Abstract
In this paper, we compute lower dimensional volumes $${\rm{Vol}}_4^{(1,1)}\;\rm{and}\;{\rm{Vol}}_6^{(2,2)}$$ about Witten deformation for 4, 6-dimensional spin manifolds with boundary respectively, and get assosiated Kastler–Kalau–Walze type theorems. We also give theoritic explaination of the gravitational action for 4, 6 dimensional manifolds with boundary by these noncommutative residues.
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