Double Ramification Cycles and the n-Point Function for the Moduli Space of Curves

(1)

Moduli Space of Curves.

White Rose Research Online URL for this paper:

http://eprints.whiterose.ac.uk/132933/

Version: Accepted Version

Article:

Buryak, A (2017) Double Ramification Cycles and the n-Point Function for the Moduli

Space of Curves. Moscow Mathematical Journal, 17 (1). pp. 1-13. ISSN 1609-3321

This is an author produced version of a paper published in the Moscow Mathematical

Journal. Materials in the Moscow Mathematical Journal may be reproduced by any means

for educational and scientific purposes without fee or permission (provided that the

customary acknowledgment of the source is given). This consent does not extend to other

kinds of copying for general distribution, for advertising or promotional purposes, or for

resale. Requests for permission for commercial use of material should be addressed to the

Independent University of Moscow.

[email protected] https://eprints.whiterose.ac.uk/ Reuse

Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item.

Takedown

If you consider content in White Rose Research Online to be in breach of UK law, please notify us by

(2)

DOUBLE RAMIFICATION CYCLES AND THE

n

-POINT FUNCTION FOR

THE MODULI SPACE OF CURVES

ALEXANDR BURYAK

Abstract. In this paper, using the formula for the integrals of theψ-classes over the double ramification cycles found by S. Shadrin, L. Spitz, D. Zvonkine and the author, we derive a new explicit formula for then-point function of the intersection numbers on the moduli space of curves.

1. Introduction

Consider the moduli space

_M

g,n

of stable complex algebraic curves of genus

g

with

n

marked

points. The class

ψ

i

∈

H

2

(

M

g,n

,

Q

) is defined as the first Chern class of the line bundle

over

_M

g,n

formed by the cotangent lines at the

i

-th marked point. We define the intersection

numbers by

h

τ

d1

τ

d2

· · ·

τ

dn

i

g

:=

Z

Mg,n

ψ

d1

1

ψ

d2 2

· · ·

ψ

dn

n

.

(1.1)

In the unstable case 2

g

₋

2 +

n

_≤

0 we define the bracket to be equal to zero. The bracket (1.1)

vanishes unless the dimension constraint

n

X

i=1

d

i

= 3

g

−

3 +

n

is satisfied. Introduce variables

t

₀

, t

₁

, t

₂

, . . .

. The celebrated conjecture of E. Witten [Wit91],

proved by M. Kontsevich [Kon92], says that for the following generating function

F

(

t

0

, t

1

, . . .

) :=

X

g≥0

n≥1

X

d1,...,dn≥0

h

τ

d1

· · ·

τ

dnig

t

d1

· · ·

t

dn

n

!

,

the exponent

e

F

_{is a tau-function of the KdV hierarchy in the variables}

_T

2i+1

=

₍₂_i_+1)!!ti

.

In this paper we discuss a different generating function for the intersection numbers (1.1).

Let

n

_≥

1. Introduce variables

x

1

, . . . , x

n

. Define

F

(

x

₁

, . . . , x

n

) :=

X

g≥0

F

g

(

x

1

, . . . , x

n

)

,

where

F

g

(

x

1

, . . . , x

n

) :=

X

d1,...,dn≥0

h

τ

d1

· · ·

τ

dn

i

g

x

d1

1

· · ·

x

dnn

.

The function

_F

(

x

₁

, . . . , x

n

) is often called the

n

-point function. There are several known

closed formulas for it [BDY15, BH07, Oko02]. In this paper we derive a simple new formula

for

_F

(

x

1

, . . . , x

n

) using the result of [BSSZ15]. We believe that our approach can be useful for

the understanding of the structure of more general Gromov-Witten invariants.

2010Mathematics Subject Classification. Primary 14H10, Secondary 14C17.

Key words and phrases. Moduli space of curves, intersection numbers.

The author was supported by grant Russian Science Foundation N16-11-10260, project ”Geometry and mathematical physics of integrable systems”. We are grateful to R. Pandharipande and S. Shadrin for useful discussions and to an anonymous referee for a number of suggestions that helped us to improve the exposition of the paper.

(3)

Let us formulate our main result. Introduce variables

a

₁

, a

₂

, . . . , a

n

. We will use the following

notations.

• Let

ζ

(

x

) :=

e

x2

−

e

− x 2

.

• For a permuation

σ

_∈

S

n

denote

a

′i

:=

a

σ(i)

and

x

′i

:=

x

σ(i)

.

• Finally,

a b

c d

=

ad

−

bc.

