\boldsymbol{s}_{mn}&-{\boldsymbol{t}^{m}}_{n}\end{pmatrix}.$ (A.10)

From the $SO(5,5)$ algebra, we can derive the following commutation relations

$\displaystyle\left[\boldsymbol{d},\boldsymbol{d}\right]$ $\displaystyle=$
$\displaystyle
0,\qquad\qquad\quad\left[\boldsymbol{d},{\boldsymbol{\tau}^{m}}_{n}\right]\ =\
0,$ (A.11) $\displaystyle\left[\boldsymbol{d},\boldsymbol{s}_{mn}\right]$
$\displaystyle=$
$\displaystyle-4\boldsymbol{s}_{mn},\quad\quad\,\,\,\left[\boldsymbol{d},\boldsymbol{h}^{mn}\right]\
=\ +4\boldsymbol{h}^{mn},$ (A.12)
$\displaystyle\left[\boldsymbol{s}_{mn},\boldsymbol{s}_{pq}\right]$
$\displaystyle=$ $\displaystyle
0,\qquad\,\qquad\left[\boldsymbol{h}^{mn},\boldsymbol{h}^{pq}\right]\ =\ 0,$
(A.13)
$\displaystyle\left[{\boldsymbol{\tau}^{m}}_{n},{\boldsymbol{\tau}^{p}}_{q}\right]$
$\displaystyle=$ $\displaystyle
2(\delta_{n}^{p}{\boldsymbol{\tau}^{m}}_{q}-\delta_{q}^{m}{\boldsymbol{\tau}^{p}}_{n}),$
(A.14)
$\displaystyle\left[{\boldsymbol{\tau}^{m}}_{n},\boldsymbol{s}_{pq}\right]$
$\displaystyle=$ $\displaystyle
2(\delta^{m}_{q}\boldsymbol{s}_{np}-\delta^{m}_{p}\boldsymbol{s}_{nq}+\frac{2}{5}\delta^{m}_{n}\boldsymbol{s}_{pq}),$
(A.15)
$\displaystyle\left[{\boldsymbol{\tau}^{m}}_{n},\boldsymbol{h}^{pq}\right]$
$\displaystyle=$ $\displaystyle
2(\delta^{p}_{n}\boldsymbol{h}^{mq}-\delta^{q}_{n}\boldsymbol{h}^{mp}-\frac{2}{5}\delta^{m}_{n}\boldsymbol{h}^{pq}),$
(A.16) $\displaystyle\left[\boldsymbol{s}_{mn},\boldsymbol{h}^{pq}\right]$
$\displaystyle=$ $\displaystyle
2(\delta^{p}_{m}{\boldsymbol{\tau}^{q}}_{n}-\delta^{q}_{m}{\boldsymbol{\tau}^{p}}_{n}-\delta^{p}_{n}{\boldsymbol{\tau}^{q}}_{m}+\delta^{q}_{n}{\boldsymbol{\tau}^{p}}_{m})-\frac{2}{5}\,\boldsymbol{d}\,\delta^{[p}_{m}\delta^{q]}_{n}$
(A.17) $\displaystyle=$ $\displaystyle
2(\delta^{p}_{m}{\boldsymbol{t}^{q}}_{n}-\delta^{q}_{m}{\boldsymbol{t}^{p}}_{n}-\delta^{p}_{n}{\boldsymbol{t}^{q}}_{m}+\delta^{q}_{n}{\boldsymbol{t}^{p}}_{m})$

in which
$\delta^{[p}_{m}\delta^{q]}_{n}=\frac{1}{2}(\delta^{p}_{m}\delta^{q}_{n}-\delta^{q}_{m}\delta^{p}_{n})$.
In the second line of (A.17), we have used (A.9) to rewrite the commutation
relation in terms of the $GL(5)$ generators. Note also that (A.14) is the
$SL(5)$ algebra. It follows that the $GL(5)$ branching rule for adjoint
representation is given by

$\underbrace{\mathbf{45}}_{\boldsymbol{t}_{MN}}\ \rightarrow\
\underbrace{\mathbf{1}^{0}}_{\boldsymbol{d}}\,\oplus\,\underbrace{\mathbf{24}^{0}}_{{\boldsymbol{\tau}^{m}}_{n}}\,\oplus\,\underbrace{\mathbf{10}^{-4}}_{\boldsymbol{s}_{mn}}\,\oplus\,\underbrace{\overline{\mathbf{10}}^{+4}}_{\boldsymbol{h}^{mn}}$
(A.18)

where the $\mathbb{R}^{+}$ weights are determined from the relations (A.11)
and (A.12).

###  A.3 Spinor

Unlike the vector, decomposition of $SO(5,5)$ spinor representation under
$GL(5)$ is not straightforward. To describe this branching rule, we begin with
the following two sets of $USp(4)\sim SO(5)$ gamma matrices satisfying

$\displaystyle\left\\{\gamma_{a},\gamma_{b}\right\\}$ $\displaystyle=$
$\displaystyle 2\delta_{ab}\mathds{1}_{4},\qquad\delta_{ab}\ =\
\text{diag}(+,+,+,+,+),$ (A.19)
$\displaystyle\left\\{\gamma_{\dot{a}},\gamma_{\dot{b}}\right\\}$
$\displaystyle=$ $\displaystyle
2\delta_{\dot{a}\dot{b}}\mathds{1}_{4},\qquad\delta_{\dot{a}\dot{b}}\ =\
\text{diag}(+,+,+,+,+)$ (A.20)

where $a,b,\ldots=1,\ldots,5$ and
$\dot{a},\dot{b},\ldots=\dot{1},\ldots,\dot{5}$ are two sets of $SO(5)$ vector
indices raised and lowered by $\delta_{ab}$ and $\delta_{\dot{a}\dot{b}}$,
respectively. For both sets of $SO(5)$ gamma matrices, we will use the
following explicit representation

$\displaystyle\gamma_{1}$ $\displaystyle=$
$\displaystyle-\sigma_{2}\otimes\sigma_{2},\quad\,\gamma_{2}\ =\
\mathds{1}_{2}\otimes\sigma_{1},\quad\gamma_{3}\ =\
\mathds{1}_{2}\otimes\sigma_{3},$ $\displaystyle\gamma_{4}$ $\displaystyle=$
$\displaystyle\sigma_{1}\otimes\sigma_{2},\qquad\gamma_{5}\ =\
\sigma_{3}\otimes\sigma_{2}$ (A.21)

where $\\{\sigma_{1},\sigma_{2},\sigma_{3}\\}$ are the usual Pauli matrices
given by

$\sigma_{1}\ =\ \begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\qquad\sigma_{2}\ =\
\begin{pmatrix}0&-i\\\ i&0\end{pmatrix},\qquad\sigma_{3}\ =\
\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$ (A.22)

Each gamma matrix is a $4\times 4$ matrix with the index structure
${(\gamma_{a})_{\alpha}}^{\beta}$ and
${(\gamma_{\dot{a}})_{\dot{\alpha}}}^{\dot{\beta}}$. Indices
$\alpha,\beta,\ldots=1,\ldots,4$ and
$\dot{\alpha},\dot{\beta},\ldots=\dot{1},\ldots,\dot{4}$ are two sets of
$USp(4)$ fundamental or $SO(5)$ spinor indices raised and lowered through two
identical $USp(4)$ symplectic forms

$\Omega_{\alpha\beta}\ =\ \Omega_{\dot{\alpha}\dot{\beta}}\ =\
\mathds{1}_{2}\otimes i\sigma_{2}$ (A.23)

satisfying

$\Omega_{\beta\alpha}\ =\ -\Omega_{\alpha\beta},\quad\Omega^{\alpha\beta}\ =\
(\Omega_{\alpha\beta})^{*},\quad\Omega_{\alpha\beta}\Omega^{\beta\gamma}\ =\
-\delta^{\gamma}_{\alpha},$ (A.24)

and similarly for $\Omega_{\dot{\alpha}\dot{\beta}}$. Therefore, the matrices
$(\gamma_{a})_{\alpha\beta}=\Omega_{\beta\gamma}{(\gamma_{a})_{\alpha}}^{\gamma}$
satisfy

$(\gamma_{a})_{\beta\alpha}\ =\
-(\gamma_{a})_{\alpha\beta},\quad\Omega^{\alpha\beta}(\gamma_{a})_{\alpha\beta}\
=\ 0,\quad((\gamma_{a})_{\alpha\beta})^{*}\ =\
\Omega^{\alpha\gamma}\Omega^{\beta\delta}(\gamma_{a})_{\gamma\delta},$ (A.25)

and similarly for
$(\gamma_{\dot{a}})_{\dot{\alpha}\dot{\beta}}=\Omega_{\dot{\beta}\dot{\gamma}}{(\gamma_{\dot{a}})_{\dot{\alpha}}}^{\dot{\gamma}}$.

The 32-dimensional $SO(5,5)$ gamma matrices,
$\tilde{\mathbf{\Gamma}}_{\underline{A}}=(\tilde{\mathbf{\Gamma}}_{a},\tilde{\mathbf{\Gamma}}_{\dot{a}})$
with $\underline{A}=1,...,10$, satisfying the Clifford algebra

$\left\\{\tilde{\mathbf{\Gamma}}_{\underline{A}},\tilde{\mathbf{\Gamma}}_{\underline{B}}\right\\}\
=\ 2\eta_{\underline{A}\underline{B}}\mathds{1}_{32}$ (A.26)

with $\eta_{\underline{A}\underline{B}}=\textrm{diag}(+,+,+,+,+,-,-,-,-,-)$
can be constructed from the $SO(5)$ gamma matrices as

$\tilde{\mathbf{\Gamma}}_{a}\ =\
(\sigma_{1}\otimes\mathds{1}_{4}\otimes\gamma_{a})\qquad\text{and}\qquad\tilde{\mathbf{\Gamma}}_{\dot{a}}\
=\ (i\sigma_{2}\otimes\gamma_{\dot{a}}\otimes\mathds{1}_{4}).$ (A.27)

The matrices $\mathds{A}$, $\mathds{B}$, and $\mathds{C}$, which respectively
realize Dirac, complex, and charge conjugation, have the following defining
properties

$(\tilde{\mathbf{\Gamma}}_{\underline{A}})^{\dagger}\ =\
-\mathds{A}\tilde{\mathbf{\Gamma}}_{\underline{A}}\mathds{A}^{-1},\qquad(\tilde{\mathbf{\Gamma}}_{\underline{A}})^{*}\
=\
-\mathds{B}\tilde{\mathbf{\Gamma}}_{\underline{A}}\mathds{B}^{-1},\qquad(\tilde{\mathbf{\Gamma}}_{\underline{A}})^{T}\
=\ -\mathds{C}\tilde{\mathbf{\Gamma}}_{\underline{A}}\mathds{C}^{-1}.$ (A.28)

In our explicit representation, the matrices $\mathds{A}$ and $\mathds{B}$ are
given by

$\mathds{A}\ =\
\tilde{\mathbf{\Gamma}}_{6}\tilde{\mathbf{\Gamma}}_{7}\tilde{\mathbf{\Gamma}}_{8}\tilde{\mathbf{\Gamma}}_{9}\tilde{\mathbf{\Gamma}}_{10}\qquad\text{and}\qquad\mathds{B}\
=\ \mathds{1}_{2}\otimes\Omega\otimes\Omega\,.$ (A.29)

The charge conjugation matrix $\mathds{C}$ can be obtained from $\mathds{A}$
and $\mathds{B}$ through the relation

$\mathds{C}\ =\ \mathds{B}^{T}\mathds{A}\,.$ (A.30)

The $SO(5,5)$ chirality matrix takes the following diagonal form

$\tilde{\mathbf{\Gamma}}_{*}\ =\
\tilde{\mathbf{\Gamma}}_{1}...\tilde{\mathbf{\Gamma}}_{10}\ =\
\text{diag}(\mathds{1}_{16},-\mathds{1}_{16}).$ (A.31)

Therefore, as seen from the definition (A.27), $SO(5,5)$ gamma matrices are
chirally decomposed as

