  We study the regularized determinant of the Laplacian as a functional on the
space of Mandelstam diagrams (noncompact translation surfaces glued from finite
and semi-infinite cylinders). A Mandelstam diagram can be considered as a
compact Riemann surface equipped with a conformal flat singular metric
$|\omega|^2$, where $\omega$ is a meromorphic one-form with simple poles such
that all its periods are pure imaginary and all its residues are real. The main
result is an explicit formula for the determinant of the Laplacian in terms of
the basic objects on the underlying Riemann surface (the prime form,
theta-functions, canonical meromorphic bidifferential) and the divisor of the
meromorphic form $\omega$. As an important intermediate result we prove a
decomposition formula of the type of Burghelea-Friedlander-Kappeler for the
determinant of the Laplacian for flat surfaces with cylindrical ends and
conical singularities.
