Sharpe for useful questions and comments.

##  Appendix A Elements of $\boldsymbol{(0,2)}$ sigma models

###  A.1 $\boldsymbol{(0,2)}$ superspace

The $(0,2)$ superspace is spanned by a complex fermionic variable $\theta^{+}$
and its conjugate ${\bar{\theta}}^{+}$ of positive chiralilty in two
dimensions and worldsheet coordinates
$\sigma=\mbox{$\frac{1}{\sqrt{2}}$}(\sigma_{1}+\sigma_{0})$ and
${\bar{\sigma}}=\mbox{$\frac{1}{\sqrt{2}}$}(\sigma_{1}-\sigma_{0})$. Using
their derivatives denoted by
$\partial_{+},\bar{\partial}_{+},\partial=\mbox{$\frac{1}{\sqrt{2}}$}(\partial_{1}+\partial_{0})$
and $\bar{\partial}=\mbox{$\frac{1}{\sqrt{2}}$}(\partial_{1}-\partial_{0})$,
respectively, super covariant derivates $D_{+}$ and
${\overline{D}}_{+}=-(D_{+})^{\dagger}$ can be defined as

$\displaystyle
D_{+}=\partial_{+}-i{\bar{\theta}}^{+}\,\partial~{},\qquad{\overline{D}}_{+}=\bar{\partial}_{+}-i\theta^{+}\,\partial~{},\qquad\big{\\{}{\overline{D}}_{+},D_{+}\big{\\}}=-2i\,\partial~{}.$
(A.1)

These super covariant derivatives anti–commute with the supercharges

$\displaystyle
Q_{+}=\partial_{+}+i{\bar{\theta}}^{+}\,\partial~{},\qquad{\overline{Q}}_{+}=\bar{\partial}_{+}+i\theta^{+}\,\partial~{}.$
(A.2)

The supercharges generate the $(0,2)$ super algebra

$\displaystyle\big{\\{}{\overline{Q}}_{+},Q_{+}\big{\\}}=2\,P~{},$ (A.3)

where $P=i\,\partial$ is the right moving momentum generator.

###  A.2 $\boldsymbol{(0,2)}$ superfields

A general $(0,2)$ superfield $G$ is a complex function of $(0,2)$ superspace
on which supersymmetry act as

$\displaystyle\delta_{\epsilon}G=(\epsilon^{+}Q_{+}+{\bar{\epsilon}}^{\,+}{\overline{Q}}_{+})G~{}.$
(A.4)

Consequently sums, products and super covariant derivatives of superfields are
again superfields.

The components of a superfield are defined by taking a number of super
covariant derivates and then set all $\theta^{+}$ and ${\bar{\theta}}^{+}$ to
zero which is denoted as $|_{+}$. A superfield $G$ is called bosonic
(fermionic) if its lowest component $G|_{+}$ is bosonic (fermionic).

There are four fundamental multiplets of $(0,2)$ supersymmetry: the chiral
multiplet, the chiral Fermi multiplet, the vector multiplet and the Fermi
gauge multiplet:

#### Chiral multiplet

A chiral multiplet $\Phi$ and its conjugate ${\overline{\Phi}}$ are bosonic
superfields defined by the chirality constraints:

$\displaystyle{\overline{D}}_{+}\Phi=0~{},\qquad D_{+}{\overline{\Phi}}=0~{}.$
(A.5)

Their components,

$\displaystyle
z=\Phi|_{+}~{},\quad\phi=\mbox{$\frac{1}{\sqrt{2}}$}D_{+}\Phi|_{+}~{},\qquad{\bar{z}}={\overline{\Phi}}|_{+}~{},\quad{\bar{\phi}}=-\mbox{$\frac{1}{\sqrt{2}}$}D_{+}{\overline{\Phi}}|_{+}~{},$
(A.6)

are a complex scalar $z$, a negative chiral (right–moving) complex spinor
$\phi$ and their conjugates ${\bar{z}}$ and ${\bar{\phi}}$.

#### Chiral Fermi multiplet

A chiral Fermi multiplet $\Lambda$ and its conjugate ${\overline{\Lambda}}$
are fermionic superfields defined by the chirality constraints:

$\displaystyle{\overline{D}}_{+}\Lambda=0~{},\qquad
D_{+}{\overline{\Lambda}}=0~{}.$ (A.7)

Their components,

$\displaystyle\lambda=\Lambda|_{+}~{},\quad
h=\mbox{$\frac{1}{\sqrt{2}}$}D_{+}\Lambda|_{+}~{},\qquad{\bar{\lambda}}=-{\overline{\Lambda}}|_{+}~{},\quad{\bar{h}}=\mbox{$\frac{1}{\sqrt{2}}$}{\overline{D}}_{+}{\overline{\Lambda}}|_{+}~{},$
(A.8)

are a positive chiral (left–moving) complex spinor $\lambda$, a complex scalar
$h$ and their conjugates ${\bar{\lambda}}$ and ${\bar{h}}$.

#### Vector multiplet

The vector multiplet $(V,A)$ consists of two real bosonic superfields $V$ and
$A$ subject to a bosonic super gauge transformation

$\displaystyle V\rightarrow
V-\mbox{$\frac{1}{2}$}\big{(}\Theta+{\overline{\Theta}}\big{)}~{},\qquad
A\rightarrow
A+\mbox{$\frac{i}{2}$}\bar{\partial}\big{(}\Theta-{\overline{\Theta}}\big{)}~{},$
(A.9)

with a chiral superfield $\Theta$ gauge parameter and its conjugate
${\overline{\Theta}}$. (Non–Abelian gauge superfields are not considered in
this work.) Their components are

$\displaystyle\Theta|_{+}=\theta=\mbox{$\frac{1}{2}$}\,a+i\,\alpha~{},\qquad\mbox{$\frac{1}{\sqrt{2}}$}D_{+}\Theta|_{+}=\zeta~{},\qquad{\overline{\Theta}}|_{+}=\theta=\mbox{$\frac{1}{2}$}\,a-i\,\alpha~{},\qquad-\mbox{$\frac{1}{\sqrt{2}}$}D_{+}{\overline{\Theta}}|_{+}={\bar{\zeta}}~{},$
(A.10)

where $a$ and $\alpha$ are real fields. The two dimensional gauge field
components are identified as