$\tilde{\mathbf{\Gamma}}_{a}\ =\
\begin{pmatrix}0&\mathds{1}_{4}\otimes\gamma_{a}\\\
\mathds{1}_{4}\otimes\gamma_{a}&0\end{pmatrix}\qquad\text{and}\qquad\tilde{\mathbf{\Gamma}}_{\dot{a}}\
=\ \begin{pmatrix}0&\gamma_{\dot{a}}\otimes\mathds{1}_{4}\\\
-\gamma_{\dot{a}}\otimes\mathds{1}_{4}&0\end{pmatrix}.$ (A.32)

Elements of the $16\times 16$ $SO(5)$ gamma matrices
$\mathds{1}_{4}\otimes\gamma_{a}$ and $\gamma_{\dot{a}}\otimes\mathds{1}_{4}$
are denoted by the following index structure

$\mathds{1}_{4}\otimes\gamma_{a}\ =\
{(\gamma_{a})_{\alpha\dot{\alpha}}}^{\beta\dot{\beta}}\ =\
{(\gamma_{a})_{\alpha}}^{\beta}\delta_{\dot{\alpha}}^{\dot{\beta}}\quad\text{and}\quad\gamma_{\dot{a}}\otimes\mathds{1}_{4}\
=\ {(\gamma_{\dot{a}})_{\alpha\dot{\alpha}}}^{\beta\dot{\beta}}\ =\
\delta_{\alpha}^{\beta}{(\gamma_{\dot{a}})_{\dot{\alpha}}}^{\dot{\beta}}\,.$
(A.33)

On the other hand, we can split a $32$-dimendional $SO(5,5)$ spinor index into
$\mathcal{A}=(A,A^{\prime})$ for $A,B,...=1,..,16$ and
$A^{\prime},B^{\prime},...=17,...,32$ so that

${(\tilde{\mathbf{\Gamma}}_{\underline{A}})_{\mathcal{A}}}^{\mathcal{B}}\ =\
\begin{pmatrix}0&{(\Gamma_{\underline{A}})_{A}}^{B^{\prime}}\\\
{(\Gamma_{\underline{A}})_{A^{\prime}}}^{B}&0\end{pmatrix}.$ (A.34)

We can then relate these two decompositions of $SO(5,5)$ spinor indices into
$A,A^{\prime}$ and a pair of $USp(4)$ indices $(\alpha\dot{\alpha})$ by using
the following transformation matrices

$\displaystyle\boldsymbol{p}_{A}^{\alpha\dot{\alpha}}$ $\displaystyle=$
$\displaystyle\delta_{A}^{\alpha}\delta_{1}^{\dot{\alpha}}+\delta_{A}^{\alpha+4}\delta_{2}^{\dot{\alpha}}+\delta_{A}^{\alpha+8}\delta_{3}^{\dot{\alpha}}+\delta_{A}^{\alpha+12}\delta_{4}^{\dot{\alpha}},$
$\displaystyle\boldsymbol{p}^{A}_{\alpha\dot{\alpha}}$ $\displaystyle=$
$\displaystyle\delta^{A}_{\alpha}\delta^{1}_{\dot{\alpha}}+\delta^{A}_{\alpha+4}\delta^{2}_{\dot{\alpha}}+\delta^{A}_{\alpha+8}\delta^{3}_{\dot{\alpha}}+\delta^{A}_{\alpha+12}\delta^{4}_{\dot{\alpha}}\,.$
(A.35)

These matrices satisfy the relations

$\boldsymbol{p}_{A}^{\alpha\dot{\alpha}}\boldsymbol{p}^{B}_{\alpha\dot{\alpha}}\
=\
\delta_{A}^{B}\qquad\text{and}\qquad\boldsymbol{p}_{A}^{\alpha\dot{\alpha}}\boldsymbol{p}^{A}_{\beta\dot{\beta}}\
=\ \delta^{\alpha}_{\beta}\delta^{\dot{\alpha}}_{\dot{\beta}}\,.$ (A.36)

We can now write chiral $SO(5,5)$ gamma matrices in terms of the $SO(5)$ ones
as

$\displaystyle{(\Gamma_{a})_{A}}^{B^{\prime}}$ $\displaystyle=$
$\displaystyle\boldsymbol{p}_{A}^{\alpha\dot{\alpha}}{(\gamma_{a})_{\alpha\dot{\alpha}}}^{\beta\dot{\beta}}\boldsymbol{p}^{B^{\prime}}_{\beta\dot{\beta}},\qquad{(\Gamma_{\dot{a}})_{A}}^{B^{\prime}}\
=\
\boldsymbol{p}_{A}^{\alpha\dot{\alpha}}{(\gamma_{\dot{a}})_{\alpha\dot{\alpha}}}^{\beta\dot{\beta}}\boldsymbol{p}^{B^{\prime}}_{\beta\dot{\beta}},$
(A.37) $\displaystyle{(\Gamma_{a})_{A^{\prime}}}^{B}$ $\displaystyle=$
$\displaystyle\boldsymbol{p}_{A^{\prime}}^{\alpha\dot{\alpha}}{(\gamma_{a})_{\alpha\dot{\alpha}}}^{\beta\dot{\beta}}\boldsymbol{p}^{B}_{\beta\dot{\beta}},\qquad{(\Gamma_{\dot{a}})_{A^{\prime}}}^{B}\
=\
-\boldsymbol{p}_{A^{\prime}}^{\alpha\dot{\alpha}}{(\gamma_{\dot{a}})_{\alpha\dot{\alpha}}}^{\beta\dot{\beta}}\boldsymbol{p}^{B}_{\beta\dot{\beta}}.$
(A.38)

To raise and lower the spinor indices $A$ and $A^{\prime}$, we use the charge
conjugation matrix which in this basis takes the form of

$\mathds{C}\ =\ \begin{pmatrix}0&\Omega\otimes\Omega\\\
-\Omega\otimes\Omega&0\end{pmatrix}.$ (A.39)

Its elements can be explicitly expressed as

$\mathds{C}_{\mathcal{A}\mathcal{B}}\ =\
\begin{pmatrix}0&\boldsymbol{c}_{AB^{\prime}}\\\
\boldsymbol{c}_{A^{\prime}B}&0\end{pmatrix}\qquad\text{and}\qquad\mathds{C}^{\mathcal{A}\mathcal{B}}\
=\ \begin{pmatrix}0&\boldsymbol{c}^{AB^{\prime}}\\\
\boldsymbol{c}^{A^{\prime}B}&0\end{pmatrix}.$ (A.40)

The $16\times 16$ matrix $\boldsymbol{c}_{A^{\prime}B}$ is antisymmetric,
$\boldsymbol{c}_{A^{\prime}B}=-\boldsymbol{c}_{BA^{\prime}}$. Similarly, the
matrix $\boldsymbol{c}^{AB^{\prime}}$ satisfying the relations

$\boldsymbol{c}_{AC^{\prime}}\boldsymbol{c}^{C^{\prime}B}=-\delta_{A}^{B}\qquad\textrm{and}\qquad\boldsymbol{c}_{A^{\prime}C}\boldsymbol{c}^{CB^{\prime}}=-\delta_{A^{\prime}}^{B^{\prime}}$
(A.41)

is also antisymmetric
$\boldsymbol{c}^{A^{\prime}B}=-\boldsymbol{c}^{BA^{\prime}}$.
By raising and lowering the $SO(5,5)$ spinor index, we can define gamma
matrices with all upper or lower indices

$\displaystyle(\tilde{\mathbf{\Gamma}}_{\underline{A}})^{\mathcal{A}\mathcal{B}}$
$\displaystyle=$
$\displaystyle\mathds{C}^{\mathcal{A}\mathcal{C}}{(\tilde{\mathbf{\Gamma}}_{\underline{A}})_{\mathcal{C}}}^{\mathcal{B}}\
=\ \begin{pmatrix}(\Gamma_{\underline{A}})^{AB}&0\\\
0&(\Gamma_{\underline{A}})^{A^{\prime}B^{\prime}}\end{pmatrix},$ (A.42)
$\displaystyle(\tilde{\mathbf{\Gamma}}_{\underline{A}})_{\mathcal{A}\mathcal{B}}$
$\displaystyle=$
$\displaystyle{(\tilde{\mathbf{\Gamma}}_{\underline{A}})_{\mathcal{A}}}^{\mathcal{C}}\mathds{C}_{\mathcal{C}\mathcal{B}}\
=\ \begin{pmatrix}(\Gamma_{\underline{A}})_{AB}&0\\\
0&(\Gamma_{\underline{A}})_{A^{\prime}B^{\prime}}\end{pmatrix}$ (A.43)

in which

$\displaystyle(\Gamma_{\underline{A}})^{AB}$ $\displaystyle=$
$\displaystyle\boldsymbol{c}^{AC^{\prime}}{(\Gamma_{\underline{A}})_{C^{\prime}}}^{B},\qquad(\Gamma_{\underline{A}})^{A^{\prime}B^{\prime}}\
=\ \boldsymbol{c}^{A^{\prime}C}{(\Gamma_{\underline{A}})_{C}}^{B^{\prime}},$
$\displaystyle(\Gamma_{\underline{A}})_{AB}$ $\displaystyle=$
$\displaystyle{(\Gamma_{\underline{A}})_{A}}^{C^{\prime}}\boldsymbol{c}_{C^{\prime}B},\qquad(\Gamma_{\underline{A}})_{A^{\prime}B^{\prime}}\
=\ {(\Gamma_{\underline{A}})_{A^{\prime}}}^{C}\boldsymbol{c}_{CB^{\prime}}.$
(A.44)

In terms of the $USp(4)$ symplectic forms $\Omega_{\alpha\beta}$ and
$\Omega_{\dot{\alpha}\dot{\beta}}$ which can be used to raise or lower
$USp(4)$ fundamental indices, we can write the matrices
$\boldsymbol{c}_{A^{\prime}B}$ and $\boldsymbol{c}^{A^{\prime}B}$ as

$\boldsymbol{c}_{A^{\prime}B}\ =\
\boldsymbol{p}_{A^{\prime}}^{\alpha\dot{\alpha}}\boldsymbol{p}_{B}^{\beta\dot{\beta}}\Omega_{\alpha\beta}\Omega_{\dot{\alpha}\dot{\beta}}\qquad\text{and}\qquad\boldsymbol{c}^{A^{\prime}B}\
=\
\boldsymbol{p}^{A^{\prime}}_{\alpha\dot{\alpha}}\boldsymbol{p}^{B}_{\beta\dot{\beta}}\Omega^{\alpha\beta}\Omega^{\dot{\alpha}\dot{\beta}}.$
(A.45)

