Infinite-dimensional Representations Research Articles

The Gelfand–Tsetlin basis for infinite-dimensional representations of

We consider the problem of determination of the Gelfand–Tsetlin basis for unitary principal series representations of the Lie algebra . The Gelfand–Tsetlin basis for an infinite-dimensional representation can be defined as the basis of common eigenfunctions of corner quantum minors of the corresponding L-operator. The construction is based on the induction with respect to the rank of the algebra: an element of the basis for is expressed in terms of a Mellin–Barnes type integral of an element of the basis for . The integration variables are the parameters (in other words, the quantum numbers) setting the eigenfunction. Explicit results are obtained for ranks 3 and 4, and the orthogonality of constructed sets of basis elements is demonstrated. For the kernel of the integral is expressed in terms of gamma-functions of the parameters of eigenfunctions, and in the case of —in terms of a hypergeometric function of the complex field at unity. The formulas presented for an arbitrary rank make it possible to obtain the system of finite-difference equations for the kernel. They include expressions for the quantum minors of L-operator via the minors of L-operator for the principal series representations, as well as formulas for action of some non-corner minors on the eigenfunctions of corner ones. The latter hold for any representation of (not only principal series) in which the corner minors of the L-operator can be diagonalized.

Alpert multi-wavelets for functional inverse problems: direct optimization and deep learning

Computational engineering models often contain unknown entities (e.g. parameters, initial and boundary conditions) that require estimation from other measured observable data. Estimating such unknown entities is challenging when they involve spatio-temporal fields because such functional variables often require an infinite-dimensional representation. We address this problem by transforming an unknown functional field using Alpert wavelet bases and truncating the resulting spectrum. Hence the problem reduces to the estimation of few coefficients that can be performed using common optimization methods. We apply this method on a one-dimensional heat transfer problem where we estimate the heat source field varying in both time and space. The observable data is comprised of temperature measured at several thermocouples in the domain. This latter is composed of either copper or stainless steel. The optimization using our method based on wavelets is able to estimate the heat source with an error between 5% and 7%. We analyze the effect of the domain material and number of thermocouples as well as the sensitivity to the initial guess of the heat source. Finally, we estimate the unknown heat source using a different approach based on deep learning techniques where we consider the input and output of a multi-layer perceptron in wavelet form. We find that this deep learning approach is more accurate than the optimization approach with errors below 4%.

Multiplicity in restricting small representations

We give a geometric criterion for the bounded multiplicity property of “small” infinite-dimensional representations of real reductive Lie groups in both induction and restrictions. In particular, for a reductive symmetric pair $(G,H)$, we determine the reductive subgroups $G'$ having the property that any irreducible $H$-distinguished admissible representations of $G$ are of bounded multiplicity when restricted to $G'$.

Quantum Rajeev–Ranken model as an anharmonic oscillator

The Rajeev–Ranken (RR) model is a Hamiltonian system describing screw-type nonlinear waves of wavenumber k in a scalar field theory pseudodual to the 1 + 1D SU(2) principal chiral model. Classically, the RR model is Liouville integrable. Here, we interpret the model as a novel 3D cylindrically symmetric quartic oscillator with an additional rotational energy. The quantum theory has two dimensionless parameters. Upon separating variables in the Schrödinger equation, we find that the radial equation has a four-term recurrence relation. It is of type [0, 1, 16] and lies beyond the ellipsoidal Lamé and Heun equations in Ince’s classification. At strong coupling λ, the energies of highly excited states are shown to depend on the scaling variable λk. The energy spectrum at weak coupling and its dependence on k in a double-scaling strong coupling limit are obtained. The semi-classical Wentzel-Kramers-Brillouin (WKB) quantization condition is expressed in terms of elliptic integrals. Numerical inversion enables us to establish a (λk)2/3 dispersion relation for highly energetic quantized “screwons” at moderate and strong coupling. We also suggest a mapping between our radial equation and the one of Zinn-Justin and Jentschura that could facilitate a resurgent WKB expansion for energy levels. In another direction, we show that the equations of motion of the RR model can also be viewed as Euler equations for a step-3 nilpotent Lie algebra. We use our canonical quantization to uncover an infinite dimensional reducible unitary representation of this nilpotent algebra, which is then decomposed using its Casimir operators.

