Graph potentials and topological quantum field theories

We introduce a new functional equation in birational geometry, whose solutions can be used to construct two-dimensional topological quantum field theories (2d TQFTs), infinite-dimensional in many interesting examples. The solutions of the equation give rise to a hierarchy of graph potentials, which, in the simplest setup, are Laurent polynomials associated with colored trivalent graphs, in such a way that the birational type of a graph potential depends only on the homotopy type of the graph. We give an efficient computational method to compute the partition functions of these TQFTs. We elaborate on the key example, related to mirror symmetric description of (Formula presented.) -character varieties, and propose how graph potentials, used to define these 2d TQFTs, can also eventually be used as a foundation for the dual construction of 4d Donaldson–Floer–Witten theories. This paper is the first in a series, and we give a survey of the applications of graph potentials in the other parts. (A similar formalism and examples were introduced independently by Kontsevich–Odesskii under the name of multiplication kernels; we hypothesize that the connection between the two can be understood via 3d mirror symmetry.).