We define the functions

P

n

(

a

1

, . . . , a

n

;

x

1

, . . . , x

n

) by

P

₁

(

a

₁

;

x

₁

) :=

1 x

1

,

P

n

(

a

1

, . . . , a

n

;

x

1

, . . . , x

n

) :=

(1.2)

=

X

σ∈Sn

σ(1)=1

x

′₂

_{· · ·}

x

′_n₋₁

ζ

a

′₁

a

′₂

x

′₁

x

′₂

ζ

a

′₁

+

a

′₂

a

′₃

x

′₁

+

x

′₂

x

′₃

· · ·

ζ

a

′₁

+

_{· · ·}

+

a

′_n₋₁

a

′_n

x

′₁

+

_{· · ·}

+

x

′_n₋₁

x

′_n

a

′₁

a

′₂

x

′₁

x

′₂

a

′₂

a

′₃

x

′₂

x

′₃

· · ·

a

′_n₋₁

a

′_n

x

′_n₋₁

x

′_n

,

n

_≥

2 .

At first sight it appears that for

n

_≥

2 the function

P

n

(

a

;

x

) has simple poles along the

hy-perplanes

_{

a

i

x

j

−

a

j

x

i

= 0

}

, but in [BSSZ15, Remark 1.6] it was shown that for

n

≥

2 the

function

P

n

(

a

;

x

) is a power series in

x

1

, . . . , x

n

with coefficients in

C

[

a

1

, . . . , a

n

]. Moreover,

by [BSSZ15, Remark 1.5], the function

P

n

(

a

;

x

) is symmetric with respect to simultaneous

permutations of the variables

a

1

, . . . , a

n

and the variables

x

1

, . . . , x

n

. Our main result is the

following theorem.

Theorem 1.1.

We have

(1.3)

_F

(

x

₁

, . . . , x

n

) =

=

e

(Pxi)3 24

(

P

x

i

)

Q

√

2 πx

i

Z

Rn

e

−

P a2_i 2_xi

_P

n

(

√

−

1 a

1

, . . . ,

√

−

1 a

n

;

x

1

, . . . , x

n

)

da

−

δ

n,1

x

2

1

−

δ

n,2

x

1

+

x

2

,

where

da

:=

da

1

· · ·

da

n

.

Let us make a few comments about the right-hand side of equation (1.3) and, in particular,

show that it is a power series in

x

₁

, . . . , x

n

. Note that for

x >

0 and

d

≥

0 we have

1 √

2 πx

Z

R

a

d

e

−a 2 2x

da

=

(

d

₋

1)!!

x

d2

,

if

d

is even

,

0 ,

if

d

is odd

.

(1.4)

Therefore, the transformation

1 Q

√

2 πx

i

Z

Rn

e

−

P a2_i 2_xi

_P

n

(

√

−

1 a

;

x

)

da

(1.5)

just means that we replace each monomial

a

d1

1

· · ·

a

dnn

in

P

n

(

a

;

x

) by

(

₋

1)

12 P

di

Y

₍

_d

i

−

1)!!

x

di 2

i

,

if all exponents

d

i

are even and by zero otherwise. The transformation (1.5) can also be

interpreted as the Laplace transformation. For

n

_≥

3, in [BSSZ15, Remark 1.6] it was shown

that the function

_xPn(a;x)

1+···+xn

is a power series in

x

1

, . . . , x

n

with coefficients from

C

[

a

1

, . . . , a

n

].

Therefore, the right-hand side of (1.3) is a power series in

x

1

, . . . , x

n

. In the case

n

= 2, again

from [BSSZ15, Remark 1.6] we know that

P2(a1,a2;x1,x2)

x1+x2

−

1

x1+x2

is a power series in

x

1

, x

2

with

(4)

DR CYCLES AND THE n-POINT FUNCTION 3

and

x

2

. For

n

= 1 we immediately see that the right-hand side of (1.3) is equal to

e x3₁ 24−1

x2₁

that

is of course a power series in

x

₁

.

Let us describe briefly the plan of the proof of Theorem 1.1. The double ramification

cy-cle DR

g

(

a

1

, . . . , a

n

) is a cohomology class in

H

2g

(

M

g,n

,

Q

). It depends on a list of integers

a

₁

, . . . , a

n

satisfying

P

a

i

= 0. In [BSSZ15] the authors derived an explicit formula for the

generating series

X

g≥0 2g−2+n>0

X

d1,...,dn≥0

Z

Mg,n

DR

g

(

a

1

, . . . , a

n

)

ψ

1d1

· · ·

ψ

ndn

!

x

d1

1

· · ·

x

dnn

.

We recall it in Section 2. Denote by

π

m

:

M

g,n

→ M

g,n−m

the forgetful map that forgets the

last

m

marked points. Let

a

₁

, . . . , a

n

and

b

1

, . . . , b

g

be arbitrary integers and let

A

:=

P

a

i

and

B

:=

P

b

j

. The push-forward

π

g∗DRg

(

a

1

, . . . , a

n

,

−

A

−

B, b

1

, . . . , b

g

) is a cohomology class of

degree 0 and it is equal to

g

!

b

2

1

b

22

· · ·

b

2g

times the unit. This observation together with the string

equation for the intersection numbers (1.1) implies that

g

!

b

2₁

_{· · ·}

b

2_g

X

x

i

F

g

(

x

1

, . . . , x

n

) =

=

X

d1,...,dn≥0

Z

Mg,n+g+1

π

g∗

DR

g

(

a

1

, . . . , a

n

,

−

A

−

B, b

1

, . . . , b

g

)

ψ

d11

· · ·

ψ

ndn

!

x

d1

1

· · ·

x

dnn

.

