Refined open intersection numbers and the Kontsevich-Penner matrix model
aa r X i v : . [ m a t h - ph ] F e b REFINED OPEN INTERSECTION NUMBERS AND THEKONTSEVICH-PENNER MATRIX MODEL
ALEXANDER ALEXANDROV, ALEXANDR BURYAK, AND RAN J. TESSLER
Abstract.
A study of the intersection theory on the moduli space of Riemann surfaces withboundary was recently initiated in a work of R. Pandharipande, J. P. Solomon and the thirdauthor, where they introduced open intersection numbers in genus 0. Their construction was latergeneralized to all genera by J. P. Solomon and the third author. In this paper we consider arefinement of the open intersection numbers by distinguishing contributions from surfaces withdifferent numbers of boundary components, and we calculate all these numbers. We then constructa matrix model for the generating series of the refined open intersection numbers and conjecturethat it is equivalent to the Kontsevich-Penner matrix model. An evidence for the conjecture ispresented. Another refinement of the open intersection numbers, which describes the distributionof the boundary marked points on the boundary components, is also discussed. Introduction
A compact Riemann surface is a compact connected complex manifold of dimension 1. Denoteby M g,l the moduli space of all compact Riemann surfaces of genus g with l marked points.P. Deligne and D. Mumford defined a natural compactification M g,l ⊂ M g,l via stable curves(with possible nodal singularities) in [DM69]. The moduli space M g,l is a non-singular complexorbifold of dimension 3 g − l . It is defined to be empty unless the stability condition2 g − l > M g,l . Foreach marking index i consider the cotangent line bundle L i → M g,l , whose fiber over a point[Σ , z , . . . , z l ] ∈ M g,l is the complex cotangent space T ∗ z i Σ of Σ at z i . Let ψ i ∈ H ( M g,l , Q ) denotethe first Chern class of L i , and write h τ a τ a · · · τ a l i cg := Z M g,l ψ a ψ a · · · ψ a l l . (1.2)The integral on the right-hand side of (1.2) is well-defined, when the stability condition (1.1) issatisfied, all the a i are non-negative integers and the dimension constraint 3 g − l = P a i holds.In all other cases h Q τ a i i cg is defined to be zero. The intersection products (1.2) are often calleddescendent integrals or intersection numbers . Let t i , i ≥
0, be formal variables and let F c ( t , t , . . . ) := X g ≥ F cg ( t , t , . . . ) , where F cg ( t , t , . . . ) := X l ≥ X a ,...,a l ≥ h τ a τ a · · · τ a l i cg Q t a i l ! . The generating series F c is called the closed free energy . The exponent τ c := exp( F c ) is called the closed partition function . Witten’s conjecture ([Wit91]), proved by M. Kontsevich ([Kon92]), saysthat the closed partition function τ c becomes a tau-function of the KdV hierarchy after the change of variables t n = (2 n + 1)!! T n +1 . Integrability immediately follows [KMMMZ92] from Kontsevich’smatrix integral representation τ c | T k = k tr Λ − k = c Λ ,M Z H M e tr H − tr H Λ dH, (1.3)where one integrates over the space of Hermitian M × M matrices, Λ = diag( λ , . . . , λ M ) is adiagonal matrix with positive real entries and c Λ ,M := (2 π ) − M M Y i =1 p λ i Y ≤ i X, ∂X ), we can canonically construct a double via Schwarz reflectionthrough the boundary. The double D ( X, ∂X ) of ( X, ∂X ) is a compact Riemann surface. Thedoubled genus of ( X, ∂X ) is defined to be the usual genus of D ( X, ∂X ). On a Riemann surfacewith boundary ( X, ∂X ), we consider two types of marked points. The markings of interior typeare points of X \ ∂X . The markings of boundary type are points of ∂X . Let M R g,k,l denote themoduli space of Riemann surfaces with boundary of doubled genus g with k distinct boundarymarkings and l distinct interior markings. The moduli space M R g,k,l is defined to be empty unlessthe stability condition 2 g − k + 2 l > M R g,k,l may have several connected components depending uponthe topology of ( X, ∂X ) and the cyclic orderings of the boundary markings. Foundational issuesconcerning the construction of M R g,k,l are addressed in [Liu02]. The moduli space M R g,k,l is a realorbifold of real dimension 3 g − k + 2 l , it is in general not compact and may be not orientablewhen g > . Since interior marked points have well-defined cotangent spaces, there is no difficulty in definingthe cotangent line bundles L i → M R g,k,l for each interior marking, i = 1 , . . . , l . Naively, onemay want to consider a descendent theory via integration of products of the first Chern classes ψ i = c ( L i ) ∈ H ( M R g,k,l , Q ) over a compactification M R g,k,l of M R g,k,l . Namely, (cid:10) τ a τ a · · · τ a l σ k (cid:11) og := 2 − g + k − Z M R g,k,l ψ a ψ a · · · ψ a l l , (1.4)when 2 X a i = 3 g − k + 2 l, and in all other cases (cid:10) τ a τ a · · · τ a l σ k (cid:11) og := 0. Note that, in particular, g + k must always be oddin order to get non-zero numbers. The new insertion σ corresponds to the addition of a boundarymarking. The coefficient in front of the integral on the right-hand side of (1.4) appears to be usefulfor the description of the new intersection numbers, that are called the open intersection numbers ,in terms of integrable systems.In genus 0 the moduli M ,k,l := M R ,k,l is canonically oriented for k odd, and one can calculate anintegral of the form R M ,k,l ψ a ψ a · · · ψ a l l , given boundary conditions for the line bundles L i . Moreprecisely, given nowhere vanishing boundary conditions s ∈ C ∞ ( E → ∂ M ,k,l ) , for E = L L ⊕ a i i , EFINED OPEN INTERSECTION NUMBERS AND THE KONTSEVICH-PENNER MATRIX MODEL 3 one may define the integral (1.4) by (cid:10) τ a τ a · · · τ a l σ k (cid:11) o := 2 − k − Z M ,k,l e ( E, s ) , (1.5)where e ( E, s ) is the relative Euler class . The result depends on the boundary conditions.In [PST14] a family of boundary conditions, called canonical boundary conditions for each bun-dle L i is constructed. It is proven that for a generic choice of canonical boundary conditions, s ij ∈ C ∞ m ( L i → ∂ M ,k,l ) , i ∈ [ l ] , j ∈ [ a i ], the boundary conditions s = L s ij is nowhere vanishingalong ∂ M ,k,l , assuming 2 P a i = 3 g − k + 2 l . Here we use the notation [ l ] for a set { , , . . . , l } and the subscript m indicates that multi-valued section, rather than sections, are used. It is thenshown that any two generic choices of canonical boundary conditions give rise to the same inte-gral (1.5). In [PST14] all open intersection numbers for doubled genus 0 were calculated, and theauthors proposed a conjectural description of the open intersection numbers in all genera. Let s be a formal variable. Define F o ( t , t , . . . , s ) := X g ≥ F og ( t , t , . . . , s ) , where F og ( t , t , . . . , s ) := X k,l ≥ X a ,...,a l ≥ (cid:10) τ a · · · τ a l σ k (cid:11) og s k Q t a i k ! l ! . The generating series F o is called the open free energy and the exponent τ o := exp( F o + F c )is called the open partition function . The conjecture of R. Pandharipande, J. P. Solomon andthe third author ([PST14]) says that the generating series F o satisfies a certain system of partialdifferential equations that is called in [PST14] the open KdV equations.In higher genus the construction of open intersection numbers needs some refinement. Firstly,the moduli space M g,k,l is in general non-orientable for g > . In order to overcome this issue,J. P. Solomon and the third author define graded spin surfaces , which are open surfaces with aspin structure and some extra structure. In [STa] the moduli of graded spin surfaces M g,k,l isdefined and is proved to be canonically oriented. When g = 0 it coincides with M R ,k,l . Canonicalboundary conditions are then constructed for the line bundles L i , and again it is proven that onecan define (cid:10) τ a τ a · · · τ a l σ k (cid:11) og := 2 − g + k − Z M g,k,l e ( E, s ) , (1.6)where e ( E, s ) is the relative Euler with respect to the canonical boundary conditions. As in g = 0 , generic choices of canonical boundary conditions give rise to the same integrals. It should bestressed that, although [STa] has not appeared yet, the moduli and boundary conditions mentionedabove are fully described in Section 2 of [Tes15].A combinatorial formula for the open intersection numbers was found in [Tes15]. The conjectureof R. Pandharipande, J. P. Solomon and the third author was proved in [BT15]. Propertiesof the open free energy F o were intensively studied in [Ale15a, Ale15b, Ale16, Bur15, Bur16,Saf16a]. In particular, in [Bur15, Bur16] the second author introduced a formal power series F o,ext ( t , t , . . . , s , s , . . . ), where s = s and s , s , . . . are new formal variables. The function F o,ext is an extension of the open free energy F o , F o,ext (cid:12)(cid:12) s ≥ =0 = F o , and, therefore, it was called the extended open free energy . The exponent τ o,ext := exp( F o,ext + F c )was called the extended open partition function . In [Bur15, Bur16] the new variables s i , i ≥ ALEXANDER ALEXANDROV, ALEXANDR BURYAK, AND RAN J. TESSLER consider them as descendents of the boundary marked points. A geometric construction of thedescendent theory for the boundary marked points, a derivation of the combinatorial formula forit, and a geometric proof of the conjecture of [Bur15] regarding the extended theory, will appearin [STb],[Tes].In [Bur16] the second author found a simple relation of the extended open partition function τ o,ext to the wave function of the Kontsevich-Witten tau-function. In [Ale15b] the first author provedthat both extended open partition function and closed partition function belong to the same familyof tau-functions, described by the matrix integrals of