The intersection numbers of the p-spin curves from random matrix theory

The intersection numbers of p-spin curves are computed through correlation functions of Gaussian ensembles of random matrices in an external matrix source. The p-dependence of intersection numbers is determined as polynomial in p; the large p behavior is also considered. The analytic continuation of intersection numbers to negative values of p is discussed in relation to SL(2,R)/U(1) black hole sigma model.


Introduction
The intersection numbers for p-spin curves appear in the generalized Kontsevich matrix model [1,2,3]. The generating function for p-spin intersection number obeys the p-th KdV equation or Gelfand-Dikii equation. In a random matrix theory, the correlation functions at the edges of the spectrum, where one can tune a degeneracy of order p, are expressed through intersection numbers [4,5,6,7]. In conformal field theory, the p-spin curve intersection theory is related to N =2 superconformal minimal theory for Lie algebra A p−1 type. It has been pointed out that it corresponds to a gauged Wess-Zumino-Witten (WZW) model of SU(2) k /U(1), where k = p − 2 is the level of the Kac-Moody algebra of Lie group SU(2) [8], and is related to SL(2,R)/U(1) black hole sigma model when k becomes negative [9].
The free energy for the p-spin curve satisfies interesting universal equations, such as a string equation, dilaton equation, and WDVV equation, so called tautological equations or universal equations, and has been studied in the connection to Gromov-Witten theory [10,11,12]. Although the p-spin curve intersection numbers can be obtained through tautological equations in a recursive way, the actual computation for higher genuses is limitted [13,14,15].
In previous papers, we have derived explicit integral formula for the pspin curve intersection numbers of the moduli space M g,n valid for all order of genus g. We have shown that they are obtained analytically for a fixed number n of marked points. Our formulation starts from simple Gaussian matrix models with an external matrix source and based upon a duality relation [5,6,7], from which one recovers a generalized Kontsevich matrix model.
The intersection numbers for the spin moduli spaces with n-marked points are obtained from the n n-point correlation functions U(s 1 , ..., s n ) of Gaussian random matrices in a scaling limit at critical edges [16,17]. In a previous article [18], we have computed explicitly the intersection numbers of moduli space of p-spin curves with one marked point, for arbitrary values of p, as polynomials in p. This allowed us to consider continuations in p ; in particular the limit p → −1 exhibits an interesting relation between the intersection numbers, and the orbifold Euler characteristics χ(M g,1 ) = ζ(1 − 2g), where ζ(x) is the Riemann zeta function) [20,21].
In this paper, we extend the evaluation of the intersection numbers beyond the one marked point (n = 1) for arbitrary p. The obtained intersection numbers are consistent with previously known results [13,14,15] for small values of p. We pursue the large p behavior, p → ∞ limit. The p-spin curve intersection theory is equivalent to gauged WZW model. For this gauged WZW theory, in which k = p − 2 appears as overall factor, the large k limit may give a semi-classical solution [9,23]. In the negative k, the gauged WZW model on SU(2)/U(1) is changed to WZW model on non-compact SL(2, R)/U(1), which is relevant to a black hole σ model [9]. We will discuss the relation between the intersection numbers and the density of state of SL(2, R)/U(1) black hole sigma model [24,25,27].

Generating function for p-spin Intersection numbers
The mathematical definition of the intersection numbers of the moduli space of p-spin curves with s-marked points is given by [3] < τ n 1 (U j 1 ) · · · τ ns (U js ) >= where U j is an operator for the primary matter field (tachyon), related to top Chern class C T (ν), and τ n is a gravitational operator, related to the first Chern class c 1 of the line bundle L i at the ith-marked point. We denote τ n (U j ) by τ n,j , and j represents the spin index (j=0,...,p-1). The problem of definition (1) has been discussed extensively [11].
In a previous paper [18], we have shown that those intersection numbers (1) are expressed trhough the correlation functions U(s 1 , ..., s n ) as coefficients of powers of s j , where s l = it l ; M is an N × N Hermitian random matrix. The bracket stands for averages with the Gaussian probability measure in which A is an N ×N external Hermitian matrix source. By an appropriate tuning of the external source matrix A, we may obtain the desired singularity, which generates the p-spin curves. The relation to the generalized Kontsevich model is discussed in § 3,4 of [18]. An exact and useful integral representation for U(s 1 , ..., s n ) is known in presence of an arbitrary external matrix source A with eigenvalues a α [19] : This representation involves contour integrals around u i = a α . In the large N limit, it is convenient to express the factors in the determinant as additional integrals. For instance, in the case of the two point correlation (n=2), after the shift u i → u i − s i 2 , s i → s i N , in the two point function, we have 1 Tuning now the a α 's, and taking the large N limit, we obtain For the three and four point correlations, similar useful formulae for the determinant part of (4) may be found in the appendices A and B of [18].

