aainstitutetext: Landau Institute for Theoretical Physics, 142432
Chernogolovka, Russiabbinstitutetext: Kharkevich Institute for Information
Transmission Problems, 127994 Moscow, Russia

# Mirror symmetry and new approach to constructing orbifolds of Gepner models

Alexander Belavin a Sergey Parkhomenko<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Motivated by the principles of the conformal bootstrap, primarily the
principle of Locality, simultaneously with the requirement of space-time
supersymmetry, we reconsider constructions of compactified superstring models.
Starting from requirements of space-time supersymmetry and mutual locality, we
construct a complete set of physical fields of orbifolds of Gepner models. To
technically implement this, we use spectral flow generators to construct all
physical fields from the chiral primary fields. The set of these spectral flow
operators forms a so-called admissible group $G_{adm}$, which defines a given
orbifold. The action of these operators produces a collection of physical
fields consistent with the action of supersymmetry generators. The selection
of mutually local fields from this collection is carried out using the mirror
group $G^{*}_{adm}$. The permutation of $G_{adm}$ and $G^{*}_{adm}$ replaces
the original orbifold with a mirror one that satisfies the same conditions as
the original one. This also implies that the resulting model is modular
invariant.

###### Keywords:

Mirror Symmetry, Calabi-Yau manifolds, Compactification

##  1 Introduction

Ten-dimensional superstring theory naturally unifies gauge quantum field
theory and quantum gravity. In CHSW  it was shown that to obtain a
4-dimensional theory with the space-time supersymmetry required for
phenomenological reasons, we must compactify 6 of the 10 dimensions on the so-
called Calabi-Yau manifold. Another equivalent way to do the same is the
compactification of $6$ dimensions on N = 2 superconformal field theory with
the central charge equal $9$ Gep .

Each of these two equivalent ways leading to the same theory has its own
advantages. The first gives a geometrical interpretation. The second allows
one to use exactly solvable N=2 SCFT models and thus obtain an explicit
solution to the problem under consideration. In this article we follow the
second way and propose a new approach to constructing orbifolds of Gepner
models built using the tensor product of $N=2$ minimal models Gep . This
approach leads to the same results as in Gep ; GrPl , based on the
simultaneous fulfillment of the requirements of supersymmetry and modular
invariance. But our initial requirements are different. Namely, instead it
requires simultaneous fulfillment of space-time supersymmetry and mutual
locality for construction of the space of all physical states of the theory.

The construction consists of the following steps. As in Gep ; BLT ; BP ; BBP ;
P , we take for the compactification the product of $N=2$ minimal models with
the full central charge of $9$ joined with four free bosonic $X^{\mu}$, four
fermionic fields $\psi^{\mu}$, as well as with BRST ghost fields.

The requirement of consistency with the action of the complete $N=2$ Virasoro
algebra, which is a subalgebra of the product of $N=2$ Virasoro algebras of
all components of the theory, imposes certain restrictions on the states of
subtheories Gep . In particular, fields from the left or right $NS$ sectors of
the theory must include products of only fields from the $NS$, not $R$ sectors
of sub-theories. The same should be done for fields from the left or right $R$
sectors. Then we find BRST-invariant massless fields in both the left and
right sectors and select from them those that are necessary to construct
space-time supersymmetry generators in both sectors. Not all physical, i.e.
BRST-invariant fields, are compatible with the action of supersymmetry
generators, or, equivalently, mutually local with respect to them. We choose
those physical fields that are mutually local with respect to supersymmetry
generators. It is equivalent to the GSO reduction.

The next step is to construct complete vertices, i.e. products of left
(holomorphic) and right (anti-holomorphic), vertices of type $(NS,NS)$, which
are mutually local. Such properties are possessed by fields $(NS,NS)$ that
satisfy the GSO conditions and are diagonal with respect to the sets of their
left and right $U(1)$ charges. However, since the space of all states must
allow the independent action of the left and the right supersymmetry
generators, it is also necessary to have twisted full vertices. That is, we
need to admit such full vertices, whose sets of $U(1)$ charges of the left and
right factors do not coincide. It is also required that the GSO conditions are
satisfied in both the left and right factors, as before. To achieve the last
requirement, as we will show below, it is necessary that the twisting between
$U(1)$ charge components in the left and right factors is restricted by the
actions of generators forming the so-called admissible group $G_{adm}$.
Technically, we implement the action of these generators using the spectral
flow transformation SS ; BP ; BBP . At last, we check the mutual locality of
the vertices constructed in this way and leave only those that satisfy this
requirement. The above-mentioned use of the spectral flow deformation turns
out to be very convenient for performing this selection. It is remarkable how
it turns out that the selection of mutually local vertices is carried out
using the mirror group $G^{*}_{adm}$. By definition, the elements of the group
$G^{*}_{adm}$ form the set of all generators that are mutually local with
respect to the generators of the group $G_{adm}$. After this, applying space-
time SUSY generators to the resulting vertices of $(NS,NS)$ type independently
in the left and right sectors, we obtain the set of vertices of $(NS,R)$,
$(R,NS)$ and $(R,R)$ types. It is checked that the resulting vertices of all
four types are mutually local. Moreover, the resulting set of physical states
of the orbifolds of Gepner models coincides with that obtained earlier in in
Gep ; GrPl . Moreover, this also implies the modular invariance of the
resulting model.

##  2 Preliminaries

We consider type $II$ superstring theory, whose matter sector is a product of
spacetime $M_{st}$ and compact $M_{int}$ sectors. The space-time degrees of
freedom of string in $4$-dimensional Minkowski space-time are bosons
$X^{\mu}(z)$, and fermions $\psi^{\mu}(z)$, $\mu=0,...,3$ with the OPE

$\displaystyle X^{\mu}(z)X^{\nu}(0)=-\eta^{\mu\nu}\log{z}+...,$
$\displaystyle\psi^{\mu}(z)\psi^{\nu}(0)=\eta^{\mu\nu}z^{-1}+...,$

and endowed with the action of $N=1$ Virasoro superalgebra with central charge
$c_{st}=6$. The currents of the algebra are given by

$\displaystyle T_{st}(z)=-\frac{1}{2}\partial X^{\mu}(z)\partial
X_{\mu}(z)-\frac{1}{2}\psi^{\mu}(z)\partial\psi_{\mu}(z),$ $\displaystyle
G_{st}(z)=\imath\psi^{\mu}(z)\partial X_{\mu}(z).$

It is convenient to bosonize fermions by introducing fields $H_{a}(z)$,
$a=1,2$ as

$H_{a}(z)H_{b}(0)=-\delta_{ab}\log{(z)}+....$ (1)
$\displaystyle\frac{1}{\sqrt{2}}(\pm\psi^{0}+\psi^{1})=\exp{[\pm\imath
H_{1}]},$
$\displaystyle\frac{1}{\sqrt{2}}(\psi^{2}\pm\imath\psi^{3})=\exp{[\pm\imath
H_{2}]}.$

Then the left part of the vertex operators of the space-time sector can be
written as

$\displaystyle V_{st}(z)=P_{st}(\partial X^{\mu},\partial
H_{a})\exp{[\imath\lambda^{a}H_{a}]}\exp{[\imath k_{\mu}X^{\mu}(z)]},$ (2)

where $P_{st}(\partial X^{\mu},\partial H_{a})$ is a polynomial of
$\partial^{n}X^{\mu}(z)$, $\partial^{m}H_{a}(z)$ and
$\lambda=(\lambda^{1},\lambda^{2})$ is the weight of $SO(1,3)$ algebra.

In $NS$ sector the weights $\lambda$ fall into two conjugacy classes. The
class of trivial representation $[0]$ contains the $\lambda$’s which are the
$SO(1,3)$ root lattice vectors. The class of fundamental vector representation
$[v]$ contains the $\lambda$’s which are the $SO(1,3)$ root lattice vectors
shifted by the highest weight vector $\omega_{1}=(1,0)$.

The weights $\lambda$ in $R$ sector also fall into other two conjugacy
classes. The class of chiral spinor representation $[\sigma]$ contains the
$\lambda$’s which are the $SO(1,3)$ root lattice vectors shifted by the
highest weight $\omega_{2}=(\frac{1}{2},\frac{1}{2})$. The class of anti-
chiral spinor representation $[\dot{\sigma}]$ contains the $\lambda$’s which
are the $SO(1,3)$ root lattice vectors shifted by the highest weight vector
$\omega_{3}=(\frac{1}{2},-\frac{1}{2})$. Completely the same picture takes
place in the right-moving part of space-time sector.

The space-time, compact and ghost sectors of supersting theory have the $N=2$
superconformal symmetry. Recall the commutation relations for the $N=2$ Super-
Virasoro algebra

$\displaystyle[L_{n},L_{m}]=(n-m)L_{n+m}+{c\over 12}(n^{3}-n)\delta_{n+m,0},$
(3) $\displaystyle[J_{n},J_{m}]={c\over 3}n\delta_{n+m,0},$
$\displaystyle[L_{n},J_{m}]=-mJ_{n+m},$
$\displaystyle\\{G^{+}_{r},G^{-}_{s}\\}=L_{r+s}+{r-s\over 2}J_{r+s}+{c\over
6}(r^{2}-{1\over 4})\delta_{r+s,0},$
$\displaystyle[L_{n},G^{\pm}_{r}]=({n\over 2}-r)G^{\pm}_{r+n},$
$\displaystyle[J_{n},G^{\pm}_{r}]=\pm G^{\pm}_{r+n}..$

The $L_{n}$ generators are the Fourier components of the stress-energy tensor
$T(z)$, $J_{n}$ are the modes of $U(1)$ current $J(z)$ and $G^{\pm}_{r}$ are
the modes of the fermionic currents $G^{\pm}(z)$. Note that the $N=1$ Virasoro
subalgebra is spanned by $L_{n}$ and $G_{r}=G^{+}_{r}+G^{-}_{r}$ generators.
In the compactification of type IIA/B superstrings, the $N=2$ Virasoro
superalgebra acts in the left and right sectors, so we have two copies of the
algebra (3).

The one-parametric family of automorphisms of $N=2$ Virasoro superalgebra SS

$\displaystyle\tilde{G}^{\pm}_{r}=U^{-t}G^{\pm}_{r}U^{t}=G^{\pm}_{r\pm t},$
(4) $\displaystyle\tilde{J}_{n}=U^{-t}J_{n}U^{t}=J_{n}+{c\over
3}t\delta_{n,0},$
$\displaystyle\tilde{L}_{n}=U^{-t}L_{n}U^{t}=L_{n}+tJ_{n}+{c\over
6}t^{2}\delta_{n,0},$

known as spectral flow, is very important in the superstring compactification
construction. It allows to relate various representations of $N=2$ Virasoro
superalgebra. In particular for $t\in\mathbb{Z}+{1\over 2}$ the spectral flow
interpolates between $NS$ and $R$ representations, while for $t\in\mathbb{Z}$
it takes $NS$ to $NS$ and $R$ to $R$.

To construct superstring physical states in orbifold compactification, it is
important to express the spectral flow operator $U^{t}$ in terms of the free
scalar field $\varphi(z)$. It is really possible because the second and third
commutation relations from (3) mean that $J_{n}$ are the Fourier modes of the
free bosonic $U(1)$-current $J(z)$, which can be written as

$J(z)=\imath\sqrt{c\over 3}{\partial\varphi\over\partial z},$ (5)

where $\varphi(z)$ is free massles scalar field. Then the relations (4) imply

$U^{t}=\exp{(\imath t\sqrt{c\over 3}\varphi(z))}.$ (6)

In Gepner’s approach Gep  the compact sector $M_{int}$ is some orbifold of the
product of $N=2$ minimal models $M_{\bf{k}}$ with the total central charge
$c_{int}=9$, obtained as a result of factorization over some so-called
admissible group $G_{adm}$

$\displaystyle M_{int}=M_{\bf{k}}/G_{adm}=(\prod_{i=1}^{r}M_{k_{i}})/G_{adm}.$
(7)

The central charge of the Minimal model $M_{k_{i}}$ is
$c_{i}=\frac{3k_{i}}{k_{i}+2}$, where $k_{i}=0,1,2,...$.

