Spectral Action for Torsion with and without Boundaries

• Published in: Communications in mathematical physics Vol. 310; no. 2; pp. 367 - 382
• Main Authors: Iochum, B., Levy, C., Vassilevich, D.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.03.2012; Springer Verlag
• Subjects: Classical and Quantum Gravitation; Complex Systems; Mathematical and Computational Physics; Mathematical Physics; Mathematics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical; Pseudodifferential Operator; Manifold; Noncommutative Geometry; Dirac Operator; Chiral Symmetry; Torsion; Spectral action; Manifolds with boundary
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-011-1406-7

We derive a commutative spectral triple and study the spectral action for a rather general geometric setting which includes the (skew-symmetric) torsion and the chiral bag conditions on the boundary. The spectral action splits into bulk and boundary parts. In the bulk, we clarify certain issues of the previous calculations, show that many terms in fact cancel out, and demonstrate that this cancellation is a result of the chiral symmetry of spectral action. On the boundary, we calculate several leading terms in the expansion of spectral action in four dimensions for vanishing chiral parameter θ of the boundary conditions, and show that θ  = 0 is a critical point of the action in any dimension and at all orders of the expansion.