International Journal of Modern Physics A

A correspondence observed by Martinec, Vafa and Warner (MVW) between the singularity theory and the N = 2 super conformal field theory is reviewed by using N = 2 SUSY quantum mechanics.

A generalization of the time-evolution operator for higher-order singular-Lagrangian systems is developed and used for the study of the relations between Lagrangian and Hamiltonian formalisms, and, more specifically, for establishing the classification of constraints on both systems.

An affine field theory may be constructed by gauging an affine algebra. The momentum integrals of the affine N = 4 Yang–Mills theory are ultraviolet finite but diverge because the sum over states is infinite. If the affine symmetry is broken by postulating a nonvanishing vacuum expectation value for that component of the scalar field lying in the L₀ direction, then the theory acquires a linear mass spectrum. This broken theory is ultraviolet finite too, but the mass spectrum is unbounded. If it is also postulated that the mass spectrum has an upper bound (say, the Planck mass), then the resulting theory appears to be altogether finite. The influence of the exotic states has been estimated and, according to the proposed scenario, is negligible below energies at which gravitational interactions become important. The final effective theory has the symmetry of a compact Lie algebra augmented by the operator L₀.

A general approach to quantizing discrete models (i.e. having a finite number of coordinates) with quadratic first-class constraints is presented in the framework of gauging linear canonical symmetries. Also, it is proposed how a natural superextension of matrix field theories (viz. orthosymplectic "zero-dimensional" matrix field theories) might emerge in this approach.

We derive an explicit realization of the SU(2) Kac–Moody algebra based on the bosonic coordinates of the Kac–Moody group manifold. Kac–Moody commutation equations are exactly evaluated for the SU(2) case. Normal-ordering is not used in these derivations.

By using thermo-field dynamics, the temperature effect in supersymmetry is discussed. The supercoherent states at zero temperature, which are determined as the eigenstates of a supersymmetric annihilation operator, are formalized. However, the finite-temperature effect can destroy the coherent behavior of the supersymmetry system whose Hamiltonian is invariant under the Bogoliubov transformations.

The fermionic constraints of the massless Casalbuoni–Brink–Schwarz superparticle in d = 10 are separated covariantly as first- and second-class constraints which are infinitely reducible. Although the reducibility conditions of the second-class constraints include the first-class ones a consistent quantization is possible. The ghost structure of the system for quantizing it in terms of the BFV–BRST methods is given and unitarity is shown.

A generalization of the Sommerfield–Sugawara formula which applies to the Virasoro operators for an arbitrarily twisted Kac–Moody algebra is given. It requires a novel but interesting modification of the normal-ordering function. The results are compared with those given previously for inner and canonical outer twists.

A gauge-invariant interacting field theory of subcritical closed strings is constructed. It is shown that for d ≤ 1 this field theory reproduces many of the features of the corresponding matrix model. Among them are the scaling dimensions of the relevant primary fields, identities involving the correlation functions of some of the redundant operators in the matrix model, and the flow between different matrix models under appropriate perturbation. In particular, it is shown that some of the constraints on the partition function derived recently by Dijkgraaf et al. and Fukuma et al. may be interpreted as Ward identities in string field theory.

We study the inequivalent quantizations of (1 + 1)-dimensional nonlinear sigma models with space manifold S¹ and target manifold X. If X is multiply connected, these models possess topological solitons. After providing a definition of "spin" and "statistics" for these solitons and demonstrating a spin-statistics correlation, we give various exmples where the solitons can have exotic statistics. In some of these models, the solitons may obey a generalized version of fractional statistics called ambistatistics. The relevance of these 2D models to the statistics of vortices in (2 + 1)-dimensional spontaneously broken gauge theories is also discussed. We close with a discussion concerning the extension of our results to higher dimensions.

We discuss a target space field theory of macroscopic loops W(ℓ,…) in two-dimensional gravity. The propagator <W(ℓ₁)W(ℓ₂)> and topology-changing amplitudes <W(ℓ₁)W(ℓ₂)W(ℓ₃)> (string interactions) are considered as off-shell Euclidean Green's functions in this field theory. In the course of the analysis, we identify a new set of operators in the c = 1 system and interpret them in two-dimensional gravity. We also identify an infinite set of new conserved charges in the c = 1 system which are associated with the special states in the theory. The analysis also shows that the eigenvalue coordinate of the matrix model and a zero mode of the Liouville field are not functionally related but are conjugate variables in an integral transform.