Bruno Anglès and Lenny Taelman have published a profound study of finite characteristic special values as listed in the title. This appeared in Proc. London Math. Soc. (3) 110 (2015) 1000-1032. The analogy with classical theory is clear throughout the paper, as well as some potentially interesting differences; these results bode very well for the future of the subject. I have written a long review of this paper as well as pointing to a very new one on the web. This is been submitted to Math Reviews/MathSciNet and appears here with their explicit permission.
Last week we (Chamseddine-Connes-van Suijlekom) posted a preprint on grand unification in the spectral Pati–Salam model which I summarize here (and here).
The paper builds on two recent discoveries in the noncommutative geometry approach to particle physics: we showed
how to obtain inner fluctuations of the metric without having to
assume the order one condition on the Dirac operator. Moreover the
original argument by classificationof finite geometriesF that can provide the fine structure of Euclidean space-time as a product M×F (where M is a usual 4-dimensional Riemannian space) has now been replaced by a much stronger uniqueness statement. This new result shows that the algebra
are the quaternions, appears uniquely when writing the higher analogue
of the Heisenberg commutation relations. This analogue is written in
terms of the basic ingredients of noncommutative geometry where one
takes a spectral point of view, encoding geometry in terms of operators
on a Hilbert space . In this way, the inverse line element is an unbounded self-adjoint operator D. The operator D is the product of the usual Dirac operator on M and a `finite Dirac operator’ on F, which is simply a hermitian matrix DF.
The usual Dirac operator involves gamma matrices which allow one to
combine the momenta into a single operator. The higher analogue of the
Heisenberg relations puts the spatial variables on similar footing by
combining them into a single operator Y
using another set of gamma matrices and it is in this process that the
above algebra appears canonically and uniquely in dimension 4.
This leads without arbitrariness to the Pati–Salam gauge group SU(2)R×SU(2)L×SU(4), together with the corresponding gauge fields and a scalar sector, all derived as inner perturbations of D.
Note that the scalar sector can not be chosen freely, in contrast to
early work on Pati–Salam unification. In fact, there are only a few
possibilities for the precise scalar content, depending on the
assumptions made on the finite Dirac operator.
where Λ is a cutoff scale and f an even and positive function. In the present case, it can be expanded using heat kernel methods,
where F4,F2,F0 are coefficients related to the function f and ak are Seeley deWitt coefficients, expressed in terms of the curvature of M
and (derivatives of) the gauge and scalar fields. This action is
interpreted as an effective field theory for energies lower than Λ.
One important feature of the spectral action is that it gives the usual Pati–Salam action with unification of the gauge couplings. Indeed, the scale-invariant term F0a4 in the spectral action for the spectral Pati–Salam model contains the terms
Normalizing this to give the Yang–Mills Lagrangian demands
which requires gauge coupling unification. This is very similar to the case of the spectral Standard Model
where there is unification of gauge couplings. Since it is well known
that the SM gauge couplings do not meet exactly, it is crucial to
investigate the running of the Pati–Salam gauge couplings beyond the
Standard Model and to find a scale Λ where there is grand
This would then be the scale at which the spectral action is valid as
an effective theory. There is a hierarchy of three energy scales: SM,
an intermediate mass scale mR where symmetry breaking occurs and which is related to the neutrino Majorana masses (1011−1013GeV), and the GUT scale Λ.
In the paper,
we analyze the running of the gauge couplings according to the usual
(one-loop) RG equation. As mentioned before, depending on the
assumptions on DF,
one may vary to a limited extent the scalar particle content,
consisting of either composite or fundamental scalar fields. We will not
limit ourselves to a specific model but consider all cases separately.
This leads to the following three figures:
Running of coupling constants for the spectral Pati–Salam model with composite Higgs fields
Running of coupling constants for the spectral Pati–Salam model with fundamental Higgs fields
Running of coupling constants for the left-right symmetric spectral Pati–Salam model
In other words, we establish grand unification for all of the scenarios with unification scale of the order of 1016 GeV, thus confirming validity of the spectral action at the corresponding scale, independent of the specific form of DF.
Uffe Haagerup was a wonderful man, with a perfect kindness and openness of mind, and a mathematician of incredible power and insight. His whole career is a succession of amazing achievements and of decisive and extremely influential contributions to the field of operator algebras, C*-algebras and von Neumann algebras. His first work (1973-80) concerned the theory of weights and more generally the modular theory of Tomita-Takesaki. UffeHaagerup began by solving a key open question,showing that semi-finite weights are indeed supremum of families of normal states. This allowed him todevelop the standard form of von Neumannalgebras, a basic result used over and over since then. I remember vividly his firstappearance in the field of operator algebras and the striking elegance, clarity and strength of his contributions. He was the first to introduce the Lp-spaces associated to a type III von Neumann algebra. These new spaces have since then received a lot of attentionboth from operator algebraists and specialists of Banach spaces and operator spaces. In a remarkable paper published in Inventiones at the end of the seventies, Haagerup was able to analyse the operator norm in the C*-algebra of the freegroup, to control it by suitable Sobolev norms in spite of the exponential growth of the group and to prove in particular that in spite of its lack of nuclearitythe reduced C*-algebra has the Banach space approximation property. This turned out in the long run to be a breakthrough of major importance. It wasextended by Haagerup and his collaborators to any discrete subgroup of a simple real rank one Lie-group. One corollary is the property RD of rapid decaywhich allows one to define the analogue of the Harish-Chandra algebra of smooth elements in the general context of Gromov hyperbolic groups. This highlynon-trivial result of Haagerup turned out to be the technical key in the proof of the Novikov conjecture for such groups. Moreover the study of discrete groups with the Haagerup property continues to play a major role in geometric group theory. The associated approximation property for the corresponding factors of type II_1 (called the Haagerup property)also played a major role in the solution by Popa of the long-standing problem of factors N non-isomorphic to M_2(N), and of exhibiting a factor with trivial fundamental group. The next fundamental contribution of Haagerup is to the classification of injective factors (1983-87). In my work onthe subject I had left one case completely open (in 76) and the classification was thus incomplete. After severalyears of extremely hard work Haagerup was able to settle this question by proving that there is up to isomorphismonly one hyperfinite factor of type III_1. This is a wonderful achievement. Then Haagerup turned to the theory of subfactors, and, once again, was able to make fundamental contributions such as his construction of new irreducible subfactors of small index > 4, and outclass the best specialists of the subject createdby Vaughan Jones at the beginning of the eighties. Another key contribution of Haagerup is to Voiculescu's Free Probability and to random matrices which he was able to apply very successfully to the theory of C*-algebras and of factors of type II1. Forinstance he proved that, in all II1factors fulfilling the approximate embedding property (which is true in all known cases)every operator T with a non-trivial Brown measure has a non-trivial closed invariant subspaceaffiliated with the von Neumann algebra generated by T. Uffe continued at the same relentless pace to produce amazing contributions and the few mentioned above only give a glimpse of his most impressive collection of breakthrough achievements.
Uffe Haagerup was a marvelous mathematician, well-known to operator algebraists, and to the general community of analysts. From my own perspective an analyst is characterized by the ability of having ”direct access to the infinite” and Uffe Haagerup possessed that quality to perfection. His disparition is a great loss for all of us.