Anomalies and curvature ofW manifolds

We study the holomorphic structure of certain complex manifolds associated withW algebras, namely, the flag manifoldsW /T andW ₁₊/T ₁₊, and the spacesW /SL(),R) andW ₁₊/GL(,R), whereT andT ₁₊ are the maximal tori inW andW ₁₊. We compute their Ricci curvature and show how the results are related to the anomaly-freedom conditions forW andW ₁₊. We discuss the relation of these manifolds with extensions of universal Teichmüller space.Supported in part by the U.S. Department of Energy, under grant DE-AS05-81ER40039Supported in part by the U.S. Department of Energy, under grant DE-FG03-84ER40168     

Quasi-classical limit of BKP hierarchy andW-infinity symmetries

Previous results on quasi-classical limit of the KP and Toda hierarchies are now extended to the BKP hierarchy. Basic tools such as the Lax representation, the Baker-Akhiezer function and the tau function are reformulated so as to fit into the analysis of quasi-classical limit. Two subalgebrasW ₁ ^B ₊ andw ₁ ^B ₊ of theW-infinity algebrasW _{1 +} andw _{1 +} are introduced as fundamental Lie algebras of the BKP hierarchy and its quasi-classical limit, the dispersionless BKP hierarchy. The quantumW-infinity algebraW ₁ ^B ₊ emerges in symmetries of the BKP hierarchy. In quasi-classical limit, theseW ₁ ^B ₊ symmetries are shown to be contracted intow ₁ ^B ₊ symmetries of the dispersionless BKP hierarchy.     

Electroweak interactions at an infinite sublayer quark level. IV. Composite model of electron,neutrino, and gauge bosons

We assume that the electron (e ^–), neutrino (v _e), and gauge bosons (W ^±,Z ⁰) are composed of only two kinds of particles, an ultimate particleu at an infinite sublayer quark level and a chargeless fermiont, such thate ^–=(u ^cp u ^cp l),V _e =(u u ^cp l,W ⁺=(u u ),W ^–=(u ^cp u ^cp andZ ⁰=(u u ^cp . It is then shown thatCP is violated in weak interactions associated with these electron, neutrino, and gauge bosons.     

A one-parameter family of hamiltonian structures for the KP hierarchy and a continuous deformation of the nonlinear WKP algebra

The KP hierarchy is hamiltonian relative to a one-parameter family of Poisson structures obtained from a generalized Adler map in the space of formal pseudodifferential symbols with noninteger powers. The resulting W-algebra is a one-parameter deformation of W_KP admitting a central extension for generic values of the parameter, reducing naturally to W _n for special values of the parameter, and contracting to the centrally extended W₁₊, W and further truncations. In the classical limit, all algebras in the one-parameter family are equivalent and isomorphic tow _KP. The reduction induced by setting the spin-one field to zero yields a one-parameter deformation of which contracts to a new nonlinear algebra of the W-type.Address after October 1993: Queen Mary and Westfield College, UK     

Eigensystem and full character formula of theW 1+∞ algebra withc = 1

By using the free field realizations, we analyze the representation theory of theW ₁₊ algebra withc = 1. The eigenvectors for the Cartan subalgebra ofW ₁₊ are parametrized by Young diagrams, and explicitly written down byW ₁₊ generators. Moreover, their eigenvalues and full character formula are also obtained.     

Moyal Deformations of Gravity via SU(∞) Gauge Theories, Branes and Topological Chern-Simons Matrix Models

Moyal noncommutative star-product deformations of higher-dimensional gravitational Einstein-Hilbert actions via lower-dimensional SU(), W gauge theories are constructed explicitly based on the holographic reduction principle. New reparametrization invariant p-brane actions and their Moyal star product deformations follows. It is conjectured that topological Chern-Simons brane actions associated with higher-dimensional knots have a one-to-one correspondence with topological Chern-Simons Matrix models in the large N limit. The corresponding large N limit of Topological BF Matrix models leads to Kalb-Ramond couplings of antisymmetric-tensor fields to p-branes. The former Chern-Simons branes display higher-spin W symmetries which are very relevant in the study of W Gravity, the Quantum Hall effect and its higher-dimensional generalizations. We conclude by arguing why this interplay between condensed matter models, higher-dimensional extensions of the Quantum Hall effect, Chern-Simons Matrix models and Gravity needs to be investigated further within the framework of W Gauge theories.     