With all these, we can eventually find the following relations

$\displaystyle(\Gamma_{a})^{AB}$ $\displaystyle=$
$\displaystyle\boldsymbol{p}^{A}_{\alpha\dot{\alpha}}\boldsymbol{p}^{B}_{\beta\dot{\beta}}(\gamma_{a})^{\alpha\dot{\alpha},\beta\dot{\beta}},\qquad(\Gamma_{\dot{a}})^{AB}=-\boldsymbol{p}^{A}_{\alpha\dot{\alpha}}\boldsymbol{p}^{B}_{\beta\dot{\beta}}(\gamma_{\dot{a}})^{\alpha\dot{\alpha},\beta\dot{\beta}},$
$\displaystyle(\Gamma_{a})^{A^{\prime}B^{\prime}}$ $\displaystyle=$
$\displaystyle\boldsymbol{p}^{A^{\prime}}_{\alpha\dot{\alpha}}\boldsymbol{p}^{B^{\prime}}_{\beta\dot{\beta}}(\gamma_{a})^{\alpha\dot{\alpha},\beta\dot{\beta}},\qquad(\Gamma_{\dot{a}})^{A^{\prime}B^{\prime}}\
=\
\boldsymbol{p}^{A^{\prime}}_{\alpha\dot{\alpha}}\boldsymbol{p}^{B^{\prime}}_{\beta\dot{\beta}}(\gamma_{\dot{a}})^{\alpha\dot{\alpha},\beta\dot{\beta}},$
$\displaystyle(\Gamma_{a})_{AB}$ $\displaystyle=$
$\displaystyle\boldsymbol{p}_{A}^{\alpha\dot{\alpha}}\boldsymbol{p}_{B}^{\beta\dot{\beta}}(\gamma_{a})_{\alpha\dot{\alpha},\beta\dot{\beta}},\qquad(\Gamma_{\dot{a}})_{AB}=\
\boldsymbol{p}_{A}^{\alpha\dot{\alpha}}\boldsymbol{p}_{B}^{\beta\dot{\beta}}(\gamma_{\dot{a}})_{\alpha\dot{\alpha},\beta\dot{\beta}},$
$\displaystyle(\Gamma_{a})_{A^{\prime}B^{\prime}}$ $\displaystyle=$
$\displaystyle\boldsymbol{p}_{A^{\prime}}^{\alpha\dot{\alpha}}\boldsymbol{p}_{B^{\prime}}^{\beta\dot{\beta}}(\gamma_{a})_{\alpha\dot{\alpha},\beta\dot{\beta}},\qquad(\Gamma_{\dot{a}})_{A^{\prime}B^{\prime}}\
=\
-\boldsymbol{p}_{A^{\prime}}^{\alpha\dot{\alpha}}\boldsymbol{p}_{B^{\prime}}^{\beta\dot{\beta}}(\gamma_{\dot{a}})_{\alpha\dot{\alpha},\beta\dot{\beta}}$
(A.46)

with

$\displaystyle(\gamma_{a})^{\alpha\dot{\alpha},\beta\dot{\beta}}$
$\displaystyle=$
$\displaystyle\Omega^{\alpha\delta}\Omega^{\dot{\alpha}\dot{\delta}}{(\gamma_{a})_{\delta\dot{\delta}}}^{\beta\dot{\beta}}\
=\ (\gamma_{a})^{\alpha\beta}\Omega^{\dot{\alpha}\dot{\beta}},$
$\displaystyle(\gamma_{\dot{a}})^{\alpha\dot{\alpha},\beta\dot{\beta}}$
$\displaystyle=$
$\displaystyle\Omega^{\alpha\delta}\Omega^{\dot{\alpha}\dot{\delta}}{(\gamma_{\dot{a}})_{\delta\dot{\delta}}}^{\beta\dot{\beta}}\
=\ \Omega^{\alpha\beta}(\gamma_{\dot{a}})^{\dot{\alpha}\dot{\beta}},$
$\displaystyle(\gamma_{a})_{\alpha\dot{\alpha},\beta\dot{\beta}}$
$\displaystyle=$
$\displaystyle\Omega_{\beta\delta}\Omega_{\dot{\beta}\dot{\delta}}{(\gamma_{a})_{\alpha\dot{\alpha}}}^{\delta\dot{\delta}}\
=\ (\gamma_{a})_{\alpha\beta}\Omega_{\dot{\alpha}\dot{\beta}},$
$\displaystyle(\gamma_{\dot{a}})_{\alpha\dot{\alpha},\beta\dot{\beta}}$
$\displaystyle=$
$\displaystyle\Omega_{\beta\delta}\Omega_{\dot{\beta}\dot{\delta}}{(\gamma_{\dot{a}})_{\alpha\dot{\alpha}}}^{\delta\dot{\delta}}\
=\ \Omega_{\alpha\beta}(\gamma_{\dot{a}})_{\dot{\alpha}\dot{\beta}}.$ (A.47)

We now transform all these results to the basis with off-diagonal $\eta_{MN}$
given in (A.1). Denoting $SO(5,5)$ gamma matrices in this basis by
$\tilde{\mathbf{\Gamma}}_{M}$, we can write the corresponding Clifford algebra
as

$\left\\{\tilde{\mathbf{\Gamma}}_{M},\tilde{\mathbf{\Gamma}}_{N}\right\\}\ =\
2\eta_{MN}\mathds{1}_{32}\,.$ (A.48)

From [30], the relation between diagonal and off-diagonal $\eta$ is given by

$\eta_{MN}\ =\
{\mathbb{M}_{M}}^{\underline{A}}\,{\mathbb{M}_{N}}^{\underline{B}}\,\eta_{\underline{A}\underline{B}}$
(A.49)

with

$\mathbb{M}\ =\
\frac{1}{\sqrt{2}}\begin{pmatrix}\mathds{1}_{5}&\mathds{1}_{5}\\\
\mathds{1}_{5}&-\mathds{1}_{5}\end{pmatrix}.$ (A.50)

We can then find the following relation between these two sets of gamma
matrices

$\tilde{\mathbf{\Gamma}}_{M}\ =\
{\mathbb{M}_{M}}^{\underline{A}}\tilde{\mathbf{\Gamma}}_{\underline{A}}$
(A.51)

with the same chiral decomposition of the form

${(\tilde{\mathbf{\Gamma}}_{M})_{\mathcal{A}}}^{\mathcal{B}}\ =\
\begin{pmatrix}0&{(\Gamma_{M})_{A}}^{B^{\prime}}\\\
{(\Gamma_{M})_{A^{\prime}}}^{B}&0\end{pmatrix}.$ (A.52)

Moreover, we can still raise and lower the chirally decomposed spinor indices
with the charge conjugation matrix given in (A.39) such that

$(\tilde{\mathbf{\Gamma}}_{M})_{\mathcal{A}\mathcal{B}}\ =\
{(\tilde{\mathbf{\Gamma}}_{M})_{\mathcal{A}}}^{\mathcal{C}}\mathds{C}_{\mathcal{C}\mathcal{B}}\
=\ \begin{pmatrix}(\Gamma_{M})_{AB}&0\\\
0&(\Gamma_{M})_{A^{\prime}B^{\prime}}\end{pmatrix}$ (A.53)

with
$(\Gamma_{M})_{AB}={(\Gamma_{M})_{A}}^{C^{\prime}}\boldsymbol{c}_{C^{\prime}B}$
and
$(\Gamma_{M})_{A^{\prime}B^{\prime}}={(\Gamma_{M})_{A^{\prime}}}^{C}\boldsymbol{c}_{CB^{\prime}}$.
We will see in the following analysis that $(\Gamma_{M})_{AB}$ play an
important role in determining specific forms of the embedding tensor.

In spinor representation, the $SO(5,5)$ generators satisfying (A.2) are given
by

${(\boldsymbol{t}_{MN})_{A}}^{B}\ =\ {(\Gamma_{MN})_{A}}^{B}.$ (A.54)

In $32\times 32$ representation, we can write

$\displaystyle{(\tilde{\mathbf{\Gamma}}_{MN})_{\mathcal{A}}}^{\mathcal{B}}$
$\displaystyle=$ $\displaystyle\
\frac{1}{2}\left({(\tilde{\mathbf{\Gamma}}_{M})_{\mathcal{A}}}^{\mathcal{C}}{(\tilde{\mathbf{\Gamma}}_{N})_{\mathcal{C}}}^{\mathcal{B}}-{(\tilde{\mathbf{\Gamma}}_{N})_{\mathcal{A}}}^{\mathcal{C}}{(\tilde{\mathbf{\Gamma}}_{M})_{\mathcal{C}}}^{\mathcal{B}}\right)$
$\displaystyle=$ $\displaystyle\ \begin{pmatrix}{(\Gamma_{MN})_{A}}^{B}&0\\\
0&{(\Gamma_{MN})_{A^{\prime}}}^{B^{\prime}}\end{pmatrix}$

with

$\displaystyle{(\Gamma_{MN})_{A}}^{B}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[{(\Gamma_{M})_{A}}^{C^{\prime}}{(\Gamma_{N})_{C^{\prime}}}^{B}-{(\Gamma_{N})_{A}}^{C^{\prime}}{(\Gamma_{M})_{C^{\prime}}}^{B}\right],$
(A.56) $\displaystyle{(\Gamma_{MN})_{A^{\prime}}}^{B^{\prime}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[{(\Gamma_{M})_{A^{\prime}}}^{C}{(\Gamma_{N})_{C}}^{B^{\prime}}-{(\Gamma_{N})_{A^{\prime}}}^{C}{(\Gamma_{M})_{C}}^{B^{\prime}}\right].$
(A.57)

It should be noted that the $SO(5,5)$ generators in spinor representation
given in (A.54) also decompose according to (A.18) and satisfy the same
algebra given in (A.11) to (A.17) for vector representation.

As pointed out in [30], the branching rules for spinor and conjugate spinor
representations of $SO(5,5)$ are respectively given by

$\mathbf{16}_{s}\ \rightarrow\
\overline{\mathbf{5}}^{+3}\,\oplus\,\mathbf{10}^{-1}\,\oplus\,\mathbf{1}^{-5}\qquad\text{and}\qquad\mathbf{16}_{c}\
\rightarrow\
\mathbf{5}^{-3}\,\oplus\,\overline{\mathbf{10}}^{+1}\,\oplus\,\mathbf{1}^{+5}.$
(A.58)

To find the corresponding decompositions of spinor indices, we define the
following transformation matrices

$\displaystyle\mathbb{T}_{Am}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2}}(\Gamma_{m})_{AB}\boldsymbol{p}^{B}_{\alpha\beta}\Omega^{\alpha\beta},$
(A.59) $\displaystyle\mathbb{T}_{A}^{mn}$ $\displaystyle=$
$\displaystyle\frac{1}{4\sqrt{2}}{(\Gamma^{mn})_{A}}^{B}\boldsymbol{p}_{B}^{\alpha\beta}\Omega_{\alpha\beta},$
(A.60) $\displaystyle\mathbb{T}_{A\ast}$ $\displaystyle=$
$\displaystyle\frac{1}{10}{({\Gamma^{m}}_{m})_{A}}^{B}\boldsymbol{p}_{B}^{\alpha\beta}\Omega_{\alpha\beta}\,.$
(A.61)

In these equations, $\boldsymbol{p}^{A}_{\alpha\beta}$ matrices are defined in
the same way as $\boldsymbol{p}^{A}_{\alpha\dot{\beta}}$ in (A.3).

We can now decompose an $SO(5,5)$ spinor in $\mathbf{16}_{s}$ representation
as

$\Psi_{A}\ =\
\mathbb{T}_{Am}\Psi^{m}+\mathbb{T}_{A}^{mn}\Psi_{mn}+\mathbb{T}_{A\ast}\Psi_{\ast}$
(A.62)

with $\Psi_{mn}=\Psi_{[mn]}$. The commutation relations between these
components and the $\mathbb{R}^{+}$ generator are given by

$\displaystyle\left[\boldsymbol{d},\mathbb{T}_{Am}\Psi^{m}\right]$
$\displaystyle=$ $\displaystyle+3\mathbb{T}_{Am}\Psi^{m},$ (A.63)
$\displaystyle\left[\boldsymbol{d},\mathbb{T}_{A}^{mn}\Psi_{mn}\right]$
$\displaystyle=$ $\displaystyle-\mathbb{T}_{A}^{mn}\Psi_{mn},$ (A.64)
$\displaystyle\left[\boldsymbol{d},\mathbb{T}_{A\ast}\Psi_{\ast}\right]$
$\displaystyle=$ $\displaystyle-5\mathbb{T}_{A\ast}\Psi_{\ast}$ (A.65)

in accord with the branching rule

$\underbrace{\mathbf{16}_{s}}_{\Psi_{A}}\ \rightarrow\
\underbrace{\overline{\mathbf{5}}^{+3}}_{\Psi^{m}}\,\oplus\,\underbrace{\mathbf{10}^{-1}}_{\Psi_{[mn]}}\,\oplus\,\underbrace{\mathbf{1}^{-5}}_{\Psi_{\ast}}\,.$
(A.66)

The inverse matrices of $\mathbb{T}_{A}$ are simply given by their complex
conjugation $\mathbb{T}^{A}=(\mathbb{T}_{A})^{-1}=(\mathbb{T}_{A})^{*}$
satisfying

