Algebraic Structure, Perturbation Theory and Galois Coaction for Exactly Solvable Quantum Field Theories

• Applicant: Dr. Alexander Hock
• Subject Area: Mathematics ; Nuclear and Elementary Particle Physics, Quantum Mechanics, Relativity, Fields
• Term: from 2021 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 465029630

A new class of exactly solvable quantum field theories (QFTs), which arises from matrix models, will be studied with respect to their mathematical properties. This type of integrable models is based on a universal structure, the so-called topological recursion. We find perturbatively the same problems as in ordinary QFT, including a factorial growth of the number of Feynman graphs and the renormalon problem. Nevertheless, the perturbation theory consisting of iterated integrals is resummable.The exact solutions of the $\phi^4$ matrix model are built from implicitly defined functions, which are explcitly solved on the noncommutative Moyal space (in 2 and 4 dimensions). The deep structure of the perturbation theory will be analysed in a way that we can construct the Hopf algebra of its renormalization procedure. Furthermore, the exact results will be compared with perturbation theory. For this purpose we will use the Galois coaction (GC) on iterated integrals, developed in 2005 (an extension of the action of the Galois group on $\bar{\mathbb{Q}}$). The transcendence conjecture of Grothendieck is assumed, which means in more detail that the GC on motivic iterated integrals and motivic periods acts equally on iterated integrals and periods. The GC can generate a huge number of Galois conjugates. It is claimed and verified in the first few perturbative orders that a QFT is closed under the GC. In other words, all Galois conjugates of Feynman amplitudes are linear combinations of other Feynman amplitudes. This conjecture gives very strong and deep mathematical constraints on a QFT, on the appearing Feynman amplitudes and on the combinatrics of Feynman graphs.This GC will be applied for the first time to an exactly solvable QFT. In doing so we will answer the question whether a QFT is closed under the GC. For that the availability of exact results on correlation functions is indispensable; the proof can never be achieved perturbatively! The recently (2019) via twisted cohomology developed extension of GC to hypergeometric functions and further special functions will play a central rôle.

• DFG Programme: WBP Fellowship
• International Connection: United Kingdom
• Host: Dr. Erik Panzer