# Johannes Walcher, Rationality of MUMs and 2-functions (Thursday, July 21)

## Rational 2-functions are abelian

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Motivations: mirror symmetry links enumerative geometry (A model) to variations of Hodge structure (B model).
On the B side, arithmetic properties naturally arise.
Do they have an interpretation on the A side?

Classical mirror symmetry involves maximal unipotent monodromy (MUM) in the moduli space of CYs.
Johannes entered this area by looking at extensions of algebraic cycles.
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2-functions: for $s \in \ZZ_{>0}$, and $s\dash$function is a power series $W\in z\QQ\fps{z}$ of the form $\sum_d n_d \Li_s(z^d)$.
Motivating examples: $g=0$ prepotential expanded around MUM in flat coordinates.
Equates generating functions for GW invariants and that for BPS invariants, $\sum a_dq^d = \sum n_d \Li_3(q^d)$.

A new feature: framing, where $Y \in 1 + z\ZZ\fps{z}$ defined an automorphism of $z\ZZ\fps{z}$.
See Lagrange inversion for finding coefficients of power series.
Theorem: if $W$ is a rational 3-function, then all of its framings are 3-functions.
Relates to some classical theorems, e.g. the Wolstenholme theorem and the Jacobsthal-Kazandzidis congruence.
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Over a number field, the $s\dash$functions form a finite rank free module. 
Muller shows that there can't be a natural choice of basis for $K$ an arbitrary number field -- if $W\in zK\fps{z}$ is rational 2-function, so $\exp(\delta W) \in K(z)$, then $K/\QQ$ must be an abelian extension.
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## MUM

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For $\pi: Y\to B$ a smooth family of Calabi-Yau threefolds of mirror quintic type, there is a $\ZZ\VHS$ (pure, weight 3).
By analog with \(p\dash \)adic Hodge theory, Kontsevich et al have a theorem that lets you conclude integrality from rationality in some special cases?
Look at monodromy of Picard-Fuchs equations.
See Bogner's "strange operator".
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Early 2000s: Almkvist-Zudilin look at Calabi-Yau differential operators, which are 4th order operators with MUM points, duality, integral fundamental periods, etc so that it could come from the integral cohomology of a CY.

See Beauville-Apery families of elliptic curves, Hadamard product.
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Idea: take a family $Y\to B$ and $C\injects Y$ some 2-cycles (curves) which is also a nice family over $B$, this gives an extension of VHSs.
Somehow classified by intermediate Jacobian?
By Griffith's transversality, the extension is fully determined by a piece in a certain $\Ext^1$ group.
Extraction of integers appears to depend on finding a good basis for 2-functions.
Speculation: analytic 2-functions with coefficients in a number field arise as solution to inhomogeneous linear ODEs with algebraic coefficients, and unravelling the enumerative interpretation (e.g. in in open GW/BPS theory) will involve natural such bases.
Maybe $K$ is the trace field of a hyperbolic 3-manifold?
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