Open $r$-spin theory in genus one, and the Gelfand-Dikii wave function

We construct the $g=1$ sector of the open $r$-spin theory, that is, an open $r$-spin theory on the moduli space of cylinders. This is the second construction of a $g>0$ open intersection theory, which includes descendents (the first is the all genus construction of the intersection theory on moduli of open Riemann surfaces with boundaries [23,30], whose $g=1$ case equals to the $r=2$ case of our construction). Unlike the construction of [30], in order to construct the $r$-spin cylinder theory we had to overcome the foundational problem of dimension jump loci, which in analogous closed theories has been treated using virtual fundamental class techniques, that are currently absent in the open setting. For this reason our construction is much more involved, and relies on the point insertion technique developed in [31,32]. We prove that the open $g=1$ potential equals, after a coordinate change, to the $g=1$ part of the Gelfand-Dikii wave function, thus confirming a conjecture of [7]. We also prove that our $g=1$ intersection numbers satisfy a $g=1$ recursion, also predicted in [7,15]. This recursion is the $g=1$ analogue of Solomon's famous $g=0$ Open WDVV equation [25], with descendents, and is also the universal $g=1$ recursion for $F$-Cohomological field theories [1]. Again, this is first geometric construction which is not the $g=1$ sector of [23,30], proven to satisfy this universal recursion.