Changes of variables in ELSV-type formulas

Transcription

1 Changes of variables in ELSV-type formulas arxiv:math/ v3 [mathag] 25 Jun 2007 Introduction Sergey Shadrin, Dimitri Zvonkine February 2, 2008 In [5] I P Goulden, D M Jackson, and R Vakil formulated a conjecture relating certain Hurwitz numbers (enumerating ramified coverings of the sphere) to the intersection theory on a conjectural Picard variety Pic g,n This variety, of complex dimension 4g 3 + n, is supposedly endowed with a natural morphism to the moduli space of stable curves M g,n The fiber over a point x M g,n lying in the open part of the moduli space is equal to the jacobian of the corresponding smooth curve C x The variety Pic g,n is also supposed to carry a universal curve C g,n with n disjoint sections s,,s n Denote by L i the pull-back under s i of the cotangent line bundle to the fiber of C g,n Then we obtain n tautological 2-cohomology classes ψ i = c (L i ) on Pic g,n We are going to use Goulden, Jackson, and Vakil s formula to study the intersection numbers of the classes ψ i on Pic g,n (if it is ever to be constructed) In particular, we prove a Witten-Kontsevich-type theorem relating the intersection theory and integrable hierarchies These equations, together with the string and dilaton equations, allow us to compute all the intersection numbers under consideration Department of Mathematics, University of Zurich, Winterthurerstrasse 90, CH-8057 Zurich, Switzerland and Department of Mathematics, Institute of System Research, Nakhimovsky prospekt 36-, Moscow 728, Russia s: sergeyshadrin@mathuzhch or shadrin@mccmeru Partly supported by the grants RFBR a, NSh , NWO-RFBR (RFBR NWO-a), by the Göran Gustafsson foundation, and by Pierre Deligne s fund based on his 2004 Balzan prize in mathematics Institut mathématique de Jussieu, Université Paris VI, 75, rue du Chevaleret, 7503 Paris, France zvonkine@mathjussieufr Partly supported by the ANR project Geometry and Integrability in Mathematical Physics ANR-05-BLAN

2 Independently of the conjecture of [5], our results can be interpreted as meaningful statements about Hurwitz numbers Our methods are close to those of M Kazarian and S Lando in [7] and make use of Hurwitz numbers We also extend the results of [7] to include the Hodge integrals over the moduli spaces, involving one λ-class The conjecture Fix n positive integers b,, b n Let d = b i be their sum Definition The number of degree d ramified coverings of the sphere by a genus g surface possessing a unique preimage of 0, n numbered preimages of with multiplicities b,,b n, and 2g +n fixed simple branch points is called a Hurwitz number and denoted by h g;b,,b n Conjecture 2 (I P Goulden, D M Jackson, R Vakil, [5]) There exists a compactification of the Picard variety over the moduli space M g,n by a smooth (4g 3 + n)-dimensional orbifold Pic g,n, natural cohomology classes Λ 2,, Λ 2g on Pic g,n of (complex) degrees 2,,2g, and an extension of the tautological classes ψ,, ψ n such that h g;b,,b n = (2g + n)! d Pic g,n Λ 2 + ± Λ 2g ( b ψ ) ( b n ψ n ) Assuming that the conjecture is true we can define τ d τ dn = ψ d ψn dn () Pic g,n By convention, this bracket vanishes unless d i = 4g 3 + n We also introduce the following generating series for the intersection numbers of the ψ-classes on Pic g,n : We denote by F(t 0, t, ) = n n! d,,d n τ d τ dn t d t dn (2) U = 2 F (3) 2 t 0 its second partial derivative While Conjecture 2 remains open, the situation should be seen in the following way The Hurwitz numbers turn out to have the unexpected property of being polynomial in variables b i (first conjectured in [4] and proved 2

3 in [5]) The coefficients of these polynomials are denoted by τ d τ dn Λ 2k (We restrict ourselves to the case k = 0 with Λ 0 = ) The conjectured relation of these coefficients with geometry is a strong motivation to study them Our goal is to find out as much as we can about the values of the bracket in the combinatorial framework, waiting for their geometrical meaning to be clarified This study was, actually, already initiated in [5] In particular, the authors proved that the values of the bracket satisfy the string and dilation equations: F t 0 = d F t = 2 d 0 t d F t d + t2 0 2, (4) (d + )t d F t d 2 F (5) By abuse of language we will usually speak of the coefficients of F as intersection numbers, implicitly assuming the conjecture to be true 2 Results We will soon see that F is related to the following generating function for the Hurwitz numbers: H(β, p, p 2, ) = g,n β 2g +n n! (2g + n)! h g;b,,bn d b,,b n p b p bn (6) Here, as before, d = b i is the degree of the coverings and 2g + n is the number of simple branch points Denote by L p the differential operator L p = bp b p b Its action on H consists in multiplying each term by its total degree d Theorem The series L 2 ph is a τ-function of the Kadomtsev Petviashvili (or KP) hierarchy in variables p i ; ie, it satisfies the full set of bilinear Hirota equations In addition, L 2 ph satisfies the linearized KP equations The proof of this theorem follows in an almost standard way from the general theory of integrable systems We will discuss it in Section 3 3

