UT-Komaba/22-1

July 2022

Quantizing multi-pronged open string junction

Masako Asano† and Mitsuhiro Kato‡

†Faculty of Science and Technology
Seikei University
Musashino-shi, Tokyo 180-8633, Japan
‡Institute of Physics

University of Tokyo, Komaba
Meguro-ku, Tokyo 153-8902, Japan

ABSTRACT

> Covariant quantization of multi-pronged open bosonic string junction is
> studied beyond static analysis. Its excited states are described by a set of
> ordinary bosons as well as some sets of twisted bosons on the world-sheet.
> The system is characterized by a certain large algebra of twisted type which
> includes a single Virasoro algebra as a subalgebra. By properly defining the
> physical states, one can show that there are no ghosts in the Hilbert space.

##  1 Introduction

Since the 1990’s, when D-branes and various string dualities were found,
string junctions have been studied by many authors in the context of
superstrings and M-theory.111We only cite lectures [1, 2] that explain their
essential points since there are so many papers and we cannot list them all.
These analyses mainly focused on the static properties such as BPS conditions
or stability, with a few exceptions (e.g. [3, 4].) String junctions are
dynamical objects formed by dynamical strings, so that one can naturally ask
their dynamical properties such as spectrum of their excited states and other
quantum features beyond the static properties.

Going back to the 1970’s, some earlier works which studied classical motions
[5, 6, 7] and simple-minded quantization [8] of string junctions appeared. In
those days string junction was considered as a model of baryon. They naively
tried to quantize 3-string junction, although they did not reach physical
spectrum largely due to the non-closed property of the constraint algebra. In
1984, ref.[9] analyzed the constraint structure more carefully and got deeper
insight on classical solutions, but still left the full quantum spectrum
undetermined.

In the present paper, the authors are going to revisit the problem. In
particular, we propose a set of physical state conditions under which we can
show there are no ghosts in the spectrum. In the following, we treat multi-
pronged open bosonic string junctions in flat spacetime. One end of each
string segment is tied together, whereas the other end is free (See Fig.1.) We
covariantly quantize the system based on so-called “old covariant quantization
(OCQ).”

Figure 1: $f$-pronged open string junction

The system is different from a system of multi free strings. Instead, string
segments are somehow interacting each other through the connecting point.
Therefore the constraints algebra governing this system is not just a multiple
of the Virasoro algebra, but rather infinite dimensional open algebra which
includes a single Virasoro algebra as a subalgebra.

Also the state space is not just a multiple of single string Fock space. One
needs to introduce some sets of twisted bosons in addition to a set of
ordinary bosons on the worldsheet. Accordingly, operator algebra of
constraints is of twisted type, namely it contains both periodic and anti-
periodic parts. Thus we find many interesting and non-trivial features worth
investigating in its own right.

The paper is organized as follows. In Section 2, we formulate $f$-pronged open
bosonic string junction starting from Nambu-Goto type action. We determine
mode expansion of the variables in the orthonormal gauge and study structure
of primary constraints. In Section 3, we investigate physical state conditions
in detail. In Section 4, we concretely determine physical spectrum and discuss
its properties. There are many things to be clarified part of which are
discussed in the final section. Some useful algebraic relations are collected
in Appnedices A and B. Some details of the physical states are given in
Appendix C. We make a remark on light-cone gauge in Appendix D.

##  2 $f$-pronged open string junction

Let us consider $f$-pronged open bosonic string junction shown in Fig.1. We
denote a coordinate variable of $i$-th string segment by
$X^{(i)\mu}(\tau,\sigma)$ $(i=1,2,\cdots,f)$, where $\mu$ $(=0,1,\cdots,D-1)$
is a space-time index and $\tau$, $\sigma$ $(\sigma\in[0,\pi])$ are the world-
sheet parameters.222We adopt $\eta^{\mu\nu}=\mbox{diag.}(-1,1,\cdots,1)$ for
the target space metric. Each string is connected to the other strings at
$\sigma=0$, i.e.

$X^{(i)\mu}(\tau,0)=X^{(j)\mu}(\tau,0)\qquad\mbox{for arbitrary $i$ and $j$,}$
(1)

while $\sigma=\pi$ is a free end. We also use a notation
$\xi^{\alpha}=(\tau,\sigma)$ $(\alpha=0,1)$ for the world-sheet parameters.
Then the Nambu-Goto type action for the system is

$S=-T\sum_{i=1}^{f}\int d\tau
d\sigma\sqrt{-\det[\partial_{\alpha}X^{(i)}{}_{\mu}(\tau,\sigma)\,\partial_{\beta}X^{(i)\mu}(\tau,\sigma)]},$
(2)

where $T(=\frac{1}{2\pi\alpha^{\prime}})$ is the string tension and the
determinant is taken with respect to the indices $\alpha,\beta$ of the partial
derivatives. In the following, we sometimes abbreviate a contraction of space-
time indices simply by dot; e.g. $A\cdot B\equiv A_{\mu}B^{\mu}$. Also denote
$\dot{X}\equiv\partial_{\tau}X$ and $X^{\prime}\equiv\partial_{\sigma}X$ as a
customary use. We can write the action more explicit way as

$S=-T\sum_{i=1}^{f}\int d\tau
d\sigma\sqrt{-(\dot{X}^{(i)}\cdot\dot{X}^{(i)})(X^{\prime(i)}\cdot
X^{\prime(i)})+(\dot{X}^{(i)}\cdot X^{\prime(i)})^{2}}.$ (3)

The action has a reparametrization invariance under the (local) transformation

$\delta
X^{(i)\mu}(\tau,\sigma)=-\epsilon^{(i)\alpha}(\tau,\sigma)\,\partial_{\alpha}X^{(i)\mu}(\tau,\sigma),$
(4)

as far as infinitesimal transformation parameters
$\epsilon^{(i)\alpha}(\tau,\sigma)$ satisfy

$\epsilon^{(i)1}(\tau,0)=0\qquad\mbox{ and
}\qquad\epsilon^{(i)1}(\tau,\pi)=0.$ (5)

Since the $i$-th term in the action only depends on the $i$-th variable
$X^{(i)\mu}$, $\epsilon^{(i)\alpha}(\tau,\sigma)$ can be taken as
independently of each $i$ except for the boundary condition

$\epsilon^{(i)0}(\tau,0)=\epsilon^{(j)0}(\tau,0)\qquad\mbox{for arbitrary $i$
and $j$,}$ (6)

which keeps the condition (1).

Taking a variation $\delta X^{(i)\mu}$ in the action, we obtain a set of
equations of motion

$\displaystyle\partial_{\tau}\left(\frac{\dot{X}^{(i)\mu}(X^{\prime(i)}\cdot
X^{\prime(i)})-X^{\prime(i)\mu}(\dot{X}^{(i)}\cdot
X^{\prime(i)})}{\sqrt{-(\dot{X}^{(i)}\cdot\dot{X}^{(i)})(X^{\prime(i)}\cdot
X^{\prime(i)})+(\dot{X}^{(i)}\cdot X^{\prime(i)})^{2}}}\right)$
$\displaystyle+\partial_{\sigma}\left(\frac{X^{\prime(i)\mu}(\dot{X}^{(i)}\cdot\dot{X}^{(i)})-\dot{X}^{(i)\mu}(\dot{X}^{(i)}\cdot
X^{\prime(i)})}{\sqrt{-(\dot{X}^{(i)}\cdot\dot{X}^{(i)})(X^{\prime(i)}\cdot
X^{\prime(i)})+(\dot{X}^{(i)}\cdot X^{\prime(i)})^{2}}}\right)=0$ (7)

and boundary conditions

$\left.\frac{X^{\prime(i)\mu}(\dot{X}^{(i)}\cdot\dot{X}^{(i)})-\dot{X}^{(i)\mu}(\dot{X}^{(i)}\cdot
X^{\prime(i)})}{\sqrt{-(\dot{X}^{(i)}\cdot\dot{X}^{(i)})(X^{\prime(i)}\cdot
X^{\prime(i)})+(\dot{X}^{(i)}\cdot
X^{\prime(i)})^{2}}}\right|_{\sigma=\pi}=0,$ (8)
$\left.\sum_{i=1}^{f}\frac{X^{\prime(i)\mu}(\dot{X}^{(i)}\cdot\dot{X}^{(i)})-\dot{X}^{(i)\mu}(\dot{X}^{(i)}\cdot
X^{\prime(i)})}{\sqrt{-(\dot{X}^{(i)}\cdot\dot{X}^{(i)})(X^{\prime(i)}\cdot
X^{\prime(i)})+(\dot{X}^{(i)}\cdot X^{\prime(i)})^{2}}}\right|_{\sigma=0}=0$
(9)

as a stationary condition for the action.

Canonical conjugate momentum for each variable $X^{(i)\mu}$ is

$P^{(i)}_{\mu}=T\frac{\dot{X}^{(i)}_{\mu}(X^{\prime(i)}\cdot
X^{\prime(i)})-X^{\prime(i)}_{\mu}(\dot{X}^{(i)}\cdot
X^{\prime(i)})}{\sqrt{-(\dot{X}^{(i)}\cdot\dot{X}^{(i)})(X^{\prime(i)}\cdot
X^{\prime(i)})+(\dot{X}^{(i)}\cdot X^{\prime(i)})^{2}}},$ (10)

from which we obtain primary constraints

$P^{(i)}_{\mu}P^{(i)\mu}+T^{2}X^{\prime(i)}_{\mu}X^{\prime(i)\mu}=0,$ (11)
$P^{(i)}_{\mu}X^{\prime(i)\mu}=0.$ (12)

The Hamiltonian given by the Legendre transform of Lagrangian in (3) vanishes.
Therefore the total Hamiltonian consists only of primary constraints
multiplied by arbitrary parameter functions $u_{1}^{(i)}(\tau,\sigma)$ and
$u_{2}^{(i)}(\tau,\sigma)$,

$H_{T}=\sum_{i=1}^{f}\int_{0}^{\pi}d\sigma\left[u_{1}^{(i)}(P^{(i)2}+T^{2}X^{\prime(i)2})+u_{2}^{(i)}P^{(i)}\cdot
X^{\prime(i)}\right].$ (13)

The above constraints are all first-class and there are no more constraints
coming from their time evolution.

Now let us take a gauge

$u_{1}^{(i)}=\frac{1}{2T},\quad u_{2}^{(i)}=0,$ (14)

which is equivalent to the so-called orthonormal gauge which imposes

$\dot{X}^{(i)}\cdot\dot{X}^{(i)}+X^{\prime(i)}\cdot X^{\prime(i)}=0,$ (15)
$\dot{X}^{(i)}\cdot X^{\prime(i)}=0,$ (16)

or alternatively

$(\dot{X}^{(i)}\pm X^{\prime(i)})^{2}=0.$ (17)

In this gauge, the equations of motion and the boundary conditions are largely
simplified as follows

$\mbox{(Eq.~{}of
motion)}\\!\qquad(\partial_{\tau}^{2}-\partial_{\sigma}^{2})X^{(i)\mu}=0,$
(18) $\mbox{(Boundary
cond.)}\qquad\partial_{\sigma}X^{(i)\mu}|_{\sigma=\pi}=0,$ (19)
$\sum_{i=1}^{f}\partial_{\sigma}X^{(i)\mu}|_{\sigma=0}=0.$ (20)

These equations combined with (1) are enough to determine the mode expansion
of $X^{(i)\mu}$. Note that in this gauge canonical momentum variable becomes
simply $P^{(i)\mu}=T\dot{X}^{(i)\mu}$. In the following, after canonically
quantizing the system, we will basically impose that physical states should
satisfy the relation

$\langle{\rm phys}|(P^{(i)}\pm TX^{\prime(i)})^{2}|{\rm phys}\rangle=0.$ (21)

Of course, in the quantized version, some central term can appear in the
constraint algebra, so we have to be careful about treating zero mode part of
constraints in operator level, which will be discussed later.

