Topological recursion for irregular spectral curves
TTOPOLOGICAL RECURSION FOR IRREGULAR SPECTRAL CURVES
NORMAN DO AND PAUL NORBURYA
BSTRACT . We study topological recursion on the irregular spectral curve xy − xy + =
0, which produces aweighted count of dessins d’enfant. This analysis is then applied to topological recursion on the spectral curve xy =
1, which takes the place of the Airy curve x = y to describe asymptotic behaviour of enumerative problemsassociated to irregular spectral curves. In particular, we calculate all one-point invariants of the spectral curve xy = C ONTENTS
1. Introduction 112. Topological recursion 662.1. Irregular spectral curves 773. Enumerating dessins d’enfant 883.1. Loop equations 883.2. Pruned dessins 993.3. Proof of topological recursion 15153.4. Polynomial behaviour of invariants 16164. Three-term recursion for dessins d’enfant 19195. Relation to the Kazarian–Zograf spectral curve 21216. The quantum curve 22227. Local irregular behaviour 25257.1. Volumes 2626Appendix A. Formulae. 2727References 27271. I
NTRODUCTION
Topological recursion developed by Eynard, Orantin and Chekhov produces invariants of a Riemann surface C equipped with two meromorphic functions x , y : C → C and a bidifferential B ( p , p ) for p , p ∈ C [33, 1313].We require the zeros of d x to be simple and refer to the data ( C , B , x , y ) as a spectral curve . For integers g ≥ n ≥
1, the invariant ω gn is a multidifferential on C or, in other words, a tensor product of meromorphic1-forms on C n . In this paper, all spectral curves will have underlying Riemann surface CP and bidifferential B = d z ⊗ d z ( z − z ) . In that case, we may specify the spectral curve parametrically via the meromorphic functions x ( z ) and y ( z ) . We call a spectral curve regular if it is non-singular at the zeros of d x — for example, if thecurve is non-singular. See Section 22 for precise definitions.The invariants ω gn of the Airy curve x = y are (total derivatives of) the following generating functions forintersection numbers of Chern classes of the tautological line bundles L i on the moduli space of stable curves Date : February 16, 2018.2010
Mathematics Subject Classification. a r X i v : . [ m a t h . G T ] D ec NORMAN DO AND PAUL NORBURY M g , n [1414].(1.1) K g , n ( z , . . . , z n ) = g − + n ∑ | d | = g − + n (cid:90) M g , n c ( L ) d · · · c ( L n ) d n n ∏ i = ( d i − ) !! z d i + i A regular spectral curve locally resembles the Airy curve x = y near zeros of d x , which are assumed to besimple. This leads to universality in the behaviour of topological recursion on regular spectral curves — theinvariants are related to intersection theory on M g , n . Three progressively more refined statements of thisrelationship are as follows.(1) Eynard and Orantin [1414] proved that the invariants ω gn behave asymptotically near a regular zero ofd x like the invariants ω gn of the Airy curve x = y at the origin. Hence, they store the intersectionnumbers appearing in equation (1.11.1).(2) Eynard [1111, 1212] pushed this further, proving that the lower order asymptotic terms of ω gn on a regularspectral curve also encode intersection numbers. These come in the form of explicit combinations ofHodge integrals on M g , n and a generalisation M ag , n , which he calls the moduli space of a -colouredstable curves.(3) For a special class of regular spectral curves, Dunin–Barkowski, Orantin, Shadrin and Spitz [1010]extended the results of Eynard, by proving that the multidifferentials ω gn encode ancestor invariantsin a cohomological field theory, which is fundamentally related to intersection theory on M g , n .In this paper, we consider irregular spectral curves that locally resemble the curve xy = x . (In Section 2.12.1, we show that any other local irregular behaviour is ill-behaved.) For such curves, thelocal behaviour of the invariants ω gn is no longer determined by the intersection numbers of equation (1.11.1).An analogue of statement (1) above holds, although we do not currently have an analogue of equation (1.11.1)to relate the invariants of the spectral curve xy = xy − xy + =
0. We then prove a three-term recursion for its 1-point invariants, and use thisto determine an exact formula for the 1-point invariants of the spectral curve xy = dessin d’enfant is a bicoloured graph embedded in a connected orientable surface, such that the complementis a union of disks. The term bicoloured means that the vertices are coloured black and white such that eachedge is adjacent to one vertex of each colour. Consequently, the underlying graph of a dessin d’enfant isnecessarily bipartite. One can interpret a dessin d’enfant as a branched cover π : Σ → P unramified over P − {
0, 1, ∞ } , often referred to as a Belyi map . The bicoloured graph is given by π − ([
0, 1 ]) ⊂ Σ , with thepoints π − ( { } ) representing black vertices and the points π − ( { } ) representing white vertices.Let B g , n ( µ , . . . , µ n ) be the set of all genus g Belyi maps π : Σ → P with ramification divisor over ∞ given by π − ( ∞ ) = µ p + · · · + µ n p n , where the points over ∞ are labelled p , . . . , p n . Two Belyi maps π : Σ → P and π : Σ → P are isomorphic if there exists a homeomorphism f : Σ → Σ that covers the identity on P and preserves the labelling over ∞ . Equivalently, one can interpret B g , n ( µ , . . . , µ n ) as the set of connectedgenus g dessins d’enfant with n labelled boundary components of lengths 2 µ , . . . , 2 µ n . By a boundarycomponent of a dessin d’enfant, we mean a cycle in the underlying graph corresponding to the boundaryof one of the labelled disks in Σ . We require an isomorphism between two dessins d’enfant to preserve thelabelling on their boundary components. Definition 1.1.
For any µ = ( µ , . . . , µ n ) ∈ Z n + , define B g , n ( µ , . . . , µ n ) = ∑ Γ ∈B g , n ( µ ) | Aut Γ | , OPOLOGICAL RECURSION FOR IRREGULAR SPECTRAL CURVES 3 where Aut Γ denotes the automorphism group of the dessin d’enfant Γ .For integers g ≥ n ≥
1, define the generating function(1.2) F g , n ( x , . . . , x n ) = ∞ ∑ µ ,..., µ n = B g , n ( µ , . . . , µ n ) n ∏ i = x − µ i i .If we let x = z + z + x i = x ( z i ) for i =
1, 2, . . . , n , then we may observe that this generatingfunction is well-behaved with respect to z , . . . , z n . In particular, Theorem 11 below implies that F g , n ( x , . . . , x n ) is a rational function of z , . . . , z n for 2 g − + n >
0, with poles only at z i = ± y = ∂∂ x F ( x ) is also rational and together with x defines a plane curve known as the spectralcurve. More precisely, Theorem 11 shows that the (total derivatives of) F g , n ( x , . . . , x n ) satisfy topologicalrecursion on the spectral curve xy − xy + =
0, given parametrically by(1.3) x = z + z + y = z + z .Furthermore, the topological recursion determines the generating functions F g , n ( x , . . . , x n ) uniquely. Theorem 1.
For g − + n > , the multidifferential (1.4) Ω g , n ( z , . . . , z n ) = ∂∂ x · · · ∂∂ x n F g , n ( x , . . . , x n ) d x ⊗ · · · ⊗ d x n is the analytic expansion of the invariant ω gn of the spectral curve (1.31.3) at the point x = · · · = x n = ∞ . From the universality property described above, the asymptotic behaviour of the generating function (1.21.2)near its pole at ( z , . . . , z n ) = (
1, . . . , 1 ) is given by 4 g − + n K g , n ( z , . . . , z n ) . More precisely, we have F g , n ( x , . . . , x n ) = g − + n ∑ | d | = g − + n (cid:90) M g , n c ( L ) d · · · c ( L n ) d n n ∏ i = ( d i − ) !! ( z i − ) d i + [ lower order poles ] .Note that the spectral curve given by equation (1.31.3) is irregular. The local behaviour near its poles ( z , . . . , z n ) = ( ±
1, . . . , ± ) is our main interest. One immediate consequence of irregularity is the novelfeature that the genus 0 generating functions F n ( x , . . . , x n ) are analytic at ( z , . . . , z n ) = ( −
1, . . . , − ) . Moregenerally, the orders of poles of F g , n ( x , . . . , x n ) at ( z , . . . , z n ) = ( −
1, . . . , − ) are independent of n . This isin contrast to ω gn having poles of order 6 g − + n , which is the case for most of the spectral curves thatappear in the literature.Properties of rational functions on the curve (1.31.3) yield a structure theorem for B g , n — see Theorem 44 — aswell as explicit formulae. For example, we have B n ( µ , . . . , µ n ) = − n ( n − ) ! | µ | ( | µ | + ) (cid:18) | µ | + n − (cid:19) n ∏ i = (cid:18) µ i µ i (cid:19) , where | µ | = n ∑ i = µ i .Another consequence of Theorem 11 is a general property of the invariants ω gn , known as the dilaton equation .For the spectral curve of interest, it implies that B g , n + ( µ , . . . , µ n ) − B g , n + ( µ , . . . , µ n ) = ( | µ | + g − + n ) B g , n ( µ , . . . , µ n ) ,where one can make sense of evaluation at µ i = B g , n . Moreover, weprove in Proposition 3.133.13 that for n positive, B g , n + m ( µ , . . . , µ n , 0, 0, . . . , 0 ) has a combinatorial meaning — itenumerates dessins d’enfant with m black vertices labelled.The enumerative problem in this paper and the associated spectral curve given by equation (1.31.3) are closelyrelated to others in the literature. Topological recursion on rational spectral curves with x = α + γ ( z + z ) describe enumeration of discrete surfaces [1414], which includes the special case of lattice points in moduli NORMAN DO AND PAUL NORBURY spaces of curves [2323], a more refined version of dessin enumeration [11, 44, 1919], and the Gromov–Witteninvariants of P [1010, 2424]. However, note that each of these examples is governed by a regular spectral curve.A quantum curve of a spectral curve P ( x , y ) = (cid:98) P ( (cid:98) x , (cid:98) y ) Z ( x , ¯ h ) =
0, where (cid:98) P ( (cid:98) x , (cid:98) y ) is a non-commutative quantisation of the spectral curve with (cid:98) x = x and (cid:98) y = ¯ h ∂∂ x . This differentialoperator annihilates a wave function Z ( x , ¯ h ) , which is a formal series in ¯ h associated to the spectral curve.The path from the quantum curve to the spectral curve is well-defined — in the semi-classical limit ¯ h → (cid:98) x and (cid:98) y . One remedy for these issues is a conjectural construction of the wave function Z ( x , ¯ h ) from the invariants ω gn of the spectral curve, suggested for example by Gukov and Sułkowski [1616]. We prove this conjecture forthe spectral curve xy − xy + = Z ( x , ¯ h ) = x − h exp (cid:20) ∞ ∑ g = ∞ ∑ n = ¯ h g − + n n ! F g , n ( x , x , . . . , x ) (cid:21) Theorem 2.
