As in the previous section where we have computed the partition function of
SU$(3)_{7}$ theory, we start from the web diagram in Figure 18(a). The two
SU(2) sub webs on the right and the two SU(2) sub webs on the left are both
overlapping, in order to compute by our proposal for topological vertex with
ON-planes, we need to swap the positions of relevant branes to make the SU(2)
sub webs separated. From the Higgsing and decoupling processes of obtaining
the SU$(3)_{7}$ theory, we know that in the Higgsing limit of Figure 18(a),
$Q_{m},Q_{c},Q_{a},Q_{n},Q_{d},Q_{e}\rightarrow\sqrt{\frac{\mathfrak{q}}{\mathfrak{t}}}$
and $Q_{s},Q_{t}\rightarrow 1$, in the decoupling limit,
$Q_{4},Q_{5},Q_{6},Q_{7}\rightarrow\infty$. As the procedures are exactly like
the procedures done in the previous section on SU$(3)_{7}$, we omit the
detailed computation process and just list the results. From the web diagram
in Figure 18(b), the Kähler parameters are expressed in terms of the gauge
theory parameters as

$\displaystyle Q_{1}=\frac{A_{1}^{2}}{A_{2}},\qquad
Q_{2}=\frac{A_{2}^{2}}{A_{1}^{4}}u,\qquad
Q_{3}=\frac{A_{2}^{2}}{A_{1}},\qquad$ (144)

where $A_{1}$ and $A_{2}$ are the Coulomb branch parameters and $u$ is the
instanton factor. Notice that $Q_{1}^{2}Q_{2}=u$ and hence any terms with
$(Q_{1}^{2}Q_{2})^{n}$ do not explicitly depend on the Coulomb branch
parameters and they are the extra factor, which will be modded out when
expressing the partition function.

The perturbative part of the partition function is

$\displaystyle Z_{\text{pert}}^{\text{SU}(3)_{9}}$
$\displaystyle=\frac{\mathcal{M}(Q_{1})\mathcal{M}(Q_{3})\mathcal{M}(Q_{1,3})\mathcal{M}(Q_{3}\frac{\mathfrak{t}}{\mathfrak{q}})}{\mathcal{M}(Q_{1}\frac{\mathfrak{t}}{\mathfrak{q}})^{3}\mathcal{M}(Q_{1,3}\frac{\mathfrak{t}}{\mathfrak{q}})^{3}}\mathcal{Z}_{0}^{4}$
$\displaystyle=\text{PE}\left[\frac{\mathfrak{q}+\mathfrak{t}}{(1-\mathfrak{q})(1-\mathfrak{t})}\left(\frac{A_{1}^{2}}{A_{2}}+A_{1}A_{2}+\frac{A_{2}^{2}}{A_{1}}\right)\right]\
.$ (145)

The instanton part of the partition function with extra factors included is

$Z_{\text{inst+extra}}^{\text{SU}(3)_{9}}=\sum_{\mu_{1},\mu_{2},\mu_{3}}Z_{\mu_{1}\mu_{2}\mu_{3}}^{\kappa=9}\mathcal{Z}_{\mu_{1}\mu_{2}\mu_{3}}^{4}\
,$ (146)

where

$Z_{\mu_{1}\mu_{2}\mu_{3}}^{\kappa=9}\equiv\frac{\mathfrak{q}^{||\mu_{2}||^{2}}\mathfrak{t}^{||\mu_{1}^{t}||^{2}+||\mu_{3}^{t}||^{2}}\prod_{i=1,2,3}\tilde{Z}_{\mu_{i}}(\mathfrak{t},\mathfrak{q})\tilde{Z}_{\mu_{i}^{t}}(\mathfrak{q},\mathfrak{t})\left({Q_{1^{2},2}}\frac{\mathfrak{t}^{2}}{\mathfrak{q}^{2}}\right)^{|\mu_{1}|}Q_{2}^{|\mu_{2}|+|\mu_{3}|}}{\mathcal{N}_{\mu_{1}\mu_{2}}(Q_{1})\mathcal{N}_{\mu_{1}\mu_{2}}(Q_{1}\frac{\mathfrak{t}}{\mathfrak{q}})\mathcal{N}_{\mu_{1}\mu_{3}}(Q_{1,3})\mathcal{N}_{\mu_{1}\mu_{3}}(Q_{1,3}\frac{\mathfrak{t}}{\mathfrak{q}})\mathcal{N}_{\mu_{2}\mu_{3}}(Q_{3})\mathcal{N}_{\mu_{2}\mu_{3}}(Q_{3}\frac{\mathfrak{t}}{\mathfrak{q}})}\
,$ (147)

