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

• Final Report Year: 2024

Final Report Abstract

In this research project, I focused on the mathematical structure and perturbation theory of exactly solvable quantum field theories (QFTs). Quantum field theory explains the structure of matter, atoms, and particles, but it is not yet fully understood mathematically. Particularly in perturbation theory, which is used in many areas of physics, the mathematical series employed are often non-convergent. My goal was to investigate the algebraic structures in specific, exactly solvable QFT models in order to broaden theoretical understanding. A key result of my research was the analysis of the Grosse-Wulkenhaar model, a scalar theory in a non-commutative space, which is considered renormalizable. In collaboration, we found an exact solution to this model, providing new insights into the structure of this and similar theories. Another focus was on the so-called Topological Recursion (TR), a universal mathematical method that is applied in various disciplines of mathematics and physics. I was able to uncover new structures and symmetries in TR, particularly the so-called x − y duality, which I co-developed during the project. This duality made it possible to derive new connections and formulas in TR, improving not only the understanding of TR itself but also its application in fields such as enumerative geometry, free probability theory, knot theory, topological string theory, and quantum field theory. Overall, these projects have not only deepened the understanding of exactly solvable QFTs but have also contributed to new discoveries in TR, which have wide-ranging applications in theoretical physics and mathematics.

Publications

• Combinatorial Dyson-Schwinger Equations of Quartic Matrix Field Theory,
  Hock, Alexander & Thürigen, Johannes
  
• From scalar fields on quantum spaces to blobbed topological recursion. Journal of Physics A: Mathematical and Theoretical, 55(42), 423001.
  Branahl, Johannes; Hock, Alexander; Grosse, Harald & Wulkenhaar, Raimar
  
• A simple formula for the x-y symplectic transformation in topological recursion. Journal of Geometry and Physics, 194, 105027.
  Hock, Alexander
  
• Complete Solution of the LSZ Model via Topological Recursion. Communications in Mathematical Physics, 401(3), 2845-2899.
  Branahl, Johannes & Hock, Alexander
  
• Laplace transform of the $x-y$ symplectic transformation formula in Topological Recursion. Communications in Number Theory and Physics, 17(4), 821-845.
  Hock, Alexander
  
• An irregular spectral curve for the generation of bipartite maps in topological recursion. Annales de l’Institut Henri Poincaré D, Combinatorics, Physics and their Interactions, 12(3), 463-473.
  Branahl, Johannes & Hock, Alexander
  
• Genus permutations and genus partitions. Enumerative Combinatorics and Applications, 5(1), Article #S2R5.
  Hock, Alexander
  
• On the x-y Symmetry of Correlators in Topological Recursion via Loop Insertion Operator. Communications in Mathematical Physics, 405(7).
  Hock, Alexander
  
• x-y duality in topological recursion for exponential variables via quantum dilogarithm. SciPost Physics, 17(2).
  Hock, Alexander
  
• Blobbed topological recursion from extended loop equations. Journal of Geometry and Physics, 212, 105457.
  Hock, Alexander & Wulkenhaar, Raimar