Path integrals and heat kernels

In this thesis we present a path integral representation for the general expression needed to compute anomalies affecting the conservation of the stress tensor (gravitational anomalies) in flat spacetime. In particular, we consider a flat spacetime with a non-abelian gauge background, and discuss the traces needed for evaluating the gravitational anomalies in theories defined in two, four and six dimensions. The peculiar property of these traces, which we refer to as generalized heat kernel traces, is that they contain the insertion of a first order differential operator, making their evaluation much more demanding than the usual traces containing insertion of terms without differential operators. We first build up a path integral for the heat kernel associated to a coloured scalar particle, i.e. a particle that interacts with a non-abelian gauge field and a non-abelian Lorentz scalar potential. This leads to the insertion of a time ordering prescription in path integral. In addition, a further coupling of the particle to an abelian gauge field allows to obtain a path integral representation for computing the heat kernel coefficients of our interest. We considered also an alternative approach to describe the colour degrees of freedom of the particle. This consists in introducing auxiliary bosonic variables on the worldline. They allow to derive a path integral which does not require the time ordering. The path integral representations of the generalized kernel trace is one of the main results of this thesis so we check their correctness by reproducing the coefficients needed for the anomaly in two and four dimensions, already computed in literature by using operator methods, further we computed the coefficient needed in six dimensions which is also a new result not present in literature, as far as we know. Discrete symmetries of our trace could in principle be studied as a future line of investigation, since they could simplify higher order computation.