A New Class of Higher Quantum Airy Structures as Modules of W( gl r ) -Algebras
aa r X i v : . [ m a t h - ph ] S e p A NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES ASMODULES OF W ( gl r ) -ALGEBRAS VINCENT BOUCHARD AND KIERAN MASTEL
Abstract.
Quantum r -Airy structures can be constructed as modules of W ( gl r )-algebrasvia restriction of twisted modules for the underlying Heisenberg algebra. In this paperwe classify all such higher quantum Airy structures that arise from modules twisted byautomorphisms of the Cartan subalgebra that have repeated cycles of the same length.An interesting feature of these higher quantum Airy structures is that the dilaton shiftsmust be chosen carefully to satisfy a matrix invertibility condition, with a natural choicebeing roots of unity. We explore how these higher quantum Airy structures may providea definition of the Chekhov, Eynard, and Orantin topological recursion for reducible alge-braic spectral curves. We also study under which conditions quantum r -Airy structuresthat come from modules twisted by arbitrary automorphisms can be extended to newquantum ( r + 1)-Airy structures by appending a trivial one-cycle to the twist withoutchanging the dilaton shifts. Introduction
The topological recursion of Chekhov, Eynard, and Orantin [9, 10, 11] is a formalismthat can be used to solve various enumerative problems involving Riemann surfaces. Itrelies on the geometry of a spectral curve, which is realized as a branched covering of P ,and constructs generating functions for enumerative invariants via residue analysis on thespectral curve.Kontsevich and Soibelman introduced the concept of quantum Airy structures in [12, 1]as an algebraic reformulation (and generalization) of the topological recursion. A quantumAiry structure is a set of second-order partial differential operators H i that satisfy certainspecific conditions. The key result is that, under these conditions, there always exists aunique solution to the differential constraints H i Z = 0, where Z has the specific form of agenerating function. It can be shown that the topological recursion of Chekhov, Eynard,and Orantin can be reformulated as a special case of quantum Airy structures [12, 1, 5].Meanwhile, the original topological recursion of Chekhov, Eynard, and Orantin wasalso generalized in [6, 7, 8]. In the original formulation, the spectral curve was requiredto be a branched covering of P with only simple ramification points. The restriction wasremoved in [6, 7, 8], to allow for branched coverings with arbitrary ramifications. However,the resulting topological recursion is not a special case of quantum Airy structures anymoreas originally formulated by Kontsevich and Soibelman.This conundrum was resolved in [4], where the concept of higher quantum Airy struc-tures was introduced. The main difference with the original quantum Airy structuresof Kontsevich and Soibelman is that the restriction that the differential operators aresecond-order is relaxed, and differential operators of arbitrary order are considered. In W ( gl r )-ALGEBRAS 2 particular, a large class of higher quantum Airy structures was constructed in [4] as mod-ules for W ( g )-algebras, where g is a Lie algebra. It was then shown that the generalizedtopological recursion of [6, 7, 8] can be reformulated as a special case of higher quantumAiry structures realized as modules of W ( gl r )-algebras.Those modules of W ( gl r )-algebras that take the form of higher quantum Airy structureswere constructed in [4] by restricting twisted modules for the Heisenberg VOA associatedto the Cartan subalgebra h of gl r (see also [2, 3, 13] for related work). As such, the con-struction relies on a choice of automorphism σ of h . In principle, arbitrary automorphismscan be considered. But it is not a priori obvious that the resulting modules may takethe form of quantum higher Airy structures. While arbitrary automorphisms were brieflyconsidered in [4], the paper mostly focused on the case where σ is either the automorphisminduced by the Coxeter element of the Weyl group ( i.e. it permutes all basis vectors of h cyclically) – see Theorem 4.9 in [4] – or the case where σ permuted all but one basisvectors of h – see Theorem 4.16. From the point of view of enumerative geometry, theformer are related to various flavours of (closed) intersection theory on M g,n (or variantsthereof), while the latter are related to open intersection theory.A natural question then is to attempt to classify all higher quantum Airy structuresthat arise as modules of W ( gl r )-algebras via the construction above, for abitrary auto-morphisms σ . This is the main motivation behind the current paper. At this stage, afull classification remains out of reach. But we make a small step in this direction inSection 3. We classify all higher quantum Airy structures that arise from automorphisms σ consisting of products of cycles of the same length. This is achieved in Theorem 3.5.One byproduct of Theorem 3.5 as that the dilaton shifts cannot simply be 1 anymore, asin [4]. In Section 3.5 we study a particularly interesting class of examples of the quantum r -Airy structures constructed in Theorem 3.5, where the dilaton shifts are taken to be rootsof unity. This is perhaps the most natural way to achieve the invertibility requirementstated in Theorem 3.5.It is also interesting to investigate the connection of these higher quantum Airy struc-tures with the topological recursion. The generalized topological recursion of [6, 7, 8] isa special case of higher quantum Airy structures for W ( gl r ) obtained by taking σ to bethe automorphism induced by the Coxeter element of the Weyl group. What should thehigher quantum Airy structures constructed from other choices of automorphisms cor-respond to? The natural guess is that they should produce a further generalization ofthe topological recursion for spectral curves that are formulated as reducible algebraiccurves. While we do not pursue this research direction in detail in this paper, we brieflyexplore this potential interpretation in Section 3.6, focusing on the higher quantum Airystructures obtained in Section 3.5 with dilaton shifts being roots of unity. It should bepossible to use these higher quantum Airy structures to formulate such a generalizationof the topological recursion in terms of residue analysis on reducible spectral curves. Wehope to probe this research direction further in the near future.We also study whether the idea of Theorem 4.16 of [4] can be generalized further. Inessence, what Theorem 4.16 is saying is that, given a quantum r -Airy structure constructedas a W ( gl r )-module for σ the fully cyclic automorphism (Theorem 4.9), one can alwaysconstruct a new quantum ( r + 1)-Airy structure as a W ( gl r +1 )-module by “appending” to σ a trivial one-cycle, with no extra dilaton shift. As the higher quantum Airy structures of NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS 3 Theorem 4.9 are related to closed intersection theory, and those of Theorem 4.16 to openintersection theory, this could be understood as some sort of open/closed correspondence.In Section 4 we prove under which conditions this idea of “appending a one-cycle”works. Namely, we start with any quantum r -Airy structure constructed from an arbitraryautomorphism σ , and we prove under which condition appending a one-cycle, with no extradilaton shift, gives rise to a new quantum ( r + 1)-Airy structure (see Theorem 4.1). Whilethe enumerative geometric side of the story is unknown at this stage, if these arbitraryquantum r -Airy structures do have an interpretation in terms of closed intersection theory,the new ones may be expected to provide an open version of the enumerative geometricproblem.We conclude with future directions in Section 5. In particular, it would be very in-teresting to investigate whether these higher quantum Airy structures for more generalautomorphisms σ have an enumerative geometric interpretation, and whether “appendingone-cycles” as in Section 4 leads to a general open/closed correspondence statement. Acknowledgements
We thank Gaetan Borot, Nitin K. Chidambaram, and Thomas Creutzig for discussions,as well as Devon Stockall and Quinten Weller for collaboration in the initial stages ofthis project. The authors acknowledge the support of the Natural Sciences and Engineer-ing Research Council of Canada (NSERC); in particular, the work of K.M. was partlysupported by a NSERC Undergraduate Student Research Award. Background
In this section we review the concept of higher quantum Airy structures, and the con-struction of [4], in order to motivate the problem addressed in this paper. We note thatthis paper is not meant to be self-contained: we simply recall the definitions and resultsthat are necessary for the remainder of the paper. For the detailed construction of higherquantum Airy structures as modules of W -algebras, the reader should consult [4].2.1. Higher Quantum Airy Structures
We start by defining higher quantum Airy structures. For simplicity, we only state theexplicit, basis-dependent definition:
Definition 2.1 ([4], Definition 2.6) . Let V be a vector space over a field K of characteristiczero. Let ( e i ) i ∈ I be a basis of V , and ( x i ) i ∈ I the corresponding basis of linear coordinates.Let ~ denote a formal parameter. Let D ~ W be the completed algebra of differential operatorson V , and define an algebra grading by assigningdeg x l = deg ~ ∂ x l = 1 , deg ~ = 2 . (2.1)A higher quantum Airy structure on V is a family of differential operators ( H k ) k ∈ I of theform H k = ~ ∂ x k − P k , (2.2)where P k ∈ D ~ W is a sum of terms of degree ≥
2. Additionally we require that the left D ~ W -ideal generated by the H k s forms a graded Lie subalgebra, i.e. there exists g k k ,k ∈ D ~ W NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS 4 such that [ H k , H k ] = ~ X k ∈ I g k k ,k H k . (2.3)We define a quantum r -Airy structure as a higher quantum Airy structure such that all P k only have terms of degree ≤ r .2.2. Higher Quantum Airy Structures as Modules of W ( gl r ) -algebras In this paper we focus on higher quantum Airy structures that are obtained as modules of W ( gl r )-algebras, following the construction of [4]. We will not go through the details of thisconstruction here, but simply highlight its main features. The reader should supplementthis paper with a careful reading of [4].Before we describe these higher quantum Airy structures we make the following com-binatorial definition. Definition 2.2.
