Wednesday, 15 July 2015

Gauge fields and strings: Loop equations


I am trying to derive Eq. (7.25) (p. 117) of Polyakov's book:



δΨ(C) = 2π0P(Fμν(x(s))expCAμdxμ)˙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:


δPexp2π0M(τ)dτ = 2π0dtP(δM(t)exp2π0M(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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