The quantum curve of xy − xy + = is given by ( (cid:98) y (cid:98) x (cid:98) y − (cid:98) y (cid:98) x + ) Z ( x , ¯ h ) = . Strictly speaking, to make sense of the action of a differential operator on a formal series in ¯ h , it is necessaryto know that all sums are finite. In Section 66, we give a more precise statement of Theorem 22, in terms of adifferential operator annihilating the formal series Z ( x , ¯ h ) = x h Z ( x , ¯ h ) ∈ Q [ ¯ h ± ][[ x − ]] .One of the main purposes of the present paper is to understand the universality exhibited by the invariantsassociated to the irregular spectral curve(1.5) xy = x , this plays the role of the Airy curve at regular zeros of d x . We expect many resultsrelating the Airy curve to invariants of spectral curves to have analogues in this setting. In particular, weexpect the invariants of our curve to be related to a new moduli space. In Section 77, we apply topologicalrecursion directly to the spectral curve given by equation (1.51.5). We calculate putative volumes of theseunidentified moduli spaces and dually, intersection numbers on them. The invariants are non-zero only forpositive genus, much like enumeration of branched covers of a torus or volumes of spaces of holomorphicdifferentials.Indirectly, we use the asymptotic behaviour of F g , n ( x , . . . , x n ) defined in equation (1.21.2) near its pole ( z , . . . , z n ) = ( −
1, . . . , − ) to study the spectral curve (1.51.5). More generally, the type of enumerativeproblem governed by (1.51.5) necessarily has no contribution in genus 0. To make this idea clearer, consider forthe moment the enumeration of non-bipartite fatgraphs with bipartite boundary components — in otherwords, boundary components of even lengths. For example, the square graph pictured below left is bipartiteand may be considered the boundary of the non-bipartite genus 1 fatgraph pictured below right.Define NB g ( µ , . . . , µ n ) to be the weighted count of connected genus g non-bipartite fatgraphs with labelledbipartite boundaries of lengths 2 µ , . . . , 2 µ n and such that the vertices are required to have valency greaterthan or equal to two — see equation (3.43.4). The valency condition on the vertices reduces the growth in µ i from exponential to polynomial. In particular, this invariant vanishes in genus zero since bipartite OPOLOGICAL RECURSION FOR IRREGULAR SPECTRAL CURVES 5 boundary components implies that the graph is bipartite for simple homological reasons. One can showthat NB g ( µ , . . . , µ n ) is quasi-polynomial in ( µ , . . . , µ n ) modulo 2. An interesting invariant is obtained bymeasuring its failure to be polynomial, which we do in the following way. Recall that if p ( µ ) is a quasi-polynomial modulo 2, then it has a natural decomposition p ( µ ) = p + ( µ ) + ( − ) µ p − ( µ ) , where p ± ( µ ) arepolynomials. For the analogous decomposition of a quasi-polynomial p ( µ , . . . , µ n ) in severable variables, itis the coefficient of ( − ) | µ | that interests us. p ( µ , . . . , µ n ) = p + ( µ , . . . , µ n ) + ( − ) | µ | p − ( µ , . . . , µ n ) + [ other terms involving powers of − ] For example, NB ( µ , . . . , µ n ) = NB ( µ ) = µ − (cid:101) ( µ ) NB ( µ , µ ) = ( µ + µ )( µ + µ − ) − (cid:101) ( | µ | )
16 ,where (cid:101) ( µ ) = [ − ( − ) µ ] . From these expressions, one can extract the non-polynomial parts p − ( µ ) = and p − ( µ , µ ) = , which determine the invariants ω and ω of the spectral curve xy =
1. More generally,the top degree part of p − g ( µ , . . . , µ n ) is equivalent to the invariant ω gn of xy = n ( n + ) B g ,1 ( n ) = ( n − )( n − ) B g ,1 ( n − ) + ( n − ) ( n − ) B g ,1 ( n − ) ,which is proven in Section 44. We remark that it has a rather different character to the topological recursion ofTheorem 11. In particular, it enables one to calculate B g ,1 recursively from B h ,1 for h ≤ g , without requiring B h , n for n ≥
2. It is analogous to the three-term recursion for fatgraphs with one face of Harer and Zagier [1717].Our three-term recursion implies a recursion satisfied by the 1-point invariants of the spectral curve xy = Theorem 3.
The 1-point invariants of the spectral curve xy = , given parametrically by x ( z ) = z and y ( z ) = z ,are ω g ( z ) = − g ( g ) ! g ! ( g − ) z − g d z .We would hope to recognise some type of intersection number in the formula of Theorem 33 analogous to the1-point invariants of the Airy curve x = y , which are given by ω g ( z ) Airy = − g ( g − ) !3 g g ! ( g − ) ! z − g d z = − g ( g − ) !! (cid:90) M g ,1 c ( L ) g − z − g d z .The spectral curve xy − xy + = y − xy + = y − xy + = ω gn = ∑ M g , n ( µ , . . . , µ n ) x − µ · · · x − µ n n , where M g , n ( µ , . . . , µ n ) = ∑ Γ ∈F g , n ( µ ) | Aut Γ | .Here, F g , n ( µ ) denotes the set of connected genus g fatgraphs — graphs embedded in a connected orientablesurface such that the complement is a union of disks — with n labelled boundary components of lengths µ , . . . , µ n . Again, we require an isomorphism between two fatgraphs to preserve the labelling on theirboundary components. Note that each bipartite fatgraph can be bicoloured in two distinct ways, therebyproducing two dessins d’enfant, so we obtain B g , n ( µ , . . . , µ n ) ≤ M g , n ( µ , . . . , 2 µ n ) .In fact, equality occurs when g =
0, although no such explicit relation exists in higher genus.
NORMAN DO AND PAUL NORBURY
One can obtain the spectral curves y − xy + = xy − xy + = y = (cid:90) − ρ ( t ) x − t d t = ∞ ∑ n = C n x n + and y = (cid:90) λ ( t ) x − t d t = ∞ ∑ n = C n x n + of the probability densities ρ ( t ) = π (cid:112) − t · [ − ] and λ ( t ) = π (cid:114) − tt · [ ] .These are known as the Wigner semicircle distribution and the Marchenko–Pastur distribution, respectively.It is elementary to show that the Catalan numbers C n = n + ( nn ) arise as moments of these probabilitydensities. C n = (cid:90) − t n ρ ( t ) d t = (cid:90) t n λ ( t ) d t Regular behaviour of spectral curves arise from so-called soft edge statistics, while the irregular behaviour ofthe spectral curve given by equation (1.31.3) arises from so-called hard edge statistics. These terms refer to thebehaviour of the associated probability densities at the endpoints of the interval of support.2. T
OPOLOGICAL RECURSION
Topological recursion takes as input a spectral curve ( C , B , x , y ) consisting of a compact Riemann surface C , abidifferential B on C , and meromorphic functions x , y : C → C . We furthermore require that the zeros of d x are simple and disjoint from the zeros of d y [1313]. A more general setup allows local spectral curves , in which C is an open subset of a compact Riemann surface. In this paper, we deal exclusively with the case when C isthe Riemann sphere CP , with global rational parameter z , endowed with the bidifferential B = d z ⊗ d z ( z − z ) . Thetwo main examples that we consider take the pair of meromorphic functions to be ( x , y ) = ( z + z + z + z ) and ( x , y ) = ( z , z ) . They are mild variants of the usual setup, since in both cases, d y has a pole at a zero ofd x . Nevertheless, as we will see below, topological recursion is well-defined in this case and retains many ofthe desired properties, while losing some others.For integers g ≥ n ≥
1, topological recursion outputs multidifferentials ω gn ( p , . . . , p n ) on C — in otherwords, a tensor product of meromorphic 1-forms on the product C n , where p i ∈ C . When 2 g − + n > ω gn ( p , . . . , p n ) is defined recursively in terms of local information around the poles of ω g (cid:48) n (cid:48) ( p , . . . , p n (cid:48) ) for2 g (cid:48) + − n (cid:48) < g − + n .Since each zero α of d x is assumed to be simple, for any point p ∈ C close to α , there is a unique pointˆ p (cid:54) = p close to α such that x ( ˆ p ) = x ( p ) . The recursive definition of ω gn ( p , . . . , p n ) uses only local informationaround zeros of d x and makes use of the well-defined map p (cid:55)→ ˆ p there. The invariants are defined as follows,for C = CP and B = d z ⊗ d z ( z − z ) with z i = z ( p i ) . Start with the base cases ω = − y ( z ) d x ( z ) and ω = d z ⊗ d z ( z − z ) .For 2 g − + n > S = {
2, . . . , n } , define(2.1) ω gn ( z , z S ) = ∑ α Res z = α K ( z , z ) (cid:20) ω g − n + ( z , ˆ z , z S ) + ◦ ∑ g + g = gI (cid:116) J = S ω g | I | + ( z , z I ) ω g | J | + ( ˆ z , z J ) (cid:21) ,where the outer summation is over the zeros α of d x and the ◦ over the inner summation means that weexclude terms that involve ω . We define K by the following formula K ( z , z ) = − (cid:82) z ˆ z ω ( z , z (cid:48) ) [ y ( z ) − y ( ˆ z )] d x ( z ) = [ y ( ˆ z ) − y ( z )] x (cid:48) ( z ) (cid:18) z − z − z − z (cid:19) d z d z , OPOLOGICAL RECURSION FOR IRREGULAR SPECTRAL CURVES 7 which is well-defined in the vicinity of each zero of d x . Note that the quotient of a differential by thedifferential d x ( z ) is a meromorphic function. The recursion is well-defined even when y has poles at thezeros α of d x . It does not use all of the information in the pair ( x , y ) , but depends only on the meromorphicdifferential y d x and the local involutions p (cid:55)→ ˆ p . For 2 g − + n >
0, the multidifferential ω gn is symmetric,with poles only at the zeros of d x and vanishing residues.In the case ( x , y ) = ( z , z ) , the differential y d x = z is analytic and non-vanishing at the zero z = x .This leads to the vanishing of its genus zero invariants, since the kernel K ( z , z ) has no pole at z = ω n has no pole at z =
0. Interesting invariants arise via ω , since a pole at z = ω ( z , ˆ z ) = d z ⊗ dˆ z ( z − ˆ z ) . Non-triviality of ω leads to non-triviality of ω gn for all g ≥ g − + n >
0, the invariants ω gn of regular spectral curves satisfy the following string equations for m =
0, 1 [1313].(2.2) ∑ α Res z = α x m y ω gn + ( z , z S ) = − n ∑ j = dz j ∂∂ z j (cid:32) x m ( z j ) ω gn ( z S ) dx ( z j ) (cid:33) They also satisfy the dilaton equation [1313](2.3) ∑ α Res z = α Φ ( z ) ω gn + ( z , z , . . . , z n ) = ( − g − n ) ω gn ( z , . . . , z n ) ,where the summation is over the zeros α of d x and Φ ( z ) = (cid:82) z y d x ( z (cid:48) ) is an arbitrary antiderivative. Thedilaton equation enables the definition of the so-called symplectic invariantsF g = ∑ α Res z = α Φ ( z ) ω g ( z ) The dilaton equation still holds for irregular spectral curves whereas the string equations no longer hold. Thefailure of the string equations can be explicitly observed for the curve xy = Irregular spectral curves.
One can classify the local behaviour of a spectral curve near a zero of d x into four types — one of these is regular and the other three are irregular. In all four cases, one can definemultidifferentials ω gn using equation (2.12.1). If α is a zero of d x , then one of the following four cases must occur.(1) Regular.
