Trace anomaly and quasi-particles in finite temperature <Emphasis Type="Italic">SU</Emphasis>(<Emphasis Type="Italic">N</Emphasis>) gauge theory

We consider deconfined matter in SU(N) gauge theory as an ideal gas of transversely polarized quasi-particle modes having a temperature-dependent mass m(T). Just above the transition temperature, the mass is assumed to be determined by the critical behavior of the energy density and the screening length in the medium. At high temperature, it becomes proportional to T as the only remaining scale. The resulting (trace anomaly based) interaction measure Δ=(ϵ−3P)/T ⁴ and energy density are found to agree well with finite temperature SU(3) lattice calculations.     

Polynomial Realization of <Emphasis Type="Italic">s</Emphasis>l<Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>(2) and Fusion Rules at Exceptional Values of <Emphasis Type="Italic">q</Emphasis>

Representations of the sℓ_q(2) algebra are constructed in the space of polynomials of real (complex) variable for q^N=1. The spin addition rule based on eigenvalues of Casimir operator is illustrated on few simplest cases and conjecture for general case is formulated.     

Symmetry Transitions in Random Matrix Theory &; <Emphasis Type="Italic">L</Emphasis>-functions

We compute the moments of the characteristic polynomials of random orthogonal and symplectic matrices, defined by averages with respect to Haar measure on SO(2N) and USp(2N), to leading order as N → ∞, on the unit circle as functions of the angle θ measured from one of the two symmetry points in the eigenvalue spectrum . Our results extend previous formulae that relate just to the symmetry points, i.e. to θ = 0. Local spectral statistics are expected to converge to those of random unitary matrices in the limit as N → ∞ when θ is fixed, and to show a transition from the orthogonal or symplectic to the unitary forms on the scale of the mean eigenvalue spacing: if θ = π y/N they become functions of y in the limit when N → ∞. We verify that this is true for the spectral two-point correlation function, but show that it is not true for the moments of the characteristic polynomials, for which the leading order asymptotic approximation is a function of θ rather than y. Symmetry points therefore influence the moments asymptotically far away on the scale of the mean eigenvalue spacing. We also investigate the moments of the logarithms of the characteristic polynomials in the same context. The moments of the characteristic polynomials of random matrices are conjectured to be related to the moments of families of L-functions. Previously, moments at the symmetry point θ = 0 have been related to the moments of families of L-functions evaluated at the centre of the critical strip. Our results motivate general conjectures for the moments of orthogonal and symplectic families of L-functions evaluated at a fixed height t up the critical line. These conjectures suggest that the symmetry of the non-trivial zeros of the L-functions influences the moments asymptotically far, on the scale of the mean zero spacing, from the centre of the critical strip. We verify that the second moments of real quadratic Dirichlet L-functions and a family of automorphic L-functions are consistent with our conjectures. JPK is supported by an EPSRC Senior Research Fellowship. BEO was supported by an Overseas Research Scholarship and a University of Bristol Research Scholarship.     

<Emphasis Type="Italic">SU</Emphasis>(3)-Instantons and <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>,<Emphasis Type="Italic">Spin</Emphasis>(7)-Heterotic String Solitons

Necessary and sufficient conditions to the existence of a hermitian connection with totally skew-symmetric torsion and holonomy contained in SU(3) are given. A formula for the Riemannian scalar curvature is obtained. Non-compact solution to the supergravity-type I equations of motion with non-zero flux and non-constant dilaton is found in dimension 6. Non-conformally flat non-compact solutions to the supergravity-type I equations of motion with non-zero flux and non-constant dilaton are found in dimensions 7 and 8. A Riemannian metric with holonomy contained in G₂ arises from our considerations and Hitchin’s flow equations, which seems to be new. Compact examples of SU(3),G₂ and Spin(7) instanton satisfying the anomaly cancellation conditions are presented.     

<Emphasis Type="Italic">SU</Emphasis>(2|1) supersymmetric mechanics as a deformation of <Emphasis Type="Italic">N</Emphasis> = 4 mechanics

We give a brief review of SU(2|1) supersymmetric quantum mechanics based on the worldline realizations of the supergroup SU(2|1) in the appropriate N = 4, d = 1 superspaces. The corresponding SU(2|1) models are deformations of standard N = 4, d = 1 models by a mass parameter m.     