Kontsevich type. Namely, the Kontsevich-Penner integral τ N | T k = k tr Λ − k := c Λ ,M Z H M e tr H − tr H Λ det N Λdet N (Λ − H ) dH (1.7)for N = 0 coincides with Kontsevich’s integral (1.3). In [Ale15b] it was shown that for N = 1it describes the extended open partition function. From this matrix integral representation itimmediately follows that the extended open partition function is a tau-function of the KP hierarchy,moreover, it is related to the closed partition function τ c by equations of the modified KP hierarchy[KMMM93]. A full set of the Virasoro and W-constrains for the tau-function, described by theKontsevich-Penner matrix integral (1.7), was derived in [Ale15b] for arbitrary N . Later theseconstraints were described by the first author [Ale16] in terms of the so-called free bosonic fields.1.1. Refined, very refined and extended refined open intersection numbers. As we al-ready discussed above, the moduli space M g,k,l may have several components depending upon thetopology of Riemann surface with boundary. For b ≥ 1, denote by M g,k,l,b the submoduli of M g,k,l that consists of isomorphism classes of surfaces with boundary with b boundary components. Sowe have the decomposition M g,k,l = G ≤ b ≤ g +1 b + g =1(mod 2) M g,k,l,b . We can decompose further. Let P ( k, b ) be the set of unordered b -tuples of non-negative integers k = ( k , . . . , k b ), k i ≥ 0, such that P k i = k . For k = ( k , . . . , k b ) ∈ P ( k, b ) let M g, ¯ k,l ⊂ M g,k,l,b be the submoduli of graded smooth Riemann surfaces with boundary of genus g , with l internalmarked points, b boundary components and k boundary marked points distributed on the boundarycomponents according to the b -tuple k . Clearly, M g,k,l,b = G ¯ k ∈ P ( k,b ) M g, ¯ k,l . It is also easy to see that if we define M g,k,l,b as the closure of M g,k,l,b in M g,k,l and M g,k,l as theclosure of M g,k,l in M g,k,l,b , then M g,k,l = G ≤ b ≤ g +1 b + g =1(mod 2) M g,k,l,b , M g,k,l,b = G ¯ k ∈ P ( k,b ) M g, ¯ k,l . In [STa] the authors defined open intersection numbers over any connected component of themoduli space M g,k,l . To be precise, they proved the following result. Theorem 1.1. Let a , . . . , a l , k be non-negative integers satisfying P a i = 3 g − k + 2 l, andlet E = P ki =1 L ⊕ a i i . Then for any connected component C of M g,k,l there exist nowhere vanishing EFINED OPEN INTERSECTION NUMBERS AND THE KONTSEVICH-PENNER MATRIX MODEL 5 canonical boundary conditions s in the sense of [PST14] , [STa] . Thus one may define the integral R C e ( E, s ) . Moreover, any two nowhere vanishing choices of the canonical boundary conditions giverise to the same integral. The theorem allows us to define refined open intersection numbers as the integrals of monomialsin psi-classes over the components M g,k,l,b of M g,k,l and very refined open intersection numbers asthe corresponding integrals over the components M g, ¯ k,l : (cid:10) τ a τ a · · · τ a l σ k (cid:11) og,b :=2 − g + k − Z M g,k,l,b e ( E, s ) , (1.8) D τ a τ a · · · τ a l σ ¯ k E og :=2 − g + k − Z M g, ¯ k,l e ( E, s ) , (1.9)where a , . . . , a l , k, E are as in Theorem 1.1 and s is a nowhere vanishing canonical multisection.These new intersection numbers are rational numbers. Let N be a positive integer. Introduce the refined open free energy F o,N by F o,N ( t , t , . . . , s ) := X g,k,l ≥ b ≥ X a ,...,a l ≥ (cid:10) τ a · · · τ a l σ k (cid:11) og,b N b s k Q t a i k ! l ! . Clearly, F o, = F o . Let q , q , . . . be formal variables. Introduce the very refined open free energy e F o by e F o ( t , t , . . . , q , q , . . . ) := X g,k,l ≥ b ≥ X k =( k ,...,k b ) ∈ P ( k,b ) a ,...,a l ≥ D τ a · · · τ a l σ k E og Q t a i Q q k j k ! l ! . Of course, the function F o,N can be easily expressed in terms of the function e F o : F o,N = e F o (cid:12)(cid:12)(cid:12) q i = Ns i . The reason, why we want to consider the refined open free energy F o,N separately, is that it admitsa natural extension, while we do not know whether the very refined open free energy e F o can beextended. The exponents τ oN := exp( F o,N + F c ) and e τ o := exp( e F o + F c ) will be called the refinedopen partition function and the very refined open partition function respectively.In this paper we generalize the result of the third author from [Tes15] and find a combinatorialformula for the very refined open intersection numbers. We also derive matrix models for therefined and the very refined open partition functions. We then show that the form of our matrixmodel for the refined open partition function τ oN suggests a natural way to add the variables s i , i ≥ 1, in it. We denote the extended function by τ o,extN and call it the extended refined open partitionfunction . This function satisfies the properties τ o,extN (cid:12)(cid:12) s ≥ =0 = τ oN , τ o,ext = τ o,ext . Therefore, it is natural to view the variables s i , i ≥ 1, in the function τ o,extN as descendents of theboundary marked points in the refined open intersection theory. We also prove that the extendedrefined open partition function τ o,extN is related to the very refined open partition function e τ o by asimple transformation. Moreover, we show that this transformation is invertible, so the collectionof functions τ o,extN , N ≥ 1, and the function e τ o are in a certain sense equivalent. Finally, weconjecture that the function τ o,extN coincides with the tau-function τ N given by the Kontsevich-Penner matrix integral (1.7) and present an evidence for the conjecture. In particular, we derive ALEXANDER ALEXANDROV, ALEXANDR BURYAK, AND RAN J. TESSLER the string and the dilaton equations for the function τ o,extN and also prove the conjecture in genus 0and 1. Remark 1.2. In [Saf16a] the author conjectured that there exists a refinement of the extendedopen partition function τ o,ext that distinguishes contributions from Riemann surfaces with differentnumbers of boundary components and that coincides with the Kontsevich-Penner tau-function τ N .Since we construct this refinement, our conjecture can be considered as a stronger version of theconjecture of B. Safnuk from [Saf16a]. Remark 1.3. Another approach to refined open intersection numbers was recently suggested byB. Safnuk in [Saf16b]. His approach is quite different to ours, because, in particular, he does notconsider boundary marked points and, moreover, he uses a different compactification of M g, ,l .His intersection numbers are given as integrals of some specific volume forms. B. Safnuk alsohas a combinatorial formula for his refined open intersection numbers and it directly gives theKontsevich-Penner matrix model. It would be interesting to obtain a direct relation between thetwo approaches.1.2. Organization of the paper. In Section 2 we show that the construction of [STa] admits arefinement that allows to define the products (1.8) and (1.9). We also prove combinatorial formulasfor the refined and the very refined open intersection numbers. In Section 3 we construct a matrixmodel for the very refined open partition function e τ oN . We then show that the specialization ofit, giving the refined open partition function, has a natural extension, where new variables canbe interpreted as descendents of boundary marked points. We prove that the extended refinedopen partition function τ o,extN is related to the very refined open partition function by a simpletransformation. We also prove the string and the dilaton equations for τ o,extN . In Section 4 weformulate our conjecture about the relation between the function τ o,extN and the Kontsevich-Pennertau-function τ N and present an evidence for it.1.3. Acknowledgements. We would like to thank Leonid Chekhov and Rahul Pandharipandefor useful discussions. The work of A.A. was supported in part by IBS-R003-D1, by the NaturalSciences and Engineering Research Council of Canada (NSERC), by the Fonds de recherche duQu´ebec Nature et technologies (FRQNT) and by RFBR grants 15-01-04217 and 15-52-50041YaF.A. B. was supported by Grant ERC-2012-AdG-320368-MCSK in the group of R. Pandharipandeat ETH Zurich and Grant RFFI-16-01-00409. R.T. is supported by Dr. Max R¨ossler, the WalterHaefner Foundation and the ETH Z¨urich Foundation.2. Very refined open intersection numbers Reviewing the proof of the combinatorial formula of [Tes15] . In order to prove acombinatorial formula for the refined open intersection numbers, we first review the proof techniquein the rather long paper [Tes15]. Throughout this subsection we shall address to places in [Tes15]. Step 1 . The starting point of [Tes15] is the following well known fact. Let M be an orbifoldwith boundary or even corners, of real dimension 2 n. Suppose E → M is a vector bundle of realrank 2 n, and s a nowhere vanishing (possibly multi-valued) section of E → ∂M. Let π : S → M be the sphere bundle associated to E, Φ an angular form and Ω an Euler form on M. In otherwords, Φ is a 2 n − S with • R π − ( p ) Φ = 1 , ∀ p ∈ M . • d Φ = − π ∗ Ω . Then we have(2.1) Z M e ( E, s ) = Z M Ω + Z ∂M s ∗ Φ . EFINED OPEN INTERSECTION NUMBERS AND THE KONTSEVICH-PENNER MATRIX MODEL 7 Step 2 . In [Tes15], Section 4 , using the theory of Jenkins-Strebel differential [Str84], with therequired modifications for graded surfaces with boundary, a combinatorial stratification of M g,k,l is constructed. The stratification, given a choice of positive perimeters p = { p , . . . , p l } , consistsof cells parameterized by metric graded ribbon graphs ( G, z ) . These are ribbon graphs with a(positive) metric on edges, l + b holes, where the last b holes, called boundaries correspond toboundary components, the i th hole for 1 ≤ i ≤ l is called a face