Narrow Results

Solutions of the Polchinski exact renormalization group equation in the scalar O(N) theory are studied. Families of regular solutions are found and their relation with fixed points of the theory is established. Special attention is devoted to the limit N = ∞, where many properties can be analyzed analytically.

Within the framework of the Exact Renormalization Group, a manifestly gauge invariant calculus is constructed for SU(N) Yang–Mills. The methodology is comprehensively illustrated with a proof, to all orders in perturbation theory, that the β function has no explicit dependence on either the seed action or details of the covariantization of the cutoff. The cancellation of these nonuniversal contributions is done in an entirely diagrammatic fashion.

This paper is a self-contained review of the loop variable approach to string theory. The Exact Renormalization Group is applied to a world sheet theory describing string propagation in a general background involving both massless and massive modes. This gives interacting equations of motion for the modes of the string. Loop variable techniques are used to obtain gauge invariant equations. Since this method is not tied to flat space–time or any particular background metric, it is manifestly background independent. The technique can be applied to both open and closed strings. Thus gauge invariant and generally covariant interacting equations of motion can be written for massive higher spin fields in arbitrary backgrounds. Some explicit examples are given.

A Legendre transform of the recently discovered conformal fixed-point equation is constructed, providing an unintegrated equation encoding full conformal invariance within the framework of the effective average action.

We consider the multiple products of relevant and marginal scalar composite operators at the Gaussian fixed-point in $D=4$ dimensions. This amounts to perturbative construction of the ${\varphi }^4$ theory where the parameters of the theory are momentum-dependent sources. Using the exact renormalization group (ERG) formalism, we show how the scaling properties of the sources are given by the short-distance singularities of the multiple products.

The exact renormalization group (ERG) is formulated implementing the decimation of degrees of freedom by means of a particular momentum integration measure. The definition of this measure involves a distribution that links this decimation process with the dimensional regularization technique employed in field theory calculations. Taking the dimension $d=4-ϵ$, the one-loop solutions to the ERG equations for the scalar field theory in this scheme are shown to coincide with the dimensionally regularized perturbative field theory calculation in the limit $ϵ\to 0^{-}$, if a particular relation between the scale parameter $\mu$ and $ϵ$ is employed. In general, it is shown that in this scheme the solutions to the ERG equations for the proper functions coincide when $ϵ\to 0^{-}$ with the complete diagrammatic contributions appearing in field theory for these functions and this theory, provided that exact relations between $\mu$ and $ϵ$ hold. In addition, a nonperturbative approximation is considered. This approximation consists in a truncation of the ERG equations, which by means of a low-momentum expansion leads to reasonable results.

A simple form of the exact renormalization group method is proposed for the study of supersymmetric gauge field theory. The method relies on the existence of ultraviolet-finite four dimensional gauge theories with extended supersymmetry. The resulting exact renormalization group equation crucially depends on the Konishi anomaly of N=1 super Yang–Mills. We illustrate our method by dealing with the NSVZ exact relation for the beta functions, the N=2 super Yang–Mills effective potential and the N=1 super Yang–Mills gluon superpotential (the so-called Veneziano–Yankielowicz potential).

We study the application of the exact renormalisation group to a many-body system consisting of fermions interacting through a short-range attractive force. This is modelled through an effective range expansion using an effective field theory inspired approach. We investigate a systematic description of many-body effects in such systems.