Double Dimer Covers on Snake Graphs from Super Cluster Expansions

Abstract

In a recent paper, the authors gave combinatorial formulas for the Laurent expansions of super λ-lengths in a marked disk, generalizing Schiffler's T -path formula. In the present paper, we give an alternate combinatorial expression for these super λ-lengths in terms of double dimer covers on snake graphs. This generalizes the dimer formulas of Musiker, Schiffler, and Williams.

Cluster Superalgebras and Stringy Integrals

“…We also defer consideration of the fourth connection to future work since φ 3 theory is not amenable to supersymmetrization, although we note that it might be interesting to look at a suitable supersymmetrization of φ 4 theory, whose tree-level amplitudes are geometrically encoded in the structure of Stokes polytopes [13]. Finally we note that there has also been recent interest in the connection between string amplitudes and cluster algebras associated to surfaces following [14,15] and work in progress by Arkani-Hamed et al It would be interesting to explore whether there is a natural way to attach cluster superalgebras to super Riemann surfaces (see for example [16,17]), and to connect those to superstring amplitudes.…”