Seminar: The Standard Model in Noncommutative Geometry

Place: IMAPP, Radboud Universiteit Nijmegen

The goal of this seminar is to understand and describe the mathematical description of the Standard Model (SM) of elementary particles, using techniques from noncommutative geometry (NCG). The main advantage of this description is that the Higgs-sector follows naturally, including a concrete (model-dependent) prediction of the mass of the Higgs.

A first proposed working plan is (references below)

  • Conventional approach to the SM: Yang-Mills theory, Spontaneous symmetry breaking. [Cottingham, Greenwood] [CM, Ch.9]
  • Gauge theory from noncommutative structures: gauge fields as inner fluctuations and Morita equivalence, gauge group. [CM, Ch.10.8]
  • The spectral action principle [CM, Ch.11]: Lagrangians associated to a NC spin manifold; examples: Einstein-Hilbert and Yang-Mills theory, description of assumptions on the 'cutoff'-function.
  • NCG of the SM [CM, Ch.12-16]: deriving the Lagrangian of the SM including Higgs from NCG; identification of gauge fields, Higgs field, fermionic fields and representations of the gauge group on these fields.
  • Phenomenology of the SM in NCG [CM, Ch.17]: prediction of the Higgs mass, postdiction mass of the top quark.

    However, this is completely open for changes suggested by the participants. Familiarity with basics of NCG is assumed; a general course on NCG is thought in spring 2010 by Klaas Landsman in the national MRI-masterclass on Arithmetic Geometry and Noncommutative Geometry, more details.

    Schedule (on Friday 10:45-12:30):

  • Friday January 22, 2010 in room HG03.084, Walter van Suijlekom, Gauge symmetries in physics.

  • Friday Febrary 5, 2010 in room HG00.633, Jord Boeijink, The Standard Model of elementary particles, following [CM,Ch.9]; (many) more details in [CG, Ch.4,5,7,11,12,14,16,19,21]. [notes]

  • Friday Febrary 26, 2010 in room HG00.633, Klaas Landsman, Spectral triples, Morita equivalence, inner fluctuations as gauge fields I [CM Ch.10].

  • Friday March 12, 2010 in room HG00.633, Klaas Landsman, Spectral triples, Morita equivalence, inner fluctuations as gauge fields II [CM Ch.10.8].

  • Wednesday March 31, 2010 (9:00!) in HG03.085, Matthias Sars, Spectral action principle [CM Ch.11].

  • Wednesday April 21, 2010 in room HG03.084 (9:00!), Thijs van den Broek Noncommutative geometry of the Standard Model [CM, Ch.12,13,14].

  • Tuesday May 4, 2010 in room HG03.084 (10:45), Koen van den Dungen Gauge bosons (W,Z,&gamma,H) as inner fluctuations [CM, Ch.15].

  • Friday May 28, 2010 (10:15!) in room HG03.082, Walter van Suijlekom, The spectral action and the Standard Model Lagrangian [CM, Ch.16-17]

    Bibliography:

    The main topic can be found in:

    [CM] A. Connes, M. Marcolli. Noncommutative geometry, quantum fields and motives. American Mathematical Society, Providence, RI; Hindustan Book Agency, New Delhi, 2008.
    [Sch] F. Scheck et al. (eds.). Noncommutative Geometry and the Standard Model of Elementary Particle Physics, Springer Lecture Notes in Physics 596.

    A physics approach to the Standard Model can be found in:

    [CL] Cheng and Li. Gauge Theory of elementary particle physics, Oxford University Press, 1988.
    [CG] W.N. Cottingham and D.A. Greenwood. An Introduction To The Standard Model Of Particle Physics. Cambridge University Press, 2007.

    General texts on noncommutative geometry are

    [Con] A. Connes. Noncommutative Geometry. Academic Press, 1984.
    [GVF] J.M. Gracia-Bondia, J.C. Varilly, H. Figueroa. Elements of Noncommutative Geometry. Birkhauser, 2000.
    [Lan] G. Landi. An Introduction to Noncommutative Spaces and their Geometries. Springer Lecture Notes in Physics 51}, Springer Verlag (Berlin Heidelberg) 1997. [arXiv:hep-th/9701078]


    For more information, please contact: N.P. Landsman or W. D. van Suijlekom.