For these values of $c_{i}$, the model $M_{k_{i}}$ is unitary and consists of
a finite number of integrable representations of the $N=2$ Virasoro
superalgebra. Since the total central charge of the product of $N=2$ minimal
models is equal $9$, this imposes the following conditions on the set $k_{i}$

$\sum_{i}\frac{k_{i}}{k_{i}+2}=3.$ (8)

If the number of minimal models $r$ is odd, this means that

$\sum_{i}\frac{1}{k_{i}+2}\in Z.$ (9)

The representations are generated from the highest primary states
$\Phi^{NS,R}_{l,q}$, where $l=0,1,...,k$, $q=-l,-l+2,...,l$. The conformal
dimension and $U(1)$ charge of the state $\Phi^{NS}_{l,q}$ are

$\Delta_{l,q}=\frac{l(l+2)-q^{2}}{4(k+2)},\ \ Q_{l,q}=\frac{q}{k+2}.$ (10)

The dimension and charge of the primary state $\Phi^{R}_{l,q}$ in $R$ sector
are

$\Delta^{R}_{l,q}=\frac{l(l+2)-(q-1)^{2}}{4(k+2)}+\frac{1}{8},\quad
Q^{R}_{l,q}=\frac{q-1}{k+2}+\frac{1}{2}.$ (11)

In the right-moving sector of the Minimal model we have the similar set of
integrable representations.

The conformal symmetry of the model $M_{\bf{k}}$ is given by the diagonal
$N=(2,2)$ Virasoro algebra acting in the tensor product of individual minimal
models $M_{k_{i}}$, whose generators

$L_{int,n}=\sum_{i}L_{(i),n},\ J_{int,n}=\sum_{i}J_{(i),n},\
G^{\pm}_{int,r}=\sum_{i}G^{\pm}_{(i),r}.$ (12)

The algebra action (12) is correctly defined on the product of only
$NS$-representations or on the product of only $R$-representations of minimal
models $M_{k_{i}}$ Gep . The primary states of the diagonal $N=2$ Virasoro
algebra in $NS$ and $R$ sectors are also products the primary states from each
minimal factor:

$\displaystyle\Phi^{NS}_{\vec{l},\vec{q}}(z)=\prod_{i}\Phi^{NS}_{l_{i},q_{i}}(z),$
(13)
$\displaystyle\Phi^{R}_{\vec{l},\vec{q}}(z)=\prod_{i}\Phi^{R}_{l_{i},q_{i}}(z).$

The left-moving vertex operators of the string in the compact factor can be
represented as

$\displaystyle
V_{int}(z)=P_{int}(T_{i},J_{i},G^{\pm}_{i})\Phi^{NS,R}_{\vec{l},\vec{q}}(z),$
(14)

where $P_{int}$ denotes the descendants contributions of the product model
$M_{\bf{k}}$.

As each factor in the product model $M_{\bf{k}}$ has $\mathbb{Z}_{k_{i}+2}$
group of symmetries ZF ; GQ ; Gep  the group of discrete symmetries of
$M_{\bf{k}}$ containes the group
$G_{tot}=\mathbb{Z}_{k_{1}+2}\times...\times\mathbb{Z}_{k_{r}+2}$.

Below we will consider orbifolds of the form $M_{\bf{k}}/G_{adm}$, where an
admissible group $G_{adm}$ by definition is any subgroup of $G_{tot}$, whose
generators include the distinguished element $\beta^{0}=(1,1,1,...,1)$, as
well as elements ${\bf{w}}=\\{w_{1},w_{2},...,w_{r}\\}$, which satisfy the
following relations Gep1

$\sum_{i}\frac{w_{i}}{(k_{i}+2)}\in\mathbb{Z}.$ (15)

In the $M_{\bf{k}}/G_{adm}$ model, the group $G_{adm}$ occurs twice as it will
be expained later. First, it generates twisted sectors by the products of
spectral flow vertices for each minimal model $M_{k_{i}}$

$U^{\bf{w}}(z)=\prod_{i}U_{i}^{w_{i}}(z),\
U_{i}=\exp{[\imath\sqrt{\frac{k_{i}}{k_{i}+2}}\phi_{i}(z)]},$ (16)

where we introduced the fields $\phi_{i}(z)$, $i=1,...,r$ for the bosonization
of left $U(1)$ currents and spectral flow operators of individual submodels
according to (5), (6). The second implementation of the group $G_{adm}$ is
given by the product of the $U(1)_{i}$ currents of $M_{k_{i}}$

$\hat{g}^{\bf{w}}=\prod_{i}g_{i}^{w_{i}}=\exp{[\imath\pi\sum_{i}w_{i}(J_{i,0}+\bar{J}_{i,0})]}.$
(17)

When constructing orbifolds of Gepner models, the first implementation
$G_{adm}$ is used to generate a complete set of vertex operators in the
twisted sector, consistent with the spacetime symmetry. As for the second
implementation, it is used to select from this set those vertex operators that
satisfy mutual locality.

It is noteworthy that the group $G_{adm}$ is naturally accompanied by the
mirror group $G^{*}_{adm}$ BH , see also BE , which is defined as

$G^{*}_{adm}=\\{{\bf{w}}^{*}\in
G_{tot},\sum_{i=1}^{r}\frac{w_{i}w^{*}_{i}}{k_{i}+2}\in\mathbb{Z}\quad\text{for
any}\quad{\bf{w}}\in G_{adm}\\}.$ (18)

The group $G^{*}_{adm}$, as it can be seen from its definition, includes the
generator $\beta^{0}=(1,1,1,...,1)$. The group $G^{*}_{adm}$ defines another
$N=(2,2)$ superconformal model $M_{\bf{k}}/G^{*}_{adm}$, which is the mirror
model DG ; LVW  of $M_{\bf{k}}/G_{adm}$, in which the roles of the groups
$G_{adm}$ and $G^{*}_{adm}$ are swapped COGP  (see also P ).

The physical states of our string theory obey the constraints given by the
Super Virasoro algebra whose currents are

$T_{mat}(z)\equiv T_{st}(z)+T_{int}(z),\ G_{mat}(z)\equiv
G_{st}(z)+G^{+}_{int}(z)+G^{-}_{int}(z)$ (19)

We will use the BRST approach, in which the physical states are defined as
BRST-invariant fields. The BRST charge in the left-moving sector includes,
along with currents $T_{mat}(z)$ and $G_{mat}(z)$ also two fermion ghosts
$b(z),c(z)$ and two bosonic ghosts $\beta(z),\gamma(z)$. The explicit
expression for the left-moving $BRST$ charge is

$Q_{BRST}=\oint dz(cT_{mat}+\gamma G_{mat}+\frac{1}{2}(cT_{gh}+\gamma
G_{gh})).$ (20)

Left-moving $bc-\beta\gamma$ ghost fields have the OPE’s

$\beta(z)\gamma(0)=-z^{-1}+...,\ \ b(z)c(0)=z^{-1}+....$ (21)

$N=1$ superconformal symmetry in the ghost factor is given by the currents

$\displaystyle T_{gh}=-\partial bc-2b\partial
c-\frac{1}{2}\partial\beta\gamma-\frac{3}{2}\beta\partial\gamma,$
$\displaystyle G_{gh}=\partial\beta c+\frac{3}{2}\beta\partial c-2b\gamma.$

$\beta-\gamma$ space of states is characterized by the vacua $V_{q}(z)$. They
are parameterized by the picture number $q$ BLT . The vacuum $V_{q}$ is
determined by

$\beta(z)V_{q}(0)\sim O(z^{q}),\ \gamma(z)V_{q}(0)\sim O(z^{-q}),$ (22)

and can be written as the free field exponential:

$V_{q}(z)=\exp{[q\phi(z)]},\quad\phi(z)\phi(0)=-\log(z)+....$ (23)

The left-moving ghost vertex operators can be represented as

$V_{gh}(z)=P_{gh}(\beta,\gamma,b,c)\exp{[q\phi]}(z),$ (24)

where $P_{gh}(\beta,\gamma,b,c)(z)$ is a polynomial of $\beta,\gamma,b,c$ and
their derivatives. The right-moving physical states are determined similarly.

For the BRST charge to be well defined, the integrand in (20) must be periodic
around zero. Therefore, the fields $\beta,\gamma$ must have the same monodromy
as the fermionic degrees of freedom of the matter. The ghost $V_{gh}$ and the
matter components $V_{st}V_{int}$ of the left-moving vertex operator at the
point $z=0$ must be in the same sector. Therefore, q must be an integer in the
$NS$-sector and half-integer in the $R$-sector. The total vertices
${\cal{V}}(z,\bar{z})$ are the products of the left-moving and the right-
moving vertices of ghost, space-time and compact factors

${\cal{V}}(z,\bar{z})=V_{gh}V_{st}V_{int}(z)\bar{V}_{gh}\bar{V}_{st}\bar{V}_{int}(\bar{z}).$
(25)

The left and the right factors of vertex operators can independently belong to
sectors $NS$ or $R$. So we have $(NS,NS)$, $(NS,R)$, $(R,NS)$, $(R,R)$ types
of the vertex operators.

##  3 Massless vertex operators and space-time supersymmetry

Imposing the BRST-invariance requirement, we find the following massless left
vertices in the sector $NS$ with the picture number $(-1)$ and in the sector
$R$ with the picture number $(-\frac{1}{2})$. We get the left-moving vertex of
massless vector boson in $(-1)$ picture

$\exp(-\phi(z))\xi_{\mu}(p)\psi^{\mu}(z)\exp(\imath
p_{\mu}X^{\mu})(z),\quad\xi_{\mu}(p)p^{\mu}=0.$ (26)

In the picture $(-\frac{1}{2})$ we find four pairs of BRST-invariant currents,
which are the vertex operators of massless fermions in the limit of momenta
equal to zero

$\displaystyle
S^{\pm}_{\sigma}(z)=\exp(-\frac{1}{2}\phi(z)+\imath{\sigma}^{a}H_{a}(z)\pm\frac{\imath}{2}\sum_{i}\frac{k_{i}\phi_{i}(z)}{\sqrt{k_{i}(k_{i}+2)}}),\sigma^{a}=\pm\frac{1}{2},\
\sum_{a=1}^{2}\sigma^{a}=\pm 1,$ (27) $\displaystyle
S^{\pm}_{\dot{\sigma}}(z)=\exp(-\frac{1}{2}\phi(z)+\imath\dot{{\sigma}}^{a}H_{a}(z)\pm\frac{\imath}{2}\sum_{i}\frac{k_{i}\phi_{i}(z)}{\sqrt{k_{i}(k_{i}+2)}}),\dot{\sigma}^{a}=\pm\frac{1}{2},\
\sum_{a=1}^{2}\dot{\sigma}^{a}=0.$

These currents are nothing else but the product of the Ramond vacua of the
three sub-theories of the string model under consideration. In particular, the
exponential factors of the internal model $M_{\bf{k}}$ are products of the
spectral flow operators for each minimal model $M_{k_{i}}$. The results in the
right sector will be similar.

It is easy to verify that each of the four currents $S^{+}_{\sigma}$,
$S^{+}_{\sigma}$, $S^{+}_{\dot{\sigma}}$ and $S^{-}_{\dot{\sigma}}$, belongs
to the two-dimensional representation of the $SO(1,3)$ Lie algebra of space-
time symmetry, namely, each of them is a Weyl spinor. Two currents
$S^{+}_{\sigma}$ and $S^{-}_{\sigma}$ have the same chirality, say left-
handed, and the other two, $S^{+}_{\dot{\sigma}}$ and $S^{-}_{\dot{\sigma}}$,
have opposite chirality, respectively right-handed. For our construction, the
mutual locality of these currents is important. A simple check shows that the
current $S^{+}_{\sigma}$ is mutually local with $S^{-}_{\dot{\sigma}}$ and
$S^{-}_{\sigma}$ is mutually local with $S^{+}_{\dot{\sigma}}$.

