KK-theory, Gauge Theory and Topological Phases, School+Workshop

KK-theory, Gauge Theory and Topological Phases School – Workshop
from 27 Feb 2017 through 10 Mar 2017

Lorentz Center Leiden

Scientific organizers:

Alan Carey (Canberra, Australia)
Steve Rosenberg (Boston, MA, USA)
Walter van Suijlekom (Nijmegen, The Netherlands)

This is a school and workshop on new developments in Kasparov theory (also referred to as KK-theory) motivated by applications to gauge theory and topological phases of matter.

The school is intended for PhD-students and postdocs working on KK-theory and/or its applications to physics. This includes young scientists who work on gauge theory or on topological phases with a strong mathematical background, and who want to learn the novel approach to these fields that KK-theory has in stock. Moreover, it is an opportunity for more senior people who are in related fields and want to learn KK-theory.

The workshop is intended for scientists working in the field and also in topological phases of matter, and particularly for the participants of the preceding school. It will attract leading experts from all parts of the world, but will also function as a platform for talented young scientists.

Website

Noncommutative Geometry and Higher Structures (Perugia)

Joint Meetings on

Noncommutative Geometry and Higher Structures

Università di Perugia, 25-29 July 2016

 

Conference website

The aim of the workshop is to bring together people working on noncommutative geometry, deformation theory and related fields, to promote new collaborations and interaction between senior scientists and students/junior researchers, and give to young mathematicians some perspectives on who is doing what in this field.The conference will take place in the mathematics department of the University of Perugia, located in Via Luigi Vanvitelli 1, Perugia.

The scientific activities of the conference will start at 2:30 PM on Monday 25 July and finish at 12:30 AM on Friday 29. To register, please send an email with your name and affiliation to Nicola Ciccoli (no registration fee is required).

 

Organizing committee:

Scientific committee:

Speakers include:

  • Iakovos Androulidakis (Univ. of Athens)
  • Paolo Antonini (SISSA – Trieste)
  • Serguei Barannikov (Univ. Diderot-Paris 7)
  • Damien Broka (Penn State)
  • Oleksandr Iena (Univ. Luxembourg)
  • Niek de Kleijn (Univ. Copenhagen)
  • Niels Kowalzig (Univ. Sapienza di Roma)
  • Giovanni Landi (Università di Trieste)
  • Camille Laurent-Gengoux (Univ. Paul-Verlaine)
  • Luigi Lunardon (Univ. Sapienza di Roma)
  • Marco Manetti (Univ. Sapienza di Roma)
  • Francesco Meazzini (Univ. Sapienza di Roma)
  • Valerio Melani (Univ. Pierre et Marie Curie Paris)
  • Chiara Pagani (Georg-August-Univ. Göttingen)
  • Francois Petit (Luxembourg Univ.)
  • Martin Schlichenmaier (Luxembourg Univ.)
  • Mathieu Stienon (Penn State University)
  • Alfonso Tortorella (Università di Firenze)
  • Ping Xu (Penn State University)

Noncommutative Geometry and Applications to Quantum Physics (CIMPA Research School — Rencontres du Vietnam)

CIMPA Research School — Rencontres du Vietnam

Noncommutative Geometry and Applications to Quantum Physics

July 12th – 22nd, 2017, Quy Nhon, Vietnam

This international CIMPA * Research School will be held at ICISE ** hosted by the Rencontres du Vietnam ***

Noncommutative Geometry (NCG) is a vivid research subject in Mathematics and Physics. The main goal of this school is to train local researchers and students in these topics and to establish strong research collaborations with colleagues, students and researchers. Leading experts in NCG will give an overview of the main well-established results, the essential tools, and some of the present active research activities:

  • Connes-Chern Character Theorem
  • Noncommutative Integration Theory (Dixmier Traces, Singular Traces…)
  • Unbounded KK
  • -theory and Kasparov Product
  • Dynamical Systems and KMS States
  • Quantum Groups
  • Fuzzy Spaces
  • Noncommutative Standard Model of Particle Physics
  • Application to the QHE…

School Schedule

  • The opening session is scheduled to take place on Thursday, July 13th;
  • the school will end on Saturday, July 22nd.
  • Lectures hours are 8:30 – 12:00 and 13:30 – 17:00.
  • Lectures will be held in the ICISE conference Center.

