Tropical BF Theory and Tropical Limits of TQFTs

TL;DR

This work introduces a tropical (anisotropic) limit of topological field theories via Maslov dequantization, yielding foliated geometries and foliation-preserving gauge symmetry. The authors construct the 2D tropical BF theory ($TBF$) by scaling $A_r\to A_r$, $A_\theta\to\hbar A_\theta$, $B\to B/\hbar$ and inserting an auxiliary field $T$, obtaining a finite-dimensional moduli space of tropicalized flat connections with transverse flatness $\partial_r A_\theta+[A_r,A_\theta]=0$ and projectability $\partial_\theta A_r=0$. They show that $\dim\mathcal{M}(\Sigma_g,G)=(g-1)\operatorname{rank}(\mathfrak{g})$ for genus $g\ge2$ by gluing sleeves of $\mathbb{TP}^1$ and refine Atiyah–Segal axioms to this foliated setting, with explicit checks for $SU(N)$ and abelian $U(1)$. The paper then extends the tropicalization to 2D Tropological Yang–Mills, higher-dimensional $TBF$ theories, and Tropical Chern–Simons, including boundary phenomena yielding a tropical WZW sector and discussing connections to JT gravity and anisotropic conformal field theories. This framework provides a tractable, spacetime-foliated approach to anisotropic topological phases and motivates future work on higher-dimensional tropicalizations and potential links to fracton-like physics.

Abstract

We study anisotropic scaling limits of topological field theories using tropical geometry. The resulting topological field theories are characterized by foliated geometries and are invariant under foliation-preserving gauge transformations. We demonstrate the tropicalization for the 2D BF theory and generalize the prescription to topological Yang-Mills and Chern-Simons theories. We call the tropical limit of the BF theory, the \textit{TBF} theory, which is an anisotropic generalization of the BF theory with an additional adjoint-valued field $T$ that enforces a projectability condition onto the leaves of the foliation. The TBF theory localizes onto the moduli space of tropicalized flat connections $\mathcal{M}(Σ_g,G)$ on a foliated Riemann surface $Σ_g$ of genus $g$. The tropical connections exhibit anisotropic behavior; their holonomy is sensitive only to the leaves of the foliation. We analyze this moduli space two distinct ways, Firstly, they are classified by leaf-wise holonomy whose dimension can be explicitly calculated for the case of tropical projective space $\mathbb{TP}^1$ by the moduli space isomorphism $\mathcal{M}\left(\mathbb{TP} ^1, G\right) \cong \operatorname{Hom}(\mathbb{Z}, G) / G$. The second way is through Kodaira-Spencer theory which gives a twisted cohomology argument to argue that $\operatorname{dim} \mathcal{M}\left(\mathbb{T} P^1, G\right)=\operatorname{rank}(\mathfrak{g})$ and we demonstrate their equivalence for the case of SU$(N)$. We show that we can glue together several $\mathbb{TP}^1$ to obtain $\operatorname{dim} \mathcal{M}\left(Σ_g, G\right)=(g-1)\operatorname{rank}(\mathfrak{g})$ for $g \geq 2$ which is precisely $\frac{1}{2}$ of the usual result through an application of a foliated refinement of the Atiyah-Segal axioms. We leave several open questions such as potential connections to JT gravity and anisotropic conformal field theory.

Figures (2)

• Figure 1: An arbitrary loop $\gamma$ on the foliated $\mathbb{CP}^1$ that represents the tropical projective space $\mathbb{T}P^1$.
• Figure 2: A Wilson loop on a foliated Riemann surface of genus 1 which is obtained from gluing a sleeve to itself. The leaves of the foliation are denoted as teal circles.