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Some idea, which leads to a non-trivial solution of the quantum four-simplex equation, is exposed in this paper. We call this idea "pentagonal algebra". Few examples of the realisation of this idea are given here, and thus few examples of $R$-matrix for the quantum four-simplex equation are presented., Comment: Misprints corrected

Recently, Bazhanov and Sergeev have described an Ising-type integrable model which can be identified as a $\sinh$-Gordon-type model with an infinite number of states but with a real parameter $q$. This model is the subject of Sklyanin's Functional Bethe Ansatz. We develop in this paper the whole technique of the FBA which includes: 1. Construction of eigenstates of an off-diagonal element of a monodromy matrix. Most important ingredients of these eigenstates are the Clebsh-Gordan coefficients of the corresponding representation. 2. Separately, we discuss the Clebsh-Gordan coefficients, as well as the Wigner's 6j symbols, in details. The later are rather well known in the theory of $3D$ indices. Thus, the Sklyanin basis of the quantum separation of variables is constructed. The matrix elements of an eigenstate of the auxiliary transfer matrix in this basis are products of functions satisfying the Baxter equation. Such functions are called usually the $Q$-operators. We investigate the Baxter equation and $Q$-operators from two points of view. 3. In the model considered the most convenient Bethe-type variables are the zeros of a Wronskian of two well defined particular solutions of the Baxter equation. This approach works perfectly in the thermodynamic limit. We calculate the distribution of these roots in the thermodynamic limit, and so we reproduce in this way the partition function of the model. 4. The real parameter $q$, which is the standard quantum group parameter, plays the role of the absolute temperature in the model considered. Expansion with respect to $q$ (tropical expansion) gives an alternative way to establish the structure of the eigenstates. In this way we classify the elementary excitations over the ground state., Comment: References updated

Faddeev' equations are a set-theoretical and an operator forms of the star-triangle equation. Known solutions of the quantum star-triangle equation, related to the Faddeev equations, are based on various forms of the modular double of the Weyl algebra including its cyclic representation. We show in this paper that Fadeev's equation also leads to a solution of the quantum star-triangle equation even in the case of a simple Weyl algebra with $|q|<1$. This paper can be seen as an addendum to the recent paper "V. Bazhanov and S. Sergeev, A distant descendant of the six-vertex model, arXiv:2310.08427"., Comment: References updated

In this paper we present a new solution of the star-triangle relation having positive Boltzmann weights. The solution defines an exactly solvable two-dimensional Ising-type (edge interaction) model of statistical mechanics where the local "spin variables" can take arbitrary integer values, i.e., the number of possible spin states at each site of the lattice is infinite. There is also an equivalent "dual" formulation of the model, where the spins take continuous real values on the circle. From algebraic point of view this model is closely related to the to the 6-vertex model. It is connected with the construction of an intertwiner for two infinite-dimensional representations of the quantum affine algebra $U_q(\widehat{sl}(2))$ without the highest and lowest weights. The partition function of the model in the large lattice limit is calculated by the inversion relation method. Amazingly, it coincides with the partition function of the off-critical 8-vertex free-fermion model., Comment: v1: 24 pages; v2: 32pages, misprints corrected, references added, sections 5 & 6 extended, new appendices added

We consider the eigenproblem for the Laplacian inside a three-dimensional revolutional domain diffeomorphic to a solid torus and construct asymptotic eigenvalues and eigenfunctions (quasimodes) of the whispering gallery-type. The whispering gallery-type asymptotics are localized near the boundary of the domain, and an explicit analytic representations in terms of Airy functions is constructed for such asymptotics. There are several different scales in the problem, which makes it possible to apply the procedure of adiabatic approximation in the form of operator separation of variables to reduce the initial problem to one-dimensional problems up to the small correction. We also discuss the relation between the constructed whispering gallery-type asymptotics and classical billiards in the corresponding domain, in particularly, such asymptotics correspond to almost integrable billiards with proper degeneracy. We illustrate the results in the case when a revolutional domain is obtained by rotation of the triangle with rounded wedges.

There is a sub-class of the solutions to Quantum Tetrahedron Equation related to the algebraical Pentagon Equation. The Quantum Tetrahedron Equation defines an evolution operator in wholly discrete three dimensional space-time. In this paper we establish the Liouville integrability of one particular quantum evolution model/classical integrable model on cubic lattice. The key feature of the model is that it has two independent quantum/classical spectral curves. In particular, on the classical level its Hamiltonian equations of motion decouple into two independent Hirota equations.

