State sum models for quantum gravity
John W. Barrett
TL;DR
This paper surveys state-sum invariants of 4-manifolds and their extensions to quantum gravity via combinatorial triangulations. It explains how a common state set on $n$-simplices and amplitudes yield the partition function $Z(M)=\sum_s \prod_σ w(s(σ))$, and reviews topological models such as Dijkgraaf-Witten, Ooguri, and Crane-Yetter and their relation to BF theory. It then describes constrained state-sum constructions for gravity with $\mathrm{SO}(4)$ and $\mathrm{SO}(3,1)$, where simple representations and a constrained $B$ field lead to the Einstein–Hilbert action via $S=\int_M e^a\wedge e^b\wedge F^{cd}\epsilon_{abcd}$, with triangle areas fixed by a Casimir and a semiclassical limit matching area-based Regge geometry. The paper also discusses matrix-model realizations and the spin-foam perspective, highlighting triangulation-independence issues and open questions about the continuum limit. Overall, the work maps current combinatorial approaches to quantum gravity, their connections to BF theory and topological invariants, and the major open challenges in achieving a satisfactory continuum theory from state sums.
Abstract
This paper reviews the construction of quantum field theory on a 4-dimensional spacetime by combinatorial methods, and discusses the recent developments in the direction of a combinatorial construction of quantum gravity.