The four supercharges, defined by the integration of these currents, define,
together with generators of Poincaré algebra operators, two $N=1$ super-
Poincaré algebras of opposite chiralities. The currents $S^{+}_{\sigma}$ and
$S^{-}_{\dot{\sigma}}$ define super generators of a $N=1$ super Poincaré for
the case left chirality as

$\displaystyle{\cal{Q}}^{(-\frac{1}{2})}_{\sigma}=\oint S^{+}_{\sigma}dz=\oint
dz\exp{[-\frac{1}{2}\phi+\imath{\sigma}^{a}H_{a}+\frac{\imath}{2}\sum_{i}\frac{k_{i}\phi_{i}}{\sqrt{k_{i}(k_{i}+2)}}]},$
(28) $\displaystyle{\cal{Q}}^{(-\frac{1}{2})}_{\dot{\sigma}}=\oint
S^{-}_{\dot{\sigma}}dz=\oint
dz\exp{[-\frac{1}{2}\phi+\imath\dot{\sigma}^{a}H_{a}-\frac{\imath}{2}\sum_{i}\frac{k_{i}\phi_{i}}{\sqrt{k_{i}(k_{i}+2)}}]}.$

Instead of the mutually local pair $S^{+}_{\sigma}$ and $S^{-}_{\dot{\sigma}}$
we can take another pair $S^{-}_{\sigma}$ and $S^{+}_{\dot{\sigma}}$. As a
result of either of these two different options, we obtain $d=4$ $N=1$
spacetime supersymmetry. For each of these two options, we have to leave from
the BRST-invariant fields only those fields that are mutually local with
respect to the chosen supercharges.

We can make such choice in both left-moving and right-moving sectors. Thus, we
obtain string models $IIB$ or $IIA$ according to whether the choice of super-
charges in the left-moving and right-moving sectors is the same or not.

##  4 Space-time supersymmetry and GSO equations

In general, the vertex operators are not mutually local with the currents of
supersymmetry operators.

In order to get SUSY algebra representation we select only the vertices which
are mutually local with SUSY currents. For the $IIB$ case this requirement
gives the GSO equations

$\displaystyle q+\sum_{a}\lambda_{a}+Q_{int}\in 2\mathbb{Z},$ (29)
$\displaystyle\bar{q}+\sum_{a}\bar{\lambda}_{a}+\bar{Q}_{int}\in 2\mathbb{Z},$

where $Q_{int}$, $\bar{Q}_{int}$ are the $U(1)$ charges of the states from the
internal sector.

For the type $IIA$ the second equation changes

$\displaystyle q+\sum_{a}\lambda_{a}+Q_{int}\in 2\mathbb{Z},$ (30)
$\displaystyle\bar{q}+\sum_{a}\bar{\lambda}_{a}-\bar{Q}_{int}\in 2\mathbb{Z},$

##  5 Construction of physical vertices.

We now move on to the construction of complete physical vertex operators,
which are products of left- and right-moving factors. Let’s start doing this
for theory $IIB$. First, we construct the corresponding set of the vertex
operators type $(NS,NS)$, which are consistent with the action of space-time
supersymmetry and satisfy the requirements of BRST invariance and mutual
locality.

To analyze the problem of mutual locality and supersymmetry, we consider only
the part of the $(NS,NS)$ vertex operators (25) which is essential for this
issue, i.e. the operators which do not include the descendants because in
$(NS,NS)$ sector the descendants can not spoil mutual locality:

${\cal{V}}^{(q,\lambda)(\bar{q},\bar{\lambda})}_{\bf{l},\bar{\bf{t}},\bf{w}}(z,\bar{z})=\exp{[q\phi+\imath\lambda^{a}H_{a}]}(z)\exp{[\bar{q}\bar{\phi}+\imath\bar{\lambda}^{a}\bar{H}_{a}]}(\bar{z})\Psi_{\bf{l},{\bar{\bf{t}}},{\bf{w}}}(z,\bar{z}).$
(31)

Here the factor $\Psi_{\bf{l},{\bar{\bf{t}}},{\bf{w}}}(z,\bar{z})$ from the
compact sector is given by application of the left-moving spectral flow
operator $U^{\bf{w}}$ to the state
$\Phi_{\bf{l},\bar{\bf{t}}}(z)\bar{\Phi}_{\bf{l},\bar{\bf{t}}}(\bar{z})$ in
the product model $M_{\bf{k}}$. This construction and the corresponding
explicit expressions are given in (48)-(51) of the Appendix.

The vectors $\bf{w}$ label the orbifolds twisted sectors, which for the
orbifolds we consider are $r$-dimensional vectors
${\bf{w}}=(w_{1},...,w_{r})$, where $w_{i}$ are integers and $\mod(\ k_{i}+2)$
defined. Thus, when ${\bf{w}=}0$ the field
$\Psi_{\bf{l},{\bar{\bf{t}}},{\bf{w}}}(z,\bar{z})$ is the diagonal product of
the left-moving and the right-moving primary states
$\Phi_{\bf{l},\bar{\bf{t}}}(z)\bar{\Phi}_{\bf{l},\bar{\bf{t}}}(\bar{z})$ from
the product model $M_{\bf{k}}$. When ${\bf{w}}\neq 0$ the field
$\Psi_{\bf{l},{\bar{\bf{t}}},{\bf{w}}}(z,\bar{z})$ is no longer the diagonal
product (but this field is still diagonal along the indices $\bf{l}$), but
spectral flow twisted product.

The reason why the twisted vertices (31) are necessary in the superstring
construction is the space-time supersymmetry requirement. Indeed, if one
applies type $IIB$ SUSY operators twice to the vertex (31) with ${\bf{w}}=0$
we obtain $(NS,NS)$ vertices (31), where ${\bf{w}}=(1,1,...,1)$. Therefore,
the minimum set of $(NS,NS)$ vertices which is consistent with the SUSY action
must be of the form (31), where ${\bf{w}}=(n,n,...,n)=n{\beta}^{0}$.

To include more general orbifolds in the construction, it is necessary to
consider twist vectors $\bf{w}$ of a more general form, but such that the
vertices (31) satisfy the GSO equations (29).

Using the expressions (48)-(51) we find that GSO equations for the vertices
(31) takes the form

$\displaystyle
q+\sum_{a}\lambda_{a}+\sum_{i=1}^{r}\frac{\bar{q}_{i}-2w_{i}}{k_{i}+2}+Q_{des}\in
2\mathbb{Z},$ (32)
$\displaystyle\bar{q}+\sum_{a}\bar{\lambda}_{a}+\sum_{i=1}^{r}\frac{\bar{q}_{i}}{k_{i}+2}+\bar{Q}_{des}\in
2\mathbb{Z},$

where $\bar{q}_{i}=l_{i}-2\bar{t}_{i}$ and $Q_{des},\bar{Q}_{des}$ are the
charges contributions of the descendants, which were omitted in (31).

We see that the requirement

$\sum_{i=1}^{r}\frac{w_{i}}{k_{i}+2}\in\mathbb{Z}$ (33)

is consistent with the GSO equations. We will see soon that this condition
follows from the mutual locality requirement. We denote the complete set of
solutions (33), including the vector $\beta^{0}=(1,1,...,1)$, as the maximal
admissible group $G^{max}_{adm}$. This set forms an additive group due to OPE
requirements. Let us fix some admissible subgroup
$G_{adm}=\\{{\bf{w}}=\sum_{a=0}^{K-1}w_{a}\beta^{a},\ \beta^{0}=(1,1,...,1),\
w_{a}\in\mathbb{Z}\\}\subset G^{max}_{adm}$, where the vectors $\beta^{a}$ are
generators of $G_{adm}$. We will use this group to construct $IIB$ superstring
compactification on the orbifold $M_{\bf{k}}/G_{adm}$.

We want to consider the question of mutual locality of $(NS,NS)$ vertex
operators (31). Then for the pair of vertex operators
${\cal{V}}^{(q_{1},\lambda_{1})(\bar{q}_{1},\bar{\lambda}_{1})}_{\bf{l}_{1},\bar{\bf{t}}_{1},\bf{w}_{1}}(z,\bar{z})$
and
${\cal{V}}^{(q_{2},\lambda_{2})(\bar{q}_{2},\bar{\lambda}_{2})}_{\bf{l}_{2},{\bar{\bf{t}}}_{2},{\bf{w}}_{2}}(0)$
we have to calculate the monodromy when the first vertex goes around the
second one. Since they are in the $(NS,NS)$ sector, the nontrivial monodromy
phase for this pair arises only from compact factors. As a result, the mutual
locality equation, as shown in the Appendix, has the form

$\sum_{i=1}^{r}\frac{w_{1i}(\bar{q}_{2i}-w_{2i})+w_{2i}(\bar{q}_{1i}-w_{1i})}{k_{i}+2}\in\mathbb{Z}.$
(34)

These equations select for each pair of vectors ${\bf{w}}_{1,2}$ the set of
$\bar{\bf{q}}_{1,2}$ which give mutually local vertices.

As one can show, from these equations it follows that the charges
$\bar{q}_{i}$ of all vertices
$\Psi^{(q,\lambda)(\bar{q},\bar{\lambda})}_{\bf{l},\bar{\bf{t}},{\bf{w}}}$
deformed by some twist ${\bf{w}}\in G_{adm}$ must satisfy the equation

$\sum_{i=1}^{r}\frac{\beta^{a}_{i}(\bar{q}_{i}-w_{i})}{k_{i}+2}\in\mathbb{Z}$
(35)

for all generators $\beta^{a}\in G_{adm}$. These equations mean that the set
of $U(1)$ charges of the mutually local vertex operators are such that they
are invariant w.r.t. the defined above operators (17)

$\hat{g}^{\bf{w}}=\exp{[\imath\pi\sum_{i}w_{i}(J_{i,0}+\bar{J}_{i,0})]},$ (36)

that is, such vertex operators (31) are $G_{adm}$-invariant.

Moreover, from the last equations it follows that the vectors
$\bf{w}^{*}\equiv\bar{\bf{q}}-\bf{w}$ form another admissible group
$G^{*}_{adm}\subset G^{max}_{adm}$ P . This group is Berglund-Krawitz-Hubsh
dual group for the Fermat type potential BH . It defines the mirror orbifold
model $M_{\bf{k}}/G^{*}_{adm}$ P .

This fact allows to represent the left $\bf{q}=\bar{\bf{q}}-2\bf{w}$ and the
right $\bar{\bf{q}}$ vectors labeling the mutually local vertices of the
twisted sector $\bf{w}$ of the model $M_{\bf{k}}/G_{adm}$ in the mirror
symmetric form

$\bf{q}={-\bf{w}+\bf{w}^{*}},\quad{\bar{\bf{q}}}={\bf{w}+\bf{w}^{*}},$ (37)

where ${\bf{w}}\in G_{adm},{\bf{w}}^{*}\in G^{*}_{adm}$.

Note that the requirement (33) we imposed earlier for the group $G_{adm}$ now
follows from (35) but for the group $G^{*}_{adm}$, since $G_{adm}$ contains
$\beta^{0}$. Hence, (33) also follows from (35) for the group $G_{adm}$
because of the mirror symmetry.

In order to get the total $BRST$ invariant vertex from (31) one needs to
recover the descendant contributions in an appropriate way.

##  6 Supersymmetry and mutual locality in $(R,NS)$, $(NS,R)$ and $(R,R)$
sectors.

In this section we generate physical $(R,NS)$, $(NS,R)$, $(R,R)$ vertices from
the $(NS,NS)$ physical vertices constructed in the previous section by
applying the type $IIB$ supercharges. We then show the mutual locality of
these vertices both with themselves and with the $(NS,NS)$ vertices we started
with.