###  2.1 Mode expansion

A general solution $X^{(i)\mu}(\tau,\sigma)$ of the equations of motion (18)
consists of left-moving and right-moving modes

$X^{(i)\mu}(\tau,\sigma)=X^{(i)\mu}_{L}(\tau+\sigma)+X^{(i)\mu}_{R}(\tau-\sigma).$
(22)

The boundary conditions also restrict the form of the functions of each mode:
Eq.(19) gives

$\dot{X}^{(i)\mu}_{L}(\tau+\pi)-\dot{X}^{(i)\mu}_{R}(\tau-\pi)=0,$ (23)

while eq.(20) gives

$\sum_{i=1}^{f}\left(\dot{X}^{(i)\mu}_{L}(\tau)-\dot{X}^{(i)\mu}_{R}(\tau)\right)=0.$
(24)

And eq.(1) leads to

$X^{(1)\mu}_{L}(\tau)+X^{(1)\mu}_{R}(\tau)=\cdots=X^{(f)\mu}_{L}(\tau)+X^{(f)\mu}_{R}(\tau),$
(25)

which tells us that each term $X^{(i)\mu}_{L}(\tau)+X^{(i)\mu}_{R}(\tau)$
equals to a certain $i$-independent function $\phi^{\mu}(\tau)$. Thus,

$X^{(i)\mu}_{R}(\tau)=-X^{(i)\mu}_{L}(\tau)+\phi^{\mu}(\tau).$ (26)

Substituting this relation into eq.(24), we have

$2\sum_{i=1}^{f}\dot{X}^{(i)\mu}_{L}(\tau)=f\dot{\phi}^{\mu}(\tau),$ (27)

which means

$\phi^{\mu}(\tau)=\frac{2}{f}\sum_{i=1}^{f}X^{(i)\mu}_{L}(\tau)+c^{\mu},$ (28)

where $c^{\mu}$ is a constant. Then the right-mover is completely determined
by the left-mover up to the constant:

$X^{(i)\mu}_{R}(\tau)=\sum_{j=i}^{f}A_{ij}X^{(j)\mu}_{L}(\tau)+c^{\mu}$ (29)

where $A_{ij}=\frac{2}{f}-\delta_{ij}$. Also the periodicity of the left-mover
is obtained through eq.(23) as

$\dot{X}^{(i)\mu}_{L}(\tau+2\pi)=\sum_{j=1}^{f}A_{ij}\dot{X}^{(j)\mu}_{L}(\tau)=\dot{X}^{(i)\mu}_{R}(\tau)$
(30)

It is easy to obtain eigenvalues and eigenvectors of the $f\times f$ matrix
$A=(A_{ij})$. Writing $A=\frac{2}{f}Z-I$ with the matrix $Z$ whose every
element is 1 ($Z_{ij}=1$) and the unit matrix $I$, we can first solve the
eigenvalue problem for $Z$. Apparently, a vector $v_{1}$ whose every element
is $\frac{1}{\sqrt{f}}$ is a normalized eigenvector of $Z$ with eigenvalue
$f$, and $f-1$ vectors $v_{a}$ ($a=2,\cdots,f$) orthogonal to $v_{1}$ and
themselves (i.e. $v_{1}\cdot v_{a}=0$, $v_{a}\cdot v_{b}=\delta_{ab}$) are
those with eigenvalue 0. Therefore, they are also the eigenvectors for $A$ as

$Av_{1}=v_{1},\qquad Av_{a}=-v_{a}\quad(a=2,\cdots,f).$ (31)

The eigenvectors $v_{i}$ are determined by

$v_{1}^{T}=\left(\frac{1}{\sqrt{f}}\,\cdots\,\frac{1}{\sqrt{f}}\right),\qquad
v_{1}\cdot v_{a}=0,\quad v_{a}\cdot v_{b}=\delta_{ab}.$ (32)

Note that one simple representation for $v_{a}$ ($a=2,\cdots,f$) can be chosen
as follows

$(v_{a})_{i}=\left\\{\begin{array}[]{ll}\frac{1}{\sqrt{a(a-1)}}&\quad(i=1,\cdots
a-1)\\\ -\sqrt{\frac{a-1}{a}}&\quad(i=a)\\\
0&\quad(i=a+1,\cdots,f).\end{array}\right.$ (33)

By defining a real orthogonal matrix $U$ whose $i$-th row is just $v_{i}$
($i=1,\cdots,f$), i.e., $U_{ij}=(v_{i})_{j}$, diagonalization of $A$ is
expressed as

$UAU^{T}=\Gamma\equiv\left(\begin{array}[]{cccc}1&&&\\\ &-1&&\\\ &&\ddots&\\\
&&&-1\end{array}\right).$ (34)

Now let us define

$Y^{i\mu}(\tau)\equiv\sum_{j=1}^{f}U_{ij}X^{(j)\mu}_{L}(\tau).$ (35)

Conversely,

$X^{(i)\mu}_{L}(\tau)=\sum_{j=1}^{f}U_{ji}Y^{j\mu}(\tau).$ (36)

Then from eq.(30),

$\dot{Y}^{i\mu}(\tau+2\pi)=\left\\{\begin{array}[]{ll}\dot{Y}^{1\mu}(\tau)&(i=1)\\\
-\dot{Y}^{i\mu}(\tau)&(i\neq 1).\end{array}\right.$ (37)

If we define also for the right-mover

$\tilde{Y}^{i\mu}(\tau)\equiv\sum_{j=1}^{f}U_{ij}X^{(j)\mu}_{R}(\tau),$ (38)

then

$\tilde{Y}^{i\mu}(\tau)=\left\\{\begin{array}[]{ll}Y^{1\mu}(\tau)+\tilde{c}^{\mu}&(i=1)\\\
-Y^{i\mu}(\tau)&(i\neq 1)\end{array}\right.$ (39)

where

$\tilde{c}^{\mu}=c^{\mu}\sum_{i=1}^{f}U_{1i}=c^{\mu}\sqrt{f}.$ (40)

Now we see from eq.(37) that $Y^{1\mu}(\tau)$ is periodic (up to constant)
with periodicity $2\pi$ whereas $Y^{a\mu}(\tau)$ ($a=2,\cdots,f$) are anti-
periodic. Therefore their mode expansions become

$Y^{1\mu}(\tau)=\frac{1}{\sqrt{2}}\left[q^{\mu}+p^{\mu}\tau+i\sum_{n\neq
0}\frac{\alpha^{\mu}_{n}}{n}e^{-in\tau}\right],$ (41) $\hskip
27.0301ptY^{a\mu}(\tau)=\frac{i}{\sqrt{2}}\sum_{r\in{\bf
Z}+\frac{1}{2}}\frac{\alpha^{a\mu}_{r}}{r}e^{-ir\tau}\qquad\mbox{($a=2,\cdots,f$).}$
(42)

Bringing them all back to the original variables, we obtain (also making
$q^{\mu}$ absorb a constant ambiguity $c^{\mu}$)

$\displaystyle X^{(i)\mu}(\tau,\sigma)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{f}}\left[q^{\mu}+p^{\mu}\tau+i\sum_{n\neq
0}\frac{\alpha^{\mu}_{n}}{n}e^{-in\tau}\cos(n\sigma)\right]$ (43)
$\displaystyle+\sqrt{2}\sum_{a=2}^{f}(v_{a})_{i}\sum_{r\in{\bf
Z}+\frac{1}{2}}\frac{\alpha^{a\mu}_{r}}{r}e^{-ir\tau}\sin(r\sigma).$

One can easily check that this expression satisfies the equations of motion
(18) and all the boundary conditions (1), (19) and (20) by noting the property
$\sum_{i=1}^{f}(v_{a})_{i}=0$ which comes from $v_{a}\cdot v_{1}=0$.

To quantize the system, we will set canonical equal-time commutation relation
in the interval $\sigma,\sigma^{\prime}\in[0,\pi]$ with the canonical
conjugate momentum
$P^{(i)\mu}(\tau,\sigma)=\frac{1}{2\pi}\dot{X}^{(i)\mu}(\tau,\sigma)$,

$\left[X^{(i)\mu}(\tau,\sigma),P^{(j)\nu}(\tau,\sigma^{\prime})\right]=i\eta^{\mu\nu}\delta_{ij}\delta(\sigma-\sigma^{\prime}).$
(44)

Here (and hereafter) we have taken $T=\frac{1}{2\pi}$ ($\alpha^{\prime}=1$)
for brevity. This relation leads to

$[\alpha^{\mu}_{n},\alpha^{\nu}_{m}]=n\delta_{n+m,0}\eta^{\mu\nu},\qquad[\alpha^{a\mu}_{r},\alpha^{b\nu}_{s}]=r\delta_{r+s,0}\delta^{ab}\eta^{\mu\nu},\qquad[q^{\mu},p^{\nu}]=i\eta^{\mu\nu},$
(45)

where $n,m\in{\bf Z}$, $r,s\in{\bf Z}+\frac{1}{2}$ and $\alpha^{\mu}_{0}\equiv
p^{\mu}$. Thus our total Fock space is described by $D$ ordinary bosons and
$(f-1)D$ twisted bosons.

###  2.2 Constraints

Let us turn to the constraints (11) and (12). They can be combined into

$(2\pi P^{(i)}\pm X^{\prime(i)})^{2}\approx 0,$ (46)

where wavy equal is used in the sense of eq.(21). Each of these can be written
in terms of the left- and right-mover

$\left(\dot{X}^{(i)}_{L}(\tau)\right)^{2}\approx
0,\qquad\left(\dot{X}^{(i)}_{R}(\tau)\right)^{2}\approx 0.$ (47)

Furthermore, in terms of the mode-diagonal variables $Y^{i\mu}$, they become

$\sum_{j,k=1}^{f}K^{i}_{jk}\,\dot{Y}^{j}(\tau)\cdot\dot{Y}^{k}(\tau)\approx
0,\qquad\sum_{j,k=1}^{f}\tilde{K}^{i}_{jk}\,\dot{Y}^{j}(\tau)\cdot\dot{Y}^{k}(\tau)\approx
0,$ (48) $K^{i}_{jk}\equiv U_{ji}U_{ki},\qquad\tilde{K}^{i}\equiv\Gamma
K^{i}\Gamma.$ (49)

Here we used matrix notation $(K^{i})_{jk}\equiv K^{i}_{jk}$ and $\Gamma$ was
defined in (34). The constraints defined by $K^{i}$ and those of
$\tilde{K}^{i}$ are interchanged when $\tau$ goes to $\tau+2\pi$ since
$\dot{Y}^{i\mu}(\tau+2\pi)=(\Gamma\dot{Y})^{i\mu}(\tau)$. If we recombine them
into those of $K^{i}+\tilde{K}^{i}$ and $K^{i}-\tilde{K}^{i}$, then each set
of constraints has definite periodicity. So one may think that there are $f$
periodic and $f$ anti-periodic independent constraints. The situation,
however, is not so simple. To see this, let us look at the structure of
$K^{i}$.

We first define the following three types of $f\times f$ symmetric matrices,

$P=\left(\begin{tabular}[]{c|ccc}$\frac{1}{f}$&&&\\\ \hline\cr&&&\\\ &&&\\\
&&&\end{tabular}\right),\qquad Q^{ij}=\left(\begin{tabular}[]{c|ccc}&&&\\\
\hline\cr&&&\\\
&&$\frac{1}{2}(v_{a})_{i}(v_{b})_{j}+\frac{1}{2}(v_{a})_{j}(v_{b})_{i}$&\\\
&&&\end{tabular}\right),$ (50)
$R^{i}=\left(\begin{tabular}[]{c|ccc}&&$\frac{1}{\sqrt{f}}(v_{b})_{i}$&\\\
\hline\cr&&&\\\ $\frac{1}{\sqrt{f}}(v_{a})_{i}$&&&\\\
&&&\end{tabular}\right).$ (51)

In other words, $(P)_{11}=\frac{1}{f}$,
$(Q^{ij})_{ab}=\frac{1}{2}(v_{a})_{i}(v_{b})_{j}+\frac{1}{2}(v_{a})_{j}(v_{b})_{i}$,
$(R^{i})_{1a}=(R^{i})_{a1}=\frac{1}{\sqrt{f}}(v_{a})_{i}$ and all other
elements are vanishing. Then $K^{i}$ and $\tilde{K}^{i}$ can be decomposed as

$K^{i}=P+Q^{ii}+R^{i},\qquad\tilde{K}^{i}=P+Q^{ii}-R^{i}.$ (52)

One can see that matrices for anti-periodic constraints
$K^{i}-\tilde{K}^{i}=2R^{i}$ are not totally independent because of the
relation $\sum_{i=1}^{f}R^{i}=0$ as a result of $\sum_{i=1}^{f}(v_{a})_{i}=0$.
We may choose $R^{a}$, for example, as independent ones. Thus, we have $f$
periodic and $f-1$ anti-periodic constraints:

$\sum_{j,k=1}^{f}(P+Q^{ii})_{jk}\,\dot{Y}^{j}(\tau)\cdot\dot{Y}^{k}(\tau)\approx
0,\qquad\sum_{j,k=1}^{f}R^{a}_{jk}\,\dot{Y}^{j}(\tau)\cdot\dot{Y}^{k}(\tau)\approx
0.$ (53)

In order to investigate constraints in more detail, we introduce so-called
Fubini-Veneziano fields $\varphi^{i\mu}(z)$ with a complex variable
$z(=e^{i\tau})$.