The Rogers-Ramanujan identity for $\!\!\phantom{|}_1\psi_1$ $$ \sum_{n\in\mathbb{Z}} \frac{(a;q)_n}{(b;q)_n} z^n\;=\; \frac{(q,b/a,az,q/az;q)_\infty}{(b,q/a,z,b/az;q)_\infty} $$ can be classified as one related to the Bethe Ansatz for ``chain length $N=1$, ground state XXZ model with an arbitrary negative spin''.

In this paper we discuss an example of classical integrable equation with rather unusual `B'-type Kadomtsev-Petviashvili (KP) soliton hierarchy.

We study the modular pair of $TQ$ equations for the quantum $\sin$-Gordon model in the framework of non-compact $\mathcal{U}_{q,q^*}(\widehat{sl}_2)$. We assume some conjectures for the thermodynamic limit allowing one to obtain its ground state.

In this paper we discuss some properties of Baxter's TQ equation for the eight-vertex elliptic Sklyanin algebra it its compact representation based on the elliptic Gamma-functions. As the main result, we establish the structure of the spectrum of the corresponding quantum integrable model.

We show that the celebrated six-vertex model of statistical mechanics (along with its multistate generalizations) can be reformulated as an Ising-type model with only a two-spin interaction. Such a reformulation unravels remarkable factorization properties for row to row transfer matrices, allowing one to uniformly derive all functional relations for their eigenvalues and present the coordinate Bethe ansatz for the eigenvectors for all higher spin generalizations of the six-vertex model. The possibility of the Ising-type formulation of these models raises questions about the precedence of the traditional quantum group description of the vertex models. Indeed, the role of a primary integrability condition is now played by the star-triangle relation, which is not entirely natural in the standard quantum group setting, but implies the vertex-type Yang-Baxter equation and commutativity of transfer matrices as simple corollaries. As a mathematical identity the emerging star-triangle relation is equivalent to the Pfaff-Saalschuetz-Jackson summation formula, originally discovered by J.~F.~Pfaff in 1797. Plausibly, all vertex models associated with quantized affine Lie algebras and superalgebras can be reformulated as Ising-type models with only two-spin interactions., Comment: 53 pages, many figures. v2,v3: minor corrections

Multiband optical photometry data of blazar S5 0716+714 obtained from 2002 to 2019 reveal stable color index change with flux variability. We analyzed this trend under variability caused by the Doppler factor change in the presence of a curved photon energy spectrum. A break in the energy spectrum of emitting electrons, caused by radiative losses, or log-parabolic electron energy distribution, or the synchrotron self-absorption acting in a compact jet part forms such the photon spectrum. We explained the observed color index change with variability by geometric effects only under the assumption that the radiating region is the synchrotron self-absorbed core and the bright optically thin jet. In this framework, we estimated the magnetic field strength in the optically thick part of the radiating region. These values correspond to other independent estimates of the magnetic field near the black hole, further supporting our assumption., Comment: Accepted for publication in ApJ. 23 pages, 21 figures

We address the spectral problem of the normal quantum mechanical operator associated to the quantized mirror curve of the toric (almost) del Pezzo Calabi--Yau threefold called local $\mathbb{P}^2$ in the case of complex values of Planck's constant., Comment: 13 pages

Motivated by applications for non-perturbative topological strings in toric Calabi--Yau manifolds, we discuss the spectral problem for a pair of commuting modular conjugate (in the sense of Faddeev) Harper type operators, corresponding to a special case of the quantized mirror curve of local $\mathbb{P}^1\times\mathbb{P}^1$ and complex values of Planck's constant. We illustrate our analytical results by numerical calculations., Comment: 23 pages, 9 figures, references added and interpretation of the numerical results of Section 6 corrected

In this paper we give an overview of exactly solved edge-interaction models, where the spins are placed on sites of a planar lattice and interact through edges connecting the sites. We only consider the case of a single spin degree of freedom at each site of the lattice. The Yang-Baxter equation for such models takes a particular simple form called the star-triangle relation. Interestigly all known solutions of this relation can be obtained as particular cases of a single "master solution", which is expressed through the elliptic gamma function and have continuous spins taking values of the circle. We show that in the low-temperature (or quasi-classical) limit these lattice models reproduce classical discrete integrable systems on planar graphs previously obtained and classified by Adler, Bobenko and Suris through the consistency-around-a-cube approach. We also discuss inversion relations, the physicical meaning of Baxter's rapidity-independent parameter in the star-triangle relations and the invariance of the action of the classical systems under the star-triangle (or cube-flip) transformation of the lattice, which is a direct consequence of Baxter's Z-invariance in the associated lattice models., Comment: Final version

