Table of contents

Čestmír Burdík, Ondřej Navrátil and Severin Pošta

Abstract

The XXV International Conference on Integrable Systems and Quantum Symmetries (ISQS-25) organized by the Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Prague and the Bogoliubov Laboratory of Theoretical Physics of the Joint Institute for Nuclear Research continues a successful series of conferences held at the Czech Technical University which began in 1992 and is devoted to problems of mathematical physics related to the theory of integrable systems, quantum groups and quantum symmetries. During the last year, the conference gathered around 200 scientists from all over the world. 44 papers of plenary lectures and contributions presented at ISQS-25 are published in the present issue of the Journal of Physics: Conference Series.

All papers published in this volume of Journal of Physics: Conference Series have been peer reviewed through processes administered by the proceedings Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing.

In this paper, the bound state solution of the modified radial Schrödinger equation is obtained for the Manning-Rosen plus Hulthén potential by implementing the novel improved scheme to surmount the centrifugal term. The energy eigenvalues and corresponding radial wave functions are defined for any l ≠ 0 angular momentum case via the Nikiforov-Uvarov (NU) and supersymmetric quantum mechanics (SUSYQM) methods. By using these two different methods, equivalent expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transformations to each other is demonstrated. The energy levels are worked out and the corresponding normalized eigenfunctions are represented in terms of the Jacobi polynomials for arbitrary l states. A closed form of the normalization constant of the wave functions is also found. It is shown that, the energy eigenvalues and eigenfunctions are sensitive to n_r radial and l orbital quantum numbers.

We present in this work, the calculations of corrections in the Newton's law of gravitation due to Kaluza-Klein gravitons in five-dimensional warped thick braneworld scenarios. We consider here a recently proposed model, namely, the hybrid Bloch brane. This model couples two scalar fields to gravity and is engendered from a domain wall-like defect. Also, two other models the so-called asymmetric hybrid brane and compact brane are considered. Such models are deformations of the ϕ⁴ and sine-Gordon topological defects, respectively. Therefore we consider the branes engendered by such defects and we also compute the corrections in their cases. In order to attain the mass spectrum and its corresponding eigenfunctions which are the essential quantities for computing the correction to the Newtonian potential, we develop a suitable numerical technique. The calculation of slight deviations in the gravitational potential may be used as a selection tool for braneworld scenarios matching with future experimental measurements in high energy collisions

Using elements of symmetry, we constructed the Noncommutative Schrödinger Equation from a representation of Exotic Galilei Group. As a consequence, we derive the Ehrenfest theorem using noncommutative coordinates. We also have showed others features of quantum mechanics in such a manifold. As an important result, we find out that a linear potential in the noncommutative Schrödinger equation is completely analogous to the ordinary case. We also worked with harmonic and anharmonic oscillators, giving corrections in the energy for each one.

A symmetry witness is a subset of the space of selfadjoint trace class operators that allows one to ascertain whether a linear map acting in that space is a symmetry transformation. This notion arises from a certain type of linear preserver problems. Precisely, a symmetry witness is a suitable set which is invariant with respect to an injective linear map in the Banach space of selfadjoint trace class operators where the quantum states live if and only if this map acts as a symmetry transformation. In particular, by a linear version of Wigner's classical theorem, the set of pure states — the rank-one projections — is a symmetry witness. Linearity entails that the usual assumption of preservation of the transition probability between pure states becomes superfluous. This result extends to every set of projections of a fixed (finite) rank, with some suitable constraint on this rank. One then obtains a classification of the sets of projections of a fixed rank that are symmetry witnesses. These symmetry witnesses are projectable. Namely, formulating the mentioned result in terms of quantum states, the sets of 'uniform' density operators of a suitable fixed rank are symmetry witnesses as well.