Moreover, this polynomial is the term of the lowest degree 3

g

₋

2 +

n

with respect to the

x

-variables in the infinite series

X

m≥0

X

d1,...,dn≥0

Z

Mg,n+g+1

π

g∗DRm

(

a

1

, . . . , a

n

,

−

A

−

B, b

1

, . . . , b

g

)

ψ

d₁1

· · ·

ψ

ndn

!

x

d1 1

· · ·

x

dn

n

.

In Section 2 we derive an explicit formula for this series and then in Section 3 prove Theorem 1.1.

In Section 4 we do an explicit computation of the one-point and the two-point functions

using our general formula.

2. Double ramification cycles

Let

n

_≥

2 and let

a

₁

, . . . , a

n

be a list of integers satisfying

P

a

i

= 0. The double ramification

cycle DR

g

(

a

1

, . . . , a

n

) is a cohomology class in

H

2g

(

M

g,n

,

Q

). If not all of

a

i

’s are equal to zero,

then the restriction

DR

g

(

a

1

, . . . , a

n

)

|

_M_g,n

to the moduli space of smooth curves

_M

g,n

⊂ M

g,n

can be defined as the Poincar´e dual to

the locus of pointed smooth curves [

C, p

1

, . . . , p

n

] satisfying

O

C

(

P

a

i

p

i

)

∼

=

O

C

. We refer the

reader, for example, to [BSSZ15] for the general definition of the double ramification cycle on

the whole moduli space

_M

g,n

. We will often consider the Poincar´e dual to the double

ramifica-tion cycle DR

g

(

a

1

, . . . , a

n

). It is an element of

H

2(2g−3+n)

(

M

g,n

,

Q

) and, abusing our notation

a little bit, it will also be denoted by DR

g

(

a

1

, . . . , a

n

). An explicit formula for the double

ram-ification cycle DR

g

(

a

1

, . . . , a

n

) in terms of tautological classes in the cohomology

H

∗

(

M

g,n

,

Q

)

was recently obtained in [JPPZ16]. In particular, it implies that the double ramification

cy-cle DR

g

(

a

1

, . . . , a

n

) depends polynomially on the parameters

a

1

, . . . , a

n

.

Let us now formulate the main result of [BSSZ15]. Let

d

1

, . . . , d

n

be non-negative integers

satisfying

P

d

i

= 2

g

−

3 +

n

. Then the integral

Z

DRg(a1,...,an)

ψ

d1

1

· · ·

ψ

dn

n

(5)

is equal to the coefficient of

x

d1

1

· · ·

x

dnn

in the generating function

P

n

(

a

1

, . . . , a

n

;

x

1

, . . . , x

n

)

ζ

(

x

1

+

· · ·

+

x

n

)

−

δ

n,2

x

1

+

x

2

.

(2.2)

As it was discussed in the introduction, this generating series is a power series in

x

1

, . . . , x

n

with

coefficients from

C

[

a

₁

, . . . , a

n

]. We see that the integral (2.1) is a polynomial in

a

1

, . . . , a

n

. Note

that

ζ

(

x

) =

P

_i_≥₀ x2i+1

22i₍₂_i_+1)!

, then from the definition (1.2) it is easy to see that the series (2.2)

has the form

P

n

(

a

1

, . . . , a

n

;

x

1

, . . . , x

n

)

ζ

(

x

1

+

· · ·

+

x

n

)

−

δ

n,2

x

1

+

x

2

=

X

j≥0 2j−2+n>0

X

i

f

i,j

(

a

)

g

i,j

(

x

)

,

where

g

i,j

(

x

)

∈

C

[

x

1

, . . . , x

n

] is a polynomial of degree 2

j

−

3 +

n

,

f

i,j

(

a

)

∈

C

[

a

1

, . . . , a

n

] and

for any fixed

j

the second summation is finite. From this we conclude that all terms in the

series (2.2) have the geometrical meaning and, thus,

X

g≥0 2g−2+n>0

X

d1,...,dn≥0

Z

DRg(a1,...,an)

ψ

d1

1

· · ·

ψ

ndn

!

x

d1

1

· · ·

x

dnn

=

P

(

a

₁

, . . . , a

n

;

x

1

, . . . , x

n

)

ζ

(

x

1

+

· · ·

+

x

n

)

−

δ

n,2

x

1

+

x

2

.

(2.3)

Now we want to generalize formula (2.3) for the integrals over the double ramification cycles

with forgotten points. We begin with the following lemma.

Lemma 2.1.

