Flat F-Manifolds, F-CohFTs, and Integrable Hierarchies

Abstract

We define the double ramification hierarchy associated to an F-cohomological field theory and use this construction to prove that the principal hierarchy of any semisimple (homogeneous) flat F-manifold possesses a (homogeneous) integrable dispersive deformation at all orders in the dispersion parameter. The proof is based on the reconstruction of an F-CohFT starting from a semisimple flat F-manifold and additional data in genus 1, obtained in our previous work. Our construction of these dispersive deformations is quite explicit and we compute several examples. In particular, we provide a complete classification of rank 1 hierarchies of DR type at the order 9 approximation in the dispersion parameter and of homogeneous DR hierarchies associated with all 2-dimensional homogeneous flat F-manifolds at genus 1 approximation.

Integrable systems of finite type from F-cohomological field theories without unit

“…The advantage of the DR hierarchy is that its equations are constructed very explicitly in terms of certain integrals over M g,n . Moreover, the DR hierarchy is defined for objects that are more general than CohFTs: for the so-called F-cohomological field theories (F-CohFTs) [ABLR21,ABLR23].…”

WDVV equations: Hamiltonian operators and symbolic computations

“…Next for the third-order operator we need its metric h i j as well as constants c i j k . We input the metric, denoted by g3, as before and similarly to Christoffel symbols for the previous metric the constants c i j k are handled as a three-dimensional filed: c_hi := Array(1 .. N, 1 .. N, 1 .. N); c_hi [2,3,1]…”

“…-First of all, in the N = 4 case the bi-Hamiltonian property has been proved only for one of the two canonical forms of η, η (1) . This was a result found in [20].…”

“…These equations and the associated geometric structures are called F-manifolds with compatible flat structures [16], or simply flat F-manifolds. They share many properties with WDVV equations and Dubrovin-Frobenius manifolds including the existence of an associated integrable dispersive hierarchy (see [1] for details). We observe, in particular, that the oriented associativity equation has an infinite hierarchy of nonlocal symmetries [22], a first-order local Hamiltonian operator of the same type as A 1 [19] and a third-order nonlocal Hamiltonian operator which is the straightforward generalization of A 2 [3,21].…”

Integrable hierarchies, Frölicher–Nijenhuis bicomplexes and Lauricella bi-flat F-manifolds

“…(1) The construction of integrable hierarchies of hydrodynamic type starting from a (1, 1)tensor field L with vanishing Nijenhuis torsion does not require that L is diagonalizable. (2) In the theory of bi-flat F-manifolds there is a natural tensor field with vanishing Nijenhuis torsion: it is the operator of multiplication by the Euler vector field. Moreover in the regular case there are special coordinates found by David and Hertling in [9] where the unit vector field, the Euler vector field and the product have a canonical form.…”