Punctures and $p$-spin curves from matrix models III.
$D_{l}$ type and logarithmic potential

Shinobu Hikami

Okinawa Institute of Science and Technology Graduate University, 1919-1
Tancha, Okinawa 904-0495, Japan. e-mail<EMAIL_ADDRESS>

Abstract  The intersection numbers for $p$ spin curves of the moduli space
$\overline{\mathcal{M}}_{g,n}$ are considered for $D_{l}$ type by a matrix
model. The asymptotic behavior of the large genus $g$ limit and large $p$
limit are derived. The remarkable features of the cases of
$p=\frac{1}{2},-\frac{1}{2},-2,-3$ are examined in the Laurent expansion for
multiple correlation functions. The strong coupling expansions for the
negative $p$ cases are considered.

Key words Random matrix model, Topological field theory, Intersection numbers

##  1 Introduction

It is well known that the generating function of the intersection numbers on
moduli space of $p$ spin stable curves becomes the $\tau$ function of the
generalized KdV hierarchies ($p$-reduced KP hierarchies), which are related to
a two dimensional gravity [1, 2]. There are now many studies of the
intersection numbers including Gelfand-Dikii pseudo differential equation [3,
4]. We have proposed a method of the calculation for this intersection numbers
by a generalized Kontsevich matrix model, which was derived by the duality and
replica method based on the random matrix theory [5, 6, 7, 8]. For one point
function, an expression for single intersection numbers has been derived
explicitly.

For the purpose of the extension to half integer spin $p$, we reformulated
this integral expressions by a new change of variables, and we have
investigated the intersection numbers of Ramond punctures for half-spin in
previous articles I, II [9, 10].

This reformulation enables us to obtain easily the intersection numbers for
integer $p$ (Neveu-Schwarz punctures) for $n$ point functions, which should be
consistent with the results obtained by the recursive method [11, 12] due to
Gelfand-Dikii equation. The evaluation of several marked points in general $p$
and for genus $g$ was obtained in the recursive calculations [12], and we show
in this article that our method of the Laurent expansion agrees with them for
the lower orders, especially for three point functions.

The ADE singularities characterized by Dynkin diagrams are important topics in
statistical physics. The $D_{l}$ singularity is represented by the algebraic
equation $y^{l-1}+yx^{2}+z^{2}=0$ ($l\geq 4$). The Coxeter number defined by
$p=2l-2$ in $D_{l}$ singularity can be interpreted as a spin $p$. The weight
system of $D_{l}$ is known as $(a,b,c;h)=(2,l-2,l-1;2(l-1))$. The simply laced
Lie algebra, $A_{l},B_{l},C_{l},D_{l},$ are related to classical Lie groups
$SU(l+1),SO(2l+1),SP(l),SO(2l)$, respectively. We have discussed
$A_{l},B_{l},C_{l},D_{l}$ cases, where HarishChandra theorem can be applied
for the random matrix models with external sources [8, 13]. These random
matrix models were applied for the non-orientable surfaces or Klein surfaces.
Recently the $D_{l}$ singularity has been discussed for the intersection
numbers of one point function [14].

The open intersection numbers have been discussed based on the logarithmic
matrix models [15, 16, 17, 18, 19, 20, 21, 22, 23], which shows the extension
of the intersection theory of Riemann surface to open Riemann surface, i.e. it
has boundaries [17, 21]. The boundaries, similar to the $D$ brane, are
represented by the logarithmic terms. The matrix model with a logarithmic
potential for the open intersection theory is written for general $p$
(generalized Airy matrix model with a logarithmic potential) [15],

$Z=\int dBe^{-\frac{c}{p+1}{\rm tr}B^{p+1}+k{\rm tr}{\rm log}B+{\rm
tr}B\Lambda}$ (1.1)

where $B$ is a Hermitian matrix and $k$ is a parameter. The matrix $\Lambda$
is an external source. This model is called as Kontsevich-Penner model, when
$p=2$.

The spin $p$ is related to $l$ as $p=2l-2$ for $D_{l}$ type singularity. We
investigated before the case of Lie algebra $so(N),sp(N)$ and found that the
correspondent one point function $u(s)$ has a logarithmic terms [13], which
makes a difference from $A_{l}$ type. These cases were discussed as a
manifestation of the feature of non-orientable surface (Klein surface) further
in [16, 8].

In this paper, we explicitly show that the intersection numbers of $D_{l}$
type can be derived from the matrix model witha logarithmic term. By the
Gaussian integral of $x$ for $y^{l-1}+x^{2}y+z^{2}$, we have $1/\sqrt{y}$
which turns to be a logarithmic potential by the exponentiation. This may give
simple explanation of the appearance of the logarithmic term in the matrix
model of $D_{l}$ singularity. In general, an oscillating integrals of
$n$-variables have asymptotic expansions with logarithmic terms, related to
Newton polygon [24] and it is well known that such expansion is related to the
resolution of the singularities. The $D_{l}$ type is related to a real
algebraic curves and the Euler characteristics $\chi$ of a real algebraic
curve is obtained in the case of $p=-1$ in $D_{l}$ type $(p=2l-2)$ [13, 25,
26].

It is known that $A_{5},A_{4},A_{3}$ (p=6,5,4) singularities correspond to
Ashkin-Teller model, 3-state Potts model and Ising model respectively [27,
28]. The $A_{p-1}$ singularity has a central charge $C=2-6/p$. The central
charge $C=\frac{1}{2}$ and critical exponent of the energy $\nu$ are
consistent with the values of Ising model. The anomalous conformal dimensions
are $\Delta_{\epsilon}=\frac{4}{p}$, $\Delta_{\phi}=\frac{p-3}{p(p-2)}$. These
dimensions and central charges agree with the well-known values of Ising
($p=4$) and 3-state Potts model ($p=5$).

For $D_{l}$ type singularities, the central charge is also given by
$C=2-\frac{6}{p}$, where $p$ is Coxeter number $p=2l-2$. The interesting
applications are found in the condensed matter physics for the topological
excitation of electron at the boundaries, as Majorana fermions [21], and the
edge excitation on the boundary as Quantum Hall effect, for instance.
Furthermore, in $D_{4}$ singularity, there are intriguing Ramond sector [29,
30], which is related to the vanishing relation for one point function of
$g=2+3m$, $m\in\mathbb{Z}$, for which we will discuss in this paper.

The spin $p$ takes a value of positive integer according to the singularity
theory. However, as discussed for $p=-1$, Euler characteristic is obtained
from the continuation from positive integer to $p=-1$ in the expression of the
intersection numbers. The cases of $p=\frac{1}{2}$ and $p=\frac{3}{2}$, which
are fractional spins, have been discussed in our previous papers [9, 10].
These cases correspond to ”fermion”, and the genus expansion of one point
function agree with the selection rule due to Riemann-Roch theorem. As a
conformal field theory (CFT) in two dimensions, $p=\frac{3}{2}$ case exists as
$\beta\gamma$ system in the supersymmetric non-linear sigma model, which also
corresponds to spanning forrest model with a central charge $C=-2$ [31]. There
is an interesting observation that for $p=\frac{1}{2}$ case, the tautological
relations become simple [32, 33, 34]. In this paper, by the results of I,II
[9, 10], we investigate the $m$ point correlation functions of
$p=\frac{1}{2}$, and find that the punctures of Ramond type appear in a
pairwise as same as $p=-2$ case.

The case $p=-2$ corresponds to the unitary matrix model of the lattice gauge
theory [35, 36, 37, 38]. The strong coupling region of this case has a
character expansion [39, 40]. We consider this character expansion [40] for
the negative $p$ case, $p=-2,-3,...$ by the $n$-point function of
$U(s_{1},...,s_{n})=<\prod_{i}{\rm tr}e^{s_{i}M}>$, which can be interpreted
as Wilson loops.

This article is organized as following: One point intersection numbers of $p$
spin curves for genus $g$ is shown in section 2 for $A_{l}$. In section 3, one
point functions of $D_{l}$ type is evaluated up to $g=11$. In section 4, one
point functions for non-integer $p$ case of $A$ type and $D$ type are
discussed for $p=\frac{3}{2},\frac{1}{2},-\frac{1}{2},-1,-2$. In section 5,
the large $g$ and large $p$ limits are discussed. The integrality of the
intersection number is discussed in the relation to Bernoulli numbers. It is
shown that the denominators of the intersection numbers and Bernoulli numbers
are same. In the limit $p\to\infty$, the intersection numbers reduce to
Bernoulli numbers [41], which are intriguingly connected to homotopy,
differential topology and number theories [42, 43, 45, 46]. In section 6, the
intersection numbers for multi marked points are evaluated, which is
consistent with the results by the recursion relations. The section 7 is
devoted to the evaluations of half spin $p=\frac{1}{2},-\frac{1}{2}$ and the
negative integers $p=-2$ and $p=-3$ cases. For the negative integer case, the
strong coupling expansion is investigated in the relation to the characters of
$U(N)$. In the section 8, we give summary and discussions.

##  2 One point function for $A_{l}$ type

Since the $p$ spin curves of the moduli space has a correspondence to
$A_{p-1}$ singularity by mirror symmetry, we use the terminology of $A_{l}$
type for the $l=p-1$ spin curves, which distinguishes the case $D$ type.

For $A_{l}$ case, the generating function of the intersection number for one
marked point is expressed as [6, 7]

$\displaystyle u(s)$ $\displaystyle=$ $\displaystyle<{\rm tr}e^{sB}>$ (2.1)
$\displaystyle=$
$\displaystyle\frac{1}{s}\int\frac{du}{2i\pi}e^{-\frac{c}{p+1}[(u+\frac{s}{2})^{p+1}-(u-\frac{s}{2})^{p+1}]}$

The generating function for the intersection numbers of $A_{l}$ ($p=l+1$) is
evaluated for small $s$ by the replacement $u=(\frac{t}{cs})^{1/p}$,

$\displaystyle u(s)$ $\displaystyle=$
$\displaystyle\frac{1}{sp\pi}\cdot\frac{1}{(cs)^{1/p}}\int_{0}^{\infty}dtt^{\frac{1}{p}-1}e^{-t}$
(2.2) $\displaystyle\times$ $\displaystyle
e^{-\frac{p(p-1)}{3!4}s^{2+\frac{2}{p}}c^{\frac{2}{p}}t^{1-\frac{2}{p}}-\frac{p(p-1)(p-2)(p-3)}{5!4^{2}}s^{4+\frac{4}{p}}c^{\frac{4}{p}}t^{1-\frac{4}{p}}+\cdots}$

This leads to

$\displaystyle u(s)$ $\displaystyle=$
$\displaystyle\frac{1}{c^{\frac{1}{p}}s^{1+\frac{1}{p}}\pi}[\Gamma(1+\frac{1}{p})-\frac{p-1}{24}z\Gamma(1-\frac{1}{p})$
(2.3)
$\displaystyle+\frac{(p-1)(2p+1)(p-3)}{5760}z^{2}\Gamma(1-\frac{3}{p})+\cdots]$

where $z=c^{\frac{2}{p}}s^{2+\frac{2}{p}}$. Writing this expansion with the
intersection numbers $<\tau_{n,j}>_{g}$ ($n$ is integer, and spin component
$j=0,1,2,...,p-2$) as

$u(s)=\sum_{g}<\tau_{n,j}>_{g}\frac{1}{\pi}\Gamma(1-\frac{j+1}{p})c^{\frac{2g-1}{p}}p^{g-1}s^{(2g-1)(1+\frac{1}{p})},$
(2.4)

one obtain the intersection numbers as a polynomial of $p$. We have a relation
of $n$ and $j$ for non-vanishing intersection numbers,

$(2g-1)(1+\frac{1}{p})=n+\frac{j+1}{p}$ (2.5)

which comes from the Riemann-Roch relation (RR) for $\tilde{s}$ marked points
by $\tilde{s}=1$

$3g-3+\tilde{s}=\sum_{i=1}^{\tilde{s}}n_{i}+(g-1)(1-\frac{2}{p})+\frac{1}{p}\sum_{i=1}^{\tilde{s}}j_{i}$
(2.6)

Thus the $s$ dependence appears for one point case ($\tilde{s}=1$) as a power
$s^{(2g-1)(1+\frac{1}{p})}=s^{n+\frac{j+1}{p}}$ in (2.4). The intersection
numbers ( in the case of $p=2$), are given by the first Chern class $c_{1}$ or
$\psi$, as

$<\tau_{n_{1}}\cdots\tau_{n_{s}}>_{g}=\int_{\bar{\mathcal{M}}_{g,s}}\psi^{n_{1}}\cdots\psi^{n_{s}}$
(2.7)

where $\bar{\mathcal{M}}_{g,s}$ is a compactified moduli space with $s$ marked
points on genus $g$ Riemann surface. The intersection numbers of one point
$\tau_{A_{l}}(g)$ (Coxeter number p= l+1) is thus given as [9, 10]

$\displaystyle<\tau_{n,j}>_{g=1}$ $\displaystyle=$
$\displaystyle\frac{p-1}{24}$ $\displaystyle<\tau_{n,j}>_{g=2}$
$\displaystyle=$ $\displaystyle\frac{(p-1)(2p+1)(p-3)}{p\cdot 5!\cdot
4^{2}\cdot 3}\frac{\Gamma(1-\frac{3}{p})}{\Gamma(1-\frac{1+j}{p})}$
$\displaystyle<\tau_{n,j}>_{g=3}$ $\displaystyle=$
$\displaystyle\frac{(p-1)(2p+1)(p-5)(8p^{2}-13p-13)}{p^{2}\cdot 7!\cdot
4^{3}\cdot 3^{2}}\frac{\Gamma(1-\frac{5}{p})}{\Gamma(1-\frac{1+j}{p})}$
$\displaystyle<\tau_{n,j}>_{g=4}$ $\displaystyle=$
$\displaystyle\frac{(p-1)(2p+1)(p-7)(72p^{4}-298p^{3}-17p^{2}+562p+281)}{p^{3}\cdot
9!4^{4}\cdot 15}$
$\displaystyle\times\frac{\Gamma(1-\frac{7}{p})}{\Gamma(1-\frac{1+j}{p})}$
$\displaystyle<\tau_{n,j}>_{g=5}$ $\displaystyle=$
$\displaystyle(p-1)(2p+1)(p-3)(p-9)(4p+3)(32p^{4}-162p^{3}+p^{2}$
$\displaystyle+326p+163)\frac{1}{p^{4}11!4^{5}3}\frac{\Gamma(1-\frac{9}{p})}{\Gamma(1-\frac{1+j}{p})}$
$\displaystyle<\tau_{n,j}>_{g=6}$ $\displaystyle=$
$\displaystyle(p-1)(2p+1)(p-11)(530688p^{8}-5830544p^{7}+16589332p^{6}$ (2.8)
$\displaystyle+8955300p^{5}-65056373p^{4}-26944928p^{3}+85178190p^{2}$
$\displaystyle+80708428p+20177107)\frac{1}{p^{5}\cdot 13!7\cdot 5\cdot
4^{6}3^{3}}\frac{\Gamma(1-\frac{11}{p})}{\Gamma(1-\frac{1+m}{p})}$

where $<\tau_{n,j}>_{g=1}=<\tau_{1,0}>_{g=1}$ [7]. In [8], $<\tau_{n,j}>$ up
to $g=9$ is evalated.

When $p=2$ (Kontsevich model), it leads to a simple expression,

$<\tau_{3g-2}>=\frac{1}{g!(24)^{g}}$ (2.9)

There appear interesting vanishing relations for $<\tau_{A_{l}}>$. For
instance,ath $p=3$ case, $<\tau_{A_{l}}>$ are vanishing at $g=2+3m$
($m\in\mathbb{Z}$). For $p=5$, $<\tau_{A_{l}}>$ are vanishing at $g=3+5m$
($m\in\mathbb{Z}$). In general odd integer $p$, $<\tau_{A_{l}}>$ are vanishing
at $g=(p+1)/2+pm$ ($m\in\mathbb{Z}$). Some of these relations can be seen in
the expressions of $<\tau_{A_{l}}>$ up to genus $g=9$ in [8]. We will see
later that $D_{4}$ ($p=6$) type has this periodicity of the vanishing relation
at $g=2+3m$ ($m\in\mathbb{Z}$) for $D_{4}$.

For the case $A_{2}$ ($p=3$), one point function $u(s)$ is given by the Airy
function as [7]

$\displaystyle u(s)$ $\displaystyle=$
$\displaystyle\frac{1}{Ns(3Ns)^{1/3}}A_{i}(\zeta)$ (2.10) $\displaystyle=$
$\displaystyle\frac{1}{Ns(3Ns)^{1/3}}\biggl{[}A_{i}(0)\biggl{(}1+\frac{1}{3!}\zeta^{3}+\frac{1\cdot
4}{6!}\zeta^{6}+\frac{1\cdot 4\cdot 7}{9!}\zeta^{9}+\cdots\biggr{)}$
$\displaystyle+A_{i}^{\prime}(0)\biggl{(}\zeta+\frac{2}{4!}\zeta^{4}+\frac{2\cdot
5}{7!}\zeta^{7}+\frac{2\cdot 5\cdot
8}{10!}\zeta^{10}+\cdots\biggr{)}\biggr{]}$

where $\zeta=-N^{2/3}(4\cdot 3^{1/3})^{-1}s^{8/3}$,
$A_{i}(0)=3^{-2/3}/\Gamma(\frac{2}{3})$ and
$A_{i}^{\prime}(0)=-3^{-1/3}/\Gamma(\frac{2}{3})$. This Airy function leads to
the intersection numbers of

$<\tau_{\frac{8g-5-j}{3},j}>_{g}=\frac{1}{(12)^{g}g!}\frac{\Gamma(\frac{g+1}{3})}{\Gamma(\frac{2-j}{3})}$
(2.11)

which shows the vanishing relations for $g=2,5,8,...$ ($g=2+3m$,
$m\in\mathbb{Z}$), for such case the value of spin $j$ takes 2. The absence of
$g=2,5,8,...$ is due to Stokes phenomena.

For $p=4$, $u(s)$ is written by the Bessel function, [8]

$\displaystyle u(s)$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{8}}e^{\frac{3}{160}s^{5}}\frac{1}{2{\rm
sin}(\frac{\pi}{4})}\biggl{[}I_{-\frac{1}{4}}(\frac{1}{32}s^{5})+I_{\frac{1}{4}}(\frac{1}{32}s^{5})\biggr{]}$
(2.12) $\displaystyle=$
$\displaystyle\frac{1}{8}\sum_{m,n=0}^{\infty}\frac{1}{m!n!\Gamma(n+\frac{5}{4})}(\frac{3}{160})^{m}(\frac{1}{64})^{2n+\frac{1}{4}}s^{5m+10n+\frac{1}{4}}$

We have Riemann-Roch relation of (2.6) for the s-point intersection numbers
$<\tau_{n_{1},j_{1}}\tau_{n_{2},j_{2}}\cdots\tau_{n_{s},j_{s}}>_{g}$. The
factor $(1-\frac{2}{p})$ is a central charge $\hat{c}=\frac{p-2}{p}$. The last
term is also charge for $p$ spin curves. Note this central charge $\hat{c}$ is
also valid for $D_{l}$ singularity, since the weight system of $D_{l}$
singularity $W=y^{l-1}+yx^{2}$, $q_{y}=\frac{1}{l-1}$,
$q_{a}=\frac{l-2}{2(l-1)}$. The central charge $\hat{c}$ is given by
$2-2q_{y}-2q_{x}=1-\frac{1}{l-1}$. Since $p=2(l-1)$, we have
$\hat{c}=1-\frac{2}{p}$ for $D_{l}$ case. Thus the Riemann-Roch relation (2.6)
is applied both for $A_{l}$ and $D_{l}$ singularities.