School website

Summer School “Coarse Index Theory”, September 26-30, 2016, Freiburg

Topic:    Index theory is a prime example of fruitful interaction between analysis, geometry, topology and operator algebras.

The index is associated to a global differential operator and is computed from the set of solutions of the associated differential equation. It turns out, however, that the index has remarkable stability properties and can often be computed a priori without solving the differential equation. This uses index theorems and the underlying topology. On the other hand, the most interesting operators are tied to the geometry and the geometry determines the set of solutions. The most powerful implementations of this idea that the relevant operators lie in operator algebras which are specific to the situation at hand. The indices are then naturally defined as elements in K-theory groups of these operator algebras.

It turns out that a particularly useful setup uses the ideas of “coarse  geometry”. The basic idea is to study (non-compact) metric spaces; but  considering only their large scale features. A lot of this can be captured  in appropriately associated C*-algebras; the coarse C*-algebras of the space (often called Roe algebra). This tool also applies to compact spaces, by passing first to their universal covering.

The corresponding manifestation of index theory in this context is “coarse   index theory” or “large scale index theory” and has many interesting  properties and applications.

The summer school will explain the relevant general background in index theory, operator algebras; and then focus on large scale geometry and index theory and its numerous applications.

Program: There will be lecture series by

  • John Roe: Coarse geometry and index theory
  • Thomas Schick: (Secondary) Coarse index and applications
  • Rudolf Zeidler: K-theory of C*-algebras

along with daily exercise and discussion sessions in the afternoon.

Schedule: Preliminary schedule

Funding: As a general rule, you are supposed to arrive with your own funding but there are also some limited funds available.

Contact: Please send an informal email to enroll until September 1, 2016, latest, to Mrs. Ursula Wöske, coarse16@math.uni-freiburg.de

Organizer: Nadine Große

Speakers: John Roe, Thomas Schick, Rudolf Zeidler

Poster: Poster Summer School 2016

Noncommutative Geometry 2016, Villa de Leyva, Colombia, June 20 – July 1, 2016

School and Conference on Noncommutative Geometry

Villa de Leyva, Colombia,  June 20 to July 1, 2016

Noncommutative Geometry is a growing and active field whose roots and branches intertwine with various areas of mathematics and physics. The school and conference on Noncommutative Geometry, Noncommutative Geometry 2016, aims at introducing advanced students and young researchers to various results and techniques from noncommutative geometry giving special attention to those which play a key role in applications. The lectures will focus on some of the main themes lying at the core of the current developments of the theory while the mini courses will provide both background and context for those themes.

Lecture Series:

Walter van Suijlekom (Radboud University Nijmegen)
Noncommutative geometry and particle physics

Farzad Fathizadeh (California Institute of Technology)
Heat kernel methods and local geometric invariants of noncommutative spaces

Özgür Ceyhan (University of Luxembourg)
Feynman integrals, associated arrangements and their motives

Gunther Cornelissen (Utrecht University)
Zeta functions in number theory and differential geometry in the light of noncommutative geometry

Bram Mesland (Leibniz Universität Hannover)
A categorical approach to spectral triples and KK-theory

Mini Courses:

Carolina Neira (Universidad Nacional de Colombia)
Hochschild cohomology and residues

Mario Velásquez  (Pontificia Universidad Javeriana)
Elements of K-theory

Andrés Reyes (Universidad de los Andes)
Basics of quantum field theory

Leonardo Cano (Universidad Sergio Arboleda)
Characteristic classes

Andrés Vargas (Pontificia Universidad Javeriana)
Spin manifolds and Dirac operators

