$0<\frac{\partial\psi_{\lambda}}{\partial\nu}(x,g(x))=\partial_{y}\psi_{\lambda}\sqrt{1+(g^{\prime}(x))^{2}}=\omega\sqrt{1+(g^{\prime}(x))^{2}}\
\ \text{on}\ \ N\setminus A,$

where $\nu=\frac{(g^{\prime}(x),1)}{\sqrt{1+(g^{\prime}(x))^{2}}}$ is the
outer normal vector to $N$. Similarly, we can show that

$\omega>0\ \ \text{on}\ \ N_{0}\setminus(0,g_{0}(0)).$

Next, we will show that

$\omega>0\ \ \text{on}\ \ \Gamma_{\lambda}.$ (4.10)

Suppose that there exists a free boundary point $X_{0}=(x_{0},y_{0})$, such
that $\omega(X_{0})=0$. Without loss of generality, we take $\nu=(0,1)$ as the
outer normal vector of $\Gamma_{\lambda}$ at $X_{0}$. Since $\Gamma_{\lambda}$
is analytic at $X_{0}$, thanks to Hopf’s lemma, one has

$\partial_{xy}\psi_{\lambda}=\frac{\partial\omega}{\partial
x}=\frac{\partial\omega}{\partial\nu}<0\ \ \text{at}\ \ X_{0}.$ (4.11)

On another hand, it follows from $|\nabla\psi_{\lambda}|^{2}=\lambda^{2}x^{2}$
on $\Gamma_{\lambda}$ that

$0=\frac{\partial(\lambda^{2}x^{2})}{\partial
s}=\frac{\partial|\nabla\psi_{\lambda}|^{2}}{\partial
s}=2\partial_{x}\psi_{\lambda}\partial_{xy}\psi_{\lambda}+2\partial_{y}\psi_{\lambda}\partial_{yy}\psi_{\lambda}=2\partial_{x}\psi_{\lambda}\partial_{xy}\psi_{\lambda},$

where $s=(0,1)$ is the tangential vector of $\Gamma_{\lambda}$ at $X_{0}$.
This contradicts to (4.11).

Recalling that $|g^{\prime}(b-0)|<+\infty$, one has

$\omega=\frac{|\nabla\psi_{\lambda}|}{\sqrt{1+(g^{\prime}(x))^{2}}}=\frac{\lambda
x}{\sqrt{1+(g^{\prime}(x))^{2}}}>0\ \ \text{at}\ \ A.$

Hence, we complete the proof of (4.9). In view of (4.10), the implicit
function theorem gives that $k_{\lambda}(x)\in C^{1}((b,\infty))$. The
analyticity of free boundary $\Gamma_{\lambda}$ gives that $k_{\lambda}(x)$ is
analytic in $(b,+\infty)$.

∎

###  4.1 Uniqueness of the compressible subsonic jet flow

In this subsection, we will consider the uniqueness of subsonic solution of
the compressible jet flow problem for any given incoming mass flux $m_{0}$.

######  Theorem 4.2.

For any given $m_{0}>0$, suppose that
$(\psi_{\lambda_{1}},\tilde{\Gamma}_{\lambda_{1}})$ and
$(\tilde{\psi}_{\lambda_{2}},\Gamma_{\lambda_{2}})$ be two subsonic solutions
to compressible impinging jet flow problem, respectively. Then
$\lambda_{1}=\lambda_{2}$ and $\psi_{\lambda_{1}}=\tilde{\psi}_{\lambda_{2}}$.

###### Proof.

We will divide the proof into two steps.

Step 1. We will show that $\lambda_{1}=\lambda_{2}$. If not, without loss of
generality, one may assume that $\lambda_{1}>\lambda_{2}$. In view of the
asymptotic behaviors of $\psi_{\lambda_{1}}$ and $\tilde{\psi}_{\lambda_{2}}$
in downstream (see Step 2 in the proof of Theorem 4.1), one has

$k_{\lambda_{1}}(x)-g_{0}(x)\sim\frac{m_{0}}{\lambda_{1}x\cos\theta}\ \
\text{and}\ \
\tilde{k}_{\lambda_{2}}(x)-g_{0}(x)\sim\frac{m_{0}}{\lambda_{2}x\cos\theta}$

for sufficiently large $x>0$, which implies

$k_{\lambda_{1}}(x)<\tilde{k}_{\lambda_{2}}(x)\ \ \text{for sufficiently large
$x>0$.}$ (4.12)

Denote $\phi_{1}=\frac{m_{0}-\psi_{\lambda_{1}}}{\Pi_{\lambda_{1}}}$ and
$\tilde{\phi}_{2}=\frac{m_{0}-\tilde{\psi}_{\lambda_{2}}}{\Pi_{\lambda_{2}}}$,
where $\Pi_{\lambda_{1}}=\rho_{\lambda_{1},cr}q_{\lambda_{1},cr}$ and
$\Pi_{\lambda_{2}}=\rho_{\lambda_{2},cr}q_{\lambda_{2},cr}$. It is easy to
check that

$\frac{|\nabla\phi_{1}|^{2}}{2x^{2}\rho^{2}}+\frac{\rho^{\gamma-1}}{\gamma-1}=\frac{\gamma+1}{2(\gamma-1)}\
\ \text{in}\ \ \Omega\cap\\{\phi_{1}>0\\},$ (4.13)

and

$\frac{|\nabla\tilde{\phi}_{2}|^{2}}{2x^{2}\rho^{2}}+\frac{\rho^{\gamma-1}}{\gamma-1}=\frac{\gamma+1}{2(\gamma-1)}\
\ \text{in}\ \ \Omega\cap\\{\tilde{\phi}_{2}>0\\},$ (4.14)

where
$\rho(t)=\frac{\rho\left(\frac{t}{\Pi^{2}_{\lambda_{1}}};\lambda^{2}_{1}\right)}{\rho_{\lambda_{1},cr}}=\frac{\rho\left(\frac{t}{\Pi^{2}_{\lambda_{2}}};\lambda^{2}_{2}\right)}{\rho_{\lambda_{2},cr}}$.
Moreover, $\rho=\rho\left(\left|\frac{\nabla\phi}{x}\right|^{2}\right)$ is
monotone decreasing with respect to
$\left|\frac{\nabla\phi}{x}\right|\in[0,1)$. Thus $\phi_{1}$ and
$\tilde{\phi}_{2}$ satisfy the following quasilinear elliptic equations,

$Q\phi_{1}=\text{div}\left(\frac{\nabla\phi_{1}}{x\rho(|\frac{\nabla\phi_{1}}{x}|^{2})}\right)=0\
\ \text{in the fluid $\Omega\cap\\{\phi_{1}>0\\}$},$

and

$Q\tilde{\phi}_{2}=\text{div}\left(\frac{\nabla\tilde{\phi}_{2}}{x\rho(|\frac{\nabla\tilde{\phi}_{2}}{x}|^{2})}\right)=0\
\ \text{in the fluid $\Omega\cap\\{\tilde{\phi}_{2}>0\\}$},$

respectively. Denote
$\tilde{\phi}_{2}^{\varepsilon}(x,y)=\tilde{\phi}_{2}(x,y-\varepsilon)$ for
$\varepsilon\geq 0$, let
$\tilde{\Gamma}_{\lambda_{2}}^{\varepsilon}:y=\tilde{k}_{\lambda_{2}}(x)+\varepsilon$
to be the free boundary of $\tilde{\phi}_{2}^{\varepsilon}$. Choose
$\varepsilon_{0}\geq 0$ to be the smallest one, such that

$\tilde{\phi}_{2}^{\varepsilon_{0}}(X)\geq\phi_{1}(X)\ \text{in $\Omega$,
and}\ \tilde{\phi}_{2}^{\varepsilon_{0}}(X_{0})=\phi_{1}(X_{0})\ \text{for
some $X_{0}\in\overline{\Omega\cap\\{\phi_{1}>0\\}}$}.$

Next, we consider the following two cases for $\varepsilon_{0}$.

Case 1. $\varepsilon_{0}=0$, then we can choose $X_{0}=A$. The strong maximum
principle gives that

$\tilde{\phi}_{2}(X)>\phi_{1}(X)\ \text{and}\ Q\phi_{1}=Q\tilde{\phi}_{2}=0\
\text{in $\Omega\cap\\{\phi_{1}>0\\}$}.$

Since $\Gamma_{\lambda_{1}}\cup N$ and $\tilde{\Gamma}_{\lambda_{2}}\cup N$
are $C^{1}$ at $A$, one has

$\frac{\lambda_{1}}{\Pi_{\lambda_{1}}}=\frac{1}{b}\frac{\partial\phi_{1}}{\partial\nu}\leq\frac{1}{b}\frac{\partial\tilde{\phi}_{2}}{\partial\nu}=\frac{\lambda_{2}}{\Pi_{\lambda_{2}}}\
\ \ \text{at $A$,}$ (4.15)

where $\nu$ is the inner normal vector of $\Gamma_{\lambda_{1}}$ and
$\tilde{\Gamma}_{\lambda_{2}}$ at $A$. After a direct computation, one has

$\frac{d}{d\lambda}\left(\frac{\Pi_{\lambda}}{\lambda}\right)=\frac{\Pi_{\lambda}(\lambda^{2}-\lambda_{cr}^{2})}{(\gamma-1)\lambda\rho^{2}_{0}\mathcal{B}(\lambda^{2})}<0\
\ \text{ for $\lambda<\lambda_{cr}$,}$

which implies that

$\frac{\Pi_{\lambda_{1}}}{\lambda_{1}}<\frac{\Pi_{\lambda_{2}}}{\lambda_{2}}\
\ \text{for any $0<\lambda_{2}<\lambda_{1}<\lambda_{cr}$}.$ (4.16)

This contradicts to (4.15).

Case 2. $\varepsilon_{0}>0$. It follows from strong maximum principle that
$X_{0}\notin\Omega\cap\\{\phi_{1}>0\\}$. In fact, if there exists a point
$X_{0}\in\Omega\cap\\{\phi_{1}>0\\}$, the continuity of $\phi_{1}(X)$ and
$\tilde{\phi}_{2}^{\varepsilon_{0}}(X)$ give that there exists a disc
$B_{r}(X_{0})\subset\Omega\cap\\{\phi_{1}>0\\}$ with $r>0$, such that

$Q\phi_{1}=Q\tilde{\phi}_{2}^{\varepsilon_{0}}=0\ \text{in $B_{r}(X_{0})$,
and}\ \tilde{\phi}_{2}^{\varepsilon_{0}}(X)\geq\phi_{1}(X)\ \text{in
$B_{r}(X_{0})$}.$

Since $\phi_{1}(X_{0})=\tilde{\phi}_{2}^{\varepsilon_{0}}(X_{0})$, the strong
maximum principle implies that
$\phi_{1}(X)\equiv\tilde{\phi}_{2}^{\varepsilon_{0}}(X)$ in $B_{r}(X_{0})$. By
using the strong maximum principle again, one has

$\phi_{1}(X)\equiv\tilde{\phi}_{2}^{\varepsilon_{0}}(X)\ \ \text{ in
$\Omega\cap\\{\phi_{1}>0\\}$},$

which leads a contradiction.

