0.1. nKdV-hierarchy. We consider the operator

YOU-CHENG CHOU

0.1. nKdV-hierarchy. We consider the operator

L = _∂

ⁿ_x⁺¹

+ _a

₁

( _x ) _∂

ⁿ_x⁻¹

+ · · · + a

n

( _x ) , For any pair

( _{α, p} ) , α = 1, . . . , n, p = 0, 1, . . .

we consider the following system of PDEs for functions a

₁

( _{x, t} ) , . . . , a

n

( _{x, t} ) :

∂

t^α,p

L = [ _L, ( _L

^nα+1+p

)

₊

] ,

where ( _L

^n+α1+p

)

+

denotes the differential part of the pseudodifferential op- erator L

^n+α1+p

.

To construct pseudodifferential operator L

^n+α1+p

, it suffice to construct L

ⁿ⁺¹¹

= _∂

_x

+ l

1

( x ) _∂

⁻_x¹

+ l

2

( x ) _∂

⁻_x²

+ · · · ^,

where ∂

⁻¹

is the inverse operator to ∂ and its commutator with function f ( _x ) is as follows:

[ _∂

⁻_x¹

, f ( _x )] = − f

x

∂

⁻_x¹

+ _f

_xx

_∂

⁻_x²

− . . . .

The last rule and equation ( _L

ⁿ⁺¹¹

)

ⁿ⁺¹

= L allow us to solve l

i

( _x ) recursively.

0.2. Constructing Solutions to the KdV Hierarchy from the Sato Grass- manian.

Definition 0.1. A Sato space is an infinite dimensional vector subspace W ⊂ C (( _z )) _{such that}

W =< _f

₁

, f

₂

, · · · >

for some f

j

( _z ) = _z

^−j

+ _αz

^−j+1

+ · · · = z

^−j

( 1 + ◦( ¹ )) _.

Let T

₁

, T

₂

, . . . be an infinite sequence of formal variables, and denote by M ( _{z; T}

₁

, T

₂

, . . . ) the function

M ( _{z; T}

₁

, T

₂

, . . . ) = _e

^{T1z−1+T2z−2+...}

.

For a Sato space W =< f

1

, f

2

, · · · > , we define the τ-function associated it as the fraction

τ

W

( _T

₁

, T

₂

, . . . ) = ( · · · ∧ M f

3

∧ M f

2

∧ M f

1

) ∧ z

⁰

∧ z

¹

∧ z

²

. . .

· · · ∧ z

⁻³

∧ z

⁻²

∧ z

⁻¹

∧ z

⁰

∧ z

¹

∧ z

²

∧ ^{. . . .}

It depends on the space W itself, and not on the specific choice of the basis f

1

, f

₂

, . . . .

1

Proposition 0.2. Let W be a Sato space such that z

⁻²

W ⊂ W. Then we have (1) τ

W

( T

1

, T

2

, . . . ) does not depend on T

_2i

for i > 0;

(2) the second order differential operators

L = _∂

²_x

+ _2∂

²_T1

_logτ

_W

( − x + T

1

, T

3

, T

5

, . . . ) satisfyies the KdV hierarchy

∂

T_2k+1

L = [ L, ( L

^2k2+1

)

+

] .

Proposition 0.3. Let W be a Sato space generated by f

_i

= z

⁻ⁱ

( 1 + ◦( 1 )) . Then for any N ≥ ^0,

det ( f

_i

( z

_j

))

detz

⁻_j ⁱ

= _τ

_W

( _T

₁

( _z

_∗

) , T

₂

( _z

_∗

) , . . . ) , where T

k

( _z

_∗

) : =

¹_k

_∑

_i^N₌₁

_z

^k_i

_.

0.3. Witten’s Conjecture.

Let M

g,n

( M

g,n

) be the moduli space of smooth (nodal) genus g, n-pointed stable curves, L

i

be line bundle on M

g,n

whose fiber at the moduli point ( _{C; x}

₁

, . . . , c

n

) is T

_x^∗_i

C and let ψ

i

: = _c

₁

( L

i

) .

