Twisted Dirac Operators and the Noncommutative Residue for Manifolds   with Boundary
  In this paper, we give two Lichnerowicz type formulas for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection. We also prove two Kastler-Kalau-Walze type theorems for twisted Dirac operators and twisted signature operators on 4-dimensional manifolds with (resp. without) boundary. 
Jian Wang and Yong Wang
math-ph math.DG math.MP