Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence
aa r X i v : . [ m a t h - ph ] F e b SPhT-T08/028
Large N expansion of convergent matrix integrals,holomorphic anomalies, and background independence
B. Eynard , Institut de Physique Th´eorique de Saclay,F-91191 Gif-sur-Yvette Cedex, France. Abstract :We propose an asymptotic expansion formula for matrix integrals, including oscil-latory terms (derivatives of theta-functions) to all orders. This formula is heuristicallyderived from the analogy between matrix integrals, and formal matrix models (combi-natorics of discrete surfaces), after summing over filling fractions. The whole oscillatoryseries can also be resummed into a single theta function. We also remark that the co-efficients of the theta derivatives, are the same as those which appear in holomorphicanomaly equations in string theory, i.e. they are related to degeneracies of Riemannsurfaces. Moreover, the expansion presented here, happens to be independent of thechoice of a background filling fraction. E-mail: [email protected] Introduction
Convergent matrix integrals of the formˆ Z = Z H n dM e − N Tr V ( M ) (1.1)are very usefull in many areas of physics (statistical physics, mesoscopic physics, quan-tum chaos,...) and in mathematics (probabilities, orthogonal polynomials,...) [35, 11].People are mostly interested in their asymptotic behavior in the large n limit (and n/N ∼ constant).There is another form of matrix integrals, called formal-matrix integrals, whichcome from combinatorics (2d quantum gravity for physicists [10, 15, 18]). They aregenerating functions for counting discrete surfaces (also called ”maps”) of given topol-ogy. Formal matrix integrals are only asymptotic series, they are not convergent ingeneral, and almost by definition, they always have a large n expansion (see [18]). Allthe terms in their large n expansion are known [17, 22], and are deeply related to alge-braic geometry and integrable systems. They have many applications to combinatorics,and string theory in physics [34, 36].In this article, we use the analogy between the two types of matrix integrals, andgeneralizing the method of [9], we propose an asymptotic formula for convergent matrixintegrals, including oscillations to all orders:ˆ Z ∼ e P g N − g F g (cid:18) Θ + 1 N ( F ′ Θ ′ + F ′′′ ′′′ ) + . . . (cid:19) ∼ e P g N − g F g X k X l i ′ X h i N P i (2 − h i − l i ) k ! l ! . . . l k ! F ( l ) h . . . F ( l k ) h k ∂ P l i Θ(1 . Z ∼ e P g N − g F g Θ N F ′ + ∞ X k =1 N − k u ( k ) , iπτ + ∞ X j =1 N − j t ( j ) ! (1.3)We also observe that the coefficients in front of the derivatives of Θ in eq.1.2, arethe same which appear in the so-called ”holomorphic anomaly equations” discoveredin the context of topological string theory [8]. In other words they are related to thecombinatorics of degeneracies of Riemann surfaces.2inally, we observe, that although we define each term of the expansion after choos-ing a reference filling fraction ǫ ∗ , the partition function is in fact independent of thatchoice. This is related to the so-called background independence problem in stringtheory, first observed by Witten [38].For the 1-hermitian matrix model (with real potential), the first term of this asymp-totic expansion ˆ Z ∼ e P g N − g F g Θ (1.4)was derived rigorously by Deift& co [14] using Riemann-Hilbert methods, and theirmethod proved the existence of a whole oscillatory series containing derivatives of theTheta-function. The same result was also obtained by heuristic physicists methods by[9]. Here, we generalize the method of [9] and we give the exact coefficient of the wholeseries.Also, in case where the genus of the Theta function is zero, there is no oscillatoryterm, and one finds the so-called topological expansion ˆ Z ∼ e P g N − g F g , which iswell known to coincide (in the sense of asymptotic formal series) with the generatingfunction for enumerating discrete surfaces [10]. In this genus zero case, the asymptoticsof the convergent matrix integral were derived by several methods and several authors[16, 30]. The coefficient of the expansion are of course the symplectic invariants of [22].For other convergent matrix models, for instance the 2-matrix model, such expan-sions were conjectured many times [23, 24], but never proved. Here, we don’t prove iteither. We merely give all the coefficients in the formula to prove, and we explain theirheuristic origin.As we said above, the heuristic origin of the formulae presented in this article, isjust the analogy between formal and convergent matrix models. Outline: • In the first section, we define the convergent matrix model on generalized paths,and write it as a sum over filling fractions. • In the second section, we consider the formal matrix model. • In the 3rd section we perform the sum over filling fractions, and we get Θ-functions. • In the 4rth section we discuss the link with degenerate Riemann surfaces andholomorphic anomaly equations. • In the 5th section we discuss the background independence problem. • Section 6 is the conclusion. 3 .1 Introductory example: 1 matrix model