For

n

_≥

2 we have

P

n

(

a

1

, . . . , a

n

;

x

1

, . . . , x

n

)

|

_x_n=0

=

ζ

(

a

n

(

x

1

+

· · ·

+

x

n−1

))

a

n

P

n−1

(

a

1

, . . . , a

n−1

;

x

1

, . . . , x

n−1

)

.

Proof.

For

n

= 2 we have

P

₂

(

a

₁

, a

₂

;

x

₁

, x

₂

) =

ζ

(

a

1

x

2

−

a

2

x

1

)

a

1

x

2

−

a

2

x

1

.

Therefore,

P

₂

(

a

₁

, a

₂

;

x

₁

, x

₂

)

_|

_x₂₌₀

=

ζ

(

a

2

x

1

)

a

2

x

1

=

ζ

(

a

2

x

1

)

a

2

P

₁

(

a

₁

;

x

₁

)

.

Suppose

n

_≥

3. Note that if we set

x

n

= 0, then the product

x

′2

· · ·

x

′n−1

on the right-hand

side of (1.2) vanishes unless

σ

(

n

) =

n

. Therefore, we obtain

P

n

(

a

1

, . . . , a

n

;

x

1

, . . . , x

n

)

|

_x_n=0

=

X

σ∈Sn

σ(1)=1, σ(n)=n

x

′₂

_{· · ·}

x

′_n₋₁

_×

×

ζ

a

′₁

a

′₂

x

′₁

x

′₂

· · ·

ζ

a

′₁

+

a

′₂

+

_{· · ·}

+

a

′_n₋₂

a

′_n₋₁

x

′₁

+

x

′₂

+

_{· · ·}

+

x

′_n₋₂

x

′_n₋₁

ζ

a

′₁

+

_{· · ·}

+

a

′_n₋₁

a

n

x

′₁

+

_{· · ·}

+

x

′_n₋₁

0 a

′

1

a

′2

x

′

1

x

′2

· · ·

a

′

n−2

a

′n−1

x

′

n−2

x

′n−1

a

′

n−1

a

n

x

′

n−1

0 =

=

ζ

(

a

n

(

x

1

+

· · ·

+

x

n−1

))

a

n

X

σ∈Sn−1

σ(1)=1

x

′₂

. . . x

′_n₋₂

ζ

a

′₁

a

′₂

x

′

1

x

′2

· · ·

ζ

a

′₁

+

a

′₂

+

_{· · ·}

+

a

′_n₋₂

a

′_n₋₁

x

′

1

+

x

′2

+

· · ·

+

x

′n−2

x

′n−1

a

′₁

a

′₂

x

′₁

x

′₂

· · ·

a

′_n₋₂

a

′_n₋₁

x

′_n₋₂

x

′_n₋₁

=

ζ

(

a

n

(

x

1

+

· · ·

+

x

n−1

))

a

n

P

n−1

(

a

1

, . . . , a

n−1

;

x

1

, . . . , x

n−1

)

.

The lemma is proved.

(6)

Lemma 2.2.

Let

n

_≥

3 ,

m

_≥

0 and

a

₁

, . . . , a

n

, b

1

, . . . , b

m

be integers satisfying

P

a

i

+

P

b

j

= 0

.

Then we have

X

g≥0

X

d1,...,dn≥0

Z

DRg(a1,...,an,b1,...,bm)

ψ

d1

1

· · ·

ψ

dn

n

!

x

d1 1

· · ·

x

dn

n

=

Q

m

i=1

S

(

b

i

X

)

S

(

X

)

X

m−1

_P

n

(

a

1

, . . . , a

n

;

x

1

, . . . , x

n

)

,

where

X

:=

P

x

i

.

Proof.

Using equation (2.3) and Lemma 2.1, we compute

X

g≥0

X

d1,...,dn≥0

Z

DRg(a1,...,an,b1,...,bm)

ψ

d1

1

· · ·

ψ

ndn

!

x

d1

1

· · ·

x

dnn

=

P

n+m

(

a

1

, . . . , a

n

, b

1

, . . . , b

m

;

x

1

, . . . , x

n+m

)

ζ

(

x

₁

+

_{· · ·}

+

x

n+m

)

xn+1=···=xn+m=0

=

1 ζ

(

X

)

m

Y

i=1

ζ

(

b

i

X

)

b

i

!

P

n

(

a

1

, . . . , a

n

;

x

1

, . . . , x

n

) =

=

Q

m

i=1

S

(

b

i

X

)

S

(

X

)

X

m−1

_P

n

(

a

1

, . . . , a

n

;

x

1

, . . . , x

n

)

.

The lemma is proved.

We keep the assumptions of the last lemma. Recall that by

π

m

:

M

g,n+m

→ M

g,n

we denote

the forgetful map that forgets the last

m

marked points. For a subset

I

_{⊂ {}

1 , . . . , m

_}

let

B

I

:=

X

i∈I

b

i

.

The following proposition generalizes formula (2.3).

Proposition 2.3.