In view of $\varepsilon_{0}>0$, it follows from (3.16) that $|X_{0}|<+\infty$,
and thus
$X_{0}\in\tilde{\Gamma}_{\lambda_{2}}^{\varepsilon_{0}}\cap\Gamma_{\lambda_{1}}$.
Moreover,

$\tilde{\phi}_{2}^{\varepsilon_{0}}(X)>\phi_{1}(X)\ \text{and}\
Q\phi_{1}=Q\tilde{\phi}_{2}^{\varepsilon_{0}}=0\ \text{in
$\Omega\cap\\{\phi_{1}>0\\}$}.$

Since $\Gamma_{\lambda_{1}}$ and
$\tilde{\Gamma}_{\lambda_{2}}^{\varepsilon_{0}}$ are analytic at $X_{0}$, it
follows from Hopf’s lemma that

$\frac{\lambda_{2}}{\Pi_{\lambda_{2}}}=\frac{1}{x}\frac{\partial\tilde{\phi}_{2}^{\varepsilon_{0}}}{\partial\nu}>\frac{1}{x}\frac{\partial\phi_{1}}{\partial\nu}=\frac{\lambda_{1}}{\Pi_{\lambda_{1}}}\
\ \ \text{at $X_{0}$,}$

where $\nu$ is the inner normal vector to
$\tilde{\Gamma}_{\lambda_{2}}^{\varepsilon_{0}}$ and $\Gamma_{\lambda_{1}}$ at
$X_{0}$, which leads a contradiction to the assumption
$\lambda_{1}>\lambda_{2}$, due to (4.16).

Hence, we obtain that $\lambda_{1}=\lambda_{2}$, and denote
$\lambda=\lambda_{1}=\lambda_{2}$ in the following.

Step 2. $\psi_{\lambda}=\tilde{\psi}_{\lambda}$. Suppose that
$\psi_{\lambda}\neq\tilde{\psi}_{\lambda}$, without loss of the generality,
one may assume that there exists some $x_{0}\in(0,+\infty)$, such that

$k_{\lambda}(x_{0})>\tilde{k}_{\lambda}(x_{0})\ \ \text{for some $x_{0}>0$.}$
(4.17)

Consider a function
$\psi_{\lambda}^{\varepsilon}(x,y)=\psi_{\lambda}(x,y-\varepsilon)$ for
$\varepsilon\geq 0$ and
$\Gamma_{\lambda}^{\varepsilon}:y=k_{\lambda}(x)+\varepsilon$ is the free
boundary of $\psi_{\lambda}^{\varepsilon}$, choosing the smallest
$\varepsilon_{0}\geq 0$ such that

$\psi_{\lambda}^{\varepsilon_{0}}(X)\leq\tilde{\psi}_{\lambda}(X)\ \text{in
$\Omega$, and}\
\psi_{\lambda}^{\varepsilon_{0}}(X_{0})=\tilde{\psi}_{\lambda}(X_{0})\
\text{for some $X_{0}\in\bar{\Omega}$}.$

It follows from (4.17) that $\varepsilon_{0}>0$, which together with the
strong maximum principle and the asymptotic behavior imply that
$X_{0}\notin\Omega\cap\\{\tilde{\psi}_{\lambda}<m_{0}\\}$ and
$X_{0}\in\Gamma_{\lambda}^{\varepsilon_{0}}\cap\tilde{\Gamma}_{\lambda}$ with
$|X_{0}|<+\infty$. Then we have

$\psi_{\lambda}^{\varepsilon_{0}}(X)<\tilde{\psi}_{\lambda}(X)\ \text{and}\
Q_{\lambda}\psi_{\lambda}^{\varepsilon_{0}}=Q_{\lambda}\tilde{\psi}_{\lambda}=0\
\text{in $\Omega\cap\\{\tilde{\psi}_{\lambda}<m_{0}\\}$}.$

Thanks to the Hopf’s lemma, one has

$\lambda=\frac{1}{x}\frac{\partial\psi_{\lambda}^{\varepsilon_{0}}}{\partial\nu}>\frac{1}{x}\frac{\partial\tilde{\psi}_{\lambda}}{\partial\nu}=\lambda\
\ \ \text{at $X_{0}$,}$

where $\nu$ is the outer normal vector of
$\Gamma_{\lambda}^{\varepsilon_{0}}\cap\tilde{\Gamma}_{\lambda}$ at $X_{0}$,
which leads a contradiction.

∎

##  5 The existence of the critical mass flux

For any sufficiently small $m_{0}>0$, we have shown that there exist a unique
$\lambda\leq\Pi_{\lambda}-4\tilde{\varepsilon}$ and a unique solution
$(u,v,\rho,\Gamma_{\lambda})$ to the free boundary problem in previous
section. One key point is that the smallness of $m_{0}$ guarantee the global
subsonicity of the compressible jet flow. In this section, we will increase
$m_{0}$ as large as possible, and obtain the critical upper bound of the
incoming mass flux $m_{0}$.

Let $\\{\varepsilon_{n}\\}_{n=1}^{\infty}$ be a strictly decreasing sequence
with $\varepsilon_{n}\downarrow 0$. Denote $\psi^{n}_{\lambda,m}(x,y)$ as the
solution of the following free boundary value problem

$\left\\{\begin{array}[]{ll}&\nabla\cdot\left(\frac{\nabla\psi}{x\rho^{n}(|\frac{\nabla\psi}{x}|^{2};\lambda^{2})}\right)=0\
\ \text{in}\ \Omega\cap\\{\psi<m\\},\\\ &\psi=0\ \text{on}\ T,\ \psi=m\
\text{on}\ N\cup\Gamma^{n}_{\lambda,m},\\\
&\frac{1}{x}\frac{\partial\psi}{\partial\nu}=\lambda\ \ \text{on}\ \
\Gamma^{n}_{\lambda,m},\end{array}\right.$ (5.1)

for any sufficiently small $m>0$ and the free boundary
$\Gamma_{\lambda,m}^{n}:y=k^{n}_{\lambda,m}(x)$ satisfies the continuous fit
condition $k^{n}_{\lambda,m}(b)=1$, where $\nu$ is the outer normal vector.
Here, $\rho^{n}(t;\lambda^{2})$ is a smooth function satisfying

$\rho^{n}(t;\lambda^{2})=\left\\{\begin{array}[]{ll}\rho(t;\lambda^{2})&\text{if}\
\ 0\leq t\leq(\Pi_{\lambda}-2\varepsilon_{n})^{2},\\\
\rho\left((\Pi_{\lambda}-\varepsilon_{n})^{2};\lambda^{2}\right)&\text{if}\ \
t\geq(\Pi_{\lambda}-\varepsilon_{n})^{2},\end{array}\right.$

and $\rho^{n}(t;\lambda)-2\rho_{1}^{n}(t;\lambda)t<\gamma_{n}<+\infty$ with
some constant $\gamma_{n}>0$.

First, we define a set

$\mathcal{K}_{n}(m)=\left\\{\psi_{\lambda,m}^{n}\mid\text{$\psi_{\lambda,m}^{n}$
is a subsonic solution to free boundary problem \eqref{e1}}\right\\},$

for any small $\varepsilon_{n}>0$ and $m>0$. For any small $m>0$ and
$\varepsilon_{n}>0$, it follows from Theorem (4.1) that there exist a
$\lambda=\lambda(m)$ and a unique subsonic solution $\psi^{n}_{\lambda(m),m}$
to the free boundary problem with

$\text{$\lambda(m)\leq\Pi_{\lambda(m)}-4\varepsilon_{n}$ and
$\sup_{X\in\Omega\cap\\{0<\psi^{n}_{\lambda(m),m}<m\\}}\left|\frac{\nabla\psi^{n}_{\lambda(m),m}}{x}\right|^{2}-\Pi_{\lambda(m)}\leq-4\varepsilon_{n}$}.$

Thus $m\in\mathcal{K}_{n}(m)$ for small $m>0$ and the set $\mathcal{K}_{n}(m)$
is not empty. The uniqueness of subsonic solution and $\lambda$ are
established in Section 4. Denote $\lambda=\lambda(m)$ and
$\psi^{n}_{\lambda(m),m}$ as the unique subsonic solution to the free boundary
problem (5.1) with

$\text{$\lambda(m)\leq\Pi_{\lambda(m)}-4\varepsilon_{n}$ and
$\sup_{X\in\Omega\cap\\{0<\psi^{n}_{\lambda(m),m}<m\\}}\left|\frac{\nabla\psi_{\lambda(m),m}^{n}}{x}\right|-\Pi_{\lambda(m)}\leq-4\varepsilon_{n}$}.$

Denote

$T_{n}(m)=\inf_{\psi_{\lambda,m}^{n}\in\mathcal{K}_{n}(m)}\left(\sup_{(x,y)\in\Omega^{n}_{\lambda,m}}\left|\frac{\nabla\psi_{\lambda,m}^{n}}{x}\right|-\Pi_{\lambda}\right),$
(5.2)

where $\Omega^{n}_{\lambda,m}=\Omega\cap\\{0<\psi^{n}_{\lambda,m}<m\\}$.

Define a set

$\begin{array}[]{rl}\Sigma_{n}=\\{m\mid&\text{for any $\tau\in(0,m)$, there
exists a subsonic solution $\psi_{\lambda,m}^{n}$ to the }\\\ &\text{free
boundary problem with $T_{n}(\tau)\leq-4\varepsilon_{n}$}\\}.\end{array}$

Along the above arguments, we have $m\in\Sigma_{n}$ for small $m>0$, which
implies that the set $\Sigma_{n}$ is non-empty. Obviously,
$\Sigma_{n}\subset\Sigma_{n+1}$.

Set

$m_{n}=\sup_{m\in\Sigma_{n}}m.$ (5.3)

The definition of $m_{n}$ implies that $m_{n}$ is monotone increasing with
respect to $n$. For $m>0$, it is easy to check that

$m=\psi_{\lambda,m}^{n}(B)-\psi_{\lambda,m}^{n}(b,g_{0}(b))\leq\sup_{X\in\Omega^{n}_{\lambda,m}}|\nabla\psi_{\lambda,m}^{n}(X)||1-g_{0}(b)|\leq\lambda_{cr}b|1-g_{0}(b)|.$
(5.4)

With the aid of (5.4), we can define

$m_{cr}=\lim_{n\rightarrow+\infty}m_{n}.$ (5.5)

######  Lemma 5.1.

For any $m\in(0,m_{n}]$, $T_{n}(m)$ is left-continuous with respect to $m$,
namely, $T_{n}(m)=\lim_{\tau\rightarrow m^{-}}T_{n}(\tau)$.

