Equivalence of the Open KdV and the Open Virasoro Equations for the Moduli Space of Riemann Surfaces with Boundary

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Equations for the Moduli Space of Riemann Surfaces with Boundary.

White Rose Research Online URL for this paper:

http://eprints.whiterose.ac.uk/132959/

Version: Accepted Version

Article:

Buryak, A (2015) Equivalence of the Open KdV and the Open Virasoro Equations for the

Moduli Space of Riemann Surfaces with Boundary. Letters in Mathematical Physics, 105

(10). pp. 1427-1448. ISSN 0377-9017

https://doi.org/10.1007/s11005-015-0789-3

eprints@whiterose.ac.uk https://eprints.whiterose.ac.uk/

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EQUATIONS FOR THE MODULI SPACE OF RIEMANN SURFACES WITH

BOUNDARY

ALEXANDR BURYAK

Abstract. In a recent paper R. Pandharipande, J. Solomon and R. Tessler initiated a study of the intersection theory on the moduli space of Riemann surfaces with boundary. The authors conjectured KdV and Virasoro type equations that completely determine all intersection num-bers. In this paper we study these equations in detail. In particular, we prove that the KdV and the Virasoro type equations for the intersection numbers on the moduli space of Riemann surfaces with boundary are equivalent.

1. Introduction

Denote by

M

g,n

the moduli space of smooth complex algebraic curves of genus

g

with

n

dis-tinct marked points. In [DM69] P. Deligne and D. Mumford defined a natural compactification

M

g,n

⊂ M

g,n

via stable curves (with possible nodal singularities). The moduli space

M

g,n

is

a nonsingular complex orbifold of dimension 3

g

−

3 +

n

.

A new direction in the study of the moduli space

M

g,n

was opened by E. Witten [Wit91].

The class

ψ

i

∈

H

2

(

M

g,n

;

C

) 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. Intersection numbers

hτ

k1

τ

k2

. . . τ

kn

i

c g

are defined as follows:

h

τ

k1

τ

k2

. . . τ

kn

i

c g

:=

Z

Mg,n

ψ

k1

1

ψ

k2

2

. . . ψ

kn n

.

The superscript

c

here signals integration over the moduli of closed Riemann surfaces. Let us

introduce variables

u, t

₀

, t

₁

, t

₂

, . . .

and consider the generating series

F

c

(

t

₀

, t

₁

, . . .

;

u

) :=

X

g≥0,n≥1 2g−2+n>0

u

2g−2

n

!

X

k1,...,kn≥0

h

τ

k1

τ

k2

. . . τ

kn

i

c

g

t

k1

t

k2

. . . t

kn

.

E. Witten ([Wit91]) proved that the generating series

F

c

_{satisfies the so-called string equation}

and conjectured that the second derivative

∂2Fc ∂t2

0

is a solution of the KdV hierarchy. Witten’s

conjecture was proved by M. Kontsevich ([Kon92]).

There is a reformulation of Witten’s conjecture due to R. Dijkgraaf, E. Verlinde and H.

Ver-linde ([DVV91]) in terms of the Virasoro algebra. They defined certain quadratic differential

operators

L

n

,

n

≥ −

1, and proved that Witten’s conjecture is equivalent to the equations

L

n

exp(

F

c

) = 0

,

(1.1)

that are called the Virasoro equations. The operators

L

n

satisfy the relation [

L

n

, L

m

] = (

n

−

m

)

L

n+m

.

In [PST14] the authors initiated a study of the intersection theory on the moduli space of

Riemann surfaces with boundary. They introduced intersection numbers on this moduli space

and completely described them in genus 0. In higher genera the authors conjectured that the

generating series of the intersection numbers satisfies certain partial differential equations that

are analagous to the string, the KdV and the Virasoro equations. In [PST14] these equations

were called the open string, the open KdV and the open Virasoro equations.

2010Mathematics Subject Classification. Primary 35Q53; Secondary 14H10.

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The open KdV equations and the open Virasoro equations provide two different ways to

describe the intersection numbers on the moduli space of Riemann surfaces with boundary. It

is absolutely non-obvious that these two descriptions are equivalent and it was left in [PST14]

as a conjecture. The main purpose of this paper is to prove this conjecture. We show that the

system of the open KdV equations has a unique solution specified by a certain initial condition

that corresponds to the simplest intersection numbers in genus 0. The main result of the paper

is the proof of the fact that this solution satisfies the open Virasoro equations. This proves that

the open KdV and the open Virasoro equations give equivalent descriptions of the intersection

numbers on the moduli space of Riemann surfaces with boundary.

1.1. Witten’s conjecture and the Virasoro equations.

In this section we review Witten’s

conjecture and its reformulation due to R. Dijkgraaf, E. Verlinde and H. Verlinde.

One of the basic properties of the generating series

F

c

is the so-called string equation ([Wit91]):

∂F

c

∂t

0

=

X

n≥0

t

n+1

∂F

c

∂t

n

+

t

2 0

2 u

2

.

(1.2)

1.1.1. KdV equations.

E. Witten conjectured ([Wit91]) that the generating series

F

c

_{is the}

logarithm of a tau-function of the KdV hierarchy. In particular, it means that it satisfies the

following system:

u

−2

2 n

+ 1

2 ∂

3

F

c

∂t

2

0

∂t

n

=

∂

2

_F

c

∂t

2 0

∂

3

F

c

∂t

2

0

∂t

n−1

+

1

2 ∂

3

F

c

∂t

3

0

∂

2

F

c

∂t

₀

∂t

n−1

+

1

8 ∂

5

F

c

∂t

4

0

∂t

n−1

,

n

≥

1 .

(1.3)

Moreover, E. Witten showed ([Wit91]) that the KdV equations (1.3) together with the string

equation (1.2) and the initial condition

F

c

|

_t_∗₌₀

= 0 uniquely determine the power series

F

c

.

1.1.2. Virasoro equations.

The Virasoro operators

L

n

, n

≥ −

1, are defined as follows:

L

n

:=

X

i≥0

(2

i

+ 2

n

+ 1)!!

2

n+1

₍₂

_i

₋

_1)!!

(

t

i

−

δ

i,1

)

∂

∂t

i+n

+

u

2

n−1

X

i=0

(2

i

+ 1)!!(2

n

−

2 i

−

1)!!

