Multivalued functionals, one-forms and deformed de Rham complex

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 $\rho_{\lambda \omega}$.