For r ∈ N we define a λ -good index set Λ r := (cid:8) ( i, m ) ∈ Z ≥ : 1 ≤ i ≤ r, m ≥ i − λ ( i ) (cid:9) (2.4)where we have defined λ ( i ) = min { s | λ + · · · + λ s ≥ i } for some integer partition λ = λ + · · · + λ l of r .With this definition out of the way, we can highlight the main features of the con-struction of [4]. The starting point is the realization that the W ( gl r )-algebra with centralcharge c = r is strongly freely generated by exactly r vectors | v i i , i = 1 , , . . . , r , in theHeisenberg VOA associated to the Cartan subalgebra h of gl r , with conformal weights1 , , . . . , r . The idea then is to construct a module Y for the W ( gl r )-algebra such that anappropriate subset of the modes of the fields H i ( z ) = Y ( | v i i , z ) takes the form of a higherquantum Airy structure.More precisely, the construction is carried through the following steps (see Section 4 in[4]). Let h be the Cartan subalgebra of gl r , and σ be an element of the Weyl group of gl r .Let | v i i , i = 1 , , . . . , r be the strong, free, generators of the W ( gl r )-algebra with centralcharge c = r .(1) We construct a σ -twisted module T of the Heisenberg VOA associated to h . Uponrestriction to the W ( gl r )-algebra (which is a sub-VOA of the Heisenberg VOA),the module becomes untwisted. The underlying vector space of T is the space offormal series in countably many variables x , x , . . . , and elements of W ( gl r ) actas differential operators (of order at most rank( gl r ) = r ) in the x k s.(2) We denote by W i ( z ) = T ( | v i i , z ) the fields of the strong, free, generators of the W ( gl r )-algebra. We pick a subalgebra of the modes W im of these fields, suchthat ( i, m ) ∈ Λ r for some partition λ of r . We call such a subalgebra λ -good .It is shown in Section 3.3 of [4] that a subalgebra of the modes W im fulfils thesubalgebra condition to be a higher quantum Airy structure (see (2.3)) if and onlyif it is λ -good for some partition λ of r .(3) For the modes W im to form a higher quantum Airy structure, they must satisfythe degree 0 and 1 term condition in (2.2). This can potentially be achieved by NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS 5 conjugation (the so-called dilaton shift ): H im = ˆ T s W im ˆ T s , ˆ T s := exp (cid:18) − Q∂ x s s (cid:19) , (2.5)for some integer s and constant Q , in conjunction with potential linear combi-nations of modes. (Note that by the Baker-Campbell-Hausdorff formula, (2.5) isequivalent to the shift x s x s − Qs (2.6)in the modes W im .) If the degree condition (2.2) can be achieved in this way, the H im form a quantum r -Airy structure.This construction was carried out in Section 4.1 of [4] for σ the automorphism of theCartan subalgebra h of gl r induced by the Coxeter element of the Weyl group, whichpermutes cyclically all r basis vectors of h . In this context, the main result is Theorem4.9 of [4], which states that, for a given r , the construction above does produce a uniquequantum r -Airy structure for each choice of integer s ∈ { , , . . . , r + 1 } such that r = ± s . The partition λ defining the appropriate subalgebra of modes is uniquely fixedby the choice of s . See Theorem 4.9 in [4] for details.But there is no reason a priori to focus on the automorphism σ induced by the Coxeterelement of the Weyl group: one could start with any automorphism σ of the Weyl group.As an example, a more general case is studied in Section 4.2.2 of [4], where σ is chosento permute the r − h and leave the last one invariant. The resultis Theorem 4.16, which states that, for a given r , the construction does again produce aunique quantum r -Airy structure, but this time for each choice of integer s ∈ { , . . . , r } such that s | r . As before, the partition λ is uniquely fixed by the choice of s . In this casehowever, a new subtelty arises: one must consider linear combinations of the conjugatedmodes H im to ensure that the degree condition (2.2) is satisfied. But for this particularchoice of σ , this can be achieved fairly easily (see Theorem 4.16).One may then ask the following questions. Does the construction outlined above pro-duce quantum r -Airy structures for all choices of automorphisms σ of the Weyl group?And if so, for what choices of integer s ? And, given a choice of σ and s , is the correspondingpartition λ uniquely fixed?In other words:Can one classify all quantum r -Airy structures that can be produced asmodules of W ( gl r )-algebras via the construction above?It turns out that the main difficulty in producing such a classification lies in step (3). Itis straightforward to construct the σ -twisted module (and its restriction to the W ( gl r )-algebra) in step (1) for arbitrary automorphisms σ : in fact, this is already done in Section4.2.1 of [4]. The classification of the subalgebra of modes that satisfy the Lie subalgebraproperty (2.3) in step (2) is already completed, as it is a purely algebraic property: it doesnot depend on the particular choice of W ( gl r )-module. What is tricky is to show that wecan bring all modes W im in a chosen subalgebra in a form that satisfies the degree condition(2.2) via conjugation and linear combinations, i.e. step (3). This is rather non-trivial.We can be a little more explicit. One can think of the degree condition (2.2) as havingthree parts:(a) All operators have no degree 0 terms; NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS 6 (b) All operators have no degree 1 terms that are coordinates x k s;(c) The degree one terms are all of the form ~ ∂ k , and all derivatives ~ ∂ k appear exactlyonce in the degree 1 term of an operator.If we have achieved conditions (a) and (b) by conjugation of the modes W im of a givensubalgebra, then what remains to be checked is that condition (c) can be satisfied bytaking linear combinations of the conjugated modes. If the algebra was finitely generated,then this problem would be equivalent to the problem of determining invertibility of afinite-dimensional matrix. However, the subalgebras of modes that we are considering areinfinite-dimensional. We are thus faced with the problem of determining invertibility ofan infinite-dimensional matrix (via countably infinite elementary row operations).The problem of inverting infinite-dimensional matrices is in general quite difficult. How-ever, if the matrix is block-diagonal, then we may invert it if and only if the blocks areinvertible, which drastically simplifies the problem.In this paper we provide a classification of quantum r -Airy structures that can beobtained via the method above for a class of automorphisms σ such that the resultinginvertibility problem is block-diagonal. More specifically, we consider the case where σ ∈ S r is an automorphism of h which is a product of cycles of the same length, andclassify all resulting modules that take the form of quantum r -Airy structures. We alsogeneralize Theorem 4.16 of [4], by studying under which conditions higher quantum Airystructures constructed from arbitrary automorphisms do produce new higher quantumAiry structures by “appending a one-cycle” with no extra dilaton shift.However, a full classification of quantum r -Airy structures obtained as modules of W ( gl r )-algebra via the recipe above for arbitrary automorphism σ remains out of reachfor the moment being. We hope to come back to this in the near future. Higher Quantum Airy Structures for σ a Product of Cycles ofthe Same Length In this section we provide a classification of higher quantum Airy structures that can beobtained as modules of W ( gl r )-algebras via the construction of the previous section, withthe automorphism σ ∈ S r consisting of products of cycles of the same length. For ourpurposes, only the cycle structure of σ matters.We make heavy use of the construction of [4]. In particular, Lemma 4.15 in Section4.2.1, which expresses the modes W im of the fields associated to the generators of the W ( gl r )-algebra in terms of the Heisenberg modes, is our starting point.3.1. Notation and Previous Results
