# Journal of Mathematical Physics

Published by American Institute of Physics
Online ISSN: 1089-7658
Print ISSN: 0022-2488
Publications
Biological systems are often subject to external noise from signal stimuli and environmental perturbations, as well as noises in the intracellular signal transduction pathway. Can different stochastic fluctuations interact to give rise to new emerging behaviors? How can a system reduce noise effects while still being capable of detecting changes in the input signal? Here, we study analytically and computationally the role of nonlinear feedback systems in controlling external noise with the presence of large internal noise. In addition to noise attenuation, we analyze derivatives of Fano factor to study systems' capability of differentiating signal inputs. We find effects of internal noise and external noise may be separated in one slow positive feedback loop system; in particular, the slow loop can decrease external noise and increase robustness of signaling with respect to fluctuations in rate constants, while maintaining the signal output specific to the input. For two feedback loops, we demonstrate that the influence of external noise mainly depends on how the fast loop responds to fluctuations in the input and the slow loop plays a limited role in determining the signal precision. Furthermore, in a dual loop system of one positive feedback and one negative feedback, a slower positive feedback always leads to better noise attenuation; in contrast, a slower negative feedback may not be more beneficial. Our results reveal interesting stochastic effects for systems containing both extrinsic and intrinsic noises, suggesting novel noise filtering strategies in inherently stochastic systems.

The multiscale approach to N-body systems is generalized to address the broad continuum of long time and length scales associated with collective behaviors. A technique is developed based on the concept of an uncountable set of time variables and of order parameters (OPs) specifying major features of the system. We adopt this perspective as a natural extension of the commonly used discrete set of time scales and OPs which is practical when only a few, widely separated scales exist. The existence of a gap in the spectrum of time scales for such a system (under quasiequilibrium conditions) is used to introduce a continuous scaling and perform a multiscale analysis of the Liouville equation. A functional-differential Smoluchowski equation is derived for the stochastic dynamics of the continuum of Fourier component OPs. A continuum of spatially nonlocal Langevin equations for the OPs is also derived. The theory is demonstrated via the analysis of structural transitions in a composite material, as occurs for viral capsids and molecular circuits.

The alpha-stable distributions introduced by Lévy play an important role in probabilistic theoretical studies and their various applications, e.g., in statistical physics, life sciences, and economics. In the present paper we study sequences of long-range dependent random variables whose distributions have asymptotic power-law decay, and which are called (q,alpha)-stable distributions. These sequences are generalizations of independent and identically distributed alpha-stable distributions and have not been previously studied. Long-range dependent (q,alpha)-stable distributions might arise in the description of anomalous processes in nonextensive statistical mechanics, cell biology, finance. The parameter q controls dependence. If q=1 then they are classical independent and identically distributed with alpha-stable Lévy distributions. In the present paper we establish basic properties of (q,alpha)-stable distributions and generalize the result of Umarov et al. [Milan J. Math. 76, 307 (2008)], where the particular case alpha=2,q[1,3) was considered, to the whole range of stability and nonextensivity parameters alpha(0,2] and q[1,3), respectively. We also discuss possible further extensions of the results that we obtain and formulate some conjectures.

In this paper, we derive full wave solutions to the problem of electromagnetic wave propagation in inhomogeneous multilayered structures of arbitrarily varying thickness. To this end, we employ generalized transforms that provide an appropriate basis for the complete expansion of the transverse components of the electromagnetic fields. The continuous parts of the wavenumber spectrum are the radiation and the lateral wave terms while the discrete part is identified as the finite set of trapped waveguide modes (surface waves). When the bounding media are characterized by perfect electric or magnetic walls (μ/∈ → 0 or ∈/μ → 0, respectively) or surface impedances, the fields are expressed exclusively in terms of an infinite set of waveguide modes. These solutions are not restricted by the approximate surface impedance concept and the sources and observation point may be located in any of the nonunifonn layers of the structure. Exact boundary conditions are imposed and the solutions satisfy the reciprocity relationships. Thus, the solutions are applicable to artificial layered structures as well as natural structures such as the inhomogeneous ionosphere and the earth's crust. These solutions can also be used to determine the scattering from objects of finite cross section in free space or embedded in the earth's crust.

