DeWitt-Virasoro construction in tensor representations

TL;DR

This paper extends the DeWitt-Virasoro construction to higher-rank tensor representations, revealing that the algebra remains consistent with Ricci-flat backgrounds despite additional spin connection terms.

Contribution

It introduces a generalized tensor representation of the DWV generators, incorporating covariant derivatives and bi-vectors, and demonstrates the algebra's invariance with Ricci-flat backgrounds.

Findings

Tensor DWV generators are constructed with covariant derivatives.

The DWV algebra remains unchanged despite additional spin connection terms.

Anomaly cancellation requires Ricci-flatness of the background.

Abstract

We generalize the DeWitt-Virasoro (DWV) construction of arXiv:0912.3987 [hep-th] to tensor representations of higher ranks. A rank-$n$ tensor state, which is by itself coordinate invariant, is expanded in terms of position eigenstates that transform as tensors of the same rank. The representation of the momentum operator in these basis states is then obtained by generalizing DeWitt's argument in Phys.Rev.85:653-661,1952. Such a representation is written in terms of certain bi-vector of parallel displacement and its covariant derivatives. With this machinery at hand we find tensor representations of the DWV generators defined in the previous work. The results differ from those in spin-zero representation by additional terms involving the spin connection. However, we show that the DWV algebra found earlier as a scalar expectation value remains the same, as required by consistency, as all the additional contributions conspire to cancel in various ways. In particular, vanishing of the anomaly term requires the same condition of Ricci-flatness for the background.