$\displaystyle
A_{\sigma}=\mbox{$\frac{1}{2}$}\big{[}\,{\overline{D}}_{+},D_{+}\big{]}V|_{+}~{},\quad
A_{\bar{\sigma}}=A|_{+}~{},$ (A.11)

which transform as

$\displaystyle A_{\sigma}\rightarrow A_{\sigma}-\partial\alpha~{},\qquad
A_{\bar{\sigma}}\rightarrow A_{\bar{\sigma}}-\bar{\partial}\alpha~{}.$ (A.12)

The super field strengths

$\displaystyle
F=-\mbox{$\frac{1}{2}$}{\overline{D}}_{+}\big{(}A-i\bar{\partial}V\big{)}~{},\qquad{\overline{F}}=\mbox{$\frac{1}{2}$}D_{+}\big{(}A+i\bar{\partial}V\big{)}~{},$
(A.13)

are super gauge invariant chiral Fermi multiplets, since by construction
${\overline{D}}_{+}F=D_{+}{\overline{F}}=0$. Consequently, their components

$\displaystyle
F|_{+}=\mbox{$\frac{1}{\sqrt{2}}$}\varphi~{},\quad{\overline{F}}|_{+}=\mbox{$\frac{1}{\sqrt{2}}$}{\bar{\varphi}}~{},\quad
D_{+}F|_{+}=\mbox{$\frac{1}{2}$}\big{(}D+i\,F_{\sigma{\bar{\sigma}}}\big{)}~{},\quad{\overline{D}}_{+}{\overline{F}}|_{+}=\mbox{$\frac{1}{2}$}\big{(}D-i\,F_{\sigma{\bar{\sigma}}}\big{)}~{},$
(A.14)

are identical in any gauge. In particular,
$D=\mbox{$\frac{1}{2}$}\,[{\overline{D}}_{+},D_{+}]A|_{+}-\partial\bar{\partial}V|_{+}$
and $F_{\sigma{\bar{\sigma}}}=F_{01}$.

The super gauge transformation can be used to set some of the components of
the vector multiplet to zero: $V|_{+}=D_{+}V|_{+}={\overline{D}}_{+}V|_{+}=0$.
In this so–called Wess–Zumino (WZ) gauge all quadratic and higher powers of
$V$ vanish. Since $V$ is a real superfield the WZ gauge does not fix the super
gauge transformations completely, there is a residual gauge transformation
with $\Theta=i\alpha$.

#### Fermi gauge multiplet

A Fermi gauge multiplet $\Sigma$ and its conjugate ${\overline{\Sigma}}$ are
complex fermionic superfields subject to fermionic super gauge transformations

$\displaystyle\Sigma\rightarrow\Sigma-\Xi~{},\qquad{\overline{\Sigma}}\rightarrow{\overline{\Sigma}}-{\overline{\Xi}}~{},$
(A.15)

where $\Xi$ is a Fermi multiplet and ${\overline{\Xi}}$ its conjugate. The
associated super field strength $\Upsilon$ and its conjugate

$\displaystyle\Upsilon={\overline{D}}_{+}\Sigma~{},\qquad{\overline{\Upsilon}}=D_{+}{\overline{\Sigma}}~{},$
(A.16)

are inert under the fermionic gauge transformations. Their components are

$\displaystyle
s=\mbox{$\frac{1}{\sqrt{2}}$}\Upsilon|_{+}~{},\qquad{\bar{s}}=\mbox{$\frac{1}{\sqrt{2}}$}{\overline{\Upsilon}}|_{+}~{},\qquad\chi=\mbox{$\frac{1}{2}$}D_{+}\Upsilon|_{+}~{},\qquad{\bar{\chi}}=\mbox{$\frac{1}{2}$}{\overline{D}}_{+}{\overline{\Upsilon}}|_{+}~{}.$
(A.17)

Using the fermionic gauge transformations, the following components of the
Fermi gauge multiplet $\Sigma$ are set to zero in the WZ–gauge:
$\Sigma|_{+}=D_{+}\Sigma|_{+}=0$.

###  A.3 Super conformal transformations and scaling dimensions

Real conformal transformations of the worldsheet coordinates

$\displaystyle\sigma\rightarrow
f(\sigma)~{},\qquad{\bar{\sigma}}\rightarrow{\bar{f}}({\bar{\sigma}})~{},$
(A.18)

are characterized by two real functions $f(\sigma)$ of $\sigma$ only and
${\bar{f}}({\bar{\sigma}})$ of ${\bar{\sigma}}$ only. Consequently, their
differential and derivatives transform

$\displaystyle\mathrm{d}\sigma\rightarrow\omega^{-1}\,\mathrm{d}\sigma~{}\qquad\mathrm{d}{\bar{\sigma}}\rightarrow{\bar{\omega}}^{-1}\,\mathrm{d}{\bar{\sigma}}~{},\qquad\partial\rightarrow\omega\,\partial~{},\qquad\bar{\partial}\rightarrow{\bar{\omega}}\,\bar{\partial}~{},$
(A.19)

where $\omega=(\partial f)^{-1}$ and
${\bar{\omega}}=(\bar{\partial}{\bar{f}})^{-1}$. Moreover, since $\theta^{+}$
is a complex parameter, there is a phase transformation, often dubbed
R–symmetry,

$\displaystyle\theta^{+}\rightarrow
e^{i\kappa}\,\theta^{+}~{},\qquad{\bar{\theta}}^{+}\rightarrow
e^{-i\kappa}\,{\bar{\theta}}^{+}~{},$ (A.20)

with $\kappa\in\mathbbm{R}$. Requiring that the algebra of the super covariant
derivatives transforms consistently with this implies:

$\displaystyle
D_{+}\rightarrow\omega^{\mbox{$\frac{1}{2}$}}e^{-i\kappa}\,D_{+}~{},\qquad{\overline{D}}_{+}\rightarrow\omega^{\mbox{$\frac{1}{2}$}}e^{+i\kappa}\,{\overline{D}}_{+}~{}.$
(A.21)

The left– and right–Weyl dimensions and the R–charge $({\cal L},{\cal
R},\text{R})$ (often collectively referred to as Weyl charges) of a general
complex superfield $G$, defined as

$\displaystyle G\rightarrow{\bar{\omega}}^{{\cal L}}\,\omega^{{\cal
R}}\,e^{i\text{R}\kappa}\,G$ (A.22)

identify how it responds to these conformal transformations. Real superfields
are necessarily inert under the R–symmetry. The Weyl and R–charges of the
superfields used in this work can be found in Table 2.

