superalgebra

and

supergeometry

# Contents

## Idea

Where a system of quantum mechanics is specified by

• a Hilbert space $\mathcal{H}$;

• a hermitean operator $H$ on $\mathcal{H}$ – the Hamiltonian;

a system of supersymmetric quantum mechanics has

• a super Hilbert space $\mathcal{H}$;

• an odd linear operator $D$ in $\mathcal{H}$, the supercharge

• such that $D \circ D = H$ is the Hamiltonian.

If we regard the Hamiltonian as the generator of the Poincare Lie algebra in one dimension – the super translation Lie algebra –, then the graded commutator $[D,D] = 2 H$ is the supersymmetry extension to the super Poincaré Lie algebra in super-dimension $(1|1)$.

The data of a system of supersymmetric quantum mechanics may also be formalized in terms of a spectral triple.

## Properties

### Relation to spinning particles

A simple but often underappreciated fact is that the worldline theory of any spinning particle is supersymmetric, and hence is supersymmetric quantum mechanics, on the worldline. In this sense relativistic supersymmetric quantum mechanics is not the exception but the rule, it is something exhibited by every fermion in the world. See at spinning particle – Worldline supersymmetryfor more on this.

Of course the bulk of the literature considers non-relativistic supersymmetric quantum mechanics. That is much less relevant in nature.

### Relation to index theory

Another fairly simple but very deep fact is that the partition function of a supersymmetric quantum mechanical system, namely the supertrace of its propagator, is equivalently what in mathematics (in index theory) is called the index of the supercharge regarded as a Fredholm operator. See the references below for more on this.

This relation is at the heart of a deep and ubiquituous role that supersymmetric quantum mechanics plays in the mathematics of K-theory and related topics (and vice versa). For a general abstract discussion of why there is such a relation see also at super algebra – Abstract idea and at super line 2-bundle.

### Relation to Morse theory

For the moment see below.

### Relation to superstrings

Supersymmetric quantum mechanics was introduced or at least became famous with (Wittem 82). As explained at the end of (Witten 85), Witten had come to consider this while looking at the point particle limit of the superstring sigma-model. The superstring sigma-model is a kind of supersymmetric quantum mechanics on loop space (see also at 2-spectral triple) and ordinary supersymmetric quantum mechanics is obtained from this in the limit of vanishing loop size (see e.g Schreiber 04). Under this identification the above discussion of index theory translates to Witten’s interpretation of the universal elliptic genus as what is now known as the Witten genus (see there for more).

One way to make this rigorously precise would be to realize the Dirac-Ramond operator of the superstring as an actual Dirac operator on smooth loop space, as originally suggested in (Witten 87b).

## References

### General

A fairly comprehensive survey and discussion of supersymmetric quantum mechanics as such, with emphasis on its relation to spectral geometry (“noncommutative geometry”) is in

Another survey is

• Fred Cooper, Avinash Khare, Uday Sukhatme, Supersymmetry and Quantum Mechanics Physics Reports Volume 251 (1995), 267-385. (arXiv:hep-th/9405029)

### Relation to Morse theory

Supersymmetric quantum mechanics gained attention with the work

which showed that Morse theory may be equivalently interpreted as the study of supersymmetric vacua in supersymmetric quantum mechanics, and which was part of what gained Witten the Fields medal 1990. In this article a certain family of deformations of superparticles on a Riemannian manifold are considered and the supersymmetric ground states are shown to be given by the Morse theory of the deformation function.

For a survey of the relation to Morse theory see for instance

• Gábor Pete, section 2 of Morse theory, lecture notes 1999-2001 (pdf)

• Rohit Jain, Supersymmetric Schrödinger operators with applications to Morse theory (pdf)

This deformed supersymmetric quantum mechanics arises as the point-particle limit of the type II superstring regarded as quantum mechanics on the loop space, a relation that is stated more explicitly at the end of

• Edward Witten, Global anomalies in string theory, in W. Bardeen and A. White (eds.) Symposium on Anomalies, Geometry, Topology, pp. 61–99. World Scientific, 1985

and then in

• Edward Witten, The Index Of The Dirac Operator In Loop Space Proc. of Conf. on Elliptic Curves and Modular Forms in Algebraic Topology Princeton (1986) (spire)

The relationon between the 2d SCFT describing the type II superstring and this deformed supersymmetric quantum mechanics on loop space has been further explored in

### Relation to index theory

The relation of the partition function of supersymmetric quantum mechanics to index theory was suggested in unpublished work of Edward Witten and formulated in

• Luis Alvarez-Gaumé, Supersymmetry and the Atiyah-Singer index theorem, Comm. Math. Phys. Volume 90, Number 2 (1983), 161-173. (Euclid)
• Ezra Getzler, Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem, Comm. Math. Phys. 92 (1983), 163-178. (pdf)
• D. Quillen, Superconnections and the Chern character Topology 24 (1985), no. 1, 89–95;

• Varghese Mathai, Daniel Quillen, Superconnections, Thom classes, and equivariant differential forms. Topology 25 (1986), no. 1, 85–110;

• Ezra Getzler, A short proof of the Atiyah-Singer index theorem, Topology 25 (1986), 111-117 (pdf)

• D. Quillen, Superconnection character forms and the Cayley transform. Topology 27 (1988), no. 2, 211–238
Revised on March 22, 2014 03:55:00 by Urs Schreiber (88.128.80.11)