k
will play a central role in our
study. Define p
k
∈ Λ
∗
by:
p
k
(λ)=
∞

i=1

(λ
i
− i +
1
2
)
k
− (−i +
1
2
)
k

+(1− 2
−k
)ζ(−k) .(0.14)
The shifted symmetric polynomials,
n

i=1

(λ
i
− i +
1
2
)
k
− (−i +
1
2
)
k

+(1− 2
−k
)ζ(−k) ,n=1, 2, 3, ,
are of degree k and are stable under restriction. Hence, p
k
is well-defined.
The shifts by
1
2
in the definition of p
k
appear arbitrary — their signifi-
cance will be clear later. The peculiar ζ-function constant term in p
k
will be
explained below.
526 A. OKOUNKOV AND R. PANDHARIPANDE
The image of p
k
in gr Λ
∗
∼
=
Λ is the usual k
th
power-sum functions. Since
the power-sums are well known to be free commutative generators of Λ, we
conclude that
Λ
∗
= Q[p
1
, p
2
, p
3
, ] .
The explanation of the constant term in (0.14) is the following. Ideally,
we would like to define p
k
by
p
k
“=”
∞

i=1
(λ
i
− i +
1
2
)
k
.(0.15)
However, the above formula violates stability and diverges when evaluated at
any partition λ. In particular, evaluation at the empty partition ∅ yields:
p
k
(∅) “=”
∞

i=1
(−i +
1
2
)
k
.(0.16)
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 by
summing the following generating series:
∞

i=1
∞

k=0
(−i +
1
2
)
k
z
k
k!
=
∞

i=1
e
z(−i+
1
2
)
=
1
z S(z)
,
where, by definition,
S(z)=
sinh(z/2)
z/2
=
∞

k=0
z
2k
2
2k
(2k + 1)!
.
The coefficients c
i
in the expansion,
1
S(z)
=
∞

i=0
c
i
z
i
,(0.17)
are essentially Bernoulli numbers. Since
(1 − 2
−k
) ζ(−k)=k! c
k+1
,
the two above regularizations are equivalent. The constants c
k
will play an
important role.
It is convenient to arrange the polynomials p
k
into a generating function:
p
k
(λ)=k![z
k
] e(λ, z) , e(λ, z)=
∞

i=0
e
z(λ
i
−i+
1
2
)
,(0.18)
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.
GROMOV-WITTEN THEORY, HURWITZ THEORY, AND COMPLETED CYCLES
527
0.4.4. The function f
µ
(λ), arising in the character formulas for Hurwitz
numbers, is shifted symmetric,
f
µ
∈ Λ
∗
,
a nontrivial result due to Kerov and Olshanski (see [23] and also [31], [33]).
Moreover, the Fourier transform (0.12) is a linear isomorphism,
φ :
∞

d=0
Z(d)  C
µ
→ f
µ
∈ Λ
∗
.(0.19)
The identification of the highest degree term of f
µ
by Vershik and Kerov ([39],
[23]) yields:
f
µ
=
1

µ
i
p
µ
+ ,(0.20)
where p
µ
=

p
µ
i
and the dots stand for terms of degree lower than |µ|.
The combinatorial interplay between the two mutually triangular linear
bases {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
C
µ
=
1

i
µ
i
φ
−1
(p
µ
) ∈
|µ|

d=0
Z(d) .
Since the basis {p
µ
} is multiplicative, a special role is played by the classes
(k)=C
(k)
,k=1, 2, ,
which we call the completed cycles. The formulas for the first few completed
cycles are:
(1) =(1) −
1
24
· () ,
(2) =(2) ,
(3) =(3) + (1, 1) +
1
12
· (1) +
7
2880
· () ,
(4) =(4) + 2 · (2, 1) +
5
4
· (2) ,
where, for example,
(1, 1) = C
(1,1)
∈Z(2) ,
is our shorthand notation for conjugacy classes.
Since f
µ
(∅) = 0 for any µ = ∅, the coefficient of the empty partition,
() = C
∅
,
in
(k) equals the constant term of
1
k
p
k
.
528 A. OKOUNKOV AND R. PANDHARIPANDE
The completion coefficients ρ
k,µ
determine the expansions of the completed
cycles,
(k)=

µ
ρ
k,µ
· (µ) .(0.21)
Formula (0.17) determining the constants,
ρ
k,∅
=(k − 1)! c
k+1
,
admits a generalization determining all the completion coefficients,
ρ
k,µ
=(k − 1)!

µ
i
|µ|!
[z
k+1−|µ|−(µ)
] S(z)
|µ|−1

S(µ
i
z) ,(0.22)
where, as before, [z
i
] stands for the coefficient of z
i
. Formula (0.22) will be
derived in Section 3.2.4
The term completed cycle is appropriate as
(k) is obtained from (k)by
adding nonnegative multiples of conjugacy classes of strictly smaller size (with
the 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 corrections
required for completion.
0.4.5. The term completed cycle was suggested in [12] when the functions
p
k
in [1], [11] were understood to count degenerations of Hurwitz coverings.
The GW/H correspondence explains the geometric meaning of the completed
cycles and, in particular, identifies the degenerate terms as contributions from
the boundary of the moduli space of stable maps.
In fact, completed cycles implicitly penetrate much of the theory of shifted
symmetric 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 of
the power-sum symmetric functions. The bases {f
µ
} and {p
µ
} are arguably
the two finalists in this contest. The relationship between these two linear
bases can be studied using various techniques; in particular, the methods of
[31], [33], [24] can be applied.
0.5. The GW/H correspondence.
0.5.1. The GW/H correspondence may be stated symbolically as:
τ
k
(ω)=
1
k!
(k +1) .(0.23)
That is, descendents of ω are equivalent to completed cycles.
GROMOV-WITTEN THEORY, HURWITZ THEORY, AND COMPLETED CYCLES
529
Let X be a nonsingular target curve. The GW/H correspondence is the
following relation between the disconnected Gromov-Witten and disconnected
Hurwitz theories:

n

i=1
τ
k
i
(ω)

•X
d
=
1

k
i
!
H
X
d

(k
1
+1), ,(k
n
+1)

,(0.24)
where the right-hand side is defined by linearity via the expansion of the com-
pleted cycles in ordinary conjugacy classes.
The GW/H correspondence, the completed cycle definition, and formula
(0.10) together yield:

n

i=1
τ
k
i
(ω)

•X
d
=

|λ|=d

dim λ
d!

2−2g(X)
n

i=1
p
k
i
+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
∞
2
V and explicitly evaluated, see Sections 3 and 5.
The GW/H correspondence naturally extends to relative Gromov-Witten
theory; 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 P
1
relative to 0 ∈ P
1
. The explicit formula (0.22) for the coeffi-
cients is a particular case of the formula for 1-point connected GW invariants
of P
1
relative to 0, ∞∈P
1
; see Theorem 2.
0.5.2. Let us illustrate the GW/H correspondence in the special case of
maps of degree 0. In particular, we will see the role played by the constant
terms in the definition of p
k
.
In the degree 0 case, the only partition λ in the sum (0.25) is the empty
partition λ = ∅. Since, by definition,
p
k
(∅)=k! c
k+1
,
the formula (0.25) yields


τ
k
i
(ω)

•X
0
=

c
k
i
+2
.
The result is equivalent to the (geometrically obvious) vanishing of all multi-
point connected invariants,
τ
k
1
(ω) ···τ
k
n
(ω)
◦X
0
=0,n>1 ,