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  Instantons in Witten's supersymmetry and Morse theory

+ 3 like - 0 dislike

I'm reading Witten's paper on supersymmetry and Morse theory and am confused about the details of the instanton calculation which he uses to define a Morse complex (beginning at page 11 of the pdf) .

Witten writes down the relevant supersymmetric Lagrangian and then states

Instanton solutions or tunneling paths in this theory would be extrema of this Lagrangian, written with a Euclidean metric and with the fermions discarded.

I get the part about the Euclidean metric, but why discard fermions? After finding the instanton solutions (which are just paths of steepest descent with regard to the Morse function. We are interested in paths connecting critical points.), he then proceeds to state that for quantum fluctuations around the classical solution,

Nonzero eigenvalues cancel between bosons and fermions, due to supersymmetry. We are left with the zero eigenvalues of the fermions. For a trajectory running from A to B, the index of the Dirac operator equals the Morse index of A minus the Morse index of B.

How does this cancellation between fermions and bosons work? What about the zero eigenvalues of the bosons? The Dirac operator here is just exterior derivation (or more precisely, the perturbed version thereof), correct? But what is meant by "the Dirac operator for a trajectory from A to B"?

A large part of the problem is that nearly all references I could find connecting Instantons and supersymmetry deal with Yang-Mills theory, or extended supersymmetry in far more complicated settings (My background in QFT is rather limited). Even then, I have never found anything which suggests "discarding the fermions" in the instanton calculation. I mostly understand the rest of the paper, but this part leaves me mystified.

This post imported from StackExchange Physics at 2015-03-08 15:16 (UTC), posted by SE-user user247679
asked Jul 21, 2014 in Theoretical Physics by user247679 (45 points) [ no revision ]

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