Table of contents

This volume contains contributions to the XII. International Symposium on Quantum Theory and Symmetries (QTS12) was at Czech Technical University in Prague, Czech Republic, from Monday July 24 until Friday July 28, 2023.

The Symposium series "Quantum Theory and Symmetries" (QTS) is a biannual meeting intended for physicists and mathematicians who are either applying symmetries to some physical model, or are studying mathematical objects that are relevant for physical applications.

The first Symposium of this series was held in Goslar (Germany) in 1999, QTS-1, then it was held in Cracow (2001) QTS-2, Cincinnati (2003) QTS-3, Varna (2005) QTS-4, Valladolid (2007) QTS-5, Lexington (2009) QTS-6, Prague (2011) QTS-7, Mexico-city (2013) QTS-8, Yerevan (2015) QTS-9, Varna (2017) QTS-10, Montreal (2019) QTS-11.

The Symposium QTS-12 was first planned to be in Moscow at the Moscow Institute of Physics and Technology in 2021, but was postponed by one year due to Covid-19, then it was postponed to 2023, July 24-28, and was held in Prague at the Czech Technical University (Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, CTU in Prague) QTS-12.

The Symposium QTS-13 is planned for the 2nd half of July 2025 to be held in Yerevan, Armenia.

The volume is organised as follows: it starts with notes in memory of H. D. Doebner. Then the invited talks at the plenary sessions and the public lecture are published followed by contributions in the parallel and poster sessions in alphabetical order.

List of Editors, Advisory committee are available in this Pdf.

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

• Type of peer review: Single Anonymous

• Conference submission management system: Morressier

• Number of submissions received: 88

• Number of submissions sent for review: 86

• Number of submissions accepted: 85

• Acceptance Rate (Submissions Accepted / Submissions Received × 100): 96.6

• Average number of reviews per paper: 1.0357142857142858

• Total number of reviewers involved: 47

• Contact person for queries:

Name: Burdík

Email: cestmir.burdik@fjfi.cvut.cz

Affiliation: Ceske vysoke uceni technicke v Praze - Ceske vysoke uceni technicke v Praze Fakulta jaderna a fyzikalne inzenyrska

In 1980 Hermann Nicolai proposed a characterization of supersymmetric theories that became known as the Nicolai map. This is a particular nonlocal and nonlinear field transformation, whose perturbative expansion is given by fermion-line trees with bosonic leaves. Quantum correlation functions can by evaluated using the inversely transformed fields in the free theory. After initial promise and excitement (fuelling the author's PhD work!), the subject all but fell dormant for 35 years. Recently however, technical progress in the construction as well as a deeper insight into the nature of the map have been achieved, from quantum mechanics to super Yang–Mills in various dimensions. I will present the Nicolai map from this modern perspective and touch on some of the current developments.

A short introduction, with extended interpretations, to the Projection Evolution approach is presented. In this model time is like the other space positions represented by an operator in the state space of a physical system. As a pedagogigal example the Minkowski spacetime is considered. Finally, a short analysis of the time reversal symmetry is presented.

In this article we show that a Berezin-type quantization can be achieved on a compact even dimensional manifold M^2d by removing a skeleton M₀ of lower dimension such that what remains is diffeomorphic to R^2d (cell decomposition) which we identify with C^d and embed in CP^d. A local Poisson structure and Berezin-type quantization are induced from CP^d. Thus we have a Hilbert space with a reproducing kernel. The symbols of bounded linear operators on the Hilbert space have a star product which satisfies the correspondence principle outside a set of measure zero. This construction depends on the diffeomorphism. One needs to keep track of the global holonomy and hence the cell decomposition of the manifold. As an example, we illustrate this type of quanitzation of the torus. We exhibit Berezin-Toeplitz quantization of a complex manifold in the same spirit as above.

We report on some new algebras we discovered that does not appear in mathematics or physics literature which we briefly name "R" algebras. Together with the real forms we developed for supersting theories shown below and dobbed as "l" algebras, both these new algebras seem to imitate split-octonionic multiplication rules we expose below. We further elaborate on Jordan algebras, magic squares, seven spheres, and supergravity, pointing to the existence of a larger theory with a 27-dimensional lattice and unifying all known superstring theories.

In the present paper we continue the project of systematic explicit construction of invariant differential operators. On the example of the non-compact group SO(p, 9 − p) (for p = 5, 6) we give the classification multiplets of indecomposable elementary representations induced from a new (relative to earlier considerations) choice of parabolic subgroup MAN so that the factor-group G/M is an Einstein manifold. This classification includes the data for the relevant invariant differential operators.

When rewriting the photon vertex of quantum electrodynamics in terms of geometrical quantities, various elements can be mapped directly to objects and properties known from classical projective geometry (PG). Elements of P⁵ when mapped to line reps in P³ exhibit their intrinsic Lorentz invariance associated to automorphisms of the Plücker-Klein quadric ${M}_{4}^{2}$, and line reps when expressed by point or plane coordinates introduce (one-parameter) pencils, or formally gl(2,), or gl(1,) which covers su(2)⊕u(1). This introduces binary forms and, using a potential approach of central forces, Schrödinger or Laplace equations and the respective special functions, as well as the projective generation of quadrics like in Dirac's approach which legitimates Clifford algebra elements as linear factors in invariant theory and the quadratic algebra to represent geometry. Physically, this identification allows for the classical concept of moments in terms of tetrahedrons which on the one hand relates to previous work on SU(4) and SU*(4) in quantum representations. On the other hand, it relates to the classical physical definitions, however, exhibiting a factor 2 between contemporary (euclidean) moments and the tetrahedral construction used in the vertex. Finally, we discuss the equilibrium conditions with respect to gauge and Yang-Mills theories in general as well as the related objects and their transformation theory.

