Combinatorics of KP hierarchy structural constants
aa r X i v : . [ h e p - t h ] J a n Combinatorics of KP hierarchy structural constants
A. Andreev a,c ∗ , A. Popolitov a,b,c † , A. Sleptsov a,b,c ‡ , A. Zhabin a,c § MIPT/TH-19/20ITEP/TH-34/20IITP/TH-21/20 a Institute for Theoretical and Experimental Physics, Moscow 117218, Russia b Institute for Information Transmission Problems, Moscow 127994, Russia c Moscow Institute of Physics and Technology, Dolgoprudny 141701, Russia
Dedicated to the memory of Sergey Mironovich Natanzon
ABSTRACT
Following Natanzon-Zabrodin, we explore the Kadomtsev–Petviashvili hierarchy as an infinite system of mutually consistent relations on the secondderivatives of the free energy with some universal coefficients. From this point of view, various combinatorial properties of these coefficients naturallyhighlight certain non-trivial properties of the KP hierarchy. Furthermore, this approach allows us to suggest several interesting directions of the KPdeformation via a deformation of these coefficients. We also construct an eigenvalue matrix model, whose correlators fully describe the universal KPcoefficients, which allows us to further study their properties and generalizations.
This paper is just the beginning of a very large program of multi-faceted study of the KP hierarchy suggested to us by SergeyNatanzon. He had his own special view of the KP hierarchy, which made it possible to see in it some new interesting structuresthat are completely invisible with other approaches. We are deeply grateful to him for numerous scientific discussions, for fuelingour interest in the KP hierarchy and for his characteristic style of discussing science.
The Kadomtsev-Petviashvili (KP) hierarchy has many different applications in modern physics and mathematics. Historicallyit was studied as equations with soliton solutions, but very soon it was discovered that partition functions and correlators ofsome field theories are solutions of the hierarchy as well. It often happens that partition function can be represented as amatrix model, which provides a connection between KP hierarchy and matrix models. Probably the most famous example isthe Kontsevich matrix model [1], which is a partition function of 2D gravity. Among other important examples lattice gaugetheories of QCD [2,3], the Ooguri-Vafa partition function for HOMFLY polynomials of any torus knot [4,5], generating functionfor simple Hurwitz numbers [6–8]. Moreover, recently interest in KP hierarchy resurgent due to fantastic rapid progress inunderstanding of superintegrable properties of a particular version of KP, the so-called BKP hierarchy [8–11].The KP hierarchy can be understood as an infinite system of compatible non-linear differential equations. All the equationsmay be encoded in the Hirota bilinear identity: I ∞ e ξ ( t ,z ) τ ( t + t − [ z − ]) τ ( t − t + [ z − ]) dz = 0 , (1)where we used a standard notation ξ ( t , z ) = ∞ X k =1 t k z k t ± [ z − ] = (cid:26) t ± z , t ± z , t ± z , . . . (cid:27) (2)Expanding the integrand near z = ∞ and calculating the coefficient in the front of z − , each coefficient in front of everymonomial of ¯t gives an equation for τ ( t ). Functions that satisfy (1) are called τ -functions. They may depend on an infinitenumber of variables t = { t , t , t , . . . } called ”times”. Previously mentioned partition and generating functions are KP τ -functions. According to the works of Kyoto school [12, 13], KP hierarchy closely related to rich mathematical structures, suchas infinite-dimensional Lie algebras, projective manifolds, symmetric functions and boson-fermion correspondence. Each of ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] τ -functions and many others have a geometric expansion over compact Riemann surfaces (genusexpansion). Genus expansion for τ -functions coincides with expansion in parameter ~ for the ~ -KP hierarchy [14]. The introduc-tion of the ~ parameter slightly modifies the hierarchy and allows one, among other things, to obtain solutions of the classicalKP hierarchy for ~ = 1 and dispersionless KP for ~ → ~ -formulation of the KP hierarchy was first studied byTakasaki and Takebe in [17, 18], where they described a method for deformation of the classical τ -function.Natanzon and Zabrodin formulated another approach [19, 20] for description of the ~ -KP. The advantage of their approachis that formal solutions for the F -function ( F = log τ ) can be explicitly expressed in terms of boundary data using universalinteger coefficients that help to define the entire ~ -KP hierarchy. Moreover, an arbitrary solution of the ~ -KP hierarchy can berestored from its boundary data, using these coefficients and their higher analogs, which are determined recursively. Namely,the set of the integer coefficients P i,j ( s , . . . , s m ), which we also call the universal KP coefficients , enters the KP equations as(see, for instance, [21]) ∂ ~ i ∂ ~ j Fij = X m ≥ ( − m +1 m X s ,...,s m ≥ P i,j ( s , . . . , s m ) ∂ x ∂ ~ s Fs . . . ∂ x ∂ ~ s m Fs m . (3)where ∂ ~ i is a ~ -deformed derivative with respect to t i , see formula (16) below. From these equation we see that P i,j ( s , . . . , s m )are one of the central ingredients of the KP equations. Definition of these coefficients can be given in combinatorial terms byenumeration of sequences of positive integers (see section 2, formula (17)).The main goal of this paper is to establish and develop the relation between combinatorics and integrability. We want tofind out how basic properties of the combinatorial coefficients P i,j ( s , . . . , s m ) affect the various properties of τ -functions. Thepurpose of the paper is to point out new interesting research directions, but we do not develop them exhaustively in this shortnote. Therefore, in many cases we stop after providing first non-trivial example, just enough to demonstrate, that a particulardirections is potentially interesting and is worth studying.The paper is organized as follows. In section 2, we introduce all the necessary definitions and theorems.Section 3 is devoted to various approaches to calculation of the combinatorial coefficients. We show that they can becalculated using an explicit formula that includes the sum of the binomial coefficients and has a clear geometric meaning. Inaddition, we consider two different generating function for the universal coefficients. One of them, up to normalization, has thesimple form of a sum over Young diagrams of length ℓ ( λ ) ≤ F ( y , y ; x ) ∼ X λ S λ ( y , y ) S λ ( x ) , (4)where S λ is Schur polynomial. This generating function becomes a τ -function of KP hierarchy itself after standard replacementof variables kt k = P i x ki , which gives us a hint on possible deformation of the universal coefficients (section 6), consideringanother solutions of KP hierarchy as generating function of new coefficients.The second generating function corresponds to the, so-called, Fay identity and, as we discuss in section 4, allows us to obtainsome restrictions on resolvents in topological recursion [22–28].In section 5 we construct a simple matrix eigenvalue model, whose correlators give the universal KP coefficients. Theform of these correlators also makes it possible to generalize the coefficients. Generalization of matrix model has the followingmotivation. There are Ward identities in matrix models which can be solved recursively, and as we expect, correspondingrecursion relations are related with recursion relations for higher analogs of universal coefficients in some sense. Furthermoregenerating function for the averages of Schur polynomials h S λ . . . S λ m i depend on the set of time variables { t (1) , . . . , t ( m ) } andin the simplest case (4) we obtain τ -function of KP hierarchy, so generalized matrix model may be somehow connected with m -component KP hierarchy.In Section 6 we discuss possible approach to KP deformation via deformation of generating functions of the combinatorialcoefficients. We suggest another deformed generating functions that have the same properties as the initial one. Such consid-eration may help to understand what is the role of the combinatorial coefficients in ~ -KP hierarchy: are they responsible forintegrability or the certain form of equations (3) is important.The last section 7 is a discussion where we list main results of this paper and questions for further research. Schur polynomials.