Let us start by fixing notation. We consider W ( gl r ). We write σ = Q nj =1 σ j ∈ S r for theautomorphism used to construct the twisted module of the underlying Heisenberg VOA,where each σ j is a cycle of length ρ , with nρ = r .In the construction of [4], there is a set of bosonic modes associated to each cycle of σ ,and a corresponding set of coordinates. We denote by K jm the bosonic modes associatedto the cycle σ j , and we introduce the quantization K j = ~ / C j , K jm = ~ ∂ x jm , K j − m = mx jm , m ∈ Z + , (3.1) NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS 7 where the C j are constants (see Remark 4.14 in [4] for the appearance of the factor of ~ / ).Lemma 4.15 in [4] gives an explicit expression for the modes W im of the fields associatedto the generators of the W ( gl r )-algebra in terms of the Heisenberg modes, as a result ofthe outlined construction for arbitrary automorphisms σ . For our choice of automorphism σ = Q nj =1 σ j , the modes take the form W im = 1 ρ i X M ⊆{ ,...,n } ρ | M | X ≤ i j ≤ ρ, j ∈ M P j ∈ I i j = i X m j ∈ Z , j ∈ M P j m j = m +1 −| M | Y j ∈ M W j,i j m j , (3.2)where the W j,i j m j are the modes of the W ( gl ρ )-module constructed from the automorphism σ j induced by the Coxeter element of the Weyl group. Those are written in terms of thebosonic modes K jm as: W j,im = 1 ρ ⌊ i/ ⌋ X ℓ =0 i ! ~ ℓ ℓ ℓ !( i − ℓ )! X p ℓ +1 ,...,p i ∈ Z P k p k = ρ ( m − i +1) Ψ ( ℓ ) ρ ( p ℓ +1 , . . . , p i ) : i Y k =2 ℓ +1 K jp k : . (3.3)We refer the reader to Definition 4.3 of [4] (and Section 4.2.1) for the definition of theΨ ( ℓ ) ρ .3.2. Dilaton Shifts (3.2) and (3.3) together express the modes W im in terms of the bosonic modes K jm . Whatremains to be shown is that we can find dilaton shifts, and possible linear combinationsof modes, so that there exists a λ -good subset of modes (for some partition λ of r ) thatsatisfy the degree condition (2.2) in order to be a quantum higher Airy structure. Forsimplicity we will restrict to the case where we apply the same dilaton shift to all sets ofbosonic modes associated to the n cycles σ j of the automorphisms σ .First, we recall from the proofs of Theorems 4.9 and 4.16 in [4] that if we do the dilatonshift K j − s K j − s − Q j (3.4)in the modes W j,im of the W ( gl ρ )-module constructed from σ j , we obtain the resultingoperators H j,im = − Q ij δ ρ ( m − i +1)+ si + Q i − j K jρm − ( ρ − s )( i − + O (2) , (3.5)where by O (2) we mean terms of order ≥ s is coprime with ρ . Indeed, let d =GCD( ρ, s ). Then the only modes K jq that appear in the degree one terms of H j,im have q divisible by d . So it will never be possible to achieve the degree condition (2.2) for aquantum r -Airy structure if s is not coprime with ρ , as some derivatives ~ ∂ x jm will neverappear in the degree one terms.We thus assume from now on, without loss of generality, that s is coprime with ρ . Thisimplies that the degree zero term will be non-zero if and only if i = ρ and m = ρ − s − Of course, in the case with ρ = 1, this is trivial, as all integers s are coprime with 1. NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS 8 So we can rewrite the shifted operators as H j,im = − Q ij δ i,ρ δ m,ρ − s − + Q i − j K jρm − ( ρ − s )( i − + O (2) . (3.6)With this under our belt we can prove the following lemma. Lemma 3.1.
Consider the modes W im of the W ( gl r ) -module constructed from an auto-morphism σ = Q nj =1 σ j , where all σ j are cycles of length ρ . The expression for the W im interms of the bosonic modes are given by (3.2) and (3.3) .Apply the same dilaton shifts K j − s → K j − s − Q j (3.7) for each set of bosonic modes, where the Q j are some (potentially zero) constants, with s an integer coprime with ρ . Then the resulting operators take the form: H k + lρm = 1 ρ k + lρ − l − δ k,ρ δ m, ( l +1)( ρ − s ) − X M ⊆{ ,...,n }| M | = l +1 Y j ∈ M ( − Q ρj ) + n X µ =1 K µρ ( m − l ( ρ − s )) − ( ρ − s )( k − Q k − µ X M ⊆{ ,... ˆ µ,...,n }| M | = l Y j ∈ M ( − Q ρj ) + O (2) , (3.8) where k ∈ { , , . . . , ρ } , l ∈ { , , . . . , n − } , and we used the standard notation that { , . . . , ˆ µ, . . . , n } stands for the set { , . . . , n } with the number µ omitted.Proof. The operators resulting from the chosen dilaton shifts are found by replacing the W j,i j m j in (3.2) by the dilaton-shifted modes H j,i j m j in (3.6). The result is: H im = 1 ρ i X M ⊆{ ,...,n } ρ | M | X ≤ i j ≤ ρ, j ∈ M P j ∈ M i j = i X m j ∈ Z , j ∈ M P j m j = m +1 −| M | Y j ∈ M (cid:16) − Q i j j δ i j ,ρ δ m j ,ρ − s − + Q i j − j K jρm j − ( ρ − s )( i j − + O (2) (cid:17) . (3.9)We are interested in the degree zero and degree one terms in H im .The degree zero term will appear when all factors in the product over j ∈ M in (3.9)contribute a degree zero term. This will happen if and only if i j = ρ and m j = ρ − s − j ∈ M . Since P j i j = i , we conclude that this will happen only if i = | M | ρ forsome integer | M | between 1 and n . Furthermore, since P j m j = m + 1 − | M | , we obtainthat this will happen only if m = | M | ( ρ − s ) − H im as H k + lρm , with 1 ≤ k ≤ ρ and 0 ≤ l ≤ n −
1. We then obtain H k + lρm = 1 ρ ( l +1)( ρ − δ k,ρ δ m, ( l +1)( ρ − s ) − X M ⊆{ ,...,n }| M | = l +1 Y j ∈ M ( − Q ρj ) + O (1) . (3.10) NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS 9 Next, we need to figure out the degree one terms. Degree one terms will appear whenall factors in the product over j ∈ M in (3.9) but one contribute a degree zero term. Let µ ∈ M , and suppose that all terms with j ∈ M and j = µ contribute a degree zero term.Thus i j = ρ and m j = ρ − s − j = µ . We use the notation i = k + lρ as above toindex the modes. Since P j i j = k + lρ , we conclude that i µ = k + lρ − ( | M | − ρ . But i µ must satisfy 1 ≤ i µ ≤ ρ : we conclude that | M | = l + 1, and hence i µ = k . Furthermore,since P j m j = m + 1 − | M | , we conclude that m µ = m − l ( ρ − s ). Putting all this together,we obtain: H k + lρm = 1 ρ k + lρ − l − δ k,ρ δ m, ( l +1)( ρ − s ) − X M ⊆{ ,...,n }| M | = l +1 Y j ∈ M ( − Q ρj ) + n X µ =1 K µρ ( m − l ( ρ − s )) − ( ρ − s )( k − Q k − µ X M ⊆{ ,... ˆ µ,...,n }| M | = l Y j ∈ M ( − Q ρj ) + O (2) . (3.11) (cid:3) A direct corollary of this lemma is the following:
Corollary 3.2.