The intersection number is defined by

< _τ

_d1

· · · τ

dn

> : =

∫

Mg,n

ψ

^d₁¹

· · · ψ

^d_nⁿ

,

where g =

^{∑ di+}₃³⁻ⁿ

. We consider the formal power series of intersection number

F ( _t

₀

, t

₁

, . . . ) = ∑

n≥0

∑

d1≥0,...,dn≥0

1

n! < _τ

_d1

· · · τ

dn

> _t

_d1

· · · t

dn

.

Theorem 0.4 (Witten’s Conjecture). e

^F

is the τ-function for KdV with respect to variables T

2i+1

=

_(2i+^ti_1)!!

Remark 0.5. F ( _t

₀

, t

₁

, . . . ) is completely determined by F ( _t

₀

, 0, 0, . . . ) =

¹₆

_t

³₀

, KdV-hierarchy, string equation and dilaton equation.

0.4. Combinatorial Model. Let X be a compact Riemann surface and ρ be a meromorphic quadratic differential on X. Locally, ρ = _ϕ ( _z )( _dz )

²

, where ϕ ( _z ) is a meromorphic function. In our case, we assume ϕ has only simple or double poles.

We define the horizontal line field as

{ ν ∈ TX | ϕ ( _z )( _dz ( _ν ))

²

> 0 } ^. Its integral curve is called horizontal trajectory.

Remark 0.6. For a generic quadratic differential, a generic horizontal trajec- tory is nonclosed.

Here we define a special kind of quadratic differential:

Definition 0.7. A Jenkins-Strebel differential is a quadratic differential with only finite nonclosed horizontal trajectory.

By a local analysis, we can see that if z

₀

zero of order d of ρ, then there are d + 2 horizontal trajectories issuing from z

₀

. If z

₀

is a simple pole, then there is a unique horizontal trajectory issuing from z

₀

. Finally, if z

₀

is a dou- ble pole with negative residue, then z

₀

is surrounded by closed horizontal trajectories. We further list some properties of Jenkins-Strebel differential that we will need later.

Proposition 0.8. Let X be a Riemann surface of finite type and ρ ( z ) = _ϕ ( z ) dz

²

be a Jenkin-Strebel differential on X. Then

• the connected component of X \ {graph of nonclosed horizontal trajectory}

is either open annulus or open disk;

• all closed horizontal trajectory in the same connected component have the same length. (We use the metric dl

²

= | ϕ ( _z ) || dz |

²

.)

Theorem 0.9. (Strebel) For any 2n + 1-tuples ( _{X; x}

₁

, . . . , x

_n

; p

₁

, . . . , p

_n

) _{, where} X is a Riemann surface of finite type, x

_i

are distinct points of X, p

_i

> 0, and n > _χ ( _X ) , there exists a unique Jenkins-Strebel differential such that

• it has double pole at x

i

and no other poles;

• connected components of X \ { graph of nonclosed horizontal trajectory } are open disks;

• the length of horizontal trajectory associated to x

i

is p

i

.

We call the unique Jenkins-Strebel differential defined above the canoni- cal Jenkins-Strebel differential.

Conversely, given an embedded graph (a graph in oriented topological surface X) with

• each valencies of vertex ≥ 3,

• face marked by x

₁

, . . . , x

_n

• fixed lengths of its edges,

• complement of embedded graph is a disjoint, union of open disks, there exists unique complex structure such that its corresponding canonical Jenkins-Strebel differential determines the given embedded graph.

Now, we define M

^comb_g,n

: = {the moduli space of genus g connected em- bedded graphs with each valencies of vertex ≥ 3, n-marked faces, fixed lengths of its edges, and complement being a disjoint union of open disks }.

Theorem 0.10. M

g,n

× R

ⁿ+

∼ = M

^combg,n

as real orbifolds.

We can further generalize the above discussion to stable curve.

Definition 0.11. For a stable curve C with given perimeter on its marked points,

the canonical Jenkins-Strebel differential on C is a quadratic differential ρ such

that:

• ρ ≡ 0 on the unmarked components;

• ρ is the canonical Jenkins-Strebel differential on the puntured marked com- ponents.

