# Where does the “Supersymmetry” in Witten's proof of the Morse inequalities come from?

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.

References:

1. E. Witten, Supersymmetry and Morse theory, J. Diff, Geom. 17, (1982) 661; Chapter 2.
• Related: Igor Prokhorenkov's notes found on Wikipedia. – Qmechanic Jul 28 '15 at 15:13
• 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. – Qmechanic Aug 10 '15 at 20:54