Quantum gravity boundary terms from the spectral action of noncommutative space

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.     

The Uncanny Precision of the Spectral Action

Noncommutative geometry has been slowly emerging as a new paradigm of geometry which starts from quantum mechanics. One of its key features is that the new geometry is spectral in agreement with the physical way of measuring distances. In this paper we present a detailed introduction with an overview on the study of the quantum nature of space-time using the tools of noncommutative geometry. In particular we examine the suitability of using the spectral action as an action functional for the theory. To demonstrate how the spectral action encodes the dynamics of gravity we examine the accuracy of the approximation of the spectral action by its asymptotic expansion in the case of the round sphere S ³. We find that the two terms corresponding to the cosmological constant and the scalar curvature term already give the full result with remarkable accuracy. This is then applied to the physically relevant case of S ³ × S ¹, where we show that the spectral action in this case is also given, for any test function, by the sum of two terms up to an astronomically small correction, and in particular all higher order terms a _2n vanish. This result is confirmed by evaluating the spectral action using the heat kernel expansion where we check that the higher order terms a ₄ and a ₆ both vanish due to remarkable cancelations. We also show that the Higgs potential appears as an exact perturbation when the test function used is a smooth cutoff function.     

Noncommutative geometry and motives: The thermodynamics of endomotives

We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of L-functions. The analogue in characteristic zero of the action of the Frobenius on ?-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative generalization of Artin motives). Semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives, with an action of the absolute Galois group. The semigroup of endomorphisms of the multiplicative group yields the Bost-Connes system, from which one obtains, through the above procedure, the desired cohomological interpretation of the zeros of the Riemann zeta function. In the last section we also give a Lefschetz formula for the archimedean local L-factors of arithmetic varieties.     

Gravity coupled with matter and the foundation of non-commutative geometry

We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds. Its unitary representations correspond to Riemannian metrics and Spin structure whileds is the Dirac propagatords=x−x=D ⁻¹, whereD is the Dirac operator. We extend these simple relations to the non-commutative case using Tomita's involutionJ. We then write a spectral action, the trace of a function of the length element, which when applied to the non-commutative geometry of the Standard Model will be shown ([CC]) to give the SM Lagrangian coupled to gravity. The internal fluctuations of the non-commutative geometry are trivial in the commutative case but yield the full bosonic sector of SM with all correct quantum numbers in this slightly non-commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group.     

Moduli Space and Structure of Noncommutative 3-Spheres

We analyze the moduli space and the structure of noncommutative 3-spheres. We develop the notion of central quadratic form for quadratic algebras, and prove a general algebraic result which considerably refines the classical homomorphism from a quadratic algebra to a cross-product algebra associated to the characteristic variety and lands in a richer cross-product. It allows to control the C*-norm on involutive quadratic algebras and to construct the differential calculus in the desired generality. The moduli space of noncommutative 3-spheres is identified with equivalence classes of pairs of points in a symmetric space of unitary unimodular symmetric matrices. The scaling foliation of the moduli space is identified to the gradient flow of the character of a virtual representation of SO(6). Its generic orbits are connected components of real parts of elliptic curves which form a net of biquadratic curves with eight points in common. We show that generically these curves are the same as the characteristic variety of the associated quadratic algebra. We then apply the general theory of central quadratic forms to show that the noncommutative 3-spheres admit a natural ramified covering π by a noncommutative three-dimensional nilmanifold. This yields the differential calculus. We then compute the Jacobian of the ramified covering π by pairing the direct image of the fundamental class of the noncommutative three-dimensional nilmanifold with the Chern character of the defining unitary and obtain the answer as the product of a period (of an elliptic integral) by a rational function. Finally, we show that the hyperfinite factor of type II₁ appears as cross-product of the field K _q of meromorphic functions on an elliptic curve by a subgroup of its Galois group ${\text{Aut}}_\mathbb{C} \left( {K_q } \right)$ .     

Cyclic Cohomology and Hopf Algebra Symmetry

Cyclic cohomology has been recently adapted to the treatment of Hopf symmetry in noncommutative geometry. The resulting theory of characteristic classes for Hopf algebras and their actions on algebras allows the expansion of the range of applications of cyclic cohomology. It is the goal of this Letter to illustrate these recent developments, with special emphasis on the application to transverse index theory, and point towards future directions. In particular, we highlight the remarkable accord between our framework for cyclic cohomology of Hopf algebras on the one hand and both the algebraic as well as the analytic theory of quantum groups on the other, manifest in the construction of the modular square.     

Renormalization in Quantum Field Theory and the Riemann--Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group

We showed in Part I that the Hopf algebra ℋ of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group G and that the renormalized theory is obtained from the unrenormalized one by evaluating at ɛ= 0 the holomorphic part γ₊(ɛ) of the Riemann–Hilbert decomposition γ₋(ɛ)^{− 1}γ₊(ɛ) of the loop γ(ɛ)∈G provided by dimensional regularization. We show in this paper that the group G acts naturally on the complex space X of dimensionless coupling constants of the theory. More precisely, the formula g ₀=gZ ₁ Z ₃ ^−3/2 for the effective coupling constant, when viewed as a formal power series, does define a Hopf algebra homomorphism between the Hopf algebra of coordinates on the group of formal diffeomorphisms to the Hopf algebra ℋ. This allows first of all to read off directly, without using the group G, the bare coupling constant and the renormalized one from the Riemann–Hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms. This shows that renormalization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of the dimensional regularization parameter ɛ. It also allows to lift both the renormalization group and the β-function as the asymptotic scaling in the group G. This exploits the full power of the Riemann–Hilbert decomposition together with the invariance of γ₋(ɛ) under a change of unit of mass. This not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group G for the full higher pole structure of minimal subtracted counterterms in terms of the residue. Received: 21 March 2000 / Accepted: 3 October 2000     

The hyperring of adèle classes

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We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adèle class space HK=AK/K^× of a global field K. After promoting F₁ to a hyperfield K, we prove that a hyperring of the form R/G (where R is a ring and G⊂R^× is a subgroup of its multiplicative group) is a hyperring extension of K if and only if G∪{0} is a subfield of R. This result applies to the adèle class space which thus inherits the structure of a hyperring extension HK of K. We begin to investigate the content of an algebraic geometry over K. The category of commutative hyperring extensions of K is inclusive of: commutative algebras over fields with semi-linear homomorphisms, abelian groups with injective homomorphisms and a rather exotic land comprising homogeneous non-Desarguesian planes. Finally, we show that for a global field K of positive characteristic, the groupoid of the prime elements of the hyperring HK is canonically and equivariantly isomorphic to the groupoid of the loops of the maximal abelian cover of the curve associated to the global field K.

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