###  A.4 Scale invariant matter actions

#### Scale invariant superspace integrals

Any real bosonic superfield $R$ can be used to form a supersymmetric invariant
by an integral over the full superspace:

$\displaystyle
S_{\text{full\,superspace}}=\int\mathrm{d}^{2}\sigma\mathrm{d}^{2}\theta^{+}\,R=\int\mathrm{d}^{2}\sigma\,{\overline{D}}_{+}D_{+}R|_{+}~{}.$
(A.23)

This action is gauge invariant if $R$ caries no gauge charges and scale
invariant if it has Weyl and R–charges $(+1,0,0)$.

Any chiral Fermi superfield $\Omega$ can be used to form a supersymmetric
invariant by an integral over the chiral superspace:

$\displaystyle
S_{\text{chiral\,superspace}}=\int\mathrm{d}^{2}\sigma\mathrm{d}\theta^{+}\,\Omega+\int\mathrm{d}^{2}\sigma\mathrm{d}{\bar{\theta}}^{+}\,{\overline{\Omega}}=\int\mathrm{d}^{2}\sigma\,\big{[}D_{+}\Omega+{\overline{D}}_{+}{\overline{\Omega}}\,\big{]}|_{+}~{}.$
(A.24)

This is gauge invariant if $\Omega$ carries no gauge charges and conformally
invariant if it has Weyl and R–charges $(+1,+\mbox{$\frac{1}{2}$},+1)$.

#### Chiral superfield action

The gauge interactions of chiral superfields $\Phi^{a}$ and their conjugates
${\overline{\Phi}}^{\underline{a}}$ with Abelian vector multiplets $(V,A)_{i}$
are parameterized by the gauge charges $(q^{i})_{a}$. In order to reduce the
abundance of indices, interpret $q\\!\cdot\\!V$ as the diagonal matrix with on
the diagonal $\sum_{i}(q^{i})_{a}V_{i}$ and interpret $\Phi$ as standing and
${\overline{\Phi}}$ as lying vectors of $\mathrm{N}_{\Phi}$ chiral superfields
and their conjugates, respectively. Their super gauge transformations read

$\displaystyle{\overline{\Phi}}\rightarrow{\overline{\Phi}}\,e^{q\cdot{\overline{\Theta}}}~{},\qquad\Phi\rightarrow
e^{q\cdot\Theta}\,\Phi~{}.$ (A.25)

Their super gauge invariant kinetic action is given by

$\displaystyle
S_{\text{chiral}}=\dfrac{i}{4}\int\mathrm{d}^{2}\sigma\mathrm{d}^{2}\theta^{+}\,\Big{[}{\overline{\Phi}}\,e^{2q\cdot
V}{\overline{\cal D}}\Phi-{\overline{\cal D}}\,{\overline{\Phi}}\,e^{2q\cdot
V}\Phi\Big{]}~{},$ (A.26)

in terms of the super gauge covariant derivatives of the chiral superfields
and their conjugates

$\displaystyle{\overline{\cal
D}}\Phi=\bar{\partial}\Phi+q\\!\cdot\\!\big{(}\bar{\partial}V+iA\big{)}\Phi~{},\qquad{\overline{\cal
D}}{\overline{\Phi}}=\bar{\partial}\,{\overline{\Phi}}+{\overline{\Phi}}\,q\\!\cdot\\!\big{(}\bar{\partial}V-iA\big{)}~{}.$
(A.27)

#### Chiral Fermi superfield action

The gauge and Fermi gauge interactions of the chiral Fermi superfields
$\Lambda^{m}$ and their conjugates ${\overline{\Lambda}}^{\underline{m}}$ with
the Fermi gauge multiplets are parameterised by the gauge charges
$(Q^{i})_{m}$ and holomorphic functions $U^{mI}(\Phi)$. The super gauge and
super fermionic gauge transformations of them read

$\displaystyle{\overline{\Lambda}}\rightarrow\big{(}{\overline{\Lambda}}+{\overline{\Xi}}\\!\cdot\\!{\overline{U}}({\overline{\Phi}})\big{)}\,e^{Q\cdot{\overline{\Theta}}}~{},\qquad\Lambda\rightarrow
e^{Q\cdot\Theta}\,\big{(}\Lambda+U(\Phi)\\!\cdot\\!\Xi\big{)}~{}.$ (A.28)

Here the notation $\big{(}U(\Phi)\\!\cdot\\!\Xi)^{m}=U^{mI}(\Phi)\,\Xi_{I}$ is
employed. The gauge charges of the holomorphic functions $U^{mI}(\Phi)$ are
$(Q^{I})_{m}$ as well. Their super gauge invariant kinetic action is given by

$\displaystyle
S_{\text{Fermi}}=-\dfrac{1}{2}\int\mathrm{d}^{2}\sigma\mathrm{d}^{2}\theta^{+}\,\big{(}{\overline{\Lambda}}+{\overline{\Sigma}}\\!\cdot\\!{\overline{U}}({\overline{\Phi}})\big{)}\,e^{2Q\cdot
V}\big{(}\Lambda+U(\Phi)\\!\cdot\\!\Sigma\big{)}~{}.$ (A.29)

#### FI actions

The Fayet–Ililopoulos (FI) action is given by the chiral superspace integral

$\displaystyle
S_{\text{FI}}=\int\mathrm{d}^{2}\sigma\mathrm{d}\theta^{+}\,W_{\text{FI}}+\text{c.c.}~{},\quad
W_{\text{FI}}=\rho(\Phi)\\!\cdot\\!F+(\kappa(\Phi)\\!\cdot\\!\Upsilon)\Lambda~{},$
(A.30)

where
$(\kappa(\Phi)\\!\cdot\\!\Upsilon)\Lambda=\kappa_{Im}(\Phi)\Upsilon_{I}\Lambda^{m}$
employing holomorphic functions $\rho_{i}(\Phi)$ and $\kappa_{Im}(\Phi)$ of
the chiral superfields $\Phi^{a}$. The lowest components of $\rho(\Phi)$,

$\displaystyle\rho_{i}|_{+}=\mbox{$\frac{1}{2}$}\,r_{i}+i\,\beta_{i}~{},\qquad{\bar{\rho}}_{i}|_{+}=\mbox{$\frac{1}{2}$}\,r_{i}-i\,\beta_{i}~{},$
(A.31)

couple to the auxiliary field $D^{i}$ and the gauge field strength
$F^{i}_{01}$, respectively:

$\displaystyle\int\mathrm{d}\theta^{+}\,\rho(\Phi)\\!\cdot\\!F+\int\mathrm{d}{\bar{\theta}}^{+}\,{\bar{\rho}}({\overline{\Phi}})\\!\cdot\\!{\overline{F}}\supset\mbox{$\frac{1}{2}$}\,r\\!\cdot\\!D-\beta\\!\cdot\\!F_{01}~{},$
(A.32)

where $\supset$ indicates that the expression on the left includes terms given
on the right.