Least squares Monte Carlo methods in stochastic Volterra rough volatility models

In stochastic Volterra rough volatility models, the volatility follows a truncated Brownian semistationary process with stochastic volatility of volatility (vol-of-vol). Recently, efficient Chicago Board Options Exchange Volatility Index (VIX) pricing Monte Carlo methods have been proposed for cases where the vol-of-vol is Markovian and independent of the volatility. Using recent empirical data, we discuss the VIX option pricing problem for a generalized framework of these models, where the vol-of-vol may depend on the volatility and/or may not be Markovian. In such a setting, the aforementioned Monte Carlo methods are not valid. Moreover, the classical least squares Monte Carlo faces exponentially increasing complexity with the number of grid time steps, while the nested Monte Carlo method requires a prohibitive number of simulations. By exploring the infinite-dimensional Markovian representation of these models, we devise a scalable least squares Monte Carlo for VIX option pricing. We apply our method first under the independence assumption for benchmarks and then to the generalized framework. We also discuss the rough vol-of-vol setting, where Markovianity of the vol-of-vol is not present. We present simulations and benchmarks to establish the efficiency of our method as well as a comparison with market data.

Asymptotics of $$\mathrm {SL}(2,{{\mathbb {C}}})$$ coherent invariant tensors

We study the semiclassical limit of a class of invariant tensors for infinite-dimensional unitary representations of $\mathrm{SL}(2,\mathbb{C})$ of the principal series, corresponding to generalized Clebsch-Gordan coefficients with $n\geq3$ legs. We find critical configurations of the quantum labels with a power-law decay of the invariants. They describe 3d polygons that can be deformed into one another via a Lorentz transformation. This is defined viewing the edge vectors of the polygons are the electric part of bivectors satisfying a (frame-dependent) relation between their electric and magnetic parts known as $\gamma$-simplicity in the loop quantum gravity literature. The frame depends on the SU(2) spin labelling the basis elements of the invariants. We compute a saddle point approximation using the critical points and provide a leading-order approximation of the invariants. The power-law is universal if the SU(2) spins have their lowest value, and $n$-dependent otherwise. As a side result, we provide a compact formula for $\gamma$-simplicity in arbitrary frames. The results have applications to the current EPRL model, but also to future research aiming at going beyond the use of fixed time gauge in spin foam models.

Symmetry and Symmetry Breaking in Physics: From Geometry to Topology

Symmetry (and group theory) is a fundamental principle of theoretical physics. Finite symmetries, continuous symmetries of compact groups, and infinite-dimensional representations of noncompact Lie groups are at the core of solid physics, particle physics, and quantum physics, respectively. The latter groups now play an important role in many branches of mathematics. In more recent years, we have been faced with the impact of topological quantum field theory (TQFT). Topology and symmetry have deep connections, but topology is inherently broader and more complex. While the presence of symmetry in physical phenomena imposes strong constraints, topology seems to be related to low-energy states and is very likely to provide information about the different dynamical trajectories and patterns that particles can follow. For example, regarding the relationship of topology to low-energy states, Hodge’s theory of harmonic forms shows that the zero-energy states (for differential forms) correspond to the cohomology. Regarding the relationship of topology to particle trajectories, a topological knot can be seen as an orbit with complex properties in spacetime. The various deformations or embeddings of the knot, performed in low or high dimensions, allow defining different equivalence classes or topological types, and interestingly, it is possible from these types to study the symmetries associated with the deformations and their changes. More specifically, in the present work, we address two issues: first, that quantum geometry deforms classical geometry, and that this topological deformation may produce physical effects that are specific to the quantum physics scale; and second, that mirror symmetry and the phenomenon of topological change are closely related. This paper was aimed at understanding the conceptual and physical significance of this connection.

Adaptive boundary control of reaction–diffusion PDEs with unknown input delay

We design an adaptive full-state feedback controller to stabilize a one-dimensional reaction–diffusion equation with unknown boundary input delay. An infinite-dimensional representation of the actuator delay is utilized to transform the system into a transport PDE cascading with a reaction–diffusion PDE. A suitable parameter update law is designed to establish local boundedness of the system trajectories and asymptotic convergence stability result using the well-known PDE backstepping technique and a Lyapunov argument. Consistent simulation results are provided to support the theoretical results.