Let’s consider the physical vertex operators $(NS,NS)$ constructed in the
previous section and apply the left supercharges to them. We obtain physical
$(R,NS)$ vertex, because the super-charges commute with $Q_{BRST}$. Modulo the
descendants factor the result of the supersymmetry action can be written in
the form

$\tilde{{\cal{V}}}^{(q,\lambda)(\bar{q},\bar{\lambda})}_{\bf{l},\bar{\bf{t}},\bf{w}}(z,\bar{z})=\exp{[(q-\frac{1}{2})\phi+\imath(\lambda^{a}+\sigma^{a})H_{a}]}(z)\exp{[\bar{q}\bar{\phi}+\imath\bar{\lambda}^{a}\bar{H}_{a}]}(\bar{z})U^{\frac{1}{2}\bf{\beta}_{0}}\Psi_{\bf{l},\bar{\bf{t}},\bf{w}}(z,\bar{z}),$
(38)

Similarly, we obtain $(NS,R)$ physical vertex by applying type $IIB$ right-
moving super-charges to some other physical $(NS,NS)$ vertex constructed in
the previous section. Modulo the descendants factor the result of the
supersymmetry action is given by

$\displaystyle\tilde{\tilde{{\cal{V}}}}^{(q,\lambda)(\bar{q},\bar{\lambda})}_{\bf{l},\bar{\bf{t}},\bf{w}}(z,\bar{z})=\exp{[q\phi+\imath\lambda^{a}H_{a}]}(z)\exp{[(\bar{q}-\frac{1}{2})\bar{\phi}+\imath(\bar{\lambda}^{a}+\bar{\sigma}^{a})\bar{H}_{a}]}(\bar{z})\bar{U}^{\frac{1}{2}\bf{\beta}_{0}}\Psi_{{\bf{l}},\bar{\bf{t}},\bf{w}}(z,\bar{z}).$
(39)

Finally, we obtain $(R,R)$ sector vertex by applying left-moving and right-
moving type $IIB$ super-charges to the physical $(NS,NS)$ vertex from the
previous section. Again, modulo the descendants factor the result is given by

$\displaystyle\begin{aligned}
&\tilde{\tilde{\tilde{{\cal{V}}}}}^{(q,\lambda)(\bar{q},\bar{\lambda})}_{\bf{l},\bar{\bf{t}},\bf{w}}(z,\bar{z})=\\\
&\exp{[(q-\frac{1}{2})\phi+\imath(\lambda^{a}+\sigma^{a})H_{a}]}(z)\exp{[(\bar{q}-\frac{1}{2})\bar{\phi}+\imath(\bar{\lambda}^{a}+\bar{\sigma}^{a})\bar{H}_{a}]}(\bar{z})U^{\frac{1}{2}\beta_{0}}\bar{U}^{\frac{1}{2}\bf{\beta}_{0}}\Psi_{{\bf{l}},\bar{\bf{t}},\bf{w}}(z,\bar{z}).\end{aligned}$
(40)

In order to prove mutual locality of the superpartners, arising in this way,
we need to calculate the monodromy phases between each of these vertices and
an arbitrary $(NS,NS)$ vertex constructed in the previous section as well as
between these vertices themselves.

Carrying out the calculation of phases one can ignore at first step the
descendants contributions and calculate the phases between the vertices in the
form (39)-(40). Then we recover descendants contributions. This is given in
the Appendix. The result shows that the phases are trivial because the
original $(NS,NS)$ vertices where mutually local and obeyed $GSO$ equations
(32).

Thus, we have constructed the physical state space of type $IIB$ superstring
compactified on the orbifold $M_{\bf{k}}/G_{adm}$. Similarly, one can
construct the physical state space of a type $IIA$ superstring.

##  7 Modular invariance

The equations (37) can be rewritten in the form

$\displaystyle\bf{q}+\bar{\bf{q}}\in G^{*}_{adm},\ \bar{\bf{q}}-\bf{q}\in
G_{adm}.$ (41)

We come thereby to the constraints for the modular invariant partition
function of the Gepner model orbifold obtained in GrPl . Therefore, the
partition function of the superstring calculated in our approach will also be
modular invariant.

##  8 Conclusion

We are grateful to D. Gepner, A. Litvinov and A. Marshakov for useful
discussions. The work is supported by RSF grant 23-12-00333.

##  9 Appendix

A.1. For the minimal model $M_{k}$ spectral flow can be used to construct all
primary fields starting from chiral or anti-chiral primary fields LVW  BBP , P
. Here we briefly discuss the construction, following mainly to the papers BBP
, P , where more details can be found.

First of all recall the definition of primary state in $NS$ sector

$\displaystyle L_{n}\Phi^{NS}_{\Delta,Q}=0,\ J_{n}\Phi^{NS}_{\Delta,Q}=0,\
n\geq 1,\ G^{\pm}_{r}\Phi^{NS}_{\Delta,Q}=0,\ r\geq{1\over 2},$ (42)
$\displaystyle L_{0}\Phi^{NS}_{\Delta,Q}=\Delta\Phi^{NS}_{\Delta,Q},\quad
J_{0}\Phi^{NS}_{\Delta,Q}=Q\Phi^{NS}_{\Delta,Q},$

Chiral primary state is subject to an additional constraint

$\displaystyle G^{+}_{-{1\over 2}}\Phi^{NS}_{\Delta,Q}=0.$ (43)

It leads to the relation between conformal dimension and $U(1)$ charge of the
chiral primary state: $Q=2\Delta$. The anti-chiral primary is defined
similarly

$\displaystyle G^{-}_{-{1\over 2}}\Phi^{NS}_{\Delta,Q}=0$ (44)

and for this field $Q=-2\Delta$.

For the Minimal model $M_{k}$, the chiral and anti-chiral primary states are
the primaries with $q=l$ and $q=-l$ correspodingly:

$\displaystyle\Phi^{c}_{l}(z)\equiv\Phi^{NS}_{l,l}(z),\
\Phi^{a}_{l}(z)\equiv\Phi^{NS}_{l,-l}(z).$ (45)

All other primary states in the Minimal model can be constructed starting for
example from chiral primaries. Namely, for the spectral flow parameter $0\leq
t\leq l$, the state

$\displaystyle\Phi_{l,t}=(UG^{-}_{-{1\over 2}})^{t}\Phi^{c}_{l}$ (46)

gives spectral flow realization of the primary state $\Phi^{NS}_{l,q}$, where
$q=l-2t$. For the spectral flow parameter $l+1\leq t\leq k+1$, the state

$\displaystyle\Phi_{l,t}=(UG^{-}_{-{1\over 2}})^{t-l-1}U(UG^{-}_{-{1\over
2}})^{l}\Phi^{c}_{l}$ (47)

gives the spectral flow realization of the primary state
$\Phi^{NS}_{\tilde{l},\tilde{q}}$, where $\tilde{l}=k-l$ and
$\tilde{q}=k+2+l-2t$.

Notice that in the expressions (46),(47) we use spectral flow parameter $t$ to
label primary states.

It is clear how to extend the spectral flow construction above for the primary
states in the product model $M_{\bf{k}}$. Moreover, the spectral flow
construction is very helpfull for the construction of the $(NS,NS)$ primary
fields of the orbifold $M_{\bf{k}}/G_{adm}$ in the twisted sectors. Indeed,
for the twisted sector ${\bf{w}}\in G_{adm}$ the twisted primary field is
given by

$\displaystyle\Psi_{\bf{l},\bar{\bf{t}},\bf{w}}(z,\bar{z})=\Phi_{\bf{l},\bar{\bf{t}}+\bf{w}}(z)\bar{\Phi}_{\bf{l},\bar{\bf{t}}}(\bar{z}),$
(48)

where

$\displaystyle\bar{\Phi}_{\bf{l},\bar{\bf{t}}}(\bar{z})=\prod_{i}\bar{\Phi}_{l_{i},\bar{t}_{i}}(\bar{z}),\
\bar{\Phi}_{l_{i},\bar{t}_{i}}(\bar{z})=(\bar{U}\bar{G}^{-}_{-{1\over
2}})_{i}^{\bar{t}_{i}}\bar{\Phi}^{c}_{l_{i}}(\bar{z}),\ 0\leq\bar{t}_{i}\leq
l_{i},$ (49)

and

$\displaystyle\Phi_{\bf{l},\bar{\bf{t}}+\bf{w}}(z)=\prod_{i=1}^{5}\Phi_{l_{i},\bar{t}_{i}+w_{i}}(z),$
(50)

where

$\displaystyle\begin{aligned}
\Phi_{l_{i},\bar{t}_{i}+w_{i}}(z)=\begin{cases}(UG^{-}_{-{1\over
2}})_{i}^{\bar{t}_{i}+w_{i}}\Phi^{c}_{l_{i}}(z),\qquad\text{if}\quad\
0\leq\bar{t}_{i}+w_{i}\leq l_{i},\\\ \\\ (UG^{-}_{-{1\over
2}})_{i}^{\bar{t}_{i}+w_{i}-l_{i}-1}U_{i}(UG^{-}_{-{1\over
2}})_{i}^{l_{i}}\Phi^{c}_{l_{i}}(z),\quad\text{if}\quad\
l_{i}+1\leq\bar{t}_{i}+w_{i}\leq k_{i}+1.\end{cases}\end{aligned}$ (51)

Applying spectral flow operators
$U^{\frac{1}{2}\beta^{0}}=\prod_{i=1}^{r}U_{i}^{\frac{1}{2}}$ and
$\bar{U}^{\frac{1}{2}\beta^{0}}=\prod_{i=1}^{r}\bar{U}_{i}^{\frac{1}{2}}$ to
the fields (48) independently we can construct primary fields of the orbifold
in $(R,NS)$, $(NS,R)$ and $(R,R)$ sectors.

A.2. Here we derive mutual locality condition (34).

To do this, we calculate the monodromy for an arbitrary pair vertices inside
sector $(NS,NS)$

$\displaystyle{\cal{V}}^{(q_{1},\lambda_{1})(\bar{q}_{1},\bar{\lambda}_{1})}_{\bf{l}_{1},\bar{\bf{t}}_{1},\bf{w}_{1}}(z,\bar{z})=\exp{[q_{1}\phi+\imath\lambda^{a}_{1}H_{a}]}(z)\exp{[\bar{q}_{1}\bar{\phi}+\imath\bar{\lambda}^{a}_{1}\bar{H}_{a}]}(\bar{z})\Psi_{\bf{l}_{1},\bar{\bf{t}}_{1},\bf{w}_{1}}(z,\bar{z}),$
$\displaystyle{\cal{V}}^{(q_{2},\lambda_{2})(\bar{q}_{2},\bar{\lambda}_{2})}_{\bf{l}_{2},{\bar{\bf{t}}}_{2},{\bf{w}}_{2}}(z,\bar{z})=\exp{[q_{2}\phi+\imath\lambda^{a}_{2}H_{a}]}(z)\exp{[\bar{q}_{2}\bar{\phi}+\imath\bar{\lambda}^{a}_{2}\bar{H}_{a}]}(\bar{z})\Psi_{\bf{l}_{2},\bar{\bf{t}}_{2},\bf{w}_{2}}(z,\bar{z}).$
(52)

It can be done by calculation of their OPE.

The fusion
${\cal{V}}^{(q_{1},\lambda_{1})(\bar{q}_{1},\bar{\lambda}_{1})}_{\bf{l}_{1},\bar{\bf{t}}_{1},\bf{w}_{1}}(z,\bar{z}){\cal{V}}^{(q_{2},\lambda_{2})(\bar{q}_{2},\bar{\lambda}_{2})}_{\bf{l}_{2},{\bar{\bf{t}}}_{2},{\bf{w}}_{2}}(0)$
gives the linear combination of vertices

${\cal{V}}^{(q_{3},\lambda_{3})(\bar{q}_{3},\bar{\lambda}_{3})}_{\bf{l}_{3},\bar{\bf{t}}_{3},\bf{w}_{3}}(0)=\exp{[q_{3}\phi+\imath\lambda^{a}_{3}H_{a}]}(0)\exp{[\bar{q}_{3}\bar{\phi}+\imath\bar{\lambda}^{a}_{3}\bar{H}_{a}]}(0)\Psi_{{\bf{l}}_{3},\bar{\bf{t}}_{3},\bf{w}_{3}}(0),$
(53)

where

$\displaystyle\begin{aligned} &q_{3}=q_{1}+q_{2},\
\bar{q}_{3}=\bar{q}_{1}+\bar{q}_{2},\\\ &\lambda_{3}=\lambda_{1}+\lambda_{2},\
\bar{\lambda}_{3}=\bar{\lambda_{1}}+\bar{\lambda}_{2},\
\bar{\bf{t}}_{3}=\bar{\bf{t}}_{1}+\bar{\bf{t}}_{2},\\\
&\bf{w}_{3}=\bf{w}_{1}+\bf{w}_{2}.\end{aligned}$ (54)

These vertices (together with the corresponding structure constants) are
accompanied by the multipliers

$z^{-q_{1}q_{2}+\lambda_{1}\cdot\lambda_{2}+\Delta_{3}-{\Delta_{1}}-{\Delta_{2}}}\bar{z}^{-\bar{q}_{1}\bar{q}_{2}+\bar{\lambda}_{1}\cdot\bar{\lambda}_{2}+\bar{\Delta}_{3}-\bar{\Delta}_{1}-\bar{\Delta}_{2}}.$
(55)

Here, $\Delta_{1,2,3},\bar{\Delta}_{1,2,3}$ are the conformal dimensions of
vertices in the compact sector.

