Abstract
In this paper, we define lower dimensional volumes of spin manifolds with boundary. We compute the lower dimensional volume V ol6(1,3) for 6-dimensional spin manifolds with boundary and derive the gravity on boundary from the noncommutative residue associated with Dirac operators. For 6-dimensional manifolds with boundary, we also get a Kastler-Kalau-Walze type theorem for a general fourth order operator.
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