2

n+1

∂

2

∂t

i

∂t

n−1−i

+

δ

n,−1

t

2₀

2 u

2

+

δ

n,0

1

16 .

They satisfy the commutation relation

[

L

n

, L

m

] = (

n

−

m

)

L

n+m

.

(1.4)

The Virasoro equations say that

L

n

exp(

F

c

) = 0

,

n

≥ −

1 .

(1.5)

For

n

=

−

1, this equation is equivalent to the string equation (1.2).

R. Dijkgraaf, E. Verlinde and H. Verlinde ([DVV91]) proved that Witten’s conjecture is

equivalent to the Virasoro equations. To be precise, they proved the following. Suppose a

power series

F

satisfies the string equation (1.2) and the KdV equations (1.3). Then

F

satisfies

the Virasoro equations (1.5). We review the proof of this fact in Appendix A.

1.2. Moduli of Riemann surfaces with boundary.

Here we briefly recall the basic

defi-nitions concerning the moduli space of Riemann surfaces with boundary. We refer the reader

to [PST14] for details.

Let ∆

∈

C

_{be the open unit disk, and let ∆ be its closure. An extendable embedding of the}

open disk ∆ in a closed Riemann surface

f

: ∆

→

C

is a holomorphic map which extends to

a holomorphic embedding of an open neighbourhood of ∆. Two extendable embeddings are

disjoint, if the images of ∆ are disjoint.

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Given a Riemann surface with boundary (

X, ∂X

), we can canonically construct the double

via the Schwartz reflection through the boundary. The double

D

(

X, ∂X

) of (

X, ∂X

) is a closed

Riemann surface. The doubled genus of (

X, ∂X

) is defined to be the usual genus of

D

(

X, ∂X

).

On a Riemann surface with boundary (

X, ∂X

), we consider two types of marked points.

The markings of interior type are points of

X\∂X

. The markings of boundary type are points

of

∂X

. Let

M

g,k,l

denote the moduli space of Riemann surfaces with boundary of doubled

genus

g

with

k

distinct boundary markings and

l

distinct interior markings. The moduli

space

M

g,k,l

is defined to be empty unless the stability condition 2

g

−

2 +

k

+ 2

l >

0 is satisfied.

The moduli space

M

g,k,l

is a real orbifold of real dimension 3

g

−

3 +

k

+ 2

l

.

The psi-classes

ψ

i

∈

H

2

(

M

g,k,l

;

C

) are defined as the first Chern classes of the cotangent line

bundles for the interior markings. The authors of [PST14] do not consider the cotangent lines

at boundary points. Naively, open intersection numbers are defined by

τ

a1

τ

a2

. . . τ

al

σ

k

o g

:=

Z

Mg,k,l

ψ

a1

1

ψ

a2

2

. . . ψ

al l

.

(1.6)

To rigorously define the right-hand side of (1.6), at least three significant steps must be taken:

• A natural compactification

M

g,k,l

⊂ M

g,k,l

must be constructed. Candidates for

M

g,k,l

are themselves real orbifolds with boundary

∂

M

g,k,l

.

• For integration over

M

g,k,l

to be well-defined, boundary conditions of the integrand

along

∂

M

g,k,l

must be specified.

• Problems with an orientation should be solved, since the moduli space

M

g,k,l

is in

general non-orientable.

The authors of [PST14] completed all these steps and rigorously defined open intersection

numbers in genus 0. Moreover, they obtained a complete description of them. In higher

genera, even though open intersection numbers are not well-defined, the authors of [PST14]

proposed a beautiful conjectural description of them that we are going to recall in the next

section.

1.3. Open KdV and open Virasoro equations.

In this section we review the KdV and the

Virasoro type equations from [PST14] for the open intersection numbers (1.6).

Introduce one more formal variable

s

and define the generating series

F

o

_by

F

o

(

t

0

, t

1

, . . . , s

;

u

) :=

X

g,k,l≥0 2g−2+k+2l>0

u

g−1

k

!

l

!

X

a1,...,al≥0

τ

a1

. . . τ

al

σ

k

o

g

t

a1

. . . t

an

s

k

_.

First of all, the authors of [PST14] conjectured the following analog of the string equation (1.2):

∂F

o

∂t

₀

=

X

i≥0

t

i+1

∂F

o

∂t

i

+

u

−1

s.

(1.7)

They call it the open string equation. The authors also proved that the following initial

condi-tion holds:

F

o

|

_t

≥1=0

=

u

−1

s

3

6 +

t

0

s

.

(1.8)

1.3.1. Open KdV equations.

The authors of [PST14] conjectured that the generating series

F

o

satisfies the following system of equations:

2 n

+ 1

2 ∂F

o

∂t

n

=

u

∂F

o

∂s

∂F

o

∂t

n−1

+

u

∂

2

_F

o

∂s∂t

n−1

+

u

2

2 ∂F

o

∂t

0

∂

2

_F

c

∂t

0

∂t

n−1

−

u

2

4 ∂

3

_F

c

∂t

2₀

∂t

n−1

,

n

≥

1 .

(1.9)

They call these equations the open KdV equations. It is clear that the open KdV

equa-tions (1.9), the initial condition (1.8) and the potential

F

c

_{uniquely determine the series}

_F

o

_.

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1.3.2. Open Virasoro equations.

In [PST14] the authors introduced the following operators:

L

n

:=

L

n

+

u

n

s

∂

n+1

∂s

n+1

+

3 n

+ 3

4 u

n

∂

n

∂s

n

,

n

≥ −

1 .

These operators satisfy the same commutation relation as the operators

L

n

:

[

L

n

,

L

m

] = (

n

−

m

)

L

n+m

.

(1.10)

In [PST14] the authors conjectured the following analog of the Virasoro equations (1.5):

L

n

exp(

F

o

+

F

c

) = 0

,

n

≥ −

1 .

(1.11)

Clearly, equations (1.11), the initial condition

F

o

_|

t∗=0

=

u

−1s3

6

and the potential

F

c

_completely

determine the series

F

o

.

1.4. Main result.

Here we formulate two main results of the paper.

Theorem 1.1.

1. The system of the open KdV equations

(1.9)

has a unique solution that

satisfies the initial condition

(1.8)

.

