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J. Harnad

Publications and source records attributed to J. Harnad.

At least 37 records · Page 2Linked to original sources
arXiv2017

Zero-temperature limit of quantum weighted Hurwitz numbers

The partition function for quantum weighted double Hurwitz numbers can be interpreted in terms of the energy distribution of a quantum Bose gas with vanishing fugacity. We compute the leading term of the partition function and the quantum weighted Hurwitz numbers in the zero temperature limit $T \rightarrow 0$, as well as the next order corrections. The leading term is shown to reproduce the case of uniformly weighted Hurwitz numbers of Belyi curves. In particular, the KP or Toda $τ$-function serving as generating function for the quantum Hurwitz numbers is shown in the limit to give the one for Belyi curves and, with suitable scaling, the same holds true for the partition function, the weights and the expectations of Hurwitz numbers.

math-ph↗
arXiv2017

Fermionic approach to weighted Hurwitz numbers and topological recursion

A fermionic representation is given for all the quantities entering in the generating function approach to weighted Hurwitz numbers and topological recursion. This includes: KP and 2D Toda $τ$-functions of hypergeometric type, which serve as generating functions for weighted single and double Hurwitz numbers; the Baker function, which is expanded in an adapted basis obtained by applying the same dressing transformation to all vacuum basis elements; the multipair correlators and the multicurrent correlators. Multiplicative and differential recursion relations are deduced for the adapted bases and their duals, and a Christoffel-Darboux type formula is derived for the pair correlator. The quantum and classical spectral curves linking this theory with the topological recursion programme are derived, as well as the generalized cut and join equations. The results are detailed for four special cases: the simple single and double Hurwitz numbers, the weakly monotone case, corresponding to signed enumeration of coverings, the strongly monotone case, corresponding to Belyi curves and the simplest version of quantum weighted Hurwitz numbers.

math-ph↗
arXiv2016

Quantum Hurwitz numbers and Macdonald polynomials

Parametric families in the centre ${\bf Z}({\bf C}[S_n])$ of the group algebra of the symmetric group are obtained by identifying the indeterminates in the generating function for Macdonald polynomials as commuting Jucys-Murphy elements. Their eigenvalues provide coefficients in the double Schur function expansion of 2D Toda $τ$-functions of hypergeometric type. Expressing these in the basis of products of power sum symmetric functions, the coefficients may be interpreted geometrically as parametric families of quantum Hurwitz numbers, enumerating weighted branched coverings of the Riemann sphere. Combinatorially, they give quantum weighted sums over paths in the Cayley graph of $S_n$ generated by transpositions. Dual pairs of bases for the algebra of symmetric functions with respect to the scalar product in which the Macdonald polynomials are orthogonal provide both the geometrical and combinatorial significance of these quantum weighted enumerative invariants.

math-ph↗
arXiv2016

Semiclassical asymptotics of quantum weighted Hurwitz numbers

This work concerns the semiclassical asymptotics of quantum weighted double Hurwitz numbers. We compute the leading term of the partition function for three versions of the quantum weighted Hurwitz numbers, as well as lower order semiclassical corrections. The classical limit $\hbar \to 0$ is shown to reproduce the simple Hurwitz numbers studied by Pandharipande and Okounkov. The KP-Toda $τ$-function serving as generating function for the quantum Hurwitz numbers is shown to converge in the classical limit to the generating function of Pandharipande and Okounkov and, with suitable scaling, so do the partition function, the weights and expectations of Hurwitz numbers.

math-ph↗
arXiv2015

Multispecies Weighted Hurwitz Numbers

The construction of hypergeometric $2D$ Toda $τ$-functions as generating functions for weighted Hurwitz numbers is extended to multispecies families. Both the enumerative geometrical significance of multispecies weighted Hurwitz numbers, as weighted enumerations of branched coverings of the Riemann sphere, and their combinatorial significance in terms of weighted paths in the Cayley graph of $S_n$ are derived. The particular case of multispecies quantum weighted Hurwitz numbers is studied in detail.

