Induced Action for Conformal Higher Spins from Worldline Path Integrals

Roberto Bonezzi

doi:10.3390/universe3030064

Abstract

Conformal higher spin (CHS) fields, yet being non unitary, provide a remarkable example of a consistent interacting higher spin theory in flat space background, that is local to all orders. The non-linear action is defined as the logarithmically UV divergent part of a one-loop scalar effective action. In this paper we take a particle model, that describes the interaction of a scalar particle to the CHS background, and compute its path integral on the circle. We thus provide a worldline representation for the CHS action, and rederive its quadratic part. We plan to come back to the subject, to compute cubic and higher vertices, in a future work.

Highlights

Four dimensional Maxwell (s = 1) theory is the first known example of a conformally invariant physical system
Massless matter lagrangians (s = 0, 1/2) have conformal symmetry in flat space, that can be enhanced to general covariance plus local Weyl symmetry when coupled to a curved spacetime metric
The action is invariant under the combined transformations δα H( x, p) = α( x, p) p2 + H( x, p), δα e = −α( x, p) e and, as we shall see the transformation of the einbein is responsible for breaking the α-symmetry at the quantum level. As it is well known from the cases of interaction with scalar, vector and gravitational backgrounds, the effective action Γ[h] can be obtained by quantizing the action (32) on the circle: Γ[h] =

Summary

Introduction

The CHS theory is power-counting renormalizable but, since Weyl and higher order algebraic symmetries are gauged, it has to be free of conformal and higher spin anomalies in order to be consistent at the quantum level. The scalar path integral with sources ∑s Js hs yields the generating functional Γ[h] of correlators of the conformal currents and, according to AdS/CFT correspondence, should be equal to the on-shell value of the, yet unknown , action of massless higher spins in AdS6. Is local and gauge invariant and can be identified as the classical non-linear action SCHS [h] for conformal higher spins [7,11]. Where in the generalized hamiltonian G ( x, p) = p2 + H( x, p) the conformal higher spin fields, contained in the p-power series expansion of H( x, p) , are treated as perturbations over the flat space background p2. Phase space worldline path integrals have been used in [79,80] in the context of non-commutative field theory

Induced Action for Conformal Higher Spins

Noether Interaction and Symmetries

Effective Action and Worldline Path Integral

The Quadratic Action

Discussion and Conclusions