The paper under review introduces a supersymmetric generalization of quantum Airy structures, discusses some of the expected properties for the generalization, and present classes of examples.
A quantum Airy structure was introduced by Kontsevich and Soibelman’s paper “Airy structures and symplectic geometry of topological recursion” as a generalization of topological recursion. This was further generalized by Borot-Bouchard-Chidambaram-Creutzig-Noshchenko’s “Higher Airy structures, W algebras and topological recursion”, which we still call a quantum Airy structure.
Let us briefly review the definition of a quantum Airy structure. Let \(V\) be a vector space over \(\mathbb{C}\). Let us discuss the theory as if \(V\) is finite-dimensional; otherwise, one has to be a bit more careful. Let \((e_i)_{i\in I}\) be a basis of \(V\) and \((x_i)_{i\in I}\) be the corresponding basis of linear coordinates of \(V\). Consider the Weyl algebra \(\mathcal{ W}_V^\hbar = \mathbb{C} \left[(x_i) _{i\in I}, (\hbar\partial_{x_i})_{i\in I} \right] [[\hbar ]]\). We set \(\deg \hbar=2\) and \(\deg x_i=1\). In particular, \(\deg \hbar\partial_{x_i}=1\). Then a quantum Airy structure (QAS) is a linear map \(L\colon V\to \mathcal W_V^\hbar \) satisfying

1.

(degree 1 condition) \(L(e_i)=L_i=\hbar \partial_{x_i} + (\text{degree }\geq 2 \text{ terms})\)

2.

(graded Lie ideal condition) \([L_i,L_j] \subset \hbar \mathcal W_V^{\hbar }\cdot \text{span}_{a\in I}(L_a) \)

The original definition of Kontsevich and Soibelman is a particularly nice case where the maximal degree of \(L_i\) is 2, that is, quadratic.
A crucial result in the subject is that for a given quantum Airy structure, there exists a unique partition function \[ F =\sum_{g,n} \sum_{i_1,\cdots,i_n} \frac{\hbar^{g-1} }{n!} F_{g,n}[i_1,\cdots,i_n] x_{i_1}\cdots x_{i_n} \in \hbar^{-1} \text{Sym}^\bullet(V^*)[[ \hbar]], \] such that

1.

for all \(i\), one has \(L_ie^F=0\), and

2.

\(F_{g,0}=0\) for all \(g\) and \(F_{0,1}=F_{0,2}=0\).

In the process of proving the existence and uniqueness, one gets the sense that the partition function should somehow be related to moduli spaces of Riemann surfaces, although precisely identifying how is not an easy question to answer. For instance, one could imagine that these are related to intersection theory of \(\overline{\mathcal{M}}_{g,n}\) or tau functions of integrable hierarchies.
The current paper generalizes a quantum Airy structure by taking a super (or \(\mathbb{Z}/2\)-graded) vector space \(V\) as the initial starting point. Then the authors prove the main result which is the existence and uniqueness of a partition function.
After restricting to the quadratic case, they provide several classes of examples. These are mainly applications/generalizations of the constructions in Anderon-Borot-Chekhov-Orantin “The ABCD of topological recursion” for (super) Lie algebras and (super) Frobenius algebras and Borot-Bouchard-Chidambaram-Creutzig-Noshchenko’s “Higher Airy structures, W algebras and topological recursion” for vertex operator (super) algebras.
The theory of super quantum Airy structures opens up a few exciting research directions: for a partial list of topics, the readers are advised to take a look at the last section where the authors discuss them.