Dirac operators with torsion and the noncommutative residue for manifolds with boundary

Abstract

In this paper, we get the Kastler-Kalau-Walze theorem associated to Dirac operators with torsion on compact manifolds with boundary. We give two kinds of operator-theoretic explanations of the gravitational action in the case of 4-dimensional compact manifolds with flat boundary. Furthermore, we get the Kastler-Kalau-Walze type theorem for four dimensional complex manifolds associated with nonminimal operators.

The moduli space of Type~A surfaces with torsion and non-singular symmetric Ricci tensor

“…The theory of connections with torsion plays an important role in string theory [1,13,15,17], they are important in almost contact geometry [14,18,25,28], they play a role in non-integrable geometries [1,2,3,7], they are important in spin geometries [19], they are useful in considering almost hypercomplex geometries [24], they appear in the study of compact solvmanifolds [12], and they have been used to study the non-commutative residue for manifolds with boundary [29]. The following result was first proved in the torsion free setting by Opozda [26] and subsequently extended to surfaces with torsion by Arias-Marco and Kowalski [4], see also [4,11,16,21,22,27] for related work.…”

Moduli spaces of type $$\mathcal {B}$$ B surfaces with torsion

“…There are, however, natural situations in which manifolds with torsion enter. We refer, for example, to work on torsion-gravity [4,11,20,21,22,36,37,45], on hyper-Kähler with torsion supersymmetric sigma models [23,24,25,43], on string theory [29,32], on almost hypercomplex geometries [38], on spin geometries [33], on B-metrics [39,44], on contact geometries [1], on almost product manifolds [41], on non-integrable geometries [2,8], on the non-commutative residue for manifolds with boundary [46], on Hermitian and anti-Hermitian geometry [40], on CR geometry [17], on Einstein-Weyl gravity at the linearized level [16], on Yang-Mills flow with torsion [30], on ESK theories [14], on double field theory [31], on BRST theory [26], and on the symplectic and elliptic geometries of gravity [12].…”

The moduli space of Type A surfaces with torsion and non-singular symmetric Ricci tensor

“…The theory of connections with torsion plays an important role in string theory [1,13,15,17], they are important in almost contact geometry [14,18,25,28], they play a role in non-integrable geometries [1,2,3,7], they are important in spin geometries [19], they are useful in considering almost hypercomplex geometries [24], they appear in the study of compact solvmanifolds [12], and they have been used to study the non-commutative residue for manifolds with boundary [29]. The following result was first proved in the torsion free setting by Opozda [26] and subsequently extended to surfaces with torsion by Arias-Marco and Kowalski [4], see also [4,11,16,21,22,27] for related work.…”