One-parameter Deformation Research Articles

On the Hadamard Product of Hopf Monoids

AbstractCombinatorial structures that compose and decompose give rise to Hopf monoids in Joyal's category of species. The Hadamard product of two Hopf monoids is another Hopf monoid. We prove two main results regarding freeness of Hadamard products. The first one states that if one factor is connected and the other is free as a monoid, their Hadamard product is free (and connected). The second provides an explicit basis for the Hadamard product when both factors are free.The first main result is obtained by showing the existence of a one-parameter deformation of the comonoid structure and appealing to a rigidity result of Loday and Ronco that applies when the parameter is set to zero. To obtain the second result, we introduce an operation on species that is intertwined by the free monoid functor with theHadamard product. As an application of the first result, we deduce that the Boolean transform of the dimension sequence of a connected Hopf monoid is nonnegative.

Anti De Sitter Deformation of Quaternionic Analysis and the Second-Order Pole

This is a continuation of a series of papers [FL1, FL2, FL3], where we develop quaternionic analysis from the point of view of representation theory of the conformal Lie group and its Lie algebra. In this paper we continue to study the quaternionic analogues of Cauchy’s formula for the second order pole. These quaternionic analogues are closely related to regularization of infinities of vacuum polarization diagrams in four-dimensional quantum field theory. In order to add some flexibility, especially when dealing with Cauchy’s formula for the second order pole, we introduce a one-parameter deformation of quaternionic analysis. This deformation of quaternions preserves conformal invariance and has a geometric realization as anti de Sitter space sitting inside the five-dimensional Euclidean space. We show that many results of quaternionic analysis – including the Cauchy-Fueter formula – admit a simple and canonical deformation. We conclude this paper with a deformation of the quaternionic analogues of Cauchy’s formula for the second order pole.

G/G gauged WZW-matter model, Bethe Ansatz for q-boson model and Commutative Frobenius algebra

We investigate the correspondence between two dimensional topological gauge theories and quantum integrable systems discovered by Moore, Nekrasov, Shatashvili. This correspondence means that the hidden quantum integrable structure exists in the topological gauge theories. We showed the correspondence between the G/G gauged WZW model and the phase model in JHEP 11 (2012) 146 (arXiv:1209.3800). In this paper, we study a one-parameter deformation for this correspondence and show that the G/G gauged WZW model coupled to additional matters corresponds to the q-boson model. Furthermore, we investigate this correspondence from a viewpoint of the commutative Frobenius algebra, the axiom of the two dimensional topological quantum field theory.

The Semi-Infinite q-Boson System with Boundary Interaction

Upon introducing a one-parameter quadratic deformation of the q-boson algebra and a diagonal perturbation at the end point, we arrive at a semi-infinite q-boson system with a two-parameter boundary interaction. The eigenfunctions are shown to be given by Macdonald's hyperoctahedral Hall-Littlewood functions of type BC. It follows that the n-particle spectrum is bounded and absolutely continuous and that the corresponding scattering matrix factorizes as a product of two-particle bulk and one-particle boundary scattering matrices.

Matter from geometry without resolution

We utilize the deformation theory of algebraic singularities to study charged matter in compactifications of M-theory, F-theory, and type IIa string theory on elliptically fibered Calabi-Yau manifolds. In F-theory, this description is more physical than that of resolution. We describe how two-cycles can be identified and systematically studied after deformation. For ADE singularities, we realize non-trivial ADE representations as sublattices of Z^N, where N is the multiplicity of the codimension one singularity before deformation. We give a method for the determination of Picard-Lefschetz vanishing cycles in this context and utilize this method for one-parameter smooth deformations of ADE singularities. We give a general map from junctions to weights and demonstrate that Freudenthal's recursion formula applied to junctions correctly reproduces the structure of high-dimensional ADE representations, including the 126 of SO(10) and the 43,758 of E_6. We identify the Weyl group action in some examples, and verify its order in others. We describe the codimension two localization of matter in F-theory in the case of heterotic duality or simple normal crossing and demonstrate the branching of adjoint representations. Finally, we demonstrate geometrically that deformations correctly reproduce the appearance of non-simply-laced algebras induced by monodromy around codimension two singularities, showing the reduction of D_4 to G_2 in an example. A companion mathematical paper will follow.

Partitioned Binary Relations

We define the category of partitioned binary relations and show that it contains many classical diagram categories, including categories of binary relations, maps, injective maps, partitions, (oriented) Brauer diagrams and (oriented) Temperley-Lieb diagrams. We construct a one-parameter deformation of the category of partitioned binary relations and show that it gives rise to classical one-parameter deformations of partition, Brauer and Temperley-Lieb categories. Finally, we describe a factorization of partitioned binary relations into a product of certain idempotents and pairs of usual binary relations.