Consider a polynomial potential V ( x ), of degree d + 1 = deg V , with complex coeffi-cients. There are many different paths γ such that the integral Z γ dx e − V ( x ) (1.5)is absolutely convergent. These are the paths which go to ∞ in a sector where Re V > V such that V ′ is arational fraction). Example: quartic potential V ( x ) = x , we have deg V = 4, i.e. there are d = 3independent paths, for example we choose: γ γ γ . In this example,we have R = γ + γ .In fact, there are d = deg V ′ homologically independent such paths. Let us choosea basis: γ , . . . , γ d (1.6)This means any (unbounded) path on which the integral R γ dx e − V ( x ) is well defined,can be decomposed on the basis: γ = d X i =1 c i γ i (1.7)By definition: Z γ dx e − V ( x ) = d X i =1 c i Z γ i dx e − V ( x ) (1.8)In this definition, the c i ’s can be arbitrary complex numbers, they don’t need to beintegers.However, if γ is itself a path, the c i ’s can take only the values +1 , − , or 0.If the c i ’s are not integers, we say that γ = P i c i γ i is a generalized path .4 .1.2 Matrix model on a generalized path Let γ be a generalized path. We define the set of Normal matrices on γ : H n ( γ ) = { M = U diag( x , . . . , x n ) U † / U ∈ U ( n ) , ∀ i x i ∈ γ } (1.9)equipped with the measure: dM = ∆( x ) dU dx . . . dx n , ∆( x ) = Y i In general H n ( γ ) is not a group, for instance the sum of two matrices in H n ( γ ) is not in H n ( γ ), and the product by a scalar is not either. Also, the ”measure” dM is not positive, in fact it is complex.The matrix integral on H n ( γ ) is defined as follows:ˆ Z ( γ ) = 1 n ! Z H n ( γ ) dM e − N Tr V ( M ) = 1 n ! Z γ n dx . . . dx n ∆( x ) Y i e − NV ( x i ) (1.11)or in other words: ˆ Z ( γ ) = X n + ... + n d = n c n . . . c n d d Z ( n /N, . . . , n d /N ) (1.12)where we have defined: Z ( n /N, . . . , n d /N ) = 1 n ! . . . n d ! Z γ n × ... × γ ndd dx . . . dx n ∆( x ) Y i e − NV ( x i ) (1.13) First, let us assume that only g + 1 ≤ d of the c i ’s are non-vanishing. We write: ∀ i = 1 , . . . , g , c i = e iπν i , c g +1 = 1 , ∀ i = g + 2 , . . . , d , c i = 0(1.14)If γ is a path, the c i ’s take the values ± 1, and thus ν i = 0 or 1 / 2. The vector ( ν , . . . , ν g )is going to be considered a charcateristic in a genus g Jacobian. Also, up to revertingthe orientation of γ i , we can always assume that if γ is a path, γ = path ⇒ ∀ i = 1 , . . . , g + 1 , c i = 1 , ⇒ ν = 0 (1.15)5 ypothesis: Our working hypothesis is that the basis paths γ , . . . , γ g +1 have been chosen sothat each Z ( n /N, . . . , n d /N ) admits a large N topological expansion:ln ( Z ( ǫ , . . . , ǫ d − )) ∼ F ( ǫ ) = ∞ X h =0 N − h F ( h ) ( ǫ ) (1.16)It is conjectured that given a (generic) potential V , and a generalized path γ , sucha ”good” basis always exists (may be not unique). In fact, for the 1-matrix modelwith polynomial potential, this can be proved a posteriori from the asymptotics of M.Bertola [6, 7]. But for more general cases, it is only a conjecture, for instance for the2-matrix model.Now, let us explain where this hypothesis comes from, and what heuristic argumentssupport it. It is well known that any integral defined in eq.1.13, satisfies an infinite set of lin-ear equations, sometimes called ”loop equations” [15], or Virasoro constraints, orSchwinger–Dyson equations, or Euler–Lagrange equations, and which just come fromintegration by parts: ∀ k ≥ − , V k .Z = 0 (1.17) V k = deg V X j =1 jt j ∂∂t k + j + 1 N k X j =0 ∂∂t j ∂∂t k − j , V ( x ) = X j t j x j (1.18)They satisfy Virasoro algebra: [ V k , V j ] = ( k − j ) V k + j (1.19) Remark: It is important to notice that, since integration by parts is independentof the integration paths (as long as there is no boundary term), both ˆ Z ( γ ) and any Z ( n /N, . . . , n d /N ) , ∀ n i , satisfy the same set of loop equations. Formal matrix integrals are defined as formal generating functions for enumeratingdiscrete surfaces (also called ”maps”, i.e. topological graphs embedded on a Riemannsurface, such that each face is a disc) of given topology. Basically, F g is the generatingfunction for counting ”maps” of genus g . The generating series:ln