arXiv subjects

Thomas Nicosanti

Publications and source records attributed to Thomas Nicosanti.

2 recordsLinked to original sources

arXiv2025

Approximating $\mathrm{SU}(2)$ Chern-Simons theory by finite group gauge theories

Motivated by some previously known facts from mathematical and physics literature, we explore certain relations between 3-dimensional topological gauge theories with continuous and finite gauge groups, commonly known as Chern-Simons (CS) and Dijkgraaf-Witten (DW) theories, respectively. Specifically, we consider the continuous and finite gauge groups to be the same algebraic group over the complex numbers and a finite field, respectively. In this paper, we focus on the $\mathrm{SU}(2)$ example and consider the relationship on the level of the corresponding partition functions on closed 3-manifolds. Mathematically, these are Witten-Reshetikhin-Turaev and DW invariants. We find that the asymptotics of the DW theory when the number of elements of the finite field is large recovers the leading asymptotics of the CS theory at large level when the 3-manifold contains no hyperbolic components. As a byproduct, we develop efficient techniques to count the number of points over finite fields $\mathbb{F}_q$ of $\mathrm{SL}(2)$ representation varieties of fundamental groups of 3-manifolds, possibly with weights pulled back from a chosen class in $H^3(\mathrm{SL}(2,\mathbb{F}_q),\mathrm{U}(1))$.

hep-th↗

arXiv2025

Summing over homology groups of 3-manifolds

We consider a toy model of a 3-dimensional topological quantum gravity. In this model, a contribution of a given 3-manifold is given by the partition function of an abelian Topological Quantum Field Theory (TQFT), with a topological boundary condition at the boundary. Using the fact that the TQFT partition function depends only on the first homology group of the 3-manifold with some additional structure, the sum over all 3-manifolds with fixed boundary can be rewritten as a sum over finitely generated abelian groups (and the extra structure). We present bounds on the universal weights in the sum, that is, the measure on the set of isomorphism classes of finitely generated abelian groups (with the extra structure) sufficient for the sum to be convergent. Moreover, with further assumptions on the measure, we argue the existence of a distribution of 2d TQFTs, such that the sum is equal to the ensemble average of their partition functions evaluated on the boundary. For a certain kind of measure, the sum over the abelian groups can be factorized into sums over abelian $p$-groups, and the analysis can be performed independently for each prime $p$.

hep-th↗