Jordan Block Structure Research Articles

Logarithmic celestial conformal field theory

We argue that the celestial conformal field theory exhibits patterns of a logarithmic conformal field theory. We uncover a Jordan block structure involving the celestial stress tensor and its logarithmic partner, a composite operator built from the stress tensor and the Liouville field. Using a limiting process whose parameter corresponds to the infrared cutoff of gravity, we perform some basic consistency checks, particularly the calculation of two-point correlators, which reveals the expected logarithmic behavior. We comment on the vanishing value of the central charge in the celestial conformal field theory and explain how the logarithmic partner is relevant for its well-behavedness. Published by the American Physical Society 2024

Jordan blocks and the Bethe ansatz III: Class 5 model and its symmetries

We study the Hilbert space of the Class 5 model described in arXiv:1904.12005. Despite being integrable, neither its transfer matrix nor its Hamiltonian are diagonalisable, meaning that the usual Algebraic Bethe Ansatz does not provide the full Hilbert space. Instead, we make use of the symmetries of the model to construct the Jordan blocks of the transfer matrix. We also show that the Hamiltonian and the transfer matrix, despite commuting, do not have the same Jordan block structure.

Kinetic Equation for Soliton Gas: Integrable Reductions

Macroscopic dynamics of soliton gases can be analytically described by the thermodynamic limit of the Whitham equations, yielding an integro-differential kinetic equation for the density of states. Under a delta-functional ansatz, the kinetic equation for soliton gas reduces to a non-diagonalisable system of hydrodynamic type whose matrix consists of several 2times 2 Jordan blocks. Here we demonstrate the integrability of this system by showing that it possesses a hierarchy of commuting hydrodynamic flows and can be solved by an extension of the generalised hodograph method. Our approach is a generalisation of Tsarev’s theory of diagonalisable systems of hydrodynamic type to quasilinear systems with non-trivial Jordan block structure.

Exceptional points of degeneracy in traveling wave tubes

The traveling wave tube (TWT) is a powerful vacuum electronic device used to amplify radio-frequency signals with numerous applications, including radar, television, and telephone satellite communications. TWT design in a nutshell comprises a pencil-like electron beam (e-beam) in vacuum interacting with a guiding slow-wave structure (SWS). In our studies here, the e-beam is represented by one-dimensional electron flow and SWS is represented by a transmission line (TL). The interaction between the e-beam and the TL is modeled by an analytic theory that generalizes the well-known Pierce model by taking into account the so-called space-charge effects, particularly electron-to-electron repulsion (debunching). Many important aspects of the analytic theory of TWTs have been already analyzed in our monograph on the subject. The focus of the studies here is on degeneracies of the TWT dispersion relations, particularly on exceptional points of degeneracy and their applications. The term exceptional point of degeneracy (EPD) refers to the property of the relevant matrix to have a nontrivial Jordan block structure. Using special parameterization, particularly suited to chosen EPD, we derive exact formulas for the relevant Jordan basis, including the eigenvectors and the so-called root vector associated with the Jordan block. Based on these studies, we develop a constructive approach to sensing of small signals.

Kakutani equivalence of unipotent flows

We study Kakutani equivalence in the class of unipotent flows acting on finite-volume quotients of semisimple Lie groups. For every such flow, we compute the Kakutani invariant of M. Ratner, the value being explicitly given by the Jordan block structure of the unipotent element generating the flow. This, in particular, answers a question of M. Ratner. Moreover, it follows that the only loosely Kronecker unipotent flows are given by (1t 01)×id acting on (SL(2,R)×G′)∕Γ′, where Γ′ is an irreducible lattice in SL(2,R)×G′ (with the possibility that G′={e}).

Slow Entropy of Some Parabolic Flows

We study nontrivial entropy invariants in the class of parabolic flows on homogeneous spaces, quasi-unipotent flows. We show that topological complexity (i.e., slow entropy) can be computed directly from the Jordan block structure of the adjoint representation. Moreover using uniform polynomial shearing we are able to show that the metric orbit growth (i.e., slow entropy) coincides with the topological one for quasi-unipotent flows (this also applies to the non-compact case). Our results also apply to sequence entropy. We establish criterion for a system to have trivial topological complexity and give some examples in which the measure-theoretic and topological complexities do not coincide for uniquely ergodic systems, violating the intuition of the classical variational principle.

Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I: well-posedness

In this paper we analyse the well-posedness of the Cauchy problem for a rather general class of hyperbolic systems with space-time dependent coefficients and with multiple characteristics of variable multiplicity. First, we establish a well-posedness result in anisotropic Sobolev spaces for systems with upper triangular principal part under interesting natural conditions on the orders of lower order terms below the diagonal. Namely, the terms below the diagonal at a distance k to it must be of order -,k. This setting also allows for the Jordan block structure in the system. Second, we give conditions for the Schur type triangularisation of general systems with variable coefficients for reducing them to the form with an upper triangular principal part for which the first result can be applied. We give explicit details for the appearing conditions and constructions for 2times 2 and 3times 3 systems, complemented by several examples.

