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.


Let me first refer you to three references pedagogically treating Instantons in quantum mechanics: 1)Riccardo Rattazzi's lecture notes treating instantons in nonsupersymmetric quantum mechanics. In these notes the anharmonic oscillator model is elaborated with great detail 2) Philip Argyres lecture notes treating instantons in supersymmetric quantum mechanics. The model considered in these lecture notes is a simplified one dimensional version Witten's model in flat space 3) Salomonson and van-Holten's original article, where they elaborate in detail the same model treated by Argyres. This article can be used to fill the gaps in Argyres' notes.

Regarding the first question:

First, one must notice that if we take a classical solution in which the fermionic coordinates are nonvanishing, then both the bosonic and fermionic coordinates acquire Grassmann components. The bosonic coordinate becomes an even Grassmann number and the fermionic and odd one. The action $S$ itself becomes an even Grassmann number. This casts a difficulty in the interpretation of $e^{-S}$ as a tunneling rate between the degenerate vacuua.

Salomonson and van-Holten (like Witten) discard the fermions (i.e., substitute zero for the fermion's "classical field"). They justify this substitution by the requirement of keeping the action finite (which is a crucial requirement because $e^{-S}$ is proportional to the transition rate between the vacuua). However, Akulov and Duplij find the general solution for the same model with a nonvanishing fermionic coordinate. They find that the Grassmann number contribution to the action identically vanishes and the action is equal to the classical action with the fermions discarded. This partly justifies the discarding of the fermions (Further justification will be given in the fermionic zero mode discussion). In addition Akulov and Duplij find that unlike the action, the Grassmann number dependence does not generally vanish in the instanton topological charge; this contribution vanishes exactly only for potentials breaking supersymmetry which is reminiscent of the vanishing of the Witten's index when supersymmetry is spontaneously broken. Furthermore Akulov and Duplij extend their analysis to a non-supersymmetric model and find that in this case the Grassmann number contribution to the action does not vanish.

The treatment of zero modes:

The fermion and boson determinants excluding the zero modes exactly cancel. As explained in Argyres, zero modes in the fermionic sector cause the partition function to vanish. However, the correction to the ground state energy entails the insertion of a supersymmetry generator (equation 4.16 in Argyres), this insertion is what is necessary to make the fermionic path integral nonvanishing, since for Grassmann variable $\int d\psi_0 = 0$ while $\int \psi_0 d\psi_0 = 1$.

Now, if we adopt the Akulov and Duplij method, the contribution of the classical fermionic field to the path integral vanishes because as already mentioned, the action does not depend on the classical fermionic variables, thus there is no insertion in the classical component, thus its contribution vanishes.

The bosonic zero mode corresponds to the instanton's collective coordinate (moduli space). Geometrically, this coordinate is the central time $t_0$ of the classical kink solution; and the solution satisfies the equations of motion for all $t_0$ values. The correct evaluation of the path integral requires to perform the bosonic integration on the nonzero modes which is finite, then integrate over the moduli space which entails the integration over $t_0$.

The path integral even in this simple case is quite cumbersome and its explicit calculation is given in Salomonson and van-Holten.

For a rigorous and detailed computation of the instanton path integral for the Witten's model, please see Alice Rogers article.


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