A Pseudo-cocycle for the Comultiplication on the Quantum <Emphasis Type="Italic">SU</Emphasis>(<Emphasis Type="Italic">2</Emphasis>) Group

We show that the comultiplication on the quantum group SU _q (2) may be obtained from that on the quantum semigroup SU ₀(2) by twisting with a unitary 2-pseudo-cocycle. Work supported by the ARC Linkage International Fellowship LX0667294, and by the Korea Research Foundation Grant (KRF-2004-041-C00024).     

Density of Complex Zeros of a System of Real Random Polynomials

We study the density of complex zeros of a system of real random SO(m+1) polynomials in m variables. We show that the density of complex zeros of this random polynomial system with real coefficients rapidly approaches the density of complex zeros in the complex coefficients case. We also show that the behavior the scaled density of complex zeros near ℝ ^m of the system of real random polynomials is different in the m≥2 case than in the m=1 case: the density approaches infinity instead of tending linearly to zero.     

On the degeneracy of <Emphasis Type="Italic">SU</Emphasis>(3)<Subscript><Emphasis Type="Italic">k</Emphasis></Subscript> topological phases

The ground state degeneracy of an SU(N) _k topological phase with n quasiparticle excitations is a relevant quantity for quantum computation, condensed matter physics, and knot theory. It is an open question to find a closed formula for this degeneracy for any N >2. Here we present the problem in an explicit combinatorial way and analyze the case N = 3. While not finding a complete closed-form solution, we obtain generating functions and solve some special cases.     

The Dirac Operator on <Emphasis Type="Italic">SU</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>(2)

We construct a 3⁺-summable spectral triple over the quantum group SU_q(2) which is equivariant with respect to a left and a right action of The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operator on the 3-dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and first-order properties need be satisfied only modulo infinitesimals of arbitrary high order.Partially supported by Polish State Committee for Scientific Research (KBN) under grant 2 P03B 022 25.Regular Associate of the Abdus Salam ICTP, Trieste.     

Contracted Hamiltonian on Symmetric Space <Emphasis Type="Italic">SU</Emphasis>(3)/<Emphasis Type="Italic">SU</Emphasis>(2) and Conserved Quantities

We study the quantum model on symmetric space SU(3)/SU(2). By using the Inonu-Wigner contraction to Lie algebra su(3), we arrive at a special case of three-body Sutherland model. It has shown that by calculating conservative quantities of this model, it has Poincare Lie algebra, too.     

Entropy for <Emphasis Type="Italic">SU</Emphasis>(3)<Subscript><Emphasis Type="Italic">c</Emphasis></Subscript> quark states

We discuss the quantum state structure using the standard model for three colored quarks in the fundamental representations of SU(3)_c making up the singlet ground state of the hadrons. This allows us to calculate a finite von Neumann entropy from the quantum reduced density matrix, which we explicitly evaluate for the quarks in a model for the meson and baryon states.Received: 9 December 2003, Revised: 23 January 2004, Published online: 8 April 2004D.E. Miller: om0@psu.eduPermanent address: Department of Physics, Pennsylvania State University, Hazleton Campus, Hazleton, Pennsylvania, 18201 USA     

<Emphasis Type="Italic">SO</Emphasis>(3,1)-Invariant Approach to Dipole Radiation

The aim of the present work is to revisit the theory of the dipole radiation, within an SO(3,1)-gauge invariant formulation, by solving the Maxwell equations. Thus, we obtain the two interconnected components, A _B , B=1,2, of the vector potential A, in terms of Hankel and Legendre polynomials. Finally, for the pure dipole-like radiation, the observables, regarded as phasors, the Umov–Poynting vector components and the well-known Larmor formula for the effective radiated power are explicitly derived.     