Following [29] we define Young diagram as a sequence of ordered positive integers λ ≥ · · · ≥ λ ℓ ( λ ) > λ = [ λ , . . . , λ ℓ ( λ ) ]; ℓ ( λ ) is the length of Young diagram. Schur polynomials S λ ( x ) are symmetric functions2epending on an arbitrary set of variables x = { x , x , . . . } and a Young diagram λ . S λ ( x , . . . , x n ) := det ≤ i,j ≤ n (cid:16) x λ j + j − i (cid:17) det ≤ i,j ≤ n (cid:16) x j − i (cid:17) (5)If n > ℓ ( λ ), then λ j are equal to zero for large enough j . Schur polynomials labeled by Young diagrams of length ℓ ( λ ) = 1 wecall symmetric Schur polynomials. Although all Schur polynomials are symmetric functions, such a name for particular Youngdiagrams is due to representation theory. Sometimes Schur polynomials are considered in variables t = { t , t , . . . } . The changefrom variables x is given via t k = 1 k X i ≥ x ki . (6)An important property of Schur polynomials that we frequently use in what follows is the Cauchy-Littlewood identity: X λ S λ ( t ) S λ ( t ) = exp ∞ X k =1 kt k t k ! (7) ~ -KP hierarchy. We briefly review the main facts about the KP equations and solutions. For the detailed explanationsee [30]. KP hierarchy is an infinite set of non-linear differential equations with the first equation given by14 ∂ F∂t = 13 ∂ F∂t ∂t − (cid:18) ∂ F∂t (cid:19) − ∂ F∂t (8)It is more common to work with τ -function τ ( t ) = exp( F ( t )) than with free energy F ( t ). We assume that τ ( t ) is at least aformal power series in times t k , and maybe it is even a convergent series. Entire set of equations of hierarchy can be written interms of τ -function using Hirota bilinear identity (1), which, in turn, is equivalent to the following functional equation( z − z ) τ [ z ,z ] τ [ z ] + ( z − z ) τ [ z ,z ] τ [ z ] + ( z − z ) τ [ z ,z ] τ [ z ] = 0 (9)where τ [ z ,...,z m ] ( t ) = τ t + m X i =1 [ z − i ] ! (10)and the shift of times is the same as in (2). Equation (9) should be satisfied for an arbitrary z , z , z . One can expand τ -functionat the vicinity of z i = ∞ and obtain partial differential equation for τ -function at every term z − k z − k z − k .All formal power series solutions of KP hierarchy can be decomposed over the basis of Schur polynomials τ ( t ) = X λ C λ S λ ( t ) . (11)Function written as a formal sum over Schur polynomials is a KP solution if and only if coefficients C λ satisfy the Pl¨uckerrelations. The first such relation is C [2 , C [ ∅ ] − C [2 , C [1] + C [2] C [1 , = 0 . (12)The simplest way to define ~ -KP hierarchy is to deform bilinear equations (9) for τ -function of the classical KP hierarchy inthe following way [19, 21]: ( z − z ) τ [ z ,z ] τ [ z ] + ( z − z ) τ [ z ,z ] τ [ z ] + ( z − z ) τ [ z ,z ] τ [ z ] = 0 τ [ z ,...,z m ] ( t ) = τ t + ~ m X i =1 [ z − i ] ! t + ~ [ z − ] = (cid:26) t + ~ z , t + ~ z , t + ~ z , . . . (cid:27) (13)By setting parameter ~ = 1 we obtain classical KP hierarchy and the limit ~ → ~ -)KP equations is the differential Fay identity:∆( z )∆( z ) F = log (cid:18) − ∆( z ) ∂ F − ∆( z ) ∂ Fz − z (cid:19) , (14)3here ∆( z ) = e ~ D ( z ) − ~ , D ( z ) = X k ≥ z − k k ∂ k . (15)KP hierarchy can be considered as an infinite set of compatible differential equations on the F -function, where F ( t ) = ~ log( τ ( t )). To describe the equations in an unfolded form we need two more definitions. First one is deformed partialderivatives ∂ ~ k which are defined via symmetric Schur polynomials in t -variables. Each t i one should replace with ~ i ∂ i : ∂ ~ k := k ~ S [ k ] ( ~ e ∂ ) , e ∂ = (cid:26) ∂ , ∂ , ∂ , . . . (cid:27) (16)Limit ~ → ∂ ~ k into usual ones ∂ k .The next definition is the main topic of our study. Let us define combinatorial coefficients P i,j ( s , . . . , s m ) as the number ofsequences ( i , . . . , i m ) and ( j , . . . , j m ) of positive integers such that i + · · · + i m = i , j + · · · + j m = j and i k + j k = s k + 1.These coefficients can also be understood as the number of matrices of size 2 × m with fixed sums over rows and columns: P i,j ( s , . . . , s m ) := (cid:18) i . . . i m j . . . j m (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i k , j k ∈ N , i + · · · + i m = ij + · · · + j m = ji k + j k = s k + 1 ∀ k ∈ , m (17)Coefficients (17) are fundamental in the following sense. They allow us to express all the KP equations in an explicit form andfully determine ~ -KP hierarchy.Following [19, Lemma 3.2] the ~ -KP hierarchy can be rewritten as the system of equations: ∂ ~ i ∂ ~ j Fij = X m ≥ ( − m +1 m X s ,...,s m ≥ P i,j ( s , . . . , s m ) ∂ x ∂ ~ s Fs . . . ∂ x ∂ ~ s m Fs m (18)for the function F ( x ; t ) = F ( t + x, t , t , . . . ). Note that sum in the r.h.s. of (18) is finite. For fixed i and j there is a restrictionon s k . Sum of all matrix elements is a sum of rows which should coincide with a sum of columns: i + j = s + · · · + s m + m .For large enough values of s k or a large number m coefficients P i,j ( s , . . . , s m ) are equal to zero.The next step is to determine all the solutions of the hierarchy. For this reason we need Cauchy-like data, which is a setof functions of variable x : ∂ ~ k F ~ ( x, t ) | t =0 = f ~ k ( x ). If we consider formal solutions, i.e. not necessarily converging series, anysolution can be expressed through Cauchy-like data using universal coefficients P ~ λ (cid:18) s . . . s m l . . . l m (cid:19) , which were mentioned before ashigher analogs of coefficients P i,j ( s , . . . , s m ).It was shown by Natanzon and Zabrodin [19, Theorem 4.3] that for an arbitrary set of smooth functions f = { f ~ ( x ) , f ~ ( x ) , . . . } there exists a unique solution F ~ ( x, t ) of the ~ -KP hierarchy with Cauchy-like data f . This solution is of the form F ~ ( x, t ) = f ~ ( x ) + X | λ |≥ f ~ λ ( x ) σ ( λ ) t ~ λ (19)where f ~ [ k ] ( x ) = f ~ k ( x ) and f ~ λ ( x ) = X m ≥ X s + l + ··· + s m + l m = | λ | ≤ s i ; 1 ≤ l i ≤ l ( λ ) − P ~ λ (cid:18) s . . . s m l . . . l m (cid:19) ∂ l x f ~ s ( x ) . . . ∂ l m x f ~ s m ( x ) (20)for l ( λ ) > σ ( λ ) = Q i ≥ m i !, where exactly m i parts of the partition λ have length i .The full recursive definition of universal coefficients P ~ λ is quite unwieldy and can be found in [19]. In this paper weare interested in simplest coefficients with l = · · · = l m = 1 and λ = [ i, j ]. They are defined as coefficients (17) with