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We provide a comprehensive summary of concepts from Calabi-Yau motivesrelevant to the computation of multi-loop Feynman integrals. From this wederive several consequences for multi-loop integrals in general, and weillustrate them on the example of multi-loop banana integrals. For example, weshow how Griffiths transversality, known from the theory of variation of mixedHodge structures, leads quite generically to a set of quadratic relations amongmaximal cut integrals associated to Calabi-Yau motives. These quadraticrelations then naturally lead to a compact expression for $l$-loop bananaintegrals in $D=2$ dimensions in terms of an integral over a period of aCalabi-Yau $(l-1)$-fold. This new integral representation generalizes in anatural way the known representations for $l\le 3$ involving logarithms withsquare root arguments and iterated integrals of Eisenstein series. In a secondpart, we show how the results obtained by some of the authors in earlier workcan be extended to dimensional regularization. We present a method to obtainthe differential equations for banana integrals with an arbitrary number ofloops in dimensional regularization without the need to solveintegration-by-parts relations. We also present a compact formula for theleading asymptotics of banana integrals with an arbitrary number of loops inthe large momentum limit. This generalizes the novel $\widehat{\Gamma}$-classintroduced by some of the authors to dimensional regularization and provides aconvenient boundary condition to solve the differential equations for thebanana integrals. As an application, we present for the first time numericalresults for equal-mass banana integrals with up to four loops and up to secondorder in the dimensional regulator.

Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives