A Pauli-Villars Path Integral in Conformal Field Theory

We present a Pauli-Villars approach to the definition of the path integral and the calculation of anomalies in two-dimensional conformal field theory. By comparing the different path integral measures, we explain the consistency, despite different origins, of the conformal anomaly in the Pauli-Villars and the Fujikawa approaches. We show that 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 more familiar operator approach, obtaining a new explanation within the context of the path integral for the anomalous transformation law and anomalous Ward identities of the latter. We provide a first direct calculation of the nontrivial contact terms arising in expectation values of certain energy-momentum products, and we use these to confirm in a simple test that the change of variables formula for the path integral is not spoiled by anomalies. Finally, we review the relationship between the conformal anomaly and the energy-momentum two-point functions in our formalism.