Hopf algebras and periods in quantum field theory

8th -10th of January 2008, Durham University

**A series of lectures by** Dirk Kreimer (IHES and Boston University)

Hopf algebras and numbers: the recursive structure of Green functions

Lecture 1: *Hopf algebra of graphs and trees*

A very elementary introduction into the Hopf algebras of graphs and trees with an emphasis how to relate graphs to trees. We will include a short reminder of the role of overlapping divergences.
(video)

Lecture 2: *Sub-Hopf algebra and Hochschild cohomology*

We study how to obtain sub Hopf algebras and their role in quantum field theory. In particular, we study the structure of a proof of renormalizability by local counterterms from the Hochschild cohomology of Hopf algebras. (video)

Lecture 3: *Dyson-Schwinger equations*

We augment perturbation theory by equations of motions, obtained from the Hochschild cohomology studied in the previous lecture. We review these equations for various renormalizable field theories, and discuss

simplifications. (video)

Lecture 4: *Quantum electrodynamics and recursive structures*

We specialize to quantum electrodynamics (QED) as a gauge theory, study ideals and co-ideals to understand the Ward identities and use all the previous lectures to boil the equations of motion down to a single ordinary differential equation which determines the beta function of QED in terms of a certain series in periods. (video)

Lecture 5: *Multiple zeta values for primitives and mixed Hodge structures*

After a review on the appearance of multiple zeta values we discuss similarities and differences between Green functions and polylogs, reporting on ongoing work with Spencer Bloch to find (limiting) mixed Hodge structures as a common language. (video)

Lecture 6: *Beyond numbers for short-distance singularities *

A very speculative review of ideas for progress beyond renormalization. (video)

**Lectures by **David Broadhurst (Open University, Milton Keynes)

Lecture 1: *Dyson-Schwinger solutions from the Hopf algebra of renormalization*

Lecture 2:* Singular values of elliptic integrals in quantum field theory*

Massless Feynman diagrams often yield multiple zeta values, which are conjectured to be periods of a mixed Tate

motive. Yet the definition of a period is much more general, encompassing for example the lemniscate constant, which comes from the first singular value for a complete elliptic integral. I shall discuss massive Feynman diagrams that yield the 15th singular value. There is a strong link between these diagrams and Green functions on diamond and cubic lattices, in condensed matter theory. (video)

**Schedule **

- 8th January 2008 (Tuesday)

10:30am: Kreimer1 *Hopf algebra of graphs and trees*

2:00pm: Kreimer 2 *Sub-Hopf algebra and Hochschild cohomology*

- 9th January 2008 (Wednesday)

10:30am: Kreimer3 *Dyson-Schwinger equations*

2:00pm: Broadhurst 1 *Dyson-Schwinger solutions from the Hopf algebra of renormalization*

4:00pm: Kreimer 4 *Quantum electrodynamics and recursive structures*

- 10th January 2008 (Thursday)

10:30am: Kreimer 5 *Multiple zeta values for primitives and mixed Hodge structures*

2:00pm: Kreimer 6 *Beyond numbers for short-distance singularities *

4:00pm: Broadhurst 2 *Singular values of elliptic integrals in quantum field theory *