Airy structures for semisimple Lie algebras

We give a complete classification of Airy structures for finite-dimensional simple Lie algebras over $\mathbb C$, and to some extent also over $\mathbb R$, up to isomorphisms and gauge transformations. The result is that the only algebras of this type which admit any Airy structures are $\mathfrak{sl}_2$, $\mathfrak{sp}_4$ and $\mathfrak{sp}_{10}$. Among these, each admits exactly two non-equivalent Airy structures. Our methods apply directly also to semisimple Lie algebras. In this case it turns out that the number of non-equivalent Airy structures is countably infinite. We have derived a number of interesting properties of these Airy structures and constructed many examples. Techniques used to derive our results may be described, broadly speaking, as an application of representation theory in semiclassical analysis.


Introduction
Quantum Airy structure is a set of differential operators of the form 1 spanning a Lie algebra g with structure constants f k ij : Airy structures were introduced in [1] as a reformulation and generalization of a system of recursive equations, referred to as the Chekhov-Eynard-Orantin topological recursion [2,3,4]. Formulated originally in the language of matrix motels, the CEO topological recursion can be rephrased more abstractly as a procedure which assigns invariants to spectral curves, i.e. Riemann surfaces equipped with certain additional geometric structre [3,4]. This turned out to be useful in the study of Hurwitz numbers [5,6,7,8,9], computation of Gromov-Witten invariants [10], in knot theory [11,12], integrable systems [13,14] and topological quantum field theories [15]. Furthermore it is connected with the subject of quantum curves [16,17,18].
It is thus conceivable that results concerning Airy structures (and their supersymmetric generalizations [19], related to supereigenvalue models and the corresponding topological recursion [20,21,22,23,24]) may find applications in some of the subjects listed above.
Every quantum Airy structure admits [1,25] a unique "free energy" F , which is a series in and x i satisfying differential equations and initial conditions F (0, ) = ∂ i F (0, 0) = ∂ i ∂ j F (0, 0) = 0. Thus the corresponding partition function Z = e −1 F may be viewed as WKB wave function 2 of a quantum system whose symmetry is generated by hamiltonians L i .
One important question about any class of mathematical objects is the classification problem, which asks for a complete list of all (up to suitably defined equivalence) objects satisfying the pertinent axioms. It is unlikely that such list of all Airy structures could ever be obtained. However, one may still hope to classify some special classes of Airy structures. In [26] study of this problem was initiated for Airy structures for which the Lie algebra g is finite-dimensional and simple. Somewhat surprisingly, only one example was found. This suggests that assumption of simplicity imposes very strong constraints. Indeed, in this work we provide a solution to this classification problem. It turns out that there exist precisely six inequivalent Airy structures, two for each of sl 2 , sp 4 and sp 10 . We construct these Airy structures explicitly. More detailed summary is given in the Subsection 5.4. Methods developed in order to obtain these results are of independent interest, because they apply also to the case of semisimple g. In this more general case classification program is not finished, but significant progress has been made in this direction.
The main ideas and methods applied to constrain and construct Airy structures can be summarized as follows. Since for the class of Airy structures under consideration there exists an unambiguous procedure of quantization, no information is lost by working with the classical hamiltonians instead of directly with quantum operators (1.1). As explained in [1], any such classical Airy structure may be obtained by expressing the moment map of a hamiltonian action of g on some affine space of dimension 2 dim g in standard coordinates centered at a regular point of −1 (0). For semisimple Lie algebras all affine representations are actually linear and completely classified. The part which is not known, to the best of our knowledge, is for which representations the locus −1 (0), also called the characteristic variety, has any regular points. To answer this question we use the fact that the set of regular points of −1 (0) is a cone with a locally transitive action of a complex Lie group with Lie algebra g. This implies that for any of its points Ω one may find a unique element J of the algebra g which is tangent to the ray of Ω. It is possible to describe many properties of J, including its spectrum. Having obtained that, we proceed to the classification. Instead of looking directly for Ω, we find all possible forms of J. Once some admissible J is found, element Ω is obtained by solving the eigenvalue equation JΩ = Ω.
The paper is organized as follows. In Section 2 we recall some basic facts about Airy structures. It contains no new results, but it introduces the language used afterwards. In Section 3 we discuss properties of the characteristic variety and of elements Ω and J. These are the main conceptual ingredients of the classification program. Section 4 concerns automorphisms and real forms of Airy structures. In Section 5 we perform explicit calculations, which culminate in the promised list of Airy structures for simple Lie algebras. Examples of application of our formalism to semisimple Lie algebras are presented in Section 6. We summarize by mentioning possible future directions in Section 7. For convenience of the reader we collect some background material in appendices. Appenix A introduces in an elementary way Lie algebra cohomology groups, which are used throughout the text. In the Appendix B we recall the notions of semisimple and regular elements of a semisimple Lie algebra. Appendix C contains a brief discussion of invariant polynomials on semisimple Lie algebras. We find relations between invariant polynomials of various types for the Lie algebra sp 10 , which is used to find the element J in this case.