Because we are in $(NS,NS)$ sector nontrivial monodromy phase arises only from
compact factors and is given by

$\exp{[\imath
2\pi(\Delta_{3}-\Delta_{1}-\Delta_{2}-\bar{\Delta}_{3}+\bar{\Delta}_{1}+\bar{\Delta}_{2})]}.$
(56)

Using (10) we find

$\Delta_{3}-\Delta_{1}-\Delta_{2}-\bar{\Delta}_{3}+\bar{\Delta}_{1}+\bar{\Delta}_{2}=\sum_{i=1}^{r}\frac{1}{4(k_{i}+2)}(q_{1i}^{2}+q_{2i}^{2}-q_{3i}^{2}-\bar{q}_{1i}^{2}-\bar{q}_{2i}^{2}+\bar{q}_{3i}^{2}).$
(57)

Taking into account (51) we find that modulo integer number the expression
above can be rewritten as

$\displaystyle\begin{aligned}
&\Delta_{3}-\Delta_{1}-\Delta_{2}-\bar{\Delta}_{3}+\bar{\Delta}_{1}+\bar{\Delta}_{2}=\\\
&\sum_{i=1}^{r}\frac{1}{4(k_{i}+2)}((\bar{q}_{1i}-2w_{1i})^{2}+(\bar{q}_{2i}-2w_{2i})^{2}-(\bar{q}_{3i}-2w_{3i})^{2}-\bar{q}_{1i}^{2}-\bar{q}_{2i}^{2}+\bar{q}_{3i}^{2})=\\\
&\sum_{i=1}^{r}\frac{1}{k_{i}+2}(w_{1i}^{2}-\bar{q}_{1i}w_{1i}+w_{2i}^{2}-\bar{q}_{2i}w_{2i}-w_{3i}^{2}+\bar{q}_{3i}w_{3i}).\\\
\end{aligned}$ (58)

Taking into account (54) we find the mutual locality condition

$\displaystyle\sum_{i=1}^{r}\frac{w_{1i}(\bar{q}_{2i}-w_{2i})+w_{2i}(\bar{q}_{1i}-w_{1i})}{k_{i}+2}\in\mathbb{Z}.$
(59)

A.3.

Here we prove mutual locality of vertices which are $IIB$ SUSY superpartners
of mutually local $(NS,NS)$ vertices.

Let us consider the fusion of two vertices
$\tilde{{\cal{V}}}^{(q_{1},\lambda_{1})(\bar{q}_{1},\bar{\lambda}_{1})}_{\bf{l}_{1},\bar{\bf{t}}_{1},\bf{w}_{1}}(z,\bar{z}){\cal{V}}^{(q_{2},\lambda_{2})(\bar{q}_{2},\bar{\lambda}_{2})}_{\bf{l}_{2},\bar{\bf{t}}_{2},\bf{w}_{2}}(0)$,
which were defined in (38) and (31). This fusion gives the linear combination
of vertices

$\tilde{{\cal{V}}}^{(q_{3},\lambda_{3})(\bar{q}_{3},\bar{\lambda}_{3})}_{\bf{l}_{3},\bar{\bf{t}}_{3},\bf{w}_{3}}(0)=\exp{[(q_{3}-\frac{1}{2})\phi+\imath(\lambda^{a}_{3}+\sigma^{a})H_{a}]}(0)\exp{[\bar{q}_{3}\bar{\phi}+\imath\bar{\lambda}^{a}_{3}\bar{H}_{a}]}(0)U^{\frac{1}{2}\beta^{0}}\Psi_{{\bf{l}}_{3},\bar{\bf{t}}_{3},\bf{w}_{3}}(0),$
(60)

where
$U^{\frac{1}{2}\beta^{0}}\Psi^{{\bf{l}}_{3}}_{\bar{\bf{t}}_{3},\bf{w}_{3}}(0)$
is $(R,NS)$ primary field and again

$\displaystyle q_{3}=q_{1}+q_{2},\ \bar{q}_{3}=\bar{q}_{1}+\bar{q}_{2},$ (61)
$\displaystyle\lambda_{3}=\lambda_{1}+\lambda_{2},\
\bar{\lambda}_{3}=\bar{\lambda_{1}}+\bar{\lambda}_{2},\
\bar{\bf{t}}_{3}=\bar{\bf{t}}_{1}+\bar{\bf{t}}_{2},$
$\displaystyle\bf{w}_{3}=\bf{w}_{1}+\bf{w}_{2}.$

These vertices are accompanied by multipliers

$z^{-(q_{1}-\frac{1}{2})q_{2}+(\lambda_{1}+s)\cdot\lambda_{2}+\Delta_{3}-{\Delta_{1}}-{\Delta_{2}}}\bar{z}^{-\bar{q}_{1}\bar{q}_{2}+\bar{\lambda}_{1}\cdot\bar{\lambda}_{2}+\bar{\Delta}_{3}-\bar{\Delta}_{1}-\bar{\Delta}_{2}}.$
(62)

The monodromy is given by

$\exp{[\imath
2\pi(\frac{1}{2}q_{2}+\sigma\cdot\lambda_{2}+\Delta_{3}-\Delta_{1}-\Delta_{2}-\bar{\Delta}_{3}+\bar{\Delta}_{1}+\bar{\Delta}_{2})]},$
(63)

where $\Delta_{1},\Delta_{3}$ are the dimensions of $R$ sector primary fields
and are given by (11), the other dimensions are given by (10). Using this we
obtain

$\displaystyle\Delta_{3}-\Delta_{1}-\Delta_{2}-\bar{\Delta}_{3}+\bar{\Delta}_{1}+\bar{\Delta}_{2}=$
$\displaystyle\sum_{i=1}^{r}\frac{1}{4(k_{i}+2)}((q_{1i}-1)^{2}+q_{2i}^{2}-(q_{3i}-1)^{2}-\bar{q}_{1i}^{2}-\bar{q}_{2i}^{2}+\bar{q}_{3i}^{2}).$

Taking into account (51) we find that modulo integer number this expression
takes the form

$\displaystyle\begin{aligned}
&\Delta_{3}-\Delta_{1}-\Delta_{2}-\bar{\Delta}_{3}+\bar{\Delta}_{1}+\bar{\Delta}_{2}=\\\
&\sum_{i=1}^{r}\frac{1}{4(k_{i}+2)}((\bar{q}_{1i}-2w_{1i})^{2}-2(\bar{q}_{1i}-w_{1i})+(\bar{q}_{2i}-2w_{2i})^{2}-(\bar{q}_{3i}-2w_{3i})^{2}+2(\bar{q}_{3i}-2w_{3i})-\\\
&\bar{q}_{1i}^{2}-\bar{q}_{2i}^{2}+\bar{q}_{3i}^{2}).\\\ \end{aligned}$

Using (61) we obtain

$\displaystyle\begin{aligned}
&\Delta_{3}-\Delta_{1}-\Delta_{2}-\bar{\Delta}_{3}+\bar{\Delta}_{1}+\bar{\Delta}_{2}=\\\
&\sum_{i=1}^{r}(\frac{w_{1i}(\bar{q}_{2i}-w_{2i})+w_{2i}(\bar{q}_{1i}-w_{1i})}{k_{i}+2}+\frac{1}{2}\frac{\bar{q}_{2i}-w_{2i}}{k_{i}+2}).\\\
\end{aligned}$

Hence, the mutual locality condition is

$\sum_{i=1}^{r}(\frac{w_{1i}(\bar{q}_{2i}-w_{2i})+w_{2i}(\bar{q}_{1i}-w_{1i})}{k_{i}+2}+\frac{1}{2}(q_{2}+\sum_{a}\lambda^{a}_{2}+\sum_{i=1}^{r}\frac{\bar{q}_{2i}-2w_{2i}}{k_{i}+2})\in\mathbb{Z}.$
(64)

The descendants which have been omitted in the vertex
${\cal{V}}^{(q_{2},\lambda_{2})(\bar{q}_{2},\bar{\lambda}_{2})}_{\bf{l}_{2},\bar{\bf{t}}_{2},\bf{w}_{2}}(0)$
correct the expression above to

$\sum_{i=1}^{r}(\frac{w_{1i}(\bar{q}_{2i}-w_{2i})+w_{2i}(\bar{q}_{1i}-w_{1i})}{k_{i}+2}+\frac{1}{2}(q_{2}+\sum_{a}\lambda^{a}_{2}+\sum_{i=1}^{r}\frac{\bar{q}_{2i}-2w_{2i}}{k_{i}+2}+Q_{2des})\in\mathbb{Z}.$
(65)

(It is easy to see that the descendants omitted in vertex
$\tilde{{\cal{V}}}^{(q_{1},\lambda_{1})(\bar{q}_{1},\bar{\lambda}_{1})}_{\bf{l}_{1},\bar{\bf{t}}_{1},\bf{w}_{1}}(z,\bar{z})$
do not contribute to the left hand side of this expression.)

Because the initial $(NS,NS)$ vertices where mutually local the first sum in
the expression above is integer so we are left with the expression

$q_{2}+\sum_{a}\lambda^{a}_{2}+\sum_{i=1}^{r}\frac{q_{2i}}{k_{i}+2}+Q_{2des}\in
2\mathbb{Z},$ (66)

which is nothing else but the first line from (29).

Similarly, the fusion of
$\tilde{\tilde{{\cal{V}}}}^{(q_{1},\lambda_{1})(\bar{q}_{1},\bar{\lambda}_{1})}_{\bf{l}_{1},\bar{\bf{t}}_{1},\bf{w}_{1}}(z,\bar{z}){\cal{V}}^{(q_{2},\lambda_{2})(\bar{q}_{2},\bar{\lambda}_{2})}_{\bf{l}_{2},\bar{\bf{t}}_{2},\bf{w}_{2}}(0)$,
where the first vertex is given by (39) gives the linear combination of
vertices

$\tilde{\tilde{{\cal{V}}}}^{(q_{3},\lambda_{3})(\bar{q}_{3},\bar{\lambda}_{3})}_{\bf{l}_{3},\bar{\bf{t}}_{3},\bf{w}_{3}}(0)=\exp{[q_{3}\phi+\imath\lambda^{a}_{3}H_{a}]}(0)\exp{[(\bar{q}_{3}-\frac{1}{2})\bar{\phi}+\imath(\bar{\lambda}^{a}_{3}+\sigma^{a})\bar{H}_{a}]}(0)\bar{U}^{\frac{1}{2}\beta^{0}}\Psi_{{\bf{l}}_{3},\bar{\bf{t}}_{3},\bf{w}_{3}}(0),$
(67)

where
$\bar{U}^{\frac{1}{2}\beta^{0}}\Psi_{{\bf{l}}_{3},\bar{\bf{t}}_{3},\bf{w}_{3}}(0)$
is $(NS,R)$ sector primary field from the compact part.