We present a novel hierarchical construction of projective spin networks of the Ponzano-Regge type from an assembling of five quadrangles up to the combinatorial 4-simplex compatible with a geometrical realization in Euclidean 4-space. The key ingredients are the projective Desargues configuration and the incidence structure given by its space-dual, on the one hand, and the Biedenharn–Elliott identity for the 6j symbol of SU(2), on the other. The interplay between projective-combinatorial and algebraic features relies on the recoupling theory of angular momenta, an approach to discrete quantum gravity models carried out successfully over the last few decades. The role of Regge symmetry–an intriguing discrete symmetry of the 6j which goes beyond the standard tetrahedral symmetry of this symbol–will be also discussed in brief to highlight its role in providing a natural regularization of projective spin networks that somehow mimics the standard regularization through a q-deformation of SU(2).

We introduce hom-associative Ore extensions as non-associative, non-unital Ore extensions with a hom-associative multiplication, as well as give some necessary and sufficient conditions when such exist. Within this framework, we also construct a family of hom-associative Weyl algebras as generalizations of the classical analogue, and prove that they are simple.

Inspired by a need of effective simulations of a system of Random Domino Automaton type defined for Bethe lattice, new variables are introduced. Main results obtained for 1-dimensional system — including a set of equations describing stationary state and relation to Motzkin numbers — are investigated in this new notion.

The aim of this paper is to study mixed-representation Green's functions from the point of view of coherent-state path integrals. This is achieved by using the machinery of generalized generating functionals of Green's functions, previously introduced by the present authors in the context of standard phase-pace path integrals. The obtained results are illustrated in the context of the linear harmonic oscillator.

We explore some consequences of a theory of internal symmetries for elementary particles constructed on exceptional quantum mechanical spaces based on Jordan algebra formulation that admit exceptional groups as gauge groups.

Let $\mathscr{P}$ be a Poisson structure on a finite-dimensional affine real manifold. Can $\mathscr{P}$ be deformed in such a way that it stays Poisson? The language of Kontsevich graphs provides a universal approach – with respect to all affine Poisson manifolds – to finding a class of solutions to this deformation problem. For that reasoning, several types of graphs are needed. In this paper we outline the algorithms to generate those graphs. The graphs that encode deformations are classified by the number of internal vertices k; for k ≤ 4 we present all solutions of the deformation problem. For k ≥ 5, first reproducing the pentagon-wheel picture suggested at k = 6 by Kontsevich and Willwacher, we construct the heptagon-wheel cocycle that yields a new unique solution without 2-loops and tadpoles at k = 8.

The properties of the intramolecular vibrational excitation (vibron) in a quasi 1D macromolecular structure are studied. It is supposed that due to the vibron interaction with optical phonon modes, a vibron might form partially dressed small polaron states. The properties of these states are investigated in dependence on the basic system parameters and temperature of a thermal bath. We also investigate the process of damping of the polaron amplitude as a function of temperature and vibron-phonon coupling strength. Two different regimes of the polaron damping are found and discussed.

We recall some basic aspects of line and line Complex representations, of symplectic symmetry emerging in bilinear point transformations as well as of Lie transfer of lines to spheres. Here, we identify SU(2) spin in terms of (classical) projective geometry and obtain spinorial representations from lines, i.e. we find a natural non-local geometrical description associated to spin. We discuss the construction of a Lagrangean in terms of line/Complex invariants. We discuss the edges of the fundamental tetrahedron which allows to associate the most real form SU(4) with its various related real forms covering SO(n,m), n + m = 6.

We study chiral trace relations in $\mathscr{N}$ = 2 supersymmetric theories. Applying localization formulae for chiral observables, we derive closed chiral trace relations relating the vacuum expectation values of chiral ring elements. In this setting, we discuss how the Ω-background breaks the polynomial nature of such relations. These results are interpreted in the light of AGT duality, thus making contact with the integrable structure of conformal field theories on Riemann surfaces.

A quantum cosmology in teleparallel gravity is presented in this article. Teleparallel gravity is used to perform such an analysis once in General Relativity (GR) the concept of gravitational energy is misleading preventing the establishment of a concise quantum cosmology. The Wheeler-DeWitt like equation is obtained using the Weyl quantization and the teleparallel expression of energy.

We test the background geometry of the BFSS model using a D4–brane probe. This proves a sensitive test of the geometry and we find excellent agreement with the D4–brane predictions based on the solution of a membrane corresponding to the D4–brane propagating on this background.