If we restrict to the subalgebra of modes H k + lρm with m ≥ l ( ρ − s ) + k − − (cid:22) sρ ( k − (cid:23) + δ k, , s ≥ , (3.12) then the H k + lρm have no degree zero terms, and the degree one terms all involve only bosonicmodes K µj with j ≥ (i.e. only derivatives ~ ∂ x µj ).Proof. This follows by direct inspection of (3.8). (cid:3)
We will focus on subalgebras of modes satisfying this condition from now. We recordfor future use the form of the modes in this case, without the degree zero terms: H k + lρm = 1 ρ k + lρ − l − n X µ =1 K µρ ( m − l ( ρ − s )) − ( ρ − s )( k − Q k − µ X M ⊆{ ,... ˆ µ,...,n }| M | = l Y j ∈ M ( − Q ρj ) + O (2) . (3.13)What remains to be shown is twofold. First, that this subalgebra of modes is a λ -goodsubalgebra, for some partition λ of r . Second, that there exists linear combinations of the H k + lρm that satisfy the degree condition (2.2).3.3. Linear Combinations of Operators
We address the second condition first. Looking at the linear terms in (3.13), we see thatfor a fixed value of q , the linear terms K µq , for µ = 1 , . . . , n , all appear together in thesame operators. This is key. What it means is that we are in a block-diagonal case.In other words, in order to show that there exists linear combinations of the operators NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS10 H k + lρm that satisfy the degree requirement (2.2), all we have to do is determine whetherthe finite-dimensional matrices of coefficients corresponding to the block of modes wherethe K µq appear (for fixed values of q ) are invertible. The existence of a quantum r -Airystructure hinges on a block diagonal matrix inversion problem. This is what we now makerigorous. Definition 3.3.
Let ( Q j ) nj =1 be a set of (possibly zero) constants, and let µ, ℓ ∈ { , . . . , n } .We define the n -by- n shift matrix : M ( Q , . . . , Q n ) µ,ℓ = X M ⊆{ ,... ˆ µ,...,n }| M | = ℓ − Y j ∈ M ( − Q ρj ) , (3.14)where we define M ( Q , . . . , Q n ) µ, = 1, so that if n = 1 we have M ( Q ) = 1 for any valueof Q . We will use the shorthand notation M µ,ℓ when the dependence on the constants Q , . . . , Q n is clear from context.Using this definition, we can rewrite (3.13) as: H k + lρm = 1 ρ k + lρ − l − n X µ =1 M µ,l +1 Q k − µ K µρ ( m − l ( ρ − s )) − ( ρ − s )( k − + O (2) . (3.15)This means that the block matrices in our block diagonal inversion problem are in fact allthe same matrix M , with its µ ’th column multiplied by the constant Q k − µ . This meansthat for Q k − µ = 0 we have reduced the problem to the inversion of one finite-dimensionalmatrix.More precisely: Lemma 3.4.
Let H k + lρm , k ∈ { , , . . . , ρ } , l ∈ { , , . . . , n − } , be the operators con-structed in Lemma 3.1. Consider the subalgebra of modes in Corollary 3.2, with m ≥ l ( ρ − s ) + k − − (cid:22) sρ ( k − (cid:23) + δ k, , (3.16) where s ≥ . Then there exists linear combinations of the operators H k + lρm that satisfythe degree condition (2.2) of a quantum r -Airy structure if and only if the shift matrix M ( Q , . . . , Q n ) is invertible, and either:(a) ρ = 1 ;(b) ρ > , s is coprime with ρ , and Q j = 0 for all j = 1 , . . . , n .Proof. We have already seen in Corollary 3.2 that the subalgebra of modes have no degreezero terms, and that the degree one terms are all derivatives ~ ∂ x µm . What remains tobe shown is that there exists linear combinations of the modes such that all derivativeoperators ~ ∂ x µm appear exactly once in the degree one terms.Consider first the case ρ = 1. Then the operators read: H lm = n X µ =1 M µ,l +1 K µm − l (1 − s ) + O (2) , (3.17)with m ≥ l (1 − s ) + 1. We then notice that, for any fixed value of q ≥
1, the modes K µq appear together in the d modes H lq + l (1 − s ) , all of which are in the subalgebra. For any q ,the matrix of coefficients is precisely the shift matrix M . Therefore, there exists linear NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS11 combinations of the operators such that all bosonic modes K µq appear exactly once in thelinear terms if and only if the shift matrix M is invertible.Now consider the case ρ >
1. The operators read: H k + lρm = 1 ρ k + lρ − l − n X µ =1 M µ,l +1 Q k − µ K µρ ( m − l ( ρ − s )) − ( ρ − s )( k − + O (2) , (3.18)with m ≥ l ( ρ − s ) + k − − (cid:22) sρ ( k − (cid:23) + δ k, . (3.19)An argument similar to the case ρ = 1 holds here as well. For any fixed value of q ≥
1, themodes K µq appear together in exactly d modes, with the matrix of coefficients given by M µ,l +1 Q k − µ . Indeed, fix a q ≥
1, and consider the modes H km (with l = 0). Then, one canalways find a unique choice of m and k such that the modes K µq appear in the linear terms H km by solving the equation ρm − ( ρ − s )( k −
1) = q , if and only if s is coprime with ρ (otherwise it would only be possible for a subset of q ’s that are multiple of GCD( ρ, s ), seethe discussion at the beginning of Section 3.2). But then the same modes K µq also appearin the linear terms of the operators H k + lρm + l ( ρ − s ) for all l ∈ { , . . . , n − } . Furthermore, theinequality (3.19) is precisely such that all these modes are included in the subalgebra. Asa result, assuming that all Q µ = 0, we can always find linear combinations of the operatorssuch that all bosonic modes K µq appear exactly once in the linear terms if and only if theshift matrix M is invertible and s is coprime with ρ .We assumed however that all Q µ = 0. Is that a necessary condition? What happens ifsome of the constants Q µ vanish? Assume that there is a vanishing constant, which wetake to be Q , without loss of generality. Then the operators with k > H k + lρm = 1 ρ k + lρ − l − n X µ =2 M µ,l +1 Q k − µ K µρ ( m − l ( ρ − s )) − ( ρ − s )( k − + O (2) . (3.20)Thus the modes K q only appear in the operators H lρm (with k = 1). But only the modes K q with q divisible by ρ appear in these operators. Therefore, in the case ρ >
1, it isnecessary for all the Q µ to be non-zero for all the bosonic modes to appear in the linearterms. (cid:3) Classification Theorem
All that remains is to determine whether the resulting subalgebras are λ -good, for somechoice of partition λ of r . The result is the main theorem of this section. We only considerthe case with n ≥ σ with more than one cycle), as the case n = 1corresponds to the quantum r -Airy structures constructed in Theorem 4.9 of [4]. Theorem 3.5.
Let H k + lρm , k ∈ { , , . . . , ρ } , l ∈ { , , . . . , n − } , n ≥ , and r = nρ ,be the operators constructed in Lemma 3.1 (that is, they are constructed as restrictions oftwisted modules of the Heisenberg algebra, where the twist is given by the automorphism σ = Q nj =1 σ j with the σ j cycles of length ρ ). Consider the subalgebra of modes in Corollary3.2, with m ≥ l ( ρ − s ) + k − − (cid:22) sρ ( k − (cid:23) + δ k, , (3.21) NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS12 where s ≥ . Then there exists linear combinations of the operators H k + lρm that form aquantum r -Airy structure if and only if P nj =1 K j = 0 , the shift matrix M ( Q , . . . , Q n ) isinvertible, and one of the following conditions is satisfied:(a) ρ = 1 , s = 1 , any number n of -cycles;(b) ρ > , s = 1 , any number n of ρ -cycles, and Q j = 0 for all j = 1 , . . . , n ;(c) ρ = 1 , s = 2 , n = 2 (two -cycles);(d) ρ > , ρ is odd, s = 2 , n = 2 (two ρ -cycles), and Q j = 0 for all j = 1 , . . . , n . Remark 3.6.
Before we prove this classification theorem, let us mention that the resultis perhaps a little bit unexpected. In the case with n = 1 considered in Section 4.1 of [4],there is a much larger choices of s that give rise to quantum r -Airy structures, namelyany s ∈ { , , . . . , r + 1 } such that r = ± s . One could have expected a similarrange of possibilities here. However, it appears to be much more constrained when n ≥ r -Airy structure only for s = 1, with theexception of the case with n = 2 ( i.e. two cycles) where s = 2 can also work. Proof.
