2025/03/20 by Emil Albrychiewicz, Albrychiewicz, Emil, Andrés Franco Valiente +1
Computer Science · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Scientific Research and Discoveries #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2503.15856
openalex publication_date 2025/03/20 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/29
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 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 M(Σg,G) on a foliated Riemann surface Σg 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 \mathbbTP1 by the moduli space isomorphism M(\mathbbTP 1, G) ≅ Hom(ℤ, G) / G. The second way is through Kodaira-Spencer theory which gives a twisted cohomology argument to argue that dim M(\mathbbT P1, G)=rank(\mathfrakg) and we demonstrate their equivalence for the case of SU(N). We show that we can glue together several \mathbbTP1 to obtain dim M(Σg, G)=(g-1)rank(\mathfrakg) for g ≥ 2 which is precisely (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.