M

^combg,n

: = { the space of stable genus g embedded graphs with vertices of valencies ≥ 3 on smooth point, at most one valency on nodal points, n- marked faces, complement being disjoint union of open disks, and fixed length of edges on marked component and no graph on unmarked compo- nents.}

To determine the relation between M

g,n

and M

^comb_g,n

, we introduce the equivalence relation as follows: Let C be a stable curve with genus g and n marked points. We can canonically decompose C as the union of two curves C = _C

⁺

∪ C

⁰

, where C

⁺

is the union of all the components of C containing marked points, and C

⁰

is the union of those containing no marked points. Let ξ

1

, . . . , ξ

u

be the points that C

⁺

has in common with C

⁰

. We say that [( _{C; x}

₁

, . . . , x

n

)] is equivalent to [( _C

^′

, x

₁^′

, . . . , x

_n^′

)] if there is a family of nodal curves { C

⁰_s

}

s∈S

over a connected base S, together with sections of smooth points τ

1

, . . . , τ

u

, with the property that ( _{C; x}

₁

, . . . , x

n

) (resp., ( _C

^′

, x

^′₁

, . . . , x

^′_n

) ) can be obtained from C

⁺

and C

_s⁰

(resp.,C

⁰_s′

) by identi- fying ξ

i

with τ

i

( _s ) _(resp.,τ

_i

( _s

^′

) ) for i = 1, . . . , u.

It is easy to check that what we just defined is an equivalence relation.

We let

Q : M

g,n

→ M

^′g,n

denote the projection via the equivalence relation. Now we can state the similar identifications for stable curves:

Theorem 0.12. H : M

^′g,n

× R

ⁿ+

→ M

^combg,n

is a homeomorphism.

0.5. Matrix Integral Model. We recall some facts about matrix integral.

Let B be a n × n positive definite symmetric matrix. We consider the integral

c

∫

Rⁿ

e

^−12(Bx,x)

∏

n i=1

dx

i

,

where c is chosen such that the integral equals 1. With this normalization we have

< _x

_i

_x

_j

> : = _c

∫

Rⁿ

x

i

x

j

e

^−12(Bx,x)

∏

n i=1

dx

i

= ( _B

⁻¹

)

_ij

. We can further generalized this computation.

Theorem 0.13 (Wick’s formula). Let f

₁

, . . . , f

_2k

be linear functions of x

1

, . . . , x

_n

. Then

< _f

₁

_f

₂

· · · f

2k

>= ∑

p1<···<pk

q1<···<q_k

< _f

_p1

_f

_q1

> · · · < f

p_k

f

q_k

> .

Let Λ = ( _Λ

_i

)

_1≤i≤N

be a diagonal matrix with Re ( _Λ

_i

) > 0 and H = ( _h

_ij

) = ( _x

_ij

+ √

− ^1y

ij

) be a Hermitian matrix. We consider the following measure on the space of Hermitian matrices

dµ

_Λ

( _H ) = _C

_Λ,N

_e

^−12trH2Λ

∏

N i=1

dx

ii

∏

i<j

dx

ij

dy

ij

, where C

_Λ,N

is chosen such that

∫

HN

dµ

_Λ

( _H ) = 1.

By direct computation, we have C

_Λ,N

= ( _2π )

^−N22

_∏

_i^N₌₁

_Λ

_i¹²

_∏

_i<j

( _Λ

_i

+ _Λ

_j

) . In coordinated x

_ii

, x

_ij

, y

_ij

, we can write tr ( _H

²

_Λ ) = ( _{Bx, x} ) , where

B =



 

 

 

 

 

 

 

 Λ

1

. ..

Λ

N

Λ

1

+ _Λ

₂

. ..

Λ

N−1

+ _Λ

_N

Λ

1

+ _Λ

₂

. ..

Λ

N−1

+ _Λ

_N



 

 

 

 

 

 

 

 .