$\displaystyle\mathbb{T}^{Am}\mathbb{T}_{An}\ =\
\delta^{m}_{n},\qquad\quad\mathbb{T}^{A}_{mn}\mathbb{T}_{A}^{pq}$
$\displaystyle=$
$\displaystyle\delta^{[p}_{m}\delta^{q]}_{n},\qquad\mathbb{T}^{A}_{\ast}\mathbb{T}_{A\ast}\
=\ 1,$ $\displaystyle\mathbb{T}^{Am}\mathbb{T}_{Anp}\ =\
0,\qquad\quad\,\mathbb{T}^{Am}\mathbb{T}_{A\ast}$ $\displaystyle=$
$\displaystyle 0,\qquad\quad\mathbb{T}^{A}_{mn}\mathbb{T}_{A\ast}\ =\ 0$
(A.67)

together with

$\mathbb{T}^{Am}\mathbb{T}_{Bm}+\mathbb{T}^{A}_{mn}\mathbb{T}_{B}^{mn}+\mathbb{T}^{A}_{\ast}\mathbb{T}_{B\ast}=\delta^{A}_{B}\,.$
(A.68)

In addition, we also note that a complex conjugation of the $SO(5,5)$ gamma
matrices is related to raising the indices
$((\Gamma_{M})_{AB})^{*}=(\Gamma^{M})^{AB}$. We can then similarly decompose a
conjugate spinor of $SO(5,5)$ transforming in $\mathbf{16}_{c}$ as follows

$\Psi^{A}\ =\
\mathbb{T}^{Am}\Psi_{m}+\mathbb{T}^{A}_{mn}\Psi^{mn}+\mathbb{T}^{A}_{\ast}\Psi_{\ast}\,.$
(A.69)

The following commutation relations

$\displaystyle\left[\boldsymbol{d},\mathbb{T}^{Am}\Psi_{m}\right]$
$\displaystyle=$ $\displaystyle-3\mathbb{T}^{Am}\Psi_{m},$ (A.70)
$\displaystyle\left[\boldsymbol{d},\mathbb{T}^{A}_{mn}\Psi^{mn}\right]$
$\displaystyle=$ $\displaystyle+\mathbb{T}^{A}_{mn}\Psi^{mn},$ (A.71)
$\displaystyle\left[\boldsymbol{d},\mathbb{T}^{A}_{\ast}\Psi_{\ast}\right]$
$\displaystyle=$ $\displaystyle+5\mathbb{T}^{A}_{\ast}\Psi_{\ast}$ (A.72)

imply the branching rule

$\underbrace{\mathbf{16}_{c}}_{\Psi^{A}}\ \rightarrow\
\underbrace{\mathbf{5}^{-3}}_{\Psi_{m}}\,\oplus\,\underbrace{\overline{\mathbf{10}}^{+1}}_{\Psi^{[mn]}}\,\oplus\,\underbrace{\mathbf{1}^{+5}}_{\Psi_{\ast}}.$
(A.73)

###  A.4 Vector-spinor

The vector-spinor of $SO(5,5)$ we are interested in is given by
$\theta^{AM}\in\mathbf{144}_{c}$, which parameterizes the embedding tensor. It
transforms according to

$\left[\boldsymbol{t}_{MN},\theta^{AP}\right]\ =\
-{(\boldsymbol{t}_{MN})_{Q}}^{P}\theta^{AQ}-{(\boldsymbol{t}_{MN})_{B}}^{A}\theta^{BP}.$
(A.74)

Here, $\theta^{AM}$ is a ($16\times 10$) matrix subject to

$(\Gamma_{M})_{AB}\,\theta^{BM}\ =\ 0$ (A.75)

which is the linear constraint required by supersymmetry, reducing $160$
components of the $\theta^{AM}$ to $144$ in $\mathbf{144}_{c}$ representation.

To determine the decomposition of the vector-spinor representation under
$GL(5)$, we first split the $SO(5,5)$ vector index $M$ as
$\theta^{AM}=(\theta^{Am},\theta^{A}_{m})$. Then, with the inverse of the
transformation matrices $\mathbb{T}_{A}$ given in (A.59) to (A.61),
$\theta^{Am}$ and $\theta^{A}_{m}$ can be further decomposed into the
following six components

$\displaystyle\theta^{Am}$ $\displaystyle=$
$\displaystyle\mathbb{T}^{An}{(\vartheta_{1})_{n}}^{m}+\mathbb{T}^{A}_{np}(\vartheta_{3})^{np,m}+\mathbb{T}^{A}_{\ast}(\vartheta_{5})^{m},$
(A.76) $\displaystyle\theta^{A}_{m}$ $\displaystyle=$
$\displaystyle\mathbb{T}^{An}(\vartheta_{2})_{nm}+\mathbb{T}^{A}_{np}(\vartheta_{4})^{np}_{m}+\mathbb{T}^{A}_{\ast}(\vartheta_{6})_{m}\,.$
(A.77)

It is straightforward to show that their commutation relations with the
$\mathbb{R}^{+}$ generators in both vector and spinor representations are
given by

$\displaystyle\left[\boldsymbol{d},\mathbb{T}^{An}{(\vartheta_{1})_{n}}^{m}\right]$
$\displaystyle=$
$\displaystyle-5\mathbb{T}^{An}{(\vartheta_{1})_{n}}^{m},\qquad\left[\boldsymbol{d},\mathbb{T}^{An}(\vartheta_{2})_{nm}\right]\
=\ -\mathbb{T}^{An}(\vartheta_{2})_{nm},$
$\displaystyle\left[\boldsymbol{d},\mathbb{T}^{A}_{np}(\vartheta_{3})^{np,m}\right]$
$\displaystyle=$
$\displaystyle-\mathbb{T}^{A}_{np}(\vartheta_{3})^{np,m},\qquad\,\left[\boldsymbol{d},\mathbb{T}^{A}_{np}(\vartheta_{4})^{np}_{m}\right]\
=\ +3\mathbb{T}^{A}_{np}(\vartheta_{4})^{np}_{m},$
$\displaystyle\left[\boldsymbol{d},\mathbb{T}^{A}_{\ast}(\vartheta_{5})^{m}\right]$
$\displaystyle=$
$\displaystyle+3\mathbb{T}^{A}_{\ast}(\vartheta_{5})^{m},\qquad\quad\left[\boldsymbol{d},\mathbb{T}^{A}_{\ast}(\vartheta_{6})_{m}\right]\
=\ +7\mathbb{T}^{A}_{\ast}(\vartheta_{6})_{m}.$

The branching rule for $\mathbf{144}_{c}$ representation is then given by

$\underbrace{\mathbf{144}_{c}}_{\theta^{AM}}\ \rightarrow\
\underbrace{\overline{\mathbf{5}}^{+3}}_{J^{m}}\,\oplus\,\underbrace{\mathbf{5}^{+7}}_{K_{m}}\,\oplus\,\underbrace{\mathbf{10}^{-1}}_{Z_{mn}}\,\oplus\,\underbrace{\mathbf{15}^{-1}}_{Y_{mn}}\,\oplus\,\underbrace{\mathbf{24}^{-5}}_{{S_{m}}^{n}}\,\oplus\,\underbrace{\overline{\mathbf{40}}^{-1}}_{U^{mn,p}}\,\oplus\,\underbrace{\overline{\mathbf{45}}^{+3}}_{W^{np}_{m}}$
(A.79)

in agreement with that given in [30]. We now explicitly construct a number of
possible embedding tensors arising from various components of the above
decomposition.

####  A.4.1 $\mathbf{24}^{-5}$ representation

With only $\vartheta_{1}\neq 0$, we have $\theta^{A}_{m}=0$, and $\theta^{Am}$
is parametrized by a $5\times 5$ matrix ${(\vartheta_{1})_{n}}^{m}$.
Therefore, the embedding tensor is given by

$\theta^{AM}=\left(\,\mathbb{T}^{An}{(\vartheta_{1})_{n}}^{m}\,,\,0\,\right).$
(A.80)

The linear constraint (A.75) implies that ${(\vartheta_{1})_{n}}^{m}$ is
traceless or

${(\vartheta_{1})_{n}}^{m}={S_{n}}^{m}$ (A.81)

for ${S_{m}}^{m}=0$. This leads to an embedding tensor in $\mathbf{24}^{-5}$
representation of $GL(5)$ given by

$\theta_{\mathbf{24}^{-5}}^{AM}=\left(\,\mathbb{T}^{An}{S_{n}}^{m}\,,\,0\,\right).$
(A.82)

####  A.4.2 $\mathbf{15}^{-1}$ representation

For $\theta^{Am}=0$, $\theta^{A}_{m}$ is parametrized by a $5\times 5$ matrix
$(\vartheta_{2})_{mn}$ which can further be decomposed in terms of symmetric
and antisymmetric parts, $Y_{mn}=Y_{(mn)}$ and $Z_{mn}=Z_{[mn]}$, as

$\theta^{A}_{m}=\mathbb{T}^{An}(\vartheta_{2})_{nm}=\mathbb{T}^{An}(Y_{nm}+Z_{nm}).$
(A.83)

The linear constraint (A.75) requires $Z_{mn}=0$, so the embedding tensor is
given by

$\theta_{\mathbf{15}^{-1}}^{AM}=\left(\,0\,,\,\mathbb{T}^{An}Y_{nm}\,\right)$
(A.84)

in $\mathbf{15}^{-1}$ representation of $GL(5)$.

####  A.4.3 $\overline{\mathbf{40}}^{-1}$ representation

With only $(\vartheta_{3})^{np,m}$ non-vanishing, we have $\theta^{A}_{m}=0$
and $\theta^{Am}$ given by

$\theta^{Am}=\mathbb{T}^{A}_{np}(\vartheta_{3})^{np,m}\,.$ (A.85)

The tensor $(\vartheta_{3})^{np,m}$ can in turn be parametrized as

$(\vartheta_{3})^{np,m}\ =\
U^{np,m}+\frac{1}{2}\varepsilon^{mnpqr}\zeta_{qr}\,.$ (A.86)

$U^{mn,p}=U^{[mn],p}$ with $U^{[mn,p]}=0$ and $\zeta_{mn}=\zeta_{[mn]}$
correspond to $\overline{\mathbf{40}}^{-1}$ and $\mathbf{10}^{-1}$
representations, respectively. The condition (A.75) requires $\zeta_{qr}=0$
resulting in the embedding tensor in $\overline{\mathbf{40}}^{-1}$
representation of the form

$\theta_{\overline{\mathbf{40}}^{-1}}^{AM}=\left(\,\mathbb{T}^{A}_{np}U^{np,m}\,,\,0\,\right).$
(A.87)

####  A.4.4 $\mathbf{10}^{-1}$ representation

Turning on $\mathbf{10}^{-1}$ irreducible part of both $(\vartheta_{2})_{nm}$
and $(\vartheta_{3})^{np,m}$ by setting $Y_{mn}=0$ and $U^{mn,p}=0$, we find
the embedding tensor of the form

$\theta_{\mathbf{10}^{-1}}^{AM}=\left(\,\frac{1}{2}\mathbb{T}^{A}_{np}\varepsilon^{mnpqr}\zeta_{qr}\,,\,\mathbb{T}^{An}Z_{nm}\,\right).$
(A.88)

The condition (A.75) is satisfied for

$\zeta_{mn}=\frac{\sqrt{2}}{3}Z_{mn}\,.$ (A.89)

Therefore, the embedding tensor in $\mathbf{10}^{-1}$ representation is given
by

$\theta_{\mathbf{10}^{-1}}^{AM}=\left(\frac{1}{3\sqrt{2}}\mathbb{T}^{A}_{np}\varepsilon^{mnpqr}Z_{qr}\,,\,\mathbb{T}^{An}Z_{nm}\,\right).$
(A.90)

####  A.4.5 ${\overline{\mathbf{45}}}^{+3}$ representation

In this case, we consider non-vanishing $(\vartheta_{4})^{np}_{m}$ which can
be decomposed into $\overline{\mathbf{45}}^{+3}$ and
$\overline{\mathbf{5}}^{+3}$ irreducible representations of the form