Z formal = ∞ X g =0 N − g F g (1.20)6eeds not be convergent, and in fact it is never convergent if the weights for ”maps”are positive. It is merely a formal series, whose only role is to encode the F g ’s.The formal matrix integrals satisfy the same loop equations, i.e. Virasoro con-straints as ˆ Z ( γ ) and Z ( n /N, . . . , n d /N ) (see [15]). In the context of combinatorics ofmaps, loop equations are known as Tutte’s equations [37], and were first obtained bycounting ”maps” recursively (removing one edge at each step).The F g ’s of formal matrix integrals have all been computed: F has been knownfor a long time, then F [12], and all the F g ’s with g ≥ N expansion of the form:ln Z = ∞ X g =0 N − g F g (1.21)can be obtained by the symplectic invariants of [22], i.e. they are encoded by a spectralcurve. Both the convergent matrix integral, and the formal matrix integral are associated toan (algebraic) spectral curve of the form: y = V ′ ( x ) − N (cid:28) Tr V ′ ( x ) − V ′ ( M ) x − M (cid:29) (1.22) • For the convergent matrix integral ˆ Z defined in eq.1.11, the average < . > istaken with respect to the measure dM e − N Tr V ( M ) . The notion of a spectral curve,comes from the orthogonal polynomials method of Dyson-Mehta [35], combined withthe theory of integrable systems [2]. The orthogonal polynomials satisfy an integrabledifferential equation of the form ~ψ ′ = D ( x ) ~ψ , where D ( x ) is a 2 × D (Jimbo-Miwa-Ueno [32]), i.e.: y = 12 Tr D ( x ) (1.23)It was proved [4] that:12 Tr D ( x ) = V ′ ( x ) − N (cid:28) Tr V ′ ( x ) − V ′ ( M ) x − M (cid:29) (1.24)7 For the formal matrix model, and more generally, for an arbitrary solution of theVirasoro constraints which has a topological expansion, the average < . > has a formalmeaning, and can be defined from the Virasoro generators V k . It is not the purposeof this article to explain where it comes from (see [18, 15]), and the spectral curve isthe algebraic equation satisfied by the ”disc amplitude”, i.e. generating function forcounting planar ”maps” with one boundary (i.e. having the topology of a discs), andit can be proved that it satisfies: y = V ′ ( x ) − P ( x ) (1.25)where P ( x ) is a polynomial of degree d − V ′′ , and with the same leadingcoefficient as V ′ . P ( x ) = ( d + 1) t d +1 x d − + d − X k =0 P k x k (1.26)The coefficients P k , are the conserved quantities in the context of integrable systems[2], whereas the coefficients of V ′ are called the Casimirs. The coefficients P k are in1-1 correspondance with the so-called ”action variables”: ǫ i = 12 iπ I A i ydx , i = 1 , . . . , d − ǫ i ’s are called filling fractions . In [22], it was proved, that given a spectral curve E ( x, y ) = 0 (1.28)(here E ( x, y ) = y − V ′ ( x ) + 4 P ( x ), i.e. in other words, given a potential V ( x ) = P d +1 k =1 t k x k and a polynomial P ( x ) = ( d + 1) t d +1 x d − + P d − k =0 P k x k , or in other wordsgiven V ′ and the filling fractions ǫ i ’s), it is possible to define an infinite sequence: F g ( E ) , g = 0 , . . . , ∞ (1.29)such that: τ ( E ) = exp ∞ X g =0 N − g F g ( E ) (1.30)is a solution of loop equations.The F g ( E ) were constructed in [22] for any spectral curve E ( x, y ) = 0, and theyhave many interesting properties, for instance they are invariant under symplectic τ ( E ) is the τ -function of an integrable hierar-chy. Their modular properties were also studied in [22] and further clarified in [19], andthey happen to be deeply related to the so-called Holomorphic anomaly equation first found in string theory [8, 1], and which relate the non-holomorphic part of thegenerating function for counting Riemann-surfaces to the contribution of degenerateRiemann surfaces (nodal surfaces). This will play a role below.Also, in [22], were defined the correlators: W ( g ) k ( z , . . . , z k ) , g = 0 , . . . , ∞ , k = 0 , . . . , ∞ , ( W ( g )0 = F g ) (1.31)which are multilinear symmetric meromorphic differential forms on the spectral curve.They also have many interesting properties, in particular they can be used to computederivatives of the F g ’s with any parameter on which E may depend. For instancederivatives with respect to filling fractions are: ∂∂ǫ j W ( g ) k ( z , . . . , z k ) = I B j W ( g ) k +1 ( z , . . . , z k , z k +1 ) (1.32)(where τ is the Riemann matrix of periods of the spectral curve, and A i ∩ B j = δ i,j isa symplectic basis of non contractible cycles, see [27, 28] for algebraic geometry). The conjecture is supported by the following facts: • Both the convergent matrix integral Z ( n N , . . . , n g +1 N , , . . . , 0) defined in eq.1.13,and the formal matrix integral