We perform a multi-wavelength polarimetric study of the quasar CTA 102 during an extraordinarily bright $\gamma$-ray outburst detected by the {\it Fermi} Large Area Telescope in September-October 2012 when the source reached a flux of F$_{>100~\mathrm{MeV}} =5.2\pm0.4\times10^{-6}$ photons cm$^{-2}$ s$^{-1}$. At the same time the source displayed an unprecedented optical and NIR outburst. We study the evolution of the parsec scale jet with ultra-high angular resolution through a sequence of 80 total and polarized intensity Very Long Baseline Array images at 43 GHz, covering the observing period from June 2007 to June 2014. We find that the $\gamma$-ray outburst is coincident with flares at all the other frequencies and is related to the passage of a new superluminal knot through the radio core. The powerful $\gamma$-ray emission is associated with a change in direction of the jet, which became oriented more closely to our line of sight ($\theta\sim$1.2$^{\circ}$) during the ejection of the knot and the $\gamma$-ray outburst. During the flare, the optical polarized emission displays intra-day variability and a clear clockwise rotation of EVPAs, which we associate with the path followed by the knot as it moves along helical magnetic field lines, although a random walk of the EVPA caused by a turbulent magnetic field cannot be ruled out. We locate the $\gamma$-ray outburst a short distance downstream of the radio core, parsecs from the black hole. This suggests that synchrotron self-Compton scattering of near-infrared to ultraviolet photons is the probable mechanism for the $\gamma$-ray production., Comment: Accepted for publication in The Astrophysical Journal

The occurrence of low-amplitude flux variations in blazars on hourly timescales, commonly known as microvariability, is still a widely debated subject in high-energy astrophysics. Several competing scenarios have been proposed to explain such occurrences, including various jet plasma instabilities leading to the formation of shocks, magnetic reconnection sites, and turbulence. In this letter we present the results of our detailed investigation of a prominent, five-hour-long optical microflare detected during recent WEBT campaign in 2014, March 2-6 targeting the blazar 0716+714. After separating the flaring component from the underlying base emission continuum of the blazar, we find that the microflare is highly polarized, with the polarization degree $\sim (40-60)\%$$\pm (2-10)\%$, and the electric vector position angle $\sim (10 - 20)$deg$\pm (1-8)$deg slightly misaligned with respect to the position angle of the radio jet. The microflare evolution in the $(Q,\,U)$ Stokes parameter space exhibits a looping behavior with a counter-clockwise rotation, meaning polarization degree decreasing with the flux (but higher in the flux decaying phase), and approximately stable polarization angle. The overall very high polarization degree of the flare, its symmetric flux rise and decay profiles, and also its structured evolution in the $Q-U$ plane, all imply that the observed flux variation corresponds to a single emission region characterized by a highly ordered magnetic field. As discussed in the paper, a small-scale but strong shock propagating within the outflow, and compressing a disordered magnetic field component, provides a natural, though not unique, interpretation of our findings., Comment: 9 pages, 4 Figures, ApJ Letter accepted

We construct $2^n$-families of solutions of the Yang-Baxter equation from $n$-products of three-dimensional $R$ and $L$ operators satisfying the tetrahedron equation. They are identified with the quantum $R$ matrices for the Hopf algebras known as generalized quantum groups. Depending on the number of $R$'s and $L$'s involved in the product, the trace construction interpolates the symmetric tensor representations of $U_q(A^{(1)}_{n-1})$ and the anti-symmetric tensor representations of $U_{-q^{-1}}(A^{(1)}_{n-1})$, whereas a boundary vector construction interpolates the $q$-oscillator representation of $U_q(D^{(2)}_{n+1})$ and the spin representation of $U_{-q^{-1}}(D^{(2)}_{n+1})$. The intermediate cases are associated with an affinization of quantum super algebras., Comment: 28 pages. Minor typo in Prop 2.1 fixed

For every quantized Lie algebra there exists a map from the tensor square of the algebra to itself, which by construction satisfies the set-theoretic Yang-Baxter equation. This map allows one to define an integrable discrete quantum evolution system on quadrilateral lattices, where local degrees of freedom (dynamical variables) take values in a tensor power of the quantized Lie algebra. The corresponding equations of motion admit the zero curvature representation. The commuting Integrals of Motion are defined in the standard way via the Quantum Inverse Problem Method, utilizing Baxter's famous commuting transfer matrix approach. All elements of the above construction have a meaningful quasi-classical limit. As a result one obtains an integrable discrete Hamiltonian evolution system, where the local equation of motion are determined by a classical Yang-Baxter map and the action functional is determined by the quasi-classical asymptotics of the universal R-matrix of the underlying quantum algebra. In this paper we present detailed considerations of the above scheme on the example of the algebra $U_q(sl(2))$ leading to discrete Liouville equations, however the approach is rather general and can be applied to any quantized Lie algebra., Comment: 36 pages, 7 figures, v2: minor corrections, references added