Preliminaries
In general it is necessary to impose additional finitness conditions on the tensors A, B, C, D and f appearing in (1.1). These are automatically satisfied if dim g is finite, which we assume from now on. We work over the field C, but our results are relevant also for Airy structures over R. Indeed, every real Airy structure admits a natural complexification. Furthermore, we discuss the concept of real forms of Airy structures in Section 4.
The classical limit of a quantum Airy structure is the set of hamiltonians with respect to the Poisson bracket defined by Classical Airy structure may be defined as a set of hamiltonians of the form (2.1) subject to relations (2.2). Every classical Airy structure may be quantized by putting D i = 1 2 B j ij +δ i with any δ satisfying f k ij δ k = 0. Thus the set of quantizations of a given classical Airy structure may be identified with the vector space H 0 (g, g * ). Choice δ = 0 corresponds to Weyl quantization. This description reduces the classification of Airy structures for a given Lie algebra to the study of their classical versions.
Classical Airy structures have a transparent geometric interpretation. Consider the common zero locus of hamiltonians i , Σ = {(x, y) ∈ C n × C n | 1 (x, y) = ... = n (x, y) = 0} (2.4) and its Zariski open subset Postulated form of i implies that the origin belongs to Σ s . Conversely, given a set of at most quadratic hamiltonians on C 2n satisfying (2.2), define Σ and Σ s as above.
Then Σ s is a Lagrangian submanifold. For any Ω ∈ Σ s one can choose a symplectic affine coordinate chart centered at Ω in which i take the form (2.1). Coordinate systems with desired properties are in one-to-one correspondence with Lagrangian complements of T Ω Σ s in C 2n . Hamiltonians corresponding to different complements are related by a change of coordinates where s is a symmetric matrix. Maps of this form are called gauge transformations. There exists an analogous notion for quantum Airy structures [1]. Transformation law for the associated partition functions is also known [26,19]. One should not fall under the impression that choice of the Lagrangian complement is completely irrelevant: some choices lead to much simpler partition functions, and transforming the partition function to other gauges is not trivial. This description allows one to define classical Airy structures in a way not referring to coordinates. We introduce also the concept of an Airy data, which can be thought of as equivalence classes of Airy structures up to gauge transformations.
Definition 1. Classical Airy structure is a quadruple (g, W, Ω, V ), where 1. g is a Lie algebra of dimension n.