The induced action ofW 3 gravity

We obtain the induced action [h, b] for chiralW ₃ gravity in thec± limit from the induced action of a gaugedSl(3,R) Wess-Zumino-Witten model by imposing constraints on the currents of the latter. In the process we find a closed gauge algebra for the gauge sector ofW ₃ gravity in which the currentsT andW become auxiliary fields. An explicit realization ofT andW in terms of the gauge fields is given. In terms of new fieldsr ands, which are a generalization of Polyakov'sf variable for ordinary gravity, the complete induced action [h, b; c±] becomes local.Work supported in part by NSF grant No. PHYS 89-08495Address after September 1, 1991: Physics Department, U.C. Berkeley, Berkeley, CA 94720, USA     

Spouge's Conjecture on Complete and Instantaneous Gelation

We investigate the stochastic counterpart of the Smoluchowski coagulation equation, namely the Marcus–Lushnikov coagulation model. It is believed that for a broad class of kernels, all particles are swept into one huge cluster in an arbitrarily small time, which is known as a complete and instantaneous gelation phenomenon. Indeed, Spouge (also Domilovskii et al. for a special case) conjectured that K(i, j)=(ij) , >1, are such kernels. In this paper, we extend the above conjecture and prove rigorously that if there is a function (i, j), increasing in both i and j such that _j=1 1/(j(i, j))< for all i, and K(i, j)ij(i, j) for all i, j, then complete and instantaneous gelation occurs. Evidently, this implies that any kernels K(i, j)ij(log(i+1)log(j+1)) , >1, exhibit complete instantaneous gelation. Also, we conjuncture the existence of a critical (or metastable) sol state: if lim _i+j ij/K(i, j)=0 and _{i, j=1} 1/K(i, j)=, then gelation time T _g satisfies 0<T _g<. Moreover, the gelation is complete after T _g.     

The Hilbert-Space Structure of Non-Hermitian Theories with Real Spectra

We investigate the quantum-mechanical interpretation of models with non-Hermitian Hamiltonians and real spectra. After describing a general framework to reformulate such models in terms of Hermitian Hamiltonians defined on the Hilbert space L ₂(-, ), we discuss the significance of the algebra of physical observables.     

Classification of all Poisson-Lie structures on an infinite-dimensional jet group

A local classification of all Poisson-Lie structures on an infinite-dimensional group G of formal power series is given. All Lie bialgebra structures on the Lie algebra G of G are also classified.     

The logarithm of the derivative operator and higher spin algebras ofW ∞ type

We use the notion of the logarithm of the derivative operator to describeW type algebras as central extensions of the algebra of differential operators. We also provide closed formulae for the truncations ofW ₁₊ to higher spin algebras withsM, for allM2. The results are extended to matrix valued differential operators, introducing a logarithmic generalization of the Maurer-Cartan cocycle.This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76Sf00098 and in part by the National Science Foundation under grants PHY-85-15857 and PHY-87-17155Address after July 1, 1992: Dept. of Mathematics, Yale University, New Haven, CTO6520, USA     

Quasifinite highest weight modules over the superW 1+∞algebra

We study quasifinite highest weight modules over the supersymmetric extension of theW ₁₊ algebra on the basis of the analysis by Kac and Radul. We find that the quasifiniteness of the modules is again characterized by polynomials, and obtain the differential equations for highest weights. The spectral flow, free field realization over the (B, C)-system, and the embedding into (|) are also presented.Address after April 1, 1994: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, JapanAddress after April 1, 1994: Uji Research Center, Yukawa Institute for Theoretical Physics, Kyoto University, Uji 611, Japan     

Dimensional crossover and finite-size scaling belowT c

Using the formalism developed in earlier work, dimensional crossover on ad-dimensional layered Ising-type system satisfying periodic boundary conditions and of sizeL is considered belowT _c (L), T _c (L) being the critical temperature of the finite-size system. Effective critical exponents _eff and _eff are shown explicitly to crossover between theird- and (d–1)-dimensional values for _L in the limitsL/ _L andL/ _L 0, respectively, _L , being the correlation length in the layers. Using anL-dependent renormalization group, the effective exponents are shown to satisfy natural generalizations of the standard scaling laws. In addition,L-dependent global scaling fields which span the entire crossover are defined and a scaling form of the equation of state in terms of them derived. All the above assertions are verified explicitly to one loop in perturbation theory, in particular effective exponents and a universal crossover equation of state are obtained and shown in the above asymptotic limits to be in good agreement with known results.     