We have

(2.4)

X

g≥0

X

d1,...,dn≥0

Z

πm∗DRg(a1,...,an,b1,...,bm)

ψ

d1

1

· · ·

ψ

dn

n

!

x

d1 1

· · ·

x

dn

n

=

1 S

(

X

)

X

I0⊔I1⊔...⊔In={1,...,m}

X

|I0|−1

n

Y

i=1

(

₋

x

i

)

|Ii|

Y

i∈I0

S

(

b

i

X

)

P

n

(

a

1

+

B

I1

, . . . , a

n

+

B

In

;

x

)

.

Proof.

For a subset

I

=

_{

i

1

, i

2

, . . . , i

|I|

} ⊂ {

1 , . . . , m

}

, where

i

1

< i

2

<

· · ·

< i

|I|

, we denote

by

b

I

the string

b

i1

, b

i2

, . . . , b

i|I|

. Clearly, formula (2.4) follows from Lemma 2.2 and the equation

Z

ψ

d1

1

· · ·

ψ

ndn

=

X

`n

i=0Ii={1,...,m}

|Ii|≤di

(

₋

1)

m−|I0|

Z

DRg

(

a1+BI₁,...,an+BIn,bI₀

)

n

Y

i=1

ψ

di−|Ii|

i

.

(2.5)

(7)

Z

ψ

d1

1

· · ·

ψ

dn

n

=

Z

(πm−1)∗DRg(a1,...,an,bm,b1,...,bm−1)

ψ

d1

1

· · ·

ψ

dn

n

(2.6)

−

X

1≤i≤n

di>0

Z

(πm−1)∗DRg(a1,...,ai+bm,...,an,b1,...,bm−1)

ψ

di−1

i

Y

j6=i

ψ

dj

j

.

For this we write

π

m

=

π

1

◦

π

m−1

, where

π

m−1

:

M

g,n+m

→ M

g,n+1

and

π

1

:

M

g,n+1

→ M

g,n

.

Then we have

Z

ψ

d1

1

· · ·

ψ

ndn

=

Z

(πm−1)∗DRg(a1,...,an,bm,b1,...,bm−1)

π

₁∗

ψ

d1

1

· · ·

ψ

ndn

.

For an integer

N

_≥

2 and a subset

J

_{∈ {}

1 , . . . , N

_}

,

_|

J

_{| ≥}

2, denote by

δ

J

0

the cohomology class

in

H

2

(

_M

g,N

,

Q

) that is Poincar´e dual to the divisor whose generic point is a nodal curve made

of one smooth component of genus 0 with the marked points labeled by the set

J

and of another

smooth component of genus

g

with the remaining marked points, joined at a separating node.

If 1

_≤

i

_≤

n

and

d

_≥

1, then we have

π

₁∗

ψ

d

i

=

ψ

di

−

δ

{i,n+1}

0

π

1∗

ψ

id−1

. Therefore,

π

∗₁

ψ

d1

1

· · ·

ψ

ndn

=

ψ

d1

1

· · ·

ψ

dnn

−

X

1≤i≤n di>0

δ

₀{i,n+1}

π

₁∗

ψ

di−1

i

Y

j6=i

ψ

dj

j

!

.

We have (see [BSSZ15] and, in particular, Section 2.1 there for the explanation of the

nota-tion

⊠

)

δ

{₀i,n+1}

_·

(

π

m−1

)∗

DR

g

(

a

1

, . . . , a

n

, b

m

, b

1

, . . . , b

m−1

) =

= (

π

m−1

)

∗

(DR

0

(

a

i

, b

m

,

−

a

i

−

b

m

)

⊠

DR

g

(

a

1

, . . . ,

a

b

i

, . . . , a

n

, b

1

, . . . , b

m−1

, a

i

+

b

m

))

.

Therefore,

Z

(πm−1)∗DRg(a1,...,an,bm,b1,...,bm−1)

δ

{₀i,n+1}

π

∗₁

ψ

di−1

i

Y

j6=i

ψ

dj

j

!

=

Z

(πm−1)∗DRg(a1,...,ai+bm,...,an,b1,...,bm−1)

ψ

di−1

i

Y

j6=i

ψ

dj

j

.

This completes the proof of equation (2.6).

It is easy to see that, applying formula (2.6)

m

times, we get equation (2.5). The proposition

is proved.

Finally, let us formulate an important geometric property of the double ramification cycle

that we will use in the next section. Let

g

_≥

0,

n

_≥

3 and

a

₁

, . . . , a

n

, b

1

, . . . , b

g

be integers

satisfying

P

a

i

+

P

b

j

= 0. Then we have (see [BSSZ15, Example 3.7])

π

g∗

DR

g

(

a

1

, . . . , a

n

, b

1

, . . . , b

g

) =

g

!

b

21

· · ·

b

2g

[

M

g,n

]

,

(2.7)

π

g∗DRm

(

a

1

, . . . , a

n

, b

1

, . . . , b

g

) = 0

,

m < g.

