Schwinger-Dyson Equations in Group Field Theories of Quantum Gravity

TL;DR

This paper explores the algebraic structures underlying Schwinger-Dyson equations in group field theories, connecting graph operations to Hopf algebras and proposing a Wilsonian flow analogue for quantum gravity models.

Contribution

It identifies a Lie algebra related to Schwinger-Dyson equations as a Hopf algebra and adapts the graph contraction operation to group field theories, advancing the mathematical framework of quantum gravity.

Findings

Identification of a Lie algebra as a Hopf algebra of Connes-Kreimer type.

Development of a Wilsonian flow analogue for effective actions.

Potential adaptation of the formalism to group field theories.

Abstract

In this talk, we elaborate on the operation of graph contraction introduced by Gurau in his study of the Schwinger-Dyson equations. After a brief review of colored tensor models, we identify the Lie algebra appearing in the Schwinger-Dyson equations as a Lie algebra associated to a Hopf algebra of the Connes-Kreimer type. Then, we show how this operation also leads to an analogue of the Wilsonian flow for the effective action. Finally, we sketch how this formalism may be adapted to group field theories.