##  3 One point function for $D_{l}$ type

The singularity theory of $D_{l}$ type is described as a two dimensional
normal singularity by the equation of $x^{l-1}+xy^{2}+z^{2}=0$ ($l\geq 4$).
The weight system is $(a,b,c;h)=(2,l-2,l-1;2(l-1))$, where $h$ is called as
Coxeter . For $D_{l}$ case, we use the spin $p$ value for Coxeter number $h$,
which is related to $l$ as $p=2l-2$. For $D_{4}$, we have spin curve of $p=6$.

The intersection numbers are extended from (2.7) to the one includes the
boundary. The tangent bundle is trivially on the boundary of the moduli space.
We need analogous term for the first Chern class on the boundary and introduce
the correspondent quantity $\sigma$ for the boundary [21].

$<\tau_{n_{1}}\cdots\tau_{n_{s}}\sigma^{m}>=\int_{\bar{\mathcal{M}}}\psi^{n_{1}}\cdots\psi^{n_{s}}$
(3.1)

where $m$ punctures on the boundary are added to (2.7). This is generalization
to open intersection numbers, and related to $D_{l}$ singularity as we will
see. Instead of working of the geometrical moduli space $\bar{\mathcal{M}}$,
we study the equivalent partition function of a matrix model as same as
$A_{l}$ case. The partition function of a matrix model is expressed by the $n$
point correlation function $u(s_{1},s_{2},...,s_{n})$ with a logarithmic
potential [8].

We have the following one point function $\hat{u}(s)$ in the integral form for
$D_{l}$, The generating function $\hat{u}(s)$ of the intersection numbers
$<\tau>$ for $D_{l}$ ($p=2l-2$) is given by

$\hat{u}(s)=\frac{1}{s}\int_{0}^{\infty}due^{-\frac{c}{(p+1)}[(u+\frac{s}{2})^{p+1}-(u-\frac{s}{2})^{p+1}]}\frac{1}{\sqrt{1-\frac{s^{2}}{4u^{2}}}}$
(3.2)

where $c=\frac{N}{p-1}\sum\frac{1}{a_{\alpha}^{p+1}}$ as shown in [7].
$a_{\alpha}$ is eigenvalues of the external source.

The last factor is absent for $A_{l}$ case. It is written as

$\displaystyle\frac{1}{\sqrt{1-\frac{s^{2}}{4u^{2}}}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\biggl{(}\sqrt{\frac{u+\frac{s}{2}}{u-\frac{s}{2}}}+\sqrt{\frac{u-\frac{s}{2}}{u+\frac{s}{2}}}\biggr{)}$
(3.3) $\displaystyle=$ $\displaystyle\frac{1}{2}\biggl{(}e^{\frac{1}{2}{\rm
log}(u-\frac{s}{2})-\frac{1}{2}{\rm log}(u+\frac{1}{2})}+e^{\frac{1}{2}{\rm
log}(u+\frac{s}{2})-\frac{1}{2}{\rm log}(u-\frac{1}{2})}\biggr{)}$

Note that if we change $u\to-u$, above term is invariant. If we write the
coefficient of the logarithm as $k$ instead of $\frac{1}{2}$, two terms are
$k$ and $-k$ coefficient, and it leads to the polynomial of even power of $k$.
This characterizes the $D_{l}$ type as we will discuss later.

The small $s$ expansion of $u(s)$ with $u=t^{\frac{1}{p}}$, becomes

$\displaystyle\hat{u}(s)=\frac{1}{N\pi
s}\frac{1}{(cs)^{1/p}}\int_{0}^{\infty}dtt^{\frac{1}{p}-1}e^{-t}[1-\frac{p(p-1)}{24}s^{2}(cs)^{2/p}t^{1-\frac{2}{p}}$
(3.4) $\displaystyle-\frac{p(p-1)(p-2)(p-3)}{5!\cdot
16}s^{4}(cs)^{4/p}t^{1-\frac{4}{p}}+\frac{p^{2}(p-1)^{2}}{2\cdot(24)^{2}}s^{4}(cs)^{4/p}t^{2-\frac{4}{p}}$
$\displaystyle-\frac{p^{3}(p-1)^{3}}{3!(24)^{3}}s^{6}(cs)^{6/p}t^{3-\frac{6}{p}}+\frac{p^{2}(p-1)^{2}(p-2)(p-3)}{5!\cdot
14\cdot 16}s^{6}(cs)^{6/p}t^{2-\frac{6}{p}}$
$\displaystyle-\frac{p^{3}(p-1)^{3}}{3!(24)^{3}}s^{6}(cs)^{6/p}t^{3-\frac{6}{p}}$
$\displaystyle-\frac{p(p-1)(p-2)(p-3)(p-4)(p-5)}{7!4^{3}}s^{6}(cs)^{6/p}t^{1-\frac{6}{p}}+\cdots]$
$\displaystyle\times$
$\displaystyle[1+\frac{1}{8}s^{2}(cs)^{2/p}t^{-\frac{2}{p}}+\frac{3}{128}s^{4}(cs)^{4/p}t^{-\frac{4}{p}}+\frac{15}{3072}s^{6}(cs)^{6/p}t^{-\frac{6}{p}}+\cdots]$

The last factor is the expansion of (3.3). The integer power of $s$, denoted
as $m$, shows the relation to genus $g$ as $2g-1=m$.

The small $s$ expansion of $\hat{u}(s)$, with the normalization factors
$\frac{1}{p^{g-1}}$ and gamma factor $1/\Gamma(1-\frac{1+j}{p})$, gives the
intersection numbers of one point case,

$\displaystyle<\tau_{1,0}>_{g=1}$ $\displaystyle=$
$\displaystyle\frac{p+2}{24}$ $\displaystyle<\tau_{n,j}>_{g=2}$
$\displaystyle=$
$\displaystyle\frac{(p+2)(2p+1)(p-6)}{5760p}\frac{\Gamma(1-\frac{3}{p})}{\Gamma(1-\frac{1+j}{p})}$
$\displaystyle<\tau_{n,j}>_{g=3}$ $\displaystyle=$
$\displaystyle\frac{(p+2)(2p+1)(8p^{3}-77p^{2}+196p+188)}{2903040p^{2}}\frac{\Gamma(1-\frac{5}{p})}{\Gamma(1-\frac{1+j}{p})}$
$\displaystyle<\tau_{n,j}>_{g=4}$ $\displaystyle=$
$\displaystyle\frac{(p+2)(2p+1)(4p^{2}-27p-22)(18p^{3}-133p^{2}+308p+332)}{1393459200p^{3}}$
$\displaystyle\times\frac{\Gamma(1-\frac{7}{p})}{\Gamma(1-\frac{1+j}{p})}$
$\displaystyle<\tau_{n,j}>_{g=5}$ $\displaystyle=$
$\displaystyle\frac{(p+2)(2p+1)(4p+3)(p-6)}{122624409600p^{4}}(32p^{5}-450p^{4}+1741p^{3}$
$\displaystyle-1642p^{2}-6788p-3288)\frac{\Gamma(1-\frac{9}{p})}{\Gamma(1-\frac{1+j}{p})}$
$\displaystyle<\tau_{n,j}>_{g=6}$ $\displaystyle=$
$\displaystyle\frac{(p+2)(2p+1)}{14!6!2^{7}\cdot
3p^{5}}(530688p^{9}-13260176p^{8}+115768820p^{7}$
$\displaystyle-412604468p^{6}+276695515p^{5}+1715374838p^{4}-2129848328p^{3}$
$\displaystyle-6843457424p^{2}-4961166736p-1156803104)\frac{\Gamma(1-\frac{11}{p})}{\Gamma(1-\frac{1+j}{p})}$
$\displaystyle<\tau_{n,j}>_{g=7}$ $\displaystyle=$
$\displaystyle\frac{(p+2)(2p+1)}{16!2^{10}3^{3}p^{6}}(2^{11}3^{3}5p^{11}-8586624p^{10}+96830032p^{9}$
$\displaystyle-488127956p^{8}+889089716p^{7}+1243914177p^{6}-5937016268p^{5}$
$\displaystyle-1741314004p^{4}+21058826784p^{3}+29690849392p^{2}$
$\displaystyle+15502250816p+2905782080)\frac{\Gamma(1-\frac{13}{p})}{\Gamma(1-\frac{1+j}{p})}$
$\displaystyle<\tau_{n,j}>_{g=8}$ $\displaystyle=$
$\displaystyle\frac{(p+2)(2p+1)(p-6)(3+4p)(5+6p)}{18!2^{15}5p^{7}}(462976p^{10}$
$\displaystyle-15295120p^{9}+179456596p^{8}-953948892p^{7}+2115904691p^{6}$
$\displaystyle+586034636p^{5}-9624755932p^{4}+5128005728p^{3}$
$\displaystyle+31236673872p^{2}+25467952320p+6355800000)\frac{\Gamma(1-\frac{15}{p})}{\Gamma(1-\frac{1+j}{p})}$
$\displaystyle<\tau_{n,j}>_{g=9}$ $\displaystyle=$
$\displaystyle(2+p)(1+2p)(16502445084498176+122234441454621184p$ (3.5)
$\displaystyle+362714955007461120p^{2}+525312614038452992p^{3}$
$\displaystyle+326733211545349216p^{4}-23742664329025152p^{5}$
$\displaystyle-100306262063206224p^{6}+8323562725999632p^{7}$
$\displaystyle+24146966038644009p^{8}-6773140965548282p^{9}$
$\displaystyle-2307841939577188p^{10}+1677474927489096p^{11}$
$\displaystyle-402367617574016p^{12}+48227812538240p^{13}$
$\displaystyle-2854331624448p^{14}+64684523520p^{15})$
$\displaystyle\times\frac{1}{271211974879377138647040000p^{8}}\frac{\Gamma(1-\frac{17}{p})}{\Gamma(1-\frac{1+j}{p})}$
$\displaystyle<\tau_{n,j}>_{g=10}$ $\displaystyle=$
$\displaystyle(2+p)(1+2p)(-24830402547748278784-209477583844413564160p$
$\displaystyle-730557543123682249216p^{2}-1318998112825968482560p^{3}$
$\displaystyle-1208129128709188488640p^{4}-333863005164255047776p^{5}$
$\displaystyle+264649746344510118240p^{6}+126068631741386072496p^{7}$
$\displaystyle-78507951575573824290p^{8}-22445884710909365885p^{9}$
$\displaystyle+21279933896679839896p^{10}-1348924974460414280p^{11}$
$\displaystyle-2661096526358202656p^{12}+1087695793798917040p^{13}$
$\displaystyle-200715490239800960p^{14}+19791837888816384p^{15}$
$\displaystyle-993551356753920p^{16}+19465064349696p^{17})$
$\displaystyle\times\frac{1}{3579998068407778230140928000000p^{9}}\frac{\Gamma(1-\frac{19}{p})}{\Gamma(1-\frac{1+j}{p})}$
$\displaystyle<\tau_{n,j}>_{g=11}$ $\displaystyle=$
$\displaystyle((-6+p)(2+p)(1+2p)(3+4p)(7+8p)$ (3.6)
$\displaystyle\times(-467169353783096832-3341630372025717504p$
$\displaystyle-9468101148218970624p^{2}-12751291781631800064p^{3}$
$\displaystyle-6632138067837797184p^{4}+1788770829880213088p^{5}$
$\displaystyle+2276784479751975200p^{6}-728458396311840240p^{7}$
$\displaystyle-535146263253641670p^{8}+300779490648921045p^{9}$
$\displaystyle+17161921882855788p^{10}-53942771498453544p^{11}$
$\displaystyle+19196939796342336p^{12}-3360970337005104p^{13}$
$\displaystyle+319429042188736p^{14}-15406349322752p^{15}$
$\displaystyle+286370611200p^{16})$
$\displaystyle\times\frac{1}{73191071620781243816214528000000p^{10}}\frac{\Gamma(1-\frac{21}{p})}{\Gamma(1-\frac{1+j}{p})}$

where $n$ and $j$ are determined by the Riemann-Roch selection rule of (2.6).

When $p=6$, which corresponds to $D_{4}$ (Coxeter number $p$ is equal to
$2l-2$ for $D_{l}$), the intersection numbers $<\tau>_{g}$ are vanishing for
$g=2+3k$, $k\in Z$. This vanishing intersection numbers can be seen explicitly
as a factor $(p-6)$ in above expression. It is interesting to observe that the
factor $(p-6)$ accompany the factor $(3+4p)$ in the $g=5,8,11$ for both
$A_{p-1}$ and $D_{l}$ cases. [8]. The existing factors $(3+4p)$ are expected
to appear in higher genus both for $A_{p-1}$ and $D_{l}$, when
$g=5+3m,(m=0,1,2,...)$. The vanishing condition of $g=2+3k,k\in Z$ for $p=6$
will be discussed at later section in more details.

We find for $p=2$ that the simple expression for one point function ($j=0$ for
$p=2$) is obtained from (3),

$<\tau_{3g-2}>=\frac{1}{g!6^{g}}$ (3.7)

Above formula for $p=2$ will be proved later in (4.9).

There is a relation to the open intersection numbers. The logarithmic
potential with a coefficient $k$ has been studied so called as Kontsevich-
Penner model or Airy matrix model with a logarithmic potential [15, 16, 18].
Putting $p=2$ and $k=\frac{1}{2}$ in the expressions of (1.1) [15, 18], the
results agree with (3). For instance, $<\tau_{1,0}>_{g=1}=\frac{p+2}{24}$
becomes $\frac{1}{6}$ for $p=2$, which agrees with
$<\tau_{1}>=\frac{1+12k^{2}}{24}=\frac{1}{6}$ with $k=\frac{1}{2}$ in [15,
18]. $<\tau_{n,j}>_{g=2}=\frac{1}{72}$ for $p=2$ agrees with
$<\tau_{4}>=\frac{1+56k^{2}+16k^{4}}{1152}=\frac{1}{72}$ for $k=\frac{1}{2}$.
For non orientable surface, taking account of projective plane
$(g=\frac{1}{2})$, Klein surface $(g=1)$, and crosscap $(g=\frac{3}{2})$,
genus $g$ is considered as a fractional (double genus) [13]. Thus for the
comparison with the expressions of $<\tau_{\frac{3g^{\prime}-1}{2}}>$ in [16,
18], we need a relation $g^{\prime}=2g-1$ i.e. the one point function of (3)
corresponds to $<\tau_{\frac{3g^{\prime}-1}{2}}>$ in [16, 18], and the result
completely agrees for the case of integer $\frac{3g^{\prime}-1}{2}$. The case
$p=2$ in $D_{l}$ type means $D_{2}$, so $D_{2}$ case has a meaning of the
Kontsevich-Penner model as an open intersection theory, although $l\geq 4$
case is discussed in the singularity theory. The terms of
$<\tau_{\frac{3g^{\prime}-1}{2}}>$, which appear with half-integer
$\frac{3g^{\prime}-1}{2}$, have expressions of odd $k$ polynomial, and they do
not appear in $D_{l}$ type in (3).

For the open intersection numbers, there appear intersection numbers such that
$<\tau_{\frac{5}{2}}>=\frac{1}{12}(k+k^{3})$. These are odd power of $k$. As
we observed in (4.9), we have sum of $k$ and $-k$ for the logarithmic
correction, therefore adding these two contributions, they are cancelled for
such $<\tau_{\frac{5}{2}}>$ intersection numbers. This is a reason that we
have no half integer indexed intersection numbers in $D_{l}$ type.

As noted in [15], such odd k intersection numbers with half integer indexed
$\tau$ are evaluated by the contour integral, and therefore can be regarded as
Ramond sectors.

Thus the intersection numbers belong to the orientable surfaces, which do not
include the case of half integer $\frac{3g^{\prime}-1}{2}$. For general $p$,
if we put $k=\frac{1}{2}$ in [15], we find also the agreement with (3).

For $p=6$, which corresponds to $D_{4}$, the intersection numbers become
$\tau_{D_{4}}=1,\frac{1}{2},0,\frac{1}{40824},\frac{13}{122472},0$ for
$g=0,1,2,3,4,5$, respectively. These values are obtained from (3) as the
coefficients of Gamma function factors. They agree with [14]. It is remarkable
that $\tau_{D_{4}}$ is vanishing at $g=2,5$, since a factor $(p-6)$ appears in
(3). This suggests $\tau_{D_{4}}$ ($p=6$ case) is vanishing periodically at
$g=2+3k$, $k\in Z$.

This periodic vanishing reallation of the intersection numbers is due to the
selection rule for spin $p$. We have from Riemann-Roch formula the relation
between the spin $p$ and the genus $g$,

$(p+1)(2g-1)=pn+j+1$ (3.8)

where $j=0,1,2,...,p-2$. For the singularity theory, $D_{l}$ has a relation

$2g-1=pn^{\prime}+m_{\alpha}$ (3.9)

where $m_{\alpha}$ is defined by the characteristic polynomial $\chi(t)$ for
the weight system $(a,b,c;h)$ as

$\chi(t)=\frac{1}{t^{h}}\frac{(t^{h}-t^{a})(t^{h}-t^{b})(t^{h}-t^{c})}{(t^{a}-1)(t^{b}-1)(t^{c}-1)}=\sum_{\alpha=1}^{l}t^{m_{\alpha}}$
(3.10)

with $l$ is Milnor number. $j$ is $m_{\alpha}-1$, which is the spin components
of spin $p$.

For $D_{4}$, the weight system becomes $(2,2,3;6)$, and $m_{\alpha}=1,3,5$
($m_{2}=m_{3}=3$, double). The relation of (3.8) is written by
$n^{\prime}=n+1$, $j=m_{\alpha}-1$ in (3.9). For $p=6$ in $D_{4}$, $g=1$
corresponds to $m_{\alpha}=1$ (j=0,n=1). $g=2$ corresponds to $m_{\alpha}=3$
($j=2,n=3$). $g=5$, $g=8$ correspond to $m_{\alpha}=3$. Thus for $g=2+3k$
($k\in\mathbb{Z}$), the exponent becomes $m_{\alpha}=3$, which is doubled.
This correspondence is related to the vanishing relation of the intersection
numbers of $D_{4}$ for $g=2+3k$.

From (3 and (3)), It is easily recognized ; (i) large $p$ behavior is same as
A-type (2), (ii) the intersection numbers of $p=-1/2$ are vanishing for $g>1$
for all order of $g$ (same as A-type), (iii) the intersection numbers of
$p=-2$ for D-type are vanishing for all genus $g>0$. These remarkable
properties will be proved in the following sections.

Since we have derived exact one point function of $D_{l}$ type, it may be
interesting to consider the negative values of $p$ and half-integer $p$ as
discussed in $A_{l}$ type [8, 9, 10]. In the next section, we examine the
negative integer values of $p$ and half-integer $p$. Interesting applications
of such non-positive integer cases were discussed in [10] for $A_{l}$ type.

##  4 One point function of $D_{l}$ type for the non-positive integer cases

We have discussed the non-positive integer spin $p$ for one marked point of
the $A_{p-1}$ type in the previous articles [9, 10]. Here we extend these
results of $A$ type to $D$ type for one marked point.

For $D_{l}$ singularity, the relation of $p=2l-2$ gives the constraint that
spin $p$ should be even integer. Since the intersection numbers are expressed
by the polynomial of $p$ as (3), the analytic continuation of $p$ to the
general values including the non-integer case is possible. Our formulation of
a matrix model allows the non-positive integer value of $p$.