thenormalization factor P ~ [ i,j ] (cid:18) s . . . s m . . . (cid:19) := ( − m +1 ijm · s . . . s m P i,j ( s , . . . , s m ) (21)The other coefficients with ℓ ( λ ) ≥ l i > Remarkable properties of combinatorial coefficients P i,j ( s , . . . , s m ) As it was claimed (18), we can rewrite all KP equations with help of certain combinatorial coefficients P i,j ( s , . . . , s m ). So itis natural to ask if there is some connection between properties of KP hierarchy and properties of these combinatorial objects.Therefore, in this section we recall the most prominent properties of the constants P i,j ( s , . . . , s m ), as well as the context aroundtheir combinatorics. We postpone the discussion of the connection with the KP till the next section.Coefficients P i,j ( s , . . . , s m ) and their n-point generalizations (23), in fact, arise in the theory of flow networks [31] and arevery well studied. Standard problem in the theory of flow networks is finding the maximum flow which gives the largest totalflow from the source to the sink. We interested here in more simple question: what is the number of different flows on the graphwhere all n sources and m sinks are connected by edges which is exactly coefficients P i ,...,i n ( s , . . . , s m ).Since there is a rich combinatorial structure of the combinatorial coefficients, there are many different ways to calculatethem, each having potential implications for our topic: explicit formula as the sum over vertices of hypercube, recursion formulaand generating function. • First of all, there is an explicit approach to calculation of the coefficients using geometric interpretation and inclusion-exclusion principle: combinatorial coefficients P i,j ( s , . . . , s m ) can be represented as the sum over vertices of m -dimensionalhypercube P i,j ( s , . . . , s m ) = δ s + ··· + s m + m,i + j X { σ k = { , }| k =1 ,...,m } ( − σ + ··· + σ m (cid:18) i − σ s − · · · − σ m s m − m − (cid:19) (22)The cube is parametrized by the sequences of zeros and unities ( σ , . . . , σ m ). Note here that we take binomial coefficients (cid:0) mk (cid:1) equal to zero if m < k or m < k <
0. (see Appendix A for the details on the derivation) • There is a natural generalization of the combinatorial coefficients in the following way. Matrices of size 2 × m aredistinguished in KP theory, but from the point of view of combinatorics one may consider the number of matrices of size n × m with fixed sums over rows and columns. P i ...i n ( s , . . . , s m ) := i (1)1 . . . i (1) m ... . . . ... i ( n )1 . . . i ( n ) m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ( l ) k ∈ N , i ( l )1 + · · · + i ( l ) m = i , ∀ l ∈ , ni (1) k + · · · + i ( n ) k = s k + n − , ∀ k ∈ , m (23)Such objects arise in the simplest flow network problem: it is the number of integer flows on complete bipartite graph [31].Note that defined coefficients are symmetric up to permutation within the set of parameters s k and within the set ofindices i l . Thus, one may consider an ordered sets of indices and parameters labeled by Young diagrams λ and µ . Moreinformation about combinatorial meaning and different applications of such coefficients, denoted as N ( λ, µ ), can be foundin [32].This interpretation in terms of the number of certain matrices (23) allows one to obtain the following recursion relations [33]: P i ...i n ( s , . . . , s m ) = X (cid:26) i n + ··· + imn = in ≤ iln ≤ sl | l =1 ,...,m (cid:27) P i ...i n − ( s − i n + 1 , . . . , s m − i nm + 1) (24)and P i ...i n ( s , . . . , s m ) = X (cid:26) im ··· + imn = sm + n − ≤ iml ≤ il − m +1 | l =1 ,...,n (cid:27) P i − i m ,...,i n − i mn ( s , . . . , s m − ) (25)Note that (24) and (25) are the same up to the symmetry between indices { i l } and parameters { s l } mentioned above. • The last approach to calculation of the combinatorial coefficients is by means of generating function. We can constructsuch function in two different ways. Both highlight some interesting properties on the KP hierarchy side which we discussin section 4. Firstly, we can write it in the following way: e G nm ( x , y ) = X i ≥ ,...,i n ≥ y i . . . y i n n X s ≥ ,...,s m ≥ x s . . . x s m m P i ...i n ( s , . . . , s m ) = m Y l =1 x l ! n Y k =1 y mk ! X λ S λ ( x ) S λ ( y ) (26)which can be rewritten more naturally with the help of shifts: i k + · · · + i mk = i k + m for k = 1 , . . . , n and i l + · · · + i ln = s l + n for l = 1 , . . . , m : 5 mn ( x , y ) = X λ S λ ( x ) S λ ( y ) = Y i,j − x i y j (27)This formula is well known [32], but we give a short calculation in Appendix B that shows how it follows from recursionrelations (24).We also consider another generating function in variables p k : H ( p ; y , y ) = X m ≥ ( − m +1 m X ij y i y j X p s . . . p s m P ij ( s , . . . , s m ) = ln y y ∞ X k =1 p k y k − y k y − y ! (28)The choice of these variables is motivated by the formula (18) where factors ∂ x ∂ s F are included in the equation in thesame way as p i into this generating function. The formula (28) can be obtained from the first generating function (27) byreplacement p k = P i x ki (more detailed calculation can be found in Appendix B) Now we discuss, what does the explicit form of the generating functions (27),(28) mean for the KP hierarchy. First of all, weargue that the generating function (27) becomes the KP tau-function after some simple change of variables, which will becomeeffective in section 6.2 where we describe possible deformations.Second of all, the other generating function (28) allows one to easily derive Fay-identity form of the KP hierarchy.We also discuss here interpretation of the combinatorial formula (20) in terms of solutions that can be restored usingtopological recursion. • Generating function (27) of the redefined coefficients can be rewritten in another variables by replacement kt k = P i x ki and k ¯ t k = P i y ki . In these variables, using Cauchy-Littlewood identity (7) we obtain: G ( x , y ) = X λ S λ ( t ) S λ ( ¯t ) = e P k kt k ¯ t k (29)which is trivially a τ -function of KP (or Toda) hierarchy where t and ¯t are corresponding times. So the generating functionfor coefficients, which defines ~ -KP, is the trivial τ -function itself. We will discuss this property in section 6 trying todeform the combinatorial coefficients. • The second generating