We show that the geometric notion of duality behind T-duality, between two string theories on different manifolds E, Ê in the sense of [3][4], is precisely that of Lie bialgebroids due to Mackenzie and Xu [11].

For the single mode the Hermite coherent states are wave packets analogous to the standard harmonic oscillator coherent states but constructed with complex Hermite polynomials substituting traditionally used complex monomials zⁿ. Hermite coherent states unify the standard coherent and squeezed states within one class of quantum states whose representatives share all properties customarily required from the standard coherent states, in particular (over)completeness. Generalization of the Hermite coherent/squeezed states to the multimode case, illustrated on the bipartite states, shows that such states exhibit entanglement which presence or absence depends on the nature of squeezing but also on the choice of description method being used.

The Grothendieck theorem considers a 'classical' quadratic form ${\mathcal{C}}$ that uses complex scalars in the unit disc, and the corresponding 'quantum' quadratic form ${\mathcal{Q}}$ that replaces the scalars with vectors in the unit ball of a Hilbert space. It shows that when ${\mathcal{C}}\le 1$ then ${\mathcal{Q}}$ might take values greater than 1, up to the complex Grothendieck constant k_G. Previous work in a quantum context, used Grothendieck's theorem with multipartite entangled systems, in contrast to the present work which uses it for a single quantum system. The emphasis in the paper is in examples with ${\mathcal{Q}}\in (1,{k}_{G})$, which is a classically forbidden region in the sense that ${\mathcal{C}}$ cannot take values in it.

Some solitary vortices to 2+1 quasi-integrable systems are discussed in the context of the planetary atmosphere. The Williams-Yamagata-Flierl (WYF) equation is one of the best candidates for the great red spot. We calculate the long-term simulation of the equation and find that the stable vortex is supported by a background zonal flow of a certain strength. The Zakharov-Kuznetsov (ZK) equation is a mimic of the WYF equation and considerably owes a great deal of its stability to the vortex. To learn more about the origin of longevity, we investigate the Painlevé test of the static ZK equation.

In this short article, we survey the importance of Villadsen algebras in the context of the classification theory, with an emphasis on its non-commutative aspects.

We investigate the T-duality relations between hyperkähler and bi-hypercomplex structures using the doubled formalism. In generalized geometry, both the hyperkähler and bi-hypercomplex structures are embedded in generalized hyperkähler structures that satisfy the split-bi-quaternion algebra. We write down the analogue of the Buscher rule, which is the T-duality transformation of the hyperkähler and bi-hypercomplex structures. As a practical example, we construct the bi-hypercomplex structure of the ${5}_{2}^{2}$-brane, known as a T-fold, from the hyperkähler structure of the Taub-NUT space using the T-duality transformation. The bi-hypercomplex structures of the T-fold have non-trivial monodromies. This results in the fact that the worldsheet instantons on the T-fold are multi-valued. We comment on the resolution of this issue using the Born sigma model.

To give a firm base to argument on quantization of an LC circuit, we derive a commutation relation of electric flux and magnetic flux from quantum electrodynamics. The electric flux is defined by integration of electric field on two-dimensional surface. The magnetic flux is defined by integration of magnetic field surrounded by one-dimensional loop. It is proved that the commutator of the electric flux and the magnetic flux is equal to a crossing number of the loop and the surface. It is also proved that the flux commutator is invariant under gauge transformations and homological deformations. It is also argued that the LC-circuit system can be used as a platform for realizing EPR entanglement.

In a recent paper (Balbino-de Freitas-Rana-FT, arXiv:2309.00965) we proved that the supercharges of the supersymmetric quantum mechanics can be statistically transmuted and accommodated into a ${Z}_{2}^{n}$-graded parastatistics. In this talk I derive the 6 = 1 + 2 + 3 transmuted spectrum-generating algebras (whose respective ${Z}_{2}^{n}$ gradings are n = 0, 1, 2) of the = 2 Superconformal Quantum Mechanics. These spectrum-generating algebras allow to compute, in the corresponding multiparticle sectors of the de Alfaro-Fubini-Furlan deformed oscillator, the degeneracies of each energy level. The levels induced by the Z₂ × Z₂-graded paraparticles cannot be reproduced by the ordinary bosons/fermions statistics. This implies the theoretical detectability of the Z₂ × Z₂-graded parastatistics.

Double field theory (DFT) is an effective theory of string theory. It has a manifest symmetry of T-duality. The gauge symmetry in DFT is related to some kind of algebroid structures, and they have a doubled structure. We focus on the gauge algebra of the O(D, D+n) gauged DFT and discuss an extension of the doubled structure. The gauge algebra of the O(D, D + n) gauged DFT has been described by the F-bracket. This bracket is related to some algebroid structures. We show that algebroids defined by the twisted C-bracket in the gauged DFT are built out of a direct sum of three Lie algebroids. They exhibit a "triple", which we call the extended double, rather than a "double" structure. We also consider the geometrical realization of these structures in a (2D + n)-dimensional manifold.

We use a recent formalism of quantum geodesics in noncommutative geometry to construct geodesic flow on the infinite chain · · · •–•–• · · ·. We find that noncommutative effects due to the discretisation of the line generically cause an initially real geodesic flow amplitude ψ (for which the density is |ψ|²) to become complex. This has been noted also for other quantum geometries and suggests that the complex nature of the wave function in quantum mechanics (and the interference effects that follow) may have its origin in a quantum/discrete nature of spacetime at the Planck scale.

In this paper we construct physical systems with ₂×₂-graded symmetries. There are two different structures:₂×₂-graded Lie algebras and₂×₂-graded Lie superalgebras. Physical models with the latter symmetries can be seen as generalizations of supersymmetric models. The systems described by ₂ × ₂-graded Lie algebras have not been investigated yet (up to our knowledge). We present examples of physical models invariant under each one of both kinds of graded structures.

