${\cal N}$ = $1$ Euler Anomaly from RG-dependent metric-Background

TL;DR

This paper derives the Euler anomaly difference in ${ m extbf{N}}=1$ supersymmetric gauge theories within the conformal window by incorporating RG-dependent metrics, linking anomaly flow to anomalous dimensions.

Contribution

It introduces a novel approach of absorbing RG flow into a metric transformation, enabling anomaly calculations via free field theory with a flow-dependent metric.

Findings

Euler anomaly difference expressed in terms of infrared anomalous dimension

Method confirms previous results using conserved current matching

Provides a geometric interpretation of RG flow in supersymmetric theories

Abstract

We consider ${\cal N}=1$ supersymmetric gauge theories in the conformal window. By applying a suitable matter superfield rescaling and a Weyl-transformation the renormalisation group running (matter and gauge field $Z$-factors) are absorbed into the metric. The latter becomes a function of the $Z$-factors. The Euler flow $\Delta a \equiv a_{\rm UV} - a_{\rm IR} |_{{\cal N}=1}$ is then obtained by free field theory computation with the non-trivial dynamics coming from expanding the Euler invariant in the flow dependent metric. The result is therefore directly obtained in terms of the infrared anomalous dimension confirming an earlier result using the matching of conserved currents.