4 Theorem 2 The series U is a τ-function for the KP hierarchy in variables T i = t i /(i )!; ie, it satisfies the full set of bilinear Hirota equations in these variables In addition, it satisfies the linearized KP equations in the same variables Theorem 2 follows from Theorem, but far from trivially, in spite of their apparent similarity Example 3 The string and the dilaton equations allow one to compute all the values of the bracket in g = 0,, 2 knowing only the following values, which can be obtained using Theorem 2: g = 0 : τ 3 0 = g = : τ 2 = 24 g = 2 : τ 6 = 920, τ 2τ 5 = , τ 3τ 4 = 920, τ 2 2 τ 4 = , τ2 τ 2 3 = 5 44, τ 3 2 τ 3 = 5 24, τ 5 2 = Acknowledgments We are grateful to M Kazarian and S Lando for stimulating discussions We also thank the Stockholm University, where the major part of this work done, for its hospitality 2 Intersection numbers and Hurwitz numbers Here we establish a link between the generating series H (for Hurwitz numbers) and F (for intersection numbers of the ψ-classes on Pic g,n ) Introduce the following linear triangular change of variables: p b = d=b β (d+)/2 ( ) d b+ (d b + )!(b )! t d (7) Thus p = β /2 t 0 β t + 2 β 3/2 t 2, p 2 = β t β 3/2 t 2, p 3 = 2 β 3/2 t 2 4

5 Let us separate the generating series H into 2 parts The unstable part, corresponding to the cases g = 0, n =, 2, is given by H unst (β, p, p 2 ) = b= p b b 2 + β 2 b,b 2 = p b p b2 b + b 2 The stable part is given by H st = H H unst The change of variables was designed to make the following proposition work Proposition 2 The change of variables (7) transforms the series H st into a series of the form β F + O(β) Proof First let β = It is readily seen that, for any d 0, d+ b= ( ) d b+ (d b + )!(b )! bψ = ψd + O(ψ d+ ) (8) as a power series in ψ Using Conjecture 2, it follows that for any d,,d n we have b,,b n b i d i + ( ) d b+ (d b + )!(b )! = h g;b,,b n (2g + n)! d Pic g,n ( Λ 2 + ± Λ 2g) n (ψ d i i + O(ψ d i+ i )) Now assume that d i = dim(pic g,n ) = 4g 3 + n Then each factor in the right-hand side contributes to the integral only through its lowest order term Therefore the right-hand side is equal to ψ d ψdn n, Pic g,n which is, up to a combinatorial factor, precisely a coefficient of F The purpose of introducing the parameter β in the change of variables (7) is precisely to isolate such terms from the others Indeed, we claim that the power of β in a term obtained by the change of variables equals dim(pic g,n ) d i i=

6 To check this, recall that the power of β in a term of H equals 2g + n by definition of H After subtracting (d + )/2 for each variable t d we obtain 2g + n/2 d i /2 = 4g 2 + n d i 2 = dim(pic g,n) d i + 2 as claimed Thus applying the change of variables to H we obtain a series with only positive (half-integer) powers of β, and the lowest order terms in β form the series βf The transformation of the partial derivatives corresponding to (7) is obtained by computing the inverse matrix It is given by p b = b d=0 β (d+)/2 (b )! (b d )! t d (9) Thus p = β /2 t 0, p 2 = β /2 t 0 + β t, p 3 = β /2 t 0 + 2β t + 2β 3/2 t 2 Proposition 22 The change of variables (7) induces the following transformations: L p L t = d 0 L 2 ph unst β t 0 (t + ) + t 2 0 (d + )t d + t d β d t d, t d Both claims of the proposition are obtained by simple computations The concinnity of this result is striking Indeed, both transforms could have contained arbitrarily large negative powers of β, but they happen to cancel out in both cases Further, L 2 ph unst is an infinite series, but after the change of variables it has become a polynomial with only three terms Most important of all, the coefficients L and L 0 of β /2 and β 0, respectively, in the operator L t are precisely the string and dilaton operators from Equations (4) and (5) This leads to the following corollaries 6

