# Andrew Harder, Hyperelliptic curves and planar 2-loop Feynman graphs (Wednesday, July 20)

## Introductory

:::{.remark}
Feynman graphs $\Gamma$: vertices, edges, and half-edges where (for us) all vertices has a unique adjacent half-edge.
Can attach a Feynman integral: to any internal edge, attach $q_e = \sum \delta_{e, \ell} k_e + \sum \delta_{e, v} p_v$ where $\delta \in \ts{0, \pm 1}$ and $I_\Gamma = c \int_{\RR^{?}} {dk \over \prod_e (q_e^2 -m_e^2) }$ for $m_e \in \CC$.
Can reduce dimension to write $G = \sum x_e (q_e^2 - m_e^2)$, look at discriminant locus for a family of quadrics? 
Yields a locus $UF=0$ for two polynomials $U$ and $F$.
Somehow get a parametric form
\[
I_\Gamma = c\int_\sigma {U^? \over F^?} \Omega_0
\]
where $\sigma$ is a cycle given by $\ts{\tv{x_1: \cdots : x_E} \st x_i \in \RR_{\geq 0}}$; this is an integral on the complement of $Z(UF) = 0$.
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:::{.theorem title="?"}
After blowing up $\PP^{\abs E - 1}$, the integral $I_\Gamma$ is well-defined and a relative period of the pair $(P\sm Z(U^bF^a), B)$ where $P$ is a toric variety and $B$ its toric boundary, and $a,b \in \ts{0, 1}$ depending on the dimension $D$.
Thus the geometry of $Z(U)$ and $Z(F)$ determine $I_\Gamma$.
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:::{.remark}
The $U$ and $F$ are defined combinatorially from the Feynman graph.
Let $\Gamma_{\in } = (V, E)$ and let $S_1$ be the set of all spanning trees of $\Gamma_{\in}$.
Let $x^T = \prod_{e\not\in T} x_e$ for $T$ any spanning tree, where we have coordinates $\ts{x_e}_{e\in E}$.
This somehow defines $U$, and $F$ is defined by looking at spanning 2-trees (subgraphs with 2 connected components, no loops, containing all vertices).
It turns out $F$ is homogeneous, so defines a projective variety.
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:::{.example title="?"}
The theta graph yields a smooth elliptic curve in $\PP^2$.
More generally, connected the two vertices with more edges yields Calabi-Yau $(n-1)\dash$folds.
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## Extensions to 2-loop graphs

:::{.remark}
2-loop graphs: two trivalent nodes connected by chains of edges:

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![](figures/2022-07-20_14-54-37.png)

What does the mixed Hodge structure on cohomology look like?
This is an important step toward understanding periods, and the answer depends heavily on $D$.
Bloch works out kite and double box graphs.
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:::{.theorem title="DHNV"}
For $(a,1,1)$ hypersurfaces, identifies when certain homology (or some of its weight-filtered pieces) are Hodge-Tate under conditions on $a$ and $D$.
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:::{.remark}
Note that a Tate HS has $h^{i, j} = 0$ for $i\neq j$, and mixed Tate is an iterated extension of such a Hodge structure. This means it's as simple as it could reasonably be.
Idea of proof: blow up a linear subspace, remove some singular hyperplanes.

Interestingly, for $D=4$ only the cases $(3,1,1)$ and $(1,1,1)$ yields a non-Hodge-Tate mixed Hodge structure.
These correspond to multiple polylogarithms.
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:::{.remark}
WIP: can say more in $(a,1,c)$ case when $a, c> 1$.
Note that the $(a,1,c)$ case can't yield K3s.

See Prym varieties?
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