Only when $\rho_{i}(\Phi)$ are super gauge invariant and $\kappa_{IA}(\Phi)$
carry the opposite charges as $\Lambda^{a}$, the FI action is gauge invariant.
This action is only invariant under fermionic gauge transformation if

$\displaystyle\kappa_{Im}(\Phi)U^{mJ}=0~{},$ (A.33)

for all $I,J$. A worldsheet variant of the Green–Schwarz mechanism involves
chiral superfield functions $\rho_{i}(\Phi)$ that transforms as shifts under
super gauge transformations.

###  A.5 None scale invariant actions

In GLSMs also actions are used that are not scale invariant. They involve
parameters of mass dimension one or two in two dimensions. For simplicity all
these parameters are assumed to be equal to $m$ or $|m|^{2}$, depending on
whether these action are chiral or full superspace integrals. Consequently,
conformal invariance is broken by these actions unless these parameters are
send to either $0$ or $\infty$. Here, the conformal limit is taken to be the
strong coupling limit $|m|\rightarrow\infty$. In a more precise analysis one
should study the renormalisation of the theory to understand if a conformal
limit exists [62, 63].

#### Gauge multiplet actions

Abelian vector multiplets $(V,A)_{i}$ have kinetic actions

$\displaystyle
S_{\text{gauge}}=\frac{1}{2|m|^{2}}\,\int\mathrm{d}^{2}\sigma\mathrm{d}^{2}\theta^{+}\,{\overline{F}}F~{}.$
(A.34)

The kinetic terms for Fermi gauge multiplets $\Sigma_{I}$ are given by

$\displaystyle
S_{\text{Fermi\,gauge}}=\frac{1}{2|m|^{2}}\,\int\mathrm{d}^{2}\sigma\mathrm{d}^{2}\theta^{+}\,{\overline{\Upsilon}}\bar{\partial}\Upsilon~{}.$
(A.35)

#### Superpotentials

To introduce gauge invariant superpotential actions, chiral superfields
$\Psi^{A}$ and Fermi superfields $\Gamma^{M}$ are needed. They are given in
Table 2. The super gauge transformations of $\Psi$ read

$\displaystyle{\overline{\Psi}}\rightarrow{\overline{\Psi}}\,e^{\textsf{q}\cdot{\overline{\Theta}}}~{},\qquad\Psi\rightarrow
e^{\textsf{q}\cdot\Theta}\,\Psi~{}.$ (A.36)

The super gauge and super fermionic gauge transformations of $\Gamma$ are
given by

$\displaystyle{\overline{\Gamma}}\rightarrow\big{(}{\overline{\Gamma}}+({\overline{\Xi}}\\!\cdot\\!{\overline{W}}({\overline{\Phi}}))\,{\overline{\Psi}}\big{)}\,e^{\textsf{Q}\cdot{\overline{\Theta}}}~{},\qquad\Gamma\rightarrow
e^{\textsf{Q}\cdot\Theta}\,\big{(}\Gamma+(\Psi\,W(\Phi)\\!\cdot\\!\Xi)\big{)}~{}.$
(A.37)

Here
$\big{[}\Psi\,(W(\Phi)\\!\cdot\\!\Xi)\big{]}^{M}=\Psi^{A}W_{A}{}^{IM}(\Phi)\Xi_{I}$
is parameterised by chiral superfield functions $W_{AI}{}^{M}(\Phi)$.

The superpotential action contains two pieces associated to the target space
geometry and the gauge bundle that supports it:

$\displaystyle
S_{SP}=m\,\int\mathrm{d}^{2}\sigma\mathrm{d}\theta^{+}\,\Big{(}P_{\text{geom}}+P_{\text{bundle}}\Big{)}+\text{c.c.}~{},\quad
P_{\text{geom}}=\Gamma\,P(\Phi)~{},\quad
P_{\text{bundle}}=\Psi\,M(\Phi)\,\Lambda~{}.$ (A.38)

Here, $\Gamma$ and $\Psi$ are interpreted as lying vectors of Fermi multiplets
$\Gamma^{M}$ and chiral multiplets $\Psi^{A}$, respectively; $P(\Phi)$ as a
standing vector of chiral superfield functions $P_{M}(\Phi)$ and $M(\Phi)$ as
a matrix of chiral superfield functions $M_{Am}(\Phi)$. This is gauge
invariant if the functions $P_{M}(\Phi)$ carry the opposite gauge charges as
$\Gamma^{M}$ and $M_{Am}(\Phi)$ the opposite gauge charges as
$\Psi^{A}\Lambda^{m}$. The superpotential action is only invariant under
fermionic gauge transformations if (3.16) holds.

The structure of the superpotential is dictated by a large extend by the Weyl
charges: The R–charge implies that $\Psi^{A}$ and $\Gamma^{M}$ can only appear
linearly in this expression. However, the superpotential is not conformal
invariant, hence the mass parameter $m$ sits out front. This implies that in
the conformal limit the superpotential has to vanish strictly.

To complete the description also kinetic terms need to be added for the field
$\Psi$ and $\Gamma$. The super gauge invariant kinetic action for $\Psi$ is
given by

$\displaystyle
S_{\text{chiral}}=\dfrac{i}{4}\int\mathrm{d}^{2}\sigma\mathrm{d}^{2}\theta^{+}\,\Big{[}{\overline{\Psi}}\,e^{2\textsf{q}\cdot
V}{\overline{\cal D}}\Psi-{\overline{\cal
D}}\,{\overline{\Psi}}\,e^{2\textsf{q}\cdot V}\Psi\Big{]}~{}.$ (A.39)

The super gauge invariant kinetic action for $\Gamma$ is given by

$\displaystyle
S_{\text{Fermi}}=-\dfrac{1}{2}\int\mathrm{d}^{2}\sigma\mathrm{d}^{2}\theta^{+}\,\big{(}{\overline{\Gamma}}+{\overline{\Sigma}}\\!\cdot\\!{\overline{W}}({\overline{\Phi}}){\overline{\Psi}}\big{)}\,e^{2\textsf{Q}\cdot
V}\big{(}\Gamma+\Psi W(\Phi)\\!\cdot\\!\Sigma\big{)}~{}.$ (A.40)

The are both scale invariant.

