- 98-16 Arthur Jaffe
- Quantum Invariants
(34K, Latex 2e)
Jan 13, 98
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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.