We consider phase space as the carrier space of canonical transformations and see that this implies, for nonbijective ones, a much subtler structure than the one commonly assumed. Discussing only problems of one degree of freedom, i.e., a phase plane, we are able to clarify this structure by analogy with the better-known situation of conformal mapping in the complex plane. Apart from the usual Riemann sheet concept, an alternate method is developed that involves the irreducible representations of the ambiguity group, i.e., the group of transformations that connects all points that are mapped on a single point by the conformal or canonical transformation. The algebra of variables then becomes a matrix algebra for which bijectiveness is retrieved.

In a 1967 paper [E. C. Zeeman, “The topology of Minkowski space,” Topology 6, 161–170 (1967)], Zeeman proposed a new topology for Minkowski space-time, physically motivated but much more complicated than the standard one. Here a detailed study is given of some properties of the Zeeman topology which had not been considered at the time. The general setting refers to Minkowski space-time of any dimension k+1. In the special case k=1, a full characterization is obtained for the compact subsets of space-time; moreover, the first homotopy group is shown to be nontrivial.

In supersymmetric quantum mechanics, the non-Abelian Berry phase is known to obey certain differential equations. Here we study N = (0, 4) systems and show that the non-Abelian Berry connection over R-4n satisfies a generalization of the self-dual Yang-Mills equations. Upon dimensional reduction, these become the tt* equations. We further study the Berry connection in N = (4, 4) theories and show that the curvature is covariantly constant. (C) 2010 American Institute of Physics. [doi:10.1063/1.3521497]

We point out that the {\em spacetime void} inferred by Castro[J. Math. Phys. 49, 042501, (2008)] results from his choice of a discontinuous radial gauge. Further since the integration constant $\alpha_0 = 2M_0$ ($G=c=1$) occurring in the vacuum Hilbert/Schwarzschild solution of a neutral "point mass" is zero [Arnowitt et al., in Gravitation: An Introduction to Current Research, ed. L. Witten, Wiley, Chap. 7, p.227; also Phys. Rev. Lett., 4, 375, (1960)]; A. Mitra, Adv. Sp. Res., 38, 2917 (2006)] Castro's gauge reduces to the well behaved and physical Hilbert gauge. Physically this means that true Hilbert/Schwarzschild black holes have unique gravitational mass M=0. Accordingly, the unphysical {\em spacetime viod} inferred by Castro is actually non-existent. Comment: Published in J. Math. Phys. General Relativity Section

In earlier work, carrying out numerical simulations of the Ricci flow of families of rotationally symmetric geometries on $S3$, we have found strong support for the contention that (at least in the rotationally symmetric case) the Ricci flow for a critical'' initial geometry - one which is at the transition point between initial geometries (on $S^3$) whose volume-normalized Ricci flows develop a singular neck pinch, and other initial geometries whose volume normalized Ricci flows converge to a round sphere - evolves into a degenerate neck pinch.'' That is, we have seen in this earlier work that the Ricci flows for the critical geometries become locally cylindrical in a neighborhood of the initial pinching, and have the maximum amount of curvature at one or both of the poles. Here, we explore the behavior of these flows at the poles, and find strong support for the conjecture that the Bryant steady solitons accurately model this polar flow. Comment: 17 pages, 5 figures

As shown in [hep-th/0406065], there exists a noncommutative deformation of the sine-Gordon model which remains (classically) integrable but features a second scalar field. We employ the dressing method (adapted to the Moyal-deformed situation) for constructing the deformed kinkantikink and breather configurations. Explicit results and plots are presented for the leading noncommutativity correction to the breather. Its temporal periodicity is unchanged. 1 Introduction and summary The sine-Gordon model is a paradigm for relativistic integrable models in 1+1 dimensions (e.g., see [1].) Its multi-soliton spectrum is well known and consists not only of multi-kink scattering configurations but also of bound states, the simplest of which is the so-called breather. It may be obtained formally by analytically continuing the kink-antikink configuration in its relative velocity