###### Proof.

For any $m\in(0,m_{n}]$, there exists a sequence $\\{\tau_{k}\\}$ with
$\tau_{k}\uparrow m$. The definition of $\Sigma_{n}$ gives that there exists a
subsonic solution $\psi^{n}_{\lambda,\tau_{k}}$ to the free boundary problem,
which satisfies that

$T_{n}(\tau_{k})=\inf_{\psi_{\lambda,\tau_{k}}^{n}\in\mathcal{K}_{n}(\tau_{k})}\left(\sup_{X\in\Omega^{n}_{\lambda,\tau_{k}}}\left|\frac{\nabla\psi_{\lambda,\tau_{k}}^{n}(X)}{x}\right|-\Pi_{\lambda}\right)\leq-4\varepsilon_{n}.$

By using the uniqueness result in Section 4, we have that
$\lambda=\lambda(\tau_{k})$ and solution
$\psi^{n}_{\lambda,\tau_{k}}=\psi^{n}_{\lambda(\tau_{k}),\tau_{k}}$ is the
unique subsonic solution to the free boundary problem (5.1). Then one has

$T_{n}(\tau_{k})=\sup_{X\in\Omega^{n}_{\lambda(\tau_{k}),\tau_{k}}}\left|\frac{\nabla\psi^{n}_{\lambda(\tau_{k}),\tau_{k}}(X)}{x}\right|-\Pi_{\lambda(\tau_{k})}\leq-4\varepsilon_{n}.$
(5.6)

By using the similar arguments in the proof of Lemma 2.5, we can take a
subsequence $\\{\tau_{k}\\}$, such that

$\lambda(\tau_{k})\rightarrow\lambda_{0}\leq\Pi_{\lambda_{0}}-4\varepsilon_{n},$

and

$\psi^{n}_{\lambda(\tau_{k}),\tau_{k}}\rightarrow\psi^{n}_{\lambda_{0},m}\
\text{weakly in $H_{loc}^{1}(\Omega)$ and uniformly in any compact subset of
$\mathbb{R}^{2}$},$

as $k\rightarrow+\infty$. Moreover, the inequality (5.6) gives that

$\lim_{\tau_{k}\rightarrow
m}T_{n}(\tau_{k})=\sup_{X\in\Omega^{n}_{\lambda_{0},m}}\left|\frac{\nabla\psi^{n}_{\lambda_{0},m}(X)}{x}\right|-\Pi_{\lambda_{0}}\leq-4\varepsilon_{n}.$

Thus $\psi^{n}_{\lambda_{0},m}$ is a subsonic solution to the free boundary
problem (5.1). Applying the uniqueness results in Section 4, we conclude that
$\lambda_{0}=\lambda(m)$ and
$\psi_{\lambda_{0},m}^{n}=\psi_{\lambda(m),m}^{n}\in\mathcal{K}_{n}(m)$. It
follows from the definition of $T_{n}(m)$ in (5.2) and the uniqueness result
in Section 4 that

$T_{n}(m)=\lim_{\tau_{k}\rightarrow m^{-}}T_{n}(\tau_{k}).$

∎

######  Lemma 5.2.

There exists a critical mass flux $m_{cr}>0$, such that for any
$m\in(0,m_{cr})$, there exist a unique $\lambda=\lambda(m)<\lambda_{cr}$ and a
unique subsonic solution $\psi_{\lambda,m}$ to the free boundary problem
(5.1), such that

$T(m)=\sup_{X\in\Omega_{\lambda,m}}\left|\frac{\nabla\psi_{\lambda,m}(X)}{x}\right|-\Pi_{\lambda}<0,$
(5.7)

where $\Omega_{\lambda,m}=\Omega\cap\\{0<\psi_{\lambda,m}<m\\}$. And $m_{cr}$
is the upper critical mass flux for the existence of subsonic solution in the
following sense: either

$T(m)\rightarrow 0\ \ \text{as}\ \ m\rightarrow m_{cr},$ (5.8)

or there is no $\sigma>0$, such that for any $m\in(m_{cr},m_{cr}+\sigma)$,
there exist a $\lambda<\lambda_{cr}$ and a subsonic solution
$\psi_{\lambda,m}$ to the free boundary problem (5.1), and

$\sup_{m\in(m_{cr},m_{cr}+\sigma)}T(m)<0.$ (5.9)

###### Proof.

For any $m\in(0,m_{cr})$, the definition of $m_{cr}$ in (5.5) gives that there
exists a $N$, such that $m<m_{n}$ for any $n>N$. Therefore, it follows from
the definition of $m_{n}$ that we have

$T_{n}(m)=\inf_{\psi^{n}_{\lambda,m}\in\mathcal{K}_{n}(m)}\left(\sup_{X\in\Omega_{\lambda,m}}\left|\frac{\nabla\psi^{n}_{\lambda,m}(X)}{x}\right|-\Pi_{\lambda}\right)\leq-4\varepsilon_{n}\
\ \text{for}\ \ n>N.$

By virtue of Theorem 4.1, we can conclude that there exist a unique
$\lambda(m)\leq\Pi_{\lambda(m)}-4\varepsilon_{n}$ and a unique subsonic
solution $\psi^{n}_{\lambda(m),m}$ to the free boundary problem (5.1), such
that

$T_{n}(m)=\sup_{X\in\Omega_{\lambda(m),m}}\left|\frac{\nabla\psi^{n}_{\lambda(m),m}(X)}{x}\right|-\Pi_{\lambda(m)}\leq-4\varepsilon_{n}.$

Taking $\psi_{\lambda,m}=\psi^{n}_{\lambda(m),m}$, then $\psi_{\lambda,m}$ is
the unique subsonic solution to the compressible impinging jet flow problem
(5.1) and $T(m)=T_{n}(m)\leq-4\varepsilon_{n}<0$.

If $\sup_{m\in(0,m_{cr})}T(m)<0$, there exists a large $N$, such that

$\sup_{m\in(0,m_{cr})}T(m)<-4\varepsilon_{n}$ (5.10)

for any $n>N$. It is easy to check that $m_{cr}\in\Sigma_{n}$, and thus
$m_{cr}\leq m_{n}$ for any $n>N$.

It follows from Lemma 5.1 that $T_{n}(m)$ is left-continuous for
$m\in(0,m_{n}]$, and thus

$T(m_{cr})=T_{n}(m_{cr})\leq-4\varepsilon_{n}.$ (5.11)

Suppose that there exists a $\sigma>0$, such that for any
$m\in(m_{cr},m_{cr}+\sigma)$, there exists a subsonic solution
$\psi_{\lambda,m}$ with $\lambda<\lambda_{cr}$ to the free boundary problem
(5.1) and

$\sup_{m\in(m_{cr},m_{cr}+\sigma)}T(m)=\sup_{m\in(m_{cr},m_{cr}+\sigma)}\left(\sup_{X\in\Omega_{\lambda,m}}\left|\frac{\nabla\psi_{\lambda,m}(X)}{x}\right|-\Pi_{\lambda}\right)<0.$
(5.12)

In view of (5.12), there exists a large $k>0$, such that

$\sup_{m\in(m_{cr},m_{cr}+\sigma)}T(m)=\sup_{m\in(m_{cr},m_{cr}+\sigma)}\left(\sup_{X\in\Omega_{\lambda,m}}\left|\frac{\nabla\psi_{\lambda,m}(X)}{x}\right|-\Pi_{\lambda}\right)\leq-4\varepsilon_{n+k}.$
(5.13)

By virtue of (5.10), (5.11) and (5.13), one has

$T_{n+k}(m_{cr}+\sigma)=T(m_{cr}+\sigma)\leq\sup_{m\in(0,m_{cr}+\sigma)}T(m)\leq
4\varepsilon_{n+k},$

for any $n>N$, which implies that $m_{cr}+\sigma\in\Sigma_{n+k}$. The
definition of $m_{n+k}$ in (5.3) gives that

$m_{n+k}\geq m_{cr}+\sigma>m_{cr}.$

This leads a contradiction to the definition of $m_{cr}$ in (5.5).

∎

##  6 The proof of the main results

Based on the previous sections, we will complete the proof of Theorem 1.3 and
Theorem 1.4 in this section.

Proof of Theorem 1.3. For any given atmosphere pressure $p_{atm}>0$, it
follows from Lemma 5.2 that there exists a critical mass flux $M_{cr}>0$, such
that for any $M_{0}\in(0,M_{cr})$, there exist a unique
$\lambda=\lambda(m_{0})<\lambda_{cr}$ and a unique subsonic solution
$(\psi_{\lambda,m_{0}},\Gamma_{\lambda,m_{0}})$ to the axially symmetric
compressible impinging flow, where

$M_{cr}=2\pi m_{cr}\ \ \text{and}\ \ M_{0}=2\pi m_{0}.$

In view of the proof of Theorem 4.1, we conclude that $\psi_{\lambda,m_{0}}$
and the free boundary $\Gamma_{\lambda,m_{0}}:y=k_{\lambda,m_{0}}(x)$ satisfy
that

$\psi_{\lambda,m_{0}}\in C^{2,\alpha}(\Omega_{0})\cap
C^{1}(\bar{\Omega}_{0}),\ k_{\lambda,m_{0}}(x)\in C^{1}([b,+\infty)),$

and

$k_{\lambda,m_{0}}(b+0)=1,~{}~{}~{}k_{\lambda,m_{0}}^{{}^{\prime}}(b+0)=g^{\prime}(b-0),\
k^{\prime}_{\lambda,m_{0}}(x)\rightarrow\tan\theta,~{}~{}~{}k_{\lambda,m_{0}}(x)-g_{0}(x)\rightarrow
0$

as $x\rightarrow+\infty$. Moreover,
$\sup_{(x,y)\in\bar{\Omega}_{0}}\frac{|\nabla\psi_{\lambda,m_{0}}|}{x\Pi_{\lambda}}<1$
and $p=p_{atm}$ on $\Gamma_{\lambda,m_{0}}$.

By virtue of the Bernoulli’s law (1.11), one has

$\frac{M^{2}_{0}}{2\pi
a^{4}\rho^{2}_{in}}+\frac{\mathcal{A}\gamma}{\gamma-1}\rho_{in}^{\gamma-1}=\frac{\lambda^{2}}{2\rho_{0}^{2}}+\frac{\mathcal{A}\gamma}{\gamma-1}\rho_{0}^{\gamma-1},\
\ \rho_{0}=\left(\frac{p_{atm}}{\mathcal{A}}\right)^{\frac{1}{\gamma}},$

which implies that the incoming pressure $p_{in}=A\rho^{\gamma}_{in}$ is
determined uniquely by $\lambda$. Moreover, the subsonicity of solution
$\psi_{\lambda,m_{0}}$ gives that $p_{in}\in(p_{1},p_{2})$, where $p_{1}$ and
$p_{2}$ satisfy that $p_{1}<p_{2}$ and

$\frac{M^{2}_{0}}{2\pi^{2}a^{4}\left(\frac{p_{1}}{\mathcal{A}}\right)^{\frac{2}{\gamma}}}+\frac{\mathcal{A}\gamma}{\gamma-1}\left(\frac{p_{1}}{\mathcal{A}}\right)^{\frac{\gamma-1}{\gamma}}=\frac{\mathcal{A}\gamma}{\gamma-1}\left(\frac{p_{atm}}{\mathcal{A}}\right)^{\frac{\gamma-1}{\gamma}}$

and

$\frac{M^{2}_{0}}{2\pi^{2}a^{4}\left(\frac{p_{2}}{\mathcal{A}}\right)^{\frac{2}{\gamma}}}+\frac{\mathcal{A}\gamma}{\gamma-1}\left(\frac{p_{2}}{\mathcal{A}}\right)^{\frac{\gamma-1}{\gamma}}=\frac{\mathcal{A}\gamma(\gamma+1)}{2(\gamma-1)}\left(\frac{p_{atm}}{\mathcal{A}}\right)^{\frac{\gamma-1}{\gamma}}.$

Denote

$u=\frac{1}{x\rho(\frac{|\nabla\psi_{\lambda,m_{0}}|^{2}}{|x|^{2}};\lambda)}\frac{\partial\psi_{\lambda,m_{0}}}{\partial
y},v=-\frac{1}{x\rho(\frac{|\nabla\psi_{\lambda,m_{0}}|^{2}}{|x|^{2}};\lambda)}\frac{\partial\psi_{\lambda,m_{0}}}{\partial
x},\ \ \Gamma=\Omega\cap\\{\psi_{\lambda,m_{0}}<m_{0}\\}$

and the density $\rho$ is determined uniquely by

$\frac{|\nabla\psi_{\lambda,m_{0}}|^{2}}{2x^{2}\rho^{2}}+\frac{\mathcal{A}\gamma}{\gamma-1}\rho^{\gamma-1}=\frac{\lambda^{2}}{2\rho_{0}^{2}}+\frac{\mathcal{A}\gamma}{\gamma-1}\rho_{0}^{\gamma-1}.$

Thus, $(u,v,\rho,\Gamma)$ satisfies the conditions (1)-(3) in Definition 1.1,
and $(u,v,\rho,\Gamma)$ is the unique subsonic solution to FBP.

