From Jack to Double Jack Polynomials via the Supersymmetric Bridge

The Calogero-Sutherland model occurs in a large number of physical contexts, either directly or via its eigenfunctions, the Jack polynomials. The supersymmetric counterpart of this model, although much less ubiquitous, has an equally rich structure. In particular, its eigenfunctions, the Jack superpolynomials, appear to share the very same remarkable combinatorial and structural properties as their non-supersymmetric version. These super-functions are parametrized by superpartitions with fixed bosonic and fermionic degrees. Now, a truly amazing feature pops out when the fermionic degree is sufficiently large: the Jack superpolynomials stabilize and factorize. Their stability is with respect to their expansion in terms of an elementary basis where, in the stable sector, the expansion coefficients become independent of the fermionic degree. Their factorization is seen when the fermionic variables are stripped off in a suitable way which results in a product of two ordinary Jack polynomials (somewhat modified by plethystic transformations), dubbed the double Jack polynomials. Here, in addition to spelling out these results, which were first obtained in the context of Macdonal superpolynomials, we provide a heuristic derivation of the Jack superpolynomial case by performing simple manipulations on the supersymmetric eigen-operators, rendering them independent of the number of particles and of the fermionic degree. In addition, we work out the expression of the Hamiltonian which characterizes the double Jacks. This Hamiltonian, which defines a new integrable system, involves not only the expected Calogero-Sutherland pieces but also combinations of the generators of an underlying affine ${\widehat{\mathfrak {sl}}_2}$ algebra.


Introduction
The AGT correspondence [3], that relates conformal blocks to the U (2) Nekrasov instanton partition function [20], has generated a boost of interest for Jack polynomials. Indeed, the latter have been shown to be key components of a new AGT-motivated basis of states in 2d-CFT [2]. More precisely, the Jack polynomials appear there in a generalized version which is indexed by a pair of partitions and decomposes into product of two Jacks with different arguments [2,19].
Here we present a somewhat analogous type of generalization of the Jack polynomials also labelled by two partitions. These new generalized Jacks arise directly from the construction of the supersymmetric counterparts of the Jack polynomials, the Jack superpolynomials [9]. 1 The latter are eigenfunctions of the supersymmetric version of the Calogero-Sutherland (CS) model [24]. It tuns out that for excited states with large fermionic degree, the eigenfunctions acquire an unexpected stability behavior. More remarkably, in this stability sector, these eigenfunctions (after a minor transformation) factorize into a product of two Jack polynomials. This factorization is highly non-trivial: there is a sort of twisting in the coupling constant (the free parameter α), which is different for the two constituent Jacks, and a reorganization of the variables (technically: a plethystic transformation). 2 The factorized form of the eigenfunctions is referred to as the "double Jack polynomials". We stress that the non-trivial structure of these double Jacks is inherited from the supersymmetric construction, which thus serves as a bridge linking the Jacks to their double version.
These peculiar properties of stability and factorization have first been observed at the level of the Macdonald generalization of the Jack superpolynomials [6]. Here we make explicit the one-parameter limit characterizing the Jacks. In addition, we unravel their underlying integrable structure by constructing the Hamiltonian for which these are eigenfunctions. 3 Somewhat unexpectedly, this Hamiltonian is built in part from the generators of the nonnegative modes of an sl 2 algebra.
The article is organized as follows. In Section 2, we briefly review the Calogero-Sutherland model and their eigenfunctions, emphasizing a Fock space representation to be used throughout the article. The supersymmetric CS model is introduced in Section 3, together with the Jack superpolynomials. For a sufficiently high fermionic degree, the supersymmetric Hamiltonian eigenvalues are shown to be decomposable into two independent parts. This points toward the splitting of the Hamiltonian into two independent CS Hamiltonians and the corresponding factorization of its eigenfunctions. The resulting double Jack polynomials are defined formally in Section 4 and exemplified for simple cases, while their corresponding Hamiltonian is derived in Section 5.

The Calogero-Sutherland model and Jack polynomials
The CS model describes a system of N identical particles of mass m lying on a circle of circumference L and interacting pairwise through the inverse of chord distance squared. Setting m = = 1 and L = 2π, the Hamiltonian reads [25]: where β is a dimensionless real coupling constant and [x j , p k ] = iδ jk . 4 To the ground state correspond the following wavefunction and eigenvalue: It is convenient to define z j = e ixj and to factor out the contribution of the ground state by redefining a gauged Hamiltonian as ψ −1 0 (H CS − E 0 )ψ 0 /β and to set β = 1/α: This is our starting point.
