Modular Invariance, Causality and the PCT-Theorem

Abstract

We describe the IOSp(D, 2∣2)-extension of the Poincaré group in the BRST- quantization of the (spinning) relativistic point particle. The Batalin-Fradkin- Vilkovisky method is used to construct the corresponding field theory, and its dimensional reduction by the Parisi-Sourlas mechanism is proven. We show that a certain element in the identity component of the SO(D, 2) subgroup of IOSp(D, 2∣2) induces the PCT-transformation in the physical subspace. We clarify the role of modular transformations (i.e., of world-line orientation-reversing diffeomorphisms) and argue that the PCT-transformation is the same as a modular transformation seen in an SO(D,2)-rotated frame.

In theories with chiral fermions, OSp(D, 2∣2) is typically broken down to O(D − 1, 1) ⊗ OSp(1, 1∣2), but modular invariance still seems to be at the heart of causality and PCT-invariance.