Motivated by some recent progress involving a non-compact gauge group, we obtain classical gauge fields using non-compact foliations of anti-de Sitter space in 4 dimensions (required dimensionality for conformal invariance of Yang–Mills theory) and transfer these to Minkowski spacetime using a series of conformal maps. This construction also builds upon some previous works involving SU(2) gauge group in that we now use its non-compact counterpart SU(1, 1) here. We note down gauge fields in both Abelian as well as non-Abelian settings and find them to be divergent at some hyperboloid, which is a hypersurface of co-dimension 1 inside the conformal boundary of AdS₄. In spite of this hurdle we find a physically relevant field configuration in the Abelian case, reproducing a known result.

Higher category theory can be employed to generalize the notion of a gauge group to the notion of a gauge n-group. This novel algebraic structure is designed to generalize notions of connection, parallel transport and holonomy from curves to manifolds of dimension higher than one. Thus it generalizes the concept of gauge symmetry, giving rise to a topological action called nBF action, living on a corresponding n-principal bundle over a spacetime manifold. Similarly as for the Plebanski action, one can deform the topological nBF action by adding appropriate simplicity constraints, in order to describe the correct dynamics of both gravity and matter fields. Specifically, one can describe the whole Standard Model coupled to gravity as a constrained 3BF or 4BF action. The split of the full action into a topological sector and simplicity constraints sector is adapted to the spinfoam quantization technique, with the aim to construct a full model of quantum gravity with matter. In addition, the properties of the gauge n-group structure open up a possibility of a nontrivial unification of all fields. An n-group naturally contains additional novel gauge groups which specify the spectrum of matter fields present in the theory, in a similar way to the ordinary gauge group that prescribes the spectrum of gauge vector bosons in the Yang-Mills theory. The presence and the properties of these new gauge groups has the potential to explain fermion families, and other structure in the matter spectrum of the theory.

Lie algebras of first-order differential operators with smooth coefficients are considered and their construction methods are reviewed. Obstacles preventing contractions of such operators are emphasized and a constructive procedure for parameterizations is proposed. A number of illustrative examples are considered.

The objective of this work is to give expressions for solutions of universal differential equation and finally, as application, provides the unique grouplike solution, by dévissage, of Knizhnik-Zamolodchikov equation satisfying asymptotic conditions, i.e. solution of KZ_n can be obtained using solution of KZ_n−1 and the noncommutative generating series of hyperlogarithms. These expressions are obtained using the algebraic combinatorics on noncommutative formal series with holomorphic coefficients and a Picard-Vessiot theory of noncommutative differential equations.

The Bethe ansatz is an analytical method to solve exactly solvable models in quantum mechanics. It has been shown that the states of the Bethe ansatz can be prepared by a deterministic quantum circuit whose quantum gates were determined numerically. We report our progress in recasting the Bethe ansatz as a deterministic quantum circuit. We present the analytical expressions of the quantum gates. Formulae rely upon diagrammatic rules that define the wave functions of the Bethe ansatz by matrix product states. Based on the analytical expressions, we prove the unitarity of the quantum gates. We use our results to clarify on the equivalence between the coordinate and algebraic Bethe ansatze in light of matrix-product states.

The non-perturbative Landau-Khalatnikov-Fradkin (LKF) transformations describe how Green functions in quantum field theory transform under a change in the photon field's linear covariant gauge parameter (denoted ξ). The transformations are framed most simply in coordinate space where they are multiplicative. They imply that information on gauge-dependent contributions from higher order diagrams in the perturbative series is contained in lower order contributions, which is useful in multi-loop calculations. We study the LKF transformations for the propagator and the vertex in both scalar and spinor QED, in some particular dimensions. A novelty of our work is to derive momentum-space integral representations of these transformations; our expressions are also applicable to the longitudinal and transverse parts of the vertex. Applying these transformations to the tree-level Green functions, we show that the one-loop terms obtained from the LKF transformation agree with the gauge dependent parts obtained from perturbation theory. Our results will be presented in more comprehensive form elsewhere.

Spectral flow is a fascinating behavior of fermion's energy levels that cross zero as a parameter varies. In fermion-soliton system, it owes the topology of the soliton. We examine the spectral flow in a (2 + 1)-dimensional P² fermionic sigma model. We employ an axisymmetric instanton solution of the P² non-linear sigma model as a background bosonic field to the Dirac fermion. The explicit form of the solutions contains two parameters concerning the size. We show that some energy levels flow from positive (negative) to negative (positive) as the parameters vary. We propose that the behavior of the energy levels can easily be understood by carefully examining the topological property of the instanton in the entire range of parameters, including the limit of zero and infinity.

Recently, a mathematical method was developed to analyze the kinetics of the protein folding process, considering it as a diffusion process described by the Fokker-Planck equation and its solution, the time-dependent probability density. This work presents the main points of the methodology, based on the application of the algebraic formalism of Supersymmetric Quantum Mechanics associated with the variational method, to analyze the symmetric tri-stable free energy potential function that describes the unfolded and folded states, as well as an intermediate state of the protein.

A multi-component P^N model's scalar electrodynamics is investigated. The model contains Q-balls/shells, which are non-topological compact solitons with time dependency e^iωt. Because of the compacton nature of solutions, Q-shells with another compact Q-ball or Q-shell inside their cavity can exist. Even if compactons do not overlap, they can interact with one another via the electromagnetic field. They look similar to the capacitor in the standard electromagnetism. We focus on the structure of such Q-capacitor with opposite charges.

