On the Stability of the O(N)-Invariant and the Cubic-Invariant Three-Dimensional N-Component Renormalization-Group Fixed Points in the Hierarchical Approximation

We compute renormalization-group fixed points and their spectrum in an ultralocal approximation. We study a case of two competing nontrivial fixed points for a three-dimensional real N-component field: the O(N)-invariant fixed point vs. the cubic-invariant fixed point. We compute the critical value N _c of the cubic ⁴-perturbation at the O(N)-fixed point. The O(N)-fixed point is stable under a cubic ⁴-perturbation below N _c; above N _c it is unstable. The Critical value comes out as 2.219435<N _c<2.219436 in the ultralocal approximation. We also compute the critical value of N at the cubic invariant fixed point. Within the accuracy of our computations, the two values coincide.     

Generalized hypergeometric functions on the torus and the adjoint representation ofU q (sl 2)

We study the homology groups with coefficient in local systems arising in the free field representation of minimal models of conformal field theory on an elliptic curve with punctures. We define an action of the quantum enveloping algebraU _q (sl ₂) on a space of relative cycles, extending results obtained previously for the sphere. Absolute cycles are identified with singular vectors. In the case of one puncture, we find that the resulting topological representation is essentially the adjoint representation.     

Heat kernel regularization of quantum fields

We discuss some consequences of the existence of a heat kernel regularization (HKR) for quantum fields. We demonstrate that HKR applies in certain examples, using methods which should be useful more generally.Supported in part by the National Science Foundation under Grant PHY/DMS 86-45122Supported in part by a German National Scholarship Foundation fellowship     

Positivity and Convergence in Fermionic Quantum Field Theory

We derive norm bounds that imply the convergence of perturbation theory in fermionic quantum field theory if the propagator is summable and has a finite Gram constant. These bounds are sufficient for an application in renormalization group studies. Our proof is conceptually simple and technically elementary; it clarifies how the applicability of Gram bounds with uniform constants is related to positivity properties of matrices associated to the procedure of taking connected parts of Gaussian convolutions. This positivity is preserved in the decouplings that also preserve stability in the case of two-body interactions.     

Renormalization group flow of a hierarchical Sine-Gordon model by partial differential equations

We use a renormalization group differential equation to rigorously control the renormalization group flow in a hierarchical lattice Sine-Gordon field theory in the Kosterlitz-Thouless phase.Supported in part by the Department of Energy under Grant DE-FG02-88ER25065     

Topological representations of the quantum groupU q(sl 2)

We define a topological action of the quantum groupU _q(sl ₂) on a space of homology cycles with twisted coefficients on the configuration space of the punctured disc. This action commutes with the monodromy action of the braid groupoid, which is given by theR-matrix ofU _q(sl ₂).Currently visiting the Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA. Supported in part by the NSF under Grant No. PHY89-04035, supplemented by funds from the NASA     

Running Coupling Expansion for the Renormalized Φ 4 4 Trajectory from Renormalization Invariance

We formulate a renormalized running coupling expansion for theΒ-function and the potential of the renormalized Φ⁴-trajectory on four-dimensional Euclidean space-time. Renormalization invariance is used as a first principle. No reference is made to bare quantities. The expansion is proved to be finite to all orders of perturbation theory. The proof includes a large-momentum bound on the connected free propagator amputated vertices.     

Supersymmetry and the spectral condition on a cylinder

We study the spectral condition for supersymmetric Wess-Zumino field theories on a cylindrical spacetime. This condition is preserved under certain ultraviolet cutoff procedures and also leads to analyticity of HKR regularized field operators in a complex neighborhood of spacetime.Supported in part by the National Science Foundation under Grant DMS/PHY86-45122.Supported in part by a German National Scholarship.     

Symanzik's improved actions from the viewpoint of the renormalization group

We investigate Symanzik's improvement program in a four-dimensional Euclidean scalar field theory with smooth momentum space cutoff. We use Wilson's renormalization group transformation to define the improved actions as a sequence of initial data for the effective action at the fundamental cutoff. This leads to a sequence of solutions to the renormalization group equation. We define the parameters of the improved actions implicitly by conditions on the effective action at a renormalization scale. The improved actions are close approximations to the continuum effective action. We prove their existence to every order of improvement and to every order of renormalized perturbation theory.Supported in part by a German National Scholarship Foundation fellowship, and by the National Science Foundation under Grant DMS/PHY 86-45122