Z formal ( ǫ , . . . , ǫ g ) satisfy the same loop equations. • Since loop equations are linear, the space of solutions is a vector space. • For given V ′ , both the convergent integral Z ( n N , . . . , n g +1 N , , . . . , Z formal ( ǫ , . . . , ǫ g ) are specified by the same number of parameters, i.e. g parameters(indeed n + . . . + n g +1 = n , so that only g of them are independent).Those observations support the idea that there exists a good basis of the vectorspace of solutions, such that each basis function is at the same time formal and conver-gent, i.e. there exists a set of basis paths γ i , such that Z ( n N , . . . , n g +1 N , , . . . , 0) admitsa topological expansion.We do not prove this conjecture in this article, but we take it as an asumption. All this can be extended to a larger context, for instance the 2-matrix model, or thechain of matrices, or the matrix model coupled to an external field.9 matrix model Consider 2 polynomial potentials V and V , such that deg V = d + 1 , deg V = d + 1. There are d × d independent paths on C × C on which the following integralis absolutely convergent: Z Z γ dx dy e − V ( x ) − V ( y )+ xy , γ = d d X i =1 c i γ i (1.33)where each γ i is a product of a path in the x − plane and a path in the y − plane.Then, similarly to the 1-matrix case, we can also define a matrix integral on ageneralized path (see [23]):ˆ Z ( γ ) = Z H n × H n ( γ ) dM dM e − N Tr ( V ( M )+ V ( M ) − M M ) (1.34)which satisfies: ˆ Z ( γ ) = X n + ... + n d = n c n . . . c n d d Z ( n /N, . . . , n d /N ) (1.35)where we have defined: Z ( n /N, . . . , n d /N ) = 1 n ! . . . n d ! Z γ n × ... × γ ndd dx ∧ dy . . . dx n ∧ dy n ∆( x )∆( y ) Y i e − N ( V ( x i )+ V ( y i ) − x i y i ) (1.36)The 2-matrix model generalized integral satisfies loop equations (which form aW-algebra instead of Virasoro), which also come from integration by parts, and areindependent of the path. In particular, each Z ( n /N, . . . , n d /N ) satisfies the same loopequations.There is also a formal 2-matrix model, which was introduced as a generating func-tion for bi-colored discrete surfaces, it was called the ”Ising model on a random lattice”[31]. Almost by definition, the formal 2-matrix model has a topological expansion:ln Z = X g N − g F g (1.37)The formal 2-matrix model satisfies the same loop equations as the convergent one,and the solution of loop equations was found in [21, 13, 22], and it was found that the F g ’s are again the symplectic invariants of [22]. matrix model with external field Z Kontsevich = Z dM e − N Tr M − M Λ (1.38)whose topological expansion is the combinatorics generating function computing inter-section numbers. Summary In all cases, there is a convergent matrix model defined on generalized paths, andthere is a formal matrix model which computes the combinatorics of some graphs. Boththe convergent and formal model obey the same set of loop equations.The formal model has a topological expansionln Z = X g N − g F g (1.39)where the F g ’s are the symplectic invariants of [22], computed for some algebraic spec-tral curve E ( x, y ) = 0. And in all cases the dimension of the homology basis of paths onwhich the integral is absolutely convergent, is the same as the genus g of the spectralcurve, i.e. the number of filling fractions: γ = g +1 X i =1 c i γ i ⇔ ǫ i = 12 iπ I A i ydx , i = 1 , . . . , g (1.40)In all those cases, the method we describe below should work. Now, assume that Z ( ǫ , . . . , ǫ d − ) has a topological asymptotic expansion:ln ( Z ( ǫ , . . . , ǫ d − )) = F ( ǫ ) = ∞ X h =0 N − h F h ( ǫ ) (2.41)Each F h must then be a solution of formal loop equations, and therefore it is given bythe formulae of [22], and therefore each F h is analytical in the ǫ i ’s.Then, we choose arbitrarily a ”prefered” filling fraction ǫ ∗ , and perform the Taylorexpansion: F h ( ǫ ) = ∞ X k =0 k ! F ( k ) h ( ǫ − ǫ ∗ ) k , F ( k ) h = ∂ k F h ∂ǫ k ( ǫ ∗ ) (2.42)11 emark: We don’t write the indices for readability, but F ( k ) h is a tensor. For read-ability we write the formulae as if there were only one variable ǫ , i.e. g = 1, but in factwe mean: F h ( ǫ ) = ∞ X k =0 k ! X i ,...,i k F ( k ) h i ,...,i k k Y j =1 ( ǫ − ǫ ∗ ) i j , F ( k ) h i ,...,i k = ∂ k F h ∂ǫ i . . . ∂ǫ i k ( ǫ ∗ )(2.43)but for simplicity we shall write eq.2.42, and we leave to the reader to restore theindices if needed.The derivatives of F g , are given by eq.1.32 (see [22]): F ( k ) h i ,...,i k = ∂ l ∂ǫ i . . . ∂ǫ i k F h = I B i . . . I B ik W ( h ) k ( z , . . . , z k ) (2.44)In particular, it is well known (see [22]), that F ′ = I B ydx (2.45)and iπ F ′′ = τ is the