The gauge-Miura correspondence establishes a map between the entire KdV and mKdV hierarchies, including positive and also negative flows, from which new relations besides the standard Miura transformation arise. We use this correspondence to classify solutions of the KdV hierarchy in terms of elementary tau functions of the mKdV hierarchy under both zero and nonzero vacua. We illustrate how interesting nonlinear phenomena can be described analytically from this construction, such as "rogue waves" of a complex KdV system that corresponds to a limit of a vector nonlinear Schrödinger equation.

Burge multipartitions are tuples of partitions that satisfy a cyclic embedding condition. When uncoloured, Burge m-multipartitions give a combinatorial model for characters of the W_m algebras (W₂ is the Virasoro algebra). When n-coloured, they generalise the "cylindric partitions" that provide a combinatorial model (and a crystal graph) for integrable characters of the affine Lie algebra $\hat{{\mathfrak{s}}{\mathfrak{l}}}(n)$.

Here, we show that the n-coloured Burge m-multipartitions yield the characters of the CFT cosets $\hat{{\mathfrak{g}}{\mathfrak{l}}}(d{)}_{m}/\hat{{\mathfrak{g}}{\mathfrak{l}}}(d-n{)}_{m}$. Having previously shown that the same combinatorial objects give the SU(m) instanton partition functions in = 2 supersymmetric gauge theories on ²/_n, we have thus established a wide-ranging extension of the AGT correspondence.

This talk is based on a collaboration with N.Macleod (Melbourne).

Integrable spin systems with potential are of great interest from the point of view of theoretical and applied physics. They make it possible to obtain accurate analytical solutions and study the properties of solitons - nonlinear wave structures that can be stable and move without distortion. The study of solitons in spin systems is of great importance not only for developing new methods for transmitting and processing information but also for developing spintronics and magnetoelectronics in general. These areas of technology are based on the use of the properties and control of the spin moment of electrons. Understanding and controlling spin dynamics in various systems opens up new possibilities for creating more efficient and powerful devices such as magnetic memories, spintronic transistors, and logic elements. This paper considers occurence of soliton in magnetic medium described by Generalized Landau-Lifshitz equation with self-consistent potential.

Ultracold gases provide an excellent platform for the realization of quantum interferometers. In the case of implementations based on Bose-Einstein condensates in double well potentials, an effective two-mode model allows to study how the interactions among particles affect the sensitivity of the interferometer. In this work we review the properties of such a model and its application to interferometric protocols, focusing on the achievable sensitivity in the presence of interactions turned on. In particular we study the full interferometric sequence when the initial state is a Twin Fock state, which is perfectly number squeezed. We found that in the presence of interactions and for certain values of the holding time in which a phase difference between the two modes is accumulated, the same sensitivity as in the non interacting case is recovered when using the population imbalance between the two wells as observable. Finally, we characterize the behaviour of the sensitivity by looking at the δ-derivative and the variance of the operator used for the measurement and studying the squeezing parameters.

Black holes in anti de Sitter spacetimes undergo phase transitions which typically lead to the existence of critical points, that can be classified using topological techniques. Availing the Bragg-Williams construction of an off-shell free energy we compute the topological charge of the Hawking-Page (HP) transition for Einstein-Born-Infeld black holes in anti de Sitter (AdS) spacetime and match the result with the confinement-deconfinement transition in the dual gauge theory, which turn out to be in perfect agreement.

We consider Bogoliubov-de Gennes equation on a metric tree graph. Formulation of the problem for arbitrary graph topology is provided. Self-adjoint vertex boundary conditions are derived. Exact solutions of the problem is obtained for quantum tree graph. A quantum graph based model for tree-branched Majorana wire network is proposed.

The aim of this contribution is twofold. First, we show that when two (or more) different quantum groups share the same noncommutative spacetime, such an 'ambiguity' can be resolved by considering together their corresponding noncommutative spaces of geodesics. In any case, the latter play a mathematical/physical role by themselves and, in some cases, they can be interpreted as deformed phase spaces. Second, we explicitly show that noncommutative spacetimes can be reproduced from 'extended' noncommutative spaces of geodesics which are those enlarged by the time translation generator. These general ideas are described in detail for the κ-Poincaré and κ-Galilei algebras.

Some measures of genuine multipartite entanglement in the three-qubit XYZ Heisenberg model are reported for a system that is embedded in a uniform magnetic field. A quantitative characterization of these measures is accomplished to determine the optimal parameters that provide maximal entanglement.

This is a brief survey of some recent results on infinitesimal symmetries of bundle gerbes and their relation to three Lie 2-algebras associated to manifolds equipped with a closed 3-form, such as 2-plectic manifolds: the Poisson Lie 2-algebra of local observables, the Lie 2-algebra of sections of an associated exact Courant algebroid, and the Atiyah Lie 2-algebra.

In the conventional Schrödinger's formulation of quantum mechanics the unitary evolution of a state ψ is controlled, in Hilbert space ${\mathcal{L}}$, by a Hamiltonian ɧ which must be self-adjoint. In the recent, "quasi-Hermitian" reformulation of the theory one replaces ɧ by its isospectral but non-Hermitian avatar H = Ω⁻¹ 𝖍Ω with Ω^†Ω = Θ ≠ I. Although acting in another, manifestly unphysical Hilbert space ${\mathcal{H}}$, the amended Hamiltonian H ≠ H^† can be perceived as self-adjoint with respect to the amended inner-product metric Θ. In our paper motivated by a generic technical "user-unfriendliness" of the non-Hermiticity of H we introduce and describe a specific new family of Hamiltonians H for which the metrics Θ become available in closed form.

