arXiv subjects

Patrick Desrosiers

Publications and source records attributed to Patrick Desrosiers.

36 records · Page 2Linked to original sources

arXiv2013

Superconformal field theory and Jack superpolynomials

We uncover a deep connection between the $\mathcal{N}=1$ superconformal field theory in 2D and eigenfunctions of the supersymmetric Sutherland model known as Jack superpolynomials (sJacks). Specifically, the singular vector at level $rs/2$ of the Kac module labeled by the two integers $r$ and $s$ are given explicitly as a sum of sJacks whose indexing diagrams are contained in a rectangle with $r$ columns and $s$ rows As a second compelling evidence for the distinguished status of the sJack-basis in SCFT, we find that the degenerate Whittaker vectors (Gaiotto states) can be expressed as a remarkably simple linear combination of sJacks. As a consequence, we are able to reformulate the supersymmetric version of the (degenerate) AGT conjecture in terms of the combinatorics of sJacks. Note that the closed-form formulas for the singular vectors and the degenerate Whittaker vectors, although only conjectured in general, have been heavily tested (in some cases, up to level 33/2). Note also that both the Neveu-Schwarz and Ramond sectors are treated.

hep-th↗

arXiv2013

Jack polynomials with prescribed symmetry and some of their clustering properties

We study Jack polynomials in $N$ variables, with parameter $α$, and having a prescribed symmetry with respect to two disjoint subsets of variables. For instance, these polynomials can exhibit a symmetry of type AS, which means that they are anti-symmetric in the first $m$ variables and symmetric in the remaining $N-m$ variables. One of our main goals is to extend recent works on symmetric Jack polynomials [arXiv:0711.3062, arXiv:1007.2692, arXiv:1303.4126] and prove that the Jack polynomials with prescribed symmetry also admit clusters of size $k$ and order $r$, that is, the polynomials vanish to order $r$ when $k+1$ variables coincide. We first prove some general properties for generic $α$, such as their uniqueness as triangular eigenfunctions of operators of Sutherland type, and the existence of their analogues in infinity many variables. We then turn our attention to the case with $α=-(k+1)/(r-1)$. We show that for each triplet $(k,r,N)$, there exist admissibility conditions on the indexing sets, called superpartitions, that guaranty both the regularity and the uniqueness of the polynomials. These conditions are also used to establish similar properties for non-symmetric Jack polynomials. As a result, we prove that the Jack polynomials with arbitrary prescribed symmetry, indexed by $(k,r,N)$-admissible superpartitions, admit clusters of size $k=1$ and order $r\geq 2$. In the last part of the article, we find necessary and sufficient conditions for the invariance under translation of the Jack polynomials with prescribed symmetry AS. This allows to find special families of superpartitions that imply the existence of clusters of size $k>1$ and order $r\geq 2$.

math.CO↗

arXiv2013

Asymptotics for products of characteristic polynomials in classical $β$-Ensembles

We study the local properties of eigenvalues for the Hermite (Gaussian), Laguerre (Chiral) and Jacobi $β$-ensembles of $N\times N$ random matrices. More specifically, we calculate scaling limits of the expectation value of products of characteristic polynomials as $N\to\infty$. In the bulk of the spectrum of each $β$-ensemble, the same scaling limit is found to be $e^{p_{1}}{}_1F_{1}$ whose exact expansion in terms of Jack polynomials is well known. The scaling limit at the soft edge of the spectrum for the Hermite and Laguerre $β$-ensembles is shown to be a multivariate Airy function, which is defined as a generalized Kontsevich integral. As corollaries, when $β$ is even, scaling limits of the $k$-point correlation functions for the three ensembles are obtained. The asymptotics of the multivariate Airy function for large and small arguments is also given. All the asymptotic results rely on a generalization of Watson's lemma and the steepest descent method for integrals of Selberg type.

math-ph↗

arXiv2012

Hermite and Laguerre Symmetric Functions Associated with Operators of Calogero-Moser-Sutherland Type

We introduce and study natural generalisations of the Hermite and Laguerre polynomials in the ring of symmetric functions as eigenfunctions of infinite-dimensional analogues of partial differential operators of Calogero-Moser-Sutherland (CMS) type. In particular, we obtain generating functions, duality relations, limit transitions from Jacobi symmetric functions, and Pieri formulae, as well as the integrability of the corresponding operators. We also determine all ideals in the ring of symmetric functions that are spanned by either Hermite or Laguerre symmetric functions, and by restriction of the corresponding infinite-dimensional CMS operators onto quotient rings given by such ideals we obtain so-called deformed CMS operators. As a consequence of this restriction procedure, we deduce, in particular, infinite sets of polynomial eigenfunctions, which we shall refer to as super Hermite and super Laguerre polynomials, as well as the integrability, of these deformed CMS operators. We also introduce and study series of a generalised hypergeometric type, in the context of both symmetric functions and 'super' polynomials.

math.QA↗

arXiv2012

Jack superpolynomials with negative fractional parameter: clustering properties and super-Virasoro ideals