###  A.6 (0,2) non–linear sigma models

The general action of a $(0,2)$ non–linear sigma model consists of two parts:
an action for the chiral superfields ${\boldsymbol{\Phi}}^{\alpha}$ ,
$\alpha=0,\ldots,3$, and Fermi multiplets ${\boldsymbol{\Lambda}}^{\mu}$ ,
$\mu=1,\ldots,16$. Here the scalar components of the chiral multiplets are
interpreted as the local coordinates of the target space manifold ${\cal M}$
and the fermionic components of the Fermi multiplets as the local coordinates
in a section of the bundle ${\cal V}$ in the same coordinate patch.

#### Torsional non–linear sigma models

The most general conformal $(0,2)$ action of the chiral multiplets

$\displaystyle
S_{\text{n.l.\,chiral}}=\frac{i}{4}\int\mathrm{d}^{2}\sigma\mathrm{d}^{2}\theta^{+}\,\Big{[}K({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})\,\bar{\partial}{\boldsymbol{\Phi}}-\bar{\partial}\boldsymbol{\overline{\Phi}}\,{\overline{K}}({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})\Big{]}~{},$
(A.41)

are parameterised in terms of a lying complex vector function
$K({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})$ with entries
$K_{\alpha}({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})$ and its
conjugate, a standing vector
${\overline{K}}({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})$ with
entries
${\overline{K}}_{\underline{\alpha}}({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})$.
These functions are defined modulo additions

$\displaystyle{\overline{K}}({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})\rightarrow{\overline{K}}({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})+{\bar{k}}(\boldsymbol{\overline{\Phi}})~{},\qquad
K({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})\rightarrow
K({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})+k({\boldsymbol{\Phi}})~{}$
(A.42)

of holomorphic vector functions $k({\boldsymbol{\Phi}})$ and
${\bar{k}}(\boldsymbol{\overline{\Phi}})$, as this would modify the full
superspace integrant by a sum of a chiral superfield and its conjugate which
vanishes. The superfield functions
$K({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})$ and
${\overline{K}}({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})$ can be
thought of as prepotentials for the metric

$\displaystyle
G_{{\underline{\alpha}}\alpha}=\mbox{$\frac{1}{2}$}\Big{(}{\overline{K}}_{{\underline{\alpha}},\alpha}+K_{\alpha,{\underline{\alpha}}}\Big{)}$
(A.43)

and the Kalb–Ramond two–form $B_{2}$

$\displaystyle
B_{{\underline{\alpha}}\alpha}=\mbox{$\frac{1}{2}$}\Big{(}{\overline{K}}_{{\underline{\alpha}},\alpha}-K_{\alpha,{\underline{\alpha}}}\Big{)}~{},\qquad
B_{\alpha\beta}=\mbox{$\frac{1}{2}$}\Big{(}K_{\alpha,\beta}-K_{\beta,\alpha}\Big{)}~{},\qquad
B_{{\underline{\alpha}}{\underline{\beta}}}=\mbox{$\frac{1}{2}$}\Big{(}{\overline{K}}_{{\underline{\alpha}},{\underline{\beta}}}-{\overline{K}}_{\beta,{\underline{\alpha}}}\Big{)}~{},$
(A.44)

combined, as can be seen by working out the kinetic action for the scalar
components of the chiral superfields. The representation of the action for the
scalar components is not unique due to $B_{2}$-field gauge transformations. A
gauge can be chosen such that the components of the $B_{2}$–field with purely
(anti–)holomorphic indices are absent. The non–vanishing components of the
gauge invariant three–form field strength $H_{3}=\mathrm{d}B_{2}$ can also be
expressed in terms of these prepotential functions:

$\displaystyle
H_{\alpha\beta{\underline{\gamma}}}=H_{\beta{\underline{\gamma}}\alpha}=H_{{\underline{\gamma}}\alpha\beta}=K_{\alpha,\beta{\underline{\gamma}}}-K_{\beta,\alpha{\underline{\gamma}}}~{},\qquad
H_{{\underline{\alpha}}{\underline{\beta}}\gamma}=H_{{\underline{\beta}}\gamma{\underline{\alpha}}}=H_{\gamma{\underline{\alpha}}{\underline{\beta}}}={\overline{K}}_{{\underline{\alpha}},{\underline{\beta}}\gamma}-{\overline{K}}_{{\underline{\beta}},{\underline{\alpha}}\gamma}~{}.$
(A.45)

if some of these components are non–zero the manifold possesses torsion.

#### Chiral superfield interactions with Fermi multiplets

The most general Weyl invariant action of Fermi multiplets is given by

$\displaystyle
S_{\text{n.l.\,Fermi}}=-\frac{1}{2}\int\mathrm{d}^{2}\sigma\mathrm{d}^{2}\theta^{+}\,\Big{\\{}\boldsymbol{\overline{\Lambda}}\,N({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})\,{\boldsymbol{\Lambda}}+\frac{1}{2}\,{\boldsymbol{\Lambda}}^{T}\,n({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})\,{\boldsymbol{\Lambda}}+\frac{1}{2}\,\boldsymbol{\overline{\Lambda}}\,{\bar{n}}({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})\,\boldsymbol{\overline{\Lambda}}^{T}~{},\Big{\\}}$
(A.46)

parameterised by an Hermitean matrix
$N({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})$ with entries
$N_{{\underline{\mu}}\nu}({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})$
assumed to be invertible and a complex anti–symmetric matrix
$n({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})$ with holomorphic indices
$n_{\mu\nu}({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})$ and its
conjugate ${\bar{n}}({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})$ with
entries
${\bar{n}}_{{\underline{\mu}}{\underline{\nu}}}({\boldsymbol{\Phi}},\boldsymbol{\overline{\Phi}})$.
They can be thought of as the prepotentials for the target space gauge fields

$\displaystyle A_{\alpha}(N)=N^{-1}N_{,\alpha}~{},\qquad
A_{\underline{\alpha}}(N)=N^{-1}N_{,{\underline{\alpha}}}~{},\qquad
A_{\underline{\alpha}}(n)=n_{,{\underline{\alpha}}}\qquad
A_{\alpha}({\bar{n}})={\bar{n}}_{,\alpha}~{}.$ (A.47)