The statements (1)-(3) of Theorem 1.3 follows from the Step 2 and Step 4 in
the proof of Theorem 4.1 directly. The final statement (4) of Theorem 1.3 is
proved in Section 5.

Hence, we complete the proof of Theorem 1.3.

Proof of Theorem 1.4. By virtue of Theorem 1.3, the subsonic solution
$(\psi_{\lambda,m_{0}},\Gamma_{\lambda,m_{0}})$ established in Theorem 1.3
satisfies that

$\nabla\psi_{\lambda,m_{0}}$ is uniformly continuous in a
$\\{\psi_{\lambda,m_{0}}<m_{0}\\}$-neighborhood of $A$, (6.1)

and

$\Gamma_{\lambda,m_{0}}\cup N$ is $C^{1}$ at $A$. (6.2)

Under the assumption that $N$ is $C^{1,\alpha}$ near $A$, with the aid of
(6.1) and (6.2), the proof of Theorem 1.4 follows from Theorem 1.1 in [ACF6]
directly.

### Acknowledgments

The authors would like to acknowledge the two anonymous reviewers for
improving this paper with their comments. Cheng was supported in part by NSFC
grant 12001387 and the Fundamental Research Funds for the Central Universities
No. YJ202046. Du was supported by NSFC grant 11971331. Zhang was supported by
NSFC grant 12001071.

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# NinjaLLM: Fast, Scalable and Cost-effective RAG using Amazon SageMaker and
AWS Trainium and Inferentia2

Tengfei Xue  NinjaTech AI  Xuefeng Li  NinjaTech AI  Roman Smirnov  NinjaTech
AI  Tahir Azim  NinjaTech AI  Arash Sadrieh  NinjaTech AI  Babak Pahlavan
NinjaTech AI

###### Abstract

Retrieval-augmented generation (RAG) techniques are widely used today to
retrieve and present information in a conversational format. This paper
presents a set of enhancements to traditional RAG techniques, focusing on
large language models (LLMs) fine-tuned and hosted on AWS Trainium /
Inferentia2 AI chips via SageMaker. These chips are characterized by their
elasticity, affordability, and efficient performance for AI compute tasks.
Besides enabling deployment on these chips, this work aims to improve tool
usage, add citation capabilities, and mitigate the risks of hallucinations and
unsafe responses due to context bias. We benchmark our RAG system’s
performance on the Natural Questions and HotPotQA datasets, achieving an
accuracy of 62% and 59% respectively, exceeding other models such as DBRX and
Mixtral Instruct.

##  1 Introduction

Retrieval-augmented generation (RAG) is a vital technique that significantly
enhances language models by integrating external knowledge dynamically. This
approach is transformative for models designed to respond accurately to
complex queries in real-time, enabling them to access and synthesize
information from broader contexts beyond their immediate training data. Such
capabilities are crucial for developing personal assistants that can handle a
diverse range of user queries with high precision and reliability. Deploying
RAG systems today is fraught with challenges. First, training and hosting
these systems, although easily doable, is expensive with Nvidia’s GPUs that
can host their large language models (LLMs).

Second, using third-party LLM APIs such as OpenAI’s GPT-4 is also costly, with
some queries costing $50 to $100 each. Finally, existing LLMs need to be
trained and prompt-engineered to answer complex queries, cite relevant data
sources, and avoid harmful and unsubstantiated answers. In this paper, we
present steps towards solving all of these problems.

First, we demonstrate the use of AWS Trainium/Inferentia2 chips [1] for
hosting LLMs as a compelling alternative to traditional Nvidia GPUs. These
chips offer elasticity, affordability, and efficient inference performance,
making them well-suited for deploying LLMs. Their design allows dynamic
resource scaling to meet demand without significant performance degradation,
which is essential for real-time applications like personal assistants.
Second, we leverage new open-weight models such as Meta’s Llama3-Instruct 70B
as an alternative to OpenAI’s models. This model enables us to support complex
reasoning over multiple documents and refine answers based on a wider context,
making it an excellent base for developing enhanced RAG systems. The
substantial size and design of this model allows it to understand and process
a vast array of information types, making it particularly adept at tasks
requiring multi-step reasoning and deep contextual understanding.

Finally, we fine-tune this model on Trainium chips to enable it to insert
citations, answer multi-hop queries, and avoid unsafe answers and
hallucinations. Combined, these enhancements enable us to achieve state-of-
the-art accuracy of 62% and 59% on the Natural Questions [2] and HotPotQA[4]
datasets, exceeding current systems like DBRX and Mixtral Instruct.

##  2 Model Enhancements

We fine-tuned Llama3-instruct 70B model with a number of enhancements. We
extended their capabilities to leverage external tools within generated
responses and added accountability and traceability features to the
information provided by the models. We trained the model on a diverse set of
data to reduce the likelihood of generating biased responses. Furthermore, we
introduced prompt engineering and model response checking mechanisms to detect
and correct hallucinations and unsafe content dynamically. To fine-tune the
model, we followed the Lima approach [3], which involved using a training
sample size of around 20 million tokens. This approach proved cost-effective,
amounting to less than $1,000. Fine-tuning focused on the format and tone of
the output while using a diverse but relatively small sample size. The model
was fully fine-tuned on a cluster of 32 TRN1 instances, with the entire
training process taking less than three hours. This method ensures the model’s
responses are not only accurate but also appropriately styled and formatted,
enhancing their overall quality and reliability. Additionally, it helps
optimize the prompt length for the model, further improving efficiency and
effectiveness. Given the sensitive nature of fine-tuning to the distribution
of training samples, several trial-and-error attempts are often required to
identify the optimal sample distribution. This aspect of the process can make
the fine-tuning effort particularly bursty, especially during the
experimentation phase. As models react differently to varied data inputs,
finding the right balance is crucial but can lead to multiple iterations of
training sessions. This bursty nature of model training necessitates access to
elastic compute resources, allowing for the dynamic scaling of computational
power to accommodate the varying demands of the training process. This
flexibility is essential to efficiently manage the computational bursts
without incurring unnecessary costs during idle times. Overall, the total cost
of these fine- tuning efforts, including the numerous trials and errors
involved in optimizing the training sample distribution, amounted to less than
$30,000. This figure underscores the importance of a well-managed and flexible
computational infrastructure to support the iterative and sometimes
unpredictable nature of model fine-tuning. The use of GPUs was not possible
due to the lack of on-demand and elastic availability, which is essential for
handling the bursty nature of fine-tuning efforts.

##  3 Model Deployment

We deploy the model using the vLLM inference engine. vLLM introduces
innovative memory management techniques to improve the serving of large
language models, particularly through features like PagedAttention and block-
level memory management. PagedAttention allows for flexible and efficient
memory usage by breaking down the attention mechanism’s memory requirements
into smaller, manageable blocks. This approach reduces internal and external
memory fragmentation, a common issue in traditional model hosting. Block-level
memory management further optimizes memory usage by dynamically allocating and
deallocating blocks based on the sequence lengths and the active decoder head,
ensuring minimal waste and maximal efficiency [5]. Furthermore, we employ
multi- bucketing to address inefficiencies in traditional model hosting,
particularly the high Time to First Token (TTFT) due to prefill operations up
to the maximum sequence length (8192 for Llama3). This technique segments
potential input sizes into different buckets (e.g., 128, 1024), allowing the
use of decoder heads tailored to the nearest bucket size greater than the
actual input length. This method significantly reduces unnecessary
computations and memory usage, as the appropriate bucket is dynamically
selected, avoiding prefill operations for the maximum token length and thereby
reducing TTFT. In conjunction with multi-bucketing, continuous batching
optimizes throughput and efficiency by processing multiple text generation
requests concurrently at the token level. This allows for efficient
utilization of accelerator resources by executing operations on different
requests simultaneously, even if they are at different stages of generation.
For example, some requests might be generating their 5th token while others
their 85th. This dynamic allows requests to join or leave the batch as they
complete their generation, without waiting for the entire batch to finish,
eliminating the need for padding requests to the same length and avoiding idle
time on the accelerator. Through the integration of multi-bucketing,
continuous batching, and vLLMs advanced memory management techniques, latency
and throughput for Llama3 model hosting are significantly improved, setting
the stage for substantial performance improvements.

##  4 Serving Infrastructure

We selected Amazon SageMaker as the primary platform for the deployment and
operational management of our machine learning models, ensuring scalability
and security. SageMaker greatly simplifies the management and auto-scaling of
models, which is crucial for efficiently handling variable computational loads
and optimizing the utilization of computational resources. Through SageMaker,
the machine learning infrastructure is defined using the AWS Cloud Development
Kit (CDK), enhancing the consistency and reproducibility of deployments. The
infrastructure-as-code approach, enabled by SageMaker, allows for models to be
deployed securely across various AWS regions. This is essential for complying
with diverse legal and regulatory requirements, ensuring that models are
deployed within the appropriate geographical and regulatory frameworks.
Moreover, built-in mechanisms for A/B testing and shadowing are provided by
SageMaker, enabling comprehensive evaluations of new models under real-world
operational conditions. This approach allows for seamless comparisons between
new model variants and existing models, ensuring that enhancements in model
performance are substantiated by rigorous, empirical evidence. TRN and INF
instances were the hardware used in the Amazon SageMaker infrastructure for
our project. Their use was instrumental in demonstrating the enhanced
performance and scalability, as well as the cost-effectiveness of serving and
fine-tuning machine learning models in a cloud environment.

##  5 Accuracy Results

We use an approach similar to [6] to measure the accuracy of our enhanced RAG
models. Specifically, we calculate accuracy by matching the model’s answer to
the expected answer, using the top 10 passages retrieved from a Wikipedia
corpus. While both [6] and our model retrieve data from the same corpus, we
perform content filtering and ranking using ColBERTv2 [7] , as opposed to bge-
large-en-v1.5 used in [6].