7 Corollary 23 We have L t F = F + 2 F + ( ) F t2 0 t β t 0 2 L t F t 0 = 2 2 F t 0 t + β ( 2 F t 2 0 t 0 ) Corollary 24 The change of variables (7) induces the following transformations of generating series: ] L 2 p H st β [U t 0 (t + ) + O β (), L 2 p H β U + O β () Here O β () is a series containing only nonnegative powers of β Proof The first result follows from Corollary 23, while the second one is obtained after a (yet another!) cancellation of the term t 0 (t + ) with the contribution of L 2 p H unst 3 Hirota equations and KP hierarchy In this section we recall some necessary facts about the Hirota and the KP hierarchies and use them to prove Theorem We start with a brief introduction to the Hirota and the KP hierarchy More details can be found, for instance, in [6] The semi-infinite wedge space W is the vector space of formal (possibly infinite) linear combinations of infinite wedge products of the form z k z k 2, with k i Z, k i = i starting from some i Consider a sequence ϕ, ϕ 2, of Laurent series ϕ i C[[z, z] such that ϕ i = z i +(lower order terms) starting from some i Then ϕ ϕ 2 is an element of W The elements that can be represented in that way are called decomposable One way to check whether an element is decomposable, is to verify if it satisfies the Plücker equations Now we will assign an element of W to any power series in variables p, p 2, The series will turn out to be a solution of the Hirota hierarchy if and only if the corresponding element of W is decomposable 7

8 To a Young diagram µ with d squares we assign the Schur polynomial s µ in variables p, p 2, defined by s µ = χ µ (σ)p σ d! σ S d Here S d is the symmetric group, σ is a permutation, χ µ (σ) is the character of σ in the irreducible representation assigned to µ, and p σ = p l p lk, where l,,l k are the lengths of cycles of σ The Schur polynomials s µ with area(µ) = d form a basis of the space of quasihomogeneous polynomials of weight d (the weight of p i being equal to i) Consider a power series τ in variables p i Decomposing it in the basis of Schur polynomials we can uniquely assign to it a (possibly infinite) linear combination of Young diagrams µ (of all areas) Now, in this linear combination we replace each Young diagram µ = (µ, µ 2, ) by the following wedge product: z µ z 2 µ 2, where (µ, µ 2, ) are the lengths of the columns of µ in decreasing order with an infinite number of zeroes added in the end We have obtained an element w τ W The bilinear Plücker equations on the coordinates of w τ happen to combine into bilinear differential equations on τ, called the Hirota equations Thus, as we said, w τ is decomposable if and only if τ is a solution of the Hirota equations Let us define these equations precisely Consider two partitions λ and µ of an integer d Denote by χ µ (λ) the character of any permutation with cycle type λ in the irreducible representation assigned to µ Denote by Aut(λ) the number of permutations of the elements of λ that preserve their values For instance Aut(7, 6, 6, 4,,,,, ) = 2! 5! Let d i = /p i Let D µ be the differential operator D µ = χ µ (λ) d λ d λk Aut(λ), λ, λ =d where k is the number of elements of λ For instance D () =, D () = d, D (2) = 2 d2 + d 2, D (,) = 2 d2 d 2, D (3) = 6 d3 + d d 2 + d 3, D (2,) = 3 d3 d 3, D (,,) = 6 d3 d d 2 + d 3 Let τ be a formal power series in p, p 2, If the constant term of τ does not vanish, we can also consider its logarithm F = lnτ 8

9 Definition 3 (see [], Proposition ) The Hirota hierarchy is the following family of bilinear differential equations: Hir i,j (τ) = D () τ D (j,i) τ D (i ) τ D (j,) τ + D (j) τ D (i,) τ, (0) for 2 i j Substituting τ = e F and dividing by τ 2 we obtain a family of equations on F It is called the Kadomtsev Petviashvili or KP hierarchy: KP i,j (F) = Hir i,j(e F ) e 2F () Finally, leaving only the linear terms in the KP hierarchy we obtain the linearized KP equations : LKP i,j (F) = linear part of KP i,j (F) (2) Example 32 Denoting the derivative with respect to p i by the index i, we have Hir 2,2 = ττ 2,2 τ 2 2 ττ,3 + τ τ τ2, 3 τ τ,, + 2 ττ,,,, KP 2,2 = F 2,2 F,3 + 2 F 2, + 2 F,,,, LKP 2,2 = F 2,2 F,3 + 2 F,,,, Hir 2,3 = ττ 2,3 τ 2 τ 3 ττ,4 + τ τ τ,τ,2 2 τ τ,,2 6 τ,,τ ττ,,,2 2 τ τ 2,2 + 2 ττ,2,2 τ,2 τ 2 2 ττ,,3 + 2 τ,τ ττ,,,, 8 τ τ,,, + 2 τ,τ,,, KP 2,3 = F 2,3 F,4 + F, F,2 + 6 F,,,2 + 2 F F 2,2 2 F F,3 + 4 F F 2, + 24 F F,,, + 2 F,2,2 2 F,,3 + 2 F,F,, + 24 F,,,,, LKP 2,3 = F 2,3 F,4 + 6 F,,,2 + 2 F,2,2 2 F,, F,,,, Remark 33 Every Hirota equation can be simplified by adding to it some partial derivatives of lower equations This, in turn, leads to simplified KP In the published version of this paper, the linearized KP equations are erronously called the dispersionless limit of KP 9