The form d y is analytic and d y ( α ) (cid:54) = α is a regular zero of d x if it is a smooth point of C . In this case, there is a pole of ω gn at α of order 6 g − + n [1313].(2) Irregular. (a) The form d y is analytic at α and d y ( α ) = ω gn ( z , . . . , z n ) loses the key property of symmetry under permuta-tions of z , . . . , z n . (Note that the symmetry of ω gn is not a priori apparent, since the recursion ofequation (2.12.1) treats z as special.) For example, if we consider the rational spectral curve givenparametrically by x ( z ) = z and y ( z ) = z , then topological recursion yields ω ( z , z , z ) = z z z (cid:104) z z + z z + z z − z z z (cid:105) .(b) The meromorphic function y has a pole at α of order greater than one.In this case, the kernel K ( z , z ) defined above has no pole at α due to the pole of y that appearsin the denominator. The residue at α in equation (2.12.1) therefore vanishes and one obtains nocontribution from a neighbourhood of α . The invariants in this case match those of the localspectral curve obtained by removing the point α .(c) The meromorphic function y has a simple pole at α .This case is the main concern of the present paper. Again, the kernel K ( z , z ) defined above hasno pole at α , but a pole of ω at α allows non-zero invariants to survive. The invariants enjoymany of the properties of the invariants for regular curves, such as symmetry of ω gn ( z , . . . , z n ) NORMAN DO AND PAUL NORBURY under permutations of z , . . . , z n . The pole of ω gn at α is now of order 2 g , which follows from thelocal analysis in Section 77.We conclude that the only interesting cases are (1) and (2c), which involve regular zeros of d x or a zero of d x at which y has a simple pole. If case (2a) is to prove interesting, then one would probably need to adjust thedefinition of topological recursion in order to recover the symmetry of the invariants.3. E NUMERATING DESSINS D ’ ENFANT
Loop equations.
Kazarian and Zograf [1919] prove that U g ( µ , . . . , µ n ) = µ · · · µ n B g , n ( µ , . . . , µ n ) satis-fies the recursion(3.1) U g ( µ , µ S ) = n ∑ j = µ j U g ( µ + µ j − µ S \{ j } ) + ∑ i + j = µ − (cid:20) U g − ( i , j , µ S ) + ∑ g + g = gI (cid:116) J = S U g ( i , µ I ) U g ( j , µ J ) (cid:21) for S = {
2, . . . , n } and the base case U ( ) =
1. The proof uses an elementary cut-and-join argument that is avariation of the Tutte recursion [1414]. Note that U ( µ ) = C µ = µ + ( µµ ) is a Catalan number. This is due to thefact that the recursion of equation (3.13.1) in the case ( g , n ) = (
0, 1 ) reproduces the Catalan recursion and initialcondition C m = ∑ i + j = m − C i C j and C = W g ( x , . . . , x n ) = ∞ ∑ µ ,..., µ n = U g ( µ , . . . , µ n ) n ∏ i = x − µ i − i satisfy loop equations W g ( x , x S ) = W g − ( x , x , x S ) + ∑ g + g = gI (cid:116) J = S W g ( x , x I ) W g ( x , x J ) (3.2) + n ∑ j = (cid:20) ∂∂ x j W g ( x , x S \{ j } ) − W g ( x S ) x − x j + x ∂∂ x j W g ( x S ) (cid:21) + δ g ,0 δ n ,1 x .The solution of the loop equations for ( g , n ) = (
0, 1 ) defines the spectral curve via the equation y = W ( x ) = ∞ ∑ µ = U ( µ ) x − µ − = ∞ ∑ µ = µ + (cid:18) µµ (cid:19) x − µ − = z + z , where x = z + z + ∏ x − µ i − i in W g ( x , x S ) is U g ( µ , µ S ) .The coefficient of ∏ x − µ i − i in W g − ( x , x , x S ) is ∑ i + j = µ − U g − ( i , j , µ S ) .The coefficient of ∏ x − µ i − i in W g ( x , x I ) W g ( x , x J ) is ∑ i + j = µ − U g ( i , µ I ) U g ( j , µ J ) .The coefficient of ∏ x − µ i − i in ∂∂ x j (cid:20) W g ( x , x S \{ j } ) − W g ( x S ) x − x j + W g ( x S ) x (cid:21) is µ j U g ( µ + µ j − µ S \{ j } ) . Thisobservation uses the fact that ∂∂ x j x − k − x − kj x − x j + x − kj x = − ∂∂ x j k − ∑ m = x m − k − x − mj = k − ∑ m = m x m − k − x − m − j . OPOLOGICAL RECURSION FOR IRREGULAR SPECTRAL CURVES 9
Remark 3.1.
The loop equations (3.23.2) are almost a special case of the loop equations appearing in the workof Eynard and Orantin [1414, Theorem 7.2]. Using their notation, equation (3.23.2) would correspond to V (cid:48) ( x ) = P ( g ) n ( x , . . . , x n ) would not be polynomial in x . Note that such a choice of V and P ( g ) n has no meaningthere.3.2. Pruned dessins.
Define b g , n ( µ ) ⊆ B g , n ( µ ) to be the set of genus g dessins without vertices of valence 1and with n labelled boundary components of lengths 2 µ , . . . , 2 µ n . We refer to such dessins without verticesof valence 1 as pruned . The notion of pruned structures has found applications for various other problems [88]and will allow us to prove polynomiality for the dessin enumeration here. Definition 3.2.
For any µ = ( µ , . . . , µ n ) ∈ Z n + , define b g , n ( µ , . . . , µ n ) = ∑ Γ ∈ b g , n ( µ ) | Aut Γ | . Proposition 3.3.
The numbers b g , n ( µ , . . . , µ n ) and B g , n ( µ , . . . , µ n ) are related by the equation (3.3) ∞ ∑ ν ,..., ν n = b g , n ( ν , . . . , ν n ) n ∏ i = z ν i i = ∞ ∑ µ ,..., µ n = B g , n ( µ , . . . , µ n ) n ∏ i = x − µ i i , where x i = z i + z i + . The two sides are analytic expansions of the generating function of equation (1.21.2) atz = · · · = z n = .Proof. The main idea is that a dessin can be created from a pruned dessin by gluing planar trees to theboundary components. The bicolouring of the vertices extends in a unique way to the additional trees.Conversely, one obtains a unique pruned dessin from a dessin via the process of pruning — in other words,repeatedly removing degree one vertices and their incident edges until no more exist.By gluing planar trees, the count b g , n ( ν , . . . , ν n ) contributes to B g , n ( ν + k , . . . , ν n + k n ) for all non-negativeintegers k , . . . , k n . The contribution is equal to b g , n ( ν , . . . , ν n ) multiplied by the number of ways to glueplanar trees with a total of k edges to boundary component 1, multiplied by the number of ways to glueplanar trees with a total of k edges to boundary component 2, and so on.The number of ways to glue k edges to a boundary component of length b can be computed as follows.It is simply the number of ways to pick rooted planar trees T , T , . . . , T b with k edges in total. There are C i rooted planar trees with i edges, where C = C = C = C =
14, . . . is the sequence of Catalannumbers. So the number of ways to choose rooted planar trees T , T , . . . , T b with k edges in total is simplythe x b coefficient of ( C + C X + C X + C X + · · · ) b = (cid:16) −√ − X X (cid:17) b = f ( X ) b . If we call this number C bk ,we obtain the following formula. B g , n ( µ , . . . , µ n ) = ∞ ∑ k ,..., k n = b g , n ( µ − k , . . . , µ n − k n ) C µ − k k · · · C µ n − k n k n In fact, one can show that C bk = bb + k ( b − + kk ) . Therefore, we have the following chain of equalities. ∞ ∑ µ ,..., µ n = B g , n ( µ , . . . , µ n ) n ∏ i = x − µ i i = ∞ ∑ µ ,..., µ n = ∞ ∑ k ,..., k n = b g , n ( µ − k , . . . , µ n − k n ) n ∏ i = C µ i − k i k i x − µ i i = ∞ ∑ ν ,..., ν n = ∞ ∑ k ,..., k n = b g , n ( ν , . . . , ν n ) n ∏ i = C ν i k i x − ν i − k i i = ∞ ∑ ν ,..., ν n = b g , n ( ν , . . . , ν n ) n ∏ i = x − ν i i ∞ ∑ k i = C ν i k i x − k i i = ∞ ∑ ν ,..., ν n = b g , n ( ν , . . . , ν n ) n ∏ i = (cid:32) f ( x − i ) x i (cid:33) ν i It remains to show that z i = f ( x − i ) x i = x i − − (cid:113) x i − x i x i = z i + z i + (cid:3) The next proposition shows that b g , n satisfies a functional recursion , in the sense that it involves only termswith simpler ( g , n ) complexity on the right hand side. This will be useful in understanding the pole structureof the generating function (1.21.2). Such a recursion is in contrast with equation (3.13.1) in the non-pruned case,which includes ( g , n ) terms on both sides of the equation. Proposition 3.4.
The pruned dessin enumeration satisfies the following recursion for ( g , n ) (cid:54) = (
0, 1 ) , (
0, 2 ) , (
0, 3 ) , (
1, 1 ) . | µ | b g , n ( µ ) = n ∑ i = ∑ p + q + r = µ i pqr (cid:20) b g − n + ( p , q , µ S \{ i } ) + stable ∑ g + g = gI (cid:116) J = S \{ i } b g , | I | + ( p , µ I ) b g , | J | + ( q , µ J ) (cid:21) + ∑ i (cid:54) = j ∑ p + q = µ i + µ j pq b g , n − ( µ S \{ i , j } ) Here, S = {
1, 2, . . . , n } and we set µ I = ( µ i , µ i , . . . , µ i k ) for I = { i , i , . . . , i k } . The word stable over thesummation indicates that we exclude all terms that involve b or b .Proof. We count dessins in b g , n ( µ ) with a marked edge. The most obvious way to count such objects is tochoose a dessin in the set, which can be accomplished in b g , n ( µ ) ways, and then to choose a suitable edge,which can be accomplished in | µ | ways. So the total number of such objects is | µ | b g , n ( µ ) , which forms theleft hand side of the recursion.To form the right hand side of the recursion, we count the same objects in the following way. For a dessin in b g , n ( µ ) with a marked edge, remove the marked edge and repeatedly remove degree 1 vertices and theirincident edges to obtain a pruned dessin. One of the following three cases must arise. Case 1.
The marked edge is adjacent to face i on both sides and its removal leaves a connected dessin.Suppose that r edges are removed in total — they necessarily form a path. The resulting pruneddessin must lie in the set b g − n + ( p , q , µ S \{ i } )) , where p + q + r = µ i .Conversely, there are pqr ways to reconstruct a marked pruned dessin in b g , n ( µ ) from a pruneddessin in the set b g − n + ( p , q , µ S \{ i } )) , where p + q + r = µ i , by adding a path of r edges. The factor r accounts for choosing a marked edge along the path. The factors p and q account for the choice ofendpoints of the path. Case 2.
The marked edge is adjacent to face i on both sides and its removal leaves the disjoint unionof two dessins.Suppose that r edges are removed in total — they necessarily form a path. Suppose that the two OPOLOGICAL RECURSION FOR IRREGULAR SPECTRAL CURVES 11 components have faces I and J . The resulting pruned dessins must lie in the sets b g , | I | + ( p , µ I )) and b g , | J | + ( q , µ J )) , where p + q + r = µ i . Furthermore, we cannot obtain a pruned dessin of type (
0, 1 ) in this manner.Conversely, there are pqr ways to reconstruct a marked pruned dessin in b g , n ( µ ) from two pruneddessins in the sets b g , | I | + ( p , µ I )) and b g , | J | + ( q , µ J )) , where p + q + r = µ i , by adding a path of r edges. The factor r accounts for choosing a marked edge along the path. The factors p and q accountfor the choice of endpoints of the path. Case 3.
