Sato's universal Grassmannian and group extensions

An extension of the symmetry group GL of Sato's universal Grassmannian GM is constructed. The extension plays a similar role to that of the central extension in the approach of Segal and Wilson to functions and KP hierarchy. Our group G contains GL_res as a subgroup and the associated function is a deformation of the usual function, leading to a deformed KP hierarchy. A relation to current algebra of Yang-Mills theory in 3+1 dimension is discussed.     

Antilocality and a Reeh-Schlieder theorem on manifolds

It is shown that the operatorA ^1/2, whereA is any positive self-adjoint extension of a positive operator of the form -Laplace-Beltrami operator +potential on ann-dimensional Riemannian manifold, is strongly antilocal. Using this result, a Reeh-Schlieder theorem for the canonical vacuum of the Klein-Gordon field propagating in ultrastatic spacetimes is derived. In a further application, we gain weaker versions of the Reeh-Schlieder theorem for more general situations.Supported by the DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

Invariant manifolds ofp-adic renormalization groups

Invariant manifolds ofp-adic renormalization groups in terms of (4-d)-expansion and dimensional renormalization are discussed.     

Projective quantum spaces

Associated to the standard SU _q (n) R-matrices, we introduce quantum spheresS _q ^2n-1 , projective quantum spaces _q ^n-1 , and quantum Grassmann manifoldsG _k( _q ⁿ ). These algebras are shown to be homogeneous spaces of standard quantum groups and are also quantum principle bundles in the sense of T. Brzeziski and S. Majid.     

On a new hyperelliptic formula

In this Letter, the correlation functions of the free fermions with odd spin structures are derived on a general hyperelliptic curve. In order to express the correlators explicitly in terms of the branch points, we construct the analogous of the Szegö kernel for the odd spin structures.     

The indecomposable representation of SO0(2, 2) on the one-particle space of the massless field in 1 + 1 dimension

In the continuum O(3) sigma model in two spatial dimensions, there are topological solitons whose size can be stabilized by adding Skyrme and potential terms. This Letter describes a lattice version, namely a natural way of modifying the 2D Heisenberg model to achieve topological stability on the lattice.     

Propagator of a particle characterized by an internal symmetry

We examine the propagators associated with particles characterized by an internal symmetry. Of particular interest is the case of spontaneously broken Kac-Moody algebra.     

On current superalgebras and super-Schwinger terms

We present a general construction of current superalgebras within the framework of quasi-free second quantization of bosons and fermions. Mathematically speaking, we given projective representations of certain Lie superalgebras realized as bounded operators on Z₂-graded Hilbert spaces and, more generally, on Grassmann algebra-modules. The super-Schwinger terms occurring correspond to Z₂-graded two-cocycles.Supported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Project No. P5588.     

Weakly coupled states on branching graphs

We consider a Schrödinger particle on a graph consisting of N links joined at a single point. Each link supports a real locally integrable potential V _j ; the self-adjointness is ensured by the type boundary condition at the vertex. If all the links are semi-infinite and ideally coupled, the potential decays as x ^–1– along each of them, is nonrepulsive in the mean and weak enough, the corresponding Schrödinger operator has a single negative eigenvalue; we find its asymptotic behavior. We also derive a bound on the number of bound states and explain how the coupling constant may be interpreted in terms of a family of squeezed potentials.     

Search for lepton flavor violation in supersymmetric models via meson decays

Considering the constraints from the experimental data on μ→eγ$\mu \to e\gamma$, μ→3e$\mu \to 3e$, μ–e$\mu \text{–}e$ conversion, etc., we analyze the lepton flavor violating decays ?(J/Ψ,?(1S))→e⁺μ⁻(μ⁺τ⁻)$?(J/\Psi ,?(1S))\to e^{+}{\mu }^{-}({\mu }^{+}{\tau }^{-})$ in the scenarios of the minimal supersymmetric extensions of Standard Model with seesaw mechanism. Numerically, there is parameter space that the LFV processes of J/Ψ(?)→μ⁺τ⁻$J/\Psi (?)\to {\mu }^{+}{\tau }^{-}$ can reach the upper experimental bounds, meanwhile the theoretical predictions on μ→eγ$\mu \to e\gamma$, μ→3e$\mu \to 3e$, μ–e$\mu \text{–}e$ conversion satisfy the present experimental bounds. For searching of new physics, lepton flavor violating processes J/Ψ(?)→μ⁺τ⁻$J/\Psi (?)\to {\mu }^{+}{\tau }^{-}$ may be more promising and effective channels.     

Sp(2) renormalization

The renormalization of general gauge theories on flat and curved space–time backgrounds is considered within the Sp(2)-covariant quantization method. We assume the existence of a gauge-invariant and diffeomorphism invariant regularization. Using the Sp(2)-covariant formalism one can show that the theory possesses gauge-invariant and diffeomorphism invariant renormalizability to all orders in the loop expansion and the extended BRST-symmetry after renormalization is preserved. The advantage of the Sp(2) method compared to the standard Batalin–Vilkovisky approach is that, in reducible theories, the structure of ghosts and ghosts for ghosts and auxiliary fields is described in terms of irreducible representations of the Sp(2) group. This makes the presentation of solutions to the master equations in more simple and systematic way because they are Sp(2)-scalars.     

