Geometry the Renormalization Group and Gravity

TL;DR

This paper explores the connection between geometry, the renormalization group, and gravity, focusing on how RG flows interpolate between different conformal representations and the geometric structure of parameter spaces.

Contribution

It introduces the concept of a 'floating' fixed point and constructs simple models to analyze the geometry of theory parameter spaces in the context of RG flows.

Findings

RG flows can interpolate between different fixed points

Parameter space of theories has a natural metric structure

Analogies between parameter space, superspace, and minisuperspace

Abstract

We discuss the relationship between geometry, the renormalization group (RG) and gravity. We begin by reviewing our recent work on crossover problems in field theory. By crossover we mean the interpolation between different representations of the conformal group by the action of relevant operators. At the level of the RG this crossover is manifest in the flow between different fixed points induced by these operators. The description of such flows requires a RG which is capable of interpolating between qualitatively different degrees of freedom. Using the conceptual notion of course graining we construct some simple examples of such a group introducing the concept of a ``floating'' fixed point around which one constructs a perturbation theory. Our consideration of crossovers indicates that one should consider classes of field theories, described by a set of parameters, rather than focus on a particular one. The space of parameters has a natural metric structure. We examine the geometry of this space in some simple models and draw some analogies between this space, superspace and minisuperspace.