math-ph↗
arXiv2015

Weighted Hurwitz numbers and hypergeometric $τ$-functions: an overview

This is an overview of recent results on the use of 2D Toda $τ$-functions as generating functions for multiparametric families of weighted Hurwitz numbers. The Bose-Fermi equivalence composed with the characteristic map provides an isomorphism between the zero charge sector of the Fermionic Fock space and the direct sum of the centers of the group algebra of the symmetric groups $S_n$. Specializing the fermionic formula to the case of diagonal group elements gives $τ$-functions of hypergeometric type, for which the expansion over products of Schur functions is diagonal, with coefficients of {\em content product} type. The corresponding abelian group action on the centre of the $S_n$ group algebra is determined by forming symmetric functions multiplicatively from a weight generating function $G(z)$ and evaluating on the Jucys-Murphy elements of the group algebra. The resulting central elements act diagonally on the basis of orthogonal idempotents and the eigenvalues $r^{G(z)}_λ$ are the {\em content product} coefficients appearing in the double Schur function expansion. Both the geometrical meaning of weighted Hurwitz numbers, as weighted sums over $n$-sheeted branched coverings, and the combinatorial one, as weighted enumeration of paths in the Cayley graph of $S_n$ generated by transpositions follow from expansion of the Cauchy-Littlewood generating functions over dual pairs of bases of the algebra of symmetric functions. The coefficients in the resulting $τ$-function expansion over products of power sum symmetric functions are the weighted Hurwitz numbers. Replacement of the Cauchy-Littlewood generating function by that for Macdonald polynomials provides $(q,t)$-deformations that yield generating functions for quantum weighted Hurwitz numbers.

math-ph↗
arXiv2015

2D Toda τ-functions as combinatorial generating functions

Two methods of constructing 2D Toda $τ$-functions that are generating functions for certain geometrical invariants of a combinatorial nature are related. The first involves generation of paths in the Cayley graph of the symmetric group $S_n$ by multiplication of the conjugacy class sums $C_λ\in C[S_n]$ in the group algebra by elements of an abelian group of central elements. Extending the characteristic map to the tensor product $C[S_n]\otimes C[S_n]$ leads to double expansions in terms of power sum symmetric functions, in which the coefficients count the number of such paths. Applying the same map to sums over the orthogonal idempotents leads to diagonal double Schur function expansions that are identified as $τ$-functions of hypergeometric type. The second method is the standard construction of $τ$-functions as vacuum state matrix elements of products of vertex operators in a fermionic Fock space with elements of the abelian group of convolution symmetries. A homomorphism between these two group actions is derived and shown to be intertwined by the characteristic map composed with fermionization. Applications include Okounkov's generating function for double Hurwitz numbers, which count branched coverings of the Riemann sphere with nonminimal branching at two points, and various analogous combinatorial counting functions.

math-ph↗
arXiv2015

Multispecies quantum Hurwitz numbers

The construction of hypergeometric 2D Toda $τ$-functions as generating functions for quantum Hurwitz numbers is extended here to multispecies families. Both the enumerative geometrical significance of these multispecies quantum Hurwitz numbers as weighted enumerations of branched coverings of the Riemann sphere and their combinatorial significance in terms of weighted paths in the Cayley graph of $S_n$ are derived.