The character problems of SU(2) and SU(1,1) are reexamined from the standpoint of a physicist by employing the Hilbert space method which is shown to yield a completely unified treatment for SU(2) and the discrete series of representations of SU(1,1). For both the groups the problem is reduced to the evaluation of an integral which is invariant under rotation for SU(2) and Lorentz transformation for SU(1,1). The integrals are accordingly evaluated by applying a rotation to a unit position vector in SU(2) and a Lorentz transformation to a unit SO(2,1) vector which is time-like for the elliptic elements and space-like for the hyperbolic elements in SU(1,1). The details of the procedure for the principal series of representations of SU(1,1) differ substantially from those of the discrete series. Comment: 31 pages, RevTeX, typos corrected. To be published in Journal of Mathematical Physics

The character of the exceptional series of representations of SU(1,1) is determined by using Bargmann's realization of the representation in the Hilbert space $H_\sigma$ of functions defined on the unit circle. The construction of the integral kernel of the group ring turns out to be especially involved because of the non-local metric appearing in the scalar product with respect to which the representations are unitary. Since the non-local metric disappears in the momentum space' $i.e.$ in the space of the Fourier coefficients the integral kernel is constructed in the momentum space, which is transformed back to yield the integral kernel of the group ring in $H_\sigma$. The rest of the procedure is parallel to that for the principal series treated in a previous paper. The main advantage of this method is that the entire analysis can be carried out within the canonical framework of Bargmann.

We derive the matrix elements of generators of unitary irreducible representations of SL(2,C) with respect to basis states arising from a decomposition into irreducible representations of SU(1,1). This is done with regard to a discrete basis diagonalized by J^3 and a continuous basis diagonalized by K^1, and for both the discrete and continuous series of SU(1,1). For completeness we also treat the more conventional SU(2) decomposition as a fifth case. The derivation proceeds in a functional / differential framework and exploits the fact that state functions and differential operators have a similar structure in all five cases. The states are defined explicitly and related to SU(1,1) and SU(2) matrix elements.

We solve the operator ordering problem for the quantum continuous integrable su(1,1) Landau-Lifshitz model, and give a prescription to obtain the quantum trace identities, and the spectrum for the higher-order local charges. We also show that this method, based on operator regularization and renormalization, which guarantees quantum integrability, as well as the construction of self-adjoint extensions, can be used as an alternative to the discretization procedure, and unlike the latter, is based only on integrable representations.

Positive discrete series representations of the Lie algebra $su(1,1)$ and the quantum algebra $U_q(su(1,1))$ are considered. The diagonalization of a self-adjoint operator (the Hamiltonian) in these representations and in tensor products of such representations is determined, and the generalized eigenvectors are constructed in terms of orthogonal polynomials. Using simple realizations of $su(1,1)$, $U_q(su(1,1))$, and their representations, these generalized eigenvectors are shown to coincide with generating functions for orthogonal polynomials. The relations valid in the tensor product representations then give rise to new generating functions for orthogonal polynomials, or to Poisson kernels. In particular, a group theoretical derivation of the Poisson kernel for Meixner-Pollaczak and Al-Salam--Chihara polynomials is obtained.

We study the natural Poisson structure on the Lie group SU(1,1) and related questions. In particular, we give an explicit description of the Ginzburg-Weinstein isomorphism for the sets of admissible elements. We also establish an analogue of Thompson's conjecture for this group.

We construct the Perelomov number coherent states for any three $su(1,1)$ Lie algebra generators and study some of their properties. We introduce three operators which act on Perelomov number coherent states and close the $su(1,1)$ Lie algebra. We show that the most general $SU(1,1)$ coherence-preserving Hamiltonian has the Perelomov number coherent states as eigenfunctions, and we obtain their time evolution. We apply our results to obtain the non-degenerate parametric amplifier eigenfunctions, which are shown to be the Perelomov number coherent states of the two-dimensional harmonic oscillator.

The osp(1,2)-covariant Lagrangian quantization of irreducible gauge theories [hep-th/9712204] is generalized to L-stage reducible theories. The dependence of the generating functional of Green's functions on the choice of gauge in the massive case is dicussed and Ward identities related to osp(1,2) symmetry are given. Massive first stage theories with closed gauge algebra are studied in detail. The generalization of the Chapline-Manton model and topological Yang-Mills theory to the case of massive fields is consedered as examples. Comment: 24 pages, AMSTEX, some minor improvements, 2 additional references

The osp(1,2)-covariant Lagrangian quantization of general gauge theories is formulated which applies also to massive fields. The formalism generalizes the Sp(2)-covariant BLT approach and guarantees symplectic invariance of the quantized action. The dependence of the generating functional of Green's functions on the choice of gauge in the massive case disappears in the limit m = 0. Ward identities related to osp(1,2) symmetry are derived. Massive gauge theories with closed algebra are studied as an example.