Vertex models, TASEP and Grothendieck polynomials

We examine the wavefunctions and their scalar products of a one-parameter family of integrable five-vertex models. At a special point of the parameter, the model investigated is related to an irreversible interacting stochastic particle system—the so-called totally asymmetric simple exclusion process (TASEP). By combining the quantum inverse scattering method with a matrix product representation of the wavefunctions, the on-/off-shell wavefunctions of the five-vertex models are represented as a certain determinant form. Up to some normalization factors, we find that the wavefunctions are given by Grothendieck polynomials, which are a one-parameter deformation of Schur polynomials. Introducing a dual version of the Grothendieck polynomials, and utilizing the determinant representation for the scalar products of the wavefunctions, we derive a generalized Cauchy identity satisfied by the Grothendieck polynomials and their duals. Several representation theoretical formulae for the Grothendieck polynomials are also presented. As a byproduct, the relaxation dynamics such as Green functions for the periodic TASEP are found to be described in terms of the Grothendieck polynomials.

On the integrability of two-dimensional models with [formula omitted] symmetry

In this paper we study the integrability of a family of models with U(1)×SU(N) symmetry. They admit fermionic and bosonic formulations related through bosonization and subsequent T-duality. The fermionic theory is just the CPN−1 sigma model coupled to a self-interacting massless fermion, while the bosonic one defines a one-parameter deformation of the O(2N) sigma model. For N=2 the latter model is equivalent to the integrable deformation of the O(4) sigma model discovered by Wiegmann. At higher values of N we find that integrability is more sporadic and requires a fine-tuning of the parameters of the theory. A special case of our study is the N=4 model, which was found to describe the AdS4×CP3 string theory in the Alday–Maldacena decoupling limit. In this case we propose a set of asymptotic Bethe ansatz equations for the energy spectrum.

THE MINIMAL κ-ENTROPY MARTINGALE MEASURE

We introduce the notion of κ-entropy (κ ∈ ℝ, |κ| ≤ 1), starting from Kaniadakis' (2001, 2002, 2005) one-parameter deformation of the ordinary exponential function. The κ-entropy is in duality with a new class of utility functions which are close to the exponential utility functions, for small values of wealth, and to the power law utility functions, for large values of wealth. We give conditions on the existence and on the equivalence to the basic measure of the minimal κ-entropy martingale measure. Moreover, we provide characterizations of its density as a κ-exponential function. We show that the minimal κ-entropy martingale measure is closely related to both the standard entropy martingale measure and the well known q-optimal martingale measures. We finally establish the convergence of the minimal κ-entropy martingale measure to the minimal entropy martingale measure as κ tends to 0.

Monodromy zeta-function of a polynomial on a complete intersection, and Newton polyhedra

For a generic (polynomial) one-parameter deformation of a complete intersection, there is defined its monodromy zeta-function. We provide explicit formulae for this zeta-function in terms of the corresponding Newton polyhedra in the case the deformation is non-degenerate with respect to its Newton polyhedra. Using this result we obtain the formula for the monodromy zeta-function at the origin of a polynomial on a complete intersection, which is an analog of the Libgober--Sperber theorem.

Laws of Zipf and Benford, intermittency, and critical fluctuations

We describe precise equivalences between theoretical descriptions of: (i) size-rank and first-digit laws for numerical data sets, (ii) intermittency at the transition to chaos in nonlinear maps, and (iii) cluster fluctuations at criticality. The equivalences stem from a common statistical-mechanical structure that departs from the usual via a one-parameter deformation of the exponential and logarithmic functions. The generalized structure arises when configurational phase space is incompletely visited such that the accessible fraction has fractal properties. Thermodynamically, the common focal expression is an (incomplete) Legendre transform between two entropy (or Massieu) potentials. The theory is in quantitative agreement with real size-rank data and it naturally includes the bends or tails observed for small and large rank.

A ONE-PARAMETER DEFORMATION OF THE NONCOMMUTATIVE LAGRANGE INVERSION FORMULA

We give a one-parameter deformation of the noncommutative Lagrange inversion formula, more precisely, of the formula of Brouder–Frabetti–Krattenthaler for the antipode of the noncommutative Faá di Bruno algebra. Namely, we obtain a closed formula for the antipode of the one-parameter deformation of this Hopf algebra discovered by Foissy.