The symmetric and triangular eigenfunctions of (2.3) are known as the Jack polynomials J (α) λ (z) [14] 5 , where the index λ stands for a partition λ = (λ 1 , λ 2 , · · · , λ N ), with the λ i 's being non-negative integers such that λ i ≥ λ i+1 . Their eigenvalues are ε 4) where [16]: Here λ ′ is the conjugate of λ obtained from λ by replacing rows by columns in its diagrammatic representation, and |λ| = i λ i is the degree of λ. We will be interested in the behavior of the wavefunction when N is large. It is thus preferable to remove the dependency in N in the eigenvalue. For this, we note that J (α) λ (z) is homogeneous in the z i 's, so that it is an eigenfunction of the momentum operator P: (2.6) Our task is achieved by redefining the Hamiltonian as Jack polynomials J 4 See [13] for an extensive and very clear presentation of the CS model. 5 For a physicist introduction to the Jack polynomials, we refer to [13,12]. A more mathematical presentation can be found in [16].
In the large N limit, it is convenient to rewrite the Hamiltonian in terms of power sums p k = z k 1 + z k 2 + · · · . SinceĤ (α) is a differential operator of order two, it is sufficient to determine its action on the product p m p n . A direct computation gives [18,4] This naturally leads to the Fock space representation The correspondence with symmetric functions, together with |0 ←→ 1, is This correspondence preserves the commutation relations. In this representation, the eigenfunctions take the form of a combination of states J For instance, up to a multiplicative constant 6 (2.14) As a side remark, we point out that it is through the correspondence (2.12) that the connection between Virasoro singular vectors and Jack polynomials is established [17,4,23,21]. The technology of Jack polynomials can even be used to derive the spectrum of the Virasoro minimal models [26,21]. These applications have recently been lifted to the sl(2) WZW model at fractional level [22].

Supersymmetric version
In order to supersymmetrize the CS model, we need to introduce anticommuting variables θ 1 , . . . , θ N and extend the CS Hamiltonian H in the following way: for two fermionic charges Q and Q † of the form where A i and A † i are fixed by the requirement of reproducing theĤ (α) term on the rhs of the above equation. This construction leads to This operator is part of the tower of conserved quantities H n , 1 ≤ n ≤ N (P = H 1 and H (α) susy = H 2 ) that reduce to the usual (gauged) CS conservation laws in the absence of anticommuting variables. But given that there are 2N degrees of freedom in the supersymmetric version, there are N extra conserved charges that vanish when all θ i = 0 [9]. The first nontrivial representative of this second tower is In the monic normalization J (α) λ = m λ + lower terms, where m λ is the monomial symmetric function, this coefficient is As a side remark, we mention that both expressions can be represented in the Fock space of a free boson, described by the modes a k , a † k (with k ≥ 1, i.e., without the zero mode) and a free fermion, whose modes are and The fermionic modes are governed by the anticommutation relations: and their correspondence with symmetric functions is 8 Now, assuming a natural triangularity condition, the common eigenfunctions of H susy are the Jack polynomials in superspace, or Jack superpolynomials, denoted by J (α) Λ (z, θ) [9]. They are homogeneous in z and in θ and invariant under the exchange of pairs (z i , θ i ) ←→ (z j , θ j ). Their labelling index Λ is a superpartition. Before displaying the eigenvalues, some notation related to superpartitions is required.
A superpartition Λ is a pair of partitions Λ = (Λ a ; Λ s ) such that Λ s is an ordinary partition Λ a is a partition with no repeated parts. (3.8) Note that the last part of Λ a is allowed to be zero. We denote by Λ * the partition obtained by reordering in non-increasing order the entries of Λ a and Λ s concatenated. The diagrammatic representation of Λ is obtained by putting dots at the end of the rows that come from Λ a (in such a way that dots never lie under an empty cell).