and $\mathcal{Z}_{\mu_{1}\mu_{2}\mu_{3}}$ is the same one that appears in
(128). We now expand this instanton part with respect to $Q_{2}$ which is the
Kähler parameter proportional to the instanton factor $u$. As discussed in the
previous section, $Z_{\mu_{1}\mu_{2}\mu_{3}}^{\kappa=9}$ is proportional to
$Q_{2}^{|\mu_{1}|+|\mu_{2}|+|\mu_{3}|}$ (or
$u^{|\mu_{1}|+|\mu_{2}|+|\mu_{3}|}$) and leads to the instanton expansion. We
then organize the sum of the $n$-instanton part $Z_{n}$ as a PE:

$\displaystyle Z_{\text{inst+extra}}^{\text{SU}(3)_{9}}$
$\displaystyle=1+\sum_{n=1}Z_{n}Q_{2}^{n}=\text{PE}\left[\sum_{n=1}f_{n}^{\kappa=9}(Q_{1},Q_{3};\mathfrak{t},\mathfrak{q})\,Q_{2}^{n}\right]\
.$ (148)

By summing over the Young diagram assignments in (146) with the corresponding
$\mathcal{Z}$’s defined in (125), order by order in $Q_{2}$, one can obtain
$f_{n}$. We list a few $f_{n}$ below. In particular, the exact one-instanton
part $f_{1}$ is given as follows:

$\displaystyle
f_{1}^{\kappa=9}=\frac{\mathfrak{t}}{(1-\mathfrak{q})(1-\mathfrak{t})\mathfrak{q}^{2}}$
$\displaystyle\bigg{(}\frac{Q_{1}^{2}\left(-\mathfrak{q}Q_{1}(1+Q_{3})+\mathfrak{t}\right)^{4}}{(1-Q_{1})(1-Q_{1}Q_{3})(\mathfrak{q}Q_{1}-\mathfrak{t})(\mathfrak{q}Q_{1}Q_{3}-\mathfrak{t})}$
(149)
$\displaystyle+\frac{(\mathfrak{q}+\mathfrak{q}Q_{1}Q_{3}-Q_{1}\mathfrak{t})^{4}}{(1-Q_{1})(1-Q_{3})(\mathfrak{q}Q_{3}-\mathfrak{t})(\mathfrak{t}Q_{1}-\mathfrak{q})}$
(150)
$\displaystyle+\frac{(\mathfrak{q}+\mathfrak{q}Q_{1}-\mathfrak{t}Q_{1}Q_{3})^{4}}{(1-Q_{3})(1-Q_{1}Q_{3})(\mathfrak{t}Q_{3}-\mathfrak{q})(\mathfrak{t}Q_{1}Q_{3}-\mathfrak{q})}\bigg{)}.$
(151)

To compare this with the result from the blowup Kim:2020hhh , we expand
$f_{1}$ with respect to $Q_{1},Q_{3}$, which gives

