Topology in Molecular Biology

11 One-Forms and Deformed de Rham Complex 193

Dεis two-dimensional disk of radiusε→0. The magnetic fieldF=Fijdxi∧
dxjvanishes outside solenoid, i.e.F≡0onM, hence

Sωα(γ)=

∫

γ

mx ̇^2

2

dt+ωα, (11.8)

whereωα=eAkdxkis an arbitrary closed 1-form onM. The cohomology
space
H^1 (M,R)=H^1 (R^2 \Dε,R)=H^1 (S^1 ,R)=R

is one dimensional and hence

ωα=

eΦα

2 π

xdy−ydx

x^2 +y^2

+dfα,

for some constantΦαand functionfαonM.
Taking the circleγ 0 =∂Dε={(εcosφ, εsinφ,0), 0 ≤φ< 2 π}we have
∫

γ 0

Aαkdxk=

1

e

∫

γ 0

ωα=Φα=

∫

Dε

F 12 dx∧dy.

Hence the constantΦαis equal to the flux of the magnetic fieldF through
the orthogonal sectionDεof our solenoid.
The formωαdetermines a representationρωαof the fundamental group of
M:

ρωα:π 1 (M)→C∗,ρωα(γ) = exp

{

2 πi

∫

γ

ωα

}

,γ∈π 1 (M).

LetSω 1 andSω 2 be two actions for our system. They are quantum-
mechanically equivalent if and only if

exp{ 2 πiSω 1 (γ)}=c(x, x′)exp{ 2 πiSω 2 (γ)},

with a phase factorc(x, x′) depending only on end pointsx, x′ ofγand
|c(x, x′)|= 1, i.e.c(x, x′) is physically unobservable. It is easy to show that
the actionsSω 1 andSω 2 are quantum-mechanically equivalent if and only if
for any loopγ∈π 1 (M) the value of the integral

∫

γ(ω^1 −ω^2 ) is integer or, in
other words, the form (ω 1 −ω 2 ) has integer-valued integrals over basic cycles
ofH 1 (M,Z).
In our caseH 1 (M,Z)=Zand the last condition is equivalent to the
following one ∫

γ 0

(ω 1 −ω 2 )=e(Φ 1 −Φ 2 )=k, k∈Z. (11.9)

Hence (one of the important observations in the Aharonov–Bohm experiment)
the fields with fluxesΦ 1 andΦ 2 , such thatΦ 1 −Φ 2 =k/e, k∈Zcannot be
distinguished by any interference effect.