Hence we have

< _x

²_ii

>

_Λ,N

: =

∫

HN

x

²_ii

dµ

_Λ

( _H ) = ¹ Λ

1

, < _x

²_ij

>

_Λ,N

=< _y

²_ij

>

_Λ,N

= ¹ Λ

i

+ _Λ

_j

^, and similarly

< _h

_ij

_h

_ji

>

_Λ,N

= ²

Λ

i

+ _Λ

_j

^, < _h

_ij

_h

_kl

>

_Λ,N

= 0 if ( _{i, j} ) ̸= ( l, k ) . Now we compute

< _e

√−1

6 tr(H³)

>

_Λ,N

=< ( 1 − ¹ 2!

1

6

²

( _tr ( _H

³

))

²

+ ¹ 4!

1

6

⁴

( _tr ( _H

³

))

⁴

) − · · · >

Λ,N

. By Wick’s formula, the right hand side can be presented as the monomial of < _h

_injn

_h

_imkm

> . Notice that each term can correspond to the gluing of 3-stars.

i

1

k

1

j

1

j

1

i

1

k

1

i

2

k

2

j

2

j

2

i

2

k

2

i

2n

k

2n

j

2n

j

2n

i

2n

k

2n

. . . t t

t t t t t t t JJJJ JJJJ J

t t t t t t t t t JJJJ JJJJ J

t t

t t

t t

t t

t

JJJJ JJJJ J

In this case an edge of the gluing corresponds to a pair < _h

_injn

_h

_imkm

> . We define the weight of an gluing the product

∏ _Λ

_i

₊ ² _Λ

_j

taken over all edges of the gluing.

Now we can rewrite the Kontsevich’s model into graph sum:

< _e

√−1

6 tr(H³)

>

_Λ,N

∼ ∑

G∈G^3,N

(

√−1 2

)

^|V(G)|

| AutG | ∏

e∈E(G)

2 Λ

e

+ _Λ

^′_e

^,

where G

^3,N

is set of equivalent class with 3-valent graphs and with N pos- sible colors Λ

1

, · · · ^, Λ

N

drawing on the face.

It can be observed by the following proposition:

Proposition 0.14.

< ( _trH )

^α1

· · · ( tr ( _H

^k

))

^αk

>

_Λ,N

= _α

₁

! · · · α

k

!2

^α2

· · · k

^αk

∑

G∈G^N

1

| AutG | ∏

e∈E(G)

2 Λ

e

+ _Λ

^′_e

^. 0.6. Proof of Witten conjecture. The first step of the proof is to give a com- binatorial formula for ψ

i

. Let π

i

: S

¹

( L

^comb_i

) → M

^combg,n

. We want to find a closed 2-form w

_i

on M

^comb_g,n

such that π

^∗_i

( _w

_i

) = _{dϕ and} ∫

S¹

ϕ |

fiber

= 1.

Fix p = ( _p

₁

, . . . , p

_n

) ∈ R

ⁿ+

, we have the commutative diagram:

M

g,n Q

||yyy yyy yy

f

I $$I I I I I I I I

M

^′_g,n ^∼h

^// M

^comb_g,n

( p )

By the way Q was constructed, the line bundle L

i

restricts to a trivial line bundle on the fibers of Q and therefore drops to a well-defined line bundle L

^′_i

^on M

^′g,n

with Q

^∗

( L

^′_i

) = L

i

. Let L

^comb_i

be the pullback of L

^′_i

^{via h}

⁻¹

^{, so} that

h

^∗

( L

^comb_i

) = L

^′_i

, f

^∗

( L

^comb_i

) = L

i

.

Now our goal becomes giving the combinatorial expression for its first Chern class.

Let | a | /Γ

a

be an orbisimplex of M

^combg,n

( _p ) , where a corresponds to an embedded graph ( _G

_a

; x

₁

, . . . , x

n

) whose i-th perimeter is equal to p

_i

and Γ

a

= Aut (( _G

_a

; x

₁

, . . . , x

_n

)) . The coordinates relative to the simplex | a | ^are the lengths

{ l

e

}

e∈E(Ga)

of the edges of G

a

. At each point x

_i

, we consider a cyclically ordered set of oriented edges of G

_a

( _e

₁

, . . . , e

_ν

)

with possible repetitions. A repetition happens when the edge bounds the same component of G

a

. We set

( w

i

)

