Simplified worldline path integrals for p-forms and type-A trace anomalies

In this work we study a simplified version of the path integral for a particle on a sphere, and more generally on maximally symmetric spaces, in the case of N=2 supersymmetries on the worldline. This quantum mechanics is generically that of a nonlinear sigma model in one dimension with two supersymmetries (N=2 supersymmetric quantum mechanics), and it is mostly used for describing spin 1 fields and p-forms in first quantisation. Here, we conjecture a simplified path integral defined in terms of a linear sigma model, rather than a nonlinear one. The use of a quadratic kinetic term in the bosonic part of the particle action should be allowed by the use of Riemann normal coordinates, while a scalar effective potential is expected to reproduce the effects of the curvature. Such simplifications have already been proven to be possible for the cases of N=0 and N=1 supersymmetric quantum mechanics. As a particular application, we employ our construction to give a simplified worldline representation of the one-loop effective action of gauge p-forms on maximally symmetric spaces. We use it to compute the first three Seeley-DeWitt coefficients, denoted by a_(p+1)(d;p), namely a_1(2;0), a_2(4;1) and a_3(6;2), that appear in the calculation of the type-A trace anomalies of conformally invariant p-form gauge potentials in d=2p+2 dimensions. The simplified model describes correctly the first two coefficients, while it seems to fail to reproduce the third one. One possible reason could be that the model is based on a conjecture about the effective potential that has been oversimplified in our analysis. Future work could improve our construction, in order to give a correct description to all orders, or alternatively disprove the possibility of having such a simplification in the full N=2 quantum mechanics.