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Ivan Avramidi
Ivan Avramidi
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Let $D$ be a Dirac operator on an even-dimensional manifold and C be the chirality endomorphism. Let $a_k$ be the coefficients of the heat kernel diagonal of the operator $D^2$ (called the local heat kernel coefficients) so that $b_k=\int_M tr a_k $. Then $Tr C a_{k}=0$ for all $k\ne n/2$, and for $k=n/2$ it is the topological invariant (the index of the Dirac operator). So, $b_{n/2}$ is not a topological invariant.

Let $D$ be a Dirac operator on an even-dimensional manifold and C be the chirality endomorphism. Let $a_k$ be the coefficients of the heat kernel diagonal of the operator $D^2$ (called the local heat kernel coefficients) so that $b_k=\int_M tr a_k $. Then $Tr C a_{k}=0$ for all $k\ne n/2$, and for $k=n/2$ it is the topological invariant (the index of the Dirac operator).

Let $D$ be a Dirac operator on an even-dimensional manifold and C be the chirality endomorphism. Let $a_k$ be the coefficients of the heat kernel diagonal of the operator $D^2$ (called the local heat kernel coefficients) so that $b_k=\int_M tr a_k $. Then $Tr C a_{k}=0$ for all $k\ne n/2$, and for $k=n/2$ it is the topological invariant (the index of the Dirac operator). So, $b_{n/2}$ is not a topological invariant.

Source Link
Ivan Avramidi
Ivan Avramidi
  • 21
  • 2

Let $D$ be a Dirac operator on an even-dimensional manifold and C be the chirality endomorphism. Let $a_k$ be the coefficients of the heat kernel diagonal of the operator $D^2$ (called the local heat kernel coefficients) so that $b_k=\int_M tr a_k $. Then $Tr C a_{k}=0$ for all $k\ne n/2$, and for $k=n/2$ it is the topological invariant (the index of the Dirac operator).