arXiv subjects

J. Harnad

Publications and source records attributed to J. Harnad.

At least 19 recordsLinked to original sources

arXiv2023

Hamiltonian structure of isomonodromic deformation dynamics in linear systems of PDE's

The Hamiltonian approach to isomonodromic deformation systems for generic rational covariant derivative operators on the Riemann sphere, having any matrix dimension $r$ and any number of isolated singularities of arbitrary Poincaré rank, is derived using the split classical rational $R$-matrix Poisson bracket structure on the dual space $L^*{\frak gl}(r)$ of the loop algebra $L{\frak gl}(r)$. Nonautonomous isomonodromic counterparts of isospectral systems are obtained by identifying the deformation parameters as Casimir elements on the phase space. These are shown to coincide with the higher Birkhoff invariants determining the local asymptotics near to irregular singular points, together with the pole loci. They appear as the negative power coefficients in the principal part of the Laurent expansion of the fundamental meromorphic differential on the associated spectral curve, while the corresponding dual spectral invariant Hamiltonians appear as the "mirror image" positive power terms in the analytic part. Infinitesimal isomonodromic deformations are generated by the sum of the Hamiltonian vector field and an explicit derivative vector field that is transversal to the symplectic foliation.

math-ph↗

arXiv2023

Hamiltonian structure of rational isomonodromic deformation systems

The Hamiltonian approach to isomonodromic deformation systems is extended to include generic rational covariant derivative operators on the Riemann sphere with irregular singularities of arbitrary Poincaré rank. The space of rational connections with given pole degrees carries a natural Poisson structure corresponding to the standard classical rational R-matrix structure on the dual space $L^*gl(r)$ of the loop algebra $Lgl(r)$. Nonautonomous isomonodromic counterparts of the isospectral systems generated by spectral invariants are obtained by identifying the deformation parameters as Casimir functions on the phase space. These are shown to coincide with the higher Birkhoff invariants determining the local asymptotics near to irregular singular points, together with the pole loci. Infinitesimal isomonodromic deformations are shown to be generated by the sum of the Hamiltonian vector field and an explicit derivative vector field that is transversal to the symplectic foliation. The Casimir elements serve as coordinates complementing those along the symplectic leaves, extended by the exponents of formal monodromy, defining a local symplectomorphism between them. The explicit derivative vector fields preserve the Poisson structure and define a flat transversal connection, spanning an integrable distribution whose leaves, locally, may be identified as the orbits of a free abelian group. The projection of the infinitesimal isomonodromic deformations vector fields to the quotient manifold under this action gives the commuting Hamiltonian vector fields corresponding to the spectral invariants dual to the Birkhoff invariants and the pole loci.

nlin.SI↗

arXiv2023

Lagrangian Grassmannians, CKP hierarchy and hyperdeterminantal relations

This work concerns the relation between the geometry of Lagrangian Grassmannians and the CKP integrable hierarchy. The Lagrange map from the Lagrangian Grassmannian of maximal isotropic (Lagrangian) subspaces of a finite dimensional symplectic vector space $V\oplus V^*$ into the projectivization of the exterior space $ΛV$ is defined by restricting the Plücker map on the full Grassmannian to the Lagrangian sub-Grassmannian and composing it with projection to the subspace of symmetric elements under dualization $V \leftrightarrow V^*$. In terms of the affine coordinate matrix on the big cell, this reduces to the principal minors map, whose image is cut out by the $2 \times 2 \times 2$ quartic {\em hyperdeterminantal} relations. To apply this to the CKP hierarchy, the Lagrangian Grassmannian framework is extended to infinite dimensions, with $V\oplus V^*$ replaced by a polarized Hilbert space $ {\mathcal H} ={\mathcal H}_+\oplus {\mathcal H}_-$, with symplectic form $ω$. The image of the Plucker map in the fermionic Fock space ${\mathcal F}= Λ^{\infty/2}{\mathcal H}$ is identified and the infinite dimensional Lagrangian map is defined. The linear constraints defining reduction to the CKP hierarchy are expressed as a fermionic null condition and the infinite analogue of the hyperdeterminantal relations is deduced. A multiparametric family of such relations is shown to be satisfied by the evaluation of the $τ$-function at translates of a point in the space of odd flow variables along the cubic lattices generated by power sums in the parameters.