In this section, we will find the remarkable coincidence of the intersection
number of $D_{l}$ type with that of $A_{p-1}$ model with a logarithmic term,
so called generalized Kontsevich-Penner model. $\bullet$ Change of variable

As discussed in [9, 30], although there is no Ramond contribution for
$A_{p-1}$ case in the positive integer $p$, there appear Ramond punctures in
$D_{l}$ type ($p=2l-2$) [10, 30]. The Ramond contribution may be obtained by
the residue of $y=0$ in the following integral representation by a change of
variable from $u$ to $y$, following the discussion of [9, 10] as

$u=\frac{i}{2}(y^{2}-\frac{1}{y^{2}})$ (4.1)

The factor of $D_{l}$ logarithmic potential becomes after the change of
variable of (4.1),

$\frac{u}{\sqrt{u^{2}-1}}=\frac{y-\frac{1}{y^{3}}}{y+\frac{1}{y^{3}}}$ (4.2)

The measure for $A_{p-1}$ type is now changed simply to $i(y+1/y^{3})dy$ due
to (4.2), which reads that the measure is $y\pm\frac{1}{y^{3}}$, $(+)$ sign
for A-type and $(-)$ sign for D-type. $\bullet$ Equivalence to generalized
Kontsevich-Penner model of open intersection numbers Extension of Airy with
logarithmic potential (Kontsevich-Penner model) to general $p$ spin case has
been investigated with the logarithmic potential with coefficient $k$ in [16,
15]. We will show that the intersection numbers of generalized Kontsevih-
Penner model with $k=\pm\frac{1}{2}$ are identical to that of $D_{l}$
($p=2l-2$) intersection numbers.

For genus $g=1$, the intersection number is

$<\tau>_{g=1}=\frac{p-1+12k^{2}}{24}$ (4.3)

With $k=\pm\frac{1}{2}$, it becomes

$<\tau>=\frac{p+2}{24}$ (4.4)

which agrees with $\tau_{D_{l}}$in (3) . For genus $g=2$, the generalized
Kontsevich-Penner model gives (Eq.(5.28) of [16]),

$<\tau>_{g=2}=\frac{1}{p(12)^{2}}[\frac{(p-1)(p-3)(2p+1)}{40}-(3p+1)k^{2}-2k^{4}]$
(4.5)

By putting $k=\pm\frac{1}{2}$, we find exactly
$<\tau>_{g=2}=\frac{1}{5760p}(p+2)(2p+1)(p-6)$ for D-type in (3).

Thus we find that the one point intersection number of $D_{l}$ type ($p=2l-2$)
is same as the intersection numbers of generalized Kontsevich-Penner matrix
model with $k=\frac{1}{2}$. One can check more higher $g$ case are consistent
with this identification. When $p=2$, the generalized open intersection
numbers are evaluated in higher orders [18], in which a parameter $N$ is same
as our $k$. The odd power terms of $k$ are cancelled by adding $\pm k$
contributions, and the results agree with $D_{l}$ type intersection numbers
with $k=\frac{1}{2}$. Note that the open intersection number is described by
the logarithmic potential with $k=1$ [15, 16, 7, 17, 22, 23].

$\bullet$ $p=1$ case We now discuss and prove the remarkable features for
specific values of $p$. The first example is $p=1$ case of $A_{l}$. The
intersection number $<\tau>$ becomes vanishing in all order of $g$ due to a
factor$(p-1)$ in (2).

$\displaystyle u(s)$ $\displaystyle=$ $\displaystyle\frac{1}{s}\int
due^{-\frac{c}{2}((u+\frac{s}{2})^{2}-(u-\frac{s}{2})^{2})}$ (4.6)
$\displaystyle=$
$\displaystyle\frac{i}{s}\oint\frac{dy}{2i\pi}(y+\frac{1}{y^{3}})e^{-\frac{ics}{2}(y^{2}-\frac{1}{y^{2}})}$
$\displaystyle=$
$\displaystyle\frac{i}{s}\oint\frac{dy}{2i\pi}(\frac{1}{-ics})\frac{d}{dy}e^{-\frac{ics}{2}(y^{2}-\frac{1}{y^{2}})}$

which becomes vanishing due to the total derivative. This is consistent with a
factor $(p-1)$ in all order in (2).

For $D_{l}$ case, the measure is a factor $(y-\frac{1}{y^{3}})$ instead of
$(y+\frac{1}{y^{3}})$,

$\displaystyle u(s)$ $\displaystyle=$
$\displaystyle\frac{i}{s}\oint\frac{dy}{2i\pi}[2y-(y+\frac{1}{y^{3}})]e^{-\frac{ics}{2}(y^{2}-\frac{1}{y^{2}})}$
(4.7) $\displaystyle=$
$\displaystyle-\sum_{m=0}^{\infty}\frac{1}{2^{2m}m!(m+1)!}s^{2m}c^{2m+1}$

This is consistent with the expression of (3) for $p=1$. The genus $g$ is
equal to $m$.

$\bullet$ $p=2$ case

For $p=2$, using the representation of $y$, one point function $u(s)$ is
expressed as

$u(s)=\frac{i}{2}e^{-\frac{5c}{24}s^{3}}\oint\frac{dy}{2i\pi}(y\pm\frac{1}{y^{3}})e^{\frac{cs^{3}}{16}(y^{4}+\frac{1}{y^{4}})}$
(4.8)

where $\pm$ means $A$ type and $D$ type, respectively. With the change of
variable $y=t^{\frac{1}{4}}$, this is written by the modified Bessel function
$I_{\nu}(z)$,

$\displaystyle u(s)$ $\displaystyle=$
$\displaystyle\frac{i}{8}e^{-\frac{5c}{24}s^{3}}\oint\frac{dt}{2i\pi}(t^{-\frac{1}{2}}\pm
t^{-\frac{3}{2}})e^{\frac{cs^{3}}{16}(t+\frac{1}{t})}$ (4.9) $\displaystyle=$
$\displaystyle\frac{i}{8}e^{-\frac{5c}{24}s^{3}}[I_{-\frac{1}{2}}(\frac{cs^{3}}{8})\pm
I_{\frac{1}{2}}(\frac{cs^{3}}{8})]$ $\displaystyle=$
$\displaystyle\frac{i}{2}e^{-\frac{5c}{24}s^{3}}\sqrt{\frac{1}{\pi(cs^{3})}}e^{\pm\frac{cs^{3}}{8}}$

By taking $c=-\frac{1}{2}$, we find the close form of for $A$ type (2.9);
$<\tau_{3g-2}>=\frac{1}{g!(24)^{g}}$, and the result of D-type (3.7);
$<\tau_{3g-2}>=\frac{1}{g!6^{g}}$. This gives a proof of (3.7).

$\bullet$ $p=-1$ When $p=-1$, one point function $u(s)$ provides Euler
characteristic $\chi(\bar{M}_{g,1})$ for $A_{l}$ type singularity [7] as

$\displaystyle u(s)$ $\displaystyle=$
$\displaystyle\frac{1}{N}\int\frac{du}{2i\pi}\biggl{(}\frac{u-\frac{1}{2}}{u+\frac{1}{2}}\biggr{)}^{N}$
(4.10) $\displaystyle=$
$\displaystyle-\frac{1}{N}\int_{0}^{\infty}\frac{dz}{2\pi}\frac{e^{-z}}{(1-e^{-z})^{2}}e^{-Nz}$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}\frac{dz}{2\pi}(\sum
B_{2n}\frac{z^{n-1}}{n!})e^{-Nz}$

where a change of variable $(u-\frac{1}{2})/(u+\frac{1}{2})=e^{-z}$ is used
and a factor $1/N^{2}$ represent the genus $g$ expansion. This $u(s)$ gives
Euler characteristic
$\chi(\bar{M}_{g,1})=\zeta(1-2g)=(-1)^{g}\frac{1}{2g}B_{2g}$, where $B_{n}$ is
a Bernoulli number
($B_{2}=\frac{1}{6},B_{4}=-\frac{1}{30},B_{6}=\frac{1}{42},...$).

For $D_{l}$ type, if we put $p=-1$ in (3) with $j=0$ (gamma function of the
denominator becomes one for $p=-1,j=0$, and numerator gamma function gives
$(2g-1)!$), , we find that $<\tau>_{g}$ as
$\frac{1}{24},-\frac{7}{960},\frac{31}{8064},...$ for $g=1,2,3,...$. These
numbers are equal to Euler characteristics $\chi$

$\chi=(1-2^{1-2g})\frac{B_{2g}}{2g}.$ (4.11)

In the limit $p\to-1$, (3.2) becomes

$\displaystyle\hat{u}(s)$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
due^{-clog(\frac{u+1}{u-1})}\frac{u}{\sqrt{u^{2}-1}}$ (4.12) $\displaystyle=$
$\displaystyle\frac{1}{2}\int du(\frac{u-1}{u+1})^{c}\frac{u}{\sqrt{u^{2}-1}}$

By the change of variable $(u-1)/(u+1)=e^{-z}$,
$du=-2e^{-z}/(1-e^{-z})^{2}dz=(-2)(e^{z/2}-e^{-z/2})^{-2}dz$,
$u/\sqrt{u^{2}-1}=\frac{1}{2}(e^{z/2}+e^{-z/2})$, it becomes after partial
integration,

$\displaystyle\hat{u}(s)=\int dz\frac{1}{e^{z/2}-e^{-z/2}}e^{-Nz}$ (4.13)

where $c$ is replaced by $N$ to make clear of genus dependence, as
$\frac{1}{N^{2g}}$ series. Above integral reduces to (4.11), with $1/24$,
7/960,… for $g=1,2,...$. This $\chi$ is same as virtual Euler characteristics,
obtained for $o(2N)$ matrix model as a non-orientable surface [8]. For $p=-1$,
there is no $\psi$ class, and only Euler class (Witten class) exists. Since
$D_{l}$ type is related to $o(2N)$ Lie algebra, it is reasonable to obtain the
result of (4.11), which is same as a virtual Euler characteristics of real
algebraic curves [26]. Indeed, we have obtained this virtual Euler
characteristic of real algebraic curves for $so(2n+1),so(2n),sp(n)$ cases as
antisymmetric matrix models [13, 8].

$\bullet$ $p$ = $-\frac{1}{2}$ It is remarkable that the intersection number
has a factor of $(2p+1)$ for $D_{l}$ type, which is same as $A_{l}$ case (2).
The reason of this factor $(2p+1)$ exists for all genus ($g>1$) is explained
by the change of variable of (4.1). The exponential factor in (3.2) becomes
simply as $e^{-c\frac{s^{1/2}}{y}}$ for $p=-1/2$. Therefore we have for
$D_{l}$ case with $p=-\frac{1}{2}$ from (4.2),

$\hat{u}(s)=\oint_{c}\frac{dy}{2i\pi}(y-\frac{1}{y^{3}})e^{-c\frac{s^{1/2}}{y}}$
(4.14)

where the integral is evaluated by a contour around $y=0$. This gives non-
vanishing term of order $s$, which means $g=1$, but the remaining terms of all
higher genus ($g>1$) should vanish due to Cauchy theorem.

$\bullet$ $p=-2$ In $A_{p-1}$ matrix model, the case $p=-2$ becomes equivalent
to the unitary matrix model (BGW) model [35, 36], and the intersection numbers
agree with BGW matrix model as shown in [40].

For $D_{l}$ ($p=2l-2$) case, there is a factor $(p+2)$ for all genus case as
in (3). This may be explained as follows [40]. We have in the $p=-2$ case
$l=(p+2)/2=0$ for $D_{l}$, namely $D_{0}$ case,

$\displaystyle\hat{u}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{s}\oint\frac{du}{2i\pi}e^{-\frac{1}{u+s}+\frac{1}{u}}\biggl{(}\frac{u+s}{u}\biggr{)}^{k}$
(4.15) $\displaystyle=$ $\displaystyle-\frac{1}{4\pi\sqrt{s}}\int
dx\frac{1}{x^{2}}e^{-\frac{4x^{2}}{1+sx^{2}}}(\frac{1-ix\sqrt{s}}{1+ix\sqrt{s}})^{k}$
$\displaystyle=$ $\displaystyle-\frac{1}{2\sqrt{\pi
s}}(-s\frac{1}{8}(4k^{2}-1)+s^{2}\frac{1}{3!2^{7}}(4k^{2}-1)(4k^{2}-9)$
$\displaystyle-s^{3}\frac{1}{5!2^{9}}(4k^{2}-1)(4k^{2}-9)(4k^{2}-25)+\cdots)$

The second line is derived by the change of variable
$u=\frac{1}{2}(-1+\frac{i}{x\sqrt{s}})$. Since the coefficient of the
logarithmic factor of $D_{l}$ is $k=\frac{1}{2}$, we find the vanishing
results of $\hat{u}(s)$ from above equation. This gives a proof for a factor
$p+2$ for the intersection numbers in all orders of D-type (3).

BGW model is considered for the unitary matrix, and its extension to O(N)
group was studied in [38]. There appears a phase transition between weak
coupling (small $s$) and a strong coupling regions (large $s$).

$\bullet$ Large $s$ expansion for $p=-2$ The large $s$ expansion was
investigated in [40] based on (4.15) for $p=-2$ of $A_{l}$ type. Due to two
terms of (1.1), the extension of the large $s$ expansion for $p=-2$ case of
$D_{l}$ type is straightforward. We obtain by the shift $u\to(u-1)/2$
following [40],

$\displaystyle\hat{u}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\oint\frac{du}{2i\pi}e^{\frac{4}{2(u^{2}-1)}}[(\frac{u+1}{u-1})^{k}+(\frac{u+1}{u-1})^{-k}]$
(4.16) $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{1}^{\infty}\frac{4^{m}}{m!s^{m}}\oint\frac{du}{2i\pi}\frac{1}{(u^{2}-1)^{m}}[(\frac{u+1}{u-1})^{k}+(\frac{u+1}{u-1})^{-k}]$

with $k=\frac{1}{2}$. Noting that

$\displaystyle\oint\frac{du}{2i\pi}\frac{1}{(u^{2}-1)^{m}}(\frac{u+1}{u-1})^{k}=-\frac{2}{\pi}{\rm
sin}\pi k\int_{1}^{\infty}dx\frac{(x+1)^{k-m}}{(x-1)^{k+m}}$
$\displaystyle=-\frac{2}{\pi}({\rm sin}\pi
k)2^{1-2m}(2m-2)!\frac{\Gamma(-k-m+1)}{\Gamma(-k+m)}$ (4.17)

we obtain $\hat{u}(s)$ in the large $s$ expansion. In the unitary matrix
model, we put $k=-N$ and the result agree with the strong coupling expansion
(character expansion) [40].

$\bullet$ Half-integer $p=\frac{1}{2}$ and $\frac{3}{2}$ For $D_{l}$ type,
with the change of variables $u\to y$ in (4.1), one point function
$\hat{u}(s)$ is

$\hat{u}(s)=\frac{1}{s}\oint\frac{dy}{2i\pi}(y-\frac{1}{y^{3}})g(y)$ (4.18)

Note that the difference between $A_{l}$ and $D_{l}$ is a factor of
$(y\pm\frac{1}{y^{3}})$, where $(+)$ is for $A_{l}$ and $(-)$ is for $D_{l}$.
The factor $g(y)$ is same as $A_{l}$ case. We have for $A_{l}$,

$u(s)=\frac{1}{s}\int\frac{du}{2i\pi}e^{-\frac{1}{(p+1)}[(u+\frac{s}{2})^{p+1}-(u-\frac{s}{2})^{p+1}]}$
(4.19)

By the change of variable of $u=\frac{i}{2}(y^{2}-y^{-2})$, it becomes after
the replace $u\to su/2$, we have for $A_{l}$ case $(+)$ and $D_{l}$ case $(-)$

$u(s)=\frac{i}{2}\int\frac{dy}{2i\pi}(y\pm\frac{1}{y^{3}})e^{-\frac{s^{p+1}}{(p+1)}(\frac{i}{4})^{p+1}\frac{1}{y^{2(p+1)}}[(y^{2}-i)^{2p+2}-(y^{2}+i)^{2p+2}]}$
(4.20)

where the exponential factor $g(y)$ is expressed as

$g(y)=e^{c^{\prime}s^{3/2}(3y-\frac{1}{y^{3}})}\hskip
5.69054pt(p=\frac{1}{2},c^{\prime}=-\frac{1}{6}i^{1/2})$ (4.21)

The half-integer $p$ case is related to Ramond sector [9, 10]. We assume
Riemann-Roch relation for $p=\frac{1}{2}$ is valid as

$3g-3+1=n+(g-1)(1-\frac{2}{p})+\frac{j}{p}$ (4.22)

with $p=\frac{1}{2}$ and $j=-1$. $j=-1$ means Ramond puncture [10]. This
selection rule indeed valid for the small $s$ expansion of $u(s)$. We have
from (4.22)

$n=6g-3$ (4.23)

which means that $u(s)$ is a power series of $s^{6g-3}$ and the intersection
number is given by the coefficient, $<\tau_{n,j}>_{g}=<\tau_{6g-3,-1}>_{g}$.

The one point function $u(s)$ for $p=\frac{1}{2}$ for $A_{l}$ is evaluated
with the integral by part,

$\displaystyle u(s)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\oint\frac{dy}{2i\pi}(y+\frac{1}{y^{3}})e^{c^{\prime}s^{3/2}(3y-\frac{1}{y^{3}})}=\frac{i}{6c^{\prime}s^{3/2}}\oint\frac{dy}{2i\pi}y\frac{d}{dy}e^{c^{\prime}s^{3/2}(3y-\frac{1}{y^{3}})}$
(4.24) $\displaystyle=$
$\displaystyle-\frac{i}{6c^{\prime}s^{3/2}}\oint\frac{dy}{2i\pi}e^{c^{\prime}s^{3/2}(3y-\frac{1}{y^{3}})}$
$\displaystyle=$ $\displaystyle
2\sum_{g=1}^{\infty}\frac{(-1)^{g}}{g!(3g-1)!}(\frac{1}{48})^{g}s^{6g-3}$

with $c^{\prime}=-i^{1/2}/6$. The summation is over genus $g$ due to (4.23).

This generating function of the intersection numbers shows the precise
agreement with (2). Note that $\Gamma(1-\frac{2g-1}{p})$ in (2) is changed to
$\Gamma(-1)$ by multiplication factor, and this $\Gamma(-1)$ is interpreted as
a normalization factor for the case of $p=\frac{1}{2}$. Then we find the
precise agreement for $p=\frac{1}{2}$ case between (4.24) and (2). The case of
$p=\frac{1}{2}$ is interpreted as a manifestation of Ramond puncture since
spin component is $j=-1$ (The denominator $\Gamma(1-\frac{1+j}{p})$ in (2)
becomes one). This fascinating result of $p=\frac{1}{2}$ will be further
discussed in the next section related to the denominator of Bernoulli numbers.
The intersection numbers are rational numbers and the denominator is common
denominator of Bernoulli numbers. In $p=\frac{1}{2}$ case, there appears
cancellation of this numbers of (5.10).

For $D_{l}$ type ($p=1/2,l=5/4$), it becomes from (4.2)

$\displaystyle\hat{u}(s)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\oint\frac{dy}{2i\pi}(y-\frac{1}{y^{3}})e^{cs^{3/2}(3y-\frac{1}{y^{3}})}$
(4.25) $\displaystyle=$
$\displaystyle\frac{i}{2s}\oint\frac{dy}{2i\pi}(y-\frac{s^{2}}{y^{3}})e^{cs(3y-\frac{s^{2}}{y^{3}})}$

This integral is written as the derivative of the exponent by $s$. Using the
same integral as (4.24), we obtain

$\displaystyle\hat{u}(s)$ $\displaystyle=$
$\displaystyle\frac{i}{6cs}\frac{d}{ds}\biggl{(}\oint\frac{dy}{2i\pi}e^{3csy-\frac{cs^{3}}{y^{3}}}\biggr{)}$
(4.26) $\displaystyle=$ $\displaystyle
2\sum_{g=1}^{\infty}\frac{(-1)^{g}(6g-1)}{g!(3g-1)!}(\frac{1}{48})^{g}s^{6g-3}$

where $c=-\frac{1}{6}i^{\frac{1}{2}}$. This result is consistent for
$p=\frac{1}{2}$ in (3) as $<\tau>_{g=1}=\frac{p+2}{24}=\frac{5}{48}$. The
expression of $p=\frac{1}{2}$ for $D_{l}$ ($l=\frac{1}{2}(p+2)=\frac{5}{2}$)
is obtained solely from the residue calculation, which means that this case is
Ramond puncture with $j=-1$ [10].

