Supersymmetric partition function hierarchies and character expansions

We construct the supersymmetric β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} and (q, t)-deformed Hurwitz–Kontsevich partition functions through W-representations and present the corresponding character expansions with respect to the Jack and Macdonald superpolynomials, respectively. Based on the constructed β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} and (q, t)-deformed superoperators, we further give the supersymmetric β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} and (q, t)-deformed partition function hierarchies through W-representations. We also present the generalized super Virasoro constraints, where the constraint operators obey the generalized super Virasoro algebra and null super 3-algebra. Moreover, the superintegrability for these (non-deformed) supersymmetric hierarchies is shown by their character expansions, i.e., ∼character\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim character$$\end{document}.


Introduction
W -representation of matrix model realizes the partition function by acting on elementary functions with exponents of the given W -operator [1].A considerable amount is already known about W -representations for matrix models.Very recently it was shown that the spectral curve can be extracted from the W -representation of matrix model [2].The spectral curve was associated with a peculiar part Ŵ spec of the Ŵ -operator.As the generalizations of matrix models from matrices to tensor, tensor models provide the analytical tool for the study of random geometries in three and more dimensions.Hence they are serious candidates for a theory of quantum gravity.The studies of W -representations for tensor models have been carried out.It was found that there are the W -representations for the Gaussian tensor model [3] and (fermionic) rainbow tensor models [4,5].Due to the W -representations, it allows the correlators of these models to be exactly calculated.In addition, there have also been attempts to investigate W -representations of supereigenvalue models.For the supereigenvalue model in the Ramond sector, its free energy depends on Grassmann couplings only up to quadratic order [6].The W -representation for this supereigenvalue model was presented in Ref. [7].The supereigenvalue model in the Neveu-Schwarz sector describes the coupling between (2, 4m) superconformal models and world-sheet supergravity [8].For the Gaussian and chiral supereigenvalue models in the Neveu-Schwarz sector, it was noted that there are the so called generalized W -representations.More precisely, they can be expressed as the infinite sums of the homogeneous operators acting on the elementary functions [9].An important feature of these models is that the compact expressions of the correlators can be derived from such generalized W -representations.
The superintegrability for matrix models has attracted much attention (see [10] and the references therein, [11]- [19]).Here the superintegrability means that for the character expansions of the matrix models, the average of a properly chosen symmetric function is proportional to ratios of symmetric functions on a proper locus, i.e., < character >∼ character.W -representations have contributed to our understanding of the superintegrability.For a wide range of superintegrable matrix models, their character expansions can be derived from the corresponding W -representations.What is also worth noticing is the Virasoro constraints for matrix models.They were used to analyze the character expansions of matrix models as well [5,20].The Hurwitz-Kontsevich (HK) matrix model with W -representation is an important superintegrable matrix model which can be used to describe the Hurwitz numbers and Hodge integrals over the moduli space of complex curves [1,21,22].Its character expansions with respect to the Schur functions can be easily derived from the W -representation.The HK partition function was extended to the β and (q, t)-deformed cases [16,23].The superintegrability for these deformed HK models was confirmed by the character expansions with respect to the Jack and Macdonald polynomials, respectively.Moreover, the superintegrable β and (q, t)-deformed partition function hierarchies have been constructed, where the β and (q, t)-deformed Hurwitz operators play a fundamental role in the W -representations.
Quite recently, it was shown that the partition function hierarchies constructed in Ref. [16] can be described by the two-matrix model that depends on two (infinite) sets of variables and an external matrix [19,24,25].Their generalizations were realized by W -representations associated with infinite commutative families of generators of w ∞ algebra.They are the special cases of skew hypergeometric Hurwitz τ -functions which are τ -functions of the Toda lattice hierarchy of the skew hypergeometric type.The multi-matrix representation as well as their β-deformations were provided in Ref. [25].W -representations for (β-deformed) multi-character partition functions were constructed in Ref. [26].They involve a generic number of sets of time variables and the integral representations for such kind of partition functions are given by tensor models and multi-matrix models with multi-trace couplings in some special cases.
The goal of this paper is to make a step towards the supersymmetric case, i.e., the superintegrability for the partition functions depending on bosonic and fermionic (or Grassmann) variables.We shall construct the supersymmetric partition function hierarchies through Wrepresentations and analyze the superintegrability.
This paper is organized as follows.In section 2, the β-deformed Hurwitz operator is extended to the supersymmetric case.We construct the supersymmetric β-deformed HK partition functions and present their character expansions with respect to the Jack superpolynomials.In sections 3, based on the β-deformed Hurwitz superoperators, we give the supersymmetric β-deformed hierarchy for the partition functions through W -representations.The superintegrability is shown by the character expansions with respect to the Jack superpolynomials.We also construct the constraints for the supersymmetric β-deformed partition functions.The remarkable feature of constraint operators is that they yield the new infinite-dimensional super algebra and null super 3-algebra.In section 4, we construct the (q, t)-deformed Hurwitz superoperators and present the supersymmetric (q, t)-deformed HK partition functions.Then the supersymmetric (q, t)-deformed partition function hierarchy is constructed through W -representations and the desired constraints are presented as well.Moreover, the superintegrability is shown by the character expansions with respect to the Macdonald superpolynomials.We end this paper with the conclusions in section 5.