We define the $osp(1,2)$ Gaudin algebra and consider integrable models described by it. The models include the $osp(1,2)$ Gaudin magnet and the Dicke model related to it. Detailed discussion of the simplest cases of these models is presented. The effect of the presence of fermions on the separation of variables is indicated. Comment: 16 pages, LaTeX, DAMTP/93-60

Born proposed a unification of special relativity and quantum mechanics that placed position, time, energy and momentum on equal footing through a reciprocity principle and extended the usual position-time and energy-momentum line elements to this space by combining them through a new fundamental constant. Requiring also invariance of the symplectic metric yields U(1,3) as the invariance group, the inhomogeneous counterpart of which is the canonically relativistic group CR(1,3) = U(1,3) *s H(1,3) where H(1,3) is the Heisenberg Group in 4 dimensions and "*s" is the semidirect product. This is the counterpart in this theory of the Poincare group and reduces in the appropriate limit to the expected special relativity and classical Hamiltonian mechanics transformation equations. This group has the Poincare group as a subgroup and is intrinsically quantum with the Position, Time, Energy and Momentum operators satisfying the Heisenberg algebra. The representations of the algebra are studied and Casimir invariants are computed. Like the Poincare group, it has a little group for a ("massive") rest frame and a null frame. The former is U(3) which clearly contains SU(3) and the latter is Os(2) which contains SU(2)*U(1). Comment: 18 pages, PDF, to be published in J. Math. Phys., Mathematica 3.0 computation files available from author at slow@austx.tandem.com

The primitive elements of the supersymmetry algebra cohomology as defined in previous work are derived for standard supersymmetry algebras in dimensions D=5,...,11 for all signatures of the related Clifford algebras of gamma matrices and all numbers of supersymmetries. The results are presented in a uniform notation along with results of previous work for D=4, and derived by means of dimensional extension from D=4 up to D=11.

The total activity of the single-seeded cellular rule 150 automaton does not follow a one-step iteration like other elementary cellular automata, but can be solved as a two-step vectorial, or string, iteration, which can be viewed as a generalization of Fibonacci iteration generating the time series from a sequence of vectors of increasing length. This allows to compute the total activity time series more efficiently than by simulating the whole spatio-temporal process, or even by using the closed expression. Comment: 4 pages (3 figs included)

In this paper we investigate the one-dimensional parabolic-parabolic Patlak-Keller-Segel model of chemotaxis. For the case when the diffusion coefficient of chemical substance is equal to two, in terms of travelling wave variables the reduced system appears integrable and allows the analytical solution. We obtain the exact soliton solutions, one of which is exactly the one-soliton solution of the Korteweg-de Vries equation.

We consider a simple modification of the 1D-Laplacian where non-mixed interface conditions occur at the boundaries of a finite interval. It has recently been shown that Schr\"odinger operators having this form allow a new approach to the transverse quantum transport through resonant heterostructures. In this perspective, it is important to control the deformations effects introduced on the spectrum and on the time propagator by this class of non-selfadjont perturbations. In order to obtain uniform-in-time estimates of the perturbed semigroup, our strategy consists in constructing stationary waves operators allowing to intertwine the modified non-selfadjoint Schr\"odinger operator with a 'physical' Hamiltonian. For small values of a deformation parameter '{\theta}', this yields a dynamical comparison between the two models showing that the distance between the corresponding semigroups is dominated by |{\theta}| uniformly in time in the L^2-operator norm.

We prove that the asymptotic behavior of the second mixed moment of the characteristic polynomials of the 1D Gaussian real symmetric band matrices coincides with those for the Gaussian Orthogonal Ensemble (GOE). Here we adapt the approach of T. Shcherbina'14, where the case of 1D Hermitian random band matrices was considered.

A lattice model of radiative decay (so-called spin-boson model) of a two level atom and at most two photons is considered. The location of the essential spectrum is described. For any coupling constant the finiteness of the number of eigenvalues below the bottom of its essential spectrum is proved. The results are obtained by considering a more general model $H$ for which the lower bound of its essential spectrum is estimated. Conditions which guarantee the finiteness of the number of eigenvalues of $H,$ below the bottom of its essential spectrum are found. It is shown that the discrete spectrum might be infinite if the parameter functions are chosen in a special form.

