On the monodromy of the deformed cubic oscillator

Abstract

We study a second-order linear differential equation known as the deformed cubic oscillator, whose isomonodromic deformations are controlled by the first Painlevé equation. We use the generalised monodromy map for this equation to give solutions to the Riemann-Hilbert problems of (Bridgeland in Invent Math 216(1):69–124, 2019) arising from the Donaldson-Thomas theory of the A$$_2$$ 2 quiver. These are the first known solutions to such problems beyond the uncoupled case. The appendix by Davide Masoero contains a WKB analysis of the asymptotics of the monodromy map.

Joyce structures on spaces of quadratic differentials I

“…Thus the question as to whether the connection ∇ J is well-defined becomes an interesting one. Note that in the one example that has been computed in detail [18], the connection ∇ J is indeed well-defined, and turns out to be quite natural.…”

“…Turning to the equation (62), note that the first term on the right-hand-side is the invariant vector field U = i z i • ∂/∂θ i on T # M,p corresponding to the value E p ∈ T M,p of the vector field (18). Exponentiating yields well-defined automorphisms exp(U/ǫ) of T # M,p given in coordinates by θ i → θ i + ǫ −1 z i .…”

“…The asymptotic property (67) should then follow from results in exact WKB analysis [39,51]. The special case g = 0, m = {7} was treated in full detail in [18].…”

“…Partial solutions to the non-linear RH problems relevant to categories of class S[A 1 ] have been obtained by Allegretti using pencils of opers on Riemann surfaces [8,9]. A complete family of solutions has been given in the particular case g = 0, m = {7} by considering opers with apparent singularities [18]. Similar pencils of opers appear in the works [22,23] under the name quantum curves.…”

“…Secondly, we are working with the conformal limit [30]; this leads to simpler geometry, which in particular cases can even be written down explicitly [18]. The relationship between the two approaches has recently been clarified [4], and there are detailed studies from both points-ofview in the case of the resolved conifold [5,6,7,16].…”