Trace Anomaly in Metric-Affine gravity

TL;DR

This paper investigates the trace anomaly in metric-affine gravity with torsion and nonmetricity, computing corrections and identifying a new hypermomentum-related anomaly using advanced heat kernel techniques.

Contribution

It introduces the first computation of the trace anomaly in metric-affine geometry including torsion and nonmetricity, revealing a novel hypermomentum anomaly.

Findings

Identifies a new hypermomentum-related anomaly due to Weyl nonmetricity.

Shows invariance under frame rescaling leads to hypermomentum-stress-energy anomaly.

No anomaly found under projective transformation in examined cases.

Abstract

We explore the trace (Weyl) anomaly within a general metric-affine geometry that includes both torsion and nonmetricity. Using the Heat Kernel method and Seeley's algorithm, we compute the Minakshisundaram coefficients for arbitrary spacetimes within this framework, incorporating the effects of the nonmetricity and torsion tensors for the first time. We then determine the corrections to the trace anomaly at one loop for the matter sector in theories invariant under conformal transformation, frame rescaling transformation, and projective transformation. We identify a new anomaly related to hypermomentum, arising from the dilation part mediated by the Weyl component of nonmetricity. As particular cases, we analyze the spin $0$ and spin $1/2$ cases, considering various couplings between matter and the gravitational sector. We demonstrate that invariance under the frame rescaling transformation results in an anomaly in the relationship between the hypermomentum and the stress-energy tensor. In contrast, under the projective transformation, no anomaly is present; specifically, there is no non-zero trace of the hypermomentum tensor in any of our concrete examples.