For $p=\frac{3}{2}$ of $A_{l}$ type, it corresponds to $\beta\gamma$ system as
discussed in [9]. There are two different punctures belongs to Neveu-Schwarz
and Ramond sectors. From (4.20), $D_{l}$ case is expressed as

$g(y)=e^{cs^{3/2}(5y^{3}-\frac{10}{y}+\frac{1}{y^{5}})}\hskip
5.69054pt(p=\frac{3}{2},c=\frac{4i}{5}(\frac{i}{4})^{5/2})$ (4.27)
$u(s)=\oint\frac{dy}{2i\pi}(y-\frac{1}{y^{3}})e^{cs^{\frac{3}{2}}[5y^{3}-\frac{10}{y}+\frac{1}{y^{5}}]}$
(4.28)

The small $s$ expansion gives $<\tau_{n,j}>_{g}s^{n+\frac{2}{3}(j+1)}$, where
$j=-1,-\frac{1}{2},0$. The spin $j=-1$ corresponds to Ramond sector, and
Neveu-Schwarz sector is $j=-\frac{1}{2},0$. The Ramond sector (R) is evaluated
by the contour integral for $A_{l}$ of $p=\frac{3}{2}$ in [9],

$u_{R}(s)=\frac{5}{2^{5}}c^{2}s^{5}-\frac{7}{3}\cdot\frac{5^{4}}{2^{16}}c^{6}s^{15}+\frac{79\cdot
11}{10!}\cdot\frac{3^{2}5^{7}}{2^{24}}c^{10}s^{25}+\cdots$ (4.29)

where $s^{(2g-1)(1+\frac{1}{p})}=s^{n+\frac{1}{p}(j+1)}$. If we take
$p=\frac{3}{2}$ and $j=-1$, then we have $s^{\frac{10}{3}g-\frac{5}{3}}$. The
first term of above equation is for $g=2$, and the second term is for $g=5$.
For $D_{l}$, $p=\frac{3}{2}$, the contribution of Ramond sector becomes

$u_{R}(s)=\frac{105}{2^{5}}c^{2}s^{5}-\frac{19\cdot 17\cdot
5^{4}}{2^{19}}c^{6}s^{15}+\frac{29\cdot 1663\cdot
5^{5}}{3^{2}2^{32}}c^{10}s^{25}+\cdots$ (4.30)

##  5 Large $p$, large $g$ limits and integrality

In a recent paper [14], the asymptotic behaviors of the intersection numbers
for $A_{p-1}$, $D_{l}$ and $E_{6}$ types for the large $g$ are discussed based
on the ordinary differential equations (ODE). In this section, we consider the
large $g$ limit of $A_{p-1}$ and $D_{l}$ types based upon the integral
representation of the intersection numbers $u(s)$, which may be simpler than
the analysis of ODE.

The exponential parts of $u(s)$ of $A_{p-1}$ and $D_{l}$ are same. We write
the exponent as a function $f(u)$, which becomes

$f(u)=c((u+\frac{s}{2})^{p+1}-(u-\frac{s}{2})^{p+1})$ (5.1)

This function $f$ is a polynomial of $u$, hence this is an algebraic relation
[14]. The one point function $u(s)$ is a series of
$s^{(1+\frac{1}{p})(2g-1)}$. Thus, large $g$ limit is equivalent to the large
$s$ limit, and this limit is obtained by a saddle point method for the
exponent $f(u)$. By the scaling $u\to\frac{s}{2}u$, $f(u)$ becomes

$f(\frac{su}{2})=c(\frac{s}{2})^{p+1}((u+1)^{p+1}-(u-1)^{p+1})$ (5.2)

We denote this exponent as $g(u)$. By the saddle point method, the first
derivative of $g$ is vanishing,

$\frac{dg}{du}=c(p+1)(\frac{s}{2})^{p+1}((u+1)^{p}-(u-1)^{p})=0$ (5.3)

which reads to

$(\frac{u+1}{u-1})^{p}=1$ (5.4)

The solution of this equation is $\frac{u+1}{u-1}=e^{2\pi i/p}$, i.e.
$u=(e^{\pi i/p}+e^{-\pi i/p})(e^{\pi i/p}-e^{-\pi i/p})$. Thus,
$f(us/2)=-cs^{p+1}(2i{\rm sin}\frac{\pi}{p})^{-p}$. The intersection number
$<\tau>_{g}$ is written by a contour integral of $t$ ($t=s^{2+2/p}$) in the
large $g$ limit apart from Gamma function of the definition in (2.4),

$<\tau>_{g}=\oint\frac{dt}{2\pi i}\frac{1}{t^{g+1}}t^{-1/2}e^{-ct^{p/2}(2i{\rm
sin}(\pi/p))^{-p}}$ (5.5)

where $t^{-1/2}$ factor comes from $1/s^{1+\frac{1}{p}}$ in the front of the
integral of $u(s)$, which corresponds to genus zero contribution. We expand
the exponential term as $\sum\frac{1}{m!}(f(su/2))^{m}$, with
$\frac{pm}{2}-\frac{1}{2}=g$. We have an asymptotic behavior in the large $g$
limit as

$<\tau>_{g}\sim\frac{2(p+1)^{\frac{1}{p}}{\rm
sin}\frac{\pi}{p}}{(\frac{2g-1}{p})!}(\frac{1}{4p(p+1)^{\frac{2}{p}}({\rm
sin}\frac{\pi}{p})^{2}})^{g}$ (5.6)

where $c=\frac{1}{p+1}$ and the normalization factor $\frac{1}{p^{g-1}}$ is
included. Thus the term of power $(-g)$ becomes $[(p+1)^{\frac{2}{p}}4p({\rm
sin}\frac{\pi}{p})^{2}]^{-g}$, which agrees with known $\frac{1}{(24)^{g}}$
for $p=2$ case.

For $D_{l}$ ($p=2l-2$) type, the exponential term is same as $A_{p-1}$. For
the large $g$ behavior, the power $g$ part $a^{g}$ becomes same.

Note that for $p=\frac{1}{2}$, there is no finite saddle point solution of
(5.4), except $u=\infty$. The asymptotic term of (5.6) diverges for
$p=\frac{1}{2k},(k\in Z$). For $p=\frac{1}{2}$, we have used the change of
variable from $u$ to $y$, and have obtained the explicit form of (4.24) for
the large $g$ limit. It is remarkable that the polynomial of $p$ of order
$p^{g}$ in the intersection numbers of genus $g$ has all real roots. Namely
the zeros of the polynomial are on the real axis.

The values of this polynomials at $p=\frac{1}{2}$ are
$-\frac{1}{2},5,(-3^{2}\cdot 5\cdot 7),(3\cdot 4\cdot 5\cdot 7\cdot 13)$ for
$g=1,2,3,4$, respectively. These numerators cancel with the denominator of
Bernoulli numbers in (5.10), and the intersection numbers of $p=\frac{1}{2}$
is expressed simply as (4.24). This fascinating feature of $p=\frac{1}{2}$
case shows that there is a characteristic topological meaning of the curves of
Riemann surface for the fermionic spins.

The intersection numbers $<\tau>_{g}$ are rational numbers since they involve
the inverse of automorphism from the orbifold. As remarked by Zagier [45],
there is a property of the integrality by multiply certain factors of two
Pochhammer symbols to the intersection numbers $<\tau>_{g}$. The denominators
of the expression of $<\tau>_{g}$ are same for $A_{p-1}$ type and $D_{l}$ type
as (2) and (3). Thus the integrality can be obtained by the application of the
same Pochhammer symbols for both cases.

As an example of integrality, with the multiplication of Pochhammer symbol
$(x)_{n}=x(x+1)\cdots(x+n-1)$ to $<\tau>_{g=5n}$, the following quantities
become [14, 45, 18].

$\displaystyle a_{n}$ $\displaystyle=$
$\displaystyle(2^{10}3^{5}5^{2})^{n}(\frac{3}{5})_{n}(\frac{4}{5})_{n}<\tau>_{g=5n}$
$\displaystyle b_{n}$ $\displaystyle=$
$\displaystyle(2^{12}3^{5}5^{4})^{n}(\frac{2}{5})_{n}(-\frac{1}{10})_{n}<\tau>_{g=5n}$
(5.7)

which are integers (apart from the normalization of $\frac{1}{p^{g-1}}$ ),
where $<\tau>_{g=5}=161/777600000$ for $A_{4}$ ($p=5$) in (2). The generating
function $\sum b_{n}t^{n}$ is algebraic, while $a_{n}$ grows exponentially
[45].

As evaluated in [41, 8], the higher spin $p$ limit ($p\to\infty$) of each
intersection numbers of a fixed genus $g$ shows interesting features, which
are expressed by Bernoulli numbers $B_{2g}$. From (2) and (3), it is easily
noticed that the values of the coefficients of highest order of $p$ become
same for $A_{p-1}$ and $D_{l}$ case for a fixed genus $g$.

${\lim_{p\to\infty}}<\tau>_{g}|_{A_{p-1}}={\lim_{p\to\infty}}<\tau>_{g}|_{D_{l}}=\frac{(-1)^{g+1}B_{2g}}{(2g)!(2g)}p^{g}+O(p^{g-1})$
(5.8)

The denominators of the expression of $<\tau>_{g}$ are same, and then
integrality should be same for $A_{p-1}$ and $D_{l}$. The result of (5.8),
which reduces to Bernoulli number $B_{n}$, has been obtained in [41], and the
relation to the partition function of black hole is discussed. With
$\sigma=\frac{s}{p}$, $u^{p+1}=x^{2}$, $u(s)$ is written in the large $p$,

$\displaystyle u(s)$ $\displaystyle=$
$\displaystyle\frac{2}{\sigma}\int\frac{dx}{2i\pi}x^{-1+\frac{2}{p}}e^{-\frac{2c}{p+1}x^{2}{\rm
sh}\frac{\sigma}{2}}$ (5.9) $\displaystyle=$
$\displaystyle\frac{2}{\sigma}\Gamma(\frac{2}{p})(\frac{2c}{p+1})^{-\frac{1}{2}}(\frac{\sigma}{2})^{-\frac{1}{p}}{\rm
exp}(-\frac{1}{p}{\rm log}\frac{{\rm sh}\frac{\sigma}{2}}{\frac{\sigma}{2}})$

Noting that ${\rm log}\frac{{\rm
sh}\frac{\sigma}{2}}{\frac{\sigma}{2}}=\sum(-1)^{n-1}\frac{B_{2n}\sigma^{2n}}{(2n)!2n}$,
the Bernoulli number $B_{2n}$ is obtained for the intersection number in the
large $p$ limit.

The notation of Bernoulli numbers are
$B_{2}=\frac{1}{6},B_{4}=-\frac{1}{30},B_{6}=\frac{1}{42},B_{8}=-\frac{1}{30},B_{10}=\frac{5}{66},B_{12}=-\frac{691}{2730},B_{14}=\frac{7}{6},B_{16}=-\frac{3617}{510}$.
The denominator of $<\tau>_{g}$ is common in the denominator of
$B_{2g}/((2g)!(2g))$ for arbitrary $p$, since the intersection number is
described by the multiplication of polynomial of $p$ to $B_{2g}/((2g)!(2g))$
for $A_{l}$ and $D_{l}$ cases.

The denominator of $B_{2g}/(2g)$ is given by [42, 43]

${\rm{denominator}}\bigl{(}\frac{B_{2g}}{2g}\bigr{)}=\prod_{p>2,(p-1)|2g}p^{1+v_{p}(2g)}$
(5.10)

The condition of the prime $p$ is $(p-1)|(2g)$, which means the $(p-1)$
divides $(2g)$, and $v_{p}(N)$ is the largest exponent $e$ such that
$p^{e}|N$. For instance, the denominator of $B_{1000}/1000$ is $2^{4}\cdot
3\cdot 5^{4}\cdot 41\cdot 101\cdot 251$. Thus the integrality of $<\tau>_{g}$
is obtained by the multiplication of the factor of the (5.10) with a factor
$(2g)!$.

The denominator of $B_{2k}/k$ counts the numbers of distinct J-class map:
$S^{m+4k-1}\to S^{m}$ as shown in (5.13) [43]. The numerator of $B_{2k}/2k$ is
related to differential topology such as characteristics. Indeed we have for
$p=-1$, Euler characteristics $\zeta(1-2g)=(-1)^{g}B_{2g}/2g$ for $A_{p-1}$,
and $(1-2^{2g-1})\frac{(-1)^{g}B_{2g}}{2g}$ for $D_{l}$ [8].

The asymptotic behavior for large $g$ of $B_{2g}$ is easily obtained by the
formula,

$\zeta(2g)=\frac{(2\pi)^{2g}}{2(2g)!}|B_{2g}|$ (5.11)

Since $\zeta(2g)=\sum\frac{1}{n^{2g}}\sim 1$ (for $g\to\infty$),
$|B_{2g}|\sim\frac{2(2g)!}{(2\pi)^{2g}}$.

The next leading term for $p\to\infty$ in (5.8) may be evaluated similar to
the derivation of (5.9). . From the lower order of $g$ in (2) and (3), the
intersection numbers are expressed for $p\to\infty$ and a fixed $g$,

$<\tau>_{g}=\frac{|B_{2g}|p^{g}}{(2g)!(2g)}[1+\frac{a_{1}(g)}{p}+\frac{a_{2}(g)}{p^{2}}+\cdots]$
(5.12)

where $a_{n}(g)$ is a polynomial of integer coefficients.

Since the denominator of $<\tau>_{g}$ is given by the denominator of
$B_{2g}/2g$ as shown above, it is interesting to note this implies the
existence of a factor in the numerator of $<\tau>_{g}$. In the case of $p=5$
($A_{4}$ type), the denominator of $B_{10}/10$ is $2^{2}\cdot 3\cdot 11$, from
the integrality of $a_{1}$ in (5) (g=5), the factor $11$ should be in the
numerator of $<\tau>_{g=5}$. Indeed the factor $2p+1$ gives 11 for $p=5$. The
factor $(2p+1)$ i.e. $11$ exists in all $<\tau>_{g}$ and it cancels with all
denominator of $<\tau>_{g=5n}$, since $B_{5g}/(5g)$ has 11 in the denominator
due to the formula of (5.10). The factor
$(2^{10}3^{5}5^{2})^{n}(\frac{3}{5})_{n}(\frac{4}{5})_{n}$ in (5) gives the
cancellation in part of the denominator of $<\tau>_{5g}$ through the
expression of Bernoulli number $B_{2g}$.

The another number of series
$c_{5n}=(\frac{4}{5})_{n}(\frac{1}{5})_{n}<\tau>_{5n}$, which is found to be
integrality in [14] gives a factor $31,41,51,61,...$ due to
$(\frac{1}{5})_{n}$. Such prime numbers 31,41,61,.. appear in the denominator
of $<\tau>_{5g}$ in higher genus $g$, and these prime numbers should be
cancelled with Pochhammer symbol of $(\frac{1}{5})_{n}$. Thus the integral
numbers $a_{n},b_{n},c_{n}$ are consistent with the integrality of
$<\tau>_{5g}$.

The relation of the numerator of Bernoulli number $B_{2k}$ to differential
topology $\Theta_{4k-1}$ is known as Milnor exotic 7 sphere. The order of
$\Theta_{4k-1}$, card($\Theta_{4k-1}$) is given by the stable homotopy
$\Pi_{4k-1}$ as [43, 42]

$card(\Theta_{4k-1})=2^{2k-3}(2^{2k-1}-1)\cdot
card(\Pi_{4k-1})\cdot\frac{B_{2k}}{2k}$ (5.13)

For $4k=2$, it becomes $28=2(2^{3}-1)\cdot 240\cdot(\frac{1}{30})/4$. The
numerator of $<\tau>_{g}$ has therefore a relation to differential topology
and homotopy theory. Exotic 7-sphere is related $E_{8}$ singularity
($x^{5}+y^{3}+z^{2}=0$) [43] and 28 different differential structures are
described by algebraic equation such that
$x_{1}^{2k-1}+x_{2}^{3}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}=0(\Sigma(2k-1,3,2,2,2))$
(k=1,2,…,28) [46].

For $E_{6}$ singularity, the intersection numbers are evaluated by the ODE
[14], and the denominators of $\tau_{E_{6}}$ are not directly expressed by
Bernoulli numbers , although it is closely related. The relation of Bernoulli
number to mapping class group $\Gamma_{g}$ and characteristic classes has been
discussed in [44]. The intersection numbers of half-integer $p$ spin curve may
have interesting further applications for topology and mapping class group.

##  6 n-point functions of $A_{l}$ types

The change of variable from $u$ to $y$ in (4.1) is useful for the evaluation
of higher point correlation functions $u(s_{1},...,s_{n})$. We have checked to
obtain the known results for the integer values of $p=3,4,5$ of [11, 12] by
the Laurent expansion of $y$.

$\bullet$ two point functions for integer $p$

The two point function $u(s_{1},s_{2})$ is written as, (for $p=\frac{1}{2}$,
see (7.11)),

$\displaystyle
u(s_{1},s_{2})=-4(\frac{s_{1}}{s_{2}})^{\frac{1}{p}}\oint\frac{dy_{1}dy_{2}}{(2i\pi)^{2}}(y_{1}+\frac{s_{1}^{2+\frac{2}{p}}}{y_{1}^{3}})(y_{2}+\frac{s_{2}^{2+\frac{2}{p}}}{y_{2}^{3}})$
$\displaystyle\times\frac{1}{y_{1}^{4}}\frac{{\rm
exp}[\sum_{i=1}^{2}(-c)(y_{i}^{2p}-\frac{p(2p+1)}{3}s_{i}^{2+\frac{2}{p}}y_{i}^{2p-4}+\frac{(2p+1)p(2p-1)(p-1)}{30}s_{i}^{4+\frac{4}{p}}y_{i}^{2p-8}+\cdots)]}{[1-\frac{s_{1}^{2+\frac{2}{p}}}{y_{1}^{4}}-(\frac{s_{1}}{s_{2}})^{\frac{1}{p}}\frac{1}{y_{1}^{2}}(y_{2}^{2}-\frac{s_{2}^{2+\frac{2}{p}}}{y_{2}^{2}})]^{2}+\frac{4s_{1}^{\frac{2}{p}}}{y_{1}^{4}}(s_{1}+s_{2})^{2}}$

For $p=\frac{1}{2}$, it becomes as (7.11) with a slight difference of notation
of $c$ (a factor 3 difference).

The selection rule of (2.6) is

$2(g-1)(1+\frac{1}{p})+2=n_{1}+n_{2}+\frac{1}{p}(j_{1}+j_{2})$ (6.2)

The two point function is expressed as

$u(s_{1},s_{2})=C\sum_{n_{i},j_{i}}<\tau_{n_{1},j_{1}}\tau_{n_{2},j_{2}}>s_{1}^{n_{1}+\frac{1}{p}(1+j_{1})}s_{2}^{n_{2}+\frac{1}{p}(1+j_{2})}$
(6.3)

where $C$ is a constant., which involves factors of gamma-function. This two
point function and the intersection numbers
$<\tau_{n_{1},j_{1}}\tau_{n_{2},j_{2}}>$ for Neveu-Schwarz punctures are
evaluated in general $p$ in [41]. Here we reconsider two point functions under
the formula by the integral $y$ in (6), which may be easily obtained in more
systematic ways for both Neveu-Schawrz and Ramond punctures. The numerator and
denominator of (6) are expanded in the small $s_{1}$ and $s_{2}$.

As an example, the case $<\tau_{2,1}\tau_{2,1}>$ of $p=4$, we find

$\displaystyle({\rm ii}.5)$ $\displaystyle
u(s_{1},s_{2})=s_{1}^{\frac{5}{2}}s_{2}^{\frac{5}{2}}\int_{0}^{\infty}dy_{1}dy_{2}e^{-c(y_{1}^{8}+y_{2}^{8})}(\frac{1}{y_{1}^{13}}(-2816y_{2}^{3}+7680y_{2}^{11})$
(6.4)
$\displaystyle+\frac{1}{y_{1}^{21}}(-28728y_{2}^{11}+27888y_{2}^{19}-2880y_{2}^{27}))=\frac{11}{240}s_{1}^{\frac{5}{2}}s_{2}^{\frac{5}{2}}[\Gamma(\frac{1}{2})]^{2}$

By multiplying a factor $\frac{1}{p}=\frac{1}{4}$, it leads to
$<\tau_{2,1}\tau_{2,1}>=\frac{11}{960}$, which agrees with [11].

