  Chern-Simons Theories with gauge super-groups appear naturally in string
theory and they possess interesting applications in mathematics, e.g. for the
construction of knot and link invariants. This paper is the first in a series
where we propose a new quantisation scheme for such super-group Chern-Simons
theories on 3-manifolds of the form $\Sigma \times \mathbb{R}$. It is based on
a simplicial decomposition of an n-punctured Riemann surface
$\Sigma=\Sigma_{g,n}$ of genus g and allows to construct observables of the
quantum theory for any g and n from basic building blocks, most importantly the
so-called monodromy algebra. In this paper we restrict to the torus case, i.e.
we assume that $\Sigma = T^2$, and to the gauge super-group G=GL(1|1). We
construct the corresponding space of quantum states for the integer level k
Chern-Simons theory along with an explicit representation of the modular group
SL(2,Z) on these states. The latter is shown to be equivalent to the
Lyubachenko-Majid action on the centre of a restricted version of the quantised
universal enveloping algebra of the Lie super-algebra gl(1|1) at the primitive
k-th root of unity.
