Topology in Molecular Biology

11 One-Forms and Deformed de Rham Complex 191

Sα(γ)=

∫

γ

(

1

2

gijx ̇ix ̇j−U+eAαkx ̇k

)

dt, (11.1)

where

x^1 =θ, x^2 =φ, F 12 dθ∧dφ=d(Aαkdxk),

∫∫

S^2

F 12 dθ∧dφ=4πs=0.

(11.2)

One can consider Feynman’s paths integral approach to the quantization
of the problem considered above. Recall that in the standard situation of
single-valued actionS, we consider the amplitude

exp{ 2 πiS(γ)},γ∈Ω(x, x′)

and the propagator

K(x, x′)=

∫

Ω(x,x′)

exp{ 2 πiS(γ)}Dγ.

For the Dirac monopole one can consider the set{S 1 ,S 2 }of local actions
whereU 1 =S^2 \PNandU 2 =S^2 \PS,byPN,PSwe denote the poles of the
sphereS^2. Taking the equatorγwith the positive orientation, one can easily
test the ambiguity of the action:

S 1 (γ)−S 2 (γ)=e

∫

γ

(A^1 kdxk−A^2 kdxk)=e

∫∫

S^2

F 12 dθ∧dφ=4πse=0.

(11.3)

The monopole is quantized if and only if the amplitude exp{ 2 πiSα(γ)}is a
single-valued functional, i.e. for an arbitrary closedγ∈U 1 ∩U 2 we have

exp{ 2 πiS 1 (γ)}=exp{ 2 πiS 2 (γ)}.

The last condition is equivalent to the following one:

4 πse=k, k∈Z. (11.4)

Generalizing the situation with the Dirac monopole Novikov [2] considered
ann-dimensional manifoldMn,n>1 with a metricgij, with a scalar potential
Uand with a two-formF of magnetic field not necessarily exact. In these
settings one can consider a set of openUα⊂Mn, such thatF=Fijdxi∧dxj
is exact onUαandMn⊂∪αUα.A1-formωα=Aαkdxk,dωα=Fijdxi∧dxj
is determined up to some closed 1-form and we can consider the set of local
actions:

Sα(γ)=

∫

γ

(

1

2

gijx ̇ix ̇j−U+eAαkx ̇k

)

dt, (11.5)