We have checked $<\tau_{1,1}\tau_{3,1}>_{g=2}=\frac{11}{4320}$ for $p=3$,
$<\tau_{1,1}\tau_{3,1}>_{g=2}=\frac{17}{1200}$ for $p=5$, which are genus
$g=2$ cases.

$\bullet$ three point functions for integer $p$

Three point function $u(s_{1},s_{2},s_{3})$ is given for $A$ type by

$\displaystyle
u(s_{1},s_{2},s_{3})=-8(\frac{s_{1}}{s_{3}})^{\frac{1}{p}}\int\frac{1}{(2i\pi)^{3}}\prod_{i=1}^{3}dy_{i}\prod_{i=1}^{3}(y_{i}+\frac{s_{i}^{2+\frac{2}{p}}}{y_{i}^{3}})e^{-c\sum(y_{i}^{2p}+\cdots)}$
$\displaystyle\times\frac{1}{y_{1}^{4}}\frac{1}{1-\frac{s_{1}^{2+\frac{2}{p}}}{y_{1}^{4}}-(\frac{s_{1}}{s_{2}})^{\frac{1}{p}}\frac{1}{y_{1}^{2}}(y_{2}^{2}-\frac{s_{2}^{2+\frac{2}{p}}}{y_{2}^{2}})-\frac{2is_{1}^{\frac{1}{p}}}{y_{1}^{2}}(s_{1}+s_{2})}$
$\displaystyle\times\frac{1}{y_{2}^{2}}\frac{1}{1-\frac{s_{2}^{2+\frac{2}{p}}}{y_{2}^{4}}-(\frac{s_{2}}{s_{3}})^{\frac{1}{p}}\frac{1}{y_{2}^{2}}(y_{3}^{2}-\frac{s_{3}^{2+\frac{2}{p}}}{y_{3}^{2}})-\frac{2is_{2}^{\frac{1}{p}}}{y_{2}^{2}}(s_{2}+s_{3})}$
$\displaystyle\times\frac{1}{1-\frac{s_{1}^{2+\frac{2}{p}}}{y_{1}^{4}}-(\frac{s_{1}}{s_{3}})^{\frac{1}{p}}\frac{1}{y_{1}^{2}}(y_{3}^{2}-\frac{s_{3}^{2+\frac{2}{p}}}{y_{3}^{2}})+\frac{2is_{1}^{\frac{1}{p}}}{y_{1}^{2}}(s_{1}+s_{3})}$
(6.5)

The exponential term is expressed after the scaling of $s$ as (6),

$e^{-y_{i}^{2p}+\frac{p(2p+1)}{3}s_{i}^{2+\frac{2}{p}}y_{i}^{2p-2}-\frac{(2p+1)p(2p-1)(p-1)}{30}s_{i}^{4+\frac{4}{p}}y_{i}^{2p-8}+\cdots}$
(6.6)

where the factor $c$ is absorbed in $y_{i}$, which follows the scaling to
$s_{i}$ as $cs^{p+1}$.

There are results of the intersection numbers of three-point for $p=3$ up to
$g=2$ for different 6 intersection numbers[11]. We evaluated these 6 cases to
verify the validity the integral representation of (6) based on the random
matrix theory.

For instance, the case of $<\tau_{1,1}\tau_{1,1}\tau_{3,0}>_{g=2}$ is
evaluated as,

$\displaystyle({\rm iii}.5)$ $\displaystyle
u(s_{1},s_{2},s_{3})=-8c^{\frac{5}{3}}s_{1}^{\frac{5}{3}}s_{2}^{\frac{5}{3}}s_{3}^{\frac{10}{3}}\int\prod
dy_{i}e^{-\sum
y_{i}^{6}}\biggl{(}-560\frac{y_{2}y_{3}^{3}}{y_{1}^{11}}-80\frac{y_{2}}{y_{1}^{11}y_{3}^{3}}$
(6.7)
$\displaystyle+\frac{1}{y_{1}^{11}y_{2}^{5}}(2860y_{3}^{3}-7700y_{3}^{9}+1470y_{3}^{15})+\frac{1}{y_{1}^{11}y_{2}^{11}}(8520y_{3}^{9}-8400y_{3}^{15}$
$\displaystyle+1225y_{3}^{21})\biggr{)}$
$\displaystyle=\frac{29}{1440}s_{1}^{\frac{5}{3}}s_{2}^{\frac{5}{3}}s_{3}^{\frac{10}{3}}(\Gamma(\frac{1}{3}))^{2}\Gamma(\frac{2}{3})$

This leads to $<\tau_{1,1}\tau_{1,1}\tau_{3,0}>_{g=2}=\frac{29}{2160}$ for
$p=3$ by the normalization of $\frac{2}{3}$.

We have correctly derived the 6 intersection numbers of three punctures of
$p=3$ in genus $g=2$ with a normalization constant $\frac{2}{3}$, which agrees
the values of [11].

Thus the method of Laurent expansion of $y$ works for the evaluation of the
higher point correlation function, which is a generating function of the
intersection number, and it provides a practical method for higher correlation
functions.

##  7 n-point functions of $p=\frac{1}{2},-\frac{1}{2},-2$, and $-3$

In this section, we consider the half integer
$p=\frac{1}{2},-\frac{1}{2},-\frac{3}{2}$ and the negative integer $p=-2,-3$
cases. Some of these one point functions have been discussd in the previous
articles [9, 10].

7-1: n-point function for $p=\frac{1}{2}$

Since $\frac{1}{p}\sum j_{i}$ is integer in this case, we are able to include
them in the integral part of $\sum n_{i}$. Then the spin component $j_{i}$ can
be chosen as arbitrary value as mod $p$. Here we take the spin component
$j_{i}$ as $-1$. Other choice may be $j=0$ or $j=-\frac{1}{2}$. All these
cases provide the shift of the integer $n$. From the continuation of Ramond
spin component $j=p-1$, the choice of $j=-\frac{1}{2}$ may be naturally
considered. However, the difference between $j=-1$ and $j=-\frac{1}{2}$ leads
the shift of the integer $n$, this choice may do not cause a serious
conclusion. For the integer $p$, the algebraic geometry has been studied [4].

One point function $u(s)$ is given by (4.24), which becomes by an integral of
part,

$u(s)=-\frac{i}{6c^{\prime}s^{3/2}}\oint\frac{dy}{2i\pi}e^{c^{\prime}s^{3/2}(3y-\frac{1}{y^{3}})}$
(7.1)

and it becomes a series of $\sum_{g}a_{g}s^{6g-3}$. $\bullet$ string equation
for $p=\frac{1}{2}$

We discuss the string equation for $p=\frac{1}{2}$. Although the usual string
equation may be not applied for this case, analogous equation about a
forgetting of a marked point can be considered.

The selection rule for $p=\frac{1}{2}$ (2.6) for $s$ marked points is,

$6g-6+s=\sum_{i=1}^{s}n_{i}+2\sum_{i=1}^{s}j_{i}$ (7.2)

Possible values of $j_{i}$ may be $0,-\frac{1}{2},-1$, for which the last term
becomes integers. The term $s_{i}^{n_{i}+\frac{1+m_{i}}{p}}$ becomes
$s_{i}^{n_{i}+2}$, $s_{i}^{n_{i}+1}$ and $s_{1}^{n_{i}}$, respectively.

By taking $j_{i}=-1$, it becomes

$6g-6+3s=\sum_{i=1}^{s}n_{i}$ (7.3)

The term of order $s_{1}^{2}s_{2}^{6g-2}$ are derived from (6), which is
considered as a string equation, since $s^{\frac{1}{p}}=s^{2}$ for
$p=\frac{1}{2}$.

$\displaystyle u(s_{1},s_{2})$ $\displaystyle=$
$\displaystyle-\frac{4s_{1}^{2}}{s_{2}^{2}}\oint\frac{dy_{1}dy_{2}}{(2i\pi)^{2}}\frac{1}{y_{1}^{3}}(y_{2}\pm\frac{s_{2}^{6}}{y_{2}^{3}})e^{c_{1}(3y_{1})+c_{2}(3y_{2}-\frac{s_{2}^{6}}{y_{2}^{3}})}$
(7.4) $\displaystyle=$
$\displaystyle\sum_{g}a_{g}s_{1}^{2}s_{2}^{6g-2}c_{1}^{2}c_{2}^{4g-2}$

The coefficient $a_{g}$ is denoted by the intersection numbers as
$<\tau_{2,-1}\tau_{6g-2,-1}>$, where we defined $n$ of $\tau_{n,-1}$ as the
power of $s^{n+\frac{1+j}{p}}$. $<\tau_{n,-1}>$ is a coefficient of $s^{n}$.
This term of order $s_{2}^{6g-2}$ is one point function in (4.24). Thus we
have a string equation,

$<\tau_{2,-1}\tau_{6g-2,-1}>=<\tau_{6g-3,-1}>$ (7.5)

in which $\tau_{2,-1}$ operates on $\tau_{6g-2,-1}$ for the change to
$\tau_{6g-3,-1}$. Note that the usual notation of $\tau_{n,j}$ is different
from above as a shift $n\to n-2$, since we took $j_{i}=-1$.

Three points function of $p=\frac{1}{2}$ is studied for $A_{l}$ case in [10].
From the selection rule of $j_{i}=-1$ ($i=1,2,3$), we have from (7.3) ;
$6g+3=n_{1}+n_{2}+n_{3}$ for
$<\tau_{n_{1},-1}\tau_{n_{2},-1}\tau_{n_{3},-1}>$.

The term of order $s_{1}^{2}$ is obtained from three point function as

$\displaystyle
u(s_{1},s_{2},s_{3})=-\frac{s_{1}^{2}}{(s_{2}s_{3})^{2}}\oint\frac{dy_{1}}{2i\pi}\frac{1}{y_{1}^{3}}e^{3c_{1}y_{1}}\oint\frac{dy_{2}dy_{3}}{(2i\pi)^{2}}(y_{2}\pm\frac{s_{2}^{6}}{y_{2}^{3}})(y_{3}+\pm\frac{s_{3}^{6}}{y_{3}^{3}})$
$\displaystyle
e^{c_{2}(3y_{2}-\frac{s_{2}^{6}}{y_{2}^{3}})+c_{3}(3y_{3}-\frac{s_{3}^{6}}{y_{3}^{3}})}\frac{1}{2i(s_{2}+s_{3})+\frac{y_{2}^{2}}{s_{2}^{2}}-\frac{s_{2}^{4}}{y_{2}^{2}}-\frac{y_{3}^{2}}{s_{3}^{2}}+\frac{s_{3}^{4}}{y_{3}^{2}}}$
(7.6)

Since two point function $u(s_{2},s_{3})$ is written as

$\displaystyle
u(s_{2},s_{3})=\frac{i}{s_{2}^{2}s_{3}^{2}(s_{2}+s_{3})}\oint\frac{dy_{2}dy_{3}}{(2i\pi)^{2}}(y_{2}\pm\frac{s_{2}^{6}}{y_{2}^{3}})(y_{3}\pm\frac{s_{3}^{6}}{y_{3}^{3}})e^{c_{2}(3y_{2}-\frac{s_{2}^{6}}{y_{2}^{3}})+c_{3}(3y_{3}-\frac{s_{3}^{6}}{y_{3}^{3}})}$
$\displaystyle\biggl{(}\frac{1}{\frac{y_{2}^{2}}{s_{2}^{2}}-\frac{s_{2}^{4}}{y_{2}^{2}}-\frac{y_{3}^{2}}{s_{3}^{2}}+\frac{s_{3}^{4}}{y_{3}^{2}}-2i(s_{2}+s_{3})}-c.c.\biggr{)}$
(7.7)

Thus we have a string equation

$<\tau_{2,-1}\tau_{n_{2},-1}\tau_{n_{3},-1}>_{g}=<\tau_{n_{2}-1,-1}\tau_{n_{3},-1}>_{g}+<\tau_{n_{2},-1}\tau_{n_{3}-1,-1}>_{g}$
(7.8)

The difference between $A_{l}$ and $D_{l}$ is due to the sign ($\pm$1) in the
representation of $y$ variable in (4.20), and the string equation of three
points function for $A_{l}$ ia valid also for $D_{l}$ case.

$\bullet$ two point function and absence of dilaton equation for
$p=\frac{1}{2}$

For integer spin $p$, a dilaton field corresponds to $\tau_{1,0}$, which
corresponds to $s^{1+\frac{1}{p}}$. When $p=\frac{1}{2}$ is inserted into this
term, it becomes $s^{3}$ term.

We have found that there is no dilaton equation for $p=\frac{1}{2}$ [10]. The
one marked point intersection number $<\tau_{1,0}>_{g=1}$ corresponds to
$<\tau_{3,-1}>_{g=1}$ in our notation. However, we do not find two point
function

$<\tau_{3}\tau_{6g-3}>_{g}=0\times<\tau_{6g-3}>_{g}$ (7.9)

which means a vanishing coefficient. Above equation plays a role of a dilaton
equation, but the coefficient becomes zero.

The two point function of $p=\frac{1}{2}$ is obtained by the expansions of
small $s_{1}$ and $s_{2}$ in a genus expansion. For this purpose, we write the
integral representation of (6) as

$\displaystyle u(s_{1},s_{2})$ $\displaystyle=$
$\displaystyle-\frac{4}{s_{1}^{2}s_{2}^{2}}\oint\frac{dy_{1}dy_{2}}{(2i\pi)^{2}}(y_{1}+\frac{s_{1}^{6}}{y_{1}^{3}})(y_{2}+\frac{s_{2}^{6}}{y_{2}^{3}})e^{c\sum(y_{i}-\frac{s_{i}^{6}}{3y_{i}^{3}})}$
(7.10)
$\displaystyle\times\frac{1}{(\frac{1}{s_{1}^{2}}(y_{1}^{2}-\frac{s_{1}^{6}}{y_{1}^{2}})-\frac{1}{s_{2}^{2}}(y_{2}^{2}-\frac{s_{2}^{6}}{y_{2}^{2}}))^{2}+4(s_{1}+s_{2})^{2}}$

If we expand in the inverse of $y_{1}$, we obtain

$\displaystyle u(s_{1},s_{2})$ $\displaystyle=$
$\displaystyle-4\frac{s_{1}^{2}}{s_{2}^{2}}\oint\frac{dy_{1}dy_{2}}{(2i\pi)^{2}}(y_{1}+\frac{s_{1}^{6}}{y_{1}^{3}})(y_{2}+\frac{s_{2}^{6}}{y_{2}^{3}})e^{c(y_{1}-\frac{s_{1}^{6}}{3y_{1}^{3}})+c(y_{2}-\frac{s_{2}^{6}}{3y_{2}^{3}})}$
(7.11) $\displaystyle\times$
$\displaystyle\frac{1}{y_{1}^{4}}[1+g+g^{2}+g^{3}+\cdots]$

where

$\displaystyle g$ $\displaystyle=$ $\displaystyle
2(\frac{s_{1}^{6}}{y_{1}^{4}}+\frac{s_{1}^{2}y_{2}^{2}}{s_{2}^{2}y_{1}^{2}}-\frac{s_{1}^{2}s_{2}^{4}}{y_{1}^{2}y_{2}^{2}})-(\frac{s_{1}^{6}}{y_{1}^{4}}+\frac{s_{1}^{2}y_{2}^{2}}{s_{2}^{2}y_{1}^{2}}-\frac{s_{1}^{2}s_{2}^{4}}{y_{1}^{2}y_{2}^{2}})^{2}$
(7.12) $\displaystyle-$
$\displaystyle\frac{4s_{1}^{4}(s_{1}+s_{2})^{2}}{y_{1}^{4}}$

By the contour integral at $y_{1}=0,y_{2}=0$, we have expanding the
exponential term proportional to $c$,

$\displaystyle u(s_{1},s_{2})=-[\frac{1}{3}c^{4}s_{1}^{2}s_{2}^{4}$
$\displaystyle+c^{8}(\frac{1}{360}s_{1}^{8}s_{2}^{4}+\frac{1}{135}s_{1}^{7}s_{2}^{5}+\frac{1}{180}s_{1}^{6}s_{2}^{6}+\frac{1}{216}s_{1}^{4}s_{2}^{8}+\frac{1}{1080}s_{1}^{2}s_{2}^{10})+O(c^{12})]$
$\displaystyle+[s_{1}\leftrightarrow s_{2}]$ (7.13)

The expression should be symmetric by adding the terms of
$s_{1}\leftrightarrow s_{2}$. The terms of order $g=2$ becomes by this
symmetrization,

$c^{8}\frac{1}{3^{3}\cdot
5}\biggl{(}(s_{1}^{8}s_{2}^{4}+s_{1}^{4}s_{2}^{8}+s_{1}^{7}s_{2}^{5}+s_{1}^{5}s_{2}^{7})+\frac{3}{2}s_{1}^{6}s_{2}^{6}+\frac{1}{8}(s_{1}^{2}s_{2}^{10}+s_{1}^{10}s_{2}^{2})\biggr{)}$
(7.14)

The first term shows the string equation; $s_{1}^{2}s_{2}^{4}\to s_{2}^{3}$.
Note that there is no $s^{3}$ term in two point function, which is supposed to
be present for the term for a dilaton equation.

From (2.6) for $p=\frac{1}{2}$, we have $6g=n_{1}+n_{2}$ for
$s_{1}^{n_{1}}s_{2}^{n_{2}}$. The first etrm of (7) is genus one, and the term
of $c^{8}$ is genus two, since $n_{1}+n_{2}=12$. It is interesting to note
that $n_{1}$ and $n_{2}$ are both even, or both odd, since $6g$ is an even
number. The terms of $s_{1}^{2}s_{2}^{10}$ is consistent with a string
equation of (7.4).

It may be easy to obtain the higher order terms. The parameter $c$ is
$c=\frac{i}{4}$. Each term of two point function is order of
$c^{4g}s_{1}^{n_{1}}s_{2}^{6g-n_{1}}$ for genus $g$, which agrees with the
selection rule of (2.6). The scaling relation between $c$ and $s$ can be seen
in (2.4). After the rescaling of $u\to\frac{s}{2}u$, the exponent becomes
$cs^{p+1}$. This leads to the scaling $cs^{\frac{3}{2}}$, and
$u(s_{1},s_{2})\sim\sum_{n=2}\sum_{g=1}^{\infty}c^{4g}s_{1}^{n}s_{2}^{6g-n}$.
(7) is consistent with this behavior.

There may be another way for the evaluation of the contour integral. For genus
one case, the last factor of (7.10) is approximated by

$\frac{1}{(\frac{y_{1}^{2}}{s_{1}^{2}}-\frac{y_{2}^{2}}{s_{2}^{2}})^{2}}=\frac{1}{(\frac{y_{1}}{s_{1}}-\frac{y_{2}}{s_{2}})^{2}(\frac{y_{1}}{s_{1}}+\frac{y_{2}}{s_{2}})^{2}}$
(7.15)

We take residues at $y_{1}=\frac{s_{1}}{s_{2}}y_{2}$ and at
$y_{1}=-\frac{s_{1}}{s_{2}}y_{2}$. Expanding the exponent in order $c^{4}$, we
find the residue of $y_{1}=\frac{s_{1}}{s_{2}}y_{2}$ and $y_{2}=0$ as

$U(s_{1},s_{2})=\frac{c^{4}}{6s_{2}^{3}}(s_{1}^{9}+2s_{1}^{8}s_{2}+s_{1}^{7}s_{2}^{2}-s_{1}^{3}s_{2}^{6}-2s_{1}^{2}s_{2}^{7}-s_{1}s_{2}^{8})$
(7.16)

and from the pole at $y_{1}=-\frac{s_{1}}{s_{2}}y_{2}$ and $y_{2}=0$ as

$u(s_{1},s_{2})=\frac{c^{4}}{6s_{2}^{3}}(-s_{1}^{9}+2s_{1}^{8}s_{2}-s_{1}^{7}s_{2}^{2}+s_{1}^{3}s_{2}^{6}-2s_{1}^{2}s_{2}^{7}+s_{1}s_{2}^{8})$
(7.17)

Adding these two contributions, by noting the cancellation of terms, we have

$u(s_{1},s_{2})=\frac{2c^{4}s_{1}^{8}}{3s_{2}^{2}}-\frac{2}{3}c^{4}s_{1}^{2}s_{2}^{4}$
(7.18)

The term of $s_{1}^{2}s_{2}^{4}$ agrees with (7) in genus one.