math-ph↗
arXiv2015

Hypergeometric τ-functions, Hurwitz numbers and enumeration of paths

A multiparametric family of 2D Toda $τ$-functions of hypergeometric type is shown to provide generating functions for composite, signed Hurwitz numbers that enumerate certain classes of branched coverings of the Riemann sphere and paths in the Cayley graph of $S_n$. The coefficients $F^{c_1, ..., c_l}_{d_1, ..., d_m}(μ, ν)$ in their series expansion over products $P_μP'_ν$ of power sum symmetric functions in the two sets of Toda flow parameters and powers of the $l+m$ auxiliary parameters are shown to enumerate $|μ|=|ν|=n$ fold branched covers of the Riemann sphere with specified ramification profiles $ μ$ and $ν$ at a pair of points, and two sets of additional branch paints, satisfying certain additional conditions on their ramification profile lengths. The first group consists of $l$ branch points, with ramification profile lengths fixed to be the numbers $(n-c_1, ..., n- c_l)$; the second consists of $m$ further groups of "coloured" branch points, of variable number, for which the sums of the complements of the ramification profile lengths within the groups are fixed to equal the numbers $(d_1, ..., d_m)$. The latter are counted with sign determined by the parity of the total number of such branch points. The coefficients $F^{c_1, ..., c_l}_{d_1, ..., d_m}(μ, ν)$ are also shown to enumerate paths in the Cayley graph of the symmetric group $S_n$ generated by transpositions, starting, as in the usual double Hurwitz case, at an element in the conjugacy class of cycle type $μ$ and ending in the class of type $ν$, but with the first $l$ consecutive subsequences of $(c_1, ..., c_l)$ transpositions strictly monotonically increasing, and the subsequent subsequences of $(d_1, ..., d_m)$ transpositions weakly increasing.

math-ph↗
arXiv2014

Finite Dimensional KP τ-functions I. Finite Grassmannians

We study τ-functions of the KP hierarchy in terms of abelian group actions on finite dimensional Grassmannians, viewed as subquotients of the Hilbert space Grassmannians of Sato, Segal and Wilson. A determinantal formula of Gekhtman and Kasman involving exponentials of finite dimensional matrices is shown to follow naturally from such reductions. All reduced flows of exponential type generated by matrices with arbitrary nondegenerate Jordan forms are derived, both in the Grassmannian setting and within the fermionic operator formalism. A slightly more general determinantal formula involving resolvents of the matrices generating the flow, valid on the big cell of the Grassmannian, is also derived. An explicit expression is deduced for the Plücker coordinates appearing as coefficients in the Schur function expansion of the τ-function.

math-ph↗
arXiv2013

Symmetric polynomials, generalized Jacobi-Trudi identities and τ-functions

An element [Φ] of the Grassmannian of n-dimensional subspaces of the Hardy space H^2, extended over the field C(x_1,..., x_n), may be associated to any polynomial basis ϕ for C(x). The Plücker coordinates S^ϕ_{λ,n}(x_1,..., x_n) of Φ, labelled by partitions λ, provide an analog of Jacobi's bi-alternant formula, defining a generalization of Schur polynomials. Applying the recursion relations satisfied by the polynomial system to the analog of the complete symmetric functions generates a doubly infinite matrix of symmetric polynomials that determine an element [H] of the Grassmannian. This is shown to coincide with [Φ], implying a set of {\it quantum Jacobi-Trudi identities} that generalize a result obtained by Sergeev and Veselov for the case of orthogonal polynomials. The symmetric polynomials S^ϕ_{λ,n}(x_1,..., x_n) are shown to be KP (Kadomtsev-Petviashvili) tau-functions in terms of the monomial sums [x] in the parameters x_a, viewed as KP flow variables. A fermionic operator representation is derived for these, as well as for the infinite sums \sum_λS_{λ,n}^ϕ([x]) S^θ_{λ,n} ({\bf t}) associated to any pair of polynomial bases (ϕ, θ), which are shown to be 2D Toda lattice τ-functions. A number of applications are given, including classical group character expansions, matrix model partition functions and generators for random processes.

math-ph↗
arXiv2012

Convolution symmetries of integrable hierarchies, matrix models and τ-functions

Generalized convolution symmetries of integrable hierarchies of KP and 2KP-Toda type multiply the Fourier coefficients of the elements of the Hilbert space $\HH= L^2(S^1)$ by a specified sequence of constants. This induces a corresponding transformation on the Hilbert space Grassmannian $\Gr_{\HH_+}(\HH)$ and hence on the Sato-Segal-Wilson τ-functions determining solutions to the KP and 2-Toda hierarchies. The corresponding action on the associated fermionic Fock space is also diagonal in the standard orthonormal base determined by occupation sites and labeled by partitions. The Plücker coordinates of the element element $W \in \Gr_{\HH_+}(\HH)$ defining the initial point of these commuting flows are the coefficients in the single and double Schur function of the associated τfunction, and are therefore multiplied by the corresponding diagonal factors under this action. Applying such transformations to matrix integrals, we obtain new matrix models of externally coupled type that are hence also KP or 2KP-Toda τ-functions. More general multiple integral representations of τfunctions are similarly obtained, as well as finite determinantal expressions for them.