2. This solution satisfies the following equation:

∂F

o

∂s

=

u

1

2 ∂F

o

∂t

0

2

+

1

2 ∂

2

_F

o

∂t

2 0

+

∂

2

_F

c

∂t

2 0

!

.

(1.12)

Theorem 1.2.

The series

F

o

_{determined by Theorem 1.1 satisfies the open Virasoro}

equa-tions

(1.11)

.

1.5. Burgers-KdV hierarchy and descendants of

s.

In Section 3 we will construct a

certain system of evolutionary partial differential equations with one spatial variable. It will be

called the Burgers-KdV hierarchy. We will prove that the series

F

o

determined by Theorem 1.1

satisfies the half of the equations of this hierarchy. The remaining flows of the Burgers-KdV

hierarchy suggest a way to introduce new variables

s

₁

, s

₂

, . . .

in the open potential

F

o

_{. These}

variables can be viewed as descendants of

s

. We hope that our idea can help to give a geometrical

construction of descendants of

s

, at least in genus 0.

1.6. Open KdV equations and the wave function of the KdV hierarchy.

In the

work [Bur14], that appeared while this paper was under consideration in the journal, we

ob-served that the open KdV equations are closely related to the equations for the wave function of

the KdV hierarchy. Using this observation our original proof of Theorem 1.1 can be simplified.

We discuss it in Section 3.7.

1.7. Acknowledgements.

The author would like to thank R. Pandharipande, J. Solomon and

R. Tessler for very useful discussions.

We would like to thank the anonymous referee for valuable remarks and suggestions that

allowed us to improve the exposition of this paper.

The author was supported by grant ERC-2012-AdG-320368-MCSK in the group of R.

Pand-haripande at ETH Zurich, by Russian Federation Government grant no. 2010-220-01-077 (ag.

no. 11.634.31.0005), the grants RFBR-10-010-00678, NSh-4850.2012.1, the Moebius Contest

Foundation for Young Scientists and ”Dynasty” foundation.

1.8. Organization of the paper.

In Section 2 we recall some basic facts about evolutionary

PDEs with one spatial variable and give a slight reformulation of Witten’s conjecture.

In Section 3 we construct the Burgers-KdV hierarchy and prove that it has a solution for

arbitrary polynomial initial conditions. We also construct a specific solution of the half of the

Burgers-KdV hierarchy that satisfies the open string equation (1.7).

Section 4 contains the proofs of Theorems 1.1 and 3.1.

In Section 5 we prove Theorem 1.2.

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2. Evolutionary partial differential equations

In this section we recall some basic facts about evolutionary PDEs with one spatial variable.

We also review the construction of the KdV hierarchy and give a slight reformulation of Witten’s

conjecture that will be useful in the subsequent sections. All this material is well-known. We

refer the reader to the book [Olv86] for the details about this subject.

2.1. Ring of differential polynomials.

Let us fix an integer

N

≥

1. Consider variables

v

i j

,

1 ≤

i

≤

N

,

j

≥

0. We will often denote

v

i

0

by

v

i

and use an alternative notation for the

variables

v

i

1

, v

i2

, . . .

:

v

_xi

:=

v

₁i

,

v

_xxi

:=

v

₂i

, . . . .

Denote by

A

v1_,v2_,...,vN

the ring of polynomials in the variables

u, u

−1

and

v

i_j

. The elements

of

A

v1_,...,vN

will be called differential polynomials.

An operator

∂

x

:

A

v1_,...,vN

→ A

_v1_,...,vN

is defined as follows:

∂

x

:=

N

X

i=1

X

s≥0

v

_si₊₁

∂

∂v

i

s

.

Consider now a sequence of differential polynomials

P

i

j

(

v, v

x

, . . .

;

u

)

∈ A

v1_,...,vN

, 1

≤

i

≤

N

,

j

≥

0. Consider the variables

v

i

as formal power series in

x, τ

0

, τ

1

, . . .

with the coefficients

from

C

_[

_{u, u}

−1

_{]. A system of evolutionary PDEs with one spatial variable is a system of the}

form:

∂v

i

∂τ

j

=

P

_ji

(

v, v

x

, . . .

;

u

)

,

1 ≤

i

≤

N,

j

≥

0 .

(2.1)

2.2. Existence of a solution.

Here we give a sufficient condition for system (2.1) to have

a solution. Let

P

1

, . . . , P

N

∈ A

v1_,...,vN

be some differential polynomials. Define an

opera-tor

V

P1_,...,PN

:

A

_v1_,...,vN

→ A

_v1_,...,vN

by

V

P1_,...,PN

:=

N

X

i=1

X

j≥0

(

∂

_xj

P

i

)

∂

∂v

i

j

.

Denote the space of all these operators by

E

v1_,...,vN

. It is a Lie algebra: for

Q

1

, Q

2

, . . . , Q

N

∈

A

v1_,...,vN

, we have

[

V

_P1_,...,PN

, V

_Q1_,...,QN

] =

V

_R1_,...,RN

,

where

R

i

=

V

_P1_,...,PN

Q

i

−

V

_Q1_,...,QN

P

i

.

Consider an operator

O

=

P

_i_≥₀

O

i

∂

xi

,

O

i

∈ A

v1_,...,vN

. We will use the following notation:

V

_P₁_,...,PN

· O

:=

X

i≥0

V

_P1_,...,PN

O

_i

∂

_xi

.

Let us again consider system (2.1). The following lemma is well-known (see e.g. [Olv86]).

Lemma 2.1.

Suppose that, for any

i, j

≥

0 , we have

[

V

_P1

i,...,PiN

, V

Pj1,...,PjN

] = 0

. Then, for an

arbitrary initial condition

v

i

_|

τj=0

=

f

i

(

x, u

)

, where

f

i

(

x, u

)

∈

C

[

x, u, u

−1

]

, system

(2.1)

has a

unique solution.

2.3. KdV hierarchy.

Consider a variable

w

and the ring

A

w

. Define differential polynomials

K

n

∈ A

w

,

n

≥

0, by the following recursion:

K

0

=

w,

∂

x

K

n

=

2 u

2

2 n

+ 1

w∂

x

+

1

2 w

x

+

1

8 ∂

3

x

K

n−1

,

for

n

≥

1 .

