MIAN-Th-18/98 NBI-HE-98-13 MPS-RR 1998-8 hep-th/9805212

The NBI matrix model of IIB Superstrings

J. Ambjørn^{a)}^{∗} and L. Chekhov^{b)}^{†}

a)The Niels Bohr Institute, Blegdamsvej 17, 2100, Copenhagen, Denmark

b)Steklov Mathematical Institute, Gubkina 8, 117966, GSP–1, Moscow, Russia

Abstract

We investigate the NBI matrix model with the potential XΛ+X^{−1} +(2η +1) log X recently proposed to describe IIB superstrings. With the proper normalization, using Virasoro constraints, we prove the equivalence of this model and the Kontsevich matrix model for η ≠ 0 and find the explicit transformation between the two models.

∗E-mail: ambjorn@nbivms.nbi.dk †E-mail: chekhov@genesis.mi.ras.ru

1 Introduction

The investigation of matrix models with an external field and a logarithmic potentials was initiated in [1, 2]. These matrix models are related to the so-called NBI matrix model, which appeared recently in the context of the IIB superstring matrix model [3, 4, 5, 6]. In [5, 6], the following (M)atrix model action was proposed:

SNBI = −\frac{α}{4} tr Y^{−1}[A_{μ}, A_{ν}]^{2} + β tr Y + 2η tr log Y − \frac{1}{2} tr \overline{Ψ}Γ^{μ}[A_{μ}, Ψ].  (1.1)

It possesses N =2 supersymmetry in the large N limit [5]. The matrix Y plays here the role of the world-sheet metric to be integrated out in order to obtain the effective action. Then, the (nonlocal) logarithmic term is a curvature term of the world-sheet metric. As was shown in [2], the relevant choice of the constant η in front of this term that leads to a non-Abelian Born–Infeld action for the string coordinates is η = −\frac{1}{4N}, N being the matrix size. However, theories for other values of η were also considered [7], and the answer for the effective action in the leading order in N was obtained in [2] for general η.

In the present paper, we derive the constraint equations for the NBI matrix model, and show their coincidence with the constraint equations for the Kontsevich matrix model, thereby proving the equivalence of the two models for nonzero η.

2 The matrix model in the large N limit

We start with the following matrix integral:

Z = ∫ dX e^{−N tr[XΛ+X^{−1}+(2η+1) log X]},  (2.1)

As shown in [2], it is related to the bosonic part of (1.1) by the following change in integration variables: Y = \frac{N}{β} X^{−1}, and by

Λ = −\frac{αβ}{4N^{2}} [A_{μ}, A_{ν}]^{2}.  (2.2)

The matrix integral (2.1) belongs to a class of generalized Kontsevich models (GKM) [8]. Such models with negative powers of the matrix X have been previously discussed in the context of c = 1 bosonic string theory [9]. In [7], the τ-function approach to such models was developed. There, the parameter η plays the role of the zeroth time in the corresponding integrable hierarchy. Moreover, at the conformal point η = 0, this model was shown [7] to have the same Schwinger–Dyson equations as the U(N) model solved in [10, 11].

For the models of this type, the large N solutions are known explicitly only in some special cases. The models with cubic potential for X [12] and the combination of the logarithmic and quadratic potentials [1] were solved by a method based on Schwinger–Dyson equations, developed first for the unitary matrix models with external field [10, 11]. The same technique, being applied to the integral (2.1), also allows one to find its large N asymptotic expansions in the closed form for arbitrary η [2].

The Schwinger–Dyson equations for (2.1) follow from the identity

\frac{1}{N^{3}} \frac{∂}{∂Λ_{jk}} \frac{∂}{∂Λ_{li}} ∫ dX \frac{∂}{∂X_{ij}} e^{−N tr[XΛ+X^{−1}+(2η+1) log X]} = 0.  (2.3)

Written in terms of the eigenvalues, these N equations read (no summation over i is implied)

[−\frac{1}{N^{2}} λ_{i} \frac{∂^{2}}{∂λ_{i}^{2}} − \frac{1}{N^{2}} ∑_{j≠i} λ_{j} \frac{1}{λ_{j} − λ_{i}} (\frac{∂}{∂λ_{j}} − \frac{∂}{∂λ_{i}} ) + \frac{1}{N} (2η − 1) \frac{∂}{∂λ_{i}} + 1] Z(λ) = 0. (2.4)

For η = 0, these formulas coincide with the corresponding formulas for the U(N) model [10, 11].