Supersymmetric β-deformed HK partition functions
i=1 Λ i and fermionic degree m is a pair of partitions written as [27] Note that the superpartitions of degree (n|0) are regular partitions.In the following, we denote Λ * = (Λ a ; Λ s ) + as the partition obtained by reordering the concatenation of entries of Λ a and Λ s in non-increasing order, Λ ⊛ = (Λ a + 1 m ; Λ s ) + , where Λ a + 1 m is the partition obtained by adding one to each entry of Λ a .
We give the supersymmetric β-deformed HK partition functions through W -representations where τ and θ are the bosonic and fermionic parameters, respectively.The Cauchy formula for the Jack superpolynomials is [32] e β ∞ k=1 ( where , m is the fermionic degree of Λ, BΛ is the set of squares in the diagram of Λ that do not belong at the same time to a fermionic row and a fermionic column, 8) and using (4), we may write the character expansions of (7) as The non-deformed partition functions follow from (7) by taking β = 1, (10) where is the Schur-Jack superpolynomial [33] and with m the fermionic degree of Λ.
Setting to zero all the fermionic variables in (10), we recover the HK model with Wrepresentation [1,21,22] where the Hurwitz operator W0 is given by ( 6) with β = 1, S λ is the Schur function.
The matrix model representation of ( 12) is given by [1] where ψ is an N × N matrix and the time variables 3 Supersymmetric β-deformed partition function hierarchy

The negative branch of hierarchy
Let us define the bosonic operators where given by replacing N in the β-deformed Hurwitz superoperators ( 5) with an arbitrary parameter u i .The actions of W (⃗ u) on the Jack superpolynomials are where The bosonic operators W There are the actions where the sum is over the superpartitions Ω satisfying Λ ⊆ Ω and We may introduce the fermionic operators By means of the action (17) and relation where the sum is over the superpartitions Ω satisfying Λ ⊆ Ω and By means of the bosonic operators ( 16) and fermionic operators (18), we give the negative branch of supersymmetric β-deformed hierarchy Since the operators 20) are defined by the nested commutators, we call a family of partition functions (20) "hierarchy" here.
In order to derive the character expansion of ( 20), let us recall the evaluation formulas for the Jack superpolynomials.We denote Λ = (Λ a ; Λ s ) as the superpartition of fermionic degree m.The evaluation E u on the power sum basis where Then the evaluation formulas on the Jack superpolynomial J Λ are given by [31] where Note that when m = 0 in (22), it gives the evaluation formula for the Jack polynomial J λ associated with the regular partition λ [34] Using the actions ( 17), (19) and evaluation formulas (22) and (24), we obtain the character expansion of (20) where λ is the regular partition, Λ is the superpartition of fermionic degree m > 0.
The supersymmetric β-deformed partition functions (20) satisfy the constraints and where the constraint operators are given by they yield null super algebra.
Multiplying by g k and then taking the sum over k in (26), we have where where l0 {g} = ∞ k=1 kg k ∂ ∂g k .Let us pause here to recall the Gaussian hermitian one-matrix model [1,35] where λ is the regular partition.
There are the well known Virasoro constraints for (31) where the Virasoro constraint operators are The character expansion of the Gaussian hermitian one-matrix model can be derived recursively from a single w-constraint [36] where l 0 {p} = ∞ k=1 kp k ∂ ∂p k .Note that there is an intrinsic connection between (35) and the Virasoro constraints (33).More precisely, (35) is equivalent to the sum of Virasoro constraints We see that the expressions of ( 29) and ( 30) are similar with (35).However, unlike the case of (35), we can not give the (super) Virasoro constraints for (20) from ( 29) and (30).
To present the desired constraints, let us introduce the operator l0 = l 0 {g} + l0 {g} − l 0 {p} − l0 {p}, where l0 {p} Thus we may further construct the constraints where ) and a ′ ∈ N, all lower indices of non-zero V-operators have to be unequal.It is interesting to note that the constraint operators (39) yield the new infinite-dimensional super algebra and the null super 3-algebra where the super 3-bracket is defined by [37] [ in which |A i j | is the parity of the superoperator A i j .
When particularized to the constraint operators Since ( 43) is the super Virasoro algebra, we call the infinite-dimensional super algebra (40) the generalized super Virasoro algebra.