$\displaystyle f_{1}^{\kappa=9}$
$\displaystyle=\frac{\mathfrak{q}+\mathfrak{t}}{(1-\mathfrak{q})(1-\mathfrak{t})}\bigg{[}1+Q_{1}+\Big{(}\frac{\mathfrak{q}}{\mathfrak{t}}+\frac{\mathfrak{t}}{\mathfrak{q}}\Big{)}Q_{3}+\Big{(}1+\frac{\mathfrak{q}^{2}}{\mathfrak{t}^{2}}+\frac{\mathfrak{t}^{2}}{\mathfrak{q}^{2}}\Big{)}Q_{3}^{2}$
(152)
$\displaystyle\\!+\\!\Big{(}1+\frac{\mathfrak{q}}{\mathfrak{t}}+\frac{\mathfrak{t}}{\mathfrak{q}}\Big{)}Q_{1}Q_{3}+\Big{(}1+\frac{\mathfrak{q}^{2}}{\mathfrak{t}^{2}}+\frac{\mathfrak{q}}{\mathfrak{t}}+\frac{\mathfrak{t}}{\mathfrak{q}}+\frac{\mathfrak{t}^{2}}{\mathfrak{q}^{2}}\Big{)}Q_{1}Q_{3}^{2}+\Big{(}1+2\frac{\mathfrak{t}}{\mathfrak{q}}+\frac{\mathfrak{t}^{2}}{\mathfrak{q}^{2}}\Big{)}Q_{1}^{2}$
(153)
$\displaystyle\\!+\\!\Big{(}1+\frac{\mathfrak{q}}{\mathfrak{t}}+\frac{\mathfrak{t}}{\mathfrak{q}}\Big{)}Q_{1}^{2}Q_{3}+\Big{(}2+\frac{\mathfrak{q}^{2}}{\mathfrak{t}^{2}}+\frac{\mathfrak{q}}{\mathfrak{t}}+\frac{\mathfrak{t}}{\mathfrak{q}}+\frac{\mathfrak{t}^{2}}{\mathfrak{q}^{2}}\Big{)}Q_{1}^{2}Q_{3}^{2}$
(154)
$\displaystyle+Q_{1}^{3}+\Big{(}\frac{\mathfrak{q}}{\mathfrak{t}}+\frac{\mathfrak{t}}{\mathfrak{q}}+\frac{\mathfrak{q}^{3}}{\mathfrak{t}^{3}}+\frac{\mathfrak{t}^{3}}{\mathfrak{q}^{3}}\Big{)}Q_{3}^{3}+\Big{(}1+\frac{\mathfrak{q}}{\mathfrak{t}}+\frac{\mathfrak{t}}{\mathfrak{q}}\Big{)}Q_{1}^{3}Q_{3}$
(155)
$\displaystyle+\Big{(}1+\frac{\mathfrak{q}^{3}}{\mathfrak{t}^{3}}+\frac{\mathfrak{q}^{2}}{\mathfrak{t}^{2}}+\frac{\mathfrak{q}}{\mathfrak{t}}+\frac{\mathfrak{t}}{\mathfrak{q}}+\frac{\mathfrak{t}^{2}}{\mathfrak{q}^{2}}+\frac{\mathfrak{t}^{3}}{\mathfrak{q}^{3}}\Big{)}Q_{3}^{3}Q_{1}+\Big{(}2+\frac{\mathfrak{q}^{2}}{\mathfrak{t}^{2}}+\frac{\mathfrak{q}}{\mathfrak{t}}+\frac{\mathfrak{t}}{\mathfrak{q}}+\frac{\mathfrak{t}^{2}}{\mathfrak{q}^{2}}\Big{)}Q_{1}^{3}Q_{3}^{2}$
(156)
$\displaystyle+\Big{(}1+\frac{\mathfrak{q}^{3}}{\mathfrak{t}^{3}}+\frac{\mathfrak{q}^{2}}{\mathfrak{t}^{2}}+\frac{2\mathfrak{q}}{\mathfrak{t}}+\frac{2\mathfrak{t}}{\mathfrak{q}}+\frac{\mathfrak{t}^{2}}{\mathfrak{q}^{2}}+\frac{\mathfrak{t}^{3}}{\mathfrak{q}^{3}}\Big{)}Q_{3}^{3}Q_{1}^{2}$
(157)
$\displaystyle+\Big{(}2+\frac{\mathfrak{q}^{3}}{\mathfrak{t}^{3}}+\frac{\mathfrak{q}^{2}}{\mathfrak{t}^{2}}+\frac{2\mathfrak{q}}{\mathfrak{t}}+\frac{2\mathfrak{t}}{\mathfrak{q}}+\frac{\mathfrak{t}^{2}}{\mathfrak{q}^{2}}+\frac{\mathfrak{t}^{3}}{\mathfrak{q}^{3}}\Big{)}Q_{1}^{3}Q_{3}^{3}\bigg{]}+\mathcal{O}(Q_{1}^{4},Q_{3}^{4})\
.$ (158)