It is convenient to set

W (λ_{i}) = \frac{1}{N} \frac{∂}{∂λ_{i}} log Z.  (2.5)

This quantity plays an important role in evaluating the large N limit.

We also introduce the eigenvalue density of the matrix Λ:

ρ(x) = \frac{1}{N} ∑_{i} δ(x − λ_{i}).  (2.6)

The density obeys the normalization condition

∫ dx ρ(x) = 1,  (2.7)

and in the large N limit it becomes a smooth function.

Simple power counting shows that the derivative of W (λ_{i}) in the first term on the lefthand side of Eq. (2.4) is suppressed by the factor 1/N and can be omitted at N = ∞. The remaining terms are as follows:

−xW^{2}(x) − ∫ dyρ(y) y \frac{W (y) − W (x)}{y − x} + (2η − 1)W (x) + 1 = 0,  (2.8)

where λ_{i} is replaced by x. Equation (2.8) can be simplified by the substitution

W̃ (x) = xW (x) − η.  (2.9)

After some transformations, using the normalization condition (2.7), we obtain

W̃^{2}(x) + x ∫ dyρ(y) \frac{W̃ (y) − W̃ (x)}{y − x} = x + η^{2}.  (2.10)

The nonlinear integral equation (2.10) can be solved with the help of the ansatz

W̃ (x) = f(x) + \frac{x}{2} ∫ dy \frac{ρ(y)}{f(y)} \frac{f(y) − f(x)}{y − x},  (2.11)

where f(x) is an unknown function to be determined by substituting (2.11) into Eq. (2.10). The asymptotic behaviors of W̃ (x) and f(x) as x → ∞ follow from eq. (2.10): W̃ (x) ∼ \sqrt{x} − 1/2, and the analytic solution with minimal set of singularities is

f(x) = \sqrt{ax + b}.  (2.12)

The parameters a and b are unambiguously determined from Eq. (2.10). We find that b = η^{2} and a is implicitly defined by

1 + \frac{1}{2} ∫ dy \frac{ρ(y)}{f(y)} = \frac{1}{\sqrt{a}},  (2.13)

or, in terms of the eigenvalues,

1 + \frac{1}{2N} ∑_{j} \frac{1}{\sqrt{aλ_{j} + η^{2}}} = \frac{1}{\sqrt{a}}.  (2.14)

Then, the answer for the integral in the large N limit reads [2]

log Z = N^{2} [(η^{2} + \frac{1}{4}) log a + \frac{4η^{2}}{\sqrt{a}} − \frac{η^{2}}{a} ]

+N ∑_{i} [\frac{2}{\sqrt{a}} \sqrt{aλ_{i} + η^{2}} + η log (λ_{i} \frac{\sqrt{aλ_{i} + η^{2}} − η}{\sqrt{aλ_{i} + η^{2}} + η} )]

−\frac{1}{2} ∑_{ij} log (\sqrt{aλ_{i} + η^{2}} + \sqrt{aλ_{j} + η^{2}}) .  (2.15)

One can verify directly that \frac{∂}{∂a} log Z = 0 and \frac{1}{N} \frac{∂}{∂λ_{i}} log Z = W (λ_{i}), as far as Eq. (2.14) holds.

3 The Kontsevich phase

We are interested in the asymptotic expansion of the model (2.1) for large Λ. Then, the expansion parameters are traces of negative powers of the external matrix Λ. Conventionally, this regime is called the Kontsevich phase of the solution.