The marked edge is adjacent to faces i and j , for i (cid:54) = j .Suppose when walking along the marked edge from the white vertex to the black vertex that face i lieson the left and face j on the right. Suppose that q edges are removed in total — they necessarily forma path. The resulting pruned dessin must lie in the set b g , n − ( p , µ S \{ i , j } )) , where p + q = µ i + µ j .Conversely, there are pq ways to reconstruct a marked pruned dessin in b g , n ( µ ) from a pruned dessinin the set b g , n − ( p , µ S \{ i , j } )) , where p + q = µ i + µ j , by adding a path of q edges. The factor q accountsfor choosing an edge along the path. The factor p arises from choosing where to glue one of the endsof the path. Note that there is a unique choice to glue in the other end to create a face of perimeter µ i on the left and a face of perimeter µ j on the right.There is a crucial subtlety that arises in the third case, which we now address. One can discern the issue byconsidering the sequence of diagrams below, in which µ i increases from left to right, relative to µ j . i j µ i ≈ µ j i j µ i > µ j ij critical ij µ i (cid:29) µ j The third case actually contributes to diagrams like the one on the far right, in which face i completelysurrounds face j , or vice versa. In fact, the edge that we remove can lie anywhere along the dashed path inthe schematic diagram. Note that this contributes to the second case, in which the marked edge is adjacent tothe face labelled i on both sides and its removal leaves the disjoint union of two connected graphs. However,observe that this surplus contribution is precisely equal to the terms from the second case that involve b , soone can compensate simply by excluding such terms. Given that we have already witnessed that b =
0, wecan restrict to the so-called stable terms in the second case, which are precisely those that do not involve b or b .Therefore, to obtain all marked dessins in b g , n ( µ ) exactly once, it is necessary to perform the reconstructionprocess in the first case for all values of i and p + q + r = µ i ;in the second case for all stable values of i , p + q + r = µ i , g + g = g , and I (cid:116) J = S \ { i } ; andin the third case for all values of i , j , and p + q = µ i + µ j .We obtain the desired recursion by summing up over all these contributions. (cid:3) Example 3.5.
Calculation of b ( µ ) builds dessins from loops of circumference p .2 µ b ( µ ) = ∑ p + q = µ p even pq = (cid:40) µ ( µ − ) µ even µ ( µ − ) µ odd Note that this is precisely the recursion of Proposition 3.43.4, using the fact that b ( µ , µ ) = δ ( µ , µ ) µ . Takingthis definition, one can also apply the recursion in the case ( g , n ) = (
0, 3 ) . Finally we obtain b ( µ , µ , µ ) = b ( µ ) = (cid:40) ( µ − ) , µ even ( µ − ) , µ odd.Calculation of b g , n for small values of g and n indicates that it is a quasi-polynomial modulo 2, although thisstructure does not follow immediately from the recursion above. In order to prove it, we use the followingasymmetric version of the recursion. Proposition 3.6.
The pruned dessin enumeration satisfies the following recursion for ( g , n ) (cid:54) = (
0, 1 ) , (
0, 2 ) , (
0, 3 ) , (
1, 1 ) . µ b g , n ( µ , µ S ) = ∑ p + q + r = µ pqr (cid:20) b g − n + ( p , q , µ S ) + stable ∑ g + g = gI (cid:116) J = S b g , | I | + ( p , µ I ) b g , | J | + ( q , µ J ) (cid:21) + ∑ i ∈ S (cid:20) ∑ p + q = µ + µ i pq b g , n − ( µ S ) (cid:12)(cid:12) µ i = p + sign ( µ − µ i ) ∑ p + q = | µ − µ i | pq b g , n − ( µ S ) (cid:12)(cid:12) µ i = p (cid:21) Here, S = {
2, . . . , n } and for I = { i , i , . . . , i k } , we set µ I = ( µ i , µ i , . . . , µ i k ) .Proof. Use the fact that both the symmetric recursion given by Proposition 3.43.4 and the asymmetric recursionhere uniquely determine all b g , n ( µ ) from the base cases b and b . So it suffices to show that the symmetricversion follows from the asymmetric version, and this can be seen by symmetrising. (cid:3) Corollary 3.7.
For ( g , n ) (cid:54) = (
0, 1 ) or (
0, 2 ) , b g , n ( µ , . . . , µ n ) is a quasi-polynomial modulo 2 of degree g − + n in µ , . . . , µ n .Proof. To show that b g , n is a polynomial in the squares, we use the following two facts, which are straightfor-ward to verify.For all non-negative integers a and b , the functions f ( ) a , b ( µ ) = ∑ p + q + r = µ p + q even p a + q b + r and f ( ) a , b ( µ ) = ∑ p + q + r = µ p + q odd p a + q b + r are odd quasi-polynomials modulo 2 in µ of degree 2 a + b + a , the functions g ( ) a ( µ , µ ) = ∑ p + q = µ + µ p even p a + q + sign ( µ − µ ) ∑ p + q = | µ − µ | p even p a + qg ( ) a ( µ , µ ) = ∑ p + q = µ + µ p odd p a + q + sign ( µ − µ ) ∑ p + q = | µ − µ | p odd p a + q are quasi-polynomials modulo 2 that are odd in µ and even in µ of degree 2 a + b ( µ , µ , µ ) and b ( µ ) are indeed even quasi-polynomialsmodulo 2 of degrees 0 and 1, respectively. Now consider b g , n satisfying 2 g − + n ≥ b g , n of lesser complexity. Then the recursion of Proposition 3.63.6 expresses µ b g , n ( µ , . . . , µ n ) as a finite linear combination of terms of the form f ( c ) a , b ( µ ) ∏ k ∈ S µ a k k and g ( c ) a ( µ , µ i ) ∏ k ∈ S \{ i } µ a k k , OPOLOGICAL RECURSION FOR IRREGULAR SPECTRAL CURVES 13 where a and b are non-negative integers and c ∈ {
0, 1 } . Upon dividing by µ , we find that b g , n ( µ , . . . , µ n ) isa quasi-polynomial modulo 2 that is even in µ , . . . , µ n . Furthermore, one can check that it is of the correctdegree. Therefore, we have proven the proposition by induction on 2 g − + n . (cid:3) g n condition on ( µ , . . . , µ n ) b g , n ( µ , . . . , µ n ) δ ( µ , µ ) µ µ + µ + µ + µ −
11 1 µ even ( µ − ) µ odd ( µ − ) µ + µ even ( µ + µ − )( µ + µ − ) µ + µ odd ( µ + µ − )( µ + µ − ) µ even ( µ − )( µ − )( µ − µ + ) µ odd ( µ − )( µ − )( µ − µ + ) The structure theorem for b g , n ( µ , . . . , µ n ) implies the following structure theorem for Ω g , n ( z , . . . , z n ) = ∂∂ x · · · ∂∂ x n F g , n ( x , . . . , x n ) d x ⊗ · · · ⊗ d x n via Proposition 3.33.3. This will play an important role in the proof of Theorem 11. Proposition 3.8.
For ( g , n ) (cid:54) = (
0, 1 ) or (
0, 2 ) , Ω g , n ( z , . . . , z n ) is a meromorphic multidifferential on the rationalcurve x = z + z + , with poles only at z i = ± . Furthermore, it satisfies the skew invariance property Ω g , n ( z , . . . , z i , . . . , z n ) = − Ω g , n ( z , . . . , z n ) , for i =
1, 2, . . . , n . Proof.
We begin with the result of Proposition 3.33.3. F g , n ( x , . . . , x n ) = ∞ ∑ ν ,..., ν n = b g , n ( ν , . . . , ν n ) n ∏ i = z ν i i The structure theorem for b g , n allows us to express this as a linear combination of terms of the form n ∏ i = f ( s i ) k i ( z i ) ,where f ( ) k ( z ) = ∑ ν even ν k z ν = (cid:18) z ∂∂ z (cid:19) k z − z and f ( ) k ( z ) = ∑ ν odd ν k z ν = (cid:18) z ∂∂ z (cid:19) k z − z .Note that f ( ) ( z ) + f ( ) ( z ) = z − z + z − z = − f ( ) ( z ) + f ( ) ( z ) = z − z + z − z = (cid:16) z ∂∂ z (cid:17) = (cid:16) w ∂∂ w (cid:17) for w = z , we have f ( s ) k ( z ) + f ( s ) k ( z ) = s =
0, 1 and k ≥
1. It follows that F g , n ( x , . . . , x n ) = − F g , n ( x , . . . , x n ) (cid:12)(cid:12) z i (cid:55)→ z i + [ terms independent of z i ] . Now apply the total derivative to both sides to obtain the skew invariance property for Ω g , n . (cid:3) Remark 3.9.
The pruned version of the enumeration of fatgraphs M g , n ( µ ) defined in the introduction givesrise to N g , n ( µ , . . . , µ n ) = ∑ Γ ∈ f g , n ( µ ) | Aut Γ | .for f g , n ( µ ) ⊆ F g , n ( µ ) defined as the subset of connected genus g fatgraphs without vertices of valence 1.This was studied in [2222, 2323] and shown to satisfy N g , n ( µ , . . . , µ n ) is a degree 6 g − + n quasi-polynomial modulo 2 N g , n ( µ , . . . , µ n ) = | µ | odd; N g , n (
0, . . . , 0 ) = χ ( M g , n ) ; andhighest coefficients of N g , n are psi-class intersection numbers on M g , n .Genus 0 fatgraphs with even length boundary components admit bipartite colourings, and there are exactlytwo ways to bicolour the vertices, so b n ( µ ) = N n ( µ ) .The N g , n satisfy a recursion similar to that in Proposition 3.43.4 and the two recursions coincide in genus 0 withall µ i even. The recursion relation does not specialise to the case of even µ i in general — for example, N ( µ ) requires N ( µ , µ , µ ) with some µ i odd. The non-bipartite enumeration mentioned in the introduction canbe obtained as(3.4) NB g ( µ , . . . , µ n ) = N g , n ( µ , . . . , 2 µ n ) − b g , n ( µ , . . . , µ n ) ,which vanishes when g =
0. It is worth noting that N g , n ( µ , . . . , 2 µ n ) is a polynomial in µ , . . . , µ n ratherthan a quasi-polynomial. Remark 3.10.
For positive genus, we have b g , n ( µ ) (cid:54) = N g , n ( µ ) in general. To check that a given fatgraph Γ is bipartite, one needs to check that β ( Γ ) independent cycles have even length, where β ( Γ ) = e ( Γ ) − v ( Γ ) + = g ( Γ ) − + n ( Γ ) is the first Betti number of Γ . Here, e , v , g , n denote the number of edges, number of vertices, genus, andnumber of boundary components, respectively.For g =
0, one can perform this check on n ( Γ ) − b n ( µ ) = N n ( µ ) , as desired. For general genus, we can check on n ( Γ ) − g more independent cycles, and eachof these imposes an independent parity condition. So we obtain the asymptotic relation b g , n ( µ ) ∼ g N g , n ( µ ) .We know that N g , n ( µ ) is polynomial in µ , . . . , µ n with leading coefficients for a + · · · + a n = g − + n given by [ µ a · · · µ a n n ] N g , n ( µ ) = g a ! · · · a n ! (cid:90) M g , n c ( L ) a · · · c ( L n ) a n .It follows that [ µ a · · · µ a n n ] b g , n ( µ ) = g − a ! · · · a n ! (cid:90) M g , n c ( L ) a · · · c ( L n ) a n . OPOLOGICAL RECURSION FOR IRREGULAR SPECTRAL CURVES 15
Proof of topological recursion.