$\varphi^{1\mu}(z)\equiv\sqrt{2}\,Y^{1\mu}(-i\ln z)=q^{\mu}-ip^{\mu}\ln
z+i\sum_{n\neq 0}\frac{\alpha^{\mu}_{n}}{n}z^{-n},$ (54)
$\varphi^{a\mu}(z)\equiv\sqrt{2}\,Y^{a\mu}(-i\ln z)=i\sum_{r\in{\bf
Z}+\frac{1}{2}}\frac{\alpha^{a\mu}_{r}}{r}z^{-r}.\qquad(a=2,\cdots,f)$ (55)

They satisfy the periodicity

$\varphi^{1\mu}(e^{2\pi i}z)=\varphi^{1\mu}(z)+2\pi
p^{\mu},\qquad\varphi^{a\mu}(e^{2\pi i}z)=-\varphi^{a\mu}(z).$ (56)

It is convenient to define $A^{i\mu}(z)\equiv i\partial_{z}\varphi^{i\mu}(z)$,
i.e.,

$A^{1\mu}(z)=\sum_{n\in{\bf Z}}\alpha^{\mu}_{n}\,z^{-n-1},\qquad
A^{a\mu}(z)=\sum_{r\in{\bf Z}+\frac{1}{2}}\alpha^{a\mu}_{r}\,z^{-r-1}.$ (57)

$A^{1\mu}(z)$ is periodic whereas $A^{a\mu}(z)$ are anti-periodic under the
replacement $z\rightarrow e^{2\pi i}z$. If we define the operator $T_{M}(z)$
for a given matrix $M$ as333Note that the term $\frac{D}{16z^{2}}{\rm
tr}(MP_{-})$ in the definition of $T_{M}(z)$ is added in order that the OPE
relations can be written in a unified manner. (See Appendix A.)

$T_{M}(z)\equiv\frac{1}{2}\sum_{i,j=1}^{f}M_{ij}:A^{i}(z)\cdot
A^{j}(z):\;+\;\frac{D}{16z^{2}}{\rm tr}(MP_{-}),$ (58)

where $:{\cal O}:$ is a normal order of ${\cal O}$ with respect to the
oscillators $\alpha^{\mu}_{n}$, $\alpha^{a\mu}_{r}$ (and
$P_{-}=\frac{1}{2}(1-\Gamma)$), then the operators corresponding to the
primary constraints eqs.(53) are given by $T_{P+Q^{ii}}(z)$ and
$T_{R^{a}}(z)$. These operators satisfy the relations

$T_{P+Q^{ii}}(e^{2\pi i}z)=T_{P+Q^{ii}}(z),\qquad T_{R^{a}}(e^{2\pi
i}z)=-T_{R^{a}}(z),$ (59)

and they have a formal Laurent expansion with integer power of $z$ and half
odd integer power respectively,

$T_{P+Q^{ii}}(z)=\sum_{n\in{\bf Z}}L^{P+Q^{ii}}_{n}z^{-n-2},$ (60)
$T_{R^{a}}(z)=\sum_{r\in{\bf Z}+\frac{1}{2}}L^{R^{a}}_{r}z^{-r-2}.$ (61)

In terms of the oscillators, each mode operator is written as follows

$L^{P+Q^{ii}}_{n}=\frac{1}{2f}\sum_{m\in{\bf
Z}}:\alpha_{n-m}\cdot\alpha_{m}:+\frac{1}{2}\sum_{a,b=2}^{f}(v_{a})_{i}(v_{b})_{i}\sum_{s\in{\bf
Z}+\frac{1}{2}}:\alpha^{a}_{n-s}\cdot\alpha^{b}_{s}:+\frac{D}{16}\frac{f-1}{f}\delta_{n,0},$
(62) $L^{R^{a}}_{r}=\frac{1}{\sqrt{f}}\sum_{b=2}^{f}(v_{b})_{a}\sum_{m\in{\bf
Z}}:\alpha^{b}_{r-m}\cdot\alpha_{m}:.$ (63)

Among the above operators, one special combination is

$\displaystyle V_{n}$ $\displaystyle\equiv$
$\displaystyle\sum_{i}L^{P+Q^{ii}}_{n}$ (64) $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{m\in{\bf
Z}}:\alpha_{n-m}\cdot\alpha_{m}:+\frac{1}{2}\sum_{a=2}^{f}\sum_{s\in{\bf
Z}+\frac{1}{2}}:\alpha^{a}_{n-s}\cdot\alpha^{a}_{s}:+\frac{D}{16}(f-1)\delta_{n,0}.$
(65)

They satisfy Virasoro algebra with central charge $Df$

$[V_{n}\,,V_{m}]=(n-m)V_{n+m}+\frac{Df}{12}n(n^{2}-1)\delta_{n+m,0}.$ (66)

Thus we have a single Virasoro algebra as a closed subalgebra of our
constraints.

##  3 Physical state condition for open string junction

###  3.1 Preliminary discussion

For the physical state condition, it is possible to impose that the positive
modes of the constraint operators should annihilate physical states just as in
the case of the old covariant quantization for string:

$\displaystyle L^{P+Q^{ii}}_{n}|{\rm phys}\rangle$ $\displaystyle=$
$\displaystyle 0,\qquad(n>0,\quad i=1,\cdots,f)$ (67) $\displaystyle
L^{R^{a}}_{r}|{\rm phys}\rangle$ $\displaystyle=$ $\displaystyle
0.\qquad\,(r>0,\quad a=2,\cdots,f)$ (68)

Note that the algebras of $L^{Q^{i-1\,i-1}-Q^{ii}}_{n}$’s and
$L^{R^{a}}_{r}$’s are not closed among themselves, as will be seen in Appendix
B. However, if $A|{\rm phys}\rangle=0$ and $B|{\rm phys}\rangle=0$, then
$[A,B]|{\rm phys}\rangle=0$, so that requiring the above conditions will be
sufficient. On the other hand, those of zero-mode operators $L^{Q^{ii}}_{0}$
must be chosen more carefully. It is not straightforward to choose appropriate
conditions for general $f$ since the set of $f-1$ operators
$L^{Q^{i-1\,i-1}-Q^{ii}}_{0}$ is not closed. We first study the special case
of $f=2$, and go into the general discussion afterward.

###  3.2 $f$=2 case

We consider the $f=2$ case where the physical object is not a junction but an
open string. In fact, in this case, there are only one positive integer and
half-integer modes oscillators $\alpha_{n}^{\mu}$’s and
$\alpha_{r}^{a=2,\mu}$’s, and the physical state conditions (67) and (68) are
reduced to the following two simple ones

$\displaystyle V_{n}|{\rm phys}\rangle$ $\displaystyle=$ $\displaystyle
0,\qquad(n>0)$ (69) $\displaystyle L^{R^{a=2}}_{r}|{\rm phys}\rangle$
$\displaystyle=$ $\displaystyle\left(\frac{1}{2}\sum_{m\in{\bf
Z}}:\alpha_{r-m}^{a=2}\cdot\alpha_{m}:\right)|{\rm
phys}\rangle=0\qquad\,(r>0)$ (70)

since $P+Q^{11}=P+Q^{22}$. Also, there is only one zero mode operator $V_{0}$
which counts the level of the state (plus $\frac{1}{2}p^{2}$ and a constant)
as usual for strings, and the zero mode condition should be taken as
$V_{0}|{\rm phys}\rangle=(a_{0}+\frac{D}{16})|{\rm phys}\rangle$ with a normal
order constant or an intercept parameter $a_{0}$. If we replace
$\alpha_{n}^{\mu}\rightarrow\frac{1}{\sqrt{2}}\tilde{\alpha}_{2n}^{\mu}$ and
$\alpha_{r}^{a=2,\mu}\rightarrow\frac{1}{\sqrt{2}}\tilde{\alpha}_{2r}^{\mu}$,
they satisfy
$[\tilde{\alpha}_{n}^{\mu},\tilde{\alpha}_{m}^{\nu}]=n\delta_{n+m,0}\eta^{\mu\nu}$.
Writing $\tilde{L}_{2n}=2(V_{n}-\frac{D}{16}\delta_{n,0})$ and
$\tilde{L}_{2n+1}=2L^{R^{a=1}}_{n+\frac{1}{2}}=-2L^{R^{a=2}}_{n+\frac{1}{2}}$,
we can easily check that $\tilde{L}_{n}$’s satisfy the Virasoro algebra with
central charge $D$, and the physical state condition can be collected as

$\left(\tilde{L}_{n}-\tilde{a}\delta_{n,0}\right)|{\rm
phys}\rangle=0\qquad(n\geq 0)$ (71)

where $\tilde{a}=2a_{0}$ and

$\tilde{L}_{n}=\frac{1}{2}\sum_{m\in{\bf
Z}}:\tilde{\alpha}_{n-m}\cdot\tilde{\alpha}_{m}:.$ (72)

This is exactly the same condition as for the old covariant quantization of
open string theory. As is well known, the constant $\tilde{a}$ should be taken
as $\tilde{a}\leq 1$ and the dimension as $D\leq 26$ in order to ensure that
there is no ghost (negative norm) state in the physical spectrum (See [10] for
example.) From the various discussions like the 1-loop level unitarity or the
modern BRS quantization, we know that the consistent choice should be
$\tilde{a}=1$ and $D=26$.

###  3.3 Zero mode condition

Now we deal with the general $f\geq 3$ case and discuss the physical spectrum
and its properties.

We have already proposed reasonable condition for positive modes of the
constraint operators as (67) and (68). Here we consider the remaining
discussion on zero modes of the constraint operators given by

$L^{P+Q^{ii}}_{0}=\frac{1}{2f}\sum_{m\in{\bf
Z}}:\alpha_{-m}\cdot\alpha_{m}:+\frac{1}{2}\sum_{a,b=2}^{f}(v_{a})_{i}(v_{b})_{i}\sum_{s\in{\bf
Z}+\frac{1}{2}}:\alpha^{a}_{-s}\cdot\alpha^{b}_{s}:+\frac{D}{16}\frac{f-1}{f}.$
(73)

These $f$ operators are all independent and the commutation relations are

$\displaystyle[L^{P+Q^{ii}}_{0},L^{P+Q^{jj}}_{0}]$ $\displaystyle=$
$\displaystyle[L^{Q^{ii}}_{0},L^{Q^{jj}}_{0}]$ (74) $\displaystyle=$
$\displaystyle-\frac{1}{f}\sum_{a,b=2}^{f}(v_{a})_{i}(v_{b})_{j}\sum_{r\in{\bf
Z}+\frac{1}{2}}r:\alpha_{-r}^{a}\cdot\alpha_{r}^{b}:\;,$

which shows that the algebra is not closed within the $f$ operators
$L^{P+Q^{ii}}_{0}$. Thus we cannot impose a condition such that a state
$|\phi\rangle$ should be a vector in the space of some finite representation
of $L^{P+Q^{ii}}_{0}$. In fact, in order to obtain a closed algebra including
all $f$ operators $L^{P+Q^{ii}}_{0}$, we have to involve infinite number of
additional operators. On the other hand, the operator
$V_{0}=\sum_{i=1}^{f}L^{P+Q^{ii}}_{0}$ in itself has a desirable property

$V_{0}|\phi\rangle=\left(N_{\rm
level}+\frac{1}{2}p^{2}+\frac{D(f-1)}{16}\right)|\phi\rangle$ (75)

where $|\phi\rangle$ is a state whose level and the squared of the momentum
are $N_{\rm level}$ and $p^{2}$ respectively. (Note that we take
$\alpha^{\prime}=1$.) We see that one reasonable physical state condition
involving the zero mode operators is

$V_{0}|{\rm phys}\rangle=\left(a_{0}+\frac{D(f-1)}{16}\right)|{\rm
phys}\rangle$ (76)

with some constant $a_{0}$. In addition, for other zero mode operators, it is
at least possible to impose the condition that $L_{0}^{P+Q^{ii}}|\phi\rangle$
is always physical if $|\phi\rangle$ is physical.