Symmetry transformations in Batalin-Vilkovisky formalism

Let us suppose that the functionalS on an odd symplectic manifold satisfies the quantum master equation e ^s = 0. We prove that in some sense every quantum observable (i.e. every functionH obeying _p (He ^s) = 0) determines a symmetry of the theory with the action functionalS. Research supported in part by NSF grant No. DMS-9201366.     

Causal Construction of the Massless Vertex Diagram

The massless one-loop vertex diagram is constructed by exploiting the causal structure of the diagram in configuration space, which can be translated directly into dispersive relations in momentum space.     

Remarks on the thermodynamics and the vacuum energy of a quantum Maxwell gas on compact and closed manifolds

The quantum Maxwell theory at finite temperature at equilibrium is studied on compact and closed manifolds in both the functional integral and Hamiltonian formalism. The aim is to shed some light onto the interrelation between the topology of the spatial background and the thermodynamic properties of the system. The quantization is not unique and gives rise to inequivalent quantum theories which are classified by θ-vacua. Based on explicit parametrizations of the gauge orbit space in the functional integral approach and of the physical phase space in the canonical quantization scheme, the Gribov problem is resolved and the equivalence of both quantization schemes is elucidated. Using zeta-function regularization the free energy is determined and the effect of the topology of the spatial manifold on the vacuum energy and on the thermal gauge field excitations is clarified. The general results are then applied to a quantum Maxwell gas on an n-dimensional torus providing explicit formulae for the main thermodynamic functions in the low- and high-temperature regimes, respectively.     

Dispersive Calculation of the Massless Multi-loop Sunrise Diagram

The massless sunrise diagram with an arbitrary number of loops is calculated in a simple but formal manner. The result is then verified by rigorous mathematical treatment. Pitfalls in the calculation with distributions are highlighted and explained. The result displays the high energy behaviour of the massive sunrise diagrams, whose calculation is involved already for the two-loop case.     

Quantization of a Hamiltonian system with an infinite number of degrees of freedom

We propose a method of quantization of a discrete Hamiltonian system with an infinite number of degrees of freedom. Our approach is analogous to the usual finite-dimensional quantum mechanics. We construct an infinite-dimensional Schrödinger equation. We show that it is possible to pass from the finite-dimensional quantum mechanics to our construction in the limit when the number of particles tends to infinity. Rigorous mathematical methods are used.     

Sigma-models having supermanifolds as target spaces

We study a topological sigma-model (A-model) in the case when the target space is an (m ₀|m ₁)-dimensional supermanifold. We prove under certain conditions that such a model is equivalent to an A-model having an (m ₀–m ₁)-dimensional manifold as a target space. We use this result to prove that in the case when the target space of A-model is a complete intersection in a toric manifold, this A-model is equivalent to an A-model having a toric supermanifold as a target space.Research supported in part by NSF grant No. DMS-9201366.     

Supercurrent coupling in the Faddeev–Skyrme model

Motivated by the sigma model limit of multicomponent Ginzburg–Landau theory, a version of the Faddeev–Skyrme model is considered in which the scalar field is coupled dynamically to a one-form field called the supercurrent. This coupled model is investigated in the general setting where physical space M$M$ is an oriented Riemannian manifold and the target space N$N$ is a Kähler manifold, and its properties are compared with the usual, uncoupled Faddeev–Skyrme model. In the case N=S²$N=S^2$, it is shown that supercurrent coupling destroys the familiar topological lower energy bound of Vakulenko and Kapitanski when M=R³$M={\mathbb{R}}^3$, and the less familiar linear bound when M$M$ is a compact 3-manifold. Nonetheless, local energy minimizers may still exist. The first variation formula is derived and used to construct three families of static solutions of the model, all on compact domains. In particular, a coupled version of the unit charge hopfion on a three-sphere of arbitrary radius is found. The second variation formula is derived, and used to analyze the stability of some of these solutions. In particular, it is shown that, in contrast to the uncoupled model, the coupled unit hopfion on the three-sphere of radius R$R$ is unstable   for all R$R$. This gives an explicit, exact example of supercurrent coupling destabilizing a stable solution of the usual Faddeev–Skyrme model, and casts doubt on the conjecture of Babaev, Faddeev and Niemi that knot solitons should exist in the low energy regime of two-component superconductors.     

An extension of the Ccayley-Hamilton theorem to the case of supermatrices

Starting from the expression for the superdeterminant of (xI - M), whereM is an arbitrary supermatrix, we propose a definition for the corresponding characteristic polynomial and we prove that each supermatrix satisfies its characteristic equation. Depending upon the factorization properties of the basic polynomials whose ratio defines the superdeterminant, we are able to construct polynomials of lower degree which are also shown to be annihilated by the supermatrix.