Ma148b Topics in Mathematical Physics: Introduction to Noncommutative Geometry Methods in Physics
Winter 2021: Caltech, Tuesday-Thursday 9:00-10:30am
Instructor:
Matilde Marcolli
(Barbara Hepworth, Forms in Movement, 1942)
Brief Course Description
This course presents an introduction to the field of Noncommutative Geometry,
with emphasis on spectral triples and the spectral action functional and
applications to theoretical physics.
Slides of Classes
Slides of classes will be posted here as the course progresses.
Summary of Lectures
- January 5: Introduction and physics motivation: geometric models in physics; mathematical toolbox: manifolds, bundles, tensors, connections, curvature;action functionals in physics: classical mechanics, symmetries and conservation, Einstein-Hilbert action, gravity coupled to matter, Yang-Mills action; key idea of perturbative QFT, standard model of elementary particles
- January 7: Introduction to Noncommutative Geometry, equivalence relations as noncommutative spaces; tools from classical geometry: vector bundles as projective modules, connections; classical spin geometry and spectral triples
- January 12: finite spectral triples, finite metric spaces, bimodules, Morita equivalence, real structures, Krajewski diagrams, Barrett's random noncommutative geometries, random matrix theory
- January 14: random matrix theory, finite spectral triples and random matrix models, recursive Dyson-Schwinger equations, surface counting and topological recursion
- January 19: gauge networks, spin networks and gauge theories on a graph, quivers and category of finite spectral triples, configuration space, lattice field theory, Wilson action; finite spectral triples and the geometry of the Standard Model of particle physics
- January 21: finite spectral triples and the Standard Model of particle physics
- January 26: almost commutative geometries, the spectral action functional and the Standard Model Lagrangian
- January 28: Higgs mass in the NCG Standard Model, Pati-Salam grand unified theories
- February 2: Spectral action and Poisson summation formula, spheres and spherical space forms, tori and Bieberbach manifolds, inflation models with slow-roll potential from the spectral action, effect of different candidate cosmic topologies
- February 4: Packed Swiss Cheese cosmology model and multifractal cosmologies, Apollonian packings of spheres, spectral triples and Dirac operators of Apollonian packings, Dirac zeta function and poles and residues, zeta regularized terms in the spectral actiona nd log-periodic terms, effect on slow-roll inflation potentials
- February 9: Heat kernel expansion of the square Dirac operator on Robertson-Walker metrics, Feynman-Kac formula, Brownian motion and Brownian bridge, complete expansion in terms of Brownian bridge integrals and Bell polynomials
- February 11: Robertson-Walker metrics and multifractal structures (packed Swiss Cheese cosmology), heat kernel expansion, zeta regularized and log-periodic terms
- February 16: Robertson-Walker heat kernel expansion, algebraic change of variables and descriptio as period integrals, motives and periods, mixed Tate motives, proof that all terms in the expansion are periods of mixed Tate motives
- February 18: Bianchi IX metrics, Dirac operator, symbol, heat kernel expansion, algebraic change of variables, motives and periods, mixed Tate property, Bianchi IX gravitational instantons and Painleve VI equations, theta characteristics and parameterization of solutions
- February 23: noncommutative tori, Morita equivalences, real multiplication, noncommutative tori and elliptic curves; quantum statistical mechanical systems, KMS states, Q-lattices and commensurability, Bost-Connes system and 1-dimensional Q-lattices, convolution algebra of 2-dimensional Q-lattices
- February 25: GL2 quantum statistical mechanical system of 2-dimensional Q-lattices, representations, partition function, KMS states, modular field and arithmetic algebra, Galois symmetries, imaginary quadratic fields and maximal abelian extension, Hilbert's 12th problem and quantum statistical mechanics
- March 2: Quantum Hall effect, electrons in solids and Bloch theory, algebraic geometry of Fermi surfaces, discretization and Haper operator, group C* algebra and Brillouin torus, magnetic field and noncommutative torus
- March 4: Student presentations
- March 9: Student presentations
Some book references
There is no official textbook for this class, but the following books will be useful references (available to download as ebooks from the Caltech library - let me know if you have problems accessing them)
- J.C. Varilly, "An introduction to noncommutative geometry", European Mathematical Society, 2006.
- M. Khalkhali, "Basic noncommutative geometry", European Mathematical Society, 2013.
- W.D. van Suijlekom, "Noncommutative geometry and particle physics", Springer, 2015.
- A. Connes and M. Marcolli, "Noncommutative geometry, quantum fields and motives", American Mathematical Society, 2008.
