Combinatorial Hopf algebra for the Ben Geloun-Rivasseau tensor field theory

The Ben Geloun-Rivasseau quantum field theoretical model is the first tensor model shown to be perturbatively renormalizable. We define here an appropriate Hopf algebra describing the combinatorics of this new tensorial renormalization. The structure we propose is significantly different from the previously defined Connes-Kreimer combinatorial Hopf algebras due to the involved combinatorial and topological properties of the tensorial Feynman graphs. In particular, the 2- and 4-point function insertions must be defined to be non-trivial only if the superficial divergence degree of the associated Feynman integral is conserved.

[1]  E. Livine,et al.  Some classes of renormalizable tensor models , 2012, 1207.0416.

[2]  A. Connes,et al.  Renormalization and motivic Galois theory , 2004, math/0409306.

[3]  Alain Connes,et al.  Renormalization in Quantum Field Theory and the Riemann–Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem , 2000 .

[4]  T. Krajewski.  Schwinger-Dyson Equations in Group Field Theories of Quantum Gravity , 2012, 1211.1244.

[5]  Adrian Tanasa,et al.  Hopf algebra of non-commutative field theory , 2007, 0707.4143.

[6]  D. O. Samary,et al.  Just Renormalizable TGFT’s on U(1)d with Gauge Invariance , 2012, Communications in Mathematical Physics.

[7]  Razvan Gurau,et al.  Colored Group Field Theory , 2009, 0907.2582.

[8]  D. O. Samary,et al.  3D Tensor Field Theory: Renormalization and One-loop $\beta$-functions , 2011, 1201.0176.

[9]  A. Connes,et al.  Noncommutative Geometry, Quantum Fields and Motives , 2007 .

[10]  D. Manchon.  Hopf algebras, from basics to applications to renormalization , 2004, math/0408405.

[11]  Sylvain Carrozza,et al.  Renormalization of Tensorial Group Field Theories: Abelian U(1) Models in Four Dimensions , 2012, 1207.6734.

[12]  V. Rivasseau,et al.  Renormalization of a SU(2) Tensorial Group Field Theory in Three Dimensions , 2014, Communications in Mathematical Physics.

[13]  Joseph Ben Geloun,et al.  A Renormalizable 4-Dimensional Tensor Field Theory , 2011, 1111.4997.

[14]  Hopf algebra of ribbon graphs and renormalization , 2001, hep-th/0112146.

[15]  Adrian Tanasa,et al.  Combinatorial Dyson-Schwinger equations in noncommutative field theory , 2009, 0907.2182.

[16]  Matilde Marcolli,et al.  Renormalization, the Riemann–Hilbert Correspondence, and Motivic Galois Theory , 2004, hep-th/0411114.

[17]  Adrian Tanasa,et al.  Algebraic structures in quantum gravity , 2009, 0909.5631.