We consider the calculation schemes in the framework of Kantorovich method that consist in the reduction of a 3D elliptic boundary-value problem (BVP) to a set of second-order ordinary differential equations (ODEs) using the parametric basis functions. These functions are solution of the 2D parametric BVP. The coefficients in the ODEs are the parametric eigenvalues and the potential matrix elements expressed by the integrals of the eigenfunctions multiplied by their first derivatives with respect to the parameter. We calculate the parametric basis functions numerically in the general case using the high-accuracy finite element method. The efficiency of the proposed calculation schemes and algorithms is demonstrated by the example of the BVP describing the bound states of helium atom.

We present a method, based on integrability, to find minimal area surfaces in hyperbolic space ending on a given contour at the boundary. The problem has physical interest since the AdS/CFT correspondence relates the area of the minimal surface to the expectation value of the Wilson loop defined by the boundary contour. We give particular solutions using the Mathieu functions and more general results using a numerical method.

Superintegrable systems with monopole interactions in flat and curved spaces have attracted much attention. For example, models in spaces with a Taub-NUT metric are well-known to admit the Kepler-type symmetries and provide non-trivial generalizations of the usual Kepler problems. In this paper, we overview new families of superintegrable Kepler, MIC-harmonic oscillator and deformed Kepler systems interacting with Yang-Coulomb monopoles in the flat and curved Taub-NUT spaces. We present their higher-order, algebraically independent integrals of motion via the direct and constructive approaches which prove the superintegrability of the models. The integrals form symmetry polynomial algebras of the systems with structure constants involving Casimir operators of certain Lie algebras. Such algebraic approaches provide a deeper understanding to the degeneracies of the energy spectra and connection between wave functions and differential equations and geometry.

We present a bi-confluent Heun potential for the Schrödinger equation involving inverse fractional powers and a repulsive centrifugal-barrier term the strength of which is fixed to a constant. This is an infinite potential well defined on the positive half-axis. Each of the fundamental solutions for this conditionally integrable potential is written as an irreducible linear combination of two Hermite functions of a shifted and scaled argument. We present the general solution of the problem, derive the exact energy spectrum equation and construct a highly accurate approximation for the bound-state energy levels.

A rigorous algebraic definition of noncommutative coverings is developed. In the case of commutative algebras this definition is equivalent to the classical definition of topological coverings of locally compact spaces. The theory has following nontrivial applications:

• Coverings of continuous trace algebras,

• Coverings of noncommutative tori,

• Coverings of the quantum SU(2) group,

• Coverings of foliations,

• Coverings of isospectral deformations of Spin – manifolds.

The theory supplies the rigorous definition of noncommutative Wilson lines.

We give an overview of recent results on the classical and quantum superfield invariants of $\mathscr{N}$ = (1, 1), 6D supersymmetric Yang-Mills theory in the off-shell $\mathscr{N}$ = (1, 0) and on-shell $\mathscr{N}$ = (1, 1), 6D harmonic superspaces.

The operator algebra is introduced based on the framework of logarithmic representation of infinitesimal generators. In conclusion a set of generally-unbounded infinitesimal generators is characterized as a module over the Banach algebra.

We consider a constructive modification of quantum-mechanical formalism. Replacement of a general unitary group by unitary representations of finite groups makes it possible to reproduce quantum formalism without loss of its empirical content. Since any linear representation of a finite group can be implemented as a subrepresentation of a permutation representation, quantum-mechanical problems can be formulated in terms of permutation groups. Reproducing quantum behavior in the framework of permutation representations of finite groups makes it possible to clarify the meaning of a number of physical concepts.

In this paper, we review the progress in the analysis of magnetic monopoles as generalized states in quantum mechanics. We show that the considered model contains rich algebraic structure that generates symmetries which have been utilized in different physical contexts. Even though are we focused on quantum mechanics in noncommutative space $R_3^\lambda$, the results can be reconstructed in ordinary quantum mechanics in R³ as well.