###  3.4 Physical state condition

From the above discussion, the most reasonable physical state condition for
the present system is to impose the following set of conditions (I), (II) and
(III) for arbitrary non-negative integer $N$, positive integer $n$ and
positive half-integer $r$:

$\displaystyle{\rm(I)}$
$\displaystyle\;L^{P+Q^{ii}}_{n}\left(\prod_{k\in\\{j_{1},\cdots,j_{N}\\}}\\!\\!\\!\\!L^{P+Q^{kk}}_{0}\right)|{\rm
phys}\rangle=0,\qquad(n>0,\quad i=1,\cdots,f)$ (77) $\displaystyle{\rm(II)}$
$\displaystyle\;L^{R^{a}}_{r}\left(\prod_{k\in\\{j_{1},\cdots,j_{N}\\}}\\!\\!\\!\\!L^{P+Q^{kk}}_{0}\right)|{\rm
phys}\rangle=0,\qquad\,(r>0,\quad a=2,\cdots,f)$ (78)
$\displaystyle{\rm(III)}$ $\displaystyle\;V_{0}|{\rm
phys}\rangle=\left(a_{0}+\frac{D(f-1)}{16}\right)|{\rm
phys}\rangle\;\;\left(\Leftrightarrow\left(N_{\rm
level}+\frac{1}{2}p^{2}-a_{0}\right)|{\rm phys}\rangle=0\right).$ (79)

In the above equations, $\prod_{k\in\\{j_{1},\cdots,j_{N}\\}}L^{P+Q^{kk}}_{0}$
is meant to be the product of $N$ $L^{P+Q^{kk}}_{0}$’s :

$\prod_{k\in\\{j_{1},\cdots,j_{N}\\}}\\!\\!\\!\\!L^{P+Q^{kk}}_{0}=L^{P+Q^{j_{1}j_{1}}}_{0}L^{P+Q^{j_{2}j_{2}}}_{0}\cdots
L^{P+Q^{j_{N}j_{N}}}_{0}$ (80)

for positive $N$, and is to be 1 for $N=0$. In what follows, assuming that the
physical state condition is indeed given by the set of these conditions, we
will identify the space of physical states.

First, we examine the properties of each of the above three conditions in
detail. The simplest condition (III) determines the relation between the level
and the mass of the states. This restriction does not affect the other two
conditions since

$[V_{0},L^{P+Q^{ii}}_{m}]=-mL^{P+Q^{ii}}_{m},\qquad[V_{0},L^{R^{a}}_{r}]=-rL^{R^{a}}_{r}$
(81)

hold for any integer $m$ or half-integer $r$. Thus we can proceed the
discussion within the space of fixed level (and mass) states separately. Now
we consider the conditions (I) and (II) for $N=0$, i.e., the conditions
$L^{P+Q^{ii}}_{n}|{\rm phys}\rangle=0$ and $L^{R^{a}}_{r}|{\rm phys}\rangle=0$
. In this case, from the commutation relations (B.7), (B.8) and (B.9), we can
prove that if we assume that (I) of $n=1,2$ and (II) of
$r=\frac{1}{2},\frac{3}{2}$ for $N=0$ are satisfied, all the other (I) and
(II) for $N=0$ are satisfied. In the process of this discussion, we can also
show that a state $|\phi_{0}\rangle$ satisfying (I) and (II) for $N=0$ always
satisfies $L_{n}^{P}|\phi_{0}\rangle=L_{n}^{Q^{ij}}|\phi_{0}\rangle=0$ for
$n\geq 2$. Next, we impose the conditions (I) and (II) for $N=1$ on a state
$|\phi_{0}\rangle$ satisfying the conditions for $N=0$. From the relations
(B.11) and (B.12), the additional conditions we have to impose on
$|\phi_{0}\rangle$ are

$L_{r}^{R^{a}}L_{0}^{P}|\phi_{0}\rangle=0,\qquad
L_{n}^{P}|\phi_{0}\rangle=0,\qquad
L_{n}^{Q^{ii}}L_{0}^{Q^{jj}}|\phi_{0}\rangle=0.\quad(r,n>0)$ (82)

In fact, it is sufficient to impose the conditions for $r=1/2$ and $n=1,2$
since the other equations result from them. By continuing the similar
discussion, we conclude that the conditions (I) and (II) are rewritten by the
following simpler set of conditions for any $i,j,j_{k}\in\\{1,2,\cdots,f\\}$
and $a\in\\{2,\cdots,f\\}$

$\displaystyle L_{n}^{P}|\phi\rangle=L_{n}^{Q^{ij}}|\phi\rangle=0,\qquad(n>0)$
(83) $\displaystyle
L_{r}^{R^{a}}\left(L_{0}^{P}\right)^{k}|\phi\rangle=0,\qquad(r>0,\;\;k=0,1,2,\cdots)$
(84) $\displaystyle
L_{n}^{Q^{ii}}\left(L_{0}^{Q^{j_{1}j_{1}}}L_{0}^{Q^{j_{2}j_{2}}}\cdots
L_{0}^{Q^{j_{N}j_{N}}}\right)|\phi\rangle=0.\qquad(n>0,\;N=0,1,2,\cdots)$ (85)

In particular, since $L_{0}^{P}=\frac{1}{2f}p^{2}+\frac{1}{f}N_{\rm
level}^{\rm(e)}$, the condition eq.(84) is reduced to the set of conditions

$L_{r}^{G^{(a)},s}|\phi\rangle=0,\qquad\left(r>0,\;s\in{\bf
Z}+\frac{1}{2},\;a=2,\cdots,f\right)$ (86)

where $L_{r}^{G^{(a)},s}=:\alpha_{s}^{a}\cdot\alpha_{r-s}:$. From the
relations (B.13), (B.18), (B.19)and (B.21), we can show that any state
satisfying (83) and (86) also satisfies the remaining condition (85).

Thus, the physical state conditions (I) and (II) are reduced to the simpler
conditions (83) and (86), which we name ${\rm(I^{\prime})}$,
${\rm(I^{\prime\prime})}$ and ${\rm(II^{\prime})}$ respectively:

$\displaystyle{\rm(I^{\prime})}$ $\displaystyle
L_{n}^{P}|\phi\rangle=0,\qquad(n>0)$ $\displaystyle{\rm(I^{\prime\prime})}$
$\displaystyle L_{n}^{Q^{ij}}|\phi\rangle=0,\qquad(n>0;\,i,j=1,\cdots,f)$
$\displaystyle{\rm(II^{\prime})}$
$\displaystyle\;L_{r}^{G^{(a)},s}|\phi\rangle=0.\qquad(r>0,\;s\in{\bf
Z}+\frac{1}{2},\;a=2,\cdots,f)$

In fact, it is sufficient to impose ${\rm(I^{\prime})}$ for $n=1,2$,
${\rm(I^{\prime\prime})}$ for $n=1$, ${\rm(II^{\prime})}$ for $r=\frac{1}{2}$
since the other conditions can be derived from them. Note that the condition
${\rm(I^{\prime})}$ and ${\rm(II^{\prime})}$ can be respectively represented
by $L_{n}^{E}|\phi\rangle=0$ and $L_{n}^{F^{(ab)}}|\phi\rangle=0$
$(a,b=2,\cdots,f)$.

##  4 Physical spectrum and the properties

We will explicitly solve the physical state condition given by
${\rm(I^{\prime})}$, ${\rm(I^{\prime\prime})}$, ${\rm(II^{\prime})}$ and
(III), and study the properties of physical spectrum. We represent a level
$N_{\rm level}$ state as $|\phi\rangle_{\\!N_{\rm level}}$ and the vacuum
state that is annihilated by all the positive frequency oscillators with
momentum $p$ as $|0,p\rangle$.

###  4.1 First three levels: $N_{\rm level}=0,\,\frac{1}{2},\,1$

We first identify the explicit form of physical states for $N_{\rm
level}=0,\frac{1}{2},1$ and find a restriction on a constant $a_{0}$.

For $N_{\rm level}=0$, there is only one state $|\phi\rangle_{0}=|0,p\rangle$
and $p^{2}$ is determined by the condition (III) as $\frac{1}{2}p^{2}=a_{0}$.

For $N_{\rm level}=\frac{1}{2}$, a general state satisfying the mass-shell
condition (III) is given by

$|\phi\rangle_{\frac{1}{2}}=\sum_{a=2}^{f}f^{a}_{\mu}\alpha^{a,\mu}_{-\frac{1}{2}}|0,p\rangle$
(87)

with $\frac{1}{2}p^{2}=a_{0}-\frac{1}{2}$. Only the non-trivial condition for
this state is given by ${\rm(II^{\prime})}$ with $r=\frac{1}{2}$ and
$s=\frac{1}{2}$. This restricts the $f-1$ coefficients vectors $f^{a}_{\mu}$
to satisfy

$p\cdot f^{a}=0.$ (88)

We see that the norm of the physical state
$|f^{a},p\rangle_{\frac{1}{2}}=f^{a}_{\mu}\alpha^{a,\mu}_{-\frac{1}{2}}|0,p\rangle$
is

${}_{\frac{1}{2}}\langle
f^{a},p|f^{a},p^{\prime}\rangle_{\frac{1}{2}}=\frac{1}{2}f^{a}\\!\cdot\\!f^{a}\,\delta^{D}\\!(p-p^{\prime})$
(89)

where $\langle 0,p|=(|0,p\rangle)^{\dagger}$ with
$(\alpha^{\mu}_{n})^{\dagger}=\alpha^{\mu}_{-n}$,
$(\alpha^{a,\mu}_{r})^{\dagger}=\alpha^{a,\mu}_{-r}$ and $\langle
0,p^{\prime}|0,p\rangle=\delta^{D}(p-p^{\prime})$. In this case, if we take
$a_{0}(=\frac{1}{2}p^{2}+\frac{1}{2})>\frac{1}{2}$, we can choose $p_{0}=0$
since $p^{2}>0$. Then, from the same discussion as in the case of the old
covariant quantization of open string theory, $f^{a}$ can be taken as a time-
like vector and the corresponding state $|f^{a},p\rangle_{\frac{1}{2}}$ has
negative norm. On the other hand, if we choose $a_{0}\leq\frac{1}{2}$,
physical states have always non-negative norm since $p^{2}\leq 0$. In
particular, for $a_{0}=\frac{1}{2}$, there are $f-1$ zero-norm physical states
$p_{\mu}\alpha^{a,\mu}_{-\frac{1}{2}}|0,p\rangle$ and the $(f-1)(D-2)$
transverse positive norm physical states. Thus, in order to ensure the no-
ghost theorem for this system, we at least have to choose

$a_{0}\leq\frac{1}{2},$ (90)

which leads to $-\frac{1}{2}p^{2}\,(=\frac{1}{2}m^{2})\,\geq N_{\rm
level}-\frac{1}{2}$. If we assume this condition, all the states with level
higher than $\frac{1}{2}$ has real mass ($m^{2}>0$). We later show that this
assumption is plausible from the discussion of $\zeta$-function regularization
calculation of normal ordering constant.

For $N_{\rm level}=1$, general states are given by

$|\phi\rangle_{1}=\Big{(}g_{\mu}\alpha_{-1}^{\mu}+\sum_{a,b}h^{ab}_{\mu\nu}\alpha^{a,\mu}_{-\frac{1}{2}}\alpha^{b,\nu}_{-\frac{1}{2}}\Big{)}|0,p\rangle$
(91)

where $h^{ab}_{\mu\nu}=h^{ba}_{\nu\mu}$ and the mass-shell condition is
$\frac{1}{2}p^{2}=a_{0}-1$. After imposing the set of non-trivial conditions
for this level, i.e., ${\rm(II^{\prime})}$ with $r=\frac{1}{2}$ and
$s=\pm\frac{1}{2}$ and ${\rm(I^{\prime\prime})}$ with $n=1$, we obtain the
general form of physical states as

$|\phi_{\rm
phys},p\rangle_{1}=\sum_{a,b}h^{ab}_{\mu\nu}\alpha^{a,\mu}_{-\frac{1}{2}}\alpha^{b,\nu}_{-\frac{1}{2}}|0,p\rangle$
(92)

where

$\eta^{\mu\nu}h^{ab}_{\mu\nu}=0,\quad
p^{\mu}h^{ab}_{\mu\nu}=0,\quad\frac{1}{2}p^{2}=a_{0}-1.$ (93)

We see that the norm of this state is calculated as

${}_{1\\!}\langle\phi_{\rm phys},p|\phi_{\rm
phys},p^{\prime}\rangle_{\\!1}=\frac{1}{2}\sum_{a,b}h^{ab}_{\mu\nu}h^{ab,\mu\nu}\,\delta^{D}\\!(p-p^{\prime})$
(94)

If we assume (90), mass-shell condition for this (and all the higher) level
ensures $p^{2}<0$, and we can choose the frame
$p_{\mu}=\delta_{\mu}{}^{0}p_{0}$ ($p_{0}\neq 0$ and $p_{i(\neq 0)}=0$). Then,
from the conditions (93), $h_{\mu\nu}^{ab}=0$ if $\mu$ or $\nu$ equals to 0,
and the norm (94) becomes positive for any non-trivial $h_{\mu\nu}^{ab}$. Note
that there is no zero-norm physical state for any $a_{0}\leq\frac{1}{2}$.