We review some aspects of the Racah algebra R(n), including the closure relations, pointing out their role in superintegrability, as well as in the description of the symmetry algebra for several models with coalgebra symmetry. The connection includes the generic model on the (n − 1) sphere. We discuss an algebraic scheme of constructing Hamiltonians, integrals of the motion and symmetry algebras. This scheme reduces to the Racah algebra R(n) and the model on the (n − 1) sphere only for the case of specific differential operator realizations. We review the method, which allows us to obtain the commutant defined in the enveloping algebra of (n) in the classical setting. The related ₃ polynomial algebra is presented for the case (3). An explicit construction of the quantization of the scheme for ₃ by symmetrization of the polynomial and the replacement of the Berezin bracket by commutator and symmetrization of the polynomial relations is presented. We obtain the additional quantum terms. These explicit relations are of interest not only for superintegrability, but also for other applications in mathematical physics.

In this paper, we consider the interaction of electrons in tilted anisotropic Dirac materials with external electric and magnetic fields with translational symmetry, for a specific non-null electric field amplitude that allows us to decouple the differential equation system arising in the eigenvalue problem. Then, the eigenstates and eigenvalues for the corresponding Hamiltonian operator can be obtained by means of supersymmetric quantum mechanics. In order to use a semi-classical approach to analyze this system, we find a family of coherent states. Finally, the properties of these states are analyzed through fidelity and the Wigner function.

The interaction between massless Dirac fermions in anisotropic Dirac materials and position-dependent electric and magnetic fields is discussed here. The effect of the electric field strength in the energy spectrum and probability density of the eigenstates of the system is analyzed. Results show that the Landau levels are dispersive and the external electric field induces a collapse on them.

The Jackiw-Pi equation, which is one of the integrable vortex equations, is studied on a torus, a compact Riemann surface of genus one. The solutions are given in terms of doubly periodic functions, i.e., the elliptic functions. We reconsider the Jackiw-Pi vortex on a torus and provide the analytical method for determining the vortex number with explicit examples.

In this study, we work on a novel Hamiltonian system which is Liouville integrable. In the integrable Hamiltonian model, conserved currents can be represented as Binomial polynomials in which each order corresponds to the integral of motion of the system. From a mathematical point of view, the equations of motion can be written as integrable second-order nonlinear partial differential equations in 1 + 1 dimensions.

We review the notion of the reduction cohomology of vertex algebras. The algebraic conditions leading to the chain property for complexes of vertex operator algebra n-point functions (with their convergence assumed) with a coboundary operator defined through reduction formulas are studied. Algebraic, geometrical, and cohomological meanings of reduction formulas and chain condition are clarified. The reduction cohomology for vertex operator algebras associated to Jacobi forms is computed. A counterpart of the Bott-Segal theorem for Riemann surfaces in terms of the reductions cohomology is proven.

There are analyzed two physically reasonable generalizations of the Kardar-Parisi-Zhang equation describing the spin glasses growth models and possessing important from physical point of view properties. The first one proved to be a completely integrable Hamiltonian dynamical system with an infinite hierarchy of commuting to each other conservation laws, and the second one proved to be linearized modulo some nonlinear constraints, imposed on its solutions.

We consider Yangians of gℓ(n) type, the L operators of Jordan-Schwinger form and corresponding Yang-Baxter relations. We discuss explicit forms of R operators and applications to Yangian symmetric correlators and operators of representation parameter permutations.

The polyhedra with A₃, B₃/C₃, H₃ reflection symmetry group G in the real 3D space are considered. The recursive rules for finding orbits with smaller radii, which provide the structures of nested polytopes, are demonstrated.

We review three different approaches to polynomial symmetry algebras underlying superintegrable systems in Darboux spaces. The first method consists of using deformed oscillator algebra to obtain finite-dimensional representations of quadratic algebras. This allow one to gain information on the spectrum of the superintegrable systems. The second method has similarities with the induced module construction approach in the context of Lie algebras and can be used to construct infinite dimensional representations of the symmetry algebras. Explicit construction of these representations is a non-trivial task due to the non-linearity of the polynomial algebras. This method allows the construction of states of the superintegrable systems beyond the reach of separation of variables. As a result, we are able to construct a large number of states in terms of Airy, Bessel and Whittaker functions which would be difficult to obtain in other ways. We also discuss the third approach which is based on the notion of commutants of subalgebras in the enveloping algebra of a Poisson algebra or a Lie algebra. This allows us to discover new superintegrable models in the Darboux spaces and to construct their integrals and symmetry algebras via polynomials in the enveloping algebras.

We discuss a class of effective extensions of the SU(2) Georgi–Glashow model and discuss its Bogomol'nyi–Prasad–Sommerfield (BPS) limit. We identify a specific subclass of these models that admit analytical solutions of the monopole type. We present some concrete examples and find that the resulting monopoles tend to have their energy concentrated not in their center, but rather in a spherical shell around it.

A notion of Quantum Motion Algebra (QMA) allows to construct quantum state spaces for various physical systems moving under a given group of motion. The main idea of QMA is a construction of a group algebra with involution generated by a group of motion G. After defining a linear and nonnegative functional on this algebra one can construct the appropriate quantum state space by means of the Gelfand-Naimark-Segal theorem. The QMA method can be also applied in the modeling of physical systems requiring additional degrees of freedom or additional constraints.

The presented paper gives a brief description of the QMA method. As an example of the QMA application, we present a model of nuclear collective pairing where the nonnegative functional is generated by a temperature dependent quantum density operator.

We introduce and study a model for the movement of surfaces, namely the conserved, restricted solid-on-solid model. The surface configurations are restricted such that the difference between the heights at adjacent sites is no more than one. In addition, the total number of particles is preserved by the dynamics of the model. Mean-field approximations are used to approximate the one-site probability function of the model in both time-independent and time-dependent regimes. In the time-dependent regime, we found that for the height at one site, the average evolves like t^1/6 and squared fluctuations like t^1/3.

We study the entanglement of quantum states associated with submanifolds of Kaehler manifolds. As a motivating example, we discuss the semiclassical asymptotics of entanglement entropy of pure states on the two dimensional sphere with the standard metric.