and is of perimeter p i , and thereare k boundary vertices which correspond to boundary marked points. z is an index for the gradedstructure, whose description is not important at the moment. The topology of the cells is defined inthe natural way using the metric. A cell M ( G ′ ,z ′ ) is a face of a cell M ( G,z ) if G ′ is obtained from G bycontracting some edges and z ′ is the degenerated graded structure. The edge contraction operationallows a compactification of the combinatorial moduli, which is a quotient of M g,k,l , generically 1 :1 . Denote this compactification by M combg,k,l ( p ) . Write also M combg,k,l = ` p ,...,p l > M combg,k,l ( p ) , and endowit with the natural topology and piecewise linear structure obtained by the graphs description. Forlater uses, write M ( G ′ ,z ′ ) = ∂ e M ( G,z ) if ( G ′ , z ′ ) is the result of contracting the edge e of G. Not only the moduli, but also the S bundles associated to the line bundles L i have a combina-torial counterpart, first obtained in [Kon92]. Using these, in [Tes15], Subsection 4 . , a combina-torial S n − bundle S = S ( E ) is constructed for any vector bundle E = L L a i i , where n = P a i . It is then shown, in Proposition 4 . , that canonical multisections used to calculate the openintersection numbers can be taken to be pull backs of canonical multisections over M combg,k,l . Callmultisections of S whose pull back is canonical combinatorial canonical . [Tes15], Lemma 4 . 42 says Lemma 2.1. For any p , . . . , p l > , Z M g,k,l e ( E, s ) = Z M combg,k,l ( p ) e ( S, s ′ ) , where s is a canonical multisection which is a pull back of the combinatorial canonical multisec-tion s ′ . Step 3 . In [Kon92] a combinatorial angular form α i and a combinatorial curvature form ω i wereconstructed, and using them a combinatorial formula for the closed numbers was obtained, byintegration over highest dimensional cells, those parameterized by trivalent ribbon graph. Themain result of [Tes15], Section 3 is an explicit formula for the angular form Φ of a bundle which isa direct sum of complex line bundles L i , in terms of their angular forms α i and curvature forms ω i , such that d Φ is the pull back of − ∧ ω i . Plugging this and (2.1) in Lemma 2.1 we get2 g + k − h τ a · · · τ a l σ k i og = Z M combg,k,l l ^ i =1 ω a i i + Z ∂ M combg,k,l ( s ′ ) ∗ Φ , where Φ is the explicit angular form for L L ⊕ a i i . Finally, this equation can be simplified by noting that only highest dimensional cells of thecombinatorial moduli and its boundary contribute to the integrals. The highest dimensional cellsof M combg,k,l are those parameterized by trivalent graded ribbon graphs. Denote their set by SR g,k,l . For any such graph, ( G, z ) write Br ( G ) for the set of bridges , that is, edges which are either internaledges between two boundary vertices or boundary edges between boundary marked points. Thehighest dimensional cells in ∂ M combg,k,l are exactly those obtained from contracting a bridge in a cellof SR g,k,l . Putting all together we obtain ([Tes15],Lemma 4.45)(2.2) 2 g + k − h τ a · · · τ a l σ k i = X ( G,z ) ∈SR g,k,l Z M ( G,z ) ( p ) l ^ i =1 ω a i i + X ( G,z ) ∈SR g,k,l e ∈ Br ( G ) Z M ∂e ( G,z ) ( p ) ( s ′ ) ∗ Φ , ALEXANDER ALEXANDROV, ALEXANDR BURYAK, AND RAN J. TESSLER where s ′ is combinatorial canonical. Step 4 . The expression (2.2) has a complicated part, the integral of ( s ′ ) ∗ Φ , since it involves themultisection s ′ . However, it turns out that the properties of canonical sections allow computing theright-hand side of (2.2) using iterative integrations by parts. The result is the integral version of thecombinatorial formula. To this end, one must first have an explicit description of the contributinggraded ribbon graphs. Definition 2.2. Let g, k, l be non-negative integers such that 2 g − k + 2 l > , A be a finite setand α : [ l ] → A a map. α, A will be implicit in the definition. A ( g, k, l ) -ribbon graph with boundary is an embedding ι : G → Σ of a connected graph G into a ( g, k, l )-surface with boundary Σ suchthat • { x i } i ∈ [ k ] ⊆ ι ( V ( G )), where V ( G ) is the set of vertices of G . We henceforth consider { x i } as vertices. • The degree of any vertex v ∈ V ( G ) \ { x i } is at least 3. • ∂ Σ ⊆ ι ( G ). • If l ≥ , then Σ \ ι ( G ) = a i ∈ [ l ] D i , where each D i is a topological open disk, with z i ∈ D i . We call the disks D i faces. • If l = 0, then ι ( G ) = ∂ Σ.The genus g ( G ) of the graph G is the genus of Σ. The number of the boundary components of G or Σ is denoted by b ( G ) and v I ( G ) stands for the number of the internal vertices. We denoteby Faces( G ) the set of faces of the graph G, and we consider α as a map α : Faces( G ) → A, by defining for f ∈ Faces( G ) , α ( f ) := α ( i ) , where z i is the unique internal marked point in f. Themap α is called the labeling of G. Denote by V BM ( G ) the set of boundary marked points { x i } i ∈ [ k ] . Two ribbon graphs with boundary ι : G → Σ , ι ′ : G ′ → Σ ′ are isomorphic, if there is an ori-entation preserving homeomorphism Φ : (Σ , { z i } , { x i } ) → (Σ ′ , { z ′ i } , { x ′ i } ) , and an isomorphism ofgraphs φ : G → G ′ , such that(1) ι ′ ◦ φ = Φ ◦ ι. (2) φ ( x i ) = x ′ i , for all i ∈ [ k ] . (3) α ′ ( φ ( f )) = α ( f ) , where α, α ′ are the labelings of G, G ′ respectively and f ∈ Faces( G ) isany face of the graph G. Note that in this definition we do not require the map Φ to preserve the numbering of the internalmarked points.A ribbon graph is critical , if • Boundary marked points have degree 2. • All other vertices have degree 3. • If l = 0 , then g = 0 and k = 3 . A (0 , , − ribbon graph with boundary is called a ghost .Consider maps K from the set of directed edges of G to Z which satisfy • K ( e ) + K (¯ e ) = 1 , where ¯ e is e with opposite orientation. • For any face f i of the graph G we have P K ( e ) = 1, where the sum is taken over thedirected edges of f i , whose direction agree with the orientation of f i . • Any directed edge of a boundary component has K = 0.A grading of a critical ribbon graph is the equivalence class of such maps modulo the relationsobtained by vertex flips. That is, K, K ′ are identified if they differ by a sequence of moves which EFINED OPEN INTERSECTION NUMBERS AND THE KONTSEVICH-PENNER MATRIX MODEL 9 12 3 4 - x x x x x x Figure 1. Critical ribbon graphs.flip all the edge assignments for the edges which touch a vertex v . Write [ K ] for the equivalenceclass of K. A graph together with a grading is called a graded graph .A metric graded graph is a graded graph ( G, [ K ]) together with a metric ℓ : Edges( G ) → R + . Let M ( G, [ K ]) be the moduli of such metrics.From now on the explicit object [ K ] will replace the abstract index z used so far.In Fig. 1 two critical ribbon graphs are shown, the right one is a ghost. We draw internal edgesas thick (ribbon) lines, while boundary edges are usual lines. Note that not all boundary verticesare boundary marked points. We draw parallel lines inside the ghost, to emphasize that the facebounded by the boundary is a special face, without a marked point inside. Definition 2.3. A nodal ribbon graph with boundary is G = ( ` i G i ) /N , where • ι i : G i → Σ i are ribbon graphs with boundary. • N ⊂ ( ∪ i V BM ( G i )) × ( ∪ i V BM ( G i )) is a set of ordered pairs of boundary marked points( v , v ), v = v , of the G i ’s which we identify.We require that • G is a connected graph, • Elements of N are disjoint as sets (without ordering).After the identification of the vertices v and v the corresponding point in the graph is called anode. The vertex v is called the legal side of the node and the vertex v is called the illegal sideof the node.The set of edges Edges( G ) is composed of the internal edges of the G i ’s and of the boundaryedges. The boundary edges are the boundary segments between successive vertices which are notthe illegal sides of nodes. For any boundary edge e we denote by m ( e ) the number of the illegalsides of nodes lying on it. The boundary marked points of G are the boundary marked pointsof G i ’s, which are not nodes. The set of boundary marked points of G will be denoted by V BM ( G )also in the nodal case.A nodal graph G = ( ` i G i ) /N is critical , if • All of its components G i are critical. • Ghost components do not contain the illegal sides of nodes.It is called odd critical if it is critical and any boundary component of G i has an odd number ofpoints that are the boundary marked points or the legal sides of nodes. 12 3 4 + - - Figure 2. A critical nodal ribbon graph. 