We first prove that cases (a)-(d) form quantum r -Airy structures. We then provethat these are the only possibilities.In all cases (a)-(d), we know from Lemma 3.4 that the degree condition (2.2) is satisfied.What we need to check is that the subalgebra of modes is a λ -good subalgebra for somechoice of partition λ of r , from which we can conclude that the subalgebra condition (2.3)is satisfied, and therefore that the operators form a quantum r -Airy structure.(a) For ρ = 1 and s = 1, we must have k = 1, and the operators read H lm = n X µ =1 M µ,l +1 K µm + O (2) , (3.22)with l ∈ { , , . . . , n − } , and m ≥ . (3.23)We need to check where this subalgebra is λ -good for some partition λ of r = n . Inother words, we are looking for a partition λ such that (refer to Definition 2.2 for thenotation): λ (1 + l ) = l. (3.24)It turns out that this is not possible, but the partition λ = 2 + n − X j =1 m ≥ − δ l . (3.26)Thus to get a λ -good subalgebra we must include the operator H = P nµ =1 K µ .As this operator does not satisfy the degree condition (2.2), we must require that P nµ =1 K µ = 0 so that the operator identically vanishes.(b) We now consider the case ρ > s = 1. The operators read: H k + lρm = 1 ρ k + lρ − l − n X µ =1 M µ,l +1 Q k − µ K µρ ( m − l ( ρ − − ( ρ − k − + O (2) , (3.27) NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS13 with m ≥ k + lρ − ( l + 1) − (cid:22) k − ρ (cid:23) + δ k, = k + lρ − ( l + 1) + δ k, , (3.28)where the equality follows since k ∈ { , , . . . , ρ } . We need to check whether thissubset is λ -good. That is, whether there exists a partition λ of r such that: λ ( k + lρ ) = l + 1 − δ k, (3.29)for k ∈ { , , . . . , ρ } and l ∈ { , , . . . , n − } . It turns out that this is, again, not quite λ -good. But the partition λ = ( ρ + 1) + n − X j =1 ρ + ( ρ −
1) (3.30)corresponds to the subalgebra of modes with m ≥ k + lρ − ( l + 1) + δ k, − δ l . (3.31)That is, to get a λ -good subalgebra we must include the operator H = P nµ =1 K µ .Again, we thus require that P nµ =1 K µ = 0, so that the operator identically vanishes.(c) We now consider the case ρ = 1 and s = 2. Then k = 1, and the operators read: H lm = n X µ =1 M µ,l +1 K µm + l + O (2) , (3.32)with m ≥ − l + 1 , (3.33)for l ∈ { , . . . , n − } . To get a λ -good subalgebra, we must have that all m ≥ λ -good subalgebra for n >
2. Now consider n = 2. In this case, this is not quite a λ -good subalgebra, but if we include, as usual,the mode H , we get the subalgebra corresponding to the partition λ = 1 + 1 . (3.34)Thus we get a quantum 2-Airy structure if we impose that P µ =1 K µ = 0.(d) We consider ρ > s = 2. We note that ρ must be odd, as it is coprime with s = 2. The operators read: H k + lρm = 1 ρ k + lρ − l − n X µ =1 M µ,l +1 Q k − µ K µρ ( m − l ( ρ − − ( ρ − k − + O (2) , (3.35)with m ≥ k + lρ − (2 l + 1) − (cid:22) ρ ( k − (cid:23) + δ k, , (3.36)for k ∈ { , , . . . , ρ } and l ∈ { , . . . , n − } . We need to determine whether thesesubalgebras are λ -good. In other words, we need to figure out whether there exists apartition λ of r such that λ ( k + lρ ) = 2 l + 1 + (cid:22) ρ ( k − (cid:23) − δ k, . (3.37)Consider first the case n = 2. To get a λ -good subalgebra, as usual we need to includethe mode H . Thus we must require that P µ =1 K µ = 0. With this mode included, it NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS14 is straightforward to check that the subalgebra is λ -good, for the partition λ of r = 2 ρ given by (recall that ρ is odd): λ = ρ + 12 + ρ + 12 + ρ −
12 + ρ − . (3.38)For n ≥ r = nρ of theform λ = ρ + 12 + ρ + 12 + ρ −
12 + ρ + 12 + . . . , (3.39)which is course a contradiction as the terms of the “partition” are not decreasing.Thus we cannot get a quantum r -Airy structure for n ≥
3, and only the n = 2 casesurvives.Now that we have proved that cases (a)-(d) form quantum r -Airy structures, whatremains is to show that these are the only possibilities. In other words, we want to showthat we cannot get λ -good partitions for the subalgebra of modes given by other choicesof s ≥ M is invertible, and either ρ = 1, or ρ >
1, in which case s must becoprime with ρ and Q j = 0 for all j = 1 , . . . , n .Consider first the case ρ = 1 (in which case k = 1), with a shift s ≥ ρ .The operators read H lm = n X µ =1 M µ,l +1 K µm − l (1 − s ) + O (2) , (3.40)with m ≥ − l ( s − , (3.41)with l ∈ { , , . . . , n − } . We know that all m must be ≥ λ -good(see Section 3.3 of [4]). But since n ≥
2, this is impossible for s ≥ H to the subalgebra). Therefore the only possible choices of s are s = 1 ,
2, whichwere considered in cases (a) and (c).Consider now the case ρ >
1, with a shift s ≥
3. The operators read H k + lρm = 1 ρ k + lρ − l − n X µ =1 M µ,l +1 Q k − µ K µρ ( m − l ( ρ − s )) − ( ρ − s )( k − + O (2) , (3.42)with m ≥ l ( ρ − s ) + k − − (cid:22) sρ ( k − (cid:23) + δ k, , (3.43)for k ∈ { , , . . . , ρ } and l ∈ { , , . . . , n − } .Let us assume first that s > ρ . Then some of the modes in the subalgebra (considerfor instance k = 2 and l = 1) will have negative m ’s. But we know from Section 3.3 of [4]that to get a λ -good subalgebra, all m must be non-negative. Thus we must have s < ρ .The question then is: for 3 ≤ s ≤ ρ −
1, can we find a partition λ of r = nρ such that λ ( k + lρ ) = sl + 1 + (cid:22) sρ ( k − (cid:23) − δ k, ? (3.44) NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS15 We can try to build such a partition. Including the mode H as usual, after tediouscalculations we see that the “partition” would look like: λ = A + s − X j =2 A j + ( A s + 1) + ( A −
1) + s − X j =2 A j + . . . , (3.45)where we defined A j = (cid:24) ρjs (cid:25) − (cid:24) ρ ( j − s (cid:25) . (3.46)For this to be a partition, the summands must be decreasing. Since all A j with j =2 , . . . , s − A −
1, we must have a := A = A = . . . = A s − = A − , (3.47)and then we must also have A s + 1 = a. (3.48)But by definition of the A j , we have: s X j =1 A j = ρ, (3.49)and thus we get: as = ρ. (3.50)But s is coprime with ρ , which is a contradiction. Therefore, there is no λ -good subalgebraof operators for s ≥ ρ >
1. The only possible choices are s = 1 and s = 2, whichwere considered in cases (b) and (d).This completes the proof of the theorem. (cid:3) An Interesting Class of Examples
An interesting feature of Theorem 3.5 is that the dilaton shifts Q j , j = 1 , . . . , n , cannot betaken to be simply 1 anymore, in contrast to Theorem 4.9 in [4]. Indeed, the shift matrix M ( Q , . . . , Q n ) must be invertible, which restricts possible choices of dilaton shifts.There is however a natural way of ensuring invertibility of the shift matrix for allquantum r -Airy structures constructed in Theorem 3.5. The idea is to let Q j = ω j , j = 1 , . . . , n , where ω is a primitve r ’th root of unity (recall that r = nρ ). We study thisinteresting class of examples in this section. They appear to be intimately connected tothe geometry of reducible spectral curves, as we briefly explore in Section 3.6.Let us start by proving a simple lemma about roots of unity, which will be necessaryto prove invertibility of the shift matrix. Lemma 3.7.