math-ph↗

arXiv2022

Tau functions, infinite Grassmannians and lattice recurrences

The addition formulae for KP $τ$-functions, when evaluated at lattice points in the KP flow group orbits in the infinite dimensional Sato-Segal-Wilson Grassmannian, give infinite parametric families of solutions to discretizations of the KP hierarchy. The CKP hierarchy may similarly be viewed as commuting flows on the Lagrangian sub-Grassmannian of maximal isotropic subspaces with respect to a suitably defined symplectic form. Evaluating the $τ$-functions at a sublattice of points within the KP orbit, the resulting discretization gives solutions both to the hyperdeterminantal relations (or Kashaev recurrence) and the hexahedron (or Kenyon-Pemantle) recurrence.

math-ph↗

arXiv2021

Fredholm Pfaffian $τ$-functions for orthogonal isospectral and isomonodromic systems

We extend the approach to $τ$-functions as Widom constants developed by Cafasso, Gavrylenko and Lisovyy to orthogonal loop group Drinfeld-Sokolov hierarchies and isomonodromic deformations systems. The combinatorial expansion of the $τ$-function as a sum of correlators, each expressed as products of finite determinants, follows from using multicomponent fermionic vacuum expectation values of certain dressing operators encoding the initial conditions and the dependence on the flow (or deformation) parameters. When reduced to the orthogonal case, these correlators become finite Pfaffians and the determinantal $τ$-functions, both in the Drinfeld-Sokolov and isomonodromic case, become squares of $τ$-functions of Pfaffian type. The results are illustrated by several examples, consisting of polynomial $τ$-functions of orthogonal Drinfeld-Sokolov type and of isomonodromic ones with four regular singular points.

math-ph↗

arXiv2021

Bilinear expansions of lattices of KP $τ$-functions in BKP $τ$-functions: a fermionic approach

We derive a bilinear expansion expressing elements of a lattice of KP $τ$-functions, labelled by partitions, as a sum over products of pairs of elements of an associated lattice of BKP $τ$-functions, labelled by strict partitions. This generalizes earlier results relating determinants and Pfaffians of minors of skew symmetric matrices, with applications to Schur functions and Schur $Q$-functions. It is deduced using the representations of KP and BKP $τ$-functions as vacuum expectation values (VEV's) of products of fermionic operators of charged and neutral type, respectively. The lattice is generated by insertion of products of pairs of charged creation and annihilation operators. The result follows from expanding the product as a sum of monomials in the neutral fermionic generators and applying a factorization theorem for VEV's of products of operators in the mutually commuting subalgebras. Applications include the case of inhomogeneous polynomial $τ$-functions of KP and BKP type.

math-ph↗

arXiv2021

Polynomial KP and BKP $τ$-functions and correlators

Lattices of polynomial KP and BKP $τ$-functions labelled by partitions, with the flow variables equated to finite power sums, as well as associated multipair KP and multipoint BKP correlation functions are expressed via generalizations of Jacobi's bialternant formula for Schur functions and Nimmo's Pfaffian ratio formula for Schur $Q$-functions. These are obtained by applying Wick's theorem to fermionic vacuum expectation value representations in which the infinite group element acting on the lattice of basis states stabilizes the vacuum.

math-ph↗

arXiv2021

Bilinear expansion of Schur functions in Schur $Q$-functions: a fermionic approach

An identity is derived expressing Schur functions as sums over products of pairs of Schur $Q$-functions, generalizing previously known special cases. This is shown to follow from their representations as vacuum expectation values (VEV's) of products of either charged or neutral fermionic creation and annihilation operators, Wick's theorem and a factorization identity for VEV's of products of two mutually anticommuting sets of neutral fermionic operators.