We construct the systems of the harmonic and Pais-Uhlenbeck oscillators, which are invariant with respect to arbitrary noncompact Lie algebras. The equations of motion of these systems can be obtained with the help of the formalism of nonlinear realizations. We prove that it is always possible to choose time and the fields within this formalism in such a way that the equations of motion become linear and, therefore, reduce to ones of ordinary harmonic and Pais-Uhlenbeck oscillators. The first-order actions, that produce these equations, can also be provided. As particular examples of this construction, we discuss the so(2, 3) and G₂₍₂₎ algebras.

We extend the relation between the Witten-Dijkgraaf-Verlinde-Verlinde equation and N = 4 supersymmetric mechanics to arbitrary curved spaces. The resulting curved WDVV equation is written in terms of the third rank Codazzi tensor. We provide the solutions of the curved WDVV equation for the so(n) symmetric conformally flat metrics. We also explicitly demonstrate how each solution of the flat WDVV equation can be lifted up to the curved WDVV solution on the conformally flat spaces.

In [1, 2], it is proved that the elastic Neumann–Poincaré operator defined on the smooth boundary of a bounded domain, which is known to be non-compact, is in fact polynomially compact. As a consequence, it is shown that the spectrum of the elastic Neumann-Poincaré operator consists of non-empty sets of eigenvalues accumulating to certain numbers determined by Lamé parameters. The purpose of this paper is to review these results and their proofs, and to discuss about some questions related to these results.

The notion of equivalence of Lie algebra realizations is revisited and the quantities stable under the equivalence transformations are proposed. As a result we formulate a practical algorithm that allows to establish the existence of equivalence between any two realizations of a Lie algebra. Several illustrative examples are considered.

We review recent developments in the construction of curvature squared invariants in off-shell $\mathscr{N}$ = (1, 0) supergravity in six dimensions.

In this proceeding, we recall the notion of quantum integrable systems on a lattice and then introduce the Sklyanin's Separation of Variables method. We sum up the main results for the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazanov-Stroganov Lax operator. These results apply as well to the spectral analysis of the lattice sine-Gordon model with open boundary conditions. The transfer matrix spectrum (both eigenvalues and eigenstates) is completely characterized in terms of the set of solutions to a discrete system of polynomial equations. We state an equivalent characterization as the set of solutions to a Baxter's like T-Q functional equation, allowing us to rewrite the transfer matrix eigenstates in an algebraic Bethe ansatz form.

The so-called 2d/4d correspondences connect two-dimensional conformal field theory (2d CFT), $\mathscr{N}$ = 2 supersymmetric gauge theories and quantum integrable systems. The latter in the simplest case of the SU(2) gauge group are nothing but the quantum-mechanical systems. In the present article we summarize our recent results and list open problems concerning an application of the aforementioned dualities in the studies of spectral problems for some Schrödinger operators with Mathieu–type periodic, periodic PT–symmetric and (Heun's) elliptic potentials.

Quantum spaces with su(2) noncommutativity can be modelled by using a family of SO(3)-equivariant differential *-representations. The quantization maps are determined from the combination of the Wigner theorem for SU(2) with the polar decomposition of the quantized plane waves. A tracial star-product, equivalent to the Kontsevich product for the Poisson manifold dual to su(2) is obtained from a subfamily of differential *-representations. Noncommutative (scalar) field theories free from UV/IR mixing and whose commutative limit coincides with the usual ϕ⁴ theory on ³ are presented. A generalization of the construction to semi-simple possibly non simply connected Lie groups based on their central extensions by suitable abelian Lie groups is discussed.

In this paper we tersely recall the main algebraic and geometric properties of the maximally superintegrable system known as "Perlick System Tipe I", considering all possible values of the relevant parameters. We will follow a classical variant of the so called factorization method, emphasizing the role played the Poisson Algebra of the constants of motion in sheding light on the geometric features of the trajectories.

In 1971, S. Smale presented a generalization of Pareto optimum he called the critical Pareto set. The underlying motivation was to extend Morse theory to several functions, i.e. to find a Morse theory for m differentiable functions defined on a manifold M of dimension ℓ. We use this framework to take a 2 × 2 Hamiltonian = (p) ∈ 2 C^∞(T^*R²) to its normal form near a singular point of the Fresnel surface. Namely we say that has the Pareto property if it decomposes, locally, up to a conjugation with regular matrices, as (p) = u^'(p)C(p)(u^'(p))^*, where u : R² → R² has singularities of codimension 1 or 2, and C(p) is a regular Hermitian matrix ("integrating factor"). In particular this applies in certain cases to the matrix Hamiltonian of Elasticity theory and its (relative) perturbations of order 3 in momentum at the origin.

