The two-dimensional conformal traintrack integral is modular, as proven by factoring its Picard-Fuchs system into Gauss hypergeometric systems via the Clingher-Doran-Malmendier gauge transformation.
Jacobian elliptic kummer surfaces and special function identities.Communications in Number Theory and Physics, 12(1):97–125
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Modularity of Feynman Integrals and Factorization of Appell F2 Systems
The two-dimensional conformal traintrack integral is modular, as proven by factoring its Picard-Fuchs system into Gauss hypergeometric systems via the Clingher-Doran-Malmendier gauge transformation.