The Jack polynomials P_λ^{(α)} at α=-(k+1)/(r-1) indexed by certain (k,r,N)-admissible partitions are known to span an ideal I^{(k,r)}_N of the space of symmetric functions in N variables. The ideal I^{(k,r)}_N is invariant under the action of certain differential operators which include half the Virasoro algebra. Moreover, the Jack polynomials in I^{(k,r)}_N admit clusters of size at most k: they vanish when k+1 of their variables are identified, and they do not vanish when only k of them are identified. We generalize most of these properties to superspace using orthogonal eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland model known as Jack superpolynomials. In particular, we show that the Jack superpolynomials P_Λ^{(α)} at α=-(k+1)/(r-1) indexed by certain (k,r,N)-admissible superpartitions span an ideal {\mathcal I}^{(k,r)}_N of the space of symmetric polynomials in N commuting variables and N anticommuting variables. We prove that the ideal {\mathcal I}^{(k,r)}_N is stable with respect to the action of the negative-half of the super-Virasoro algebra. In addition, we show that the Jack superpolynomials in {\mathcal I}^{(k,r)}_N vanish when k+1 of their commuting variables are equal, and conjecture that they do not vanish when only k of them are identified. This allows us to conclude that the standard Jack polynomials with prescribed symmetry should satisfy similar clustering properties. Finally, we conjecture that the elements of {\mathcal I}^{(k,2)}_N provide a basis for the subspace of symmetric superpolynomials in N variables that vanish when k+1 commuting variables are set equal to each other.

math-ph↗

arXiv2011

Selberg Integrals, Super hypergeometric functions and Applications to $β$-Ensembles of Random Matrices

We study a new Selberg-type integral with $n+m$ indeterminates, which turns out to be related to the deformed Calogero-Sutherland systems. We show that the integral satisfies a holonomic system of $n+m$ non-symmetric linear partial differential equations. We also prove that a particular hypergeometric function defined in terms of super Jack polynomials is the unique solution of the system. Some properties such as duality relations, integral formulas, Pfaff-Euler and Kummer transformations are also established. As a direct application, we evaluate the expectation value of ratios of characteristic polynomials in the classical $β$-ensembles of Random Matrix Theory.

math-ph↗

arXiv2011

Evaluation and normalization of Jack superpolynomials

Two evaluation formulas are derived for the Jack superpolynomials. The evaluation formulas are expressed in terms of products of fillings of skew diagrams. One of these formulas is nothing but the evaluation formula of the Jack polynomials with prescribed symmetry, which thereby receives here a remarkably simple formulation. Among the auxiliary results required to establish the evaluation formulas, the determination of the conditions ensuring the non-vanishing coefficients in a Pieri-type rule for Jack superpolynomials is worth pointing out. An important application of the evaluation formulas is a new derivation of the combinatorial norm of the Jack superpolynomials. We finally mention that the introduction of a simpler version of the dominance ordering on superpartitions is fundamental to establish our results.

math.CO↗

arXiv2009

Supermatrix models, loop equations, and duality

We study integrals over Hermitian supermatrices of arbitrary size $p+q$, that are parametrized by an external field $X$ and a source $Y$, of respective size $m+n$ and $p+q$. We show that these integrals exhibit a simple topological expansion in powers of a formal parameter $\hbar$, which can be identified with $1/(p-q)$. The loop equation and the associated spectral curve are also obtained. The solutions to the loop equation are given in terms of the symplectic invariants introduced in arXiv:math-ph/0702045. The symmetry property of the latter objects allows us to prove a duality that relates supermatrix models in which the role of $X$ and $Y$ are interchanged.

math-ph↗

arXiv2008

Duality in random matrix ensembles for all Beta

Gaussian and Chiral Beta-Ensembles, which generalise well known orthogonal (Beta=1), unitary (Beta=2), and symplectic (Beta=4) ensembles of random Hermitian matrices, are considered. Averages are shown to satisfy duality relations like {Beta,N,n} <--> {4/Beta,n,N} for all Beta>0, where N and n respectively denote the number of eigenvalues and products of characteristic polynomials. At the edge of the spectrum, matrix integrals of the Airy (Kontsevich) type are obtained. Consequences on the integral representation of the multiple orthogonal polynomials and the partition function of the formal one-matrix model are also discussed. Proofs rely on the theory of multivariate symmetric polynomials, especially Jack polynomials.

math-ph↗

arXiv2006

Orthogonality of Jack polynomials in superspace

Jack polynomials in superspace, orthogonal with respect to a ``combinatorial'' scalar product, are constructed. They are shown to coincide with the Jack polynomials in superspace, orthogonal with respect to an ``analytical'' scalar product, introduced in hep-th/0209074 as eigenfunctions of a supersymmetric quantum mechanical many-body problem. The results of this article rely on generalizing (to include an extra parameter) the theory of classical symmetric functions in superspace developed recently in math.CO/0509408

math-ph↗

arXiv2006

A note on biorthogonal ensembles

We consider ensembles of random matrices, known as biorthogonal ensembles, whose eigenvalue probability density function can be written as a product of two determinants. These systems are closely related to multiple orthogonal functions. It is known that the eigenvalue correlation functions of such ensembles can be written as a determinant of a kernel function. We show that the kernel is itself an average of a single ratio of characteristic polynomials. In the same vein, we prove that the type I multiple polynomials can be expressed as an average of the inverse of a characteristic polynomial. We finally introduce a new biorthogonal matrix ensemble, namely the chiral unitary perturbed by a source term.