(2.2)

It is a non-trivial fact that the right-hand side of (2.2) lies in the image of the operator

∂

x

(see e.g. [MJD00]). So the recursion (2.2) determines a differential polynomial

K

n

up to a

(7)

Consider now the variable

w

as a power series in variables

x, t

₁

, t

₂

, . . .

with the coefficients

from

C

_[

_{u, u}

−1

_{]. The KdV hierarchy is the following system of partial differential equations:}

∂w

∂t

n

=

∂

x

K

n

,

n

≥

1 .

(2.3)

Another form of Witten’s conjecture says that the second derivative

∂2Fc ∂t2

0

is a solution of

the KdV hierarchy (2.3). Here we identify

x

with

t

0

. This form of Witten’s conjecture is

equivalent to the form that was stated in Section 1.1 ([Wit91]). Let us formulate it precisely.

Let

w

(

x

=

t

0

, t

1

, t

2

, . . .

;

u

) be the solution of the KdV hierarchy (2.3) specified by the initial

condition

w

|

t≥1=0

=

u

−2

_x

_{. Let}

S

:=

∂

∂t

₀

−

X

n≥0

t

n+1

∂

∂t

n

.

It is easy to show that

Sw

=

u

−2

and there exists a unique power series

F

(

t

0

, t

1

, . . .

;

u

) such

that

w

=

∂_∂t2F2 0

,

SF

=

t20

2u2

and

F

|

t∗=0

= 0. Moreover, we have ([Wit91])

∂

2

_F

∂t

0

∂t

n

=

K

n

|

_wi₌∂i+2F ∂ti₀+2

and, therefore,

F

satisfies system (1.3).

3. Burgers-KdV hierarchy

In this section we construct the Burgers-KdV hierarchy and prove that its flows satisfy the

commutation relation from Lemma 2.1. This guarantees that the hierarchy has a solution for

arbitrary polynomial initial conditions. We also construct a specific solution of the half of

the Burgers-KdV hierarchy that satisfies the open string equation (1.7). Finally, we discuss

a relation of the Burgers-KdV hierarchy to the equations for the wave function of the KdV

hierarchy.

3.1. Construction.

Consider an extra variable

v

and the ring

A

v,w

. Define differential

poly-nomials

R

n

, Q

n

∈ A

v,w

,

n

≥

0, as follows:

R

₀

=

v

x

,

R

n

=

2 u

2

2 n

+ 1

1

2 ∂

2

x

+

v

x

∂

x

+

v

_x2

+

v

xx

2 +

w

R

n−1

+

1

2 v

x

K

n−1

+

3

4 ∂

x

K

n−1

,

for

n

≥

1;

Q

₀

=

u

v

_x2

+

v

xx

2 +

w

,

Q

n

=

u

2

n

+ 1

1

2 ∂

2

x

+

v

x

∂

x

+

v

2

x

+

v

xx

2 +

w

Q

n−1

,

for

n

≥

1 .

We call the Burgers-KdV hierarchy the following system:

∂v

∂t

n

=

R

n

,

n

≥

1;

∂w

∂t

n

=

∂

x

K

n

,

n

≥

1;

∂v

∂s

n

=

Q

n

,

n

≥

0;

∂w

∂s

n

= 0

,

n

≥

0 .

We see that

w

is just a solution of the KdV hierarchy. The simplest equation for

v

is

∂v

∂s

₀

=

u

v

2_x

+

v

xx

2 +

w

.

(8)

The system

∂v

∂t

n

=

R

n

,

n

≥

1;

∂w

∂t

n

=

∂

x

K

n

,

n

≥

1;

∂v

∂s

=

u

v

_x2

+

v

xx

2 +

w

;

∂w

∂s

= 0

.

will be called the half of the Burgers-KdV hierarchy.

Another result of the paper is the following theorem.

Theorem 3.1.

Let

F

o

_{be the power series determined by Theorem 1.1. Then the pair}

_v

₌

F

o

_{, w}

₌

∂2Fc ∂t2

0

satisfies the half of the Burgers-KdV hierarchy.

Here we again identify

x

with

t

0

.

3.2. Commutativity of the flows.

We are going to prove the following proposition.

Proposition 3.2.

All operators

V

Ri,∂xKi

and

V

Qi,0

commute with each other.

The proof of the proposition will occupy Sections 3.3-3.5. The plan is the following. First,

we consider the differential polynomials

R

e

i

:=

R

i

|

wj=0

,

Q

e

i

:=

Q

i

|

wj=0

∈ A

v

and show that

the operators

V

_Ri_e

, V

_Qi_e

∈ E

v

pairwise commute. We do it in Section 3.3. Then in Section 3.4

we prove that the operators

V

Ri,∂xKi

and

V

Qi,0

commute with

V

Q0,0

. Finally, in Section 3.5 we

deduce that all operators

V

Ri,∂xKi

and

V

Qi,0

commute with each other.

3.3. Burgers hierarchy.

Define an operator

B

by

B

:=

∂

x

+

v

x

. It is easy to see that

1

2 B

2

₌

1

2 ∂

2

x

+

v

x

∂

x

+

v

2_x

+

v

xx

2 .

Therefore, we have

e

R

i

=

u

2i

(2

i

+ 1)!!

B

2i

v

x

,

Q

e

i

=

u

2i+1

2

i

₍

_i

_{+ 1)!}

B

2i

v

2x

+

v

xx

2 .

We can easily recognize here the differential polynomials that describe the flows of the Burgers

hierarchy (see e.g. [Olv86]), up to multiplication by a constant. The fact that the

opera-tors

V

_Ri_e

, V

_Qi_e

∈ E

v

commute with each other is well-known (see e.g. [Olv86]).

3.4. Commutators

[

V

Q0,0

, V

Rn,∂xKn

]

and

[

V

Q0,0

, V

Qn,0

]

.

Let

P

:=

v

2

x

+

v

xx

2 ,

P

∗

:=

X

∂P

∂v

i

∂

_xi

=

1

2 ∂

2

x

+

v

x

∂

x

.

