  We discuss some applications of the Morse-Novikov theory to some problems in
modern physics, where appears a non-exact closed 1-form $\omega$ (a
multi-valued functional). We focus mainly our attention to the cohomology of
the de Rham complex of a compact manifold $M^n$ with a deformed differential
$d_{\omega}=d +\lambda \omega$. Using Witten's approach to the Morse theory one
can estimate the number of critical points of $\omega$ in terms of the
cohomology of deformed de Rham complex with sufficiently large values of
$\lambda$ (torsion-free Novikov's inequalities). We show that for an
interesting class of solvmanifolds this cohomology can be computed as the
cohomology of the corresponding Lie algebra $\mathfrak{g}$ associated with the
one-dimensional representation $\rho_{\lambda \omega}$.