Here an important note is in order. In [2], we did not discuss which branch of the root— positive or negative—should be chosen in (2.14), since both choices led to the same answer for the integral in the large N limit (2.15). However, in what follows, we must fix this sign.

The Kontsevich phase is the strong coupling regime where the expansion in negative powers of λ_{i} is to be performed. Then we see that the dependence is only on λ_{i}^{−n−1/2}, n = 0, 1, . . ..

As the first step, we perform the phase analysis for the toy case where all λ_{i}’s coincide, so (2.14) becomes

\frac{a}{(1 − \sqrt{a})^{2}} = 4(aλ + η^{2}), a > 0, λ > 0.  (3.1)

Then, obviously, the sign of the square term in (2.14) is negative for a > 1 and positive for 0 < a < 1.

Algebraically, there always (except if η = 0) exist two solutions to (3.1): one with 0 < a < 1 and another with a > 1. For the Kontsevich phase to be possible, we demand that the expansion in terms of the so-called times

τ_{k} = \frac{1}{2k − 1} tr Λ^{−k+1/2}, k = 1, 2, ….  (3.2)

should make sense. So, we assume that

|\frac{η^{2}}{aλ}| < 1.  (3.3)

Then, from (3.1) and (3.3), we obtain the following phase diagram:

[figure: no Kontsevich; phase; ∃ Kontsevich; phase for; \sqrt{a} > 1; ∃ two; Kontsevich; phases for \sqrt{a} > 1; and \sqrt{a} < 1; \frac{1}{2\sqrt{2}}; \frac{1}{2\sqrt{2}}; η; \sqrt{λ}]

4 Constraint equations in the Kontsevich phase

Now we write (2.4) in terms of the relevant times (3.2). Here, to obtain rigorous results, the normalizing factor is necessary. From the theory of the generalized Kontsevich model [8], the proper expression, which has no explicit dependence on the matrix size N, reads

Z(\{τ_{n}\}) = \frac{∫ DX e^{ΛX+V (X)}}{e ^{tr ΛX_{0}+V (X_{0})} det^{−1/2} (\frac{δ}{δX} ⊗ \frac{δ}{δX} V (X_{0}))}.  (4.1)

Here X_{0} is the stationary point, Λ + V′(X_{0}) = 0, and the determinant in the normalizing factor comes from the quasi-classical integration.

Let us choose the new variables

λ_{i} = z_{i}^{2} − (η + 1/2)^{2}.  (4.2)

Then the normalizing factor reads

exp \{−N ∑_{i}[2z_{i} − 2η log(z_{i} + η + 1/2)] − \frac{1}{2} ∑_{i,j} log(z_{i} + z_{j})\} .  (4.3)

Note that if we perform the standard Itzykson–Zuber integration, expression (4.1) for integral (2.1) becomes

Z(\{τ_{n}\}) = \frac{det_{{1≤i≤N \atop 0≤l≤N−1}} ||ξ_{i}^{2ηN−l}K_{−2ηN−l}(Nξ_{i})||}{∏_{i} e^{−2Nz_{i}}(z_{i} + η + 1/2)^{2ηN}(2z_{i})^{−1/2}Δ(z)},  ξ_{i} ≡ \sqrt{λ_{i}},  (4.4)

where K_{ν}(x) are Macdonald functions, K_{ν}(x) = ∫_{−∞}^{∞} ds e^{−2x cosh s+νs}, and (4.4) does not resemble too much the corresponding expression for the Kontsevich matrix model where these Airy functions stand instead of K_{ν}(x). To find the large N asymptotic behavior of (4.4), we use the constraint equation method.

