$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\left(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{12}(\phi_{1}+\phi_{2})+\epsilon_{+}^{2}(\phi_{1}+\phi_{2})\right),$
(138) $\displaystyle 6\,\mathcal{F}$
$\displaystyle=6m(\phi_{1}^{2}-\phi_{1}\phi_{2}+\phi_{2}^{2})+3\kappa\phi_{1}\phi_{2}(\phi_{1}-\phi_{2})+8\phi_{1}^{3}-3\phi_{1}^{2}\phi_{2}-3\phi_{1}\phi_{2}^{2}+8\phi_{2}^{3}\,,$

where $m$ is the gauge coupling. The GV-invariant of this theory can be
factorized as

$\displaystyle Z_{GV}(\phi_{i},m;\epsilon_{1},\epsilon_{2})$
$\displaystyle=\mathcal{Z}_{\mathrm{pert}}(\phi_{i};\epsilon_{1},\epsilon_{2})\cdot\mathcal{Z}_{\mathrm{inst}}(\phi_{i},m;\epsilon_{1},\epsilon_{2})\
,$
$\displaystyle\mathcal{Z}_{\mathrm{pert}}(\phi_{i};\epsilon_{1},\epsilon_{2})$
$\displaystyle=\operatorname{PE}\quantity[-\frac{1+p_{1}p_{2}}{(1-p_{1})(1-p_{2})}\quantity(e^{-(2\phi_{1}-\phi_{2})}+e^{-(2\phi_{2}-\phi_{1})}+e^{-(\phi_{1}+\phi_{2})})]\
,$
$\displaystyle\mathcal{Z}_{\mathrm{inst}}(\phi_{i},m;\epsilon_{1},\epsilon_{2})$
$\displaystyle=\sum_{k=0}^{\infty}q^{k}Z_{k}(\phi;\epsilon_{1},\epsilon_{2})\
,$ (139)

with the instanton fugacity $q=e^{-m}$ and $Z_{0}=1$.

We can then write the blowup equation for the $SU(3)_{\kappa}$ theory as

$\displaystyle\sum_{k,k^{\prime}=0}^{\infty}q^{k+k^{\prime}}\Lambda_{k^{\prime}}\hat{Z}_{k}$
$\displaystyle=\sum_{\vec{n}\in\mathbb{Z}^{2}}\sum_{k_{1},k_{2}=0}^{\infty}(-1)^{\absolutevalue{\vec{n}}}e^{-V}\frac{\mathcal{Z}_{\mathrm{pert}}^{(N)}\mathcal{Z}_{\mathrm{pert}}^{(S)}}{\mathcal{Z}_{\mathrm{pert}}}(q\,p_{1}^{B_{m}})^{k_{1}}\hat{Z}^{(N)}_{k_{1}}\cdot(q\,p_{2}^{B_{m}})^{k_{2}}\hat{Z}^{(S)}_{k_{2}}\
.$ (140)

Again, the blowup equation at the $k$-th order can be written as

$\displaystyle\Lambda_{k}(m;\epsilon_{1},\epsilon_{2})+\hat{Z}_{k}(\phi_{i},\epsilon_{1},\epsilon_{2})=$
$\displaystyle~{}p_{1}^{kB_{m}}\hat{Z}_{k}(\phi_{i},\epsilon_{1},\epsilon_{2}-\epsilon_{1})+p_{2}^{kB_{m}}\hat{Z}_{k}(\phi_{i},\epsilon_{1}-\epsilon_{2},\epsilon_{2})$
$\displaystyle+(\text{terms with }Z_{r<k}\text{ and }\Lambda_{r<k})\ .$ (141)

The $\Lambda_{0}$ factor is then fixed to be 1 from the zeroth order
expansion. In $q^{1}$ order, the blowup equation is given by

$\displaystyle\Lambda_{1}+\hat{Z}_{1}(\phi_{i},\epsilon_{1},\epsilon_{2})=~{}$
$\displaystyle
p_{1}^{B_{m}}\hat{Z}_{1}(\phi_{i},\epsilon_{1},\epsilon_{2}-\epsilon_{1})+p_{2}^{B_{m}}\hat{Z}_{1}(\phi_{i},\epsilon_{1}-\epsilon_{2},\epsilon_{2})$
$\displaystyle+\sum_{\vec{n}\in
S_{1}}q^{-1}e^{-V}\frac{\mathcal{Z}_{\mathrm{pert}}^{(N)}\mathcal{Z}_{\mathrm{pert}}^{(S)}}{\mathcal{Z}_{\mathrm{pert}}}\
,$ (142)

where the last term comes from the contribution with $k_{1}=k_{2}=0$ and
$n_{1}^{2}-n_{1}n_{2}+n_{2}^{2}=1$ which is solved for $(n_{1},n_{2})\in
S_{1}=\\{\pm(1,1),\pm(1,0),\pm(0,1)\\}$.

The three choices of $B_{m}$ given in (137) provide three linearly independent
algebraic equations for $\hat{Z}_{k}(\phi_{i},\epsilon_{1},\epsilon_{2})$ at
each $k$-th order. They allow us to find a closed form of
$\hat{Z}_{k}(\phi_{i},\epsilon_{1},\epsilon_{2})$. Thus, the $k$-instanton
partition function $Z_{k}(\phi_{i},\epsilon_{1},\epsilon_{2})$ for the
$SU(3)_{\kappa}$ theory at $|\kappa|\leq 7$ can be computed by recursively
solving three blowup equations with $B_{m}$’s in (137). In Kim:2019uqw , the
partition functions for $|\kappa|\leq 3$ were computed in this manner and
checked against the results from the ADHM calculations.

The instanton partition function for the $SU(3)_{4}$ theory can be computed by
solving the blowup equations similarly to the $\absolutevalue{\kappa}\leq 3$
cases. Using the effective prepotential (138) and consistent magnetic fluxes
(137), we solve the blowup equations in terms of volumes of primitive 2-cycles

$\displaystyle\operatorname{vol}(f_{1})=2\phi_{1}-\phi_{2}\,,\quad\operatorname{vol}(f_{2})=-\phi_{1}+2\phi_{2}\,,\quad\operatorname{vol}(e_{2})=-\phi_{1}+\phi_{2}+m\,.$
(143)

Instead of giving an explicit form of the partition function, we list some BPS
states in Table 5.101010As the BPS spectrum in Table 5 is expressed as the
expansion of 2-cycles, one can reconstruct the partition function given in
terms of dynamical variables, by substituting the 2-cycles with (143).
Likewise, one can also rewrite the partition function given in terms of
dynamical variables as the BPS spectrum, by first converting them into the
2-cycles and then expanding the 2-cycles.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,0,0)$ |  $(0,0)$ | $(1,0,1)$ |  $(0,1)$
$(1,0,2)$ |  $(0,2)$ | $(1,0,3)$ |  $(0,3)$
$(1,1,0)$ |  $(0,0)$ | $(1,1,1)$ |  $(0,0)\oplus(0,1)$
$(1,1,2)$ |  $(0,1)\oplus(0,2)$ | $(1,1,3)$ |  $(0,2)\oplus(0,3)$
$(1,2,1)$ |  $(0,1)$ | $(1,2,2)$ |  $(0,0)\oplus(0,1)\oplus(0,2)$
$(1,2,3)$ |  $(0,1)\oplus(0,2)\oplus(0,3)$ | $(1,3,2)$ |  $(0,1)\oplus(0,2)$
$(1,3,3)$ |  $(0,0)\oplus(0,1)\oplus(0,2)\oplus(0,3)$ | $(2,0,2)$ |  $(0,\frac{5}{2})$
$(2,0,3)$ |  $(0,\frac{5}{2})\oplus(0,\frac{7}{2})\oplus(\frac{1}{2},4)$ | $(2,1,2)$ |  $(0,\frac{3}{2})\oplus(0,\frac{5}{2})$
$(2,1,3)$ |  $(0,\frac{3}{2})\oplus 3(0,\frac{5}{2})\oplus 2(0,\frac{7}{2})\oplus(\frac{1}{2},3)\oplus(\frac{1}{2},4)$ | $(2,2,2)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(0,\frac{5}{2})$
$(2,2,3)$ |  $(0,\frac{1}{2})\oplus 3(0,\frac{3}{2})\oplus 4(0,\frac{5}{2})\oplus 2(0,\frac{7}{2})\oplus(\frac{1}{2},2)\oplus(\frac{1}{2},3)\oplus(\frac{1}{2},4)$ | $(2,3,2)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(0,\frac{5}{2})$
$(2,3,3)$ |  $3(0,\frac{1}{2})\oplus 4(0,\frac{3}{2})\oplus 4(0,\frac{5}{2})\oplus 2(0,\frac{7}{2})\oplus(\frac{1}{2},1)\oplus(\frac{1}{2},2)\oplus(\frac{1}{2},3)\oplus(\frac{1}{2},4)$ | $(3,0,3)$ |  $(0,3)\oplus(\frac{1}{2},\frac{9}{2})$
$(3,1,3)$ |  $(0,2)\oplus 2(0,3)\oplus(0,4)\oplus(\frac{1}{2},\frac{7}{2})\oplus(\frac{1}{2},\frac{9}{2})$ | $(3,2,3)$ |  $(0,1)\oplus 2(0,2)\oplus 3(0,3)\oplus(0,4)\oplus(\frac{1}{2},\frac{5}{2})\oplus(\frac{1}{2},\frac{7}{2})\oplus(\frac{1}{2},\frac{9}{2})$
$(3,3,3)$ |  $\\!(0,0)\oplus 2(0,1)\oplus 3(0,2)\oplus 3(0,3)\oplus(0,4)\oplus(\frac{1}{2},\frac{3}{2})\oplus(\frac{1}{2},\frac{5}{2})\oplus(\frac{1}{2},\frac{7}{2})\oplus(\frac{1}{2},\frac{9}{2})\\!$
Table 5: BPS spectrum of $SU(3)_{4}$ for $d_{i}\leq 3$. Here,
$\mathbf{d}=(d_{1},d_{2},d_{3})$ labels the BPS states with charge
$d_{1}e_{2}+d_{2}f_{1}+d_{3}f_{2}$.

The $SU(3)_{5}$ theory is dual to the $Sp(2)_{\pi}$ gauge theory, so they can
provide a non-trivial check for the validity of the blowup equations. In the
geometry (130) with $\kappa=5$, this duality is realized as the base-fiber
duality exchanging two curve classes $e_{2}$ and $f_{2}$ in $\mathbb{F}_{0}$.
The map between the parameters in the $SU(3)_{5}$ theory and those in the
$Sp(2)_{\pi}$ theory can be easily found from the geometric realization. The
volumes of 2-cycles in the $SU(3)$ frame are

$\displaystyle\operatorname{vol}(f_{1})=2\phi_{1}-\phi_{2},\quad\operatorname{vol}(f_{2})=-\phi_{1}+2\phi_{2},\quad\operatorname{vol}(e_{2})=-2\phi_{1}+2\phi_{2}+m\
,$ (144)

while those in the $Sp(2)$ frame after exchanging $e_{2}$ and $f_{2}$ are

$\displaystyle\operatorname{vol}(f_{1})=2\phi_{1}-\phi_{2},\quad\operatorname{vol}(f_{2})=-2\phi_{1}+2\phi_{2},\quad\operatorname{vol}(e_{2})=-\phi_{1}+2\phi_{2}+m\
.$ (145)

From this, we find a natural map between the parameters in two dual frames as
Gaiotto:2015una ; Hayashi:2016abm ; Hayashi:2018lyv

$\displaystyle\phi_{1}^{SU}$ $\displaystyle=\phi_{1}^{Sp}+\frac{1}{3}m^{Sp}\
,\quad\phi_{2}^{SU}=\phi_{2}^{Sp}+\frac{2}{3}m^{Sp}\ ,\quad
m^{SU}=-\frac{2}{3}m^{Sp}\ .$ (146)

The effective prepotential of the $Sp(2)_{\pi}$ theory on the Coulomb branch
is written as

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\left(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{12}(\phi_{1}+\phi_{2})+\epsilon_{+}^{2}(\phi_{1}+\phi_{2})\right)\
,$ $\displaystyle 6\mathcal{F}$
$\displaystyle=6m(2\phi_{1}^{2}-2\phi_{1}\phi_{2}+\phi_{2}^{2})+8\phi_{1}^{3}+12\phi_{1}^{2}\phi_{2}-18\phi_{1}\phi_{2}^{2}+8\phi_{2}^{3}\
,$ (147)

where $m$ is the $Sp(2)$ gauge coupling. One can easily check that this
$\mathcal{E}$ of the $Sp(2)_{\pi}$ theory coincides with that of the
$SU(3)_{5}$ theory in (138) under the above parameter map up to terms
independent of the dynamical Kähler parameters. The perturbative GV-invariants
of the $Sp(2)_{\pi}$ gauge theory is

$\displaystyle\mathcal{Z}_{\rm pert}={\rm
PE}\left[-\frac{1+p_{1}p_{2}}{(1-p_{1})(1-p_{2})}\left(e^{-2\phi_{1}}+e^{-\phi_{2}}+e^{-(2\phi_{2}-2\phi_{1})}+e^{-(2\phi_{1}-\phi_{2})}\right)\right]\
.$ (148)

In the $Sp(2)$ frame, there are three (and more) sets of consistent fluxes
respecting the $Sp(2)$ structure:

$\displaystyle n_{1},n_{2}\in\mathbb{Z}\ ,\quad B_{m}=-1,0,1\,.$ (149)

The three sets of unity blowup equations from these magnetic fluxes can be
easily solved and the solution provides a closed expression of the instanton
partition function of the $Sp(2)_{\pi}$ theory at each instanton order. We
checked that the result perfectly matches the BPS states captured by the
$SU(3)_{5}$ calculation under the parameter map (146) in the Kähler parameter
expansion. Instead of giving explicit forms of the instanton partition
functions, we list BPS spectrum up to 3-instanton order in Table 6.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,0,0)$ |  $(0,\frac{1}{2})$ | $(1,0,1)$ |  $(0,\frac{3}{2})$
$(1,0,2)$ |  $(0,\frac{5}{2})$ | $(1,1,0)$ |  $(0,\frac{1}{2})$
$(1,1,1)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})$ | $(1,1,2)$ |  $(0,\frac{3}{2})\oplus(0,\frac{5}{2})$
$(1,2,0)$ |  $(0,\frac{1}{2})$ | $(1,2,1)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})$
$(1,2,2)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(0,\frac{5}{2})$ | $(2,0,1)$ |  $(0,\frac{5}{2})$
$(2,0,2)$ |  $(0,\frac{5}{2})\oplus(0,\frac{7}{2})\oplus(\frac{1}{2},4)$ | $(2,1,1)$ |  $(0,\frac{3}{2})\oplus(0,\frac{5}{2})$
$(2,1,2)$ |  $(0,\frac{3}{2})\oplus 3(0,\frac{5}{2})\oplus 2(0,\frac{7}{2})\oplus(\frac{1}{2},3)\oplus(\frac{1}{2},4)$ | $(2,2,1)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(0,\frac{5}{2})$
$(2,2,2)$ |  $(0,\frac{1}{2})\oplus 3(0,\frac{3}{2})\oplus 4(0,\frac{5}{2})\oplus 2(0,\frac{7}{2})\oplus(\frac{1}{2},2)\oplus(\frac{1}{2},3)\oplus(\frac{1}{2},4)$ | $(3,0,1)$ |  $(0,\frac{7}{2})$
$(3,0,2)$ |  $(0,\frac{5}{2})\oplus(0,\frac{7}{2})\oplus 2(0,\frac{9}{2})\oplus(\frac{1}{2},4)\oplus(\frac{1}{2},5)\oplus(1,\frac{11}{2})$ | $(3,1,1)$ |  $(0,\frac{5}{2})\oplus(0,\frac{7}{2})$
$(3,1,2)$ |  $(0,\frac{3}{2})\oplus 3(0,\frac{5}{2})\oplus 5(0,\frac{7}{2})\oplus 3(0,\frac{9}{2})\oplus(\frac{1}{2},3)\oplus 3(\frac{1}{2},4)\oplus 2(\frac{1}{2},5)\oplus(1,\frac{9}{2})\oplus(1,\frac{11}{2})$ | $(3,2,1)$ |  $(0,\frac{3}{2})\oplus(0,\frac{5}{2})\oplus(0,\frac{7}{2})$
$(3,2,2)$ |  $(0,\frac{1}{2})\oplus 3(0,\frac{3}{2})\oplus 8(0,\frac{5}{2})\oplus 7(0,\frac{7}{2})\oplus 4(0,\frac{9}{2})\oplus(\frac{1}{2},2)\oplus 3(\frac{1}{2},3)\oplus 4(\frac{1}{2},4)\oplus 2(\frac{1}{2},5)\oplus(1,\frac{7}{2})\oplus(1,\frac{9}{2})\oplus(1,\frac{11}{2})$
Table 6: BPS spectrum of $SU(3)_{5}$ for $d_{1}\leq 3$ and $d_{2},d_{3}\leq
2$. Here, $\mathbf{d}=(d_{1},d_{2},d_{3})$ labels the BPS states with charge
$d_{1}e_{2}+d_{2}f_{1}+d_{3}f_{2}$.

In the case of $\kappa=6$, one should be careful about the $\Lambda$ factor.
When background magnetic flux is $B_{m}=3/2$, the last term of (3.2.2) in the
Kähler parameter expansion contains a $\phi_{i}$ independent term :

$\displaystyle-p_{1}^{1/2}p_{2}^{1/2}\ \in\ \sum_{\vec{n}\in
S_{1}}q^{-1}e^{-V}\frac{\mathcal{Z}_{\mathrm{pert}}^{(N)}\mathcal{Z}_{\mathrm{pert}}^{(S)}}{\mathcal{Z}_{\mathrm{pert}}}\
.$ (150)

This term should be absorbed into $\Lambda_{1}$ on the left side of (3.2.2).
Thus we have

$\displaystyle\Lambda_{1}=0\quad\text{for}\quad B_{m}=\pm\tfrac{1}{2}\
,\qquad\Lambda_{1}=-p_{1}^{1/2}p_{2}^{1/2}\quad\text{for}\quad
B_{m}=\tfrac{3}{2}\ .$ (151)

Then the solution $Z_{1}(\phi_{i};\epsilon_{1},\epsilon_{2})$ does not contain
$\phi_{i}$ independent terms in the expansion. Similarly, the last term of the
blowup equation at $q^{2}$ order

$\displaystyle\Lambda_{2}(m;\epsilon_{1},\epsilon_{2})+\hat{Z}_{2}(\phi_{i},\epsilon_{1},\epsilon_{2})=$
$\displaystyle~{}p_{1}^{2B_{m}}\hat{Z}_{2}(\phi_{i},\epsilon_{1},\epsilon_{2}-\epsilon_{1})+p_{2}^{2B_{m}}\hat{Z}_{2}(\phi_{i},\epsilon_{1}-\epsilon_{2},\epsilon_{2})$
$\displaystyle+(\text{terms from }\hat{Z}_{1}\text{ and }\Lambda_{1})$ (152)

contains $\phi_{i}$ independent terms for three magnetic fluxes
$B_{m}=-\tfrac{1}{2},\tfrac{1}{2},\tfrac{3}{2}$ as

$\displaystyle\frac{-2p_{1}p_{2}}{(1-p_{1})(1-p_{2})(p_{1}-p_{2})^{2}}\,,\ \
\frac{-2p_{1}^{2}p_{2}^{2}}{(1-p_{1})(1-p_{2})(p_{1}-p_{2})^{2}}\,,\ \
\frac{-F(p_{1},p_{2})}{(1-p_{1})(1-p_{2})(p_{1}-p_{2})^{2}}$ (153)

respectively, where

$\displaystyle F(p_{1},p_{2})=p_{1}p_{2}$
$\displaystyle(-p_{1}^{2}-p_{1}^{4}+2p_{1}p_{2}+p_{1}^{3}p_{2}+p_{1}^{4}p_{2}-p_{2}^{2}$
$\displaystyle+2p_{1}^{2}p_{2}^{2}-p_{1}^{3}p_{2}^{2}+p_{1}p_{2}^{3}-p_{1}^{2}p_{2}^{3}-p_{2}^{4}+p_{1}p_{2}^{4})\
.$ (154)

After absorbing these terms into $\Lambda_{2}$, the blowup equation at $q^{2}$
order is solved consistently. However, when we take the plethystic logarithm
and extract single particle states from the resulting partition function, it
turns out that the partition function contain unexpected $\phi_{i}$
independent term given by

$\displaystyle-\frac{p_{1}p_{2}q^{2}}{(1-p_{1})^{2}(1-p_{2})^{2}}\ \in\
\operatorname{PLog}\quantity[1+qZ_{1}(\phi_{i};\epsilon_{1},\epsilon_{2})+q^{2}Z_{2}(\phi_{i};\epsilon_{1},\epsilon_{2})]\
,$ (155)

where $\operatorname{PLog}$ denotes the plethystic logarithm. The PE of this
term provides some $\phi_{i}$ independent terms and again we put them into the
$\Lambda_{2}$ factor. After all, we find that the $\Lambda_{2}$ factor is
given by

$\displaystyle\Lambda_{2}=0\quad\text{for }B_{m}=\pm\tfrac{1}{2},$
$\displaystyle\Lambda_{2}=-p_{1}p_{2}(p_{1}+p_{2})\quad\text{for}\quad
B_{m}=\tfrac{3}{2}\ .$ (156)