2.
W is an affine space of dimension 2n equipped with a translation invariant symplectic form ω and a g-action ξ : g → Γ(T W ) on W which is hamiltonian with at most quadratic moment map : W → g * . 3. Ω is an element of Σ s = {q ∈ −1 (0)| d | q : T q W → g * has rank n}.
Triple (g, W, Ω) satisfying points 1 − 3 of the above list is called an Airy datum. We say that Airy datum is nontrivial if n > 0.
We will frequently use the following characterization of the set Σ s . Proposition 1. Let (g, W ) be as in the Definition 1. Suppose that g = [g, g]. Then Proof. ⊆ : Follows from the preceding discussion (for any g). Definition 2. Homomorphism of Airy structures (g, W, Ω, V ) → (g , W , Ω , V ) is a pair (φ, f ), where φ : g → g is a homomorphism of Lie algebras and f is an affine Poisson map W → W , subject to the following conditions: If (φ, f ) satisfies only conditions 1 and 2, we say that it is a homomorphism of Airy data. In other words, the following diagram is required to be commutative: is a homomorphism of Airy data, then φ and f are surjective. Moreover we have where ξ is the g -action on W . In particular φ is uniquely determined by f .
Proof. f is an affine Poisson map, so it is surjective. In particular dim(g) ≥ dim(g ). Evaluating the differential of at Ω we get Since the rank of d | Ω is equal to dim(g), the rank of φ t is at least dim(g). Now choose T ∈ g, q ∈ W and a holomorphic function g on W . Equation = φ t • • f implies that we have ξ(T )(f * g ) = f * (ξ (φ(T ))(g)), or in other words Since g was arbitrary, formula (2.10) follows. The last statement is a consequence of the formula (2.10) and the fact that the g -action ξ is faithful.
In view of the Proposition 2, one could abuse notation and refer to f itself as a morphism (g, W, Ω) → (g , W , Ω ). We choose not to do so, because it is unclear how to rephrase point 1 in the Definition 2 without referring to φ.
Let (g, W, Ω) be an Airy datum and let G be a simply-connected Lie group with Lie algebra g. The g-action on W exponentiates to an affine action of G, which preserves −1 (0) and Σ s . By the Jacobian criterion, Σ s is a nonsingular subvariety of W of dimension n. In particular it is a complex manifold with finitely many connected components. The G-orbits in Σ s are open in Σ s , so they coincide with the connected components. For any q ∈ Σ s , let Stab(q) = {g ∈ G|g · q = q} and Orb(q) = {g · q ∈ Σ s |g ∈ G}. Mapping G g → g · q ∈ Orb(q) is a universal cover, with fiber π 1 (Orb(q)) ∼ = Stab(q). This means that to get a rather complete picture of the topology of Σ s , it is sufficient to find the connected components and compute the corresponding stabilizers. This task is relevant for the classification program for several reasons. If (g, W, Ω ) is another Airy datum with Ω ∈ Orb(Ω), there exists g ∈ G such that g · Ω = Ω . Then (Ad g , g) is an isomorphism (g, W, Ω) → (g, W, Ω ). Secondly, group Stab(Ω) is an invariant of Airy data. It allows to distinguish non-isomorphic Airy data with isomorphic g and W . Finally, Airy structure and its partition function may be regarded as a quantization of Orb(Ω). Thus it is desirable to know what these spaces look like.
We finish this section with an elementary discussion of products of Airy data. We remark that Airy datum constructed in the Definition 3 is indeed a product (with obvious projection maps) in the category of Airy data. There is also an analogous notion for classical and quantum Airy structures, but we shall not use it.
Definition 3. Given two Airy data (g i , W i , Ω i ), i = 1, 2 we define the product with the moment map on W 1 ×W 2 given by (q 1 , q 2 )(T 1 , Definition 4. Airy datum is said to be indecomposable if it is nontrivial and not isomorphic to a product of two nontrivial Airy data.
Proposition 3. Every Airy datum is isomorphic to a product of finitely many indecomposable Airy data.
Proof. By induction on dimension.
3 Semisimple Lie algebras -general facts Definition 5. Airy datum (g, W, Ω) is said to be homogeneous if W is a linear representation of g with purely quadratic moment map. Proof. For any λ ∈ C × we have λΩ ∈ Σ s . In particular λΩ belongs to the path component of Σ s containing Ω. Since this set coincides with the orbit of Ω, we have Recall that the null cone N (W ) of a representation W of a group G is defined as the common zero locus of all G-invariant polynomials on W homogeneous of positive degree. It is a basic object of interest in the classical invariant theory. Proposition 4 implies that for any homogeneous Airy datum (g, W, Ω) we have an inclusion Σ s ⊆ N (W ). There exist classes of representations for which the structure of the null cone is well understood. This makes Proposition 4 useful in constraining homogeneous Airy data. We will now show that assumption of homogeneity leads to no loss of generality in the case of semisimple Lie algebras.
Proposition 5. Every Airy datum (g, W, Ω) with semisimple g is isomorphic to a homogeneous Airy datum.
Proof. This fact was established in [26]. For completness we give two other proofs.
Choose a basis in g and a symplectic affine coordinate system in W centered at Ω such that the moment map is represented by polynomials of the form (2.1). Decompose i = y i + Q i . Then Q i are homogeneous of degree two and satisfy {Q i , Q j } = f k ij Q k , so they furnish a linear representation of g on the linear span of x i and y i . Furthermore relations (2.2) imply the cocycle condition By the Whitehead's lemma (discussed in the appendix A), there exist coefficients a j , b j such that Now consider the affine automorphism of W given by Generators i are mapped to Q i + i , where i are some constants. Relations (2.2) then imply that f k ij k = 0, so = 0. We have found a new affine coordinate chart in which i are purely quadratic, so the generated G-action is linear.
One may also avoid the use of Lie algebra cohomology 3 . Instead we choose a maximal compact subgroup K ⊆ G. Using averaging techniques we may find a fixed point of the action of K on W . This fixed point is then also a fixed point for the action of whole G, by holomorphicity of the G-action. Expressing the moment map in coordinates centered at the fixed point we get vanishing linear term. Then commutation relations imply that the constant term also vanishes.
From now on, we restrict attention to homogeneous Airy data (g, W, Ω) whenever g is semisimple. Our next step is to briefly review facts about symplectic representations of semisimple Lie algebras essential for further discussion.
Fix a semisimple Lie algebra g. Choose a Cartan subalgebra h ⊆ g and a set of positive roots ∆ + . We let Λ ⊆ h * be the lattice of integral weights, C ⊆ Λ ⊗ Z R ⊆ h * the (closed) fundamental Weyl chamber and Λ + = Λ∩C the set of dominant integral weights. We shall also consider the dual lattice Λ * = {H ∈ h|∀µ ∈ Λ µ(H) ∈ Z} ⊆ h. In other words, Λ * consists of these elements of h which have integral eigenvalues in all finite-dimensional representation of g. For any λ ∈ Λ + , denote the highest weight module with highest weight λ by V λ . Each λ ∈ Λ + has one of the following mutually exclusive properties: Any finite-dimensional representation W of g decomposes as W = λ∈Λ V ⊕m λ λ , with m λ ∈ N vanishing for all but finitely many λ. It follows from the Schur's lemma that W admits an invariant symplectic form if and only if m λ is even for V λ of real type and m λ = m λ * for V λ of complex type. In this situation the symplectic structure on W is unique up to a g-module isomorphism. The g-action is automatically hamiltonian, with the moment map uniquely determined as (q)(T ) = 1 2 ω(T q, q) for q ∈ W and T ∈ g. Vector space W decomposes as a direct sum of its weight spaces, W = ⊕ µ∈Λ W µ with Proposition 6. If g is a semisimple Lie algebra, then there are finitely many isomorphism classes of Airy data of the form (g, W, Ω).
Proof. The number of isomorphism classes of W is finite, symplectic form ω is unique up to isomorphism and the moment map is uniquely determined by W and ω. Once g, W , ω and are fixed, space Σ s has finitely many connected components.
We shall say that a symplectic g-module W is admissible if there exists an Airy datum of the form (g, W, Ω). This is true if and only if the corresponding set Σ s is nonempty. It turns out that many symplectic g-modules of dimension 2 dim g are not admissible. To rule them out, we will need to better understand properties of the element Ω. The first steps in this direction are the following statements: Proposition 7. Let (g, W, Ω) be an Airy datum with g semisimple. Then 2. Suppose that the contrary is true. We write Ω = (Ω 1 , Ω 2 ) ∈ g⊕g. Let p : g×g → g be a Lie polynomial 4 and k > 0 a natural number and consider the function φ is continuous, g-invariant and φ(0) = 0. Therefore φ(Ω) = 0 by Proposition 4. Since p and k were arbitrary, we conclude that for any element T of the Lie subalgebra n ⊆ g generated by Ω 1 , Ω 2 , traces of all powers of ad T vanish. Thus ad T is a nilpotent endomorphism of g, and hence also of the invariant subspace n ⊆ g. Since T ∈ n was arbitrary, n is a nilpotent Lie algebra. In particular its center Z(n) is nontrivial.
Proposition 8. Let (g, W, Ω) be a homogeneous Airy datum. There exists a unique J ∈ g such that JΩ = Ω. If g is semisimple, J is a semisimple element of g.
This proves the existence of J. If J ∈ g satisfies J Ω, then J = J (since the annihilator of Ω in g is trivial). Now assume that g is semisimple and let J = J ss + J n be the Jordan-Chevalley decomposition of J. Then J ss Ω = Ω and J n Ω = 0, so J ss = J.
Recall [29] that every semisimple element of a semisimple Lie algebra g belongs to some (not necessarily unique) Cartan subalgebra of g. Moreover action of the group of inner automorphisms of g on the set of Cartan subalgebras is transitive. Therefore we may fix a Cartan subalgebra h ⊆ g. Every Airy datum (g, W, Ω) is isomorphic to one such that J ∈ h. From now on, we restrict attention to Airy data of this form. The next step is to further constrain the element J. 4. J is rational, in the sense that J ∈ Λ * ⊗ Z Q.
Proof. 1. Obvious. 2. Suppose otherwise. Then there exists a nonzero H ∈ h such that µ(H) = 0 for each µ ∈ Ξ, so HΩ = 0. Contradiction. 3. By construction, J determines H and Ξ. J is the unique element T ∈ h such that µ(T ) = 1 for every µ ∈ H. Since every basis of h is contained in a unique hyperplane, one may reconstruct H from Ξ as the unique hyperplane containing Ξ. 4. Let Ξ = {µ 1 , ..., µ m }. J is uniquely determined by the affine system of equations µ i (J) = 1. Since µ i belong to Λ ⊆ Λ ⊗ Z Q = (Λ * ⊗ Z Q) * , this system has a solution in Λ * ⊗ Z Q ⊆ h. Since solution of this system considered in h is unique, J ∈ Λ * ⊗ Z Q. 5. By the previous point, J must belong to the dual cone of some Weyl chamber. Since the Weyl group acts transitively on the set of Weyl chambers, we may assume that J lies in the dual cone of the fundamental Weyl chamber. 6. Assume otherwise. Then g α annihilates Ω. Contradiction.
It is natural to ask if point 4 of the above Proposition can be strengthened, i.e. if J belongs to the lattice Λ * . One of the examples constructed in the Subsection 5.1 shows that this is not necessarily true even if g is simple. This leads to the concept of the denominator of J, which is defined as the smallest positive integer denom(J) such that denom(J) · J ∈ Λ * . Similarly for the point 5, one can ask if condition α(J) ≥ 0 can be replaced by a strict inequality. This happens to be true for all simple Lie algebras, but there exist Airy structures for semisimple Lie algebras for which J is orthogonal to some root of g, i.e. such that J is not a regular element of g. For the benefit of the reader we recall the definition and properties of regular elements of a semisimple Lie algebra in the Appendix B.
For fixed g and W the number of weights of W is finite, so points 2 and 3 of Proposition 9 determine J up to a finite ambiguity. This ambiguity is reduced by imposing the additional condition α(J) ≥ 0 for every root α. Many of the remaining candidates for J may be excluded by the following fact. Proposition 10. Let (g, W, Ω) be an Airy datum with g semisimple and let λ 1 , ..., λ n be the eigenvalues of ad J . Then 1. Each λ i is a rational number.
3. Spectrum of J acting in W takes the form Proof. 1. Special case of 4. in Proposition 9. 2. Follows from the fact that J is an infinitesimal symplectomorphism.
We proceed inductively. First notice that ker(J − 1 − λ 1 ) + ker(J + 1 + λ 1 ) is a symplectic subspace of W , so we can find f 1 with ω(e i , f 1 ) = δ i1 and Jf 1 = −(1 + λ 1 )f 1 . Now suppose that we have found {f 1 , ..., f k } for some 1 ≤ k < n. Applying the same argument to the orthogonal complement of the symplectic subspace spanned by It is of interest to classify indecomposable Airy data for semisimple Lie algebras. This doesn't reduce to classification of Airy data for simple Lie algebras. Indeed, explicit examples of indecomposable Airy data for semisimple Lie algebras which are not simple are presented in Section 6. Here we derive a simple criterion for indecomposability and prove uniqueness of indecomposable factors. Definition 6. Let (g, W, Ω) be an Airy datum with g semisimple. We define its associated graph by taking the simple factors of g as vertices, with an edge between two simple factors g and g if and only if W contains an irreducible submodule on which both g and g act nontrivially.
Proposition 11. Airy datum (g, W, Ω) with g semisimple is indecomposable if and only if its associated graph is connected.
Proof. Clearly (g, W, Ω) is indecomposable if its associated graph G is connected. Now suppose that G is not connected. Then we may decompose g = g 1 × g 2 (with both factors nonzero), W = W 1 ⊕ W 2 . In this situation Σ s is the product of the corresponding sets for (g 1 , W 1 ) and (g 2 , W 2 ), so also Ω factorizes.
We remark that formation of the associated graph is a contravariant functor from the category of Airy structures for semisimple Lie algebras to the category of graphs.
is an isomorphism. Then m = n and (possibly after a permutation) there exist Proof. We identify factors of n i=1 g i with their images in m i=1 g i through φ. Using the fact that simple factors of a semisimple Lie algebra are uniquely determined and functoriality of the associated graph construction we see that (after a permutation) we have m = n and g i = g i . Then clearly f = n i=1 f i for some module isomorphisms We close this section with a remark that in all examples of Airy data (g, W, Ω) constructed in this paper Ω is a cyclic vector for W . We have not managed to decide if this is always true for g semisimple. Below we prove a weaker statement. Proposition 13. Let (g, W, Ω) be a nontrivial Airy datum with g semisimple. Then the submodule of W generated by Ω has dimension strictly greater than dim g.
Proof. Let W ⊆ W be the submodule generated by Ω. Since W contains the Lagrangian subspace T Ω Σ s , we have dim W ≥ dim g. Suppose that this inequality , which leads to an absurd chain of equalities 4 Automorphisms of Airy data Proposition 14. Suppose that (g, W, Ω) is an Airy datum with g semisimple. Then Stab(Ω) is a finite group. Moreover Ad g (J) = J for any g ∈ Stab(Ω).
We remark that Proposition 14 is false if the assumption of semisimplicity of g is dropped. In general G does not come equipped with a canonical structure of an algebraic variety. Even if such structure exists, it may happen that the G-action on W is not algebraic. This is the case in some of the examples of Airy data discussed in [25], in which Stab(Ω) was found to be infinite cyclic.
1. If f = g for some g ∈ G, then φ = Ad g . In particular every inner automorphism is almost inner.

