Super Higher-Teichmüller Geometry and Loop Amplitudes
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| Format: | Preprint |
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2025
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| _version_ | 1866908614086623232 |
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| author | Song, Chaoming |
| author_facet | Song, Chaoming |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22769 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Super Higher-Teichmüller Geometry and Loop Amplitudes Song, Chaoming Mathematical Physics High Energy Physics - Theory 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. |
| title | Super Higher-Teichmüller Geometry and Loop Amplitudes |
| topic | Mathematical Physics High Energy Physics - Theory |
| url | https://arxiv.org/abs/2510.22769 |