arXiv subjects
Publications and source records attributed to Alain Connes.
This paper contains two parts. The first is the solution of a challenge question, proposed by Etienne Ghys, on the determination of all maps from rational numbers to themselves such that the difference quotient (f(x)-f(y))/(x-y) is always a square. The second is the computer determination, done with the help of Stephane Gaubert, of a function of primes which plays a key role in the first part as a generalization of a result of Carlitz.
We describe the Riemann-Roch strategy which consists of adapting in characteristic zero Weil's proof, of RH in positive characteristic, following the ideas of Mattuck, Tate and Grothendieck. As a new step in this strategy we implement the technique of tropical descent that allows one to deduce existence results in characteristic one from the Riemann-Roch result over the complex numbers. In order to deal with arbitrary distribution functions this technique involves the results of Bohr, Jessen and Tornehave on almost periodic functions. Our main result is the construction, at the adelic level, of a complex lift of the adele class space of the rationals. We interpret this lift as a moduli space of elliptic curves endowed with a triangular structure. The equivalence relation yielding the noncommutative structure is generated by isogenies. We describe the tight relation of this complex lift with the GL(2)-system. We construct the lift of the Frobenius correspondences using the Witt construction in characteristic one.
We complete the proof of the Trace Theorem in the quantized calculus for quasi-Fuchsian group which was stated and sketched, but not fully proved, on pp. 322-325 in the book "Noncommutative Geometry" of the first author.
This article develops several main results for a general theory of homological algebra in categories such as the category of sheaves of idempotent modules over a topos. In the analogy with the development of homological algebra for abelian categories the present paper should be viewed as the analogue of the development of homological algebra for abelian groups. Our selected prototype, the category Bmod of modules over the Boolean semifield B is the replacement for the category of abelian groups. We show that the semi-additive category Bmod fulfills analogues of the axioms AB1 and AB2 for abelian categories. By introducing a precise comonad on Bmod we obtain the conceptually related Kleisli and Eilenberg-Moore categories. The latter category Bmod^s is simply Bmod in the topos of sets endowed with an involution and as such it shares with Bmod most of its abstract categorical properties. The three main results of the paper are the following. First, when endowed with the natural ideal of null morphisms, the category Bmod^s is a semi-exact, homological category in the sense of M. Grandis. Second, there is a far reaching analogy between Bmod^s and the category of operators in Hilbert space, and in particular results relating null kernel and injectivity for morphisms. The third fundamental result is that, even for finite objects of Bmod^s the resulting homological algebra is non-trivial and gives rise to a computable Ext functor. We determine explicitly this functor in the case provided by the diagonal morphism of the Boolean semiring into its square.
We give a survey of our joint ongoing work with Ali Chamseddine, Slava Mukhanov and Walter van Suijlekom. We show how a problem purely motivated by "how geometry emerges from the quantum formalism" gives rise to a slightly noncommutative structure and a spectral model of gravity coupled with matter which fits with experimental knowledge. This text will appear as a contribution to the volume: "Foundations of Mathematics and Physics one century after Hilbert". Editor: Joseph Kouneiher. Collection Mathematical Physics, Springer 2017
We consider the Laplacian associated with a general metric in the canonical conformal structure of the noncommutative two torus, and calculate a local expression for the term a_4 that appears in its corresponding small-time heat kernel expansion. The final formula involves one variable functions and lengthy two, three and four variable functions of the modular automorphism of the state that encodes the conformal perturbation of the flat metric. We confirm the validity of the calculated expressions by showing that they satisfy a family of conceptually predicted functional relations. By studying these functional relations abstractly, we derive a partial differential system which involves a natural action of cyclic groups of order two, three and four and a flow in parameter space. We discover symmetries of the calculated expressions with respect to the action of the cyclic groups. In passing, we show that the main ingredients of our calculations, which come from a rearrangement lemma and relations between the derivatives up to order four of the conformal factor and those of its logarithm, can be derived by finite differences from the generating function of the Bernoulli numbers and its multiplicative inverse. We then shed light on the significance of exponential polynomials and their smooth fractions in understanding the general structure of the noncommutative geometric invariants appearing in the heat kernel expansion. As an application of our results we obtain the a_4 term for noncommutative four tori which split as products of two tori. These four tori are not conformally flat and the a_4 term gives a first hint of the Riemann curvature and the higher dimensional modular structure.