These vertices (together with the corresponding structure constants) are
accompanied by multipliers

$z^{-q_{1}q_{2}+\lambda_{1}\cdot\lambda_{2}+\Delta_{3}-{\Delta_{1}}-{\Delta_{2}}}\bar{z}^{-(\bar{q}_{1}-\frac{1}{2})\bar{q}_{2}+(\bar{\lambda}_{1}+\sigma)\cdot\bar{\lambda}_{2}+\bar{\Delta}_{3}-\bar{\Delta}_{1}-\bar{\Delta}_{2}}.$
(68)

The monodromy is given by

$\exp{[\imath
2\pi(-\frac{1}{2}\bar{q}_{2}-\sigma\cdot\bar{\lambda}_{2}+\Delta_{3}-\Delta_{1}-\Delta_{2}-\bar{\Delta}_{3}+\bar{\Delta}_{1}+\bar{\Delta}_{2})]},$
(69)

where $\bar{\Delta}_{1},\bar{\Delta}_{3}$ are the dimensions of $R$ sector
primary fields and are given by (11), the other dimensions are given by (10).
Using this we obtain

$\displaystyle\Delta_{3}-\Delta_{1}-\Delta_{2}-\bar{\Delta}_{3}+\bar{\Delta}_{1}+\bar{\Delta}_{2}=$
$\displaystyle\sum_{i=1}^{r}\frac{1}{4(k_{i}+2)}(q_{1i}^{2}+q_{2i}^{2}-q_{3i}^{2}-(\bar{q}_{1i}-1)^{2}-\bar{q}_{2i}^{2}+(\bar{q}_{3i}-1)^{2}).$

Taking into account (51) we find that modulo integer number this expression
takes the form

$\displaystyle\Delta_{3}-\Delta_{1}-\Delta_{2}-\bar{\Delta}_{3}+\bar{\Delta}_{1}+\bar{\Delta}_{2}=$
$\displaystyle\sum_{i=1}^{r}\frac{1}{4(k_{i}+2)}((\bar{q}_{1i}-2w_{1i})^{2}+(\bar{q}_{2i}-2w_{2i})^{2}-(\bar{q}_{3i}-2w_{3i})^{2}-$
$\displaystyle(\bar{q}_{1i}-1)^{2}-\bar{q}_{2i}^{2}+(\bar{q}_{3i}-1)^{2}).$

Using (61) we obtain

$\displaystyle\Delta_{3}-\Delta_{1}-\Delta_{2}-\bar{\Delta}_{3}+\bar{\Delta}_{1}+\bar{\Delta}_{2}=$
$\displaystyle\sum_{i=1}^{r}(\frac{w_{1i}(\bar{q}_{2i}-w_{2i})+w_{2i}(\bar{q}_{1i}-w_{1i})}{k_{i}+2}-\frac{1}{2}\frac{\bar{q}_{2i}}{k_{i}+2}).$

Hence, the mutual locality condition is

$\sum_{i=1}^{r}(\frac{w_{1i}(\bar{q}_{2i}-w_{2i})+w_{2i}(\bar{q}_{1i}-w_{1i})}{k_{i}+2}-\frac{1}{2}(\bar{q}_{2}+\sum_{a}\bar{\lambda}^{a}_{2}+\sum_{i=1}^{r}\frac{\bar{q}_{2i}}{k_{i}+2})\in\mathbb{Z}.$
(70)

The descendants which have been omitted in the vertex
${\cal{V}}^{(q_{2},\lambda_{2})(\bar{q}_{2},\bar{\lambda}_{2})}_{\bf{l}_{2},\bar{\bf{t}}_{2},\bf{w}_{2}}(0)$
correct the expression above to

$\sum_{i=1}^{r}(\frac{w_{1i}(\bar{q}_{2i}-w_{2i})+w_{2i}(\bar{q}_{1i}-w_{1i})}{k_{i}+2}-\frac{1}{2}(\bar{q}_{2}+\sum_{a}\bar{\lambda}^{a}_{2}+\sum_{i=1}^{r}\frac{\bar{q}_{2i}}{k_{i}+2}+\bar{Q}_{2des})\in\mathbb{Z}.$
(71)

Hence, the mutual locality condition is nothing else but the second line from
(29)):

$\displaystyle\bar{q}_{2}+\sum_{a}\bar{\lambda}^{a}_{2}+\sum_{i=1}^{r}\frac{\bar{q}_{2i}}{k_{i}+2}+\bar{Q}_{2des}\in
2\mathbb{Z}.$

Similarly, one can prove the mutual locality of a vertex operators of other
types.

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# Ab initio calculation of molecular pentaquark magnetic moments in heavy
pentaquark chiral perturbation theory

Hao-Song Li<EMAIL_ADDRESS>School of Physics, Northwest University,
Xian 710127, China  Shaanxi Key Laboratory for Theoretical Physics Frontiers,
Xi an 710127, China  Peng Huanwu Center for Fundamental Theory, Xi an 710127,
China

###### Abstract

We propose a new heavy pentaquark chiral perturbation theory for the recently
observed hidden-charm pentaquark states by LHCb Collaboration. With the ab
initio constructed chiral Lagrangians, we present a parameter-free calculation
of the octet molecular pentaquark magnetic moments up to one-loop level. We
improve the quark model description of the data when we include the leading
SU(3) breaking effects coming from the one-loop corrections. Without any
experimental inputs, our predictions are so simple and unique that we regard
them as a theoretical benchmark to be compared with experiments as well as
other theoretical models.

The hidden-charm pentaquark state was firstly observed by LHCb collaboration
in 2015 LHCb:2015yax , which appear to be inconsistent with the predictions of
the conventional quark model Gell-Mann:1964ewy . Since then, experimental
collaborations have shown great interest to the pentaquark states, an
increasing amount of data has produced many pentaquark states. In 2019,
$P^{N}_{\psi}(4312)^{{}^{+}}$, $P^{N}_{\psi}(4440)^{{}^{+}}$ and
$P^{N}_{\psi}(4457)^{{}^{+}}$ were reported in the updated analyses of the
LHCb collaboration LHCb:2019kea . In 2020, the LHCb Collaboration observed the
hidden-charm strange pentaquark state $P_{\psi{}s}^{\Lambda}(4459)^{0}$
through an amplitude analysis of the $\Xi^{-}_{b}\to{}J/\psi{}\Lambda{}K^{-}$
decay LHCb:2020jpq . In 2021, $P^{N}_{\psi}(4337)^{{}^{+}}$ was observed by
the LHCb Collaboration in the $B_{s}^{0}\to J/\psi p\bar{p}$ decay
LHCb:2021chn . Recently, the LHCb collaboration observed a new structure
$P_{\psi{}s}^{\Lambda}(4338)^{0}$ in the $B^{-}\to{}J/\psi{}\Lambda\bar{p}$
decay LHCb:2022ogu .

With the discovery of so many hidden-charm pentaquark states in the
experiments, theorists have made intense speculations on the nature of these
states Deng:2022vkv ; Wang:2019nvm ; Xiao:2019gjd ; Liu:2020hcv ; Peng:2020hql
; Zhu:2021lhd ; Du:2021bgb ; Chen:2021tip ; Yang:2021pio ; Ozdem:2021btf . Due
to their proximity to various charmed meson-baryon thresholds, those
candidates of the hidden-charm pentaquark states can be explained as the
molecular states. It is reasonable to speculate that the full molecular
multiplet of the hidden-charm molecular pentaquarks will be observed in the
coming future.

The hidden-charm molecular pentaquark states are composed of the corresponding
singly charmed baryons and anti-charmed mesons. Considering the direct product
$3\otimes 3\otimes 3=1\oplus 8_{1}\oplus 8_{2}\oplus 10$ and that the
$P^{N}_{\psi}$ pentaquark states are in the isospin $I=1/2$ configuration
Chen:2015loa ; Chen:2015moa , we construct the chiral lagrangians for octet
hidden-charm pentaquark states of $SU(3)$ flavor symmetry in this work.

To determine the nature of the pentaquark states, it is crucial to understand
their internal structures. The hadron magnetic moments provide valuable
insight in describing the inner structures of hadrons and understanding the
mechanism of strong interactions at low-energy. We first investigated the
magnetic moments of the pentaquark state with quark model in Ref. Gao:2021hmv
; Guo:2023fih , however it is difficult to include the Goldstone boson cloud
effect.

In fact, chiral perturbation theory (ChPT) Weinberg:1978kz  is quite helpful
to analyze the low-energy interactions. To consider the chiral corrections of
pentaquark we develop the heavy pentaquark chiral perturbation theory (HPChPT)
similar to heavy baryon chiral perturbation theory (HBChPT) Jenkins:1990jv ;
Jenkins:1992pi ; Bernard:1992qa ; Bernard:1995dp  which offers a useful power
counting and ease of calculation of Feynman diagrams. In the heavy pentaquark
limit, the pentaquark field $B$ can be decomposed into the large component $P$
and the small component $L$

$\displaystyle B=e^{-iM_{P}v\cdot x}(P+L),$ $\displaystyle P=e^{iM_{P}v\cdot
x}\frac{1+v/}{2}B,~{}L=e^{iM_{P}v\cdot x}\frac{1-v/}{2}B,$

where $M_{P}$ is the mass of pentaquark state and $v_{\mu}=(1,\vec{0})$ is the
velocity of the pentaquark state. The octet hidden-charm molecular pentaquark
$P_{n}$ reads

$P_{n}=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}{{}P_{\psi
s}^{\Sigma}}^{0}+\frac{1}{\sqrt{6}}{{}P_{\psi s}^{\Lambda}}^{0}&{{}P_{\psi
s}^{\Sigma}}^{+}&{{}P_{\psi}^{N}}^{+}\\\ {{}P_{\psi
s}^{\Sigma}}^{-}&-\frac{1}{\sqrt{2}}{{}P_{\psi
s}^{\Sigma}}^{0}+\frac{1}{\sqrt{6}}{{}P_{\psi
s}^{\Lambda}}^{0}&{{}P_{\psi}^{N}}^{0}\\\ {{}P_{\psi ss}^{N}}^{-}&{{}P_{\psi
ss}^{N}}^{+}&\frac{2}{\sqrt{3}}{{}P_{\psi s}^{\Lambda}}^{0}\\\
\end{array}\right).$

The flavor wave functions of the octet hidden-charm molecular pentaquark under
$SU(3)$ symmetry are shown in Table 1 (details in Ref. Li111 ).

Table 1: $8_{1}$ and $8_{2}$ flavor wave functions States | Flavor | Wave functions
---|---|---
${P_{\psi}^{N+}}$ | $8_{1}$ | $-\sqrt{\frac{1}{3}}\Sigma_{c}^{+}\bar{D}^{(*)0}+\sqrt{\frac{2}{3}}\Sigma_{c}^{++}D^{(*)-}$
| $8_{2}$ | $\Lambda_{c}^{+}\bar{D}^{(*)0}$
${P_{\psi}^{N0}}$ | $8_{1}$ | $\sqrt{\frac{1}{3}}\Sigma_{c}^{+}D^{(*)-}-\sqrt{\frac{2}{3}}\Sigma_{c}^{0}\bar{D}^{(*)0}$
| $8_{2}$ | $\Lambda_{c}^{+}D^{(*)-}$
${P_{\psi s}^{\Sigma+}}$ | $8_{1}$ | $\sqrt{\frac{1}{3}}\Xi_{c}^{\prime+}\bar{D}^{(*)0}-\sqrt{\frac{2}{3}}\Sigma_{c}^{++}D_{s}^{(*)-}$
| $8_{2}$ | $\Xi_{c}^{+}\bar{D}^{(*)0}$
${P_{\psi s}^{\Sigma 0}}$ | $8_{1}$ | $\sqrt{\frac{1}{6}}\Xi_{c}^{\prime+}D^{(*)-}+\sqrt{\frac{1}{6}}\Xi_{c}^{\prime 0}\bar{D}^{(*)0}-\sqrt{\frac{2}{3}}\Sigma_{c}^{+}D_{s}^{(*)-}$
| $8_{2}$ | $\sqrt{\frac{1}{2}}\Xi_{c}^{+}D^{(*)-}+\sqrt{\frac{1}{2}}\Xi_{c}^{0}\bar{D}^{(*)0}$
${P_{\psi s}^{\Lambda 0}}$ | $8_{1}$ | $\sqrt{\frac{1}{2}}\Xi_{c}^{\prime+}D^{(*)-}-\sqrt{\frac{1}{2}}\Xi_{c}^{\prime 0}\bar{D}^{(*)0}$
| $8_{2}$ | $\sqrt{\frac{1}{6}}\Xi_{c}^{+}D^{(*)-}-\sqrt{\frac{1}{6}}\Xi_{c}^{0}\bar{D}^{(*)0}-\sqrt{\frac{2}{3}}\Lambda_{c}^{+}D_{s}^{(*)-}$
${P_{\psi s}^{\Sigma-}}$ | $8_{1}$ | $\sqrt{\frac{1}{3}}\Xi_{c}^{\prime 0}D^{(*)-}-\sqrt{\frac{2}{3}}\Sigma_{c}^{0}D_{s}^{(*)-}$
| $8_{2}$ | $\Xi_{c}^{0}D^{(*)-}$
${P_{\psi ss}^{N0}}$ | $8_{1}$ | $\sqrt{\frac{1}{3}}\Xi_{c}^{\prime+}D_{s}^{(*)-}-\sqrt{\frac{2}{3}}\Omega_{c}^{0}\bar{D}^{(*)0}$
| $8_{2}$ | $\Xi_{c}^{+}D_{s}^{(*)-}$
${P_{\psi ss}^{N-}}$ | $8_{1}$ | $\sqrt{\frac{1}{3}}\Xi_{c}^{\prime 0}D_{s}^{(*)-}-\sqrt{\frac{2}{3}}\Omega_{c}^{0}D^{(*)-}$
| $8_{2}$ | $\Xi_{c}^{0}D_{s}^{(*)-}$

In this work, we investigate the magnetic moments of spin-$\frac{1}{2}$ octet
hidden-charm molecular pentaquark family in HPChPT. We explicitly consider the
molecular pentaquark intermediate states in the loop calculation. We use quark
model to determine the corresponding low energy constants (LECs) and calculate
the magnetic moments order by order. We follow the standard power counting
scheme, the chiral order $D_{\chi}$ of a given diagram is given by
Ecker:1994gg

$D_{\chi}=4N_{L}-2I_{M}-I_{P}+\sum_{n}nN_{n},$ (1)

where $N_{L}$ is the number of loops, $I_{M}$ is the number of internal pion
lines, $I_{P}$ is the number of internal molecular pentaquark lines and
$N_{n}$ is the number of the vertices from the $n$th order Lagrangians.

