Consider maps t↦xi(t) from circle to some Riemannian (spin) manifold and lagrangian
L=12gij(x)∂txi∂txj+12gijψj(δik∂t+Γimk∂txm)ψk,
where
ψk are real Grassmann variables. This is supersymmetric under
δxi=ϵψi,δψi=ϵ∂txi.
We want to compute
Tr(−)Fe−βH=∫periodic[dx][dψ]exp(−∫β0dtL),
in the limit
β→0.
My question is: to see that the lagrangian for quadratic fluctuations around constant configurations ξi=xi−xi0, ηi=ψi−ψi0 (namely the one surviving in β→0 limit) is
L(2)=12gij(x0)∂tξi∂tξj−14Rijklξi∂tξjψk0ψl0+i2ηa∂tηa,
what are the right substitutions to make, besides using Riemann normal coordinates and vielbein
eaiebjηab=gij?
References: http://inspirehep.net/record/195891 or http://inspirehep.net/record/190192