Proof of Theorem 11.
The proof that the loop equations (3.23.2) imply topological recursion for the spectral curve x = z + z + y = z + z is standard. If we write y = W ( x ) , then (3.23.2) implies for 2 g − + n > W g ( x , x S ) − y W g ( x , x S ) = W g − ( x , x , x S ) + stable ∑ g + g = gI (cid:116) J = S W g ( x , x I ) W g ( x , x J )+ n ∑ j = (cid:20) W ( x , x j ) W g ( x , x S \{ j } ) + ∂∂ x j W g ( x , x S \{ j } ) x − x j − ∂∂ x j W g ( x S ) x − x j + x ∂∂ x j W g ( x S ) (cid:21) ⇒ − z + z W g ( x , x S ) d x ⊗ d x S = W g − ( x , x , x S ) d x ⊗ d x S + stable ∑ g + g = gI (cid:116) J = S W g ( x , x I ) W g ( x , x J ) d x ⊗ d x S + n ∑ j = (cid:20) W ( x , x j ) W g ( x , x S \{ j } ) + ∂∂ x j W g ( x , x S \{ j } ) x − x j − ∂∂ x j W g ( x S ) x − x j + x ∂∂ x j W g ( x S ) (cid:21) d x ⊗ d x S The word stable over the summation indicates that we exclude all terms that involve W ( x ) or W ( x , x i ) .From Section 3.23.2, we know that Ω g ( z , . . . , z n ) = W g ( x , . . . , x n ) d x · · · d x n is a meromorphic multidifferen-tial on the rational curve x = z + z +
2, with poles only at z i = ± Ω g ( z , . . . , z i , . . . , z n ) = − Ω g ( z , . . . , z i , . . . , z n ) , for i =
1, 2, . . . , n .Put ω ( z , w ) = d z d w ( z − w ) and note that (cid:20) W ( x , x j ) + ( x − x j ) (cid:21) d x d x j = ω ( z , z j ) − ω ( z , z j ) .Hence, Ω g ( z , z S ) − z + z d x = Ω g − ( z , z , z S ) + stable ∑ g + g = gI (cid:116) J = S Ω g ( z , z I ) Ω g ( z , z J )+ n ∑ j = (cid:20) ω ( z , z j ) − ω ( z , z j ) (cid:21) Ω g ( z , z S \{ j } ) − ∂∂ x j n ∑ j = W g ( x S ) x j ( x − x j ) x d x ⊗ d x S = − Ω g − ( z , z , z S ) − stable ∑ g + g = gI (cid:116) J = S Ω g ( z , z I ) Ω g ( z , z J ) − ∂∂ x j n ∑ j = W g ( x S ) x j ( x − x j ) x d x ⊗ d x S − n ∑ j = ω ( z , z j ) Ω g ( z , z S \{ j } ) − ω ( z , z j ) Ω g ( z , z S \{ j } ) .A rational differential is a sum of its principal parts. Recall that the principal part of a meromorphic differential h ( z ) with respect to the rational parameter z at α ∈ C is [ h ( z )] α : = Res w = α h ( w ) d wz − w = negative part of the Laurent series of h ( z ) at α . Hence, express Ω g ( z , . . . , z n ) as a sum of its principal parts thus. Ω g ( z , z S ) = − ∑ α = ± Res z = α d zz − z + z − z x ( z ) (cid:20) Ω g − ( z , z , z S ) + stable ∑ g + g = gI (cid:116) J = S Ω g ( z , z I ) Ω g ( z , z J )+ n ∑ j = ω ( z , z j ) Ω g ( z , z S \{ j } ) + ω ( z , z j ) Ω g ( z , z S \{ j } ) (cid:21) The third term is annihilated by the residue, since1 + z − z ∂∂ x j n ∑ j = W g ( x S ) x j ( x − x j ) x d x d x S is analytic at z = ±
1. This follows from the fact that 1 + z − z d xx = − d zz is analytic at z = ± z (cid:55)→ z of all terms allows us to express this as Ω g ( z , z S ) = ∑ α = ± Res z = α (cid:18) d zz − z − d zz − z (cid:19) + z − z x ( z ) (cid:20) Ω g − ( z , z , z S ) + stable ∑ g + g = gI (cid:116) J = S Ω g ( z , z I ) Ω g ( z , z J )+ n ∑ j = ω ( z , z j ) Ω g ( z , z S \{ j } ) + ω ( z − , z j ) Ω g ( z , z S \{ j } ) (cid:21) = ∑ α = ± Res z = α K ( z , z ) (cid:20) Ω g − ( z , z , z S ) + ◦ ∑ g + g = gI (cid:116) J = S Ω g ( z , z I ) Ω g ( z , z J ) (cid:21) .Here, K ( z , z ) = (cid:18) d zz − z − d zz − z (cid:19) + z − z x ( z ) and the ◦ over the inner summation means that we exclude termsthat involve Ω . This completes the proof that Ω g ( z , . . . , z n ) = ω gn ( z , . . . , z n ) for the spectral curve C givenby equation (1.31.3). (cid:3) Polynomial behaviour of invariants.
It is possible to solve the recursion (3.13.1) explicitly in low genus.
Proposition 3.11.
In genus 0, we have the explicit formula (3.5) B n ( µ , . . . , µ n ) = − n ( | µ | − )( | µ | − ) · · · ( | µ | − n + ) n ∏ i = (cid:18) µ i µ i (cid:19) , for n ≥ B g , n ( µ , . . . , µ n ) — namely, that it is a polynomial multiplied by an explicitseparable part. Theorem 4.
For ( g , n ) (cid:54) = (
0, 1 ) or (
0, 2 ) ,B g , n ( µ , . . . , µ n ) = p g , n ( µ , . . . , µ n ) n ∏ i = c g ( µ i ) , where p g , n is a polynomial of degree g − + n + ng andc g ( µ ) = ( µ − g ) ! µ ! ( µ − g ) ! = (cid:18) µµ (cid:19) − g g ∏ k = µ − k + The right hand expression for c g ( µ ) allows for evaluation at µ =
0, 1, . . . , g − . OPOLOGICAL RECURSION FOR IRREGULAR SPECTRAL CURVES 17
Remark 3.12.
The two unstable cases ( g , n ) = (
0, 1 ) and (
0, 2 ) do in fact satisfy a similar structure result ifone allows polynomials of negative degree. B ( µ ) = µ ( µ + ) (cid:18) µ µ (cid:19) B ( µ , µ ) = ( µ + µ ) (cid:18) µ µ (cid:19)(cid:18) µ µ (cid:19) Proof.
The proof simply uses the structure of meromorphic functions on the spectral curve xy − xy + = W ( x , . . . , x n ) = ∞ ∑ µ ,..., µ n = U ( µ , . . . , µ n ) n ∏ i = x − µ i − i is an expansion of a function that is rational in z i for x i = z i + z i +
2, with poles only at z i = n −
4. Furthermore, the principal part at z i = z i (cid:55)→ z i for each i =
1, 2, . . . , n . (Notethat since there is only one pole, W ( x , . . . , x n ) is equal to its principal part at z i = i =
1, 2, . . . , n .)Eynard and Orantin [1313] show that there would also be poles at z i = − n − under the assumptionthat y is analytic at the zeros of d x . However, that assumption does not hold here, since y has a pole at z = − V n be the vector space of meromorphic differentials of a single variable with a pole only at z = − n , and (with principal part) skew invariant under z (cid:55)→ z . It has dimension n , because a basisfor this vector space can be obtained by taking the principal part at z = − s − k d s , for k =
1, 2, . . . , n and s the local coordinate defined by x = + s .The expansion of any ξ ∈ V n at x = ∞ can be understood from the following fundamental expansion. ξ ( z ) : = ∞ ∑ k = (cid:18) kk (cid:19) x − k = (cid:114) xx − = + z − z (A different choice of (cid:113) xx − replaces z by z on the right hand side.) Since ξ ( z ) = − ξ ( z ) , then d ξ ∈ V .Consider the operator − x dd x = z ( z + ) − z dd z .It preserves skew invariance under z (cid:55)→ z , since x ( z ) = x ( z ) . Any function with only poles at z = − z = ∞ remains a regular point. Hence, for the meromorphic function ξ n ( z ) = (cid:18) − x dd x (cid:19) n − ξ ( z ) = ∞ ∑ k = k n − (cid:18) kk (cid:19) x − k ,we have d ξ n ∈ V n . In particular, the dimension n vector space V n is spanned by differentials with expansionaround x = ∞ given by ∞ ∑ k = p ( k ) (cid:18) kk (cid:19) x − k − d x , where p ( k ) is a polynomial of degree at most n with p ( ) = W ( x , . . . , x n ) d x · · · d x n = ∞ ∑ µ ,..., µ n = U ( µ , . . . , µ n ) n ∏ i = x − µ i − i d x i is a linear combination of monomials in the single variable differentials, and the total order of the polecorresponds to the total degree of the polynomial. Hence, this proves the theorem in the genus 0 case.For higher genus, Theorem 11 shows that the generating function W g ( x , . . . , x n ) = ∞ ∑ µ ,..., µ n = U g ( µ , . . . , µ n ) n ∏ i = x − µ i − i is an expansion of a function which is rational in z i for x i = z i + z i +
2, with poles only at z i = g + n − z i = − g . Furthermore, the principal part at z i = ± z i (cid:55)→ z i for each i =
1, 2, . . . , n .The operator − dd x = z − z dd z introduces poles at z = −
1. An order 2 g − z = − (cid:18) − dd x (cid:19) g p ( k ) (cid:18) kk (cid:19) x − k = k ( k + ) · · · ( k + g − ) p ( k ) (cid:18) kk (cid:19) x − k − g = q ( m ) (cid:18) m − gm − g (cid:19) x − m .A pole of order 2 g − p i ( m ) for i =
0, 1, . . . , g , there exists a polynomial p ( m ) such that p ( m ) ( m − g ) ! m ! ( m − g ) ! = p g ( m ) (cid:18) m − gm − g (cid:19) + p g − ( m ) (cid:18) m − g + m − g + (cid:19) + · · · + p ( m ) (cid:18) mm (cid:19) ,which proves the theorem. (cid:3) Proof of Proposition 3.113.11.
The proof uses the structure theorem and an elementary relation known as thedivisor equation. By Theorem 44 B n ( µ , . . . , µ n ) = p ( µ , . . . , µ n ) n ∏ i = (cid:18) µ i µ i (cid:19) , for n ≥ divisor equation is easy to prove combinatorially, by considering the result of doubling any edgeof a dessin.(3.6) B n + ( µ , . . . , µ n ) = | µ | · B n ( µ , . . . , µ n ) In terms of the polynomial part of B n , it implies that(3.7) p ( µ , . . . , µ n ) = | µ | · p ( µ , . . . , µ n ) .It is easy to check that the recursion (3.73.7) is satisfied by p ( µ , . . . , µ n ) = − n ( | µ | − )( | µ | − ) · · · ( | µ | − n + ) .A degree n − n variables p ( µ , . . . , µ n ) is uniquely determined by its evaluationat one variable, say µ = a . Hence, equation (3.73.7) uniquely determines p ( µ , . . . , µ n , µ n + ) from p ( µ , . . . , µ n ) and inductively, from the initial condition p ( µ , µ , µ ) = . (This last fact is equivalent to b ( µ , µ , µ ) = p ( µ , . . . , µ n ) = − n ( | µ | − )( | µ | − ) · · · ( | µ | − n + ) is indeed the uniquesolution and the proposition is proven. (cid:3) The definition of B g , n ( µ , . . . , µ n ) requires all µ i to be positive. However, the polynomial structure of B g , n ( µ , . . . , µ n ) allows one to evaluate at µ i =
0. The following proposition gives a combinatorial meaningto such evaluation.
Proposition 3.13.