In this way, we can solve the blowup equation iteratively while absorbing all
$\phi_{i}$ independent terms into the $\Lambda$ factor. The resulting BPS
spectrum of the $SU(3)_{6}$ theory is given in Table 7.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,0,0)$ |  $(0,0)$ | $(1,0,1)$ |  $(0,1)$
$(1,0,2)$ |  $(0,2)$ | $(1,0,3)$ |  $(0,3)$
$(1,1,1)$ |  $(0,0)\oplus(0,1)$ | $(1,1,2)$ |  $(0,1)\oplus(0,2)$
$(1,1,3)$ |  $(0,2)\oplus(0,3)$ | $(1,2,0)$ |  $(0,0)$
$(1,2,1)$ |  $(0,0)\oplus(0,1)$ | $(1,2,2)$ |  $(0,0)\oplus(0,1)\oplus(0,2)$
$(1,2,3)$ |  $(0,1)\oplus(0,2)\oplus(0,3)$ | $(1,3,1)$ |  $(0,1)$
$(1,3,2)$ |  $(0,1)\oplus(0,2)$ | $(1,3,3)$ |  $(0,0)\oplus(0,1)\oplus(0,2)\oplus(0,3)$
$(2,0,2)$ |  $(0,\frac{5}{2})$ | $(2,0,3)$ |  $(0,\frac{5}{2})\oplus(0,\frac{7}{2})\oplus(\frac{1}{2},4)$
$(2,1,2)$ |  $(0,\frac{3}{2})\oplus(0,\frac{5}{2})$ | $(2,1,3)$ |  $(0,\frac{3}{2})\oplus 3(0,\frac{5}{2})\oplus 2(0,\frac{7}{2})\oplus(\frac{1}{2},3)\oplus(\frac{1}{2},4)$
$(2,2,1)$ |  $(0,\frac{1}{2})$ | $(2,2,2)$ |  $(0,\frac{1}{2})\oplus 2(0,\frac{3}{2})\oplus(0,\frac{5}{2})$
$(2,2,3)$ |  $(0,\frac{1}{2})\oplus 3(0,\frac{3}{2})\oplus 5(0,\frac{5}{2})\oplus 2(0,\frac{7}{2})\oplus(\frac{1}{2},2)\oplus(\frac{1}{2},3)\oplus(\frac{1}{2},4)$ | $(2,3,1)$ |  $(0,\frac{1}{2})$
$(2,3,2)$ |  $2(0,\frac{1}{2})\oplus 2(0,\frac{3}{2})\oplus(0,\frac{5}{2})$ | $(2,3,3)$ |  $3(0,\frac{1}{2})\oplus 5(0,\frac{3}{2})\oplus 5(0,\frac{5}{2})\oplus 2(0,\frac{7}{2})\oplus(\frac{1}{2},1)\oplus(\frac{1}{2},2)\oplus(\frac{1}{2},3)\oplus(\frac{1}{2},4)$
$(3,0,3)$ |  $(0,3)\oplus(\frac{1}{2},\frac{9}{2})$ | $(3,1,3)$ |  $(0,2)\oplus 2(0,3)\oplus(0,4)\oplus(\frac{1}{2},\frac{7}{2})\oplus(\frac{1}{2},\frac{9}{2})$
$(3,2,2)$ |  $(0,2)$ | $(3,2,3)$ |  $(0,1)\oplus 3(0,2)\oplus 5(0,3)\oplus(0,4)\oplus(\frac{1}{2},\frac{5}{2})\oplus 2(\frac{1}{2},\frac{7}{2})\oplus(\frac{1}{2},\frac{9}{2})$
$(3,3,2)$ |  $(0,1)\oplus(0,2)$ | $(3,3,3)$ |  $(0,0)\oplus 3(0,1)\oplus 7(0,2)\oplus 6(0,3)\oplus(0,4)\oplus(\frac{1}{2},\frac{3}{2})\oplus 2(\frac{1}{2},\frac{5}{2})\oplus 2(\frac{1}{2},\frac{7}{2})\oplus(\frac{1}{2},\frac{9}{2})$
Table 7: BPS spectrum of the $SU(3)_{6}$ theory for $d_{i}\leq 3$. Here,
$\mathbf{d}=(d_{1},d_{2},d_{3})$ labels the BPS states with charge
$d_{1}e_{2}+d_{2}f_{1}+d_{3}f_{2}$.

When $\kappa=7$, the fiber-base duality of $\mathbb{F}_{0}$ in the geometry
(130) gives the $G_{2}$ gauge theory description. The $G_{2}$ instanton
partition function can be calculated using the ADHM-like method in Kim:2018gjo
or using the topological vertex in Hayashi:2018bkd . Here, we will present the
partition function computation of this theory using blowup equations in both
the $SU(3)$ and $G_{2}$ descriptions. In the $SU(3)$ frame, the volumes of
2-cycles in the geometry are

$\displaystyle\operatorname{vol}(f_{1})$ $\displaystyle=2\phi_{1}-\phi_{2}\,,$
$\displaystyle\operatorname{vol}(f_{2})$
$\displaystyle=-\phi_{1}+2\phi_{2}\,,$
$\displaystyle\operatorname{vol}(e_{2})$
$\displaystyle=-3\phi_{1}+2\phi_{2}+m\ .$ (157)

On the other hand, in the $G_{2}$ frame after exchanging the fiber and the
base of $\mathbb{F}_{0}$, the volumes are

$\displaystyle\operatorname{vol}(f_{1})$ $\displaystyle=2\phi_{1}-\phi_{2}\,,$
$\displaystyle\operatorname{vol}(f_{2})$
$\displaystyle=-3\phi_{1}+2\phi_{2}\,,$
$\displaystyle\operatorname{vol}(e_{2})$ $\displaystyle=-\phi_{1}+2\phi_{2}+m\
.$ (158)

The parameters in these two descriptions are thus mapped each other as
Hayashi:2018lyv

$\displaystyle\phi_{1}^{SU}$
$\displaystyle=\phi_{1}^{G_{2}}+\frac{1}{3}m_{0}^{G_{2}}\,,\quad\phi_{2}^{SU}=\phi_{2}+\frac{2}{3}m_{0}^{G_{2}}\,,\quad
m^{SU}=-\frac{1}{3}m^{G_{2}}\,.$ (159)

In the $G_{2}$ description, the effective prepotential on the Coulomb branch
where $2\phi_{1}-\phi_{2}>0$ and $2\phi_{2}-3\phi_{1}>0$ with
$\phi_{1},\phi_{2}>0$ is given by

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\left(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{12}(\phi_{1}+\phi_{2})+\epsilon_{+}^{2}(\phi_{1}+\phi_{2})\right)\,,$
$\displaystyle 6\mathcal{F}$
$\displaystyle=6m\quantity(3\phi_{1}^{2}-3\phi_{1}\phi_{2}+\phi_{2}^{2})+8\phi_{1}^{3}+18\phi_{1}^{2}\phi_{2}-24\phi_{1}\phi_{2}^{2}+8\phi_{2}^{2}\,.$
(160)

One can check that this agrees with the $SU(3)_{7}$ effective prepotential
under the above parameter map up to terms independent of $\phi_{i}$. The
perturbative GV-invariants of the $G_{2}$ gauge theory is

$\displaystyle\mathcal{Z}_{\rm pert}$ $\displaystyle={\rm
PE}\bigg{[}-\frac{1+p_{1}p_{2}}{(1-p_{1})(1-p_{2})}\left(e^{-\phi_{1}}+e^{-\phi_{2}}+e^{-(3\phi_{1}-\phi_{2})}+e^{-(2\phi_{1}-\phi_{2})}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\left.+e^{-(\phi_{2}-\phi_{1})}+e^{-(2\phi_{2}-3\phi_{1})}\right)\bigg{]}\
.$ (161)

There are again three sets of consistent magnetic fluxes respecting the
$G_{2}$ structure:

$\displaystyle n_{1},n_{2}\in\mathbb{Z}\ ,\quad B_{m}=-1,0,1\ .$ (162)

We computed the unity blowup equations built from these fluxes and checked
that two results from the $SU(3)_{7}$ theory and the $G_{2}$ theory perfectly
agree with each other in the Kähler parameter expansion. The BPS spectrum of
this theory can be found in Table 8.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,0,0)$ |  $(0,\frac{1}{2})$ | $(1,0,1)$ |  $(0,\frac{3}{2})$
$(1,0,2)$ |  $(0,\frac{5}{2})$ | $(1,1,0)$ |  $(0,\frac{1}{2})$
$(1,1,1)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})$ | $(1,1,2)$ |  $(0,\frac{3}{2})\oplus(0,\frac{5}{2})$
$(1,2,0)$ |  $(0,\frac{1}{2})$ | $(1,2,1)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})$
$(1,2,2)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(0,\frac{5}{2})$ | $(2,0,1)$ |  $(0,\frac{5}{2})$
$(2,0,2)$ |  $(0,\frac{5}{2})\oplus(0,\frac{7}{2})\oplus(\frac{1}{2},4)$ | $(2,1,1)$ |  $(0,\frac{3}{2})\oplus(0,\frac{5}{2})$
$(2,1,2)$ |  $(0,\frac{3}{2})\oplus 3(0,\frac{5}{2})\oplus 2(0,\frac{7}{2})\oplus(\frac{1}{2},3)\oplus(\frac{1}{2},4)$ | $(2,2,1)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(0,\frac{5}{2})$
$(2,2,2)$ |  $(0,\frac{1}{2})\oplus 3(0,\frac{3}{2})\oplus 4(0,\frac{5}{2})\oplus 2(0,\frac{7}{2})\oplus(\frac{1}{2},2)\oplus(\frac{1}{2},3)\oplus(\frac{1}{2},4)$ | $(3,0,1)$ |  $(0,\frac{7}{2})$
$(3,0,2)$ |  $(0,\frac{5}{2})\oplus(0,\frac{7}{2})\oplus 2(0,\frac{9}{2})\oplus(\frac{1}{2},4)\oplus(\frac{1}{2},5)\oplus(1,\frac{11}{2})$ | $(3,1,1)$ |  $(0,\frac{5}{2})\oplus(0,\frac{7}{2})$
$(3,1,2)$ |  $(0,\frac{3}{2})\oplus 3(0,\frac{5}{2})\oplus 5(0,\frac{7}{2})\oplus 3(0,\frac{9}{2})\oplus(\frac{1}{2},3)\oplus 3(\frac{1}{2},4)\oplus 2(\frac{1}{2},5)\oplus(1,\frac{9}{2})\oplus(1,\frac{11}{2})$ | $(3,2,1)$ |  $(0,\frac{3}{2})\oplus(0,\frac{5}{2})\oplus(0,\frac{7}{2})$
$(3,2,2)$ |  $(0,\frac{1}{2})\oplus 3(0,\frac{3}{2})\oplus 8(0,\frac{5}{2})\oplus 7(0,\frac{7}{2})\oplus 4(0,\frac{9}{2})\oplus(\frac{1}{2},2)\oplus 3(\frac{1}{2},3)\oplus 4(\frac{1}{2},4)\oplus 2(\frac{1}{2},5)\oplus(1,\frac{7}{2})\oplus(1,\frac{9}{2})\oplus(1,\frac{11}{2})$
Table 8: BPS spectrum of the $SU(3)_{7}$ theory for $d_{1}\leq 3$ and
$d_{2},d_{3}\leq 2$. Here, $\mathbf{d}=(d_{1},d_{2},d_{3})$ labels the BPS
states with charge $d_{1}e_{2}+d_{2}f_{1}+d_{3}f_{2}$.

####  3.2.3 6d minimal $SU(3)$ SCFT on a circle with $\mathbb{Z}_{2}$ twist:
5d $SU(3)_{9}$

Elliptic genera of many 6d theories, for instance, E-string, M-string, or 6d
SCFTs compactified on a circle without twist, have been evaluated using the
blowup equations Gu:2018gmy ; Gu:2019dan ; Gu:2019pqj ; Gu:2020fem . In this
subsection, we will demonstrate with a simple example of how to bootstrap BPS
partition functions (or elliptic genera) for 6d SCFTs on a circle with outer
automorphism twists by solving the blowup equations.

We will consider a circle compactification of the 6d minimal $SU(3)$ SCFT with
$\mathbb{Z}_{2}$ automorphism twist. This theory is dual to the 5d $SU(3)$
gauge theory at CS-level $\kappa=9$ Jefferson:2017ahm ,

(163)

We follow the notations in Bhardwaj:2019fzv  to denote 6d theories. It is
geometrically engineered by gluing $\mathbb{F}_{10}$ and $\mathbb{F}_{0}$
Jefferson:2018irk :

(164)

In geometry, the duality is simply the exchange of the base and the fiber
classes in $\mathbb{F}_{0}$. We will solve the blowup equations for this
theory in both 5d and 6d frames and compare the results.

In the 5d $SU(3)_{9}$ theory, the effective prepotential $\mathcal{E}$ is
given by

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\left(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{12}(\phi_{1}+\phi_{2})+\epsilon_{+}^{2}(\phi_{1}+\phi_{2})\right),$
(165) $\displaystyle 6\mathcal{F}$
$\displaystyle=6m\big{(}\phi_{1}^{2}-\phi_{1}\phi_{2}+\phi_{2}^{2}\big{)}+8\phi_{1}^{3}+24\phi_{1}^{2}\phi_{2}-30\phi_{1}\phi_{2}^{2}+8\phi_{2}^{2}\
,$

where $m$ is the gauge coupling. The GV-invariant of this theory takes the
same form of (139). It is convenient to use volumes of three independent
2-cycles in the geometry (164) as the basis of states in the GV-invariant,

$\displaystyle\operatorname{vol}(f_{1})=2\phi_{1}-\phi_{2},\quad\operatorname{vol}(f_{2})=-\phi_{1}+2\phi_{2},\quad\operatorname{vol}(e_{2})=-4\phi_{1}+2\phi_{2}+m\
.$ (166)

Unlike the other $SU(3)_{\kappa}$ theories with $|\kappa|\leq 7$, this theory
has only one set of consistent magnetic fluxes respecting the $SU(3)$
structure:

$\displaystyle n_{1},n_{2}\in\mathbb{Z}\ ,\quad B_{m}=0\ .$ (167)

This set of fluxes gives rise to a unity blowup equation. As we discussed
already, a single unity blowup equation is enough to compute the BPS
invariants. By solving the unity blowup equation, we find the BPS spectrum of
the $SU(3)_{9}$ theory listed in Table 9.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,0,0)$ |  $(0,\frac{1}{2})$ | $(1,0,1)$ |  $(0,\frac{3}{2})$
$(1,0,2)$ |  $(0,\frac{5}{2})$ | $(1,1,0)$ |  $(0,\frac{1}{2})$
$(1,1,1)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})$ | $(1,1,2)$ |  $(0,\frac{3}{2})\oplus(0,\frac{5}{2})$
$(1,2,1)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})$ | $(1,2,2)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(0,\frac{5}{2})$
$(2,0,1)$ |  $(0,\frac{5}{2})$ | $(2,0,2)$ |  $(0,\frac{5}{2})\oplus(0,\frac{7}{2})\oplus(\frac{1}{2},4)$
$(2,1,1)$ |  $(0,\frac{3}{2})\oplus(0,\frac{5}{2})$ | $(2,1,2)$ |  $(0,\frac{3}{2})\oplus 3(0,\frac{5}{2})\oplus 2(0,\frac{7}{2})\oplus(\frac{1}{2},3)\oplus(\frac{1}{2},4)$
$(2,2,1)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(0,\frac{5}{2})$ | $(2,2,2)$ |  $(0,\frac{1}{2})\oplus 3(0,\frac{3}{2})\oplus 4(0,\frac{5}{2})\oplus 2(0,\frac{7}{2})\oplus(\frac{1}{2},2)\oplus(\frac{1}{2},3)\oplus(\frac{1}{2},4)$
$(3,0,1)$ |  $(0,\frac{7}{2})$ | $(3,0,2)$ |  $(0,\frac{5}{2})\oplus(0,\frac{7}{2})\oplus 2(0,\frac{9}{2})\oplus(\frac{1}{2},4)\oplus(\frac{1}{2},5)\oplus(1,\frac{11}{2})$
$(3,1,1)$ |  $(0,\frac{5}{2})\oplus(0,\frac{7}{2})$ | $(3,1,2)$ |  $(0,\frac{3}{2})\oplus 3(0,\frac{5}{2})\oplus 5(0,\frac{7}{2})\oplus 3(0,\frac{9}{2})\oplus(\frac{1}{2},3)\oplus 3(\frac{1}{2},4)\oplus 2(\frac{1}{2},5)\oplus(1,\frac{9}{2})\oplus(1,\frac{11}{2})$
$(3,2,1)$ |  $(0,\frac{3}{2})\oplus(0,\frac{5}{2})\oplus(0,\frac{7}{2})$ | $(3,2,2)$ |  $(0,\frac{1}{2})\oplus 3(0,\frac{3}{2})\oplus 8(0,\frac{5}{2})\oplus 7(0,\frac{7}{2})\oplus 4(0,\frac{9}{2})\oplus(\frac{1}{2},2)\oplus 3(\frac{1}{2},3)\oplus 4(\frac{1}{2},4)\oplus 2(\frac{1}{2},5)\oplus(1,\frac{7}{2})\oplus(1,\frac{9}{2})\oplus(1,\frac{11}{2})$
Table 9: BPS spectrum of $SU(3)_{9}$ theory for $d_{1}\leq 3$ and
$d_{2},d_{3}\leq 2$. Here, $\mathbf{d}=(d_{1},d_{2},d_{3})$ labels BPS states
with charge $d_{1}e_{2}+d_{2}f_{1}+d_{3}f_{2}$ cycle.

We now perform a similar computation in the perspective of the 6d $SU(3)$
gauge theory with $\mathbb{Z}_{2}$ twist. In the 6d frame, the volumes of
2-cycles in the geometry are

${\rm vol}(f_{1})=\frac{\tau}{4}-\phi_{1}\ ,\quad{\rm vol}(f_{2})=2\phi_{1}\
,\quad{\rm vol}(e_{2})=3\phi_{0}+2\phi_{1}-\frac{\tau}{2}\ .$ (168)

The effective prepotential in this frame is given in (2.3) with a shift
$\phi_{0}\rightarrow\phi_{0}-\frac{1}{16}\tau$.

The GV-invariant for this 6d theory can be written as

$\displaystyle
Z_{GV}(\Phi,\phi_{1},\tau;\epsilon_{1,2})=\mathcal{Z}_{\mathrm{pert}}(\phi_{1},\tau;\epsilon_{1,2})\quantity(1+\sum_{k=1}^{\infty}e^{-k\Phi}Z_{k}(\phi_{1},\tau,\epsilon_{1,2}))\
,$ (169)

where $Z_{k}(\phi_{1},\tau,\epsilon_{1,2})$ is the $k$-string elliptic genus
and $\Phi\equiv 3\phi_{0}-\tau/2$ which is the string number fugacity. Here,
the perturbative part can be read off from the 6d $SU(3)$ vector multiplets
under the $\mathbb{Z}_{2}$ automorphism twist, which is given by

$\displaystyle\mathcal{Z}_{\mathrm{pert}}=\operatorname{PE}\quantity[-\frac{1+p_{1}p_{2}}{(1-p_{1})(1-p_{2})}\frac{1}{1-q}\quantity(e^{-2\phi_{1}}+(q^{1/4}+q^{3/4})(e^{-\phi_{1}}+e^{\phi_{1}})+qe^{2\phi_{1}})],$
(170)

where $q=e^{-\tau}$. See Appendix A for more details.

There are three sets of consistent magnetic fluxes preserving the affine
$A^{(2)}_{2}$ algebra structure:

$\displaystyle n_{1}\in\mathbb{Z}\ ,\quad B_{\tau}=0\ ,\quad
n_{0}\in\mathbb{Z}+B_{0}\ \ {\rm with}\ \ B_{0}=0,1/3,2/3\ .$ (171)

Then the elliptic genus can be found by solving three unity blowup equations
from these three flux sets. For example, the blowup equations at 1-string
order are

$\displaystyle\Lambda(\tau;\epsilon_{1},\epsilon_{2})\hat{Z}_{1}(\phi_{1},\tau;\epsilon_{1},\epsilon_{2})$
$\displaystyle=\sum_{\vec{n}=(n,2n)}e^{-V}\frac{\hat{\mathcal{Z}}_{\mathrm{pert}}^{(N)}\hat{\mathcal{Z}}_{\mathrm{pert}}^{(S)}}{\hat{\mathcal{Z}}_{\mathrm{pert}}}\quantity(p_{1}^{3n+B_{0}}\hat{Z}_{1}(\phi_{1},\epsilon_{1},\epsilon_{2}-\epsilon_{1})+p_{2}^{3n+B_{0}}\hat{Z}_{1}(\phi_{1},\epsilon_{1}-\epsilon_{2},\epsilon_{2}))$
$\displaystyle\qquad+\sum_{\vec{n}=(n,2n\pm
1)}e^{\Phi}e^{-V}\frac{\hat{\mathcal{Z}}_{\mathrm{pert}}^{(N)}\hat{\mathcal{Z}}_{\mathrm{pert}}^{(S)}}{\hat{\mathcal{Z}}_{\mathrm{pert}}}\
.$ (172)

Here the prefactor can be computed by collecting all $\phi_{i}$ independent
terms in the blowup equation as follows:

$\displaystyle\Lambda(\tau;\epsilon_{1},\epsilon_{2})=\sum_{\vec{n}=(n,2n)}e^{-V}\
.$ (173)

The solution at 1-instanton string is then given by

$\displaystyle Z_{1}(\phi_{1},\tau;\epsilon_{1},\epsilon_{2})$
$\displaystyle=\frac{e^{-2\phi_{1}}p_{1}p_{2}(1+p_{1}p_{2})}{(1-p_{1})(1-p_{2})(e^{-2\phi_{1}}-p_{1}p_{2})(1-e^{-2\phi_{1}}p_{1}p_{2})}$
$\displaystyle\quad+\frac{e^{-\phi_{1}}(1+e^{-2\phi_{1}})p_{1}p_{2}(1+p_{1}p_{2})}{(1-p_{1})(1-p_{2})(e^{-2\phi_{1}}-p_{1}p_{2})(1-e^{-2\phi_{1}}p_{1}p_{2})}q^{1/4}$
$\displaystyle\quad+\frac{e^{-2\phi_{1}}(1+2p_{1}p_{2}+2p_{1}^{2}p_{2}^{2}+p_{1}^{3}p_{2}^{3})}{(1-p_{1})(1-p_{2})(e^{-2\phi_{1}}-p_{1}p_{2})(1-e^{-2\phi_{1}}p_{1}p_{2})}q^{1/2}+\cdots\
.$ (174)

This is in perfect agreement with the BPS spectrum of the $SU(3)_{9}$ theory
given in Table 9 and also topological vertex as well as the ADHM calculations
Kim:2021cua .