It is common practice to describe elementary particles by irreducible unitary representations of the Poincaré group. In the same way, multi-particle systems can be described by irreducible unitary representations of the Poincaré group. Representations of the Poincaré group are characterised by fixed eigenvalues of two Casimir operators corresponding to a fixed mass and a fixed angular momentum. In multi-particle systems (of massive spinless particles), fixing these eigenvalues leads to correlations between the particles. In the quasi-classical approximation of large quantum numbers, these correlations take on the structure of a gravitational interaction described by the field equations of conformal gravity. A theoretical value of the corresponding gravitational constant is calculated. It agrees with the empirical value used in the field equations of general relativity.

Various generalisations (group theoretical, nonlinear, discrete,...) of Glauber-Sudarshan coherent states are presented in a unified way, with their statistical properties and their role in Quantum Mechanics, Quantum Optics, Signal Analysis, and quantization with or without Planck constant.

In a Supersymetric Quantum Mechanics framework, the Dirac equation describing a Dirac material in the presence of electromagnetic fields is solved. Considering parallel static non-uniform electromagnetic fields, the Dirac equation is transformed into a two-dimensional system of equations. By means of variable separation, we can define one-dimensional eigenfunctions, which are solutions for two pairs of supersymmetric partner Schrödinger-like Hamiltonians. For Pöschl-Teller-like quantum potentials, we look for conditions that guarantee the existence of bound states, and determine an analytic zero-mode solution for the Dirac equation.

Carroll's group is shown as a group of transformations in a 5-dimensional space () obtained from the embedding of the Euclidean space into a (4,1)-de Sitter space. Three of the five dimensions of are related to ³, and the other two to mass and time. A covariant formulation of Caroll's group is established in phase space. The Landau problem was studied. Finally, the negative parameter of the Wigner function is calculated.

In the framework of quantum mechanics constructed over a quadratic extension of the field of p-adic numbers, we consider an algebraic definition of physical states. Next, the corresponding observables, whose definition completes the statistical interpretation of the theory, are introduced as SOVMs, a p-adic counterpart of the POVMs associated with a quantum system over the complex numbers. Differently from the standard complex setting, the space of all states of a p-adic quantum system has an affine — rather than convex — structure. Thus, a symmetry transformation may be defined, in a natural way, as a map preserving this affine structure. We argue that the group of all symmetry transformations of a p-adic quantum system has a richer structure wrt the case of standard quantum mechanics over the complex numbers.

In this article we deal with the special matrices used in the analysis of the Boolean quadratic forms. We define basic notation and terminology, like Boolean independence or Boolean cumulants, which are useful to formulate the matrix problem. We check for which matrices appropriately constructed quadratic forms are Boolean independent. We show that the size of the matrix matters in this problem. We present the methods useful to find matrices for which corresponding Boolean quadratic forms are or are not Boolean independent.

Quantum Toda lattice may solved by means of the Representation Theory of semisimple Lie groups, or alternatively by using the technique of the Quantum Inverse Scattering Method. A comparison of the two approaches sheds a new light on Representation Theory and leads to a number of challenging questions.

A geometrisation scheme internal to the category of Lie supergroups is discussed for the supersymmetric de Rham cocycles on the super-Minkowski group which determine the standard super-p-brane dynamics with that target, and interpreted within Cartan's approach to the modelling of orbispaces of group actions by homotopy quotients. The ensuing higher geometric objects are shown to carry a canonical equivariant structure for the action of a discrete subgroup of , which results in their descent to the corresponding orbifolds of and in the emergence of a novel class of superfield theories with defects.

We present a procedure to extract generalised eigenvectors of a non-diagonalisable matrix by considering a diagonalisable perturbation and computing the non-diagonalisable limit of its eigenvectors. As an example, we show how to obtain a subset of the spectrum of the eclectic spin chain from the spectrum of a twisted (3) spin chain.

In previous papers we have shown how Schrödinger's equation which includes an electromagnetic field interaction can be deduced from a fluid dynamical Lagrangian of a charged potential flow that interacts with an electromagnetic field. The quantum behaviour was derived from Fisher information terms which were added to the classical Lagrangian. It was thus shown that a quantum mechanical system is drived by information and not only electromagnetic fields.

This program was applied also to Pauli's equations by removing the restriction of potential flow and using the Clebsch formalism. Although the analysis was quite successful there were still terms that did not admit interpretation, some of them can be easily traced to the relativistic Dirac theory. Here we repeat the analysis for a relativistic flow, pointing to a new approach for deriving relativistic quantum mechanics.

Three generations of fermions with SU(3)_C symmetry are represented algebraically in terms of the algebra of sedenions, , generated from the octonions, , via the Cayley-Dickson process. Despite significant recent progress in generating the Standard Model gauge groups and particle multiplets from the four normed division algebras, an algebraic motivation for the existence of exactly three generations has been difficult to substantiate. In the sedenion model, one generation of leptons and quarks with SU(3)_C symmetry is represented in terms of two minimal left ideals of ℓ(6), generated from a subset of all left actions of the complex sedenions on themselves. Subsequently, the finite group S₃, which are automorphisms of but not of , is used to generate two additional generations. The present paper highlight the key aspects and ideas underlying this construction.

This study is devoted to the discovery of super-stable points and cycles in antiferromagnetic Ising and Ising-Heisenberg models with spin 1 on diamond chains with nodal-nodal interactions. These phenomena are important for understanding the complex behavior of magnetic systems. We specifically investigate their connection with magnetization plateaus, which serve as critical indicators of the model's characteristics. Employing the recurrence relations technique, we derive multidimensional rational mappings that give insights about the statistical properties of the models. Carefully examining the stability properties of these mappings, in particular, by analyzing the maximum Lyapunov exponent, we have revealed the complex relationship between the magnetization plateau and dynamic properties. Throughout our extensive research, we have comprehensively studied the existence and behavior of super-stable points and cycles for various parameter configurations in spin-1 models on the diamond chains. By highlighting the basic properties of dynamics and stability, our research advances a fundamental understanding of complex magnetic systems and their fascinating properties.