#### From (0,2) GLSMs to (0,2) NLSMs

By integrating out the gauge superfields $(0,2)$ GLSMs can be related to
$(0,2)$ NLSMs. In particular, the equations of motion of $A$ lead to the
constraints (3.27) in the conformal limit. Then, by applying partial
integrations on the derivative $\bar{\partial}$ in the remaining ($A$
independent) terms in the FI–interaction (A.30) and combining them with the
remaining kinetic terms of the chiral multiplets (A.26), these actions can be
cast in the form of the NLSM action (A.41) with the prepotentials

$\displaystyle K_{a}=\big{(}{\overline{\Phi}}\,e^{2q\cdot
V}\big{)}_{a}+2\,\rho_{,a}\\!\cdot\\!V~{},\qquad{\overline{K}}_{\underline{a}}=\big{(}e^{2q\cdot
V}\Phi\big{)}_{\underline{a}}+2\,{\bar{\rho}}_{,{\underline{a}}}\\!\cdot\\!V~{}.$
(A.48)

To see if these prepotentials for the metric and the B–field possess torsion,
we compute the anti–symmetrised derivative

$\displaystyle K_{[a,b]}=K_{a,b}-K_{b,a}=\big{(}{\overline{\Phi}}\,e^{2q\cdot
V}q\\!\cdot\\!V_{,[b}\big{)}{}_{a]}+2\,\rho_{,[a}\\!\cdot\\!V_{,b]}~{}.$
(A.49)

This expression can be simplified by taking the partial derivative w.r.t.
$\Phi^{a}$ of equation (3.27) and after that contracting it with $V_{i,b}$.
This gives

$\displaystyle\big{(}{\overline{\Phi}}\,e^{2q\cdot
V}q\\!\cdot\\!V_{,b}\big{)}{}_{a}+2{\overline{\Phi}}\,e^{2q\cdot
V}(q\\!\cdot\\!V_{,a})(q\\!\cdot\\!V_{,b})\Phi=\rho_{,a}\\!\cdot\\!V_{,b}~{},$
(A.50)

hence anti–symmetrised:

$\displaystyle\big{(}{\overline{\Phi}}\,e^{q\cdot
V}q\\!\cdot\\!V_{,[b}\big{)}{}_{a]}=\rho_{,[a}\\!\cdot\\!V_{,b]}~{}.$ (A.51)

From which in general it may be concluded, that there will be torsion if the
FI–functions $\rho_{i}(\Phi)$ are not constant

$\displaystyle K_{[a,b]}=3\,\rho_{,[a}\\!\cdot\\!V_{,b]}~{}.$ (A.52)

From this the three–form $H$ expression (3.28) follows immediately.

##  Appendix B Anomalies in two dimensional GLSMs

###  B.1 Chiral anomaly

Let $\psi$ be a Dirac fermion in two dimensions and ${\bar{\psi}}$ its
conjugate. Consider the chiral transformation

$\displaystyle\psi\rightarrow
e^{i\alpha\,\frac{\mathbbm{1}+{\tilde{\gamma}}}{2}}\,\psi~{},\qquad{\bar{\psi}}\rightarrow{\bar{\psi}}\,e^{-i\alpha\,\frac{\mathbbm{1}+{\tilde{\gamma}}}{2}}~{}.$
(B.1)

Here ${\tilde{\gamma}}=\gamma^{0}\gamma^{1}$ is the chirality operator in two
dimensions satisfying ${\tilde{\gamma}}^{2}=\mathbbm{1}$. The anti–symmetrised
product product of two gamma matrices is proportion to this operator:

$\displaystyle\gamma^{\mu\nu}=\mbox{$\frac{1}{2}$}\,[\gamma^{\mu},\gamma^{\nu}]=\epsilon^{\mu\nu}\,{\tilde{\gamma}}~{},$
(B.2)

where $\epsilon^{\mu\nu}=-\epsilon^{\nu\mu}$ is the anti–symmetric epsilon
tensor in two with the normalisation $\epsilon^{01}=1$. The Dirac operator of
this fermion is assumed to couple chirally to a gauge field $A_{\mu}$:

$\displaystyle D/\hskip 0.86108pt=\partial/\hskip
0.86108pt+i\,A/\,\frac{\mathbbm{1}+{\tilde{\gamma}}}{2}~{},$ (B.3)

where $A/\hskip 0.86108pt=\gamma^{\mu}A_{\mu}$ as usual. Note that

$\displaystyle D/\hskip
0.86108pt^{2}=D^{2}+\frac{i}{2}\,{\tilde{\gamma}}\,\epsilon^{\mu\nu}F_{\mu\nu}~{},$
(B.4)

where $iF_{\mu\nu}=[D_{\mu},D_{\nu}]$ is the invariant gauge field strength or
expressed as a two form

$\displaystyle
F_{2}=\mbox{$\frac{1}{2}$}\,F_{\mu\nu}\mathrm{d}\sigma^{\mu}\mathrm{d}\sigma^{\nu}=\mbox{$\frac{1}{2}$}\,\epsilon^{\mu\nu}F_{\mu\nu}\,\mathrm{d}^{2}\sigma=F_{01}\,\mathrm{d}^{2}\sigma~{}.$
(B.5)

If the path integral measure

$\displaystyle{\cal D}\psi{\cal D}{\bar{\psi}}\rightarrow{\cal D}\psi{\cal
D}{\bar{\psi}}\,e^{i{\cal A}_{\text{chiral}}}$ (B.6)

is not invariant under this transformation, the chiral transformation is said
to be anomalous. The anomaly can be expressed as the trace

$\displaystyle{\cal A}_{\text{chiral}}=\mbox{Tr}[\alpha\,{\tilde{\gamma}}]$
(B.7)

over both the full Hilbert and spinor space. This trace needs to be
regularised. In case of anomalies a standard procedure is to use Fujikawa’s
regularisation

$\displaystyle{\cal
A}_{\text{chiral}}=\int\mathrm{d}^{2}\sigma\,\alpha\,\text{tr}\langle
x|{\tilde{\gamma}}\,e^{D/^{2}/M^{2}}|x\rangle~{},$ (B.8)

where $M$ is a regulator mass taken to be infinitely large. Using a plane wave
expansion with a momentum variable $p$, scaling it as $p\rightarrow M\,p$ and
keeping only the leading terms this expression can be evaluated to