In terms of z-variables, (2.4) becomes

[−\frac{1}{N^{2}} (z_{i}^{2} − (η + 1/2)^{2})\frac{1}{2z_{i}} \frac{∂}{∂z_{i}} (\frac{1}{2z_{i}} \frac{∂}{∂z_{i}} )

−\frac{1}{N^{2}} ∑_{j≠i} \frac{z_{j}^{2} − (η + 1/2)^{2}}{z_{j}^{2} − z_{i}^{2}} (\frac{1}{2z_{j}} \frac{∂}{∂z_{j}} − \frac{1}{2z_{i}} \frac{∂}{∂z_{i}} ) + \frac{2η − 1}{N} \frac{1}{2z_{i}} \frac{∂}{∂z_{i}} + 1] Z(z_{i}) = 0 (4.5)

When pushing the normalizing factor (4.3) through derivatives w.r.t. z_{j}-variables, we replace (∂_{i} ≡ \frac{∂}{∂z_{i}} )

∂_{i} → ∂_{i} − 2N + 2ηN \frac{1}{z_{i} + η + 1/2} − ∑_{j} \frac{1}{z_{i} + z_{j}} .  (4.6)

Let us introduce the new times

t_{n+1} = \frac{1}{2n − 1} ∑_{i} \frac{1}{z_{i}^{2n−1}} + δ_{n,0} \frac{N}{η + 1/2},  n = 0, 1, …,  (4.7)

which differs slightly from the conventional ones defines above by (3.2). As is easily checked, they are related by a lower triangular transformation, so they are equivalent from the view point of phase transitions and critical behavior.

Then, the constraint equations for Z({t_{n}}) are obtained after some tedious algebra which we omit here. Collecting all coefficients to the term \frac{1}{z_{i}^{2s+4}} ≡ \frac{∂}{∂z_{i}} t_{s+2}, we obtain

L̃_{s}Z({t_{n}}) = 0,  s ≥ −1,  (4.8)

where

L̃_{s} = δ_{s,−1} [−\frac{1}{16N^{2}} + t_{1}^{2} \frac{(η + 1/2)^{2}}{4N^{2}} ] + δ_{s,0} \frac{(η + 1/2)^{2}}{16N^{2}}

−\frac{1}{2N^{2}} ∑_{{p=0 \atop p+s≥0}}^{∞} [−(η + 1/2)^{2}(2p + 1)t_{p+1} + (2p − 1)(1 − δ_{p,0})t_{p}] \frac{∂}{∂t_{p+s+1}}

−\frac{1}{N} (\frac{∂}{∂t_{s+2}} − \frac{η + 1/2}{2}(1 − δ_{s,−1})\frac{∂}{∂t_{s+1}} )

−\frac{1}{4N^{2}} ∑_{k=1}^{s+1} \frac{∂}{∂t_{k}} \frac{∂}{∂t_{s+2−k}} + \frac{(η + 1/2)^{2}}{4N^{2}} ∑_{k=1}^{s} \frac{∂}{∂t_{k}} \frac{∂}{∂t_{s+1−k}} .  (4.9)

In the convenient conformal field theory notation, (4.9) becomes

L̃_{s} = V_{s+1} − (η + 1/2)^{2}V_{s} + \frac{4ηN}{η + 1/2}α_{2s+3}.  (4.10)

where

α_{2n+1}|_{n≥0} ≡ \frac{∂}{∂t_{n+1}} ,  (4.11)
α_{−2n−1}|_{n≥0} ≡ (2n + 1)t_{n+1},  (4.12)
[α_{−2k−1}, α_{2q+1}] = −(2k + 1)δ_{k,q},  (4.13)

and the operators V_{s} in the free-field representation are quadratic in α_{i},

V_{q} ≡ ∑_{a,b} δ_{q,a+b+1}:α_{2a+1}α_{2b+1}: + 1/4δ_{s,0} ,  (4.14)

where the normal ordering : : means that all α_{a} with positive indices are on the right of all α_{b} with negative indices.

