Random matrix models generalize to Group Field Theories (GFT) whose Feynman graphs are dual to higher dimensional topological spaces. The perturbative development of the usual GFT’s is rather involved combinatorially and plagued by topological singularities (which we discuss in great detail in this paper), thus very difficult to control and unsatisfactory.

Colored Group Field Theory

We identify a large class R of three-dimensional N = 2 superconformal
field theories. This class includes the effective theories T_M of M5-branes
wrapped on 3-manifolds M, discussed in previous work by the authors,
and more generally comprises theories that admit a UV description as
abelian Chern–Simons-matter theories with (possibly non-perturbative)
superpotential. Mathematically, class R might be viewed as an extreme
quantum generalization of the Bloch group; in particular, the equivalence
relation among theories in class R is a quantum-field-theoretic “2 to 3
move.” We proceed to study the supersymmetric index of theories in
class R, uncovering its physical and mathematical properties, including
relations to algebras of line operators and to 4d indices. For 3-manifold
theories T_M, the index is a new topological invariant, which turns out to
be equivalent to non-holomorphic SL(2,ℂ) Chern–Simons theory on M
with a previously unexplored “integration cycle.”

https://www.intlpress.com/site/pub/files/_fulltext/journals/atmp/2013/0017/0005/ATMP-2013-0017-0005-a003.pdf

3-Manifolds and 3d Indices

1. Introduction 2. Riemannian geometry background 3. The maximum principle 4. Comments on existence theory for parabolic PDE 5. Existence theory for the Ricci flow 6. Ricci flow as a gradient flow 7. Compactness of Riemannian manifolds and flows 8. Perelman's W entropy functional 9. Curvature pinching and preserved curvature properties under Ricci flow 10. Three-manifolds with positive Ricci curvature and beyond.

/pdf/lectures-on-the-ricci-flow-2twqhias46.pdf

Lectures on the Ricci Flow

In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs play an important role. Furthermore we study the algebraic theory of LSAs such as structure theory, radical theory, cohomology theory and the classification of simple LSAs. We also discuss applications to faithful Lie algebra representations.

https://arxiv.org/pdf/math-ph/0509016.pdf

Left-symmetric algebras, or pre-Lie algebras in geometry and physics

Let K⩾1 be a parameter. A K-approximate group is a finite set A in a (local) group which contains the identity, is symmetric, and such that A⋅A is covered by K left translates of A. The main result of this paper is a qualitative description of approximate groups as being essentially finite-by-nilpotent, answering a conjecture of H. Helfgott and E. Lindenstrauss. This may be viewed as a generalisation of the Freiman-Ruzsa theorem on sets of small doubling in the integers to arbitrary groups. We begin by establishing a correspondence principle between approximate groups and locally compact (local) groups that allows us to recover many results recently established in a fundamental paper of Hrushovski. In particular we establish that approximate groups can be approximately modeled by Lie groups. To prove our main theorem we apply some additional arguments essentially due to Gleason. These arose in the solution of Hilbert’s fifth problem in the 1950s. Applications of our main theorem include a finitary refinement of Gromov’s theorem, as well as a generalized Margulis lemma conjectured by Gromov and a result on the virtual nilpotence of the fundamental group of Ricci almost nonnegatively curved manifolds.

The structure of approximate groups

Three-dimensional geometry and topology. Volume 1, by William P. Thurston, (edited by Sylvio Levy). Princeton Mathematical Series, 35. £29.95. 1997. ISBN 0-691-08304-5 (Princeton University Press).

I. M. James

Papers

Three-dimensional geometry and topology. Volume 1, by William P. Thurston, (edited by Sylvio Levy). Princeton Mathematical Series, 35. £29.95. 1997. ISBN 0-691-08304-5 (Princeton University Press).