The positive branch of hierarchy
Let us turn to construct the bosonic operators and fermionic operators where (n 1 , n 2 ) ∈ N 2 \ (0, 0).There are the actions where the sum is over the superpartitions Γ satisfying Γ ⊆ Λ, We note that in terms of variables x i and θ i , W (1,0) 2 (0) is given by It is the gauged Hamiltonian (with ω = 0) for the supersymmetric rational CMS model [38].By means of the operators ( 44) and (45), we give the positive branch of supersymmetric β-deformed hierarchy where λ and µ are regular partitions, Λ and Γ are superpartitions of fermionic degree m > 0 and m ′ > 0, respectively.There are the constraints for (48) Wk Z n 1 ,n 2 {⃗ u; p, p|g, g|h, h} = 0, where the constraint operators are given by they also yield null super algebra.Let l0 = l 0 {h} + l0 { h} + l 0 {p} + l0 {p}, we then have Thus we may construct the generalized super Virasoro constraints where ) and a ′ ∈ N.These constraint operators satisfy the generalized super Virasoro algebra (40) and null super 3-algebra (41).
We have given the desired supersymmetric β-deformed hierarchy.The non-deformed hierarchy follows from the β = 1 case and the Jack superpolynomials in the character expansions are replaced by the Schur-Jack superpolynomials.Setting to zero all the fermionic variables in the supersymmetric β-deformed hierarchy, we arrive at the results in Refs.[25,26] It is noted that the β-deformed eigenvalue models in Refs.[39,40] are special cases of (54a).Furthermore, the (skew) hypergeometric Hurwitz τ -functions [19,24,25] correspond to the β = 1 case in (54).
4 Supersymmetric (q, t)-deformed partition function hierarchy 4.1 Supersymmetric (q, t)-deformed HK partition functions The Macdonald superpolynomials P Λ are eigenfunctions of two family of commuting superoperators [41] where S m and S m c are groups of permutations of ∂ ∂θ j θ j .There are the actions where E λ (z; q, t) where (qx i −x j )(tx i −x j ) θ I ϱ I , ϱ I picks up the coefficient of θ I in a superpolynomial, i.e., ϱ I θ J = δ IJ , τ i is the q-shift operator such that x i → qx i and x j → x j for i ̸ = j.
Here we have used the evaluation formulas [43] E u (P Λ ) where m is the fermionic degree of Λ, and n(λ) = i (i − 1)λ i .When m = 0 in (68), it gives the evaluation formula for the Macdonald polynomial P λ associated with the regular partition λ [34] For the supersymmetric (q, t)-deformed partition functions (67), there are the generalized super Virasoro constraints and where the constraint operators are given by Lα and ) and a ′ ∈ N. The constraint operators (73) and (74) satisfy the generalized super Virasoro algebra (40) and null super 3-algebra (41).
We list the first several partition functions of (67) as follows: (75) It is clear that the β-deformed cases in the previous section can be recovered by taking the limit t = q β , q → 1 in (67).Furthermore, setting to zero all the fermionic variables in (67), we arrive at the results in Ref. [23] Ẑ−n {⃗ u; p, g} =

Conclusions
We have extended the β and (q, t)-deformed Hurwitz operators to the supersymmetric cases.
Then we gave the supersymmetric β and (q, t)-deformed HK partition functions ( 7) and (60) through W -representations.The superintegrability was shown by the character expansions with respect to the Jack and Macdonald superpolynomials, respectively.Based on the β-deformed Hurwitz superoperators (5), we have constructed a series of bosonic and fermionic operators by the nested commutators.Then we gave the supersymmetric βdeformed hierarchy (see negative and positive branches (20) and ( 48)) for the partition functions through W -representations, where the W -operators in W -representations were given by the constructed bosonic and fermionic operators.By providing the corresponding character expansions with respect to the Jack superpolynomials, the superintegrablity for supersymmetric β-deformed partition function hierarchy has been confirmed, i.e., < character >∼ character.Moreover, when β = 1, the Jack superpolynomials in the character expansions reduce to the Schur-Jack superpolynomials.The superintegrablity for the reduced supersymmetric hierarchy still holds.In addition, we have constructed the generalized super Virasoro constraints ( 38) and (52) for the supersymmetric β-deformed partition functions, where the constraint operators obey the generalized super Virasoro algebra and null super 3-algebra.From the algebraic point of view, the new infinite-dimensional super (3-)algebras presented in this paper are also interesting in their own right due to the super higher algebraic structures.Searching for other realizations of such super (3-)algebras and more applications would be interesting.
Similarly, based on the (q, t)-deformed superoperators (62), we have also constructed the supersymmetric (q, t)-deformed partition function hierarchy (67) through W -representations and presented the generalized super Virasoro constraints (71) and (72).The superintegrability was shown by the character expansions with respect to the Macdonald superpolynomials.When we set all the fermionic variables to zero in the supersymmetric (β and (q, t)-deformed) partition functions presented in this paper, they coincide with the known results in the literature [23,25,26].Our results may shed new light on supereigenvalue and supermatrix models.For further research, it would be interesting to search for the integral representations for the supersymmetric (β and (q, t)-deformed) partition functions with W -representations.