  The goal of the present paper is to calculate the determinant of the Dirac
operator with a mass in the cylindrical geometry. The domain of this operator
consists of functions that realize a unitary one-dimensional representation of
the fundamental group of the cylinder with $n$ marked points. The determinant
represents a version of the isomonodromic $\tau$-function, itroduced by M.
Sato, T. Miwa and M. Jimbo. It is calculated by comparison of two sections of
the $\mathrm{det}^*$-bundle over an infinite-dimensional grassmannian. The
latter is composed of the spaces of boundary values of some local solutions to
Dirac equation. The principal ingredients of the computation are the formulae
for the Green function of the singular Dirac operator and for the so-called
canonical basis of global solutions on the 1-punctured cylinder. We also derive
a set of deformation equations satisfied by the expansion coefficients of the
canonical basis in the general case and find a more explicit expression for the
$\tau$-function in the simplest case $n=2$.
