Ma148b Topics in Mathematical Physics: Introduction to Noncommutative Geometry Methods in Physics

Winter 2021: Caltech, Tuesday-Thursday 9:00-10:30am

(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.

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