For m and n positive integers, B g , n + m ( µ , . . . , µ n , 0, 0, . . . , 0 ) enumerates dessins in B g , n ( µ , . . . , µ n ) with m distinct black vertices labelled.Proof. Intuitively, µ i = n + n +
2, . . . , n + m . More precisely, we can use Theorem 11to write the dilaton equation (2.32.3) equivalently as B g , n + ( µ , . . . , µ n ) − B g , n + ( µ , . . . , µ n ) = ( | µ | + g − + n ) B g , n ( µ , . . . , µ n ) . OPOLOGICAL RECURSION FOR IRREGULAR SPECTRAL CURVES 19
Together with the divisor equation (3.63.6), we have B g , n + ( µ , . . . , µ n ) = ( | µ | − g + − n ) B g , n ( µ , . . . , µ n ) = V · B g , n ( µ , . . . , µ n ) ,where V = | µ | − g + − n is the number of vertices of any dessin in the set B g , n ( µ , . . . , µ n ) . So theexpression V · B g , n ( µ , . . . , µ n ) can be interpreted as the enumeration of dessins in B g , n ( µ , . . . , µ n ) with onevertex labelled. Due to the symmetry between black and white vertices, one can interpret the expression V · B g , n ( µ , . . . , µ n ) as the enumeration of dessins in B g , n ( µ , . . . , µ n ) with one black vertex labelled. Nowapply this relation m times to the expression B g , n + m ( µ , . . . , µ n , 0, 0, . . . , 0 ) to obtain the desired result. (cid:3) The previous proposition does not apply when n =
0, but we conjecture that B g , m (
0, . . . , 0 ) = − m χ ( M g , m ) .This would be a consequence of the dilaton equation and the m = B g ,1 ( ) = χ ( M g ,1 ) . We have notyet proven that B g ,1 ( ) = χ ( M g ,1 ) , although we have verified it numerically for small values of g . It shouldfollow from the three-term recursion of the next section, using a method analogous to that of Harer andZagier [1717]. 4. T HREE - TERM RECURSION FOR DESSINS D ’ ENFANT
Harer and Zagier calculated the virtual Euler characteristics of moduli spaces of smooth curves via theenumeration of fatgraphs with one face [1717]. They define (cid:101) g ( n ) to be the number of ways to glue the edges ofa 2 n -gon in pairs and obtain an orientable genus g surface. Equivalently, (cid:101) g ( n ) is equal to 2 n multiplied bythe number of genus g fatgraphs with one face and n edges, counted with the usual weight | Aut Γ | . Throughthe analysis of a Hermitian matrix integral, they arrive at the following three-term recursion for fatgraphswith one face. ( n + ) (cid:101) g ( n ) = ( n − ) (cid:101) g ( n − ) + ( n − )( n − )( n − ) (cid:101) g − ( n − ) There are myriad combinatorial results concerning the enumeration of fatgraphs with one face, and many ofthese have analogues for the enumeration of dessins with one face. For example, the first equation below,which appears in [1717] with C ( n , z ) in place of F n ( z ) , gives a formula for the polynomial generating function offatgraphs with one face and n edges. The second is known as Jackson’s formula [1818] and gives the polynomialgenerating function for dessins with one face and n edges. F n ( z ) = ∞ ∑ g = (cid:101) g ( n ) z n + − g = ( n ) !2 n n ! n ∑ r = r (cid:18) nr (cid:19)(cid:18) zr + (cid:19) (4.1) G n ( z ) = ∞ ∑ g = U g ( n ) z n + − g = n ! n − ∑ r , s = (cid:18) n − r , s (cid:19)(cid:18) zr + (cid:19)(cid:18) zs + (cid:19) (4.2)Here, we use the notation ( n − r , s ) = ( n − ) ! r ! s ! ( n − − r − s ) ! with the convention that if r + s > n −
1, then the expressionis equal to zero.To the best of our knowledge, the following analogue of the Harer–Zagier three-term recursion for dessinsdoes not appear in the literature. It will play an important role in our calculation of the 1-point invariants ofthe spectral curve xy = Theorem 5.
The following recursion holds for all g ≥ and n ≥ , where we set U ( ) = . (4.3) ( n + ) U g ( n ) = ( n − ) U g ( n − ) + ( n − ) ( n − ) U g − ( n − ) Proof.
We begin with the following observation of Bernardi and Chapuy [22, Theorem 5.3]. G n ( z ) = ( n − ) ! n ! ∑ i + j = n − F i ( z )( i ) ! F j ( z )( j ) ! .To see this, we simply substitute the expressions from equation (4.14.1) and (4.24.2) on both sides. n ! n − ∑ r , s = (cid:18) n − r , s (cid:19)(cid:18) zr + (cid:19)(cid:18) zs + (cid:19) = ( n − ) ! n ! ∑ i + j = n − i i ! 12 j j ! i ∑ r = r (cid:18) ir (cid:19)(cid:18) zr + (cid:19) j ∑ s = s (cid:18) js (cid:19)(cid:18) zs + (cid:19) = ( n − ) ! n ! ∑ i + j = n − i ! j ! n − ∑ r , s = r + s − n + (cid:18) ir (cid:19)(cid:18) js (cid:19)(cid:18) zr + (cid:19)(cid:18) zs + (cid:19) = ( n − ) ! n ! n − ∑ r , s = r ! s ! (cid:18) zr + (cid:19)(cid:18) zs + (cid:19) n − − r − s ∑ k = r + s − n + k ! ( n − − r − s − k ) !The two sides are equal since the inner summation on the right hand side simplifies to ( n − − r − s ) ! . Itimmediately follows that G n ( z ) = n ! ( n − ) ! [ t n + ] E ( z , t ) ,where E ( z , t ) is the generating function E ( z , t ) = ∞ ∑ n = F n ( z )( n ) ! t n + = ∞ ∑ n = ∞ ∑ k = k n n ! (cid:18) nk (cid:19)(cid:18) zk + (cid:19) t n + = t ∞ ∑ k = k k ! (cid:18) zk + (cid:19) ∞ ∑ n = k ( t /2 ) n ( n − k ) ! = t e t /2 ∞ ∑ k = (cid:18) zk + (cid:19) t k k ! .This expansion implies that E ( z , t ) = zM − z ,1/2 ( t ) , where M denotes the Whittaker function. It follows that E ( z , t ) satisfies the following second order differential equation, which one can verify directly from theexpansion above. ∂ ∂ t E ( z , t ) − (cid:18) + zt (cid:19) E ( z , t ) = ∂ ∂ t E ( z , t ) = (cid:18) zt + (cid:19) ∂∂ t E ( z , t ) − zt E ( z , t ) By collecting terms in the t -expansion of both sides, we obtain ( n + ) n ( n − ) [ t n + ] E ( z , t ) = nz [ t n ] E ( z , t ) + ( n − ) [ t n − ] E ( z , t ) − z [ t n ] E ( z , t ) ,which is equivalent to ( n + ) G n ( z ) = ( n − ) z G n − ( z ) + ( n − ) ( n − ) G n − ( z ) .Now we use U g ( n ) = [ z n + − g ] G n ( z ) to deduce that ( n + ) U g ( n ) = ( n − ) U g ( n − ) + ( n − ) ( n − ) U g − ( n − ) . (cid:3) The three-term recursion for enumeration of dessins with one face is equivalent to a recursion for thegenerating functions F g ,1 ( x ) . From this recursion, one can extract the highest order coefficients of the poles at z = −
1. This is enough to prove Theorem 33, which states that the 1-point invariants of the spectral curve xy = ω g ( z ) = − g ( g ) ! g ! ( g − ) z − g d z . OPOLOGICAL RECURSION FOR IRREGULAR SPECTRAL CURVES 21
Proof of Theorem 33.
As in Section 3.13.1, we use the notation W g ( x ) to denote W g ( x ) d x = d F g ,1 ( x ) = ∞ ∑ n = U g ( n ) x − n − d x .Then equation (4.34.3) is equivalent to the differential equation (cid:20) x − x dd x − x (cid:21) W g ( x ) = (cid:20) d d x + x d d x + x dd x (cid:21) W g − ( x ) .Write ω g ( z ) = w g ( z ) d z = W g ( x ) d x , where w g ( z ) is a rational function of z with poles at z = ±
1. Thedifferential equation above can be written in terms of w g ( z ) and w g − ( z ) . Extract the highest order terms ofthe principal part at z = − (cid:20) dd z − (cid:21) w g ( z ) = (cid:20) d d z + + z d d z − ( + z ) dd z + ( + z ) (cid:21) w g − ( z ) + [ lower order terms ] .By “lower order terms”, we mean terms with lower order poles at z = −
1. This becomes an exact differentialequation for the 1-point invariants ω g ( z ) = v g ( z ) d z of the spectral curve xy =
1. For g ≥
1, we havedd z v g ( z ) = (cid:20) d d z + z d d z − z dd z + z (cid:21) v g − ( z ) ,with the boundary condition v g ( ∞ ) =
0. This comes from the fact that the differential v g ( z ) d z has no poleat z = ∞ for g ≥
1. Given the initial condition v ( z ) = −
2, the system has a unique solution. Since v ( z ) ishomogeneous in z , each v g ( z ) is homogeneous of degree − g . If we write v g ( z ) = a g z − g , then the previousequation implies that − g a g = − ( g − )( g − ) a g − .From the initial condition a = −
2, we obtain the solution a g = − g ( g ) ! g ! ( g − ) . It follows that the 1-pointinvariants of the spectral curve xy = ω g ( z ) = v g ( z ) d z = − g ( g ) ! g ! ( g − ) z − g d z . (cid:3)
5. R
ELATION TO THE K AZARIAN –Z OGRAF SPECTRAL CURVE
Kazarian and Zograf [1919] study a more refined version of our enueration of dessins d’enfant. They define N k , (cid:96) ( µ ) to be the weighted sum of connected Belyi covers with k points above 0, (cid:96) points above 1 andramification prescribed by µ = ( µ , . . . , µ n ) above infinity. Hence, we have the relation B g , n ( µ ) = ∑ k + (cid:96) = | µ | + − g − n N k , (cid:96) ( µ ) .They prove that the generating function W KZg ( s , u , v , x , . . . , x n ) = ∑ µ ∑ k + (cid:96) = | µ | + − g − n N k , (cid:96) ( µ ) s | µ | u k v (cid:96) x µ · · · x µ n n satisfies topological recursion on the regular spectral curve C given by(5.1) xy + ( − s + ( u + v ) x ) y + uvx = u → v limit to be related to the unrefined count — however, the curve (5.15.1) is ill-behaved inthe limit since ω gn →
0. This can be seen from x = s y ( y + u ) ( y + v ) ⇒ d x = s uv − y ( y + u ) ( y + v ) d y , so in the u → v limit d x has a single zero since one of the two zeros of d x cancels with a pole. We can choosea family of rational parametrisations that fix the poles and zeros of d x and hence counteract the collision ofzeros and poles. x = √ uvs ( u − v ) (cid:16) z + z (cid:17) + u + vs ( u − v ) y = −√ uv z √ u + √ vz √ v + √ u = − u + u − v + (cid:113) vu z The meromorphic functions x (cid:48) = s ( v − u ) (cid:104) √ uv (cid:16) z + z (cid:17) + u + v (cid:105) and y (cid:48) = (cid:113) vu z + (cid:113) vu z define a spectral curve C (cid:48) that yields equivalent invariants ω gn ( C ) = ω gn ( C (cid:48) ) . The invariants are preservedunder the transformations y (cid:55)→ y + v and ( x , y ) (cid:55)→ (( v − u ) x , yv − u ) , since x and y appear in the recursion onlyvia the combination y d x . The limit u → v still causes the ω gn to degenerate to zero, but now in a controlledway. In fact, we have ω gn ( C ) = ( v − u ) g − + n ω gn ( C (cid:48)(cid:48) ) ,where C (cid:48)(cid:48) is the spectral curve given by x (cid:48)(cid:48) = √ uv (cid:16) z + z (cid:17) + u + v and y (cid:48)(cid:48) = (cid:113) vu z + (cid:113) vu z .This resembles our original spectral curve (1.31.3), although C (cid:48)(cid:48) is regular for u (cid:54) = v and becomes irregular onlywhen u = v . We have not been able to prove Theorem 11 via this limit. The qualitative difference betweenregular and irregular curves may explain this.6. T HE QUANTUM CURVE
Recall from Theorem 11 that the topological recursion applied to the spectral curve x = z + z + y = z + z produces the invariants ω gn ( z , . . . , z n ) = ∂∂ x · · · ∂∂ x n F g , n ( x , . . . , x n ) d x ⊗ · · · ⊗ d x n , for 2 g − + n > free energiesF g , n ( x , . . . , x n ) = ∞ ∑ µ ,..., µ n = B g , n ( µ , . . . , µ n ) n ∏ i = x − µ i i are natural generating functions for the enumeration of dessins of type ( g , n ) . By exception, we modify F by defining F ( x ) = − log x + ∞ ∑ µ = B ( µ ) x − µ .The logarithmic term appearing in the definition of F ( x ) is consistent with the fact that U ( ) = ∂∂ x · · · ∂∂ x n F g , n ( x , . . . , x n ) = ( − ) n ∑ µ ,..., µ n U g ( µ , . . . , µ n ) n ∏ i = x − µ i − i . OPOLOGICAL RECURSION FOR IRREGULAR SPECTRAL CURVES 23
From these, one defines the wave function as follows, which differs from the expression given in Section 11due to the adjustment of F . Z ( x , ¯ h ) = exp (cid:20) ∞ ∑ g = ∞ ∑ n = ¯ h g − + n n ! F g , n ( x , x , . . . , x ) (cid:21) Theorem 6.