One may evaluate the residue of (7.10) by taking 8 poles of $y_{1}$. There are
8 poles, except a pole of numerator $y_{1}=0$,

$y_{1}=\pm\frac{is_{1}s_{2}^{2}}{2y_{2}}\pm\frac{s_{1}y_{2}}{2s_{2}}\pm\frac{s_{1}\sqrt{-s_{2}^{6}\pm
4is_{1}s_{2}^{2}y_{2}^{2}\pm 2is_{2}^{3}y_{2}^{2}+y_{2}^{4}}}{2s_{2}y_{2}}$
(7.19)

These residues for 8 poles are simply expressed as

$\displaystyle\oint\frac{dy_{1}}{2i\pi}\frac{(y_{1}+\frac{s_{1}^{6}}{y_{1}^{3}})(y_{2}+\frac{s_{2}^{6}}{y_{2}^{3}})}{(\frac{1}{s_{1}^{2}}(y_{1}^{2}-\frac{s_{1}^{6}}{y_{1}^{2}})-\frac{1}{s_{2}^{2}}(y_{2}^{2}-\frac{s_{2}^{6}}{y_{2}^{2}}))^{2}+4(s_{1}+s_{2})^{2}}$
$\displaystyle=\pm
i\frac{s_{1}^{2}(s_{2}^{6}+y_{2}^{4})}{8(s_{1}+s_{2})y_{2}^{3}}$ (7.20)

Adding the contribution of 8 poles, there is a cancellation. Thus the
contribution comes only from the pole of $y_{1}=y_{2}=0$.

$\bullet$ three-point function for $p=\frac{1}{2}$ We have three point
function of $p=\frac{1}{2}$ by putting $p=\frac{1}{2}$ in (6),

$\displaystyle
u(s_{1},s_{2},s_{3})=-8\frac{s_{1}^{2}}{s_{3}^{2}}\oint\prod_{i=1}^{3}\frac{dy_{i}}{2i\pi}\prod_{i=1}^{3}(y_{i}+\frac{s_{i}^{6}}{y_{i}^{3}})e^{c\sum(3y_{i}-\frac{s_{i}^{6}}{y_{i}^{3}})}$
$\displaystyle\times(\frac{1}{y_{1}^{4}y_{2}^{2}})\frac{1}{1-\frac{s_{1}^{6}}{y_{1}^{4}}-\frac{s_{1}^{2}}{s_{2}^{2}y_{1}^{2}}(y_{2}^{2}-\frac{s_{2}^{6}}{y_{2}^{2}})-\frac{2is_{1}^{2}}{y_{1}^{2}}(s_{1}+s_{2})}$
$\displaystyle\times\frac{1}{1-\frac{s_{2}^{6}}{y_{2}^{4}}-\frac{s_{2}^{2}}{s_{3}^{2}y_{2}^{2}}(y_{3}^{2}-\frac{s_{3}^{6}}{y_{3}^{2}})-\frac{2is_{2}^{2}}{y_{2}^{2}}(s_{2}+s_{3})}$
$\displaystyle\times\frac{1}{1-\frac{s_{1}^{6}}{y_{1}^{4}}-\frac{s_{1}^{2}}{s_{3}^{2}y_{1}^{2}}(y_{3}^{2}-\frac{s_{3}^{6}}{y_{3}^{2}})+\frac{2is_{1}^{2}}{y_{1}^{2}}(s_{1}+s_{3})}$
(7.21)

where $c=-\frac{1}{6}i^{1/2}$. Expanding the denominators in the power series
of $s_{i}$, we obtain by the evaluation of residues,

$\displaystyle
u(s_{1},s_{2},s_{3})=-8\frac{s_{1}^{2}}{s_{3}^{2}}\biggl{(}\frac{27}{4}c^{4}s_{3}^{6}+i\frac{81}{4}c^{6}(s_{1}^{2}s_{2}s_{3}^{6}+3s_{2}^{3}s_{3}^{6}+3s_{2}^{2}s_{3}^{7})+O(c^{8})\biggr{)}$

These terms are order of $c^{4g+2}s^{6g+3}$ according to the scaling relation
between $c$ and $s$ as $cs^{3/2}$. After the contour integration of $y_{1}$,
the remaining term is odd for the exchange of
$(s_{2},y_{2})\leftrightarrow(s_{3},y_{3})$. For this order of $c^{4}$, the
contour integral of $y_{1}$ is factorized as

$\displaystyle U(s_{1},s_{2},s_{3})$ $\displaystyle=$
$\displaystyle-8s_{1}^{2}\oint\frac{dy_{1}}{2i\pi}\frac{y_{1}}{y_{1}^{4}}e^{c(3y_{1}-\frac{s_{1}^{6}}{y_{1}^{3}})}\oint\frac{dy_{2}dy_{3}}{(2i\pi)^{2}}(y_{2}+\frac{s_{2}^{6}}{y_{2}^{3}})(y_{3}+\frac{s_{3}^{6}}{y_{3}^{3}})$
(7.23) $\displaystyle\times$
$\displaystyle\frac{e^{c(3y_{2}-\frac{s_{2}^{6}}{y_{2}^{3}})+c(3y_{3}-\frac{s_{3}^{6}}{y_{3}^{3}})}}{s_{3}^{2}y_{2}^{2}-\frac{s_{2}^{6}s_{3}^{2}}{y_{2}^{2}}-s_{2}^{2}y_{3}^{2}+\frac{s_{2}^{2}s_{3}^{6}}{y_{3}^{2}}-2is_{2}^{2}s_{3}^{2}(s_{2}+s_{3})}$

The denominator becomes $(s_{3}^{2}y_{2}^{2}-s_{2}^{2}y_{3}^{2})^{-1}$ in this
order, and it is odd for the exchange
$(s_{2},y_{2})\leftrightarrow(s_{3},y_{3})$. Therefore the first term of order
$c^{4}$ is vanishing by adding the symmetric terms of $s_{2}\leftrightarrow
s_{3}$. Absence of order $c^{4}$ term agrees with (2.6), which implies the
order of $c^{4g+2}s^{6g+3}$ for a genus $g$ term.

The term of order $c^{6}$ in (7) contains $i=(-1)^{1/2}$, which comes from the
term of $i(s_{i}+s_{j})$ in the denominator of (7). Note that
$c=-\frac{1}{6}i^{3/2}$, $ic^{6}$ is real number. The term of order $c^{6}$ is
also vanishing by the following reason. We have expanded the large
$y_{1}>y_{2}>y_{3}$. We have to consider also the case of $y_{3}>y_{2}$. For
this case, we exchange $y_{2}\leftrightarrow y_{3}$ as well as
$s_{2}\leftrightarrow s_{3}$ with the complex conjugate of the term
$-2i(s_{2}+s_{3})$, and this leads to the opposite sign of the term of $c^{6}$
with the exchange $s_{2}\leftrightarrow s_{3}$. Thus after adding this
symmetric term, we obtain the cancellation of (7) and vanishing result of
$c^{6}$ term. The fact that there is no three point function suggests the
punctures are pairwise, and the vanishing three point function gives the
simple structure of the spin $p=\frac{1}{2}$ case, which may be related to the
simplification for Faber conjecture [33] as discussed in [10, 34].

$\bullet$ four-point function for $p=\frac{1}{2}$ The selection rule for four
point function is $6g+6=\sum_{i=1}^{4}n_{i}$. The connected four point
function $u(s_{1},s_{2},s_{3},s_{4})$ is expressed as in [10] with the same
change of variables $u_{i}\to\frac{i}{2}(y_{i}^{2}-\frac{1}{y_{i}^{2}})$, by
assuming order of smallness $s_{1}<s_{2}<s_{3}<s_{4}$,

$\displaystyle
u(s_{1},s_{2},s_{3},s_{4})=-16\frac{s_{1}^{2}}{s_{4}^{2}}\oint\prod_{i=1}^{4}(y_{i}+\frac{s_{i}^{6}}{y_{i}^{3}})e^{c\sum_{i=1}^{4}(3y_{i}-\frac{s_{i}^{6}}{y_{i}^{3}})}$
$\displaystyle\times$
$\displaystyle\frac{1}{y_{1}^{2}-\frac{s_{1}^{6}}{y_{1}^{2}}-\frac{s_{1}^{2}}{s_{2}^{2}}(y_{2}^{2}-\frac{s_{2}^{6}}{y_{2}^{2}})-2is_{1}^{2}(s_{1}+s_{2})}\frac{1}{y_{2}^{2}-\frac{s_{2}^{6}}{y_{2}^{2}}-\frac{s_{2}^{2}}{s_{3}^{2}}(y_{3}^{2}-\frac{s_{3}^{6}}{y_{3}^{2}})-2is_{2}^{2}(s_{2}+s_{3})}$
$\displaystyle\times$
$\displaystyle\frac{1}{y_{3}^{2}-\frac{s_{3}^{6}}{y_{3}^{2}}-\frac{s_{3}^{2}}{s_{4}^{2}}(y_{4}^{2}-\frac{s_{4}^{6}}{y_{4}^{2}})-2is_{3}^{2}(s_{3}+s_{4})}\frac{1}{y_{1}^{2}-\frac{s_{1}^{6}}{y_{1}^{2}}-\frac{s_{1}^{2}}{s_{4}^{2}}(y_{4}^{2}-\frac{s_{4}^{6}}{y_{4}^{2}})+2is_{1}^{2}(s_{1}+s_{4})}$

Since the four point function is cyclic about $y_{i}$ ($i=1,...,4$), we take
$y_{1}$ and $y_{3}$ are large, and expansion of $1/y_{1}$ and $1/y_{3}$ in
(7). Similar to the three point function, we find that the contour integrals
of (7) are vanishing for the terms of order $c^{i}$ ($i=1,...,7$). The non-
vanishing term appears at the order $c^{8}$ with a result of

$u(s_{1},s_{2},s_{3},s_{4})=-729c^{8}s_{1}^{2}s_{3}^{2}s_{2}^{4}s_{4}^{4}+O(c^{12}s^{18})$
(7.25)

where $c=-\frac{1}{6}i^{1/2}$. This is consistent with the selection rule of
$6g+6=\sum_{i=1}^{4}n_{i}$, where $n_{i}$ is a power of $s_{i}$. In this
order, the genus $g$ is one, and it is consistent with the scaling of
$c^{4g+4}s^{6g+6}=c^{8}s^{12}$. The result of (7.25) is obtained for
$y_{1}>\\{y_{2},y_{4}\\}$ and $y_{3}>\\{y_{2},y_{4}\\}$. The total expression
is obtained by the permutation of the variables$\\{s_{1},...,s_{4}\\}$.

In this four point function, the numbers of punctures are even numbers (i.e.
it is four). The punctures have the spin indices $j=-1$ as a Ramond puncture,
and according to this spin indices, the selection rule
$6g+6=\sum_{i=1}^{4}n_{i}$ is obtained. Thus our explicit result of the four
point function is consistent with the conjecture that Ramond punctures are
pairwise.

The term of (7.25) is consistent with the successive string equations, which
give the reduction of $s_{1}^{2}s_{3}^{2}s_{2}^{4}s_{4}^{4}\to
s_{1}^{2}s_{2}^{3}s_{4}^{4}\to s_{3}^{2}s_{4}^{4}\to s_{4}^{3}$. Note that
$<\tau_{3}>$ is non-vanishing one point intersection number.

For $n$-point function ($n>1$), this Ramond puncture pairing conjecture states
that if $n$ is odd, the $n$-point function is vanishing, and if $n$ is even,
it is non-vanishing. For $n$-point function ($n\geq 6$), the contour integral
becomes similar to $n=4$. Assuming $y_{j}$ ($j$ is odd) is large, and expand
in the inverse power of $y_{j}$, we obtain the $n$-point function
$u(s_{1},...,s_{n})$ from the evaluation of the residues of $y_{i}$ at
$y_{i}=0$. Through these residual calculations, we find the selection rule of
(2.6) for $p=\frac{1}{2}$ is valid with the Ramond values of spin component
$j=-1$, and Ramond punctures ($j=-1$) appear as pair-wise form.

$\bullet$ continuation from integer $p$ to $p=\frac{1}{2}$ For the
continuation to $p=\frac{1}{2}$, the spin component $j$ is assumed to be (-1)
[9, 10]. From the selection rule of (2.6) , we have for $p=\frac{1}{2}$

$6g-6+3=\sum_{i=1}^{3}n_{i}+2\sum_{i=1}^{3}j_{i}$ (7.26)

For general $p$, the three point function
$<\tau_{0,0}\tau_{0,0}\tau_{0,p-2}>t_{0,0}t_{0,0}t_{0,p-2}$ is genus zero
term, which corresponds to
$s_{1}^{\frac{1}{p}}s_{2}^{\frac{1}{p}}s_{3}^{\frac{p-1}{p}}$ term. Putting
$p=\frac{1}{2}$, this term becomes $s_{1}^{2}s_{2}^{2}s_{3}^{-1}$. This term
is obtained from the contour integral in (7) by expanding the second
denominator about $2is_{2}^{2}s_{3}/y_{2}^{2}$ as

$u(s_{1},s_{2},s_{3})_{g=0}=-8s_{1}^{2}s_{2}^{2}s_{3}^{-1}\oint\prod_{i=1}^{3}\frac{dy_{i}}{2i\pi}\frac{y_{3}^{3}}{y_{1}^{3}y_{2}^{3}}e^{c\sum
y_{i}}$ (7.27)

However the contour integral of $y_{3}$ gives zero around $y_{3}=0$. Thus we
have no 3-point function of $p=\frac{1}{2}$ at genus zero. This result is
consistent with the conjecture of the punctures of Ramond type, which should
be paired.

The string equation of $p=\frac{1}{2}$ has been found in [10]. It is transform
from $t_{0,0}\to t_{2}$, $(t_{2}=s^{2})$.

For $u(s)$ is given by (4.24).

$u(s)=\frac{c^{2}}{48}s^{3}-\frac{1}{5!}(\frac{1}{48})^{2}c^{6}s^{9}+\cdots$
(7.28)

This expression is continuation of $u(s)$ for the integer $p$, with
$p=\frac{1}{2}$, since for general $p$, it is given by [7] as

$u(s)=\frac{1}{s^{1+\frac{1}{p}}}[\Gamma(1+\frac{1}{p})-\frac{p-1}{24}y\Gamma(1-\frac{1}{p})+\frac{(p-1)(p-3)(1+2p)}{5!4^{2}3}y^{2}\Gamma(1-\frac{3}{p})+\cdots]$
(7.29)

with $y=s^{2+\frac{2}{p}}$. There appears a divergence in the limit of
$p=\frac{1}{2}$ for $\Gamma$-function, and with the normalization of this
factor, (7.29) becomes equivalent to (7.28) as noted in Eq.(18) of [10].

From the results of (7), $u(s_{1},s_{2})$ is expressed as

$\displaystyle
u(s_{1},s_{2})=-\frac{1}{3}c^{4}s_{1}^{2}s_{2}^{4}+c^{8}(\frac{1}{1080}s_{1}^{2}s_{2}^{10}+\frac{1}{216}s_{1}^{4}s_{2}^{8}+\frac{1}{180}s_{1}^{6}s_{2}^{6}+\frac{1}{135}s_{1}^{7}s_{2}^{5}$
$\displaystyle+\frac{1}{360}s_{1}^{8}s_{2}^{4})$
$\displaystyle-c^{12}(\frac{s_{1}^{2}s_{2}^{16}}{3265920}+\frac{s_{1}^{4}s_{2}^{14}}{204120}+\frac{7s_{1}^{6}s_{2}^{12}}{291600}+\frac{s_{1}^{7}s_{2}^{11}}{48600}+\frac{313s_{1}^{8}s_{2}^{10}}{8164800}+\frac{s_{1}^{9}s_{2}^{9}}{22680}$
$\displaystyle+\frac{443s_{1}^{10}s_{2}^{8}}{16329600}+\frac{11s_{1}^{11}s_{2}^{7}}{510300}+\frac{169s_{1}^{12}s_{2}^{6}}{8164800}+\frac{19s_{1}^{13}s_{2}^{5}}{2041200}+\frac{s_{1}^{14}s_{2}^{4}}{583200})+O(c^{16})$

This expression is interpreted as a continuation of $p\to\frac{1}{2}$.
$s_{2}^{4}$ corresponds to $s_{2}^{\frac{2}{p}}$ with $p=\frac{1}{2}$. Since
$s_{2}^{\frac{2}{p}}$ is $t_{0,1}$, $s^{4}$ corresponds to $t_{0,1}$. The term
$s^{5}$ corresponds to $t_{1,1}$.

The intersection number for $s_{1}^{\frac{2}{p}}s_{2}^{4+\frac{2}{p}}$ is
given

$<\tau_{0,1}\tau_{4,1}>_{g=2}=\frac{(p-1)(p+2)(p-2)}{2880p}\to\frac{1}{768}$
(7.31)

This expressesion agrees for the value of [11, 12], $\frac{1}{320}$ and
$\frac{7}{1200}$ for $p=4$ and $p=5$. (7.31) corresponds to
$s_{1}^{\frac{2}{p}}s_{2}^{4+\frac{2}{p}}\to s_{1}^{4}s_{2}^{8}$ for
$p=\frac{1}{2}$. The term, which gives $s_{1}^{4}s_{2}^{8}$ in the limit
$p\to\frac{1}{2}$, may exists also from another term $<\tau_{2,0}\tau_{2,2}>$
with $s_{1}^{2+\frac{1}{p}}s_{2}^{2+\frac{3}{p}}$. Therefore,
$s_{1}^{4}s_{2}^{8}$ is obtained from the limit of different terms of general
$p$. The term $s_{1}^{2+\frac{1}{p}}s_{2}^{2+\frac{3}{p}}$ is as

$u(s_{1},s_{2})=-\frac{1}{576}(p+1)(p-1)\Gamma(1-\frac{1}{p})\Gamma(1-\frac{3}{p})s_{1}^{2+\frac{2}{p}}s_{2}^{2+\frac{1}{p}}$
(7.32)

which by the change $s_{1}\leftrightarrow s_{2}$, it becomes
$s_{1}^{4}s_{2}^{8}$ for $p\to\frac{1}{2}$.

For $s_{1}^{6}s_{2}^{6}$ term in (7), we have

$<\tau_{0,2}\tau_{4,0}>_{g=2}=\frac{(p-1)(p-3)(2p+11)}{5760p}\to\frac{1}{192}$
(7.33)

which corresponds to $s_{1}^{\frac{3}{p}}s_{2}^{4+\frac{1}{p}}\to
s_{1}^{6}s_{2}^{6}$ for $p=\frac{1}{2}$. The values of these intersection
numbers agree with the intersection numbers from (7).

For $<\tau_{2,1}\tau_{2,1}>_{g=2}$ corresponds to
$s_{1}^{2+\frac{2}{p}}s_{2}^{2+\frac{2}{p}}\to s_{1}^{6}s_{2}^{6}$, which
degenerates to (7.33). Thus the continuation to $p=\frac{1}{2}$ is not unique.