We have
6. Suppose that g is semisimple and J is regular. Then Stab(Ω) is contained in the subgroup e h ⊆ G generated by h. In particular Stab(Ω) is abelian.
Recall that real structure on a complex vector space V is an antilinear involution Conversely, given a real subspace V ⊆ V with V ⊗ R C there exists a unique real structure σ on V such that V = V σ . Now let g be a complex Lie algebra. Antilinear involution σ on g is said to be a real structure of g if it is a homomorphism of real Lie algebras. In this situation g σ is a real Lie algebra. If the Killing form on g σ is negative-definite, we say that σ is a compact real form. In this situation g is semisimple and g σ is the Lie algebra of a simply-connected compact Lie group G σ . Let W be a representation of g. Real structure K on W is said to be compatible with σ if K(T q) = σ(T )K(q) for T ∈ g, q ∈ W , or equivalently if W K is a representation of g σ and W = W K ⊗ R C as a g σ -module. In this situation we shall abuse the notation by denoting the involution K simply by σ. We remark that real structures on affine representations of g may also be defined, but by Proposition 5 we shall not need them here. Definition 8. Let A = (g, W, Ω) be a homogeneous Airy datum. A real structure on A is a real structure σ on g together with a compatible real structure σ on W such that σ(Ω) = Ω.
Proposition 16. Let σ be a real structure on a nontrivial homogeneous Airy datum (g, W, Ω). Then σ is not compact.
As illustrated by examples in Sections 5 and 6, noncompact real forms do exist, at least for some Airy data.

