Massey products, A_∞-algebras, differential equations, and Chekanov homology

Abstract

We consider (classical and generalized) Massey products on the Chekanov homology of a Legendrian knot, and we prove that they are invariant under Legendrian isotopies. We also construct a minimal A ∞ -algebra structure on the Chekanov algebra of a Legendrian knot, we prove that this structure is invariant under Legendrian isotopy, and we observe that its higher multiplications allow us to find representatives for classical Massey products. Finally, we consider differential equations: we remark that the Massey product Legendrian invariants admit a "dynamical interpretation", in the sense that they provide solutions for a Maurer-Cartan equation posed on an infinite-dimensional bigraded Lie algebra, and we show how to set up and solve a (twisted) KadomtsevPetviashvili hierarchy of equations starting from the Chekanov algebra of a Legendrian knot.

On $(t_2,t_3)-$Zakharov-Shabat equations of generalized Kadomtsev-Petviashvili hierarchies

“…The classical KP-II equation on C ∞ (S 1 , R) can be deduced from it by an adequate substitution procedure, as explained in [1]. In order to work in generalized settings, we stop our computations at system ( 10)- (11), because the substitution procedure just mentioned may not be available in them.…”

“…in which we have used that the first equation implies that ∂u −1,t2 = 0. Now we set 2u −1 = u, t 2 = y and t 3 = t. Equation ( 10) yields (13) u = 4∂ −1 y ∂u −2 , and therefore equation (11) becomes…”

“…In these two references we can already see that the equations deduced from the (t 2 , t 3 )−ZS equations can be deeply different from the classical KP equation, see also Section 3.3 below. Other examples of "non-standard" KP hierarchies can be found in [20,9,11]; we also mention [10] for non-standard commutative examples, and [4,21,19] as representative works on a KP hierarchy posed on algebras equipped with a Moyal multiplication. Now, the construction of general solutions to the KP system for differential algebras A has been systematically studied in [2,10]: in these references, the issues of existence and uniqueness of solutions, and well-posedness of the corresponding Cauchy problem, are formulated and stated in frameworks as general as possible.…”

On the Cauchy problem for a Kadomtsev-Petviashvili hierarchy on non-formal operators and its relation with a group of diffeomorphisms

“…In Mulase's words, "the KP system is the master equation for the largest possible family of iso-spectral deformations of arbitrary ordinary differential operators", see [37,Section 3] and [38]; see also [8,Corollary 6.2.8] for another expression of this universality. Solutions to KP can be recovered from quantum field theory and algebraic geometry among other fields, see for instance [20,35,37] and references therein, and (1.1) can be posed for instance in contact geometry, see [34].…”