Ma 192 a: Geometry and Arithmetic of Quantum Fields

* The book "Feynman Motives" based on the lectures of this course will soon appear published by World Scientific

Brief Course Description

List of Topics

• The problem of Renormalization in Quantum Field Theory
• The BPHZ method of perturbative renormalization
• Commutative Hopf algebras and affine group schemes
• Hopf algebras and perturbative renormalization
• The Connes-Kreimer Hopf algebra of Feynman graphs
• The geometry of Dimensional Regularization
• Renormalization as a Birkhoff factorization
• Parametric Feynman integrals
• Graph hypersurfaces and Feynman motives
• Algebro-geometric properties of graph hypersurfaces
• Graph hypersurfaces, Feynman integrals, and periods
• Feynman motives, singularities, and Hodge theory
• The renormalization group and the Gross-'t Hooft relations
• Beta function and iterated integrals
• Equisingular connections
• The Riemann-Hilbert correspondence
• The Tannakian formalism
• The category of flat equisingular vector bundles
• Cartier's "cosmic Galois group" and mixed Tate motives
• Basic notions of noncommutative geometry
• The formalism of spectral triples
• Noncommutative spectral manifolds and real structures
• The finite geometry of the Standard Model of elementary particles
• The spectral action principle
• The Standard Model with neutrino mixing from the spectral action
• Anomalies and noncommutative geometry

Lectures

• Lecture 1, Tuesday September 30: Overview
• Lecture 2, Thursday October 2: Feynman integrals: a toy model
• Lecture 3, Tuesday Oct 7: Lagrangian, functional integrals, effective action (check later for more notes on Dim Reg and the Bogolyubov recursion)
• Lecture 4 Thursday Oct 9: Parametric Feynman integrals and graph hypersurfaces
• Lecture 5 Tuesday Oct 14: Affine group schemes and commutative Hopf algebras (notes coming up soon)
• Lecture 6 Thursday Oct 16: The Connes-Kreimer Hopf algebra and the Birkhoff factorization of loops (notes coming up soon)
• Lecture 7 Tuesday Oct 21: Introduction to motives, pure motives
• Lecture 8 Thursday Oct 23: Introduction to motives, mixed motives
• Lecture 9 Tuesday Oct 28: The Grothendieck group of varieties, classes of graph hypersurfaces
• Lecture 10 Thursday Oct 30: The class of the banana graph hypersurfaces, simplifications in the Grothendieck group between graph hypersurfaces

Grading

Research Seminar

The first seminar will be on Thursday October 16 at 4:15 pm.

Titles and abstracts of seminars will be posted here.

• Thursday October 16: Tobias Fritz "Convexity and Quantum Mechanics"
• Thursday October 23: Nikolay Ivankov "Covering spaces and Noncommutative Geometry"
• Tuesday November 4, 2:30 pm: Rafael Torres-Ruiz "4-manifolds"
• Thursday November 13, 4:15 pm: Michel van Garrel "Isomorphism Classes of Elliptic Curves with Complex Multiplication defined over a Number Field with the same Hecke Character"
• EXTRA SEMINAR: Monday November 17, 3:30 pm SLN 159: Bram Mesland "KK-theory and limit sets"
• Thursday November 20, 4:15: Ozgur Ceyhan "Open-closed string theory and enumerative geometry"
• EXTRA SEMINAR: Monday, November 24, 3:30 pm SLN 159: Branimir Cacic "Finite spectral triples"
• Thursday Dec 4, 4:15: Dapeng Zhang "Canonical spectral triple for Riemannian manifold via Morita equivalence"

Suggested reading material for seminar presentations

• Hopf algebras in physics: Hopf algebra structures in particle physics (Stefan Weinzierl)
• Renormalization and motives: Quantum Fields and Motives (Connes-Marcolli)
• Feynman motives and graph hypersurfaces: Motives associated to graphs (Spencer Bloch)
• Kontsevich and Zagier, "Periods", in "Mathematics unlimited - 2001 and beyond", pp.771-808, Springer, 2001.
• D.Kreimer The residues of quantum fields - numbers we should know
• Multiple zeta values in QFT: Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops (Broadhurst and Kreimer)
• Feynman integrals and periods: Periods and Igusa Zeta functions (Belkale and Brosnan)
• Feynman integrals and periods: Periods and Feynman integrals (Christian Bogner, Stefan Weinzierl)
• Renormalization, Hopf algebras, and gauge theories: Renormalization of gauge fields using Hopf algebras (van Suijlekom)
• Renormalization, Hopf algebras, and gauge theories: Representing Feynman graphs on BV-algebras (van Suijlekom)
• Renormalization and Rota-Baxter algebras: Rota-Baxter Algebras in Renormalization of Perturbative Quantum Field Theory (Kurusch Ebrahimi-Fard, Li Guo)
• Hopf algebras and Euler-Zagier sums in particle physics: Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals (Sven Moch, Peter Uwer, Stefan Weinzierl)
• Differential Galois Theory and Tannakian formalism: Differential Galois theory, universal rings and universal groups (van der Put)
• Motivic fundamental groups for mixed Tate motives: Groupes fondamentaux motiviques de Tate mixte (Deligne-Goncharov)
• A.Connes, "Geometry from the spectral point of view". Lett. Math. Phys. 34 (1995), no. 3, 203-238.
• Spectral action in NCG: The Spectral Action Principle (Chamseddine-Connes)
• Spectral action in NCG: Inner fluctuations of the spectral action (Chamseddine-Connes)

Textbook

* A.Connes, M.Marcolli: Noncommutative Geometry, Quantum Fields and Motives. American Mathematical Society, Colloquium Publications Vol.55, January 2008.

Useful General Bibliographical References

• A.Zee, "Quantum Field Theory in a nutshell" Princeton University Press.

• Claude Itzykson and Jean-Bernard Zuber "Quantum Field Theory", Dover.
• James D Bjorken and Sidney D. Drell "Relativistic Quantum Mechanics" and "Relativistic Quantum Fields", McGraw-Hill.

• W.C.Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, Springer Verlag, 1979.

• J.Collins, Renormalization, Cambridge University Press, 1984.

• Y.Andre' "Une Introduction aux Motifs" Societe' Mathematique de France, 2005.