$(\vartheta_{4})^{np}_{m}\ =\ W^{np}_{m}+J^{[n}\delta^{p]}_{m}$ (A.91)

with $W^{np}_{m}=W^{[np]}_{m}$ satisfying $W^{nm}_{m}=0$. The linear
constraint (A.75) requires $J^{m}=0$ leading to the embedding tensor in
$\overline{\mathbf{45}}^{+3}$ representation given by

$\theta_{{\overline{\mathbf{45}}}^{+3}}^{AM}=\left(\,0\,,\,\mathbb{T}^{A}_{np}W^{np}_{m}\,\right).$
(A.92)

####  A.4.6 $\overline{\mathbf{5}}^{+3}$ representation

We now consider non-vanishing $\overline{\mathbf{5}}^{+3}$ components from
both $(\vartheta_{5})^{m}$ and $(\vartheta_{4})^{m}$ in terms of which the
embedding tensor is given by

$\theta_{\overline{\mathbf{5}}^{+3}}^{AM}=\left(\,\mathbb{T}^{A}_{\ast}\xi^{m}\,,\,\mathbb{T}^{A}_{np}J^{[n}\delta^{p]}_{m}\,\right).$
(A.93)

This satisfies the linear constraint (A.75) for

$\xi^{m}=-\frac{2\sqrt{2}}{5}J^{m}.$ (A.94)

Therefore, we find the embedding tensor in $\overline{\mathbf{5}}^{+3}$
representation given by

$\theta_{\overline{\mathbf{5}}^{+3}}^{AM}=\left(\,-\frac{2\sqrt{2}}{5}\mathbb{T}^{A}_{\ast}J^{m}\,,\,\mathbb{T}^{A}_{np}J^{[n}\delta^{p]}_{m}\,\right).$
(A.95)

####  A.4.7 $\mathbf{5}^{+7}$ representation

Finally, we consider $(\vartheta_{6})_{m}$ corresponding to $\mathbf{5}^{+7}$
representation. This can be parameterized by an arbitrary $GL(5)$ vector of
the form $(\vartheta_{6})_{m}=K_{m}$. The corresponding embedding tensor is
also in $\mathbf{5}^{+7}$ representation and takes the form

$\theta_{\mathbf{5}^{+7}}^{AM}=\left(\,0\,,\,\mathbb{T}^{A}_{\ast}K_{m}\,\right)$
(A.96)

which automatically satisfies the condition (A.75).
We end this appendix by giving the full parametrization of the embedding
tensor $\theta^{AM}=(\theta^{Am},\theta^{A}_{m})$ under $GL(5)$

$\displaystyle\theta^{Am}$ $\displaystyle=$
$\displaystyle\mathbb{T}^{An}{S_{n}}^{m}+\mathbb{T}^{A}_{np}\left(U^{np,m}+\frac{1}{3\sqrt{2}}\varepsilon^{mnpqr}Z_{qr}\right)-\frac{2\sqrt{2}}{5}\mathbb{T}^{A}_{\ast}J^{m},$
(A.97) $\displaystyle\theta^{A}_{m}$ $\displaystyle=$
$\displaystyle\mathbb{T}^{An}(Y_{nm}+Z_{nm})+\mathbb{T}^{A}_{np}(W^{np}_{m}+J^{[n}\delta^{p]}_{m})+\mathbb{T}^{A}_{\ast}K_{m}\,.$
(A.98)

##  Appendix B Symplectic-Majorana-Weyl spinors in six dimensions

In this appendix, we collect conventions and fundamental relations involving
irreducible spinors in six-dimensional space-time used throughout this work.
In six dimensions, there exist Dirac spinors with $16$ real components. The
Dirac spinors are reducible and can be decomposed into two irreducible Weyl
spinors of opposite chirality with $8$ real components each. The six-
dimensional Clifford algebra is defined by the relation

$\\{\hat{\gamma}_{\hat{\mu}}\hat{\gamma}_{\hat{\nu}}+\hat{\gamma}_{\hat{\nu}}\hat{\gamma}_{\hat{\mu}}\\}=2\eta_{\hat{\mu}\hat{\nu}}\mathds{1}_{8}.$
(B.1)

Here, $\hat{\gamma}_{\hat{\mu}}$ are $8\times 8$ Dirac matrices and
$\eta_{\hat{\mu}\hat{\nu}}=\text{diag}(-1,+1,+1,+1,+1,+1)$ with
$\hat{\mu},\hat{\nu},...=0,1,...,5$ being six-dimensional flat space-time
indices. We will use the following explicit representation of the gamma
matrices

$\displaystyle\hat{\gamma}_{0}=\sigma_{1}\otimes
i\sigma_{2}\otimes\mathds{1}_{2},\qquad\hat{\gamma}_{1}=\sigma_{2}\otimes\mathds{1}_{2}\otimes\sigma_{2},\qquad\hat{\gamma}_{2}=\sigma_{1}\otimes\sigma_{1}\otimes\mathds{1}_{2},$
$\displaystyle\hat{\gamma}_{3}=\sigma_{2}\otimes\mathds{1}_{2}\otimes\sigma_{1},\qquad\hat{\gamma}_{4}=\sigma_{2}\otimes\mathds{1}_{2}\otimes\sigma_{3},\qquad\hat{\gamma}_{5}=\sigma_{1}\otimes\sigma_{3}\otimes\mathds{1}_{2}\,.$
(B.2)

In this representation, the Dirac, complex, and charge conjugation matrices
are respectively given by

$\hat{\mathds{A}}=\hat{\gamma}_{0},\qquad\hat{\mathds{B}}=-i\hat{\gamma}_{3}\hat{\gamma}_{4},\quad\hat{\mathds{C}}=i\hat{\gamma}_{0}\hat{\gamma}_{3}\hat{\gamma}_{4}\,.$
(B.3)

They satisfy the following relations

$(\hat{\gamma}_{\hat{\mu}})^{\dagger}\ =\
-\hat{\mathds{A}}\hat{\gamma}_{\hat{\mu}}\hat{\mathds{A}}^{-1},\qquad(\hat{\gamma}_{\hat{\mu}})^{*}\
=\
-\hat{\mathds{B}}\hat{\gamma}_{\hat{\mu}}\hat{\mathds{B}}^{-1},\qquad(\hat{\gamma}_{\hat{\mu}})^{T}\
=\ -\hat{\mathds{C}}\hat{\gamma}_{\hat{\mu}}\hat{\mathds{C}}^{-1}$ (B.4)

together with the identities

$\hat{\mathds{B}}^{T}=\hat{\mathds{C}}\hat{\mathds{A}}^{-1},\qquad\hat{\mathds{B}}^{*}\hat{\mathds{B}}=-\mathds{1}_{8},\qquad\hat{\mathds{C}}^{T}=-\hat{\mathds{C}}^{-1}=-\hat{\mathds{C}}^{\dagger}=\hat{\mathds{C}}\,.$
(B.5)

The chirality operator $\hat{\gamma}_{\ast}$ can be defined as

$\hat{\gamma}_{\ast}=\hat{\gamma}_{0}\hat{\gamma}_{1}\hat{\gamma}_{2}\hat{\gamma}_{3}\hat{\gamma}_{4}\hat{\gamma}_{5}=\text{diag}(\mathds{1}_{4},-\mathds{1}_{4})\quad\text{with}\quad\hat{\gamma}_{\ast}\hat{\gamma}_{\ast}=\mathds{1}_{8}\,.$
(B.6)

The diagonal form of $\hat{\gamma}_{\ast}$ implies that a Dirac spinor $\Psi$
can be chirally decomposed as

$\Psi=\Psi_{+}+\Psi_{-}\qquad\text{with}\qquad\mathbb{P}_{\pm}\Psi_{\pm}=\pm\Psi_{\pm}$
(B.7)

where the projection operators are given by

$\mathbb{P}_{\pm}=\frac{1}{2}(\mathds{1}_{8}\pm\hat{\gamma}_{\ast}).$ (B.8)

Therefore, we can define two irreducible Weyl spinors $\psi_{+}$ and
$\chi_{-}$ from the Dirac spinor $\Psi$ by

$\Psi=\begin{pmatrix}\psi_{+}\\\
\chi_{-}\end{pmatrix},\qquad\Psi_{+}=\begin{pmatrix}\psi_{+}\\\
0\end{pmatrix},\qquad\Psi_{-}=\begin{pmatrix}0\\\ \chi_{-}\end{pmatrix}.$
(B.9)

Although the second property in (B.5) implies that a reality condition cannot
be imposed on the Dirac or Weyl spinors, we can define a symplectic-Majorana-
Weyl spinor of the form

$\Psi_{+\alpha}=\begin{pmatrix}\psi_{+\alpha}\\\
0\end{pmatrix}\qquad\text{and}\qquad\Psi_{-\dot{\alpha}}=\begin{pmatrix}0\\\
\chi_{-\dot{\alpha}}\end{pmatrix}$ (B.10)

with $\Psi_{+\alpha}$ and $\Psi_{-\dot{\alpha}}$ satisfying the pseudo-reality
condition given by

$\Psi_{+}^{\alpha}=(\Psi_{+\alpha})^{*}=\Omega^{\alpha\beta}\hat{\mathds{B}}\Psi_{+\beta}\qquad\text{and}\qquad\Psi_{-}^{\dot{\alpha}}=(\Psi_{-\dot{\alpha}})^{*}=\Omega^{\dot{\alpha}\dot{\beta}}\hat{\mathds{B}}\Psi_{-\dot{\beta}}\,.$
(B.11)

$\Omega^{\alpha\beta}$ and $\Omega^{\dot{\alpha}\dot{\beta}}$ are symplectic
forms of the two $USp(4)$ factors in the R-symmetry $USp(4)\times USp(4)$
under which $\Psi_{+\alpha}$ and $\Psi_{-\dot{\alpha}}$ transform separately.

##  Appendix C Truncation ansatze

In this appendix, we collect some useful formulae for a consistent truncation
of seven-dimensional $SO(5)$ gauged supergravity on a circle ($S^{1}$), giving
rise to $SO(5)$ gauged supergravity in six dimensions. This truncation has
been constructed in [28].

The truncation ansatze for the seven-dimensional metric, scalar, vector, and
tensor fields are respectively given by

$\displaystyle d\hat{s}_{7}^{2}$ $\displaystyle=$ $\displaystyle
e^{\frac{\sigma}{\sqrt{10}}}ds_{6}^{2}+e^{-\frac{4\sigma}{\sqrt{10}}}(dz+A_{(1)})^{2},$
(C.1) $\displaystyle{\hat{\Pi}_{I}}^{\ i}(x^{\mu},z)$ $\displaystyle=$
$\displaystyle{\Pi_{I}}^{i}(x^{\mu}),$ (C.2)
$\displaystyle{\hat{B}_{(1)I}}^{\quad\ J}$ $\displaystyle=$
$\displaystyle{B_{(1)I}}^{J}+{B_{(0)I}}^{J}(dz+A_{(1)}),$ (C.3)
$\displaystyle{\hat{S}_{(3)I}}$ $\displaystyle=$
$\displaystyle{S_{(3)I}}+{S_{(2)I}}(dz+A_{(1)})$ (C.4)

in which hatted quantities refer to seven-dimensional fields. $x^{\mu}$ are
six-dimensional space-time coordinates, and $z$ is the coordinate on $S^{1}$.
Here, $I,J,...=1,...,5$ are vector indices of the gauge group $SO(5)$ while
$i,j,...=1,...,5$ are vector indices for the local composite $SO(5)_{c}$.
There are $(1+14+10)$ scalars denoted by
$\\{\sigma,\,{\Pi_{I}}^{i},\,{B_{(0)I}}^{J}\\}$ in the six-dimensional theory.
These are given by the dilaton scalar field from the matric ansatz (C.1),
fourteen scalars parametrizing $SL(5)/SO(5)_{c}$ coset, and ten axionic scalar
fields from the truncation ansatz of vector fields (C.3). There are also
$(10+1)$ vectors $\\{{B_{(1)I}}^{J},\,A_{(1)}\\}$ together with five two-form
potentials $S_{(2)I}$. The five three-form potentials $S_{(3)I}$ do not
contribute to the Lagrangian of the $SO(5)$ gauged theory in six dimensions
since the seven-dimensional self-duality condition allows to eliminate them in
favor of the two-form potentials.