The term $s^{\frac{2}{p}}$ and $s^{2+\frac{1}{p}}$ become same $s^{4}$ in the
limit $p\to\frac{1}{2}$. The terms $s^{\frac{2}{p}}$, $s^{2+\frac{2}{p}}$ and
$s^{4+\frac{1}{p}}$ become $s^{6}$ in the limit of $p\to\frac{1}{2}$.

7-2: n-point function of $p=-\frac{1}{2}$

For $p=-\frac{1}{2}$, the selection rule becomes for $k$-marked points

$2g-2+k=-\sum_{i=1}^{k}n_{i}$ (7.34)

where we take $j_{i}=-1$. This requires the negative power $n_{i}$ of $s$ for
higher genus $g$, which leads to the inverse power of $s_{i}$ in $m$-point
function $u(s_{1},...,s_{m})$. As discussed in [9, 10], one point function
$u(s)$ is a contour integral,

$\displaystyle u(s)$ $\displaystyle=$ $\displaystyle
s\oint\frac{dy}{2i\pi}(y+\frac{1}{s^{2}y^{3}})e^{-\frac{c}{y}}$ (7.35)
$\displaystyle=$ $\displaystyle\frac{1}{2}c^{2}s$

This term belongs to genus $g=0$ order, which leads to $n_{1}=1$ from (7.34)
as $s^{n_{1}}$. The coefficient $c^{2}$ is given by the scaling
$(cs^{p+1})^{2}=c^{2}s$. From the expansion of the intersection numbers in
(2), the higher order terms are vanishing for $p=-\frac{1}{2}$ by the existent
factor $(2p+1)$ for $g>1$. In the genus $g=1$ order, the intersection number
$<\tau_{1,0}>=\frac{p-1}{24}$ in (2.3) for the integer $p$ suggests the
nonvanishing value for $p=-\frac{1}{2}$. This order is given by the second
term of the integrand. By the change of $y\to\frac{1}{t}$, (7.35) is written
as

$u(s)=s\int\frac{dt}{2i\pi}(-\frac{1}{t^{2}})(\frac{1}{t}+\frac{t^{3}}{s^{2}})e^{-ct}$
(7.36)

The second term of the integrand becomes by the integration as
$-\frac{1}{c^{2}s}$ $(c=(\frac{1}{4})^{p}=2)$, which is a continuation of
$\frac{p-1}{24}=-\frac{1}{16}$. There is a factor $\Gamma(3)=2$ in (2.3). Thus
genus $g=1$ term is consistent with (2.3).

Two point function of (generalp) is written for $p=-\frac{1}{2}$ as

$\displaystyle u(s_{1},s_{2})$ $\displaystyle=$
$\displaystyle-4(\frac{s_{2}}{s_{1}})^{2}\oint\frac{dy_{1}dy_{2}}{(2i\pi)^{2}}\prod_{i=1}^{2}(y_{i}+\frac{1}{s_{i}^{2}y_{i}^{3}})e^{-c\sum\frac{1}{y_{i}}}$
(7.37)
$\displaystyle\frac{1}{y_{1}^{4}}\frac{1}{(1-\frac{1}{s_{1}^{2}y_{1}^{4}}-(\frac{s_{2}}{s_{1}})^{2}\frac{1}{y_{1}^{2}}(y_{2}^{2}-\frac{1}{s_{2}^{2}y_{2}^{2}}))^{2}+\frac{4}{s_{1}^{4}y_{1}^{4}}(s_{1}+s_{2})^{2}}$

The contour integral is evaluated for the poles of $y_{1}=y_{2}=0$. By the
expansion of the exponential term as
$\sum(-c(\frac{1}{y_{1}}+\frac{1}{y_{2}}))^{k}/k!$, we obtain

$\displaystyle u(s_{1},s_{2})=-4s_{2}^{2}(\frac{1}{4}c^{4}-\frac{c^{8}}{2\cdot
6!}(s_{1}^{2}+8s_{1}s_{2}+12s_{2}^{2})$ $\displaystyle+\frac{c^{12}}{2\cdot
10!}(s_{1}^{4}+24s_{1}^{3}s_{2}+161s_{1}^{2}s_{2}^{2}+384s_{1}s_{2}^{3}+288s_{2}^{4})$
$\displaystyle-\frac{c^{16}}{2\cdot
14!}(s_{1}^{6}+48s_{1}^{5}s_{2}+715s_{1}^{4}s_{2}^{2}+4528s_{1}^{3}s_{2}^{3}+13441s_{1}^{2}s_{2}^{4}+18304s_{1}s_{2}^{5}$
$\displaystyle+9152s_{2}^{6})+O(c^{20}))+(s_{1}\leftrightarrow s_{2})$ (7.38)

Each term is written as $c^{4g}s_{1}^{|n_{1}|}s_{2}^{|n_{2}|}$,
$n_{1}+n_{2}=2g$. For $p=-\frac{1}{2}$, the selection rule becomes for two
point function as ($m_{i}=-1$)

$-2g=n_{1}+n_{2}$ (7.39)

Thus $n_{1},n_{2}$ are negative integers for $s_{1}^{-n_{1}}s_{2}^{-n_{2}}$.
The expansion of (7) is similar to the weak coupling expansion of $p=-2$ case.

By the change of variable $y_{i}=\frac{1}{t_{i}}$, ($i=1,2$), it becomes by
$t_{i}\to t_{i}/c$,

$\displaystyle u(s_{1},s_{2})$ $\displaystyle=$
$\displaystyle-4(\frac{s_{2}}{s_{1}})^{2}\oint\frac{dt_{1}dt_{2}}{(2i\pi)^{2}}(t_{1}+\frac{t_{1}^{5}}{c^{4}s_{1}^{2}})e^{-t_{1}}(\frac{1}{t_{2}^{3}}+\frac{t_{2}}{c^{4}s_{2}^{2}})e^{-t_{2}}$
(7.40)
$\displaystyle\frac{1}{(1-\frac{t_{1}^{4}}{c^{4}s_{1}^{2}}-(\frac{s_{2}}{s_{1}})^{2}t_{1}^{2}(\frac{1}{t_{2}^{2}}-\frac{t_{2}^{2}}{c^{4}s_{2}^{2}}))^{2}+\frac{4t_{1}^{4}}{c^{4}s_{1}^{4}}(s_{1}+s_{2})^{2}}$

From the selection rule of (7.34), in the case two point function $n=2$, we
have

$2g=-(n_{1}+n_{2})$ (7.41)

for $u(s_{1},s_{2})=\sum c_{n_{1},n_{2}}s_{1}^{n_{1}}s_{2}^{n_{2}}$. The
values of $n_{1}$ and $n_{2}$ can be both positive and negative integers. The
expansion of (7.40) is

$\displaystyle u(s_{1},s_{2})$ $\displaystyle=$
$\displaystyle-4(\frac{s_{2}}{s_{1}})^{2}\oint\frac{dt_{1}dt_{2}}{(2i\pi)^{2}}e^{-t_{1}-t_{2}}\frac{t_{1}t_{2}}{(t_{2}^{2}-\frac{s_{2}^{2}}{s_{1}^{2}}t_{1}^{2})^{2}}\biggl{(}1+\frac{t_{1}^{4}}{c^{4}s_{1}^{2}}+\frac{t_{2}^{4}}{c^{4}s_{2}^{2}}$
(7.42)
$\displaystyle+2\frac{t_{1}^{2}}{c^{4}s_{1}^{2}}\frac{t_{1}^{2}-t_{2}^{2}}{1-\frac{s_{2}^{2}}{s_{1}^{2}}\frac{t_{1}^{2}}{t_{2}^{2}}}-4\frac{(s_{1}+s_{2})^{2}}{c^{4}s_{1}^{4}}\frac{t_{1}^{4}}{(1-\frac{s_{2}^{2}}{s_{1}^{2}}\frac{t_{1}^{2}}{t_{2}^{2}})^{2}}+O(\frac{1}{c^{8}})\biggr{)}$

where the dependence of $c$ is obtained from the scaling to $s$ as a
combination of $c^{2}s$.  7-3: Strong coupling of $p=-2$

$\bullet$ $p=-2$ and unitary matrix model It was pointed out that for $p=-2$
corresponds to unitary matrix model [37, 40]. The unitary matrix model has two
phases of the weak coupling and string coupling phases [35, 36].

For one point function, we have for large $s$,

$\displaystyle
u(s)=\frac{i}{2}\oint\frac{dy}{2i\pi}(y+\frac{1}{y^{3}})e^{\frac{c}{s}\frac{y^{4}}{(y^{4}+1)^{2}}}$
$\displaystyle=\frac{2}{s}+\frac{2}{s^{2}}+\frac{2}{s^{3}}+\frac{5}{3s^{4}}+\frac{7}{6s^{5}}+\frac{7}{10s^{6}}+\frac{11}{30s^{7}}+\frac{11\cdot
13}{840s^{8}}+\frac{11\cdot 13}{2016s^{9}}$ $\displaystyle+\frac{11\cdot
13\cdot 17}{2^{5}3^{4}5\cdot 7s^{10}}+\frac{13\cdot 17\cdot
19}{2^{5}3^{4}5^{2}\cdot 7s^{11}}+\frac{13\cdot 17\cdot
19}{2^{6}3^{4}5^{2}\cdot 11s^{12}}+\frac{17\cdot 19\cdot
23}{2^{7}3^{5}5^{2}\cdot 11s^{13}}+\cdots$

where $c=(\frac{1}{4})^{p}=2^{4}$. The contour integral reduces to the
calculation of the residue at $y=(-1)^{1/4}$. This expansion is written by the
modified Bessel function as

$\displaystyle u(s)$ $\displaystyle=$
$\displaystyle=\frac{1}{s}\sum_{k=0}^{\infty}\frac{2^{2k+1}\Gamma(k+\frac{1}{2})}{\Gamma(\frac{1}{2})\Gamma(k+2)k!}\frac{1}{s^{k}}$
(7.44) $\displaystyle=$
$\displaystyle\frac{2}{s}F(\frac{1}{2},2;\frac{4}{s})=\frac{2}{s}e^{\frac{2}{s}}[I_{0}(\frac{2}{s})-I_{1}(\frac{2}{s})]$

where $F(\alpha,\gamma;z)$ is a confluent hypergeometric function, and
$I_{0}(z)$ is a modified Bessel function. The crossover to the weak coupling
of small $s$ is at $s=2$, which corresponds to $s=\frac{1}{\Lambda}=2$. The
external source is $\Lambda$. The small $s$ expansion is given by (2) through
the intersection numbers of $p=-2$ [40]. We have

$\displaystyle u(s)$ $\displaystyle=$
$\displaystyle\frac{1}{s}\oint\frac{du}{2i\pi}e^{-\frac{1}{u+s}+\frac{1}{u}}$
(7.45) $\displaystyle=$
$\displaystyle-\frac{1}{4\pi\sqrt{s}}\int\frac{dx}{x^{2}}e^{-\frac{4x^{2}}{1+x^{2}s}}$
$\displaystyle=$
$\displaystyle-\frac{1}{4\pi\sqrt{s}}\sum_{n=0}^{\infty}\frac{(2n-1)!!}{2^{n}n!}\Gamma(n-\frac{1}{2})(\frac{s}{4})^{n}$

By noting that $(2n-1)!!=2^{n}\Gamma(n+\frac{1}{2})/\sqrt{\pi}$, the last sum
is written as

$u(s)=-\frac{1}{4\pi^{\frac{3}{2}}\sqrt{s}}\sum_{n=0}^{\infty}\frac{\Gamma(n+\frac{1}{2})\Gamma(n-\frac{1}{2})}{n!}(\frac{s}{4})^{n}$
(7.46)

This weak coupling expansion of one point function has been derived and the
comparison with the unitary matrix model has been discussed in [40].

The model $p=-2$ has naturally a logarithmic potential, which becomes
equivalent to unitary $U(N)$ matrix model. The strong coupling expansion in
(7.44) is evaluated by the character expansion of $U(N)$ group, which
character $\chi$ satisfies

$\int
dU\chi_{r}(AU)\chi_{r^{\prime}}(U^{\dagger}A^{\dagger})=\delta_{rr^{\prime}}\frac{\chi_{r}(AA^{\dagger})}{d_{r}}$
(7.47)

The character of $U(N)$ is expressed by

$\chi_{n_{1},n_{2},...}=\frac{{\rm det}[t_{i}^{n_{j}+N-j}]}{{\rm
det}[t_{i}^{N-j}]}$ (7.48)

where $t_{i}$ is eigenvalue of $U$. The dimension $d_{r}$ is equal to
$d_{r}=\chi_{r}(1)$. For the eigenvalue 1, ($t_{i}=1$),

$d_{[2,0]}=\chi_{[2,0]}(1)=\frac{1}{2}(N^{2}+N),\hskip
8.53581ptd_{[1,1]}=\chi_{[1,1]}(1)=\frac{1}{2}(N^{2}-N)$ (7.49)

The integration of unitary matrix $U$ is written as

$\int dU{\rm tr}(AU){\rm tr}(BU^{\dagger})=C_{1}{\rm tr}(AB)$ (7.50)

where $A$ and $B$ are arbitrary matrices,

$\displaystyle\int DU{\rm tr}(A_{1}U){\rm tr}(A_{2}U){\rm
tr}(B_{1}U^{\dagger}){\rm tr}(B_{2}U^{\dagger})=C_{1^{2}}[{\rm
tr}(A_{1}B_{1}){\rm tr}(A_{2}B_{2})$ $\displaystyle+{\rm tr}(A_{1}B_{2}){\rm
tr}(A_{2}B_{1})]+C_{2}[{\rm tr}(A_{1}B_{1}A_{2}B_{2})+{\rm
tr}(A_{1}B_{2}A_{2}B_{1})]$ (7.51)

Generalization of this equation for $n$-times trace is described by
$C_{l_{1},...,l_{n}}$.

The coefficient of the single trace is $C_{n}=C_{n,0,...0}$, which are

$\displaystyle C_{1}=1,\hskip 5.69054ptC_{2}=-\frac{1}{N^{2}-1},\hskip
5.69054ptC_{3}=\frac{4}{(N^{2}-1)(N^{2}-4)},$ $\displaystyle
C_{4}=-\frac{30}{(N^{2}-1)(N^{2}-4)(N^{2}-9)},$ $\displaystyle
C_{5}=\frac{336}{(N^{2}-1)(N^{2}-4)(N^{2}-9)(N^{2}-16)}$ (7.52)

The general formula of $C_{n}$ is [39]

$\displaystyle C_{n}$ $\displaystyle=$
$\displaystyle(-1)^{n}\frac{[(n-1)!]^{3}}{n}\frac{1}{(N^{2}-1)(N^{2}-4)(N^{2}-9)\cdots(N^{2}-(n-1)^{2})}$
(7.53) $\displaystyle\times$
$\displaystyle\sum_{q=0}^{n-1}\frac{1}{q!q!(n-q-1)!(n-q-1)!}$

The coefficient of $1/s^{n}$ in (7) is $2C_{n}$, and agrees with $N=0$ [40,
39].

It is interesting to note that when we change the measure
$(y+\frac{1}{y^{3}})$ to $(y-\frac{1}{y^{3}})$, i.e. D-type, one point
function is vanishing i.e. $u(s)=0$. The replacement of this measure
$y+1/y^{3}$ by $y$, the values of $u(s)$ becomes half of (7).

For the derivation of the $N$ dependence, we make a change of variable $u$ to
$z$ as

$\frac{u-1}{u+1}=e^{-z},\hskip 8.53581ptu=\frac{1+e^{-z}}{1-e^{-z}}$ (7.54)

and

$du=-2\frac{e^{-z}}{(1-e^{-z})^{2}}$ (7.55)

Thus we obtain

$\displaystyle U(s)$ $\displaystyle=$ $\displaystyle\frac{N}{2}\int
due^{\frac{4c}{s(u^{2}-1)}}(\frac{u-1}{u+1})^{N}$ (7.56) $\displaystyle=$
$\displaystyle-N\int_{0}^{\infty}dz\sum_{n=0}^{\infty}\frac{1}{n!}e^{-Nz}[\frac{c}{s}e^{z}(1-e^{-z})^{2}]^{n}\frac{e^{-z}}{(1-e^{-z})^{2}}$

The leading term ($c=0$) gives the Euler characteristics $\zeta(1-2g)$ [7],
the next order is $\frac{c}{s}$, and the second order becomes

$N\frac{c^{2}}{2s^{2}}\int_{0}^{\infty}dze^{z}(1-e^{-z})^{2}e^{-Nz}=\frac{c^{2}}{s^{2}}\frac{1}{N^{2}-1}$
(7.57)

The next order is

$\displaystyle-\frac{N}{6}(\frac{c}{s})^{3}\int dze^{2z}(1-e^{-z})^{4}e^{-Nz}$
$\displaystyle=-\frac{N}{6}(\frac{c}{s})^{3}[\frac{1}{N-2}-\frac{4}{N-1}+\frac{6}{N}-\frac{4}{N+1}+\frac{1}{N+2}]$
$\displaystyle=-4N(\frac{c}{s})^{3}\frac{1}{N(N^{2}-1)(N^{2}-4)}$ (7.58)

Extending these evaluations, we find from (7.56)

$U(s)=\sum_{n=1}^{\infty}\frac{(2n-2)!}{n!s^{n}}\frac{1}{\prod_{l=1}^{n-1}(N^{2}-l^{2})}$
(7.59)

where $c=1$. The number $(2n-2)!/n!$ coincides with the numerator of $C_{n}$
in (7). Thus we find that one point function $U(s)$, which is a single trace
operator, is a generating function of the coefficient $C_{n}$ in (7.53).

For two point function $U(s_{1},s_{2})$ with a logarithmic term is written as

$\displaystyle U(s_{1},s_{2})=\int
du_{1}du_{2}e^{\frac{4c}{s_{1}(u_{1}^{2}-1)}+\frac{4c}{s_{2}(u_{2}^{2}-1)}}(\frac{u_{1}+1}{u_{1}-1})^{N}(\frac{u_{2}+1}{u_{2}-1})^{N}$
$\displaystyle\times\frac{1}{(s_{2}u_{2}-s_{1}u_{1}+(s_{1}+s_{2}))(s_{2}u_{2}-s_{1}u_{1}-(s_{1}+s_{2}))}$
(7.60)

Using the change of variables in (7.54), one obtain

$\displaystyle U(s_{1},s_{2})=s_{1}s_{2}\int
dz_{1}dz_{2}e^{-Nz_{1}-Nz_{2}}\frac{e^{-z_{1}}}{(1-e^{-z_{1}})^{2}}\frac{e^{-z_{2}}}{(1-e^{-z_{2}})^{2}}$
$\displaystyle\times\frac{1}{s_{2}(1+\frac{2e^{-z_{2}}}{1-e^{-z_{2}}})-s_{1}(1+\frac{2e^{-z_{1}}}{1-e^{-z_{1}}})+(s_{1}+s_{2})}$
$\displaystyle\times\frac{1}{s_{2}(1+\frac{2e^{-z_{2}}}{1-e^{-z_{2}}})-s_{1}(1+\frac{2e^{-z_{1}}}{1-e^{-z_{1}}})-(s_{1}+s_{2})}$
$\displaystyle\times
e^{\frac{c}{s_{1}}e^{z_{1}}(1-e^{-z_{1}})^{2}+\frac{c}{s_{2}}e^{z_{2}}(1-e^{-z_{2}})^{2}}$
(7.61)

Expanding the two denominators for $s_{2}>s_{1}$,

$\frac{1}{4s_{2}^{2}}\sum_{m=0}^{\infty}\sum_{l=0}^{\infty}(\frac{1-e^{-z_{2}}}{1-e^{-z_{1}}}e^{-z_{1}})^{m}(\frac{1-e^{-z_{2}}}{1-e^{-z_{1}}}e^{z_{2}})^{l}(\frac{s_{1}}{s_{2}})^{m+l}e^{z_{2}}(1-e^{-z_{2}})^{2}$
(7.62)

with a measure of $du_{i}=e^{-z_{i}}/(1-e^{-z_{i}})^{2}dz_{i}$, we obtain the
strong coupling expansion of $1/s_{1}^{n_{1}}s_{2}^{n_{2}}$,

$\displaystyle U(s_{1},s_{2})=(s_{1}s_{2})\int
dz_{1}dz_{2}e^{-Nz_{1}-Nz_{2}}\frac{e^{-z_{1}}}{(1-e^{-z_{1}})^{2}}\frac{e^{-z_{2}}}{(1-e^{-z_{2}})^{2}}$
$\displaystyle\times\frac{1}{4s_{2}^{2}}\sum_{m=0}^{\infty}\sum_{l=0}^{\infty}(\frac{1-e^{-z_{2}}}{1-e^{-z_{1}}}e^{-z_{1}})^{m}(\frac{1-e^{-z_{2}}}{1-e^{-z_{1}}}e^{z_{2}})^{l}(\frac{s_{1}}{s_{2}})^{m+l}e^{z_{2}}(1-e^{-z_{2}})^{2}$
$\displaystyle\times
e^{\frac{c}{s_{1}}e^{z_{1}}(1-e^{-z_{1}})^{2}+\frac{c}{s_{2}}e^{z_{2}}(1-e^{-z_{2}})^{2}}$
(7.63)

The leading term is given by Euler characteristics $\zeta(1-2g)$,

$\displaystyle U(s_{1},s_{2})=\frac{1}{4s_{2}^{2}}\int
dz_{1}dz_{2}e^{-Nz_{1}-Nz_{2}}\frac{e^{-z_{1}}}{(1-e^{-z_{1}})^{2}}$
$\displaystyle=\frac{1}{4s_{2}^{2}N^{2}}\sum_{n=0}\frac{1}{N^{2n}}(-1)^{n-1}\frac{B_{n}}{2n}$
$\displaystyle=\frac{1}{4s_{2}^{2}N^{2}}\sum_{n}\frac{1}{N^{2n}}\zeta(1-2n)$
(7.64)

where we have to count $s_{1}\leftrightarrow s_{2}$, and $B_{n}$ is Bernoulli
number $B_{1}=\frac{1}{6},B_{2}=\frac{1}{30},B_{3}=\frac{1}{42},...$. Note
this term has no $s_{1}$ dependence.