function allows us to write the analog of the Fay identity in the following way: it gives us generatingfunction for all KP equations (18) by replacement p k → ∂∂ ~ k Fk : ∂ ~ i ∂ ~ j Fij = h y i y j i ln y y ∞ X k =1 ∂∂ ~ k Fk y k − y k y − y ! (30)From the other hand the Fay identity for ~ -KP hierarchy has the form (14). Now, using replacement z i → y i and the factthat ∂ = ∂ x = ∂ we obtain exactly (30). The explicit derivation of (30) from Fay identity can be found in [19].As we can see here, combinatorial properties of the coefficients P i,j ( s , . . . , s m ) in the form (24) lead to the generatingfunction (28) which is exactly Fay identity in terms of KP hierarchy. • Let us now turn to the question of restrictions which explicit form of KP equations imposes on the topological recursion.Many solutions of the KP hierarchy (e.g., simple Hurwitz numbers [6], Hermitian matrix model [34–36], Kontsevich τ -function [1]) allows one to construct multi-differentials, which are related by the so-called spectral curve topologicalrecursion [22–28]. The initial data for the recursion procedure are 1-point and 2-point function of genus 0 which areexpected to be independent. However, naively, from formula (20) it follows that two-point functions f ~ λ ,λ can be expressedvia one-point functions f ~ k .Let us recall main concepts of the topological recursion. This approach firstly arose in the theory of matrix models whereall correlators have natural genus expansion [37,38]. In such theories we consider the following correlators which are calledresolvents: W n ( p , . . . , p n ) = (cid:28) Tr 1 p − X . . .
Tr 1 p n − X (cid:29) Conn (31)6here we integrate over matrices X with some measure and Conn means we consider connected diagrams only. They alsohave some genus expansion W n = X g ~ g W g,n (32)Topological recursion allows us to recover all resolvents in the genus g = n from g < n resolvents if we know the initialdata: spectral curve, W , and W , .Moreover, in many cases where topological recursion is applicable, the logarithm of partition function F = ~ log( Z ) turnsout to be a solution of ~ -KP. We can also represent resolvents via F in the following way W ( p , . . . , p s ) = − ∂∂V ( p ) . . . ∂∂V ( p s ) F (cid:12)(cid:12)(cid:12) t =0 ,x =0 (33)where ∂∂V ( p ) = − ∞ X j =1 p j +1 ∂∂t j (34)is the loop insertion operator [26].Returning to the Natanzon-Zabrodin formulation of KP hierarchy we can consider the Cauchy-like data as genus zeroresolvents since in the limit ~ → F ~ =0 ( x, t ) = f ~ =00 ( x ) + X | λ |≥ f ~ =0 λ ( x ) σ ( λ ) t λ (35)and W ( p , . . . , p n ) = ( − n X λ ,...,λ n ≥ p λ +11 . . . p λ n +1 n ∂ . . . ∂ n F (cid:12)(cid:12)(cid:12) t =0 ,x =0 , ~ =0 = ( − n X λ ≥···≥ λ n ≥ p λ +11 . . . p λ n +1 n f ~ =0 λ (cid:12)(cid:12)(cid:12) x =0 (36)Now it is clear that KP hierarchy imposes some restrictions since this formula connects two point resolvents with functions ∂f k | x =0 which in terms of W corresponds to two-point resolvents in the following way. Let W ( p , p ) = X λ ≥ λ ≥ p λ +11 p λ +12 ω λ λ (37)then f ~ =0 λ λ = ω λ λ and ω λ = ∂f ~ =0 λ | x =0 . It is possible now to write nontrivial condition on two-point resolvents using(20) for ℓ ( λ ) = 2: ω λ ,λ λ λ = h y λ y λ i ln y y ∞ X k =1 ω k, k y k − y k y − y ! (38)Summarizing, the combinatorial view on KP hierarchy allows us to obtain a nontrivial condition on solutions of KPhierarchy that admits recovering using topological recursion. This equation means that we can express all genus zerotwo-point resolvents using only ω k, data. It would be very interesting to see whether these KP restrictions are relatedwith the decomposition property ( [39, Lemma 4.1]), which under certain mild assumptions holds for W , . This questionis left for further research. In this section we provide a complete description of combinatorial coefficients P i ,...,i n ( s , . . . , s m ) in terms of an eigenvaluemodel. The model is an integral over eigenvalues of a matrix with a simple measure. Combinatorial coefficients appear to becertain correlators in the model, i.e. averages of product of m symmetric Schur polynomials h S s − . . . S s m − i . An arbitrarycorrelator in the model may be expressed with the help of the full basis of observables. The basis is obtained as a naturalgeneralization of combinatorial coefficients P i ,...,i n ( s ) with one parameter s and coincides with a subset of Kostka numbers.Partition function of the model can be calculated explicitly. The common property of matrix models is the existence of Wardidentities that might be solved recursively. In this model Ward identities give new recursion relations on combinatorial coefficients P i ,...,i n ( s , . . . , s m ). 7his model takes the simplest form for slightly modified coefficients P i ,...,i n ( s , . . . , s m ) with symmetric definition for bothlower indices i k and integers s j : i ( k )1 + · · · + i ( k ) m = i k + m − , ≤ k ≤ ni (1) j + · · · + i ( n ) j = s j + n − , ≤ j ≤ m (39)Note that such a definition differs from (23) by shift of i k . However, both definitions provide coefficients that are in one-to-onecorrespondence by the shift of indices, so we denote them as P i ,...,i n ( s , . . . , s m ) as well.Let us introduce an eigenvalue model Z n ( t ) = 1(2 πi ) n I dz · · · I dz n n Y k =1 z − i k k ! exp ∞ X k =1 t k Tr Z k ! , (40)where z k are complex variables, integration contours are unit circles and Z is diagonal matrix Z = diag( z , . . . , z n ). UsingCauchy-Littlewood identity (7), we rewrite it in the form Z n ( t ) = X λ ( πi ) n I dz · · · I dz n n Y k =1 z − i k k ! S λ ( Z ) ) S λ ( t ) ≡ X λ h S λ i S λ ( t ) , (41)which can be understood as a generating function for correlators h S λ i . Combinatorial coefficients P i ,...,i n ( s , . . . , s m ) appear tobe correlators of specific form in such eigenvalue model.Any combinatorial coefficient P i ,...,i n ( s , . . . , s m ) can be represented as an average of m symmetric Schur polynomials: P i ,...,i n ( s , . . . , s m ) = 1(2 πi ) n I dz · · · I dz n n Y k =1 z − i k k ! m Y j =1 S s j − ( Z ) ≡ h S s − . . . S s m − i (42)Although this integral seems complicated, it i fact, has simple meaning of extracting certain coefficient in front of certain powersof z -variables of integrand: P i ,...,i n ( s , . . . , s m ) = [ z i − . . . z i n − n ] (cid:16)Q mj =1 S s j − ( Z ) (cid:17) . This formula can be obtained as follows.Restrictions (39) allow us to represent the definition of combinatorial coefficients as a sum over product of delta-symbols (eachrestriction corresponds to one delta-symbol): P i ,...,i n ( s , . . . , s m ) = X i (1)1 ≥ ...i (1) m ≥ · · · X i ( n )1 ≥ ...i ( n ) m ≥ n Y k =1 δ i (1)1 + ··· + i (1) m ,i k + m − ! m Y j =1 δ i (1) j + ··· + i ( n ) j ,s j + n − (43)Delta-symbols are replaced with contour integrals with the help of simple relation δ n,m = 12 πi I dzz n − m − . (44)We change the first n delta-symbols to integrals in such way. The obtained expression is of the form P i ,...,i n ( s , . . . , s m ) = 1(2 πi ) n I dz · · · I dz n n Y k =1 z − i k k ! m Y j =1 X i (1) j ≥ · · · X i ( n ) j ≥ z i (1) j − . . . z i ( n ) j − n δ i (1) j + ··· + i ( n ) j ,s j + n − (45)The expression in square brackets can be evaluated independently for each j and depends only on s j . It is equal to Schurpolynomial S s j − ( z , . . . , z n ). Detailed calculations are presented in Appendix C. Thus, we proved formula (42).Eigenvalue model (41) is a natural generalization of coefficients P i ,...,i n ( s ) = h S s − i , i.e. one may consider averages of anarbitrary Schur polynomial h S λ i , not only symmetric ones. Any other coefficients such as P i ,...,i n ( s , . . . , s m ) = h S s − . . . S s m − i or their natural generalizations h S λ . . . S λ m i can be expressed in terms of linear combinations of h S λ i : product of Schurpolynomials is decomposed in linear combination of single Schur polynomials with Littlewood-Richardson coefficients [29]. So,correlators h S λ i form the appropriate full basis in the space of observables of the model.Moreover, correlators h S λ i coincide with Kostka numbers. One of the definitions of Kostka numbers K λ,µ is the decompositionof Schur polynomial into the sum over monomial symmetric functions m λ ( z , . . . , z n ) or, equivalently, into the sum over all weakcompositions α of n [32]: S λ ( z , . . . , z n ) = X µ K λ,µ m µ ( z , . . . , z n ) = X α K λ,α z α , (46)8here z α denotes the monomial z α . . . z α n n . The simple form of average (41) exactly coincides with coefficient in front of onemonomial in Schur polynomial decomposition: h S λ ( Z ) i = [ z i − . . . z i n − n ] S λ ( Z ). The latter one is the Kostka number K λ, e α ,where e α = ( i − , . . . , i n − h S λ ( Z ) i = K λ, e α . (47)The set of basis observables in the eigenvalue model is a subset of Kostka numbers. All correlators in the model may be expressedwith the help of Kostka numbers.The complete information about eigenvalue model is given by an explicit expression for generating function (41). It is possibleto calculate not only Z n ( t ) but also more general generating function: Z n ( t (1) , . . . , t ( m ) ) = X λ · · · X λ m h S λ . . . S λ m i S λ ( t (1) ) . . . S λ m ( t ( m ) ) , (48)where t ( k ) is an infinite vector of times t ( k ) = ( t ( k )1 , t ( k )2 , t ( k )3 , . . . ) for each k . First of all, Cauchy-Littlewood identity (7) allowsus to evaluate sums over partitions λ , . . . , λ m and obtain an expression similar to (40): Z n ( t (1) , . . . , t ( m ) ) = 1(2 πi ) n I dz · · · I dz n n Y k =1 z − i k k ! m Y k =1 exp ( ∞ X l =1 t ( k ) l ( z l + · · · + z ln ) )! (49)If we change the sum over z α in the exponent into product of exponents and, in its turn, the product of exponents into sumover t αm in the exponent we obtain the following expression Z n ( t (1) , . . . , t ( m ) ) = n Y k =1 I dz k πi z − i k k exp ( ∞ X l =1 ( t (1) l + · · · + t ( m ) l ) z lk ) , (50)where it is possible to calculate each integral. The given exponent is a generating series for symmetric Schur polynomials invariables t (1) + · · · + t ( m ) [29], so contour integral is exactly S i k − for each k and we obtain the product of n Schur polynomials: Z n ( t (1) , . . . , t ( m ) ) = n Y k =1 S i k − ( t (1) + · · · + t ( m ) ) . (51)The particular case of m = 1 is the eigenvalue model (41). Generating function of the form (51) is not very useful to restorecoefficients h S λ . . . S λ n i since one has to differentiate it with operator S ( ˜ ∂ (1) ) . . . S ( ˜ ∂ ( n ) ) at t (1) = · · · = t ( n ) = 0. However, itcontains the product of Schur polynomials, which seems similar to the Frobenius formula. The difference between them is thatFrobenius formula contains sum over products of Schur polynomials. One may hope that adding times in Schur polynomials asin (51) leads to some good properties.One more question which arises while studying matrix models is the question about any recursion relations. On the one handwe already mentioned recursion relations (24) and (25). On the other hand matrix model always has Ward identities, whichsometimes can be solved recursively. It turns out that recursion relations obtained from the eigenvalue model are different fromboth (24) and (25). Eigenvalue model (40) is provided with Ward identities that give new recursion relations different from(24), (25). We introduce new recursion relations for combinatorial coefficients P i ,...,i n ( s , . . . , s m ) with an arbitrary parameters i , . . . , i n and s , . . . , s m : P i ,...,i n ( s , . . . , s m ) = 1 i − m X k =1 s k − X l =1 P i − s k + l,i ,...,i n ( s , . . . , s k − , l, s k +1 , . . . , s m ) . (52)As usual for matrix models, Ward identities are obtained with the help of change of variables under the integral that doesnot change the entire integral. In the case of P i ,...,i n ( s , . . . , s m ) = h S s − . . . S s m − i in the form (42) change of variables isdilatation of the first variable z → (1 + q ) z , q = 0. Integration contour is around z = 0, so it is a possible change of variablesand deformed integral is independent on q . The explicit calculations of deformed integral and proof of (52) are presented inAppendix C.We finish this section with brief review of the obtained results. We provide complete description of combinatorial coefficients P i ,...,i n ( s , . . . , s m ) in terms of an eigenvalue model (40). Combinatorial coefficients are certain correlators in the model (42)– averages of symmetric Schur polynomials h S s − . . . S s m − i . The full basis of observables is the set of averages of arbitrarySchur polynomials h S λ i . It is a certain subset of Kostka numbers (47). Generating function Z n ( t ) can be calculated explicitly(51). Ward identities give new recursion relations (52) for combinatorial coefficients P i ,...,i n ( s , . . . , s m ).9 Towards deformations of KP hierarchy
