### 2.8 Heat kernel and the Sommerfeld formula

Consider for concreteness a quantum bosonic field described by a field operator so that the
partition function is . Then, the effective action defined as
where parameter is a UV cutoff, is expressed in terms of the trace of the heat kernel
. The latter is defined as a solution to the heat equation
In order to calculate the effective action we use the heat kernel method. In the context of manifolds
with conical singularities this method was developed in great detail in [69, 101]. In the Lorentz invariant
case the invariance under the abelian symmetry plays an important role. The heat kernel
(where we omit the coordinates other than the angle ) on regular flat space then depends
on the difference . This function is periodic with respect to . The heat
kernel on a space with a conical singularity is supposed to be periodic.
It is constructed from the periodic quantity by applying the Sommerfeld formula [207]
That this quantity still satisfies the heat kernel equation is a consequence of the invariance under the
abelian isometry . The contour consists of two vertical lines, going from to
and from to and intersecting the real axis between the poles of the
: , and , , respectively. For the integrand in Eq. (22) is
a -periodic function and the contributions of these two vertical lines cancel each other.
Thus, for a small angle deficit the contribution of the integral in Eq. (22) is proportional to
.