14. FIELD THEORIES ON NON-COMPACT MANIFOLDS 83 be the N th partial sum of this asymptotic expansion (where we have intro- duced a cut-off ψ(x, y) so that KN t is zero outside of a small neighbourhood of the diagonal). Then, we require that for all compact subsets C M × M × X, ∂k t ( Kt KN t ) C,l,m = O(tN−dim M/2−l/2−k ). where l,m refers to the norm on Γ(M, E∨ Dens(M)) E A where we differentiate l times on M × M, m times on X, and take the supremum over the compact subset C. These estimates are the same as the ones satisfied by the actual heat kernel on a compact manifold M, as detailed in (BGV92). The fake heat kernel Kt satisfies the heat equation up to a function which vanishes faster than any power of t. This implies that the asymptotic expansion t− dim M/2 e−x−y 2 /t i≥0 tiΦi(x, y). of Kt must be a formal solution to the heat equation (in the sense described in (BGV92), Section 2.5). This characterizes the functions Φi(x, y) (defined in a neighbourhood of the diagonal) uniquely. Lemma 14.3.2. Let Kt, Kt be two fake heat kernels. Then Kt Kt extends across t = 0 to a smooth kernel, that is, Kt Kt E ! E A C∞(R≥0). Further, Kt Kt vanishes to all orders at t = 0. Proof. This is clear from the existence and uniqueness of the small t asymptotic expansion. Lemma 14.3.3. A fake heat kernel always exists. Proof. The techniques of (BGV92) allow one to construct a fake heat kernel by approximating it with the partial sums of the asymptotic expan- sion. 14.4. Let us suppose that M is an open subset of a compact manifold N, and that the free field theory E on M is restricted from one, say F , on N. Then, the restriction of the heat kernel for F to M is an element Kt E ! 1 E1 C∞(R 0 ) A , which satisfies all the axioms of a fake heat kernel except that of requiring proper support. If Ψ C∞(M × M) is a smooth function, with proper support, which takes value 1 on a neighbourhood of the diagonal, then ΨKt E ! 1 E1 C∞(R 0 ) A is a fake heat kernel.
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