As $Q_{1}^{2}Q_{2}=u$ does not depend on the Coulomb branch parameters, the
term proportional to $Q_{1}^{2}$ in (158) contributes to the extra factor.
Separating out such extra factor, we rewrite (158) in terms of the spin state
$[j_{l},j_{r}]$ defined in (85),

$\displaystyle f_{1}^{\kappa=9}$
$\displaystyle=[0,\tfrac{1}{2}]+[0,\tfrac{3}{2}]Q_{3}+[0,\tfrac{5}{2}]Q_{3}^{2}+[0,\tfrac{1}{2}]Q_{1}+\Big{(}[0,\tfrac{1}{2}]+[0,\tfrac{3}{2}]\Big{)}Q_{1}Q_{3}$
(159)
$\displaystyle+\Big{(}[0,\tfrac{3}{2}]+[0,\tfrac{5}{2}]\Big{)}Q_{1}Q_{3}^{2}+\Big{(}[0,\tfrac{1}{2}]+[0,\tfrac{3}{2}]\Big{)}Q_{1}^{2}Q_{3}+\Big{(}[0,\tfrac{1}{2}]+[0,\tfrac{3}{2}]+[0,\tfrac{5}{2}]\Big{)}Q_{1}^{2}Q_{3}^{2}$
(160)
$\displaystyle+[0,\tfrac{7}{2}]Q_{3}^{3}+\Big{(}[0,\tfrac{5}{2}]+[0,\tfrac{7}{2}]\Big{)}Q_{1}Q_{3}^{3}+\Big{(}[0,\tfrac{3}{2}]+[0,\tfrac{5}{2}]+[0,\tfrac{7}{2}]\Big{)}Q_{1}^{2}Q_{3}^{3}$
(161)
$\displaystyle+[0,\tfrac{1}{2}]Q_{1}^{3}+\Big{(}[0,\tfrac{1}{2}]+[0,\tfrac{3}{2}]\Big{)}Q_{1}^{3}Q_{3}+\Big{(}[0,\tfrac{1}{2}]+[0,\tfrac{3}{2}]+[0,\tfrac{5}{2}]\Big{)}Q_{1}^{3}Q_{3}^{2}$
(162)
$\displaystyle+\Big{(}[0,\tfrac{1}{2}]+[0,\tfrac{3}{2}]+[0,\tfrac{5}{2}]+[0,\tfrac{7}{2}]\Big{)}Q_{1}^{3}Q_{3}^{3}+\mathcal{O}(Q_{1}^{4},Q_{3}^{4})+(\text{extra
factor})\ .$ (163)

This agrees with the blowup result in Kim:2020hhh  where the terms up to
$Q_{1}^{2}Q_{3}^{2}$ are presented at the first order of $Q_{2}$.

The contribution at order $Q_{2}^{2}$ leads to two-instanton partition
function. Summing over Young diagrams in (146) with the corresponding
$\mathcal{Z}$’s in (127) and combining (148) and (151), we can obtain the
exact form of $f_{2}$. As the exact form of $f_{2}$ is rather long111111The
exact form of $f_{2}$ for the SU(3)9 theory is presented in a Mathematica file
attached as an ancillary file., we expand the form with respect to
$Q_{1},Q_{3}$ up to $Q_{1}^{2}Q_{3}^{2}$ to compare it with the known result,
as follows:

