Unified treatment of fermions and bosons

Dirac's matrices can be interpreted as an 8-rank covariant antisymmetric tensor field on an 11-dimensional manifold (space-time ×S ⁷) enforcing a linkage between the Lorentz transformation and rotations ofS ⁷, conferring spinorial properties on any quantity having an index in the inner spaceS ⁷.     

(1+2)-Dimensional model of the early universe

An anisotropic cosmological model is obtained by solving (1+3)-dimensional field equations. The topology of the model isR ¹ M ² S ¹, whereR ¹ is the real line (time axis),M ² is 2-dimensional space, andS ¹ is the circle. Employing the method of Kaluza-Klein type compactification onS ¹ and one-loop quantum correction to scalar fields, an effective (1+2)-dimensional gravity is obtained. The resulting (1+2)-dimensional cosmological model of the early universe is derived.     

A multidimensional radiation-filled universe

We consider a radiation-filled universe which possesses the product symmetry: (N-dimensional space of constant curvature) × (n sphere). The solutions of all the types, within this class, to the classical field equations are given. In the case of theN-dimensional space of zero or negative curvature constant, the solutions exhibit a tendency to approach asymptotically the Kasner-like state in which theN-dimensional subspace expands while then sphere shrinks to the final singularity. Our conclusions based on the phase-diagram method are in agreement with the results concerning the ^N × S ⁿ universe calculated by Sahdev with the help of numerical methods.     

Symmetries of Einstein-Yang-Mills fields and dimensional reduction

LetE be a manifold on which a compact Lie groupS acts simply (all orbits of the same type);E can be written locally asM×S/I,M being the manifold of orbits (space-time) andI a typical isotropy group for theS action. We study the geometrical structure given by anS-invariant metric and anS-invariant Yang Mills field onE with gauge groupR. We show that there is a one to one correspondence between such structures and quadruplets of fields defined solely onM; _v is a metric onM,h are scalar fields characterizing the geometry of the orbits (internal spaces), ⁱ are other scalar fields (Higgs fields) characterizing theS invariance of the Lie(R)-valued Yang Mills field and is a Yang Mills field for the gauge groupN(I)|I×Z((I)),N(I) being the normalizer ofI inS, is a homomorphism ofI intoR associated to theS action, andZ((I)) is the centralizer of(I) inR. We express the Einstein-Yang-Mills Lagrangian ofE in terms of the component fields onM. Examples and model building recipes are given.     

Gravity from Lie Algebroid Morphisms

Inspired by the Poisson Sigma Model and its relation to 2d gravity, we consider models governing morphisms from T to any Lie algebroid E, where is regarded as a d-dimensional spacetime manifold. We address the question of minimal conditions to be placed on a bilinear expression in the 1-form fields, S^ij(X)A_iAj, so as to permit an interpretation as a metric on . This becomes a simple compatibility condition of the E-tensor S with the chosen Lie algebroid structure on E. For the standard Lie algebroid E=TM the additional structure is identified with a Riemannian foliation of M, in the Poisson case E=T^*M with a sub-Riemannian structure which is Poisson invariant with respect to its annihilator bundle. (For integrable image of S, this means that the induced Riemannian leaves should be invariant with respect to all Hamiltonian vector fields of functions which are locally constant on this foliation). This provides a huge class of new gravity models in d dimensions, embedding known 2d and 3d models as particular examples.     

Classical solutions of the quantum Yang-Baxter equation

The classical analogue is developed here for part of the construction in which knot and link invariants are produced from representations of quantum groups. Whereas previous work begins with a quantum group obtained by deforming the multiplication of functions on a Poisson Lie group, we work directly with a Poisson Lie groupG and its associated symplectic groupoid. The classical analog of the quantumR-matrix is a lagrangian submanifold in the cartesian square of the symplectic groupoid. For any symplectic leafS inG, induces a symplectic automorphism ofS×S which satisfies the set-theoretic Yang-Baxter equation. When combined with the flip map exchanging components and suitably implanted in each cartesian powerS ⁿ , generates a symplectic action of the braid groupB _n onS ⁿ . Application of a symplectic trace formula to the fixed point set of the action of braids should lead to link invariants, but work on this last step is still in progress.Research partially supported by NSF Grant DMS-90-01089Research partially supported by NSF Grant DMS 90-01956 and Research Foundation of University of Pennsylvania     