In previous sections we discussed the appearance of coefficients P i,j ( s , . . . , s m ) in the KP equations (18). In this section weare interested in a possible generalization of the theory mentioned above, i.e. in the deformation of KP equations.As it often happens, various deformations help to understand the underlying structure of the formula, find out which partsare essential and which can be deformed. We try to reveal what role do the combinatorial coefficients P i,j ( s , . . . , s m ) play inequations (18). Integrability might be determined by combinatorial coefficients or might be a consequence of the particularform of equations. We deform only the combinatorial coefficients leaving the form of equations unchanged. It turns out thatan arbitrary deformation is not possible, equations (18) contain some restrictions that come from the fact that some of theequations should be fulfilled trivially. These restrictions appear even before the question about compatibility of obtained systemof differential equations. However, there is a hopeful deformation direction.The idea of deformation is based on the fact that we know explicit expression for the generating function of coefficients P i,j ( s , . . . , s m ) of the form (27). Let us deform this generating function. Deformed coefficients P ( def ) i,j ( s , . . . , s m ) are obtainedas coefficients in the expansion of the deformed generating function similarly to the original ones. At first glance, deformationof the generating function can be done in many ways. For example, we know that generating function (27) is a KP τ -function.So, one can try to let the new generating function be another KP τ -function of a similar form, i.e. τ -function of hypergeometrictype [2, 40]. Another way of deformation of generating function is the replacement of Schur polynomials with some otherpolynomials, which are considered as deformed Schur polynomials, for example, MacDonald polynomials [29]. These two typesof generating functions we consider below.First of all, let us examine which equations in (18) are trivial. It is obvious that equations (18) are symmetric due topermutations i ↔ j . Therefore, we can consider only ordered pair of indices i > j , or, equivalently, equations are labeled by allYoung diagrams of length 2. In the case of i = n and j = 1: ∂ ~ n ∂ F = X s ≥ s = n P n, ( s ) ∂ ∂ ~ s F + X s ,s ≥ s + s = n − ( − s s P n, ( s , s ) ∂ ∂ ~ s F · ∂ ∂ ~ s F + higher m | {z } =0 (53)Since one of the indices is equal to 1, there is no matrix of size 2 × m with m ≥ n equation (53) reduces to ∂ ∂ ~ n F = P n, ( n ) ∂ ∂ ~ n F (54)For any n it is easy to calculate that P n, ( n ) = 1, thus, equations (54) are hold trivially.When we replace coefficients P i,j ( s , . . . , s m ) with the deformed ones P ( def ) i,j ( s , . . . , s m ), the latter ones are calculatedvia deformed generating function in the following way. Since P i,j ( s , . . . , s m ) = [ x i − x j − y s − . . . y s m − m ] G ( x , y ), deformedcoefficients are obtained similarly as: P ( def ) i,j ( s , . . . , s m ) = [ x i − x j − y s − . . . y s m − m ] G ( def ) ( x , y ) (55)The deformed equations in the case of i = n, j = 1 are of the form ∂ ∂ ~ n F = P ( def ) n ( n ) ∂ ∂ ~ s F (56)and again should be fulfilled trivially. Thus, we have the condition on the deformed coefficients: P ( def ) n ( n ) = 1 , ∀ n ∈ N ⇔ [ x k y k ] G ( def ) ( x , y ) = 1 , ∀ k ∈ N ∪ { } (57)This condition we consider as a necessary condition for the deformed generating function. Let us consider the generating function for simple Hurwitz numbers as a new generating function for combinatorial coefficients.This generating function is a member of the set of hypergeometric τ -functions [41] and can be written as: G H ( x , y ) = X λ e u C ( λ ) S λ ( x ) S λ ( y ) (58)10here C ( λ ) is an eigenvalue of the second Casimir operator [6] ( C ( λ ) = P ℓ ( λ ) i =1 λ i ( λ i − i + 1)). The first few terms of thegenerating function are G H ( x , y ) = 1 + ( x + x )( y + y ) + e u ( x + x x + x )( y + y y + y ) + e − u ( x x )( y y ) + . . . (59)Already in the second order [ x y ] G H ( x , y ) = e u , thus, such deformed coefficients violate necessary condition (57). We concludethat Hurwitz numbers is a bad choice for deformed combinatorial coefficients. ( q, t ) -deformation Although smart ( q, t )-deformation of KP hierarchy that possesses an underlying structure of some algebra and solutions like( q, t )-deformed matrix models [42, 43] is still unknown, we make an attempt to construct ( q, t )-deformed KP equations. Let usconsider the sum over MacDonald polynomials as the deformed generating function for combinatorial coefficients: G ( q,t ) ( x , y ) = X λ M λ ( x ) M λ ( y ) , (60)Necessary condition (57) is fulfilled at least for the first few polynomials ( P ( q,t ) n ( n ) = 1 for n = 2,3,4,5), so it is possible that itholds for an arbitrary n .The first non-trivial equation of ( q, t )-deformed hierarchy is ( i = 2 , j = 2): ∂ ~ ∂ ~ F = 43 (cid:18) q − t − qt (cid:19) ∂ ∂ ~ F − (cid:0) ∂ F (cid:1) (61)The second non-trivial equation of ( q, t )-deformed hierarchy is ( i = 3 , j = 2): ∂ ~ ∂ ~ F = 32 (cid:18) q − t )( q + 1)1 − q t (cid:19) ∂ ∂ ~ F − (cid:0) ∂ F (cid:1) (cid:0) ∂ ∂ ~ F (cid:1) (62)Both equations become equations of classical KP hierarchy in the limit q = t . However the question about compatibility ofdeformed differential equations is still open and deserves a separate study.Unfortunately, generating function (60) does not satisfy equations (61) and (62). Thus, it cannot be considered as a trivial τ -function of the deformed hierarchy similarly to (27), which is a trivial τ -function of non-deformed KP. However, the formof the equations remains the same as classical KP hierarchy: each term contains at least two derivatives. Thus, any linearcombination of times t k is a solution of these equations. A possible candidate for the deformed trivial τ -function comes fromthe modification of Cauchy-Littlewood identity (7) for MacDonald polynomials [29]: X λ C λ C ′ λ M λ ( t