We introduce a homomorphism from the quantum affine algebras $U_q(D^{(2)}_{n+1}), U_q(A^{(2)}_{2n}), U_q(C^{(1)}_{n})$ to the $n$-fold tensor product of the $q$-oscillator algebra ${\mathcal A}_q$. Their action commute with the solutions of the Yang-Baxter equation obtained by reducing the solutions of the tetrahedron equation associated with the modular and the Fock representations of ${\mathcal A}_q$. In the former case, the commutativity is enhanced to the modular double of these quantum affine algebras., Comment: 11 pages, minor corrections

In this paper we construct a three-dimensional (3D) solvable lattice model with non-negative Boltzmann weights. The spin variables in the model are assigned to edges of the 3D cubic lattice and run over an infinite number of discrete states. The Boltzmann weights satisfy the tetrahedron equation, which is a 3D generalisation of the Yang-Baxter equation. The weights depend on a free parameter 0

We prove a recently conjectured star-star relation, which plays the role of an integrability condition for a class of 2D Ising-type models with multicomponent continuous spin variables. Namely, we reduce this relation to an identity for elliptic gamma functions, previously obtained by Rains., Comment: 8 pages, 3 figures

The Zamolodchikov model describes an exact relativistic factorized scattering theory of straight strings in (2+1)-dimensional space-time. It also defines an integrable 3D lattice model of statistical mechanics and quantum field theory. The three-string S-matrix satisfies the tetrahedron equation which is a 3D analog of the Yang-Baxter equation. Each S-matrix depends on three dihedral angles formed by three intersecting planes, whereas the tetrahedron equation contains five independent spectral parameters, associated with angles of an Euclidean tetrahedron. The vertex weights are given by rather complicated expressions involving square roots of trigonometric function of the spectral parameters, which is quite unusual from the point of view of 2D solvable lattice models. In this paper we consider a particular four-parameter specialization of the tetrahedron equation when one of its vertices goes to infinity and the tetrahedron itself degenerates into an infinite prism. We show that in this limit all the vertex weights in the tetrahedron equation can be represented as meromorphic functions on an elliptic curve. Moreover we show that a special reduction of the tetrahedron equation in this case leads precisely to an example of the tetrahedral Zamolodchikov algebra, previously constructed by Korepanov. This algebra plays important role for a "layered" construction of the Shastry's R-matrix and the 2D S-matrix appearing in the problem of the ADS/CFT correspondence for N=4 SUSY Yang-Mills theory in four dimensions. Possible applications of our results in this field are briefly discussed., Comment: 18 pages, 3 figures; v2: minor changes, a reference added; v3: minor changes

We present new photometric observations of Supernova (SN) 2003ie starting one month before discovery, obtained serendipitously while observing its host galaxy. With only a weak upper limit derived on the mass of its progenitor (<25 M_sun) from pre-explosion studies, this event could be a potential exception to the "red supergiant (RSG) problem" (the lack of high mass RSGs exploding as Type IIP supernovae). However, this is true only if SN2003ie was a Type IP event, something which has never been determined. Using recently derived core collapse SN light curve templates, as well as by comparison to other known SNe, we find that SN2003ie was indeed a likely Type IIP event. However, it is found to be a member of the faint Type IIP class. Previous members of this class have been shown to arise from relatively low mass progenitors (<12 M_sun). It therefore seems unlikely that this SN had a massive RSG progenitor. The use of core collapse SN light curve templates is shown to be helpful in classifying SNe with sparse coverage. These templates are likely to become more robust as large homogeneous samples of core collapse events are collected., Comment: Submitted to AJ

It is known that a solution of the tetrahedron equation generates infinitely many solutions of the Yang-Baxter equation via suitable reductions. In this paper this scheme is applied to an oscillator solution of the tetrahedron equation involving bosons and fermions by using special 3d boundary conditions. The resulting solutions of the Yang-Baxter equation are identified with the quantum R matrices for the spin representations of B^{(1)}_n, D^{(1)}_n and D^{(2)}_{n+1}., Comment: 17 pages, 7 figures, minor misprint corrected