##  4 Rank 1 theories

In this section, we will compute the BPS partition functions of rank-1 5d
theories by bootstrapping with the blowup equations. All rank-1 theories can
be obtained via RG flows, which integrate out massive hypermultiplets, from
three KK theories that come from 6d SCFTs compactified on a circle
with/without twists. When a theory admits solvable blowup equations with
consistent magnetic fluxes, all the IR theories descending from this UV theory
by RG flows are guaranteed to have solvable blowup equations. So we will show
that the blowup equations for all the rank-1 KK theories can be solved and the
solutions are consistent with the BPS spectra computed by other methods. This
will prove that the BPS spectra of all rank-1 5d theories can be computed by
solving the blowup equations. We will also compute the BPS partition function
of a new rank-1 theory, which we call the local $\mathbb{P}^{2}+1\mathbf{Adj}$
theory, obtained from the $\mathcal{N}=2$ $SU(2)$ gauge theory at $\theta=\pi$
by integrating out an instantonic hypermultiplet.

###  4.1 KK theories

There are three rank-1 KK theories: the $SU(2)$ gauge theory with 8
fundamental hypermultiplets, the $\mathcal{N}=2$ $SU(2)$ gauge theory at
$\theta=0,\pi$. We discuss them one by one.

####  4.1.1 $SU(2)+8\mathbf{F}$

The $SU(2)$ gauge theory with 8 fundamental hypermultiplets is a KK theory
arising from the 6d rank-1 E-string theory compactified on a circle
Witten:1995gx ; Ganor:1996mu ; Seiberg:1996bd .

(175)

The BPS partition function for this theory was computed in Hwang:2014uwa ;
Kim:2014dza  using the ADHM instanton string construction and also in
Kim:2015jba ; Kim:2017jqn  based in its 5-brane webs using topological vertex.
It was also shown in Gu:2019pqj  that the elliptic genus of the 6d E-string
theory is consistent with the blowup equation.

We shall here solve the blowup equation in perspective of the 5d gauge theory.
The prepotential on the Coulomb branch is given by

$\displaystyle
6\mathcal{F}=6m_{0}\phi^{2}+8\phi^{3}-\frac{1}{2}\sum_{i=1}^{8}\Big{(}(\phi+m_{i})^{3}+(\phi-
m_{i})^{3}\Big{)}\ ,$ (176)

where $m_{0}$ is the gauge coupling and $m_{i=1,\cdots,8}$ are the mass
parameters of fundamental hypermultiplets. Collecting the gauge/gravitational
and the gauge/$SU(2)_{R}$ Chern-Simons terms, the effective prepotential on
the $\Omega$-background is expressed as

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{48}\bigg{(}4\phi-\sum_{i=1}^{8}\Big{(}(\phi+m_{i})+(\phi-
m_{i})\Big{)}\bigg{)}+\epsilon_{+}^{2}\phi)$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}+\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{4}\phi+\epsilon_{+}^{2}\phi)\,.$
(177)

The perturbative part of the GV-invariant is

$\displaystyle\mathcal{Z}_{\rm pert}(\phi,m_{i};\epsilon_{1,2})={\rm
PE}\left[-\frac{1+p_{1}p_{2}}{(1-p_{1})(1-p_{2})}e^{-2\phi}+\frac{\sqrt{p_{1}p_{2}}}{(1-p_{1})(1-p_{2})}\sum_{i=1}^{8}e^{-(\phi\pm
m_{i})}\right]\,.$ (178)

For this theory, we find a set of the consistent magnetic fluxes

$\displaystyle n\in\mathbb{Z}\,,\quad B_{m_{0}}=0\,,\quad B_{m_{i}}=1/2\ \
{\rm for}\ 1\leq i\leq 8\,,$ (179)

which leads to a unity blowup equation. The unity blowup equation with the
magnetic fluxes can be solved and the solution is summarized in Table 10. This
result matches the elliptic genus of the rank-1 E-string theory computed in
Kim:2014dza .

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,1)$ |  $128(0,0)$ | $(1,2)$ |  $128(0,\frac{1}{2})$
$(1,3)$ |  $128(0,1)$ | $(2,1)$ |  $576(0,0)\oplus 16(\frac{1}{2},\frac{1}{2})$
$(2,2)$ |  $1942(0,\frac{1}{2})\oplus(\frac{1}{2},0)\oplus 121(\frac{1}{2},1)\oplus(1,\frac{3}{2})$ | $(2,3)$ |  $560(0,0)\oplus 4960(0,1)\oplus 16(\frac{1}{2},\frac{1}{2})\oplus 576(\frac{1}{2},\frac{3}{2})\oplus 16(1,2)$
Table 10: BPS spectrum of $SU(2)+8\mathbf{F}$ theory for $d_{1}\leq 2$ and
$d_{2}\leq 3$. Here, $\mathbf{d}=(d_{1},d_{2})$ labels BPS states with charge
$d_{1}m_{0}+d_{2}\phi$. For convenience, we set all flavor mass parameters
$m_{i=1,\cdots,8}$ to zero.

####  4.1.2 $SU(2)_{0}+1\mathbf{Adj}$

The compactification of the 6d rank-1 M-string theory on a circle gives rise
to the 5d $SU(2)$ gauge theory at $\theta=0$ with an adjoint hypermultiplet
preserving $\mathcal{N}=2$ supersymmetry,

(180)

The validity of the blowup equation for this theory was previously checked in
Gu:2019pqj .

The cubic prepotential on the Coulomb branch where $2\phi\pm m_{1}\geq 0$ with
$\phi>0$ is

$\displaystyle
6\mathcal{F}=6m_{0}\phi^{2}+8\phi^{3}-\frac{1}{2}\quantity((2\phi+m_{1})^{3}+(2\phi-
m_{1})^{3})\ ,$ (181)

where $m_{0}$ is the gauge coupling and $m_{1}$ is the mass parameter of the
adjoint hypermultiplet. The effective prepotential on $\Omega$-background is
then given by

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{48}\Big{(}4\phi-\big{(}(2\phi+m_{1})+(2\phi-
m_{1})\big{)}\Big{)}+\epsilon_{+}^{2}\phi)$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}+\epsilon_{+}^{2}\phi)\,.$
(182)

The perturbative part of the GV-invariant is

$\displaystyle\mathcal{Z}_{\rm pert}(\phi,m_{1};\epsilon_{1,2})={\rm
PE}\left[-\frac{1+p_{1}p_{2}}{(1-p_{1})(1-p_{2})}e^{-2\phi}+\frac{\sqrt{p_{1}p_{2}}}{(1-p_{1})(1-p_{2})}e^{-(2\phi\pm
m_{1})}\right]\,.$ (183)

One can formulate a unity blowup equation for this theory using the following
consistent magnetic fluxes.

$\displaystyle n\in\mathbb{Z}\,,\quad B_{m_{0}}=0\,,\quad B_{m_{1}}=1/2\,.$
(184)

The solution to the blowup equation is presented in Table 11. The result
indeed matches the $\mathcal{N}=2$ $SU(2)_{0}$ instanton partition function
computed using the ADHM calculus in Hwang:2014uwa .

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,2,-2)$ |  $(0,\frac{1}{2})$ | $(1,2,-1)$ |  $(0,0)\oplus(0,1)\oplus(\frac{1}{2},\frac{1}{2})$
$(1,2,0)$ |  $2(0,\frac{1}{2})\oplus(\frac{1}{2},0)\oplus(\frac{1}{2},1)$ | $(1,4,-2)$ |  $(0,\frac{3}{2})$
$(1,4,-1)$ |  $(0,1)\oplus(0,2)\oplus(\frac{1}{2},\frac{3}{2})$ | $(1,4,0)$ |  $2(0,\frac{3}{2})\oplus(\frac{1}{2},1)\oplus(\frac{1}{2},2)$
$(2,2,-3)$ |  $(0,0)$ | $(2,2,-2)$ |  $2(0,\frac{1}{2})\oplus(\frac{1}{2},0)\oplus(\frac{1}{2},1)$
$(2,2,-1)$ |  $4(0,0)\oplus 2(0,1)\oplus 3(\frac{1}{2},\frac{1}{2})\oplus(\frac{1}{2},\frac{3}{2})\oplus(1,1)$ | $(2,2,0)$ |  $5(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus 3(\frac{1}{2},0)\oplus 3(\frac{1}{2},1)\oplus(1,\frac{1}{2})\oplus(1,\frac{3}{2})$
$(2,4,-3)$ |  $(0,1)\oplus(0,2)\oplus(\frac{1}{2},\frac{3}{2})$ | $(2,4,-2)$ |  $2(0,\frac{1}{2})\oplus 4(0,\frac{3}{2})\oplus(0,\frac{5}{2})\oplus 3(\frac{1}{2},1)\oplus 3(\frac{1}{2},2)\oplus(1,\frac{3}{2})\oplus(1,\frac{5}{2})$
$(2,4,-1)$ |  $(0,0)\oplus 7(0,1)\oplus 6(0,2)\oplus 3(\frac{1}{2},\frac{1}{2})\oplus 8(\frac{1}{2},\frac{3}{2})\oplus 3(\frac{1}{2},\frac{5}{2})\oplus 2(1,1)\oplus 3(1,2)\oplus(1,3)\oplus(\frac{3}{2},\frac{5}{2})$ | $(2,4,0)$ |  $5(0,\frac{1}{2})\oplus 10(0,\frac{3}{2})\oplus 3(0,\frac{5}{2})\oplus(\frac{1}{2},0)\oplus 8(\frac{1}{2},1)\oplus 8(\frac{1}{2},2)\oplus(\frac{1}{2},3)\oplus(1,\frac{1}{2})\oplus 4(1,\frac{3}{2})\oplus 3(1,\frac{5}{2})\oplus(\frac{3}{2},2)\oplus(\frac{3}{2},3)$
Table 11: BPS spectrum of $SU(2)_{0}+1\mathbf{Adj}$ theory for $d_{1}\leq 2$
and $d_{2}\leq 4$. Here, $\mathbf{d}=(d_{1},d_{2},d_{3})$ labels the BPS
states with charge $d_{1}m_{0}+d_{2}\phi+d_{3}m_{1}$. The states related by
the symmetry $d_{3}\leftrightarrow-d_{3}$ are omitted in the table.

####  4.1.3 $SU(2)_{\pi}+1\mathbf{Adj}$

The 5d KK-theory $SU(2)_{\pi}+1\mathbf{Adj}$ is obtained by a circle
compactification of the 6d $\mathcal{N}=(2,0)$ $A_{2}$ theory with
$\mathbb{Z}_{2}$ outer automorphism twist Tachikawa:2011ch ,

(185)

This theory has no conventional geometric description Bhardwaj:2019fzv .

The perturbative spectrum cannot distinguish whether $\theta=0$ or
$\theta=\pi$ for the $SU(2)$ gauge group. Thus the effective prepotential and
the perturbative GV-invariant of this theory are the same as (4.1.2) and
(183), respectively, for the $SU(2)_{0}+1{\bf Adj}$ theory. Naively, this
means that the blowup equation of this theory is the same as that of the
theory at $\theta=0$. However, it turns out that there are two independent
blowup equations from the same effective prepotential but distinguished by two
different sets of consistent magnetic fluxes. Instead of the fluxes given in
(184) for the theory at $\theta=0$, this theory at $\theta=\pi$ has another
set of consistent magnetic fluxes quantized as

$n\in\mathbb{Z}+1/2\,,\quad B_{m_{0}}=0\,,\quad B_{m_{1}}=1/2\,.$ (186)

These magnetic fluxes provide a unity blowup equation. Some leading BPS
invariants we computed using the blowup equation are listed in Table 12, which
are indeed different from the BPS spectrum of the theory at $\theta=0$. This
result also matches the instanton partition function using the ADHM quantum
mechanics in Hwang:2014uwa . This is our first example showing that our
bootstrap approach can be applied for the BPS spectrum computation for non-
geometric theories.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,1,-2)$ |  $(0,0)$ | $(1,1,-1)$ |  $(0,\frac{1}{2})\oplus(\frac{1}{2},0)$
$(1,1,0)$ |  $2(0,0)\oplus(\frac{1}{2},\frac{1}{2})$ | $(1,3,-2)$ |  $(0,1)$
$(1,3,-1)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(\frac{1}{2},1)$ | $(1,3,0)$ |  $2(0,1)\oplus(\frac{1}{2},\frac{1}{2})\oplus(\frac{1}{2},\frac{3}{2})$
$(2,2,-2)$ |  $(0,\frac{1}{2})\oplus(\frac{1}{2},1)$ | $(2,2,-1)$ |  $(0,0)\oplus 2(0,1)\oplus 2(\frac{1}{2},\frac{1}{2})\oplus(\frac{1}{2},\frac{3}{2})\oplus(1,1)$
$(2,2,0)$ |  $3(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(\frac{1}{2},0)\oplus 3(\frac{1}{2},1)\oplus(1,\frac{1}{2})\oplus(1,\frac{3}{2})$ | $(2,4,-3)$ |  $(0,0)\oplus(0,2)\oplus(\frac{1}{2},\frac{3}{2})$
$(2,4,-2)$ |  $2(0,\frac{1}{2})\oplus 3(0,\frac{3}{2})\oplus(0,\frac{5}{2})\oplus(\frac{1}{2},0)\oplus 2(\frac{1}{2},1)\oplus 3(\frac{1}{2},2)\oplus(1,\frac{3}{2})\oplus(1,\frac{5}{2})$ | $(2,4,-1)$ |  $4(0,0)\oplus 4(0,1)\oplus 6(0,2)\oplus 3(\frac{1}{2},\frac{1}{2})\oplus 7(\frac{1}{2},\frac{3}{2})\oplus 3(\frac{1}{2},\frac{5}{2})\oplus 2(1,1)\oplus 3(1,2)\oplus(1,3)\oplus(\frac{3}{2},\frac{5}{2})$
$(2,4,0)$ |  $5(0,\frac{1}{2})\oplus 8(0,\frac{3}{2})\oplus 3(0,\frac{5}{2})\oplus 3(\frac{1}{2},0)\oplus 6(\frac{1}{2},1)\oplus 8(\frac{1}{2},2)\oplus(\frac{1}{2},3)\oplus(1,\frac{1}{2})\oplus 4(1,\frac{3}{2})\oplus 3(1,\frac{5}{2})\oplus(\frac{3}{2},2)\oplus(\frac{3}{2},3)$
Table 12: BPS spectrum of $SU(2)_{\pi}+1\mathbf{Adj}$ theory for $d_{1}\leq 2$
and $d_{2}\leq 4$. Here, $\mathbf{d}=(d_{1},d_{2},d_{3})$ labels BPS states
with charge $d_{1}m_{0}+d_{2}\phi+d_{3}m_{1}$. The states related by the
symmetry $d_{3}\leftrightarrow-d_{3}$ are omitted in the table.

###  4.2 5d SCFTs: Non-Lagrangian theories

The BPS partition functions of all the rank-1 5d SCFTs which have no gauge
theory descriptions, can be computed by solving the blowup equations. To show
this we will consider two simple examples: a local $\mathbb{P}^{2}$ theory and
a local $\mathbb{P}^{2}+1{\bf Adj}$ theory.

####  4.2.1 Local $\mathbb{P}^{2}$

This theory is a rank-1 SCFT with no mass parameter, which is engineered by
compactification of M-theory on a local $\mathbb{P}^{2}$ embedded in a CY
3-fold. This theory is also called the $E_{0}$ theory. The blowup equation for
the SCFT of a local $\mathbb{P}^{2}$ was solved in Huang:2017mis . We shall
briefly review this.

The effective prepotential of this theory on the $\Omega$-background is simply
given by

$\displaystyle\mathcal{E}=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(9\phi^{3}-\frac{(\epsilon_{1}^{2}+\epsilon_{2}^{2})\phi}{8}+\epsilon_{+}^{2}\phi)\,.$
(187)

The CY 3-fold of a local $\mathbb{P}^{2}$ contains only one primitive curve
class $\ell$ with $\ell^{2}=+1$. The volume of this curve is
$\operatorname{vol}(\ell)=3\phi$. Thus the magnetic fluxes that couple to the
curve should be quantized as

$\displaystyle n\in\mathbb{Z}\pm 1/6\quad{\rm or}\quad n\in\mathbb{Z}+1/2\,.$
(188)

These are all the consistent magnetic fluxes. The blowup equation with
$n\in\mathbb{Z}+1/2$ is a vanishing blowup equations, whereas the other
choices of fluxes lead to unity blowup equations Huang:2017mis , and they are
all solvable. The solution to these blowup equations is summarized in Table
13.

$d$ |  $\oplus N_{j_{l},j_{r}}^{d}(j_{l},j_{r})$ | $d$ |  $\oplus N_{j_{l},j_{r}}^{d}(j_{l},j_{r})$
---|---|---|---
$1$ |  $(0,1)$ | $2$ |  $(0,\frac{5}{2})$
$3$ |  $(0,3)\oplus(\frac{1}{2},\frac{9}{2})$ | $4$ |  $\\!(0,\frac{5}{2})\oplus(0,\frac{9}{2})\oplus(0,\frac{13}{2})\oplus(\frac{1}{2},4)\oplus(\frac{1}{2},5)\oplus(\frac{1}{2},6)\oplus(1,\frac{11}{2})\oplus(\frac{3}{2},7)\\!$
Table 13: BPS spectrum of a local $\mathbb{P}^{2}$ for $d\leq 4$. Here, $d$
labels the BPS states wrapping the degree $d$ curve in $\mathbb{P}^{2}$.

####  4.2.2 Local $\mathbb{P}^{2}+1\mathbf{Adj}$

The local $\mathbb{P}^{2}$ \+ 1$\mathbf{Adj}$ (or
$``SU(2)_{\pi}+1\mathbf{Adj}-1\mathbf{F}"$) theory was first proposed in
Bhardwaj:2019jtr . This theory can be obtained by an RG flow from the UV
$SU(2)_{\pi}+1{\bf Adj}$ theory after integrating out an instantonic
hypermultiplet.

One way to see the existence of this theory is as follows. Recall that the
SCFT of a local $\mathbb{P}^{2}$ (or the $E_{0}$ theory) can be obtained by
blowing down an exceptional curve in a Hirzebruch surface $\mathbb{F}_{1}$.
This corresponds to integrating out an instantonic hypermultiplet in the pure
$SU(2)_{\pi}$ theory. In this regard, the $E_{0}$ theory can be thought of as
“$SU(2)_{\pi}-1\mathbf{F}$” since we are “subtracting” a hypermultiplet.
Likewise, the $SU(2)_{\pi}+1\mathbf{Adj}$ theory we discussed in section 4.1.3
has a hypermultiplet with charge ${\bf d}=(1,1,-2)$ at the 1-instanton sector
and integrating out this hypermultiplet induces an RG flow to a consistent
rank-1 SCFT with one mass parameter. We will call this IR SCFT the local
$\mathbb{P}^{2}+1\mathbf{Adj}$ theory. This theory is a non-Lagrangian theory.
Here $+1\mathbf{Adj}$ simply means this theory contains a hypermultiplet
originating from the adjoint hypermultiplet in its parent $SU(2)_{\pi}+1{\bf
Adj}$ theory. The adjoint hypermultiplet in this SCFT can also be integrated
out to give the SCFT of a local $\mathbb{P}^{2}$.

This theory is also a non-geometric theory. However, there is a 5-brane web
for this theory with an $O7^{+}$-plane where the hypermultiplet associated
with an $O7^{+}$-plane is in the symmetric representation of $SU(2)$ that is
equivalent to the adjoint of the UV $SU(2)$ gauge symmetry. A simple way to
construct the corresponding 5-brane web is to attach an $O7^{+}$-plane to a
5-brane web for the pure $SU(2)_{\pi}$ theory, as depicted in Figure 2. (See
also Appendix LABEL:sec:appendix2 for more details of obtaining a 5-brane web
for the $SU(2)_{\pi}+1\mathbf{Adj}$ theory.)

The hypermultiplet which we will integrate out is the $(p,q)$-string state
associated with the edge of the length $-\phi-m_{0}+2m_{1}$ in the brane web.
Integrating out this hypermultiplet corresponds to taking the infinite length
limit of the edge $-\phi-m_{0}+2m_{1}\rightarrow\infty$ while the lengths of
other edges are kept finite. After this, we will get a 5-brane web for the
local $\mathbb{P}^{2}+``1\mathbf{Adj}"$ theory. In this 5-brane web diagram,
one can also see that by taking another limit $m_{1}-2\phi\rightarrow\infty$,
after taking the limit $-\phi-m_{0}+2m_{1}\rightarrow\infty$, the diagram
reduces to the brane web for a local $\mathbb{P}^{2}$, which is consistent
with the expected RG flow for this theory.

${3\phi+m_{0}-2m_{1}}$${-\phi-m_{0}+2m_{1}}$${m_{1}-2\phi}$$O7^{+}$ Figure 2:
A 5-brane web for $SU(2)_{\pi}+1\mathbf{Adj}$, where a flop transition
associated with the $e$ curve in $\mathbb{P}^{2}$ is performed.