We construct the scaling site S by implementing the extension of scalars on the arithmetic site, from the smallest Boolean semifield to the tropical semifield of positive real numbers. The obtained semiringed topos is the Grothendieck topos semi-direct product of the Euclidean half-line and the monoid of positive integers acting by multiplication, endowed with the structure sheaf of piecewise affine, convex functions with integral slopes. We show that the points of this topos coincide with the adele class space of the rationals and that this latter space inherits the geometric structure of a tropical curve. We restrict this construction to the periodic orbit of the scaling flow associated to each prime and obtain a quasi-tropical structure which turns this orbit into a variant C of the classical Jacobi description of an elliptic curve. On C, we develop the theory of Cartier divisors, determine the structure of the quotient of the abelian group of divisors by the subgroup of principal divisors, develop the theory of theta functions, and prove the Riemann-Roch formula which involves real valued dimensions, as in the type II index theory. We show that one would have been led to the same definition of the scaling site by analyzing the well known results on the localization of zeros of analytic functions involving Newton polygons in the non-archimedean case and the Jensen's formula in the complex case.
We show that the basic categorical concept of an S-algebra as derived from the theory of Segal's Gamma-sets provides a unifying description of several constructions attempting to model an algebraic geometry over the absolute point. It merges, in particular, the approaches using monoids, semirings and hyperrings as well as the development by means of monads and generalized rings in Arakelov geometry. The assembly map determines a functorial way to associate an S-algebra to a monad on pointed sets. The notion of an S-algebra is very familiar in algebraic topology where it also provides a suitable groundwork to the definition of topological cyclic homology. The main contribution of this paper is to point out its relevance and unifying role in arithmetic, in relation with the development of an algebraic geometry over symmetric closed monoidal categories.
We investigate the semi-ringed topos obtained by extension of scalars from the arithmetic site of our previous work, by replacing the smallest Boolean semifield by the tropical semifield of real numbers with the max-plus operations. The obtained site is the semi-direct product of the Euclidean half-line by the action of the monoid of positive integers by multiplication. Its points are the same as the points of the arithmetic site over the tropical semifield of real numbers, and coincide with the quotient of the adele class space of Q by the action of the maximal compact subgroup of the idele class group. The structure sheaf of the scaling topos endows it with a natural structure of tropical curve over the arithmetic topos. The restriction of this structure to the periodic orbits of the scaling flow gives, for each prime p, an analogue of an elliptic curve whose Jacobian is a cyclic group of order p-1. The Riemann-Roch formula holds and involves real valued dimensions and real degrees for divisors.
The Riemann hypothesis is, and will hopefully remain for a long time, a great motivation to uncover and explore new parts of the mathematical world. After reviewing its impact on the development of algebraic geometry we discuss three strategies, working concretely at the level of the explicit formulas. The first strategy is "analytic" and is based on Riemannian spaces and Selberg's work on the trace formula and its comparison with the explicit formulas. The second is based on algebraic geometry and the Riemann-Roch theorem. We establish a framework in which one can transpose many of the ingredients of the Weil proof as reformulated by Mattuck, Tate and Grothendieck. This framework is elaborate and involves noncommutative geometry, Grothendieck toposes and tropical geometry. We point out the remaining difficulties and show that RH gives a strong motivation to develop algebraic geometry in the emerging world of characteristic one. Finally we briefly discuss a third strategy based on the development of a suitable "Weil cohomology", the role of Segal's Gamma-rings and of topological cyclic homology as a model for "absolute algebra" and as a cohomological tool.
We analyze the running at one-loop of the gauge couplings in the spectral Pati-Salam model that was derived in the framework of noncommutative geometry. There are a few different scenario's for the scalar particle content which are determined by the precise form of the Dirac operator for the finite noncommutative space. We consider these different scenarios and establish for all of them unification of the Pati-Salam gauge couplings. The boundary conditions are set by the usual RG flow for the Standard Model couplings at an intermediate mass scale at which the Pati-Salam symmetry is broken.
We introduce the Arithmetic Site: an algebraic geometric space deeply related to the non-commutative geometric approach to the Riemann Hypothesis. We prove that the non-commutative space quotient of the adele class space of the field of rational numbers by the maximal compact subgroup of the idele class group, which we had previously shown to yield the correct counting function to obtain the complete Riemann zeta function as Hasse-Weil zeta function, is the set of geometric points of the arithmetic site over the semifield of tropical real numbers. The action of the multiplicative group of positive real numbers on the adele class space corresponds to the action of the Frobenius automorphisms on the above geometric points. The underlying topological space of the arithmetic site is the topos of functors from the multiplicative semigroup of non-zero natural numbers to the category of sets. The structure sheaf is made by semirings of characteristic one and is given globally by the semifield of tropical integers. In spite of the countable combinatorial nature of the arithmetic site, this space admits a one parameter semigroup of Frobenius correspondences obtained as sub-varieties of the square of the site. This square is a semi-ringed topos whose structure sheaf involves Newton polygons. Finally, we show that the arithmetic site is intimately related to the structure of the (absolute) point in non-commutative geometry.
