Arthur Jaffe
Quantum Invariants
(34K, Latex 2e)
ABSTRACT. Consider the partition function Z(Q,a,g). In this
paper we give an elementary proof that this is an invariant.
This is what we mean: assume that Q is a self-adjoint
operator acting on a Hilbert space, and that the operator Q
is odd with respect to a grading gamma of the Hilbert space .
Assume that a is an operator that is even with respect to
the grading and whose square equals I. Suppose further that
the heat kernel generated by H=Q^2 has a finite trace, and
that U(g) is a unitary group representation that commutes
with gamma, with Q, and with a. Define the differential
da=[Q,a]. Then Z(Q,a,g) is an invariant in the following
sense: if the operator Q(lambda) depends differentiably on
a parameter lambda, and if da satisfies a suitable bound,
(we specify the regularity conditions in Section XI)
then Z(Q,a,g) is independent of lambda. Once we have set
up the proper framework, a short calculation in Section IX
shows that the derivative of Z with respect to lambda vanishes.
These considerations apply to non-commutative geometry,
to super-symmetric quantum theory, to string theory, and to
generalizations of these theories to underlying quantum spaces.