math-ph↗
arXiv2011

Multiple sums and integrals as neutral BKP tau functions

We consider multiple sums and multi-integrals as tau functions of the BKP hierarchy using neutral fermions as the simplest tool for deriving these. The sums are over projective Schur functions $Q_α$ for strict partitions $α$. We consider two types of such sums: weighted sums of $Q_α$ over strict partitions $α$ and sums over products $Q_αQ_γ$. In this way we obtain discrete analogues of the beta-ensembles ($β=1,2,4$). Continuous versions are represented as multiple integrals. Such sums and integrals are of interest in a number of problems in mathematics and physics.

math-ph↗
arXiv2010

Schur function expansions of KP tau functions associated to algebraic curves

The Schur function expansion of Sato-Segal-Wilson KP tau-functions is reviewed. The case of tau-functions related to algebraic curves of arbitrary genus is studied in detail. Explicit expressions for the Plücker coordinate coefficients appearing in the expansion are obtained in terms of directional derivatives of the Riemann theta function or Klein sigma function along the KP flow directions. Using the fundamental bi-differential, it is shown how the coefficients can be expressed as polynomials in terms of Klein's higher genus generalizations of Weierstrass' zeta and P functions. The cases of genus two hyperelliptic and genus three trigonal curves are detailed as illustrations of the approach developed here.

math-ph↗
arXiv2008

Approaches to Open Access in Scientific Publishing

Approaches to scientific journal publishing that provide free access to all readers are challenging the standard subscription-based model. But in domains that have a well-functioning system of publicly accessible preprint repositories like arXiv, Open Access is already effectively available. In physics, such repositories have long coexisted constructively with refereed, subscription based journals. Trying to replace this by a system based on journals whose revenue is derived primarily from fees charged to authors is unlikely to provide a better guarantee of Open Access, and may be in conflict with the maintenance of high quality standards.

physics.soc-ph↗
arXiv2008

Superharmonic Perturbations of a Gaussian Measure, Equilibrium Measures and Orthogonal Polynomials

This work concerns superharmonic perturbations of a Gaussian measure given by a special class of positive weights in the complex plane of the form $w(z) = \exp(-|z|^2 + U^μ(z))$, where $U^μ(z)$ is the logarithmic potential of a compactly supported positive measure $μ$. The equilibrium measure of the corresponding weighted energy problem is shown to be supported on subharmonic generalized quadrature domains for a large class of perturbing potentials $U^μ(z)$. It is also shown that the $2\times 2$ matrix d-bar problem for orthogonal polynomials with respect to such weights is well-defined and has a unique solution given explicitly by Cauchy transforms. Numerical evidence is presented supporting a conjectured relation between the asymptotic distribution of the zeroes of the orthogonal polynomials in a semi-classical scaling limit and the Schwarz function of the curve bounding the support of the equilibrium measure, extending the previously studied case of harmonic polynomial perturbations with weights $w(z)$ supported on a compact domain.

math-ph↗
arXiv2007

Trouble with Physics?

This is a review of Lee Smolin's "The Trouble with Physics". The main gist of the review is that the physics of the past three decades has been rich with new discoveries in a large number of domains. The Standard Model, while providing a successful framework for the electroweak and strong interactions still needs to be developed further to fully account fot the mass of strong interaction data collected over the decades. A multitude of new phenomena have been discovered however, or predicted theoretically, in the physics of very low temperatures (e.g. fractional Hall effect, Bose-Einstein condensation) and quantum macroscopic effects (liquid crystals, high temperature superconductivity), some of which remain to be fully explained. Therefore if there is any "Trouble with Physics", as implied by the author of this book, it mainly concerns speculative work on Superstring theory, which may or may not turn out to agree with the results of observation. The subject as a whole does not appear to be in trouble.

physics.hist-ph↗