Gauging the Higher-Spin-Like Symmetries by the Moyal Product. II

Continuing the study of the Moyal Higher Spin Yang–Mills theory started in our previous paper we provide a detailed discussion of matter coupling and the corresponding tree-level amplitudes. We also start the investigation of the spectrum by expanding the master fields in terms of ordinary spacetime fields. We note that the spectrum can be consistent with unitarity while still preserving Lorentz covariance, albeit not in the usual way, but by employing an infinite-dimensional unitary representation of the Lorentz group.

Quantum groups and functional relations for arbitrary rank

The quantum integrable systems associated with the quantum loop algebras Uq(L(sll+1)) are considered. The factorized form of the transfer operators related to the infinite dimensional evaluation representations is found and the determinant form of the transfer operators related to the finite dimensional evaluation representations is obtained. The master TQ- and TT-relations are derived. The operatorial T- and Q-systems are found. The nested Bethe equations are obtained.

Ompleteness of SoV Representation for SL(2,R) Spin Chains

This work develops a new method, based on the use of Gustafson's integrals and on the evaluation of singular integrals, allowing one to establish the unitarity of the separation of variables transform for infinite-dimensional representations of rank one quantum integrable models. We examine in detail the case of the SL(2,R) spin chains.

Aspects of holography of Taub-NUT- AdS4 spacetimes

In this paper we consider aspects of the holographic interpretation of Taub-NUT-${\mathrm{AdS}}_{4}$. We review our earlier results which show that ${\mathrm{TNAdS}}_{4}$ gives rise to a holographic three-dimensional conformal fluid having constant vorticity. We then study the holographic relevance of the Misner string by considering bulk scalar fluctuations. The scalar fluctuations organize naturally into representations of the $SU(2)\ifmmode\times\else\texttimes\fi{}\mathbb{R}$ isometry algebra. If we require the string's invisibility we obtain a Dirac-like quantization relating the frequency of the scalar field modes to the NUT charge. As the latter quantity determines the total vorticity flux of the boundary fluid, we argue that such an assumption allows for a holographic interpretation of ${\mathrm{TNAdS}}_{4}$ as a nondissipative superfluid whose excitations are quantized vortices. Alternatively, if we regard the Misner string as a physical object, as has recently been advocated for thermodynamically, the aforementioned quantization conditions are removed, and we find that ${\mathrm{TNAdS}}_{4}$ corresponds to a holographic fluid whose dissipative properties are probed as usual by the complex quasinormal modes of the bulk fluctuations. We show that such quasinormal modes are, perhaps surprisingly, organized into infinite-dimensional nonunitary representations of the isometry algebra.

Inversion of Rankin–Cohen operators via Holographic Transform

L’analyse des problèmes de branchement des restrictions des représentations fait émerger le concept de transformation de brisure de symétrie et celui de transformation holographique. Les opérateurs de brisure de symétrie diminuent le nombre de variables dans les modèles géométriques tandis que les opérateurs holographiques l’augmentent. Plusieurs développements en série ou intégrale de l’analyse classique sont des cas particuliers de telles transformations.

Representations of the Lie Superalgebra osp(1|2n) with Polynomial Bases

We study a particular class of infinite-dimensional representations of $\mathfrak{osp}(1|2n)$. These representations $L_n(p)$ are characterized by a positive integer $p$, and are the lowest component in the $p$-fold tensor product of the metaplectic representation of $\mathfrak{osp}(1|2n)$. We construct a new polynomial basis for $L_n(p)$ arising from the embedding $\mathfrak{osp}(1|2np) \supset \mathfrak{osp}(1|2n)$. The basis vectors of $L_n(p)$ are labelled by semi-standard Young tableaux, and are expressed as Clifford algebra valued polynomials with integer coefficients in $np$ variables. Using combinatorial properties of these tableau vectors it is deduced that they form indeed a basis. The computation of matrix elements of a set of generators of $\mathfrak{osp}(1|2n)$ on these basis vectors requires further combinatorics, such as the action of a Young subgroup on the horizontal strips of the tableau.

