Non-perturbative Group field theory from combinatorial Dyson-Schwinger equations and their algebraic structure II

• Applicant: Dr. Johannes Thürigen
• Subject Area: Nuclear and Elementary Particle Physics, Quantum Mechanics, Relativity, Fields
• Term: from 2019 to 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 418838388

• Final Report Year: 2025

Final Report Abstract

When spacetime shows features of discreteness in the quantum regime of gravity, the semiclassical limit to the observed continuum geometries is a challenge in various proposals for such a fundamental theory of quantum gravity. Here we have considered tensorial group field theories (TGFT) in which this continuum limit is expected to arises non-perturbatively at criticality, similar to matrix models. To bridge the gap from the well understood perturbative theory to such non-perturbative regime we have worked out the (Hopf-)algebraic structure of TGFT renormalization in detail. This is already of high interest as it gives an algorithm to compute amplitudes and unveals the algebraic structure behind the intricate Feynman diagrammatics of this combinatorially non-local theory. We have further investigated to what extend Dyson- Schwinger equations based on this algebra allow to find non-perturbative solutions. An expected reduction of complexity due to symmetries could not be found; to the contrary, analysis of such Ward identities have shown that only a symmetry-reduced ”isotropic” sector of TGFT has been considered in the literature so far, and that TGFT allows for a much richer theory space in general. To explore this, we have applied the functional renormalization group method and found a new class of ”anisotropic” non- Gaussian fixed points. We expect these critical points to correspond to novel types of continuum geometry which uncloses a new arena of continuum limits in quantum gravity.

Publications

• Phase transitions in TGFT: functional renormalization group in the cyclic-melonic potential approximation and equivalence to O(N) models. Journal of High Energy Physics, 2020(12).
  Pithis, Andreas G. A. & Thürigen, Johannes
  
• (No) phase transition in tensorial group field theory. Physics Letters B, 816, 136215.
  Pithis, Andreas G.A. & Thürigen, Johannes
  
• Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme. Symmetry, Integrability and Geometry: Methods and Applications.
  Thürigen, Johannes
  
• Renormalization in Combinatorially Non-Local Field Theories: The Hopf Algebra of 2-Graphs. Mathematical Physics, Analysis and Geometry, 24(2).
  Thürigen, Johannes
  
• One-matrix differential reformulation of two-matrix models. Reviews in Mathematical Physics, 34(08).
  Brunekreef, Joren; Lionni, Luca & Thürigen, Johannes
  
• Mean-Field Phase Transitions in Tensorial Group Field Theory Quantum Gravity. Physical Review Letters, 130(14).
  Marchetti, Luca; Oriti, Daniele; Pithis, Andreas G. A. & Thürigen, Johannes
  
• Primitive asymptotics in φ4 vector theory, submitted to Ann. H. Poincare D
  P. Balduf & J. Thürigen, H.
  
• QFT with tensorial and local degrees of freedom: Phase structure from functional renormalization. Journal of Mathematical Physics, 65(3).
  Ben, Geloun Joseph; Pithis, Andreas G. A. & Thürigen, Johannes
  
• Combinatorial Dyson-Schwinger Equations of Quartic Matrix Field Theory, submitted to J. Noncom. Geom.
  A. Hock & J. Thürigen
  
• New fixed points from melonic interactions. Physics Letters B, 860, 139218.
  Juliano, Leonardo & Thürigen, Johannes