_|a|

= ∑

1≤s<t≤ν−1

d ( l

es

p

i

) ∧ d ( l

et

p

i

)

Lemma 0.15. For each x

i

and p ∈ R

ⁿ+

,

[ _w

_i

] = _c

₁

( L

^comb_i

) ∈ H

²

( M

^combg,n

( _p )) . In particular,

[ _f

^∗

( _w

_i

)] = _c

₁

( L

i

) ∈ H

²

( M

g,n

) . Now we can rewrite the intersection number

< _τ

_d1

, . . . , τ

dn

>=

∫

Mg,n

ψ

₁^d1

· · · ψ

_n^dn

=

∫

M^combg,n (p)

w

^d₁¹

· · · w

^d_nⁿ

=

∫

M^combg,n (p)

w

^d₁¹

· · · w

^d_nⁿ

The last equality is true since the boundary is measure zero.

Let Ω = _∑

ⁿ_i=1

_p

²_i

_w

_i

.

∫

Rⁿ_≥0

e

^{−∑ λipi}

( ^∫

M^combg,n (p)

Ω

^d

d!

)

dp

1

· · · dp

n

= ∑

d1+···dn=d

< _τ

_d1

· · · τ

dn

>

∏

n i=1

2d

_i

!

d

i

! λ

⁻_i ^2(di+1)

= ( 1 ) , where Re( λ

i

) > 0 and d = 3g − ³ + n.

We use the combinatorial theorem due to Kontsevich.

Theorem 0.16.

^Ω_d!^d

dp

1

∧ · · · ∧ dp

n

= 2

^2n+5g−5

dl

e₁

∧ · · · ∧ dl

e_6g−6+3n

. We have

( 1 ) =

∫

Rⁿ_≥0

e

^{−∑ λipi}

( ^∫

M^comb_g,n (p)

2

^2n+5g−5

dl

e₁

∧ · · · ∧ dl

e6g−6+3n

)

= ∑

G∈G^3,cg,n

2

^2n+5g−5

| AutG |

∫

|a(G)|

e

^{−∑ λipi}

dl

e1

∧ · · · ∧ dl

e6g−6+3n

,

where G

g,n^3,c

is the isomorphism class of connected 3-valent embedded graph with genus g and n-marked points. We further do some change of vari- ables.

∑

n i=1

λ

i

p

i

= ∑

e∈E(G)

( _λ

_e

+ _λ

^′_e

) _l

_e

,

where λ

e

and λ

^′_e

are the perimeter of the two faces adjacent to the edge e.

Now we have the relation ( ∗)

G

∑

^3,cg,n

2

^−|V(G)|

| ^Aut ( _G ) | ∏

e∈E(G)

2

λ

e

+ _λ

^′_e

= ∑

d1+···+dn=d

< _τ

_d1

· · · τ

dn

>

∏

n i=1

( 2d

_i

− ¹ ) !!

λ

^2d_i ⁱ⁺¹

.

Theorem 0.17. Let F ( _t

₀

, t

₁

, · · · ) = ∑

n≥0

∑

d₁,...,dn≥0

1

n! < _τ

_d1

· · · τ

dn

> _t

_d1

· · · t

dn

. Set Λ = diag ( _Λ

₁

, . . . Λ

N

) _{with Re} ( _Λ

_i

) > 0 and t

_i

( _Λ ) = −( ²ⁱ − ¹ ) !!Tr ( _Λ

⁻²ⁱ⁻¹

) _. Then

F ( _t

₀

( _Λ ) , t

₁

( _Λ ) , . . . ) = ∑

G∈G^3,c,N

(

√−1 2

)

^|V(G)|

| ^Aut ( _G ) | ∏

e∈E(G)

2 Λ

e

+ _Λ

^′_e

^,

where G

^3,c,N

is the isomorphism class of connected 3-valent embedded graph with N-possible colors Λ

1

, · · · ^, Λ

N

drawing on the face.

Corollary 0.18. F ( _t

₀

( _Λ ) , t

₁

( _Λ ) , · · · ) is the asymptotic expansion of log < _e

√−1

6 trH³

>

_Λ,N

_{as Λ}

⁻¹

→ ^0.