$\displaystyle f_{2}^{\kappa=9}$
$\displaystyle=\frac{1}{(1-\mathfrak{q})(1-\mathfrak{t})}\bigg{[}\Big{(}\mathfrak{q}+\frac{\mathfrak{q}^{3}}{\mathfrak{t}^{2}}+\frac{\mathfrak{q}^{2}}{\mathfrak{t}}+\mathfrak{t}+\frac{\mathfrak{t}^{2}}{\mathfrak{q}}+\frac{\mathfrak{t}^{3}}{\mathfrak{q}^{2}}\Big{)}Q_{3}+\Big{(}1+2\mathfrak{q}+\mathfrak{q}^{2}+\frac{\mathfrak{q}^{4}}{\mathfrak{t}^{4}}+\frac{\mathfrak{q}^{3}}{\mathfrak{t}^{3}}$
(164)
$\displaystyle+\frac{\mathfrak{q}^{4}}{\mathfrak{t}^{3}}+\frac{\mathfrak{q}^{5}}{\mathfrak{t}^{3}}+\frac{\mathfrak{q}^{2}}{\mathfrak{t}^{2}}+2\frac{\mathfrak{q}^{3}}{\mathfrak{t}^{2}}+\frac{\mathfrak{q}^{4}}{\mathfrak{t}^{2}}+\frac{\mathfrak{q}}{\mathfrak{t}}+2\frac{\mathfrak{q}^{2}}{\mathfrak{t}}+\frac{\mathfrak{q}^{3}}{\mathfrak{t}}+2\mathfrak{t}+\frac{\mathfrak{t}}{\mathfrak{q}}+\mathfrak{q}\mathfrak{t}+\mathfrak{t}^{2}+\frac{\mathfrak{t}^{2}}{\mathfrak{q}^{2}}+2\frac{\mathfrak{t}^{2}}{\mathfrak{q}}+\frac{\mathfrak{t}^{3}}{\mathfrak{q}^{3}}$
(165)
$\displaystyle+2\frac{\mathfrak{t}^{3}}{\mathfrak{q}^{2}}+\frac{\mathfrak{t}^{3}}{\mathfrak{q}}+\frac{\mathfrak{t}^{4}}{\mathfrak{q}^{4}}+\frac{\mathfrak{t}^{4}}{\mathfrak{q}^{3}}+\frac{\mathfrak{t}^{4}}{\mathfrak{q}^{2}}+\frac{\mathfrak{t}^{5}}{\mathfrak{q}^{3}}\Big{)}Q_{3}^{2}+\Big{(}2\mathfrak{q}+\frac{\mathfrak{q}^{3}}{\mathfrak{t}^{2}}+2\frac{\mathfrak{q}^{2}}{\mathfrak{t}}+2\mathfrak{t}+2\frac{\mathfrak{t}^{2}}{\mathfrak{q}}+\frac{\mathfrak{t}^{3}}{\mathfrak{q}^{2}}\Big{)}Q_{1}Q_{3}$
(166)
$\displaystyle+\Big{(}2\\!+\\!6\mathfrak{q}+2\mathfrak{q}^{2}+\frac{\mathfrak{q}^{4}}{\mathfrak{t}^{4}}+2\frac{\mathfrak{q}^{3}}{\mathfrak{t}^{3}}+2\frac{\mathfrak{q}^{4}}{\mathfrak{t}^{3}}+\frac{\mathfrak{q}^{5}}{\mathfrak{t}^{3}}+2\frac{\mathfrak{q}^{2}}{\mathfrak{t}^{2}}+5\frac{\mathfrak{q}^{3}}{\mathfrak{t}^{2}}+2\frac{\mathfrak{q}^{4}}{\mathfrak{t}^{2}}+2\frac{\mathfrak{q}}{\mathfrak{t}}+6\frac{\mathfrak{q}^{2}}{\mathfrak{t}}+2\frac{\mathfrak{q}^{3}}{\mathfrak{t}\\!}$