Formulas for monodromy

Given a family X of complex varieties degenerating over a punctured disk, one is interested in computing related invariants called the motivic nearby fiber and the refined limit mixed Hodge numbers, both of which contain information about the induced action of monodromy on the cohomology of a fiber of X. Our first main result is that the motivic nearby fiber of X can be computed by first stratifying X into locally closed subvarieties that are nondegenerate in the sense of Tevelev, and then applying an explicit formula on each piece of the stratification that involves tropical geometry. Our second main result is an explicit combinatorial formula for the refined limit mixed Hodge numbers in the case when X is a family of nondegenerate hypersurfaces. As an application, given a complex polynomial, then, under appropriate conditions, we give a combinatorial formula for the Jordan block structure of the action of monodromy on the cohomology of the Milnor fiber, generalizing a famous formula of Varchenko for the associated eigenvalues. In addition, we give a formula for the Jordan block structure of the action of monodromy at infinity.

On the Jordan Block Structure of a Product of Long and Short Root Elements in Irreducible Representations of Algebraic Groups of Type B r

The behavior of a product of commuting long and short root elements of the group of type B r in p-restricted irreducible representations is investigated. For such representations with certain local properties of highest weights, it is shown that the images of these elements have Jordan blocks of all a priori possible sizes. For a p-restricted representation with highest weight a1ω1 +· · ·+arωr, this fact is proved when aj ≠ p − 1 for some j < r − 1 and one of the following conditions holds: (1) $$ {a}_r\ne p-1\kern0.75em and\kern0.5em {\displaystyle \sum_{i=1}^{r-2}{a}_i\ge p-1;} $$ and (2) $$ 2{a}_{r-1}+{a}_r<p,{\displaystyle \sum_{i=1}^{r-2}r-3}{a}_i\ne 0\;for\;2{a}_{r-1}+{a}_r=p-2\; or\;p-1\; and\;{\displaystyle \sum_{i=1}^{r-3}{a}_i\ne 0} $$ or (r−3) (p-1) for a r = p−1.

Stability of Periodic Traveling Waves for Nonlinear Dispersive Equations

We study the stability and instability of periodic traveling waves for Korteweg--de Vries-type equations with fractional dispersion and related, nonlinear dispersive equations. We show that a local constrained minimizer for a suitable variational problem is nonlinearly stable to period preserving perturbations, provided that the associated linearized operator enjoys a Jordan block structure. We then discuss when the linearized equation admits solutions exponentially growing in time.

Hamiltonian Operators of Dubrovin-Novikov Type in 2D

First order Hamiltonian operators of differential-geometric type were introduced by Dubrovin and Novikov in 1983, and thoroughly investigated by Mokhov. In 2D, they are generated by a pair of compatible flat metrics \({g}\) and \({\tilde g}\) which satisfy a set of additional constraints coming from the skew-symmetry condition and the Jacobi identity. We demonstrate that these constraints are equivalent to the requirement that \({\tilde g}\) is a linear Killing tensor of g with zero Nijenhuis torsion. This allowed us to obtain a complete classification of n-component operators with n ≤ 4 (for n = 1, 2 this was done before). For 2D operators the Darboux theorem does not hold: the operator may not be reducible to constant coefficient form. All interesting (non-constant) examples correspond to the case when the flat pencil \({g, \tilde g}\) is not semisimple, that is, the affinor \({\tilde g g^{-1}}\) has non-trivial Jordan block structure. In the case of a direct sum of Jordan blocks with distinct eigenvalues, we obtain a complete classification of Hamiltonian operators for any number of components n, revealing a remarkable correspondence with the class of trivial Frobenius manifolds modelled on H *(CP n-1).

Modulational Instability and Variational Structure

We study the modulational instability of periodic traveling waves for a class of Hamiltonian systems in one spatial dimension. We examine how the Jordan block structure of the associated linearized operator bifurcates for small values of the Floquet exponent to derive a criterion governing instability to long wavelengths perturbations in terms of the kinetic and potential energies, the momentum, the mass of the underlying wave, and their derivatives. The dispersion operator of the equation is allowed to be nonlocal, for which Evans function techniques may not be applicable. We illustrate the results by discussing analytically and numerically equations of Korteweg‐de Vries type.

On the local diffeomorphism and submanifolds of matrices with fixed Jordan block structure

In this paper a local diffeomorphism, which allows to investigate different matrix sub- manifolds is introduced. With its help the submanifold of complex matrices with fixed structure of Jordan blocks is described. Then, explicit formulas of some special perturbations of Jordan structure are given.