For (cid:98) x = x and (cid:98) y = − ¯ h ∂∂ x , we have the equation [ (cid:98) y (cid:98) x (cid:98) y − (cid:98) x (cid:98) y + ] Z ( x , ¯ h ) = which is the quantum curve corresponding to the spectral curve xy − xy + = . In order to interpret Theorem 66, we need to make precise what we mean by this equation, given that the¯ h -expansion of Z ( x , ¯ h ) is not well-defined. One way to do this is to express the wave function as Z ( x , ¯ h ) = x − h Z ( x , ¯ h ) ,where the term x − h comes from the exceptional logarithmic term in the definition of F . So we interpretTheorem 66 as(6.1) x h [ (cid:98) y (cid:98) x (cid:98) y − (cid:98) x (cid:98) y + ] x − h Z ( x , ¯ h ) = ⇒ (cid:20) x ¯ h ∂ ∂ x + ¯ h ( ¯ h − + x ) ∂∂ x + x − (cid:21) Z ( x , ¯ h ) = Z ( x , ¯ h ) has an expansion in x − with coefficients that are Laurent poly-nomials in ¯ h — in other words, Z ( x , ¯ h ) ∈ Q [ ¯ h ± ][[ x − ]] . So the rigorous statement of Theorem 66 is viaequation (6.16.1), in terms of a differential operator annihilating the formal series Z ( x , ¯ h ) ∈ Q [ ¯ h ± ][[ x − ]] .In fact, we will explicitly calculate the coefficients of Z ( x , ¯ h ) in the x − -expansion and use this to derive thequantum curve. The strategy is to interpret the coefficients of Z ( x , ¯ h ) combinatorially using the followingobservations about the definition of the wave function Z ( x , ¯ h ) .The expression F g , n ( x , x , . . . , x ) counts dessins not with respect to the tuple of boundary lengths, butwith respect to the sum of the boundary lengths. This is precisely the number of edges in the dessin.The term ¯ h g − + n n ! ignores the labels of the boundary components and organises the count of dessinsby Euler characteristic rather than by genus.The exponential in the definition of Z ( x , ¯ h ) passes from a count of connected dessins to a count ofdisconnected dessins, via the usual exponential formula. Proposition 6.1.
The modified wave function Z ( x , ¯ h ) is an element of Q [ ¯ h ± ][[ x − ]] . Furthermore, it can be expressedas Z ( x , ¯ h ) = + ∞ ∑ e = ¯ h e (cid:104) ¯ h − ( ¯ h − + )( ¯ h − + ) · · · ( ¯ h − + e − ) (cid:105) x − e . Proof.
First, consider the logarithm of the modified wave function.log Z ( x , ¯ h ) = ∞ ∑ g = ∞ ∑ n = ¯ h g − + n n ! F g , n ( x , x , . . . , x )= ∞ ∑ g = ∞ ∑ n = ¯ h g − + n n ! ∞ ∑ µ ,..., µ n = B g , n ( µ , . . . , µ n ) x − ( µ + ··· + µ n ) = ∞ ∑ v = ∞ ∑ e = f ( v , e ) ¯ h e − v x − e Here, f ( v , e ) denotes the weighted count of connected dessins with v vertices, e edges, and unlabelledboundary components. To obtain this last expression, we have used the fact that v − e = g − + n and µ + · · · + µ n = e for any dessin. The factor n ! accounts for the fact that we are now considering dessins withunlabelled faces. Note that we exclude from consideration the dessins consisting of an isolated vertex. Next, we use the exponential formula to pass from the connected count to its disconnected analogue. Z ( x , ¯ h ) = + ∞ ∑ v = ∞ ∑ e = f • ( v , e ) ¯ h e − v x − e Here, f • ( v , e ) denotes the weighted count of possibly disconnected dessins with v vertices, e edges, andunlabelled faces. We furthermore require that no connected component consists of an isolated vertex.Now note that f • ( v , e ) is equal to e ! multiplied by the number of triples ( σ , σ , σ ) of permutations in thesymmetric group S e such that σ σ σ = id and c ( σ ) + c ( σ ) = v . Here, we use c ( σ ) to denote the number ofdisjoint cycles in the permutation σ . However, this is clearly equal to e ! multiplied by the number of pairs ( σ , σ ) of permutations in S e such that c ( σ ) + c ( σ ) = v . Recall that the Stirling number of the first kind [ nk ] counts the number of permutations in S n with k disjoint cycles. So we have deduced that f • ( v , e ) = e ! ∑ a + b = v (cid:20) ea (cid:21)(cid:20) eb (cid:21) .It is evident from this formula that for fixed e , we require 2 ≤ v ≤ e to have f • ( v , e ) (cid:54) =
0. Therefore, themodified wave function Z ( x , ¯ h ) is indeed an element of Q [ ¯ h ± ][[ x − ]] .Now we simply use the fact that the generating function for Stirling numbers of the first kind is given by n ∑ k = (cid:20) nk (cid:21) x k = x ( x + )( x + ) · · · ( x + n − ) .Use this in the expression for the modified wave function as follows. Z ( x , ¯ h ) = + ∞ ∑ v = ∞ ∑ e = e ! ∑ a + b = v (cid:20) ea (cid:21)(cid:20) eb (cid:21) ¯ h e − v x − e = + ∞ ∑ e = e ! ∞ ∑ a = (cid:20) ea (cid:21) ¯ h − a ∞ ∑ b = (cid:20) eb (cid:21) ¯ h − b ¯ h e x − e = + ∞ ∑ e = ¯ h e e ! (cid:104) ¯ h − ( ¯ h − + )( ¯ h − + ) · · · ( ¯ h − + e − ) (cid:105) x − e (cid:3) The quantum curve for the enumeration of dessins is now a straightforward consequence of the previousproposition.
Proof of Theorem 66.
We use Proposition 6.16.1 to derive the quantum curve. Start by writing Z ( x , ¯ h ) = ∞ ∑ e = a e ( ¯ h ) x − e , where a e ( ¯ h ) = ¯ h e e ! (cid:104) ¯ h − ( ¯ h − + )( ¯ h − + ) · · · ( ¯ h − + e − ) (cid:105) .Then take the relation ( e + ) a e + ( ¯ h ) = ¯ h ( ¯ h − + e ) a e ( ¯ h ) , multiply both sides by x − e − , and sum over all e . ∞ ∑ e = ( e + ) a e + ( ¯ h ) x − e − = ∞ ∑ e = ¯ h ( ¯ h − + e ) a e ( ¯ h ) x − e − ∞ ∑ e = e a e ( ¯ h ) x − e = ¯ h − ∞ ∑ e = a e ( ¯ h ) x − e − + ∞ ∑ e = e a e ( ¯ h ) x − e − + ¯ h ∞ ∑ e = e a e ( ¯ h ) x − e − − x ∂ Z ∂ x = ¯ h − x − Z − ∂ Z ∂ x + ¯ h ∂∂ x (cid:20) x ∂ Z ∂ x (cid:21) Now use the product rule on the final term and rearrange the equation to obtain the desired quantum curve,as expressed in equation (6.16.1). (cid:3) The numbers f • ( v , e ) appear in the triangle of numbers given by sequence A246117 in the OEIS. There, the number f • ( v , e ) is describedas the number of parity-preserving permutations in S e with v cycles. A parity-preserving permutation p on the set {
1, 2, . . . , n } is onethat satisfies p ( i ) ≡ i ( mod 2 ) for i =
1, 2, . . . , n . OPOLOGICAL RECURSION FOR IRREGULAR SPECTRAL CURVES 25
Remark 6.2.
In the semi-classical limit, the quantum curve differential operator becomes a multiplicationoperator. The limit is obtained by sending ¯ h → S m ( x ) = ∑ g − + n = m − n ! F g , n ( x , x , . . . , x ) ⇒ Z ( x , ¯ h ) = exp (cid:20) ∞ ∑ m = ¯ h m − S m ( x ) (cid:21) .Then we have lim ¯ h → exp (cid:20) − h S ( x ) (cid:21) ( (cid:98) y (cid:98) x (cid:98) y − (cid:98) x (cid:98) y + ) Z ( x , ¯ h )= lim ¯ h → exp (cid:20) − h S ( x ) (cid:21) ( (cid:98) y (cid:98) x (cid:98) y − (cid:98) x (cid:98) y + ) exp (cid:20) h S ( x ) (cid:21) exp (cid:20) ∞ ∑ m = ¯ h m − S m ( x ) (cid:21) = lim ¯ h → (cid:16) xS (cid:48) ( x ) − xS (cid:48) ( x ) + + ¯ hS (cid:48) ( x ) − ¯ h (cid:17) exp (cid:20) ∞ ∑ m = ¯ h m − S m ( x ) (cid:21) .For this expression to vanish, we must have xS (cid:48) ( x ) − xS (cid:48) ( x ) + =
0, which is precisely the spectral curvegiven by equation (1.31.3) since y = F (cid:48) ( x ) = S (cid:48) ( x ) . Remark 6.3.