Simple Lie algebras -classification
Proposition 17. We list isomorphism classes of symplectic representations of simple Lie algebras whose admissibility is not ruled out by the Proposition 7. Whenever g is a classical Lie algebra, we denote the tautological representation by F . In the case of symplectic algebras, we let Λ k 0 F , k ∈ N be the subspace of these elements of Λ k F whose any contraction with the symplectic form of F vanishes.
• g 2 : F ⊕4 , where F is the unique irreducible representation of dimension 7.
• f 4 : F ⊕4 , where F is the unique irreducible representation of dimension 26.
Proof. First note [30, p. 217-218] that the only simple Lie algebras g which admit an irreducible symplectic representation of dimension at most 2 dim(g) are sl 6 , so 11 , so 12 , so 13 , e 7 and the symplectic Lie algebras. Furthermore for n ≥ 6 the only irreducible symplectic representation of sp 2n of desired dimension is the tautological representation. As for irreducible representations which are not symplectic, it is sufficient to consider those of dimension at most dim(g). Complete list of such representations is given in [31, p. 414, 531-532]. Having established which representations may appear in the decomposition of W , one has to find all ways to add them together to get a representation of dimension 2 dim(g). The end result of this calculation is the table above.
Our next goal is to determine which representations among those listed in the Proposition 17 are admissible. The following fact rules out all but finitely many candidates.
Proof. We present the proof for g = sp 2k . The second case is handled analogously. Suppose that (g, W, Ω) is an Airy datum. Write Ω = (Ω 1 , ..., Ω 2k+1 ), with Ω i ∈ F . Proposition 4 implies that elements Ω i are pairwise orthogonal with respect to the symplectic form of F . Therefore they are contained in some Lagrangian subspace L ⊆ F . It is easy to check that there exists a nonzero element T ∈ g annihilating L.
Most of the remaining representations are ruled out by the following construction. If g is simple, its invariant bilinear form is unique up to scale. Thus for any representation W there is a real 5 number ind(W ) (called the index of W ) such that tr W (T S) = ind(W )tr g (ad T ad S ) for any T, S ∈ g. Proof. Let λ 1 , ..., λ n be the eigenvalues of ad J . By Proposition 10 we have where we used n i=1 λ i = 0. Rearrangement of this equation yields (5.1). Since the eigenvalues of ad J are rational and not all equal to zero, tr g (ad 2 J ) > 0. Similarly, dim(g) > 0. Therefore equation (5.1) enforces that ind(W ) − 2 > 0.
Computation of indices of representations listed in Proposition 17 excludes all simple Lie algebras except of sl 2 , sp 4 and sp 10 . Each of these algebras admits two non-isomorphic Airy data, as we will demonstrate by explicit calculations.

Lie algebra sl 2
Due to the isomorphism sl 2 ∼ = sp 2 , admissibility of the representation F ⊕3 is excluded by the Proposition 18. We will show that F ⊕ Sym 3 F and Sym 5 F are admissible, and that there exist two isomorphism classes of Airy data for sl 2 .
Let H, X, Y be the standard basis [31] of sl 2 . These elements satisfy By construction, Ω ± ∈ Σ s . It's easy to check that Stab(Ω ± ) = 0 and that there exists no element g ∈ SL 2 such that g · Ω + = Ω − . Therefore we conclude that Even though Σ s is disconnected, Airy data corresponding to distinct connected components are still isomorphic. Indeed, the two connected components of Σ s are interchanged by the g-module automorphism W (u, v) → (u, −v) ∈ W .
Case W = Sym 5 F is handled similarly, with the result that one can take J = H 3 , Ω = e 11112 . Space Σ s is connected, but in this case the stabilizer of Ω is nontrivial: In contrast to the previous example, Stab(Ω) is not a normal subgroup of SL 2 . Thus Orb(Ω) is not a Lie group. Nevertheless, W is admissible and we have Σ s ∼ = SL 2 Z 3 . We remark that this Airy datum was constructed for the first time in [26].
We remark that Aut(g, W, Ω) = Inn(g, W, Ω) for Airy data constructed in this section. This happens to be true for all Airy data for simple Lie algebras. For semisimple Lie algebras both Aut(g,W,Ω) AInn(g,W,Ω) and AInn(g,W,Ω) Inn(g,W,Ω) may be nontrivial, as demonstrated by examples in Section 6.
Airy data admit a real structure σ with g σ = sl 2 (R). It is defined by σ(Z) = Z for Z ∈ {H, X, Y }, σ(e i ) = e i for i ∈ {1, 2} and extended to other representations by demanding that σ is a homomorphism of the tensor algebra.