Riemann matrix of periods (see [27, 28] for introduction toalgebraic geometry) of the specral curve E , i.e.12 iπ ∂ F ∂ǫ i ∂ǫ j = τ i,j = τ j,i = I B i du j (2.46)where du j is the normalized basis of holomorphic differentials [27, 28] on E : I A i du j = δ i,j (2.47)And thus we have (formally): Z ( ǫ ) = Z ( ǫ ∗ ) e iπN ( ǫ − ǫ ∗ ) τ ( ǫ − ǫ ∗ ) e iπN ζ ( ǫ − ǫ ∗ ) X k X l i X h i N P i (2 − h i ) k ! l ! . . . l k ! F ( l ) h . . . F ( l k ) h k ( ǫ − ǫ ∗ ) P l i (2.48)where we the sum carries only on l i ≥ − h i − l i < i .One should notice that the exponential is now at most quadratic in the ǫ ’s. Now we are going to perform the sum of eq.1.12:ˆ Z ( γ ) = X n + ... + n g +1 = n c n . . . c n g +1 g +1 Z ( n /N, . . . , n d /N ) (3.49)12here γ = X i c i γ i , c i = e iπ ν i (3.50)Since the filling fractions ǫ i = n i N take integer values (up to a 1 /N factor), we haveto perform a sum of exponentials of square of integers. Such sums are called thetafunctions . They play a key role in algebraic geometry. Let us recall a few properties[27, 28]. We define the Θ-function:Θ( u, t ) = X n ∈ Z g e ( n − Nǫ ∗ ) u e ( n − Nǫ ∗ ) t ( n − Nǫ ∗ ) e iπ n ν (3.51)It clearly satisfies: ∂ Θ ∂t = ∂ Θ ∂u (3.52)It is related to the usual Jacobi-theta function:Θ( u, t ) = θ − Nǫ ∗ ,ν ( u iπ , tiπ ) e iπνNǫ ∗ (3.53)where ( − N ǫ ∗ , ν ) is called the characteristics. The Jacobi theta function with charac-teristics ( a, b ) is defined by: θ a,b ( u, τ ) = X n e iπ ( n + a )( u + b ) e iπ ( n + a ) τ ( n + a ) = θ , ( u + b + τ a, τ ) e iπaτa e iπa ( u + b ) (3.54)It takes a phase after translation along an integer lattice period n + τ m : θ a,b ( u + n + τ m, τ ) = e iπ ( an − mb ) θ a,b ( u, τ ) e − iπmu e − iπmτm (3.55) We thus have: ˆ Z ( γ ) ∼ X n e iπ n ν Z formal ( n /N ) ∼ X n c n . . . c n g g Z ( n /N, . . . , n g /N, , . . . , 0) (3.56)The sum carries on integers n i ≥ P i n i = n . Therefore n g +1 = n − P gi =1 n i is notindependent from the others. Another remark, is that in that sum we expect that onlythe vicinity of some extremal n i will dominate the sum, and that values of the n i ’s far13rom the extremum should give an exponentially small contribution. That asumptionallows to extend the sum to n i ∈ Z .Then, we use the Taylor expansion of eq.2.48, and we find (again we use tensorialnotations):ˆ Z ( γ ) ∼ Z ( ǫ ∗ ) X n ∈ Z g e iπ P i ν i n i e iπ ( n − Nǫ ∗ ) τ ( n − Nǫ ∗ ) e iπNζ ( n − Nǫ ∗ ) X k X l i > X h i > − li N P i (2 − h i − l i ) k ! l ! . . . l k ! F ( l ) h . . . F ( l k ) h k ( n − N ǫ ∗ ) P l i (3 . Z ( γ ) ∼ Z ( ǫ ∗ ) X k X l i > X h i > − li N P i (2 − h i − l i ) k ! l ! . . . l k ! F ( l ) h . . . F ( l k ) h k ∂ P l i Θ ∂u P l i (cid:12)(cid:12)(cid:12)(cid:12) u = NF ′ ,t = iπτ (3.58)This formula is the main result presented in this article.For instance the first few terms in powers of N − are:ˆ Z ( γ ) ∼ Z ( ǫ ∗ ) (cid:16) Θ + 1 N ( F ′ Θ ′ + F ′′′ ′′′ )+ 1 N ( F ′′ ′′ + ( F ′ ) ′′ + F ′′′′ 24 Θ (4) + F ′ F ′′′ (4) + ( F ′′′ ) 72 Θ (6) )+ . . . (cid:17) (3 . The expansion of formula .3.58 can be resummed into a single Θ-function. We want towrite it as: ˆ Z ( γ ) = Z ( ǫ ∗ ) Θ( u, t ) (3.60)where u = N F ′ + ∞ X h =1 N − h u ( h ) , t = iπτ + ∞ X h =1 N − h t ( h ) (3.61)For instance, one easily finds the first orders: u (1) = F ′ + F ′′′ ′′′ ( u (0) , iπτ )Θ ′ ( u (0) , iπτ ) (3.62) t (1) = F ′′ F ′′′′ 24 Θ ′′′′ ( u (0) , iπτ )Θ ′′ ( u (0) , iπτ ) + F ′ F ′′′ (cid:18) Θ (4) Θ ′′ − Θ ′′′ Θ ′ (cid:19) + ( F ′′′ ) (cid:18) Θ (6) Θ ′′ − Θ ′′′ Θ ′ (cid:19) (3.63)14he Taylor expansion of eq.3.60 reads (and we use eq.3.52):ˆ Z ( γ ) = Z ( ǫ ∗ ) Θ( N F ′ + 1 N u (1) + . . . , iπτ + 1 N t (1) + . . . )= Z ( ǫ ∗ ) X m,n ( m + n )! m ! n ! ( u − u (0) ) m ( t − t (0) ) n ∂ m +2 n ∂u m +2 n Θ( u (0) , t (0) )= Z ( ǫ ∗ ) X m,n X k ,...,k m X j ,...,j n ( m + n )! N m − P k i − P j i m ! n ! u ( k ) . . . u ( k m ) t ( j ) . . . t ( j n ) ∂ m +2 n ∂u m +2 n Θ( u (0) , t (0) ) (3.64)and now we identify the powers of N with equation.3.58. For any given p > 0, we musthave: X k ,...,k m X j ,...,j n ( m + n )! m ! n ! u ( k ) . . . u ( k m ) t ( j ) . . . t ( j n ) ∂ m +2 nu Θ( u (0) , t (0) )= X r X l i X h i r ! l ! . . . l r ! F ( l ) h . . . F ( l r ) h r ∂ P l i u Θ( u (0) , t (0) ) (3.65)where in the first sum, the indices are such that p = 2 m X i =1 k i + 2 n X i =1 j i − m , k i > , j i > p = r X i =1 (2 h i + l i − , l i > , − h i − l i < u ( k ) and t ( l ) recursively in a unique way.Indeed, assume that we have already computed u (1) , . . . , u ( q − and t (1) , . . . , t ( q − .Choose p = 2 q − u ( q ) Θ ′ ( u (0) , t (0) )= X r X l i X h i r ! l ! . . . l r ! F ( l ) h . . . F ( l r ) h r ∂ P l i u Θ( u (0) , t (0) ) − X k ,...,k m X j ,...,j n ( m + n )! m ! n ! u ( k ) . . . u ( k m ) t ( j ) . . . t ( j n ) ∂ m +2 nu Θ( u (0) , t (0) )(3 . q − P ri =1 (2 h i + l i − , l i > , − h i − l i < 0, andin the second sum we have 2 q − P mi =1 k i + 2 P ni =1 j i − m , which implies k i < q and j i < q , i.e. all the terms in the RHS are known from the recursion hypothesis. Wehave thus determined u ( q ) . Then, let p = 2 q , we have: t ( q ) Θ ′′ ( u (0) , t (0) ) 15 X r X l i X h i r ! l ! . . . l r ! F ( l ) h . . . F ( l r ) h r ∂ P l i u Θ( u (0) , t (0) ) − X k ,...,k m X j ,...,j n ( m + n )! m ! n ! u ( k ) . . . u ( k m ) t ( j ) . . . t ( j n ) ∂ m +2 nu Θ( u (0) , t (0) )(3 . q = P ri =1 (2 h i + l i − , l i > , − h i − l i < 0, andin the second sum we have ( m, n ) = (0 , q = 2 P mi =1 k i + 2 P ni =1 j i − m , whichimplies k i ≤ q and j i < q , i.e. all the terms in the RHS are known from the recursionhypothesis. We have thus determined t ( q ) .Therefore we have:ˆ Z ( γ ) = Z ( ǫ ∗ ) Θ N F ′ + X k N − k u ( k ) , iπτ + X j N − j t ( j ) ! (3.70)It would be interesting to understand how this formula matches the tau-functionobtained from integrability properties [2]. One may observe that all the terms with even powers of N in formula eq.3.58 have al-ready appeared in another context, in topological string theory [34], and more preciselythe so called ”holomorphic anomaly equations” [8].Holomorphic anomaly equations are about modular invariance versus holomorphic-ity. Let us briefly sketch the idea. String theory partition functions represent ”integrals”over the set of all Riemann surfaces with some conformal invariant weight. In otherwords, they are integrals over moduli spaces of Riemann surfaces of given topology, andtopological strings are integrals with a topological weight, they compute intersectionnumbers (see [36, 34] for introduction to topological strings).Moduli spaces can be compactified by adding their ”boundaries”, which corre-spond to degenerate Riemann surfaces (for instance when a non contractible cyclegets pinched). The integrals have thus boundary terms, which can be representedby δ -functions, and δ -functions are not holomorphic. In other words, string theorypartition functions contain non-holomorphic terms which count degenerate Riemannsurfaces.On the other hand, if one decides to integrate only on non-degenerate surfaces, onegets holomorphic patition functions, but not modular invariant, because the boundaries16f the moduli spaces are associated to a choice of pinched cycles. Modular invariantmeans independent of a choice of cycles.To summarize, the holomorphic partition function is obtained after a choice ofboundaries, i.e. a choice of a symplectic basis of non contractible cycles A i ∩ B j = δ i,j ,and cannot be modular invariant. The modular invariance is restored by adding theboundaries, but this breaks holomorphicity.There is thus a relationship between holomorphicity and modular invariance.Let F g be the partition function corresponding to the moduli space of non-degenerate Riemann surfaces of genus g , i.e. F g is holomorphic but not modularinvariant (it assumes a choice of a basis of cycles A i , B i , i = 1 , . . . , g ), and let ˆ F g be the partition function including degenerate surfaces, i.e. non holomorphic but mod-ular invariant. The holomorphic anomaly equation discovered by [8], states that: ∂ ˆ F g = 12 ∂κ ˆ F ′′ g − + g − X h =1 ˆ F ′ h ˆ F ′ g − h ! (4.71)where κ is the Zamolodchikov K¨ahler metric symmetric matrix: κ = ( τ − τ ) − (4.72)It was found in [8, 1, 19] that:ˆ Z = e P g N − g ˆ F g = e P g N − g F g X l X k X l i > X h i > − li N P i (2 − h i − l i ) k ! l ! . . . l k ! F ( l ) h . . . F ( l k ) h k (2 l − κ l δ l − P l i (4 . l − κ l F ( l ) h . . . F ( l k ) h k (4.74)means in fact a sum of (2 l − l indicesof the matrix κ , with the 2 l indices of the tensors F ( l i ) h i .For example to order N − , i.e. g = 2 we have:ˆ F = F + κ (cid:18) F ′′ F ′ ) (cid:19) + 3 κ (cid:18) F ′′′′ 4! + 2 F ′ F ′′′ (cid:19) + 15 κ (cid:18) ( F ′′′ ) (cid:19) (4.75)where the last term 15 κ ( F ′′′ ) contains two topologically inequivalent types of pairingsamong the indices:15 κ ( F ′′′ ) → X i ,i ,i ,i ,i ,i κ i ,i κ i ,i κ i ,i ∂ F ∂ǫ i ∂ǫ i ∂ǫ i ∂ F ∂ǫ i ∂ǫ i ∂ǫ i κ i ,i κ i ,i κ i ,i ∂ F ∂ǫ i ∂ǫ i ∂ǫ i ∂ F ∂ǫ i ∂ǫ i ∂ǫ i (4.76)This equation can be diagrammatically represented as follows [1]:ˆ F = + 12 + 12 + 18 + 12+ 18 + 112 (4.77)where each term represents a possible degeneracy of a genus 2 Riemann surface (imagineeach link contracted to a point). The