In the construction of spectral manifolds in noncommutative geometry, a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of real scalar fields naturally appears and implies, by equality with the index formula, the quantization of the volume. We first show that this condition implies that the manifold decomposes into disconnected spheres which will represent quanta of geometry. We then refine the condition by involving the real structure and two types of geometric quanta, and show that connected spin-manifolds with large quantized volume are then obtained as solutions. The two algebras M_2(H) and M_4(C) are obtained which are the exact constituents of the Standard Model. Using the two maps from M_4 to S^4 the four-manifold is built out of a very large number of the two kinds of spheres of Planckian volume. We give several physical applications of this scheme such as quantization of the cosmological constant, mimetic dark matter and area quantization of black holes.
We define the universal thickening of the field of real numbers. This construction is performed in three steps which parallel the universal perfection, the Witt construction and a completion process. We show that the transposition of the perfection process at the real archimedean place is identical to the "dequantization" process and yields Viro's tropical real hyperfield. Then we prove that the archimedean Witt construction in the context of hyperfields allows one to recover a field from a hyperfield, and we obtain the universal pro-infinitesimal thickening of the field of real numbers. We provide the real analogues of several algebras used in the construction of the rings of p-adic periods. We supply the canonical decomposition of elements in terms of Teichmuller lifts, we make the link with the Mikusinski field of operational calculus and compute the Gelfand spectrum of the archimedean counterparts of the rings of p-adic periods. In the second part of the paper we discuss the complex case and its relation with the theory of oscillatory integrals in quantum physics.
Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation appears in two versions, one sided and two sided. It implies the quantization of the volume. In the one-sided case it implies that the manifold decomposes into a disconnected sum of spheres which will represent quanta of geometry. The two sided version in dimension 4 predicts the two algebras M_2(H) and M_4(C) which are the algebraic constituents of the Standard Model of particle physics. This taken together with the non-commutative algebra of functions allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model. We show that any connected Riemannian Spin 4-manifold with quantized volume >4 (in suitable units) appears as an irreducible representation of the two-sided commutation relations in dimension 4 and that these representations give a seductive model of the "particle picture" for a theory of quantum gravity in which both the Einstein geometric standpoint and the Standard Model emerge from Quantum Mechanics. Physical applications of this quantization scheme will follow in a separate publication.
The assumption that space-time is a noncommutative space formed as a product of a continuous four dimensional manifold times a finite space predicts, almost uniquely, the Standard Model with all its fermions, gauge fields, Higgs field and their representations. A strong restriction on the noncommutative space results from the first order condition which came from the requirement that the Dirac operator is a differential operator of order one. Without this restriction, invariance under inner automorphisms requires the inner fluctuations of the Dirac operator to contain a quadratic piece expressed in terms of the linear part. We apply the classification of product noncommutative spaces without the first order condition and show that this leads immediately to a Pati-Salam SU(2)_{R}x SU(2)_{L}x SU(4) type model which unifies leptons and quarks in four colors. Besides the gauge fields, there are 16 fermions in the (2,1,4)+(1,2,4) representation, fundamental Higgs fields in the (2,2,1), (2,1,4) and (1,1,1+15) representations. Depending on the precise form of the order one condition or not there are additional Higgs fields which are either composite depending on the fundamental Higgs fields listed above, or are fundamental themselves. These additional Higgs fields break spontaneously the Pati-Salam symmetries at high energies to those of the Standard Model.
We determine the points of the epicyclic topos which plays a key role in the geometric encoding of cyclic homology and the lambda operations. We show that the category of points of the epicyclic topos is equivalent to projective geometry in characteristic one over algebraic extensions of the infinite semifield of max-plus integers. An object of this category is a pair of an algebraic extension of the semifield and an archimedean semimodule over this extension. The morphisms are projective classes of semilinear maps between semimodules. The epicyclic topos sits over the arithmetic topos which we recently introduced and the fibers of the associated geometric morphism correspond to the cyclic site. In two appendices we review the role of the cyclic and epicyclic toposes as the geometric structures supporting cyclic homology and the lambda operations.
We show that the non-commutative geometric approach to the Riemann zeta function has an algebraic geometric incarnation: the "Arithmetic Site". This site involves the tropical semiring viewed as a sheaf on the topos which is the dual of the multiplicative semigroup of positive integers. We prove that the set of points of the arithmetic site over the maximal compact subring of the tropical semifield is the non-commutative space quotient of the adele class space of Q by the action of the maximal compact subgroup of the idele class group. We realize the Frobenius correspondences in the square of the "Arithmetic Site" and compute their composition. This note provides the algebraic geometric space underlying the non-commutative approach to RH.