Loop Equations from Differential Systems on Curves

WKB solutions of difference equations and reconstruction by the topological recursion

“…The proof is straightforward from (3-7) and (3)(4)(5)(6)(7)(8). We first observe that M 0 (x) has indeed the correct behavior and that the claim is also correct for M 1 (x) that is given by:…”

“…Note in particular that the setting (1-8) may be seen as the exponentiated version of a standard linear differential system and therefore results presented in this article are likely to extend to general connections on some Lie groups G (in the case of P 1 , the Lie group is G = SL 2 (C)). Therefore, the article may also be seen as a first step to generalize results of [8] to problems directly defined on the Lie group rather than on the associated Lie algebra. The main steps and results of the article are:…”

“…Projecting (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) at order k provides the following equation:…”

“…Using the two compatible systems (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16), we obtain that it must satisfy the equations: The first equation is standard in the theory of differential systems while the second one is just the exponentiated form of the first one. Technically, since L(x; ) and D(x; ) can be reconstructed from one another, the former two equations are completely equivalent.…”

“…Then by induction, if we assume the proposition to be correct for all j ≤ k, we may use (3)(4)(5)(6)(7)(8) to prove the right behavior of M k+1 (x). Indeed, we first observe that M 0 is only involved through its derivatives, hence the factor diag(1, 0) will not play any role.…”