Multivalued functionals, one-forms and deformed de Rham complex

2005/12/26 by Dmitri V. Millionschikov, Millionschikov, Dmitri V.
Mathematics · Physics and Astronomy · #17B30 #17B56 #57T15 #58A12 #Algebraic Topology (math.AT) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AT #math.MP #msc:17B30 #msc:17B56 #msc:57T15 #msc:58A12

paper · pdf · doi:10.48550/arxiv.math/0512572

arxiv created 2005/12/26 · arxiv updated 2009/12/01

Abstract

We discuss some applications of the Morse-Novikov theory to some problems in modern physics, where appears a non-exact closed 1-form ω (a multi-valued functional). We focus mainly our attention to the cohomology of the de Rham complex of a compact manifold Mⁿ with a deformed differential d_ω=d +λω. Using Witten's approach to the Morse theory one can estimate the number of critical points of ω in terms of the cohomology of deformed de Rham complex with sufficiently large values of λ (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 \mathfrakg associated with the one-dimensional representation ρ_λω.

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