(2.8)

The last property implies that the coefficient of a monomial

x

d1

1

· · ·

x

dnn

in the generating

se-ries (2.4) is zero, if

P

d

i

<

3 m

−

3 +

n

. We will use this observation in the next section.

3. Proof of Theorem 1.1

(8)

3.1. Reduction to the case

n

_≥

3 .

The intersection numbers

_h

τ

d1

· · ·

τ

dn

i

g

satisfy the string

equation

h

τ

₀

τ

d1

· · ·

τ

dn

i

g

=

X

1≤i≤n di>0

*

τ

di−1

Y

j6=i

τ

dj

+

g

,

with one exceptional case

_h

τ

₀3

_i0

= 1. The string equation implies that for

n

_≥

2 we have

F

(

x

1

, . . . , x

n

)

|

xn=0

= (

x

1

+

· · ·

+

x

n−1

)

F

(

x

1

, . . . , x

n−1

) +

δ

n,3

.

(3.1)

Denote the right-hand side of (1.3) by

_G

(

x

1

, . . . , x

n

). Let us show that the function

G

also

satisfies equation (3.1). We compute

1 Q

n i=1

√

2 πx

i

Z

Rn

e

−

Pn i=1

a2_i 2_xi

_P

n

√

−

1 a

₁

, . . . ,

√

₋

1 a

n

;

x

1

, . . . , x

n

da

₁

_{· · ·}

da

n

xn=0

=

1 (2

π

)

n2

Z

Rn

e

−Pni=1 a2_i

2

P

_n

√

−

x

₁

a

₁

, . . . ,

√

−

x

_n

a

_n

;

x

₁

, . . . , x

_n

da

₁

· · ·

da

_n

xn=0

by Lemma 2.1

=

x

1

+

· · ·

+

x

n−1

(2

π

)

n2

Z

Rn

e

−Pni=1 a2_i

2

P

_n

−1

√

−

x

1

a

1

, . . . ,

√

−

x

n−1

a

n−1

;

x

1

, . . . , x

n−1

da

₁

_{· · ·}

da

n

=

x

1

_Q

+

_n₋

· · ·

₁

+

x

n−1

i=1

√

2 πx

i

Z

Rn−1

e

−

Pn−1 i=1

a2_i 2_xi

_P

n−1

√

−

1 a

₁

, . . . ,

√

₋

1 a

n−1

;

x

1

, . . . , x

n−1

da

₁

_{· · ·}

da

n−1

.

Thus, we obtain

G

(

x

₁

, . . . , x

n

)

|

_x_n=0

= (

x

1

+

· · ·

+

x

n−1

)

G

(

x

1

, . . . , x

n−1

) +

δ

n,3

.

(3.2)

We see that if equation (1.3) holds for

n

=

n

₀

, then it holds for all

n

_≤

n

₀

. Hence, it is sufficient

to prove equation (1.3) for

n

_≥

3.

3.2. Case

n

_≥

3 .

We assume that

n

_≥

3. Let

g

_≥

0 and consider arbitrary integers

a

1

, . . . , a

n

and

b

1

, . . . , b

g

. Let

A

:=

n

X

i=1

a

i

,

B

:=

g

X

i=1

b

i

,

X

:=

n

X

i=1

x

i

.

Equations (2.7) and (3.1) imply that

g

!

b

2₁

_{· · ·}

b

2_g

X

_F

g

(

x

1

, . . . , x

n

) =

g

!

b

12

· · ·

b

2g

F

g

(

x

1

, . . . , x

n

, x

n+1

)

|

xn+1=0

=

X

d1,...,dn≥0

Z

πg∗DRg(a1,...,an,−A−B,b1,...,bg)

ψ

d1

1

· · ·

ψ

ndn

!

x

d1

1

· · ·

x

dnn

(3.3)

independently of

a

₁

, . . . , a

n

. Moreover, by the remark at the end of Section 2, expression (3.3)

is equal to the term of the lowest degree 3

g

₋

2 +

n

with respect to the

x

-variables in the series

X

m≥0

X

d1,...,dn≥0

Z

πg∗DRm(a1,...,an,−A−B,b1,...,bg)

ψ

d1

1

· · ·

ψ

dn

n

!

x

d1 1

· · ·

x

dn

n

.

(3.4)

Proposition 2.3 and Lemma 2.1 imply that

X

m≥0

X

d1,...,dn≥0

Z

πg∗DRm(a1,...,an,−A−B,b1,...,bg)

ψ

d1

1

· · ·

ψ

ndn

!

x

d1

1

· · ·

x

dnn

=

(3.5)

=

S

((

−

A

−

B

)

X

)

S

(

X

)

X

`n

i=0Ii={1,...,g}

X

|I0|

n

Y

i=1

(

₋

x

i

)

|Ii|

Y

i∈I0

(9)

In the last expression the multiplication by

S((−_SA₍−_XB₎)X)

doesn’t change the lowest degree term

with respect to the

x

-variables. Therefore, we obtain

g

!

