Buryak-Okounkov formula for the n-point function

Buryak-Okounkov formula for the -point function and a new proof of the Witten conjecture

Alexander Alexandrov Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang 37673, Korea; and Institute for Theoretical and Experimental Physics, 25 Bolshaya Cheryomushkinskaya Ulitsa, Moscow 117218, Russia alex@ibs.re.kr Francisco Hernández Iglesias Korteweg-de Vriesinstituut voor Wiskunde, Universiteit van Amsterdam, Postbus 94248, 1090GE Amsterdam, Nederland f.hernandeziglesias@uva.nl  and  Sergey Shadrin Korteweg-de Vriesinstituut voor Wiskunde, Universiteit van Amsterdam, Postbus 94248, 1090GE Amsterdam, Nederland s.shadrin@uva.nl
Abstract.

We identify the formulas of Buryak and Okounkov for the -point functions of the intersection numbers of psi-classes on the moduli spaces of curves. This allows us to combine the earlier known results and this one into a principally new proof of the famous Witten conjecture / Kontsevich theorem, where the link between the intersection theory of the moduli spaces and integrable systems is established via the geometry of double ramification cycles.

1. Introduction

The symbol denotes the intersection number . It can be non-zero only if , , , , and . Witten conjectured [WittenConj] that the generating function of these intersection numbers defined as

is the logarithm of the unique tau-function of the Korteweg-de Vries (KdV) hierachy that in addition satisfies the string equation,

The string equation is easy to prove, see [WittenConj], so the main part of the conjecture is the equations of the KdV hierarchy.

This conjecture was first proved by Kontsevich in [Kontsevich] using the Strebel-Penner ribbon graph model of the moduli space of curves, and later on more proofs have appeared. Mirzakhani [mirzakhani] used symplectic reduction for the Weil-Peterson volumes of the moduli space, and Okounkov and Pandharipande [okounkovpand] and Kazarian and Lando [kazarianlando] used the ELSV formula that connects the intersection theory and Hurwitz numbers. There are more papers where the Witten conjecture / Kontsevich theorem is proved (see e. g. [OkounkovMain, MulaseSafnuk, Kazarian, KimLiu, ChenLiLiu, Witten]), but on the geometric side they all use one of the ideas mentioned above: the Strebel-Penner ribbon graph model, symplectic reduction, or the ELSV formula for Hurwitz numbers.

In this paper we give a new proof of the Witten conjecture based on a completely different geometric idea than any of the earlier existing proofs: the intersection theory of double ramification cycles. More precisely, the full proof that we explain here consists of four big steps, where three of them were already available in the literature, and the fourth missing one is the main subject of this paper:

  1. In [BSSZ] Buryak et al. fully described the intersection numbers of the monomials of psi-classes with the double ramification cycles.

  2. In [BuryakMain] Buryak used the previous result and a relation between the double ramification cycles and the fundamental cycles of the moduli spaces of curves to describe explicitly the so-called -point function , .

  3. In [OkounkovMain] Okounkov proved a different explicit formula for the -point functions and he showed in Section 3 of op. cit. that the generating function of their coefficients is the logarithm of the string tau-function of the KdV hierarchy.

  4. In this paper we identify Buryak’s and Okounkov’s formulas for the -point function, and this makes the sequence of papers [BSSZ] [BuryakMain] the present paper [OkounkovMain, Section 3] a new proof of the Witten conjecture.

Let us say a few words about the geometric techniques used in [BSSZ] and [BuryakMain]. A double ramification cycle , , is the class of a certain compactification of the locus of the isomorphism classes of smooth curves with marked points such that is the divisor of a meromorphic function . These cycles inherit rich geometry of the space of maps to and this allows to express the psi-classes restricted to these cycles in terms of the double ramification cycles of smaller dimension, which is in principle enough to compute all intersection numbers of psi-classes with the double ramification cycles. Next, observe that under the projection that forgets marked points the push-forward of a double ramification cycle is a multiple of the fundamental cycle of . This relates the intersection numbers of psi-classes on double ramification cycles to . There is, of course, a long way from these computational ideas to nice closed formulas derived in [BSSZ] and [BuryakMain].

This idea of computation of the intersection numbers has been used in a number of earlier papers, cf. [ShaUMN03, ShaIMRN03, Sha05, Sha06, Bursha], and these papers might serve a good source of examples of particular computations. In particular, an explicit algorithm for the computation of all intersection numbers is given in [ShaZvo08]. We have to warn the reader, however, that there are different versions of double ramification cycles used in the literature depending on what particular compactification of the space of maps is used in the construction, cf. a discussion in [BSSZ, Section 2.3].