Table 1 presents the results of our analysis compared to published results for
other existing systems:

| NQ | HotPotQA
---|---|---
GPT 4 Turbo | 63.90% | 62.90%
Ninja LLM | 62.22% | 58.84%
Prompted | 61.52% | 58.31%
LlaMA 3 |  |
DBRX | 60.00% | 55.00%
Mixtral | 59.10% | 54.20%
GPT 3.5 Turbo | 57.70% | 53.00%
LLama2-70B | 56.50% | 54.70%
Table 1: Accuracy of Different Models on Natural Questions and HotPotQA

The results from the benchmark tests show that our enhanced model, Ninja LLM,
performs remarkably well, achieving 62.22% on the NQ Open and 58.84% on the
HotPotQA benchmarks. While it trails the leading GPT 4 Turbo model, it
outperforms other competitive models such as DBRX and Mixtral Instruct,
demonstrating the efficacy of our enhancements and the robustness of the
LLaMA3-instruct 70B base model fine-tuned using AWS Trainium via Amazon
SageMaker. The accuracy improvement is particularly notable in the HotPotQA
benchmark, where complex multi-hop reasoning is required, underscoring the
benefits of our model enhancements.

##  6 Future Work

We plan to explore more advanced techniques for inference, such as speculative
decoding [8] and flash attention [9], to further enhance the model. These
approaches have the potential to significantly reduce latency and increase
throughput for complex decoding tasks and will be particularly beneficial for
extending the model’s applications to agentic workflows and code generation
tasks, where rapid and accurate planning is essential. Using these techniques,
we expect to improve the model’s ability to autonomously generate and refine
plans and code, supporting more sophisticated and context-aware interactions.
This direction will not only push the boundaries of the model’s capabilities
in question-answering but also expand its applicability to a broader range of
AI-driven tasks. Furthermore, we observed a small accuracy gap compared to
leading models like GPT-4 Turbo, indicating potential areas for further
optimization, particularly in refining and expanding training datasets.

##  7 Conclusion

In this study, we presented several enhancements to the LLaMA-3 model within a
retrieval-augmented generation (RAG) framework, aimed at improving its
performance on complex question-answering tasks. We achieved accuracies of
62.22% on the Natural Questions (NQ) Open and 58.84% on HotPotQA by using our
enhanced LLaMA-3 RAG model, demonstrating notable improvements over baseline
models. The model’s enhanced capabilities of multi-hop reasoning and deep
contextual analysis, essential for accurate and reliable question answering,
are highlighted by these results. The model’s computational efficiency was
significantly optimized by the integration of multi-bucketing and continuous
batching techniques. Multi-bucketing reduces unnecessary computations by
dynamically selecting optimal bucket sizes, thereby improving the Time to
First Token (TTFT) References and responsiveness of the model. Continuous
batching helps increase throughput by processing multiple text generation
requests concurrently, optimizing the utilization of computational resources
and adapting efficiently to variable input lengths. On the model itself, we
focused on the safety and reliability of its outputs. We enhanced its
robustness and trustworthiness by training on a diverse dataset and
implementing mechanisms to detect and correct hallucinations and biases. These
improvements are crucial for maintaining the integrity of responses,
especially in sensitive applications where user trust is paramount. However,
despite these advancements, we observed a slight accuracy gap compared to
leading models like GPT-4 Turbo, indicating potential areas for further
optimization, particularly in refining the RAG mechanism and expanding
training datasets.

## References

  * [1] Nafea Bshara. AWS Trainium: The Journey for Designing and Optimization Full Stack ML Hardware. In Proceedings of the 29th ACM International Conference on Architectural Support for Programming Languages and Operating Systems, Volume 3, pp. 4–4, 2024.
  * [2] Tom Kwiatkowski, Jennimaria Palomaki, Olivia Redfield, Michael Collins, Ankur Parikh, Chris Alberti, Danielle Epstein, Illia Polosukhin, Jacob Devlin, Kenton Lee, et al. Natural Questions: A Benchmark for Question Answering Research. Transactions of the Association for Computational Linguistics, 7:453–466, 2019.
  * [3] Chunting Zhou, Pengfei Liu, Puxin Xu, Srinivasan Iyer, Jiao Sun, Yuning Mao, Xuezhe Ma, Avia Efrat, Ping Yu, Lili Yu, and others. “Lima: Less is more for alignment”. Advances in Neural Information Processing Systems, volume 36, 2024.
  * [4] Zhilin Yang, Peng Qi, Saizheng Zhang, Yoshua Bengio, William W. Cohen, Ruslan Salakhutdinov, Christopher D. Manning. HotpotQA: A Dataset for Diverse, Explainable Multi-hop Question Answering. arXiv preprint arXiv:1809.09600, 2018.
  * [5] W. Kwon, Z. Li, S. Zhuang, Y. Sheng, L. Zheng, C. H. Yu, J. Gonzalez, H. Zhang, and I. Stoica. Efficient memory management for large language model serving with paged attention. In Proceedings of the 29th Symposium on Operating Systems Principles, pp. 611-626, 2023.
  * [6] The Mosaic Research Team. Introducing dbrx: A New State-of-the-Art Open LLM. 2024. URL https://www.databricks.com/blog/introducing-dbrx-new-state-art-open-llm. Accessed on April 26, 2024.
  * [7] K. Santhanam, O. Khattab, J. Saad-Falcon, C. Potts, and M. Zaharia. Colbertv2: Effective and efficient retrieval via lightweight late interaction. arXiv preprint arXiv:2112.01488, 2021.
  * [8] Tianle Cai, Yuhong Li, Zhengyang Geng, Hongwu Peng, Jason D. Lee, Deming Chen, and Tri Dao. Medusa: Simple LLM Inference Acceleration Framework with Multiple Decoding Heads. arXiv preprint arXiv:2401.10774, 2024.
  * [9] Tri Dao, Dan Fu, Stefano Ermon, Atri Rudra, and Christopher R’e. Flashattention: Fast and Memory-Efficient Exact Attention with IO-Awareness. Advances in Neural Information Processing Systems, 35:16344–16359, 2022.


# GLSM resolutions of torsional heterotic
$\boldsymbol{\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}$ orbifolds

Stefan Groot Nibbelink  School of Engineering and Applied Sciences, Rotterdam
University of Applied Sciences,
G.J. de Jonghweg 4 - 6, 3015 GG Rotterdam, the Netherlands
Research Centre Innovations in Care, Rotterdam University of Applied Sciences,
Postbus 25035, 3001 HA Rotterdam, the Netherlands
<EMAIL_ADDRESS>
###### Abstract

Heterotic toriodal $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds may
possess discrete torsion between the two defining orbifold twists in the form
of additional cocycle factors in their one–loop partition functions. Using
Gauged Linear Sigma Models (GLSMs) the consequences of discrete torsion can be
uncovered when the orbifold is smoothed out by switching on appropriate blowup
modes. Here blowup modes with twisted oscillator excitations are chosen to
reproduce bundles that are close to the standard embedding without torsion.
The orbifold resolutions with discrete torsion are distinguished from
resolutions without torsion, since they require NS5–branes at their
exceptional cycles.

###### keywords:

Gauged linear sigma models, orbifolds, smooth compactifications, discrete
torsion, heterotic string.

14 November 2023

PACS numbers: 11.15.-q, 11.25.-w, 11.30.Pb, 12.10.-g

##  1 Introduction

String theory may provide the ultimate theory of nature unifying both
gravitational and gauge interactions among chiral fermions. Heterotic string
compactifications using $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds of
six–dimensional tori $T^{6}$ are among the most studied string constructions
to date with appealing phenomenological properties like three generations of
quarks and leptons with the potential to provide realistic Yukawa
interactions. Such orbifold models have been studied both using the free
fermionic formulation, see e.g. refs.[1, 2, 3, 4, 5, 6] , or using bosonic
constructions [7, 8, 9, 10] as well as their smooth resolutions[11] .

One of the theoretically appealing features of toroidal
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds is that they admit an exact
conformal field theory description. In particular, their one loop partition
function can be written down explicitly as a sum over untwisted and twisted
orbifold sectors. A well–known feature of
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds is that they may possess
discrete torsion[12, 13] . In the model with discrete torsion precisely the
conjugate twisted states survive the orbifold projections as compared to the
model without torsion.

This proceedings investigates what happens to the discrete torsion between
orbifold twists if one fully resolves the orbifolds. Gauged Linear Sigma
Models (GLSMs) are used to do so, as they are a convenient framework
interpolating between the singular orbifold limit and smooth
compactifications. This work is therefore closely related to ref. [14] by the
author. (See also the thesis [15] .) The main difference is that in that
reference non–oscillator blowup modes were considered, while in this
proceedings twisted states with oscillators are used as the blowup modes.
Apart from investigating how general the results of that reference are, this
investigation is particular interesting as standard embedding resolutions make
use of oscillator blowup modes.

The proceedings is structured as follows: Section 2 gives a brief introduction
to $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds without and with discrete
torsion and labels the twisted states with oscillators. Section 3 describes
resolutions of the compact $T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ using
a $(0,2)$ GLSM formalism starting from a triple elliptic curve to parameterise
the six–torus. Section 4 gives the GLSM description of the resolved orbifold
with torsion. The proceedings is completed with a discussion section comparing
the results obtained here with existing literature.

##  2 Discrete torsion on $\boldsymbol{\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}$
orbifolds

This section describes some essential properties of heterotic
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds necessary to investigate
their resolutions using GLSM techniques. A more complete review of heterotic
orbifolds can be found, e.g., in refs.[16, 17, 18, 19, 20] in general and in,
e.g., refs.[21, 22, 23, 24, 10] for $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$
orbifolds in particular.

The standard $T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold is defined
by a six–dimensional torus $T^{6}$ with complex coordinates $z=(z_{u})$, where
$u=1,2,3$ labels the three two–torus directions, on which two twists act as

$\displaystyle z\rightarrow e^{2\pi i\,v_{g}}\,z~{},\qquad
v_{g}=t_{1}\,v_{1}+t_{2}\,v_{2}~{},$ (1)

where the twist vectors are taken to be

$\displaystyle
v_{1}=\big{(}0,\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$}\big{)}~{},\qquad
v_{2}=\big{(}{\,\mbox{-}}\mbox{$\frac{1}{2}$},0,\mbox{$\frac{1}{2}$}\big{)}~{}.$
(2)

The labels, $t_{1},t_{2}=0,1,$ distinguish the untwisted ($t_{1}=t_{2}=0$) and
three twisted sectors. These orbifold twists are extended to act as shifts on
the gauge lattice via the gauge shift embedding

$\displaystyle V_{g}=t_{1}\,V_{1}+t_{2}\,V_{2}~{}.$ (3)

The orbifold standard embedding takes the gauge shift vectors identical to the
twist vectors augmented with the appropriate number of zero entries so that
their total number of entries equals 16:

$\displaystyle
V_{1}=\big{(}0,\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}~{},\qquad
V_{2}=\big{(}{\,\mbox{-}}\mbox{$\frac{1}{2}$},0,\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}~{}$
(4)

embedded in the $E_{8}\times E_{8}$ root lattice.