Lie algebra sp 4
We will now consider the Lie algebra g = sp 4 . Representations F ⊕5 and F ⊕4 are ruled out by Proposition 18. We will show that Sym 3 F is also not admissible, while F ⊗ F admits two non-isomorphic Airy data. Representation F is a codimension one direct summand in Λ 2 F . Thus we put e ij = e i ∧ e j for 1 ≤ i < j ≤ 4. Scalar product on Λ 2 F is defined by (e ij , e kl ) = 2ω(e i , e k )ω(e j , e l ) − 2ω(e i , e l )ω(e j , e k ). (5.12) We define also η = e 13 − e 24 . Set {e 12 , e 23 , e 34 , e 14 , η} is a basis of F . Finally, the symplectic form on F ⊗ F is defined first on decomposable tensors and extended to the whole space by bilinearity. Weight diagrams for the most basic representations of g are presented in Figure 1. We shall also consider slightly more complicated representations Sym 3 F and F ⊗ F . It will be important that the latter is reducible. More precisely, contraction with the symplectic form yields a nonzero g-module epimorphism tr : F ⊗ F → F . Kernel of this map is an irreducible representation, which we denote by F ⊥ .
Examination of the weight diagrams of the adjoint representation and of Sym 3 F (see Figure 2) shows that the only possible forms of J not excluded by the . Proposition 19 yields tr g (ad 2 J ) = 40 3 , which is not true for any of the candidates. Thus Sym 3 F is not admissible. In the case of F ⊗ F , the only candidates for J are 3H 1 + H 2 and H 1 + H 2 . Proposition 19 gives tr g (ad 2 J ) = 120. This is satisfied for 3H 1 + H 2 . Spectral test is also passed: with some s, t, u, v ∈ C. We must have v = 0, for otherwise H 2 Ω = 0. Furthermore we have tr(Ω) = (u − s − t)e 2 . Thus if we had u − s − t = 0, submodule of W generated by Ω would be a proper symplectic subspace, and hence couldn't contain a Lagrangian subspace. We conclude that u−s−t = 0. By passing to another vector related by the action of diagonal matrices in Sp 4 , we may put u = 1 + s + t and v = 1. The next step is to compute elements of W obtained by acting on Ω with elements of g. We list them in the order of decreasing eigenvalue of J (consecutive eigenvalues are 7, 5, 3, 3, 1, 1, −1, −1, −3, −5): The only nontrivial scalar products between vectors listed above are 6 : ω(X 12 Ω, Z 12 Ω) = ω(H 2 Ω, V 2 Ω) = 4 + 8s + 8t + 4s 2 + 12st + 4t 2 . (5.17) All these scalar products vanish if and only if (s, t) is chosen as − 4 5 , 1 5 or (0, −1). Vectors (5.16) are linearly independent in both cases. This means that we have found two Airy data, with Ω of one of the following forms: We claim that Airy data (g, W, Ω 1 ) and (g, W, Ω 2 ) are not isomorphic. Indeed, suppose that (φ, f ) : (g, W, Ω 1 ) → (g, W, Ω 2 ) is an isomorphism. Every automorphism of g is inner, so φ = Ad D for some D ∈ Sp 4 . Clearly Ad D (J) = J. Since J is a diagonal matrix with distinct eigenvalues, this implies that D is diagonal. Origin of W is the unique point where vanishes to second order, so = φ t • • f implies that f (0) = 0, i.e. f is a linear map. On the other hand we have = Ad t D • • D, so T = D −1 • f is a g-module automorphism. Using Schur's lemma and (5.19) we see that no map of the form f = D • T with T ∈ End g (W ) carries Ω 1 to Ω 2 , which completes the proof.
It follows from the previous paragraph that the orbits of Ω 1 and Ω 2 are distinct. To complete the computation of Σ s , notice that any element of Stab(Ω i ), i = 1, 2 commutes with J, so it has to be diagonal. Given this information, it is easy to check that stabilizers of Ω 1 and Ω 2 are trivial. Therefore we have (5.20) We note that the element J belongs to an sl 2 triple embedded in g, which is unique up to automorphism of the form Ad D with diagonal D ∈ Sp 4 . For example we can put J + = U 2 +X 12 and J − = 4V 2 +3X 21 . Then [J, J ± ] = ±2J ± , [J + , J − ] = J. Regarding g and W as sl 2 -modules, they decompose as We remark that Airy data {(g, W, Ω i )} 2 i=1 admit no nontrivial automorphisms. However they do admit a real structure σ with g σ = sp 4 (R). Its construction is analogous to that in the Subsection 5.1.

Lie algebra sp 10
The last simple Lie algebra to consider is sp 10 . The only candidate for W is Λ 3 0 F . We will use trace techniques to completely determine J. Once this is done, we will find Ω. Due to the large number of variables, calculations are difficult to carry out manually. We have performed them using symbolic algebra software.
We ask if (g, W, Ω ± ) are isomorphic. Since the element J is regular and W is irreducible, it is sufficient to check if there exists a diagonal element D ∈ Sp 10 such that DΩ + = Ω − or DΩ + = −Ω − . This is a system of equations for the five independent matrix elements of D. One may show that no solution exists, so that the two Airy structures are non-isomorphic. In particular Σ s has two connected components. This is striking, because there surely exists such isomorphism if we give up linearity over C. In our standard bases it is given by the complex conjugation.
Finally, let's compute stabillizers of Ω ± . Once again, since J is regular, it is sufficient to consider diagonal elements of Sp 10 . It turns out that the equation DΩ ± = Ω ± has three solutions, so Stab(Ω ± ) ∼ = Z 3 . Therefore we have (5.33)

Summary
Sp 10 Z 3 Table 1: List of isomorphism classes of Airy structures for simple Lie algebras.
We have found that there exist six isomorphism classes of Airy data (g, W, Ω) with simple g. This is summarized in Table 1. It is worth noticing that sp 4 admits two non-isomorphic Airy data for which all invariant characteristics, such as the conjugacy class of J or of the subgroup Stab(Ω) ⊆ G, coincide. This raises the question if there are other invariants which can be used to distinguish the two Airy data. Unfortunately, we have not found any. Similar statement applies to sp 10 , but in this case lack of desired invariants is explained by the fact that the two Airy data are related by complex conjugation.

