Superconformal quantum field theory in curved spacetime

TL;DR

This paper develops a framework for superconformal quantum field theories on curved four-manifolds, analyzing classical symmetries, quantum preservation of supersymmetry, and the structure of renormalization and cohomology in curved spacetime.

Contribution

It constructs superconformal theories on curved backgrounds, establishes criteria for quantum symmetry preservation, and proves the one-loop exactness of the beta-function.

Findings

Classical superconformal symmetry is extended to curved spacetime.

Conditions for quantum superconformal invariance are derived.

The beta-function is shown to be one-loop exact.

Abstract

By conformally coupling vector and hyper multiplets in Minkowski space, we obtain a class of field theories with extended rigid conformal supersymmetry on any Lorentzian four-manifold admitting twistor spinors. We construct the conformal symmetry superalgebras which describe classical symmetries of these theories and derive an appropriate BRST operator in curved spacetime. In the process, we elucidate the general framework of cohomological algebra which underpins the construction. We then consider the corresponding perturbative quantum field theories. In particular, we examine the conditions necessary for conformal supersymmetries to be preserved at the quantum level, i.e. when the BRST operator commutes with the perturbatively defined S-matrix, which ensures superconformal invariance of amplitudes. To this end, we prescribe a renormalization scheme for time-ordered products that enter the perturbative S-matrix and show that such products obey certain Ward identities in curved spacetime. These identities allow us to recast the problem in terms of the cohomology of the BRST operator. Through a careful analysis of this cohomology, and of the renormalization group in curved spacetime, we establish precise criteria which ensure that all conformal supersymmetries are preserved at the quantum level. As a by-product, we provide a rigorous proof that the beta-function for such theories is one-loop exact. We also briefly discuss the construction of chiral rings and the role of non-perturbative effects in curved spacetime.