Exactly the same idea of computation of the intersection numbers of -classes is proposed in [costello, Section 9]. It is mentioned in [costello, Section 1.3] that for the further applications of the results of that paper a required first step is to give a new proof of Witten’s conjecture [WittenConj] using the technique developed there. So, it is precisely what the present paper (combined with [BSSZ], [BuryakMain], and [OkounkovMain]) does.

Finally, to conclude the introduction, let us mention that the -point functions for the intersection numbers of psi-classes have recently been studied from different points of view, see [EOr, LiuXu, Zhou, bertoladubyang, ZhouEmergent, BH0704, BH0709, AMMP]. The comparison of different formulas and recursive relations for their coefficients is very interesting and usually highly non-trivial, and this paper can also be considered as a step towards unification (see also [ZhouEmergent]) of the variety of formulas for the -point functions.

1.1. Organization of the paper

In Section 2 we recall the formulas of Buryak and Okounkov and some statements about these formulas that we use in this paper, and state our main results. In Section 3 we derive an equivalent form of the Buryak formula. In Section 4 we prove that the principal terms in Buryak and Okounkov formulas coincide. In Section 5 we prove that all other terms, namely, the so-called diagonal terms needed for a regularization of the principal ones, also coincide in Buryak and Okounkov formulas.

1.2. Acknowledgments

We thank G. Carlet and R. Kramer for useful discussions. A. A. was supported by IBS-R003-D1 and by RFBR grant 17-01-00585. F. H. I. and S. S. were supported by the Netherlands Organization for Scientific Research. A.A. wishes to thank the KdV Institute for its kind hospitality.

2. Buryak and Okounkov formulas

In this section we recall the formulas for the -point functions in [BuryakMain] and [OkounkovMain]. It is convenient to append the intersection numbers by two unstable cases and . Namely, we assume by definition that and , and we add these terms to and , respectively.

2.1. Formula of Buryak

Let . Define the function by and for we have

(1)

Though it is not obvious from the definition, is a formal power series in all its variables, which is invariant with respect to the diagonal action of the symmetric group on and , see [BSSZ, Remarks 1.5 and 1.6].

Define the function as the Gaussian integral

Theorem 2.1 (Buryak [BuryakMain]).

For we have .

2.2. Formula of Okounkov

Define the function as

where denotes . Then the function defined as

is invariant under the -action on .

Denote by the set of all partitions of the set into a disjoint union of unordered subsets , for all . Let , . Define the function as

and the function as

Theorem 2.2 (Okounkov [OkounkovMain]).

The generating function of the coefficients of , , is the logarithm of the string tau-function of the KdV hierarchy.

2.3. Main theorem

We are ready to state our main result.

Theorem 2.3.

We have: , .

The rest of the paper is devoted to the proof of this theorem. An immediate corollary of Theorems 2.12.2, and 2.3 is the following:

Corollary 2.4.

The Witten conjecture is true, that is, the function is the string tau-function of the KdV hierarchy.

As we explain in the introduction, the real importance of this new proof of the Witten conjecture is that it uses a new way to relate the intersection theory of the moduli space of curves to the theory of integrable hierarchies, based on geometry of double ramification cycles. Otherwise, though Theorem 2.3 is interesting by itself, the identity has an alternative proof in [OkounkovMain, Section 2].

3. Buryak formula revisited

Our first goal is to translate the cumbersome formula of Buryak into something more manageable. Let and .

Proposition 3.1.

For we have:

(2)

It is clearly true for and we prove it below for . Now the function is manifestly invariant with respect to the diagonal action of the symmetric group on and .

Corollary 3.2.

We have:

(3)

3.1. Proof of Proposition 3.1

Assume that . Expanding the definition of the function allows us to rewrite Equation (1) for as

(4)

3.1.1. Exponential terms in the numerators

In order to identify Equations (2) and (4), we consider for each particular fixed sequence of signs , , all terms in Equations (2) and (4) where the numerator is equal to , , and prove that the total coefficient of coincides in both formulas. The symbols , , are understood in the rest of the proof as just formal variables satisfying the relations .

Let denote the set . For and we define

It is a convenient way to keep track of signs in the exponential terms in the numerators of (2) and (4). It is easy to see that

  • In Equation (2) the numerators are indexed by , for all ;

  • In Equation (4) the numerators are indexed by , for all such that and for all .

So, we have to obtain a full description of all , and as above such that .

3.1.2. Notation for the symmetric group

Decompose as , where denotes the subgroup of permutations such that .

Denote by , , the cyclic permutation . Consider the subset defined as . The following lemma implies that it is in fact a disjoint union.

Lemma 3.3.

We have: , and

Proof.

Observe that . Hence it is enough to show that (which is obvious) and , .

