International Journal of Modern Physics A

We extend the notion of a Kac–Moody algebra defined on the S¹ circle to super Kac–Moody algebras defined on M × G_N, M being a smooth closed compact manifold of dimension greater than one, and G_N the Grassman algebra with N generators. We compute all the central extensions of these algebras. Then, we determine for each such algebra the derivation algebra constructed from the M × G_N diffeomorphisms. The twists of such super Kac–Moody algebras as well as the generalization to noncompact surfaces are partially studied. Finally, we apply our general construction to the study of conformal and superconformal algebras, as well as area-preserving diffeomorphisms algebra and its supersymmetric extension.

The U(1) Kac–Moody character functions are derived from a path integral expression. It is shown that point-split regularization of the Virasoro generator also exactly regularizes the path integral. An exact derivation of the Weyl–Kac denominator is then given for an arbitrary Lie group using the semiclassical approximation and the results of the U(1) calculation.

The effective scalar theory is constructed for the nonlinear Schrödinger (NS) matrix equation, which gives us the tool for investigating the two-point correlator and the partition function for some (classical) NS-like systems. The spatial dependence of the correlator coincides with the free correlator in all quantization schemes. For the classical NS system the effective two-matrix model is found (throughout this paper both the model with compactified space variables x ∈ S¹ and the model corresponding to noncompactified x ∈ ℝ are implied).

Starting from a given S-matrix of an integrable quantum field theory in 1 + 1 dimensions, and knowledge of its on-shell quantum group symmetries, we describe how to extend the symmetry to the space of fields. This is accomplished by introducing an adjoint action of the symmetry generators on fields, and specifying the form factors of descendents. The braiding relations of quantum field multiplets is shown to be given by the universal ℛ-matrix. We develop in some detail the case of infinite-dimensional Yangian symmetry. We show that the quantum double of the Yangian is a Hopf algebra deformation of a level zero Kac–Moody algebra that preserves its finite-dimensional Lie subalgebra. The fields form infinite-dimensional Verma module representations; in particular, the energy–momentum tensor and isotopic current are in the same multiplet.

Similarly to the Virasoro algebra, the Neveu–Schwarz algebra has a discrete series of unitary irreducible highest weight representations. These are labeled by the values of (the central charge) and of the highest weight h_pq = [(p (m + 2) − qm)² − 4]/(8m (m + 2)) where m, p, q are some integers. The Verma modules constructed with these values (c, h) are not irreducible, however, as they contain two Verma submodules, each generated by a singular vector ψ_p,q (of weight h_pq + pq/2) and ψ_{m−p, m+2−q} (of weight h_pq + (m−p)(m+2−q)/2), respectively. We give an explicit expression for these singular vectors whenever one of its indices is 1.

The Klein–Gordon equation is derived from a generalized Markov principle in a five-dimensional space in which the Minkowski space is embedded as a subspace. Compactification of the parametric fifth dimension results in a nonlinear equation whose linearized version is the Klein–Gordon equation.

We investigate at the quantum level the nonlinear transformation relating the string operators (zero modes and oscillators) and Fock space states before and after the collision with gravitational shock waves. This throws light on the rôle of the space–time geometry in this problem. We do all the treatment for a general shock wave space–time of any localized source. We compute the exact expectation values of the total number (N) and mass (M²) operators and show that they are finite, which generalize our previous results in the Aichelburg–Sexl geometry. We study the energy-momentum tensor of the string and compute the exact expectation values of all its components. We analyze vacuum polarization and quadratic fluctuations. All these physical magnitudes are finite. We express all of them in terms of exact integral representations in which the role of the real pole singularities characteristic of the tree level string spectrum (real mass resonances) are clearly exhibited. The presence of such poles is not at all related to the structure of the space–time geometry (which may or may not be singular).

The superconformal master equation contains a large set of solvable fermionic constructions which live on an infinite class of 2-dimensional simplicial complexes. All the constructions have rational central charge, and irrational conformal weights are expected in the generic construction.

We consider a Chern–Simons model in the Landau gauge and introduce, following Symanzik ideas, a plane boundary which modifies the propagators. The BRS symmetry, restored at the classical level by a suitable choice of boundary counterterms becomes anomalous due to the lower dimensionality boundary breakings. The anomaly is recognized as the central charge of an algebra of conserved chiral currents defined on the boundary itself.

The manifestly covariant canonical operator formalism of two-dimensional topological gravity is formulated. Its unitarity is confirmed by means of constructing the Kugo–Ojima's quartets. A number of new symmetries are found by adopting a particular gauge fixing condition. These symmetries correspond to the "choral symmetry" generated by the 4N-dimensional Poincaré-like superalgebra in the ordinary N-dimensional quantum gravity.

Useful expressions relating the metric components g_rr and g_tt of the static isotropic space–time, and the scalar curvature R, with the general Lagrangian are obtained. A generalized metric of gravity is introduced, which is essentially nonsingular, and reducible to cosmological and hence Schwarzschild geometries by imposing a weak-field constraint.