Schedule of Workshop 2: Noncommutative Geometry

Thursday, July 31

09:00-10:00 Henri Moscovici: Twisted spectral triples of conformal type
10:00-11:00 Mathai Varghese: Noncommutative sigma models
11:00-11:30 Coffee break
11:30-12:30 Ali Chamseddine: Classification of discrete noncommutative geometries
12:30-14:15 Lunch break
14:15-15:15 Giovanni Landi: The noncommutative projective plane
15:15-15:45 Coffee break
15:45-16:45 Chiara Pagani: Noncommutative fibrations and instantons
16:45-17:45 Fabien Vignes-Tourneret: Quantum field theory on the degenerate Moyal space

Abstracts:

Hossein Abbaspour: (Algebraic) String bracket as a Poisson bracket
I will explain how the Poisson bracket of Maurer-Cartan moduli space corresponds to the Chas-Sullivan/Goldman String bracket on the equivariant homology of free loop space and the negative cyclic cohomology of a DGA with an appropriate trace function. Among the examples of the latter are, the deformation complex of a flat connection or a holomorphic structure on a complex bundle on a Calabi-Yau. (Joint work with T. Tradler & M. Zeinalian).

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Abishek Banerjee: Periodicity in Cyclic Cohomology and Monodromy at Archimedean Infinity
The cohomology of the "fibre at infinity" of an arithmetic variety can be computed by means of a complex first introduced by Consani. At archimedean infinity, this complex replaces Steenbrink’s complex for the cohomology of the universal fibre of a degeneration over a disc. The nearby cycles complex associated to this degeneration carries a monodromy operator N and we can show that the graded pieces of the filtration on the cohomology of the nearby cycles complex by Ker(Nj), j ≥ 0, are isomorphic to the cyclic homology of a sheaf of differential operators (using some results of Wodzicki). Further, we can show that, under this isomorphism, the periodicity operator in cyclic homology coincides with the (logarithm of) the monodromy on the nearby cycles complex.
In this talk, we will do the same at archimedean infinity, where we have to work with "global sections" rather than with sheaves, and therefore show that there is a natural map from the cyclic cohomology of the ring of differential operators to the graded pieces of a filtration on the cohomology of the fibre at infinity, and that in this framework, the periodicity operator in cyclic cohomology is again the counterpart of the monodromy operator on Consani’s complex. The switch between cyclic homology and cohomology is a consequence of the fact that the monodromy operators on the nearby cycles complex and on Steenbrink’s complex are equal only up to homotopy. This is followed up by defining a complex with monodromy that plays the role of a nearby cycles complex for the fibre at infinity. Again, the monodromy operator on the latter is equivalent to the monodromy on Consani’s complex up to homotopy.
Finally, we consider the long exact sequence of Connes and Karoubi involving the algebraic, topological and relative K-theories of a Frechet Algebra. This long exact sequence lies above the periodicity sequence in cyclic homology. In this talk, we will construct the same sequence for the K-theories of the sheaf of differential operators, using the cohomologies of simplicial sheaves as defined by Brown and Gersten. This long exact sequence then lies above the periodicity sequence of cyclic (hyper)cohomologies.

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Moulay Benameur: Index, eta and rho on foliated bundles (joint work with P. Piazza)
A leafwise Dirac type operator on a foliated bundle yiels a regular self-adjoint operator Dm on the maximal Connes-Skandalis Hilbert module. We shall first explain how the functional calculus of Dm encodes both the leafwise calculus and the monodromy calculus in the corresponding von Neumann algebras. When the foliation is endowed with a holonomy invariant Borel Measure, we then explain the compatibility of various traces and determinants. As an application of this analysis, we shall end the talk by giving an outline of the K-theory proof of the leafwise homotopy invariance of the measured rho invariant under appropriate assumptions. This last result extends to foliated bundles a nice theorem due to N. Keswani.

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Ali Chamseddine: Classification of discrete noncommutative geometries
Assuming that space-time is a product of a continuous four-dimensional manifold time a discrete space F, we classify the irreducible geometries F consistent with imposing reality and chiral conditions on spinors. Remarkably we find that the noncommutative geometry of the standard model results almost uniquely, with all the necessary details.

