Super Higher-Teichmüller Geometry and Loop Amplitudes

TL;DR

The work constructs a supersymmetric extension of the FG cluster ensemble, producing a super higher–Teichmüller geometry with a mutation–covariant loop–fiber framework. A canonical logarithmic superform $\Omega_{\text{super}}^{(L)}$ and its super period $P_{\text{super}}=\int_{C}\Omega_{\text{super}}^{(L)}$ encode loop amplitudes for planar $\mathcal{N}=4$ SYM in a triangulation–independent way, splitting into Chen iterated integrals (even) and an invariant BCFW delta (odd). The construction enforces Steinmann factorization and cluster adjacency, while a boundary quotient isolates IR–finite observables; quantization via a quantum super torus and RTT–Yangian layers points toward an integrable underpinning. The framework generalizes to arbitrary bordered surfaces and other split Lie supergroups, offering a unified geometric basis for supersymmetric loop amplitudes and potential nonplanar extensions.

Abstract

We construct a supersymmetric extension of the Fock-Goncharov cluster ensemble associated with a split basic classical Lie supergroup $G$ and a marked bordered surface $S$. The resulting structure defines a super higher-Teichmüller geometry: a split super--thickening of $(\mathscr A_{G,S}, \mathscr X_{G,S})$ equipped with a mutation atlas preserving a canonical super log-symplectic form. Each super seed carries an integer weight matrix $W$ encoding Cartan weights of an abelian odd slice, transforming by the column $g$--vector rule and giving rise to a flat logarithmic superconnection and a canonical super volume form. On this geometric foundation we define a canonical logarithmic superform $Ω_{\mathrm{super}}^{(L)}$ on a loop fibration $π_L : \mathscr X^{(L)}_{G,S} \!\to\! \mathscr X_{G,S}$ as the relative lift of the base super volume. For $G = PGL(4|4)$, the corresponding super period $P_{\mathrm{super}} = \int_{C} Ω_{\mathrm{super}}^{(L)}$ encodes the loop amplitude data of planar $N = 4$ super Yang--Mills, expressed through a unified and triangulation-independent formula that satisfies Steinmann and cluster adjacency, with the even sector given by Chen iterated integrals and the odd sector captured by an invariant BCFW delta.