arXiv subjects
Publications and source records attributed to Matilde Marcolli.
We classify graph C*-algebras, namely, Cuntz-Krieger algebras associated to the Bass-Hashimoto edge incidence operator of a finite graph. This is done by a purely graph theoretical calculation of the K-theory and the position of the unit therein.
We describe the construction of theta summable and finitely summable spectral triples associated to Mumford curves and some classes of higher dimensional buildings. The finitely summable case is constructed by considering the stabilization of the algebra of the dual graph of the special fiber of the Mumford curve and a variant of the Antonescu-Christensen spectral geometries for AF algebras. The information on the Schottky uniformization is encoded in the spectral geometry through the Patterson-Sullivan measure on the limit set. Some higher rank cases are obtained by adapting the construction for trees.
This text is written for the volume of the school/conference "Noncommutative Geometry 2005" held at IPM Tehran. It gives a survey of methods and results in noncommutative geometry, based on a discussion of significant examples of noncommutative spaces in geometry, number theory, and physics. The paper also contains an outline (the ``Tehran program'') of ongoing joint work with Consani on the noncommutative geometry of the adeles class space and its relation to number theoretic questions.
In this paper we give a survey of some models of the integer and fractional quantum Hall effect based on noncommutative geometry. We begin by recalling some classical geometry of electrons in solids and the passage to noncommutative geometry produced by the presence of a magnetic field. We recall how one can obtain this way a single electron model of the integer quantum Hall effect. While in the case of the integer quantum Hall effect the underlying geometry is Euclidean, we then discuss a model of the fractional quantum Hall effect, which is based on hyperbolic geometry simulating the multi-electron interactions. We derive the fractional values of the Hall conductance as integer multiples of orbifold Euler characteristics. We compare the results with experimental data.
This is a survey of our results on the relation between perturbative renormalization and motivic Galois theory. The main result is that all quantum field theories share a common universal symmetry realized as a motivic Galois group, whose action is dictated by the divergences and generalizes that of the renormalization group. The existence of such a group was conjectured by P. Cartier based on number theoretic evidence and on the Connes-Kreimer theory of perturbative renormalization. The group provides a universal formula for counterterms and is obtained via a Riemann-Hilbert correspondence classifying equivalence classes of flat equisingular bundles, where the equisingularity condition corresponds to the independence of the counterterms on the mass scale.
We construct a quantum statistical mechanical system which generalizes the Bost-Connes system to imaginary quadratic fields K of arbitrary class number and fully incorporates the explict class field theory for such fields. This system admits the Dedekind zeta function as partition function and the Idele class group as group of symmetries. The extremal KMS states at zero temperature intertwine this symmetry with the Galois action on the values of the states on the arithmetic subalgebra. We also give an interpretation of the original BC system and of the GL(2) system in terms of Shimura varieties, which motivates the construction for imaginary quadratic fields. The geometric notion underlying the construction is that of commensurability of K-lattices.
We establish a precise relation between Galois theory in its motivic form with the mathematical theory of perturbative renormalization (in the minimal subtraction scheme with dimensional regularization). We identify, through a Riemann-Hilbert correspondence based on the Birkhoff decomposition and the t'Hooft relations, a universal symmetry group (the "cosmic Galois group" suggested by Cartier), which contains the renormalization group and acts on the set of physical theories. This group is closely related to motivic Galois theory. We construct a universal singular frame of geometric nature, in which all divergences disappear. The paper includes a detailed overview of the work of Connes-Kreimer and background material on the main quantum field theoretic and algebro-geometric notions involved. We give a complete account of our results announced in math.NT/0409306.
This is the text of a series of five lectures given by the author at the "Second Annual Spring Institute on Noncommutative Geometry and Operator Algebras" held at Vanderbilt University in May 2004. It is meant as an overview of recent results illustrating the interplay between noncommutative geometry and arithmetic geometry/number theory.
We investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain `` motivic Galois group'', which is uniquely determined and universal with respect to the set of physical theories. The renormalization group can be identified canonically with a one parameter subgroup. The group is obtained through a Riemann-Hilbert correspondence. Its representations classify equisingular flat vector bundles, where the equisingularity condition is a geometric formulation of the fact that in quantum field theory the counterterms are independent of the choice of a unit of mass. As an algebraic group scheme, it is a semi-direct product by the multiplicative group of a pro-unipotent group scheme whose Lie algebra is freely generated by one generator in each positive integer degree. There is a universal singular frame in which all divergences disappear. When computed as iterated integrals, its coefficients are certain rational numbers that appear in the local index formula of Connes-Moscovici. When working with formal Laurent series over the field of rational numbers, the data of equisingular flat vector bundles define a Tannakian category whose properties are reminiscent of a category of mixed Tate motives.
