190 D.V. Millionschikov
where byHtωp(M,R) we denote thep-th cohomology group of the de Rham
complex (Λ∗(M),d+tω) with respect to the new deformed differential d +tω.
Taking a complex parameterλone can identifyHλω∗ (M,C) with the co-
homologyHρ∗λω(Mn,C) with coefficients in the local systemρλωof groupsC,
where
ρλω(γ) = exp∫
γλω, γ∈π 1 (M).Alania in [7] studiedHρ∗λω(Mn,C) of a class of nilmanifoldsMn.Heproved
thatHρ∗λω(Mn,C) is trivial ifλω= 0. The proof was based on the Nomizu
theorem [8] that reduces the problem in the computation in terms of the
corresponding nilpotent Lie algebra. It was remarked in [9, 10] that triviality of
Hρ∗λω(G/Γ,R), withλω= 0 follows from Dixmier’s theorem [11], namely:for
a nilmanifoldG/Γthe cohomologyHω∗(G/Γ,R)coincides with the cohomology
Hω∗(g)associated with the one-dimensional representation of the Lie algebra
ρω:g→R,ρω(ξ)=ω(ξ)and henceHω∗(G/Γ,R)=Hω∗(g)=0.
Applying Hattori’s theorem [12] one can observe that the isomorphism
Hω∗(G/Γ,R)∼=Hω∗(g)still holds on for compact solvmanifoldsG/Γ with completely solvable Lie
groupG. The calculations show that the cohomologyHω∗(G/Γ,R)canbe
non-trivial for certain values [ω]∈H^1 (G/Γ,R). However there exist only a
finite number of such values.
Let us consider a finite subsetΩG/ΓinH^1 (G/Γ,R)∼=H^1 (g):
ΩG/Γ={αi 1 +···+αis| 1 ≤i 1 <···<is≤n, s=1,...,n},where the set{α 1 ,...,αn}of closed 1-forms is in fact the set of the weights of
completely reducible representation associated to the adjoint representation
ofg. It was proved in [9]:if−[ω]∈/ΩG/Γ,then the cohomologyHω∗(G/Γ,R)
is trivial.
11.2 Dirac Monopole, Multi-Valued Actions
and Feynman Quantum Amplitude
The notion of multi-valued functional originates from topological study of the
quantization process of the motion of a charged particle in the field of a Dirac
monopole [13]. The Kirchhoff–Thomson equations for free motion of solids
in a perfect noncompressible liquid also can be reduced to the theory of a
charged particle on the sphereS^2 with some metricgαβin a potential fieldU
and in an effective magnetic fieldF=F 12 with a non-zero flux 4πsthrough
S^2. Locally (in some domainUα) on the sphere we have the following formula
for the actionSα(γ):