The full one loop partition function is formed out of modular covariant
blocks. These blocks are labeled by constructing elements $g,h$ of the
orbifold group, defining the distinct sectors, and the projecting elements
$g^{\prime},h^{\prime}$, implementing orbifold projections. These elements are
summed to form the full modular invariant partition function. By this
construction the orbifold partition function is uniquely defined up to certain
discrete torsion phases[12, 13] . In particular, the discrete torsion between
the orbifold twists reads

$\displaystyle\Phi^{\\!\times}{}^{t_{1},t_{2}}_{t_{1}^{\prime},t_{2}^{\prime}}=e^{\pi
i\,\varepsilon^{\\!\times}(t_{1}^{\phantom{\prime}}t_{2}^{\prime}-t_{2}^{\phantom{\prime}}t_{1}^{\prime})}~{}.$
(5)

Discrete torsion is absent for $\varepsilon^{\\!\times}=0$ and present for
$\varepsilon^{\\!\times}=1$.

Instead of introducing this non–trivial discrete torsion phase in the full
partition function, one can equivalently consider a so–called brother model
[25] with gauge shift vectors

$\displaystyle
V^{\\!\times}_{1}=-V_{1}=\big{(}0,{\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}~{},\qquad
V^{\\!\times}_{2}=-V_{2}=\big{(}\mbox{$\frac{1}{2}$},0,{\,\mbox{-}}\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}~{},$
(6)

that are opposite to (4) and therefore differ by $E_{8}\times E_{8}$ lattice
vectors.

The shifted momenta

$\displaystyle p_{g}=p+v_{g}~{},\qquad P_{g}=P+V_{g}~{},$ (7)

identify the states in the orbifold spectrum up to possible twisted
left–moving oscillator excitations ${\tilde{\alpha}}{}^{u}_{-1/2}$ and
$\overline{{\tilde{\alpha}}}{}^{u}_{-1/2}$. Here $p\in V_{4}\oplus S_{4}$ and
$P\in(O_{8}\oplus S_{8})\otimes(O_{8}\oplus S_{8})$, where $V_{n}$, $O_{n}$
and $S_{n}$ denote the vector, adjoint and spinor lattices in $n$ dimensions,
respectively.

The shifted momenta of massless states $|p_{g},P_{g}\rangle$ with possible
oscilators in the physical spectrum are characterised by the requirements

$\displaystyle\frac{1}{2}\,p_{g}^{2}=\frac{1}{2}-\delta
c_{g}~{},\qquad\frac{1}{2}\,P_{g}^{2}=1-\delta
c_{g}-\omega_{g}\cdot\widetilde{N}_{g}-{\bar{\omega}}_{g}\cdot\overline{\widetilde{N}}_{g}~{},$
(8)

where the orbifold vacuum shift

$\displaystyle\delta c_{g}=\frac{1}{2}\sum_{u}\omega_{g,u}(1-\omega_{g,u})$
(9)

is defined in terms of $\omega_{g,u}\equiv(v_{g})_{u}$ and
${\bar{\omega}}_{g,u}\equiv-(v_{g})_{u}$ which satisfy the inequalities:
$0<\omega_{g,u},{\bar{\omega}}_{g,u}\leq 1$. Finally, the number operators
$(\widetilde{N}_{g})_{u}$ and $(\overline{\widetilde{N}}_{g})_{u}$ count the
number of left–moving oscillators acting on the state. Only the states that
survive the orbifold projection conditions,

$\displaystyle P_{g}\cdot V_{g^{\prime}}-p_{g}\cdot
v_{g^{\prime}}\equiv\frac{1}{2}\big{(}V_{g}\cdot V_{g^{\prime}}-v_{g}\cdot
v_{g^{\prime}}\big{)}+\big{(}\overline{\widetilde{N}}_{g}-\widetilde{N}_{g}\big{)}\cdot
v_{g^{\prime}}+\frac{\varepsilon^{\\!\times}}{2}\big{(}t_{1}^{\phantom{\prime}}t_{2}^{\prime}-t_{2}^{\phantom{\prime}}t_{1}^{\prime}\big{)}~{},$
(10)

are part of the physical orbifold spectrum. The last term indicates that
discrete torsion only affect the twisted sectors: the resulting twisted
spectra are each others charge conjugates. For the purpose of this proceedings
only the twisted states with oscillator excitations in the three twisted
sectors with $t=(t_{1},t_{2}):(1,0),(0,1)$ and $(1,1)$ are given in Table 2.

There exists a number of different
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds depending on their
underlying six–torus lattice, see e.g. refs.[26, 27, 24, 28, 29] ; here the
orbifold with Hodge numbers $(51,3)$ is considered only. Therefore, each
twisted sector of this orbifold has support at 16 fixed two–tori. The fixed
two–tori of sector (1,0) are labelled by $y,z=1,\ldots,4$, which determine
their positions in the second and third two–tori directions. Similarly, the
positions of the fixed two–tori of sector (0,1) are labelled by
$x,z=1,\ldots,4$ in the first and third two–tori directions, and the positions
of the fixed two–tori of sector (1,1) are labelled by $x,y=1,\ldots,4$ in the
first and second two–tori directions.

This table lists the shifted momenta $p_{g},P_{g}$ of bosonic twisted states
with left–moving oscillator excitations (complex conjugated oscillators are
not made explicit here) and indicates whether they are in the physical
spectrum without or with torsion, $\epsilon^{\\!\times}=0$ or $1$,
respectively.  Sector Bosonic twisted states $|p_{g},P_{g}\rangle$ with
oscillator excitations Repr. $\varepsilon^{\\!\times}=0$ $1$ $(1,0)$
${\tilde{\alpha}}_{-\mbox{$\frac{1}{2}$}}^{2}\Big{|}\big{(}0,\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$}\big{)}\big{(}0,{\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}\Big{\rangle};{\tilde{\alpha}}_{-\mbox{$\frac{1}{2}$}}^{3}\Big{|}\big{(}0,\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$}\big{)}\big{(}0,\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}\Big{\rangle}$
$4\,(1)$ in out
${\tilde{\alpha}}_{-\mbox{$\frac{1}{2}$}}^{2}\Big{|}\big{(}0,\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$}\big{)}\big{(}0,\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}\Big{\rangle};{\tilde{\alpha}}_{-\mbox{$\frac{1}{2}$}}^{3}\Big{|}\big{(}0,\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$}\big{)}\big{(}0,{\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}\Big{\rangle}$
$4\,(1)$ out in $(0,1)$
${\tilde{\alpha}}_{-\mbox{$\frac{1}{2}$}}^{1}\Big{|}\big{(}\mbox{$\frac{1}{2}$},0,\mbox{$\frac{1}{2}$}\big{)}\big{(}{\,\mbox{-}}\mbox{$\frac{1}{2}$},0,\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}\Big{\rangle};{\tilde{\alpha}}_{-\mbox{$\frac{1}{2}$}}^{3}\Big{|}\big{(}\mbox{$\frac{1}{2}$},0,\mbox{$\frac{1}{2}$}\big{)}\big{(}\mbox{$\frac{1}{2}$},0,{\,\mbox{-}}\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}\Big{\rangle}$
$4\,(1)$ in out
${\tilde{\alpha}}_{-\mbox{$\frac{1}{2}$}}^{1}\Big{|}\big{(}\mbox{$\frac{1}{2}$},0,\mbox{$\frac{1}{2}$}\big{)}\big{(}\mbox{$\frac{1}{2}$},0,{\,\mbox{-}}\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}\Big{\rangle};{\tilde{\alpha}}_{-\mbox{$\frac{1}{2}$}}^{3}\Big{|}\big{(}\mbox{$\frac{1}{2}$},0,\mbox{$\frac{1}{2}$}\big{)}\big{(}{\,\mbox{-}}\mbox{$\frac{1}{2}$},0,\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}\Big{\rangle}$
$4\,(1)$ out in $(1,1)$
${\tilde{\alpha}}_{-\mbox{$\frac{1}{2}$}}^{1}\Big{|}\big{(}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},0\big{)}\big{(}{\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},0,0^{5}\big{)}\big{(}0^{8}\big{)}\Big{\rangle};{\tilde{\alpha}}_{-\mbox{$\frac{1}{2}$}}^{2}\Big{|}\big{(}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},0\big{)}\big{(}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},0,0^{5}\big{)}\big{(}0^{8}\big{)}\Big{\rangle}$
$4\,(1)$ in out
${\tilde{\alpha}}_{-\mbox{$\frac{1}{2}$}}^{1}\Big{|}\big{(}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},0\big{)}\big{(}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},0,0^{5}\big{)}\big{(}0^{8}\big{)}\Big{\rangle};{\tilde{\alpha}}_{-\mbox{$\frac{1}{2}$}}^{2}\Big{|}\big{(}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},0\big{)}\big{(}{\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},0,0^{5}\big{)}\big{(}0^{8}\big{)}\Big{\rangle}$
$4\,(1)$ out in

##  3 GLSMs for standard embedding resolutions of
$\boldsymbol{T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}$

A GLSM description of the resolution of toroidal orbifold
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ in the standard embedding can be
obtained by[30] :

  1. 1.

Start from a GLSM description of the underlying six–torus $T^{6}$ compatible
with the $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold symmetries.

  2. 2.

Add so–called exceptional gaugings to introduce the orbifold actions and
define the exceptional cycles.

This program is executed in a $(0,2)$ language, as the torsion models
constructed later cannot be described using $(2,2)$ GLSMs. Therefore, first
$(0,2)$ GLSMs encoding the geometry are determined. Then by insisting on
implicit $(2,2)$ worldsheet supersymmetry, additional superfields, fermionic
gaugings and superpotentials are inferred.

###  3.1 $T^{6}$ GLSM with ${\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}$ symmetries

Gauge charges of the $(0,2)$ Superfields of a $T^{6}$ GLSM with
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ symmetries.  Sfield $\Phi_{u\,1}$
$\Phi_{u\,2}$ $\Phi_{u\,3}$ $\Phi_{u\,4}$ $\Gamma^{\phantom{\prime}}_{u}$
$\Gamma_{u}^{\prime}$ $\Lambda_{u\,1}$ $\Lambda_{u\,2}$ $\Lambda_{u\,3}$
$\Lambda_{u\,4}$ $\Psi^{\phantom{\prime}}_{u}$ $\Psi_{u}^{\prime}$
$R_{u^{\prime}}$ $\mbox{$\frac{1}{2}$}\delta_{u^{\prime}u}$
$\mbox{$\frac{1}{2}$}\delta_{u^{\prime}u}$
$\mbox{$\frac{1}{2}$}\delta_{u^{\prime}u}$
$\mbox{$\frac{1}{2}$}\delta_{u^{\prime}u}$ $-\delta_{u^{\prime}u}$
$-\delta_{u^{\prime}u}$

Following ref. [30, 14] toroidal orbifolds $T^{6}$ with
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ symmetries can be described by the
superfields given in Table 3.1. The chiral superfields $\Phi_{u\,x}$ and the
fermi superfields $\Gamma_{u},\Gamma^{\prime}_{u}$ encode the geometry. The
fermi superfields $\Lambda_{u\,x}$ and the chiral superfields
$\Psi_{u},\Psi^{\prime}_{u}$ describe the gauge bundle in the $(0,2)$
language. By $(2,2)$ supersymmetry the charges of Table 3.1 imply the
fermionic gauge symmetries