New Infinite Series of Einstein Metrics on Sphere Bundles from AdS Black Holes

A new infinite series of Einstein metrics is constructed explicitly on S²×S³, and the non-trivial S³-bundle over S², containing infinite numbers of inhomogeneous ones. They appear as a certain limit of 5-dimensional AdS Kerr black holes. In the special case, the metrics reduce to the homogeneous Einstein metrics studied by Wang and Ziller. We also construct an inhomogeneous Einstein metric on the non-trivial S^d–2-bundle over S² from a d-dimensional AdS Kerr black hole. Our construction is a higher dimensional version of the method of Page, which gave an inhomogeneous Einstein metric on      

The dynamics of multidimensional cosmological models in the presence of quantum effects. II. The low-temperature case

The dynamical system method is applied to the dynamics of multidimensional cosmological models with the topology FRLW×T ^D (D-dimensional torus). Stability and typical states of the metric are discussed. It is shown that quantum effects in low-temperature approximations, when added to the monopole configuration electromagnetic fields, give a correct mechanism for the dynamical reduction if the cosmological constant is nonzero and if the microspace is anisotropic.     

Determinants of dirac boundary value problems over odd-dimensional manifolds

We present a canonical construction of the determinant of an elliptic selfadjoint boundary value problem for the Dirac operatorD over an odd-dimensional manifold. For 1-dimensional manifolds we prove that this coincides with the -function determinant. This is based on a result that elliptic self-adjoint boundary conditions forD are parameterized by a preferred class of unitary isomorphisms between the spaces of boundary chiral spinor fields. With respect to a decompositionS ¹=X ⁰X ¹, we explain how the determinant of a Dirac-type operator overS ¹ is related to the determinants of the corresponding boundary value problems overX ⁰ andX ¹.     

A poisson—lie structure on the diffeomorphism group of a circle

Starting from the Gelfand-Fuks-Virasoro cocycle on the Lie algebraX(S ¹) of the vector fields on the circleS ¹ and applying the standard procedure described by Drinfel'd in a finite dimension, we obtain a classicalr-matrix (i.e. an elementr X(S ¹) X(S ¹) satisfying the classical Yang-Baxter equation), a Lie bialgebra structure onX(S ¹), and a sort of Poisson-Lie structure on the group of diffeomorphisms. Quantizations of such Lie bialgebra structures may lead to quantum diffeomorphism groups.Research supported by the Erwin Schrödinger International Institute for Mathematical Physics.     

Noncommutative Sine-Gordon Model

The sine-Gordon model may be obtained by dimensional and algebraic reduction from (2+2)-dimensional self-dual U(2) Yang-Mills through a (2+1)-dimensional integrable U(2) sigma model. It is argued that the noncommutative (Moyal) deformation of this procedure should relax the algebraic reduction from U(2) U(1) to U(2)U(1) × U(1). The result are novel noncommutative sine-Gordon equations for a pair of scalar fields. The dressing method is outlined for constructing its multi-soliton solutions. Finally, the tree-level amplitudes demonstrate that this model possesses a factorizable and causal S-matrix in spite of its time-space noncommutativity.     

Long-time behavior of Navier-Stokes flow on a two-dimensional torus excited by an external sinusoidal force

In this paper we study the Navier-Stokes flow on the two-dimensional torusS ¹ ×S ¹ excited by the external force (k ² sinky, 0) and find the long-time behavior for the flow starting from some states, whereS ¹=[0,2](mod 2). Especially for the casek=2, it follows from an analysis and computation that the Navier-Stokes flow with the initial state cos(mx+ny) or sin(mx+ny) will likely evolve through at most one step bifurcation to either a steady-state solution or a time-dependent periodic solution for any Reynolds number and integersm andn.     

Stabilization of the Hierarchy in Brane-World Scenarios

We consider a class of warped brane models with topology M ₄ × × S¹/Z ₂, where is a D₂-dimensional compact manifold, and two branes are placed at the orbifold fixed points. In a scenario where supersymmetry is broken not far below the cutoff scale, the hierarchy between the electroweak and the Planck scales is generated by a combination of the redshift and the large volume effects. We evaluate the effective potential induced by bulk scalar fields in these models and show that it can stabilize the moduli and the hierarchy without fine-tuning, provided that the internal space is flat. We also comment on the relation between these models and the five-dimensional scalar-tensor models that describe them classically when the compactification scale is small.     

