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 $\rho_{\lambda \omega}$.
@article{0512572,
author = {Millionschikov, Dmitri V.},
title = {Multivalued functionals, one-forms and deformed de Rham complex},
journal = {arXiv},
volume = {2005},
number = {0},
year = {2005},
language = {en},
url = {http://dml.mathdoc.fr/item/0512572}
}
Millionschikov, Dmitri V. Multivalued functionals, one-forms and deformed de Rham complex. arXiv, Tome 2005 (2005) no. 0, . http://gdmltest.u-ga.fr/item/0512572/