Chiral anomalies in curved space

The subject of this thesis is a combinatoric method developed by the author to calculate the gravitational contributions to the anomalies in the chiral currents of spin 1/2 and spin 3/2 fields in arbitrary space-time dimensions. Using general arguments it is possible to reduce the work involved in finding either of these contributions to the evaluation of a single one-loop Feynman diagram. It is a straightforward matter to calculate the loop momentum integral in this diagram, but one is then faced with the daunting task of summing the remaining function of the external momenta over all permutations of the external graviton legs. Chapter 4 outlines a notation by which the quantities relevant to this sum may be described. This notation is then used to show that recurrence relations exist between certain of the quantities in different dimensions. By solving the recurrence relations one finally arrives, with the aid of contour integral methods, at expressions for the spin 1/2 and spin 3/2 anomalies in terms of Bernoulli numbers. The calculation of the spin 3/2 anomaly is complicated by the presence of gauge degrees of freedom in the Rarita-Schwinger tensor-spinor field. In the conventional Rarita-Schwinger formulation of spin 3/2 theory both the calculation of the spin 3/2 anomaly and a proof of its gauge independence are practically impossible due to the involved forms of the propagator and vertices. However, the Rarita-Schwinger formulation is not the only formulation of spin 3/2 field theory. In fact there exists a one-parameter family of possible formulations. As it happens, one of these formulations is particularly suited to a calculation of the spin 3/2 anomaly, while in another of them the gauge-independence of the anomaly is made manifest. I therefore adopted these two formulations when calculating the spin 3/2 anomaly and demonstrating its gauge-independence, and the work of this thesis is based on the assumption that the spin 3/2 anomaly remains the same in different formulations. Since all formulations within the one-parameter family may be reached from the Rarita-Schwinger formulation via linear transformations of the field variable, this assumption is entirely reasonable.