(167)
$\displaystyle+\\!6\mathfrak{t}+\\!2\frac{\mathfrak{t}}{\mathfrak{q}}+\\!2\mathfrak{q}\mathfrak{t}+\\!2\mathfrak{t}^{2}+\\!2\frac{\mathfrak{t}^{2}}{\mathfrak{q}^{2}}+\\!6\frac{\mathfrak{t}^{2}}{\mathfrak{q}}+\\!2\frac{\mathfrak{t}^{3}}{\mathfrak{q}^{3}}+\\!5\frac{\mathfrak{t}^{3}}{\mathfrak{q}^{2}}+\\!2\frac{\mathfrak{t}^{3}}{\mathfrak{q}}+\\!\frac{\mathfrak{t}^{4}}{\mathfrak{q}^{4}}+\\!2\frac{\mathfrak{t}^{4}}{\mathfrak{q}^{3}}+\\!2\frac{\mathfrak{t}^{4}}{\mathfrak{q}^{2}}+\\!\frac{\mathfrak{t}^{5}}{\mathfrak{q}^{3}}\Big{)}Q_{1}Q_{3}^{2}$
(168)
$\displaystyle+\Big{(}3\mathfrak{q}+\frac{\mathfrak{q}^{3}}{\mathfrak{t}^{2}}+2\frac{\mathfrak{q}^{2}}{\mathfrak{t}}+3\mathfrak{t}+2\frac{\mathfrak{t}^{2}}{\mathfrak{q}}+\frac{\mathfrak{t}^{3}}{\mathfrak{q}^{2}}\Big{)}Q_{1}^{2}Q_{3}+\Big{(}3+10\mathfrak{q}+3\mathfrak{q}^{2}+\frac{\mathfrak{q}^{4}}{\mathfrak{t}^{4}}+2\frac{\mathfrak{q}^{3}}{\mathfrak{t}^{3}}+2\frac{\mathfrak{q}^{4}}{\mathfrak{t}^{3}}$
(169)
$\displaystyle+\frac{\mathfrak{q}^{5}}{\mathfrak{t}^{3}}+3\frac{\mathfrak{q}^{2}}{\mathfrak{t}^{2}}+6\frac{\mathfrak{q}^{3}}{\mathfrak{t}^{2}}+2\frac{\mathfrak{q}^{4}}{\mathfrak{t}^{2}}+3\frac{\mathfrak{q}}{\mathfrak{t}}+9\frac{\mathfrak{q}^{2}}{\mathfrak{t}}+3\frac{\mathfrak{q}^{3}}{\mathfrak{t}}+10\mathfrak{t}+3\frac{\mathfrak{t}}{\mathfrak{q}}+3\mathfrak{q}\mathfrak{t}+3\mathfrak{t}^{2}+3\frac{\mathfrak{t}^{2}}{\mathfrak{q}^{2}}+9\frac{\mathfrak{t}^{2}}{\mathfrak{q}}$
(170)
$\displaystyle+2\frac{\mathfrak{t}^{3}}{\mathfrak{q}^{3}}+6\frac{\mathfrak{t}^{3}}{\mathfrak{q}^{2}}+3\frac{\mathfrak{t}^{3}}{\mathfrak{q}}+\frac{\mathfrak{t}^{4}}{\mathfrak{q}^{4}}+2\frac{\mathfrak{t}^{4}}{\mathfrak{q}^{3}}+2\frac{\mathfrak{t}^{4}}{\mathfrak{q}^{2}}+\frac{\mathfrak{t}^{5}}{\mathfrak{q}^{3}}\Big{)}Q_{1}^{2}Q_{3}^{2}\bigg{]}+\mathcal{O}(Q_{1}^{3},Q_{3}^{3})\
,$ (171)

