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.

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