To calculate the chiral corrections to the magnetic moment of
spin-$\frac{1}{2}$ pentaquark state, we follow the basic definitions of the
pseudoscalar mesons and the spin-$\frac{1}{2}$ baryon chiral effective
Lagrangians in Refs. Bernard:1995dp ; Li:2017cfz  to construct the relevant
chiral Lagrangians. The pseudoscalar meson fields read

$\phi=\left(\begin{array}[]{ccc}\pi^{0}+\frac{1}{\sqrt{3}}\eta&\sqrt{2}\pi^{+}&\sqrt{2}K^{+}\\\
\sqrt{2}\pi^{-}&-\pi^{0}+\frac{1}{\sqrt{3}}\eta&\sqrt{2}K^{0}\\\
\sqrt{2}K^{-}&\sqrt{2}\bar{K}^{0}&-\frac{2}{\sqrt{3}}\eta\end{array}\right).$

The chiral connection and axial vector field read Bernard:1995dp ;
Scherer:2002tk :

$\displaystyle\Gamma_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[u^{\dagger}(\partial_{\mu}-ir_{\mu})u+u(\partial_{\mu}-il_{\mu})u^{\dagger}\right],$
(2) $\displaystyle u_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}i\left[u^{\dagger}(\partial_{\mu}-ir_{\mu})u-u(\partial_{\mu}-il_{\mu})u^{\dagger}\right],$
(3)

where $u^{2}=\mathit{U}=\exp(i\phi/f_{0})$, $r_{\mu}=l_{\mu}=-eQA_{\mu}$,
$Q=\rm{diag}(2/3,-1/3,-1/3)$. We use the pseudoscalar meson decay constants
$f_{\pi}\approx$ 92.4 MeV, $f_{K}\approx$ 113 MeV and $f_{\eta}\approx$ 116
MeV.

The octet hidden-charm molecular pentaquark states have three spin
configurations $J^{P}(J_{b}^{P_{b}}\otimes J_{m}^{P_{m}})$:
$\frac{1}{2}^{-}(\frac{1}{2}^{+}\otimes 0^{-})$,
$\frac{1}{2}^{-}(\frac{1}{2}^{+}\otimes 1^{-})$, and
$\frac{3}{2}^{-}(\frac{1}{2}^{+}\otimes 1^{-})$, $J_{b}^{P_{b}}$ and
$J_{m}^{P_{m}}$ correspond to the angular momentum and parity of baryon and
meson, respectively. We only consider the
$J^{P}=\frac{1}{2}^{-}(\frac{1}{2}^{+}\otimes 0^{-})$ pentaquark state in this
work.

The free Lagrangian of pentaquark state reads

$\mathcal{L}_{0}^{(1)}={\rm Tr}[\bar{P}(iv\cdot D)P],$ (4) $\displaystyle
D_{\mu}P$ $\displaystyle=$ $\displaystyle\partial_{\mu}P+\Gamma_{\mu}P.$ (5)

The interaction Lagrangians of $8_{1}$ and $8_{2}$ flavor pentaquark states
read

$\mathcal{L}_{\rm int1}=2g_{1}{\rm
Tr}(\bar{P}_{1}S_{\mu}\\{u^{\mu},P_{1}\\})+2f_{1}{\rm
Tr}(\bar{P}_{1}S_{\mu}[u^{\mu},P_{1}]),$ (6) $\mathcal{L}_{\rm
int2}=2g_{4}{\rm Tr}(\bar{P}_{2}S_{\mu}\\{u^{\mu},P_{2}\\})+2f_{4}{\rm
Tr}(\bar{P}_{2}S_{\mu}[u^{\mu},P_{2}]),$ (7)

where $P_{1,2}$ represent the $8_{1,2}$ flavor pentaquark states, $S_{\mu}$ is
the covariant spin-operator. The $\phi PP$ coupling constants $f_{1,4}$ and
$g_{1,4}$ are estimated in our another work in Ref. Li111 . For the
pseudoscalar mesons masses, we use $m_{\pi}=0.140$ GeV, $m_{K}=0.494$ GeV, and
$m_{\eta}=0.550$ GeV.

We also need the lowest order $\mathcal{O}(p^{2})$ Lagrangian contributes to
the magnetic moments of the $8_{1,2}$ flavor pentaquark states at the tree
level

$\displaystyle\mathcal{L}_{\mu 1}^{(2)}$ $\displaystyle=$ $\displaystyle
b_{p1}{\rm
Tr}(\frac{-i}{4M_{B}}\bar{P}_{1}[S^{\mu},S^{\nu}]\\{F_{\mu\nu}^{+},P_{1}\\})$
(8) $\displaystyle+b_{p2}{\rm
Tr}(\frac{-i}{4M_{B}}\bar{P}_{1}[S^{\mu},S^{\nu}][F_{\mu\nu}^{+},P_{1}]),$
$\displaystyle\mathcal{L}_{\mu 2}^{(2)}$ $\displaystyle=$ $\displaystyle
b_{p3}{\rm
Tr}(\frac{-i}{4M_{B}}\bar{P}_{2}[S^{\mu},S^{\nu}]\\{F_{\mu\nu}^{+},P_{2}\\})$
(9) $\displaystyle+b_{p4}{\rm
Tr}(\frac{-i}{4M_{B}}\bar{P}_{2}[S^{\mu},S^{\nu}][F_{\mu\nu}^{+},P_{2}]).$

The coefficients $b_{p(1,2,3,4)}$ are LECs. The chirally covariant QED field
strength tensor $F_{\mu\nu}^{\pm}$ is defined as

$\displaystyle F_{\mu\nu}^{\pm}$ $\displaystyle=$ $\displaystyle
u^{\dagger}F_{\mu\nu}^{R}u\pm uF_{\mu\nu}^{L}u^{\dagger},$ $\displaystyle
F_{\mu\nu}^{R}$ $\displaystyle=$
$\displaystyle\partial_{\mu}r_{\nu}-\partial_{\nu}r_{\mu}-i[r_{\mu},r_{\nu}],$
$\displaystyle F_{\mu\nu}^{L}$ $\displaystyle=$
$\displaystyle\partial_{\mu}l_{\nu}-\partial_{\nu}l_{\mu}-i[l_{\mu},l_{\nu}],$

We use the nucleon mass $M_{B}=0.938$ GeV, so that we can express the magnetic
moments directly in nuclear magnetons.

Figure 1: Feynman diagrams where octet pentaquark state is denoted by the solid line. The dot represent $\mathcal{O}(p^{2})$ coupling. The dashed and wiggly lines represent the pseudoscalar meson and photon respectively. Table 2: The coefficients of the loop corrections to the pentaquark magnetic moments from Figs. 1(b). States | $P_{\psi}^{N+}$ | $P_{\psi}^{N0}$ | $P_{\psi s}^{\Sigma+}$ | $\ \ \ P_{\psi s}^{\Sigma 0}\ \ \ $ | $\ \ \ P_{\psi s}^{\Lambda 0}\ \ \ $ | $P_{\psi s}^{\Sigma-}$ | $P_{\psi ss}^{N0}$ | $P_{\psi ss}^{N-}$
---|---|---|---|---|---|---|---|---
$\zeta_{1}=\zeta_{2}$ | $\frac{1}{3}$ | $-\frac{2}{3}$ | $\frac{1}{3}$ | $\frac{1}{3}$ | $-\frac{1}{3}$ | $\frac{1}{3}$ | $-\frac{2}{3}$ | $\frac{1}{3}$
$\kappa_{1}=\kappa_{2}$ | 1 | 0 | 1 | 0 | 0 | -1 | 0 | -1
$\beta_{1}^{\pi}$ | $2(f_{1}+g_{1})^{2}$ | $-2(f_{1}+g_{1})^{2}$ | $4f_{1}^{2}+\frac{4}{3}g_{1}^{2}$ | $0$ | $0$ | $-4f_{1}^{2}-\frac{4}{3}g_{1}^{2}$ | $2(g_{1}-f_{1})^{2}$ | $-2(g_{1}-f_{1})^{2}$
$\beta_{1}^{K}$ | $4f_{1}^{2}+\frac{4}{3}g_{1}^{2}$ | $2(g_{1}-f_{1})^{2}$ | $2(f_{1}+g_{1})^{2}$ | $4f_{1}g_{1}$ | $-4f_{1}g_{1}$ | $-2(g_{1}-f_{1})^{2}$ | $-2(f_{1}+g_{1})^{2}$ | $-2(f_{1}+g_{1})^{2}$
$\beta_{2}^{\pi}$ | $2(f_{4}+g_{4})^{2}$ | $-2(f_{4}+g_{4})^{2}$ | $4f_{4}^{2}+\frac{4}{3}g_{4}^{2}$ | $0$ | $0$ | $-4f_{4}^{2}-\frac{4}{3}g_{4}^{2}$ | $2(g_{4}-f_{4})^{2}$ | $-2(g_{4}-f_{4})^{2}$
$\beta_{2}^{K}$ | $4f_{4}^{2}+\frac{4}{3}g_{4}^{2}$ | $2(g_{4}-f_{4})^{2}$ | $2(f_{4}+g_{4})^{2}$ | $4f_{4}g_{4}$ | $-4f_{4}g_{4}$ | $-2(g_{4}-f_{4})^{2}$ | $-2(f_{4}+g_{4})^{2}$ | $-2(f_{4}+g_{4})^{2}$

There are six Feynman diagrams contributing to the pentaquark magnetic moments
to one-loop order shown in Fig. 1. The tree-level Lagrangians in Eq. (9)
contribute to $8_{1,2}$ flavor pentaquark magnetic moments at leading order
(LO) as shown in diagram (a)

$\displaystyle\mu_{P1}^{(\rm tree)}$ $\displaystyle=$
$\displaystyle\zeta_{1}b_{p1}+\kappa_{1}b_{p2},$ (10)
$\displaystyle\mu_{P2}^{(\rm tree)}$ $\displaystyle=$
$\displaystyle\zeta_{2}b_{p3}+\kappa_{2}b_{p4},$ (11)

where $\mu_{P(1,2)}^{(\rm tree)}$ represent the LO $8_{1,2}$ flavor pentaquark
magnetic moments, the coefficients $\zeta_{1,2}$ and $\kappa_{1,2}$ for
$8_{1,2}$ flavor pentaquark are listed in Table 2. Due to the SU(3)-symmetry,
the LO contributions show the Coleman-Glashow relations similar to the nucleon
octet.

The diagram (c) vanishes in the heavy pentaquark mass limit. The diagrams (d),
(e) and (f) contribute to the pentaquark magnetic moments to next-to-next-to-
leading order. Thus, only diagrams (b) account for the next-to-leading order
(NLO) corrections which are induced by the corresponding breaking in the
pseudoscalar meson masses. The NLO corrections to the pentaquark magnetic
moments can be expressed as

$\displaystyle\mu_{P(1,2)}^{(\rm loop)}$ $\displaystyle=$
$\displaystyle\sum_{\phi=\pi,K}\frac{-m_{\phi}M_{B}\beta_{(1,2)}^{\phi}}{64\pi
f_{\phi}^{2}}$ (12)

$\mu_{P(1,2)}^{(\rm loop)}$ represent the loop corrections to the $8_{1,2}$
flavor pentaquark magnetic moments. We collect the coefficients
$\beta_{1,2}^{\phi}$ explicit expressions in Table 2.