The issue discussed is a thermodynamic version of the Bern-Kosower master amplitude formula, which contains all necessary one-loop Feynman diagrams. It is demonstrated how the master amplitude at finite values of temperature and chemical potential can be formulated within the framework of the world-line formalism. In particular we present an elegant method how to introduce a chemical potential for a loop in the master formula. Various useful integral formulae for the master amplitude are then obtained. The non-analytic property of the master formula is also derived in the zero temperature limit with the value of chemical potential kept finite.

We investigate the concept of superconformal symmetry in six dimensions, applied to the interacting theory of (2,0) tensor multiplets and self-dual strings. The action of a superconformal transformation on the superspace coordinates is found, both from a six-dimensional perspective and by using a superspace with eight bosonic and four fermionic dimensions. The transformation laws for all fields in the theory are derived, as well as general expressions for the transformation of on-shell superfields. Superconformal invariance is shown for the interaction of a self-dual string with a background consisting of on-shell tensor multiplet fields, and we also find an interesting relationship between the requirements of superconformal invariance and those of a local fermionic kappa-symmetry. Finally, we try to construct a superspace analogue of the Poincare dual to the string world-sheet and consider its properties under superconformal transformations. Comment: 31 pages, LaTeX. v2: clarifications and minor corrections

In this paper, we address the issue of quaternionic Toledo invariant to study the character variety of two dimensional complex hyperbolic uniform lattices into $SU(4,2)$. We construct four distinct representations to prove that the character variety contains at least four distinct components. We also address the holomorphic horizontal liftability to various period domains of $SU(4,2)$.

The family F of all potentials V(x) for which the Hamiltonian H in one space dimension possesses a high order Lie symmetry is determined. A sub-family F', which contains a class of potentials allowing a realization of so(2,1) as spectrum generating algebra of H through differential operators of finite order, is identified. Furthermore and surprisingly, the families F and F' are shown to be related to the stationary KdV hierarchy. Hence, the "harmless" Hamiltonian H connects different mathematical objects, high order Lie symmetry, realization of so(2,1)-spectrum generating algebra and families of nonlinear differential equations. We describe in a physical context the interplay between these objects. Comment: 15 pages, LaTeX

We revise the unireps. of $U(2,2)$ describing conformal particles with continuous mass spectrum from a many-body perspective, which shows massive conformal particles as compounds of two correlated massless particles. The statistics of the compound (boson/fermion) depends on the helicity $h$ of the massless components (integer/half-integer). Coherent states (CS) of particle-hole pairs ("excitons") are also explicitly constructed as the exponential action of exciton (non-canonical) creation operators on the ground state of unpaired particles. These CS are labeled by points $Z$ ($2\times 2$ complex matrices) on the Cartan-Bergman domain $\mathbb D_4=U(2,2)/U(2)^2$, and constitute a generalized (matrix) version of Perelomov $U(1,1)$ coherent states labeled by points $z$ on the unit disk $\mathbb D_1=U(1,1)/U(1)^2$. Firstly we follow a geometric approach to the construction of CS, orthonormal basis, $U(2,2)$ generators and their matrix elements and symbols in the reproducing kernel Hilbert space $\mathcal H_\lambda(\mathbb D_4)$ of analytic square-integrable holomorphic functions on $\mathbb D_4$, which carries a unitary irreducible representation of $U(2,2)$ with index $\lambda\in\mathbb N$ (the conformal or scale dimension). Then we introduce a many-body representation of the previous construction through an oscillator realization of the $U(2,2)$ Lie algebra generators in terms of eight boson operators with constraints. This particle picture allows us for a physical interpretation of our abstract mathematical construction in the many-body jargon. In particular, the index $\lambda$ is related to the number $2(\lambda-2)$ of unpaired quanta and to the helicity $h=(\lambda-2)/2$ of each massless particle forming the massive compound.

