Topics in Physical Mathematics by Kishore Marathe

Author:Kishore Marathe
Language: eng
Format: epub
Publisher: Springer London, London

(7.31)

with H0 = id. It is customary to write formally . We note that for z ∈ C, Re(z) > > 0, the operator D− z is well-defined and is of trace class. This complex power of the operator D, its zeta function ζD, and the heat family are related by

(7.32)

where Γ(z) is the usual gamma function. The operators Ht are themselves of trace class and are given by convolution with the heat kernel KD. Heat equation was used by Patodi in his study of the index theorem. It can also be used to study harmonic forms and the Hodge decomposition theorem, (see, for example, [214]). It is the detailed study of the heat kernel that enters into the computation of the semiclassical measure when applied to suitable Laplace operators on Λk(M, ad(P)).

The calculations of the semiclassical expectation are quite involved. We state the result for the expectations of | k | and λ in the case that M = S4 and G = SU(2) and refer to [174] for further details.

Theorem 7.1

Let M = S 4 and G = SU(2). Let denote the moduli space of k = 1 instantons over M. Then for any smooth, bounded, gauge-invariant function Φ, the semiclassical partition function

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