Mathematical aspects of QFT

Principal investigators: Olaf Hohm, Dirk Kreimer

Doctoral researchers: Tomas Codina, Allison Pinto

Hopf algebras and Feynman graphs

In this research area the focus is on providing a solid mathematical basis for the algebraic structure of quantum field theory regarded as a series over Feynman graphs, in a way that allows to combine this algebraic structure with analytic and algebra-geometric techniques to allow for progress.

Key aspects of this approach

  • Underly the process of renormalization
  • Allow the classification of equations of motion in quantum field theory
  • Classification of gauge symmetries through co-ideals in the Hopf algebra
  • New approach to physics beyond perturbation theory
  • Rich structure as graph complexes relating to geometric group theory
  • Dyson-Schwinger as fixed-point equation in Hochschild cohomology
  • Classification of renormalization schemes stable under field diffeomorphisms
  • Co-actions and analytic structure from algebra
The cubical chain complex associates to each graph variations 
in accordance with Cutkosky cuts

L-algebras in Quantum Field Theory

This aspect of the research program is concerned with higher algebraic structures in QFT and, more generally, in theoretical and mathematical physics. 

Lie groups and Lie algebras are the structures underlying most of contemporary fundamental physics, from the standard model of particle physics to Einstein’s theory of general relativity. Recent developments have shown that higher algebraic structures in form of
L-algebras could play an important role for the physics of the 21st century. 

  • L-algebras are a generalisation of Lie algebras satisfying higher Jacobi identities involving potentially infinitely many higher brackets. It has become clear recently that L-algebras can function naturally as the organisational principle for arbitrary field theories
  • A powerful machinery for L-algebras exists —homotopy transfer—that has recently been argued to govern dualities of QFT 
  • There are rich sub-theories of full string theory that include massive string modes in a duality covariant formulation that are not captured by the usual spacetime theories such as supergravity. The explicit construction of such theories and the determination of their underlying L-algebras is the focus of current research
  • Higher-derivative α-corrections in string inspired gravitational theories could play a role in resolving the big-bang singularity or in providing a non-perturbative mechanism to generate de Sitter vacua.
    L-algebras within the framework of double field theory can help in understanding the symmetry principles of these corrections
Graphical presentation of higher Jacobi identity

Integration into the RTG

This research topic integrates well into the RTG as it provides mathematical foundations for phenomenology, scattering amplitudes, the AdS/CFT correspondence and for perturbative expansions in terms of Feynman diagrams.
In addition, Dyson-Schwinger equations provide a link to the non-perturbative physics of lattice field theory. The spacetime formulation provided by double field theory is a promising framework in order to understand the principles underlying the “double copy” construction for the computation of amplitudes.