10 and LKP equations For instance, we have Hir 2,3 Hir 2,2 = ττ 2,3 τ 2 τ 3 ττ,4 + τ τ p 2 τ,τ,2 2 τ τ,,2 6 τ,,τ ττ,,,2, KP 2,3 2 F KP 2,2 KP 2,2 = F 2,3 F,4 + F, F,2 + 2 p 6 F,,,2, LKP 2,3 LKP 2,2 = F 2,3 F,4 + 2 p 6 F,,,2 Thus we obtain a simplified hierarchy which is, of course, equivalent to the initial one Sometimes it is the equations of this simplified hierarchy that are called Hirota equations However, for our purposes it is easier to use the equations as we defined them Proof of Theorem We will actually prove that for any function c = c(β), the series c + L 2 ph satisfies the Hirota hierarchy Let us show that the element of W assigned to c + L 2 ph is decomposable Consider the following Laurent series in z: ϕ = cz + n 0 β n(n+)/2 z n, ϕ i = z i e (i )β z i for i 2 The coefficients of these series are shown in the matrix below e 0β e 6β e 3β e β c e β e 2β e 3β e 4β We claim that expanding the wedge product ϕ ϕ 2 and replacing every Young diagram by the corresponding Schur polynomial we obtain the series c + L 2 ph The proof goes as in [7] We introduce the so-called cut-and-join operator A = 2 i,j= [ ] 2 (i + j)p i p j + ijp i+j p i+j p i p j 0

11 Then L p H satisfies the equation (L p H)/β = A(L p H) (see [3]) Since the operator L p commutes both with A and with /β, the series L 2 p H satisfies the same equation The Schur polynomials s λ are eigenvectors of A The eigenvalue corresponding to a Young diagram λ equals f λ = λi (λ 2 i 2i+), where λ i are the column lengths This allows one to reconstitute the whole series L 2 ph starting with its β-free terms L 2 ph β=0 : if then L 2 ph β=0 = c λ s λ L 2 p H = c λ s λ e f λβ It is apparent from the form of the above matrix that the coefficients of s λ in the expansion are nonzero only in two cases: (i) for the empty diagram, where the coefficient equals c; (ii) for the hook Young diagrams λ = hook(a, b) with column lengths a +,,,, }{{} b For a Young diagram like that, the coefficient of s hook(a,b) equals For the β-free terms we have ( ) b e [a(a+)/2 b(b+)/2]β L 2 ph β=0 = i p i = a,b 0( ) b s hook(a,b), the second equality being an exercise in the representation theory To this we add the remark that a(a + )/2 b(b + )/2 is precisely the eigenvalue f λ for λ = hook(a, b) It follows that the series corresponding to ϕ ϕ 2 equals L 2 ph as claimed Thus we have proved that the series c+l 2 ph satisfies the Hirota hierarchy The claim about the LKP equations is a simple corollary of that Indeed, it follows from Definition 3 that for any series G we have Hir i,j ( + G) Hir i,j (G) = LKP i,j (G) Thus, from the fact that both L 2 ph and +L 2 ph satisfy the Hirota equations it follows immediately that L 2 ph satisfies the linearized KP equations

12 4 Hirota equations and the change of variables Now we are going to study the effect of the change of variables (7) on the Hirota hierarchy and prove Theorem 2 In Section 3 we assigned to each Young diagram µ an operator D µ and used these operators as building blocks to define the Hirota equations It turns out that the change of variables (7) acts on D µ by biting off the corners of µ From this we will deduce that each Hirota equation becomes, after the change of variables, a linear combination of lower Hirota equations As in Theorem 2, we rescale the variable t i by setting t i = i!t i+ Then we have ( ) i = β /2 ( ) i + β ( ) i + + β i/2 (3) p i 0 T T 2 i T i instead of Eq (9) Consider a linear differential operator D with constant coefficients in variables p i Denote /p i by d i and consider d i as a new set of variables Introduce the differential operator S = i id i d i+ in these variables Then we have the following lemma Lemma 4 Assume D is a quasi-homogeneous polynomial in variables d i with total weight n Then applying the change of variables (3) to D viewed as a differential operator is equivalent to applying the differential operator β n/2 e S/ β to D viewed as a polynomial in variables d i The proof is a simple check What we actually want is to apply the change of variables (3) to the operators D µ defined in Section 3 Indeed, these operators are the building blocks of the Hirota equations (Definition 3) The answer is given below in Proposition 43 Definition 42 A square of a Young diagram µ is called a corner if, when we erase it, we obtain another Young diagram In other words, a corner is a square with coordinates (i, µ i ) such that either µ i+ < µ i or µ i is the last column of µ If (i, µ i ) is a corner of µ we will denote by µ i the diagram obtained by erasing this corner 2

