BKP and CKP hierarchies via orbifold Saito theory

Semisimple Dubrovin-Frobenius manifolds can be used to construct integrable hierarchies, following the work of Dubrovin-Zhang and Buryak. Examples of such hierarchies include the Kac-Wakimoto hierarchies, the KP hierarchy, among others. In all these examples, the Saito theory of isolated singularities played a crucial role. In this note, we show that the BKP and CKP hierarchies can likewise be constructed from Dubrovin-Frobenius manifolds. This new construction, however, utilizes the orbifold version of Saito theory for isolated singularities endowed with a symmetry group.

💡 Research Summary

The paper “BKP and CKP hierarchies via orbifold Saito theory” by Alexey Basalaev presents a novel construction of the BKP and CKP integrable hierarchies using the framework of semisimple Dubrovin‑Frobenius manifolds, but with a crucial twist: the orbifold version of Saito’s theory for isolated singularities equipped with a symmetry group.

The introduction reviews the well‑established link between cohomological field theories and integrable systems, recalling Witten’s conjecture, its proofs, and the subsequent development by Dubrovin‑Zhang and Buryak that associates an integrable hierarchy to any semisimple Dubrovin‑Frobenius manifold. It notes that for ADE‑type singularities the associated Dubrovin‑Frobenius manifolds reproduce the Kac‑Wakimoto hierarchies (including the KP hierarchy for the A‑series). The natural question posed is whether similar constructions exist for the BKP and CKP hierarchies, which are obtained from KP by imposing a single additional Lax constraint.

Section 2 introduces the orbifold Saito theory. For the A‑type singularity (f_A = x^{N+1}/(N+1) + y^2) and the D‑type singularity (f_D = x^{N-1}/(N-1) + xy^2), the author considers the pairs ((f, G)) where (G = \mathbb{Z}/2\mathbb{Z}) acts by sign changes on the unfolding parameters. These pairs are called Landau‑Ginzburg orbifolds. The associated G‑graded Dubrovin‑Frobenius manifold has an invariant sector obtained by restricting to the G‑invariant part of the Saito residue pairing and product. The paper shows that this invariant sector is a natural Dubrovin‑Frobenius submanifold: the product restricts without mixing with the orthogonal complement, and the Euler vector field and metric remain non‑degenerate.

The author then writes down explicit potentials (F_A) and (F_D) in flat coordinates using the combinatorial formulas of Noumi‑Yamada (functions (\psi^{(1)}) and (\psi^{(2)})). The potentials satisfy the quasi‑homogeneity condition (E\cdot F = (3-\delta)F) with appropriate Euler fields. By setting all even‑indexed flat coordinates to zero in the A‑type case, one obtains the B‑type Frobenius manifold (M_{B_N}); similarly, setting the last coordinate to zero in the D‑type case yields the same B‑type manifold. These embeddings are shown to be natural submanifolds via a (\mathbb{Z}/2\mathbb{Z}) symmetry argument, which is precisely the orbifold action.

Section 3 discusses integrable systems derived from stabilizing series of potentials. A series ({F_N}) is called stabilizing if second derivatives with respect to low‑index variables become independent of (N) once the index sum is far below (N). The author recalls that the A‑type series stabilizes to give the dispersionless KP hierarchy, while the D‑type series stabilizes to the dispersionless limit of a one‑component reduced two‑component BKP hierarchy. The dispersionless equations are written as \