Conformal Invariance = Finiteness and Beta Deformed N=4 SYM Theory

TL;DR

This paper demonstrates that in certain supersymmetric gauge theories, conformal invariance and finiteness are equivalent, and applies this to beta deformed N=4 SYM, showing conformal invariance can be achieved for any complex deformation parameter.

Contribution

It establishes the equivalence of conformal invariance and finiteness in supersymmetric gauge theories and applies this to beta deformed N=4 SYM, allowing for arbitrary complex deformations.

Findings

Conformal invariance implies finiteness in the studied theories.

Finiteness implies conformal invariance, establishing their equivalence.

Conformal invariance can be achieved for any complex deformation parameter in beta deformed N=4 SYM.

Abstract

We claim that if by a choice of the couplings the theory can be made conformally invariant (vanishing of the beta functions) it is automatically finite and vice versa. This is demonstrated by explicit example in supersymmetric gauge theory. The formalism is then applied to the beta deformed ${\cal N}=4$ SYM theory and it is shown that the requirement of conformal invariance = finiteness can be achieved for any complex parameter of deformations.