In [F. Finster, “Derivation of local gauge freedom from a measurement principle”, Preprint funct-an/9701002 (1997)] it was suggested to link the physical gauge principle with quantum mechanical measurements of the position variable. In the present paper, the author extends this concept to relativistic quantum mechanics and applies it to the Dirac equation. For Dirac spinors, he obtains local U(2,2) gauge freedom. His main result is that this U(2,2) symmetry allows a natural description of both electrodynamics and general relativity as a classical gauge theory. This is shown by deriving a U(2,2) spin connection from the Dirac operator and analyzing the geometry of this connection. Although he develops the subject from a particular point of view, this paper can be used as an introduction to the Dirac theory in curved space-time.

We show that subsingular vectors exist in Verma modules over W(2,2), and present a subquotient structure of these modules. We prove conditions for irreducibility of a tensor product of intermediate series module with the highest weight module. Relations to intertwining operators over vertex operator algebra associated to W(2,2) is discussed. Also, we study a tensor product of intermediate series and highest weight module over the twisted Heisenberg-Virasoro algebra, and present series of irreducible modules with infinite-dimensional weight spaces.

We study the minimal unitary representation (minrep) of SO(4,2) over an Hilbert space of functions of three variables, obtained by quantizing its quasiconformal action on a five dimensional space. The minrep of SO(4,2), which coincides with the minrep of SU(2,2) similarly constructed, corresponds to a massless conformal scalar in four spacetime dimensions. There exists a one-parameter family of deformations of the minrep of SU(2,2). For positive (negative) integer values of the deformation parameter \zeta one obtains positive energy unitary irreducible representations corresponding to massless conformal fields transforming in (0,\zeta/2) ((-\zeta/2,0)) representation of the SL(2,C) subgroup. We construct the supersymmetric extensions of the minrep of SU(2,2) and its deformations to those of SU(2,2|N). The minimal unitary supermultiplet of SU(2,2|4), in the undeformed case, simply corresponds to the massless N=4 Yang-Mills supermultiplet in four dimensions. For each given non-zero integer value of \zeta, one obtains a unique supermultiplet of massless conformal fields of higher spin. For SU(2,2|4) these supermultiplets are simply the doubleton supermultiplets studied in arXiv:hep-th/9806042. Comment: Revised with an extended introduction and additional references. Typos corrected. 49 pages; Latex file

We show that the support of an irreducible weight module over the $W$-algebra $W(2, 2)$, which has an infinite dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the the $W$-algebra $W(2, 2)$, having a nontrivial finite dimensional weight space, is a Harish-Chandra module (and hence is either an irreducible highest or lowest weight module or an irreducible module of the intermediate series). Comment: 10 pages

For every $p \geq 2$, we obtained an explicit construction of a family of $\mathcal{W}(2,2p-1)$-modules, which decompose as direct sum of simple Virasoro algebra modules. Furthermore, we classified all irreducible self-dual $\mathcal{W}(2,2p-1)$-modules, we described their internal structure, and computed their graded dimensions. In addition, we constructed certain hidden logarithmic intertwining operators among two ordinary and one logarithmic $\mathcal{W}(2,2p-1)$-modules. This work, in particular, gives a mathematically precise formulation and interpretation of what physicists have been referring to as "logarithmic conformal field theory" of central charge $c_{p,1}=1-\frac{6(p-1)^2}{p}, p \geq 2$. Our explicit construction can be easily applied for computations of correlation functions. Techniques from this paper can be used to study the triplet vertex operator algebra $\mathcal{W}(2,(2p-1)^3)$ and other logarithmic models.

In this paper we continue the study of representation theory of formal distribution Lie superalgebras initiated in q-alg/9706030. We study finite Verma-type conformal modules over the N=2, N=3 and the two N=4 superconformal algebras and also find explicitly all singular vectors in these modules. From our analysis of these modules we obtain a complete list of finite irreducible conformal modules over the N=2, N=3 and the two N=4 superconformal algebras. Comment: 36 pages, no figures, LaTeX format