We consider certain forms of nonlinear partial differential equations that arise in Financial Mathematics. Our central aim is to derive mappings that connect such equations with linear equations. We use point and contact transformations and also Hopf-Cole transformations. The concept of infinite-dimensional (Lie or contact) symmetries admitted by linearizable equations is discussed.

The main subject considered in this paper is the quantum magnetic properties and entanglement of an octanuclear nickel phosphonate-based cage. We measured the temperature-dependent magnetic susceptibility of this cage, which indicates the coexistence of both antiferromagnetic and ferromagnetic interactions between the magnetic centers, Ni ions with spin 1. This observation prompted us to theoretically investigate the magnetic properties of such compounds. Our theoretical calculations were compared with the experimental results for magnetic susceptibility. Additionally, we explored the magnetization plateaus and magnetic susceptibility from an external magnetic field at low temperatures. We determined the thermal entanglement (negativity) and the logarithmic negativity for the octanuclear nickel phosphonate-based cage. The observed correlation between magnetization plateau jumps and magnetic susceptibility peaks in relation to the external magnetic field suggests the need for further experimental measurements.

We study the phase-space properties of the 3-level Lipkin-Meshkov-Glick as paradigmatic case of critical, parity symmetric, N-quDit systems undergoing a quantum phase transition in the thermodynamic limit N→ ∞. We generalize U(2) spin coherent states to U(D) (quDits), and define the coherent state representation Q_ψ (Husimi function) of a symmetric N- quDit state |ψ〉 in the phase space P^{D −1} = U(D)/[U(1) × U(D − 1)]. This allows us to define parity adapted U(D) coherent states (𝕔-DCATs), which reproduce accurately the lowest energy Hamiltonian eigenstates obtained by numerical diagonalization. We visualize precursors of the QPTs by plotting localization measures (Husimi function and its moments) of the parity adapted U(D) coherent states and the numerical Hamiltonian eigenstates for a finite number of particles.

We study generalized (doubled) structures in 2D-dimensional Born geometries in which T-duality symmetry is manifestly realized. We show that spacetime structures of Kähler, hyperkähler, bi-hermitian and bi-hypercomplex manifolds are implemented in Born geometries as generalized (doubled) structures. We find that the Born structures and the generalized Kähler (hyperkähler) structures appear as subalgebras of bi-quaternions × and split-tetra-quaternions × Sp. We investigate the nature of T-duality for the worldsheet instantons in Born sigma models. This manuscript is based on the original paper [1].

In this paper, we recall Richardson's solution of the reduced BCS model, its relationship with the Gaudin model, and the known implementation of these models in conformal field theory. The CFT techniques applied here are based on the use of the free field realization, or more precisely, on the calculation of saddle-point values of Coulomb gas integrals representing certain (perturbed) WZW conformal blocks. We identify the saddle-point limit as the classical limit of conformal blocks. We show that this observation implies a new method for calculating classical conformal blocks and can be further used in the study of quantum integrable models.

We discuss the one-dimensional Schrödinger equation for a harmonic oscillator with a finite step at the origin and/or an external field described by a ramp function. The first half of this paper is a partial review of our recent work. The latter half is devoted to an extension of the problem, i.e., imposing an external linear field on the negative half line. The solvability of the problem via the Hermite polynomials is discussed. We demonstrate that a harmonic oscillator with a step and a ramp can have one eigenstate whose wavefunction is expressed in terms of Hermite polynomials of different orders. Explicit examples are also provided at appropriate places in the text.

We advocate that the dual picture of spacetime noncommutativity, i.e. the existence of a curved momentum space, could be a way out to solve some of the open conceptual problems in the field, such as the basis dependence of observables. In this framework, we show how to build deformed Klein–Gordon and Dirac equations. In addition, we give an outlook of how one could define quantum field theories, both free and interacting ones.

In this article, using the method of differential constraints and Lie group analysis, new exact kink-like solutions are obtained for certain families of nonlinear Schrödinger equations with cubic-quintic nonlinearity. The foregoing solutions are presented in terms of the Lambert W function.

First we recall a method of computing scalar products of eigenfunctions of a Sturm-Liouville operator. This method is then applied to Macdonald and Gegenbauer functions, which are eigenfunctions of the Bessel, resp. Gegenbauer operators. The computed scalar products are well defined only for a limited range of parameters. To extend the obtained formulas to a much larger range of parameters, we introduce the concept of a generalized integral. The (standard as well as generalized) integrals of Macdonald and Gegenbauer functions have important applications to operator theory. Macdonald functions can be used to express the integral kernels of the resolvent (Green functions) of the Laplacian on the Euclidean space in any dimension. Similarly, Gegenbauer functions appear in Green functions of the Laplacian on the sphere and the hyperbolic space. In dimensions 1,2,3 one can perturb these Laplacians with a point potential, obtaining a well defined self-adjoint operator. Standard integrals of Macdonald and Gegenbauer functions appear in the formulas for the corresponding Green functions. In higher dimensions the Laplacian perturbed by point potentials does not exist. However, the corresponding Green function can be generalized to any dimension by using generalized integrals.