For supersymmetric domain wall solutions considered in this work, we can set
$A_{(1)}={B_{(1)I}}^{J}={S_{(2)I}}=0$. Using the domain wall ansatz for the
matric from (3.8) together with $\varphi=\frac{\sigma}{2\sqrt{10}}$, we find
that (C.1) becomes

$\displaystyle e^{2\hat{A}}dx^{2}_{1,5}+d\hat{r}^{2}$ $\displaystyle=$
$\displaystyle
e^{2A+2\varphi}dx^{2}_{1,4}+e^{-8\varphi}dz^{2}+e^{2\varphi}dr^{2}$ (C.5)
$\displaystyle=$ $\displaystyle
e^{\frac{8A}{5}}(dx^{2}_{1,4}+dz^{2})+e^{2\varphi}dr^{2}$

In the second line, we have substituted $\varphi=-\frac{A}{5}$ from the domain
wall solutions given here. This is also necessary for $dx_{1,4}^{2}$ and
$dz^{2}$ to form a six-dimensional flat space-time matching $dx_{1,5}^{2}$ on
the left hand side. We can also see the relations between the warped factors
$\hat{A}=\frac{4A}{5}$ and the radial coordinates $d\hat{r}=e^{\varphi}dr$.

The ansatz (C.2) implies that the scalars parametrizing $SL(5)/SO(5)_{c}$
coset in seven- and six-dimensional supersymmetric domain walls are the same
since they are independent of $z$ and depend only on the corresponding radial
coordinates

${\hat{\Pi}_{I}}^{\ i}(\hat{r})={\Pi_{I}}^{i}(r).$ (C.6)

For ten axionic scalars ${B_{(0)I}}^{J}$, which are called shift scalars in
this work, we can see from (C.3) that they give rise to non-vanishing vector
fields in seven dimensions

${\hat{B}_{(1)I}}^{\quad\ J}={B_{(0)I}}^{J}dz.$ (C.7)

Therefore, domain wall solutions with non-vanishing axionic scalars obtained
in this work cannot be obtained from an $S^{1}$ reduction of any domain wall
solutions in seven dimensions.

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# An Autoregressive Text-to-Graph Framework for Joint
Entity and Relation Extraction

Urchade Zaratiana1,2, Nadi Tomeh2, Pierre Holat1,2, Thierry Charnois2

###### Abstract

In this paper, we propose a novel method for joint entity and relation
extraction from unstructured text by framing it as a conditional sequence
generation problem. In contrast to conventional generative information
extraction models that are left-to-right token-level generators, our approach
is span-based. It generates a linearized graph where nodes represent text
spans and edges represent relation triplets. Our method employs a transformer
encoder-decoder architecture with pointing mechanism on a dynamic vocabulary
of spans and relation types. Our model can capture the structural
characteristics and boundaries of entities and relations through span
representations while simultaneously grounding the generated output in the
original text thanks to the pointing mechanism. Evaluation on benchmark
datasets validates the effectiveness of our approach, demonstrating
competitive results. Code is available at https://github.com/urchade/ATG.

## Introduction

Joint entity and relation extraction is a fundamental task in Natural Language
Processing (NLP), serving as the basis for various high-level applications
such as Knowledge Graph construction (Ye et al. 2022b) and question answering
(Chen et al. 2017). Traditionally, this task was tackled via pipeline models
that independently trained and implemented entity recognition and relation
extraction, often leading to error propagation (Brin 1999). Deep learning has
led to the creation of end-to-end models, allowing for the use of shared
representations and joint optimization of loss functions for both tasks
(Wadden et al. 2019; Wang and Lu 2020; Zhao et al. 2021; Zhong and Chen 2021;
Yan et al. 2021). Despite this advancement, these models essentially remain
pipeline-based, with entity and relation predictions executed by separate
classification heads, thereby ignoring potential interactions between these
tasks.

Recent advancements have seen a shift towards “real” end-to-end solutions,
where the prediction of entities and relations is intertwined, accomplished
through autoregressive models. These models treat the joint entity-relation
task as a process of generating plain text, employing augmented languages to
encode and decode structural information (Paolini et al. 2021; Lu et al. 2022;
Liu et al. 2022; Fei et al. 2022). While these models have achieved remarkable
performance, we argue that they also expose room for improvement, especially
in terms of grounding the output in the input text.

In this paper, we present an autoregressive transformer encoder-decoder model
that generates a linearized graph instead of generating plain text. Our model
makes use of a pointing mechanism (Vinyals, Fortunato, and Jaitly 2015) on a
dynamic vocabulary of spans and relations, providing explicit grounding in the
original text. In fact, without grounding, models can generate output that are
semantically coherent but contextually detached from the input. Our pointing
mechanism mitigates this issue by ensuring that the decoder’s outputs,
specifically the entity spans, are directly tied to the input text.
Furthermore, by generating spans and relations directly from the text, rather
than producing standalone plain text, our model encode the structural
characteristics and boundaries of entities/spans more accurately, which can be
missed by previous generative information extraction models. The cornerstone
of our solution is the explicit enumeration of all spans111Up to a certain
length in practice. at the encoder’s output, making them readily available to
the decoder. Although the number of spans can be extensive, we note that when
bounding the span size, our model’s vocabulary is typically smaller than that
of traditional language models (Devlin et al. 2019; Raffel et al. 2019), as
discussed in subsequent sections.

Moreover, as previous generative IE models operate at the token level, they
scatter the information regarding an entity’s span and its boundaries over
multiple decoding steps. In contrast, generating an entity and its type in our
approach is accomplished in a single decoding step, resulting in shorter
sequence (Figure 5). Additionally, our method naturally ensures the well-
formedness of the output while some generative IE models that produce text,
often require non-regular constraints. As an example, in TANL (Paolini et al.
2021), if a portion of the generated sentence has an invalid format, that
segment is discarded. Such issues are readily addressed in our model since the
vocabulary can be fully controlled.

Figure 1: Linearization for Information Graph Generation. The input text is
mapped into an information extraction graph. The graph consists of entities
and relation triplets, which are generated sequentially by first producing
entity spans (represented by start word, end word, and entity type) followed
by relation triplets (head entity, tail entity, and relation type).

We evaluated our model on three benchmark datasets for joint entity and
relation extraction: CoNLL 2004, SciERC, and ACE 05. Our model demonstrated
competitive performance on all datasets. Our contributions can be summarized
as follows:

  * •

We propose a novel method for joint entity and relation extraction by framing
it as a conditional sequence generation problem. Our approach generates a
linearized graph representation, where nodes represent entity spans and edges
represent relation triplets.

  * •

Our model employs a transformer encoder-decoder architecture with a pointing
mechanism on a dynamic vocabulary of spans and relation types. This allows to
capture the structural characteristics and boundaries of entities and
relations while grounding the generated output in the original text.

  * •

We demonstrate the effectiveness of our approach through extensive evaluations
on benchmark datasets, including CoNLL 2004, SciERC, and ACE 05. Our model
achieves state-of-the-art results on CoNLL 2004 and SciERC, surpassing
previous comparable models in terms of Entity F1 scores and Relation F1
scores.

## Task Definition

We address the task of joint entity and relation extraction from text as a
graph generation approach. Our proposed model generates nodes and edges as a
single sequence, effectively integrating both entity and relation extraction
into a unified framework. Formally, the task can be defined as follows: Given
an input text sequence ${\bm{x}}=\\{x_{1},x_{2},...,x_{L}\\}$, where $x_{i}$
represents the $i$-th token in the sequence, our objective is to generate a
linearized graph representation $\mathbf{y}=\\{y_{1},y_{2},...,y_{M}\\}$,
where $y_{j}$ represents a token in the generated sequence. As shown in Figure
1, in Each token $y_{j}$ can take one of three forms:

  * •

Entity span: The token $y_{j}$ represents an entity span, defined as
$y_{j}=(s_{j},e_{j},t_{j})$, where $s_{j}$ and $e_{j}$ denote the starting and
ending positions of the entity span, and $t_{j}$ denotes the type of the
entity.

  * •

Relation type: The token $y_{j}$ represents a relation type between two
entities, such as Work_For relation.

  * •

Special token: The token $y_{j}$ represents special tokens used in the
generation process, such as <SEP> to separate entities and relations or the
<END> to stop the generation.

Figure 2: Illustration of the architecture of our model, ATG. (left) The
Encoder takes in the input sequence $X$ and generates representations of the
tokens ${\bm{H}}$ and spans ${\bm{S}}$. (middle) The Decoder then generates
the next token conditioned on the previous tokens and the input representation
${\bm{H}}$. (right) The vocabulary matrix used for decoding consists of the
concatenation of span embeddings ${\bm{S}}$, learned relation type embeddings,
and special token embeddings.

The template employed in our model, we refer to ATG (Autoregrestive Text-to-
Graph), is depicted in Figure 1. It starts by generating the entities,
followed by a <SEP> token, and ends with the relation triplets, each
consisting of head node, tail node and edge/relation type. During training, we
try two distinct orderings for the graph linearization: sorted and random. As
shown in the Figure 1, the sorted linearization organizes entities and
relations based on their positions in the original text, while the random
linearization randomly shuffles entities and relations order.

## Model Architecture

Our model, ATG, employed an encoder-decoder architecture, which processes the
input text sequence and produces a linearized graph as illustrated in the
Figure 2.

### Encoder

The encoder in ATG utilizes a transformer layer that takes an input text
sequence $\mathbf{x}$ and outputs token representations
$\mathbf{H}\in\mathbb{R}^{L\times D}$, where $D$ is the model dimension.

### Vocabulary Construction

#### Dynamic vocabulary

To enable the pointing mechanism in our decoder, we construct a dynamic
vocabulary matrix $\mathbf{E}$ that includes embeddings for spans, special
tokens, and relation types. While special tokens and relation type embeddings
are randomly initialized and updated during training, the span embeddings are
dynamically computed (Zaratiana et al. 2023), i.e their representations depend
on the input sequence. More specifically, the embedding of a span
$(\text{start},\text{end},\text{type})$ is computed as follows:

$\displaystyle\mathbf{S}_{[\text{start},\text{end},\text{type}]}=\mathbf{W}_{\text{type}}^{T}[\mathbf{h}_{\text{start}}\odot\mathbf{h}_{\text{end}}]$
(1)

In this equation, $[\odot]$ represents a concatenation operation;
$\mathbf{h}_{\text{start}}$ and $\mathbf{h}_{\text{end}}$ denote the
representations of tokens at the start and end positions, respectively.
$\mathbf{W}_{\text{type}}\in\mathbb{R}^{2D\times D}$ is a weight matrix
associated with the entity type (i.e, there is a $\mathbf{W}_{\text{type}}$
for each entity types in a datasets). Finally, the vocabulary embedding matrix
$\mathbf{E}$ is formed by stacking all the span embeddings $\mathbf{S}$,
special token embeddings $\mathbf{T}$, and relation type embeddings
$\mathbf{R}$.

#### Vocabulary size

The size of the vocabulary matrix $\mathbf{E}\in\mathbb{R}^{V\times D}$ is
$V=L\times K\times C+R+T$, where $L$ represents the sequence length, $K$ the
maximum span size, $C$ the number of entity types, $R$ the number of
relations, and $T$ the number of special tokens (<START>, <END>, and <SEP>).
Let’s take the CoNLL 2004 dataset as an example to illustrate this. This
dataset has the following characteristics: $K=12$, $C=4$, $R=5$, and $T=3$.
Considering a sentence of length 114 (which is the maximum length in the
training set), the resulting vocabulary size would be 5480. This size is
considerably smaller when compared with the vocabulary size of a typical
language model, which usually hovers around 30,000 distinct tokens.