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Katia Consani: The integral BC-endomotive and its reduction mod. p.
The talk will focus on the definition and the properties of an integral model for the noncommutative space which supports the "BC-dynamical system", including the description of its reduction at rational primes.

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Alexander Gorokhovsky: Algebraic index theorem for Fourier integral operators.
In their work on integral transforms V. Guillemin and S. Sternberg were led to consider algebras of Fourier integral operators. We construct the analogous objects in the context of formal deformations and discuss the algebraic index theorem in this context. This is joint work with P. Bressler, R. Nest and B. Tsygan

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Harald Grosse: Renormalizable noncommutative Quantum Field Theory
Deformation of space-time does not cure all problems of QFT. In addition to the infrared and ultraviolet problems a mixing of both occurs. Our first solution found in collaboration with Raimar Wulkenhaar consist in adding one additional operator, which violates translation symmetry. At the self duality point the model may be constructed soon. Recently we found a way to add a nonlocal operator respecting translational symmetry which implies also renormalizability.

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Atabey Kaygun: Connes-Moscovici Characteristic map and uniqueness of cup products in Hopf-cyclic cohomology
Starting from a conceptual definition of Hopf-cyclic cohomology with arbitrary twisting coefficients, we will explain how one can construct pairings and cup products in Hopf-cyclic cohomology. One of the best known examples of such pairings in NCG is Connes-Moscovici characteristic map. We will show that various cup products and pairings defined in the literature are essentially the same.

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Masoud Khalkhali: The algebra of twisted formal pseudodifferential symbols
Motivated by the notion of a "twisted spectral triple", introduced by Connes and Moscovici, we shall introduce a new algebra of twisted formal pseudodifferential symbols and show that there is an analogue of the Adler-Manin noncommutative residue on such algebras. There is also a higher dimensional version. I will also introduce a very general method of producing examples (joint work with F. Fathizadeh). Time permitting, I shall also discuss a separate work on "Hopf cyclic cohomology in braided monoidal categories" (joint work with A. Pourkia, posted at http://arxiv.org/abs/0807.3890).

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Thomas Krajewski: Algebraic aspects of Wilsonian renormalization and some combinatorial applications
In this talk, we present an algebraic formalism inspired by Butcher’s B-series in numerical analysis and the Connes-Kreimer approach to perturbative renormalization. We first define power series of non linear operators and propose several applications, among which the perturbative solution of a fixed point equation using the non linear geometric series. Then, following Polchinski, we show how perturbative renormalization works for a non linear perturbation of a linear differential equation that governs the flow of effective actions. Furthermore, we define a general Hopf algebra of Feynman diagrams adapted to iterations of background field effective action computations. As a simple combinatorial illustrations, we show how these techniques can be used to recover the universality of the Tutte polynomial and its relation to the q-state Potts model as well as the hook length formula presented by A. Postnikov. As a more sophisticated example, we use ordered diagrams with decorations and external structures to solve the Polchinski’s exact renormalization group equation. This is based on arXiv:0806.4309.

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Giovanni Landi: The noncommutative projective plane
Aiming at the construction of physical models, we describe the noncommutative geometry of the quantum projective plane via a suitable spectral triple giving a covariant calculus.

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Sergey Neshveyev: KMS-states for quasi-free dynamics on Cuntz-Pimsner algebras
I will review my work with Marcelo Laca on KMS-states on Cuntz-Pimsner algebras. A continuous one-parameter group of unitary isometries of a right Hilbert C*-bimodule induces a quasi-free dynamics on the Cuntz-Pimsner C*-algebra of the bimodule and on its Toeplitz extension. The restriction of such a dynamics to the algebra of coefficients of the bimodule is trivial, and the corresponding KMS-states of the Toeplitz-Cuntz-Pimsner and Cuntz-Pimsner C*-algebras are characterized in terms of traces on the algebra of coefficients.
This generalizes and sheds light onto various earlier results about KMS-states of the gauge actions on Cuntz algebras, Cuntz-Krieger algebras, and crossed products by endomorphisms. We also obtain a more general characterization for the case in which the inducing isometries are not unitary, and accordingly, the restriction of the quasi-free dynamics to the algebra of coefficients is nontrivial.