Galilean anti-de-Sitter spacetime in Romans theory

The Romans type IIA theory is the only known example of 10-dimensional maximal supergravity where (tensor) fields are explicitly massive. We provide an example of a non-relativistic anti-de-Sitter NRadS₄×S⁶${\mathit{NRadS}}_4\times S^6$ background as a solution in massive type IIA. A compactification of which on S⁶$S^6$ gives immediately the prototype NRadS background in D=4$D=4$ which is proposed to be dual to ‘cold atoms’ or unitary fermions on a wire.     

Perturbed sine-Gordon model: S=1/2 anisotropicxy chain with a longitudinal magnetic field

The thermodynamic properties and the longitudinal dynamical spin-correlation functionS ^zz (q, ) of theS=1/2 anisotropicxy chain with a longitudinal magnetic field are calculated in the continuum limit. For low fields we have an out-of-plane distortion of the sine-Gordon solitons present in the zero field case, while for high fields a completely different regime, characterized by a central peak forS ^zz (q, ), is found.Also Unità di Firenze GNSM-CISM     

The extended phase space of the BRS approach

The origin of the classical BRS symmetry is discussed for the case of a first class constrained system consisting of a 2n-dimensional phase spaceS with free action of a Lie gauge groupG of dimensionm. The extended phase spaceS _ext of the Fradkin-Vilkovisky approach is identified with a globally trivial vector bundle overS with fibreL*(G)L(G), whereL(G) is the Lie algebra ofG andL*(G) its dual. It is shown that the structure group of the frame bundle of the supermanifoldS _ext is the orthosymplectic group OSp(m,m; 2n), which is the natural invariance group of the super Poisson bracket structure on the function spaceC (S _ext). The action of the BRS operator is analyzed for the caseS=R ²ⁿ with constraints given by pure momenta. The breaking of the osp(m,m; 2n)-invariance down to sp(2n–2m) occurs via an intermediate osp(m; 2n–m). Starting from a (2n+2m)-dimensional system with orthosymplectic invariance, different choices for the BRS operator correspond to choosing different 2n-dimensional constraint supermanifolds inS _ext, which in general characterize different constrained systems. There is a whole family of physically equivalent BRS operators which can be used to describe a particular constrained system.     

Unbounded Entropy in Spacetimes with Positive Cosmological Constant

In theories of gravity with a positive cosmological constant, we consider product solutions with flux, of the form (A)dS _p ×S ^q . Most solutions are shown to be perturbatively unstable, including all uncharged dS _p ×S ^q spacetimes. For dimensions greater than four, the stable class includes universes whose entropy exceeds that of de Sitter space, in violation of the conjectured N-bound. Hence, if quantum gravity theories with finite-dimensional Hilbert space exist, the specification of a positive cosmological constant will not suffice to characterize the class of spacetimes they describe.     

Higher-dimensional cosmological solutions with action of scalar and metric fields

An investigation is made of higher-dimensional (D5) cosmological solutions with action of scalar and metric fields for which a matter term is added. We restrict our attention to the most symmetric solutions with the structureM ^D–2×S ². We present the variant cosmological solutions for the symmetry breaking patternGSU(2)×U(1) (type IA, IIA) and patternGSO(3) (type IB, IIB). InD=6 case type IA is interesting for cosmology, which corresponds to a conformally invariant theory.     

Equations for the Magnetization Modes in a System Having a Quadruple Nucleus with the Spin S = 1

Within the framework of the method of a density matrix, differential equations are obtained that describe the evolution of the modes of magnetization <Î₊>, <Î_{+ z} >, and <Î_{+ z} ²> for a spin system containing a nucleus with a 1/2 spin connected scalarly with the quadruple nucleus of the spin S = 1 in the presence of crosscorrelation between the dipole ISinteractions, anisotropy of the chemical shift of the I and S nuclei, and quadruple interaction of the S nucleus. The equations are used for analysis of the evolution of the intensities of each line in the triplet of the Raman spectrum of the 1/2 spin. Expressions for the times of crossrelaxation and the times of transverse relaxation of each of the lines of the triplet are obtained.