In ChPT, the LECs should be fitted through experimental inputs. As there are
not any experimental data of the pentaquark magnetic moments so far, in this
work, we use quark model to determine the LO tree level magnetic moment. The
magnetic moments in the quark model are the matrix elements of the following
operator,

$\vec{\mu}=\sum_{i}\mu_{i}\vec{\sigma}^{i},$

where $\mu_{i}$ is the magnetic moment of the quark:

$\mu_{i}={e_{i}\over 2m_{i}},\quad i=u,d,s,c.$

With the wave functions of the octet hidden-charm molecular pentaquark states
in Table 1 (details in Ref. Li111 ), we give the LO pentaquark magnetic
moments in the second and third columns in Table 3. When considering SU(3)
flavor symmetry ($m_{u}=m_{d}=m_{s}$) and heavy quark symmetry
($m_{c}=\infty$), we obtain the LECs $b_{p1}=0.93$, $b_{p2}=1.55$,
$b_{p3}=b_{p4}=0$. When we take the constituent quark masses as input:
$m_{u}=m_{d}=0.336$ GeV, $m_{s}=0.540$ GeV, $m_{c}=1.660$ GeV, we obtain the
LO pentaquark magnetic moments shown in the second row in Table 4. Thus, up to
NLO, there exist only four LECs: $f_{1,4}$, $g_{1,4}$ at this order.

Table 3: The $8_{1}$ and $8_{2}$ flavor pentaquark magnetic moments at tree level (in unit of $\mu_{N}$). States |  $8_{1}$ LO |  $8_{2}$ LO
---|---|---
$P_{\psi}^{N+}$ | $\frac{2}{9}(5\mu_{u}+\mu_{d})-\frac{1}{3}\mu_{c}$ | $\mu_{c}$
$P_{\psi}^{N0}$ | $\frac{2}{9}(\mu_{u}+5\mu_{d})-\frac{1}{3}\mu_{c}$ | $\mu_{c}$
$P_{\psi s}^{\Sigma+}$ | $\frac{2}{9}(5\mu_{u}+\mu_{s})-\frac{1}{3}\mu_{c}$ | $\mu_{c}$
$P_{\psi s}^{\Sigma 0}$ | $\frac{1}{9}(5\mu_{u}+5\mu_{d}+2\mu_{s})-\frac{1}{3}\mu_{c}$ | $\mu_{c}$
$P_{\psi s}^{\Lambda 0}$ | $\frac{1}{3}(\mu_{u}+\mu_{d}+2\mu_{s})-\frac{1}{3}\mu_{c}$ | $\mu_{c}$
$P_{\psi s}^{\Sigma-}$ | $\frac{2}{9}(5\mu_{d}+\mu_{s})-\frac{1}{3}\mu_{c}$ | $\mu_{c}$
$P_{\psi ss}^{N0}$ | $\frac{2}{9}(\mu_{u}+5\mu_{s})-\frac{1}{3}\mu_{c}$ | $\mu_{c}$
$P_{\psi ss}^{N-}$ | $\frac{2}{9}(\mu_{d}+5\mu_{s})-\frac{1}{3}\mu_{c}$ | $\mu_{c}$

To determine the coupling constants $g_{1}$ and $f_{1}$, similar to the
procedure employed for the nucleon, we consider the $\pi_{0}$ meson decay of
$P_{\psi}^{N^{+}}$ and ${P_{\psi ss}^{N^{0}}}$. The Lagrangians for the
$\pi_{0}$ decay of ${{}P_{\psi}^{N^{+}}}$ and ${P_{\psi ss}^{N^{0}}}$ with the
spin configuration $J^{P}=\frac{1}{2}^{-}(\frac{1}{2}^{+}\otimes 0^{-})$ at
the hadron level read

$\displaystyle\mathcal{L}_{P_{\psi}^{N+}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(g_{1}+f_{1}){{}\bar{P}_{\psi}^{N+}}\gamma^{\mu}\gamma_{5}\partial_{\mu}\Phi{{}P_{\psi}^{N+}},$
$\displaystyle\mathcal{L}_{{P_{\psi ss}^{N0}}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(f_{1}-g_{1}){{}\bar{P}_{\psi
ss}^{N0}}\gamma^{\mu}\gamma_{5}\partial_{\mu}\Phi{{}P_{\psi ss}^{N0}}.$

At the hadron level, the $\pi_{0}$ hadron interaction read

$\displaystyle\langle{P_{\psi}^{N+}};\pi_{0}|\frac{g_{1}+f_{1}}{f_{\pi}}{{}\bar{P}_{\psi}^{N+}}\partial_{z}{\pi_{0}}{{}P_{\psi}^{N+}}|{P_{\psi}^{N+}}\rangle\sim\frac{g_{1}+f_{1}}{2}\frac{q_{z}}{f_{\pi}},$
$\displaystyle\langle{P_{\psi
ss}^{N0}};\pi_{0}|\frac{f_{1}-g_{1}}{f_{\pi}}{\bar{P}_{\psi
ss}^{N0}}\partial_{z}{\pi_{0}}{P_{\psi ss}^{N0}}|{P_{\psi
ss}^{N0}}\rangle\sim\frac{f_{1}-g_{1}}{2}\frac{q_{z}}{f_{\pi}}.$

At the quark level, the $\pi_{0}$ quark interaction read

$\displaystyle\langle{P_{\psi}^{N+}},+\frac{1}{2};~{}\pi_{0}|\mathcal{L}_{quark}|{{}P_{\psi}^{N+}},+\frac{1}{2}\rangle\sim\frac{4}{9}\frac{q_{z}}{f_{\pi}}g_{q},$
$\displaystyle\langle{P_{\psi
ss}^{N0}},+\frac{1}{2};~{}\pi_{0}|\mathcal{L}_{quark}|{P_{\psi
ss}^{N0}},+\frac{1}{2}\rangle\sim\frac{1}{9}\frac{q_{z}}{f_{\pi}}g_{q}.$

Compare with the axial charge of the nucleon $g_{A}$,

$\displaystyle\frac{\frac{1}{2}g_{A}}{\frac{5}{6}g_{q}}=\frac{\frac{g_{1}+f_{1}}{2}}{\frac{4}{9}g_{q}}=\frac{\frac{f_{1}-g_{1}}{2}}{\frac{1}{9}g_{q}}.$

Table 4: The pentaquark magnetic moments to NLO (in unit of $\mu_{N}$). States | $\ \ \ P_{\psi}^{N+}\ \ \ $ | $\ \ \ P_{\psi}^{N0}\ \ \ $ | $\ \ \ P_{\psi s}^{\Sigma+}\ \ \ $ | $\ \ \ P_{\psi s}^{\Sigma 0}\ \ \ $ | $\ \ \ P_{\psi s}^{\Lambda 0}\ \ \ $ | $\ \ \ P_{\psi s}^{\Sigma-}\ \ \ $ | $\ \ \ P_{\psi ss}^{N0}\ \ \ $ | $\ \ \ P_{\psi ss}^{N-}\ \ \ $
---|---|---|---|---|---|---|---|---
$8_{1}$ LO | 1.74 | -0.75 | 1.81 | 0.26 | -0.20 | -1.29 | -0.36 | -0.98
$8_{1}$ NLO | -0.21 | 0.06 | -0.22 | -0.07 | 0.07 | 0.07 | 0.16 | 0.15
$8_{1}$ Total | 1.53 | -0.69 | 1.59 | 0.19 | -0.13 | -1.22 | -0.20 | 0.83
$8_{2}$ LO | 0.38 | 0.38 | 0.38 | 0.38 | 0.38 | 0.38 | 0.38 | 0.38
$8_{2}$ NLO | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$8_{2}$ Total | 0.38 | 0.38 | 0.38 | 0.38 | 0.38 | 0.38 | 0.38 | 0.38

We obtain $f_{1}=\frac{1}{3}g_{A}=0.42$ and $g_{1}=\frac{1}{5}g_{A}=0.25$.
Similarly, we can also obtain $f_{4}=g_{4}=0$. With $f_{1,4}$ and $g_{1,4}$,
we obtain the chiral loop corrections to the pentaquark magnetic moments. Thus
we obtain the numerical results of the NLO pentaquark magnetic moments in
Table 4.

For $8_{2}$ flavor pentaquark states, it is interesting to notice that the NLO
contributions to the magnetic moments are all zero due to $f_{4}=g_{4}=0$ in
quark model. Thus, the $8_{2}$ flavor pentaquark magnetic moments are all
equal to $\mu_{c}=0.38\mu_{N}$. At present, the magnetic moments of the
recently observed pentaquark states have not been measured in the experiment,
our calculation shows that if the magnetic moment of the pentaquark equal to
$\mu_{c}=0.38\mu_{N}$, the pentaquark state belong to
$J^{P}=\frac{1}{2}^{-}(\frac{1}{2}^{+}\otimes 0^{-})$ $8_{2}$ flavor
pentaquark states whose wave functions are shown in Table 1. We are looking
forward to further progresses in experiment.

For $8_{1}$ flavor pentaquark states, in order to study the convergence of the
chiral expansion, we separate the LO from NLO contributions for the $8_{1}$
flavor pentaquark magnetic moments (in unit of $\mu_{N}$), we also compare the
$8_{1}$ flavor pentaquark magnetic moments with the corresponding baryon-octet
magnetic moments in PDG ParticleDataGroup:2022pth

$\displaystyle\mu_{P1_{\psi}^{N+}}$ $\displaystyle=$ $\displaystyle
1.74(1-0.12)=1.53=0.55\mu_{p},$ $\displaystyle\mu_{P1_{\psi}^{N0}}$
$\displaystyle=$ $\displaystyle-0.75(1-0.08)=-0.69=0.36\mu_{n},$
$\displaystyle\mu_{P1_{\psi s}^{\Sigma+}}$ $\displaystyle=$ $\displaystyle
1.81(1-0.12)=1.59=0.65\mu_{\Sigma^{+}},$ $\displaystyle\mu_{P1_{\psi
s}^{\Sigma 0}}$ $\displaystyle=$ $\displaystyle 0.26(1-0.27)=0.19,$
$\displaystyle\mu_{P1_{\psi s}^{\Lambda 0}}$ $\displaystyle=$
$\displaystyle-0.20(1-0.35)=-0.13=0.21\mu_{\Lambda},$
$\displaystyle\mu_{P1_{\psi s}^{\Sigma-}}$ $\displaystyle=$
$\displaystyle-1.29(1-0.05)=-1.22=1.05\mu_{\Sigma^{-}},$
$\displaystyle\mu_{P1_{\psi ss}^{N0}}$ $\displaystyle=$
$\displaystyle-0.36(1-0.44)=-0.20=0.16\mu_{\Xi^{0}},$
$\displaystyle\mu_{P1_{\psi ss}^{N-}}$ $\displaystyle=$
$\displaystyle-0.98(1-0.15)=-0.83=1.27\mu_{\Xi^{-}}.$

It is obvious that the convergence of the chiral expansion is quite good. The
reason is accessible, since the axial charges of $8_{1}$ flavor pentaquark are
generally lower compared to that of the nucleon ($f_{1}=\frac{1}{3}g_{A}$,
$g_{1}=\frac{1}{5}g_{A}$), thus the NLO contributions for the $8_{1}$ flavor
pentaquark magnetic moments which are proportional to $g_{A}^{2}$ are greatly
suppressed compared to the NLO contributions for the baryon-octet magnetic
moments. The $8_{1}$ flavor pentaquark magnetic moments and the baryon-octet
magnetic moments present a certain degree of positive correlation.

In summary, we have investigated the magnetic moments for the octet hidden-
charm molecular pentaquark family to NLO in the framework of HPChPT. We
construct the chiral lagrangians of the octet pentaquark states and determine
all free parameters needed for the low energy dynamics of octet pentaquark
with the quark model. Thus, we report on the first global study of octet
molecular pentaquark magnetic moments up to one-loop order.

Our calculations indicate that for $8_{1}$ flavor pentaquark magnetic moments,
the convergence of the chiral expansion is pretty good, for $8_{2}$ flavor
pentaquark states, the pentaquark magnetic moments are all equal to
$\mu_{c}=0.38\mu_{N}$ even to NLO. Moreover, these calculations depend only on
the spin-flavor wave functions of the pentaquark states. Without any
experimental inputs, our predictions are so simple and unique that we regard
them as a theoretical benchmark to be compared with experiments as well as
other theoretical models. We are looking forward to further progresses and
hope our results may be useful for future experimental measurement.

## ACKNOWLEDGMENTS

This project is supported by the National Natural Science Foundation of China
under Grants No. 11905171 and No. 12047502. This work is also supported by the
Natural Science Basic Research Plan in Shaanxi Province of China (Grant No.
2022JQ-025).

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