###  4.2 Physical states for an arbitrary level $N_{\rm level}$

We now investigate the physical state condition and obtain general form of
physical states for an arbitrary level $N_{\rm level}$. Fortunately, physical
state condition we set is strong enough to completely identify the space of
physical states for any level. The result is the following: Any state
satisfying the physical state condition has the form

$h^{a_{1}a_{2}\cdots
a_{K}}_{[\mu^{M}_{1}\mu^{M}_{2}\cdots\mu^{M}_{f}\cdots\cdots\mu^{2}_{1}\mu^{2}_{2}\cdots\mu^{2}_{f}\mu^{1}_{1}\mu^{1}_{2}\cdots\mu^{1}_{f}]\sigma_{1}\sigma_{2}\cdots\sigma_{K}}\left(\prod_{m=1}^{M}\alpha_{-m}^{\mu^{m}_{1}}\alpha_{-(m-1/2)}^{2,\mu^{m}_{2}}\cdots\alpha_{-(m-1/2)}^{f,\mu^{m}_{f}}\right)\prod_{i=1}^{K}\alpha_{-\frac{1}{2}}^{a_{i},\sigma_{i}}|0,p\rangle$
(95)

or

$\displaystyle h^{a_{1}a_{2}\cdots
a_{K}}_{[\mu^{M+1}_{2}\cdots\mu^{M+1}_{f}\mu^{M}_{1}\mu^{M}_{2}\cdots\mu^{M}_{f}\cdots\cdots\mu^{2}_{1}\mu^{2}_{2}\cdots\mu^{2}_{f}\mu^{1}_{1}\mu^{1}_{2}\cdots\mu^{1}_{f}]\sigma_{1}\sigma_{2}\cdots\sigma_{K}}$
$\displaystyle\hskip
28.45274pt\times\alpha_{-(M+1/2)}^{2,\mu^{M+1}_{2}}\cdots\alpha_{-(M+1/2)}^{f,\mu^{M+1}_{f}}\left(\prod_{m=1}^{M}\alpha_{-m}^{\mu^{m}_{1}}\alpha_{-(m-1/2)}^{2,\mu^{m}_{2}}\cdots\alpha_{-(m-1/2)}^{f,\mu^{m}_{f}}\right)\prod_{i=1}^{K}\alpha_{-\frac{1}{2}}^{a_{i},\sigma_{i}}|0,p\rangle.$
(96)

Here, $M$ and $K$ are non-negative integers. Also, $h_{[\cdots],\cdots}$ is a
tensor field with all the spacetime indices $\mu_{a}^{m}$ within the bracket
$[\cdots]$ are anti-symmetric. The tensor field $h_{\cdots}$ given in eq.(95)
should satisfy the relations

$\displaystyle p^{\mu_{a}^{m}}h^{a_{1}a_{2}\cdots
a_{K}}_{[\mu_{1}^{M}\mu_{2}^{M}\cdots\mu^{1}_{1}\mu^{1}_{2}\cdots\mu^{1}_{f}]\sigma_{1}\sigma_{2}\cdots\sigma_{K}}=p^{\sigma_{i}}h^{a_{1}a_{2}\cdots
a_{K}}_{[\mu_{1}^{M}\mu_{2}^{M}\cdots\mu^{1}_{1}\mu^{1}_{2}\cdots\mu^{1}_{f}]\sigma_{1}\sigma_{2}\cdots\sigma_{K}}=0$
(97) $\displaystyle\eta^{\sigma_{i}\mu^{1}_{k}}h^{a_{1}a_{2}\cdots
a_{K}}_{[\mu_{1}^{M}\mu_{2}^{M}\cdots\mu^{1}_{1}\mu^{1}_{2}\cdots\mu^{1}_{f}]\sigma_{1}\sigma_{2}\cdots\sigma_{K}}=\eta^{\sigma_{i}\sigma_{j}}h^{a_{1}a_{2}\cdots
a_{K}}_{[\mu_{1}^{M}\mu_{2}^{M}\cdots\mu^{1}_{1}\mu^{1}_{2}\cdots\mu^{1}_{f}]\sigma_{1}\sigma_{2}\cdots\sigma_{K}}=0,$
(98) $\displaystyle h^{a_{1}a_{2}\cdots
a_{K}}_{[\mu_{1}^{M}\mu_{2}^{M}\cdots\mu^{1}_{1}\mu^{1}_{2}\cdots\mu^{1}_{f}]\sigma_{1}\sigma_{2}\cdots\sigma_{K}}\quad\mbox{is
symmetric under a permutation
$(a_{i},\sigma_{i})\,\leftrightarrow\,(a_{j},\sigma_{j})$ }$ (99)

for any $i$ and $j$ ($i\neq j$). Also, $h_{\cdots}$ given in eq.(96) satisfies
the corresponding relations. Note that the level of the state (95) is

$\frac{K}{2}+\sum_{m=1}^{M}\left(m+(f-1)(m-1/2)\right)=\frac{1}{2}(K+fM(M+1)-Mf+M),$
(100)

and that of (96) is $\frac{1}{2}(K+(M+1)(Mf+f-1))$. The proof that all the
physical states are given by the states of the form (95) or (96) is given in
Appendix C.

As a non-trivial example, we represent all the physical states for $f=3$ and
$N_{\rm level}=5$. There are three types of physical states:

$\displaystyle
h_{[\mu^{2}_{2}\mu^{2}_{3}\mu^{1}_{1}\mu^{1}_{2}\mu^{1}_{3}]}\alpha_{-\frac{3}{2}}^{2,\mu^{2}_{2}}\alpha_{-\frac{3}{2}}^{3,\mu^{2}_{3}}\alpha_{-1}^{\mu^{1}_{1}}\alpha_{-\frac{1}{2}}^{2,\mu^{1}_{2}}\alpha_{-\frac{1}{2}}^{3,\mu^{1}_{3}}|0,p\rangle,$
(101) $\displaystyle h^{a_{1}a_{2}\cdots
a_{6}}_{[\mu^{1}_{1}\mu^{1}_{2}\mu^{1}_{3}]\sigma_{1}\sigma_{2}\cdots\sigma_{6}}\alpha_{-1}^{\mu^{1}_{1}}\alpha_{-\frac{1}{2}}^{2,\mu^{1}_{2}}\alpha_{-\frac{1}{2}}^{3,\mu^{1}_{3}}\alpha_{-\frac{1}{2}}^{a_{1},\sigma_{1}}\alpha_{-\frac{1}{2}}^{a_{2},\sigma_{2}}\cdots\alpha_{-\frac{1}{2}}^{a_{6},\sigma_{6}}|0,p\rangle,$
(102) $\displaystyle h^{a_{1}a_{2}\cdots
a_{10}}_{\sigma_{1}\sigma_{2}\cdots\sigma_{10}}\alpha_{-\frac{1}{2}}^{a_{1},\sigma_{1}}\alpha_{-\frac{1}{2}}^{a_{2},\sigma_{2}}\cdots\alpha_{-\frac{1}{2}}^{a_{10},\sigma_{10}}|0,p\rangle$
(103)

where $a_{i}=2$ or $3$. Each field satisfies the following relations

$\displaystyle
p^{\mu^{2}_{2}}h_{[\mu^{2}_{2}\mu^{2}_{3}\mu^{1}_{1}\mu^{1}_{2}\mu^{1}_{3}]}=0,$
(104) $\displaystyle p^{\mu^{1}_{1}}h^{a_{1}a_{2}\cdots
a_{6}}_{[\mu^{1}_{1}\mu^{1}_{2}\mu^{1}_{3}]\sigma_{1}\sigma_{2}\cdots\sigma_{6}}=p^{\sigma_{i}}h^{a_{1}a_{2}\cdots
a_{6}}_{[\mu^{1}_{1}\mu^{1}_{2}\mu^{1}_{3}]\sigma_{1}\sigma_{2}\cdots\sigma_{6}}=0,$
$\displaystyle\qquad\eta^{\mu^{1}_{1}\sigma_{i}}h^{a_{1}a_{2}\cdots
a_{6}}_{[\mu^{1}_{1}\mu^{1}_{2}\mu^{1}_{3}]\sigma_{1}\sigma_{2}\cdots\sigma_{6}}=\eta^{\sigma_{i}\sigma_{j}}h^{a_{1}a_{2}\cdots
a_{6}}_{[\mu^{1}_{1}\mu^{1}_{2}\mu^{1}_{3}]\sigma_{1}\sigma_{2}\cdots\sigma_{6}}=0,$
(105) $\displaystyle p^{\sigma_{i}}h^{a_{1}a_{2}\cdots
a_{10}}_{\sigma_{1}\sigma_{2}\cdots\sigma_{10}}=0,\quad\eta^{\sigma_{i}\sigma_{j}}h^{a_{1}a_{2}\cdots
a_{10}}_{\sigma_{1}\sigma_{2}\cdots\sigma_{10}}=0$ (106)

and the appropriate symmetric properties.

###  4.3 Properties of general physical states

Now that we have fully identified the possible form of the physical states, we
investigate and summarize their general properties.

First, we consider the constant $a_{0}$ which appears in eq.(76) and has not
been determined yet. In the case of string theory, the corresponding constant
can be identified by calculating the zero-point energy of the sum of all
physical degrees of freedom and the result is confirmed by the discussion of
the BRS quantization method. For our system, the similar calculation can be
performed if we assume the number of physical degrees of freedom is the same
as that of the transverse degrees of freedom. The sum of the zero-point energy
corresponding to the degrees of freedom for one space-time direction for our
system is given by

$\sum_{n=0}^{\infty}n+(f-1)\sum_{n=0}^{\infty}\left(n+\frac{1}{2}\right)$
(107)

multiplied by $-\frac{1}{2}$. This type of summation can be performed by using
the $\zeta$-function regularization method and the result is given by

$\sum_{n=0}^{\infty}n\rightarrow-\frac{1}{12},\qquad\sum_{n=0}^{\infty}\left(n+\frac{1}{2}\right)\rightarrow\frac{1}{24}.$
(108)

By using the result,

$\sum_{n=0}^{\infty}n+(f-1)\sum_{n=0}^{\infty}\left(n+\frac{1}{2}\right)\rightarrow\frac{f-3}{24},$
(109)

and if we assume the number of physical degrees of freedom is same as that of
the transverse degrees of freedom, which is the case for the string theory,
$a_{0}$ is given by eq.(109) multiplied by $-\frac{D-2}{2}$ as

$a_{0}=-(D-2)\frac{f-3}{48}.$ (110)

For any $f\geq 3$, and for the string case $f=2$ and $D=26$, this result
indeed satisfies the relation $a_{0}\leq\frac{1}{2}$ which is necessary for
all level $N_{\rm level}=\frac{1}{2}$ physical states to have positive norm.
In particular, for $D=26$, $a_{0}=-\frac{f-3}{2}$, and the on-shell condition
(76) restricts

$-\frac{1}{2}p^{2}\left(=\frac{1}{2}m^{2}\right)=N_{\rm level}+\frac{f-3}{2},$
(111)

which means that all states satisfying the on-shell condition should have
integer or half-integer $-\frac{1}{2}p^{2}$ for every $f$ including the string
$f=2$ case. In particular, $m^{2}=f-3$ for $N_{\rm level}=0$ and $m^{2}=f-2$
for $N_{\rm level}=\frac{1}{2}$. Thus, physical spectrum for $f\geq 3$ is
limited to the massive one except for $f=3$ and $N_{\rm level}=0$ in which
case the ground state $|0,p\rangle$ is massless and physical.