The number of bound states of singular oscillating potentials

For a spherically symmetric potential such that rVL ¹(a, ), a>0, and is such that, if we define W=– _r V(t) d(t), W belongs to L ¹ (0, ) and rW0 as r0, we show that the number of bound states in any partial-wave satisfies the bound n2 ₀ r W ² dr. It was shown in a previous paper [1] that this class of potentials is regular from the point of view of abstract scattering theory as well as from the time-independent theory and the Jost function approach. We show also that, for large values of the coupling constant, n(gV) has the asymptotic behaviour C _±g ₀ W(r) dr as g±.     

Spectral properties of certain composition operators arising in statistical mechanics

By applying the theory of linear positive operators in a Banach space we derive spectral properties of certain composition operators in the Banach spaceA () of holomorphic functions over some domain . Examples of such operators are provided by the so called generalized transfer matrices of classical one-dimensional lattice systems.     

Random and cooperative sequential adsorption on infinite ladders and strips

We analyze various processes where particles are added irreversibly and sequentially at the sites of infinite ladders or broader strips (i.e., on terraces) of adsorption sites. For sufficiently narrow strips or ladders, exact solution in closed form is possible for a variety of processes. Often this is most naturally achieved by mapping the process onto an equivalent one-dimensional process typically involvingcompetitive adsorption. We demonstrate this procedure for sequential adsorption with nearest-neighbor exclusion on a 2× square ladder. For other select processes on strips slightly too broad for exact solution, almost exact analysis is possible exploiting an empty-site shielding property. In this way, we determine a jamming coverage of 0.91556671 for random sequential adsorption of dimers on a 2× square ladder. For broader strips, we note that the complexity of these problems quickly approaches that for × lattices.     

Trapping and cascading of eigenvalues in the large coupling limit

We consider eigenvaluesE of the HamiltonianH =–+V+W,W compactly supported, in the limit. ForW0 we find monotonic convergence ofE to the eigenvalues of a limiting operatorH (associated with an exterior Dirichlet problem), and we estimate the rate of convergence for 1-dimensional systems. In 1-dimensional systems withW0, or withW changing sign, we do not find convergence. Instead, we find a cascade phenomenon, in which, as , each eigenvalueE stays near a Dirichlet eigenvalue for a long interval (of lengthO( )) of the scaling range, quickly drops to the next lower Dirichlet eigenvalue, stays there for a long interval, drops again, and so on. As a result, for most large values of the discrete spectrum ofH is close to that ofE , but when reaches a transition region, the entire spectrum quickly shifts down by one. We also explore the behavior of several explicit models, as .Max Kade Foundation FellowPartially supported by USNSF under Grant DMS-8416049On leave of absence from Department of Mathematics and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA. Partially supported by USNSF under Grant DMS-8620231 and the Case Institute of Technology, RIG     

Electroweak interactions at an infinite sublayer quark level

In a previous paper, we proposed the infinite sublayer quark model, in which there exists an infinite number of quarksu and antiquarksu ^c at an infinite sublayer level. By applying the standard model of the electroweak interactions to the weak isospin doublet (u ,u ^c ) ^T it is shown that there exists only one gauge fieldW _µ ³ , from which the electromagnetic fieldA _µ=W _µ ³ cos _w and the neutral vector boson fieldZ _µ ⁰ =W _µ ³ sin _w are derived.     

Variance calculations and the Bessel kernel

In the Laguerre ensembleof n xN Hermitian matrices, it is of interest both theoretically and for applications to quantum transport problems to compute the variance of a linear statistic, denoted var_N f, asN . Furthermore, this statistic often contains an additional parameter a for which the limit is most interesting and most difficult to compute numerically. We derive exact expressions for both lim_N var_N f and lim , lim_N var_N f.     

Asymptotic and essentially singular solutions of the Feigenbaum equation

For suitably defined largeN, we express Feigenbaum's equation as a singular Schroder functional equation whose solution is obtained using a scaling ansatz. In the limit of infiniteN certain self-consistency conditions on the scaled Schroder solution lead to an essentially singular solution of Feigenbaum's equation with a length scale factor of 0.0333 and. a limiting feigenvalue of 30.50, in agreement with Eckmann and Wittwer's value of =0.0333831... and their conjectured estimate of 30.