3
Intersection numbers for p = 3 with two marked points The intersection numbers are obtained as coefficients of the power series in s 1 , s 2 of U(s 1 , s 2 ). In a previous paper [18], for p=3, we have computed the intersection numbers with two marked points or genus one case (g=1) starting from (7). As an example, we compute the p = 3 case up to genus 3. The general expansion with the condition, is applied to the special case n = 2, p = 3 The gamma functions in (7) represent the spin factors. After rescaling of the parameters, in which a = (s 1 /s 2 ) 1/3 , and the Airy function is The Airy function satisfies the differential equation The genus one case (g=1) has been studied in [7]. If one expands the hyperbolic sine function and the Airy functions in (9) up to relevant orders, we find a sum of six terms which, for g = 2, involve the following integrals: A repeated use of (11) plus integrations by parts allows us to write all these integrals in terms of which cannot be reduced to A i (0) or A ′ i (0). For instance one finds and so on. However, all the T-dependence cancels when we sum up all the terms relevant to g = 2 in U(s 1 , s 2 ). For instance the sum of all terms of order s 16 3 2 is given by and we add up the six terms and expand in powers of s 1 to the relevant orders we find From these results, we obtain the intersection numbers Rather than computing the exact dependence in a of the terms proportional to s 16/3 2 and then re-expand in a to obtain the various terms of (17), we may proceed in a simpler way by expanding and we then recover (18).
In the genus-three case (g=3), we have again ten distinct integrals J 1 −J 10 for the terms of order s 8 2 a m , in the small s 1 , s 2 expansion of (9).
The genus 3 contribution for U(s 1 , s 2 ) is then expressed as the sum of four terms, U (1) -U (4) . The term U (1) , which is related to J 1 , is The term U (2) , which is related to J 2 , is The term U (3) , which is related to J 3 , is The term U (4) from the sum of the contributions of J 4 to J 10 . We have with L = A i (0)A ′ i (0), and K 1 ,K 2 are given below. We have also the following relations between J 1 , J 2 and J 3 , Thus U (4) becomes Since a = (s 1 /s 2 ) 1/3 , the above expression for U(s 1 , s 2 ) is a symmetric function in s 1 and s 2 . Denoting and dividing U(s 1 , s 2 ) by 1/g p , i.e. 1/27 in this case, we obtain the intersection numbers < τ m 1 ,j 1 τ m 2 ,j 2 > g=3 as the coefficient of t m 1 ,j 1 t m 2 ,j 2 in U(s 1 , s 2 ) taking into account the spin factors. The following spin factor appears as a over all factor in U(s 1 , s 2 ) at genus 3.

Intersection numbers for p > 3
For higher multicritical points the algebra is similar, except that we have to deal with generalized Airy functions. For instance for p = 4 instead of A i (x) we have to work with φ(x) defined as Then, similarly where a = (s 1 /s 2 ) 1/4 . In complete analogy with the p = 3 case, a repeated use of integration by parts and of (33) leads to the expansion of U(s 1 , s 2 ). In the genus one case, which provide the j = 0, j = 2 spin factors respectively. Replacing s 1 , s 2 by t m , j, (s m+(1+j)/p = t m,j ), Multiplying by a factor 1 p g , we obtain the intersection numbers as coefficients of (37) for p = 4 in the genus one case, For g = 2, p = 4, from the term s For general p we have to deal with the generalized Airy functions φ(x) for p > 2, which satisfy the differential equation, where φ (p) (x) means the p-th derivative of φ(x). The generalized Airy function has an integral representation, As examples of what the method can provide we give a few results : for the case p = 5, we obtain For the case p = 6, For the case p = 7, < τ 0,2 τ 1,5 > g=1 =< τ 1,2 τ 0,5 > g=1 = 1 42 < τ 0,4 τ 1,3 > g=1 =< τ 0,3 τ 1,4 > g=1 = 1 28