The linear (homogeneous and inhomogeneous) (k, N, N-k) supermultiplets of the N-extended one-dimensional Supersymmetry Algebra induce D-module representations for the N=2,4,8 superconformal algebras. For N=2, the D-module representations of the A(1,0) superalgebra are obtained. For N=4 and scaling dimension \lambda=0, the D-module representations of the A(1,1) superalgebra are obtained. For $\lambda\neq 0$, the D-module representations of the D(2,1;\alpha) superalgebras are obtained, with $\alpha$ determined in terms of the scaling dimension $\lambda$ according to: $\alpha=-2\lambda$ for k=4, i.e. the (4,4) supermultiplet, $\alpha=-\lambda$ for k=3, i.e. (3,4,1), and $\alpha=\lambda$ for k=1, i.e. (1,4,3). For $\lambda\neq 0$ the (2,4,2) supermultiplet induces a D-module representation for the centrally extended sl(2|2) superalgebra. For N=8, the (8,8) root supermultiplet induces a D-module representation of the D(4,1) superalgebra at the fixed value $\lambda=1/4$. A Lagrangian framework to construct one-dimensional, off-shell, superconformal invariant actions from single-particle and multi-particles D-module representations is discussed. It is applied to explicitly construct invariant actions for the homogeneous and inhomogeneous N=4 (1,4,3) D-module representations (in the last case for several interacting supermultiplets of different chirality).

An error in the paper [J. Math. Phys. 43, 6343 (2002); math-ph/0207009] is corrected. Further explanation is given.

I consider the $\mathbb{Z}_\lambda,$ $\lambda$ prime free-bosonic permutation orbifolds as interacting physical string systems at $\hat{c} = 26\lambda$. As a first step, I introduce twisted tree diagrams which confirm at the interacting level that the physical spectrum of each twisted sector is equivalent to that of an ordinary $c=26$ closed string. The untwisted sectors are surprisingly more difficult to understand, and there are subtleties in the sewing of the loops, but I am able to propose provisional forms for the full modular-invariant cosmological constants and one-loop diagrams with insertions.

$E_6$ is an attractive group for unification model building. However, the complexity of a rank 6 group makes it non-trivial to write down the structure of higher dimensional operators in an $E_6$ theory in terms of the states labeled by quantum numbers of the Standard Model gauge group. In this paper, we show the results of our computation of the Clebsch-Gordan coefficients for the products of the {\bf 27} with irreducible representations of higher dimensionality: ${\bf 78}$, ${\bf 351}$, ${\bf 351^\prime}$, ${\bf \ol{351}}$, and ${\bf \ol{351^\prime}}$. Application of these results to $E_6$ model building involving higher dimensional operators is straightforward.

We consider an isotropic two dimensional harmonic oscillator with arbitrarily time-dependent mass $M(t)$ and frequency $\Omega(t)$ in an arbitrarily time-dependent magnetic field $B(t)$. We determine two commuting invariant observables (in the sense of Lewis and Riesenfeld) $L,I$ in terms of some solution of an auxiliary ordinary differential equation and an orthonormal basis of the Hilbert space consisting of joint eigenvectors $\phi_\lambda$ of $L,I$. We then determine time-dependent phases $\alpha_\lambda(t)$ such that the $\psi_\lambda(t)=e^{i\alpha_\lambda}\phi_\lambda$ are solutions of the time-dependent Schr\"odinger equation and make up an orthonormal basis of the Hilbert space. These results apply, in particular to a two dimensional Landau problem with time-dependent $M,B$, which is obtained from the above just by setting $\Omega(t) \equiv 0$. By a mere redefinition of the parameters, these results can be applied also to the analogous models on the canonical non-commutative plane.

In this paper, we study a linearized two-dimensional Euler equation. This equation decouples into infinitely many invariant subsystems. Each invariant subsystem is shown to be a linear Hamiltonian system of infinite dimensions. Another important invariant besides the Hamiltonian for each invariant subsystem is found, and is utilized to prove an unstable disk theorem'' through a simple Energy-Casimir argument. The eigenvalues of the linear Hamiltonian system are of four types: real pairs ($c,-c$), purely imaginary pairs ($id,-id$), quadruples ($\pm c\pm id$), and zero eigenvalues. The eigenvalues are computed through continued fractions. The spectral equation for each invariant subsystem is a Poincar\'{e}-type difference equation, i.e. it can be represented as the spectral equation of an infinite matrix operator, and the infinite matrix operator is a sum of a constant-coefficient infinite matrix operator and a compact infinite matrix operator. We have obtained a complete spectral theory.