We will now compute the partition function using the chamber for the brane web
of the $SU(2)_{\pi}+1{\bf Adj}$ theory depicted in Figure 2. It is convenient
to reparameterize the parameters in the brane web as follows:

$\displaystyle\tilde{\phi}=\phi+\frac{1}{3}m_{0}-\frac{2}{3}m_{1}\,,\quad\tilde{m}=\frac{2}{3}m_{0}-\frac{1}{3}m_{1}\,,$
(189)

so that the volumes of 2-cycles in Figure 2 become

$\displaystyle 3\phi+m_{0}-2m_{1}=3\tilde{\phi}\,,\quad
m_{1}-2\phi=\tilde{m}-2\tilde{\phi}\,,$ $\displaystyle-\phi-
m_{0}+2m_{1}=-\tilde{\phi}+2m_{0}-4\tilde{m}\,.$ (190)

In order to get the IR local $\mathbb{P}^{2}+1\mathbf{Adj}$ theory, we take
the limit $m_{0}\to\infty$ while $\tilde{\phi}$ and $\tilde{m}$ are kept
finite. The cubic prepotential and the mixed gravitational Chern-Simons level
of the IR theory can be obtained from those of the parent theory as

$\displaystyle\mathcal{F}_{IR}=\mathcal{F}_{UV}+\frac{1}{6}(\phi+m_{0}-2m_{1})^{3}\,,\quad
C^{G}_{IR}=C^{G}_{UV}-2(\phi+m_{0}-2m_{1})\,,$ (191)

whereas the mixed gauge/$SU(2)_{R}$ level $C^{R}$ remains the same along the
RG flow. Thus the effective prepotential of a local
$\mathbb{P}^{2}+1\mathbf{Adj}$ on the extended Coulomb branch where
$\tilde{\phi}>0$ and $\tilde{m}-2\tilde{\phi}\geq 0$ becomes

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\frac{3}{2}\tilde{\phi}^{3}-\frac{(\epsilon_{1}^{2}+\epsilon_{2}^{2})\tilde{\phi}}{8}+\epsilon_{+}^{2}\tilde{\phi})\,,$
(192)

up to the terms independent of $\tilde{\phi}$. Interestingly, this
prepotential is the same as that of a local $\mathbb{P}^{2}$. However, they
are different theories because the $\mathbb{P}^{2}+1\mathbf{Adj}$ theory
contains an additional hypermultiplet coming from the adjoint hypermultiplet
of the parent theory, while the $E_{0}$ theory has no hypermultiplet. The GV-
invariant for this hypermultiplet is thus a key input for solving the blowup
equation

$\mathcal{Z}_{\rm
hyper}(\tilde{\phi},\tilde{m};\epsilon_{1,2})=\operatorname{PE}\quantity[\frac{\sqrt{p_{1}p_{2}}}{(1-p_{1})(1-p_{2})}e^{-(\tilde{m}-2\tilde{\phi})}]\,.$
(193)

The feasible magnetic fluxes that one can turn on are

$\displaystyle n\in\mathbb{Z}+1/6\,,\quad B_{\tilde{m}}=-1/6\,.$ (194)

The blowup equation with this flux choice is a unity blowup equation. We solve
this equation and find the BPS spectrum of the local
$\mathbb{P}^{2}+1\mathbf{Adj}$ theory listed in Table 14. As expected, this
result also agrees with the RG flow from the BPS spetrum in the
$SU(2)_{\pi}+1{\bf Adj}$ theory given in Table 12 in the limit
$-\tilde{\phi}+2m_{0}-4\tilde{m}\rightarrow\infty$ while $\tilde{\phi}$ and
$\tilde{m}$ are kept finite.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,-2)$ |  $(0,0)$ | $(1,1)$ |  $(0,\frac{1}{2})\oplus(\frac{1}{2},0)$
$(1,4)$ |  $(0,0)\oplus(0,2)\oplus(\frac{1}{2},\frac{3}{2})$ | $(1,7)$ |  $(0,\frac{3}{2})\oplus(0,\frac{5}{2})\oplus(0,\frac{7}{2})\oplus(\frac{1}{2},2)\oplus(\frac{1}{2},3)\oplus(\frac{1}{2},4)\oplus(1,\frac{7}{2})$
$(2,2)$ |  $(0,\frac{1}{2})\oplus(\frac{1}{2},1)$ | $(2,5)$ |  $2(0,1)\oplus(0,2)\oplus(0,3)\oplus(\frac{1}{2},\frac{1}{2})\oplus(\frac{1}{2},\frac{3}{2})\oplus 2(\frac{1}{2},\frac{5}{2})\oplus(1,2)\oplus(1,3)$
$(2,8)$ |  $2(0,\frac{1}{2})\oplus 2(0,\frac{3}{2})\oplus 6(0,\frac{5}{2})\oplus 3(0,\frac{7}{2})\oplus 4(0,\frac{9}{2})\oplus 2(\frac{1}{2},1)\oplus 4(\frac{1}{2},2)\oplus 6(\frac{1}{2},3)\oplus 6(\frac{1}{2},4)\oplus 2(\frac{1}{2},5)\oplus(1,\frac{3}{2})\oplus 3(1,\frac{5}{2})\oplus 5(1,\frac{7}{2})\oplus 4(1,\frac{9}{2})\oplus(1,\frac{11}{2})\oplus(\frac{3}{2},3)\oplus 2(\frac{3}{2},4)\oplus 2(\frac{3}{2},5)\oplus(2,\frac{9}{2})\oplus(2,\frac{11}{2})$
Table 14: BPS spectrum of $\mathbb{P}^{2}+1\mathbf{Adj}$ theory with
$d_{1}\leq 2$ and $d_{2}\leq 9$. Here, $\mathbf{d}=(d_{1},d_{2})$ labels the
BPS states with charge $d_{1}\tilde{m}+d_{2}\tilde{\phi}$.

##  5 Rank 2 theories

In this section, we will compute the BPS spectra of all rank-2 KK theories by
employing our bootstrap approach. The computations will then guarantee that
all rank-2 5d and 6d QFTs have solvable blowup equations and their BPS spectra
can be obtained by the bootstrap method. To support our claim, we will present
computations of the BPS spectra of three non-geometric 5d SCFTs: $SU(3)_{8}$,
$\mathbb{P}^{2}\cup\mathbb{F}_{3}+1{\bf Sym}$, and
$\mathbb{P}^{2}\cup\mathbb{F}_{6}+1{\bf Sym}$.

###  5.1 KK theories

In this subsection, we solve the blowup equations for all rank-2 5d KK
theories classified in Jefferson:2017ahm ; Jefferson:2018irk ;
Bhardwaj:2019fzv .

####  5.1.1 $Sp(2)+3\mathbf{\Lambda}^{2}$

The 5d $Sp(2)$ gauge theory with 3 anti-symmetric hypermultiplets
($Sp(2)+3\mathbf{\Lambda}^{2}$) is the KK-theory arising from the twisted
compactification of the 6d $SU(3)$ gauge theory with 6 fundamental
hypermultiplets,

(195)

This theory has a geometric description Jefferson:2018irk  as

(196)

and the fiber-base duality of $\mathbb{F}_{0}^{3}$ exchanges two dual
descriptions: the 5d $Sp(2)$ gauge theory with 3 anti-symmetric hypers and the
6d $SU(3)$ gauge theory with 6 fundamentals on a circle with $\mathbb{Z}_{2}$
twist.

In 5d $Sp(2)$ gauge theory description, the cubic prepotential on the chamber
described by the above geometry is

$\displaystyle 6\mathcal{F}$
$\displaystyle=8\phi_{1}^{3}+12\phi_{1}^{2}\phi_{2}-18\phi_{1}\phi_{2}^{2}+8\phi_{2}^{3}+6m_{0}(2\phi_{1}^{2}-2\phi_{1}\phi_{2}+\phi_{2}^{2})$
(197)
$\displaystyle\quad-\frac{1}{2}\sum_{i=1}^{3}\quantity((\phi_{2}+m_{i})^{3}+(2\phi_{1}-\phi_{2}+m_{i})^{3}+(-2\phi_{1}+\phi_{2}+m_{i})^{3}+(\phi_{2}-m_{i})^{3})\,,$

where $m_{0}$ is the gauge coupling and $m_{i=1,2,3}$ are mass parameters for
the 3 antisymmetric hypermultiplets. The full effective prepotential on the
$\Omega$-background is

$\displaystyle\mathcal{E}=\frac{1}{\epsilon_{1}\epsilon_{2}}\left(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{48}(4\phi_{1}-2\phi_{2})+\epsilon_{+}^{2}(\phi_{1}+\phi_{2})\right)\,.$
(198)

The volumes of the primitive 2-cycles in the geometry (196) in terms of the
$Sp(2)$ gauge theory parameters can be written as

$\displaystyle\operatorname{vol}(f_{1})=2\phi_{1}-\phi_{2}\,,\quad$
$\displaystyle\operatorname{vol}(f_{2})=-2\phi_{1}+2\phi_{2}\,,$
$\displaystyle\operatorname{vol}(e_{2})=-4\phi_{1}+2\phi_{2}+m_{0}\,,\quad$
$\displaystyle\operatorname{vol}(x_{i})=-2\phi_{1}+\phi_{2}+m_{i}\quad(i=1,2,3)\,.$
(199)

From this information, we can choose a set of magnetic fluxes as

$\displaystyle n_{i}\in\mathbb{Z}\,,\quad B_{m_{0}}=0\,,\quad
B_{m_{i}}=1/2\quad(i=1,2,3)\,,$ (200)

which gives rise to a unity blowup equation. The unity blowup equation can be
easily solved and the resulting BPS spectrum is summarized in Table 15.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,0,0,-1)$ |  $3(0,0)$ | $(1,0,0,0)$ |  $(0,\frac{1}{2})$
$(1,0,1,-3)$ |  $(0,0)$ | $(1,0,1,-2)$ |  $3(0,\frac{1}{2})$
$(1,0,1,-1)$ |  $3(0,1)$ | $(1,0,1,0)$ |  $(0,\frac{3}{2})$
$(1,1,0,0)$ |  $(0,\frac{1}{2})$ | $(1,1,1,-2)$ |  $3(0,\frac{1}{2})$
$(1,1,1,-1)$ |  $3(0,0)\oplus 3(0,1)$ | $(1,1,1,0)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})$
$(2,0,1,-3)$ |  $(0,1)$ | $(2,0,1,-2)$ |  $3(0,\frac{3}{2})$
$(2,0,1,-1)$ |  $3(0,2)$ | $(2,0,1,0)$ |  $(0,\frac{5}{2})$
$(2,1,1,-3)$ |  $(0,0)\oplus(0,1)$ | $(2,1,1,-2)$ |  $3(0,\frac{1}{2})\oplus 3(0,\frac{3}{2})$
$(2,1,1,-1)$ |  $3(0,1)\oplus 3(0,2)$ | $(2,1,1,0)$ |  $(0,\frac{3}{2})\oplus(0,\frac{5}{2})$
Table 15: BPS spectrum of $Sp(2)+3\mathbf{\Lambda}^{2}$ for $d_{1}\leq 2$,
$d_{2}\leq 1$, $d_{3}\leq 1$. Here, $\mathbf{d}=(d_{1},d_{2},d_{3},d_{4})$
labels the BPS states with charge
$d_{1}e_{2}+d_{2}f_{1}+d_{3}f_{2}+d_{4}x_{i}$ and $d_{4}$ counts collective
degrees of all anti-symmetric hypermultiplets.

Now we consider the 6d $SU(3)$ gauge theory. Under the $\mathbb{Z}_{2}$ twist,
the vector multiplet and the hypermultiplets in 6d are decomposed into the
representations of the invariant subalgebra $\mathfrak{su}(2)$ as

$\displaystyle\mathbf{8}\text{ of }\mathfrak{su}(3)\ $ $\displaystyle\to\
\mathbf{3}_{0}\oplus\mathbf{2}_{1/4}\oplus\mathbf{2}_{3/4}\oplus\mathbf{1}_{1/2}\text{
of }\mathfrak{su}(2)\ ,$ $\displaystyle\mathbf{3}\oplus\bar{\mathbf{3}}\text{
of }\mathfrak{su}(3)\ $ $\displaystyle\to\
\mathbf{2}_{0}\oplus\mathbf{1}_{1/4}\oplus\mathbf{2}_{1/2}\oplus\mathbf{1}_{3/2}\text{
of }\mathfrak{su}(2)\ ,$ (201)

where the subscripts denote the shifted KK charges due to the $\mathbb{Z}_{2}$
twist. Collecting the 1-loop contributions from these KK states and the
classical Green-Schwarz contributions, we obtain the full effective
prepotential on the $\Omega$-background as

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\bigg{(}\mathcal{E}_{\mathrm{tree}}+\mathcal{F}_{\text{1-loop}}+\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{24}\phi_{1}+\epsilon_{+}^{2}\phi_{1}\bigg{)}\
,$ $\displaystyle\mathcal{E}_{\mathrm{tree}}$
$\displaystyle=\tau\phi_{0}^{2}+2\phi_{0}\quantity(\phi_{1}^{2}-\frac{1}{2}\sum_{i=1}^{3}m_{i}^{2}+\frac{3}{2}\epsilon_{+}^{2})\
,$ $\displaystyle\mathcal{F}_{\text{1-loop}}$
$\displaystyle=\frac{5}{6}\phi_{1}^{3}-\frac{3}{16}\tau\phi_{1}^{2}-\frac{1}{2}\sum_{i=1}^{3}m_{i}^{2}\phi_{1}\
.$ (202)

Here, the Kähler parameters in the 6d gauge theory description can be
converted to the parameters in the 5d $Sp(2)$ gauge theory description simply
by the following reparametrization:

$\displaystyle\phi_{0}\to\phi_{1}+\frac{m_{0}}{16}-\frac{1}{4}\sum_{i=1}^{3}m_{i}\,,\quad\phi_{1}\to\phi_{2}-2\phi_{1}+\frac{m_{0}}{2}\,,\quad\tau\to
2m_{0}\,,\quad m_{i}\to m_{i}-\frac{m_{0}}{2}\,.$ (203)

One can easily check that the effective prepotential of the 6d theory with
this reparametrization reproduces that of the 5d theory given above up to
terms independent of $\phi_{i}$.

It is convenient in the 6d perspective to use the 6d parameters shifted as

$\displaystyle\phi_{0}\to\phi_{0}-\frac{5}{32}\tau-\frac{1}{4}\sum_{i=1}^{3}m_{i}\,,\quad\phi_{1}\to\phi_{1}-2\phi_{0}\,.$
(204)

The volumes of 2-cycles in the geometry (196) can be written in terms of the
shifted 6d parameters as

$\displaystyle\operatorname{vol}(f_{1})=2\phi_{0}-\phi_{1}+\frac{\tau}{4}\,,\quad$
$\displaystyle\operatorname{vol}(f_{2})=-4\phi_{0}+2\phi_{1}\,,$
$\displaystyle\operatorname{vol}(e_{2})=-2\phi_{0}+2\phi_{1}-\frac{\tau}{2}\,,\quad$
$\displaystyle\operatorname{vol}(x_{i})=-2\phi_{0}+\phi_{1}+m_{i}\,.$ (205)

One finds a pair of consistent magnetic fluxes as

$\displaystyle n_{1}\in\mathbb{Z}\,,\quad B_{\tau}=0\,,\quad
B_{m_{i}}=1/2\,,\quad n_{0}\in\mathbb{Z}+B_{0}\quad{\rm with}\quad
B_{0}=0,1/2\,,$ (206)

which lead to unity blowup equations. We checked that the solutions to the
blowup equations match the result in Table 15 from the 5d $Sp(2)$ gauge
theory.

####  5.1.2 $SU(3)_{4}+6\mathbf{F}$,
$Sp(2)+2\mathbf{\Lambda}^{2}+4\mathbf{F}$, $G_{2}+6\mathbf{F}$

This theory is a KK theory coming from the 6d $SU(3)$ gauge theory with 12
fundamental hypermultiplets on a circle with $\mathbb{Z}_{2}$ twist. It has
three different 5d gauge theory descriptions: $SU(3)_{4}+6\mathbf{F}$,
$Sp(2)+2\mathbf{\Lambda}^{2}+4\mathbf{F}$, and $G_{2}+6\mathbf{F}$,

(207)

This theory is geometrically engineered by gluing $\mathbb{F}_{2}$ and
$\mathbb{F}_{0}^{6}$ as Jefferson:2018irk

(208)

We shall solve the blowup equations by using the $SU(3)$ and $G_{2}$ frames.
We first compute the cubic prepotential of the $SU(3)$ gauge theory on the
chamber $\phi_{2}\geq\phi_{1}>0$ as

$\displaystyle 6\mathcal{F}$
$\displaystyle=8\phi_{1}^{3}-3\phi_{1}^{2}\phi_{2}-3\phi_{1}\phi_{2}^{2}+8\phi_{2}^{2}+12\phi_{1}\phi_{2}(\phi_{1}-\phi_{2})$
(209)
$\displaystyle\quad-\frac{1}{2}\sum_{i=1}^{6}\Big{(}(\phi_{2}-m_{i})^{3}+(\phi_{2}-\phi_{1}+m_{i})^{3}+(\phi_{1}+m_{i})^{3}\Big{)}+6m_{0}(\phi_{1}^{2}-\phi_{1}\phi_{2}+\phi_{2}^{2})\,,$

where $m_{0}$ is the $SU(3)$ gauge coupling and $m_{i=1,\cdots,6}$ are mass
parameters of 6 fundamentals. The full effective prepotential is

$\displaystyle\mathcal{E}=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{48}(4\phi_{1}-8\phi_{2})+\epsilon_{+}^{2}(\phi_{1}+\phi_{2}))\,.$
(210)

A choice of magnetic fluxes

$\displaystyle n_{i}\in\mathbb{Z}\,,\quad B_{m_{0}}=-1/2\,,\quad
B_{m_{i}}=1/2\quad(1\leq i\leq 6)\,$ (211)

provides a solvable unity blowup equation. We solve the blowup equation and
find the BPS spectrum of the $SU(3)_{4}+6{\bf F}$ theory listed in Table 16.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,-\frac{4}{3},\frac{1}{3})$ |  $(0,\frac{1}{2})$ | $(1,-\frac{4}{3},\frac{4}{3})$ |  $(0,\frac{3}{2})$
$(1,-1,0)$ |  $6(0,0)$ | $(1,-1,1)$ |  $6(0,1)$
$(1,-1,2)$ |  $6(0,2)$ | $(1,-\frac{2}{3},\frac{2}{3})$ |  $15(0,\frac{1}{2})$
$(1,-\frac{2}{3},\frac{5}{3})$ |  $15(0,\frac{3}{2})$ | $(1,-\frac{1}{3},\frac{1}{3})$ |  $20(0,0)\oplus(0,\frac{1}{2})$
$(1,-\frac{1}{3},\frac{4}{3})$ |  $(0,\frac{1}{2})\oplus 20(0,1)\oplus(0,\frac{3}{2})$ | $(1,0,1)$ |  $6(0,0)\oplus 15(0,\frac{1}{2})\oplus 6(0,1)$
$(1,0,2)$ |  $6(0,1)\oplus 15(0,\frac{3}{2})\oplus 6(0,2)$ | $(1,\frac{1}{3},\frac{2}{3})$ |  $6(0,0)\oplus 15(0,\frac{1}{2})$
$(1,\frac{1}{3},\frac{5}{3})$ |  $\\!15(0,\frac{1}{2})\oplus 6(0,1)\oplus 15(0,\frac{3}{2})\\!$ | $(1,\frac{2}{3},\frac{1}{3})$ |  $20(0,0)\oplus(0,\frac{1}{2})$
$(1,\frac{2}{3},\frac{4}{3})$ |  $20(0,0)\oplus 2(0,\frac{1}{2})\oplus 20(0,1)\oplus(0,\frac{3}{2})$ | $(1,1,0)$ |  $6(0,0)$
$(1,1,1)$ |  $6(0,0)\oplus 15(0,\frac{1}{2})\oplus 6(0,1)$ | $(1,1,2)$ |  $6(0,0)\oplus 15(0,\frac{1}{2})\oplus 6(0,1)\oplus 15(0,\frac{3}{2})\oplus 6(0,2)$
$(2,-2,1)$ |  $15(0,\frac{3}{2})$ | $(2,-\frac{5}{3},\frac{2}{3})$ |  $20(0,1)$
$(2,-\frac{4}{3},\frac{1}{3})$ |  $15(0,\frac{1}{2})$ | $(2,-1,0)$ |  $6(0,0)$
$(2,-1,1)$ |  $6(0,0)\oplus 15(0,\frac{1}{2})\oplus 96(0,1)\oplus 15(0,\frac{3}{2})\oplus 6(\frac{1}{2},\frac{3}{2})$ | $(2,-\frac{2}{3},\frac{2}{3})$ |  $20(0,0)\oplus 37(0,\frac{1}{2})\oplus 20(0,1)\oplus(\frac{1}{2},1)$
$(2,-\frac{1}{3},\frac{1}{3})$ |  $12(0,0)\oplus 15(0,\frac{1}{2})$ | $(2,0,1)$ |  $102(0,0)\oplus 66(0,\frac{1}{2})\oplus 102(0,1)\oplus 15(0,\frac{3}{2})\oplus 6(\frac{1}{2},\frac{1}{2})\oplus 6(\frac{1}{2},\frac{3}{2})$
Table 16: BPS spectrum of the $SU(3)_{4}+6\mathbf{F}$ theory for
$(d_{1}=1,d_{2}\leq 1,d_{3}\leq 2)$ and $(d_{1}=2,d_{2}\leq 0,d_{3}\leq 1)$
where $\mathbf{d}=(d_{1},d_{2},d_{3})$ labels the BPS states with charge
$d_{1}m_{0}+d_{2}\alpha_{1}+d_{3}\alpha_{2}$ for simple roots
$\alpha_{1}=2\phi_{1}-\phi_{2}$, $\alpha_{2}=-\phi_{1}+2\phi_{2}$ of
$\mathfrak{su}(3)$ algebra, and 6 flavor charges are blindly summed over.

