About a quantum field theory for 3D gravity (Q862021)

This is a survey paper arisen from two talks given by the author at the Seminario Matematico e Fisico Milano and the Colloquium of the Institut Mathématique de Jussieu (Paris). The author outlines some aspects of a dilogarithmic quantum field theory (DQFT) for a \(2+1\) bordism category based on \(3\)-manifolds equipped with principal flat \(\text{PSL}(2,\mathbb C)\) bundles. The classical 3D (pure) gravity concerns the study of Riemannian or Lorentzian \(3\)-manifolds of constant curvature. The sign of the curvature coincides with the sign of the cosmological constant. The Lorentzian spacetimes are time oriented. Along the world lines of ``particles'' there are concentrated singularities of the metric. The 3D-quantum filed theory is equivalent to the representation of some \((2+1)\)-bordism theory in the tensorial category of complex linear spaces. The author has introduced a family \(\mathcal{D}_N\) (where \(N\geq 1\) is an odd integer) of so called dilogarithmic quantum field theories, for a suitable \((2+1)\)-bordism category based on oriented compact \(3\)-manifolds \(Y\), which include properly embedded \(1\)-dimensional framed links \(L\), and are equipped with flat connections on principal \(\text{PSL}(2,\mathbb C)\) bundles on \(Y\backslash L\) having arbitrary holonomy at the meridians of the link components. The author gives some geometric motivations supporting the idea that these quantum field theories should be pertinent to 3D gravity, and describes the building blocks of DQFT: the matrix dilogarihms.