$\displaystyle\delta\Lambda_{u\,x}=\mbox{$\frac{1}{2}$}\,\Phi_{u\,x}\,\Xi_{u}~{},\qquad\delta\Gamma_{u}=-\Psi_{u}\,\Xi_{u}~{},\qquad\delta\Gamma_{u}^{\prime}=-\Psi_{u}^{\prime}\,\Xi_{u}~{},$
(11)

where $\Xi_{u}$ are the Fermi superfield gauge parameters. The elliptic curve
descriptions of the three two tori are cut out by the F–term constraints
resulting from the superpotential

$\displaystyle P_{\text{geom
$T^{6}$}}=\sum_{u}\Big{(}\kappa_{u}\,\Phi_{u\,1}^{2}+\Phi_{u\,2}^{2}+\Phi_{u\,3}^{2}\Big{)}\Gamma^{\phantom{\prime}}_{u}+\Big{(}\Phi_{u\,1}^{2}+\Phi_{u\,2}^{2}+\Phi_{u\,4}^{2}\Big{)}\Gamma_{u}^{\prime}~{},$
(12)

where the complex structures $\tau_{u}$ of the three two–tori are
parameterised

$\displaystyle\kappa_{u}=\frac{\wp_{\tau_{u}}(\mbox{$\frac{\tau_{u}}{2}$})-\wp_{\tau_{u}}(\mbox{$\frac{1}{2}$})}{\wp_{\tau_{u}}(\mbox{$\frac{1+\tau_{u}}{2}$})-\wp_{\tau_{u}}(\mbox{$\frac{1}{2}$})}$
(13)

in terms of the Weierstrass $\wp$ function. By $(2,2)$ supersymmetry there is
an accompanying superpotential

$\displaystyle P_{\text{bundle $T^{6}$}}=$
$\displaystyle~{}2\sum_{u}\Big{(}\kappa_{u}\,\Phi_{u\,1}\Lambda_{u\,12}+\Phi_{u\,2}\Lambda_{u\,2}+\Phi_{u\,3}\Lambda_{u\,3}\Big{)}\Psi^{\phantom{\prime}}_{u}$
$\displaystyle+$
$\displaystyle~{}2\sum_{u}\Big{(}\Phi_{u\,1}\Lambda_{u\,1}+\Phi_{u\,2}\Lambda_{u\,2}+\Phi_{u\,4}\Lambda_{u\,3}\Big{)}\Psi_{u}^{\prime}~{}.$
(14)

This expression is obtained by replacing the superfields
$\Gamma_{u},\Gamma_{u}^{\prime}$ by $\Psi_{u},\Psi_{u}^{\prime}$ and applying
the functional operator

$\displaystyle\sum_{u,x}\Lambda_{u\,x}\dfrac{\delta\phantom{\Phi_{u\,x}}}{\delta\Phi_{u\,x}}~{}.$
(15)

###  3.2 Maximal full resolution GLSM for
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$

The maximal full resolution GLSM111There exist a whole family of other
resolution GLSMs in which not all exceptional cycles have exceptional gaugings
associated to them individually. Instead a number of exceptional cycles are
described using the same exceptional gauging. This physically means that the
sizes of these exceptional cycles are identical. The GLSM with the least
number of gaugings which still resolves all singularities is called the
minimal full resolution GLSM. For details see ref.[30] . for the toroidal
orbifold $T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ has three ordinary
gaugings $R_{1}$, $R_{2}$ and $R_{3}$ to define three two–tori and $3\cdot
16=48$ exceptional gaugings $E_{1,yz}$, $E_{2,xz}$ and $E_{3,xy}$ associated
to the exceptional cycles. This gives the most complete description of this
orbifold resolution in the sense that all the Kähler parameters associated to
the volumes of the exceptional cycles are made explicit. The full charge table
is given in Table 3.2.

Gauge charges of the maximal full resolution GLSM for
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$.  Sfield $\Phi_{1\,x}$
$\Phi_{2\,y}$ $\Phi_{3\,z}$ $\Gamma_{1},\Gamma_{1}^{\prime}$
$\Gamma_{2},\Gamma_{2}^{\prime}$ $\Gamma_{3},\Gamma_{3}^{\prime}$
$\Phi_{1\,yz}^{\prime}$ $\Phi_{2\,xz}^{\prime}$ $\Phi_{3\,xy}^{\prime}$
$\Lambda_{1\,x}$ $\Lambda_{2\,y}$ $\Lambda_{3\,z}$
$\Psi_{1},\Psi_{1}^{\prime}$ $\Psi_{2},\Psi_{2}^{\prime}$
$\Psi_{3},\Psi_{3}^{\prime}$ $\Lambda_{1\,yz}^{\prime}$
$\Lambda_{2\,xz}^{\prime}$ $\Lambda_{3\,xy}^{\prime}$ $R_{1}$ $\frac{1}{2}$
$0$ $0$ $-1$ $0$ $0$ $0$ $0$ $0$ $R_{2}$ $0$ $\frac{1}{2}$ $0$ $0$ $-1$ $0$
$0$ $0$ $0$ $R_{3}$ $0$ $0$ $\frac{1}{2}$ $0$ $0$ $-1$ $0$ $0$ $0$
$E_{1\,y^{\prime}z^{\prime}}$ 0 $\mbox{$\frac{1}{2}$}\delta_{y^{\prime}y}$
$\mbox{$\frac{1}{2}$}\delta_{z^{\prime}z}$ $0$ $0$ $0$
$-\delta_{y^{\prime}y}\delta_{z^{\prime}z}$ $0$ $0$
$E_{2\,x^{\prime}z^{\prime}}$ $\mbox{$\frac{1}{2}$}\delta_{x^{\prime}x}$ $0$
$\mbox{$\frac{1}{2}$}\delta_{z^{\prime}z}$ $0$ $0$ $0$ $0$
$-\delta_{x^{\prime}x}\delta_{z^{\prime}z}$ $0$ $E_{3\,x^{\prime}y^{\prime}}$
$\mbox{$\frac{1}{2}$}\delta_{z^{\prime}z}$
$\mbox{$\frac{1}{2}$}\delta_{y^{\prime}y}$ $0$ $0$ $0$ $0$ $0$ $0$
$-\delta_{x^{\prime}x}\delta_{y^{\prime}y}$

This charge table is in line with the description of local orbifold
resolutions using twisted states with oscillators as proposed in ref.[31] :
the shifted momenta $p_{g}$, given in Table 2 for the various twisted sectors,
determine the exceptional charges $E_{r\,xy}$ of the chiral multiplets
$\Phi_{u\,x}$ completed with novel chiral multiplets $\Phi_{r\,xy}^{\prime}$
with charge $-1$ for each of the three twisted sectors $r$ located at the
fixed two–tori labeled by $x,y$. The GLSM charges of the fermi multiplets
cannot directly be determined by the shifted momenta $P_{g}$ given in Table 2.
To obtain the charges of the fermi multiplets $\Lambda_{u\,x}$ one has to add
$+1$ to the negative entry of $P_{g}$ and add novel fermi multiplets
$\Lambda_{r\,xy}^{\prime}$ with charge $-1$. The choice of these signs is such
that the charges of the fermi multiplets $\Lambda_{u\,x}$ and
$\Lambda_{r\,xy}^{\prime}$ match those of the chiral multiplets $\Phi_{u\,x}$
and $\Phi_{r\,xy}^{\prime}$. In order that the number of fermionic degrees of
freedom remain the same, the exceptional gaugings have to be accompanied by
fermionic gauge transformations. Given the charge assignment in Table 3.2 the
following specific form of the fermionic gaugings

$\displaystyle\begin{array}[]{ccc}\delta\Lambda_{1\,x}=\mbox{$\frac{1}{2}$}\,\Phi_{1\,x}\Big{(}\Xi_{1}+\sum\limits_{z}\Xi_{2\,xz}+\sum\limits_{y}\Xi_{3\,xy}\Big{)}~{},&&\delta\Lambda_{1\,yz}^{\prime}=-\Phi_{1\,yz}^{\prime}\,\Xi_{1\,yz}~{},\\\
\delta\Lambda_{2\,y}=\mbox{$\frac{1}{2}$}\,\Phi_{2\,y}\Big{(}\Xi_{2}+\sum\limits_{z}\Xi_{1\,yz}+\sum\limits_{x}\Xi_{3\,xy}\Big{)}~{},&&\delta\Lambda_{2\,xz}^{\prime}=-\Phi_{2\,xz}^{\prime}\,\Xi_{2\,xz}~{},\\\
\delta\Lambda_{3\,z}=\mbox{$\frac{1}{2}$}\,\Phi_{3\,z}\Big{(}\Xi_{3}+\sum\limits_{y}\Xi_{1\,yz}+\sum\limits_{x}\Xi_{2\,xz}\Big{)}~{},&&\delta\Lambda_{3\,xy}^{\prime}=-\Phi_{3\,xy}^{\prime}\,\Xi_{3\,xy}~{},\\\
\delta\Gamma_{u}=-\Psi_{u}\,\Xi_{u}~{},\qquad\delta\Gamma_{u}^{\prime}=-\Psi_{u}^{\prime}\,\Xi_{u}~{},&&\end{array}$
(20)

is dictated by $(2,2)$ supersymmetry. In addition, there are a number of
neutral fermi multiplet $\Lambda^{0}_{p}$, $p=1,\ldots,13$, to complete the
heterotic theory, corresponding to the $5+8=13$ zero entries in the shifted
momenta $P_{g}$ given in Table 2.

One may wonder whether the proposed exceptional charge assignment of the fermi
multiplets $\Lambda_{y\,x}$ and $\Lambda_{r\,xy}^{\prime}$ could not both be
opposite. Indeed, one might as well add $-1$ to the positive entry of $P_{g}$
and add a novel fermi multiplet with charge $+1$. However, by considering the
whole parameter space of bundle deformations one realises that the choice made
above is singled out. For simplicity this argumentation is presented here for
a local resolution GLSM for
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ with chiral superfields
$\Phi_{u}$, $\Phi_{r}^{\prime}$ and fermi multiplets $\Lambda_{u}$,
$\Lambda_{r}^{\prime}$ and $\Lambda_{p}^{0}$ only222In fact, if this logic
would be followed for the maximal full resolution GLSM considered in this
proceedings, one would run out of fermi multiplets in the heterotic theory.
However, the argument would work with the minimal full resolution GLSM
presented in ref.[14] as it has the same number of exceptional gaugings as the
local GLSM used in the argument here.. The local version of the fermionic
gauge transformations (20) are a specific choice of the general form given in
equation (46) of ref.[31] , which for these fields read in general

$\displaystyle\delta\Lambda_{u}=\Phi_{u}\beta_{u\,s}\,\Xi_{s}~{},\qquad\delta\Lambda_{r}^{\prime}=\Phi_{r}^{\prime}\beta_{r\,s}\,\Xi_{s}~{},\qquad\delta\Lambda_{p}^{0}=\beta_{p\,s}\,\Xi_{s}~{},$
(21)

as long as the coefficients $\beta_{u\,s}$, $\beta_{r\,s}$ and $\beta_{p\,s}$
are linear independent. These coefficients parameterise the bundle
deformations. The extreme deformations, for which

$\displaystyle\beta_{u\,s}=\beta_{r\,s}=0\quad\text{and}\quad\beta_{p\,s}=\beta_{s}\,\delta_{p\,s}~{},\quad\beta_{s}\neq
0~{},$ (22)

the fermionic gauge transformations do not involve the fermi superfields
$\Lambda_{u}$ and $\Lambda_{r}^{\prime}$ anymore and they just trivially
remove three neutral fermi multiplets $\Lambda_{s}^{0}$, $s=1,2,3$. These
deformations would thus be interpreted as line bundle GLSMs. However, the
fermi multiplet charge assignments of line bundle GLSMs are uniquely dictated
by the shifted momenta of non–oscillator states[31] , as discussed at length
in ref.[14] . For the local resolution models without torsion they are those
given above.

