Exact Results in Quantum Theory

I will discuss the quantization of quadratic Lagrangians in an external potential focusing on the example of some massive Gaussian Free Field with position-dependent mass viewed as some external potential.We show how to make sense of the effective action by a renormalization obtained by subtracting only local counterterms and we will give some geometrical applications. Then we shall relate our results to some unpublished conjecture from Quillen (30th april 1989) on determinant line bundles.