prefactor is 1 / Aut , i.e. the inverse of the numberof automorphisms, for instance in the last graph we have a Z symmetry by exchangingthe 2 spheres, and a σ symmetry from permuting the 3 endpoints of the edges, i.e.12 = Z × σ ) automorphisms.Formally, eq.4.73 is very similar to eq.3.58, with the identification:(2 k − κ k → Θ (2 k ) (4.78) proof: eq.4.73 is the Wick theorem expansion of the following integral [1, 19]: Z ( ǫ ∗ , κ ) = e P h N − h F h ( ǫ ∗ ,κ ) = Z dη e F ( η ) − N ( η − ǫ ∗ ) F ′ − N ( η − ǫ ∗ ) F ′′ ( η − ǫ ∗ ) − N iπ ( η − ǫ ∗ ) κ − ( η − ǫ ∗ ) = Z ( ǫ ∗ ) Z dη e P l> P h> − l/ N − hl ! F ( l ) h ( η − ǫ ∗ ) l − N iπ ( η − ǫ ∗ ) κ − ( η − ǫ ∗ ) = Z ( ǫ ∗ ) X k X l i > X h i > − l i / N P i − h i k ! l ! . . . l k ! F ( l ) h . . . F ( l k ) h k Z dη ( η − ǫ ∗ ) P l i e − N iπ ( η − ǫ ∗ ) κ − ( η − ǫ ∗ ) (4.79)i.e. Z ( ǫ ∗ , κ ) = Z ( ǫ ∗ ) X k X l i X h i N P i (2 − h i − l i ) k ! l ! . . . l k ! F ( l ) h . . . F ( l k ) h k ∂ P l i f∂u P l i (cid:12)(cid:12)(cid:12)(cid:12) u =0 ,t = − κ − (4 . f ( u, t ) is nearly the same as Θ except that we have an integral instead of a sumover integers: f ( u, t ) = Z dǫ e N ( ǫ − ǫ ∗ ) u e N ( ǫ − ǫ ∗ ) t ( ǫ − ǫ ∗ ) e iπN ǫν = e iπN ǫ ∗ ν Z dǫ e Nǫ ( u +2 iπν ) e N ǫtǫ = e iπN ǫ ∗ ν e − ( u +2 iπν ) t − ( u +2 iπν ) . ∂f∂t = ∂ f∂u (4.82)It is clear that: ∂ l +1 f∂u P l i (cid:12)(cid:12)(cid:12)(cid:12) u =0 ,t = − κ − = 0 , ∂ l f∂u P l i (cid:12)(cid:12)(cid:12)(cid:12) u =0 ,t = − κ − = (2 l − κ l (4.83)which proves our claim eq.4.78.This analogy between convergent integrals obtained by summing over filling frac-tions, and holomorphic anomaly equations is puzzling, and it would be worth gettingsome understanding of that fact. So far, ǫ ∗ was chosen arbitrary, and eq.3.58, eq.3.70 and the property 4.78 hold in-dependently of the choice of ǫ ∗ . Indeed ˆ Z ( γ ) does not depend at all on a choice of ǫ ∗ . If we take eq.3.58 as a definition of a string theory partition function, it seems atfirst sight that it depends on ǫ ∗ , but in fact it does not. Those facts are related to theso-called ”background independence” problem in string theory [38].From [6], it should be expected that if we choose ǫ ∗ such that the spectral curvehas the Boutroux property:Boutroux property : ∀C , Re I C ydx = 0 (5.84)then, the formal series P g N − g F g as well as the Θ-sums in eq.3.58 and eq.3.70, shouldbe convergent series, and thus we really have an asymptotic expansion instead of onlyan asymptotic series. However, this fact is not proved yet (except for the 1-matrixmodel).Boutroux curves in particular, are such that: ǫ ∗ = 12 iπ I A ydx ∈ R g , Re F ′ = Re I B ydx = 0 (5.85)Boutroux curves can be obtained as follows: Notice that Re F ′′ = − π Im τ < − Re F is a convex function on ǫ ∗ ∈ R g , therefore it has a minimum in each cell ofthe moduli space. The minimum clearly satisfies eq.5.85. In other words there should19e one Boutroux curve in each cell of the moduli space of the spectral curve. One mayexpect that each cell corresponds to a possible connectivity pattern of the generalizedpath γ .Notice that if the potentials are real, and the filling fraction ǫ ∗ is real, then F isreal as well, and the Boutroux condition becomes F ′ = 0. In this article, we have improved the asymptotic (conjectured) formula of [9] for matrixintegrals to all orders. We have also found an interesting connection between thisexpansion and combinatoric geometry of degenerate Riemann surfaces, through theholomorphic anomaly equation.The relationship between higher genus g > g = 0)spectral curve formal matrix models. This works tends to show that higher genusspectral curves have to do with nodal surfaces. This relationship needs to be furtherinvestigated. Acknowledgments We would like to thank M. Mari˜no for careful reading of the manuscript, and M.Bertola, T. Grava, N. Orantin for useful and fruitful discussions on this subject. Thiswork is partly supported by the Enigma European network MRT-CT-2004-5652, by theANR project G´eom´etrie et int´egrabilit´e en physique math´ematique ANR-05-BLAN-0029-01, by the Enrage European network MRTN-CT-2004-005616, by the EuropeanScience Foundation through the Misgam program, by the French and Japaneese gov-ernments through PAI Sakurav, by the Quebec government with the FQRNT. References [1] M. Aganagic, V. Bouchard and A. Klemm, “Topological strings and (almost)modular forms,” arXiv:hep-th/0607100.[2] O Babelon, D Bernard, M Talon, ”Introduction to Classical Integrable Systems”,Cambridge University Press, 2003. 