Free field realization of the [formula omitted] algebra for the [formula omitted] system, integrals of motion and characters

We introduce a free field realization of the central extension of the Lie algebra Dq of difference operators on the circle in terms of the fermionic η–ξ system. This realization admits a nontrivial Jordan block structure. We also review the free field realization of W1+∞ algebra, and point out some relations between its generators of weight zero and the local integrals of motion of Bazhanov, Lukyanov and Zamolodchikov. Finally we compute the finitized characters, and the continuum characters of the Local Integrals of Motion, and find out and interesting analogy with the generating functions for the counting of branched covers of elliptic curves.

On unitary subsectors of polycritical gravities

We study higher-derivative gravity theories in arbitrary space-time dimension d with a cosmological constant at their maximally critical points where the masses of all linearized perturbations vanish. These theories have been conjectured to be dual to logarithmic conformal field theories in the (d-1)-dimensional boundary of an AdS solution. We determine the structure of the linearized perturbations and their boundary fall-off behaviour. The linearized modes exhibit the expected Jordan block structure and their inner products are shown to be those of a non-unitary theory. We demonstrate the existence of consistent unitary truncations of the polycritical gravity theory at the linearized level for odd rank.

PT-symmetric sinusoidal optical lattices at the symmetry-breaking threshold

The $\mathcal{PT}$-symmetric potential ${V}_{0}[\mathrm{cos}(2\ensuremath{\pi}x/a)+i\ensuremath{\lambda}\mathrm{sin}(2\ensuremath{\pi}x/a)]$ has a completely real spectrum for $\ensuremath{\lambda}\ensuremath{\leqslant}1$ and begins to develop complex eigenvalues for $\ensuremath{\lambda}&gt;1$. At the symmetry-breaking threshold $\ensuremath{\lambda}=1$ some of the eigenvectors become degenerate, giving rise to a Jordan-block structure for each degenerate eigenvector. In general this is expected to result in a secular growth in the amplitude of the wave. However, it has been shown in a recent paper by Longhi, by numerical simulation and by the use of perturbation theory, that for a broad initial wave packet this growth is suppressed, and instead a saturation leading to a constant maximum amplitude is observed. We revisit this problem by explicitly constructing the Bloch wave functions and the associated Jordan functions and using the method of stationary states to find the dependence on the longitudinal distance $z$ for a variety of different initial wave packets. This allows us to show in detail how the saturation of the linear growth arises from the close connection between the contributions of the Jordan functions and those of the neighboring Bloch waves.

Perturbation Theory for PT-Symmetric Sinusoidal Optical Lattices at the Symmetry-Breaking Threshold

The PT symmetric potential V0[cos(2πx/a) + iλ sin(2πx/a)] has a completely real spectrum for λ ≤ 1, and begins to develop complex eigenvalues for λ &gt; 1. At the symmetry-breaking threshold λ = 1 some of the eigenvectors become degenerate, giving rise to a Jordan-block structure for each degenerate eigenvector. In general this is expected to give rise to a secular growth in the amplitude of the wave. However, it has been shown in a recent paper by Longhi, by numerical simulation and by the use of perturbation theory, that for an initial wave packet this growth is suppressed, giving instead a constant maximum amplitude. We revisit this problem by developing the perturbation theory further. We verify that the results found by Longhi persist to second order, and with different input wave packets we are able to see the seeds in perturbation theory of the phenomenon of birefringence first discovered by El-Ganainy et al.

Exceptional points in quantum and classical dynamics

We note that when a quantum system involves exceptional points, i.e. the special values of parameters where the Hamiltonian loses its self-adjointness and acquires the Jordan block structure, the corresponding classical system also exhibits singular behaviour associated with the restructuring of classical trajectories. A system with the crypto-Hermitian Hamiltonian H = (p2 + z2)/2 − igz5 and hyper-elliptic classical dynamics is studied in detail. Analogies with supersymmetric Yang–Mills dynamics are elucidated.

Steady State of Stochastic Sandpile Models

We study the steady state of the abelian sandpile models with stochastic toppling rules. The particle addition operators commute with each other, but in general these operators need not be diagonalizable. We use their abelian algebra to determine their eigenvalues, and the Jordan block structure. These are then used to determine the probability of different configurations in the steady state. We illustrate this procedure by explicitly determining the numerically exact steady state for a one dimensional example, for systems of size $\le12$, and also study the density profile in the steady state.

Small quadratic elements in representations of the special linear group with large highest weights

For almost all p-restricted irreducible representations of the group An(K) in characteristic p > 0 with highest weights large enough with respect to p, the Jordan block structure of the images of small quadratic unipotent elements in these representations is determined. It is proved that if φ is an irreducible p-restricted representation of An(K) with highest weight $$m_1 \omega _1 + ... + m_n \omega _n ,\sum\limits_{i = 1}^n {m_i \geqslant p - 1} $$ , not too few of the coefficients mi are less than p − 1, and n is large enough with respect to the codimension of the fixed subspace of an element z under consideration, then φ(z) has blocks of all sizes from 1 to p. Bibliography: 15 titles.