Figure 3: Input/ouptut of the decoder. The process starts with the special
token <START> and continues until the <END> token is generated. To separate
the generation of nodes and edges, a special token <SEP> is used. At each
position, the decoder takes in the sum of the embedding of the current token,
absolute position embedding, and structural embedding.

### Decoder

The decoder is a causal transformer trained to predict the next token in the
sequence, akin to traditional language modeling. However, it is important to
note that the vocabulary of our decoder consists of entity spans, relation
types, and special tokens, rather than plain text. The decoder conditions its
predictions on the previously generated tokens $y_{<j}$ using self-attention
and on the input token representations $\mathbf{H}$ using cross-attention.
This enables the decoder to attend to relevant information from both the
previously generated tokens and the input text. Through attention
visualizations, as depicted in Figure 8 and 9, we observed that the model
effectively harnesses both sources of information. Finally, The training
objective aims to maximize the following conditional probability:

$p({\bm{y}}|{\bm{x}})=\prod_{j=1}^{M}p(y_{j}|{\bm{y}}_{<j},{\bm{H}})$ (2)

This is achieved during the training by minimizing the negative log-likelihood
of a reference sequence obtained by linearizing the reference IE graph.
Details about the decoder input and output are given in the subsequent
paragraphs.

#### Decoder input embedding

The embedding step feeds the previous decoder outputs $y_{1},\ldots,y_{i-1}$
into the model using the vocabulary matrix $\mathbf{E}$, along with positional
and structural embeddings as shown in Figure 3. This process can be expressed
as follows:

$\begin{split}{\bm{z}}_{1},\ldots,{\bm{z}}_{i-1}=&{\bm{E}}[y_{1},\ldots,y_{i-1}]\\\
+&{\bm{E}}_{pos}[1,\ldots,i-1]\\\
+&{\bm{E}}_{struct}[y_{1},\ldots,y_{i-1}]\end{split}$ (3)

Here, ${\bm{E}}[y_{1},\ldots,y_{i-1}]$ corresponds to the token embeddings,
which may corresponds to spans, relation types, or special tokens embeddings
depending on the nature of $y_{<i}$. The matrix ${\bm{E}}_{pos}$ represents
absolute positional embedding. It allows to capture the positional information
of the decoder outputs from $y_{1}$ to $y_{i-1}$. Additionally, the structural
embedding ${\bm{E}}_{struct}$ serve as indicators to guide the model in
generating specific elements. In particular, they provide information about
whether the model should generate a Node (before <SEP> tokens), Tail, Head
nodes, or Relation types (as illustrated in Figure 3). Both the absolute
position encoding and the structural embedding are randomly initialized and
updated during training. In summary, by combining these embeddings, ATG can
capture the semantic, positional and structural information about the
linearized graph.

Figure 4: State-Transition diagram for constrained decoding. This diagram
illustrates the state-based decision process used during the inference phase,
which ensures the generation of a correct graph. Each state is represented by
a node, and directed edges indicate valid actions. We use the same color code
as the structural embedding in Figure 3.

#### Decoder output

We define $\tilde{{\bm{z}}_{i}}$ as the hidden state at the last position of
the decoder output sequence obtained by feeding the previous output embedding
and the encoder outputs ${\bm{H}}$ (for cross-attention) to the decoder, i.e,

$\displaystyle\tilde{{\bm{z}}_{i}}=\mathcal{D}ecoder({\bm{z}}_{1},\ldots,{\bm{z}}_{i-1};{\bm{H}})\mathbf{[-1]}$
(4)

Then, to compute the probability distribution over the dynamic vocabulary for
generating the next token, $y_{i}$, our model employs the softmax function on
the dot product between the dynamic vocabulary embedding matrix ${\bm{E}}$ and
$\tilde{{\bm{z}}_{i}}$:

$\displaystyle
p(y_{i}|{\bm{y}}_{<i},{\bm{H}})=\frac{\exp{\bm{E}}^{T}\tilde{{\bm{z}}_{i}}}{\sum_{k=1}^{V}(\exp{\bm{E}}^{T}\tilde{{\bm{z}}_{i}})_{k}}$
(5)

The probabilities generated by this formulation allow the model to select the
appropriate token from the vocabulary for generating a span, special token, or
relation type.

#### Constrained decoding

During inference, we sample from the model by enforcing constraints that
preserve the well-formedness of the output graph. More specifically, during
inference, we feed our model with the <START> token, and the generation
process is guided by state-transition constraints as outlined in Figure 4.
This structured approach ensures that each step in the generation aligns with
the defined template, thereby maintaining the well-formedness of the output
and allowing the production of valid IE graphs. All generations start with the
start token Start state and continue until the <END> token is sampled. In
practice, we also add other constraints: in state 1, we prevent the repetition
of already generated spans, and in state 3, we ensure the tail span is
different from the head. Furthermore, it is also possible to incorporate
domain knowledge into the prediction. For instance, if the type of a head
entity is PER and the tail entity is ORG, the relation can be constrained to
be Work_For (CoNLL 04 dataset).

### Training with Sentence Augmentation

In our work, we observed oversmoothing (Kulikov, Eremeev, and Cho 2022), where
the model prematurely generates the <EOS>, i.e a bias towards short sequences
(Murray and Chiang 2018; Xuewen et al. 2021). We found this bias to harm the
recall of the tasks as the generation terminates before predicting all the
entities/relations. To counteract this, we propose sentence augmentation,
drawing inspiration from Pix2seq (Chen et al. 2022), who encounter a similar
problem for their generative object detection model. This approach forms an
augmented training sample $s_{\text{{aug}}}$ by randomly concatenating
sentences from the training set $\mathcal{D}=\\{s_{1},s_{2},...,s_{N}\\}$:

$s_{\text{{aug}}}=\bigoplus_{k=1}^{n}s_{i_{k}},\quad
i_{k}\sim\mathcal{U}(1,N)\text{, }n\sim\mathcal{U}(1,B)$ (6)

Here, $\mathcal{U}$ denotes the uniform distribution, and $B$ is a
hyperparameter indicating the maximum number of sentences that can be
concatenated. By applying this sentence augmentation technique during
training, the model is exposed to diverse and longer output sequence lengths,
reducing the risk of premature generation of the <EOS> token and thus
improving recall. We perform an ablation study of the effect of sentence
augmentation in our experiments section, showing that it largely improves the
overall performance of our model.

Figure 5: Linearization for different models. In contrast to existing
approaches (TANL (Paolini et al. 2021), UIE (Lu et al. 2022), LasUIE (Fei et
al. 2022)), our proposed model, ATG, generates spans (along with
relation/special tokens) instead of text tokens, which allows for a shorter
output sequence, richer (span-level) representation and fully controlled
decoding.

## Experimental Setup

### Datasets

Dataset | $|\mathcal{E}|$ | $|\mathcal{R}|$ | # Train | # Dev | # Test
---|---|---|---|---|---
ACE05 | 7 | 6 | 10,051 | 2,424 | 2,050
CoNLL 04 | 4 | 5 | 922 | 231 | 288
SciERC | 6 | 7 | 1,861 | 275 | 551
Table 1: The statistics of the datasets. We use ACE04, ACE05, SciERC, and
CoNLL 04 for evaluating end-to-end relation extraction.

We evaluated our model on three benchmark English datasets for joint entity-
relation extraction, namely SciERC (Luan et al. 2018), CoNLL 2004 (Carreras
and Màrquez 2004), and ACE 05 (Walker et al. 2006). The statistics of the
dataset is reported on Table 1.

#### ACE 05

is collected from a variety of domains, such as newswire, online forums and
broadcast news. It provides a diverse set of entity types such as Persons
(PER), Locations (LOC), Geopolitical Entities (GPE), and Organizations (ORG),
along with intricate relation types that include ART (Artifact relationships),
GEN-AFF (General affiliations), and PER-SOC (Personal social relationships).
This dataset is particularly notable for its complexity and wide coverage of
entity and relation types, making it a robust benchmark for evaluating the
performance of IE models.

#### CoNLL 2004

is an annotated corpus collected from newswires and focuses on general
entities such as People, Organizations, and Locations, and relations like
Work_For and Live_in.

#### ScIERC

is a dataset that comes with entity, coreference, and relation annotations for
a collection of documents from 500 AI paper abstracts. The dataset defines
scientific term types and relation types specifically designed for AI domain
knowledge graph construction.

Models | SciERC | ACE 05 | CoNLL 2004
---|---|---|---
ENT | REL | REL+ | ENT | REL | REL+ | ENT | REL | REL+
DYGIE++ (Wadden et al. 2019) | 67.5 | 48.4 | – | 88.6 | 63.4 | – | – | – | –
Tab-Seq (Wang and Lu 2020) | – | – | – | 89.5 | – | 64.3 | 90.1 | 73.8 | 73.6
PURE (Zhong and Chen 2021) | 66.6 | 48.2 | 35.6 | 88.7 | 66.7 | 63.9 | – | – | –
PFN (Yan et al. 2021) | 66.8 | – | 38.4 | 89.0 | – | 66.8 | – | – | –
UniRE (Wang et al. 2021) | 68.4 | – | 36.9 | 89.9 | – | 66.0 | – | – | –
TablERT (Ma, Hiraoka, and Okazaki 2022) | – | – | – | 87.8 | 65.0 | 61.8 | 90.5 | 73.2 | 72.2
GENERATIVE
HySPA (Ren et al. 2021) | – | – | – | 88.9 | 68.2 | – | – | – | –
TANL (Paolini et al. 2021) | – | – | – | 89.0 | – | 63.7 | 90.3 | – | 70.0
ASP (Liu et al. 2022) † | – | – | – | 91.3 | 72.7 | 70.5† | 90.3 | – | 76.3
UIE (Lu et al. 2022) | – | – | 36.5 | – | – | 66.6 | – | 75.0 | –
LasUIE (Fei et al. 2022) | – | – | – | – | – | 66.4 | – | 75.3 | –
ATG (Our model) | 69.7 | 51.1 | 38.6 | 90.1 | 68.7 | 66.2 | 90.5 | 78.5 | 78.5
Table 2: Comparison of our proposed model with state-of-the-art methods.
Results are reported in terms of Entity (ENT) F1, Relation (REL) F1, and
Strict Relation (REL+) F1 scores. The best scores are shown in bold, and the
second-best scores are underlined. † Italic scores use undirected evaluation
for relation extraction and thus are not strictly comparable to our results.

### Evaluation Metrics

For the NER task, we adopt a span-level evaluation requiring precise entity
boundaries and type predictions. To evaluate relations, we use two metrics:
(1) Boundaries evaluation (REL) necessitates the correct prediction of entity
boundaries and relation types; (2) Strict evaluation (REL+) additionally
require accurate entity type prediction. We report the micro-averaged F1
score.

### Baselines

We succinctly and briefly describe here the baseline that we compared with our
model, which we separate into two categories: Span-based/table-filling and
generative IE.

#### Span-based and table-filling

DyGIE++ (Wadden et al. 2019) is a model that uses a pretrained transformer to
compute contextualized representations and employs graph propagation to update
the representations of spans for prediction. Tab-Seq (Wang and Lu 2020)
tackles the task of joint information extraction by treating it as a table
filling problem. PURE (Zhong and Chen 2021) is a pipeline model for the
information extraction task that learns distinct contextual representations
for entities and relations. PFN (Yan et al. 2021) introduces methods that
model two-way interactions between tasks by partitioning and filtering
features. UniRE (Wang et al. 2021) proposes a joint entity and relation
extraction model that eliminates the separation of label spaces for entity
detection and relation classification. Their model uses a unified classifier
to predict labels for each cell in a table of word pairs. In TablERT (Ma,
Hiraoka, and Okazaki 2022), entities and relations are treated as tables, and
the model utilizes two-dimensional CNNs to effectively capture and model local
dependencies.

