look familiar. This spaceHis well known to mathematicians, as the complex-
valued differential forms onRd, often written Ω∗(Rd), where here the∗denotes
an index taking values from 0 (the 0-forms, or functions) tod(thed-forms).
In the theory of differential forms, it is well known that one has an operatord
on Ω∗(Rd) with square zero, called the de Rham differential. Using the inner
product onRd, a Hermitian inner product can be put on Ω∗(Rd) by integration,
and thendhas an adjointδ, also of square zero. The Laplacian operator on
differential forms is
= (d+δ)^2
The supersymmetric quantum system we have been considering corresponds
precisely to this, once one conjugatesd,δas follows
Q+=e−W(q)deW(q), Q−=eW(q)δe−W(q)In mathematics, the interest in differential forms mainly comes from the
fact that they can be constructed not just onRd, but on a general differentiable
manifoldM, with a corresponding construction ofd,δ,operators. In Hodge
theory, one studies solutions of
ψ= 0
(these are called “harmonic forms”) and finds that the dimension of the space of
solutionsψ∈Ωk(M) gives a topological invariant called thekth Betti number
of the manifoldM.
33.4 For further reading
For a reference at the level of these notes, see [31]. For more details about
supersymmetric quantum mechanics see the quantum mechanics textbook of
Tahktajan [90], and lectures by Orlando Alvarez [1]. These references also
describe the relation of these systems to the calculation of topological invariants,
a topic pioneered in Witten’s 1982 paper on supersymmetry and Morse theory
[102].