this naif picture: (1) First, usually there are fluctuations belonging to the continuous spectrum. To cope with this problem one put the system in a normalization box and impose periodic boundary conditions on the fields. Equivalently, a toric variety is taken as space and only at the end the volume is allowed to go to infinity. (2) Second, much more dangerous is the existence of massless particles and/or zero mode fluctuations. These long range fluctuations do not disappear in the low temperature (long time) regime and use of the high temperature asymptotics is made dubious. Barvinsky and Vilkovisky proposed to introduce non-local terms to treat this problem in covariant perturbation theory, see e.g. Reference [@Barvinsky1990:npb], an idea that was put at work by Gusev and Zelnikov [@Gusev2000:prd] to compute the effective action in dilatonic two-dimensional gravity. Recall that effective actions are related to determinants of elliptic PDO, susceptible of being regularized by means of the derivative of the spectral zeta function at the origin of the $s$-complex plane.

E. Weinberg in [@Weinberg1979:prd] showed an index theorem in the open $\mathbb{R}^2$-plane for the deformation operator arising from the linear perturbations of the first-order partial differential equations satisfied by self-dual/BPS vortices. The 