Semisimple Lie algebras -examples
Proposition 20. The number of isomorphism classes of indecomposable Airy data (g, W, Ω) with g semisimple is countably infinite.
family of mutually non-isomorphic indecomposable Airy data in the Subsection 6.4.
The only semisimple Lie algebra of rank 2 which is not simple is sl 2 ×sl 2 . Airy data for this algebra are classified in the Subsection 6.1. Already in this case the number of non-isomorphic Airy data is quite large. Therefore we don't carry out analogous computations for Lie algebras of rank 3. Instead we will decide which algebras admit at least one Airy datum. This is facilitated by the following criterion.
Let (g, W, Ω) be an Airy datum with g = g × g semisimple. Let G be a simplyconnected Lie group with Lie algebra g . Put n = dim(g ) . H 0 (g , W ) is a symplectic submodule of W , so W = W ⊕ H 0 (g , W ), where W is the orthogonal complement of H 0 (g , W ). By construction H 0 (g , W ) = 0. Let be the moment map for the gaction on W and define Σ s = {q ∈ −1 (0)| d | q has rank n }. We have an important inclusion Σ s ⊆ Σ s × H 0 (g , W ). (6.1) Σ s is a coisotropic submanifold of W of dimension 2 dim(W ) − n ≥ n . This statement is particularly useful when this inequality is saturated. In this situation Σ s is Lagrangian with a locally transitive G -action. In particular there exists J ∈ g such that J Ω is the projection of Ω onto W . More importantly, existence of an Airy datum of the form (g , W , Ω ) is necessary for Σ s = ∅.
Proof. Put g = g 2 . Inspection of the list of irreducible representations of g 2 shows that the only possible forms of W are F ⊕4 and g ⊕2 2 , where F is the unique irreducible representation of dimension 7. Bound dim(W ) ≥ 2 dim(g ) is saturated in both cases, so Σ s = ∅ follows from the fact that g 2 admits no Airy data.
It happens to be true that also sl 3 × sl 2 doesn't admit any Airy data, but in this case more complicated argument, presented in the Subsection 6.2, is required. Lie algebra sp 4 × sl 2 admits three decomposable Airy data and at least one indecomposable -see Subsection 6.3. This exhausts the list of semisimple Lie algebras with two simple factors and rank 3. The only remaining Lie algebra of rank 3 is 3 i=1 sl 2 . This one does admit an indecomposable Airy datum 7 which is a special case of the construction presented in the Subsection 6.4.

Lie algebra sl 2 × sl 2
We will now present the list of indecomposable Airy data for g = sl 2 × sl 2 , up to isomorphism. We omit details of calculations, which are analogous to previous sections. Canonical generators of the first (resp. second) copy of sl 2 will be denoted by H, X, Y (resp. H, X, Y ). Similarly, their fundamental modules are denoted by F and F . They are generated by e 1 , e 2 and e 1 , e 2 , respectively.
1. W = (Sym 4 F ⊗ F ) ⊕ F . In this case Σ s has two connected components, which turn out to correspond to isomorphic Airy data. One may take J = 1 2 H + H. Vectors Ω corresponding to the two connected components take the form Stabilizers of Ω ± are isomorphic to Z 4 and we have We remark that this is the first example considered in this paper in which Stab(Ω) is not contained in the one-parameter subgroup of G generated by J.

Lie algebra sl 3 × sl 2
In this subsection we consider the Lie algebra sl 3 × sl 2 . Let F , F be the defining representations of sl 3 and sl 2 , respectively. We denote the standard basis of sl 2 by H, X, Y . Since sl 3 admits no Airy data, we must have dim(H 0 (sl 3 , W )) < 6. It follows that the only possible forms of W are 10) Consider the possibility that W = W 1 . Since the projection of Ω onto F must be nonzero, J is necessarily of the form J = J + H with some J ∈ sl 3 . The index of W considered as a representation of sl 3 is equal to 5 3 , so tr W (J 2 ) = 14 + 5 3 tr(ad 2 J ). On the other hand tr W (J 2 ) = 38 + 2 tr(ad 2 J ), by Proposition 10. Comparing the two results we get tr(ad 2 J ) = −72, which is impossible. Hence W = W 1 is ruled out. Representation W 2 may also be excluded by similar reasoning. In this case we have J = J + λ H with λ ∈ {1, 1 3 }. Repetition of the calculation presented above gives tr(ad 2 J ) < 0 in both cases. For W 3 value λ = 1 3 is excluded for similar reasons, but possibility of λ = 1 remains. In this situation we have tr(ad J ) 2 = 10 + tr R ( H 2 ). Second relation is clearly inconsistent, so also W 4 is ruled out. Therefore sl 3 × sl 2 admits no Airy structures.

Lie algebra sp 4 × sl 2
In this subsection we shall confine ourselves to presenting a single example of an indecomposable Airy datum for the Lie algebra 8 g = sp 4 × sl 2 . We use the same notation for generators and bases relevant for the sp 4 algebra as in subsection 5.2 while, for the sl 2 algebra, the notation used parallels the one in subsection 6.2.
The pertinent Airy datum exists for J = H 1 + H 2 + H and the module where F is the defining representations of sp 4 , F denotes the defining representations of sl 2 (and not, as in subsection 5.2, the definig representation of so 5 ) and Λ 2 0 F denotes the codimension 1 subspace of these elements of Λ 2 F whose any contraction with the symplectic form of F vanishes. Since This gives Ω = (αe 12 ⊗ e 2 + (β + e 14 + β − e 23 + β 0 η) ⊗ e 1 , 0, 0) + (0, (a 1 e 1 + a 2 e 2 ) ⊗ e 12 + (a 3 e 3 + a 4 e 4 ) ⊗ e 11 , 0) (6.16) + (0, 0, u e 112 ) with complex parameters α, β 0 , β ± , a i , i = 1, . . . 4 and u. Requiring that {T Ω} T ∈g is an isotropic subspace of W we get the following set of equations: In this subsection we shall consider the Lie algebra g = N i=1 sl where each sl (i) 2 is an independent copy of the sl 2 algebra with defining representation denoted by F (i) . Standard generators of sl (i) 2 will be denoted by H (i) , X (i) and Y (i) . Analogous notation will be used for representations of sl There exists a family of Airy structures for g where Vectors H (i) Ω, X (i) Ω and Y (i) Ω turn out to be linearly independent and orthogonal with respect to the g invariant symplectic form on W if α i = −α 2 i−1 , i = 2, . . . , N and α 2 N −1 1 + 1 = 0.
We shall discuss in detail the simplest case where all α i = 0, i.e.
A Lagrangian complement of T Ω Σ = lin{H j Ω, X j Ω, Y j Ω} can be constructed as It is the immediate to check that vectors Decomposing W w = n j=−n a=0,± α a −j e a j + β a j f a j 9 The construction for even N is completely analogous.
particularly interesting to find a relation between them and fields in which topological recursion has found applications, or with some quantum systems studied in physics. Perhaps that could shed some light on the striking fact that Airy structures for semisimple Lie algebras are so much constrained. We believe that derivation of integral representations of partition functions could be particularly illuminating.
Having classified Airy structures for simple Lie algebras, it is natural to ask for an extension to semisimple Lie algebras. As shown by presented examples, in this case the number of distinct Airy structures is infinite. However it is finite for any given semisimple Lie algebra, so it could be that this problem is manageable. Some difficulties do arise, though. Firstly, there are many Lie algebras and representations to consider. It is not clear to us how to generate a complete list. Secondly, for a given representation of a Lie algebra of high rank the number of cases one has to consider in order to find the possible forms of J is large. We avoided this step in the derivation of the sp 10 Airy structures by deriving the only consistent form of J directly from its spectral properties and relations between invariant polynomials. This method typically breaks down for Lie algebras with more than one simple factor. Indeed, each simple factor contributes its own set of invariant polynomials, so we get more unknowns than equations to solve. Some new restrictions on J would have to be derived in order to make this method viable.
Last but not least, it would be interesting to partially extend our results to more general classes of Airy structures. Besides allowing more general Lie algebras, one could also consider Lie superalgebras with semisimple even part. If it is possible to generalize some of our findings to infinite-dimensional Airy structures, that could have direct consequences for classical topological recursion. Kac-Moody algebras generalize simple Lie algebras in a natural way and have direct connections with conformal field theory and integrable systems, so they would be interesting to study in this context.