$\displaystyle{\cal
A}_{\text{chiral}}=\int\mathrm{d}^{2}\sigma\,\alpha\int\frac{\mathrm{d}^{2}p}{(2\pi)^{2}}e^{-p^{2}}\text{tr}\Big{[}{\tilde{\gamma}}\,\mbox{$\frac{i}{2}$}\,{\tilde{\gamma}}\,\epsilon^{\mu\nu}F_{\mu\nu}\Big{]}$
(B.9)

where all the $M$ dependence dropped out (after taking the limit
$M\rightarrow\infty$). Using the Gaussian integral

$\displaystyle\int\mathrm{d}^{2}p\,e^{-p^{2}}=\pi~{},$ (B.10)

the chiral anomaly can be expressed as

$\displaystyle{\cal
A}_{\text{chiral}}=\int\mbox{$\frac{i}{2}$}\,\alpha\,\frac{F_{2}}{2\pi}~{}.$
(B.11)

###  B.2 Super gauge anomalies

The result for the chiral anomaly above can be used for chiral gauge theories
as well where then the parameter $\alpha$ is interpreted as the gauge
parameter of a $U(1)$ symmetry. For left–moving charged fermion the result can
immediately be taken over, while for a right–moving fermion the expression
will have an additional minus sign. If we have a set of left– and right–moving
fermions with charges $Q_{i}$ and $q_{i}$ under a number of $U(1)$ gauge
symmetries, the result generalises to

$\displaystyle{\cal
A}_{\text{gauge}}=\int\mbox{$\frac{i}{2}$}\,\alpha^{i}\,\mathcal{A}_{ij}\,\frac{F^{j}_{2}}{2\pi}~{},$
(B.12)

where the anomaly matrix is given by

$\displaystyle\mathcal{A}_{ij}=Q_{i}\cdot Q_{j}-q_{i}\cdot q_{j}~{}.$ (B.13)

Here the dot product indicates the sum over all charged left and right
fermions present in the theory. Assuming the existence of a supersymmetric
regulator, the general form of super gauge anomalies in two dimensions can be
written as

$\displaystyle
S_{\text{anom}}=\int\mathrm{d}^{2}\sigma\mathrm{d}\theta^{+}\,\frac{1}{4\pi}\sum_{i,j}\mathcal{A}_{ij}\,\Theta^{i}F^{j}+\int\mathrm{d}^{2}\sigma\mathrm{d}{\bar{\theta}}^{+}\,\frac{1}{4\pi}\sum_{i,j}\mathcal{A}_{ij}\,{\overline{\Theta}}^{i}{\overline{F}}^{j}~{}.$
(B.14)

##  Appendix C Charge matrices

In Section 4 a number of so–called charge matrices are used to perform certain
computations. In a given patch of a given phase of the GLSM a number of
charged superfields are necessarily non–zero. Their charge matrices are given
by:

$\displaystyle{\cal
Q}_{(O)}=\begin{pmatrix}~{}{\,\mbox{-}}1~{}&~{}0~{}&~{}0~{}\\\
0&{\,\mbox{-}}1&0\\\ 0&0&{\,\mbox{-}}1\end{pmatrix}~{},\qquad{\cal
Q}_{(S)}=\begin{pmatrix}~{}0~{}&~{}\mbox{$\frac{1}{2}$}~{}&~{}\mbox{$\frac{1}{2}$}~{}\\\
\mbox{$\frac{1}{2}$}&0&\mbox{$\frac{1}{2}$}\\\
\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}&0\end{pmatrix}~{},$ (C.1a)
$\displaystyle{\cal
Q}_{(11)}=\begin{pmatrix}\,{\,\mbox{-}}1~{}&~{}0~{}&~{}0~{}\\\
\mbox{$\frac{1}{2}$}&0&\mbox{$\frac{1}{2}$}\\\
\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}&0\end{pmatrix}~{},\quad{\cal
Q}_{(12)}=\begin{pmatrix}~{}{\,\mbox{-}}1~{}&~{}0~{}&~{}0~{}\\\
0&\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}\\\
\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}&0\end{pmatrix}~{},\quad{\cal
Q}_{(13)}=\begin{pmatrix}~{}{\,\mbox{-}}1~{}&~{}0~{}&~{}0~{}\\\
0&\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}\\\
\mbox{$\frac{1}{2}$}&0&\mbox{$\frac{1}{2}$}\end{pmatrix}~{},$ (C.1b)
$\displaystyle{\cal
Q}_{(21)}=\begin{pmatrix}\mbox{$\frac{1}{2}$}&0&\mbox{$\frac{1}{2}$}\\\
~{}0~{}&~{}{\,\mbox{-}}1~{}&~{}0~{}\\\
\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}&0\end{pmatrix}~{},\quad{\cal
Q}_{(22)}=\begin{pmatrix}0&\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}\\\
~{}0~{}&~{}{\,\mbox{-}}1~{}&~{}0~{}\\\
\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}&0\end{pmatrix}~{},\quad{\cal
Q}_{(23)}=\begin{pmatrix}0&\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}\\\
~{}0~{}&~{}{\,\mbox{-}}1~{}&~{}0~{}\\\
\mbox{$\frac{1}{2}$}&0&\mbox{$\frac{1}{2}$}\end{pmatrix}~{},$ (C.1c)
$\displaystyle{\cal
Q}_{(31)}=\begin{pmatrix}\mbox{$\frac{1}{2}$}&0&\mbox{$\frac{1}{2}$}\\\
\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}&0\\\
~{}0~{}&~{}0~{}&~{}{\,\mbox{-}}1~{}\end{pmatrix}~{},\quad{\cal
Q}_{(32)}=\begin{pmatrix}0&\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}\\\
\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}&0\\\
~{}0~{}&~{}0~{}&~{}{\,\mbox{-}}1~{}\end{pmatrix}~{},\quad{\cal
Q}_{(33)}=\begin{pmatrix}0&\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}\\\
\mbox{$\frac{1}{2}$}&0&\mbox{$\frac{1}{2}$}\\\
~{}0~{}&~{}0~{}&~{}{\,\mbox{-}}1~{}\end{pmatrix}~{}.$ (C.1d)

Their transposed inverse are:

