Tropical BF Theory and Tropical Limits of TQFTs

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 $\mathcal{M}(\Sigma_g,G)$ on a foliated Riemann surface $\Sigma_g$ 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 $\operatorname{dim} \mathcal{M}\left(\Sigma_g, G\right)=(g-1)\operatorname{rank}(\mathfrak{g})$ 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.