<Emphasis Type="Italic">SU</Emphasis>(2) x <Emphasis Type="Italic">SU</Emphasis>(2) nonlocal quark model with confinement

The nonlocal version of the SU(2) x SU(2) symmetric four-quark interaction of the NJL type is considered. Each of the quark lines contains the form factors. These form factors remove the ultraviolet divergences in quark loops. The additional condition on the quark mass function m(p) ensures the absence of the poles in the quark propagator (quark confinement). The constituent-quark mass m(0) is expressed through the cut-off parameter , MeV in the chiral limit. These parameters are fixed by the experimental value of the weak pion decay and allow us to describe the mass of the light scalar meson, the strong decay and the D/S ratio in the decay in satisfactory agreement with the experimental data.Received: 16 June 2003, Revised: 28 July 2003, Published online: 18 December 2003PACS: 14.40.-n Mesons - 11.10.Lm Nonlinear or nonlocal theories and models - 12.39.Ki Relativistic quark model     

Nested bethe ansatz for <Emphasis Type="Italic">y</Emphasis>(<Emphasis Type="Italic">gl</Emphasis>(n)) open spin chains with diagonal boundary conditions

In this proceeding we present the nested Bethe ansatz for open spin chains of XXX-type, with arbitrary representations (i.e. “spins”) on each site of the chain and diagonal boundary matrices (K ⁺(u), K ⁻(u)). The nested Bethe ansatz applies for a general K ⁻(u), but a particular form of the K ⁺(u) matrix. We give the eigenvalues, Bethe equations and the form of the Bethe vectors for the corresponding models. The Bethe vectors are expressed using a trace formula     

Three-Qubit Entangled Embeddings of <Emphasis Type="Italic">CPT</Emphasis> and Dirac Groups within <Emphasis Type="Italic">E</Emphasis><Subscript>8</Subscript> Weyl Group

In quantum information context, the groups generated by Pauli spin matrices, and Dirac gamma matrices, are known as the single qubit Pauli group ℘, and two-qubit Pauli group ℘₂, respectively. It has been found (Socolovsky, Int. J. Theor. Phys. 43: 1941, 2004) that the CPT group of the Dirac equation is isomorphic to ℘. One introduces a two-qubit entangling orthogonal matrix S basically related to the CPT symmetry. With the aid of the two-qubit swap gate, the S matrix allows the generation of the three-qubit real Clifford group and, with the aid of the Toffoli gate, the Weyl group W(E ₈) is generated (Planat, Preprint , 2009). In this paper, one derives three-qubit entangling groups [(P)\tilde]\tilde{\mathcal{P}} and [(P)\tilde]₂\tilde{\mathcal{P}}_{2}, isomorphic to the CPT group ℘ and to the Dirac group ℘₂, that are embedded into W(E ₈). One discovers a new class of pure three-qubit quantum states with no-vanishing concurrence and three-tangle that we name CPT states. States of the GHZ and CPT families, and also chain-type states, encode the new representation of the Dirac group and its CPT subgroup.     

Multiple two-variable <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">q-L</Emphasis>-function and its behavior at <Emphasis Type="Italic">s</Emphasis> = 0

The objective of this paper is to construct a multiple p-adic q-L-function of two variables which interpolates multiple generalized q-Bernoulli polynomials. By using this function, we solve a question of Kim and Cho. We also define a multiple partial q-zeta function which is related to the multiple q-L-function of two variables. Finally, we give a finite-sum representation of the multiple p-adic q-L-function of two variables and prove a multiple q-extension of the generalized formula of Diamond and Ferrero-Greenberg.     

A reanalysis of finite temperature <Emphasis Type="Italic">SU</Emphasis>(<Emphasis Type="Italic">N</Emphasis>) gauge theory

We revise the SU(N _c ), N _c =3,4,6, lattice data on pure gauge theories at finite temperature by means of a quasi-particle approach. In particular, we focus on the relation between the effective mass of the quasi-particle and the order of the deconfinement transition, the scaling of the interaction measure with N²_c -1N^{2}_{c} -1, the role of gluon condensate, and the screening mass.     

Spherical Symmetric Solution in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) Model Around Charged Black Hole

A static, asymptotically flat, spherically symmetric solutions is investigated in f(R) theories of gravity for a charged black hole. We have studied the weak field limit of f(R) gravity for the some f(R) model such as f(R)=R+ε h(R). In particular, we consider the case lim  _R→0 h(R)/h′(R)→0 and find the space time metric for f(R)=R+[(m⁴)/(R)]f(R)=R+{\mu^{4}\over R} and f(R)=R ^1+ε theories of gravity far away a charged mass point.