Infinite hierarchies of nonlocal symmetries for the oriented associativity equations

The associativity equations, also known as the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations [1, 2], and the related geometric structures, namely, the Frobenius manifolds [3, 4, 5, 6, 7, 8], have recently attracted considerable attention because of their manifold applications in physics and mathematics. More recently, the oriented associativity equations, a generalization of the WDVV equations, and the related geometric structures, F -manifolds, see e.g. [7, 8, 9, 10, 11], have also become a subject of intense study. These equations have first appeared in [6] (see Proposition 2.3) as the equations for the displacement vector. The oriented associativity equations also describe isoassociative deformations of commutative associative algebras [12], cf. also [13, 14]. The oriented associativity equations (1) admit a gradient reduction (17) which is nothing but the WDVV equations stripped of the so-called quasihomogeneity condition and the condition (18) expressing existence of the unit element in the related associative algebra. Equations (17) naturally arise in differential geometry and in theory of hydrodynamic-type systems and were extensively studied in this context, see e.g. [15, 16, 17, 18, 19, 20, 21, 22, 23] and references therein. There is a considerable body of work on symmetry properties of the WDVV equations, see e.g. [24, 25, 26, 27] for the point symmetries of the WDVV and generalized WDVV equations and [15, 28, 29, 30] and references therein for the higher symmetries and (bi-)Hamiltonian structures for the WDVV equations, as well as equations (17), in three and four independent variables. Although the approach of [15, 28, 29, 30] in principle could [15] be generalized to the WDVV equations in more than four independent variables, this was not done yet. Thus, to the best of our knowledge, higher symmetries of the WDVV equations and of the oriented associativity equations in arbitrary dimensions were never fully explored. The results of Dubrovin (see Lecture 6 of [4]), Mironov and Morozov [24], and, most recently, of Chen, Kontsevich, and Schwarz [26] strongly suggest that one could construct plenty of symmetries for the WDVV equations and, by extension, for the oriented associativity equations, from the eigenfunctions of auxiliary spectral problems for these systems. In the present paper we show that these very eigenfunctions indeed are (infinitesimal) nonlocal symmetries for the oriented associativity equations (1) and the gradient reduction (17) thereof. This is rather unusual per se, as for large classes of spectral problems symmetries turn out to be quadratic [31] rather than linear in the solutions of auxiliary linear problems. Moreover, expanding the eigenfunctions in question into the formal Taylor series with respect to the spectral parameter, we construct infinite hierarchies of nonlocal higher symmetries for (1) and (17) of (1), see Theorem 1 and Corollaries 2–5 below for details.