  Spin Hurwitz numbers count ramified covers of a spin surface, weighted by the
size of their automorphism group (like ordinary Hurwitz numbers), but signed
$\pm 1$ according to the parity of the covering surface. These numbers were
first defined by Eskin-Okounkov-Pandharipande in order to study the moduli of
holomorphic differentials on a Riemann surface. They have also been related to
Gromov-Witten invariants of of complex 2-folds by work of Lee-Parker and
Maulik-Pandharipande. In this paper, we construct a (spin) TQFT which computes
these numbers, and deduce a formula for any genus in terms of the combinatorics
of the Sergeev algebra, generalizing the formula of
Eskin-Okounkov-Pandharipande. During the construction, we describe a procedure
for averaging any TQFT over finite covering spaces based on the finite path
integrals of Freed-Hopkins-Lurie-Teleman.