Here we recall some properties of Airy function. We first study its as- ymptotic behaviour by stationary phase method.

a ( _y ) =

∫

_∞

−∞

e

√−1(x³/3−xy)

dx

∼ ^∫

U(y^1/2)

e

^{√−1(x3/3−xy)}

dx +

∫

U(−y^1/2)

e

^{√−1(x3/3−xy)}

dx

∼ ^const · ∑

±y^1/2

y

⁻⁴³

e

⁻²

√−1

3 y³²

f

1

( _y

⁻¹²

) , and similarly we have

a

^j−1

( _y ) =

∫

_∞

−∞

( − √

− 1x )

^j−1

_e

√−1(x³/3−xy)

dx

∼ ^const · ∑

±y^1/2

y

⁻⁴³

e

⁻²

√−1

3 y³²

f

j

( _y

⁻¹²

) , where f

_j

( _y ) = _y

^−j

( 1 + ◦( 1 )) .

We have similar expression for matrix Airy function:

A ( _Y ) =

∫

HN

e

√−1tr(X³/3−XY)

dX

∼ ∑

Y^1/2

∫

U(Y¹²)

e

^{√−1tr(X3/3−XY)}

dX

= ∑

Y^1/2

∫

U(0)

e

√−1tr((X+Y^1/2)³/3−(X+Y^1/2)Y)

dX

= ∑

Y^1/2

e

⁻²

√−1 3 trY^3/2

∫

U(0)

e

^{√−1tr(X3/3−X2Y1/2)}

dX

∼ const · ∑

Y^1/2

e

⁻²

√−1

3 trY^3/2

Y

_i^−1/4

∏

i<j

( Y

_i^1/2

+ Y

_j^1/2

)

^−1/2

e

^{F(t˜0(Y1/2),··· )}

− ( 1 ) ,

where ˜t

_i

( Y

^1/2

) = 2

^−(2i+1)/3

( 2i − ¹ ) !!tr ( Y

^−i−1/2

) . We can express matrix

Airy function in another form.

Lemma 0.19. If Φ is a conjugacy invariant function on H

N

, then for any diagonal real matrix Y,

∫

HN

Φ ( _X ) _e

⁻

√−1trXY

dX

= ( − ² π √

− ¹ )

^N(N−1)/2

detY

_i^j−1

∫

_∞

−∞

· · · ^∫

^∞

−∞

Φ ( _D ) _e

^{−√−1trDY}

detD

_i^j−1

dD

1

· · · dD

N

Now we have A ( _Y ) = ( − 2π √

− 1 )

^N(N−1)/2

detY

_i^j−1

∫

_∞

−∞

· · · ^∫

^∞

−∞

∏

i

e

√−1tr(D³_i/3−D_iY_i)

detD

_i^j−1

dD

1

· · · dD

N

∼ const · ∑

Y^1/2

e

⁻²

√−1

3 trY^3/2 N

∏

i=1

Y

_i⁻³⁴

det f

_j

( _Y

_i^−1/2

) detY

_i^j−1

− ( 2 ) . We can compare ( 1 ) and ( 2 ) . Then we get

e

^{F(t0(Λ),...)}

∼ ^det ( _f

_j

( _Λ

_i

)) det ( _Λ

⁻_i ^j

) ^.

We conclude that e

^{F(t0(Λ),...)}

safisties KdV hierarchy with respect to vari- ables:

T

2i+1

= ¹

2i + 1 TrΛ

⁻²ⁱ⁻¹

= ( − 1 )

²ⁱ⁺¹

_t

_i

( 2i + 1 ) !! .

Finally, notice that T

_i

→ c

ⁱ

T

i

also satisfies KdV hierarchy. This proves Wit- ten conjecture.

R EFERENCES

[1] Enrico Arbarello, maurizio Cornalba, Pillip A. Griffiths: Geometry of Algebraic Curves, vol. 2, chap. 20.

[2] S. K. Lando, A.K. Zvonkin: Graphs on Surfaces and their Applications, chap.3, 4.

[3] M. Kontsevich: Intersection Theory on the Moduli Space of Curves and the Matrix

Airy Function.