12 3 4 + - + Figure 3. A non-critical nodal ribbon graph.A graded (odd) critical nodal graph ( G, [ K ]) is a critical (odd) ribbon graph with gradingsassociated to each component G i . A nodal ribbon graph with boundary is naturally embedded into the nodal surface Σ = ( ` i Σ i ) /N .The (doubled) genus of Σ is called the genus of the graph. The notions of an isomorphism andmetric are also as in the non-nodal case. Write M ( G, [ K ]) for the moduli of metrics on ( G, [ K ]) . Remark 2.4. The genus of a closed, and in particular doubled, nodal surface Σ is the genus ofthe smooth surface obtained by smoothing all nodes of Σ . In Fig. 2 there is a critical nodal graph of genus 0, with 5 boundary marked points, 6 internalmarked points, three components, one of them is a ghost, two nodes, where a plus sign is drawnnext to the legal side of a node and a minus sign is drawn next to the illegal side.In Fig. 3 a non-critical nodal graph is shown. Here there is some vertex of degree 4 , thecomponents do not satisfy the parity condition and the ghost component has an illegal node.Let SR mg,k,l ( g SR mg,k,l ) be the set of isomorphism classes of graded (odd) critical nodal ribbongraphs with boundary of genus g , with k boundary marked points, l faces and together with abijective labeling α : Faces( G ) ∼ → [ l ], and m nodes.Denote by e R mg,k,l the set of isomorphism classes of odd critical nodal ribbon graphs with boundaryof genus g , with k boundary marked points, l faces and together with a bijective labeling α :Faces( G ) ∼ → [ l ] , and m nodes. EFINED OPEN INTERSECTION NUMBERS AND THE KONTSEVICH-PENNER MATRIX MODEL 11 Definition 2.5. An effective bridge in a graded critical graph ( G, [ K ]) is a bridge e with m ( e ) = 0 . We denote their set by Br eff ( G ). The graph ∂ e ( G, [ K ]), the result of contracting of the edge e of( G, [ K ]) , which has one node N more than G has, can also be made critical nodal by declaringthe side of N which corresponds to e to be legal, if K ( e ) = 0 , and otherwise declare the other sideof N to be legal. Denote the resulting graph by B ∂ e ( G, [ K ]) . The operation B is called the baseoperation . Definition 2.6. For a metric graded ribbon graph G, define W G := Y e ∈ Edges( G ) ℓ m ( e ) e ( m ( e ) + 1)! , f W G := Y e ∈ Edges( G ) ℓ m ( e ) e m ( e )!( m ( e ) + 1)! . Definition 2.7. An l − set is a map L : [ n ] → [ l ] . The size of L is n. A subset of an l − set is therestriction map L : A → [ l ] , A ⊆ [ n ] . It can canonically identified with a map L ′ : [ | A | ] → [ l ] , hence can be thought as an l − set on its own right. We write L ′ ⊆ L, and set (cid:0) Lm (cid:1) for the set ofall (cid:0) nm (cid:1) l − subsets of L of size m. Definition 2.8. Any l − set defines a vector bundle E L := L L L ( i ) , defined both on the moduli andon the combinatorial moduli. Let S L be the associated combinatorial sphere bundle. Let Φ L bethe associated explicit angular form, mentioned in Step 3 above, and defined in [Tes15, Section 3].Its curvature form is ω L = V i ∈ [ n ] ω L ( i ) . Lemma 2.9. Write n = k +2 l +3 g − Let C ⊆ SR mg,k,l be a set of graphs and let C ′ ⊆ SR m +1 g,k,l be theset of graphs obtained by applying for any graph in C and any effective bridge e of it, first the edgecontraction ∂ e and then the base operation B . Suppose C is closed in the following sense: for anygraph ( G, [ K ]) in SR mg,k,l \ C and any effective bridge e of it we have B ( ∂ e G ) / ∈ C ′ . Then X ( G, [ K ]) ∈ C X e ∈ Br eff ( G ) X L ′ ∈ ( Ln − m ) Z M ∂e ( G, [ K ]) W G Φ L ′ == X ( G, [ K ]) ∈ C X L ′ ∈ ( Ln − m − ) Z M ( G, [ K ]) W G ω L ′ + X e ∈ Br eff ( G ) Z M ∂e ( G, [ K ]) W G Φ L ′ . This lemma is the global version of the combination of Lemmas 6.7 and 6.8 of [Tes15] (therea local version is given, in terms of a single graph, rather than a set C, and in terms of a single l − subset of it, rather than summing over all subsets).Applying Lemma 2.9 iteratively to C = SR mg,k,l , and using some parity observation (Proposi-tion 6.13 in [Tes15]) give the integrated form of the combinatorial formula, [Tes15], Theorem 6.12. Theorem 2.10. For integers a , . . . , a l ≥ which sum to n = k +2 l +3 g − , let L be any l − set with E L = L L ⊕ a i i , then g + k − h τ a · · · τ a l σ k i og = X m ≥ X ( G, [ K ]) ∈ g SR mg,k,l X L ′ ∈ ( Ln − m ) Z M ( G, [ K ]) ( p ) W G ω L ′ . A straightforward corollary is (equation (35) in [Tes15]) Corollary 2.11. g + k − X P li =1 a i = n Y p a i i h τ a · · · τ a l σ k i og = X m ≥ X ( G, [ K ]) ∈ g SR mg,k,l X L ′ ∈ ( Ln − m ) Z M ( G, [ K ]) ( p ) f W G ¯ ω n − m ( n − m )! , where ¯ ω = P i p i ω i . Note that in the last theorem and corollary there is no more dependence on the choice of themultisection. Step 5 . The last step is to perform Laplace transform to the integrated formula described above.This is the content of [Tes15, Sections 6.2, 6.3]. The only difficulty in the calculation of the Laplacetransform of Z M ( G, [ K ]) ( p ) f W G ¯ ω n − m ( n − m )!for a given ( G, [ K ]) ∈ g SR ∗ g,k,l is to show l ^ i =1 dp i ∧ ¯ ω n − m ( n − m )! : ^ e ∈ Edges( G ) dℓ e = ± Y i g ( Gi )+ b ( Gi ) − + v I ( G i ) , and to understand the signs. Here G i are the components of G . This is the content of Section 6.2in [Tes15].After understanding the sign and the ratio of forms, the Laplace transform calculations arestraightforward and give(2.3) Z p i ∈ R + ^ dp i exp (cid:16) − X λ i p i (cid:17) Z M ( G, [ K ]) ( p ) f W G ¯ ω n − m ( n − m )! = ± Q i v I ( G i )+ g ( Gi )+ b ( Gi ) − | Aut( G, [ K ]) | Y e ∈ Edges( G ) λ ( e ) , where λ ( e ) := λ i + λ j , if e is an internal edge between faces i and j ; m +1) (cid:0) mm (cid:1) λ − m − i , if e is a boundary edge of face i and m ( e ) = m ;1 , if e is a boundary edge of a ghost . (2.4)Summing over the different gradings K, and using the results of Section 6.2 regarding the signsgive(2.5) X [ K ] is a grading for G Z p i ∈ R + ^ dp i exp (cid:16) − X λ i p i (cid:17) Z M ( G, [ K ]) ( p ) f W G ¯ ω n − m ( n − m )! == Q i v I ( G i )+ g ( Gi )+ b ( Gi ) − | Aut( G ) | Y e ∈ Edges( G ) λ ( e ) . Summing over all graphs, the resulting combinatorial formula is Theorem 2.12. Fix g, k, l ≥ such that g − k + 2 l > . Let λ , . . . , λ l be formal variables.Then we have (2.6) 2 g + k − X a ,...,a l ≥ h τ a τ a · · · τ a l σ k i og l Y i =1 a i (2 a i − λ a i +1 i == X G = ( ` i G i ) /N ∈ e R ∗ g,k,l Q i v I ( G i )+ g ( G i )+ b ( G i ) − | Aut( G ) | Y e ∈ Edges( G ) λ ( e ) . EFINED OPEN INTERSECTION NUMBERS AND THE KONTSEVICH-PENNER MATRIX MODEL 13 A combinatorial formula for the refined and very refined numbers. In order to writea combinatorial formula for the more refined numbers, first note Observation 2.13. Let ( G ′ , [ K ′ ]) ∈ SR mg,k,l be an arbitrary graph, then there exists a graph ( G, [ K ]) ∈ SR g,k,l , called the smoothing of ( G ′ , [ K ′ ]) and a sequence ( e j ) mj =1 of bridges of G suchthat B ∂ e m · · · B ∂ e ( G, [ K ]) = ( G ′ , [ K ′ ]) . Moreover, if [ ˜ K ′ ] is another graded structure on G ′ then the smoothing of ( G ′ , [ ˜ K ′ ]) is some ( G, [ ˜ K ]) with the same G. Thus, the number of boundaries and partitions of boundary points of the smoothingof a graph ( G ′ , [ K ′ ]) is well-defined and independent of the graded structure. The proof is immediate, the operation B remembers the cyclic order of the illegal nodes on eachboundary edge, hence remembers the topology of the graph on which B was applied. The edgecontraction is easily inverted on the level of graphs, and the value of K on the contracted bridgecan be read from knowing which side of the node the B operation declared to be illegal. The secondpart of the observation follows from the fact that the different gradings on G ′ do not change theway we invert ∂ e .Note that Steps 1–3 of the previous section work without change for the more refined numbers,giving us(2.7) 2 g + k − h τ a · · · τ a l σ k i og,b = X ( G,z ) ∈SR g,b,k,l Z M ( G,z ) ( p ) l ^ i =1 ω a i i + X ( G,z ) ∈SR g,b,k,l e ∈ Br ( G ) Z M ∂e ( G, [ K ]) ( p ) ( s ′ ) ∗ Φ , where SR mg,b,k,l is the subset of SR mg,k,l made of graphs whose smoothing has b boundary compo-nents, and s ′ is again combinatorial canonical. Define similarly g SR mg,b,k,l , R mg,b,k,l and e R mg,b,k,l . Define SR mg, ¯ k,l , g SR mg,k,l , R mg,k,l , e R mg,k,l , accordingly, for graphs which correspond to a partition ¯ k of boundarymarked points. Then acting similarly for the very refined numbers yields(2.8) 2 g + k − h τ a · · · τ a l σ ¯ k i og = X ( G,z ) ∈SR g, ¯ k,l Z M ( G,z ) ( p ) l ^ i =1 ω a i i + X ( G,z ) ∈SR g, ¯ k,l e ∈ Br ( G ) Z M ∂e ( G, [ K ]) ( p ) ( s ′ ) ∗ Φ , where s ′ is again combinatorial canonical.Step 4 requires some modification. Observation 2.13 allows us to apply Lemma 2.9 to the sets C obtained by taking an arbitrary ( G, [ K ]) ∈ SR g,k,l and creating all elements of SR mg,k,l obtainedfrom it by contracting bridges and applying B . Using Lemma 2.9 iteratively now gives Theorem 2.14. For integers a , . . . , a l ≥ which sum to n = k +2 l +3 g − , let L be any l − set with E L = L L ⊕ a i i , then g + k − h τ a · · · τ a l σ k i og,b = X m ≥ X ( G, [ K ]) ∈ g SR mg,b,k,l X L ′ ∈ ( Ln − m ) Z M ( G, [ K ]) ( p ) W G ω L ′ , and g + k − X P li =1 a i = n Y p a i i h τ a · · · τ a l σ k i og,b = X m ≥ X ( G, [ K ]) ∈ g SR mg,b,k,l X L ′ ∈ ( Ln − m ) Z M ( G, [ K ]) ( p ) f W G ¯ ω n − m ( n − m )! , where ¯ ω = P i p i ω i . Similarly, under the same assumptions, Theorem 2.15. g + k − h τ a · · · τ a l σ ¯ k i og = X m ≥ X ( G, [ K ]) ∈ g SR mg,k,l X L ′ ∈ ( Ln − m ) Z M ( G, [ K ]) ( p ) W G ω L ′ , and g + k − X P li =1 a i = n Y p a i i h τ a · · · τ a l σ k i og = X m ≥ X ( G, [ K ]) ∈ g SR mg,k,l X L ′ ∈ ( Ln − m ) Z M ( G, [ K ]) ( p ) f W G ¯ ω n − m ( n − m )! . Step 5 follows without change, since the Laplace transform is performed cell-by-cell, and thensummed over gradings, we see that for the refined numbers it holds that Theorem 2.16. Fix g, k, l ≥ such that g − k + 2 l > . Let λ , . . . , λ l be formal variables.Then we have (2.9) 2 g + k − X a ,...,a l ≥ h τ a τ a · · · τ a l σ k i og,b l Y i =1 a i (2 a i − λ a i +1 i == X G = ( ` i G i ) /N ∈ e R ∗ g,b,k,l Q i v I ( G i )+ g ( G i )+ b ( G i ) − | Aut( G ) | Y e ∈ Edges( G ) λ ( e ) . (2.10) 2 g + k − X a ,...,a l ≥ h τ a τ a · · · τ a l σ ¯ k i og l Y i =1 a i (2 a i − λ a i +1 i == X G = ( ` i G i ) /N ∈ e R ∗ g, ¯ k,l Q i v I ( G i )+ g ( G i )+ b ( G i ) − | Aut( G ) | Y e ∈ Edges( G ) λ ( e ) . Matrix models In this section we present matrix models for the very refined and the extended refined openpartition functions and study their properties. In Section 3.1 we briefly recall the derivation ofthe matrix model for the open partition function τ o . Then