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BKP and CKP hierarchies via orbifold Saito theory

Alexey Basalaev

TL;DR

The paper constructs BKP and CKP hierarchies from semisimple Dubrovin--Frobenius manifolds using orbifold Saito theory for ADE singularities, establishing that BKP and CKP arise as invariant-sector subhierarchies of Landau--Ginzburg orbifolds. It shows that the invariant sectors of the orbifold DF-manifolds for $(A_{2N-1},\mathbb{Z}/2\mathbb{Z})$ and $(D_N,\mathbb{Z}/2\mathbb{Z})$ reproduce BKP and CKP, respectively, via a precise isomorphism with the corresponding A- and D-type DF-structures and via natural submanifold constraints. The work connects Fay-form and dispersionless limits of KP-type hierarchies to the orbifold Saito construction, and extends to dispersionfull hierarchies through known reductions and theorems that relate dispersionless limits to full BKP/CKP. This provides a globally geometric realization of BKP/CKP from Landau--Ginzburg orbifolds, enriching the toolkit for generating integrable hierarchies from Frobenius-manifold data and offering a new mirror-symmetric perspective on these classic integrable systems.

Abstract

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.

BKP and CKP hierarchies via orbifold Saito theory

TL;DR

The paper constructs BKP and CKP hierarchies from semisimple Dubrovin--Frobenius manifolds using orbifold Saito theory for ADE singularities, establishing that BKP and CKP arise as invariant-sector subhierarchies of Landau--Ginzburg orbifolds. It shows that the invariant sectors of the orbifold DF-manifolds for and reproduce BKP and CKP, respectively, via a precise isomorphism with the corresponding A- and D-type DF-structures and via natural submanifold constraints. The work connects Fay-form and dispersionless limits of KP-type hierarchies to the orbifold Saito construction, and extends to dispersionfull hierarchies through known reductions and theorems that relate dispersionless limits to full BKP/CKP. This provides a globally geometric realization of BKP/CKP from Landau--Ginzburg orbifolds, enriching the toolkit for generating integrable hierarchies from Frobenius-manifold data and offering a new mirror-symmetric perspective on these classic integrable systems.

Abstract

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.
Paper Structure (23 sections, 6 theorems, 48 equations)

This paper contains 23 sections, 6 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Dubrovin--Frobenius manifolds
  4. A and D type Dubrovin--Frobenius manifolds
  5. $A_N$ and $D_N$ Frobenius potentials
  6. AN case
  7. DN case
  8. Dubrovin--Frobenius submanifold
  9. B type Dubrovin--Frobenius manifold
  10. Integrable systems
  11. Stabilizing potentials
  12. Dispersionless system
  13. Dispersionfull systems
  14. Dubrovin--Frobenius submanifolds and subflows
  15. KP, BKP and CKP
  16. ...and 8 more sections

Key Result

Theorem 1.1

BKP hierarchy is the invariant sector hierarchy of the series of Landau--Ginzburg orbifolds $(A_{2N+1},\mathbb{Z}/2\mathbb{Z})$. CKP hierarchy is the invariant sector hierarchy of the series of Landau--Ginzburg orbifolds $(D_{N},\mathbb{Z}/2\mathbb{Z})$.

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 2.1: cf. D96S1ST
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Theorem 4.1: Krichever, Zabrodin
  • Theorem 5.1