The latter fact we can prove by induction. For we see that . Assume we know that for any the product is equal to for some . Then for any we have:

where . Thus . ∎

3.1.3. versus

The full description of the correspondences between , , and , , , is given by the following lemma.

Lemma 3.4.

(1) For any , , there exists a such that .

(2) For any the only combination of , where and , such that is given by , , .

(3) For any , , the complete list of the combinations , where and , such that is indexed by the sequences , where

3.1.4. Comparison of the coefficients

The symbols , , are understood in the rest of the proof as just formal variables satisfying the relations and for all . For , , the symbols denotes

Up to a factor (which is a common factor for (2) and (4)), the coefficient of in (2) is equal to . Up to the same factor, the coefficient of is equal to .

Lemma 3.5.

For any , , we have:

(5)

Lemma 3.4 and Lemma 3.5 together imply that the right hand side of Equation (2) is equal to the right hand side of Equation (4), which completes the proof of Proposition 3.1.

3.2. Technical lemmas

In this section we prove Lemma 3.4 and Lemma 3.5 used in the proof of Proposition 3.1.

3.2.1. Proof of Lemma 3.4

The proof is based on several observations. First, observe the left invariance of the identities for :

Lemma 3.6.

We have: implies for any .

Proof.

Direct inspection of signs. ∎

Second, we have uniqueness:

Lemma 3.7.

The equality considered as an equation for has at most one solution.

Proof.

Assume we have two solutions, and , that is, . Applying Lemma 3.6 twice, we obtain: . Hence . ∎

Finally, we can solve this equation:

Lemma 3.8.

For any , we have , where .

Proof.

We prove it by induction on . The base case is trivial. Assume we know it for . Then, for we have:

Since acts only on , it doesn’t affect the second sum and the part of the first sum for . Since it is a cycle, the only terms when and hold simultaneously are the terms with . Hence this total expression is equal to

Now we are ready to prove Lemma 3.4. The first statement follows from Lemmas 3.6 and 3.8. Then, note that Lemmas 3.8 and 3.7 imply that the equality can hold only for , where (and if ). Hence .

3.2.2. Proof of Lemma 3.5

First, observe that the basic properties of imply the following identity that we’ll use in the proof (one can prove it by induction on , for instance):

(6)

Second, observe that Equation (5) is invariant under the left products with any , so it is sufficient to prove it for . We, however, prove a more general statement. Namely, for any we prove that

This can be proved by induction on , with the case being obvious. Assume this statement is proved for . Then for we have (the computation is completely analogous in the cases and , so we perform it only in the first case):

Here the second equality is the induction assumption, and the final equality follows from Equation (6).

4. The principal terms

Recall a reformulation of the formula for proposed in [OkounkovMain, Equation (3.3)]:

(7)

The idea behind this formula is that the whole expression for can be considered as the regularization of its principal part, which is the first summand on the right hand side of Equation (7), by the terms that are Laplace transforms of distributions supported on the diagonals, see [OkounkovMain, Sections 2.6.3 and 3.1.4].

The formula of Buryak, in the form of Equation (3), can also be represented as the sum of its principal part and the regularizing terms supported on the diagonals. Firstly, we interpret the integrals as Cauchy principal values in order to interchange and in Equation (3). We obtain:

(8)

Here the expressions under the sign of the integral have poles along the diagonals defined as , . Recall the integrals should be understood as the Cauchy principal value integrals, that is, we exclude the tubular neighborhood of the divisor of poles of the radius , integrate, and take the limit of the resulting expression. Similarly to Okounkov’s formula, they can be decomposed into a principal part without poles and a diagonal part by applying the Sokhotski-Plemelj formula.

Lemma 4.1.

The right hand side of Equation (8) decomposes in a similar way to the right hand side of Equation (7), that is, into a sum of its principal part and some diagonal regularization terms. The principal parts of the right hand sides of Equations (7) and (8) are equal.

Proof.

Fix and consider the corresponding summand on the right hand of Equation (8). We apply the following change of the variables :

With this change of variables we have:

and

Thus, the right hand side of Equation (8) is equal to

(9)

where in the second line denotes . The diagonal terms are half-residues arising as a result of translating the contour of the ’s back to , removing the diagonal singularities in the process. An explicit expression for the diagonal terms will be computed in the next section using the Sokhotski-Plemelj formula.

Remark 4.2.

Let us note that Equation (9) is similar to the expressions for the -point functions obtained by Brézin and Hikami in [BH0704, BH0709].

Since we got a sum over , as in the principal part of the right hand side of Equation (7), it is sufficient to prove for each that the corresponding summands are equal. Without loss of generality we can assume that . Then we have to prove that

(10)

or, equivalently, if we cancel the common factors and rescale by , we have to prove that

(11)