From eq.(111), we can also study the relation between mass and spin for each
level. In general, the highest spin state for level $N$ is given by the state
of the form

$h_{(\sigma_{1}\sigma_{2}\cdots\sigma_{2\\!N})}^{a_{1}a_{2}\cdots{a_{2\\!N}}}\alpha_{-\frac{1}{2}}^{a_{1},\sigma_{1}}\alpha_{-\frac{1}{2}}^{a_{2},\sigma_{2}}\cdots\alpha_{-\frac{1}{2}}^{a_{2\\!N},\sigma_{2\\!N}}|0,p\rangle.$
(112)

Any other physical state for the level has spin less than that of this state:
$0\leq J\leq 2N(=J_{\max})$. Thus, for arbitrary $f\geq 3$, the relation
between mass $m$ and spin $J$ is represented by

$J\leq m^{2}-f+3\,,$ (113)

which is also applied to the $f=2$ open string case. (Remember that we take
$\alpha^{\prime}=1$.)

Next, we will show the No-ghost theorem that ensures all the physical states
have positive norm under the assumption $a_{0}\leq\frac{1}{2}$ for general
$f\geq 3$. We have already shown the theorem is indeed met for $N_{\rm
level}\leq 1$ in section 4.1. As for the discussion on general $N_{\rm
level}$, we first note that any on-shell physical state has
$-\frac{1}{2}p^{2}>0$ for $f\geq 3$ and $N_{\rm level}>0$ under the assumption
$a_{0}\leq\frac{1}{2}$. Then, choosing the momentum frame as
$p_{\mu}=\eta_{\mu 0}p^{0}$, we see that any component of the tensor field
$h_{\nu_{1}\nu_{2}\cdots}$ corresponding to any physical state of the form
(95) or (96) always vanishes if any space-time index $\nu_{j}=0$. That is, any
non-zero physical state (95) or (96) includes only space-like oscillators
$\alpha_{-n}^{i}$ and $\alpha_{-s}^{ai}$. From the properties of the
commutation relations of the oscillators, we easily see that all such states
have positive norm. Thus we have proven the No-ghost theorem.

From the above discussion, we see that there is no zero norm physical state
unlike the case of string theory. Zero norm physical states for string theory
play an important role for the theory to be equipped with gauge symmetry. For
$f\geq 3$, there seems to be no gauge degrees of freedom in the physical
spectrum.

##  5 Discussion

We have revisited the covariant quantization problem of $f$-pronged open
string junction. Its excitation is described by a set of ordinary bosons as
well as $f-1$ sets of twisted bosons on the world sheet. The constraints form
an open algebra with operators obeying both periodic and anti-periodic
boundary conditions. We have, for the first time, succeeded in giving the
physical state condition and identify the physical states which indeed have
positive norm. Several remarks and/or questions are in order.

One may wonder whether our result of physical states coincides with light-cone
gauge which is familiar in the ordinary single string case. This question is
not so simple to be answered because taking a light-cone gauge is a non-
trivial issue for string junction as described in the Appendix D. Indeed,
naive truncation to the transverse oscillators gives slightly different number
of states in each level. For example, in $N_{\rm level}=1$ (with $D\geq 3$,
$f\geq 2$) case the number of our physical states is

$\frac{1}{2}(f-1)(D-2)[(f-1)(D-2)+1]+(f-1)^{2}(D-2),$ (114)

while naive truncation to the transverse mode gives

$\frac{1}{2}(f-1)(D-2)[(f-1)(D-2)+1]+(D-2).$ (115)

The number of constraints seems enough to eliminate light-cone degrees of
freedom, but structure of physical states is not the same as naive truncation
to the transverse oscillators.

As is explained, the constraint algebra is not closed. If we include all
operators newly appeared in the commutator one by one, then we eventually
obtain a very large algebra. It is interesting to understand this algebra and
to interpret our physical states in terms of it. In fact, by construction our
physical states in each level will become some sort of representation of a
zero-mode subalgebra of that. If we define

$B^{(p)}_{ab}=\sum_{r\in{\bf
Z}+\frac{1}{2}}r^{p-1}:\alpha^{a}_{-r}\cdot\alpha^{b}_{r}:\qquad p=1,2,\cdots$
(116)

which appears in the zero mode part of the above mentioned large algebra, they
satisfy

$\left[B^{(p)}_{ab}\,,\,B^{(q)}_{cd}\right]=\delta_{bc}B^{(p+q)}_{ad}+(-1)^{p-1}\delta_{ac}B^{(p+q)}_{bd}+(-1)^{q-1}\delta_{bd}B^{(p+q)}_{ac}+(-1)^{p+q-2}\delta_{ad}(-1)^{q-1}B^{(p+q)}_{bc},$
(117) $B^{(p)}_{ab}=(-1)^{p-1}B^{(p)}_{ba}.$ (118)

Note that $B^{(1)}_{ab}=2L_{0}^{F^{(ab)}}-\frac{D}{8}\delta^{ab}$ in terms of
the operator defined in (A.16). This algebra is isomorphic to a twisted (${\bf
Z}_{>0}$)-graded version of gl$(f-1)$, i.e., $\\{E^{+}_{ab}\otimes
u^{2k-1},E^{-}_{ab}\otimes u^{2k}\,|\,k=1,2,\cdots;\,E^{\pm}_{ab}=\pm
E^{\pm}_{ba}\in{\rm gl}(f-1);\,u\in{\bf C}\\}$. We hope this point will be
clarified in the future.

Our analysis in the present paper has been within so-called old covariant
quantization (OCQ). One can find other fine structures from new covariant or
BRST quantization which will be our next task. After getting it done, we are
also able to construct free field theory for string junction. The authors
previously studied extended string field theory [12, 13] where multiple string
Fock spaces are utilized to describe massless higher spin modes with massive
tower, just as closed string field theory can be formulated by doubled Fock
space of open string with suitable matching condition. Fock space structure
there is very similar to the current one, so that the adaptation of the
formalism to the string junction may be straightforward.

Another future task is to consider interactions. In [7] single string emission
vertex from the free end of a string segment was considered. There may be more
varieties of interactions some of which may need introduction of other type of
junction. We need to classify them anyway.

## Acknowledgements

The work was supported in part by JSPS KAKENHI Grant Number JP21K03579 (M.A.),
JP20K03966 and JP21K03537 (M.K.).

##  Appendix Appendix A  OPE for $T_{M}(z)$

We study the OPE properties of the operators $T_{M}(z)$ defined by eq.(58) for
general $M$. The product of two operators $A^{i\mu}(z)$ and $A^{i\mu}(w)$ is
rewritten by normal-ordered form by using the commutation relations (45) as

$A^{1\mu}(z)A^{1\nu}(w)=\frac{1}{(z-w)^{2}}\eta^{\mu\nu}\;+\;:A^{1\mu}(z)A^{1\nu}(w):$
(A.1)

and

$A^{a\mu}(z)A^{b\nu}(w)=\left(\frac{1}{(z-w)^{2}}+\epsilon(z,w)\right)\delta^{ab}\eta^{\mu\nu}\;+\;:A^{a\mu}(z)A^{b\nu}(w):$
(A.2)

where

$\epsilon(z,w)=\frac{1}{2\sqrt{zw}(\sqrt{z}+\sqrt{w})^{2}}\;.$ (A.3)

By using these relations, the product of $T_{M}(z)$ and $T_{N}(w)$ for general
$f\times f$ symmetric matrices $M$ and $N$ can be calculated as

$\displaystyle T_{M}(z)T_{N}(w)$ $\displaystyle\sim$
$\displaystyle\frac{D}{2}\frac{1}{(z-w)^{4}}{\rm tr}(M\circ
N)+\frac{2}{(z-w)^{2}}T_{M\circ N}(w)+\frac{1}{z-w}\partial_{w}T_{M\circ
N}(w)$ (A.4)
$\displaystyle+\frac{1}{4}\frac{1}{z-w}\left([M,N]\right)_{ij}\left(:(\partial_{w}A^{i}(w))\cdot
A^{j}(w):-:A^{i}(w)\cdot\partial_{w}A^{j}(w):\right)$

where

$M\circ N\equiv\frac{1}{2}(MN+NM).$ (A.5)

Note that in the right hand side of eq.(A.4), there remain terms which cannot
be represented only by $T$ unless $[M,N]=0$.

We investigate the properties of (A.4) for general $M$ and $N$. First, note
that any symmetric $f\times f$ matrix $M$ can be expanded by the base matrices
$H^{AB}=H^{BA}=(H^{AB})^{T}$ ($A,B=1,2,\cdots,f$)

$(H^{(AB)}{})_{ij}=\frac{1}{2}\left(\delta_{iA}\delta_{jB}+\delta_{iB}\delta_{jA}\right).$
(A.6)

We also use the following expressions

$E=H^{11},\qquad F^{(ab)}=H^{ab},\qquad G^{(a)}=H^{(1a)}$ (A.7)

where $a=2,\cdots,f$. We can divide these base matrices $H^{(AB)}$ into two
classes and define

${\cal M}^{+}={\rm Span}\\{H^{11},H^{ab}\\},\qquad{\cal M}^{-}={\rm
Span}\\{H^{1a}\\}.$ (A.8)

For any matrix $M^{\pm}\in{\cal M}^{\pm}$, from the definition of $T_{M}(z)$
given by eq.(58), the following relation holds:

$T_{M^{\pm}}(e^{2\pi i}z)=\pm T_{M^{\pm}}(z).$ (A.9)

The mode expansion of $T_{M^{\pm}}(z)$ is given by

$T_{M^{+}}(z)=\sum_{n\in{\bf Z}}L^{M^{+}}_{n}z^{-n-2},\qquad
T_{M^{-}}(z)=\sum_{r\in{\bf Z}+\frac{1}{2}}L^{M^{-}}_{r}z^{-r-2}.$ (A.10)

Now that we have prepared the appropriate base matrices $H^{AB}$ and the mode
expansion for $T_{M}(z)$, we can calculate the commutation relation
$[L^{M}_{\xi},L^{N}_{\eta}]$ by operating

$\oint_{C_{0}}\frac{dw}{2\pi i}\oint_{C_{w}}\frac{dz}{2\pi
i}\,z^{\xi+1}w^{\eta+1}$ (A.11)

on eq.(A.4) for any two matrices $M$ and $N$. Here $C_{0}$ and $C_{w}$
represent contour integration around $w=0$ and $z=w$ respectively. Note that
$\xi$ (or $\eta$) is an integer if the matrix $M$ (or $N$) belongs to ${\cal
M}^{+}$, and a half-integer if $M$ (or $N$) belongs to ${\cal M}^{-}$. After
performing the integration calculation, we obtain the commutation relation

$\displaystyle[L^{M}_{\xi},L^{N}_{\eta}]$ $\displaystyle=$
$\displaystyle(\xi-\eta)L_{\xi+\eta}^{M\circ N}+\frac{D}{12}\mathrm{Tr}(M\circ
N)(\xi^{3}-\xi)\delta_{\xi+\eta,0}$ (A.12)
$\displaystyle\qquad+\frac{1}{4}\left([M,N]\right)_{ij}\sum_{\zeta}(2\zeta-\xi-\eta):\alpha_{\xi+\eta-\zeta}^{i}\cdot\alpha^{j}_{\zeta}:\;.$

Here $\zeta$ is an integer for $j=1$ and and a half-integer for $j=a$.

In the following, we explicitly present the commutation relations for each
pair of base matrices $H^{(AB)}$ (or $E$, $F^{(ab)}$ and $G^{(a)}$) after
collecting the related useful relations.

The base matrices satisfy the relations

$\displaystyle H^{(AB)}\circ H^{(CD)}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(\delta_{AC}H^{BD}+\delta_{AD}H^{BC}+\delta_{BC}H^{AD}+\delta_{BD}H^{AC}\right),$
(A.13) $\displaystyle\left[H^{(AB)},H^{(CD)}\right]_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(\delta_{AC}(\delta_{Bi}\delta_{Dj}-\delta_{Bj}\delta_{Di})+\delta_{AD}(\delta_{Bi}\delta_{Cj}-\delta_{Bj}\delta_{Ci})\right.$
(A.14)
$\displaystyle\qquad\left.+\delta_{BC}(\delta_{Ai}\delta_{Dj}-\delta_{Aj}\delta_{Di})+\delta_{BD}(\delta_{Ai}\delta_{Cj}-\delta_{Aj}\delta_{Ci})\right).$

The mode operators for base matrices are given by

$L_{n}^{E}=\frac{1}{2}\sum_{m\in{\bf Z}}:\alpha_{n-m}\cdot\alpha_{m}:,$ (A.15)
$L_{n}^{F^{(ab)}}=\frac{1}{2}\sum_{r\in{\bf
Z}+\frac{1}{2}}:\alpha^{a}_{n-r}\cdot\alpha^{b}_{r}:\;+\frac{D}{16}\delta^{ab}\delta^{n,0},$
(A.16) $L_{r}^{G^{(a)}}=\frac{1}{2}\sum_{m\in{\bf
Z}}:\alpha^{a}_{r-m}\cdot\alpha_{m}:.$ (A.17)

Commutation relations are summarized as follows.

$[L_{m}^{E},L_{n}^{E}]=(m-n)L_{m+n}^{E}+\delta_{m+n,0}\frac{D}{12}m(m^{2}-1),$
(A.18) $\displaystyle[L_{m}^{F^{(a_{1}a_{2})}},L_{n}^{F^{(b_{1}b_{2})}}]$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{2}\sum_{j=1}^{2}\delta^{a_{i+1},b_{j+1}}\left\\{\frac{1}{4}(m-n)L_{m+n}^{F^{a_{i}b_{j}}}\right.$
(A.19)
$\displaystyle\qquad\qquad\qquad\left.+\frac{1}{8}r\left(:\alpha^{a_{i}}_{m+n-r}\cdot\alpha^{b_{j}}_{r}:-:\alpha^{b_{j}}_{m+n-r}\cdot\alpha^{a_{i}}_{r}:\right)\right\\}$
$\displaystyle+\;\delta_{m+n,0}\left(\delta^{a_{1},b_{1}}\delta^{a_{2},b_{2}}+\delta^{a_{1},b_{2}}\delta^{a_{2},b_{1}}\right)\frac{D}{24}m\left(m^{2}-1\right),$
$\displaystyle[L_{r}^{G^{(a)}},L_{s}^{G^{(b)}}]$ $\displaystyle=$
$\displaystyle\frac{1}{4}(r-s)\left(L_{r+s}^{F^{(ab)}}+\delta^{a,b}L_{r+s}^{E}\right)+\frac{1}{8}\sum_{t\in{\bf
Z}+\frac{1}{2}}t\left(:\alpha^{a}_{r+s-t}\cdot\alpha^{b}_{t}:-:\alpha^{b}_{r+s-t}\cdot\alpha^{a}_{t}:\right)$
(A.20)
$\displaystyle\qquad+\delta_{r+s,0}\delta^{ab}\frac{D}{24}r\left(r^{2}-1\right),$
$[L_{n}^{E},L_{s}^{G^{(a)}}]=\frac{1}{2}\sum_{m\in{\bf
Z}}(n-m):\alpha^{a}_{s+n-m}\cdot\alpha_{m}:\,,\qquad\qquad$ (A.21)
$\displaystyle[L_{n}^{F^{(ab)}},L_{s}^{G^{(c)}}]$ $\displaystyle=$
$\displaystyle\delta^{a,c}\,\frac{1}{4}\sum_{m\in{\bf
Z}}(m-s):\alpha^{b}_{s+n-m}\cdot\alpha_{m}:$ (A.22)
$\displaystyle+\delta^{b,c}\,\frac{1}{4}\sum_{m\in{\bf
Z}}(m-s):\alpha^{a}_{s+n-m}\cdot\alpha_{m}:\;.$

Note that $i$ and $j$ are taken as mod 2 values in eq.(A.19). We see that the
operators $L_{n}^{E}$ form the Virasoro algebra with central charge $D$. On
the other hand, the set of all the operators $L_{n}^{E}$, $L_{n}^{F^{(ab)}}$
and $L_{r}^{G^{(a)}}$ does not form a closed algebra because of the last term
of eq.(A.12).

Commutation relations for general matrices are obtained from the above
relations by using

$L_{\xi}^{M+N}=L_{\xi}^{M}+L_{\xi}^{N}$ (A.23)

which follows from

$T_{M}(z)+T_{N}(z)=T_{M+N}(z).$ (A.24)

##  Appendix Appendix B  Algebra of $L_{\xi}^{M}$ and the related useful
relations

We further investigate the properties of the algebra given by the operators
$L_{\xi}^{M}$ obtained by the mode expansion of $T_{M}(z)$ especially for
$M=P,Q^{ij},R^{i}$. The matrices $P$, $Q^{ij}$ and $R^{i}$ defined in eqs.(50)
and (51) are expanded by the base matrices $E$, $F^{(ab)}$ and $G^{(a)}$ as

$\displaystyle P$ $\displaystyle=$ $\displaystyle\frac{1}{f}E,$ (B.1)
$\displaystyle Q^{ij}$ $\displaystyle=$
$\displaystyle\sum_{a=2}^{f}\sum_{b=2}^{f}(v_{a})_{i}(v_{b})_{j}F^{(ab)},$
(B.2) $\displaystyle R^{i}$ $\displaystyle=$
$\displaystyle\sum_{a=2}^{f}\frac{2}{\sqrt{f}}(v_{a})_{i}G^{(a)}.$ (B.3)

We see that the relations

$\sum_{i=1}^{f}Q^{ij}=0,\quad\sum_{i=1}^{f}R^{i}=0$ (B.4)

hold since

$\sum_{i=1}^{f}(v_{a})_{i}=0,\qquad\sum_{i=1}^{f}(v_{a})_{i}(v_{b})_{i}=\delta_{ab}.$
(B.5)

Note that the following relation is also useful.

$\sum_{a=2}^{f}(v_{a})_{i}(v_{a})_{j}=\delta_{ij}-\frac{1}{f}\;\left(\;={\rm
Tr}\,Q^{ij}\;\right)\;.$ (B.6)

From the general formula of commutation relations (A.12), We can derive the
following set of relations that are useful for analyzing the physical state
conditions.

$\displaystyle\frac{1}{2}[L_{m}^{Q^{ii}},L_{n}^{Q^{jj}}]+\frac{1}{2}[L_{m}^{Q^{jj}},L_{n}^{Q^{ii}}]$
$\displaystyle=$
$\displaystyle(m-n)\left(\delta_{ij}-\frac{1}{f}\right)L_{m+n}^{Q^{ij}}$ (B.7)
$\displaystyle+\,\delta_{m+n,0}\left(\delta^{ij}-\frac{1}{f}\right)^{2}\frac{D}{12}m\left(m^{2}-1\right),$
$\displaystyle\frac{1}{2}[L_{r}^{R^{i}},L_{s}^{R^{j}}]+\frac{1}{2}[L_{r}^{R^{j}},L_{s}^{R^{i}}]=(r-s)\left(\frac{1}{f}L_{r+s}^{Q^{ij}}+\left(\delta_{ij}-\frac{1}{f}\right)L_{r+s}^{P}\right)$
$\displaystyle\hskip
199.16928pt+\delta_{r+s,0}\frac{1}{f}\left(\delta_{ij}-\frac{1}{f}\right)\frac{D}{6}r\left(r^{2}-1\right),$
(B.8)
$\displaystyle\frac{1}{f}[L_{n}^{Q^{ij}},L_{s}^{R^{k}}]+\frac{1}{2}\left(\delta_{jk}-\frac{1}{f}\right)[L_{n}^{P},L_{s}^{R^{i}}]+\frac{1}{2}\left(\delta_{ik}-\frac{1}{f}\right)[L_{n}^{P},L_{s}^{R^{j}}]$
$\displaystyle\hskip
85.35826pt=\frac{1}{2f}(n-s)\left\\{\left(\delta_{jk}-\frac{1}{f}\right)L_{n+s}^{R^{i}}+\left(\delta_{ik}-\frac{1}{f}\right)L_{n+s}^{R^{j}}\right\\}.$
(B.9)

Also, note that

$[L^{P}_{m}\,,L^{P}_{n}]=(m-n)\frac{1}{f}L^{P}_{m+n}+\delta_{m+n,0}\frac{D}{12f^{2}}m(m^{2}-1).$
(B.10)

Next, we give several useful commutation relations between zero mode operators
and non-zero mode operators:

$\displaystyle[L_{r}^{R^{j}},L_{0}^{P+Q^{ii}}]=\left(\delta_{ij}-\frac{1}{f}\right)rL_{r}^{R^{i}}+\frac{1}{2}\left(2-\delta_{ij}f\right)[L_{r}^{R^{i}}+L_{r}^{R^{j}},L_{0}^{P}],$
(B.11)
$[L_{m}^{P+Q^{ii}},L_{0}^{P+Q^{jj}}]=\frac{1}{f}mL_{m}^{P}+[L_{m}^{Q^{ii}},L_{0}^{Q^{jj}}],$
(B.12) $\displaystyle[L_{m\neq
0}^{Q^{ij}},L_{0}^{Q^{kk}}]=\frac{1}{2}\frac{\delta_{jk}f-1}{\delta_{ik}f-1}\,[L_{m}^{Q^{ii}},L_{0}^{Q^{kk}}]+\frac{1}{2}\frac{\delta_{ik}f-1}{\delta_{jk}f-1}\,[L_{m}^{Q^{jj}},L_{0}^{Q^{kk}}].$
(B.13)

In particular, since
$[L_{m}^{Q^{ii}},L_{0}^{Q^{ii}}]=(1-\frac{1}{f})mL_{m}^{Q^{ii}}$,

$[L_{m}^{P+Q^{ii}},L_{0}^{P+Q^{ii}}]=\frac{1}{f}mL_{m}^{P}+\left(1-\frac{1}{f}\right)mL_{m}^{Q^{ii}}.$
(B.14)

Further, if we define the operators

$L_{n}^{E,m}=\;:\alpha_{n-m}\cdot\alpha_{m}:,\qquad
L_{n}^{F^{(ab)},r}=\;:\alpha^{a}_{n-r}\cdot\alpha^{b}_{r}:,\qquad
L_{r}^{G^{(a)},s}=\;:\alpha^{a}_{s}\cdot\alpha_{r-s}:,$ (B.15)

we can form a closed algebra (with central extension) by using them as
generators by noting that

$L_{n}^{E,m}=L_{n}^{E,n-m},\qquad L_{n}^{F^{(ab)},r}=L_{n}^{F^{(ba)},n-r}$
(B.16)

and

$L_{n}^{E}=\frac{1}{2}\sum_{m\in{\bf Z}}L_{n}^{E,m},\qquad
L_{n}^{F^{(ab)}}=\frac{1}{2}\sum_{r\in{\bf
Z}+\frac{1}{2}}L_{n}^{F^{(ab)},r},\qquad
L_{r}^{G^{(a)}}=\frac{1}{2}\sum_{s\in{\bf Z}+\frac{1}{2}}L_{r}^{G^{(a)},s}.$
(B.17)

For example, the right-hand sides of eqs.(A.19)-(A.22) can be written by using
the operators given in (B.15). Also, the following relations are useful for
analyzing the physical state conditions:

$[L_{r}^{R^{j}},L_{0}^{Q^{ii}}]=\frac{1}{\sqrt{f}}\left(\delta_{ij}-\frac{1}{f}\right)\sum_{a=2}^{f}(v_{a})_{i}\sum_{s\in{\bf
Z}+\frac{1}{2}}sL_{r}^{G^{(a)},s},$ (B.18)
$[L_{r}^{G^{(a)},s},L_{0}^{Q^{ii}}]=\frac{1}{\sqrt{f}}(v_{a})_{i}\sum_{b=2}^{f}(v_{b})_{i}sL_{r}^{G^{(b)},s},$
(B.19)

and

$\displaystyle[L_{m(\neq 0)}^{Q^{ii}},L_{0}^{Q^{jj}}]$ $\displaystyle=$
$\displaystyle\left(\delta_{ij}-\frac{1}{f}\right)\sum_{a,b}(v_{a})_{i}(v_{b})_{j}\\!\sum_{s\in{\bf
Z}+\frac{1}{2}}(m-s)L_{m}^{F^{(ab)},m-s}$ (B.20) $\displaystyle=$
$\displaystyle\left(\delta_{ij}-\frac{1}{f}\right)\left(L_{m}^{Q^{ij}}+\sum_{s\in{\bf
Z}+\frac{1}{2}}[L_{m-\frac{1}{2}}^{G^{(a)},s},\;L_{\frac{1}{2}}^{G^{(b)},m-s}\,]\right)$
(B.21)

where

$[L_{r}^{G^{(a)},s},\;L_{m-r}^{G^{(b)},t}\,]=\delta_{m,s+t}(r-s)\,L_{m}^{F^{(ab)},m-s}+\delta^{a,b}\delta_{s+t,0}\,s\,L_{m}^{E,r-s}.$
(B.22)

##  Appendix Appendix C  General solutions of physical state condition

We give a proof that the space of states satisfying the physical state
condition given in section 3.4 is spanned by the states of the form (95) and
(96) with the conditions (97)- (99). The condition we have to impose is the
set of relations ${\rm(I^{\prime})}$ for $n=1,2$, ${\rm(I^{\prime\prime})}$
for $n=1$, ${\rm(II^{\prime})}$ (or (86)) for $r=\frac{1}{2}$ and ${\rm(III)}$
as we discussed in the section 3.4.

