Algebraic Structures in Quantum Field Theory (Spring 2004)

Welcome to the seminar on Algebraic structures in Quantum Field Theory! The seminar is organized by Ettore Aldrovandi. Please send an email to contact me.

The seminar meets on Fridays, 4:30-5:30pm in 104 LOV after the Mathematics Colloquium.

The goal of this seminar is to explore the works of Connes, Kreimer, Cartier, Goncharov, Weinzierl (and others) on the occurrence of Hopf Algebras in Quantum Field Theory, and how this might be connected to several topics: renormalization, Galois symmetries, Polyzeta functions and multiple Polylogarithms.


References

This is a collection of references that should serve to orient people as to what this series of seminars is all about. It is by no means complete, so email me if you have suggestions for additional entries.

Hopf Algebras

QFT

Polylogarithms, Polyzêtas, Galois


Speakers, Titles and Abstracts

January 16 Organizational meeting
January 23 No meeting (Departmental ext. review)
January 30 Ettore Aldrovandi, FSU Introduction to Hopf Algebras I

We will introduce Hopf Algebras and their properties, and begin studying some examples. We will also cover some required background notions.

February 6 Ettore Aldrovandi, FSU Introduction to Hopf Algebras II

We explore in more details the structure of Hopf Algebras, introduce duality and more examples.

February 13 Ettore Aldrovandi, FSU Introduction to Hopf Algebras III

We refine the concept of Hopf Algebra by adding the notion of antipode, patterned after the inverse map in a group, and that of bialgebra. We also explore some more examples, hopefully including the Enveloping Algebra of a Lie Algebra.

February 20 Ettore Aldrovandi, FSU The Hopf Algebra of rooted trees

We start exploring one of the cornerstones of Connes and Kreimer's approach: the Hopf Algebra associated to the alphabet of rooted trees. This will also serve as an additional example in our ongoing introduction to Hopf Algebras.

February 27 Ettore Aldrovandi, FSU The Hopf Algebra of rooted trees II

We continue exploring Connes and Kreimer's Hopf Algebra generated by rooted trees, in particular the constructions of the coproduct and the antipode.

March 5 Meeting cancelled
March 12 Spring break
March 19 Matilde Marcolli, MPI & FSU Quantum Statistical Mechanics of Q-lattices II

In this joint work with Alain Connes we generalize the Bost-Connes dynamical system with arithmetic spontaneous symmetry breaking to a system for GL2 of adèles. The underlying noncommutative space is the set of 2-dimensional Q-lattices up to scaling, modulo commensurability.

Note: this is the second part of a two-parts talk started in the Algebra and its applications seminar.

March 26 Ettore Aldrovandi, FSU The Hopf Algebra of Rooted trees III

We will finish talking about the coproduct and the antipode. We will also try to look into the relation of the Hopf Algebra of rooted trees with other relevant objects, in particular the Lie algebra of formal vector fields in one variable.

April 2 Laura Reina, Physics Dept., FSU Rooted trees and Feynman diagrams

We discuss Connes and Kreimer's toy quantum field theory to illustrate the correspondence between Feynman diagrams, renormalization, and the Hopf algebra of rooted trees.

April 9 Laura Reina, Physics Dept., FSU Rooted trees and Feynman diagrams II

We discuss the realization of the subtraction scheme via the antipode map in the Hopf Algebra of rooted trees. We discuss explicit examples in the framework of Connes and Kreimer's toy quantum field model.

April 16 Meeting cancelled
April 23 Ettore Aldrovandi, FSU The Hopf Algebra of Polyzêtas

We discuss multiple Zeta functions and related objects, multiple Polylogarithms, Z-sums, iterated integrals, and their algebraic properties. We aim in particular at describing a notable Hopf algebra structure related to the one of rooted trees and to interesting Lie algebras.


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