On the other hand, the full effective prepotential in the $G_{2}$ gauge theory
in the same chamber is

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{48}(4\phi_{1}-8\phi_{2})+\epsilon_{+}^{2}(\phi_{1}+\phi_{2}))\,,$
$\displaystyle 6\mathcal{F}$
$\displaystyle=8\phi_{1}^{3}+18\phi_{1}^{2}\phi_{2}-24\phi_{1}\phi_{2}^{2}+8\phi_{2}^{3}+6m_{0}(3\phi_{1}^{2}-3\phi_{1}\phi_{2}+\phi_{2}^{2})$
$\displaystyle\quad-\frac{1}{2}\sum_{i=1}^{6}\Big{(}(\phi_{1}+m_{i})^{3}+(-\phi_{1}+\phi_{2}+m_{i})^{3}+(2\phi_{1}-\phi_{2}+m_{i})^{3}$
$\displaystyle\qquad\qquad+(-2\phi_{1}+\phi_{2}+m_{i})^{3}+(-\phi_{1}+\phi_{2}-m_{i})^{3}+(\phi_{1}-m_{i})^{3}+m_{i}^{3}\Big{)}\,.$
(212)

Here, $m_{0}$ is the $G_{2}$ gauge coupling and $m_{i=1,\cdots,6}$ are mass
parameters of the $G_{2}$ fundamental hypers.

Under the duality between the $SU(3)$ and $G_{2}$ descriptions, the Kähler
parameters in two theories have a natural map Hayashi:2018lyv  given by

$\displaystyle\phi_{1}^{SU}$
$\displaystyle=\phi_{1}^{G_{2}}+\frac{1}{3}m_{0}^{G_{2}}-\frac{1}{3}\sum_{j=1}^{6}m_{j}^{G_{2}}\,,\quad\phi_{2}^{SU}=\phi_{2}^{G_{2}}+\frac{2}{3}m_{0}^{G_{2}}-\frac{2}{3}\sum_{j=1}^{6}m_{j}^{G_{2}}\
,$ (213) $\displaystyle m_{0}^{SU}$
$\displaystyle=\frac{2}{3}m_{0}^{G_{2}}-\frac{1}{6}\sum_{j=1}^{6}m_{j}^{G_{2}}\,,\quad
m_{i}^{SU}=-\frac{1}{3}m_{0}^{G_{2}}-m_{i}^{G_{2}}+\frac{1}{3}\sum_{j=1}^{6}m_{j}^{G_{2}}\quad(1\leq
i\leq 6)\,.$

The magnetic flux quantization is satisfied by

$\displaystyle n_{i}\in\mathbb{Z}\,,\quad B_{m_{0}}=0\,,\quad
B_{m_{i}}=1/2\quad(1\leq i\leq 6)\,.$ (214)

This choice of magnetic fluxes provides a solvable unity blowup equation. We
check that the solution to the equation matches the BPS spectrum of the
$SU(3)$ theory in Table 16 under the above parameter map.

####  5.1.3 $SU(3)_{\frac{3}{2}}+9\mathbf{F}$,
$Sp(2)+1\mathbf{\Lambda}^{2}+8\mathbf{F}$

This theory comes from the circle compactification of the 6d rank-2 E-string
theory. This theory can also be described by two 5d gauge theories:
$SU(3)_{3/2}+9\mathbf{F}$ and $Sp(2)+1\mathbf{\Lambda}^{2}+8\mathbf{F}$,

(215)

The geometric construction for this theory is given by

(216)

The validity of the blowup equations for the 6d rank-2 E-string was checked in
Gu:2019pqj . Here, we will solve the blowup equations from the perspective of
two dual 5d gauge theories. First, the full effective prepotential of the
$SU(3)$ theory on the Coulomb branch where $\phi_{2}\geq\phi_{1}>0$ is

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{48}\quantity(4\phi_{1}-14\phi_{2})+\epsilon_{+}^{2}(\phi_{1}+\phi_{2}))\,,$
$\displaystyle 6\mathcal{F}$
$\displaystyle=8\phi_{1}^{3}-3\phi_{1}^{2}\phi_{2}-3\phi_{1}\phi_{2}^{2}+8\phi_{2}^{2}+\frac{9}{2}\phi_{1}\phi_{2}(\phi_{1}-\phi_{2})+6m_{0}(\phi_{1}^{2}-\phi_{1}\phi_{2}+\phi_{2}^{2})$
$\displaystyle\quad-\frac{1}{2}\sum_{i=1}^{9}\quantity((\phi_{2}-m_{i})^{3}+(\phi_{2}-\phi_{1}+m_{i})^{3}+(\phi_{1}+m_{i})^{3})\,,$
(217)

where $m_{0}$ is the $SU(3)$ gauge coupling and $m_{i=1,\cdots,9}$ are mass
parameters of the fundamental hypermultiplets. A set of consistent magnetic
fluxes

$\displaystyle n_{i}\in\mathbb{Z}\,,\quad B_{m_{0}}=-3/4\,,\quad
B_{m_{i}}=1/2\quad(1\leq i\leq 9)\,.$ (218)

provides a solvable unity blowup equation. By solving the blowup equation, we
obtain the BPS spectrum given in Table 17.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,-1,0)$ |  $(0,0)$ | $(1,-1,1)$ |  $(0,1)$
$(1,-1,2)$ |  $(0,2)$ | $(1,-\frac{2}{3},\frac{2}{3})$ |  $9(0,\frac{1}{2})$
$(1,-\frac{2}{3},\frac{5}{3})$ |  $9(0,\frac{3}{2})$ | $(1,-\frac{1}{3},\frac{1}{3})$ |  $36(0,0)$
$(1,-\frac{1}{3},\frac{4}{3})$ |  $36(0,1)$ | $(1,0,1)$ |  $(0,0)\oplus 84(0,\frac{1}{2})\oplus(0,1)$
$(1,0,2)$ |  $(0,1)\oplus 84(0,\frac{3}{2})\oplus(0,2)$ | $(1,\frac{1}{3},\frac{2}{3})$ |  $126(0,0)\oplus 9(0,\frac{1}{2})$
$(1,\frac{1}{3},\frac{5}{3})$ |  $9(0,\frac{1}{2})\oplus 126(0,1)\oplus 9(0,\frac{3}{2})$ | $(1,\frac{2}{3},\frac{1}{3})$ |  $45(0,0)$
$(1,\frac{2}{3},\frac{4}{3})$ |  $36(0,0)\oplus 126(0,\frac{1}{2})\oplus 36(0,1)$ | $(1,1,0)$ |  $(0,0)\oplus(0,\frac{1}{2})$
$(1,1,1)$ |  $85(0,0)\oplus 85(0,\frac{1}{2})\oplus(0,1)$ | $(1,1,2)$ |  $(0,0)\oplus 84(0,\frac{1}{2})\oplus 85(0,1)\oplus 84(0,\frac{3}{2})\oplus(0,2)$
$(2,-1,1)$ |  $84(0,1)$ | $(2,-\frac{2}{3},\frac{2}{3})$ |  $126(0,\frac{1}{2})$
$(2,-\frac{1}{3},\frac{1}{3})$ |  $126(0,0)$ | $(2,0,1)$ |  $85(0,0)\oplus 1219(0,\frac{1}{2})\oplus 85(0,1)\oplus 84(\frac{1}{2},1)$
$(2,\frac{1}{3},\frac{2}{3})$ |  $801(0,0)\oplus 135(0,\frac{1}{2})\oplus 36(\frac{1}{2},\frac{1}{2})$ | $(2,\frac{2}{3},\frac{1}{3})$ |  $\\!162(0,0)\oplus 9(0,\frac{1}{2})\oplus 9(\frac{1}{2},0)\\!$
$(2,1,0)$ |  $(0,0)\oplus(0,1)\oplus(\frac{1}{2},\frac{1}{2})$ | $(2,1,1)$ |  $1465(0,0)\oplus 1387(0,\frac{1}{2})\oplus 168(0,1)\oplus(0,2)\oplus 84(\frac{1}{2},0)\oplus 83(\frac{1}{2},\frac{1}{2})\oplus 84(\frac{1}{2},1)\oplus(\frac{1}{2},\frac{3}{2})\oplus(1,1)$
Table 17: BPS spectrum of the $SU(3)_{3/2}+9\mathbf{F}$ theory for $d_{1}=1$,
$d_{2}\leq 1$, $d_{3}\leq 2$ and $d_{1}=2$, $d_{2},d_{3}\leq 1$. All flavor
mass parameters $m_{i=1,\cdots,9}$ are turned off and
$\mathbf{d}=(d_{1},d_{2},d_{3})$ labels the BPS states with charge
$d_{1}m_{0}+d_{2}\alpha_{1}+d_{3}\alpha_{2}$ for the simple roots $\alpha_{1}$
and $\alpha_{2}$ of $\mathfrak{su}(3)$ algebra.

On the other hand, in the $Sp(2)$ theory, the full effective prepotential in
the same chamber is

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{48}\quantity(4\phi_{1}-14\phi_{2})+\epsilon_{+}^{2}(\phi_{1}+\phi_{2}))\,,$
(219) $\displaystyle 6\mathcal{F}$
$\displaystyle=8\phi_{1}^{3}+12\phi_{1}^{2}\phi_{2}-18\phi_{1}\phi_{2}^{2}+8\phi_{2}^{3}+m_{0}(2\phi_{1}^{2}-2\phi_{1}\phi_{2}+\phi_{2}^{2})$
$\displaystyle\quad-\frac{1}{2}\sum_{i=1}^{8}\quantity((\phi_{1}+m_{i})^{3}+(-\phi_{1}+\phi_{2}+m_{i})^{3}+(-\phi_{1}+\phi_{2}-m_{i})^{3}+(\phi_{1}-m_{i})^{3})$
$\displaystyle\quad-\frac{1}{2}\Big{(}(\phi_{2}+m_{9})^{3}+(2\phi_{1}-\phi_{2}+m_{9})^{3}+(-2\phi_{1}+\phi_{2}+m_{9})^{3}+(\phi_{2}-m_{9})^{3}\Big{)}\,,$

where $m_{0}$ is the $Sp(2)$ gauge coupling, $m_{i=1,\cdots,8}$ are mass
parameters of the $Sp(2)$ fundamentals and $m_{9}$ denotes the mass parameter
of the antisymmetric hyper.

The Kähler parameters in the $Sp(2)$ theory are mapped to those in the $SU(3)$
theory as

$\displaystyle\phi_{1}^{Sp}$
$\displaystyle=\phi_{1}^{SU}+\frac{1}{2}m_{0}^{SU}+\frac{1}{4}\sum_{j=1}^{8}m_{j}^{SU}-\frac{1}{4}m_{9}^{SU},\
\phi_{2}^{Sp}=\phi_{2}^{SU}+m_{0}^{SU}+\frac{1}{2}\sum_{j=1}^{8}m_{j}^{SU}-\frac{1}{2}m_{9}^{SU}\
,$ $\displaystyle m_{0}^{Sp}$
$\displaystyle=\frac{3}{2}m_{0}^{SU}+\frac{1}{4}\sum_{i=1}^{9}m_{i}^{SU},\
m_{i}^{Sp}=\frac{1}{2}m_{0}^{SU}-m_{i}^{SU}+\frac{1}{4}\sum_{j=1}^{8}m_{j}^{SU}-\frac{1}{4}m_{9}^{SU}\
(1\leq i\leq 7)\ ,$ $\displaystyle m_{8}^{Sp}$
$\displaystyle=-\frac{1}{2}m_{0}^{SU}+m_{8}^{SU}-\frac{1}{4}\sum_{j=1}^{8}m_{j}^{SU}+\frac{1}{4}m_{9}^{SU},\
\ m_{9}^{Sp}=m_{0}^{SU}+\frac{1}{2}\sum_{j=1}^{9}m^{SU}_{j}\,.$ (220)

In the $Sp(2)$ frame, we find a set of consistent magnetic fluxes

$\displaystyle n_{i}\in\mathbb{Z}\,,\quad B_{m_{0}}=0\,,\quad
B_{m_{i}}=1/2\quad(1\leq i\leq 9)\,.$ (221)

This leads to a solvable unity blowup equation. We have checked that the
solution to the blowup equation agrees with the BPS spectrum of the dual
$SU(3)$ gauge theory given in Table 17.

The instanton partition function of the
$Sp(2)+1\mathbf{\Lambda}^{2}+8\mathbf{F}$ theory can also be calculated using
the ADHM construction introduced in Aharony:1997pm ; Kim:2012gu ;
Hwang:2014uwa . We checked that our result from the blowup formula is in
perfect agreement with the ADHM result in the Kähler parameter expansion.

####  5.1.4 $SU(3)_{0}+10\mathbf{F}$, $Sp(2)+10\mathbf{F}$

The 5d $SU(3)_{0}+10\mathbf{F}$ theory and the $Sp(2)+10\mathbf{F}$ theory are
two dual 5d descriptions for the KK theory arising from the circle
compactification of the 6d $Sp(1)$ gauge theory with 10 fundamental
hypermultiplets,

(222)

This theory is geometrically engineered by gluing $\mathbb{F}_{6}^{10}$ and
$\mathbb{F}_{0}$ Jefferson:2018irk .

(223)

The blowup equation for this theory from the 6d gauge theory perspective was
checked to be satisfied by substituting the known one-string elliptic genus
into the blowup equation in Gu:2020fem .

We will now compute the BPS spectrum of this theory using the blowup equations
in the 5d gauge theory descriptions. First, the full prepotential of the
$SU(3)$ gauge theory on the chamber $\phi_{2}\geq\phi_{1}>0$ is

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{48}\quantity(4\phi_{1}-16\phi_{2})+\epsilon_{+}^{2}(\phi_{1}+\phi_{2}))\,,$
$\displaystyle 6\mathcal{F}$
$\displaystyle=8\phi_{1}^{3}-3\phi_{1}^{2}\phi_{2}-3\phi_{1}\phi_{2}^{2}+8\phi_{2}^{2}+6m_{0}(\phi_{1}^{2}-\phi_{1}\phi_{2}+\phi_{2}^{2})$
$\displaystyle\quad-\frac{1}{2}\sum_{i=1}^{10}\quantity((\phi_{2}-m_{i})^{3}+(\phi_{2}-\phi_{1}+m_{i})^{3}+(\phi_{1}+m_{i})^{3})\,.$
(224)

We find a set of consistent magnetic fluxes

$\displaystyle n_{i}\in\mathbb{Z}\,,\ \ B_{m_{0}}=0\,,\ \ B_{m_{i}}=1/2\ \
(1\leq i\leq 5)\,,\ \ B_{m_{i}}=-1/2\ \ (6\leq i\leq 10)\ ,$ (225)

which provides a solvable unity blowup equation.

In the $Sp(2)$ gauge theory, on the other hand, the full effective
prepotential on the same chamber takes the form of

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{48}(4\phi_{1}-16\phi_{2})+\epsilon_{+}^{2}(\phi_{1}+\phi_{2}))\,,$
$\displaystyle 6\mathcal{F}$
$\displaystyle=8\phi_{1}^{3}+12\phi_{1}^{2}\phi_{2}-18\phi_{1}\phi_{2}^{2}+8\phi_{2}^{3}+m_{0}(2\phi_{1}^{2}-2\phi_{1}\phi_{2}+\phi_{2}^{2})$
$\displaystyle\quad-\frac{1}{2}\sum_{i=1}^{10}\quantity((\phi_{1}\pm
m_{i})^{3}+(-\phi_{1}+\phi_{2}\pm m_{i})^{3})\,,$ (226)

where $m_{0}$ is the $Sp(2)$ gauge coupling and $m_{i=1,\cdots,10}$ are mass
parameters of the fundamental hypers. One possible choice of consistent
magnetic fluxes in the $Sp(2)$ theory is

$\displaystyle n_{i}\in\mathbb{Z}\,,\quad B_{m_{0}}=0\,,\quad
B_{m_{i}}=1/2\quad(1\leq i\leq 10)\ .$ (227)

The Kähler parameters in two 5d theories are mapped to each other by the
relations Hayashi:2016abm

$\displaystyle\phi_{1}^{Sp}$
$\displaystyle=\phi_{1}^{SU}+\frac{1}{2}m_{0}^{SU}+\frac{1}{4}\sum_{i=1}^{10}m_{i}^{SU}\,,\
\
\phi_{2}^{Sp}=\phi_{2}^{SU}+m_{0}^{SU}+\frac{1}{2}\sum_{i=1}^{10}m_{i}^{SU}\,,\
\ m_{0}^{Sp}=m_{0}^{SU}\,,$ $\displaystyle m_{i}^{Sp}$
$\displaystyle=\frac{1}{2}m_{0}^{SU}-m_{j}^{SU}+\frac{1}{4}\sum_{j=1}^{10}m_{j}^{SU}\quad(1\leq
i\leq 9)\,,\ \
m_{10}^{Sp}=-\frac{1}{2}m_{0}^{SU}-\frac{1}{4}\sum_{j=1}^{9}m_{j}^{SU}+\frac{3}{4}m_{10}^{SU}.$
(228)

We list in Table 18 some leading BPS states computed by solving the blowup
equation in the $Sp(2)+10\mathbf{F}$ theory. We checked that this solution
agrees with the result from the $SU(3)_{0}+10\mathbf{F}$ description and also
agree with the ADHM result for the $Sp(2)+10\mathbf{F}$ theory computed in
Yun:2016yzw  as well as topological vertex Hayashi:2016abm  in series
expansion of Kähler parameters.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,1,1)$ |  $512(0,0)$ | $(1,1,\frac{3}{2})$ |  $512(0,\frac{1}{2})$
$(1,1,2)$ |  $512(0,1)$ | $(1,2,\frac{3}{2})$ |  $512(0,\frac{1}{2})$
$(1,2,2)$ |  $512(0,0)\oplus 512(0,1)$ | $(2,0,-1)$ |  $(0,\frac{1}{2})$
$(2,0,-\frac{1}{2})$ |  $20(0,0)$ | $(2,0,\frac{1}{2})$ |  $20(0,0)$
$(2,0,1)$ |  $(0,\frac{1}{2})$ | $(2,1,-1)$ |  $(0,\frac{3}{2})$
$(2,1,-\frac{1}{2})$ |  $20(0,1)$ | $(2,1,0)$ |  $192(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(\frac{1}{2},0)\oplus(\frac{1}{2},1)$
$(2,1,\frac{1}{2})$ |  $1180(0,0)\oplus 20(0,1)\oplus 20(\frac{1}{2},\frac{1}{2})$ | $(2,1,1)$ |  $192(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(\frac{1}{2},0)\oplus(\frac{1}{2},1)$
$(2,2,-1)$ |  $(0,\frac{5}{2})$ | $(2,2,-\frac{1}{2})$ |  $20(0,2)$
$(2,2,0)$ |  $(0,\frac{1}{2})\oplus 192(0,\frac{3}{2})\oplus(0,\frac{5}{2})\oplus(\frac{1}{2},1)\oplus(\frac{1}{2},2)$ | $(2,1,\frac{1}{2})$ |  $20(0,0)\oplus 1180(0,1)\oplus 20(0,2)\oplus 20(\frac{1}{2},\frac{1}{2})\oplus 20(\frac{1}{2},\frac{3}{2})$
$(2,2,1)$ |  $5230(0,\frac{1}{2})\oplus 192(0,\frac{3}{2})\oplus(0,\frac{5}{2})\oplus 192(\frac{1}{2},0)\oplus 193(\frac{1}{2},1)\oplus(\frac{1}{2},2)\oplus(1,\frac{1}{2})\oplus(1,\frac{3}{2})$
Table 18: BPS spectrum of the $Sp(2)+10\mathbf{F}$ theory for $d_{1}=1$,
$d_{2},d_{3}\leq 2$ and $d_{1}=2$, $d_{2}\leq 2$, $d_{3}\leq 1$. Here,
$\mathbf{d}=(d_{1},d_{2},d_{3})$ labels BPS states with charge
$d_{1}m_{0}+d_{2}\alpha_{1}+d_{3}\alpha_{2}$, where
$\alpha_{1}=2\phi_{1}-\phi_{2}$, $\alpha_{2}=-2\phi_{1}+2\phi_{2}$ are the
simple roots of $\mathfrak{sp}(2)$ algebra, and 10 flavor charges are blindly
summed over.

####  5.1.5 $SU(2)\times SU(2)+2\,\mathbf{bi\text{-}F}$

We will now give an example for 5d quiver gauge theories. The 5d theory with
$SU(2)\times SU(2)$ gauge group and 2 bi-fundamental hypermultiplets
($\mathbf{bi\text{-}F}$) is the KK theory obtained from a circle
compactification of the 6d $SU(2)$ gauge theory with 4 fundamental
hypermultiplets.

The full effective prepotential of the $SU(2)\times SU(2)$ gauge theory on the
Coulomb branch where $\phi_{2}\geq\phi_{1}>0$ is given by

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{12}\quantity(\phi_{1}-\phi_{2})+\epsilon_{+}^{2}(\phi_{1}+\phi_{2}))\,,$
(229) $\displaystyle 6\mathcal{F}$
$\displaystyle=8\phi_{1}^{3}+8\phi_{2}^{3}+6m_{1}\phi_{1}^{2}+6m_{2}\phi_{2}^{2}-\frac{1}{2}\sum_{i=3}^{4}\Big{(}(\phi_{1}+\phi_{2}\pm
m_{i})^{3}+(-\phi_{1}+\phi_{2}\pm m_{i})^{3}\Big{)}\ ,$

where $m_{1}$ and $m_{2}$ are the gauge couplings for two $SU(2)$ gauge groups
respectively, while $m_{3}$ and $m_{4}$ are the mass parameters of two bi-
fundamentals.