The following formulas will be very useful for us:

V

Q0,0

· B

=

u

[

P

∗

, B

] +

uw

x

,

1

2 B

2

₌

_P

∗

+

P. Let us prove that [

V

Q0,0

, V

Qn,0

] = 0 or, equivalently,

V

Q0,0

Q

n

−

V

Qn,0

Q

0

= 0. We have

(

n

+ 1)

u

−2

(

V

Q0,0

Q

n

−

V

Qn,0

Q

0

) =

V

Q0,0

1

2 B

2

₊

_w

Q

n−1

−

uP

_∗

1

2 B

2

₊

_w

Q

n−1

=

u

2 [

P

∗

, B

2

_]

_Q

n−1

+

u

2 (

w

x

B

+

B

◦

w

x

)

Q

n−1

+

1

2 B

2

₊

_w

(

V

Q0,0

Q

n−1

)

−

u

2 (

P

∗

◦

B

2

₎

_Q

n−1

−

u

(

P

∗

w

)

Q

n−1

−

uw

x

∂

x

Q

n−1

−

uwP

∗

Q

n−1

=

1

2 B

2

₊

_w

V

Q0,0

Q

n−1

−

V

Qn−1,0

Q

0

.

Continuing in the same way we get

V

Q0,0

Q

n

−

V

Qn,0

Q

0

=

u

2n

(

n

+ 1)!

1

2 B

2

₊

_w

n

(9)

Let us prove that [

V

Q0,0

, V

Rn,∂xKn

] = 0. Since

K

n

doesn’t depend on

v

i

, we have

V

Q0,0

∂

x

K

n

=

0. Thus, it remains to prove that

V

Q0,0

R

n

−

V

Rn,∂xKn

Q

0

= 0. We proceed by induction on

n

.

So assume that

V

Q0,0

R

n−1

−

V

Rn−1,∂xKn−1

Q

0

= 0.

We have

u

−3

2 n

+ 1

2 V

Q0,0

R

n

=

u

−1

_V

Q0,0

1

2 B

2

₊

_w

R

n−1

+

1

2 v

x

K

n−1

+

3

4 ∂

x

K

n−1

=

1

2 [

P

∗

, B

2

_{] +}

_w

x

B

+

B

◦

w

x

R

n−1

+

u

−1

1

2 B

2

₊

_w

(

V

Q0,0

R

n−1

)

+

1

2 (

∂

x

P

+

w

x

)

K

n−1

=

1

2 [

P

∗

, B

2

_]

_R

n−1

+

w

x

BR

n−1

+

1

2 w

xx

R

n−1

+

1

2 B

2

₊

_w

(

P

_∗

R

n−1

+

∂

x

K

n−1

) +

1

2 (

∂

x

P

+

w

x

)

K

n−1

=

P

∗

1

2 B

2

₊

_w

R

n−1

+

1

2 B

2

₍

_∂

x

K

n−1

) +

w∂

x

K

n−1

+

1

2 (

∂

x

P

+

w

x

)

K

n−1

.

On the other hand, we have

u

−3

2 n

+ 1

2 V

Rn,∂xKn

Q

0

=

P

∗

1

2 B

2

₊

_w

R

n−1

+

1

2 v

x

K

n−1

+

3

4 ∂

x

K

n−1

+

2 n

+ 1

2 u

−2

_∂

x

K

n

.

Therefore, we get

u

−3

2 n

+ 1

2 V

Q0,0

R

n

−

u

−3

2 n

+ 1

2 V

Rn,∂xKn

Q

0

=

w∂

x

K

n−1

+

1

2 w

x

K

n−1

+

1

8 ∂

3

x

K

n−1

−

2 n

+ 1

2 u

−2

_∂

x

K

n

= 0

.

This completes our proof.

3.5. All commutators.

Let

V

₁

and

V

₂

be any two operators from the set

{

V

Ri,∂xKi

}

i≥1

∪

{

V

Qi,0

}

i≥0

. We have to prove that [

V

1

, V

2

] = 0. We begin with the following lemma.

Lemma 3.3.

Let

[

V

₁

, V

₂

] =

V

T1,T2

. Then

T

2

= 0

and

T

1

|

w∗=0

= 0

.

Proof.

Let us prove that

T

₂

= 0. Clearly, we have to do it, only if

V

₁

, V

₂

∈ {

V

Ri,∂xKi

}

i≥1

. Let

V

₁

=

V

Ri,∂xKi

and

V

2

=

V

Rj,∂xKj

. Since the differential polynomials

K

l

don’t depend on

v

∗

, we

have

T

₂

=

V

Ri,∂xKi

∂

x

K

j

−

V

Rj,∂xKj

∂

x

K

i

=

V

∂xKi

∂

x

K

j

−

V

∂xKj

∂

x

K

i

.

Here we consider the operators

V

∂xKi

and

V

∂xKj

as elements of the space

E

w

. The last

expres-sion is equal to 0, because the differential polynomials

∂

x

K

l

describe the flows of the KdV

hierarchy (see e.g. [Olv86]).

Let us prove that

T

1

|

w∗=0

= 0. Let

R

e

i

:=

R

i

|

w∗=0

and

Q

e

i

:=

Q

i

|

w∗=0

. We consider the

operators

V

_Ri_e

and

V

_Qi_e

as elements of

E

v

. Obvioulsy, we have

(

V

Qi,0

R

j

)

|

_w_∗₌₀

=

V

Qie

R

e

j

and (

V

Qi,0

Q

j

)

|

w∗=0

=

V

Qie

Q

e

j

.

Since

∂

x

K

i

|

_w_∗₌₀

= 0, we get

(

V

Ri,∂xKi

R

j

)

|

_w_∗₌₀

=

V

Rie

R

e

j

and (

V

Ri,∂xKi

Q

j

)

|

w∗=0

=

V

Rie

Q

e

j

.

In Section 3.3 we showed that the operators

V

_Ri_e

and

V

_Qj_e

pairwise commute. Thus,

T

1

|

w∗=0

= 0.

The lemma is proved.

(10)

Lemma 3.4.

Suppose

T

∈ A

v,w

is an arbitrary differential polynomial such that

T

|

_w_∗₌₀

= 0

and

[

V

Q0,0

, V

T,0

] = 0

. Then

T

= 0

.

Proof.