in Section 3.2 we show how to modifyit in order to control the distribution of boundary marked points on boundary components of aRiemann surface with boundary. As a result, we obtain a two-matrix model for the very refinedopen partition function e τ o . In Section 3.3 we give a construction of the extended refined openpartition function τ o,extN and present simple transformations that relate it to the function e τ o . InSection 3.4 we analyze the Feynman diagram expansion of the matrix integral for τ o,extN and thenin Sections 3.5, 3.6 derive the string and the dilaton equations for τ o,extN .It will be useful for the future to rewrite formula (2.10) in the following way. For a graph G = ( ` i G i ) /N ∈ e R ∗ g,k,l introduce a combinatorial constant c ( G ) by c ( G ) := Q i c ( G i ), where c ( G i ) := ( , if G i is a ghost , e I ( G i ) − v I ( G i ) − v B ( G i ) − v BM ( G i )+ b ( G i ) , otherwise , (3.1)and e I ( G i ) denotes the number of internal edges in G i , v B ( G i ) is the number of boundary trivalentvertices and v BM ( G i ) is the number of boundary marked points in G i . Then for any g, k, l ≥ EFINED OPEN INTERSECTION NUMBERS AND THE KONTSEVICH-PENNER MATRIX MODEL 15 b ≥ k ∈ P ( k, b ) we have X a ,...,a l ≥ h τ a τ a · · · τ a l σ k i og l Y i =1 (2 a i − λ a i +1 i = X G = ( ` i G i ) /N ∈ e R ∗ g,k,l c ( G ) | Aut( G ) | Y e ∈ Edges( G ) λ ( e ) . (3.2)3.1. Open partition function. Let M ≥ 1. Consider positive real numbers λ , . . . , λ M ∈ R > and the diagonal matrix Λ := diag( λ , . . . , λ M ) . Let c Λ ,M := (2 π ) − M M Y i =1 p λ i Y ≤ i 1. Here the degree is introduced by putting deg(tr( H Λ − d H Λ − d · · · H Λ − d r )) := r + 2 P ri =1 d i .Note that the integral c Λ ,N Z H N P a,b,m e − tr H Λ dH is zero, if m is odd, and is a rational function in λ , . . . , λ N of degree − m , if m is even. The integralon the right-hand side of (3.3) is understood as the term-wise integral of (3.4) with respect to ourGaussian probability measure on H M . We refer the reader to [BT15] for a more detailed discussion.Let us briefly recall the derivation of formula (3.3). It is obtained from the combinatorialformula (2.6), rewritten similarly to (3.2), using the standard matrix models technique. An oddcritical nodal ribbon graph with boundary can be obtained from the disjoint union of criticalnon-nodal ribbon graphs with boundary by gluing boundary marked points. Since the sides ofeach node of the nodal graph are marked by plus or minus, we should assign pluses and minuses • + • + • − s s − tr ( H Λ − H Λ − H Λ − H Λ − ) Figure 4. Boundary piece tr( H ) •• • ++ + s Figure 5. Trivalent star and a ghostto the boundary marked points of the critical non-nodal ribbon graphs with boundary. A collarneighborhood of a boundary component of a critical non-nodal ribbon graph with boundary, thatis not a ghost, is a circle with ribbon half-edges attached to it and also with boundary markedpoints (see Fig. 4). Such a circle with a configuration of ribbon half-edges and marked points willbe called a boundary piece. We see that our odd critical nodal ribbon graph with boundary isobtained by • gluing a set of trivalent stars (see Fig. 5) and boundary pieces, • taking the disjoint union with a number of ghost components (see Fig. 5), and • gluing each boundary marked point coming with minus to a boundary marked point comingwith plus.Remember also that, according to the definition of an odd critical nodal ribbon graph with bound-ary, the number of boundary marked points coming plus on each boundary piece should be odd.We obtain that the trivalent stars give the contribution e tr H in the matrix model (3.3). Let G (Λ , s − ) := X m ≥ − m m + 1 (cid:18) mm (cid:19) s m − Λ − m − = 2Λ + p Λ − s − . Then boundary pieces giveexp tr "X k ≥ k (cid:18) H + s G (Λ , s − ) (cid:19) k − X k ≥ k (cid:18) H − s G (Λ , s − ) (cid:19) k = det Λ + p Λ − s − − H + s Λ + p Λ − s − − H − s . The ghost components give the factor e s in (3.3). The application of the operator e ∂ ∂s∂s − andsetting s − = 0 correspond to gluing each boundary marked point coming with minus to a boundarymarked point coming with plus.3.2. Very refined open partition function. Let us construct now a matrix model for the veryrefined open partition function e τ o . Let N ≥ 1. In addition to the space H M of Hermitian matrices,we consider the space Mat N,N ( C ) of complex N × N matrices. We consider it as a real vector spaceof dimension 2 N . For a matrix Z ∈ Mat N,N ( C ) denote by z i,j , 1 ≤ i, j ≤ N , its entries. Define a EFINED OPEN INTERSECTION NUMBERS AND THE KONTSEVICH-PENNER MATRIX MODEL 17 • + + −− + −• •• • ++ ••− + − −• Figure 6. Inserting external ribbon edgesvolume form dZ on Mat N,N ( C ) by dZ := Y ≤ i,j ≤ N d (Re z i,j ) d (Im z i,j ) . Consider the Gaussian probability measure on Mat N,N ( C ) given by the form1(2 π ) N e − tr ZZ t dZ. Let θ i,j , 1 ≤ i, j ≤ N , be complex variables andΘ :=( θ i,j ) ≤ i,j ≤ N ∈ Mat N,N ( C ) ,q m (Θ) := tr Θ m , m ≥ . Theorem 3.1. We have e τ o | t i = t i (Λ) q i = q i (Θ) = c Λ ,M (2 π ) N Z H M × Mat N,N ( C ) e − tr H Λ − tr ZZ t e tr H + tr Z + tr Z t Θ × (3.5) × det Λ ⊗ id N + q Λ ⊗ id N − id M ⊗ Z t − H ⊗ id N + id M ⊗ Z Λ ⊗ id N + q Λ ⊗ id N − id M ⊗ Z t − H ⊗ id N − id M ⊗ Z dHdZ. Proof. We now use the combinatorial formula (3.2). As we explained in the previous section, anodd critical nodal ribbon graph with boundary is obtained by gluing trivalent stars (Fig. 5) andboundary pieces (Fig. 4), adding ghost components (Fig. 5) and then gluing boundary points tocreate nodes. The problem now is to control the distribution of the boundary marked points onthe boundary components in a smoothing the resulting nodal ribbon graph with boundary. Ouridea is the following. Consider the nodal surface with boundary that is associated with our nodalribbon graph with boundary. Consider a small neighborhood of a boundary node of this surface.At this node two small pieces of boundary components meet. Then, instead of gluing these twopieces at one point, we connect them by a small ribbon edge (see Fig. 6). The new ribbon edgewill be called an external ribbon edge. In Fig. 6 we fill the external ribbon edges by dots in order ++ − tr( H Λ − H Λ − H Λ − H Λ − ) tr( Z Z t ) Figure 7. Boundary piece with external ribbon half-edges++ + tr( Z ) Figure 8. Ghost component with external ribbon half-edges P ≤ i,j ≤ N θ i,j z i,j − Figure 9. External ribbon half-edge corresponding to a marked pointto distinguish them with the usual internal ribbon edges. Doing this procedure at each node, weobtain a non-nodal surface with boundary, which is a smoothing of the initial nodal surface. Notethat each half of an external ribbon edge is marked by plus or minus.Note that the resulting non-nodal surface can be glued from elementary pieces in the followingway. Again we have trivalent stars (Fig. 5). Then we have boundary pieces similar to what we havein the previous section, but now we want to replace each boundary marked point by an externalribbon half-edge, marked by plus or minus (see Fig. 7). In the same way we replace the ghostcomponent from Fig. 5 by the ghost component with external ribbon half-edges (see Fig. 8). Inorder to have marked points we have to introduce an external ribbon half-edge marked by minus(see Fig. 9). Now, in order to obtain our non-nodal surface with external ribbon edges, we glue aset of elementary pieces of four types (Fig. 5, 7, 8, 9) according to the following rules: • An internal ribbon half-edge should be glued to an internal ribbon half-edge. • An external ribbon half-edge with some sign should be glued to an external ribbon half-edgewith an opposite sign.For a polynomial P ( Z ) ∈ C [ z ij , z kl ] let h P ( Z ) i := 1(2 π ) N Z P ( Z ) e − tr ZZ t dZ. Then we have h z i,j , z k,l i = h z i,j , z k,l i = 0 , (3.6) h z i,j , z k,l i = 2 δ i,k δ j,l . (3.7) EFINED OPEN INTERSECTION NUMBERS AND THE KONTSEVICH-PENNER MATRIX MODEL 19 Formulas (3.6) and (3.7) show that our Gaussian probability measure on Mat N,N ( C ) is the correctmeasure to control gluings of external ribbon half-edges with signs. To each elementary piece fromFig. 5, 7, 8, 9 we assign a function on H M × Mat N,N ( C ) in the way shown on these figures. Onlythe case of boundary pieces with external ribbon half-edges needs explanations. The function,corresponding to such a piece, is the product of a function on H M and a function on Mat N,N ( C ).The function of H M is obtained in the same way as in the previous section with the only differencethat we forget about the variables s and s − . Concerning a function on Mat N,N ( C ), we go aroundthe boundary piece in the clockwise direction and look at the external ribbon half-edges that wemeet. If an external ribbon half-edge is marked by plus then we assign to it the matrix Z and ifit is marked by minus then we assign to it the matrix Z t . Then the function on Mat N,N ( C ) is thetrace of the product of these matrices taken according to their order in the clockwise direction. So,the resulting function on H M × Mat N,N ( C ) is the product of two traces. Note that the product ofthe traces of two matrices is the trace of their tensor product. We obtain that all boundary pieceswith external ribbon half-edges give the following contribution to the matrix model for e τ o :exp tr "X k ≥ k (cid:18) H ⊗ id N + id M ⊗ Z G (Λ , Z t ) (cid:19) k − X k ≥ k (cid:18) H ⊗ id N − id M ⊗ Z G (Λ , Z t ) (cid:19) k , (3.8)where id M and id N are the identity matrices in the spaces H M and Mat N,N ( C ), respectively, and G (Λ , Z t ) := X m ≥ − m m + 1 (cid:18) mm (cid:19) Λ − m − ⊗ ( Z t ) m = 2Λ ⊗ id N + q Λ ⊗ id N − id M ⊗ Z t . We see that the expression (3.8) is equal todet Λ ⊗ id N + q Λ ⊗ id N − id M ⊗ Z t − H ⊗ id N + id M ⊗ Z Λ ⊗ id N + q Λ ⊗ id N − id M ⊗ Z t − H ⊗ id N − id M ⊗ Z . Finally, the trivalent stars give the contribution e tr H in the matrix model (3.5), the ghost compo-nents with external ribbon half-edges give e tr Z and the external ribbon half-edges correspondingto marked points give e tr Z t Θ . The theorem is proved. (cid:3) Using this theorem, we can obtain a matrix model for the refined open partition function τ oN inthe following way: τ oN | t i = t i (Λ) = (cid:18) e τ o | t i = t i (Λ) q i = q i (Θ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Θ= s id N =(3.9) = c Λ ,M (2 π ) N Z H M × Mat N,N ( C ) e − tr H Λ − tr ZZ t e tr H + tr Z + s tr Z t ×× det Λ ⊗ id N + q Λ ⊗ id N − id M ⊗ Z t − H ⊗ id N + id M ⊗ Z Λ ⊗ id N + q Λ ⊗ id N − id M ⊗ Z t − H ⊗ id N − id M ⊗ Z dHdZ. Extended refined open partition function. The extended open partition function τ o,ext ∈ Q [[ t , t , . . . , s , s , . . . ]], introduced in [Bur15, Bur16], is uniquely determined by the following equations: τ o,ext (cid:12)(cid:12) s ≥ =0 = τ o , (3.10) ∂∂s n τ o,ext = 1( n + 1)! ∂ n +1 ∂s n +1 τ o,ext , n ≥ . (3.11)Note that equation (3.9) gives a formula for τ o that is slightly different to the initial formula (3.3), τ o | t i = t i (Λ) = c Λ ,M π Z H M × C e − tr H Λ − zz e tr H + z + sz det Λ + √ Λ − z − H + z Λ + √ Λ − z − H − z dHd z, (3.12)where d z := d (Re z ) d (Im z ). Formulas (3.10) and (3.11) imply that τ o,ext (cid:12)(cid:12) t i = t i (Λ) == c Λ ,M π Z H M × C e − tr H Λ − zz e tr H + z det Λ + √ Λ − z − H + z Λ + √ Λ − z − H − z e P i ≥ − i − i +1)! s i z i +1 dHd z. Let s i (Λ) := 2 i i ! tr Λ − i − , i ≥ . It is easy to see that e P i ≥ − i − i +1)! s i z i +1 (cid:12)(cid:12)(cid:12)(cid:12) s i = s i (Λ) = e P i ≥ zi +1( i +1) tr Λ − i − = e − tr log(1 − z Λ − ) == det 1 √ − z Λ − = det Λdet √ Λ − z . So, we get τ o,ext (cid:12)(cid:12) t i = t i (Λ) s i = s i (Λ) =(3.13) = c Λ ,M π Z H M × C e − tr H Λ − zz e tr H + z det Λ + √ Λ − z − H + z Λ + √ Λ − z − H − z det Λdet √ Λ − z dHd z. This formula together with equation (3.9) motivates us to introduce a formal power series τ o,extN ∈ C [[ t , t , . . . , s , s , . . . ]] by τ o,extN (cid:12)(cid:12) t i = t i (Λ) s i = s i (Λ) = c Λ ,M (2 π ) N Z H M × Mat N,N ( C ) e − tr H Λ − tr ZZ t e tr H + tr Z × (3.14) × det Λ ⊗ id N + q Λ ⊗ id N − id M ⊗ Z t − H ⊗ id N + id M ⊗ Z Λ ⊗ id N + q Λ ⊗ id N − id M ⊗ Z t − H ⊗ id N − id M ⊗ Z det Λ N dHdZ det q Λ ⊗ id N − id M ⊗ Z t . The uniqueness of a power series with this property is obvious. However, the existence of sucha series is not trivial. In order to prove it we will define a formal power series τ o,extN using thefunction e τ o and then prove that it satisfies equation (3.14).For a given N ≥ τ o,extN ∈ Q [[ t , t , . . . , s , s , . . . ]] by τ o,extN ( t , t , . . . , s , s , . . . ) := 1(2 π ) N Z Mat N,N ( C ) e τ o | q i = q i ( Z ) e P i ≥ − i − i +1)! s i tr( Z t ) i +1 e − tr ZZ t dZ. (3.15) Lemma 3.2. The function τ o,extN satisfies equation (3.14) . EFINED OPEN INTERSECTION NUMBERS AND THE KONTSEVICH-PENNER MATRIX MODEL 21 Proof. Note that det Λ N det q Λ ⊗ id N − id M ⊗ Z t = 1det q id M ⊗ id N − Λ − ⊗ Z t = e P i ≥ − i − i +1)! s i (Λ) tr( Z t ) i +1 . (3.16)Then the lemma follows from Theorem 3.1 and the elementary formula:1(2 π ) N Z Mat N,N ( C ) Q (Θ) e tr Z t Θ e − tr ΘΘ t d Θ = Q ( Z ) , where Q (Θ) ∈ C [ θ i,j ] is an arbitrary polynomial. (cid:3) For a finite value of N the transform, defined by the right hand side of (3.15) is not invertible.However, if we know τ o,extN for all N ≥ 1, we can find e τ o . Let us consider the space U N of unitary N × N matrices. Then we introduce the volume form on U N , which is proportional to the Haarmeasure and normalized by Z U N dU = 1 . Let p , p , . . . and p ′ , p ′ , . . . be formal variables. Lemma 3.3. If f N ( p ′ , p ′ , . . . ) := 1(2 π ) N Z Mat N,N ( C ) g | p i = i tr Z i e P i ≥ − i p ′ i tr( Z t ) i e − tr ZZ t dZ for some g ∈ C [[ p , p , . . . ]] , then g | p i = i tr A i = Z U N f N | p ′ i = i tr U i e tr U t A dU, A ∈ Mat N,N ( C ) . Proof. The Schur functions s λ ( p , p , . . . ), labeled by partitions λ = { λ ≥ λ ≥ λ ≥ . . . } ,constitute a basis in the space of formal series in the variables p , p , . . . . Recall that they can bedefined by s λ := det( h λ i − i + j ) ≤ i,j ≤ l ( λ ) , where the polynomials h k ( p , p , . . . ), k ∈ Z , are defined by X i ≥ h i z i = e P i ≥ p i z i for k ≥ h k := 0 for k < 0. Thus, it is enough to prove the lemma for g = s λ ,where l ( λ ) ≤ N . (If l ( λ ) > N then both g | p i = i tr Z i and f N are equal to zero.) For any matrix A ∈ Mat N,N ( C ) let p i ( A ) := 1 i tr A i , i ≥ . For any partition µ and matrices A, B ∈ Mat N,N ( C ) we have the following formula [Ale11, eq. (39)],1(2 π ) N Z Mat N,N ( C ) s λ ( p ∗ ( ZA )) s µ (cid:18) p ∗ (cid:18) Z t B (cid:19)(cid:19) e − tr ZZ t dZ = s λ ( p ∗ ( AB )) s λ (1 , , , . . . ) δ λ,µ . Then for any unitary matrix U we can compute1(2 π ) N Z Mat N,N ( C ) s λ ( p ∗ ( Z )) e − tr ZZ t + P i ≥ − ii tr U i tr( Z t ) i dZ == 1(2 π ) N Z Mat N,N ( C ) s λ ( p ∗ ( Z )) X µ s µ ( p ∗ ( U )) s µ (cid:16) p ∗ (cid:16) / Z t (cid:17)(cid:17) e − tr ZZ t dZ == s λ ( p ∗ (id N )) s λ (1 , , , . . . ) s λ ( p ∗ ( U )) [Ale11, Sec. 1.1] = C N ( λ ) s λ ( p ∗ ( U )) , where C N ( λ ) = N Y i =1 ( λ i + N − i )!( N − i )! . On the other hand, for any partition µ and matrices A, B ∈ Mat N,N ( C ) we have [Ale11, eq. (31)] Z U N s λ ( p ∗ ( U A )) s µ ( p ∗ ( U t B )) dU = s λ ( p ∗ ( AB )) s λ ( p ∗ (id N )) δ λ,µ . Therefore, we obtain Z U N s λ ( p ∗ ( U )) e tr U t A dU = Z U N s λ ( p ∗ ( U )) X k ≥ p ( U t A ) k k ! dU == Z U N s λ ( p ∗ ( U )) X µ s µ (1 , , , . . . ) s µ ( p ∗ ( U t A )) dU = 1 C N ( λ ) s λ ( p ∗ ( A )) . This completes the proof of the lemma. (cid:3) In particular, we have e τ o | q i = q i ( A ) = Z U N τ o,extN (cid:12)(cid:12) s i = i ! tr U i +1 e tr U t A dU. Equations (3.9), (3.13), (3.14) and (3.16) imply that τ o,extN (cid:12)(cid:12) s ≥ =0 = τ oN , τ o,ext = τ o,ext . We conjecture that there exists a geometric construction of boundary descendents in the refinedopen intersection theory giving the extended refined open partition function τ o,extN .The function F o,ext,N := log τ o,extN − F c will be called the extended refined open free energy .3.4. Feynman diagram expansion of the extended matrix model. Introduce the extendedrefined open intersection numbers by h τ a · · · τ a l σ c · · · σ c k i o,ext,N := ∂ l + k F o,ext,N ∂t a · · · ∂t a l ∂s c · · · ∂s c k (cid:12)(cid:12)(cid:12)(cid:12) t ∗ = s ∗ =0 . (3.17)From (3.15) it follows that the intersection number h τ a · · · τ a l σ c · · · σ c k i o,ext,N is actually a polyno-mial in N with rational coefficients. So, it is well-defined for all values of N , not necessarily positiveintegers. Therefore, the extended refined open partition function τ o,extN is also well-defined for all EFINED OPEN INTERSECTION NUMBERS AND THE KONTSEVICH-PENNER MATRIX MODEL 23 ••• • −−− − s tr( Z t ) Figure 10. Exceptional graph and the corresponding diagram for the extendedmatrix modelvalues of N . We want to write a combinatorial formula for the extended refined open intersectionnumbers similar to (2.9). Let us write the matrix model (3.14) for τ o,extN in the following way: τ o,extN (cid:12)(cid:12) t i = t i (Λ) = c Λ ,M (2 π ) N Z H M × Mat N,N ( C ) e − tr H Λ − tr ZZ t e tr H + tr Z × (3.18) × det Λ ⊗ id N + q Λ ⊗ id N − id M ⊗ Z t − H ⊗ id N + id M ⊗ Z Λ ⊗ id N + q Λ ⊗ id N − id M ⊗ Z t − H ⊗ id N − id M ⊗ Z e P i ≥ − i − i +1)! s i tr( Z t ) i +1 dHdZ. We see that this matrix model is obtained from (3.9) simply by adding the factor e P i ≥ − i − i +1)! s i tr( Z t ) i +1 in the integrand. Doing the Feynman diagram expansion of (3.18) one can easily see that thereis a combinatorial formula for the intersection numbers (3.17) similar to (2.9), where we allowodd critical nodal ribbon graphs with boundary to have certain exceptional components. Let usformulate it precisely.Recall that a ( g, k, l )-ribbon graph with boundary is called critical, if • Boundary marked points have degree 2. • All other vertices have degree 3. • If l = 0, then g = 0 and k = 3.We will call a ( g, k, l )-ribbon graph with boundary exceptional , if g = l = 0 and k ≥ 1. Obviously,for each k ≥ k = 1 or k = 2 are strictly speaking not stable. Note also that a critical(0 , , k = 3. However, our idea is to distinguish them. Speaking formally, to a (0 , , G = ( ` i G i ) /N will be called extended critical , if • It does not have boundary marked points. • All of its components G i are critical or exceptional. • Ghost components do not contain the illegal sides of nodes. • Exceptional components do not contain the legal sides of nodes.The fact, that we do not allow boundary marked points now, may look surprising, but one cannote that an exceptional component with k = 1 can be easily interpreted as a boundary markedpoint. An extended critical nodal ribbon graph with boundary G = ( ` i G i ) /N is called odd if anyboundary component of each non-exceptional G i has an odd number of the legal sides of nodes.Denote by e R extl the set of odd extended critical nodal ribbon graphs with boundary with l internal faces. For a graph G = ( ` i G i ) /N ∈ e R extl introduce the following notations. Denote by b ( G )the number of boundary components in a smoothing of the nodal surface associated with G . Let c ( G ) := Q i c ( G i ), where c ( G i ) is defined by (3.1) if G i is non-exceptional and c ( G i ) := 1 m ! , if G i