203] M. Bertola, B. Eynard, J. Harnad, Differential systems for bi-orthogonal polynomi-als appearing in two-matrix models and the associated Riemann-Hilbert problem,Comm. Math. Phys. 243 no.2 (2003) 193-240, nlin.SI/0208002.[4] M. Bertola, B. Eynard, J. Harnad, Partition functions for Matrix Models andIsomonodromic Tau Functions, J. Phys. A Math. Gen. 36 No 12 (28 March 2003)3067-3083, nlin.SI/0204054.[5] M. Bertola, Bilinear semi-classical moment functionals and their integral repre-sentation, arXiv:math/0205160.[6] M. Bertola, Boutroux curves with external potential: equilibrium measures with-out a minimization problem, arXiv: 0705.3062, 2007.[7] M. Bertola, M. Y. Mo, Commuting difference operators, spinor bundles and theasymptotics of pseudo-orthogonal polynomials with respect to varying complexweights, arXiv:math-ph/0605043.[8] M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, “Holomorphic anomalies intopological field theories,” Nucl. Phys. B , 279 (1993) [arXiv:hep-th/9302103].[9] G. Bonnet, F. David, B. Eynard, Breakdown of Universality in multi-cut matrixmodels, J. Phys A33 (2000) 6739, cond-mat/0003324.[10] E. Br´ezin, C. Itzykson, G. Parisi and J. B. Zuber, “Planar Diagrams,” Commun.Math. Phys. , 35 (1978).[11] P. Bleher, Lectures on random matrix models. The Riemann-Hilbert approach,arXiv:0801.1858v1 [math-ph].[12] L. Chekhov, Genus one corrections to multi-cut matrix model solutions, Theor.Math. Phys. (2004) 1640-1653, hep-th/0401089.[13] L. Chekhov, B. Eynard and N. Orantin, “Free energy topological expansion forthe 2-matrix model,” JHEP , 053 (2006) [arXiv:math-ph/0603003].[14] P. A. Deift, T. Kriecherbauer, K. T-R. McLaughlin, S. Venakides, and Z. Zhou,Uniform asymptotics for polynomials orthogonal with respect to varying exponen-tial weights and applications to universality questions in random matrix theory.Commun. Pure Appl. Math. 52 (1999), 1335-1425.[15] P. Di Francesco, P. Ginsparg, J. Zinn-Justin, “2D Gravity and Random Matrices”, Phys. Rep. , 1 (1995). 2116] N.M. Ercolani and K.D.T-R McLaughlin, Asymptotics of the partition functionfor random matrices via Riemann-Hilbert techniques and applications to graphicalenumeration. Int. Math. Res. Not. 14 (2003), 755-820.[17] B. Eynard, “Topological expansion for the 1-hermitian matrix model correlationfunctions,” arXiv:hep-th/0407261.[18] B. Eynard, “Formal matrix integrals and combinatorics of maps,” arXiv:math-ph/0611087.[19] B. Eynard, M. Marino, N. Orantin, Holomorphic anomaly and matrix models,hep-th/0702110, JHEP 089P 0307.[20] L. Chekhov, B. Eynard, Hermitean matrix model free energy: Feynman graphtechnique for all genera, JHEP 009P 0206/5, hep-th/0504116.[21] B. Eynard and N. Orantin, “Topological expansion of the 2-matrix model cor-relation functions: Diagrammatic rules for a residue formula,” JHEP , 034(2005) [arXiv:math-ph/0504058].[22] B. Eynard and N. Orantin, “Invariants of algebraic curves and topological ex-pansion”, Communications in Number Theory and Physics, Vol 1, Number 2,p347-452, arXiv:math-ph/0702045.[23] B. Eynard, The 2-matrix model, biorthogonal polynomials, Riemann-Hilbertproblem, and algebraic geometry, math-ph/0504034, habilitation `a diriger lesrecherches, universit´e Paris 7.[24] B. Eynard, Eigenvalue distribution of large random matrices, from one matrix toseveral coupled matrices, Nuc. Phys. B506,3 633-664 (1997), cond-mat/9707005.[25] B. Eynard, N. Orantin, Weil-Petersson volume of moduli spaces, Mirzakhani’srecursion and matrix models, math-ph: arXiv:0705.3600.[26] B. Eynard, Recursion between Mumford volumes of moduli spaces, math-ph:arXiv:0706.4403.[27] H.M. Farkas, I. Kra, ”Riemann surfaces” 2nd edition, Springer Verlag, 1992.[28] J.D. Fay, ”Theta functions on Riemann surfaces”, Springer Verlag, 1973.[29] A. Fokas, A. Its, A. Kitaev, ”The isomonodromy approach to matrix models in2D quantum gravity”, Commun. Math. Phys. 147, 395-430 (1992).2230] A. Guionnet, E. Maurel Segara,[31] V. A. Kazakov, “Ising model on a dynamical planar random lattice: Exact solu-tion,” Phys. Lett. A , 140 (1986).[32] M. Jimbo, T. Miwa and K. Ueno, ”Monodromy Preserving Deformation of LinearOrdinary Differ ential Equations with Rational Coefficients I.”, Physica 2D, 306-352 (1981).[33] M.Kontsevitch, “Intersection theory on the moduli space of curves and the matrixAiry function”, Funk. Anal. Prilozh. (1991) 50-57; Max-Planck Institut preprintMPI/91-47, MPI/91-77.[34] M. Mari˜no, “Les Houches lectures on matrix models and topological strings,”arXiv:hep-th/0410165.[35] M.L. Mehta, Random Matrices ,2nd edition, (Academic Press, New York, 1991).[36] M. Vonk, “A mini-course on topological strings,” arXiv:hep-th/0504147.[37] W.T. Tutte, “A census of planar maps”, Can. J. Math.15