A unity blowup equation can be formulated with a set of the consistent
magnetic fluxes given by

$\displaystyle n_{i}\in\mathbb{Z}\ ,\quad B_{m_{1}}=0\ ,\quad B_{m_{2}}=0\
,\quad B_{m_{3,4}}=1/2\ .$ (230)

The solution to the blowup equation is given in Table 19. We also computed the
partition function of this theory using the ADHM method and confirmed that the
result agrees with the BPS spectrum in Table 19.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,0,1,-1,-1)$ |  $2(0,0)$ | $(1,0,1,1,-1)$ |  $2(0,0)$
$(1,0,2,-2,0)$ |  $(0,\frac{1}{2})$ | $(1,0,2,0,-2)$ |  $(0,\frac{1}{2})$
$(1,0,2,0,0)$ |  $4(0,\frac{1}{2})$ | $(1,0,2,2,0)$ |  $(0,\frac{1}{2})$
$(1,1,-1,1,-1)$ |  $2(0,0)$ | $(1,1,0,2,-2)$ |  $4(0,\frac{1}{2})$
$(1,1,0,2,0)$ |  $10(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(\frac{1}{2},0)\oplus(\frac{1}{2},1)$ | $(1,1,1,-1,-1)$ |  $2(0,0)$
$(1,1,1,1,-3)$ |  $2(0,0)$ | $(1,1,1,1,-1)$ |  $14(0,0)\oplus 4(0,1)\oplus 2(\frac{1}{2},\frac{1}{2})$
$(1,1,2,0,-2)$ |  $4(0,\frac{1}{2})$ | $(1,1,2,0,0)$ |  $10(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(\frac{1}{2},0)\oplus(\frac{1}{2},1)$
$(1,1,2,2,-2)$ |  $10(0,\frac{1}{2})\oplus 5(0,\frac{3}{2})\oplus(\frac{1}{2},0)\oplus(\frac{1}{2},1)$ | $(1,1,2,2,0)$ |  $25(0,\frac{1}{2})\oplus 14(0,\frac{3}{2})\oplus(0,\frac{5}{2})\oplus 4(\frac{1}{2},0)\oplus 5(\frac{1}{2},1)\oplus(\frac{1}{2},2)$
Table 19: BPS spectrum of the $SU(2)\times SU(2)$ theory with $2$
bifundamentals. Here, $\mathbf{d}=(d_{1},d_{2},d_{3},d_{4},d_{5})$ labels the
BPS states with charge
$d_{1}m_{1}+d_{2}m_{2}+d_{3}\phi_{1}+d_{4}\phi_{2}+d_{5}m_{i}$ where $d_{5}$
counts collective degrees of two bi-fundamental matters with masses
$m_{i=3,4}$. There is a symmetry exchanging
$(d_{1},d_{2},d_{3},d_{4},d_{5})\leftrightarrow(d_{2},d_{1},d_{4},d_{3},d_{5})$
and $d_{5}\leftrightarrow-d_{5}$, so we list BPS states up to
$(d_{3},d_{4})\leq(2,2)$ and $d_{5}\leq 0$ for $(1,0)$ and $(1,1)$ instantons.

####  5.1.6 $SU(3)_{0}+1\mathbf{Adj}$

The 5d $SU(3)_{0}$ gauge theory with one adjoint hypermultiplet is equivalent
to a circle reduction of the 6d $\mathcal{N}=(2,0)$ $A_{2}$ theory
Bhardwaj:2019fzv ,

(231)

The cubic prepotential of this theory on the chamber where
$2\phi_{2}>\phi_{1}>\frac{1}{2}\phi_{2}$ is

$\displaystyle 6\mathcal{F}$
$\displaystyle=8\phi_{1}^{3}-3\phi_{1}^{2}\phi_{2}-3\phi_{1}\phi_{2}^{2}+8\phi_{2}^{3}+6m_{0}(\phi_{1}^{2}-\phi_{1}\phi_{2}+\phi_{2}^{2})$
$\displaystyle\quad-\frac{1}{2}\Big{(}(\phi_{1}+\phi_{2}\pm
m_{1})^{3}+(-\phi_{1}+2\phi_{2}\pm m_{1})^{3}+(2\phi_{1}-\phi_{2}\pm
m_{1})^{3}\Big{)},$ (232)

where $m_{0}$ is the gauge coupling and $m_{1}$ is the mass parameter of the
adjoint matter. The effective prepotential is

$\displaystyle\mathcal{E}=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}+\epsilon_{+}^{2}(\phi_{1}+\phi_{2}))\
.$ (233)

We find that a set of consistent magnetic fluxes

$\displaystyle n_{i}\in\mathbb{Z}\ ,\quad B_{m_{0}}=0\ ,\quad B_{m_{1}}=1/2\ $
(234)

gives rise to a solvable unity blowup equation. The BPS spectrum obtained by
solving the blowup equation is listed in Table 20. We also computed the
instanton partition function for $SU(3)_{0}+1\mathbf{Adj}$ independently using
the ADHM construction and confirmed that our result agrees with the result
from the ADHM method.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,0,1,-2)$ |  $(0,\frac{1}{2})$ | $(1,0,1,-1)$ |  $(0,0)\oplus(0,1)\oplus(\frac{1}{2},\frac{1}{2})$
$(1,0,1,0)$ |  $2(0,\frac{1}{2})\oplus(\frac{1}{2},0)\oplus(\frac{1}{2},1)$ | $(1,0,2,-2)$ |  $(0,\frac{3}{2})$
$(1,0,2,-1)$ |  $(0,1)\oplus(0,2)\oplus(\frac{1}{2},\frac{3}{2})$ | $(1,0,2,0)$ |  $2(0,\frac{3}{2})\oplus(\frac{1}{2},1)\oplus(\frac{1}{2},2)$
$(1,1,1,-3)$ |  $(0,0)$ | $(1,1,1,-2)$ |  $3(0,\frac{1}{2})\oplus(\frac{1}{2},0)$
$(1,1,1,-1)$ |  $5(0,0)\oplus 2(0,1)\oplus 3(\frac{1}{2},\frac{1}{2})$ | $(1,1,1,0)$ |  $6(0,\frac{1}{2})\oplus 4(\frac{1}{2},0)\oplus 2(\frac{1}{2},1)$
$(1,1,2,-3)$ |  $(0,1)$ | $(1,1,2,-2)$ |  $2(0,\frac{1}{2})\oplus 2(0,\frac{3}{2})\oplus(\frac{1}{2},1)$
$(1,1,2,-1)$ |  $(0,0)\oplus 5(0,1)\oplus(0,2)\oplus 2(\frac{1}{2},\frac{1}{2})\oplus 2(\frac{1}{2},\frac{3}{2})$ | $(1,1,2,0)$ |  $4(0,\frac{1}{2})\oplus 4(0,\frac{3}{2})\oplus(\frac{1}{2},0)\oplus 4(\frac{1}{2},1)\oplus(\frac{1}{2},2)$
$(1,2,2,-3)$ |  $(0,0)\oplus(0,1)$ | $(1,2,2,-2)$ |  $4(0,\frac{1}{2})\oplus 3(0,\frac{3}{2})\oplus(\frac{1}{2},0)\oplus(\frac{1}{2},1)$
$(1,2,2,-1)$ |  $5(0,0)\oplus 7(0,1)\oplus 2(0,2)\oplus 4(\frac{1}{2},\frac{1}{2})\oplus 3(\frac{1}{2},\frac{3}{2})$ | $(1,2,2,0)$ |  $8(0,\frac{1}{2})\oplus 6(0,\frac{3}{2})\oplus 4(\frac{1}{2},0)\oplus 6(\frac{1}{2},1)\oplus 2(\frac{1}{2},2)$
$(2,0,1,-3)$ |  $(0,0)$ | $(2,0,1,-2)$ |  $2(0,\frac{1}{2})\oplus(\frac{1}{2},0)\oplus(\frac{1}{2},1)$
$(2,0,1,-1)$ |  $4(0,0)\oplus 2(0,1)\oplus 3(\frac{1}{2},\frac{1}{2})\oplus(\frac{1}{2},\frac{3}{2})\oplus(1,1)$ | $(2,0,1,0)$ |  $5(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus 3(\frac{1}{2},0)\oplus 3(\frac{1}{2},1)\oplus(1,\frac{1}{2})\oplus(1,\frac{3}{2})$
$(2,1,1,-3)$ |  $5(0,0)\oplus 2(0,1)\oplus 3(\frac{1}{2},\frac{1}{2})$ | $(2,1,1,-2)$ |  $14(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus 9(\frac{1}{2},0)\oplus 7(\frac{1}{2},1)\oplus 2(1,\frac{1}{2})$
$(2,1,1,-1)$ |  $22(0,0)\oplus 14(0,1)\oplus 22(\frac{1}{2},\frac{1}{2})\oplus 4(\frac{1}{2},\frac{3}{2})\oplus 4(1,0)\oplus 5(1,1)$ | $(2,1,1,0)$ |  $31(0,\frac{1}{2})\oplus 5(0,\frac{3}{2})\oplus 21(\frac{1}{2},0)\oplus 17(\frac{1}{2},1)\oplus 9(1,\frac{1}{2})\oplus 3(1,\frac{3}{2})$
Table 20: BPS spectrum of $SU(3)_{0}+1\mathbf{Adj}$ theory for $d_{1}=1$,
$d_{2},d_{3}\leq 2$ and $d_{1}=2$, $d_{2},d_{3}\leq 1$. Here
$\mathbf{d}=(d_{1},d_{2},d_{3},d_{4})$ labels the BPS states with charge
$d_{1}m_{0}+d_{2}\alpha_{1}+d_{3}\alpha_{2}+d_{4}m_{1}$, where $\alpha_{1}$
and $\alpha_{2}$ are simple roots of $\mathfrak{su}(3)$ algebra. The states
related by the symmetries $d_{2}\leftrightarrow d_{3}$ and
$d_{4}\leftrightarrow-d_{4}$ are omitted.

####  5.1.7 $Sp(2)_{0}+1\mathbf{Adj}$

There are two $Sp(2)_{\theta}$ gauge theories with an adjoint hypermultiplet
in 5d distinguished by two distinct theta angles $\theta=0$ or $\theta=\pi$.
In this subsection, we discuss the case with $\theta=0$.

The 5d $Sp(2)_{0}+1\mathbf{Adj}$ theory is the KK-theory obtained by the
$\mathbb{Z}_{2}$ twisted compactification of the 6d $\mathcal{N}=(2,0)$
$A_{3}$ theory Tachikawa:2011ch ,

(235)

This is a non-geometric theory. This theory on the other hand has a 5-brane
web realization with an $O7^{+}$ plane (or frozen singularity) which will be
discussed in Appendix LABEL:sec:appendix2.

The BPS spectrum of this theory can be obtained by solving the blowup equation
as follows. The full effective prepotential of this theory on the chamber
where $2\phi_{1}>\phi_{2}>\phi_{1}>0$ is

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{48}\quantity(4\phi_{1}-2\phi_{2})+\epsilon_{+}^{2}(\phi_{1}+\phi_{2}))\,,$
$\displaystyle 6\mathcal{F}$
$\displaystyle=8\phi_{1}^{3}+12\phi_{1}^{2}\phi_{2}-18\phi_{1}\phi_{2}^{2}+8\phi_{2}^{3}+m_{0}(2\phi_{1}^{2}-2\phi_{1}\phi_{2}+\phi_{2}^{2})$
(236) $\displaystyle\quad-\frac{1}{2}\quantity((2\phi_{1}\pm
m_{1})^{3}+(\phi_{2}\pm m_{1})^{3}+(-2\phi_{1}+2\phi_{2}\pm
m_{1})^{3}+(\pm(2\phi_{1}-\phi_{2})+m_{1})^{3}),$

where $m_{0}$ is the gauge coupling and $m_{1}$ is the adjoint mass parameter.
One can choose the consistent magnetic fluxes as

$\displaystyle n_{i}\in\mathbb{Z}\ ,\quad B_{m_{0}}=0\ ,\quad B_{m_{1}}=1/2\
,$ (237)

and formulate a unity blowup equation. The solution to the blowup equation is
given in Table 21. We checked that this result matches the instanton partition
function of the 5d $\mathcal{N}=2$ $Sp(2)_{0}$ gauge theory computed in
Nekrasov:2004vw ; Shadchin:2004yx  using the ADHM construction.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,0,1,-2)$ |  $(0,\frac{1}{2})$ | $(1,0,1,-1)$ |  $(0,0)\oplus(0,1)\oplus(\frac{1}{2},\frac{1}{2})$
$(1,0,1,0)$ |  $2(0,\frac{1}{2})\oplus(\frac{1}{2},0)\oplus(\frac{1}{2},1)$ | $(1,0,2,-2)$ |  $(0,\frac{3}{2})$
$(1,0,2,-1)$ |  $(0,1)\oplus(0,2)\oplus(\frac{1}{2},\frac{3}{2})$ | $(1,0,2,0)$ |  $2(0,\frac{3}{2})\oplus(\frac{1}{2},1)\oplus(\frac{1}{2},2)$
$(1,1,1,-3)$ |  $(0,0)$ | $(1,1,1,-2)$ |  $2(0,\frac{1}{2})\oplus(\frac{1}{2},0)$
$(1,1,1,-1)$ |  $4(0,0)\oplus(0,1)\oplus 2(\frac{1}{2},\frac{1}{2})$ | $(1,1,1,0)$ |  $4(0,\frac{1}{2})\oplus 3(\frac{1}{2},0)\oplus(\frac{1}{2},1)$
$(2,0,1,-2)$ |  $2(0,\frac{1}{2})\oplus(\frac{1}{2},0)\oplus(\frac{1}{2},1)$ | $(2,0,1,-1)$ |  $4(0,0)\oplus 2(0,1)\oplus 3(\frac{1}{2},\frac{1}{2})\oplus(\frac{1}{2},\frac{3}{2})\oplus(1,1)$
$(2,0,1,0)$ |  $5(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus 3(\frac{1}{2},0)\oplus 3(\frac{1}{2},1)\oplus(1,\frac{1}{2})\oplus(1,\frac{3}{2})$ | $(2,1,0,-2)$ |  $(0,\frac{1}{2})$
$(2,1,0,-1)$ |  $(0,0)\oplus(0,1)\oplus(\frac{1}{2},\frac{1}{2})$ | $(2,1,0,0)$ |  $2(0,\frac{1}{2})\oplus(\frac{1}{2},0)\oplus(\frac{1}{2},1)$
$(2,1,1,-3)$ |  $2(0,0)\oplus(\frac{1}{2},\frac{1}{2})$ | $(2,1,1,-2)$ |  $6(0,\frac{1}{2})\oplus 4(\frac{1}{2},0)\oplus 2(\frac{1}{2},1)\oplus(1,\frac{1}{2})$
$(2,1,1,-1)$ |  $11(0,0)\oplus 5(0,1)\oplus 10(\frac{1}{2},\frac{1}{2})\oplus(\frac{1}{2},\frac{3}{2})\oplus 2(1,0)\oplus 2(1,1)$ | $(2,1,1,0)$ |  $14(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus 11(\frac{1}{2},0)\oplus 7(\frac{1}{2},1)\oplus 4(1,\frac{1}{2})\oplus(1,\frac{3}{2})$
Table 21: BPS spectrum of the $Sp(2)_{0}+1\mathbf{Adj}$ theory for $d_{1}=1,2$
and $d_{2}\leq 1$, $d_{3}\leq 1$. Here, $\mathbf{d}=(d_{1},d_{2},d_{3},d_{4})$
labels the BPS states with charge
$d_{1}m_{0}+d_{2}\alpha_{1}+d_{3}\alpha_{2}+d_{4}m_{1}$ where
$\alpha_{1}=2\phi_{1}-\phi_{2}$, $\alpha_{2}=-2\phi_{1}+2\phi_{2}$ are simple
roots of $\mathfrak{sp}(2)$ algebra. The theory has a symmetry exchanging
$d_{4}\leftrightarrow-d_{4}$ which provides BPS states with flipped charge
$d_{4}\rightarrow-d_{4}$.

####  5.1.8 $SU(3)_{\frac{3}{2}}+1\mathbf{Sym}$, $Sp(2)_{\pi}+1\mathbf{Adj}$

The 5d $Sp(2)_{\pi}+1\mathbf{Adj}$ theory is the KK-theory obtained by
$\mathbb{Z}_{2}$ twisted compactification of the 6d $\mathcal{N}=(2,0)$
$A_{4}$ theory. This theory is also dual to the $SU(3)_{3/2}+1\mathbf{Sym}$
theory Jefferson:2017ahm ,

(238)

This theory has no geometric construction, but it can be realized by a 5-brane
web with an $O7^{+}$ plane as discussed in Hayashi:2018lyv . See also Appendix
LABEL:sec:appendix2 for more details.

In the $SU(3)$ description, the effective prepotential on the Coulomb branch
where $\phi_{2}\geq\phi_{1}>0$ is given by

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{48}\quantity(4\phi_{1}-2\phi_{2})+\epsilon_{+}^{2}(\phi_{1}+\phi_{2}))\,,$
$\displaystyle 6\mathcal{F}$
$\displaystyle=8\phi_{1}^{3}-3\phi_{1}^{2}\phi_{2}-3\phi_{1}\phi_{2}^{2}+8\phi_{2}^{2}+\frac{9}{2}\phi_{1}\phi_{2}(\phi_{1}-\phi_{2})+m_{0}(\phi_{1}^{2}-\phi_{1}\phi_{2}+\phi_{2}^{2})$
$\displaystyle\quad-\frac{1}{2}\big{(}(2\phi_{1}+m_{1})^{3}+(\phi_{2}+m_{1})^{3}+(-2\phi_{1}+2\phi_{2}+m_{1})^{3}$
$\displaystyle\qquad\qquad+(-\phi_{1}+\phi_{2}-m_{1})^{3}+(\phi_{1}-m_{1})^{3}+(2\phi_{2}-m_{1})^{3}\big{)}\
,$ (239)

where $m_{0}$ is the gauge coupling and $m_{1}$ is the mass parameter of the
symmetric matter. The following magnetic fluxes give a solvable unity blowup
equation for this theory,

$\displaystyle n_{i}\in\mathbb{Z}\,,\quad B_{m_{0}}=-1/4\,,\quad
B_{m_{1}}=1/2\ .$ (240)

On the other hand, the blowup equation from the $Sp(2)$ theory perspective is
rather subtle. The effective prepotentials and the 1-loop GV-invariant of
$Sp(2)_{\pi}+1\mathbf{Adj}$ theory are the same as those of the $\theta=0$
case. Therefore the form of the blowup equation for the $Sp(2)_{\theta}$
theory at $\theta=\pi$ is expected to be determined by another choice of
magnetic fluxes different from the fluxes used for the theory at $\theta=0$ in
the previous subsection. Indeed, there exists another set of consistent
magnetic fluxes given by

$\displaystyle n_{1}\in\mathbb{Z}+1/2\,,\quad n_{2}\in\mathbb{Z}\,,\quad
B_{m_{0}}=0\,,\quad B_{m_{1}}=1/2\ .$ (241)

Rather surprisingly, we notice that the blowup equation coming from this set
of fluxes has two distinct solutions. While solving the blowup equation, we
always find two independent solutions at each order in the expansion. For
example, $N_{(0,0)}^{(1,1,1,-3)}$ at 1-instanton order has two solutions,
either $1$ or $0$. Interestingly, it turns out that these two solutions of the
single blowup equation correspond to the $\theta=0$ and $\theta=\pi$ cases
respectively. Indeed, we checked up to 3-instantons that the instanton
partition functions of the $\mathcal{N}=2\ Sp(2)_{\theta}$ gauge theories both
at $\theta=0,\pi$ computed using their ADHM constructions satisfy the same
blowup equation formulated with the fluxes in (241). This example tells us
that two $Sp(N)$ gauge theories with different theta angles can be
distinguished by flux choices leading to different blowup equations, or by
distinct solutions of a single blowup equation.