math-ph↗

arXiv2006

Asymptotic correlations for Gaussian and Wishart matrices with external source

We consider ensembles of Gaussian (Hermite) and Wishart (Laguerre) $N\times N$ hermitian matrices. We study the effect of finite rank perturbations of these ensembles by a source term. The rank $r$ of the perturbation corresponds to the number of non-null eigenvalues of the source matrix. In the perturbed ensembles, the correlation functions can be written in terms of kernels. We show that for all $N$, the difference between the perturbed and the unperturbed kernels is a degenerate kernel of size $r$ which depends on multiple Hermite or Laguerre functions. We also compute asymptotic formulas for the multiple Laguerre functions kernels in terms multiple Bessel (resp. Airy) functions. This leads to the large $N$ limiting kernels at the hard (resp. soft) edge of the spectrum of the perturbed Laguerre ensemble. Similar results are obtained in the Hermite case.

math-ph↗

arXiv2006

Relationships between $τ$-function and Fredholm determinant expressions for gap probabilities in random matrix theory

The gap probabilities at the hard and soft edges of scaled random matrix ensembles with orthogonal symmetry are known in terms of $τ$-functions. Extending recent work relating to the soft edge, it is shown that these $τ$-functions, and their generalizations to contain a generating function parameter, can be expressed as Fredholm determinants. These same Fredholm determinants also occur in exact expressions for gap probabilities in scaled random matrix ensembles with unitary and symplectic symmetry.

math-ph↗

arXiv2006

Hermite and Laguerre $β$-ensembles: asymptotic corrections to the eigenvalue density

We consider Hermite and Laguerre $β$-ensembles of large $N\times N$ random matrices. For all $β$ even, corrections to the limiting global density are obtained, and the limiting density at the soft edge is evaluated. We use the saddle point method on multidimensional integral representations of the density which are based on special realizations of the generalized (multivariate) classical orthogonal polynomials. The corrections to the bulk density are oscillatory terms that depends on $β$. At the edges, the density can be expressed as a multiple integral of the Konstevich type which constitutes a $β$-deformation of the Airy function. This allows us to obtain the main contribution to the soft edge density when the spectral parameter tends to $\pm\infty$.

math-ph↗

arXiv2005

Classical symmetric functions in superspace

We present the basic elements of a generalization of symmetric function theory involving functions of commuting and anticommuting (Grassmannian) variables. These new functions, called symmetric functions in superspace, are invariant under the diagonal action of the symmetric group on the sets of commuting and anticommuting variables. In this work, we present the superspace extension of the classical bases, namely, the monomial symmetric functions, the elementary symmetric functions, the completely symmetric functions, and the power sums. Various basic results, such as the generating functions for the multiplicative bases, Cauchy formulas, involution operations as well as the combinatorial scalar product are also generalized.

math.CO↗

arXiv2004

Jack superpolynomials: physical and combinatorial definitions

Jack superpolynomials are eigenfunctions of the supersymmetric extension of the quantum trigonometric Calogero-Moser-Sutherland. They are orthogonal with respect to the scalar product, dubbed physical, that is naturally induced by this quantum-mechanical problem. But Jack superpolynomials can also be defined more combinatorially, starting from the multiplicative bases of symmetric superpolynomials, enforcing orthogonality with respect to a one-parameter deformation of the combinatorial scalar product. Both constructions turns out to be equivalent. This provides strong support for the correctness of the various underlying constructions and for the pivotal role of Jack superpolynomials in the theory of symmetric superpolynomials.

hep-th↗

arXiv2003

Explicit formulas for the generalized Hermite polynomials in superspace

We provide explicit formulas for the orthogonal eigenfunctions of the supersymmetric extension of the rational Calogero-Moser-Sutherland model with harmonic confinement, i.e., the generalized Hermite (or Hi-Jack) polynomials in superspace. The construction relies on the triangular action of the Hamiltonian on the supermonomial basis. This translates into determinantal expressions for the Hamiltonian's eigenfunctions.

hep-th↗

arXiv2003

Generalized Hermite polynomials in superspace as eigenfunctions of the supersymmetric rational CMS model

We present an algebraic construction of the orthogonal eigenfunctions of the supersymmetric extension of the rational Calogero-Moser-Sutherland model with harmonic confinement. These eigenfunctions are the superspace extension of the generalized Hermite (or Hi-Jack) polynomials. The conserved quantities of the rational supersymmetric model are related to their trigonometric relatives through a similarity transformation. This leads to a simple expression between the corresponding eigenfunctions: the generalized Hermite superpolynomials are written as a differential operator acting on the corresponding Jack superpolynomials. As an aside, the maximal superintegrability of the supersymmetric rational Calogero-Moser-Sutherland model is demonstrated.

hep-th↗