13 Proposition 43 We have SD µ = (i,µ i )=corner of µ (µ i i) D µ i Proof Let µ be a Young diagram with d squares and λ a partition of d Assume that λ has k parts Then, for i k, denote by λ+ i the partition of d obtained from λ by replacing λ i by λ i + Writing down explicitly the action of S on D µ one finds that the assertion of the proposition is equivalent to the following identity: (i,µ i )=corner of µ (µ i i) χ µ i (λ) = k λ i χ µ (λ + i ) (4) i= To prove this identity we need a short digression into the representation theory of the symmetric group as presented in [0] Consider the subgroup S d S d consisting of the permutations that fix the last element d The irreducible representation of S d assigned to µ is then also a representation of S d, although not necessarily irreducible It turns out that this representation is isomorphic to µ i, corners of µ where, by abuse of notation, µ i stands for the irreducible representation of S d assigned to this Young diagram Further, consider the following element of the group algebra CS d : X = (, d) + (2, d) + + (d, d) This element (the sum of all transpositions involving d) is called the first Jucys Murphey-Young element It obviously commutes with the subgroup S d Therefore its eigenspaces in the representation µ coincide with the irreducible subrepresentations of S d, ie, they are also in one-to-one correspondence with the corners of µ The eigenvalue corresponding to the corner (i, µ i ) equals µ i i (see [0]) Using this information, let us choose a permutation σ S d with cycle type λ and compute in two different ways the character where σ X CS d χ µ (σ X), 3

14 First way Both σ and X leave invariant the irreducible subrepresentations of S d For X such a subrepresentation is an eigenspace with eigenvalue µ i i The character of σ in the same subrepresentation equals χ µ i (λ) We obtain the left-hand side of Identity (4) Second way Let us see what happens when we multiply σ by X Each transposition in X increases the length of precisely one cycle of σ by This is equivalent to increasing one of the λ i s by Moreover if the ith cycle of σ has length λ i, it will be touched by a transposition from X exactly λ i times Thus we obtain the right-hand side of Identity (4) This completes the proof Proposition 44 The change of variables (3) transforms the Hirota equation Hir i,j into an equation of the form +j c i,j )/2 Hir β(i i,j 2 i i, 2 j j i j for some rational constants c i,j The constant c i,j of the leading term equals Proof By Definition 3 the equation Hir i,j has the form Hir i,j (τ) = D () τ D (j,i) τ D (i ) τ D (j,) τ + D (j) τ D (i,) τ According to Lemma 4, applying the change of variables to the equation is the same as applying to each D µ in this expression the operator β (i+j)/2 e S/ β To simplify the computations, consider the flow e ts applied to Hir i,j We will compute the derivative of this flow with respect to t If D is the vector space of all polynomials in variables d i, then Hir i,j lies in D D The flow e ts acts as e ts e ts, while its derivative with respect to t is S + S We will prove that S+S applied to Hir i,j is a linear combination of lower Hirota equations (i < i, j < j) Since this is true for all i, j, when we integrate the flow we see that Hir i,j will have changed by a linear combination of lower Hirota equations It remains to apply S + S to Hir i,j To do that we use Proposition 43 If i < j we obtain ] D () [(i 2)D (j,i ) + (j )D (j,i) + 0 D (j,i) D (i ) (j )D (j,) (i 2)D (i 2) D (j,) + D (j) (i 2)D (i 2,) + (j )D (j ) D (i,) 4

15 If i = j we obtain = (i 2) Hir i,j + (j ) Hir i,j D () (i 2)D (i,i ) + 0 D (i,i) D (i ) (i )D (i,) (i 2)D (i 2) D (i,) + D (i) (i 2) D (i 2,) + (i )D (i ) D (i,) This completes the proof = (i 2) Hir i,i Remark 45 The family of equations given in Proposition 44 is equivalent to the Hirota hierarchy Indeed, the equations of the Hirota hierarchy can be obtained from these equations by linear combinations and vice versa Proof of Theorem 2 The series c + L 2 ph satisfies the Hirota equations by Theorem Therefore, by Proposition 44 and Remark 45, the series obtained from it under the change of variables (7) also satisfies the Hirota hierarchy According to Corollary 24, this new series has the form c + β U + O β () Taking c = c / β and considering the lowest order terms in β we obtain that c + U satisfies the Hirota hierarchy for any constant c It follows that U satisfies the linearized KP hierarchy Appendix: On Hodge integrals In this section we will use the change of variables suggested in [7] to study Hodge integrals over the moduli spaces of curves We consider the integrals involving a unique λ-class and arbitrary powers of ψ-classes A Hurwitz numbers and Hodge integrals Here we study intersection theory on moduli spaces rather than on Picard varieties We follow the same path as in Sections and 2, but with different intersection numbers and Hurwitz numbers Our aim is to extend the results of [7] Instead of () we define the following brackets τ d τ dn (k) = ψ d ψn dn λ k (5) M g,n 5

