Coordinate-invariant Path Integral Methods in Conformal Field Theory

TL;DR

This paper introduces a coordinate-invariant path integral approach for 2D conformal field theory using Pauli-Villars measures, offering advantages over traditional methods and clarifying the nature of conformal anomalies and energy-momentum tensors.

Contribution

It develops a novel, coordinate-invariant path integral framework for conformal field theory, contrasting it with operator formalism and previous path integral approaches, and explains the conformal anomaly within this context.

Findings

Path integral measure is conformally invariant and reparametrization invariant.

The energy-momentum tensor is a true coordinate-invariant tensor in this approach.

The formalism confirms the validity of change of variables in path integrals and clarifies the conformal anomaly's origin.

Abstract

We present a coordinate-invariant approach, based on a Pauli-Villars measure, to the definition of the path integral in two-dimensional conformal field theory. We discuss some advantages of this approach compared to the operator formalism and alternative path integral approaches. We show that our path integral measure is invariant under conformal transformations and field reparametrizations, in contrast to the measure used in the Fujikawa calculation, and we show the agreement, despite different origins, of the conformal anomaly in the two approaches. The natural energy-momentum in the Pauli-Villars approach is a true coordinate-invariant tensor quantity, and we discuss its nontrivial relationship to the corresponding non-tensor object arising in the operator formalism, thus providing a novel explanation within a path integral context for the anomalous Ward identities of the latter. We provide a direct calculation of the nontrivial contact terms arising in expectation values of certain energy-momentum products, and we use these to perform a simple consistency check confirming the validity of the change of variables formula for the path integral. Finally, we review the relationship between the conformal anomaly and the energy-momentum two-point functions in our formalism.