The answer is quite simple. You should use eqs.(6.8) in the paper. You put them into the last term of eq.(7.11) then, a straightforward integration, I mean something like δτ→τ, δAˉw→Aˉw and so on, should do the job.
So, let us consider (note that in your post there is a wrong sign)
δS=(2π)2i(−Twwδτ+Tˉwˉwδˉτ+τ2πJIwδAIˉw+τ2π˜JIˉwδ˜AIw)constant .
(here "constant" means that only the zero mode is retained) and the corresponding eqs.(6.8) in Kraus' review
Tww=−k8π+18πA2w+18π˜A2w ,Tˉwˉw=−˜k8π+18πA2ˉw+18π˜A2ˉw ,JIw=i2kIJAJw ,˜JIˉw=i2˜kIJ˜AJˉw .
By substitution one has
δS=(2π)2i[−(−k8π+18πA2w+18π˜A2w )δτ+(−˜k8π+18πA2ˉw+18π˜A2ˉw )δˉτ
+iτ22πkIJAJwδAIˉw+iτ22π˜kIJ˜AJˉwδ˜AIw]constant.
From this you get immediately the result when you note that the variation with respect to the gauge field just cancels out the τ1 contribution, having ˉτ−τ that comes from the squared terms, and is recovered upon integration.
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