b

2₁

_{· · ·}

b

2_g

X

_F

g

(

x

1

, . . . , x

n

) =

=





X

`n

i=0Ii={1,...,g}

X

|I0|

n

Y

i=1

(

₋

x

i

)

|Ii|

Y

i∈I0

S

(

b

i

X

)

P

n

(

a

1

+

B

I1

, . . . , a

n

+

B

In

;

x

)





3g−2+n

=





X

`n

i=0Ii={1,...,g}

X

|I0|

n

Y

i=1

(

₋

x

i

)

|Ii|

Y

i∈I0

S

(

b

i

X

)

P

n

(

B

I1

, . . . , B

In

;

x

)





3g−2+n

,

where by [

_·

]

d

we denote the degree

d

part with respect to the

x

-variables. Equivalently, we can

write

(3.6)

g

!

X

_F

g

(

x

1

, . . . , x

n

) =

=





Coef

b2₁···b2 g

X

`n

i=0Ii={1,...,g}

X

|I0|

n

Y

i=1

(

₋

x

i

)

|Ii|

Y

i∈I0

S

(

b

i

X

)

P

n

(

B

I1

, . . . , B

In

;

x

)





3g−2+n

.

From the definition (1.2) we see that the series

P

n

(

a

1

, . . . , a

n

;

x

1

, . . . , x

n

) satisfies the following

homogeneity property:

P

n

(

λ

−1

a

1

, . . . , λ

−1

a

n

;

λx

1

, . . . , λx

n

) =

λ

n−2

P

n

(

a

1

, . . . , a

n

;

x

1

, . . . , x

n

)

.

Therefore, the expression in the square brackets on the right-hand side of (3.6) has automatically

degree 3

g

₋

2 +

n

in the

x

-variables. Since

S

(

z

) = 1 +

z2

24

+

. . .

, we have

Coef

b2 1···b2g

Y

i∈I0

S

(

b

i

X

)

P

n

(

B

I1

, . . . , B

In

;

x

)

!

=

X

2|I0|

24

|I0|

Coef

Q

i /∈I₀b2i

P

n

(

B

I1

, . . . , B

In

;

x

)

.

Note that

Coef

Q_{i /}_∈_I

0b 2 i

n

Y

i=1

B

di

Ii

=

(Q

n

i=1

di!

2di/2

,

if

d

i

= 2

|

I

i

|

,

0 ,

otherwise

.

Therefore, we get

Coef

b2 1···b2g

X

`n

i=0Ii={1,...,g}

X

|I0|

n

Y

i=1

(

₋

x

i

)

|Ii|

Y

i∈I0

S

(

b

i

X

)

P

n

(

B

I1

, . . . , B

In

;

x

) =

=

X

`n

i=0Ii={1,...,g}

X

3|I0|

24

|I0|

n

Y

i=1

(

₋

x

i

)

|Ii|

(2

_|

I

i

|

)!

2

|Ii|

Coef

Qn i=1a

2|_Ii|

i

P

n

(

a

;

x

) =

=

X

m0,m1,...,mn≥0

m0+···+mn=g

g

!

m

0

!

· · ·

m

n

!

X

3

24

m0

Y

n

i=1

(

₋

x

i

)

mi

(2

m

i

)!

2

mi

Coef

Qn i=1a

2_mi

i

P

n

(

a

;

x

) =

=

X

m0,m1,...,mn≥0

m0+···+mn=g

g

!

m

₀

!

X

3

24

m0

Y

n

i=1

((

₋

x

i

)

mi

(2

m

i

−

1)!!) Coef

Qn i=1a

2_mi

(10)

Finally, we obtain

X

_F

(

x

1

, . . . , x

n

) =

X

g≥0

X

_F

g

(

x

1

, . . . , x

n

) =

=

X

m0,m1,...,mn≥0

1 m

₀

!

X

3

24

m0

Y

n

i=1

((

₋

x

i

)

mi

(2

m

i

−

1)!!) Coef

Qn i=1a

2_mi i

P

n

(

a

;

x

)

by (1.4)

=

e

X3 24

Q

√

2 πx

i

Z

Rn

e

−

P a2_i 2_xi

_P

n

√

−

1 a

;

x

da.

This completes the proof of the theorem.

4. Examples

For

n

= 1 Theorem 1.1 immediately gives the well-known formula (see e.g. [FP00])

F

(

x

1

) =

e

x

3 1 24

−

1 x

2 1

.

Suppose

n

= 2. We have

P

2

(

a

1

, a

2

;

x

1

, x

2

) =

ζ

(

a

₁

x

₂

₋

a

₂

x

₁

)

a

1

x

2

−

a

2

x

1

=

S

(

a

1

x

2

−

a

2

x

1

) =

X

n≥0

(

a

₁

x

₂

₋

a

₂

x

₁

)

2n

(2

n

+ 1)!2

2n

=

X

n≥0

X

m1+m2=2n

(

a

₁

x

₂

)

m1

₍

₋

_a

2

x

1

)

m2

(2

n

+ 1)2

2n

_m

1

!

m

2

!

