Where does the "Supersymmetry" in Witten's proof of the Morse inequalities (original paper and outline of proof for mathematicians) come from? Hopefully someone can provide an intuitive understanding? I am a math student, so I do not have a very extensive knowledge of physics.


TL;DR: It is the wedge product $\wedge$ and the exterior derivative/differential $d$ (which squares to zero $d^2=0$) that give rise to Grassmann-odd elements and supersymmetry.

More concretely, Ref. 1 first writes down a (non-relativistic) SUSY algebra ${\cal A}$

$$\tag{10} \{Q_1,Q_1\}_+~=~2H~=~\{Q_2,Q_2\}_+, \qquad \{Q_1,Q_2\}_+~=~0, $$

spanned by two Grassmann-odd elements $Q_i$ and one Grassmann-even element $H$,

$$\tag{9} Q_1 ~:=~d+d^{\ast}, \qquad Q_2 ~:=~i(d-d^{\ast}), \qquad H~:=~\Delta~=~\{d,d^{\ast}\}_+. $$

Here $d^{\ast}\equiv\delta $ is the codifferential (=adjoint of the exterior differential $d$) and $\Delta$ is the Laplace-de Rham operator on exterior forms.

The relevant SUSY algebra ${\cal A}_t$ is a twisted/conjugated version of eq. (9) and (10), where $d$ and $d^{\ast}$ are replaced with

$$\tag{11} d_t ~:=~ e^{-ht}de^{ht}, \qquad d_t^{\ast} ~=~ e^{ht}d^{\ast}e^{-ht},$$

respectively. Here $h$ is a (real-valued) Morse function, and $t\in\mathbb{R}$ a real parameter. See Ref. 1 for further details.


  1. E. Witten, Supersymmetry and Morse theory, J. Diff, Geom. 17, (1982) 661; Chapter 2.
  • $\begingroup$ Related: Igor Prokhorenkov's notes found on Wikipedia. $\endgroup$ – Qmechanic Jul 28 '15 at 15:13
  • $\begingroup$ More References: K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, Mirror Symmetry, 2003; Section 10.4. The pdf file is available here. $\endgroup$ – Qmechanic Aug 10 '15 at 20:54

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