The Hahn oscillator and a discrete Fourier–Hahn transform

We define the quadratic algebra which is a one-parameter deformation of the Lie algebra extended by a parity operator. The odd-dimensional representations of (with representation label j, a positive integer) can be extended to representations of . We investigate a model of the finite one-dimensional harmonic oscillator based upon this algebra . It turns out that in this model the spectrum of the position and momentum operator can be computed explicitly, and that the corresponding (discrete) wavefunctions can be determined in terms of Hahn polynomials. The operation mapping position wavefunctions into momentum wavefunctions is studied, and this so-called discrete Fourier–Hahn transform is computed explicitly. The matrix of this discrete Fourier–Hahn transform has many interesting properties, similar to those of the traditional discrete Fourier transform.

Fivebranes and resolved deformed G 2 manifolds

We study supergravity solutions corresponding to fivebranes wrapped on a three-sphere inside a G_2 holonomy manifold. By changing a parameter the solutions interpolate between a G_2 manifold X_i \cong S^3 x R^4 with flux on a three-sphere and a distinct G_2 manifold X_j \cong S^3 x R^4 with branes on another three-sphere. Therefore, these realise a G_2 geometric transition purely in the supergravity context. We can add D2 brane charge by applying a simple transformation to the initial solution and we obtain one-parameter deformations of warped G_2 holonomy backgrounds. These solutions suggest a connection between the N=1 Chern-Simons theory on the fivebranes and the field theory dual to D2 branes and fractional NS5 branes, transverse to the G_2 manifold S^3 x R^4.

Non-Kähler heterotic rotations

We study a supersymmetry-preserving solution-generating method in heterotic supergravity. In particular, we use this method to construct one-parameter non-Kahler deformations of Calabi-Yau manifolds with a U(1) isometry, in which the complex structure remains invariant. We explain how to obtain corresponding solutions to heterotic string theory, up to first order in \alpha', by a modified form of the standard embedding. In the course of the paper we also show that Abelian heterotic supergravity embeds into type II supergravity, and note that the solution-generating method in this context is related to a dipole-type deformation when there is a field theory dual.

截面箭图代数的Gerstenhaber 括号积

Based on the minimal projective bimodule resolution, the Gerstenhaber bracket products on the Hochschild cohomology spaces for truncated quiver algebras are described explicitly in terms of parallel paths. As a consequence, for truncated basic cycle algebra Λ, each element in HH2(Λ) defines a noncommutative Poisson structure on Λ, and in turn defines the first multiplication of a one-parameter deformation of Λ.

A one-parameter deformation of the Farahat–Higman algebra

We show, by introducing an appropriate basis, that a one-parameter family of Hopf algebras introduced by Foissy [L. Foissy, Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations, Adv. Math. 218 (1) (2008) 136–162] interpolates between the Faà di Bruno algebra and the Farahat–Higman algebra. Its structure constants in this basis are deformations of the top connection coefficients, for which we obtain analogues of Macdonald’s formulas.

Global stability of special multi-flags

In this paper, we study special multi-flags on manifolds. A special multiflag is a certain nested sequence of subbundles of the tangent bundle which are derived by Lie brackets. A property of a special multi-flag is characterized by the existence of a completely integrable subdistribution of corank one in the largest distribution in the sequence, which is a so-called covariant subdistribution. It is proved that a one-parameter deformation of a special multi-flag on a compact manifold can be described by a family of global diffeomorphisms of the underlying manifold, if the covariant subdistribution of the largest distribution in the flag is preserved.

Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles

We consider a family of chiral non-Hermitian Gaussian random matrices in the unitarily invariant symmetry class. The eigenvalue distribution in this model is expressed in terms of Laguerre polynomials in the complex plane. These are orthogonal with respect to a non-Gaussian weight including a modified Bessel function of the second kind, and we give an elementary proof for this. In the large n limit, the eigenvalue statistics at the spectral edge close to the real axis are described by the same family of kernels interpolating between Airy and Poisson that was recently found by one of the authors for the elliptic Ginibre ensemble. We conclude that this scaling limit is universal, appearing for two different non-Hermitian random matrix ensembles with unitary symmetry. As a second result we give an equivalent form for the interpolating Airy kernel in terms of a single real integral, similar to representations for the asymptotic kernel in the bulk and at the hard edge of the spectrum. This makes its structure as a one-parameter deformation of the Airy kernel more transparent.

Scaling Relation for Excitation Energy under Hyperbolic Deformation

We introduce a one-parameter deformation for one-dimensional (1D) quantum lattice models, the hyperbolic deformation, where the scale of the local energy is proportional to cosh lambda j at the j-th site. Corresponding to a 2D classical system, the deformation does not strongly modify the ground state. In this situation, the effective Hamiltonian of the quantum system shows that the quasi particle is weakly bounded around the center of the system. By analyzing this binding effect, we derive scaling relations for the mean-square width <w^2> of confinement, the energy correction with respect to the excitation gap \Delta, and the deformation parameter $\lambda$. This finite-size scaling allows us to investigate excitation gap of 1D non-deformed bulk quantum systems.