Let n ∈ Z + , with n ≥ , and µ, ℓ ∈ { , , . . . , n } . Let θ be a primitive n -throot of unity. Then X M ⊆{ ,... ˆ µ,...,n }| M | = ℓ − Y j ∈ M ( − θ j ) = θ µ ( ℓ − . (3.51) Proof.
Vieta’s formula gives: X M ⊆{ ,...,n }| M | = ℓ − Y j ∈ M θ j = 0 . (3.52) NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS16 We can separate on the left-hand-side contributions from subsets that include the index µ ∈ { , . . . , n } . Rearranging, we get: X M ⊆{ ,..., ˆ µ,...,n }| M | = ℓ − Y j ∈ M θ j = − θ µ X M ⊆{ ,..., ˆ µ,...,n }| M | = ℓ − Y j ∈ M θ j . (3.53)Doing this iteratively, we conclude that X M ⊆{ ,..., ˆ µ,...,n }| M | = ℓ − Y j ∈ M θ j = ( − ℓ − θ µ ( ℓ − , (3.54)from which the statement of the lemma follows. (cid:3) An immediate corollary is the following:
Corollary 3.8.
Let ρ, n ∈ Z + with n ≥ , and µ, ℓ ∈ { , , . . . , n } . Let r = nρ , and ω bea primitive r -th root of unity. Let Q j = ω j for j = 1 , , . . . , n , and define θ = ω ρ , whichis a primitive n -th root of unity. Then the shift matrix (see Definition 3.3) M ( ω, ω , . . . , ω n ) µ,ℓ = X M ⊆{ ,... ˆ µ,...,n }| M | = ℓ − Y j ∈ M ( − ω ρj ) = ω ρµ ( ℓ − = θ µ ( ℓ − . (3.55) Note that the shift matrix M can be written explicitly as the n × n Vandermonde matrix: M = θ θ . . . θ ( n − θ θ . . . θ n − ... ... ... . . . ... θ ( n − θ n − . . . θ ( n − n − . . . (3.56) In particular, it is invertible.
The upshot is that we have constructed a large class of quantum r -Airy structures withinteresting potential interpretations. Consider a quantum r -Airy structure constructedas in Theorem 3.5, that is, as a W ( gl r )-module descending from a twisted module of theunderlying Heisenberg algebra with the twist given by an automorphism σ = Q nj =1 σ j , witheach cycle σ j of length ρ . Here r = nρ . Then choose the dilaton shifts Q j , j = 1 , . . . , n to be given by Q j = ω j , where ω is a primitive r -th root of unity. This choice of dilatonshifts always satisfy the invertibility condition in Theorem 3.5, and, assuming that we arein one of the cases specified in the theorem, we obtain a quantum r -Airy structure.We will come back to a potentially interesting interpretation for this class of quantum r -Airy structures in Section 3.6. Meanwhile, let us write down in detail one of the simplesthigher quantum Airy structures in this class, to make things explicit. Example 3.9.
We write down in detail the operators of the quantum 4-Airy structureobtained as a module of the W ( gl )-algebra via restriction of a twisted module of theunderlying Heisenberg algebra, where the twist results from the automorphism σ = σ σ with two 2-cycles. We consider the case with s = 1, which is part of the family (b) inTheorem 3.5. To satisfy the invertibility condition of the shift matrix, we take the dilatonshifts Q j , j = 1 , Q = i , Q = i = −
1, where i = √−
1, as in Corollary3.8.
NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS17 We let K jm , j = 1 ,
2, be the bosonic modes associated to the two 2-cycles, and let W j,im , j = 1 , i = 1 ,
2, be the modes of the two W ( gl )-modules associated to the cycles σ j , j = 1 ,
2. The modes W im , i = 1 , . . . , W ( gl ) can be written in terms ofthose as: W m = W , m + W , m ,W m = 12 W , m + 12 W , m + X m ,m ∈ Z m + m = m − W , m W , m ,W m = 12 X m ,m ∈ Z m + m = m − (cid:0) W , m W , m + W , m W , m (cid:1) ,W m = 14 X m ,m ∈ Z m + m = m − W , m W , m , (3.57)with the subalgebra condition m ≥ (cid:4) i +12 (cid:5) . We then implement the dilaton shifts: K j − K j − − i j , j = 1 , . (3.58)After the shift, the modes of the two W ( gl )-modules become: H j, m = K j m ,H j, m = i j K j m − − ( − j δ m, + 12 X p ,p ∈ Z p + p =2( m − (cid:0) δ | p δ | p − (cid:1) : K jp K jp : − ~ δ m, , (3.59)for j = 1 ,
2. Finally, replacing the W j,im in (3.57) by the shifted H j,im , which implementsthe dilaton shifts on the W ( gl )-module, yields the operators of the resulting quantum4-Airy structure. We write down explicitly the resulting expanded form of H m , H m , and H m , but leave out H m for brevity, as its expanded form is rather long: NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS18 H m = K m + K m , (3.60)2 H m = iK m − − K m − + 12 X p ,p ∈ Z p + p =2( m − (cid:0) δ | p δ | p − (cid:1) (cid:0) : K p K p : + : K p K p : (cid:1) + 2 X m ,m ∈ Z m + m = m − K m K m − ~ δ m, , (3.61)4 H m = − K m − + K m − − ~ (cid:0) K m − + K m − (cid:1) − X m ,m ∈ Z m + m = m − K m K m − + 2 i X m ,m ∈ Z m + m = m − K m − K m + X m ,m ∈ Z m + m = m − X p ,p ∈ Z p + p =2( m − (cid:0) δ | p δ | p − (cid:1) K m : K p K p :+ X m ,m ∈ Z m + m = m − X p ,p ∈ Z p + p =2( m − (cid:0) δ | p δ | p − (cid:1) K m : K p K p : , (3.62)8 H m = − iK m − − K m − + O (2) , (3.63)with the subalgebra of mode given by m ≥ (cid:4) i +12 (cid:5) . As required by Theorem 3.5, we alsoimpose that H = K + K = 0 , (3.64)but each K j does not have to vanish independently.3.6. Higher Quantum Airy Structures and Topological Recursion
It is interesting to try to connect the construction of higher quantum Airy structures inTheorem 3.5 to the Chekhov, Eynard, and Orantin topological recursion. It is shownin [4] that the generalized topological recursion of [6, 7, 8] can be reformulated as aspecial case of higher quantum Airy structures realized as W ( gl r )-modules, originatingfrom twisted modules of the underlying Heisenberg algebra with the twist given by theautomorphism induced by the Coxeter element of the Weyl group. Indeed, this was theoriginal motivation for the study of higher quantum Airy structures [4]. Note that theoriginal topological recursion of Chekhov, Eynard, and Orantin then corresponds to thespecial case of (quadratic) quantum Airy structures originally studied by Kontsevich andSoibelman [1, 12].The higher quantum Airy structures that are relevant for the generalized topologicalrecursion of [6, 7, 8] are those of Theorem 4.9 [4], which are indexed by an integer r ≥ s ∈ { , . . . , r + 1 } such that r = ± s (in particular, r and s arecoprime). They are constructed as W ( gl r )-modules, with dilaton shift K − s K − s − . (3.65)Recall that the topological recursion relies on the geometry of a spectral curve. It isshown in [4] that those quantum r -Airy structures encapsulate the same information as NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS19 the topological recursion of [6, 7, 8] on the so-called ( r, s )-spectral curves, which are realizedas the algebraic curves r r − s x r − s y r − ( − r = 0 (3.66)in standard polarization (for the meaning of “standard polarization” here, see Section 5.1in [4]). One can also think of these spectral curves in parametric form, as being given bythe two following rational functions on P : x = 1 r z r , y = − z r − s . (3.67)Following through the steps of the correspondence established in Section 5 of [4], one seesthat the value of the dilaton shift can be extracted as follows. One constructs the one-form ω , ( z ) = y ( z ) dx ( z ) = − z s − dz. (3.68)The index m of the mode K m that is shifted should be one more than the exponent of thepower of z in ω , , and the shift should be the coefficient. For instance, if one consideredthe quantum r -Airy structure of Theorem 4.9 but with dilaton shift K − s K − s − Q, Q = 