Before proving the lemma let us introduce several notations. A partition

λ

is a sequence

of non-negative integers

λ

1

, . . . , λ

r

such that

λ

1

≥

λ

2

≥

. . .

≥

λ

r

. Note that our terminology is

slightly non-standard, because we allow zeroes in

λ

. Let

l

(

λ

) :=

r

and

|

λ

|

:=

P

r_i₌₁

λ

i

. The set

of all partitions will be denoted by

P

. For a partition

λ

let

v

λ

:=

Q

l(λ)

i=1

v

λi

and

w

λ

:=

Q

l(λ)

i=1

w

λi

.

Consider any differential polynomial

Q

∈ A

v,w

. Let

Q

=

P

λ,µ∈P

d

λ,µ

v

λ

w

µ

, where

d

λ,µ

∈

C

_[

_{u, u}

−1

_{]. Let}

Gr

i

Q

:=

X

l(µ)=i

d

λ,µ

v

λ

w

µ

,

Gr

ij

Q

:=

X

l(µ)=i |µ|=j

d

λ,µ

v

λ

w

µ

.

The equation [

V

Q0,0

, V

T,0

] = 0 means that

V

Q0,0

T

−

V

T,0

Q

0

= 0. Let

T

=

P

λ,µ∈P l(µ)≥1

c

λ,µ

v

λ

w

µ

,

where

c

λ,µ

∈

C

[

u, u

−1

]. We have

V

T,0

Q

0

=

u

v

x

∂

x

T

+

1

2 ∂

2

x

T

,

(3.1)

V

Q0,0

T

=

u

X

λ,µ∈P l(µ)≥1

X

i≥0

c

λ,µ

∂v

λ

∂v

i

(

∂

_xi

P

+

w

i

)

w

µ

.

(3.2)

Suppose

T

6 = 0. Let

i

₀

be the minimal

i

such that

Gr

i

T

6 = 0. From the condition

T

|

w∗=0

= 0

it follows that

i

₀

≥

1. Let

j

₀

be the maximal

j

such that

Gr

j_i

0

T

6 = 0. From (3.1) it is easy to

see that

Gr

j0+2

i0

(

V

T,0

Q

0

) =

u

2 X

λ,µ∈P l(µ)=i0,|µ|=j0

c

λ,µ

v

λ

∂

x2

(

w

µ

)

6 = 0

.

On the other hand, from (3.2) it obviously follows that

Gr

j0+2

i0

(

V

Q0,0

T

) = 0. This contradiction

proves the lemma.

3.6. String solution.

In this section we construct a specific solution of the half of the

Burgers-KdV hierarchy that satisfies the open string equation.

Proposition 3.5.

Consider the half of the Burgers-KdV hierarchy. Let us specify the following

initial data for the hierarchy:

v

|

_t_≥₁₌₀_,s₌₀

= 0

and

w

|

_t_≥₁₌₀_,s₌₀

=

u

−2

x.

(3.3)

Then the solution of the hierarchy satisfies the open string equation

∂v

∂t

₀

=

X

n≥0

t

n+1

∂v

∂t

n

+

u

−1

s.

We remind the reader that we identify

x

with

t

₀

.

Proof.

Recall that

S

:=

∂ ∂t0

−

P

n≥0

t

n+1_∂tn∂

. We have to prove that

Sv

=

u

−1

s

. We have

Sw

=

u

−2

_,

_w

₌

∂2_F

∂t2 0

and

K

n

=

∂

2_F

∂t0∂tn

(see Section 2.3), therefore,

SK

n

=

K

n−1

, for

n

≥

1. Let

us put

K

₋₁

:=

u

−2

_{, so the last equation is also valid for}

_n

_{= 0.}

It is easy to see that

v|

t≥1=0

=

u

−1s3

6

+

u

−1

_t

0

s

. Hence,

(

Sv

)

|

t≥1=0

=

u

−1

_s.

(11)

For

n

≥

1, we have

∂

∂t

n

(

Sv

) =

S

∂v

∂t

n

−

∂v

∂t

n−1

=

2 u

2

2 n

+ 1

(

Sv

)

x

∂

x

+ (

Sv

)

x

v

x

+

1

2 (

Sv

)

xx

+

u

−2

∂v

∂t

n−1

+

1

2 B

2

₊

_w

S

∂v

∂t

n−1

+

1

2 (

Sv

)

x

K

n−1

+

1

2 v

x

K

n−2

+

3

4 ∂

x

K

n−2

−

∂v

∂t

n−1

=

2 u

2

2 n

+ 1

(

Sv

)

x

∂

x

+ (

Sv

)

x

v

x

+

1

2 (

Sv

)

xx

∂v

∂t

n−1

+

1

2 B

2

₊

_w

∂

∂t

n−1

(

Sv

) +

1

2 (

Sv

)

x

K

n−1

.

This system together with the initial condition (3.4) uniquely determines the power series

Sv

.

It is easy to see that

Sv

=

u

−1

s

satisfies the system. Proposition 3.5 is proved.

3.7. Relation to the wave function of the KdV hierarchy.

Consider the operator

L

:=

∂

_x2

+ 2

w.

Recall that the KdV hierarchy can be written in the so-called Lax form:

∂

∂t

n

L

=

u

2n

(2

n

+ 1)!!

h

(

L

n+12

)

+

, L

i

,

n

≥

1 .

Here we use the language of pseudo-differential operators. We briefly review it in [Bur14] and

we refer the reader to the book [Dic03] for a detailed introduction to this subject. Introduce

variables

t

n

with

n

∈

1₂

+

Z

≥0

and let

t

1

2+k

=

s

k

,

k

∈

Z

_≥₀

_{. In [Bur14] we prove that the}

Burgers-KdV hierarchy is equivalent to the following system of evolutionary PDEs for the functions

w

and

φ

=

e

v

_:

∂

∂t

n

L

=

u

2n

(2

n

+ 1)!!

h

(

L

n+12

)

+

, L

i

,

n

∈

1

2 Z

≥1

;

(3.5)

∂

∂t

n

φ

=

u

2n

(2

n

+ 1)!!

(

L

n+1

2

)

₊

φ,

n

∈

1

2 Z

≥1

.

