Tropical BF Theory and Tropical Limits of TQFTs · arXivDesk
2503.15856Mar 20, 2025Update title and abstract to better reflect content, fix typos and expand discussion about tropical limit of matrix models. 37 pages
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
Nearby in the stack
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
TP1
by the moduli space isomorphism
M(TP1,G)≅Hom(Z,G)/G
. The second way is through Kodaira-Spencer theory which gives a twisted cohomology argument to argue that
dimM(TP1,G)=rank(g)
and we demonstrate their equivalence for the case of SU
(N)
. We show that we can glue together several
TP1
to obtain
dimM(Σg,G)=(g−1)rank(g)
for
g≥2
which is precisely
21
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.