Using the holographic machinery built up in a previous work, we show that the hidden SL(2,R) symmetry of a scalar quantum field propagating in a Rindler spacetime admits an enlargement in terms of a unitary positive-energy representation of Virasoro algebra, with central charge c=1, defined in the Fock representation. The Virasoro algebra of operators gets a manifest geometrical meaning if referring to the holographically associated QFT on the horizon: It is nothing but a representation of the algebra of vector fields defined on the horizon equipped with a point at infinity. All that happens provided the Virasoro ground energy h vanishes and, in that case, the Rindler Hamiltonian is associated with a certain Virasoro generator. If a suitable regularization procedure is employed, for h=1/2, the ground state of that generator corresponds to thermal states when examined in the Rindler wedge, taking the expectation value with respect to Rindler time. This state has inverse temperature 1/(2beta), where beta is the parameter used to define the initial SL(2,R) unitary representation. (As a consequence the restriction of Minkowski vacuum to Rindler wedge is obtained by fixing h=1/2 and 2beta=beta_U, the latter being Unruh's inverse temperature). Finally, under Wick rotation in Rindler time, the pair of QF theories which are built up on the future and past horizon defines a proper two-dimensional conformal quantum field theory on a cylinder.

Langmuir-Blodgett films (LB-films) consist from few LB-monolayers which are high structured nanomaterials that are very promising materials for applications. We use a geometrical approach to describe structurization into LB-monolayers. Consequently, we develop on the 1-jet space J^1([0,\infty),R^2) the single-time Lagrange geometry (in the sense of distinguished (d-) connection, d-torsions and an abstract anisotropic electromagnetic-like d-field) for the Lagrangian governing the 2D-motion of a particle of monolayer. One assumed that an expansion near singular points for the constructed geometrical Lagrangian theory describe phase transitions to LB-monolayer. Trajectories of particles in a field of the electrocapillarity forces of monolayer have been calculated in a resonant approximation utilizing some Jacobi equations. A jet geometrical Yang-Mills energy is introduced and some physical interpretations are given.

It is known that the chiral part of any 2d conformal field theory defines a 3d topological quantum field theory: quantum states of this TQFT are the CFT conformal blocks. The main aim of this paper is to show that a similar CFT/TQFT relation exists also for the full CFT. The 3d topological theory that arises is a certain square'' of the chiral TQFT. Such topological theories were studied by Turaev and Viro; they are related to 3d gravity. We establish an operator/state correspondence in which operators in the chiral TQFT correspond to states in the Turaev-Viro theory. We use this correspondence to interpret CFT correlation functions as particular quantum states of the Turaev-Viro theory. We compute the components of these states in the basis in the Turaev-Viro Hilbert space given by colored 3-valent graphs. The formula we obtain is a generalization of the Verlinde formula. The later is obtained from our expression for a zero colored graph. Our results give an interesting holographic'' perspective on conformal field theories in 2 dimensions. Comment: 29+1 pages, many figures

The Hamilton-Jacobi formalism generalized to 2-dimensional field theories according to Lepage's canonical framework is applied to several relativistic real scalar fields, e.g. massless and massive Klein-Gordon, Sinh and Sine-Gordon, Liouville and $\phi^4$ theories. The relations between the Euler-Lagrange and the Hamilton-Jacobi equations are discussed in DeDonder and Weyl's and the corresponding wave fronts are calculated in Carath\'eodory's formulation. Unlike mechanics we have to impose certain integrability conditions on the velocity fields to guarantee the transversality relations and especially the dynamical equivalence between Hamilton-Jacobi wave fronts and families of extremals embedded therein. B\"acklund Transformations play a crucial role in solving the resulting system of coupled nonlinear PDEs. Comment: LaTeX, 44 pages, 3 figures, submitted to: Journal of Mathematical Physics; replaced due to mailer problems

Finite-dimensional reductions of the 2D dispersionless Toda hierarchy, constrained by the string equation'' are studied. These include solutions determined by polynomial, rational or logarithmic functions, which are of interest in relation to the `Laplacian growth'' problem governing interface dynamics. The consistency of such reductions is proved, and the Hamiltonian structure of the reduced dynamics is derived. The Poisson structure of the rationally reduced dispersionless Toda hierarchies is also derived

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• The University of York
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• Johannes Kepler University Linz
• Deutsches Elektronen-Synchrotron