Topological Lattice Models in Four Dimensions

TL;DR

This paper introduces a four-dimensional topological lattice model based on group $G$, extending the Ponzano-Regge model, with invariance under topological moves and potential applications to knot and link invariants in four dimensions.

Contribution

The paper constructs a 4D topological lattice model using group $G$, demonstrating its invariance under Alexander moves and relating it to $BF$ theory, with a $q$-deformed version and operator realizations.

Findings

Partition function invariant under Alexander moves

Model related to 4D $BF$ theory

Potential for new 4D knot and link invariants

Abstract

We define a lattice statistical model on a triangulated manifold in four dimensions associated to a group $G$. When $G=SU(2)$, the statistical weight is constructed from the $15j$-symbol as well as the $6j$-symbol for recombination of angular momenta, and the model may be regarded as the four-dimensional version of the Ponzano-Regge model. We show that the partition function of the model is invariant under the Alexander moves of the simplicial complex, thus it depends only on the piecewise linear topology of the manifold. For an orientable manifold, the model is related to the so-called $BF$ model. The $q$-analogue of the model is also constructed, and it is argued that its partition function is invariant under the Alexander moves. It is discussed how to realize the 't Hooft operator in these models associated to a closed surface in four dimensions as well as the Wilson operator associated to a closed loop. Correlation functions of these operators in the $q$-deformed version of the model would define a new type of invariants of knots and links in four dimensions.