The map between Kähler parameters of the $SU(3)$ and $Sp(2)$ theories is

$\displaystyle\phi_{1}^{SU}$
$\displaystyle=\phi_{1}^{Sp}+\frac{1}{3}\quantity(m_{0}^{Sp}-3m_{1}^{Sp})\,,\
\ \phi_{2}^{SU}=\phi_{2}^{Sp}+\frac{2}{3}\quantity(m_{0}^{Sp}-3m_{1}^{Sp})\,,$
$\displaystyle m_{0}^{SU}$ $\displaystyle=m_{0}^{Sp}-\frac{1}{2}m_{1}^{Sp}\,,\
\ m_{1}^{SU}=-\frac{2}{3}m_{0}^{Sp}+m_{1}^{Sp}.$ (242)

Under this map, the BPS spectrum from the solution to the blowup equation in
the $SU(3)$ description agrees with that from the dual $Sp(2)$ description.
Some leading BPS states of the $SU(3)$ theory are listed in Table 22. We also
confirmed that this result agrees with the ADHM result for the
$Sp(2)_{\pi}+1{\bf Adj}$ theory.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,-1,0,\frac{3}{2})$ |  $(0,0)$ | $(1,-1,1,\frac{3}{2})$ |  $(0,1)$
$(1,-\frac{2}{3},\frac{2}{3},\frac{5}{2})$ |  $(0,\frac{1}{2})$ | $(1,-\frac{1}{3},\frac{1}{3},\frac{1}{2})$ |  $(0,\frac{1}{2})\oplus(\frac{1}{2},0)$
$(1,0,1,\frac{3}{2})$ |  $2(0,0)\oplus 2(0,1)\oplus(\frac{1}{2},\frac{1}{2})$ | $(1,\frac{1}{3},\frac{2}{3},-\frac{1}{2})$ |  $2(0,0)\oplus(\frac{1}{2},\frac{1}{2})$
$(1,\frac{1}{3},\frac{2}{3},\frac{5}{2})$ |  $(0,\frac{1}{2})$ | $(1,\frac{2}{3},\frac{1}{3},-\frac{5}{2})$ |  $(0,0)$
$(1,\frac{2}{3},\frac{1}{3},\frac{1}{2})$ |  $(0,\frac{1}{2})\oplus(\frac{1}{2},0)$ | $(1,1,0,-\frac{3}{2})$ |  $(0,\frac{1}{2})$
$(1,1,0,\frac{3}{2})$ |  $(0,0)$ | $(1,1,1,-\frac{3}{2})$ |  $2(0,\frac{1}{2})\oplus(\frac{1}{2},0)$
$(1,1,1,\frac{3}{2})$ |  $2(0,0)\oplus 2(0,1)\oplus(\frac{1}{2},\frac{1}{2})$ | $(2,-1,1,3)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(\frac{1}{2},1)$
$(2,-\frac{2}{3},\frac{2}{3},1)$ |  $(0,\frac{1}{2})\oplus(\frac{1}{2},1)$ | $(2,-\frac{2}{3},\frac{2}{3},4)$ |  $(0,0)$
$(2,-\frac{1}{3},\frac{1}{3},2)$ |  $2(0,0)\oplus(\frac{1}{2},\frac{1}{2})$ | $(2,0,1,0)$ |  $2(0,0)\oplus 3(0,1)\oplus 2(\frac{1}{2},\frac{1}{2})\oplus(\frac{1}{2},\frac{3}{2})\oplus(1,1)$
$(2,0,1,3)$ |  $5(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus 2(\frac{1}{2},0)\oplus 2(\frac{1}{2},1)$ | $(2,\frac{1}{3},\frac{2}{3},1)$ |  $6(0,\frac{1}{2})\oplus 4(\frac{1}{2},0)\oplus 2(\frac{1}{2},1)\oplus(1,\frac{1}{2})$
$(2,\frac{1}{3},\frac{2}{3},4)$ |  $(0,0)$ | $(2,\frac{2}{3},\frac{1}{3},-1)$ |  $2(0,\frac{1}{2})\oplus 2(\frac{1}{2},0)$
$(2,\frac{2}{3},\frac{1}{3},2)$ |  $2(0,0)\oplus(\frac{1}{2},\frac{1}{2})$ | $(2,1,0,0)$ |  $(0,0)\oplus(0,1)\oplus(\frac{1}{2},\frac{1}{2})$
$(2,1,1,0)$ |  $11(0,0)\oplus 9(0,1)\oplus(0,2)\oplus 10(\frac{1}{2},\frac{1}{2})\oplus 2(\frac{1}{2},\frac{3}{2})\oplus 2(1,0)\oplus 2(1,1)$ | $(2,1,1,3)$ |  $5(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus 2(\frac{1}{2},0)\oplus 2(\frac{1}{2},1)$
Table 22: BPS spectrum of the $SU(3)_{3/2}+1\mathbf{Sym}$ for $d_{1}\leq 2$
and $d_{2},d_{3}\leq 1$. Here, $\mathbf{d}=(d_{1},d_{2},d_{3},d_{4})$ labels
the BPS states with charge
$d_{1}m_{0}+d_{2}\alpha_{1}+d_{3}\alpha_{2}+d_{4}m_{1}$ for simple roots
$\alpha_{1}$ and $\alpha_{2}$ of $\mathfrak{su}(3)$ algebra.

####  5.1.9 $SU(3)_{0}+1\mathbf{Sym}+1\mathbf{F}$

The 5d $SU(3)$ gauge theory at CS-level $0$ with a symmetric and a fundamental
hypermultiplets is the KK-theory obtained by a twisted compactification of the
6d rank-2 $(A_{1},A_{1})$ conformal matter theory introduced in
DelZotto:2014hpa ,

(243)

This is another non-geometric rank-2 theory that can be realized by a brane
web with $O7^{+}$ plane. The associated brane web will be given in Appendix
LABEL:sec:appendix2.

We can compute the BPS spectrum of this theory using the blowup formula. The
effective prepotential of this theory on the chamber $\phi_{2}\geq\phi_{1}>0$
is given by

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\bigg{(}\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{48}(4\phi_{1}-4\phi_{2})+\Big{(}\frac{\epsilon_{1}+\epsilon_{2}}{2}\Big{)}^{2}(\phi_{1}+\phi_{2})\bigg{)}\,,$
$\displaystyle 6\mathcal{F}$
$\displaystyle=8\phi_{1}^{3}-3\phi_{1}^{2}\phi_{2}-3\phi_{1}\phi_{2}^{2}+8\phi_{2}^{3}+6m_{0}(\phi_{1}^{2}-\phi_{1}\phi_{2}+\phi_{2}^{2})$
$\displaystyle\quad-\frac{1}{2}\big{(}(2\phi_{1}\\!+\\!m_{1})^{3}+(\phi_{2}\\!+\\!m_{1})^{3}+(-2\phi_{1}\\!+\\!2\phi_{2}+m_{1})^{3}+(-\phi_{1}\\!+\\!\phi_{2}-m_{1})^{3}+(\phi_{1}\\!-\\!m_{1})^{3}\big{)}$
$\displaystyle\quad-\frac{1}{2}\big{(}(\phi_{1}+m_{2})^{3}+(-\phi_{1}+\phi_{2}+m_{2})^{3}+(\phi_{2}-m_{2})^{3}\big{)},$
(244)

where $m_{0}$ is the gauge coupling, $m_{1}$ and $m_{2}$ are mass parameters
of the symmetric and fundamental hypers, respectively. One finds a set of
consistent magnetic fluxes

$\displaystyle n_{i}\in\mathbb{Z}\,,\quad B_{m_{0}}=0\,,\quad
B_{m_{1}}=1/2\,,\quad B_{m_{2}}=1/2\ ,$ (245)

which provides a solvable unity blowup equation. The BPS spectrum from the
solution of the blowup equation is given in Table 23.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,-\frac{2}{3},\frac{2}{3},\frac{3}{2},\frac{1}{2})$ |  $(0,\frac{1}{2})$ | $(1,-\frac{1}{3},\frac{1}{3},\frac{3}{2},-\frac{1}{2})$ |  $(0,0)$
$(1,-\frac{1}{3},\frac{1}{3},\frac{5}{2},\frac{1}{2})$ |  $(0,0)$ | $(1,0,1,\frac{1}{2},\frac{1}{2})$ |  $\\!(0,0)\oplus(0,1)\oplus(\frac{1}{2},\frac{1}{2})\\!$
$(1,0,1,\frac{5}{2},-\frac{1}{2})$ |  $(0,\frac{1}{2})$ | $(1,\frac{1}{3},-\frac{1}{3},-\frac{5}{2},-\frac{1}{2})$ |  $(0,0)$
$(1,\frac{1}{3},-\frac{1}{3},-\frac{3}{2},\frac{1}{2})$ |  $(0,0)$ | $(1,\frac{1}{3},\frac{2}{3},-\frac{5}{2},-\frac{1}{2})$ |  $(0,0)$
$(1,\frac{1}{3},\frac{2}{3},-\frac{3}{2},\frac{1}{2})$ |  $(0,0)$ | $(1,\frac{1}{3},\frac{2}{3},\frac{1}{2},-\frac{1}{2})$ |  $(0,\frac{1}{2})\oplus(\frac{1}{2},0)$
$(1,\frac{1}{3},\frac{2}{3},\frac{3}{2},\frac{1}{2})$ |  $2(0,\frac{1}{2})\oplus(\frac{1}{2},0)$ | $(1,\frac{2}{3},-\frac{2}{3},-\frac{3}{2},-\frac{1}{2})$ |  $(0,\frac{1}{2})$
$(1,\frac{2}{3},\frac{1}{3},-\frac{3}{2},-\frac{1}{2})$ |  $2(0,\frac{1}{2})\oplus(\frac{1}{2},0)$ | $(1,\frac{2}{3},\frac{1}{3},-\frac{1}{2},\frac{1}{2})$ |  $(0,\frac{1}{2})\oplus(\frac{1}{2},0)$
$(1,\frac{2}{3},\frac{1}{3},\frac{3}{2},-\frac{1}{2})$ |  $(0,0)$ | $(1,\frac{2}{3},\frac{1}{3},\frac{5}{2},\frac{1}{2})$ |  $(0,0)$
$(1,1,0,-\frac{5}{2},\frac{1}{2})$ |  $(0,\frac{1}{2})$ | $(1,1,0,-\frac{1}{2},-\frac{1}{2})$ |  $\\!(0,0)\oplus(0,1)\oplus(\frac{1}{2},\frac{1}{2})\\!$
$(1,1,1,-\frac{5}{2},\frac{1}{2})$ |  $(0,\frac{1}{2})$ | $(1,1,1,-\frac{1}{2},-\frac{1}{2})$ |  $3(0,0)\oplus(0,1)\oplus 2(\frac{1}{2},\frac{1}{2})$
$(1,1,1,\frac{1}{2},\frac{1}{2})$ |  $3(0,0)\oplus(0,1)\oplus 2(\frac{1}{2},\frac{1}{2})$ | $(1,1,1,\frac{5}{2},-\frac{1}{2})$ |  $(0,\frac{1}{2})$
Table 23: BPS spectrum of the $SU(3)_{0}+1\mathbf{Sym}+1\mathbf{F}$ theory.
Here, $\mathbf{d}=(d_{1},d_{2},d_{3},d_{4},d_{5})$ labels the BPS states with
charge $d_{1}m_{0}+d_{2}\alpha_{1}+d_{3}\alpha_{2}+d_{4}m_{1}+d_{5}m_{2}$
where $\alpha_{1}$ and $\alpha_{2}$ are simple roots of $\mathfrak{su}(3)$.

####  5.1.10 $SU(3)_{\frac{15}{2}}+1\mathbf{F}$, $G_{2}+1\mathbf{Adj}$

The last rank-2 KK theory is the theory obtained by $\mathbb{Z}_{3}$ twisted
compactification of the 6d $\mathcal{N}=(2,0)$ $D_{4}$ theory. This theory has
two 5d gauge theory descriptions: one is the $G_{2}$ gauge theory with an
adjoint hypermultiplet Tachikawa:2011ch  and another one is the
$SU(3)_{15/2}+1\mathbf{F}$ theory Bhardwaj:2019jtr ,

(246)

It can be geometrically described by

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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-110.16068pt}{-3.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\mathbb{F}_{8}{}\big{|}_{1}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-80.97156pt}{9.13867pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{${}_{e}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.4958pt}{-3.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\mathbb{F}^{1}_{0}{}\big{|}_{2}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-19.6669pt}{10.05257pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{${}_{f+3h}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ .$ (247)

However, this geometry is not shrinkable and there is no known shrinkable
geometric phase for this theory Jefferson:2018irk . Nonetheless, it is
expected that a series of flop transitions give rise to a phase for the
unitary KK theory Bhardwaj:2019jtr . As we will see below, this expectation is
consistent with the BPS spectrum which we can compute by solving the blowup
equations.

We shall solve the blowup equations from both the $G_{2}$ and $SU(3)$ theory
perspectives. The effective prepotential for the $SU(3)$ theory on the Coulomb
branch where $\phi_{2}\geq\phi_{1}>0$ is given by

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{48}\big{(}4\phi_{1}+2\phi_{2}\big{)}+\epsilon_{+}^{2}\quantity(\phi_{1}+\phi_{2}))\,,$
$\displaystyle 6\mathcal{F}$
$\displaystyle=8\phi_{1}^{3}-3\phi_{1}^{2}\phi_{2}-3\phi_{1}\phi_{2}^{2}+8\phi_{2}^{3}+\frac{45}{2}\phi_{1}\phi_{2}(\phi_{1}-\phi_{2})+6m_{0}(\phi_{1}^{2}-\phi_{1}\phi_{2}+\phi_{2}^{2})$
$\displaystyle\quad-\frac{1}{2}\Big{(}(\phi_{2}-m_{1})^{3}+(\phi_{2}-\phi_{1}+m_{1})^{3}+(\phi_{1}+m_{1})^{3}\Big{)}\,,$
(248)

where $m_{0}$ is the $SU(3)$ gauge coupling and $m_{1}$ is the mass parameter
of the fundamental hyper.

In the $G_{2}$ gauge theory, the effective prepotential in the same chamber is

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\bigg{(}\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{48}(4\phi_{1}+2\phi_{2})+\epsilon_{+}^{2}(\phi_{1}+\phi_{2})\bigg{)}\,,$
$\displaystyle 6\mathcal{F}$
$\displaystyle=8\phi_{1}^{3}+18\phi_{1}^{2}\phi_{2}-24\phi_{1}\phi_{2}^{2}+8\phi_{2}^{3}+6m_{0}(3\phi_{1}^{2}-3\phi_{1}\phi_{2}+\phi_{2}^{2})$
$\displaystyle\quad-\frac{1}{2}\big{(}(\phi_{2}\pm
m_{1})^{3}+(\pm(3\phi_{1}-\phi_{2})+m_{1})^{3}+(\pm\phi_{1}+m_{1})^{3}+(\pm(\phi_{2}-\phi_{1})+m_{1})^{3}$
$\displaystyle\qquad\qquad+(\pm(-3\phi_{1}+2\phi_{2})+m_{1})^{3}+(\pm(2\phi_{1}-\phi_{2})+m_{1})^{3})\big{)},$
(249)

where $m_{0}$ is the $G_{2}$ gauge coupling and $m_{1}$ is the mass parameter
of the adjoint matter.

Under the duality, the Kähler parameters in the $G_{2}+1\mathbf{Adj}$ and in
the $SU(3)_{15/2}+1\mathbf{F}$ theories are mapped to each other as follows:

$\displaystyle\phi_{1}^{G_{2}}$
$\displaystyle=\phi_{1}^{SU}+m_{0}^{SU}-\frac{1}{2}m_{1}^{SU}\,,\ \
\phi_{2}^{G_{2}}=\phi_{2}^{SU}+2m_{0}^{SU(3)}-m_{1}^{SU}\,,$ (250)
$\displaystyle m_{0}^{G_{2}}$
$\displaystyle=5m_{0}^{SU}+\frac{3}{2}m_{1}^{SU}\,,\ \
m_{1}^{G_{2}}=2m_{0}^{SU}.$ (251)

One can check with this map that the effective prepotentials of two dual gauge
theories match well up to constant terms.

We first solve the blowup equation from the perspective of $G_{2}$ gauge
theory. A set of consistent magnetic fluxes

$\displaystyle n_{i}\in\mathbb{Z}\,,\quad B_{m_{0}}=0\,,\quad B_{m_{1}}=1/2\ $
(252)

provides a solvable blowup equation. By solving the blowup equation, we find
the BPS spectrum of the $G_{2}$ gauge theory given in Table 24.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,0,1,0)$ |  $2(0,\frac{1}{2})\oplus(\frac{1}{2},0)\oplus(\frac{1}{2},1)$ | $(1,0,1,1)$ |  $(0,0)\oplus(0,1)\oplus(\frac{1}{2},\frac{1}{2})$
$(1,0,1,2)$ |  $(0,\frac{1}{2})$ | $(1,0,2,0)$ |  $2(0,\frac{3}{2})\oplus(\frac{1}{2},1)\oplus(\frac{1}{2},2)$
$(1,0,2,1)$ |  $(0,1)\oplus(0,2)\oplus(\frac{1}{2},\frac{3}{2})$ | $(1,0,2,2)$ |  $(0,\frac{3}{2})$
$(1,1,1,0)$ |  $4(0,\frac{1}{2})\oplus 3(\frac{1}{2},0)\oplus(\frac{1}{2},1)$ | $(1,1,1,1)$ |  $4(0,0)\oplus(0,1)\oplus 2(\frac{1}{2},\frac{1}{2})$
$(1,1,1,2)$ |  $2(0,\frac{1}{2})\oplus(\frac{1}{2},0)$ | $(1,1,1,3)$ |  $(0,0)$
$(1,1,2,0)$ |  $4(0,\frac{1}{2})\oplus 4(0,\frac{3}{2})\oplus(\frac{1}{2},0)\oplus 4(\frac{1}{2},1)\oplus(\frac{1}{2},2)$ | $(1,1,2,1)$ |  $(0,0)\oplus 5(0,1)\oplus(0,2)\oplus 2(\frac{1}{2},\frac{1}{2})\oplus 2(\frac{1}{2},\frac{3}{2})$
$(1,1,2,2)$ |  $2(0,\frac{1}{2})\oplus 2(0,\frac{3}{2})\oplus(\frac{1}{2},1)$ | $(1,1,2,3)$ |  $(0,1)$
$(1,2,1,0)$ |  $4(0,\frac{1}{2})\oplus 3(\frac{1}{2},0)\oplus(\frac{1}{2},1)$ | $(1,2,1,1)$ |  $4(0,0)\oplus(0,1)\oplus 2(\frac{1}{2},\frac{1}{2})$
$(1,2,1,2)$ |  $2(0,\frac{1}{2})\oplus(\frac{1}{2},0)$ | $(1,2,1,3)$ |  $(0,0)$
$(1,2,2,0)$ |  $8(0,\frac{1}{2})\oplus 4(0,\frac{3}{2})\oplus 4(\frac{1}{2},0)\oplus 5(\frac{1}{2},1)\oplus(\frac{1}{2},2)$ | $(1,2,2,1)$ |  $5(0,0)\oplus 6(0,1)\oplus(0,2)\oplus 4(\frac{1}{2},\frac{1}{2})\oplus 2(\frac{1}{2},\frac{3}{2})$
$(1,2,2,2)$ |  $4(0,\frac{1}{2})\oplus 2(0,\frac{3}{2})\oplus(\frac{1}{2},0)\oplus(\frac{1}{2},1)$ | $(1,2,2,3)$ |  $(0,0)\oplus(0,1)$
$(2,0,1,0)$ |  $5(0,\frac{1}{2})\oplus 1(0,\frac{3}{2})\oplus 3(\frac{1}{2},0)\oplus 3(\frac{1}{2},1)\oplus(1,\frac{1}{2})\oplus(1,\frac{3}{2})$ | $(2,0,1,1)$ |  $4(0,0)\oplus 2(0,1)\oplus 3(\frac{1}{2},\frac{1}{2})\oplus(\frac{1}{2},\frac{3}{2})\oplus(1,1)$
$(2,0,1,2)$ |  $2(0,\frac{1}{2})\oplus(\frac{1}{2},0)\oplus(\frac{1}{2},1)$ | $(2,0,1,3)$ |  $(0,0)$
$(2,1,1,0)$ |  $\\!12(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus 10(\frac{1}{2},0)\oplus 6(\frac{1}{2},1)\oplus 4(1,\frac{1}{2})\oplus(1,\frac{3}{2})\\!$ | $(2,1,1,1)$ |  $\\!10(0,0)\oplus 4(0,1)\oplus 9(\frac{1}{2},\frac{1}{2})\oplus(\frac{1}{2},\frac{3}{2})\oplus 2(1,0)\oplus 2(1,1)\\!$
$(2,1,1,2)$ |  $5(0,\frac{1}{2})\oplus 4(\frac{1}{2},0)\oplus 2(\frac{1}{2},1)\oplus(1,\frac{1}{2})$ | $(2,1,1,3)$ |  $2(0,0)\oplus(\frac{1}{2},\frac{1}{2})$
Table 24: BPS spectrum of the $G_{2}+1\mathbf{Adj}$ for $d_{1}=1$,
$d_{2},d_{3}\leq 2$ and $d_{1}=2$, $d_{2},d_{3}\leq 1$. Here,
$\mathbf{d}=(d_{1},d_{2},d_{3},d_{4})$ labels the BPS states with charge
$d_{1}m_{0}+d_{2}\alpha_{1}+d_{3}\alpha_{2}+d_{4}m_{1}$, where
$\alpha_{1}=2\phi_{1}-\phi_{2}$ and $\alpha_{2}=-3\phi_{1}+2\phi_{2}$ are
simple roots of $G_{2}$. The theory has a symmetry exchanging
$d_{4}\leftrightarrow-d_{4}$.

In the $SU(3)$ description, one can use the same magnetic fluxes to formulate
a unity blowup equation. The duality map (250) implies that the above fluxes
in the $G_{2}$ perspective are converted into the magnetic fluxes in the
$SU(3)$ gauge theory given by

$\displaystyle n_{1}\in\mathbb{Z}+1/3\,,\quad n_{2}\in\mathbb{Z}+2/3\,,\quad
B_{m_{0}}=1/4\,,\quad B_{m_{1}}=-5/6\ .$ (253)

We checked that the blowup equation with these fluxes in the $SU(3)$
perspective can be solved and the solution perfectly agrees with the result
from the $G_{2}+1\mathbf{Adj}$ description.

The BPS spectrum tells us how to move on to the unitary phase where all the
BPS states have non-negative masses in the UV limit. First note that the
geometric phase (247) is non-shrinkable due to an exceptional curve $f_{2}-x$
whose volume cannot be non-negative while keeping volumes of all other curves
non-negative at the same time. This implies that we need to flop the $f_{2}-x$
curve in the UV limit. Unfortunately, this transition leads to a phase which
cannot be described by a smooth CY 3-fold.

However, we are able to trace all the phase transitions toward the unitary
phase relying on the BPS spectrum we computed. To move on to the unitary
phase, we need to flop 4 curves associated with 4 hypermultiplets with the
masses, consecutively as,

$m_{1}-3\phi_{1}+\phi_{2}\ \ \rightarrow\ \ m_{1}-\phi_{1}\ \ \rightarrow\ \
m_{1}+\phi_{1}-\phi_{2}\ \ \rightarrow\ \ m_{1}+3\phi_{1}-2\phi_{2}$ (254)

in the $G_{2}$ description. In the $SU(3)$ description, these hypermultiplets
are all instantonic states. In the geometry (247), the flop transitions
performed on the exceptional curves as

$f_{2}-x\ \ \rightarrow\ \ f_{1}+f_{2}-x\ \ \rightarrow\ \ 2f_{1}+f_{2}-x\
\rightarrow\ \ 3f_{1}+f_{2}-x\ ,$ (255)

will lead to the unitary phase smoothly connected to the UV fixed point. One
can check that in the final phase all the BPS states have non-negative masses.
This result provides a strong evidence that the flop transitions in (255) in
the geometric description (247) are physically well-established transitions,
though the phases after the flop transitions are non-geometrical.

