A nonperturbative form of the spectral action principle in noncommutative geometry (original) (raw)
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Physical Review Letters, 2007
We study the boundary terms of the spectral action of the noncommutative space, defined by the spectral triple dictated by the physical spectrum of the standard model, unifying gravity with all other fundamental interactions. We prove that the spectral action predicts uniquely the gravitational boundary term required for consistency of quantum gravity with the correct sign and coefficient. This is a remarkable result given the lack of freedom in the spectral action to tune this term.
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The spectral triple approach to noncommutative geometry allows one to develop the entire standard model (and supersymmetric extensions) of particle physics from a purely geometry stand point and thus treats both gravity and particle physics on the same footing. The bosonic sector of the theory contains a modification to Einstein-Hilbert gravity, involving a nonconformal coupling of curvature to the Higgs field and conformal Weyl term (in addition to a nondynamical topological term). In this paper we derive the weak field limit of this gravitational theory and show that the production and dynamics of gravitational waves are significantly altered. In particular, we show that the graviton contains a massive mode that alters the energy lost to gravitational radiation, in systems with evolving quadrupole moment. We explicitly calculate the general solution and apply it to systems with periodically varying quadrupole moments, focusing in particular on the the well know energy loss formula for circular binaries.
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Field theories on deformed spaces suffer from the IR/UV mxing and renormalization is generically spoiled. In work with R. Wulkenhaar, one of us realized a way to cure this desease by adding one more marginal operator. We review these ideas, show the application to φ 3 models and use heat kernel expansion methods for a scalar field theory coupled to an external gauge field on a θ-deformed space and derive noncommutative gauge actions.
Quantum Gravity Boundary Terms from Spectral Action of Noncommutative Space
Eprint Arxiv 0705 1786, 2007
We study the boundary terms of the spectral action of the noncommutative space, defined by the spectral triple dictated by the physical spectrum of the standard model, unifying gravity with all other fundamental interactions. We prove that the spectral action predicts uniquely the gravitational boundary term required for consistency of quantum gravity with the correct sign and coefficient. This is a remarkable result given the lack of freedom in the spectral action to tune this term.
A novel approach to non-commutative gauge theory
Journal of High Energy Physics
We propose a field theoretical model defined on non-commutative space-time with non-constant non-commutativity parameter Θ(x), which satisfies two main requirements: it is gauge invariant and reproduces in the commutative limit, Θ → 0, the standard U(1) gauge theory. We work in the slowly varying field approximation where higher derivatives terms in the star commutator are neglected and the latter is approximated by the Poisson bracket, −i[f, g] ≈ {f, g}. We derive an explicit expression for both the NC deformation of Abelian gauge transformations which close the algebra [δ f , δ g ]A = δ {f,g} A, and the NC field strength F , covariant under these transformations, δ f F = {F , f }. NC Chern-Simons equations are equivalent to the requirement that the NC field strength, F , should vanish identically. Such equations are non-Lagrangian. The NC deformation of Yang-Mills theory is obtained from the gauge invariant action, S = F 2. As guiding example, the case of su(2)-like non-commutativity, corresponding to rotationally invariant NC space, is worked out in detail.
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Physics Letters B, 2010
It is known that gauge transformation of the Kalb-Ramond field B μν with vector parameter Λ μ is symmetry of the closed string world-sheet action. It fails at the end points of the open string and can be restored by introducing Maxwell field A μ. We show that the same conclusion valid for space-time equations of motion, because they are conditions for conformal invariance of the world-sheet action. This is example how the symmetries of space-time theory can be investigated using properties of σ-model energy-momentum tensor. We also show that the reducible part of the closed string symmetry transformation, with Λ μ = ∂ μ λ, turns to irreducible part of the open string one. As well as Maxwell field A μ , the parameter λ is nontrivial only on the string endpoints. We show that after quantization the symmetry transformations of the background fields B μν and A μ turn to the modified symmetries defined in terms of Moyal star product. The modified Λ μ-transformation related different regularizations as well as Seiberg-Witten map.