cannot be reproduced and we were forced to restrict the Mellin transform to a finite range near the high temperature regime. The contributions of the low energy fluctuations to the spectral zeta function are thus almost suppressed. In this sense the question about if the quantum fluctuations induce forces between the BPS vortices remained unsolved because of the lack of control on the previously mentioned source of errors.

The idea to repair this difficulty was to include in the heat kernel expansion a (non local) term that takes care of the effect of zero modes surviving in the low temperature range. The new term induced by the zero modes depends of an arbitrary a priori function of the (fictitious) temperature which is chosen by demanding two properties: (1) The known behaviour of the heat kernel not only at high but also at low temperature are reproduced. (2) The solution of the recurrence relations implied by the asymptotic expansion is minimally perturbed by the arbitrary function. This modification allowed a much more precise evaluation of the Mellin’s transform of the heat trace to obtain the spectral zeta extending the integration interval to all the temperature range. By this token we are 