I am trying to derive Eq. (7.25) (p. 117) of Polyakov's book:
δΨ(C) = ∫2π0P(Fμν(x(s))exp∮CAμdxμ)˙xνδxμ(x)ds,
where the non-abelian phase factor around a closed loop C is defined as
Ψ(C) = Pexp(∮Aμdxμ)=Pexp(∫2π0Aμ˙xμds).
It seems that he is using the relation given on p. 116:
δPexp∫2π0M(τ)dτ = ∫2π0dtP(δM(t)exp∫2π0M(τ)dτ).
Matching with (7.25) I find δAν=Fμνδxμ. This relation seems to be saying that if I change the position of the loop at the parameter s by δxμ(s) then the vector potential changes by δAν(x(s))=Fμν(x(s))δxμ(s).
I don't know how to derive this relation. Is it legitimate?
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