Yangian-Invariant Fishnet Integrals in Two Dimensions as Volumes of Calabi-Yau Varieties

We argue that ℓ-loop Yangian-invariant fishnet integrals in two dimensions are connected to a family of Calabi-Yau ℓ folds. The value of the integral can be computed from the periods of the Calabi-Yau, while the Yangian generators provide its Picard-Fuchs differential ideal. Using mirror symmetry, we can identify the value of the integral as the quantum volume of the mirror Calabi-Yau. We find that, similar to what happens in string theory, for ℓ=1 and 2 the value of the integral agrees with the classical volume of the mirror, but starting from ℓ=3, the classical volume gets corrected by instanton contributions. We illustrate these claims on several examples, and we use them to provide for the first time results for 2- and 3-loop Yangian-invariant train track integrals in two dimensions for arbitrary external kinematics.