Geometry and arithmetic of integrable hierarchies of KdV type. I. Integrality

For each of the simple Lie algebras $\mathfrak{g}=A_l$, $D_l$ or $E_6$, we show that the all-genera one-point FJRW invariants of $\mathfrak{g}$-type, after multiplication by suitable products of Pochhammer symbols, are the coefficients of an algebraic generating function and hence are integral. Moreover, we find that the all-genera invariants themselves coincide with the coefficients of the unique calibration of the Frobenius manifold of $\mathfrak{g}$-type evaluated at a special point. For the $A_4$ (5-spin) case we also find two other normalizations of the sequence that are again integral and of at most exponential growth, and hence conjecturally are the Taylor coefficients of some period functions.