Here is an example: Λ = (4, 2, 0; 3, 2, 1, 1) ←→ (4, 3, 2, 2, 1, 1, 0) ←→ A superpartition is equally well described by the pair Λ * and Λ ⊛ , where the latter is the partition obtained by replacing dots by boxes, e.g., in the example above, (3.10) Finally, the bosonic degree of a superpartition is the number of boxes of Λ * and the fermionic degree, generally denoted by m, is the number of dots in the diagram of Λ, that is, the number of parts of Λ a . 7 In a supersymmetric context, the modes of the partner free fermion should pertain to the Neveu-Schwarz sector, hence be halfintegers. This can be achieved by redefining (b k , b † k ) as (b k+1/2 , b † k+1/2 ) in the relation (3.7) below. However, this precision is not required in the present context. 8 Via such free-field representation, the Jack superpolynomials have been shown to be related to the super-Virasoro singular vectors [11,1].
We are now in position to give the eigenvalues of H In the supersymmetric case, we are not only interested in the large N limit but also in the large m limit (actually, in the large m and N − m limits). We thus want to extract from the above two eigenvalues, their dependence on m which is somewhat hidden. For this, we first notice that when m is large (relative to the size of Λ, an estimation that is made precise in (3.22)), there are circles in every possible positions in the diagram of Λ. 9 As such, the circles can be ignored and we observe that Λ * differs slightly from its core δ (m) = (m − 1, m − 2, · · · , 1, 0). In the diagrammatic representation of Λ * , the deviations to the core are located at the top right and at the bottom left of the diagram. We thus see that the superpartition can be disentangled into its fermionic core plus two small partitions λ and µ such that Λ = (λ + δ (m) ; µ) [6]. 10 For instance, for m = 8, we have It is clear that Λ is fully characterized by m and the pair (λ, µ) (whose total degree is much less than that of Λ). The main advantage of this diagrammatic decomposition is that it implies readily that when m is large the conjugate of Λ is Λ ′ = (µ ′ + δ (m) ; λ ′ ).
Let us reformulate the eigenvalues in terms of the data λ, µ and m. For the I We can easily remove the dependency in m in the eigenvalue by redefining I (α) susy as follows: (3.14) This subtraction is well defined since M is also a conserved quantity. The modified eigenvalue reads then The eigenvalues of H susy can also be reformulated in terms of λ, µ and m, again keeping in mind that this is valid only for sufficiently large m. Observe that 11 and similarly for Λ * ′ = (µ ′ + δ (m) ) ∪ λ ′ , where ℓ(µ) is the length of the partition µ (the number of non-zero parts). Fortunately, the calculation of n(Λ * ) (and n(Λ * ′ )) is independent of the precise relationship between the indices i 9 Here is a technical precision that could safely be skipped. There are unimportant exceptions to the statement that when m is sufficiently large (meaning larger or equal to its lower bound, which is |λ| + |µ| for λ and µ defined below), there are dots in every possible positions. That all allowed slots are filled by dots is true when m ≥ ℓ(λ) + 1 + µ 1 , ℓ(λ) being the length of the partition λ.
Since |λ| + |µ| + 1 ≥ ℓ(λ) + 1 + µ 1 , the statement is always true for instance when m ≥ |λ| + |µ| + 1. 10 For the + operation, the parts add up. For example, we have (3, 1) + (4, 2, 2) = (7, 3, 2). 11 For the ∪ operation, the rows of the second partition are inserted into the first one; for instance (3, 1) ∪ (4, 2, 2) = (4, 3, 2, 2, 1). and j, k in the above notation if we use the second expression of n(λ) given in (2.5). Let us first consider: where in the last step, we use the first expression in (2.5). For the computation of n(Λ * ), we simply replace λ and µ by µ ′ and λ ′ respectively in the previous expression to get: Combining these two expressions yields susy eigenvalue is then simplyε These two expectations are indeed verified: the eigenfunctions both stabilize and factorize (after a certain transformation that will be explained in eq. (3.24)) 12 for [6]: Let us consider a simple example. For (λ, µ) = ( , ), the m = 1, 2, 3, 4 eigenfunctions read respectively Clearly, the m = 1 (< |λ| + |µ| = 2) wavefunction does not belong to the stable sector. For m ≥ 2, the coefficients and the a † content of each term are always the same. This is the stability property. 12 When the Jack superpolynomial is expressed in terms of the variables (x, θ) rather than in modes, the transformation is simply Although they stabilize, the eigenfunctions still depend on m. However, consider the map 25) and s λ is the Schur function. Observe that p n (y, z) is simply p n in the variables y 1 , y 2 , . . . , y m , z 1 , z 2 , . . . , z N −m . This maps the above eigenfunctions corresponding to the values m = 2, 3, 4 to (inserting the proper normalization) The stability has now been lifted to the full structure of the eigenfunction.
