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  susy QM and Atiyah-Singer index theorem

+ 3 like - 0 dislike
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Consider maps txi(t) from circle to some Riemannian (spin) manifold and lagrangian
L=12gij(x)txitxj+12gijψj(δikt+Γimktxm)ψk,


where ψk are real Grassmann variables. This is supersymmetric under
δxi=ϵψi,δψi=ϵtxi.

We want to compute
Tr()FeβH=periodic[dx][dψ]exp(β0dtL),

in the limit β0.

My question is: to see that the lagrangian for quadratic fluctuations around constant configurations ξi=xixi0, ηi=ψiψi0 (namely the one surviving in β0 limit) is
L(2)=12gij(x0)tξitξj14Rijklξitξjψk0ψl0+i2ηatηa,


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

asked Nov 4, 2014 in Theoretical Physics by jj_p (150 points) [ revision history ]
edited Nov 5, 2014 by jj_p

I'm a bit confused. There are infinite number of constant configurations, so what exactly are x0 and ψ0?

Right, just fix one of them, and expand around it; the path integral measure splits as [dx][dψ]=[dx0][dψ0][dξ][dη].

To be honest I can't see how [dx][dψ]=[dx0][dψ0][dξ][dη]. Let's take lattice regularization then the integral is just a finite dimensional multiple integral, then clearly for each fixed x0 and ψ0, we have [dx][dψ]=[dξ][dη], and then shouldn't [dx0][dψ0][dξ][dη]=some volume[dx][dψ]?

What I meant is: expand x=x0+ξ around constant configuration x0(t)=x0, and the same for fermions. Then the measure splits as [dx]=N[dx0][dξ], where N is unknown normalization; my point is that, whatever β-dependent substitution you make, it should leave that measure invariant. Do you agree?

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