For s, t ≥ −1, L̃_{s} satisfy the algebra

[L̃_{s}, L̃_{t}] = 4(s − t)(L̃_{s+t+1} − (η + 1/2)^{2}L̃_{s+t}),  (4.15)

from which the Virasoro algebra can be obtained by the lower-triangle replacement

L_{s} ≡ L̃_{s} + ∑_{k=1}^{∞}(η + 1/2)^{−2k}L̃_{s+k}.  (4.16)

Performing replacement (4.16) and rescaling V_{s}, we obtain the Virasoro algebra in terms of the generators

L_{s} = −(η + 1/2)^{2}V_{s} + 4ηN ∑_{k=2}^{∞} \frac{1}{(η + 1/2)^{2k−3}} ∂_{s+k}.  (4.17)

Amazingly, if we manage to remove the derivative terms in (4.17), then the constraints we obtain will be just constraints of the Kontsevich matrix model [13, 14]. To remove these terms, we shift all of the higher times, leaving the times t_{1} and t_{2} unshifted,

t̃_{k} ≡ t_{k} − \frac{4ηN}{(η + 1/2)^{2k+1}} \frac{1}{2(2k + 1)}, k ≥ 3,  t̃_{1,2} = t_{1,2}.  (4.18)

Explicitly, in terms of the new times,

L_{−1} = t̃_{1}^{2} + 2 ∑_{m=1}^{∞}(2m + 1)t̃_{m+1} \frac{∂}{∂t̃_{m}} − \frac{4ηN}{(η + 1/2)^{3}} ∂_{1}  (4.19)
L_{s} = ∑_{m=1}^{s} \frac{∂}{∂t̃_{m}} \frac{∂}{∂t̃_{s−m}} + 2 ∑_{m=1}^{∞}(2m − 1)t̃_{m} \frac{∂}{∂t̃_{m+s}} + \frac{1}{4}δ_{s,0} − \frac{4ηN}{(η + 1/2)^{3}} \frac{∂}{∂t̃_{s+2}} . (4.20)

This is nothing but the Virasoro algebra that appears in the Kontsevich matrix model. Therefore, we have proven the equivalence between the model (2.1) and the Kontsevich matrix model [15].

5 A large N limits comparison

Let us explicitly compare the model (2.1) after the time changing (4.18) and the Kontsevich model with the partition function

Z(M) = \frac{∫ dX exp − tr γ \{MX^{2}/2 − iX^{3}/6\}}{∫ dX exp − tr γMX^{2}/2}, M = diag \{m_{1}, …, m_{N}\}.  (5.1)

Let us consider the constraint equation (2.14). We have

\frac{1}{2n − 1} tr M^{−2n+1} = t̃_{n} = \frac{1}{2n − 1} tr Z^{−2n+1} + \{ \begin{matrix}\frac{N}{η+1/2}, & n = 1, \\ 0, & n = 2, \\ −\frac{1}{2(2n−1)} \frac{4ηN}{(η+1/2)^{2n−1}} , & n ≥ 3, \end{matrix}  (5.2)

i.e.,

tr M^{−2n+1} = tr Z^{−2n+1} − \frac{1}{2} \frac{4ηN}{(η + 1/2)^{2n−1}} + \frac{1}{2} \frac{4ηN}{(η + 1/2)^{3}} δ_{n,2} + 2Nδ_{n,1}.  (5.3)

Then we obtain, in terms of these shifted variables, (± depends on the branch of the square root),

\frac{1}{\sqrt{a}} = 1 ± \frac{1}{2N} ∑_{j} \frac{1}{\sqrt{az_{j}^{2} − a(η + 1/2)^{2} + η^{2}}}

= 1 ± \frac{1}{2N\sqrt{a}} ∑_{j} ∑_{n=0}^{∞} \frac{(2n − 1)!!}{(2n)!!} \frac{1}{z_{j}^{2n+1}} ((η + 1/2)^{2} − η^{2}/a)^{n}

= 1 ± \frac{1}{2N\sqrt{a}} [∑_{j} \frac{1}{\sqrt{m_{j}^{2} + η^{2}/a − (η + 1/2)^{2}}}

+2N\sqrt{a} − 2N − \frac{ηN}{η + 1/2} + \frac{η^{3}N}{(η + 1/2)^{3}a}] .  (5.4)

Assuming the minus sign and denoting

(η + 1/2)^{2} − η^{2}/a ≡ 2s,  −\frac{2ηN}{(η + 1/2)^{3}} ≡ γ,  (5.5)

we obtain

∑_{i=1}^{N} \frac{1}{\sqrt{m_{i}^{2} − 2s}} = −γs,  (5.6)

i.e., the constraint equation of the Kontsevich model itself [13, 14].