#### Generative IE

HySPA (Ren et al. 2021) is a model for text-to-graph extraction that has
linear space and time complexity using a Hybrid span generator. TANL (Paolini
et al. 2021) treat the joint IE task as translation from plain text to
augmented natural languages by fine-tuning a T5 model (Raffel et al. 2019).
This model has been further extended by UIE (Lu et al. 2022) and LasUIE (Fei
et al. 2022), which both proposed better linearization and additional
pretraining to enhance results. Finally, ASP (Liu et al. 2022) handles entity
and relation extraction by encoding the target structure as a series of
structure-building actions, using a conditional language model to predict
these actions.

### Hyperparameter Settings

Our model, ATG, employs a transformer encoder-decoder (Vaswani et al. 2017)
architecture. We train it for a maximum of 70k steps using AdamW (Loshchilov
and Hutter 2017) optimizer. We use learning rate warmup for the first 10% of
training and then decay to 0. The base learning rates are 3e-5 for the
encoder, 7e-5 for the decoder, and 1e-4 for other projection layers. Unlike
other generative IE models that utilize pretrained encoder-decoder
architectures, often relying on large models such as T5 (Raffel et al. 2019),
we initialize ATG’s encoder with pre-trained transformer encoders, while the
decoder is randomly initialized. In our preliminary experiments, we observed
that initializing ATG with a pretrained encoder-decoder led to suboptimal
performance. We hypothesize that this is due to our decoder’s utilization of a
dynamic vocabulary, whereas existing pretrained encoder-decoder models have a
fixed token vocabulary, which creates a large discrepancy. We use DeBERTa (He,
Gao, and Chen 2021) for Conll-04 and ACE 05 and SciBERT (Beltagy, Lo, and
Cohan 2019) for SciERC dataset. Across all configurations, the number of
decoder layers is set to 6, though we noted that even a single layer can be
enough in certain cases. The sentence augmentation hyperparameter $B$ is set
to 5.

## Main Results

Table 2 presents the main results of our experiments, along with comparable
approaches from the literature. ATG demonstrates strong performance across all
datasets. On the SciERC dataset, ATG achieves the highest scores across all
metrics. It outperforms the second-best result by 0.2 in REL+ and surpasses
the best generative approach, UIE (Lu et al. 2022), by 2.1 points. On ACE 05,
ATG provide a competitive performance, securing the second-highest scores. The
reported top-performing model, ASP (Liu et al. 2022), operates under a
relaxed, undirected relation evaluation, thereby limiting a fair comparison of
results (Taillé et al. 2021). On the CoNLL 2004 dataset, ATG exhibits its
superiority by outperforming the second-best result by 2.2 in terms of REL+.
Overall, across all three datasets, our proposed model either holds the top
position or showcases strong competitive performance.

Figure 6: Investigation of the effect of different choices on model
performance (REL+). (Left) Effect of the number of decoder layers, (Center)
Impact of Sentence Augmentation, (Right) Study of different values of top-p
for Nucleus Sampling.

## Ablation Studies

#### Number of decoder layers

The number of decoder layers impact is illustrate on the Figure 6. It has a
varying impact on performance across the datasets. In SciERC, increasing the
number of decoder layers leads to a gradual improvement in the performance,
reaching a peak of 38.6 at 6 layer. For ACE 05, the score shows a slight
improvement from 65.3 to 66.2 as the number of decoder layers increases from 1
to 6. For the CoNLL 2004 dataset, the score fluctuates with different numbers
of decoder layers, achieving already strong performance with only a single
layer. Overall, the choice of the number of decoder layers can have a
noticeable impact on REL+ performance but the effect may vary across datasets.

#### Sentence augmentation

The effect of sentence augmentation size on REL+ performance is illustrated in
Figure 6. The results reveal that increasing the number of sentence
augmentations always improves performance across all datasets, except for
CoNLL, where achieving state-of-the-art (SOTA) results is possible with just a
size of 2. However, the absence of sentence augmentation leads to a
significant decrease in REL+, proving its importance.

#### Nucleus sampling

The impact of different top p values in nucleus sampling (Holtzman et al.
2019) on the performance (REL+) is shown in Figure 6. The scores across all
datasets demonstrate a relatively stable trend, with minor variations observed
as the top p value changes. This can be attributed to the application of
constrained decoding, which ensures that the output remains well-formed.
However, the lowest values of top p, corresponding to greedy decoding,
consistently deliver the best performance.

| SciERC | ACE 05 | CoNLL 2004
---|---|---|---
Full | 38.6 | 66.2 | 78.5
\- Pos | 36.4 | 66.0 | 78.3
\- Struct | 36.1 | 65.8 | 78.4
\- Both | 35.4 | 65.4 | 78.0
Table 3: Effect of positional (Pos) and structural embedding (Struct) on REL+.
Figure 7: Impact of sequence ordering on REL+.

#### Positional and structural smbeddings

Table 3 illustrates the importance of positional and structural encoding on
the Relation F1 score. When employing both encoding, ATG achieves the best
performance across all datasets (38.6 for SciERC, 66.2 for ACE 05, and 78.5
for CoNLL 2004). Excluding positional encoding causes only slight performance
drops, since the span representations may contain some positional information.
Omitting structural encoding leads to similar, but slightly larger drops.
Finally, when both are removed, the scores decrease the most, indicating their
importance for the task.

#### Sequence ordering

Figure 7 compares the effects of sorted and random sequence ordering across
different datasets. The results clearly show that the sorted ordering approach
consistently outperforms the random one. The difference in performance is
particularly significant on SciERC and ACE 05, with improvements of 4.7 and
4.9, respectively. On the CoNLL 04 dataset, although the sorted approach still
leads, the difference narrows to 1. Interestingly, we initially hypothesized
that random ordering would deliver better performance, given that any
generation errors in a sorted order could be difficult to rectify.

## Interpretability Analysis

### Attention Maps

Here we analyze the attention of the model during the decoding step, which
allow us to explain some of the model’s decision. We investigate both the
self-attention (Fig. 8) and cross-attention (Fig. 9).

Figure 8: Decoder Self-Attention Visualization. This figure illustrates the
attention patterns among elements in the generated sequence.

#### Self-attention

The self-attention map, shown in Figure 8, depicts the distribution of
attention across preceding tokens during generation. One notable observation
is the model’s tendency to focus on the head and tail entities that comprise
the relation when predicting relation types. For example, when predicting the
Work_For relation, the model allocates most of its attention weight to the
tokens (0,1,Peop) and (4,5,Org).

Figure 9: Decoder Cross-Attention Visualization. This map shows how each
target element in the decoder interacts with and utilizes the original input
text.

#### Cross-attention

The cross-attention map in Figure 9 indicates the specific areas in the input
sequence that the decoded tokens attend to during generation. For entity
labels in the output sequence such as (0,1,Peop), (4,5,Org), and (7,7,Loc), we
can observe higher attention scores for the words Alain, McGill, and Montreal,
respectively, in the input sequence. This indicates that the model tends to
focuses on the beginning of each entity span when generating these entities in
the output sequence. Furthermore, when predicting tail entities for relations,
significant attention is directed toward the prepositions ’at’ and ’in’ in the
input sequence. This suggests that the model has learned to associate these
prepositions with specific relations between entities.

### Learned Structure Embedding

To investigate the impact of the learned structure embedding on model
performance, we analyze the similarity between the structure embeddings
learned during training, depicted in Figure 10. Notably, we observe a
consistent pattern across datasets: the embeddings for the Head and Tail
exhibit a high negative correlation. This finding may suggest that the model
learns to differentiate between the Head and Tail entities, capturing their
distinct characteristics.

Figure 10: Structure embedding similarity. This map shows the cosine
similarity between pairs of structure embedding.

However, we do not have a clear interpretation of this phenomenon. Moreover,
we also report structure embedding values (over the 512 embedding dimensions)
in Figure 11. We observe that the structure embeddings $E_{Head}$ and
$E_{Tail}$ exhibit higher values than the others, which may suggest that
predicting the head and tail entities is the most challenging for the model.

Figure 11: Structure embedding values. This shows the values taken by the
learned structure embeddings.

## Related Works

### Classification-based IE

In the field of information extraction (IE), traditional pipeline models have
been used, consisting of separate stages for entity recognition and relation
extraction (Roth and Yih 2004). Entity recognition is performed to identify
mentioned entities (Chiu and Nichols 2015; Lample et al. 2016; Zaratiana et
al. 2022a, b, 2023), followed by relation extraction to determine the
relationships between these entities (Zelenko, Aone, and Richardella 2002;
Bach and Badaskar 2007; Lin et al. 2016; Wu, Bamman, and Russell 2017).
However, this approach suffers from error propagation, where mistakes in
entity recognition can negatively impact the accuracy of relation extraction
(Brin 1999; Roth and Yih 2004; Nadeau and Sekine 2007). To address these
challenges, there has been a shift towards end-to-end models that jointly
optimize both entity recognition and relation extraction. This joint
optimization aims to harness the interplay between the two tasks, thereby
enhancing overall performance (Sun et al. 2021; Zhao et al. 2021; Ye et al.
2022a). Noteworthy directions in this domain include table-filling methods
(Wang and Lu 2020; Ma, Hiraoka, and Okazaki 2022), span pair classification
(Eberts and Ulges 2019; Wadden et al. 2019), set prediction (Sui et al. 2020),
augmented sequence tagging mechanisms (Ji et al. 2020), fine-grained triplet
classification (Shang, Huang, and Mao 2022), and the use of unified labels for
the task (Wang et al. 2021).

### Generative IE

Recent advancements in generative Information Extraction (IE) emphasize the
use of language models (LMs) to produce entities and relations, either as text
or as a sequence of actions (Paolini et al. 2021; Lu et al. 2022; Nayak and Ng
2020; Liu et al. 2022; Fei et al. 2022; Wan et al. 2023). Typically, these
models employ pretrained encoder-decoder architectures, such as T5 (Raffel et
al. 2019) or BART (Lewis et al. 2020), to encode an input text and
subsequently decode it into a structured output. Their primary advantage over
non-generative methods is their ability to seamlessly integrate tasks by
treating them as a unified generation process. A comprehensive review of this
approach is available in (Ye et al. 2022b; Xu et al. 2023). Beyond entity and
relation extraction, generative models have also found applications in other
IE tasks, including entity linking (Cao et al. 2021), event extraction (Li,
Ji, and Han 2021), and document-level relation extraction (Giorgi, Bader, and
Wang 2022).

### Constrained Decoding

In the generative IE paradigm, the model can in principle generate any
sequence over the LM’s vocabulary if the decoder is not constrained in some
way. One might resort to controlled to bias the model and guide the generation
(Li et al. 2022; Kumar, Paria, and Tsvetkov 2022; Amini, Du, and Cotterell
2023). These approaches still generate a sequence over LM’s vocabulary that
needs to be mapped to an output graph using some kind of a parser that
analyzes the output and extracts a well-formed structure (Paolini et al.
2021). Another approach incorporates constraints explicitly into the decoding
algorithm to restrict the LM’s vocabulary to allowed tokens. For instance, Cao
et al. (2021) and Josifoski et al. (2022) use constrained beam search to force
the output to a set of allowed entities and relations from a knowledge base
schema. Another solution is to build a custom decoder that is constrained to a
tailored vocabulary and decoding algorithm guaranteed to produce a linearized
well-formed structure. Liu et al. (2022) and Lu et al. (2022) use a
specialized decoder with some sort of explicit grammar over a specific
vocabulary to restrict the output to valid sequences. Our work falls in this
category which can be formalized as generating words from a formal language
described by a grammar over a specialized alphabet (Willard and Louf 2023;
Geng et al. 2023).

## Conclusion

In conclusion, our autoregressive text-to-graph framework for joint entity and
relation extraction has demonstrated its effectiveness in achieving state-of-
the-art or competitive results on multiple benchmark datasets. By directly
generating a linearized graph representation instead of plain text, ATG
successfully captures the structural characteristics, boundaries, and
interactions of entities and relations. Moreover, the pointing mechanism on
dynamic vocabulary provides robust grounding in the original text, which also
allows our model’s decoding to be fully controllable.

## Acknowledgments

This work was granted access to the HPC resources of IDRIS under the
allocation 2023-AD011014472 made by GENCI.

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