k ) M λ ( t k ) = exp ∞ X k =1 [ β ] q kt k t k ! (63)where C λ = Y ( i,j ) ∈ λ [ βArm λ ( i, j ) + Leg λ ( i, j ) + 1] q , C ′ λ = Y ( i,j ) ∈ λ [ βArm λ ( i, j ) + Leg λ ( i, j ) + β ] q (64)Here [ x ] q denotes the quantum number, t = q β and Arm λ ( i, j ) , Leg λ ( i, j ) are notations of combinatorial objects such as armsand legs of the Young diagram λ (for the detailed description of these objects see, for example, [44]). The F -function is alogarithm of (63) and is just a linear combination of times t k for fixed parameters t k . Therefore it satisfies deformed equations(61), (62) and might be a possible candidate for a trivial τ -function.This approach contains some hopeful directions that will be considered in more details elsewhere. Right now generatingfunction (60) seems as a possible choice for deformed combinatorial coefficients. In this paper we presented a combinatorial view on the ~ -KP hierarchy based on Natanzon-Zabrodin approach with universalcombinatorial coefficients P ij ( s , . . . , s m ). We showed that studying of the combinatorial coefficients naturally highlights certainproperties of the KP hierarchy: • generating function (27) is the KP τ -function by itself and generating function (28) gives Fay identity (30). Theseproperties give an idea about possible deformations of KP hierarchy from the combinatorial point of view: we expect thatdeformation of generating function (27) will lead to some interesting deformations of KP hierarchy.11 generating function (28) and form of solutions (19) gives information about conditions on Cauchy-like data that correspondsto genus zero resolvents in topological recursion for ~ -KP solutions. In particular, this may be used as a quick test forputative spectral curves for enumerative problems, known to be KP integrable. • combinatorial coefficients P ij ( s , . . . , s m ) have complete description in terms of quite simple eigenvalue matrix model (40).This approach allows us to describe non-trivial recursion relation (52) on the combinatorial coefficients. This matrixmodel may be used in studying KP hierarchy in terms of the combinatorial coefficients and it gives new questions aboutinterpretation of corresponding averages in terms of KP hierarchy.The aim of this paper is to demonstrate that combinatorial approach to KP hierarchy is instrumental in giving motivationand insights for further study of emergent properties of KP. Here we list some questions that appear naturally when applyingthis approach: • The question about combinatorial deformation of KP hierarchy is still open: can we deform combinatorial coefficients inequations (18) in such a way that we obtain an integrable hierarchy? (Discussed in section 6) • What do coefficients h S λ . . . S λ n i mean in terms of combinatorial objects or KP hierarchy? (Discussed in section 5) • It is easy to generalize combinatorial definition of the coefficients replacing matrices by tensors. For example, the numberof three-tensors with fixed sums over two of three indices is called a Kronecker coefficient, which has a lot of differentapplications [45, 46]. It is natural to ask, is there any integrable hierarchy formulated via Kronecker coefficients in thesame way as the ~ -KP? • How to write a matrix model for such generalizations and how do Ward identities in this model looks like? • According to [19] it is possible to recover any formal solution of ~ -KP from Cauchy-like data (20) using higher coefficients P ~ λ (cid:18) s . . . s m l . . . l m (cid:19) . Is there any simple combinatorial description for these coefficients? Are they connected with Kroneckernumbers in some way? Or may be there is some matrix model generating these coefficients.We hope to address some, or all, of these intriguing questions in the future. Acknowledgements
This work was funded by the Russian Science Foundation (Grant No.20-71-10073). We are grateful to Sergey Fomin andAnton Zabrodin for very useful discussions and remarks. Our special acknowledgement is to Sergey Natanzon for a formulationof the problem and for inspiring us to work on this project.
Appendix A. Explicit calculation of P i,j ( s , . . . , s m ) We start here from the sum that follows from the definition: P i,j ( s , . . . , s m ) = X { ≤ i k | k =1 ,...,m } X { ≤ j k | k =1 ,...,m } δ i + ··· + i m = i δ j + ··· + j m = j δ i + j = s +1 . . . δ i m + j m = s m +1 (65)Resolving equations i k + j k = s k + 1 we obtain: P i,j ( s , . . . , s m ) = δ s + ··· + s m + m,i + j X { ≤ i l ≤ s l | l =1 ,...,m } δ i + ··· + i m ,i (66)Sum in the r.h.s. has geometric interpretation as the section of m -dimensional parallelogram R s ,...,s m = { i k | ≤ i k ≤ s k , k =1 , . . . , m } by m − i + · · · + i m = i . In order to calculate this sum we use inclusion-exclusion principle for m -dimensional ”quadrants” Q a ,...,a m = { i k | a k ≤ i k , k = 1 , . . . , m } . Contribution from m -dimensional parallelogram R s ,...,s m then expressed as the sum over all ”quadrants” with vertices coinciding with vertices of R s ,...,s m : R Conts ,...,s m = X { σ k = { , }| k =1 ,...,m } ( − σ + ··· + σ m Q Cont σ s ,..., σ m s m (67)where set of variables σ k enumerate all vertices. 12he next step is to calculate contribution of ”quadrant” Q Cont ,..., , which is just a number of ordered partitions of i : Q Cont ,..., = X ≤ i k δ i + ··· + i m ,i = (cid:18) i − m − (cid:19) . (68)Shifting of ”quadrant” Q ..., ,... → Q ..., s k ,... is equivalent to shifting i → i − s , so for the contribution of Q σ s ,..., σ m s m wehave the following formula: Q σ s ,..., σ m s m = (cid:18) i − σ s − · · · − σ m s m − m − (cid:19) (69)Combining now (69) and (67) we obtain: R Conts ,...,s m = X { σ k = { , }| k =1 ,...,m } ( − σ + ··· + σ m (cid:18) i − σ s − · · · − σ m s m − m − (cid:19) (70) Appendix B. Calculation of generating functions
