Finiteness Properties of the N=4 Super-Yang--Mills Theory in Supersymmetric Gauge

TL;DR

This paper proves the renormalizability and ultraviolet finiteness of the N=4 super-Yang--Mills theory using shadow fields and twisted representations, highlighting its superconformal invariance and finiteness of BPS operators.

Contribution

It introduces shadow fields and twisted representations to establish renormalizability and finiteness of N=4 SYM independently of regularization methods.

Findings

Proves the cancellation of the beta function.

Shows UV finiteness of 1/2 BPS operators at all orders.

Demonstrates superconformal invariance features.

Abstract

With the introduction of shadow fields, we demonstrate the renormalizability of the N=4 super-Yang--Mills theory in component formalism, independently of the choice of UV regularization. Remarkably, by using twisted representations, one finds that the structure of the theory and its renormalization is determined by a subalgebra of supersymmetry that closes off-shell. Starting from this subalgebra of symmetry, we prove some features of the superconformal invariance of the theory. We give a new algebraic proof of the cancellation of the $\beta$ function and we show the ultraviolet finiteness of the 1/2 BPS operators at all orders in perturbation theory. In fact, using the shadow field as a Maurer--Cartan form, the invariant polynomials in the scalar fields in traceless symmetric representations of the internal R-symmetry group are simply related to characteristic classes. Their UV finiteness is a consequence of the Chern--Simons formula.