math-ph↗

arXiv2020

Isotropic Grassmannians, Plücker and Cartan maps

This work is motivated by the relation between the KP and BKP integrable hierarchies, whose $τ$-functions may be viewed as sections of dual determinantal and Pfaffian line bundles over infinite dimensional Grassmannians. In finite dimensions, we show how to relate the Cartan map which, for a vector space $V$ of dimension $N$, embeds the Grassmannian ${\mathrm {Gr}}^0_V(V+V^*)$ of maximal isotropic subspaces of $V+ V^*$, with respect to the natural scalar product, into the projectivization of the exterior space $Λ(V)$, and the Plücker map, which embeds the Grassmannian ${\mathrm {Gr}}_V(V+ V^*)$ of all $N$-planes in $V+ V^*$ into the projectivization of $Λ^N(V + V^*)$. The Plücker coordinates on ${\mathrm {Gr}}^0_V(V+V^*)$ are expressed bilinearly in terms of the Cartan coordinates, which are holomorphic sections of the dual Pfaffian line bundle ${\mathrm {Pf}}^* \rightarrow {\mathrm {Gr}}^0_V(V+V^*, Q)$. In terms of affine coordinates on the big cell, this is equivalent to an identity of Cauchy-Binet type, expressing the determinants of square submatrices of a skew symmetric $N \times N$ matrix as bilinear sums over the Pfaffians of their principal minors.

math-ph↗

arXiv2020

Constellations and $τ$-functions for rationally weighted Hurwitz numbers

Weighted constellations give graphical representations of weighted branched coverings of the Riemann sphere. They were introduced to provide a combinatorial interpretation of the $2$D Toda $τ$-functions of hypergeometric type serving as generating functions for weighted Hurwitz numbers in the case of polynomial weight generating functions. The product over all vertex and edge weights of a given weighted constellation, summed over all configurations, reproduces the $τ$-function. In the present work, this is generalized to constellations in which the weighting parameters are determined by a rational weight generating function. The associated $τ$-function may be expressed as a sum over the weights of doubly labelled weighted constellations, with two types of weighting parameters associated to each equivalence class of branched coverings. The double labelling of branch points, referred to as "colour" and "flavour" indices, is required by the fact that, in the Taylor expansion of the weight generating function, a particular colour from amongst the denominator parameters may appear multiply, and the flavour labels indicate this multiplicity.

math-ph↗

arXiv2020

Generating weighted Hurwitz numbers

Multicurrent correlators associated to KP $τ$-functions of hypergeometric type are used as generating functions for weighted Hurwitz numbers. These are expressed as formal Taylor series and used to compute generic, simple, rational and quantum weighted single Hurwitz numbers.

math-ph↗

arXiv2020

Matrix model generating function for quantum weighted Hurwitz numbers

The KP $τ$-function of hypergeometric type serving as generating function for quantum weighted Hurwitz numbers is used to compute the Baker function and the corresponding adapted basis elements, expressed as absolutely convergent Laurent seriesin the spectral parameter. These are equivalently expressed as Mellin-Barnes integrals, analogously to Meijer $G$-functions, but with an infinite product of $Γ$-functions as integral kernel. A matrix model representation is derived for the $τ$-function evaluated at trace invariants of an externally coupled matrix.

math-ph↗

arXiv2020

Weighted Hurwitz numbers and topological recursion

The KP and 2D Toda tau-functions of hypergeometric type that serve as generating functions for weighted single and double Hurwitz numbers are related to the topological recursion programme. A graphical representation of such weighted Hurwitz numbers is given in terms of weighted constellations. The associated classical and quantum spectral spectral curves are derived, and these are interpreted combinatorially in terms of the graphical model. The pair correlators are given a finite Christoffel-Darboux representation and determinantal expressions are obtained for the multipair correlators. The genus expansion of the multicurrent correlators is shown to provide generating series for weighted Hurwitz numbers of fixed ramification profile lengths. The WKB series for the Baker function is derived and used to deduce the loop equations and the topological recursion relations in the case of polynomial weight functions.

math-ph↗

arXiv2020

Rationally weighted Hurwitz numbers, Meijer $G$-functions and matrix integrals

The quantum spectral curve equation associated to KP $τ$-functions of hypergeometric type serving as generating functions for rationally weighted Hurwitz numbers is solved by generalized hypergeometric series. The basis elements spanning the corresponding Sato Grassmannian element are shown to be Meijer $G$-functions, or their asymptotic series. Using their Mellin integral representation the $τ$-function, evaluated at the trace invariants of an externally coupled matrix, is expressed as a matrix integral.

