Super Higher-Teichmüller Geometry and Loop Amplitudes

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 $({\mathcal{A}}_{G,S},{\mathcal{X}}_{G,S})$ 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 ${\mathrm{\Omega }}_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{e}\mathrm{r}}^{(L)}$ on a loop fibration ${\pi }_L:{\mathcal{X}}_{G,S}^{(L)}\text{\hspace{0.17em}⁣}\to \text{\hspace{0.17em}⁣}{\mathcal{X}}_{G,S}$ as the relative lift of the base super volume. For $G=PGL(4\mathrm{\mid }4)$, the corresponding super period $P_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{e}\mathrm{r}}={\int }_C{\mathrm{\Omega }}_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{e}\mathrm{r}}^{(L)}$ 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.