16 for k + d i = 3g 3 + n (otherwise the bracket vanishes) Instead of (2) we will use the generating series F (k) (t 0, t, ) = n n! We can also regroup these series into a unique series d,,d n τ d τ dn (k) t d t dn (6) F (z; t 0, t, ) = k 0( ) k z k F (k) Instead of the Hurwitz numbers of Definition we now use different Hurwitz numbers Fix n positive integers b,,b n Let d = b i be their sum Definition A6 The number of degree d ramified coverings of the sphere by a genus g surface possessing n numbered preimages of with multiplicities b,,b n, and d + n + 2g 2 fixed simple branch points is called a Hurwitz number and denoted by h g;b,,b n We introduce the following generating series for the these numbers: H(β, p, p 2, ) = g,n n! β d+n+2g 2 (d + n + 2g 2)! b,,b n h g;b,,b n p b p bn It is divided in two parts The unstable part, corresponding to g = 0, n =, 2 equals H unst = b β b bb 2 p b + b! 2 b,b 2 β b +b 2 b b b b 2 2 (b + b 2 )b!b 2! p b p b2 The stable part equals H st = H H unst Finally, instead of Conjecture 2 we use the so-called ELSV formula proved in [2] Theorem 3 (The ELSV formula [2]) We have h g;b,,b n = (d + n + 2g 2)! n i= b b i i b i! M g,n λ + λ 2 ± λ g ( b ψ ) ( b n ψ n ) As before, it turns out that the series F and H are related via a change of variables based on Equation (8) However the change of variables is different 6

17 from (7) due to (i) the factors b b i i /b i! in the ELSV formula and (ii) to a different relation between the number of simple branch points and the dimension of the Picard/moduli space Namely, following [7], we let p b = d=b ( ) d b+ (d b + )! b b β b (2d+)/3 t d (7) Thus p = β 4/3 t 0 β 6/3 t + 2 β 8/3 t 2, p 2 = 2 β 9/3 t 2 β /3 t 2 +, p 3 = 9 β 4/3 t 2 This change of variables transforms H into a series in variables t 0, t,, and β 2/3 We will also need a more detailed version of Equation (8): d+ b= ( ) d b+ (d b + )!(b )! bψ = ψd + a d,d+k ψ d+k, where a d,d+k are some rational constants that actually happen to be integers For instance, k= ψ ψ + 2ψ = + ψ + ψ 2 +, = ψ + 3ψ 2 + 7ψ 3 +, /2 ψ 2ψ + /2 3ψ = ψ2 + 6ψ ψ 4 + Using these constants we introduce the following differential operators: L = L 2 = n=0 n=0 a n,n+ t n+ t n, a n,n+2 t n+2 t n + 2! n,n 2 =0 a n,n + a n2,n 2 + t n +t n2 + 2 t n t n2, 7

18 L 3 = a n,n+3 t n a n,n t n=0 n 2! +2 a n2,n 2 + t n +2t n2 + t n,n 2 =0 n t n2 + 2 a n,n 2! + a n2,n 2 +2 t n +t n2 +2 t n,n 2 =0 n t n2 + 3 a n,n 3! + a n2,n 2 + a n3,n 3 + t n +t n2 +t n3 +, t n t n2 t n3 n,n 2,n 3 =0 and so on We can also regroup these operators in a unique operator L = + zl + z 2 L 2 + These operators and the change of variables (7) were designed to make the following proposition work Proposition A7 Performing the change of variables (7) on the series H st and replacing β 2/3 by z we obtain the series LF Proof Using the ELSV formula one can check that the change of variables (7) transforms H into the series n,g n! d,d n ( β 2/3 λ + β 4/3 λ 2 ) M g,n n i= (ψ d i + β 2/3 a di,d i + ψ d i+ + ) The proposition follows A2 Hierarchies and operators Proposition A8 We have L = e l, l = zl + z 2 l 2 + is a first order linear differential operator: l k = α n,n+k t n+k t n Proof Consider the operators l k and l as above with indeterminate coefficients α n,n+k Consider the expansion of e l and denote by a n,n+k the coefficient of t n+k /t n in this expansion We have a n,n+ = α n,n+, 8