Other Reading Material
Suggested reading material including both material covered in the lectures and possible suggestions for student presentations
- pdf
A.H.Chamseddine, A.Connes, "The Spectral Action Principle",
Commun.Math.Phys.186:731-750,1997
- pdf B.Cacic,
"A reconstruction theorem for almost-commutative spectral triples",
Lett. Math. Phys. 100 (2012), no. 2, 181-202
- pdf
W.D.van Suijlekom, "Renormalizability conditions for almost-commutative geometries", Ann. Henri Poincare' 15 (2014), no. 5, 985-1011
- pdf E.Gesteau, "Renormalizing Yukawa interactions in the standard model with matrices and noncommutative geometry", J. Phys A Math. Theor. 54 (2021) 035203 (18pp)
- pdf
M.Khalkhali, N.Pagliaroni, "Phase transition in random noncommutative geometries" arXiv:2006.02891
- pdf
S.Azarfar, M.Khalkhali, "Random Finite Noncommutative Geometries and Topological Recursion", arXiv:1906.09362
- pdf
B.Iochum, C.Levy, "Spectral triples and manifolds with boundary",
J. Funct. Anal. 260 (2011), no. 1, 117-134
- pdf
M.A.Kurkov, F.Lizzi, M.Sakellariadou, A.Watcharangkool,
"Spectral action with zeta function regularization", Phys.Rev. D91 (2015) 6, 065013
- pdf
W.Nelson, J.Ochoa, M.Sakellariadou
"Gravitational Waves in the Spectral Action of Noncommutative Geometry"
Phys.Rev.D82:085021,2010
- pdf
A.H.Chamseddine, A.Connes, "The Uncanny Precision of the Spectral Action",
Commun.Math.Phys.293:867-897,2010
- pdf
W.D. van Suijlekom, "Perturbations and operator trace functions",
J.Funct.Anal. 260 (2011), no. 8, 2483-2496
- pdf
F.Fathizadeh, A.Ghorbanpour, M.Khalkhali, "Rationality of Spectral Action
for Robertson-Walker Metrics", JHEP 12 (2014) 064
- pdf F.Fathizadeh, Y.Kafkoulis, M.Marcolli, "Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson-Walker Cosmologies", Ann. Henri Poincare, 21 (2020), no. 4, 1329-1382
- pdf F.Fathizadeh, M.Marcolli, "Periods and motives in the spectral action of Robertson-Walker spacetimes", Comm. Math. Phys. 356 (2017), no. 2, 641-671
- pdf
W.Kalau, M.Walze, "Gravity, Non-Commutative Geometry and the Wodzicki Residue",
J.Geom.Phys. 16 (1995) 327-344
- pdf
F.Fathizadeh, M.Khalkhali, "The Algebra of Formal Twisted Pseudodifferential
Symbols and a Noncommutative Residue", Lett.Math.Phys. 94 N.1 (2010) 41-61
- pdf
J.W.Barrett, "Matrix geometries and fuzzy spaces as finite spectral triples",
arXiv:1502.05383
- pdf
G.V.Dunne, "Heat Kernels and Zeta Functions on Fractals",
J.Phys. A45 (2012) 374016
- pdf
S.Brain, B.Mesland, W.D.van Suijlekom, "Gauge Theory for Spectral Triples
and the Unbounded Kasparov Product", arXiv:1306.1951
- pdf
T. Krajewski, "Classification of Finite Spectral Triples", J.Geom.Phys. 28 (1998) 1-30
- pdf
C.Connes, M.Marcolli, N.Ramachandran, "KMS states and complex multiplication",
Selecta Math. (N.S.) 11 (2005), no. 3-4, 325-347
- pdf
M.Laca, N.Larsen, S.Neshveyev, "On Bost-Connes type systems for number fields",
J. Number Theory 129 (2009), no. 2, 325-338
- pdf
C.Consani, M.Marcolli, "Quantum statistical mechanics over function fields"
J. Number Theory 123 (2007), no. 2, 487-528
- pdf
A.Connes, C.Consani, M.Marcolli, "Noncommutative geometry and motives:
the thermodynamics of endomotives". Adv. Math. 214 (2007), no. 2, 761-831
- pdf
A.Connes, C.Consani, M.Marcolli, "The Weil proof and the geometry of
the adeles class space", in Algebra, Arithmetic, and Geometry: in honor
of Yu. I. Manin. Vol. I, 339-405, Progr. Math., 269, Birkhaeuser, 2009
- pdf
Yu.I.Manin, M.Marcolli, "Continued fractions, modular symbols, and
noncommutative geometry", Selecta Math. (N.S.) 8 (2002), no. 3, 475-521
- pdf
Yu.I.Manin, "Real Multiplication and noncommutative geometry", in
The legacy of Niels Henrik Abel, 685-727, Springer, 2004
Schedule of Presentations
- March 4: Adam Artymowicz,
Noncommutative geometry of the Integer Quantum Hall Effect
- March 4: Sunghyuk Park,
Phase transition in random noncommutative geometries
- March 9: Aru Mukherjea,
Heat kernel and zeta functions on fractals
- March 9: Dan Rostovtsev,
Dirac spectra and cosmic topology
- March 9: Angus Gruen (asynchrounous), TBA