The Fermi superfields $\Gamma_{1},\Gamma_{1}^{\prime}$,
$\Gamma_{2},\Gamma_{2}^{\prime}$ and $\Gamma_{3},\Gamma_{3}^{\prime}$ feature
in the superpotential to define the three underlying two–tori, see (12). In
light of the exceptional gaugings the superpotential is extended to

$\displaystyle P_{\text{geom\ res}}=\frac{1}{2}\sum\limits_{r\neq s\neq
u}\prod\limits_{y,z}\Big{\\{}\big{(}\\!\kappa_{u}\,\Phi_{u\,1}^{2}\Phi_{r\,1z}^{\prime}\Phi_{s\,1y}^{\prime}+\Phi_{u\,2}^{2}\Phi_{r\,2z}^{\prime}\Phi_{s\,2y}^{\prime}+\Phi_{u\,3}^{2}\Phi_{r\,3z}^{\prime}\Phi_{s\,3y}^{\prime}\big{)}\Gamma_{u}$
$\displaystyle+\big{(}\Phi_{u\,1}^{2}\
\Phi_{r\,1z}^{\prime}\Phi_{s\,1y}^{\prime}+\Phi_{u\,2}^{2}\Phi_{r\,2z}^{\prime}\Phi_{s\,2y}^{\prime}+\Phi_{u\,4}^{2}\Phi_{r\,4z}^{\prime}\Phi_{s\,4y}^{\prime}\big{)}\Gamma_{u}^{\prime}\Big{\\}}$
(23)

together with a bundle superpotential $P_{\text{bundle\ res}}$. It can be
obtained from the geometry superpotential in a similar fashion as (3.1) was
obtained by replacing and superfields $\Gamma_{u},\Gamma_{u}^{\prime}$ by
$\Psi_{u},\Psi_{u}^{\prime}$ and applying the functional operator

$\displaystyle\sum_{u,x}\Lambda_{u\,x}\dfrac{\delta\phantom{\Phi_{u\,x}}}{\delta\Phi_{u\,x}}+\sum_{r,x,y}\Lambda_{r\,xy}^{\prime}\dfrac{\delta\phantom{\Phi_{r\,xy}^{\prime}}}{\delta\Phi_{r\,xy}^{\prime}}~{},$
(24)

which extends the operator (15). Given that (23) is already a bulky
expression, the resulting bundle superpotential is not explicitly given here.

Notice that the resulting GLSM is very similar as the maximal full resolution
GLSM discussed in ref. [14]. However, there the local resolutions could be
thought of as line bundle backgrounds since there were no fermionic gaugings
associated to the exceptional gaugings and no novel fermi multiplets
$\Lambda_{r\,xy}^{\prime}$ were added.

##  4 GLSMs resolutions of
$\boldsymbol{T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}$ with torsion

###  4.1 Construction of GLSM associated to the torsion orbifold

The effect of discrete torsion is that the opposite charged twisted states
survive the orbifold projections. This implies that the exceptional gauge
charges $E_{r\,xy}$ of the fermi multiplets $\Lambda_{u\,x}^{\\!\\!\times}$
and $\Lambda_{r\,xy}^{\\!\\!\times\,\prime}$ in the GLSM associated to the
orbifold with torsion are precisely opposite to those of $\Lambda_{u\,x}$ and
$\Lambda_{r\,xy}^{\prime}$ in the GLSM associated to the orbifold model
without torsion, while all other superfields (in particular $\Phi_{u\,x}$ and
$\Phi_{r\,xy}^{\prime}$) and all other charges (in particular the $R_{u}$
charges of $\Lambda_{u\,x}$ and $\Lambda_{r\,xy}^{\prime}$) are kept the same.
Indeed, by the same local deformation argument as presented in the previous
section, the generic bundle defined by the fermionic gauge transformations can
be deformed to line bundle backgrounds, which are now generated by twisted
states without oscillators with precisely opposite shifted momenta, hence the
corresponding GLSM charges are all inverted.

Here, the superscript $\times$ indicates that the fermi superfields are those
in the model obtained from the orbifold with torsion. Hence, the torsion
resolution GLSM is a genuine $(0,2)$ model that cannot be lifted to a $(2,2)$
theory. To obtain the expressions for the fermionic gauge transformations and
the superpotentials, the following replacements

$\displaystyle\Lambda_{u\,x}\rightarrow\prod_{y}\Phi_{r\,xy}^{\prime-1}\prod_{z}\Phi_{s\,xz}^{\prime-1}\Lambda_{u\,x}^{\\!\\!\times}~{},\qquad\Lambda_{r\,xy}^{\prime}\rightarrow\Phi_{r\,xy}^{\prime
2}\Lambda_{r\,xy}^{\\!\\!\times\,\prime}~{},$ (25)

with $r\neq s\neq u$, should be applied to the appropriate expressions in the
non–torsion theory. Applying this replacement to the fermionic transformation
rules (20) leads, in particular, to the transformations

$\displaystyle\delta\Lambda_{r\,xy}^{\\!\\!\times\,\prime}=-\Phi_{r\,xy}^{\prime\,-1}\,\Xi_{r\,xy}~{},$
(26)

which are not holomorphic at the loci $\Phi_{r\,xy}^{\prime}=0$. However, as
discussed next, these loci are avoided because of logarithmic singularities
that arise at them due to field dependent FI–terms.

###  4.2 Cancellation of mixed anomalies

The sign–flipped $E_{r\,xy}$–gauge charges do not affect the anomaly
cancellation conditions for the $R_{u},R_{u^{\prime}}$ gaugings as well as the
expectional gaugings $E_{r\,xy},E_{r^{\prime}\,x^{\prime}y^{\prime}}$ as these
pure and mixed anomalies are even under these sign-flips. However, the mixed
$R_{u}E_{r\neq u\,xy}$–anomalies do not vanish

$\displaystyle{\cal A}_{u~{}r\,xy}={\cal
A}_{r\,xy~{}u}=\mbox{$\frac{1}{2}$}\cdot\mbox{$\frac{1}{2}$}-(\mbox{$\frac{1}{2}$})\cdot(-\mbox{$\frac{1}{2}$})=\mbox{$\frac{1}{2}$}~{},$
(27)

$u\neq r$. This can be easily confirmed using Table 3.2 taking into account
that the $E_{r\,xy}$ charges of fermi multiplets
$\Lambda_{u\,x}^{\\!\\!\times}$ and $\Lambda_{r\,xy}^{\\!\\!\times}$ have
opposite signs. Hence, the GLSMs associated to the
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold models without or with
torsion are genuinely physically distinct.

In terms of the super field strengths $F_{u}$ and $F_{r\,xy}$ of the $R_{u}$–
and $E_{r\,xy}$–gaugings, the field dependent FI–terms [32, 33]

$\displaystyle W_{\text{FI\,anom}}=\frac{1}{4\pi}\sum_{u\neq
r,x,y}\Big{(}\frac{1-c_{u~{}r\,xy}}{2}\,\log(N_{r\,xy})\,F_{u}+\frac{c_{u~{}r\,xy}}{2}\,\log(N_{u})\,F_{r\,xy}\Big{)}~{},$
(28)

cancel these mixed anomalies provided that the chiral superfield funtions
$N_{r\,xy}$ and $N_{u}$ have negative unit charge under the $E_{r\,xy}$– and
$R_{u}$–gaugings, respectively, and all other gauge charges are zero. Since by
counter terms can shift the two–dimensional mixed anomalies around, these
field dependent FI–terms contain arbitrary coefficients $c_{u~{}r\,xy}$. The
explicit expressions of these functions $N_{r\,xy}$ and $N_{u}$ involve
various parameters that encode the positions of the NS5–branes [32, 33] . A
particularly simple representation is obtained by taking all
$c_{u~{}r\,xy}=0$, so that the functions $N_{u}$ become irrelevant, and set

$\displaystyle N_{r\,xy}=\Phi_{r\,xy}^{\prime}~{}.$ (29)

For this choice the NS5–branes are all located at the resolved exceptional
divisors $E_{r\,xy}$.

##  5 Discusions

This proceedings studied the effect of discrete torsion between the two
orbifold twists of the $T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold
in blowup using a $(0,2)$ GLSM formalism. To this end first the twisted states
that generate the resolution were identified in the orbifold theory. These
states with oscillator excitations were chosen which, in blowup, are described
by GLSMs with exceptional gaugings accompanied by fermionic gauge symmetries.
Consequently, the model without torsion effectively possesses $(2,2)$
supersymmetry.

Since the effect of the discrete torsion is that opposite charged twisted
states survive the orbifold projections, the fermi multiplets in the
resolution GLSMs without and with torsion have opposite exceptional gauge
charges. Even though this might seem insignificant, it results in a number of
mixed anomalies between the exceptional gaugings and the gaugings that are
used in the definition of the elliptic curve descriptions of the underlying
$T^{6}$. These mixed anomalies can be cancelled by field dependent FI–terms
implementing a worldsheet version of the Green–Schwarz mechanism. Since these
FI–terms become logarithmically divergent if some (combination of) chiral
superfields vanish and hence signify that the heterotic worldsheet is
interacting with NS5–branes. For a specific choice of parameters encoding
different realisations of these field dependent FI–terms, these NS5–branes are
located at the exceptional cycles that appeared in the resolution process.

This proceedings is closely related to the publication ref. [14] by the
author. In that work twisted states without oscillator excitations were
considered to generate the resolutions, while in this work twisted states with
oscillators were used. As a consequence, in that work only the minimal full
resolution GLSM could be considered as otherwise one would need more charged
fermi superfields than the theory possesses. This problem did not occur in
this work as GLSMs, generated by twisted states with oscillators, require
fermionic gaugings accompanying the exceptional gaugings of the local blowups.
Therefore, the aforementioned problem did not arise in the present work for
the maximal full resolution GLSM. As far as the distinction between the GLSMs
associated with the orbifold without and with torsion goes, the findings in
this proceedings and ref. [14] essentially coincide: the choice of twisted
states without or with oscillator excitation did not seem to affect the
torsion resolution GLSMs in a significant way. In both cases mixed anomalies
arise which can be cancelled by very similar field dependent FI–interactions
indicating that NS5–branes are present.

## Acknowledgments

The author would like to thank the organisers of the workshop GLSM@30 and the
Simons Center for their kind invitation and for hosting this very pleasant
workshop and the opportunity to contribute to the proceedings of this meeting.
In addition, the author is indebted to Alon Faraggi and Martin Hurtado Heredia
for the inspiring collaboration on which a large part of this work has been
based. Finally, the author would like to thank Benjamin Percival for carefully
reading the manuscript and providing valuable comments.

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