0 , (3.69)it would correspond to topological recursion on the spectral curve x = 1 r z r , y = − Qz r − s , (3.70)that is, on the algebraic curve r r − s x r − s y r − ( − Q ) r = 0 . (3.71)The coprime condition between r and s is crucial here: it ensures that the spectral curve,as an algebraic curve, is irreducible. The current formulation of topological recursion isonly defined if the spectral curve is irreducible.From the point of view of topological recursion, a natural question then is whether itis possible to generalize the definition of topological recursion to allow reducible algebraicspectral curves. We claim that the higher quantum Airy structures that we construct inthis paper may give precisely such a generalization.Let us be a little more precise. We consider the quantum r -Airy structures constructedin Theorem 3.5, originating from twisted modules of the underlying Heisenberg algebrawith automorphisms given by σ = Q nj =1 σ j , with each σ j a cycle of length ρ , and n ≥ r = nρ . We use the dilaton shifts by r -th roots of unity explored in Corollary 3.8: K j − s K j − s − ω j , j = 1 , . . . , n, (3.72)where ω is a primitive r -th root of unity. We focus on the case ρ > ρ = 1is untwisted, and does not appear to be directly connected to the standard formulation oftopological recursion). According to Theorem 3.5, we have two choices: either s = 1, or s = 2, n = 2, and ρ is odd.We claim that the case s = 1 should give an explicit formulation of topological recursionon the following reducible algebraic spectral curve: ρ r − n x r − n y r − ( − r = 0 . (3.73)We note that this spectral curve is certainly reducible, as r = nρ , and thus r and n arenever coprime (since n ≥ NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS20 In fact, substituting r = nρ , we can rewrite this curve in reduced form as: ρ r − n x r − n y r − ( − r = n Y j =1 (cid:0) ρ ρ − x ρ − y ρ − ( − ω j ) ρ (cid:1) = 0 , (3.74)where ω is a primitive r -th root of unity. It has precisely n components. It is now clearwhy we expect our quantum r -Airy structures to be connected to this spectral curve. Eachcomponent is a ( ρ, Q = ω j . This is precisely whatour construction is doing, with each cycle σ j of the automorphism σ corresponding to anirreducible component of the reducible spectral curve.More precisely, as topological recursion is not currently defined for reducible spectralcurves, our claim is that: The quantum r -Airy structures of Theorem 3.5, with ρ ≥ , n ≥ , s = 1 ,and the dilaton shifts being given by roots of unity as in Corollary 3.8, mayprovide a definition of topological recursion on the reducible ( r, n ) -spectralcurves (3.73) with n | r . What about the other class of Theorem 3.5, with n = 2, s = 2, and ρ odd? Followingthe same logic, it should define topological recursion on the reducible algebraic curves(here r = 2 ρ ): ρ r − x r − y r − ( − r = Y j =1 (cid:0) ρ ρ − x ρ − y ρ − ( − ω j ) ρ (cid:1) = 0 , (3.75)where ω is a primitive 2 ρ -th root of unity. In other words, we claim that: The quantum r -Airy structures of Theorem 3.5, with ρ ≥ , ρ odd n = 2 , s = 2 , and the dilaton shifts being given by roots of unity as in Corollary3.8, may provide a definition of topological recursion on the reducible ( r, -spectral curves (3.73) with r even but ∤ r . Remark 3.10.
What is surprising however is that, with this construction, we do notrecover all reducible ( r, n )-spectral curves, but only these two particular families. It isunclear to us what is special about these families of reducible spectral curves, and whyother families do not appear to have counterparts in our construction of quantum r -Airystructures. Example 3.11.
As an example, according to our claim, the quantum 4-Airy structurestudied in Example 3.9 should correspond to the reducible (4 , y x − Y j =1 (cid:0) y x − ( − i j ) (cid:1) = 0 (3.76) Appending 1-Cycles
In Section 3, we provided a classification of higher quantum Airy structures that arise asmodules of W ( gl r )-algebras following the method of [4], for arbitrary automorphisms σ that are products of n cycles of the same length. One can think of this construction as anatural generalization of Theorem 4.9 of [4], which considers the case n = 1 (i.e. σ is theautomorphism induced by the Coxeter element of the Weyl group). NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS21 Theorem 4.9 was also generalized in a different direction in [4]. Theorem 4.16 studiedhigher quantum Airy structures that can be obtained from automorphisms σ that permuteall but one of the basis vectors of the Cartan subalgebra. Moreover, in this context theextra one-cycle did not come with an extra dilaton shift. Thus, one may think of thisresult as follows. Given a quantum r -Airy structure constructed as a module of the W ( gl r )-algebra as in Theorem 4.9, one can always construct a new quantum ( r + 1)-Airystructure as a module of the W ( gl r +1 )-algebra by “appending” to it a one-cycle, with noextra dilaton shift. This is, in essence, what Theorem 4.16 is doing.In this section we investigate the question of whether, given a quantum r -Airy structureconstructed as W ( gl r )-module for an arbitrary automorphism σ , we can always construct anew quantum ( r + 1)-Airy structure as a W ( gl r +1 )-module by appending to σ a one-cycle,with no extra dilaton shift.Our result is the following theorem. Theorem 4.1.
Let the H im , i = 1 , . . . , r , be the operators of a quantum r -Airy structureobtained as a W ( gl r ) -module descending from a twisted module of the underlying Heisen-berg algebra, with the twist given by an automorphism σ = Q nj =1 σ j , where the σ j arecycles of length ρ j respectively (and thus r = P nj =1 ρ j ), and dilaton shifts K j − s j K j − s j − Q j , j = 1 , . . . , n, (4.1) for some positive integers s j . Then there exists a quantum ( r +1) -Airy structure realized asa W ( gl r +1 ) -module descending from a twisted module of the underlying Heisenberg algebra,with the twist ˜ σ given by the automorphism σ with a one-cycle appended, and the dilatonshifts still given by (4.1) , if:(a) Q j = 0 for all j ∈ { , . . . , n } , and;(b) There exists a partition ˜ λ of r + 1 such that ˜ λ ( i ) = λ ( i ) for all i = 1 , . . . , r and ˜ λ ( r + 1) = P nj =1 s j . Proof.
Let K r +1 m be the set of bosonic modes associated to the extra one-cycle in ˜ σ . Theoperators ˜ H jm for the quantum ( r + 1)-Airy structure are constructed as follows:˜ H m = K r +1 m + H m ,r i − ˜ H im = H im + r X m ,m ∈ Z m + m = m − K r +1 m H i − m , i = 2 , . . . , r,r r ˜ H r +1 m = r X m ,m ∈ Z m + m = m − K r +1 m H rm . (4.2)We need to determine under which conditions do the operators ˜ H jm satisfy the degreecondition (2.2) and the subalgebra condition (2.3).Let us consider the degree condition first. As the H im form a higher quantum Airystructure, we see that the ˜ H im will have no degree zero terms, as long as the subalgebra ofmodes for ˜ H im for i = 1 , . . . , r is the same (or possibly smaller) as for H im . But it shouldn’tbe smaller, otherwise some linear terms in the H im would not appear in the linear termsfor the ˜ H im . Thus we conclude that ˜ λ ( i ) = λ ( i ) for all i ≤ r . We are using the notation from Definition 2.2.
NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS22 Now consider the operator ˜ H r +1 m . Does it have linear terms? It will for the modes H rm that have degree zero terms. It is fairly straightforward to calculate that H rm = A n Y j =1 ( − Q j ) ρ j δ m ,r − P j s j − + O (1) , (4.3)for some number A . As a result, ˜ H r +1 m has a linear term of the form r r ˜ H r +1 m = rA n Y j =1 ( − Q j ) ρ j K r +1 m − r + P j s j + O (2) . (4.4)From this, it follows that all Q j must be non-zero, otherwise the ˜ H r +1 m would have nolinear terms, and we wouldn’t get a quantum ( r + 1)-Airy structure. Moreover, to ensurethat all the bosonic modes K r +1 k with k ≥ H r +1 m , and thatno bosonic modes K r +1 k with k ≤ m ≥ r + 1 − P j s j for˜ H r +1 m . If we can impose this inequality, then we have isolated all the modes K r +1 k with k ≥
1, and by linear combinations we can remove the modes K r +1 k from the linear termsof all other ˜ H im with 1 ≤ i ≤ r . We conclude that the degree condition (2.2) is satisfied.It remains to be checked whether the subalgebra condition (2.3) will be satisfied if weimpose that m ≥ i − λ ( i ) for the modes ˜ H im , i = 1 , . . . , r , and m ≥ r + 1 − P j s j for ˜ H r +1 m .This subset of modes will satisfy the subalgebra condition (2.3) if and only if it is ˜ λ -goodfor some partition ˜ λ of r + 1. This will happen if there exists a partition ˜ λ of r + 1 suchthat: ˜ λ ( i ) = λ ( i ) , i = 1 , . . . , r (4.5)˜ λ ( r + 1) = X j s j . (4.6)This concludes the proof of the theorem. (cid:3) As an example, we can apply this theorem to the higher quantum Airy structuresconstructed in Theorem 3.5.
Corollary 4.2.
Consider the quantum r -Airy structures constructed in Theorem 3.5.Assuming that all dilaton shifts Q j , j = 1 , . . . , n are non-zero, a one-cycle can be appendedto the quantum r -Airy structures in cases (a) and (b) to produce new quantum ( r + 1) -Airystructures, but not for cases (c) and (d).Proof. This follows by inspection of the partitions. For case (a), the partition is λ = 2 + n − X j =1 . (4.7)Then ˜ λ = λ + 1 is a partition of r + 1 such that ˜ λ ( i ) = λ ( i ) for all i = 1 , . . . , r and˜ λ ( r + 1) = n = P nj =1 s j , since s j = 1 for all j = 1 , . . . , n .For case (b), the partition is λ = ( ρ + 1) + n − X j =1 ρ + ( ρ − . (4.8) NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS23 Then ˜ λ = ( ρ + 1) + n − X j =1 ρ + ρ (4.9)is a partition of r + 1 such that ˜ λ ( i ) = λ ( i ) for all i = 1 , . . . , r , and ˜ λ ( r + 1) = n = P nj =1 s j ,since s j = 1 for all j = 1 , . . . , n .However, this doesn’t work for cases (c) and (d). For case (c), where n = r = 2, thepartition is λ = 1 + 1. The only possibility for ˜ λ then is ˜ λ = 1 + 1 + 1, but then ˜ λ (3) = 3which is not equal to P j =1 s j = 4 as s = s = 2.Similarly, for case (d), where n = 2, the partition is λ = ρ + 12 + ρ + 12 + ρ −
12 + ρ − . (4.10)The only choice for ˜ λ is then ˜ λ = λ + 1, but then ˜ λ ( r + 1) = 5, which is not equal to P j =1 s j = 4 as s = s = 2. (cid:3) Future Directions
In this paper we made a first step towards a classification of higher quantum Airy struc-tures constructed as W ( gl r )-modules following the method of [4], by classifying those thatarise from twisted modules of the underlying Heisenberg algebra with the twist correspond-ing to an automorphism with arbitrary cycles of the same length. We also studied thequestion of when new higher quantum Airy structures can be constructed by “appendinga one-cycle with no dilaton shift”.A few open questions immediately come to mind: • It would be very interesting to complete the classification for arbitrary automor-phisms σ . The key insight that the degree condition (2.2) can be thought of as amatrix inversion problem could prove useful. • The proposed interpretation of our quantum r -Airy structures as defining topo-logical recursion on reducible spectral curves in Section 3.6 deserves to be studiedfurther. For instance, a residue formulation of topological recursion on reduciblespectral curves could potentially be extracted from our quantum r -Airy structures.The fact that only particular families of reducible spectral curves seem to havecounterparts in our classification of quantum r -Airy structures is also intriguingand deserves further investigation. • The quantum r -Airy structures constructed as W ( gl r )-modules for fully cyclicautomorphisms, as in Theorem 4.9 of [4], have natural interpretations in enu-merative geometry. They are known to produce generating functions for variousflavours of (closed) intersection theory on M g,n (or variants thereof). It wouldbe interesting to explore whether the quantum r -Airy structures for more generalautomorphisms, such as those constructed in Theorem 3.5, also have interestingenumerative geometric interpretations. • The idea of “appending a one-cycle with no dilaton shift” to a higher quantumAiry structure has a compelling interpretation in enumerative geometry, for theparticular families of higher quantum Airy structures studied in [4]. Indeed, whilethe higher quantum Airy structures for fully cyclic automorphisms from Theo-rem 4.9 are connected to various flavours of closed intersection theory on M g,n NEW CLASS OF HIGHER QUANTUM AIRY STRUCTURES AS MODULES OF W ( gl r )-ALGEBRAS24 (or variants thereof), the corresponding higher quantum Airy structures obtainedby appending a one-cycle, as in Theorem 4.16, are related to the open versionof the appropriate intersection theory. “Appending a one-cycle” to the higherquantum Airy structures may then be understood as some sort of open/closedcorrespondence. If an enumerative geometric interpretation for higher quantumAiry structures for arbitrary automorphisms is found, it would be fascinating tosee whether such an open/closed correspondence holds for the general procedureof appending a one-cycle studied in Theorem 4.1. References [1] J. E. Andersen, G. Borot, L. O. Chekhov and N. Orantin, “The ABCD of topological recursion,” arXiv:1703.03307 .[2] B. Bakalov and T. Milanov, “ W N +1 -constraints for singularities of type A N arXiv:0811.1965 .[3] B. Bakalov and T. Milanov, “ W -constraints for the total descendant potential of a simple singularity,”Compositio Math., 149(5):840–888, 2013 [ arXiv:1203.3414 ].[4] G. Borot, V. Bouchard, N. K. Chidambaram, T. Creutzig and D. Noshchenko, “Higher Airy Struc-tures, W Algebras and Topological Recursion,” arXiv:1812.08738 .[5] G. Borot, “Lecture notes on topological recursion and geometry,” arXiv:1705.09986 .[6] V. Bouchard and B. Eynard, “Think globally, compute locally,” JHEP 02(143) (2013)[ arXiv:1211.2302 ].[7] V. Bouchard and B. Eynard, “Reconstructing WKB from topological recursion,” Journal de l’Ecolepolytechnique – Mathematiques, 4 (2017) 845-908 [ arXiv:1606.04498 ].[8] V. Bouchard, J. Hutchinson, P. Loliencar, M. Meiers, and M. Rupert, “A generalized topologicalrecursion for arbitrary ramification,” Annales Henri Poincar´e, 15(1):143–169, 2014 [ arXiv:1208.6035 ].[9] L.O. Chekhov and B. Eynard, “Matrix eigenvalue model: Feynman graph technique for all genera,”JHEP 0612:026, 2006 [ arXiv:math-ph/0604014 ].[10] B. Eynard and N. Orantin. “Invariants of algebraic curves and topological expansion,” Commun.Number Theory Phys., 1(2):347–452 (2007) [ arXiv:math-ph/0702045 ].[11] B. Eynard and N. Orantin. “Topological recursion in random matrices and enumerative geometry,”J. Phys. A: Mathematical and Theoretical, 42(29) (2009) [ arXiv:0811.3531 ].[12] M. Kontsevich and J. Soibelman, “Airy structures and symplectic geometry of topological recursion,” arXiv:1701.09137 .[13] T. Milanov, “ W -algebra constraints and topological recursion for A N -singularity,” Int. J. Math.,27(1650110), 2016 (with an Appendix by Danilo Lewanski) [ arXiv:1603.00073 ]. Department of Mathematical & Statistical Sciences, University of Alberta, 632 CAB,Edmonton, Alberta, Canada T6G 2G1
E-mail address : [email protected] Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, On-tario, Canada N2L 3G1
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