Renormalization of Gauge Theories and Gravity

The renormalization of gauge theories and, eventually, gravity is one of the biggest current challenges in mathematical physics. I will start this talk by discussing the Hopf algebraic approach to renormalizable gauge theories, which results in the following two aspects:

 

1) Avoiding gauge anomalies, which translates to the validity of the corresponding Slavnov—Taylor identities: A theorem by van Suijlekom (2007), improved by myself (2022), states that they can be represented via Hopf ideals.

 

2) Transversality of the formal Green's function, which can be implemented combinatorially via cancellation identities: In an ongoing project, I aim to implement these via an appropriate Feynman graph cohomology, similar to the one constructed by Kreimer et al. (2013).

 

Specifically, I aim to relate both aspects in said project by extending the renormalization Hopf algebra to a differential-graded renormalization Hopf module, as outlined in my doctoral thesis, cf. arXiv:2210.17510 [hep-th]. Finally, I will address the application of these constructions to Quantum General Relativity, which is non-renormalizable by power counting. To overcome this issue, I will close with a recent conjecture of mine for the appropriate UV-completion thereof: This is a deformation of the Einstein—Hilbert Lagrange density, which is renormalizable, and reproduces General Relativity in the long-scale limit.