The Quaternionic Geometry of 4D Conformal Field Theory

TL;DR

This paper develops a quaternionic geometric framework for 4D conformal field theory, paralleling 2D theories, and introduces Fueter analyticity, quaternionic line bundles, and new expressions for key physical quantities.

Contribution

It introduces a quaternionic geometric formulation of 4D conformal field theory, including Fueter analyticity and quaternionic expressions for fundamental objects.

Findings

Quaternionic formalism parallels 2D conformal theory

Explicit quaternionic expressions for energy-momentum tensor and gauge currents

Analysis of operator product expansions in free fields

Abstract

We show that 4--dimensional conformal field theory is most naturally formulated on Kulkarni 4--folds, i. e. real 4--folds endowed with an integrable quaternionic structure. This leads to a formalism that parallels very closely that of 2--dimensional conformal field theory on Riemann surfaces. In this framework, the notion of Fueter analyticity, the quaternionic analogue of complex analyticity, plays an essential role. Conformal fields appear as sections of appropriate either harmonic real or Fueter holomorphic quaternionic line bundles. In the free case, the field equations are statements of either harmonicity or Fueter holomorphicity of the relevant conformal fields. We obtain compact quaternionic expressions of such basic objects as the energy-momentum tensor and the gauge currents for some basic models in terms of Kulkarni geometry. We also find a concise expression of the conformal anomaly and a quaternionic 4--dimensional analogue of the Schwarzian derivative describing the covariance of the quantum energy-momentum tensor. Finally, we analyse the operator product expansions of free fields.