# Amanda Francis, Hanging gardens and divisors from Adinkras (Thursday, July 21)

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Physics motivations: standard model has 5 bosons and 12 fermions, and supersymmetry switches and matches these in pairs.
Form an algebra from supersymmetry operators $Q_i$ transforming bosons to fermions, where $Q_i \phi = \pm (iH)^s \psi$, where $H$ is a Hamiltonian.
Visualize these with Adinkras (etymology: Ghanaian art): a bipartite graph (bosons/fermions), $Q_i$ operators represented by colored edges, 
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Chromotopology: bipartite, $N\dash$regular with a consistent edge coloring (each vertex has $N$ incident edges, one of each color), and other conditions.
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Somehow get an algebraic (Riemann) surface $X$ from this.
Match up with dessin d'enfants?

If a height function $h$ on a triangular mesh sends adjacent vertices to distinct values, this is a discrete Morse function.
So we can triangulate these graphs by adding new vertices and new fractional heights to get Morse functions.
Build divisors from critical points of $h$ by defining multiplicities $m_v$; this yields a Morse divisor $D_h = \sum_{v\in M} -m_v [v]$.
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The Jacobian of $X$ carries interesting data.
Neat fact: the Belyi map $X\to \PP^1$ factors through a beach ball $B_N$, and the map $X\to B_N$ is less ramified.
See Carvacho-Hidalgo-Quispe: $N=5$ hypercube curve is the generalized Fermat curve, a complete intersection $X \injects \PP^4$.
Remainder of talk: discussion of counting height functions on $N\dash$hypercubes.
The number blows up very quickly!
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