(3.6)

Equations (3.6) coincide with the equations for the wave function of the KdV hierarchy (see

e.g. [Dic03]). The commutativity of the flows of the system (3.5)-(3.6) is actually well-known

(see e.g. [Dic03]) and the proof is simple. Let us recall it. Consider the ring

A

φ,w

. Let

T

n

:=

u

2n

(2n+1)!!

(

L

n+1₂

₎

+

φ

∈ A

φ,w

. We have to check that the operators

V

Tn,∂xKn

∈ E

φ,w

,

n

∈

1

2 Z

≥1

,

pairwise commute. Here we, by definition, put

K

n

= 0, for

n

∈

1₂

+

Z

≥0

. Let

m, n

∈

1₂

Z

≥1

and

[

V

Tm,∂xKm

, V

Tn,∂xKn

] =

V

P1,P2

.

The fact that

P

2

= 0 follows from the commutativity of the flows of the KdV hierarchy. For

P

1

we have

P

₁

=

V

Tm,∂xKm

T

n

−

V

Tn,∂xKn

T

m

=

u

2m+2n

(2

m

+ 1)!!(2

n

+ 1)!!

×

h

(

L

m+12

)

₊

, L

n+12

i

+

+ (

L

n+1

2

)

₊

(

L

m+12

)

₊

−

h

(

L

n+12

)

₊

, L

m+12

i

+

−

(

L

m+1

2

)

₊

(

L

n+12

)

₊

φ

=

u

2m+2n

(2

m

+ 1)!!(2

n

+ 1)!!

h

(

L

m+12

)

₊

,

(

L

n+12

)

₋

i

+

−

h

(

L

n+12

)

₊

, L

m+12

i

+

(12)

Since

_h

(

L

n+12

)

₊

, L

m+ 1 2

i

+

=

−

h

(

L

n+12

)

₋

, L

m+ 1 2

i

+

=

−

h

(

L

n+12

)

₋

,

(

L

m+ 1 2

)

₊

i

+

,

we conclude that

P

1

= 0. The commutativity of the flows of the Burgers-KdV hierarchy is

proved.

4. Proofs of Theorems 1.1 and 3.1

The proof of Theorems 1.1 and 3.1 goes as follows. Let a pair (

v, w

) be the solution of the

half of the Burgers-KdV hierarchy with the initial condition (3.3). Let us show that the series

F

o

₌

_v

_{is a solution of the open KdV equations (1.9). Indeed, we have}

2 n

+ 1

2 ∂v

∂t

n

=

u

2

1

2 ∂

2

x

+

v

x

∂

x

+

v

_x2

+

v

xx

2 +

w

∂v

∂t

n−1

+

1

2 v

x

K

n−1

+

3

4 ∂

x

K

n−1

=

u

2

v

_x2

+

v

xx

2 +

w

∂v

∂t

n−1

+

u

2

∂

∂t

n−1

v

2_x

+

v

xx

2 +

w

+

u

2

2 v

x

K

n−1

−

u

2

4 ∂

x

K

n−1

=

u

∂v

∂s

∂v

∂t

n−1

+

u

∂

2

_v

∂s∂t

n−1

+

u

2

2 v

x

K

n−1

−

u

2

4 ∂

x

K

n−1

.

It remains to note that

K

n

=

∂

2_Fc

∂t0∂tn

and we see that Theorems 1.1 and 3.1 are proved.

5. Proof of Theorem 1.2

Denote by

a

i,j

the number

(2i+1)!!(2₂i+j+2j+1)!!

. Let

τ

:= exp(

F

o

+

F

c

) and

τ

o

:= exp(

F

o

). In order

to save some space we will use the subscript

n

for the partial derivative by

t

n

and the subscript

s

for the partial derivative by

s

. The proof of the theorem is based on the following lemma.

Lemma 5.1.

We have

1.

L0τ

τ

−

(

uF

o

s

+

u∂

s

)

L−_τ1τ

= 0

.

2. If

n

≥

0 , then

L

n+1

τ

−

(

uF

o

s

+

u∂

s

)

L

n

τ

=

u

2

4 (2

n

+ 1)!!

2

n

F

c

0,n

+

u

4

n−1

X

0

a

i,n−1−i

F

₀c,i

F

c

0,n−1−i

(5.1)

+

u

2

2 (2

n

+ 1)!!

2

n+2

F

o n

F

o 0

+

u

2

n−1

X

0

a

i,n−1−i

F

io

u

2

2 F

o

0

∂

x

−

u

2

4 ∂

2

x

F

_nc₋₁₋_i

+

u

2

2 (2

n

+ 1)!!

2

n+2

F

o

0,n

+

u

4

n−1

X

0

a

i,n−1−i

F

0o,i

F

0c,n−1−i

−

u

n+1

4 ∂

n+1

s

τ

o

τ

o

.

Proof.

Let us prove point

1 . From the usual Virasoro equations (1.5) it follows that

L

−1

τ

=

−

∂F

o

∂t

₀

+

X

n≥0

t

n+1

∂F

o

∂t

n

+

u

−1

s,

L

0

τ

=

−

3

2 ∂F

o

∂t

₁

+

X

n≥0

2 n

+ 1

2 t

n

∂F

o

∂t

n

+

s

∂F

o

∂s

+

3

4 .

Using the open KdV equations (1.9) we get

L

0

τ

−

(

uF

o

s

+

u∂

s

)

L

−1

τ

=

−

u

2

2 F

o

0

F

c

0,0

+

u

2

4 F

c

0,0,0

+

1

2 t

0

F

o

0

+

X

n≥0

t

n+1

u

2

2 F

o 0

F

c

0,n

−

u

2

4 F

c

0,0,n

+

sF

_so

+

3

4 −

(

uF

o

(13)

By the string equation (1.2), the last expression is equal to zero.

Let us prove point

2 . Let

L

(1)_n

:=

X

i≥0

(2

i

+ 2

n

+ 1)!!

2

n+1

₍₂

_i

₋

_1)!!

(

t

i

−

δ

i,1

)

∂

∂t

i+n

,

L

(2)_n

:=

u

2

n−1

X

i=0

a

i,n−1−i

∂

2

∂t

i

∂t

n−1−i

,

L

(3)_n

:=

u

n

s

∂

n+1

∂s

n+1

+

3 n

+ 3

4 u

n

∂

n

∂s

n

.

