The Heat Kernel Method
The goal of the heat kernel method is to express (2.40) as an integral over the fixed point set M γ in M of the transformation γ. Here M γ = M if γ is the identity. The method is based on the following observations about arbitrary elliptic differential operators D, acting on sections of a smooth vector bundle F over a compact manifold M, which admits a a direct sum decomposition F = F + ⊕ F − . In our case, F = E ⊗ L, with the splitting F ± = E ± ⊗ L, and D is the spin-c Dirac operator. For the required facts about trace class operators, see for instance Hörmander [42, Sec. 19.1], or Duistermaat .
KeywordsVector Bundle Integral Operator Heat Kernel Integral Kernel Trace Class Operator
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