Recent Developments of the Lauricella String Scattering Amplitudes and Their Exact SL(K + 3,C) Symmetry

In this review, we propose a new perspective to demonstrate the Gross conjecture regarding the high-energy symmetry of string theory. We review the construction of the exact string scattering amplitudes (SSAs) of three tachyons and one arbitrary string state, or the Lauricella SSA (LSSA), in the 26D open bosonic string theory. These LSSAs form an infinite dimensional representation of the SL(K+3,C) group. Moreover, we show that the SL(K+3,C) group can be used to solve all the LSSAs and express them in terms of one amplitude. As an application in the hard scattering limit, the LSSA can be used to directly prove the Gross conjecture, which was previously corrected and proved by the method of the decoupling of zero norm states (ZNS). Finally, the exact LSSA can be used to rederive the recurrence relations of SSA in the Regge scattering limit with associated SL(5,C) symmetry and the extended recurrence relations (including the mass and spin dependent string BCJ relations) in the nonrelativistic scattering limit with the associated SL(4,C) symmetry discovered recently.

Quantum spin systems and supersymmetric gauge theories. Part I

The relation between supersymmetric gauge theories in four dimensions and quantum spin systems is exploited to find an explicit formula for the Jost function of the N site mathfrak{sl} 2X X X spin chain (for infinite dimensional complex spin representations), as well as the SLN Gaudin system, which reduces, in a limiting case, to that of the N-particle periodic Toda chain. Using the non-perturbative Dyson-Schwinger equations of the supersymmetric gauge theory we establish relations between the spin chain commuting Hamiltonians with the twisted chiral ring of gauge theory. Along the way we explore the chamber dependence of the supersymmetric partition function, also the expectation value of the surface defects, giving new evidence for the AGT conjecture.

Flavour and Colour of Quarks in Spin Topological Space

An assumption that all the six flavour quarks are attributed to be the components of a same, a common isospin multiplets space named STS is proposed. Base on Pauli Exclusion Principle, every quark is assigned to different flavour marks in STS. Every flavour quark possesses its own colour spectral line array specially appointed. The collection of colour spectral line arrays of the six flavour quarks constructs together the CSDF, Colour Spectrum Diagram of Flavour, further baryons and mesons could be constructed from CSDF. STS, Spin Topological Space is a math frame with infinite dimensional matrix representation for spin angular momentum. Flavours is an isospin angular momentum coupling phenomena of the three-colour-quarks.

Special functions associated with K -types of degenerate principal series of Sp ( n , C )

We study K -types of degenerate principal series of Sp ( n , C ) by using two realizations of these infinite-dimensional representations. The first model we use is the classical compact picture; the second model is conjugate to the noncompact picture via an appropriate partial Fourier transform. In the first case, we find a family of K -finite vectors that can be expressed as solutions of specific hypergeometric differential equations; the second case leads to a family of K -finite vectors whose expressions involve Bessel functions.

A Fock Model and the Segal-Bargmann Transform for the Minimal Representation of the Orthosymplectic Lie Superalgebra osp(m,2|2n)

The minimal representation of a semisimple Lie group is a 'small' infinite-dimensional irreducible unitary representation. It is thought to correspond to the minimal nilpotent coadjoint orbit in Kirillov's orbit philosophy. The Segal-Bargmann transform is an intertwining integral transformation between two different models of the minimal representation for Hermitian Lie groups of tube type. In this paper we construct a Fock model for the minimal representation of the orthosymplectic Lie superalgebra $\mathfrak{osp}(m,2|2n)$. We also construct an integral transform which intertwines the Schr\"odinger model for the minimal representation of the orthosymplectic Lie superalgebra $\mathfrak{osp}(m,2|2n)$ with this new Fock model.

Random walks on Ramanujan complexes and digraphs

The cutoff phenomenon was recently confirmed for random walks on Ramanujan graphs by the first author and Peres. In this work, we obtain analogs in higher dimensions, for random walk operators on any Ramanujan complex associated with a simple group G over a local field F . We show that if T is any k -regular G -equivariant operator on the Bruhat–Tits building with a simple combinatorial property (collision-free), the associated random walk on the n -vertex Ramanujan complex has cutoff at time log _k n . The high-dimensional case, unlike that of graphs, requires tools from non-commutative harmonic analysis and the infinite-dimensional representation theory of G . Via these, we show that operators T as above on Ramanujan complexes give rise to Ramanujan digraphs with a special property ( r -normal), implying cutoff. Applications include geodesic flow operators, geometric implications, and a confirmation of the Riemann Hypothesis for the associated zeta functions over every group G, previously known for groups of type \widetilde A_n and \widetilde C_2 .