The next term in the strong coupling is the term of order $1/s_{1}s_{2}$ as we
will see later for $N=0$ in (7). In the strong coupling region,
$U(s_{1},s_{2})$ is expanded in the inverse power of $s_{1}$ and $s_{2}$. The
pole of $(1-e^{-z_{1}})$ in (7) should be cancelled with the expansion of the
exponent in the last term. Then, similar to one point function $U(s)$, two
point function is evaluated in the form of characters of $U(N)$ as shown in
[40], which used a different method as shown here.

We have from (7) two point function for $p=-2$,

$\displaystyle U(s_{1},s_{2})$ $\displaystyle=$
$\displaystyle\frac{c}{4s_{2}N^{2}}+\frac{c^{2}}{4s_{1}s_{2}}\frac{1}{N^{2}(N^{2}-1)}+\frac{c^{3}}{s_{1}^{2}s_{2}}\frac{1}{N^{2}(N^{2}-1)(N^{2}-4)}$
(7.65)
$\displaystyle+\frac{30c^{4}}{4s_{1}^{3}s_{2}}\frac{1}{N^{2}(N^{2}-1)(N^{2}-4)(N^{2}-9)}+O(\frac{1}{s_{1}^{n}s_{2}^{2}})$

The coefficient $C_{1,n}$ is [40]

$\displaystyle C_{1,1}=\frac{1}{N^{2}-1},\hskip
8.53581ptC_{1,2}=-\frac{12}{(N^{2}-1)(N^{2}-4)}$ $\displaystyle
C_{1,3}=\frac{120}{(N^{2}-1)(N^{2}-4)(N^{2}-9)}$ $\displaystyle
C_{1,4}=-\frac{1680}{(N^{2}-1)(N^{2}-4)(N^{2}-9)(N^{2}-16)}$ (7.66)

The expression of $U(s_{1},s_{2})$ of (7) in a strong coupling agrees with the
coefficients of $C_{n_{1},n_{2}}$.

This shows that the $n$ point function $U(s_{1},...,s_{n})$ of $p=-2$ case in
the inverse $s$ expansion is a generating function of the coefficient
$C_{l_{1},...,l_{n}}$ of unitary integral of the multi-trace products.

We now turn to the change of variable $y$. We here use the expression of (6)
for $p=-2$. It becomes for two point function,

$\displaystyle
u(s_{1},s_{2})=-4s_{1}s_{2}\oint\frac{dy_{1}dy_{2}}{(2i\pi)^{2}}(\frac{1+y_{1}^{4}}{y_{1}^{3}})(\frac{1+y_{2}^{4}}{y_{2}^{3}})e^{-\frac{c}{s_{1}}\frac{y_{1}^{4}}{(1+y_{1}^{4})^{2}}-\frac{c}{s_{2}}\frac{y_{2}^{4}}{(1+y_{2}^{4})^{2}}}$
$\displaystyle\times\frac{1}{(s_{1}(y_{1}^{2}-\frac{1}{y_{1}^{2}})-s_{2}(y_{2}^{2}-\frac{1}{y_{2}}))^{2}+4(s_{1}+s_{2})^{2}}$
(7.67)

For the large $s$ expansion, the expansion parameter is coupled to $c$ as
$\frac{c}{s}$ by a scaling. The exponent of the integrand has a pole at
$1+y_{i}^{4}=0$, which leads to
$y_{i}=\pm(-1)^{\frac{1}{4}},y_{i}=\pm(-1)^{\frac{3}{4}}$. This pole appears
for $p\leq-\frac{3}{2}$, since the term of exponent is
$(y^{2}+\frac{1}{y^{2}})^{2p+2}$. Taking the pair of poles
$y_{1}=\pm(-1)^{\frac{1}{4}},y_{2}=\pm(-1)^{\frac{1}{4}}$ or
$y_{1}=\pm(-1)^{\frac{3}{4}},y_{2}=\pm(-1)^{\frac{3}{4}}$, two point function
in the large $s$ is evaluated with a contour integral as,

$\displaystyle
u(s_{1},s_{2})=\frac{c^{2}}{2^{6}}\frac{1}{s_{1}s_{2}}-\frac{c^{3}}{2^{10}}(\frac{1}{s_{1}s_{2}^{2}}+\frac{1}{s_{1}^{2}s_{2}})+\frac{c^{4}}{3\cdot
2^{15}}(\frac{5}{s_{1}s_{2}^{3}}+\frac{7}{s_{1}^{2}s_{2}^{2}}+\frac{5}{s_{1}^{3}s_{2}})$
$\displaystyle-\frac{c^{5}}{2^{20}\cdot
3}(\frac{7}{s_{1}s_{2}^{4}}+\frac{13}{s_{1}^{2}s_{2}^{3}}+\frac{13}{s_{1}^{3}s_{2}^{2}}+\frac{7}{s_{1}^{4}s_{2}})$
$\displaystyle+\frac{c^{6}}{2^{24}\cdot
15}(\frac{21}{s_{1}s_{2}^{5}}+\frac{49}{s_{1}^{2}s_{2}^{4}}+\frac{66}{s_{1}^{3}s_{2}^{3}}+\frac{49}{s_{1}^{4}s_{2}^{2}}+\frac{21}{s_{1}^{5}s_{2}})$
$\displaystyle-\frac{c^{7}}{2^{28}\cdot
15}(\frac{11}{s_{1}s_{2}^{6}}+\frac{31}{s_{1}^{2}s_{2}^{5}}+\frac{53}{s_{1}^{3}s_{2}^{4}}+\frac{53}{s_{1}^{4}s_{2}^{3}}+\frac{31}{s_{1}^{5}s_{2}^{2}}+\frac{11}{s_{1}^{6}s_{2}})+O(c^{8})$
(7.68)

where $c=(\frac{1}{4})^{p}=2^{4}$. This result is consistent with [40], where
there is a factor $N$, which comes from the logarithmic potential. Above
expansion coincides with the result of $N=0$ in [40], since we do not take the
logarithmic potential in (7). From this derivation, it can be found that the
large $s$ region is related to the pole at the solution of $y^{4}+1=0$. This
pole exists for $p=-2$, while this pole does not exists for the positive
integer $p$. It may be interesting to discuss $p=-3/2$ and $p=-3$ cases, which
seem to be related to the interesting phase of the gravity. The relation to
the black hole for $p\to-\infty$ has been discussed in [8].

For two point function in the weak coupling region, we use the expression of
(6) with the value of $p=-2$. Since the leading term in the exponent is
$-c\frac{1}{y_{i}^{4}}$, we make a change of variable
$\frac{1}{y_{i}^{4}}=t_{i}$. Then the exponent becomes
$-ct_{i}/(1+s_{i}t_{i})^{2}$. The two point function $u(s_{1},s_{2})$ is
written as

$\displaystyle u(s_{1},s_{2})$ $\displaystyle=$
$\displaystyle-\frac{1}{4}(\frac{s_{2}}{s_{1}})^{\frac{1}{2}}\oint\frac{dt_{1}dt_{2}}{(2i\pi)^{2}}t_{1}^{-\frac{1}{2}}t_{2}^{-\frac{3}{2}}(1+s_{1}t_{1})(1+s_{2}t_{2})e^{-c\sum\frac{t_{i}}{(1+s_{i}t_{i})^{2}}}$
(7.69) $\displaystyle\times$ $\displaystyle\frac{1}{1-g}$

where $g$ is

$g=2s_{1}t_{1}+2(\frac{s_{2}t_{1}}{s_{1}t_{2}})^{\frac{1}{2}}(1-s_{2}t_{2})-[s_{1}t_{1}+(\frac{s_{2}t_{1}}{s_{1}t_{2}})^{\frac{1}{2}}(1-s_{2}t_{2})]^{2}-\frac{4t_{1}}{s_{1}}(s_{1}+s_{2})^{2}$
(7.70)

By the expansion of $1/(1-g)=\sum g^{n}$, we evaluate the terms of order
$s_{2}^{n_{2}}s_{1}^{n_{1}}$. The selection rulein (2.6) becomes for two
marked points of $p=-2$,

$g+1+\frac{1}{2}(j_{1}+j_{2})=n_{1}+n_{2}$ (7.71)

From (7.69), we find the terms in the lower orders similar to (2.1)-(2.5) in
the previous section. For $g=1$, $j_{1}=j_{2}=0$, we obtain the term
$s_{1}^{\frac{1}{2}}s_{2}^{\frac{1}{2}}$ as

$\displaystyle u(s_{1},s_{2})$ $\displaystyle=$ $\displaystyle
c^{-1}s_{1}^{\frac{1}{2}}s_{2}^{\frac{1}{2}}\int\frac{dt_{1}st_{2}}{(2i\pi)^{2}}\frac{1}{2t_{2}^{\frac{3}{2}}}(t_{1}^{\frac{1}{2}}-t_{1}^{\frac{3}{2}})e^{-t_{1}-t_{2}}$
(7.72) $\displaystyle=$
$\displaystyle-\frac{1}{16\pi}s_{1}^{\frac{1}{2}}s_{2}^{\frac{1}{2}}$

For the term of order $s_{1}s_{2}$, which is $g=2$, $j_{1}=j_{2}=-1$,

$\displaystyle u(s_{1},s_{2})$ $\displaystyle=$ $\displaystyle
c^{-2}s_{1}s_{2}\int\frac{dt_{1}dt_{2}}{(2i\pi)^{2}}e^{-t_{1}-t_{2}}\frac{1}{t_{2}^{2}}(-7t_{1}^{2}+13t_{1}^{3}-t_{1}^{4})$
(7.73) $\displaystyle=$ $\displaystyle-\frac{5}{64\pi}s_{1}s_{2}$

where the integral of $t_{2}$ is a residue evaluation, and the integral
$t_{1}$ is the exponential integral for $0<t_{1}<\infty$. The coefficient $c$
is $c=4^{-p}=4^{2}$.

The terms $s_{1}^{n_{1}}s_{2}^{n_{2}}$ ($n_{1},n_{2}$: integers) belong to
Ramond sector ($j_{1}=j_{2}=-1$), and they appear in a pair. The terms of
$s_{1}^{n_{1}+\frac{1}{2}}s_{2}^{n_{2}+\frac{1}{2}}$ belong to Neveu-Schwarz
sectors with the spin components $j_{1}=j_{2}=0$ in $u(s_{1},s_{2})\sim
s_{1}^{n_{1}-\frac{1+j_{1}}{2}}s_{2}^{n_{2}-\frac{1+j_{2}}{2}}$.

7-4: Strong coupling of $p=-3$ For $p=-3$ case, the strong coupling expansion
of the inverse of $s$ is

$u(s)=\frac{i}{2}\oint\frac{dy}{2i\pi}(y+\frac{1}{y^{3}})e^{\frac{c}{s^{2}}\frac{y^{6}(y^{4}-1)}{(y^{4}+1)^{4}}}$
(7.74)

This large $s$ expansion is given by the residue at $y=(-1)^{1/4}$,

$\displaystyle u(s)=\frac{1}{s^{4}}+\frac{3}{8s^{8}}+\frac{13}{2^{6}\cdot
3s^{12}}+\frac{17\cdot 19}{2^{10}\cdot 3^{2}\cdot
5s^{16}}+\frac{23}{2^{14}\cdot 3s^{20}}$ $\displaystyle+\frac{29\cdot
31}{2^{17}\cdot 3^{2}\cdot 5\cdot 7s^{24}}+\frac{29\cdot 31\cdot
37}{2^{20}\cdot 3^{5}\cdot 5^{2}\cdot 7s^{28}}+\frac{37\cdot 41\cdot
43}{2^{25}\cdot 3^{4}\cdot 5^{2}\cdot 7^{2}s^{32}}$
$\displaystyle+\frac{37\cdot 41\cdot 43\cdot 47}{2^{30}\cdot 3^{6}\cdot
5^{3}\cdot 7\cdot 11s^{36}}+\frac{41\cdot 43\cdot 47\cdot
53}{2^{33}3^{8}5^{3}\cdot 7\cdot 13s^{40}}+\frac{47\cdot 53\cdot 59\cdot
61}{2^{36}3^{7}5^{4}\cdot 7\cdot 11\cdot 13s^{44}}$ $\displaystyle+\cdots$
(7.75)

with $c=2^{6}$. These terms of order of $\frac{1}{s^{2k}}$ is expressed as

$\displaystyle u(s)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\frac{(6n-5)!!}{(4n-1)!!(2n-3)!!(n-1)!(n-1)!}\frac{3}{n2^{2n-2}}\frac{1}{s^{4n}}$
(7.76) $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{(6n+1)!(2n+1)}{(4n+4)!(3n)!n!}\frac{8}{4^{n}s^{4n+4}}$

With a logarithmic potential, $p=-3$ one point function $U(s)$ becomes

$U(s)=\oint\frac{du}{2i\pi}e^{\frac{c}{s^{2}}(\frac{1}{(u+1)^{2}}-\frac{1}{(u-1)^{2}})}(\frac{u+1}{u-1})^{N}$
(7.77)

Similar to $p=-2$ case, we make a change of variables of (7.54). Then we have

$\displaystyle U(s)$ $\displaystyle=$ $\displaystyle-2\int
dze^{-Nz}\frac{e^{-z}}{(1-e^{-z})^{2}}e^{-\frac{c}{4s^{2}}e^{2z}(1-e^{-z})^{3}(1+e^{-z})}$
(7.78) $\displaystyle=$ $\displaystyle-2\int
dz\frac{e^{-z}}{(1-e^{-z})^{2}}e^{\frac{c}{4s^{2}}[-2e^{-z}+e^{-2z}-e^{2z}+2e^{z}]}e^{-Nz}$

This gives a strong coupling expansion. The term of order $\frac{c}{s^{2}}$ is

$-(\frac{c}{s^{2}})\frac{1}{N^{2}-1}$

The term of order $\frac{c^{2}}{s^{4}}$ is

$\displaystyle-(\frac{c}{4s^{2}})^{2}\int
dze^{-Nz}e^{3z}(1-e^{-z})^{4}(1+e^{-z})^{2}$
$\displaystyle=-(\frac{c}{4s^{2}})^{2}\frac{48(2N^{2}-3)}{N(N^{2}-1)(N^{2}-4)(N^{2}-9)}$
(7.79)

The order of $\frac{c^{3}}{s^{6}}$ in $U(s)$ for $p=-3$ becomes

$\frac{8965}{4}(\frac{c^{3}}{s^{6}})\frac{N^{2}-7}{(N^{2}-1)(N^{2}-4)(N^{2}-9)(N^{2}-16)(N^{2}-25)}$

The order of $\frac{c^{4}}{s^{8}}$ becomes

$-9450\frac{c^{4}}{s^{8}}\frac{2N^{4}-38N^{2}+63}{N(N^{2}-1)(N^{2}-4)(N^{2}-9)(N^{2}-16)(N^{2}-25)(N^{2}-36)(N^{2}-49)}$

The denominator is same as $p=-2$ case, but the numerator is different from
$p=-2$ case. There is no term of order $\frac{1}{s^{2n+1}}$. The terms of
order $\frac{1}{s^{4n}}$ have a factor $\frac{1}{N}$ and there is no such
factor for other terms. When $N\to 0$, $U(s)$ becomes in the power series of
$\frac{1}{s^{4n}}$. Indeed the expansion of (7) agrees with the expansion of
(7.78), and the coefficient is consistent with $c=2$.

##  8 Summary and discussions

In this article, we have extended the results of previous articles I and II
[9, 10] to $D_{l}$ type and to the multiple correlation functions of $p$ spin
curve, specially for the non-positive integer values of $p$.

The agreement with the values, evaluated by the Gelfand Dikii equations [11,
12], has been shown in the Laurent expansions of $y$ variable.

The intersection number of one marked point was examined for the large $p$ and
large genus $g$. The analysis reveals interesting relations between the
intersection numbers and the number theory through Bernoulli numbers. The
intersection numbers are shown to be expressed as Bernoulli numbers ( for
$p\to\infty$) multiplied a polynomial of $p$. Therefore, the denominator of
the intersection numbers have common values of the denominator of Bernoulli
numbers. For $p=\frac{1}{2}$ spin case (fermionic), this denominator is
cancelled by the numerator, and the intersection numbers are simply expressed
as (4.24).

This integral representation enables us to continue the integer $p$ to the
non-positive integer $p$, like $p=\frac{1}{2}$, $p=-\frac{1}{2}$ and $p=-2$
for which we evaluated $n$ point functions explicitly as discussed in I and II
[9, 10].

Since the central charge is given by $C=2-\frac{6}{p}$, the case
$p=-\frac{1}{2}$ corresponds to $C=14$ for instance. Such extension of the
central charge $C$ ($C>1$) is interesting from the view point of conformal
field theory, since it goes over a barrier at $C=1$. The CFT for $C>1$ has
attracted interest for the case $1<C<26$, which is related to 2d gravity
coupled to matter field of the central charge $C$ [47], and in this region,
the behavior like a branched polymer is expected. The behavior of quantum
Liouville theory is discussed recently in the probabilistic approach [48]. The
conformal field theory of $p$ spin curves in the area of $1<C$, $p<0$ is
interesting with respect to gauge theory and quantum gravity in higher
dimensions and further studies are desirable.

The $D_{l}$ type singularity has been investigated, under the new
representation of the contour integrals in the variables of $y_{i}$ for
$m$-point correlation functions. The difference between $A_{l}$ and $D_{l}$
types is characterized by the factor, which appears as the different measure
of the contour integrals. We have shown that $D_{l}$ type is obtained by the
logarithmic term, which is expected from the supersymmetric random matrices
[20].

Acknowledgement The author thank Edouard Brézin for the encouragement and
joint works, which appeared as I, II [9, 10]. He also thank Andreani Petrou
for the discussions of knot polynomials. This work is supported by JSPS
Kakenhi 19H01813.

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