.

Therefore, we obtain

e

(x1+x2) 3 24

2 π

√

x

₁

x

₂

Z

R2

e

−

a2₁ 2x₁−

a2₂ 2x₂

_P

2

(

√

−

1 a

1

,

√

−

1 a

2

;

x

1

, x

2

)

da

=

e

(x₁₊x₂₎3 24

2 π

√

x

1

x

2

X

n≥0

X

m1+m2=n

Z

R2

e

−

a2₁ 2x1−

a2₂ 2x2

(

₋

1)

n

_a

2m1

2

a

21m2

x

21m1

x

22m2

2

2n

₍₂

_n

_{+ 1)(2}

_m

1

)!(2

m

2

)!

da

=

e

(x1+x2) 3

24

X

n≥0

X

m1+m2=n

(

₋

1)

n

_x

m1

2

x

m12

x

21m1

x

22m2

2

3n

₍₂

_n

_{+ 1)}

_m

1

!

m

2

!

=

e

x 3 1+x32

24

e

x₁x₂₍x₁₊x₂₎

8

X

n≥0

(

₋

1)

n

₍

_x

1

x

2

(

x

1

+

x

2

))

n

8

n

₍₂

_n

_{+ 1)}

_n

_!

=

e

x 3 1+x32

24

X

n≥0

x

1

x

2

(

x

1

+

x

2

)

8

n

_X

m1+m2=n

(

₋

1)

m2

m

₁

!

m

₂

!(2

m

₂

+ 1)

.

Let

C

n

:=

P

m1+m2=n

(−1)m₂

m1!m2!(2m2+1)

. For

n

≥

1 we compute

C

n

=

1 n

!

Z

1

0

(1

₋

y

2

)

n

dy

=

2 (

n

₋

1)!

Z

1

0

y

2

(1

₋

y

2

)

n−1

dy

=

₋

2 nC

n

+ 2

C

n−1

.

Therefore,

C

n

=

₂_n2₊₁

C

n−1

, and, since

C

0

= 1, we get

C

n

=

2

n

(2n+1)!!

. Hence, we obtain

e

(x1+x2) 3 24

2 π

√

x

1

x

2

Z

R2

e

−

a2₁ 2x1−

a2₂

2x2

P

₂

(

√

−

1 a

₁

,

√

−

1 a

₂

;

x

₁

, x

₂

)

da

=

e

x3₁₊x3₂

24

X

n≥0

n

!

(2

n

+ 1)!

_x

1

x

2

(

x

1

+

x

2

)

2

n

(11)

As a result, we get the following formula for the two-point function:

F

(

x

₁

, x

₂

) =

e

x3₁₊x3₂ 24

(

x

1

+

x

2

)

X

n≥0

n

!

(2

n

+ 1)!

_x

1

x

2

(

x

1

+

x

2

)

2

n

−

_x

1

+

x

2

.

This formula was found by R. Dijkgraaf (see e.g. [FP00]).

References

[BDY15] M. Bertola, B. Dubrovin, D. Yang. Correlation functions of the KdV hierarchy and applications to intersection numbers over Mg,n. Physica D. Nonlinear Phenomena 327 (2016), 30–57.

[BH07] E. Br´ezin, S. Hikami.Vertices from replica in a random matrix theory. Journal of Physics. A. Math-ematical and Theoretical 40 (2007), no. 45, 13545–13566.

[BS11] A. Buryak, S. Shadrin. A new proof of Faber’s intersection number conjecture. Advances in Mathe-matics 228 (2011), 22–42.

[BSSZ15] A. Buryak, S. Shadrin, L. Spitz, D. Zvonkine.Integrals of ψ-classes over double ramification cycles. American Journal of Mathematics 137 (2015), no. 3, 699–737.

[FP00] C. Faber, R. Pandharipande.Logarithmic series and Hodge integrals in the tautological ring. With an appendix by Don Zagier. Dedicated to William Fulton on the occasion of his 60th birthday. Michigan Mathematical Journal 48 (2000), 215–252.

[JPPZ16] F. Janda, R. Pandharipande, A. Pixton, D. Zvonkine.Double ramification cycles on the moduli spaces of curves. arXiv:1602.04705.

[Kon92] M. Kontsevich.Intersection theory on the moduli space of curves and the matrix Airy function. Com-munications in Mathematical Physics 147 (1992), no. 1, 1–23.

[Oko02] A. Okounkov.Generating functions for intersection numbers on moduli spaces of curves. International Mathematics Research Notices 2002, no. 18, 933–957.

[Wit91] E. Witten.Two-dimensional gravity and intersection theory on moduli space. Surveys in differential geometry (Cambridge, MA, 1990), 243–310, Lehigh Univ., Bethlehem, PA, 1991.

Department of Mathematics, ETH Z¨urich, Switzerland, and

Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Russian Fed-eration