Tadpoles and commutative spectral triples

Sitarz

2014

Commun. Math. Phys.

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Section: About the Dimension Spectrummentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Heat Trace and Spectral Action on the Standard Podleś Sphere

Eckstein

Sitarz

2014

Commun. Math. Phys.

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“…We follow the boundary conditions introduced in [4] and which have been considered in [5,7,28,29,38]: choose…”

Section: The Spectral Triple For Chiral Bag Boundary Conditionmentioning

confidence: 99%

Spectral Action for Torsion with and without Boundaries

Lévy

Vassilevich

2012

Commun. Math. Phys.

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.

“…66,[76][77][78]81,95 Spectral triples associated to manifolds with boundary have been considered in. 14,18,18,58,59,61 The main difficulty is precisely to put nice boundary conditions to the operator D to still get a selfadjoint operator and then, to define a compatible algebra A. This is probably a must to obtain a result in a noncommutative Hamiltonian theory in dimension 1+3.…”

Section: On the Physical Meaning Of The Asymptotics Of Spectral Actionmentioning

confidence: 99%

Spectral Geometry

Geometric, Algebraic and Topological Methods for Quantum Field Theory

2013

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The goal of these lectures is to present the few fundamentals of noncommutative geometry looking around its spectral approach. Strongly motivated by physics, in particular by relativity and quantum mechanics, Chamseddine and Connes have defined an action based on spectral considerations, the so-called spectral action.The idea here is to review the necessary tools which are behind this spectral action to be able to compute it first in the case of Riemannian manifolds (Einstein-Hilbert action). Then, all primary objects defined for manifolds will be generalized to reach the level of noncommutative geometry via spectral triples, with the concrete analysis of the noncommutative torus which is a deformation of the ordinary one.The basics of different ingredients will be presented and studied like, Dirac operators, heat equation asymptotics, zeta functions and then, how to get within the framework of operators on Hilbert spaces, the notion of noncommutative residue, Dixmier trace, pseudodifferential operators etc. These notions are appropriate in noncommutative geometry to tackle the case where the space is swapped with an algebra like for instance the noncommutative torus.