We first consider the condition (86) for $r=\frac{1}{2}$, that is,
$L_{\frac{1}{2}}^{G^{(a)},s}|\phi\rangle=0$. Since

$[L_{r}^{G^{(a)},s},\alpha_{-t}^{b,\mu}]=s\delta^{a,b}\delta_{s,t}\alpha_{r-s}^{\mu},\qquad[L_{r}^{G^{(a)},s},\alpha_{-t+\frac{1}{2}}^{\mu}]=(r-s)\delta_{s,r-t+\frac{1}{2}}\alpha_{s}^{a,\mu},$
(C.1)

the operator $L_{\frac{1}{2}}^{G^{(a)},s}$ gives a non-trivial effect only on
a state including the oscillator $\alpha_{-s}^{a,\mu}$ for $s>0$, and
$\alpha_{s-1/2}^{\mu}$ for $s<0$. In fact, $L_{\frac{1}{2}}^{G^{(a)},s}$
changes the oscillator as $\alpha_{-s}^{a,\mu}\rightarrow
s\alpha_{-s+\frac{1}{2}}^{\mu}$ for $s>0$, and
$\alpha_{s-\frac{1}{2}}^{\mu}\rightarrow(\frac{1}{2}-s)\alpha_{s}^{a,\mu}$ for
$s<0$. From this property, possible combination of oscillators
$\alpha_{-s}^{a,\mu}$ and $\alpha_{-s+\frac{1}{2}}^{\mu}$ (for $s>0$) or
$\alpha_{s-\frac{1}{2}}^{\mu}$ and $\alpha_{s}^{a,\mu}$ (for $s<0$) within a
state $|\phi\rangle$ satisfying the condition
$L_{\frac{1}{2}}^{G^{(a)},s}|\phi\rangle=0$ can be determined as follows. For
$s\geq\frac{3}{2}$, if there is any $\alpha_{-s}^{a,\mu_{a}}$ in the state
$|\phi\rangle$, then, in order to satisfy
$L_{\frac{1}{2}}^{G^{(a)},s}|\phi\rangle=0$, there also should be $f-1$
different type of oscillators $\alpha_{-s}^{b,\mu_{b}}$ $(b\neq a)$ and
$\alpha_{-s+\frac{1}{2}}^{\mu}$, all of which should form the following anti-
symmetric combination:

$\alpha_{-s}^{2,[\mu_{2}}\alpha_{-s}^{3,\mu_{3}}\cdots\alpha_{-s}^{f,\mu_{f}}\alpha_{-s+\frac{1}{2}}^{\mu]}.$
(C.2)

Also no other mode $-s$ oscillator $\alpha_{-s}^{a,\mu^{\prime}}$ could be
added in the state. Similarly, for $-s\leq-\frac{1}{2}$, if there is any
$\alpha_{-s-\frac{1}{2}}^{\mu}$ in the state $|\phi\rangle$, then, in order to
satisfy $L_{\frac{1}{2}}^{G^{(a)},-s}|\phi\rangle=0$, there should be $f-1$
oscillators $\alpha_{-s}^{a,\mu_{a}}$ ($a=2,\cdots f$) and no other
$\alpha_{-s-\frac{1}{2}}^{\mu^{\prime}}$ within the state, and the $f$
oscillators should form the combination

$\alpha_{-s-\frac{1}{2}}^{[\mu}\alpha_{-s}^{2,\mu_{2}}\alpha_{-s}^{3,\mu_{3}}\cdots\alpha_{-s}^{f,\mu_{f}]}.$
(C.3)

Note that in this case any number of extra $\alpha_{-s}^{b,\nu}$’s can be
included within the state. Finally, for $s=\frac{1}{2}$, to satisfy
$L_{\frac{1}{2}}^{G^{(a)},-s}|\phi\rangle=0$, $|\phi\rangle$ can include any
number $K$ of oscillators $\alpha_{-\frac{1}{2}}^{a_{i},\mu_{i}}$
($i=1,\cdots,K$). However, if we write the corresponding combination as

$h^{a_{1}a_{2}\cdots
a_{K}}_{\mu_{1}\mu_{2}\cdots\mu_{K}}\alpha_{-\frac{1}{2}}^{a_{1},\mu_{1}}\alpha_{-\frac{1}{2}}^{a_{2},\mu_{2}}\cdots\alpha_{-\frac{1}{2}}^{a_{K},\mu_{K}},$
(C.4)

the coefficient should satisfy $p^{\mu_{1}}h^{a_{1}a_{2}\cdots
a_{K}}_{\mu_{1}\mu_{2}\cdots\mu_{K}}=0$. From the above discussion, we see
that any state satisfying the condition ${\rm(II^{\prime})}$ for
$r=\frac{1}{2}$, i.e., $L_{\frac{1}{2}}^{G^{(a)},-s}|\phi\rangle=0$ for all
$s\in{\bf Z}+\frac{1}{2}$, must have the form (95) and (96) with the condition
$p^{\sigma_{i}}h^{a_{1}a_{2}\cdots
a_{K}}_{[\mu_{1}^{M}\mu_{2}^{M}\cdots\mu^{1}_{1}\mu^{1}_{2}\cdots\mu^{1}_{f}]\sigma_{1}\sigma_{2}\cdots\sigma_{K}}=0$.

Next, we impose the condition ${\rm(I^{\prime\prime})}$ for $n=1$ on a state
$|\phi_{{\rm(II^{\prime})}_{\\!\frac{1}{2}}}\rangle$ satisfying
${\rm(II^{\prime})}$ for $r=\frac{1}{2}$. The condition ${\rm(II^{\prime})}$
can be rewritten as the following simpler form

$L_{n}^{F^{(ab)}}|\phi\rangle=0\qquad(a,b=2,\cdots f)$ (C.5)

where $L_{n}^{F^{ab}}$ is given by (A.16), and the condition for
$r=\frac{1}{2}$ is explicitly given by

$\displaystyle
L_{\frac{1}{2}}^{F^{(ab)}}|\phi_{{\rm(II^{\prime})}_{\\!\frac{1}{2}}}\rangle$
$\displaystyle=$
$\displaystyle\left(\frac{1}{2}\alpha_{\frac{1}{2}}^{a}\cdot\alpha_{\frac{1}{2}}^{b}+\alpha_{-\frac{1}{2}}^{(a}\cdot\alpha_{\frac{3}{2}}^{b)}+\alpha_{-\frac{3}{2}}^{(a}\cdot\alpha_{\frac{5}{2}}^{b)}+\cdots\right)|\phi_{{\rm(II^{\prime})}_{\\!\frac{1}{2}}}\rangle$
(C.6) $\displaystyle=$
$\displaystyle\frac{1}{2}\alpha_{\frac{1}{2}}^{a}\cdot\alpha_{\frac{1}{2}}^{b}|\phi_{{\rm(II^{\prime})}_{\\!\frac{1}{2}}}\rangle=0.$
(C.7)

Here the second equality comes from the properties of
$|\phi_{{\rm(II^{\prime})}_{\\!\frac{1}{2}}}\rangle$. This gives the traceless
condition on each pair of indices corresponding to the coefficients of
$\alpha_{-\frac{1}{2}}^{a\mu}$’s as (98).

Finally, we consider the condition ${\rm(I^{\prime})}$ for $n=1,2$ on
$|\phi_{{\rm(II^{\prime})}_{\\!\frac{1}{2}}}\rangle$. From the condition
${\rm(II^{\prime})}$, the integer mode oscillators of a state
$|\phi_{{\rm(II^{\prime})}_{\\!\frac{1}{2}}}\rangle$ has to have a form

$|\phi_{{\rm(II^{\prime})}_{\\!\frac{1}{2}}}\rangle=\alpha_{-1}^{[\mu_{1}^{1}}\alpha_{-2}^{\mu_{1}^{2}}\cdots\alpha_{-M}^{\mu_{1}^{M}]}\;\left(\mbox{\scriptsize
half-integer mode oscillators}\right)|0,p\rangle.$ (C.8)

Since $L_{n}^{P}=\frac{1}{f}L_{n}^{E}$ and

$[L_{n}^{E},\alpha_{-m}^{\nu}]=m\alpha_{n-m}^{\nu},$ (C.9)

the relations

$\displaystyle L_{1}|\phi_{{\rm(II^{\prime})}_{\\!\frac{1}{2}}}\rangle$
$\displaystyle=$
$\displaystyle\alpha_{0}^{[\mu_{1}^{1}}\alpha_{-2}^{\mu_{1}^{2}}\cdots\alpha_{-M}^{\mu_{1}^{M}]}\;\left(\mbox{\scriptsize
half-integer mode oscillators}\right)|0,p\rangle,$ (C.10) $\displaystyle
L_{2}|\phi_{{\rm(II^{\prime})}_{\\!\frac{1}{2}}}\rangle$ $\displaystyle=$
$\displaystyle
2\alpha_{-1}^{[\mu_{1}^{1}}\alpha_{0}^{\mu_{1}^{2}}\alpha_{-3}^{\mu_{1}^{3}}\cdots\alpha_{-M}^{\mu_{1}^{M}]}\;\left(\mbox{\scriptsize
half-integer mode oscillators}\right)|0,p\rangle$ (C.11)

hold. This gives the condition on the coefficient as

$p^{\mu_{1}^{1}}h^{a_{1}a_{2}\cdots
a_{K}}_{[\mu_{1}^{M}\mu_{2}^{M}\cdots\mu^{1}_{1}\mu^{1}_{2}\cdots\mu^{1}_{f}]\sigma_{1}\sigma_{2}\cdots\sigma_{K}}=0$
(C.12)

(or the equivalent condition $p^{\mu_{1}^{2}}h^{a_{1}a_{2}\cdots
a_{K}}_{[\mu_{1}^{M}\mu_{2}^{M}\cdots\mu^{1}_{1}\mu^{1}_{2}\cdots\mu^{1}_{f}]\sigma_{1}\sigma_{2}\cdots\sigma_{K}}=0$.)
This concludes the proof.

##  Appendix Appendix D Remark on the light-cone gauge

Here we give a remark on the light-cone gauge. Let us first remind how the
light-cone gauge was taken for the ordinary string. After the orthonormal
gauge is taken, we utilize residual gauge degrees of freedom which preserve
the gauge condition to make light-cone oscillators $\alpha_{n}^{+}=0$. Then
the Virasoro condition $L_{n}=0$ together with $\alpha_{n}^{+}=0$ become
second-class constraint which we can solve it for the remaining light-cone
oscillators $\alpha_{n}^{-}$ in terms of the transverse modes (or eliminate
$\alpha_{n}^{\pm}$ with Dirac bracket.) Thus only transverse variables are
left with us.

Now let us turn to our string junction case. We can easily find
reparametrization transformation (4) which preserves the orthonormal gauge
condition (17). They should satisfy

$\partial_{+}\epsilon^{(i)-}=0,\quad\partial_{-}\epsilon^{(i)+}=0,$ (D.13)

where $\pm$ stands for the worldsheet light-cone directions. The boundary
conditions (5) and (6) restrict them further

$\epsilon^{(i)\pm}(\tau,\sigma)=\epsilon(\tau\pm\sigma),$ (D.14)

where $\epsilon(\tau)$ is an $i$-independent function which satisfies
$\epsilon(\tau)=\epsilon(\tau+2\pi)$. Then we can use this degree of freedom
to make $\alpha_{n}^{+}=0$ as in the string case. So we left with $2f-1$ sets
of oscillators $\alpha_{n}^{-},\alpha_{r}^{a+},\alpha_{r}^{a-}$ other than
transverse modes. Our remaining constraints are
$V_{n},R_{r}^{a},L_{n}^{Q^{a-1\,a-1}-Q^{a\,a}}$ whose number of degrees of
freedom is also $2f-1$ in total. These are second-class constraints together
with $\alpha_{n}^{+}=0$. Thus if we were able to solve them in terms of
transverse oscillators, then in principle we could reach the light-cone gauge.
But these relations are so complicated that we have not succeeded to
explicitly solve them yet. Indeed, even in classical level, the authors in [9]
concluded that it is not possible, though their treatment of remaining second-
class constraints is not clear.

Thus, although the number of physical degrees of freedom is likely to coincide
with that of light-cone gauge, the structure of physical state may not be the
same as the naive truncation to the transverse modes.

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