A Lie algebra cohomology
We present an ad hoc definition of the first two Lie algebra cohomology groups, sufficient for our purposes. For a more conceptual treatment of the subject, see [33].
Let g be a Lie algebra and M -a representation of g. The zeroth cohomology group H 0 (g, M ) of g valued in M is defined as the space of all element of M annihilated by g. To define the first cohomology group, we let Z 1 (g, M ) be the vector space of linear maps (called cocycles) γ : g → M such that γ([T, S]) = T γ(S) − Sγ(T ), and B 1 (g, M ) the space of linear maps (called coboundaries) g → M of the form γ(T ) = T m for some m ∈ M . Every coboundary is a cocycle, so it makes sense to put H 1 (g, M ) = Z 1 (g,M ) B 1 (g,M ) . The following fact is used in this work: Proposition 22 (Whitehead). Let g be a finite-dimensional semisimple Lie algebra and M -a finite-dimensional g-module. Then H 1 (g, M ) = 0.

B Semisimple and regular elements
Let W be a finite-dimensional vector space and let T ∈ End(W ). We say that T is semisimple if for every T -invariant subspace V ⊆ W there exists a complementary T -invariant subspace V , so that W = V ⊕ V . Since we restrict attention to vector spaces over C, operator T is semisimple if and only if it is diagonalizable.
Proposition 23 (Jordan-Chevalley decomposition). Let T be a linear operator on a finite-dimensional vector space W . There exist unique linear operators T ss , T n on W such that T ss is semisimple, T n is nilpotent, T ss T n = T n T ss and T = T ss + T n . Furthermore there exist polynomials p, q ∈ C[t] such that T ss = p(T ), T n = q(T ). In particular every T -invariant subspace of W is T ss -and T n -invariant.
Proposition 24. Let g be a finite-dimensional, semisimple Lie algebra. For any T ∈ g there exist unique T ss , T n ∈ g such that (ad T ) ss = ad Tss , (ad T ) n = ad Tn . Moreover T ss (resp. T n ) acts as a semisimple (resp. nilpotent) operator in every finite-dimensional g-module.
Let g be a finite-dimensional, semisimple Lie algebra. Element T ∈ g is said to be semisimple (resp. nilpotent) if T = T ss (resp. T = T n ). Set of semisimple elements of g is nonempty and Zariski open, hence dense. It coincides with the union of all Cartan subalgebras of g.
Rank of g is defined as the greatest integer r such that the characteristic polynomial of ad T vanishes at zero with multiplicity at least r for every T ∈ g. Element T ∈ g is said to be regular if its characteristic polynomial vanishes at zero with multiplicity exactly r. By construction, the set of regular elements of g is nonempty and Zariski open. One can show that it is contained in the set of semisimple elements. If T ∈ g is a regular element, then the commutant {T ∈ g|[T, T ] = 0} is the unique Cartan subalgebra of g which contains T . Now suppose that some Cartan subalgebra h ⊆ g is chosen. Element T ∈ h is regular in g if and only if α(T ) = 0 for every root α.
We remark that some authors define T to be regular if the dimension of its commutant is equal to r. Elements with this property are not necessarily semisimple. However, the two notions do coincide for semisimple elements.

C Invariant polynomials on sp 10
Let G be a complex semisimple Lie group with Lie algebra g. Choose a Cartan subalgebra h ⊂ g and let W be the corresponding Weyl group. Denote the algebra of G-invariant polynomial functions on g by C[g] G and the algebra of W-invariant polynomial functions on h by C[h] W . If φ ∈ C[g] G , then the restriction φ| h belongs to C[h] W . In other words, we have a homomorphism res : (C.1) We claim that res is injective. Indeed, suppose that φ| h = 0. If T ∈ g is semisimple, then the G-orbit of T intersects h nontrivially, so φ(T ) = 0. Since the set of semisimple elements is dense and φ is continuous, we must have φ = 0.
Now let's specialize to g = sp 10 . We use the standard choice of h, basis in g and basis is h * described in [31]. Weyl group takes the form W = S 5 Z 5 2 , (C.2) with S 5 acting on {L i } 5 i=1 by permutations and Z 5 2 generated by the five reflections L i → −L i . It follows that elements of C[h] W are symmetric polynomials in {L 2 i } 5 i=1 . Therefore by the fundamental theorem of symmetric polynomials [35], functions with k ∈ {1, ..., 5} are algebraically independent generators of C[h] W . Hence they extend uniquely to invariant polynomials on g and we have In particular the dimension of the space of invariant polynomials on g of degree 2k is equal to the number of partitions of k.
Values of coefficients α i may be found by evaluating this equation on any sufficiently large set of elements of h and solving a system of linear equations. Their exact values will be of no use for us, but by computing them allows to check that {Q g 2k } 5 k=1 generate the algebra C[g] G . By dimensionality reasons they have to be algebraically independent, so we have