We describe how to construct and compute unambiguously path integrals for
particles moving in a curved space, and how these path integrals can be used to
calculate Feynman graphs and effective actions for various quantum field
theories with external gravity in the framework of the worldline formalism. In
particular, we review a recent application of this worldline approach and
discuss vector and antisymmetric tensor fields coupled to gravity. This
requires the construction of a path integral for the N=2 spinning particle,
which is used to compute the first three Seeley-DeWitt coefficients for all
p-form gauge fields in all dimensions and to derive exact duality relations.Comment: 16 pages, to appear in the proceedings of "Path Integrals 2005: from
Quantum Information to Cosmology" (Prague, June 6-10, 2005), references adde