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Ryszard Nest: Universal Coefficient Theorems for Kirchberg’s KK-theory
We will describe a general approach to construct Universal Coefficient Theorems for Kirchberg’s KK-theory for C*-algebras over finite topological spaces and discuss two examples: one where filtrated K-theory is enough to get a UCT, and one where it is not and where we have to add another invariant to get a UCT.
Joint work with Ralf Meyer.

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Chiara Pagani: Noncommutative fibrations and instantons
We describe two deformations of the SU(2) Hopf fibration over the four sphere. In the first construction the total space arises from the quantum symplectic group Spq(2) and the structure group is the quantum group SUq(2). The base space is obtained by means of a suitable projection which provides a deformation of an instanton bundle over the classical 4-sphere. The second construction is in the framework of isospectral deformations, and in this case we describe the construction of a noncommutative family of instantons parametrized by a quantum quotient. Joint work with Giovanni Landi, Cesare Reina, Walter van Suijlekom.

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Sylvie Paycha: Weighted infinite dimensional spaces and renormalised Feynman integrals
Elliptic operators such as Dirac or Laplace type operators turn out to be very useful when investigating infinite dimensional spaces modelled on spaces of sections of some vector bundle over a closed manifold. We call such a space together with an elliptic operator, a weighted space. In quantum field theory (QFT), this data amounts to a free field theory with the action functional given by the quadratic form associated with the elliptic operator. In the language of non commutative geometry, a weighted space with operator of Dirac type gives rise to a Hilbert module. We discuss how in the context of QFT, embedding the weight into a holomorphic germ of weights leads to Feynman integrals on holomorphic germs of symbols, which can then be renormalised.

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Markus Pflaum: Cyclic cocycles on the weyl algebra and higher index theorems for orbifolds
First, we describe how to construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space. Using this cyclic cocycle we then construct an explicit, local quasi-isomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof. We prove an algebraic higher index theorem by computing the pairing between such cyclic cocycles and the K-theory of the formal deformation quantization. Furthermore, we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold. As an application, we obtain the analytic higher index theorem of Connes-Moscovici and its extension to orbifolds.

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John Phillips: An Index Theory for Certain Gauge Invariant KMS States on C*-Algebras
We present, by examples, an index theory appropriate to algebras without trace. In particular, our examples include the Cuntz algebras and a larger class of unital seperable C*-algebras that generate all injective IIIλ-factors for 0 < λ < 1. These algebras are denoted by 0λ and include the Cuntz algebras: 01/n = 0n. Our main result is an index theorem (formulated in terms of spectral flow) using a twisted cyclic cocycle where the twisting comes from the modular automorphism group for a natural gauge (circle action) invariant KMS state. We introduce a modified K1-group for these algebras that we can pair with this twisted cocycle. As a corollary we obtain a noncommutative geometry interpretation for Araki’s notion of relative entropy in these examples. This is a joint work with Alan Carey and Adam Rennie.

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Bahram Rangipour: The characteristic classes in Hopf cyclic cohomology.
We briefly recall the Hopf cyclic cohomology with coefficients and its application in index theory. We then compute a complete basis for the classes of this cohomology for the Connes-Moscovici Hopf algebra. The new apparatus used in our discussions is an analogue of Weil complex of Lie algebras in Hopf cyclic theory.

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Adam Rennie: Index theory for KMS states
The index theory for KMS states rests on several constructions; two in equivariant KK-theory and an analytic construction building on the work of Laca and Neshveyev. I will outline these constructions and how they lead to a local index formula in twisted cyclic cohomology. Various features of interest, such as twisted eta cocycles, relative entropy and various possible areas for development will be described. This is joint work with Carey, Nesahveyev, Nest and Phillips.