In this work, we present Lax pair for two-dimensional complex modified Korteweg-de Vries and Maxwell-Bloch (cmKdV-MB) system with the time-dependent coefficient. Dark and bright soliton solutions for the cmKdV-MB system with variable coefficient are received by Darboux transformation. Moreover, the determinant representation of the one-fold and two-fold Darboux transformation for the cmKdV-MB system with time-dependent coefficient is presented.

The amplitude for the singlet channels in the 4-point function of the fundamental field in the conformal field theory of the 2d O(n) model is studied as a function of n. For a generic value of n, the 4-point function has infinitely many amplitudes, whose landscape can be very spiky as the higher amplitude changes its sign many times at the simple poles, which generalize the unique pole of the energy operator amplitude at n = 0. In the stadard parameterization of n by angle in unit of π, we find that the zeros and poles happen at the rational angles, forming a hierarchical tree structure inherent in the Poincaré disk. Some relation between the amplitude and the Farey path, a piecewise geodesic that visits these zeros and poles, is suggested. In this hierarchy, the symmetry of the congruence subgroup Γ(2) of SL(2, ) naturally arises from the two clearly distinct even/odd classes of the rational angles, in which one respectively gets the truncated operator algebras and the logarithmic 4-point functions.

Using binary coding of orbit we introduce a finite level (N) surface over the initial value domain D of 2-dim AKP. It gives a tiling of D by base ribbons. The scheme of the one-time map is studied and the properness of the tiling is proved. This analysis in turn resolves the long standing puzzle in AKP—the non-uniqueness issue of a PO for a given code. We argue that the unique existence of a periodic orbit (PO) for a given binary code generally holds (for inverse anisotropy parameter γ < 8/9) but there is a remarkable exception in which a ribbon with a certain code escapes from shrinking at large N and embodies the Broucke-type stable PO (S). It comes along the bifurcation of an unstable PO (U): U(R) → S(R) + U^'(NR) (R for retracing and NR for non-retracing). An analysis based on orbit topology clarifies the pattern of the bifurcation; we give a conjecture that it occurs among odd rank Y-symmetric POs.

The method of construction of classical integrable systems on quiver varieties is considered in the case of cyclic quiver. For two types of framing we obtained integrable spin generalisations of the Calogero–Moser systems associated with the complex reflection groups S_n ⋉ (/m)ⁿ. This method gives Lax matrices of the systems in explicit form. In some particular cases the first Hamiltonians of the system are written explicitly. The paper is based on the work of the author with Oleg Chalykh (the University of Leeds).

We investigate quantum corrections in $\mathscr{N}$ = 1 non-Abelian supersymmetric gauge theories, regularized by higher covariant derivatives. In particular, by the help of the Slavnov–Taylor identities we prove that the vertices with two ghost legs and one leg of the quantum gauge superfield are finite in all orders. This non-renormalization theorem is confirmed by an explicit one-loop calculation. By the help of this theorem we rewrite the exact NSVZ β-function in the form of the relation between the β-function and the anomalous dimensions of the matter superfields, of the quantum gauge superfield, and of the Faddeev–Popov ghosts. Such a relation has simple qualitative interpretation and allows suggesting a prescription producing the NSVZ scheme in all loops for the theories regularized by higher derivatives. This prescription is verified by the explicit three-loop calculation for the terms quartic in the Yukawa couplings.

Main result is the multiplicative formula for universal R-matrix for Quantum Double of Yangian of strange Lie superalgebra Q_n type. We introduce the Quantum Double of the Yangian of the strange Lie superalgebra Q_n and define its PBW basis. We compute the Hopf pairing for the generators of the Yangian Double. From the Hopf pairing formulas we derive a factorized multiplicative formula for the universal R-matrix of the Yangian Double of the Lie superalgebra Q_n. After them we obtain coefficients in this multiplicative formula for universal R-matrix.