math-ph↗

arXiv2020

Weighted Hurwitz numbers, $τ$-functions and matrix integrals

The basis elements spanning the Sato Grassmannian element corresponding to the KP $τ$-function that serves as generating function for rationally weighted Hurwitz numbers are shown to be Meijer $G$-functions. Using their Mellin-Barnes integral representation the $τ$-function, evaluated at the trace invariants of an externally coupled matrix, is expressed as a matrix integral. Using the Mellin-Barnes integral transform of an infinite product of $Γ$ functions, a similar matrix integral representation is given for the KP $τ$-function that serves as generating function for quantum weighted Hurwitz numbers.

math-ph↗

arXiv2018

Weighted Hurwitz numbers and topological recursion: an overview

Multiparametric families of hypergeometric $τ$-functions of KP or Toda type serve as generating functions for weighted Hurwitz numbers, providing weighted enumerations of branched covers of the Riemann sphere. A graphical interpretation of the weighting is given in terms of constellations mapped onto the covering surface. The theory is placed within the framework of topological recursion, with the Baker function at ${\bf t} ={\bf 0}$ shown to satisfy the quantum spectral curve equation, whose classical limit is rational. A basis for the space of formal power series in the spectral variable is generated that is adapted to the Grassmannian element associated to the $τ$-function. Multicurrent correlators are defined in terms of the $τ$-function and shown to provide an alternative generating function for weighted Hurwitz numbers. Fermionic VEV representations are provided for the adapted bases, pair correlators and multicurrent correlators. Choosing the weight generating function as a polynomial, and restricting the number of nonzero "second" KP flow parameters in the Toda $τ$-function to be finite implies a finite rank covariant derivative equation with rational coefficients satisfied by a finite "window" of adapted basis elements. The pair correlator is shown to provide a Christoffel-Darboux type finite rank integrable kernel, and the WKB series coefficients of the associated adjoint system are computed recursively, leading to topological recursion relations for the generators of the weighted Hurwitz numbers.

math-ph↗

arXiv2018

Asymptotics of quantum weighted Hurwitz numbers

This work concerns both the semiclassical and zero temperature asymptotics of quantum weighted double Hurwitz numbers. The partition function for quantum weighted double Hurwitz numbers can be interpreted in terms of the energy distri- bution of a quantum Bose gas with vanishing fugacity. We compute the leading semi- classical term of the partition function for three versions of the quantum weighted Hurwitz numbers, as well as lower order semiclassical corrections. The classical limit $\hbar \ra 0$ is shown to reproduce the simple single and double Hurwitz numbers studied by Pandharipande and Okounkov [20,22]. The KP-Toda $τ$-function that serves as generating function for the quantum Hurwitz numbers is shown to have the $τ$-function of [20,22] as its leading term in the classical limit, and, with suitable scaling, the same holds for the partition function, the weights and expectations of Hurwitz numbers. We also compute the zero temperature limit $T \ra 0$ of the partition function and quantum weighted Hurwitz numbers. The KP or Toda $τ$-function serving as generating function for the quantum Hurwitz numbers are shown to give the one for Belyi curves in the zero temperature limit and, with suitable scaling, the same holds true for the partition function, the weights and the expectations of Hurwitz numbers.

math-ph↗

arXiv2017

Generating functions for weighted Hurwitz numbers

Double Hurwitz numbers enumerating weighted $n$-sheeted branched coverings of the Riemann sphere or, equivalently, weighted paths in the Cayley graph of $S_n$ generated by transpositions are determined by an associated weight generating function. A uniquely determined $1$-parameter family of 2D Toda $τ$-functions of hypergeometric type is shown to consist of generating functions for such weighted Hurwitz numbers. Four classical cases are detailed, in which the weighting is uniform: Okounkov's double Hurwitz numbers, for which the ramification is simple at all but two specified branch points; the case of Belyi curves, with three branch points, two with specified profiles; the general case, with a specified number of branch points, two with fixed profiles, the rest constrained only by the genus; and the signed enumeration case, with sign determined by the parity of the number of branch points. Using the exponentiated quantum dilogarithm function as weight generator, three new types of weighted enumerations are introduced. These determine {\em quantum} Hurwitz numbers depending on a deformation parameter $q$. By suitable interpretation of $q$, the statistical mechanics of quantum weighted branched covers may be related to that of Bosonic gases. The standard double Hurwitz numbers are recovered in the classical limit.

math-ph↗