$\displaystyle{\cal
Q}_{(O)}^{-T}=\begin{pmatrix}~{}{\,\mbox{-}}1~{}&~{}0~{}&~{}0~{}\\\
0&{\,\mbox{-}}1&0\\\ 0&0&{\,\mbox{-}}1\end{pmatrix}~{},\qquad{\cal
Q}_{(S)}^{-T}=\begin{pmatrix}~{}{\,\mbox{-}}1~{}&~{}1~{}&~{}1~{}\\\
1&{\,\mbox{-}}1&1\\\ 1&1&{\,\mbox{-}}1\end{pmatrix}~{},$ (C.2a)
$\displaystyle{\cal
Q}_{(11)}^{-T}=\begin{pmatrix}\,{\,\mbox{-}}1~{}&~{}1~{}&~{}1~{}\\\ 0&0&2\\\
0&2&0\end{pmatrix}~{},\quad{\cal
Q}_{(12)}^{-T}=\begin{pmatrix}\,{\,\mbox{-}}1~{}&~{}1~{}&\,{\,\mbox{-}}1~{}\\\
0&0&2\\\ 0&2&{\,\mbox{-}}2\end{pmatrix}~{},\quad{\cal
Q}_{(13)}^{-T}=\begin{pmatrix}\,{\,\mbox{-}}1~{}&\,{\,\mbox{-}}1~{}&~{}1~{}\\\
0&2&0\\\ 0&{\,\mbox{-}}2&2\end{pmatrix}~{},$ (C.2b) $\displaystyle{\cal
Q}_{(21)}^{-T}=\begin{pmatrix}0&0&2\\\
~{}1~{}&\,{\,\mbox{-}}1~{}&\,{\,\mbox{-}}1~{}\\\
2&0&{\,\mbox{-}}2\end{pmatrix}~{},\quad{\cal
Q}_{(22)}^{-T}=\begin{pmatrix}0&0&2\\\ ~{}1~{}&\,{\,\mbox{-}}1~{}&~{}1~{}\\\
2&0&0\end{pmatrix}~{},\quad{\cal
Q}_{(23)}^{-T}=\begin{pmatrix}{\,\mbox{-}}2&0&2\\\
\,{\,\mbox{-}}1~{}&\,{\,\mbox{-}}1~{}&~{}1~{}\\\ 2&0&0\end{pmatrix}~{},$
(C.2c) $\displaystyle{\cal Q}_{(31)}^{-T}=\begin{pmatrix}2&{\,\mbox{-}}2&0\\\
0&2&0\\\
~{}1~{}&\,{\,\mbox{-}}1~{}&\,{\,\mbox{-}}1~{}\end{pmatrix}~{},\quad{\cal
Q}_{(32)}^{-T}=\begin{pmatrix}{\,\mbox{-}}2&2&0\\\ 2&0&0\\\
\,{\,\mbox{-}}1~{}&~{}1~{}&\,{\,\mbox{-}}1~{}\end{pmatrix}~{},\quad{\cal
Q}_{(33)}^{-T}=\begin{pmatrix}0&2&0\\\ 2&0&0\\\
~{}1~{}&~{}1~{}&\,{\,\mbox{-}}1~{}\end{pmatrix}~{}.$ (C.2d)

The charge matrices associated to the superfields that define a given patch
read:

$\displaystyle\widetilde{{\cal
Q}}_{(O)}=\begin{pmatrix}~{}0~{}&~{}\mbox{$\frac{1}{2}$}~{}&~{}\mbox{$\frac{1}{2}$}~{}\\\
\mbox{$\frac{1}{2}$}&0&\mbox{$\frac{1}{2}$}\\\
\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}&0\end{pmatrix}~{},\qquad\widetilde{{\cal
Q}}_{(S)}=\begin{pmatrix}~{}{\,\mbox{-}}1~{}&~{}0~{}&~{}0~{}\\\
0&{\,\mbox{-}}1&0\\\ 0&0&{\,\mbox{-}}1\end{pmatrix}~{},$ (C.3a)
$\displaystyle\widetilde{{\cal
Q}}_{(11)}=\begin{pmatrix}0&\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}\\\
~{}0~{}&~{}{\,\mbox{-}}1~{}&~{}0~{}\\\
0&0&{\,\mbox{-}}1\end{pmatrix}~{},\quad\widetilde{{\cal
Q}}_{(12)}=\begin{pmatrix}\mbox{$\frac{1}{2}$}&0&\mbox{$\frac{1}{2}$}\\\
~{}0~{}&~{}{\,\mbox{-}}1~{}&~{}0~{}\\\
0&0&{\,\mbox{-}}1\end{pmatrix}~{},\quad\widetilde{{\cal
Q}}_{(13)}=\begin{pmatrix}\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}&0\\\
~{}0~{}&~{}{\,\mbox{-}}1~{}&~{}0~{}\\\ 0&0&{\,\mbox{-}}1\end{pmatrix}~{},$
(C.3b) $\displaystyle\widetilde{{\cal
Q}}_{(21)}=\begin{pmatrix}~{}{\,\mbox{-}}1~{}&~{}0~{}&~{}0~{}\\\
0&\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}\\\
0&0&{\,\mbox{-}}1\end{pmatrix}~{},\quad\widetilde{{\cal
Q}}_{(22)}=\begin{pmatrix}~{}{\,\mbox{-}}1~{}&~{}0~{}&~{}0~{}\\\
\mbox{$\frac{1}{2}$}&0&\mbox{$\frac{1}{2}$}\\\
0&0&{\,\mbox{-}}1\end{pmatrix}~{},\quad\widetilde{{\cal
Q}}_{(22)}=\begin{pmatrix}~{}{\,\mbox{-}}1~{}&~{}0~{}&~{}0~{}\\\
\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}&0\\\
0&0&{\,\mbox{-}}1\end{pmatrix}~{},$ (C.3c) $\displaystyle\widetilde{{\cal
Q}}_{(31)}=\begin{pmatrix}~{}{\,\mbox{-}}1~{}&~{}0~{}&~{}0~{}\\\
~{}0~{}&~{}{\,\mbox{-}}1~{}&~{}0~{}\\\
0&\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}\\\
\end{pmatrix}~{},\quad\widetilde{{\cal
Q}}_{(32)}=\begin{pmatrix}~{}{\,\mbox{-}}1~{}&~{}0~{}&~{}0~{}\\\
~{}0~{}&~{}{\,\mbox{-}}1~{}&~{}0~{}\\\
\mbox{$\frac{1}{2}$}&0&\mbox{$\frac{1}{2}$}\\\
\end{pmatrix}~{},\widetilde{{\cal
Q}}_{(33)}=\begin{pmatrix}~{}{\,\mbox{-}}1~{}&~{}0~{}&~{}0~{}\\\
~{}0~{}&~{}{\,\mbox{-}}1~{}&~{}0~{}\\\
\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}&0\\\ \end{pmatrix}~{}.$ (C.3d)

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