Using the Virasoro equations (1.5) we get

L

n+1

τ

−

(

uF

o

s

+

u∂

s

)

L

n

τ

=

L

(1)

n+1

τ

o

τ

o

−

(

uF

o

s

+

u∂

s

)

L

(1)n

τ

o

τ

o

(A)

+

L

(2)

n+1

τ

o

τ

o

−

(

uF

o

s

+

u∂

s

)

L

(2)n

τ

o

τ

o

(B)

+

u

2

n

X

0

a

i,n−i

F

ic

F

o n−i

−

u

2

₍

_uF

o

s

+

u∂

s

)

n−1

X

0

a

i,n−1−i

F

ic

F

o n−1−i

(C)

+

L

(3)

n+1

τ

o

τ

o

−

(

uF

o

s

+

u∂

s

)

L

(3)n

τ

o

τ

o

.

(D)

Using the open KdV equations (1.9) and the Virasoro equations (1.5) we can compute that

(A) =

∞

X

0

(2

i

+ 2

n

+ 1)!!

2

n+1

₍₂

_i

₋

_1)!!

(

t

i

−

δ

i,1

)

u

2

2 F

o

0

∂

x

−

u

2

4 ∂

2

x

F

_nc₊_i

=

−

u

2

2 (2

n

+ 1)!!

2

n+1

F

o

0

F

c n

|

{z

}

∗

−

u

4

n−1

X

0

a

i,n−1−i

F

₀o

(

F

₀c,i,n−1−i

| {z }

∗∗

+ 2

F

₀c_,i

F

_nc₋₁₋_i

|

{z

}

∗∗∗

)

+

u

2

4 (2

n

+ 1)!!

2

n

F

c

0,n

+

u

4

8

n−1

X

0

a

i,n−1−i

(

F

0c,0,i,n−1−i

|

{z

}

••

+2

F

₀c_,i

F

₀c_,n₋₁₋_i

+2

F

₀c_,₀_,i

F

_nc₋₁₋_i

|

{z

}

•

)

.

For expression (B) we have

(B) =

u

2

n

X

0

a

i,n−i

(

F

i,no −i

+

F

o i

F

o n−i

)

−

u

2

2 (

uF

o

s

+

u∂

s

)

n−1

X

0

a

i,n−1−i

(

F

i,no −1−i

+

F

o i

F

o

n−1−i

) =

=

u

2

2 (2

n

+ 1)!!

2

n+2

F

o

0,n

+

u

4

n−1

X

0

a

i,n−1−i

F

0o,i

F

c

0,n−1−i

+

u

2

n−1

X

0

a

i,n−1−i







u

2

2 F

o

0

∂

x

| {z }

∗∗

−

u

2

4 ∂

2 x

| {z }

••







F

_i,nc ₋₁₋_i

+

u

2

2 (2

n

+ 1)!!

2

n+2

F

o n

F

o 0

+

u

2

n−1

X

0

a

i,n−1−i

F

io

u

2

2 F

o

0

∂

x

−

u

2

4 ∂

2

x

F

_nc₋₁₋_i

.

Computing (C) in a similar way we get

(C) =

u

2

(2

n

+ 1)!!

2

n+2

F

c n

F

o

0

|

{z

}

∗

+

u

2

n−1

X

0

a

i,n−1−i

F

ic







u

2

2 F

o

0

∂

x

| {z }

∗∗∗

−

u

2

4 ∂

2 x

| {z }

•





(14)

It is easy to compute that

(D) =

−

u

n+1

4 ∂

_sn+1

τ

o

τ

o

.

We have marked the terms that cancel each other in the total sum (A)+(B)+(C)+(D).

Col-lecting the remaining terms we get (5.1).

From the commutation relation (1.10) it follows that

L

n

=

(−1) n−2

(n−2)!

ad

n−2

L1

L

2

, for

n

≥

3. Thus,

it is sufficient to prove the open Virasoro equations (1.11) only for

n

=

−

1 ,

0 ,

1 ,

2. By Theorem 3.1 and Proposition 3.5, the series

F

o

_{satisfies the open string equation (1.7).}

Thus,

L

−1

τ

= 0. By Lemma 5.1,

L

0

τ

=

τ

(

uF

so

+

u∂

s

)

L−1 τ τ

= 0.

Substituting

n

= 0 in (5.1) we get

L

1

τ

−

(

uF

o

s

+

u∂

s

)

L

0

τ

=

u

4 u

F

₀c_,₀

+

1

2 (

F

o

0

)

2

+

1

2 F

o

0,0

−

F

_so

by Theorem 1.1

=

0 .

Therefore,

L

1

τ

= 0.

Finally, let us write equation (5.1) for

n

= 1. We get

L

2

τ

−

(

uF

o

s

+

u∂

s

)

L

1

τ

=

3 u

2

8 F

c

0,1

+

u

4

16 F

c

0,0

F

0c,0

+

3 u

2

16 F

o

1

F

0o

+

u

2

8 F

o

0

u

2

2 F

o

0

∂

x

−

u

2

4 ∂

2

x

F

₀c

(5.2)

+

3 u

2

16 F

o

0,1

+

u

4

16 F

o

0,0

F

0c,0

−

u

2

4 (

F

o s

)

2

₊

_F

o s,s

.

Denote

∂2_Fc

∂t2₀

by

w

and

F

o

_by

_v

_{. We have}

_F

c

0,1

=

K

1

=

w

2

+

1

12

w

xx

. By the open KdV

equations (1.9), we have

F

₁o

=

u

2

v

_x3

3 +

v

x

v

xx

+

v

xxx

3 +

v

x

w

+

w

x

2 .

By Theorem 1.1,

F

o s

=

u

v2

x+vxx

2

+

w

. Substituting these expressions in the right-hand side

of (5.2), after somewhat lengthy computations, we get zero. Hence,

L

2

τ

= 0. The theorem is

proved.

Appendix

A. Virasoro equations for the moduli of closed Riemann surfaces

In this section we revisit the proof of the equivalence of the usual KdV and the Virasoro

equations for the moduli space of stable curves. There is a reason to do it. In all papers,

that we found, the proof is presented in a way more suitable for physicists. So we decided to

rewrite it in a more mathematical style and also to make it more elementary. We follow the