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Elmar Schrohe: Index Theory for Boundary Value Problems via Continuous Fields of C*-Algebras
For a smooth manifold X with boundary we construct a semigroupoid T-X and a continuous field C*r(T-X) of C*-algebras which extend Connes’ construction of the tangent groupoid.
We show the asymptotic multiplicativity of semiclassical pseudodifferential boundary value problems with smoothing symbols and compute the K-theory of the associated symbol algebra.
We then use these results to derive a deformation theoretic index theorem for boundary value problems in Boutet de Monvel’s calculus using ideas of Elliott-Natsume-Nest.

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Thomas Schick: Smooth K-theory and refined index theorems
"Smooth refinements of cohomology theories" have found a lot of interest in mathematics and mathematical physics. The original example are "Cheeger-Simons characters" or "Deligne cohomology" which refine ordinary cohomology with integer coefficients. Hopkins-Singer have abstractly constructed refinements of arbitrary cohomology theories.
In the talk, joint work with Ulrich Bunke will be presented about an explict geometric model for smooth K-theory, the cycles made up of families of index problems with geometric data.
We show how this model can be used to construct all relevant operations on the theory (product, integration,..). We establish a refinement of the Atiyah-Singer index theorem for families. Finally, we discuss equivariant generalizations of the theory.

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Thomas Schücker: The noncommutative standard model and its post- and predictions
A review of the standard model of electromagnetic, weak and strong forces will be presented together with its unification with gravity in noncommutative geometry. The emphasis will be on testable post- and predictions, in particular the one of the Higgs-mass. We also survey the other Higgs-mass predictions in the literature.

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Boris Sternin and Anton Savin: Nonlocal elliptic problems and noncommutative geometry
We consider nonlocal elliptic operators associated with a discrete group of automorphisms of a smooth manifold. Such operators appear in many problems of noncommutative geometry (e.g., elliptic theory on noncommutative torus (Connes) or on deformations of function algebras on toric manifolds (Connes-Landi)). Our main result is a cohomological formula computing the Fredholm index of such operators. This is joint work with Vladimir Nazaikinskii.

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Richard Szabo: Quiver gauge theory and noncommutative vortices
We construct explicit BPS and non-BPS solutions of the Yang-Mills equations on noncommutative spaces R2nθ x G/H which are manifestly G-symmetric. Given a G-representation, by twisting with a particular bundle over G/H, we obtain a G-equivariant U(k) bundle with a G-equivariant connection over R2nθ x G/H. The U(k) Donaldson-Uhlenbeck-Yau equations on these spaces reduce to vortex-type equations in a particular quiver gauge theory on R2nθ. Seiberg-Witten monopole equations are particular examples. The noncommutative BPS configurations are formulated with partial isometries, which are obtained from an equivariant Atiyah-Bott-Shapiro construction. They can be interpreted as D0-branes inside a space-filling brane-antibrane system.

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Fabien Vignes-Tourneret: Quantum field theory on the degenerate Moyal space
I will review the recent advances in the field of renormalisation on noncommutative spaces. I will then focus on quantum field theory on a Moyal space with both commutative and noncommutative coordinates.

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Raimar Wulkenhaar: A spectral triple for harmonic oscillator Moyal space
Scalar field theory on harmonic oscillator Moyal space is expected to exist as a 4-dimensional non-perturbative quantum field theory. We show that the noncommutative geometry behind this type of models is in fact a non-compact spectral triple. In particular, the Mehler heat kernel for the harmonic oscillator Hamiltonian allows us to compute the dimension spectrum as 4-N. All residues are local, i.e. integrated Moyal products. This is further support for the conjecture that residues of operator zeta-functions are the noncommutative geometrical counterpart for locality in physics.

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Yi-Jun Yao: Rankin-Cohen Deformations
Rankin-Cohen brackets and the corresponding deformation questions appeared first in the theory of modular forms, but got involved with noncommutative geometry in the past few years. We will try to explain some results related to this topic as well as (operator algebraic) motivations of these studies.

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