In other rigorously known instances of the topological recursion/quantum curve paradigmwhere the spectral curve is polynomial, the quantum curve is often obtained using the normal ordering ofoperators that places differentiation operators to the right of multiplication operators. For example,the spectral curve y − xy + = (cid:98) y − (cid:98) x (cid:98) y + xy + y + = (cid:98) x (cid:98) y + (cid:98) y + y a − xy + = a -hypermaps has quantum curve (cid:98) y a − (cid:98) x (cid:98) y + h term in the following way, where we use thecommutation relation [ (cid:98) x , (cid:98) y ] = ¯ h . (cid:98) P ( (cid:98) x , (cid:98) y ) = (cid:98) y (cid:98) x (cid:98) y − (cid:98) x (cid:98) y + = ( (cid:98) x (cid:98) y − ¯ h ) (cid:98) y − (cid:98) x (cid:98) y + = ( (cid:98) x (cid:98) y − (cid:98) x (cid:98) y + ) − ¯ h (cid:98) y Remark 6.4.
The statementlim ¯ h → ¯ h dd x log Z ( x , ¯ h ) = y = (cid:90) λ ( t ) x − t d t , where λ ( t ) = π (cid:114) − tt · [ ] agrees with equation (3.4) of [1515], where Z ( x , ¯ h ) is replaced by the expectation (cid:104) det ( x − A ) (cid:105) of a matrixintegral over positive definite Hermitian matrices. This confirms the known fact in the physics literature thatthe wave function corresponds to the expectation (cid:104) det ( x − A ) (cid:105) .7. L OCAL IRREGULAR BEHAVIOUR
The asymptotic behaviour of ω gn near zeros of d x is governed by the local behaviour of the curve C there [1414].The usual assumption is that the local behaviour is described by x = y which, as a global curve, hasinvariants ω gn that store tautological intersection numbers over the compactified moduli space of curves M g , n . Here, we also consider the local behaviour described by xy = x = z and y = z . We include the factor of in x simply to reduce powers of 2 in the resulting invariants. One can calculateinvariants via topological recursion and obtain ω n =
0, for n ≥ ω n = − ( n − ) ! n ∏ i = d z i z i ω n = − ( n + ) ! n ∏ i = d z i z i n ∑ i = z i .If we write ω gn = ∑ u g ( µ , . . . , µ n ) n ∏ i = d z i z µ i + i ,then the coefficients satisfy the recursion(7.2) u g ( µ , µ S ) = n ∑ j = µ j u g ( µ + µ j − µ S \{ j } ) + ∑ i + j = µ − (cid:20) u g − ( i , j , µ S ) + ∑ g + g = gI (cid:116) J = S u g ( i , µ I ) u g ( j , µ J ) (cid:21) ,for S = {
2, . . . , n } . We impose the base cases u ( µ , . . . , µ n ) = µ , . . . , µ n and u g ( µ , . . . , µ n ) = µ , . . . , µ n are even. In low genus, the recursion is solved by u (
1, . . . , 1 ) = − ( n − ) ! and u ( µ , . . . , µ n ) = u (
3, 1, . . . , 1 ) = − ( n + ) ! and u ( µ , . . . , µ n ) = u (
5, 1, . . . , 1 ) = − ( n + ) ! u (
3, 3, 1, . . . , 1 ) = − ( n + ) ! and u ( µ , . . . , µ n ) = u g ( µ , . . . , µ n ) is non-zero only if µ is a partition of 2 g − + n with only odd parts. This suggestsa possible relationship with connected branched covers of the torus with n branch points of ramificationorders µ , . . . , µ n . By the Riemann–Hurwitz formula, such a cover is necessarily of genus g .7.1. Volumes.
One can associate polynomials V g ( L , . . . , L n ) to the curve (7.17.1), which are dual to the ancestorinvariants u g ( µ , . . . , µ n ) . We refer to them as volumes, since they have properties that resemble the Kontse-vich volumes associated to the cell decomposition of the moduli space of curves [2020]. These polynomialssatisfy L (cid:2) V g ( L , . . . , L n ) (cid:3) = (cid:90) ∞ · · · (cid:90) ∞ V g ( L , . . . , L n ) ∏ exp ( − z i L i ) · L i d L i = ω gn ( z , . . . , z n ) .Note that L ( L k ) = ( k + ) ! z k + .We highlight several properties of these volumes.(1) For S = {
2, . . . , n } , we have the recursion2 L V g ( L , L S ) = n ∑ j = (cid:20) ( L j + L ) V g ( L j + L , L S \{ j } ) − ( L j − L ) V g ( L j − L , L S \{ j } ) (cid:21) + (cid:90) L d x · x ( L − x ) (cid:20) V g − ( x , L − x , L S ) + ∑ g + g = gI (cid:116) J = S V g ( x , L I ) V g ( L − x , L J ) (cid:21) .(2) The volume V g ( L , . . . , L n ) is a degree 2 g − L , . . . , L n . OPOLOGICAL RECURSION FOR IRREGULAR SPECTRAL CURVES 27 (3) The volume V g depends on n in a mild way — we have V g ( L , . . . , L n ) = ( g − + n ) ! ∑ µ (cid:96) g − C g ( µ ) m µ ( L ) ,where the summation is over partitions µ of g −
1, the expression m µ ( L ) denotes the monomialsymmetric function in L , . . . , L n , and C g ( µ ) are constants.(4) There exists the following dilaton equation for the volumes. V g ( L , . . . , L n , 0 ) = ( g − + n ) V g ( L , . . . , L n ) (5) One can calculate the following formulae. V ( L ) = − · ( n − ) ! V ( L ) = − · · ( n + ) ! ∑ L i V ( L ) = − · ( n + ) ! (cid:18) ∑ L i + ∑ L i L j (cid:19) V g ( L ) = − g (cid:18) gg (cid:19) ( g − + n ) ! ( g − ) ! ∑ L g − i + · · · The recursion may help to answer the question: volumes of what ?A PPENDIX
A. F
ORMULAE .In the following table, we use the notation introduced earlier. c g ( µ ) = ( µ − g ) ! µ ! ( µ − g ) ! = (cid:18) µµ (cid:19) − g g ∏ k = µ − k + g n B g , n ( µ , . . . , µ n ) ∏ c g ( µ i ) µ ( µ + ) ( µ + µ ) n − n ( | µ | − )( | µ | − ) · · · ( | µ | − n + ) ( µ − )( µ − ) ( µ µ + µ µ + µ µ − µ − µ − µ µ − µ µ + µ + µ + µ µ − µ − µ + ) ( µ − )( µ − )( µ − )( µ − )( µ − µ + ) ( µ − )( µ − )( µ − )( µ − )( µ − )( µ − )( µ − µ + µ − µ + ) R EFERENCES[1] Ambjørn, Jan and Chekhov, Leonid.
The matrix model for dessins d’enfants. arXiv:1404.4240arXiv:1404.4240[2] Olivier Bernardi and Guillaume Chapuy.
A bijection for covered maps, or a shortcut between Harer–Zagier’s and Jackson’s formulas.
J.Comb. Theory, Ser. A (6), (2011), 1718–1748.[3] Chekhov, Leonid and Eynard, Bertrand.
Hermitian matrix model free energy: Feynman graph technique for all genera.
J. High EnergyPhys. (3), (2006), 014.[4] Di Francesco, P.
Rectangular matrix models and combinatorics of colored graphs.
Nuc. Phys. B , (2003), 461–496.[5] Do, Norman; Dyer, Alastair and Mathews, Daniel.
Topological recursion and a quantum curve for monotone Hurwitz numbers. arXiv:1408.3992arXiv:1408.3992[6] Do, Norman and Manescu, David.
Quantum curves for the enumeration of ribbon graphs and hypermaps.
To appear in Commun. NumberTheory Phys. (4), (2014). [7] Do, Norman and Norbury, Paul. Counting lattice points in compactified moduli spaces of curves.
Geometry & Topology , (2011),2321–2350.[8] Do, Norman and Norbury, Paul. Pruned Hurwitz numbers. arXiv:1312.7516arXiv:1312.7516[9] P. Dunin–Barkowski; N. Orantin; A. Popolitov and S. Shadrin.
Combinatorics of loop equations for branched covers of sphere. arXiv:1412.1698arXiv:1412.1698[10] P. Dunin–Barkowski; N. Orantin; S. Shadrin and L. Spitz.
Identification of the Givental formula with the spectral curve topological recursionprocedure.
Comm. Math. Phys. , (2014), 669–700.[11] Eynard, Bertrand.
Invariants of spectral curves and intersection theory of moduli spaces of complex curves.
Commun. Number TheoryPhys. (3), (2014), 541–588.[12] Eynard, Bertrand. Intersection numbers of spectral curves. arXiv:1104.0176arXiv:1104.0176[13] Eynard, Bertrand and Orantin, Nicolas.
Invariants of algebraic curves and topological expansion.
Commun. Number Theory Phys. (2),(2007), 347–452.[14] Eynard, Bertrand and Orantin, Nicolas. Topological recursion in enumerative geometry and random matrices.
J. Phys. A: Math. Theor. (29), (2009) 293001.[15] P. J. Forrester and D.-Z. Liu. Raney distributions and random matrix theory. arXiv:1404.5759arXiv:1404.5759[16] Gukov, Sergei and Sułkowski, Piotr.
A-polynomial, B-model, and quantization.
J. High Energy Phys. (2), (2012), 070.[17] Harer, John and Zagier, Don.
The Euler characteristic of the moduli space of curves.
Invent. Math. (1986), 457–485.[18] D. M. Jackson. Some combinatorial problems associated with products of conjugacy classes of the symmetric group.
J. Comb. Theory, Ser. A, (2), (1988), 363–369.[19] Kazarian, Maxim and Zograf, Peter. Virasoro constraints and topological recursion for Grothendieck’s dessin counting. arXiv:1406.5976arXiv:1406.5976[20] Kontsevich, Maxim.
Intersection theory on the moduli space of curves and the matrix Airy function.
Comm. Math. Phys. , (1992), 1–23.[21] Mulase, Motohico and Sułkowski, Piotr.
Spectral curves and the Schr¨odinger equations for the Eynard–Orantin recursion. arXiv:1210.3006arXiv:1210.3006[22] Norbury, Paul.
Counting lattice points in the moduli space of curves.
Math. Res. Lett. , (2010), 467–481.[23] Norbury, Paul. String and dilaton equations for counting lattice points in the moduli space of curves. Trans. AMS. , (2013), 1687–1709.[24] Norbury, Paul and Scott, Nick.
Gromov–Witten invariants of P and Eynard–Orantin invariants. Geometry & Topology , (2014),1865–1910.S CHOOL OF M ATHEMATICAL S CIENCES , M
ONASH U NIVERSITY , VIC 3800 A
USTRALIA D EPARTMENT OF M ATHEMATICS AND S TATISTICS , U
NIVERSITY OF M ELBOURNE , VIC 3010 A
USTRALIA
E-mail address ::