On the Dimensional Reduction Procedure

TL;DR

This paper investigates the validity of the dimensional reduction procedure in Euclidean spaces, showing that it holds exactly for symmetric spaces and approximately for general cases through short-time heat-kernel asymptotics, with implications for anomalies.

Contribution

It demonstrates the conditions under which the dimensional reduction procedure yields exact or approximate results, clarifying its limitations and implications for anomalies.

Findings

Exact heat-kernel density equals reduced density for symmetric spaces.

Heat-kernels coincide up to two leading terms in short-time expansion.

Results have implications for understanding dimensional-reduction anomalies.

Abstract

The issue related to the so-called dimensional reduction procedure is revisited within the Euclidean formalism. First, it is shown that for symmetric spaces, the local exact heat-kernel density is equal to the reduced one, once the harmonic sum has been succesfully performed. In the general case, due to the impossibility to deal with exact results, the short time heat-kernel asymptotics is considered. It is found that the exact heat-kernel and the dimensionally reduced one coincide up to two non trivial leading contributions in the short time expansion. Implications of these results with regard to dimensional-reduction anomaly are discussed.