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Rational reductions of the 2D-Toda hierarchy and mirror symmetry. (English) Zbl 1368.81139

The 2D Toda hierarchy is a classical example of an integrable non-linear dynamical system that provides a unifying framework for a variety of problems in enumerative geometry, combinatorics of matrix integrals, physics of melting crystals, etc. The paper constructs an infinite family of its rational symmetry reductions, dubbed RR2T, and gives some non-trivial applications to mirror symmetry. Each RR2T is characterized by the bidegree \((a,b)\), with \(a,b>0\), corresponding to a factorization of 2D Toda Lax operators as \(L_1^a=AB^{-1}\), \(L_2^b=BA^{-1}\), where \(A\) and \(B\) are the upper and lower diagonal difference operators. Previously known special cases of RR2T include: a relativistic generalization of 1D Toda hierarchy studied by Gibbons and Kuperschmidt of bidegree \((a,1)\), the Ablowitz-Ladik hierarchy, of bidegree \((1,1)\), the Frenkel’s \(q\)-deformed Gelfand-Dickey hierarchy, and a lattice analog of the KdV hierarchy for the degenerate limit \(b=0\). The reductions are purely kinematic, the submanifold where the Lax operators factorize comes with an infinite dimensional degeneration of the (second) Poisson tensor of the parent hierarchy, and is independent of the form of particular Hamiltonians.
The semi-classical Lax-Sato formalism for the dispersionless Takasaki-Takebe limit gives rise to old and new solutions to the WDVV equations as families of semi-simple, Frobenius dual-type structures on a genus zero double Hurwitz space with flat identity. This entails Dubrovin-Novikov bi-Hamiltonian structures in some special cases, and even a tri-Hamiltonian one for \(a=b\). And for all \(a,b\) it furnishes a 1D B-model description in terms of logarithmic Landau-Ginzburg models. The Frobenius dual-type structures have Frobenius manifolds isomorphic to the \((\mathbb{C}^*)^2\) orbifold cohomology of local \(\mathbb{P}^1\) orbifolds with two stacky points of orders \(a\) and \(b\). Equivalently, one has an isomorphism to the equivariant cohomology of their minimal toric resolutions (toric trees).
This provides a novel mirror theorem for these targets, proved in the paper, with applications to wall-crossings in the Gromov-Witten theory. Moreover, the authors conjecture that the full descendent Gromov-Witten potential is the \(\tau\) function of RR2T, which they verify in genus zero and one by establishing a Miura equivalence between the dispersive expansion of the RR2T to the second order, and a Dubrovin-Zhang type formalism applied to the local theory of orbifold \(\mathbb{P}^1\). Flat sections of the A-model’s Dubrovin connection are proved to be the multivariate hypergeometric functions of type \(F_D\).

MSC:

81R12 Groups and algebras in quantum theory and relations with integrable systems
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
14H70 Relationships between algebraic curves and integrable systems
14J33 Mirror symmetry (algebro-geometric aspects)
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
17B80 Applications of Lie algebras and superalgebras to integrable systems
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