Also, let us recall the answer in the large N limit for the Kontsevich model [13]. If we reconstruct the dependence on the coupling constant γ, then it reads (in the original notation)

F_{0} = γ^{2} \frac{s^{3}}{6} + \frac{γ}{3} ∑_{i=1}^{N} \{m_{i}^{3} − (m_{i}^{2} − 2s)^{3/2} − 3s\sqrt{m_{i}^{2} − 2s}\}

−\frac{1}{2} ∑_{i,j=1}^{N} log \frac{\sqrt{m_{i}^{2} − 2s} + \sqrt{m_{j}^{2} − 2s}}{m_{i} + m_{j}},  (5.7)

where s is determined from the constraint equation (5.6).

Let us compare the answer (5.7) with formula (2.15) while accounting for the normalizing condition (4.3). Then, in variables z_{i}, assuming the minus sign in the constraint (2.14), we have

F̃_{0} = N^{2}η^{2} log a + \frac{4N^{2}η^{2}}{\sqrt{a}} − \frac{N^{2}η^{2}}{a}

+ N ∑_{i} 2 (z_{i} − \sqrt{z_{i}^{2} − 2s}) + Nη ∑_{i} log (\frac{z_{i}^{2} − (η + 1/2)^{2}}{(z_{i} + η + 1/2)^{2}} ) (\frac{−\sqrt{z_{i}^{2} − 2s} − η/\sqrt{a}}{−\sqrt{z_{i}^{2} − 2s} + η/\sqrt{a}})

− \frac{1}{2} ∑_{i,j} log (\frac{\sqrt{z_{i}^{2} − 2s} + \sqrt{z_{j}^{2} − 2s}}{z_{i} + z_{j}} ) .  (5.8)

After a tedious algebra, taking into account (5.3), we obtain

F̃_{0} = F_{0} + 2N^{2}η^{2} log \frac{η}{η + 1/2} + N^{2} [\frac{7}{3}η^{2} + η − \frac{1}{4}] ,  (5.9)

where F_{0} is given by (5.7) with the substitution (5.5). The difference between F̃_{0} and F_{0} is just irrelevant constant terms. This again proves that the two matrix models under consideration coincide.

6 Higher genus expressions

In this section, we set the Kontsevich coupling constant γ = 1. The higher genus contributions in the Kontsevich model are expressed in terms of the so-called moments I_{k},

I_{0} = −\frac{1}{N} ∑_{i=1}^{N} \frac{1}{\sqrt{m_{i}^{2} − 2s}},  I_{k} = \frac{∂^{k}}{∂s^{k}} I_{0}, k > 0.  (6.1)

Then, for g > 1, we have [14]

F_{g} = _{∑_{k=2}(k−1)l_{k}=3g−3}^{∑}⟨τ_{2}^{l_{2}}τ_{3}^{l_{3}} ⋯ τ_{3g−2}^{l_{3g−2}}⟩_{g} \frac{1}{(1 − I_{1})^{2(g−1)+∑ l_{p}}} \frac{I_{2}^{l_{2}}}{l_{2}!} \frac{I_{3}^{l_{3}}}{l_{3}!} ⋯ \frac{I_{3g−2}^{l_{3g−2}}}{l_{3g−2}!},  (6.2)

i.e., F_{g} is a finite sum of monomials in I_{k}/(1 − I_{1})^{(2k+1)/3} with coefficients being the intersection indices on the moduli space [15]. For g = 1, F_{1} = \frac{1}{24} log \frac{1}{1−I_{1}} .