19 a n,n+2 = α n,n α n,n+ α n+,n+2, a n,n+3 = α n,n α n,n+ α n+,n α n,n+2 α n+2,n α n,n+ α n+,n+2 α n+2,n+3, and so on Note that these equalities allow one to determine the coefficients α unambiguously knowing the coefficients a Now consider the coefficient of a monomial p t ni +k i t i= ni in the same expansion of e l It is equal to Aut{(n, k ),, (n p, k p )} p! and we claim that this sum can be factorized as Aut{(n, k ),,(n p, k p )} p! αni,n i +k i + higher order terms, ani,n i +k i Indeed, suppose that we have already chosen a power of l, say l q and a term in each of the q factors that contribute to the coefficient of p t ni +k i i= t ni Now we must choose some additional power l r of l and a term in each of the r factors that will contribute to the coefficient of t np+k p = t n+k t np t n Moreover, we must choose the positions of the r new factors among the q that are already chosen This can be done in ( ) q + r q ways (The operator t n+k /t n acts by replacing t n by t n+k Thus the r terms in question divide the segment [n, n + k] into r parts and should be ordered in a uniquely determined way) 9

20 In the end we must divide the coefficient thus obtained by (q + r)! since we are looking at e l Thus we obtain a coefficient of q! r! for any choice of r terms Now, if q = 0 what we have finally obtained is precisely the expression for a n,n+k For a general q we will therefore obtain the same expression for a n,n+k divided by q! Thus we have proved that a n,n+k = a np,n p+k p can be factored out in the coefficient of p t ni +k i t ni i= The same is true for a ni,n i +k i for all i Thus the coefficient is the product of a ni,n i +k i as claimed In other words, we showed that the coefficients of exp(l ) coincide with those of L Conjecture A9 The operators l k have the form ( ) n + k + l k = c k t n k + n 0 for some sequence of rational constants c k t n+k The sequence c k seems quite irregular and starts as follows:, 2, 2, 2 3, 2, 3 4, 6, 29 4, 493 2, 27 6, , , 60 Now we will establish a hierarchy of partial differential equations satisfied by F We use Propositions A7 and A8 together with the following fact Theorem 4 (See [9, 7]) The series H satisfies the KP hierarchy in variables p, p 2, This theorem is proved like Theorem 2 by noticing that exp(h) satisfies the cut-and-join equation Applying the change of variables (7) to the KP equations we will obtain partial differential equations satisfied by L F These equations can, of course, be considered as equations on F, since the coefficients of L are known However, the equations thus obtained are infinite, ie, with an infinite number of 20

21 terms Our goal is to prove that we can combine them in a way that leads to finite differential equations The derivatives /p b are expressed via /t d by computing the inverse of the matrix of the change of variables (7) We have b = b b p b d=0 β b+(2d+)/3 (b d )! t d (8) Thus = β 4/3, p t 0 p 2 = 2β 7/3 t 0 + 2β 9/3 t, p 3 = 9 2 β0/3 t 0 + 9β 2/3 t + 9β 4/3 t 2 Now using Theorem 4 and Proposition A7 we will transform the KP hierarchy into a system of equations on F We will illustrate the procedure on the example of KP 2,2 We know that KP i,j (H) = 0 For i = j = 2 this means 2 H p H p p ( 2 H p 2 ) H 2 p 4 = 0 Using H = H st + H unst and the explicit expression of H unst we transform KP i,j into a (finite) equation KP i,j on H st For instance, for i = j = 2, we obtain 2 H st p H st p p ( 2 H st p 2 ) H st + H st 2 p 4 2 β22 p 2 = 0 Applying the change of variables (8) and replacing β 2/3 by z we transform this into an equation KP i,j on LF For i = j = 2 we have 2 (LF) t 0 t + 2 ( 2 (LF) t 2 0 ) 2 + ( ) 4 (LF) +z 4 2 (LF) 9 2 (LF) = 0 2 t 4 0 t 2 t 0 t 2 In principle, we could have stopped here However, in this form the equation is only useful to study the z-free part F (0) of F, which was done in [7] Indeed the operators L i for i are composed of infinitely many terms 2

22 This means that if we develop the above equation and take its coefficient of z, we will obtain an infinite equation on F (0) and F () Such an equation is quite useless if we want to compute F () Therefore we continue with the following theorem (recall that L = e l ): Theorem 5 Consider the expression e l KP i,j (e l F ) as a series in z Then its coefficient of z k is a finite differential equation on F (0),,F (k) Example A0 The coefficient of z in e l KP 2,2 (e l F ) gives the following equation: 2 F () t 0 t + 2 F (0) t F () t F () 2 t F (0) 3 2 F (0) 2 2 F (0) t 0 t 2 t 2 t F (0) 4 F (0) t 0 t 3 t 3 0 t = 0 Assuming that we know F (0), this equation, together with the string and the dilaton equations, allow us to compute all the coefficients of F (), that is, all Hodge integrals involving λ Proof of Theorem 5 Let Q be a linear differential operator (in variables t d ) whose coefficients are polynomials in z Then e l Qe l = Q + [Q,l] + 2 [[Q,l],l] + is a series in z whose coefficients are finite differential operators We will denote this series by Q Now suppose we have several linear operators Q,,Q r as above Since l is a first order operator, we obtain e l Q (e l F ) Q r (e l F ) = Q (F ) Q r (F ) This is, once again, a series in z whose coefficients are finite differential equations on the F (k) The theorem now follows from the fact that every equation KP i,j is a finite linear combination of expressions of the form Q (e l F ) Q r (e l F ) 22