We examine certain nonassociative deformations of quantum mechanics and gravity in three dimensions related to the dynamics of electrons in uniform distributions of magnetic charge. We describe a quantitative framework for nonassociative quantum mechanics in this setting, which exhibits new effects compared to ordinary quantum mechanics with sourceless magnetic fields, and the extent to which these theoretical consequences may be experimentally testable. We relate this theory to noncommutative Jordanian quantum mechanics, and show that its underlying algebra can be obtained as a contraction of the alternative algebra of octonions. The uncontracted octonion algebra conjecturally describes a nonassociative deformation of three-dimensional quantum gravity induced by magnetic monopoles, which we propose is realised by a non-geometric Kaluza-Klein monopole background in M-theory.

The methods of spectral geometry are useful for investigating the metric aspects of noncommutative geometry and in these contexts require extensive use of pseudo-differential operators. In a foundational paper, Connes showed that, by direct analogy with the theory of pseudo-differential operators on finite-dimensional real vector spaces, one may derive a similar pseudo-differential calculus on noncommutative n-tori, and with the development of this calculus came many results concerning the local differential geometry of noncommutative tori for n=2,4, as shown in the groundbreaking paper in which the Gauss–Bonnet theorem on the noncommutative two-torus is proved and later papers. Certain details of the proofs in the original derivation of the calculus were omitted, such as the evaluation of oscillatory integrals, so we make it the objective of this paper to fill in all the details. After reproving in more detail the formula for the symbol of the adjoint of a pseudo-differential operator and the formula for the symbol of a product of two pseudo-differential operators, we extend these results to finitely generated projective right modules over the noncommutative n-torus. Then we define the corresponding analog of Sobolev spaces and prove equivalents of the Sobolev and Rellich lemmas.

Based on first principles solutions in a unified framework of quantum mechanics and electromagnetism we predict the presence of a universal attractive depolarisation radiation (DR) Lorentz force (F) between quantum entities, each being either an IED matter particle or light quantum, in a polarisable dielectric vacuum. Given two quantum entities i = 1, 2 of either kind, of characteristic frequencies $\nu _i^0$, masses $m_i^0 = h\nu _i^0/{c^2}$ and separated at a distance r⁰, the solution for F is $F = - {\cal G}m_1^0m_2^0/{\left( {{r^2}} \right)^2}$, where ${\cal G} = \chi _0^2{e^4}/12{\pi ^2} \in _0^2{\rho _\lambda };{\chi _0}$ is the susceptibility and π_λ is the reduced linear mass density of the vacuum. This force F resembles in all respects Newton's gravity and is accurate at the weak F limit; hence ℊ equals the gravitational constant G. The DR wave fields and hence the gravity are each propagated in the dielectric vacuum at the speed of light c; these can not be shielded by matter. A test particle µ of mass m⁰ therefore interacts gravitationally with all of the building particles of a given large mass M at r⁰ apart, by a total gravitational force F = −GMm⁰/(r⁰)² and potential V = −∂F/∂r⁰. For a finite V and hence a total Hamiltonian H = m⁰c² + V, solution for the eigenvalue equation of µ presents a red-shift in the eigen frequency ν = ν⁰(1 − GM/r⁰c²) and hence in other wave variables. The quantum solutions combined with the wave nature of the gravity further lead to dilated gravito optical distance r = r⁰/(1 − GM/r⁰c²) and time t = t⁰/(1 − GM/r⁰c²), and modified Newton's gravity and Einstein's mass energy relation. Applications of these give predictions of the general relativistic effects manifested in the four classical test experiments of Einstein's general relativity (GR), in direct agreement with the experiments and the predictions given based on GR.

We will give a short reminder for vertex operator algebra notion and corresponding characters. Then we discuss algebraic methods for explicit computation of the partition and correlation functions. We then illustrate general ways to find number theory identities for related modular forms by specific examples of modular form relations arising from our construction. Finally, we present new results concerning identities for prime forms on genus g Riemann surfaces and genus two n-point functions for vertex operator algebra characters.

We present the component structure of the superconformal gravity invariants in six dimensions, which was recently elaborated in arXiv:1701.08163.