Geometry and arithmetic of integrable hierarchies of KdV type. I. Integrality
Abstract
For each of the simple Lie algebras $\mathfrak{g}=A_l$, $D_l$ or $E_6$, we show that the allgenera onepoint 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 allgenera 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 $A_4$ (5spin) 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2021
 arXiv:
 arXiv:2101.10924
 Bibcode:
 2021arXiv210110924D
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematical Physics;
 Mathematics  Differential Geometry;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 v2: added a note and several footnotes specifying the role of the deceased first author in this collaborative research, corrected typos, and added or updated a few references. 56 pages