Now we rewrite the expression (6.2) via the moments of the model (2.1). Let us introduce the new moments J_{k}

J_{k} = −(2k − 1)!!\frac{1}{N} ∑_{j=1}^{N} \frac{1}{(aλ_{j} + η^{2})^{k+1/2}} , k = 0, 1, 2, ….  (6.3)

We are interested only in the transformation law for I_{k} with k > 0 since the only dependence on the moment I_{0} is via the constraint equation (5.6). Then, Eq. (5.3) implies

(I_{1} − 1) → a^{3/2}(J_{1} + 2/η^{2}),
I_{k} → a^{k+1/2}(J_{k} + 2/η^{2k}),

i.e., the expression (6.2) becomes

F_{g}^{NBI} = _{∑_{k=2}(k−1)l_{k}=3g−3}^{∑}⟨τ_{2}^{l_{2}}τ_{3}^{l_{3}} ⋯ τ_{3g−2}^{l_{3g−2}}⟩_{g} \frac{1}{(J_{1} + \frac{2}{η^{2}} )^{2(g−1)+∑ l_{p}}} ∏_{k=2}^{3g−2} \frac{(J_{k} + \frac{2}{η^{2k}})^{l_{k}}}{l_{k}!} , g > 1

(6.4)

and

F_{1}^{NBI} = \frac{1}{24} log[a^{3/2}(J_{1} + 2/η^{2})].  (6.5)

Therefore, expression (2.15) for genus zero, taking into account the normalizing factor (4.3) and the expressions (6.5), (6.4) completely determine the partition function of the model (2.1) for all genera. These expansions are, however, ill-defined for η → 0, which corresponds to the U(N) model, and for η ∼ 1/N, which corresponds to the model defined in [5].

7 Remarks

1. The last observation above is related to the initial constraints (4.8). Note that we can consider a “minimal reduction,” where all times t_{k} but the time t_{1} are equal to zero. Then this partition function is entirely determined from the action of L̃_{−1} on Z,

(\frac{4η}{η + 1/2} + 2t_{1}) \frac{∂F}{∂t_{1}} |_{t_{2}=t_{3}=...=0} − N^{2}(η + 1/2)^{2}t_{1}^{2} + 1/4 = 0,

where

t_{1} ≡ \frac{1}{N} ∑_{i=1}^{N} z_{i}^{−1} + (η + 1/2)^{−1}|_{t_{2}=t_{3}=...=0}= \frac{1}{N} ∑_{i=1}^{N} λ_{i}^{−1/2} + (η + 1/2)^{−1},

i.e.,

\frac{∂F}{∂t_{1}} |_{t_{2}=t_{3}=...=0} = \frac{η + 1/2}{4} [N^{2}((η + 1/2)t_{1} − 2η) + \frac{(2ηN)^{2} − 1/4}{(η + 1/2)t_{1} + 2η} ] .  (7.1)

Remark that F becomes a polynomial in t_{1} for

η = ±\frac{1}{4N} ,

and this is exactly the point corresponding to the IIB superstring model [5].

On the other hand, from expressions (6.4), (6.5) it is clear that in this case all J_{k} = 0 for k > 0, so there is no t_{1}-dependent contributions from g > 1 and only \frac{1}{16} log a comes from the genus one term. One can compare with expression for F_{1} coming from (7.1) and, taking into account the constraint equation (2.14), one finds an exact coincidence of the two expressions, i.e.,

F_{1}|_{t_{2}=t_{3}=...=0} = −1/8 log(2 + 1/N tr λ^{−1/2}) = \frac{1}{16} log a.

2. The string susceptibility in the large N limit can be obtained by differentiating twice w.r.t. the string coupling constant η. One should check that after the first differentiation of (2.15), the stationary condition still holds, so the total derivative coincides with the partial one and, as a result, we have

\frac{d^{2} log Z}{dη^{2}} = \frac{∂^{2} log Z}{∂η^{2}} = 2N^{2}(log a + 3).  (7.2)

Acknowledgement

J. A. acknowledges the support of MaPhySto, funded by a grant from the Danish National Research Foundation. L. Ch. is grateful to the Niels Bohr Institute for the hospitality. The work of L. Ch. was supported in part by the Russian Foundation for Basic Research, Grant No. 96-01-00344.

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