We give here an approach to calculation of generating functions.In order to obtain the G -generating function (27) it is convenient to use recursion relation (24). Let us substitute (24) intothe generating function: e G nm ( x , y ) = X i ≥ ,...,i n ≥ y i . . . y i n n X s ≥ ,...,s m ≥ x s . . . x s m m X (cid:26) i n + ··· + imn = in ≤ iln ≤ sl | l =1 ,...,m (cid:27) P i ...i n − ( s − i n + 1 , . . . , s m − i mn + 1) (71)The next step is to swap two sums on the right and rewrite each x s l l as x i nl − x s l − i nl +1 l : e G nm ( x , y ) = X i ≥ ,...,i n ≥ y i . . . y i n n X (cid:26) i n + ··· + imn = in ≤ iln | l =1 ,...,m (cid:27) x i n − . . . x i mn − m X iln ≤ sll =1 ,...,m x s − i n +11 . . . x s m − i mn +1 m P i ...i n − ( s − i n +1 , . . . , s m − i mn +1)(72)After replacement s ′ l = s l − i ln + 1 for l = 1 , . . . , m we obtain simple recursion relation: e G nm ( x , y ) = X ik ≥ k =1 , ˙ ,m y i . . . y i n n X (cid:26) i n + ··· + imn = in ≤ iln | l =1 ,...,m (cid:27) x i n − . . . x i mn − m F n − ( x , y ) = e G ( n − m ( x , y ) m Y l =1 y n (1 − x l y n ) (73)where sums over i k are independent and each of them is geometric progression. It is easy now to write the entire generatingfunction. e G nm ( x , y ) = e G m ( x , y ) m Y l =1 n Y k =2 y k (1 − x l y k ) , (74)where according to our definition of coefficients: P i ( s , . . . , s m ) = δ s + ··· + s m ,i (75)and hence e G m ( x , y ) = X i ≥ y i X s ≥ ,...,s m ≥ x s . . . x s m m δ s + ··· + s m ,i = m Y l =1 x l y (1 − x l y ) . (76)Finally, the generating function is of the form: e G nm ( x , y ) = m Y l =1 x l n Y k =1 y k (1 − x l y k ) = m Y l =1 x l ! n Y k =1 y mk ! X λ S λ ( x ) S λ ( y ) (77)Now, using this result we can calculate the second generating function (28). The main idea is to make replacement p k = P i x ki : H ( p ; y , y ) = X m ≥ ( − m +1 m X ij y i y j X s ,...,s m X i x s i ! . . . X i m x s m i m ! P ij ( s , . . . , s m ) (78)13sing generating function ˜ G m we obtain X m ≥ ( − m +1 m X i x s i ! . . . X i m x s m i m ! m Y l =1 (cid:18) x l y y (1 − y x l )(1 − y x l ) (cid:19) . (79)It can be rewritten as the product X m ≥ ( − m +1 m m X l =1 x l y y (1 − y x l )(1 − y x l ) ! m = y y y − y m X l =1 (cid:18) − y x l − − y x l (cid:19)! m (80)and expanding geometric progression we obtain function in p i variables: X m ≥ ( − m +1 m y y y − y ∞ X k =1 (cid:0) y k p k − y k p k (cid:1)! m = y y ∞ X k =1 p k y k − y k y − y ! m (81)Now summing over m we obtain the generating function. Appendix C. Eigenvalue model calculations
Firstly, we show that expression in brackets in (45) is a Schur polynomial. Thus, we prove formula (42). It is obvious thatit can be calculated for each j independently, so we do not write index j in the proof. The expression in brackets is equal to z s + n − n z . . . z n − s X i (1) =1 · · · s X i ( n − =1 s + n − − i (1) −···− i ( n − X i ( n − =1 (cid:18) z z n (cid:19) i (1) . . . (cid:18) z n − z n (cid:19) i ( n − ≡ A s − (82)Let us denote the expression as A s − and calculate its generating series A ( ξ ) = ∞ X s =1 A s − ξ s − . (83)To perform the calculation we need to swap sum over s with the other ( n −
1) sums over i ( k ) . All the possible values of indicesare inside an n -dimensional semi-infinite triangle, and, as usual, changing the order of sums changes the order in which we moveinside this triangle with new restrictions on the indices. After swapping the sums one obtains the following expression A ( ξ ) = ∞ X i (1) =1 · · · ∞ X i ( n − =1 ∞ X s = i (1) + ··· + i ( n − − n +2 z s + n − n z . . . z n − (cid:18) z z n (cid:19) i (1) . . . (cid:18) z n − z n (cid:19) i ( n − ξ s − , (84)which is now easy to calculate. One has to calculate infinite geometric progressions: A ( ξ ) = ∞ X i (1) =1 (cid:18) z z n (cid:19) i (1) · · · ∞ X i ( n − =1 (cid:18) z n − z n (cid:19) i ( n − · z n − n z . . . z n − ( z n ξ ) i (1) + ··· + i ( n − − n +2 ξ (1 − ξz n ) == ∞ X i (1) =1 z ξ ( z ξ ) i (1) ! . . . ∞ X i ( n − =1 z n − ξ ( z n − ξ ) i ( n − ! · − ξz n = n Y α =1 − ξz α (85)The last expression in (85) is exactly a generating function for symmetric Schur polynomials [29], thus, each A s − is equal toSchur polynomial S s − , which proves (42).Secondly, we explicitly make clear the derivation of recursion relations (52) using the same technique as for Ward identitiesin common matrix models. Let us rescale the first variable under the integral z → (1 + q ) z . There is no singularities exceptpoint z = · · · = z n = 0, so such a change of variables preserves the value of the integral: I ( q ) = 1(2 πi ) n I (1 + q ) dz · · · I dz n (1 + q ) − i n Y k =1 z − i k k ! m Y j =1 S s j − ((1 + q ) z , . . . , z n ) (86)14his expression is independent on q , so the derivative is equal to zero ∂I∂q = 0. We calculate the derivative at the point q = 0.Derivative acts on each Schur polynomial independently, so let us first calculate the derivative for only one Schur polynomial: ∂I ( q ) ∂q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q =0 = (1 − i ) P i ,...,i n ( s ) + I dz · · · I dz n n Y k =1 z − i k k ! ∂∂q S s − ((1 + q ) z , . . . , z n , , . . . ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q =0 = 0 . (87)To calculate the derivative of Schur polynomial we use the generating function A ( ξ ) as in (85), where we rescale the first variable: A ( q, ξ ) = ∞ X j =0 S j ((1 + q ) z , . . . , z n ) ξ j = 11 − (1 + q ) z ξ n Y α =2 − z α ξ . (88)Now it is possible to calculate the derivative of the obtained expression. ∂A ( q, ξ ) ∂q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q =0 = z ξ − z ξ n Y α =1 − z α ξ ! = ∞ X j =0 ∞ X p =0 S j z p +11 ξ j + p +1 (89)Let us change the summation indices in the last expression: a = j + p + 1. ∂A ( q, ξ ) ∂q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q =0 = ∞ X a =1 ξ a a − X p =0 S p z a − p ! (90)If we compare it with the expression (88), we obtain the following result ∂S a ∂q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q =0 = a − X p =0 S p z a − p , (91)which we substitute into formula (87):(1 − i ) P i ,...,i n ( s ) + I dz · · · I dz n n Y k =1 z − i k k ! s − X p =0 S p z a − p ! = 0 . (92)Let us simplify the last expression so it can be rewritten only through combinatorial coefficients:0 = (1 − i ) P i ,...,i n ( s ) + s − X p =0 I dz · · · I dz n z − i + s − − p n Y k =2 z − i k k ! S p == (1 − i ) P i ,...,i n ( s ) + s − X p =1 P i − s + p,i ,...,i n ( p ) (93)Now it is easy to do the same calculations for many parameters s , . . . , s m . Derivative acts on each Schur polynomial labeledby these parameters independently. The result is formula (52). References [1] M. Kontsevich, “Intersection theory on the moduli space of curves and the matrix Airy function,”
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