Alexander Cardona (Universidad de los Andes)
Index theorems


Scientific committee:
Matilde Marcolli (California Institute of Technology)
Özgür Ceyhan (University of Luxembourg)

Organising committee:
Jorge Plazas  (Pontificia Universidad Javeriana)
Mario Velásquez  (Pontificia Universidad Javeriana)
Fernando Novoa  (Pontificia Universidad Javeriana)
Diana Guevara  (Pontificia Universidad Javeriana)

Ciencias45-1escudoujaveriana

Masterclass on unbounded KK-theory in Copenhagen

At the end of a summer there will be a master class in Copenhagen on unbounded KK-theory and the more analytic aspects of non commutative geometry. During the master class there will be lecture series:
  • Matthias Lesch (University of Bonn)
    “Sums of regular self-adjoint operators”
  • Adam Rennie (University of Wollongong)
    “Applications of KK-theory in non-commutative geometry and physics”
  • Fedor Sukochev (University of New South Wales)
    “Introduction to Double Operator Integration and Quantum Differentiability of Essentially Bounded Functions on Euclidean Space”
The master class will take place 22-26/8 at the mathematics department in Copenhagen. More information can be found on:
There is funding for local expenses and possibilities for participants to contribute with talks. The deadline for registration is August 1st. If you want to stay in the shared accommodation booked by the department it is July 1st.

Grand Unification in the Spectral Pati-Salam Model

Today we (Chamseddine-Connes-van Suijlekom) posted a preprint on grand unification in the spectral Pati–Salam model which I summarize 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 classification of finite geometries \(F\) that can provide the fine structure of Euclidean space-time as a product \(M\times F\) (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

\(
M_{2}(\mathbb{H})\oplus M_{4}(\mathbb{C})
\)

where \(\mathbb{H}\) 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 \(\mathcal{H}\). 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 \(D_{F}\). 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 \(SU(2)_{R}\times SU(2)_{L}\times SU(4)\), 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.

From the spectral action principle, the dynamics and interactions are described by the spectral action,

\(
\mathrm{tr}(f(D/\Lambda))
\)

where \(\Lambda\) is a cutoff scale and \(f\) an even and positive function. In the present case, it can be expanded using heat kernel methods,

\(\mathrm{tr}(f(D/\Lambda))\sim F_{4}\Lambda^{4}a_{0}+F_{2}\Lambda^{2}%
a_{2}+F_{0}a_{4}+\cdots
\)

where \(F_{4},F_{2},F_{0}\) are coefficients related to the function \(f\) and \(a_{k}\) 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 \(\Lambda\).

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 \(F_{0}a_{4}\) in the spectral action for the spectral Pati–Salam model contains the terms

\(
\frac{F_{0}}{2\pi^{2}}\int\left( g_{L}^{2}\left( W_{\mu\nu L}^{\alpha
}\right) ^{2}+g_{R}^{2}\left( W_{\mu\nu R}^{\alpha}\right) ^{2}%
+g^{2}\left( V_{\mu\nu}^{m}\right) ^{2}\right) .
\)

Normalizing this to give the Yang–Mills Lagrangian demands

\(
\frac{F_{0}}{2\pi^{2}}g_{L}^{2}=\frac{F_{0}}{2\pi^{2}}g_{R}^{2}=\frac{F_{0}%
}{2\pi^{2}}g^{2}=\frac{1}{4},
\)

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 \(\Lambda\) where there is grand
unification:

\(
g_{R}(\Lambda)=g_{L}(\Lambda)=g(\Lambda).
\)

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 \(m_{R}\) where symmetry breaking occurs and which is related to the neutrino Majorana masses (\(10^{11}-10^{13}\)GeV), and the GUT scale \(\Lambda\).

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 \(D_{F}\), 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 composite Higgs fields

PSrunningNoOrder1

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.

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 \(10^{16}\) GeV, thus confirming validity of the spectral action at the corresponding scale, independent of the specific form of \(D_{F}\).