The first-order formalism of relativistic wave equations, such as the Dirac (spin s = 1=2) and the Kemmer (s = 1) equations, are generalized to the Bhabha (one s) equation, where the generalized gamma matrices for space-time dimension (3+1) are labeled with not only s but also s' ∈ {s, s − 1,... s − ⎣s⎦}. We show that for s' = s, the generators (D, P_μ, K_μ, S_μν ) constructed from the generalized gamma matrices satisfy the conformal algebra, in which we find that the null-eigenstates |s_±〉 with respect to P_μ and K_μ as P_μ|s₊〉 = 0 = K_μ|s₋〉 have the two following properties: the dimension of the eigenspace for|s_±〉 is given by 2s + 1, and ${\widehat{s}}^{2}|s\pm \rangle =s(s+1)|s\pm \rangle$, where $\widehat{s}$ represents the spin magnitude. In this sense, we can regard |s_±〉 as physical states for a massive particle.

For the chiral oscillator described by a second order and degenerate Lagrangian with special Euclidean group of symmetries, we show, by cotangent bundle Hamiltonian reduction, that reduced equations are Lie-Poisson on dual of oscillator algebra, the central extension of special Euclidean algebra in two dimensions.

The effective potential for the axial mode of gravitational wave on noncommutative Schwarzschild background is presented. Noncommutativity is introduced via deformed Hopf algebra of diffeomorphisms by means of a semi-Killing Drinfeld twist. The analysis is performed up to the first order in perturbation of the metric and noncommutativity parameter. This results in a modified Regge-Wheeler potential with the strongest differences in comparison to the classical Regge-Wheeler potential being near the horizon.

It is shown that the two-body Coulomb problem in the Sturm representation leads to a new two-dimensional, exactly-solvable, superintegrable quantum system in curved space with a g⁽²⁾ hidden algebra and a cubic polynomial algebra of integrals. The two integrals are of orders two and four, they are made from two components of the angular momentum and from the modified Laplace-Runge-Lenz vector, respectively. It is demonstrated that the cubic polynomial algebra is an infinite-dimensional subalgebra of the universal enveloping algebra U_g(2).

In this talk, we discuss an extension of the Marsden–Weinstein–Meyers symplectic reduction theorem to multisymplectic manifolds, and an adaptation of the Śniatycki–Weinstein, Dirac and Arms–Gotay–Cushman Poisson algebra reduction theorems to L_∞-algebras of multisymplectic observables. This is based on joint work with A Miti and L Ryvkin.

We revisit the Atiyah-Hitchin manifold using the generalized Legendre transform approach. Originally it is examined by Ivanov and Roček, and it has been further explored by Ionaș, with a particular focus on calculating the explicit forms of the Kähler potential and the Kähler metric. Notably, there exists a distinction between the former study and the latter. In the framework of the generalized Legendre transform approach, a Kähler potential is formulated through the contour integration of a specific function with holomorphic coordinates. It's essential to note that the choice of the contour in the latter differs from that in the former. This discrepancy in contour selection may result in variations in both the Kähler potential and, consequently, the Kähler metric. Our findings demonstrate that the former exclusively yields the real Kähler potential, aligning with its defined properties. In contrast, the latter produces a complex Kähler potential. We present the derivation of the Kähler potential and metric for the Atiyah-Hitchin manifold in terms of holomorphic coordinates, considering the contour specified by Ivanov and Roček.

In this note we present in a compact form the recently obtained classification of topological Lie quasi-bialgebra and non-degenerate Lie bialgebra structures over the Lie algebra 〚x〛, for a simple Lie algebra .

Vortices in the nonlinear equations, including Zakharov-Kuznetsov (ZK) equation and the regularized long-wave (RLW) equation are studied. The Physics-Informed Neural Networks solve these equations in the forward process and obtain the solutions. In the inverse process, the proper equations can successfully be derived from a given training data. However, between the ZK equation and the RLW equation, sometimes serious misidentification occurs. In order to improve the resolution of the identification, we introduce two methods: a friction method and deformations of the initial profile which offers a nice discrimination of the equations.

The formula ⋆ mod ō(ħ^k) of Kontsevich's star-product with harmonic propagators was known in full at ħ^k⩾6 since 2018 for generic Poisson brackets, and since 2022 also at k = 7 for affine brackets. We discover that the mechanism of associativity for the star-product up to ō(ħ⁶) is different from the mechanism at order 7 for both the full star-product and the affine star-product. Namely, at lower orders the needed consequences of the Jacobi identity are immediately obtained from the associator mod ō(ħ⁶), whereas at order ħ⁷ and higher, some of the necessary differential consequences are reached from the Kontsevich graphs in the associator in strictly more than one step.

We are interested in decay estimates of the ground state (or the low energy eigenstates), outside the potential wells, for a semi-classical Magnetic Schrödinger operator with smooth coefficients P_A(x, hD_x) = (hD_x − µA(x))² + V (x) on L²(R^d). We shall essentially consider the case where µ is large. This kind of estimates, in case of Schrödinger operator without a magnetic field, have been studied by Agmon [1], also in the case of a Riemannian manifold M. Agmon estimates hold true for any h, but are particularly useful in the limit h → 0 when studying tunneling.

We determine second order Lagrangians for (2 + 1)-dimensional generalized Boussinesq equations and we discuss some aspects concerning conservation laws associated with invariance properties of their extended 'full' equivalents, in particular of Krupka–Betounes type. Such equivalents are constructed by means of a recursive formula involving geometric integration by parts formulae.

The theory of Lie–Hamilton systems is used to construct generalized time-dependent SIS epidemic Hamiltonians with a variable infection rate from the 'book' Lie algebra. Although these are characterized by a set of non-autonomous nonlinear and coupled differential equations, their corresponding exact solution is explicitly found. Moreover, the quantum deformation of the book algebra is also considered, from which the corresponding deformed SIS Hamiltonians are obtained and interpreted as perturbations in terms of the quantum deformation parameter of previously known SIS systems. The exact solutions for these deformed systems are also obtained.