The p dependence of the intersection numbers
In a previous article [18], we have considered the intersection numbers with one marked point for arbitrary p, and found results such as with n = 2g − 1 + 2g−2−j p . In the large p limit, the intersection numbers < τ n,j > g behave as with B g is a Bernouilli number, B 1 = 1 6 , B 2 = 1 30 , B 3 = 1 42 , B 4 = 1 30 . Note the well known relation to ζ(2g) as We have derived (46) from U(s) in the large p limit. The one point function U(s) has the following expression [16], With s = σ p , and u p+1 = x 2 , we have Thus we obtain This may be written as and expanding the exponent in 1 p , we find If we use the expansion and drop the factors ( 2c p+1 ) − 1 p , and (σ/2) −1/p which are close to one in the large p limit, we obtain Since the intersection numbers < τ n,j > g are related to U(s) by [16] U(s) = g < τ n,j > g 1 Nπ with (p + 1)(2g − 1) = pn + j + 1, we have rederived the large p behavior of (46). From (53), taking a derivative with respect to σ, gives, Using this relation one obtains The di-gamma function ψ(z) has the following expression, From (57) and (58) we find thus in the large p limit, The last integral is related to the density of states. In (2), s is replaced by s = it, and if we replace z by iE, and take the imaginary part, we obtain the density of states ρ(E). After integration by parts, we obtain We will discuss this expression in connection to the density of states of the SL(2, R)/U(1) black hole sigma model in the next section.
Next we consider the two point correlation function U(s 1 , s 2 ). For general p, U(s 1 , s 2 ) is expressed as The exponent of (61) follows from the binomial expansion, and we use c(p + 1) = 1, u p s = v p /p. As in the case of p=3, polynomials in a (21) give the intersection numbers. Therefore we expand (61) in power series of a. At lowest order in a, we obtain two terms from (61), From U 2 we find a term proportional to as Since s Similarly one obtains the g=2 terms from the coefficients of a m s For the particular values of p = 3, 4, 5, the above expressions agree with the previous results (18),(39) and (42) for the genus two case.
From the a 5 s term, one finds which is valid for p ≥ 6.
In the large p limit, the three terms of (68) become equal, and coincide with the result for the one point intersection number (46).
Note that (69) is order p, and is negligible compared to (68).
From the terms a m s in the small a expansion of U(s 1 , s 2 ), we obtain the g=3 (genus 3) terms. In the case m=1, we have This is identical to < τ 5,4 > g=3 in (45). The identity follows from the string equation, in which the insertion of τ 0,0 reduces the intersection number from s to s − 1 marked points: In our formulation, this string equation follows from the integral representation for the intersection numbers, when one collects the terms proportional to a. By explicit calculation of two marked points, we verified this string equation. It might be possible to verify this string equation for n-marked points by the taking account of the term of a.
From a 2 s 6+ 6 p 2 , we have for p > 5, From a 3 s In the large p limit, these g=3 terms exhibit same behavior as in (46), Analytic continuation to negative p One may analytically continue the integral representations of the correlation functions to negative values of p. This continuation was already examined in [18], and we recall some of the results here : where c = N Expanding the exponent, we obtain (77) This integrals yield Gamma functions after the replacement u = ( t cs ) 1/p , with y = c 2 p s 2+ 2 p . From this expansion, we obtain the intersection numbers for one marked point as (45). The intersection number < τ n,j > g is obtained from the term y g Γ(1 − 1 p − j p ) in (78). The continuation to p < 0 is straightforward. The t-integral in (78) can be changed to v by t = 1 v ,(0 < v < ∞), and one obtains the small s expansion for negative p. Therefore the expression for the intersection numbers (45) can be analytically continued to negative p. This analytic continuation can also be done for two marked points, since we have computed them in the previous sections for general p. For instance, from (45), we have the intersection numbers for p = −3, In a previous article [18], we have computed the intersection numbers < τ 1,0 > g for the case of p = −1 from U(s), which provides the orbifold Euler characteristics χ(M g,1 ) with one marked point, with the Bernoulli number B g , (B 1 = 1 6 , B 2 = 1 30 , B 3 = 1 42 , ...). The spoint orbifold Euler characteristics χ(M g,s ) may be obtained from the dilaton equation: Since the Euler characteristics with s marked points is < τ 1,0 · · · τ 1,0 > g , the dilaton equation yields from (80), This agrees with previous results obtained in [20,21,22]. For p = −2, we have considered previously the equivalence with the unitary matrix model in a matrix source [26].
The central charge of the gauged Wess-Zumino-Witten model with symmetry SU(2) k /U(1) is Changing p to p = −p ′ , k to k = −k ′ (p < 0,k < 0), we have p ′ = k ′ − 2, and the central charge C is given by The analytic continuation to negative p yields a gauged WZW model for SL(2, R) k ′ /U(1) . It is known that this model represents a black hole σ model [9], in particular for the value k ′ = 9 4 (p = − 1 4 ), for which the central charge C becomes 26.
The density of states for the SL(2, R)/U(1) black hole has been studied in [24,25,27], in which ǫ is a regularization factor, and m = 1 2 (n − kw),m = − 1 2 (kw + n) are eigenvalues of J 3 0 andJ 3 0 in CFT (J 3 0 −J 3 o = n, J 3 0 +J 3 0 = −kw). If we neglect m ,m, and the 1 2 terms in the large E limit, we obtain ρ(E) = 1 π logǫ + 1 2πi This expression agrees with (60), obtained from the intersection numbers for large p. We have scaled s = σ/p, and the expression (60) is valid for small s. Therefore, the Fourier transform of U(s) gives the large E behavior, in which the terms m,m and 1/2 in (85) can be neglected.

Discussion
In this article, we have shown that the correlation functions U(s 1 , s 2 , · · · , s n ) of a Gaussian matrix model in a tuned external source, provide the intersection numbers for p-spin curves. For instance, from the two point function U(s 1 , s 2 ), in the case of p=3, the intersection numbers are computed up to genus 3, We have also computed the intersection numbers for general p. They are given by power series in a, a = ( s 1 s 2 ) 1 p . Then we have considered the large p behavior for the two point functions. The density of states ρ(E) becomes a di-gamma function in the large p limit, and this expression agrees with the density of states of a SL(2, R) k /U(1) WZW model, which has been studied in the context of two dimensional black hole solutions. The n-point correlation functions U(s 1 , · · · , s n ) are known through the determinant of a kernel for the p-spin curve case. It will be interesting to investigate further the detailed comparison of those correlation functions, between SL(2, R) k /U(1) WZW theory and the intersection numbers for negative p-spin curves .