23 Thus every equation KP i,j and every power of z gives us a finite differential equation on the functions F (k) Below we describe some facts concerning these equations that we have observed but not proved For any F (k) and any (d, d ) (0, 0) by taking linear combinations of the equations in question we can obtain an equation of the form F (k) t d t d = terms with more than 2 derivations For homogeneity reasons the sum of indices in these derivatives will be smaller than d +d Therefore we use similar equations with smaller d +d to simplify the right-hand part by substitutions After a finite number of substitutions we will obtain an expression of F (k) /t d t d exclusively via partial derivatives with respect to t 0 Moreover, these expressions themselves be organized into equations on F : F 0, = F 0,2 = F, = F 0,3 = F,2 = ( 2 F 2 0,0 + ) ( 2 F 0,0,0,0 z 24 F 2 0,0,0 + ) 720 F 0,0,0,0,0,0 ( +z F 0,0,0F 0,0,0,0, F 2 0,0,0,0 + ) F 0,0,0,0,0,0,0,0 +, ( 6 F 3 0,0 + 2 F 0,0F 0,0,0, F 2 0,0,0 + ) 240 F 0,0,0,0,0,0 ( z 24 F 0,0F 2 0,0, F 0,0F 0,0,0,0,0, F 0,0,0F 0,0,0,0, F 2 0,0,0,0 + ) 7560 F 0,0,0,0,0,0,0,0 +, ( 3 F 3 0,0 + 6 F 0,0F 0,0,0, F 2 0,0,0 + ) 44 F 0,0,0,0,0,0 ( z 2 F 0,0F 2 0,0, F 0,0F 0,0,0,0,0, F 0,0,0F 0,0,0,0, F 2 0,0,0,0 + ) 4320 F 0,0,0,0,0,0,0,0 +, ( 24 F 4 0, F 2 0,0F 0,0,0, F 0,0F 2 0,0, F 0,0F 0,0,0,0,0, F 0,0,0F 0,0,0,0, F 2 0,0,0,0 + ) 6720 F 0,0,0,0,0,0,0,0 +, ( 8 F 4 0,0 + 8 F 2 0,0 F 0,0,0,0 + 2 F 0,0F 2 0,0, F 0,0F 0,0,0,0,0,0 23

24 + 60 F 0,0,0F 0,0,0,0, F 2 0,0,0,0 + ) 2880 F 0,0,0,0,0,0,0,0 + Not thoroughly unexpectedly, it turns out that the free terms of these equations form the well-known Korteweg de Vries (or KdV) hierarchy on F (0) The first equation above (expressing F 0, ), together with the string and dilaton equations, is sufficient to determine the values of all Hodge integrals involving a single λ-class This approach seems to be simpler than the method of [8] based on the study of double Hurwitz numbers References [] B A Dubrovin, S M Natanzon Real theta-function solutions of the Kadomtsev-Petviashvili equation (Russian) Izv Akad Nauk SSSR Ser Mat bf 52 (988), no 2, , 446; translation in Math USSR-Izv 32 (989), no 2, [2] T Ekedahl, S Lando, M Shapiro, A Vainshtein Hurwitz numbers and intersections on moduli spaces of curves Invent Math 46 (200), no 2, [3] I P Goulden, D M Jackson Transitive factorizations into transpositions and holomorphic mappings on the sphere Proc Amer Math Soc 25 (997), no, 5 60 [4] I P Goulden, D M Jackson The number of ramified coverings of the sphere by the double torus, and a general form for higher genera J Combin Theory A 88 (999), [5] I P Goulden, D M Jackson, R Vakil Towards the geometry of double Hurwitz numbers Adv Math 98 (2005), no, [6] V G Kac, A K Raina Bombay lectures on highest weight representations of infinite-dimensional Lie algebras Advanced Series in Mathematical Physics, 2 World Scientific Publishing Co, Inc, Teaneck, NJ, 987 [7] M E Kazarian, S K Lando An algebro-geometric proof of Witten s conjecture Max-Planck Institute preprint MPIM (2005), 4 p 24

25 [8] Y-S Kim Computing Hodge integrals with one λ-class arxiv: math-ph/05008 (2005), 30 p [9] A Okounkov Toda equations for Hurwitz numbers Math Res Lett 7 (2000), no 4, [0] A M Vershik, A Okounkov A new approach to representation theory of symmetric groups II Zap Nauchn Sem S-Peterburg Otdel Mat Inst Steklov (POMI) 307 (2004), Teor Predst Din Sist Komb i Algoritm Metody 0, 57 98, 28 (in Russian); English translation in J Math Sci (N Y) 3 (2005), no 2,