But in addition, the map (3.24) captures the factorization property suggested by the form of the eigenvalues. Using the Pieri rule for Schur functions [5,16] to express the sum of the last two terms in a product form, s 1,1 (y) + s 2 (y) = s 1 (y) s 1 (y), (3.27) we see that J (1), (1) can also be written in a product form This is a simple illustration of the announced factorization.
A more formal characterization of J (α) λ,µ (y, z), which we call the double Jack polynomials, is as follows [6]. They are the unique bi-symmetric functions such that J (α) λ,µ (y, z) = s λ (y)s µ (z) + smaller terms (4.7) and J The triangularity condition that specifies the "smaller terms" refers to the double version of the dominance ordering: while the orthogonality condition refers to the scalar product s λ (y) p µ (y, z), s ν (y) p κ (y, z) = δ λν δ µκ z µ α ℓ(µ) (4.10) with z µ = i≥1 i nµ(i) n µ (i)!, n µ (i) being the multiplicity of the part i in µ. Observe that this scalar product has the form •, • = ·, · y Schur ·, · y,z Jack . (4.11) where ·, · y Schur is the scalar product with respect to which the Schur functions s λ (y) are orthonormal while ·, · y,z Jack is the scalar product with respect to which the Jack polynomials J and Note that p n [X] and p n (z) are considered to be independent. Being the sum of two independent integrable Hamiltonians, H D trivially characterizes a new integrable model. However, the above splitting of H D is not very interesting since it is hard to give a physical meaning to the power-sums p n [X], p n (z) and their derivatives. The structure of the scalar product (4.10) points toward a more interesting choice of variables, namely p n (y) and p n (y, z), whose adjoints are n∂ pn(y) and nα∂ pn(y,z) respectively. With the change of variables is thus p n [X] = α α + 1 p n (y) + 1 α + 1 p n (y, z) p n (z) = p n (y, z) − p n (y), (5.5) which gives (using the chain rule in two variables) These expressions are readily checked by verifying that they satisfy the commutation relations: For these manipulations, we stress that p n (y) and p n (y, z) are considered to be independent, meaning: [∂ pn(y) , p m (y, z)] = [∂ pn(y,z) , p m (y)] = 0. (5.9) We then substitute (5.5) and (5.6) into (α + 1)H 1 + H 2 . The result, obtained after straightforward manipulations, is best rewritten in terms of two independent sets of bosonic modes defined as together with a † n = p n (y, z) and a n = nα∂ pn(y,z) (⇒ [a k , a † ℓ ] = kαδ k,ℓ ). (5.11) The resulting form of H D is It turns out that H D can be reexpressed as (5.14) Note thatĤ (α) y,z isĤ (α) in the variables y 1 , y 2 , . . . , y m , z 1 , z 2 , . . . , z N −m , so that: Similarly,Ĥ y isĤ (α) in the variables y 1 , y 2 , . . . , y m but evaluated at α = 1: Next, it is simple to check that α[Q 1 ,Ĥ y ] yields the second line in (5.12). Therefore, parts of the constituents of H D have a direct interpretation in terms of variables. However, this is not the case for Q 1 and Q 2 . Note that the action of Q 1 amounts to exchanging the a and A modes (which thereby appears to be a remnant of the action of a supersymmetric charge). Nevertheless, it turns out that Q 1 and Q 2 have a nice Lie algebraic interpretation. More precisely, both are combinations of the generators of an underlying affine sl 2 algebra (whose existence is not surprising in the presence of two independent infinite sets of bosonic modes). It is straightforward to verify that the operators We thus get that Q 1 = √ α(f (0) − e (0) ) and Q 2 = h (1) − h (0) (5.19) and, as such, H D is built from a special intertwining ofĤ This intertwining pattern is expected to hold for all the conserved quantities of the double CS model. Consider for instance the two conserved quantities of degree 1 20) whose eigenvalues are respectively (3.15) and |λ|+|µ|. As for H D , the conserved quantity I D can be written in terms of the usual conserved quantities of the two CS models specified byĤ (α) y,z andĤ (1) y , and the (nonnegative-mode) generators of sl 2 : I D = (α − 1)P y + [Q 1 , P y ], (5.22) where P y is the momentum operator P in the variables y 1 , . . . , y m . Similarly, we have P D = P y + P y,z . (5.23)