Archimedean cohomology provides a cohomological interpretation for the calculation of the local L-factors at archimedean places as zeta regularized determinant of a log of Frobenius. In this paper we investigate further the properties of the Lefschetz and log of monodromy operators on this cohomology. We use the Connes-Kreimer formalism of renormalization to obtain a fuchsian connection whose residue is the log of the monodromy. We also present a dictionary of analogies between the geometry of a tubular neighborhood of the ``fiber at arithmetic infinity'' of an arithmetic variety and the complex of nearby cycles in the geometry of a degeneration over a disk, and we recall Deninger's approach to the archimedean cohomology through an interpretation as global sections of a analytic Rees sheaf. We show that action of the Lefschetz, the log of monodromy and the log of Frobenius on the archimedean cohomology combine to determine a spectral triple in the sense of Connes. The archimedean part of the Hasse-Weil L-function appears as a zeta function of this spectral triple. We also outline some formal analogies between this cohomological theory at arithmetic infinity and Givental's homological geometry on loop spaces.
This is the first installment of a paper in three parts, where we use noncommutative geometry to study the space of commensurability classes of Q-lattices and we show that the arithmetic properties of KMS states in the corresponding quantum statistical mechanical system, the theory of modular Hecke algebras, and the spectral realization of zeros of L-functions are part of a unique general picture. In this first chapter we give a complete description of the multiple phase transitions and arithmetic spontaneous symmetry breaking in dimension two. The system at zero temperature settles onto a classical Shimura variety, which parameterizes the pure phases of the system. The noncommutative space has an arithmetic structure provided by a rational subalgebra closely related to the modular Hecke algebra. The action of the symmetry group involves the formalism of superselection sectors and the full noncommutative system at positive temperature. It acts on values of the ground states at the rational elements via the Galois group of the modular field.
We make some brief remarks on the relation of the mixmaster universe model of chaotic cosmology to the geometry of modular curves and to noncommutative geometry. In particular we consider a class of solutions with bounded number of cycles in each Kasner era and describe their dynamical properties, which we relate to the noncommutative geometry of a moduli space of such solutions, given by a Cuntz-Krieger C*-algebra.
In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the ``closed fibers at infinity''. Manin described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic handlebody endowed with a Schottky uniformization. In this paper we consider arithmetic surfaces over the ring of integers in a number field, with fibers of genus $g\geq 2$. We use Connes' theory of spectral triples to relate the hyperbolic geometry of the handlebody to Deninger's Archimedean cohomology and the cohomology of the cone of the local monodromy $N$ at arithmetic infinity as introduced by the first author of this paper.
In this survey, written for the proceedings of the VII meeting of the CNTA held in May 2002 in Montreal, we describe how Connes' theory of spectral triples provides a unified view, via noncommutative geometry, of the archimedean and the totally split degenerate fibers of an arithmetic surface.
We construct spectral triples associated to Schottky--Mumford curves, in such a way that the local Euler factor can be recovered from the zeta functions of such spectral triples. We propose a way of extending this construction to the case where the curve is not k-split degenerate.
For a rational homology 3-sphere $Y$ with a $\spinc$ structure $\s$, we show that simple algebraic manipulations of our construction of equivariant Seiberg-Witten Floer homology lead to a collection of variants which are topological invariants. We establish exact sequences relating them, we show that they satisfy a duality under orientation reversal, and we explain their relation to ou previous construction of equivariant Seiberg-Witten Floer (co)homologies. We conjecture the equivalence of these versions of equivariant Seiberg-Witten Floer homology with the Heegaard Floer invariants introduced by Ozsváth and Szabó.
This note is a brief overview of the results of math.AG/0205306. We use Connes' theory of spectral triples to provide a connection between Manin's model of the dual graph of the fiber at infinity of an Arakelov surface and the cohomology of the mapping cone of the local monodromy.
We establish a surgery formula for 3-dimensional Seiberg-Witten monopoles under (+1) Dehn surgery on a knot in a homology 3-sphere. (substantial revision)