###  5.2 Rank 2 5d SCFTs

####  5.2.1 $\mathbb{P}^{2}\cup\mathbb{F}_{3}$

The local $\mathbb{P}^{2}\cup\mathbb{F}_{3}$ theory is a rank-2 analog of the
SCFT engineered by a local $\mathbb{P}^{2}$. It is a non-Lagrangian theory
with no mass parameter. Its geometric construction is represented by

$\displaystyle\leavevmode\hbox to129.76pt{\vbox
to21.74pt{\pgfpicture\makeatletter\hbox{\hskip 100.5173pt\lower-8.90189pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-71.1319pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-97.1843pt}{-3.06888pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\evaluated{\mathbb{P}^{2}}_{1}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-66.6081pt}{9.50812pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{${}_{\ell}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{8.23013pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\evaluated{\mathbb{F}_{3}}_{2}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.99422pt}{9.13867pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{${}_{e}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ .$ (256)

The volumes of two primitive 2-cycles are given by

$\displaystyle\operatorname{vol}(\ell)=3\phi_{1}-\phi_{2}\
,\quad\operatorname{vol}(f_{2})=-\phi_{1}+2\phi_{2}\ ,$ (257)

where $\ell$ is a curve with $\ell^{2}=1$ in $\mathbb{P}^{2}$ and $f_{2}$ is
the fiber in $\mathbb{F}_{3}$. The geometry can be obtained by blowing down an
exceptional cycle of $\mathbb{F}_{1}$ in geometry
$\mathbb{F}_{1}-\mathbb{F}_{3}$, which implies that this theory can be
obtained by an RG flow from the pure $SU(3)_{2}$ theory by integrating out an
instantonic hypermultiplet. The web diagram for this theory is depicted in
Figure 3.

(a)

(b)

(c)

Figure 3: A 5-brane web construction for $\mathbb{P}^{2}\cup\mathbb{F}_{3}$.
Starting from the web diagram of $SU(3)_{2}$ theory (a), flop transition for
an instantonic hypermultiplet gives (b). Integrating out the hypermultiplet
gives a web diagram of $\mathbb{P}^{2}\cup\mathbb{F}_{3}$ depicted in (c).

The effective prepotential on the $\Omega$-background is

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{48}(6\phi_{1}+4\phi_{2})+\epsilon_{+}^{2}(\phi_{1}+\phi_{2}))\
,$ $\displaystyle 6\mathcal{F}$
$\displaystyle=9\phi_{1}^{3}-9\phi_{1}^{2}\phi_{2}+3\phi_{1}\phi_{2}^{2}+8\phi_{2}^{3}\
.$ (258)

For a unity blowup equation, we choose the magnetic fluxes as

$\displaystyle n_{1}\in\mathbb{Z}+1/5\ ,\quad n_{2}\in\mathbb{Z}+1/10\ ,$
(259)

which assign half-integral fluxes for $\ell$ and integral fluxes for $f_{2}$.
The BPS spectrum obtained by solving the blowup equation is given in Table 25.
We checked that the result matches the spectrum after taking an RG flow from
the BPS spectrum of the pure $SU(3)_{2}$ theory by integrating out the
instantonic hypermultiplet.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(0,1)$ |  $(0,\frac{1}{2})$ | $(1,0)$ |  $(0,1)$
$(1,1)$ |  $(0,0)\oplus(0,1)$ | $(1,2)$ |  $(0,1)$
$(1,3)$ |  $(0,2)$ | $(2,0)$ |  $(0,\frac{5}{2})$
$(2,1)$ |  $(0,\frac{3}{2})\oplus(0,\frac{5}{2})$ | $(2,2)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(0,\frac{5}{2})$
$(2,3)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(0,\frac{5}{2})$ | $(3,0)$ |  $(0,3)\oplus(\frac{1}{2},\frac{9}{2})$
$(3,1)$ |  $(0,2)\oplus 2(0,3)\oplus(0,4)\oplus(\frac{1}{2},\frac{7}{2})\oplus(\frac{1}{2},\frac{9}{2})$ | $(3,2)$ |  $(0,1)\oplus 2(0,2)\oplus 3(0,3)\oplus(0,4)\oplus(\frac{1}{2},\frac{5}{2})\oplus(\frac{1}{2},\frac{7}{2})\oplus(\frac{1}{2},\frac{9}{2})$
$(3,3)$ |  $\\!(0,0)\oplus 2(0,1)\oplus 3(0,2)\oplus 3(0,3)\oplus(0,4)\oplus(\frac{1}{2},\frac{3}{2})\oplus(\frac{1}{2},\frac{5}{2})\oplus(\frac{1}{2},\frac{7}{2})\oplus(\frac{1}{2},\frac{9}{2})\\!$
Table 25: BPS spectrum of $\mathbb{P}^{2}\cup\mathbb{F}_{3}$ for $d_{i}\leq
3$. Here, $\mathbf{d}=(d_{1},d_{2})$ labels the BPS states with charge
$d_{1}\ell+d_{2}f_{2}$.

####  5.2.2 $\mathbb{P}^{2}\cup\mathbb{F}_{6}$

The local $\mathbb{P}^{2}\cup\mathbb{F}_{6}$ theory is another non-Lagrangian
rank-2 theory with no mass parameter. Its geometric construction is given by

$\displaystyle\leavevmode\hbox to129.76pt{\vbox
to21.74pt{\pgfpicture\makeatletter\hbox{\hskip 100.5173pt\lower-8.90189pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-71.1319pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-97.1843pt}{-3.06888pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\evaluated{\mathbb{P}^{2}}_{1}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-68.00809pt}{9.50812pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{${}_{2\ell}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{8.23013pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\evaluated{\mathbb{F}_{6}}_{2}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.99422pt}{9.13867pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{${}_{e}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ .$ (260)

The volumes of two primitive 2-cycles are

$\displaystyle\operatorname{vol}(\ell)=3\phi_{1}-2\phi_{2}\,,\quad\operatorname{vol}(f_{2})=-\phi_{1}+2\phi_{2}$
(261)

where $\ell$ is the curve class in $\mathbb{P}^{2}$ and $f_{2}$ is the fiber
in $\mathbb{F}_{6}$. This theory can be obtained by an integrating out an
instantonic hypermultiplet of the pure $SU(3)_{4}$ theory. The web diagram and
the RG flow of this theory are given in Figure 4.

(a)

(b)

(c)

Figure 4: A 5-brane web construction for $\mathbb{P}^{2}\cup\mathbb{F}_{6}$.
Starting from the web diagram of $SU(3)_{4}$ theory (a), flop transition of
two instantonic hypermultiplets gives (b). Integrating out the hypermultiplet
gives a web diagram of $\mathbb{P}^{2}\cup\mathbb{F}_{6}$ depicted in (c).

The effective prepotential is

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{48}(6\phi_{1}+4\phi_{2})+\epsilon_{+}^{2}(\phi_{1}+\phi_{2}))\,,$
$\displaystyle 6\mathcal{F}$
$\displaystyle=9\phi_{1}^{3}-18\phi_{1}^{2}\phi_{2}+12\phi_{1}\phi_{2}^{2}+8\phi_{2}^{3}\
.$ (262)

We choose the magnetic fluxes

$\displaystyle n_{1}\in\mathbb{Z}+1/4\,,\quad n_{2}\in\mathbb{Z}+1/8\ ,$ (263)

to get a solvable unity blowup equation. The BPS spectrum obtained by solving
the blowup equation is given in Table 26. We checked that the result matches
the spectrum after taking an RG flow from the BPS spectrum of the pure
$SU(3)_{4}$ theory given in Table 5 by integrating out the instantonic
hypermultiplet.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(0,1)$ |  $(0,\frac{1}{2})$ | $(1,0)$ |  $(0,1)$
$(1,1)$ |  $(0,0)\oplus(0,1)$ | $(1,2)$ |  $(0,1)$
$(2,0)$ |  $(0,\frac{5}{2})$ | $(2,1)$ |  $(0,\frac{3}{2})\oplus(0,\frac{5}{2})$
$(2,2)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(0,\frac{5}{2})$ | $(2,3)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(0,\frac{5}{2})$
$(3,0)$ |  $(0,3)\oplus(\frac{1}{2},\frac{9}{2})$ | $(3,1)$ |  $(0,2)\oplus 2(0,3)\oplus(0,4)\oplus(\frac{1}{2},\frac{7}{2})\oplus(\frac{1}{2},\frac{9}{2})$
$(3,2)$ |  $(0,1)\oplus 2(0,2)\oplus 3(0,3)\oplus(0,4)\oplus(\frac{1}{2},\frac{5}{2})\oplus(\frac{1}{2},\frac{7}{2})\oplus(\frac{1}{2},\frac{9}{2})$ | $(3,3)$ |  $(0,0)\oplus 2(0,1)\oplus 3(0,2)\oplus 3(0,3)\oplus(0,4)\oplus(\frac{1}{2},\frac{3}{2})\oplus(\frac{1}{2},\frac{5}{2})\oplus(\frac{1}{2},\frac{7}{2})\oplus(\frac{1}{2},\frac{9}{2})$
Table 26: BPS spectrum of $\mathbb{P}^{2}\cup\mathbb{F}_{6}$ for $d_{i}\leq
3$. Here, $\mathbf{d}=(d_{1},d_{2})$ labels the BPS state with charge
$d_{1}\ell+d_{2}f_{2}$.

####  5.2.3 $\mathbb{P}^{2}\cup\mathbb{F}_{3}+1\mathbf{Sym}$

The $\mathbb{P}^{2}\cup\mathbb{F}_{3}+1\mathbf{Sym}$ theory is a rank-2 analog
of the local $\mathbb{P}^{2}+{\bf 1Adj}$ theory. As proposed in
Bhardwaj:2019jtr , we can obtain this theory from the UV
$Sp(2)_{\pi}+1\mathbf{Adj}$ theory or $SU(3)_{3/2}+1\mathbf{Sym}$ theory by
integrating out an instantonic hypermultiplet. See Appendix
LABEL:sec:app-P2_U_F3+1Sym for their 5-brane constructions.

More precisely, there is a hypermultiplet with
$\mathbf{d}=(1,\frac{2}{3},\frac{1}{3},-\frac{5}{2})$ which has mass
$m_{0}+\phi_{1}-\frac{5}{2}m_{1}$ in the UV parent theory, as listed in Table
22. We flop this hypermultiplet and integrate it out to get the
$\mathbb{P}^{2}\cup\mathbb{F}_{3}+1\mathbf{Sym}$ theory in IR. This RG flow is
realized in the 5-brane web in Figure 5 as taking a limit
$-m_{0}-\phi_{1}+\frac{5}{2}m_{1}\to\infty$ while the lengths of other edges
are kept finite.

$-m_{0}-\phi_{1}+\frac{5}{2}m_{1}$$m_{0}+3\phi_{1}-\phi_{2}-\frac{5}{2}m_{1}$$-\phi_{1}+2\phi_{2}$$m_{1}-2\phi_{2}$$O7^{+}$
Figure 5: A 5-brane web for $SU(3)_{3/2}+1\mathbf{Sym}$ and $Sp(2)_{\pi}+1{\bf
Adj}$, where a flop transition for the edge with length
$\phi_{1}+m_{0}-\frac{5}{2}m_{1}$ is performed.

To compute the effective prepotential and the BPS partition function of this
theory, it is convenient to redefine the parameters in the web diagram in
Figure 5 as

$\displaystyle\tilde{\phi}_{1}=\phi_{1}+\frac{2}{5}m_{0}-m_{1}\,,\quad\tilde{\phi}_{2}=\phi_{2}+\frac{m_{0}}{5}-\frac{m_{1}}{2}\,,\quad\tilde{m}=\frac{2}{5}m_{0}\
.$ (264)

Then the volumes of 2-cycles in Figure 5 become

$\displaystyle-
m_{0}-\phi_{1}+\frac{5}{2}m_{1}=-\tilde{\phi}_{1}-\frac{3}{2}\tilde{m}+\frac{3}{2}m_{1}\,,\quad-\phi_{1}+2\phi_{2}=-\tilde{\phi}_{1}+2\tilde{\phi}_{2}\,,$
$\displaystyle
m_{0}+3\phi_{1}-\phi_{2}-\frac{5}{2}m_{1}=3\tilde{\phi}_{1}-\tilde{\phi}_{2}\,,\quad
m_{1}-2\phi_{2}=\tilde{m}-2\tilde{\phi}_{2}\ .$ (265)

Hence, the RG flow corresponds to the limit $m_{1}\to\infty$, while keeping
$\tilde{\phi}_{i}$ and $\tilde{m}$ finite.

The cubic prepotential and mixed gravitational Chern-Simons term in the IR
$\mathbb{P}^{2}\cup\mathbb{F}_{3}+1\mathbf{Sym}$ theory can be obtained from
those of the UV $SU(3)_{3/2}+1\mathbf{Sym}$ theory as follows:

$\displaystyle\mathcal{F}_{IR}=\mathcal{F}_{UV}+\frac{1}{6}\quantity(m_{0}+\phi_{1}-\frac{5}{2}m_{1})^{3}\,,\quad
C_{IR}=C_{UV}-2\quantity(m_{0}+\phi_{1}-\frac{5}{2}m_{1})\,,$ (266)

whereas the mixed gauge/$SU(2)_{R}$ Chern-Simons term remains the same. In
terms of the parameters in (264), the effective prepotential in the IR theory
takes the form

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}_{IR}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{48}(6\tilde{\phi}_{1}+4\tilde{\phi}_{2})+\epsilon_{+}^{2}(\tilde{\phi}_{1}+\tilde{\phi}_{2}))\,,$
$\displaystyle 6\mathcal{F}_{IR}$
$\displaystyle=9\tilde{\phi}_{1}^{3}-9\tilde{\phi}_{1}^{2}\tilde{\phi}_{2}+3\tilde{\phi}_{1}\tilde{\phi}_{2}^{2}+8\tilde{\phi}_{2}^{3}\
,$ (267)

up to constant terms. One may notice that the prepotential is the same as that
of $\mathbb{P}^{2}\cup\mathbb{F}_{3}$. However, this theory has an additional
hypermultiplet coming from the symmetric matter of the parent theory. The GV-
invariant for this hypermultiplet

$\displaystyle\mathcal{Z}_{\mathrm{hyper}}(\tilde{\phi}_{i},\tilde{m};\epsilon_{1,2})=\operatorname{PE}\quantity[\frac{(p_{1}p_{2})^{1/2}}{(1-p_{1})(1-p_{2})}e^{-(\tilde{m}-2\tilde{\phi}_{2})}]$
(268)

is an additional input for the blowup equation of the IR theory.

We find a set of consistent magnetic fluxes

$\displaystyle\tilde{n}_{1}\in\mathbb{Z}-3/5\,,\quad\tilde{n}_{2}\in\mathbb{Z}-3/10\,,\quad
B_{\tilde{m}}=-1/10$ (269)

which leads to a unity blowup equation. The solution of the blowup equation
provides the BPS spectrum of the theory, summarized in Table 27. This solution
matches the spectrum that we can obtain indirectly from the RG flow of the BPS
spectrum in $SU(3)_{3/2}+1\mathbf{Sym}$ in Table 22 after integrating out the
hypermultiplet.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,0,0)$ |  $(0,0)$ | $(1,0,1)$ |  $(0,0)$
$(1,0,2)$ |  $(0,0)$ | $(1,1,1)$ |  $(0,\frac{1}{2})$
$(1,1,2)$ |  $2(0,\frac{1}{2})\oplus(\frac{1}{2},0)$ | $(1,1,3)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(\frac{1}{2},1)$
$(1,2,1)$ |  $(0,2)$ | $(1,2,2)$ |  $\\!(0,0)\oplus(0,1)\oplus 2(0,2)\oplus(\frac{1}{2},\frac{3}{2})\\!$
$(1,2,3)$ |  $2(0,0)\oplus 3(0,1)\oplus 2(0,2)\oplus(\frac{1}{2},\frac{1}{2})\oplus(\frac{1}{2},\frac{3}{2})$ | $(1,3,1)$ |  $(0,\frac{5}{2})\oplus(0,\frac{7}{2})\oplus(\frac{1}{2},4)$
$(1,3,2)$ |  $2(0,\frac{3}{2})\oplus 4(0,\frac{5}{2})\oplus 3(0,\frac{7}{2})\oplus(\frac{1}{2},2)\oplus 2(\frac{1}{2},3)\oplus 2(\frac{1}{2},4)\oplus(1,\frac{7}{2})$ | $(1,3,3)$ |  $2(0,\frac{1}{2})\oplus 6(0,\frac{3}{2})\oplus 8(0,\frac{5}{2})\oplus 4(0,\frac{7}{2})\oplus(\frac{1}{2},1)\oplus 4(\frac{1}{2},2)\oplus 4(\frac{1}{2},3)\oplus 2(\frac{1}{2},4)\oplus(1,\frac{5}{2})\oplus(1,\frac{7}{2})$
$(2,1,3)$ |  $2(0,0)\oplus(\frac{1}{2},\frac{1}{2})$ | $(2,2,3)$ |  $(0,\frac{1}{2})\oplus(0,\frac{3}{2})\oplus(\frac{1}{2},1)$
$(2,3,2)$ |  $(0,3)$ | $(2,3,3)$ |  $(0,1)\oplus 3(0,2)\oplus 3(0,3)\oplus(\frac{1}{2},\frac{3}{2})\oplus 2(\frac{1}{2},\frac{5}{2})\oplus(\frac{1}{2},\frac{7}{2})\oplus(1,3)$
Table 27: BPS spectrum of $\mathbb{P}^{2}\cup\mathbb{F}_{3}+1\mathbf{Sym}$ for
$d_{1}\leq 2$, $d_{2},d_{3}\leq 3$. Here, $\mathbf{d}=(d_{1},d_{2},d_{3})$
labels the BPS state with charge
$d_{1}(\tilde{m}-2\tilde{\phi}_{2})+d_{2}(3\tilde{\phi}_{1}-\tilde{\phi}_{2})+d_{3}(-\tilde{\phi}_{1}+2\tilde{\phi}_{2})$.
The $d_{1}=0$ sector is the same with that of
$\mathbb{P}^{2}\cup\mathbb{F}_{3}$.

####  5.2.4 $\mathbb{P}^{2}\cup\mathbb{F}_{6}+1\mathbf{Sym}$

The theory $\mathbb{P}^{2}\cup\mathbb{F}_{6}+1\mathbf{Sym}$ is a non-
Lagrangian theory obtained from the KK-theory
$SU(3)_{0}+1\mathbf{Sym}+1\mathbf{F}$ by integrating out the fundamental
hypermultiplet and an instantonic hypermultiplet together Bhardwaj:2019jtr .
The corresponding 5-brane web is given in Figure 6. This theory can also be
obtained from a non-perturbative Higgsing on $SU(4)_{0}+1\mathbf{Sym}$. See
Appendix LABEL:sec:app-P^2F_6+1Sym for 5-brane construction and
generalizations.

From $SU(3)_{0}+1\mathbf{Sym}+1\mathbf{F}$, we can first integrate out the
fundamental matter. The IR theory then becomes the
$SU(3)_{-1/2}+1\mathbf{Sym}$ theory, whose effective prepotential is given by

$\displaystyle\mathcal{E}_{SU(3)}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}_{SU(3)}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{24}(2\phi_{1}+2\phi_{2}-3m_{1})+\epsilon_{+}^{2}(\phi_{1}+\phi_{2}))\,,$
$\displaystyle 6\mathcal{F}_{SU(3)}$
$\displaystyle=8\phi_{1}^{3}-3\phi_{1}^{2}\phi_{2}-3\phi_{1}\phi_{2}^{2}+8\phi_{2}^{3}-\frac{3}{2}\phi_{1}\phi_{2}(\phi_{1}-\phi_{2})+6m_{0}(\phi_{1}^{2}-\phi_{1}\phi_{2}+\phi_{2}^{2})$
$\displaystyle\quad-\frac{1}{2}\Big{(}(2\phi_{1}+m_{1})^{3}+(\phi_{2}+m_{1})^{3}+(-2\phi_{1}+2\phi_{2}+m_{1})^{3}$
$\displaystyle\qquad\quad+(\phi_{1}-\phi_{2}+m_{1})^{3}+(-\phi_{1}+m_{1})^{3}+(-2\phi_{1}+m_{1})^{3}\Big{)}\
,$ (270)

where $m_{0}$ is the gauge coupling and $m_{1}$ is the mass parameter of the
symmetric matter.

This theory has a set of consistent magnetic fluxes

$\displaystyle n_{i}\in\mathbb{Z}\,,\quad B_{m_{0}}=-1/4\,,\quad
B_{m_{1}}=1/2\ ,$ (271)

which gives a unity blowup equation. Solving the blowup equation yields the
BPS spectrum of this theory.

Among the BPS states in $SU(3)_{-1/2}+1\mathbf{Sym}$ theory, there are
instantonic hypermultiplets whose masses are
$m_{0}+\phi_{1}-\phi_{2}-\frac{5}{2}m_{1}$ and
$m_{0}+\phi_{2}-\frac{5}{2}m_{1}$. In terms of
$SU(3)_{0}+1\mathbf{Sym}+1\mathbf{F}$ theory given in Table 23, they
correspond to
$\mathbf{d}=(1,\frac{1}{3},-\frac{1}{3},-\frac{5}{2},-\frac{1}{2})$ and
$\mathbf{d}=(1,\frac{1}{3},\frac{2}{3},-\frac{5}{2},-\frac{1}{2})$,
respectively. We can perform flop transitions for these two hypermultiplets to
get a 5-brane web shown in Figure 6.

$O7^{+}$$-m_{0}-\phi_{2}+\frac{5}{2}m_{1}$$-\phi_{1}+2\phi_{2}$$m_{0}+3\phi_{1}-2\phi_{2}-\frac{5}{2}m_{1}$$m_{1}-2\phi_{2}$
Figure 6: A 5-brane web for $SU(3)_{-1/2}+1\mathbf{Sym}$, after flopping the
edges with lengths $m_{0}+\phi_{1}-\phi_{2}-\frac{5}{2}m_{1}$ and
$m_{0}+\phi_{2}-\frac{5}{2}m_{1}$.

The $SU(3)_{-1/2}+1\mathbf{Sym}$ theory has an RG flow to the