is an exceptional graph with m + 1 boundary vertices, m ≥ . For m ≥ m ( G ) the number of exceptional components G i with exactly m + 1boundary vertices. The set of edges Edges( G ) is composed of the internal edges of the G i ’s andof the boundary edges. The boundary edges are the boundary segments in non-exceptional G i ’sbetween successive legal sides of nodes. For an edge e ∈ Edges( G ) the function λ ( e ) is definedby the old formula (2.4). The Feynman diagram expansion of the matrix model (3.18) gives thefollowing formula for the intersection numbers (3.17):(3.19) X a ,...,a l ≥ X m ≥ X c ,...,c m ≥ h τ a · · · τ a l σ c · · · σ c m i o,ext,N l Y i =1 (2 a i − λ a i +1 i Q mj =1 s c j m ! == X G = ( ` i G i ) /N ∈ e R extl c ( G ) | Aut( G ) | N b ( G ) Y e ∈ Edges( G ) λ ( e ) Y m ≥ s exc m ( G ) m . String equation.Proposition 3.4. We have the string equation ∂∂t − X i ≥ t i +1 ∂∂t i − X i ≥ s i +1 ∂∂s i − t − N s ! τ o,extN = 0 . (3.20) Proof. We will use formula (3.18). Denote I := e tr H + tr Z − tr H Λ − tr ZZ t ,I := det Λ ⊗ id N + q Λ ⊗ id N − id M ⊗ Z t − H ⊗ id N + id M ⊗ Z Λ ⊗ id N + q Λ ⊗ id N − id M ⊗ Z t − H ⊗ id N − id M ⊗ Z ,I := e P i ≥ − i − i +1)! s i tr( Z t ) i +1 ,Z M,N := τ o,extN (cid:12)(cid:12) t i = t i (Λ) = c Λ ,M (2 π ) N Z H M × Mat N,N ( C ) I I I dHdZ. Our approach is a modification of the diagrammatic method of E. Witten ([Wit92]) that he usedfor a proof of the Virasoro equations for the closed partition function τ c . First of all, note that − X i ≥ t i +1 ∂∂t i ! τ o,extN (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t j = t j (Λ) = M X i =1 λ i ∂∂λ i Z M,N =(3.21) = t (Λ) Z M,N + c Λ ,M (2 π ) N Z H M × Mat N,N ( C ) − I H Λ − + M X i =1 λ i ∂I ∂λ i ! I I dHdZ EFINED OPEN INTERSECTION NUMBERS AND THE KONTSEVICH-PENNER MATRIX MODEL 25 C C + − Figure 11. Graphs that dominate for λ → ∞ and − X i ≥ s i +1 ∂∂s i − N s ! τ o,extN (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t j = t j (Λ) = − X i ≥ s i +1 ∂∂s i − N s ! Z M,N =(3.22) = c Λ ,M (2 π ) N Z H M × Mat N,N ( C ) − X i ≥ − i i ! s i tr( Z t ) i ! I I I dHdZ == c Λ ,M (2 π ) N Z H M × Mat N,N ( C ) I I ( − N X i =1 ∂I ∂z i,i dHdZ. The only non-trivial step in the proof is to express the derivative ∂τ o,extN ∂t (cid:12)(cid:12)(cid:12) t i = t i (Λ) , as a matrixintegral. Let us prove that ∂τ o,extN ∂t (cid:12)(cid:12)(cid:12)(cid:12) t i = t i (Λ) = c Λ ,M (2 π ) N Z H M × Mat N,N ( C ) (tr H + tr Z ) I I I dHdZ. (3.23)The t derivative corresponds to an extra insertion of τ on the left-hand side of (3.17). Wewant to consider the generating function from the left-hand side of (3.19) with an extra insertionof τ . In order to get it from the right-hand side of (3.19), we have to sum over graphs G =( ` i G i ) /N ∈ e R extl +1 with a distinguished face, which we call C , labeled with a variable λ , thenconsider the behavior for λ → ∞ and extract the coefficient of λ . The coefficient of λ comesprecisely from graphs, where the face C has only one edge. The structure of the neighborhood ofthe distinguished face in such graphs is indicated in Fig. 11. We see that there are two cases. Inthe first case, the edge of our face is internal. In the second case, the edge of the face is boundary.Then, automatically, the face belongs to a component G i of type (0 , , τ c . Thecontribution of this picture in our situation is computed in exactly the same way, as in [Wit92], andit gives the first term tr H in the brackets on the right-hand side of (3.23). Consider the secondpicture in Fig. 11. A graph outside the dotted lines can be an arbitrary odd extended criticalnodal ribbon graph with an additional distinguished illegal ”half” of a node. The part inside thedotted lines gives λ . So, in order to get the contribution of the second picture, we should sumover all exteriors. It is easy to see that this sum gives the second term tr Z in the brackets on theright-hand side of (3.23). Computations (3.21), (3.22) and (3.23) show that the string equation (3.20) is equivalent to theequation Z H M × Mat N,N ( C ) "(cid:18) − tr H Λ − H + tr Z (cid:19) I I + I M X i =1 λ i ∂I ∂λ i − I N X i =1 ∂I ∂z i,i I dHdZ = 0 . (3.24)Note that " M X i =1 λ i (cid:18) ∂∂λ i + ∂∂h i,i (cid:19) + 2 N X j =1 ∂∂z j,j I = 0 . Therefore, equation (3.24) is equivalent to Z H M × Mat N,N ( C ) "(cid:18) − tr H Λ − H + tr Z (cid:19) I I − I M X i =1 λ i ∂I ∂h i,i − N X i =1 ∂ ( I I ) ∂z i,i I dHdZ = 0 . (3.25)Applying the relations0 = Z H M × Mat N,N ( C ) M X i =1 λ i ∂ ( I I I ) ∂h i,i dHdZ == Z H M × Mat N,N ( C ) "(cid:18) tr H Λ − − tr H (cid:19) I I + I M X i =1 λ i ∂I ∂h i,i I dHdZ, Z H M × Mat N,N ( C ) N X j =1 ∂ ( I I I ) ∂z j,j dHdZ == Z H M × Mat N,N ( C ) " − tr Z I I + N X i =1 ∂ ( I I ) ∂z i,i I dHdZ, we see that equation (3.25) is true. The string equation (3.20) is proved. (cid:3) Dilaton equation.Proposition 3.5. We have the dilaton equation ∂∂t − X n ≥ n + 13 t n ∂∂t n − X n ≥ n + 23 s n ∂∂s n − N − ! τ o,extN = 0 . (3.26) Proof. We have(3.27) − X i ≥ i + 13 t i ∂∂t i ! τ o,extN (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t j = t j (Λ) = 13 X λ i ∂∂λ i Z M,N == c Λ ,M (2 π ) N Z H M × Mat N,N ( C ) (cid:20)(cid:18) M − tr H Λ6 (cid:19) I I I + 13 I I X λ i ∂I ∂λ i (cid:21) dHdZ. It is also easy to see that − X i ≥ i + 23 s i ∂∂s i ! τ o,extN (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t j = t j (Λ) = c Λ ,M (2 π ) N Z H M × Mat N,N ( C ) (cid:20) − I I X z i,j ∂I ∂z i,j (cid:21) dHdZ. (3.28) EFINED OPEN INTERSECTION NUMBERS AND THE KONTSEVICH-PENNER MATRIX MODEL 27 C C C C C Figure 12. Graphs of internal type that contribute in order λ As in the proof of the string equation, the only non-trivial step here is the computation of the t derivative. Let us prove that ∂τ o,extN ∂t (cid:12)(cid:12)(cid:12)(cid:12) t i = t i (Λ) = c Λ ,M (2 π ) N Z H M × Mat N,N ( C ) (cid:20)(cid:18) 13 tr H − tr H Λ + tr H Λ + M (cid:19) I I I (3.29) + I I (cid:18) − X λ i ∂I ∂h i,i + 12 X z i,j ∂I ∂z i,j + X h i,j ∂I ∂h i,j (cid:19) (3.30) + (cid:18) 112 tr Z + 14 tr ZZ t (cid:19) I I I (cid:21) dHdZ. (3.31)We want to compute the generating series from the left-hand side of (3.19) with an extra insertionof τ . In order to get it from the right-hand side of (3.19), we have to sum over graphs G =( ` i G i ) /N ∈ e R extl +1 with a distinguished face, which we call C , labeled with a variable λ , andthen pick out the coefficient of λ . The coefficient of λ can only come from graphs, where theface C has at most three edges. The structure of such graphs in indicated in Fig. 12 and Fig. 13.We see that there are 10 cases and we divide them in two types. Graphs of internal type arethose graphs where all the edges of the face C are internal and graphs of boundary type are thosegraphs where at least one edge of the face C is boundary. The diagrams inside the dotted linesin the top row in Fig. 12 are pieces of arbitrary larger graphs, while the graphs in the bottom rowin Fig. 12 are special ribbon graphs corresponding to closed Riemann surfaces of genus 0 and 1respectively. The five pictures in Fig. 12 already appeared in [Wit92] in the diagrammatic proofof the dilaton equation for τ c . The contribution of these pictures in our situation is computed inexactly the same way, as in [Wit92], and it gives the five terms in the integrand in line (3.29).Let us consider graphs of boundary type. Let us look at the first picture in Fig. 13. Supposethat the face adjacent to C is labeled with λ i . Then the diagram inside the dotted lines gives12 1 λ ( λ + λ i ) = 12 (cid:18) λ − λ i λ + . . . (cid:19) . C ,λ λ i C ,λ + − λ i C λ j λ i + − + − + − C + − − + C Figure 13. Graphs of boundary type that contribute in order λ So, the coefficient of λ is − λ i . Note that the graph outside the dotted lines can be an arbitrary oddextended critical ribbon graph with boundary with a distinguished face labeled by λ i and havinga boundary edge. Now it is easy to see that the first picture in Fig. 13 gives the term − P λ i ∂I ∂h i,i in line (3.30). Consider now the second picture in Fig. 13. The part inside the dotted lines gives14 1 λ ( λ + λ i ) = 14 1 λ + . . . . So, the coefficient of λ is . Now we may shrink the interior of the dotted lines to a point andsum over all possible exteriors. This gives the term P z i,j ∂I ∂z i,j in line (3.30). In the third picturein Fig. 13 the interior of the dotted lines gives12 1 λ ( λ + λ i )( λ + λ j ) = 12 1 λ + . . . . Therefore, the coefficient of λ is , and this picture corresponds to the term P h i,j ∂I ∂h i,j inline (3.30). One can also easily see that the two pictures in the bottom row in Fig. 13 corre-spond to the two terms in the integrand in line (3.31). Thus, formula (3.29) for the t derivativeis proved.Formulas (3.27), (3.28) and (3.29) imply that the dilaton equation (3.26) is equivalent to Z H M × Mat N,N ( C ) (cid:20)(cid:18) 13 tr H − 76 tr H Λ + tr H Λ + 23 M + 112 tr Z + 14 tr ZZ t − N (cid:19) I I I (3.32) + I I (cid:18) X λ i ∂I ∂λ i − X λ i ∂I ∂h i,i + 12 X z i,j ∂I ∂z i,j + X h i,j ∂I ∂h i,j (cid:19) − I I X z i,j ∂I ∂z i,j (cid:21) dHdZ = 0 . EFINED OPEN INTERSECTION NUMBERS AND THE KONTSEVICH-PENNER MATRIX MODEL 29 Using the relation0 = Z H M × Mat N,N ( C ) X ∂∂z i,j ( z i,j I I I ) dHdZ == Z H M × Mat N,N ( C ) (cid:20)(cid:18) N − 12 tr ZZ t (cid:19) I I I + I I X z i,j ∂I ∂z i,j + I I X z i,j ∂I ∂z i,j (cid:21) dHdZ, we see that equation (3.32) is equivalent to Z H M × Mat N,N ( C ) (cid:20)(cid:18) 13 tr H − 76 tr H Λ + tr H Λ + 23 M + 112 tr Z − 112 tr ZZ t + 16 N (cid:19) I I I (3.33)+ I I (cid:18) X λ i ∂I ∂λ i − X λ i ∂I ∂h i,i + 12 X z i,j ∂I ∂z i,j + X h i,j ∂I ∂h i,j + 23 X z i,j ∂I ∂z i,j (cid:19)(cid:21) dHdZ = 0 . Note that (cid:18)X λ i ∂∂λ i + 2 X z i,j ∂∂z i,j + X h i,j ∂∂h i,j + X z i,j ∂∂z i,j (cid:19) I = 0 . This relation simplifies (3.33) in the following way, Z H M × Mat N,N ( C ) (cid:20)(cid:18) 13 tr H − 76 tr H Λ + tr H Λ + 23 M + 112 tr Z − 112 tr ZZ t + 16 N (cid:19) I I I (3.34) + I I (cid:18) − X λ i ∂I ∂h i,i + 16 X z i,j ∂I ∂z i,j + 23 X h i,j ∂I ∂h i,j (cid:19)(cid:21) dHdZ = 0 . Using now the relation0 = Z H M × Mat N,N ( C ) X ∂∂z i,j ( z i,j I I I ) dHdZ == Z H M × Mat N,N ( C ) (cid:20)(cid:18) N − 12 tr ZZ t + 12 tr Z (cid:19) I I I + I I X z i,j ∂I ∂z i,j (cid:21) dHdZ, we obtain that (3.34) is equivalent to Z H M × Mat N,N ( C ) (cid:20)(cid:18) 13 tr H − 76 tr H Λ + tr H Λ + 23 M (cid:19) I I I (3.35) + I I (cid:18) − X λ i ∂I ∂h i,i + 23 X h i,j ∂I ∂h i,j (cid:19)(cid:21) dHdZ = 0 . Finally, using the relations0 = Z H M × Mat N,N ( C ) X ∂∂h i,j ( h i,j I I I ) dHdZ == Z H M × Mat N,N ( C ) (cid:20)(cid:18) M + 12 tr H − tr H Λ (cid:19) I I I + I I X h i,j ∂I ∂h i,j (cid:21) dHdZ, Z H M × Mat N,N ( C ) X λ i ∂∂h i,i ( I I I ) dHdZ == Z H M × Mat N,N ( C ) (cid:20)(cid:18) 12 tr H Λ − tr H Λ (cid:19) I I I + I I X λ i ∂I ∂h i,i (cid:21) dHdZ, we see that equation (3.35) is true. This completes the proof of the dilaton equation. (cid:3) Main conjecture In this section we formulate a conjectural relation of the extended refined open partition func-tion τ o,extN to the Kontsevich-Penner tau-function τ N from (1.7). In the case N = 1 we show how torelate directly our matrix model (3.14) to the Kontsevich-Penner matrix model. We also discussmore evidence for the conjecture. In particular, we show that the conjecture is true in genus 0and 1.4.1. Kontsevich-Penner matrix model and the partition function τ o,extN . Let T k , k ≥ 1, beformal variables. Recall that the Kontsevich-Penner tau-function τ N is defined as a unique formalpower series in the variables T , T , . . . satisfying equation (1.7) for each M ≥ 1. It is not hardto see (see Section 4.3.2 below) that τ N is a formal power series in T , T , . . . with the coefficientsthat are polynomials in N with rational coefficients. Therefore, similarly to τ o,extN , the function τ N is well-defined for all values of N , not necessarily positive integers. Remark 4.1. Note that in [Ale15a] our variables T k are denoted by t k . Note also that we writethe Kontsevich-Penner matrix integral in a way slightly different from [Ale15a] (see formula (1.1)there). In order to identify formula (1.1) from [Ale15a] with the right-hand side of (1.7), one hasto make the shift Φ Φ + Λ and then the variable change Φ = − H .In [Ale15a, Ale15b] the first author proved that τ o,ext = τ | T i +1 = ti (2 i +1)!! ,T i +2 = si i +1( i +1)! . (4.1)We propose the following conjecture. Conjecture 4.2. For any N we have τ o,extN = τ N | T i +1 = ti (2 i +1)!! ,T i +2 = si i +1( i +1)! . Case N = In [Ale15a, Ale15b] the relation between τ o,ext and τ was established with thehelp of some properties of the integrable hierarchies. In this section we prove directly that for N = 1 the integral representation (3.14) for the generating series of the extended refined openintersection numbers indeed coincides with the Kontsevich-Penner matrix integral (1.7).Let ˜ Z M := 1 c Λ ,M τ o,ext (cid:12)(cid:12)(cid:12)(cid:12) t i = t i (Λ) ,s i = s i (Λ) . Then from (3.13) we have˜ Z M = 12 π Z H M × C e tr H − tr H Λ − | z | + z det Λ + B − H + z Λ + B − H − z det Λ B dHd z, (4.2)where B := √ Λ − ¯ z. Let us use the identity, valid for arbitrary formal series f of two variables: Z C d z e − | z | f (¯ z, z ) = Z ∞−∞ dx Z ∞−∞ dy e ixy f ( − iy, x ) . Remark 4.3. This relation can be considered as a simplest example of the more general relationbetween a complex matrix model and a Hermitian two-matrix model. EFINED OPEN INTERSECTION NUMBERS AND THE KONTSEVICH-PENNER MATRIX MODEL 31 This identity allows us to rewrite (4.2) as˜ Z M = 12 π Z H M dH Z ∞−∞ dx Z ∞−∞ dy e tr H − tr H Λ+ ixy + x det Λ + A − H + x Λ + A − H − x det Λ A , where A := p Λ + 2 iy is a diagonal matrix A = diag( a , . . . , a M ). Let us change the variable of integration H H + Λ + A. Then 16 H − H Λ H + 12 H A + iyH + 16 (Λ + A ) ( A − Z M = 12 π Z H M dH Z ∞−∞ dx Z ∞−∞ dy ×× exp (cid:18) 16 tr (Λ + A ) ( A − H + x iy ( x + tr H ) + 12 tr H A (cid:19) det H − xH + x det Λ A . The Harish-Chandra-Itzykson-Zuber formula for the unitary matrix integral, dependent on twodiagonal matrices V = diag( v , v , . . . , v M ) , W = diag( w , w , . . . , w M )yields Z U M e tr UV U t W dU = M − Y k =1 k ! ! det Mi,j =1 e v i w j Q
H C ¯ C t (cid:19) , where H is an M × M Hermitian matrix and C is a complex vector. Sincetr ˜Φ = tr H + 3 ¯ C t HC and tr ˜Φ Λ ∗ = tr H Λ + ¯ C t Λ C we have ˜ Z M = det Λ(2 π ) M Z H M × C M e tr (cid:16) H − H (cid:17) − ¯ C t (Λ − H ) C dH M Y i =1 dC i = Z H M e tr H − tr H Λ det Λdet(Λ − H ) dH. Remark 4.4. We expect that a similar argument can be applied for any positive integer N .4.3. Further evidence. EFINED OPEN INTERSECTION NUMBERS AND THE KONTSEVICH-PENNER MATRIX MODEL 33 String and dilaton equations. String and dilaton equations for the Kontsevich-Penner modelwere derived in [BH12, Ale15a] (In a more general setup of the Generalized Kontsevich Modelthe string equation in terms of the eigenvalues of the external matrix was derived already in[KMMM93]). They coincide with the equations for the extended refined open partition function,derived in Sections 3.5 and 3.6.4.3.2. Genus expansion. Let F KP,N := log τ N − F c | t i =(2 i +1)!! T i +1 , h θ a · · · θ a n i KP,N := ∂ n F KP,N ∂T a · · · ∂T a n (cid:12)(cid:12)(cid:12)(cid:12) T ∗ =0 , n ≥ , a , . . . , a n ≥ . Then Conjecture 4.2 is equivalent to the equation h τ a · · · τ a l σ c · · · σ c k i o,ext,N = h θ a +1 · · · θ a l +1 θ c +2 · · · θ c k +2 i KP,N Q i (2 a i + 1)!! Q j c j +1 ( c j + 1)! . (4.3)Let us insert genus parameters on the both sides of this equation. Let us look at the combinatorialformula (3.19). An elementary computation shows that for a graph G ∈ e R extl we have − deg Y e ∈ Edges( G ) λ ( e ) + X m ≥ (2 m + 2)exc m ( G ) = 3 g ( G ) − l + X m ≥ exc m ( G ) ! , where deg denotes the degree of a rational function in λ , . . . , λ l . This implies that a graph G ∈ e R extl contributes only to intersection numbers DQ li =1 τ a i Q kj =1 σ c j E o,ext,N with P (2 a i + 1) + P (2 c j + 2) = 3( g ( G ) − l + k ). For g ≥ DQ li =1 τ a i Q kj =1 σ c j E o,ext,Ng to be equal to DQ li =1 τ a i Q kj =1 σ c j E o,ext,N , if P (2 a i + 1) + P (2 c j + 2) = 3( g − l + k ), and to be equal to 0otherwise. Note that for a graph G ∈ e R extl the parity of b ( G ) is opposite to the parity of g ( G ) andalso b ( G ) ≤ g ( G ) + 1. Thus, DY τ a i Y σ c j E o,ext,Ng is ( an odd polynomial in N of degree ≤ g + 1 , if g is even , an even polynomial in N of degree ≤ g + 1 , if g is odd . (4.4)In particular, DY τ a i Y σ c j E o,ext,Ng = DY τ a i Y σ c j E o,ext, g N g +1 , for g = 0 , . (4.5)Let us now look at the numbers h θ a · · · θ a n i KP,N . For n ≥ R KPn the set of criticalribbon graphs with boundary, but with no boundary marked points and n internal faces togetherwith a bijective labeling α : Faces( G ) ∼ → [ n ]. Doing the Feynman diagram expansion of theKontsevich-Penner matrix model (1.7) (see [Saf16b]), one gets that X a ,...,a n ≥ h θ a · · · θ a n i KP,N n Y i =1 a i λ a i i = X G ∈R KPn e I ( G ) − v I ( G ) | Aut( G ) | N b ( G ) Y e ∈ Edges( G ) λ ( e ) , n ≥ . (4.6)We see that, similarly to the intersection numbers (3.17), the number h θ a · · · θ a n i KP,N is a poly-nomial in N with rational coefficients. It is easy to see that a graph G ∈ R KPn contributes only tointersection numbers h θ a · · · θ a n i KP,N with P a i = 3( g ( G ) − n ). So, for a non-negative integer g we define h θ a · · · θ a n i KP,Ng to be equal to h θ a · · · θ a n i KP,N , if P a i = 3( g − n ), and to be equalto 0 otherwise. The combinatorial formula (4.6) immediately implies that DY θ a i E KP,Ng is ( an odd polynomial in N of degree ≤ g + 1 , if g is even , an even polynomial in N of degree ≤ g + 1 , if g is odd . (4.7)Therefore, DY θ a i E KP,Ng = DY θ a i E KP, g N g +1 , for g = 0 , . (4.8)Properties (4.4) and (4.7) agree with the conjectural equation (4.3). Also these propertiestogether with equation (4.1) imply that Conjecture 4.2 is true for N = − 1. Equations (4.5)and (4.8) together with (4.1) imply that the equation h τ a · · · τ a l σ c · · · σ c k i o,ext,Ng = h θ a +1 · · · θ a l +1 θ c +2 · · · θ c k +2 i KP,Ng Q i (2 a i + 1)!! Q j c j +1 ( c j + 1)!(4.9)is true for g = 0 and g = 1.We have also checked equation (4.9) in several cases in genus 2. References [Ale11] A. Alexandrov. Matrix models for random partitions. Nuclear Phys. B 851 (2011), no. 3, 620–650.[Ale15a] A. Alexandrov. Open intersection numbers, matrix models and MKP hierarchy . Journal of High EnergyPhysics (2015), no. 3, 042, front matter+13 pp.[Ale15b] A. Alexandrov. Open intersection numbers, Kontsevich-Penner model and cut-and-join operators . Journalof High Energy Physics (2015), no. 8, 028, front matter+24 pp.[Ale16] A. Alexandrov. 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Alexandrov:Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang 37673, Republic ofKorea,CRM, Universit´e de Montr´eal, Montreal, Canada,Department of Mathematics and Statistics, Concordia University, Montreal, Canada,and ITEP, Moscow, Russian Federation E-mail address : [email protected] A. Buryak:Department of Mathematics, ETH Zurich, Switzerland E-mail address : [email protected] R. J. Tessler:Institute for Theoretical Studies, ETH Zurich, Switzerland E-mail address ::