which can be rewritten as

$\displaystyle f_{2}^{\kappa=9}$
$\displaystyle=[0,\tfrac{5}{2}]Q_{3}+\Big{(}[0,\tfrac{5}{2}]+[0,\tfrac{7}{2}]+[\tfrac{1}{2},4]\Big{)}Q_{3}^{2}+\Big{(}[0,\tfrac{3}{2}]+[0,\tfrac{5}{2}]\Big{)}Q_{1}Q_{3}$
(172)
$\displaystyle+\Big{(}[0,\tfrac{3}{2}]+3[0,\tfrac{5}{2}]+2[0,\tfrac{7}{2}]+[\tfrac{1}{2},3]+[\tfrac{1}{2},4]\Big{)}Q_{1}Q_{3}^{2}+\Big{(}[0,\tfrac{1}{2}]+[0,\tfrac{3}{2}]+[0,\tfrac{5}{2}]\Big{)}Q_{1}^{2}Q_{3}$
(173)
$\displaystyle+\Big{(}[0,\tfrac{1}{2}]+3[0,\tfrac{3}{2}]+4[0,\tfrac{5}{2}]+2[0,\tfrac{7}{2}]+[\tfrac{1}{2},2]+[\tfrac{1}{2},3]+[\tfrac{1}{2},4]\Big{)}Q_{1}^{2}Q_{3}^{2}+\mathcal{O}(Q_{1}^{3},Q_{3}^{3})\
,$

which agrees with the two-instanton part from the blowup result Kim:2020hhh .
Notice that there is no extra factor in (172), which is because the extra
factor at order $Q_{2}^{2}$ comes from terms proportional to $Q_{1}^{4}$.

The instanton partition function at higher order $n\geq 3$ can be obtained in
the same way. It is to repeat the same computation with
$|\mu_{1}|+|\mu_{2}|+|\mu_{3}|=n$. One can also express the partition function
in terms of the gauge theory parameters by substituting the Kähler parameters
with the Coulomb branch parameters and instanton factor in (144), after
dropping the extra factors.

##  4 Conclusion

In this paper, we generalized refined topological vertex formalism so that it
is applicable for 5-brane webs with ON-planes. 5-brane system with an ON-plane
describes a D-type quiver gauge theory. In order for a 5-brane web with an ON-
plane to correctly account for a D-type quiver, there should be no
bifundamental contributions between two gauge theories belonging to bivalent
nodes of a D-type quiver. To this end, we proposed a new vertex factor
$C^{\text{R}}$ for the reflected vertices over an ON-plane and also new
associated edge factors. Such new vertex and edge factors ensures the
resulting partition functions correctly capture the BPS spectrum of a D-type
quiver gauge theory. It is also worthy of noting that one can use only this
new vertex factor and edge factor for usual 5-brane webs to obtain the same
partition function obtained with the conventional vertex and edge factors as
shown in Appendix B. This means that there are two different types of vertex
and edge factors, yielding the same partition function. It may also support
our proposal that introducing the new vertex and edge factors associated with
the 5-brane web reflected over an ON-plane would be a natural procedure to
distinguish original domain from the reflected domain of 5-brane system with
an ON-plane.

Through the Higgsing, 5-brane configurations with ON-plane(s) can give rise to
5-brane webs for SU(3) gauge theories at higher Chern-Simons level. We
computed the refined partition function for SU(3)7 and SU(3)9 theories and
confirmed that the results perfectly agree with the blowup computation
Kim:2020hhh . We also checked our proposal against 6d E-string theory on a
circle, which requires two ON-planes, which also agree with other known
results. These are the first examples that the refined partition functions are
obtained based on 5-brane webs with ON-plane. We also present the exact form
of one- and two-instanton partition functions of these theories. For theories
of higher rank gauge groups or of more complicated matter that can be
described by a 5-brane web with an ON-plane Hayashi:2015vhy , one can apply
our new vertex and edge factors to obtain their partition functions in a
straightforward way.

From the perspective of the S-duality, 5-brane configuration with an O5-plane
can be understood as an S-dual configuration of that with an ON-plane. It is
natural to generalize our proposal to 5-brane systems with an O5-plane, which
describes 5d SO/Sp gauge theories. Though our new $C^{\text{R}}$ is compatible
with O5-plane, the preferred direction is assigned along the edges associated
with W-bosons, and the resulting partition function become an expansion of
Coulomb branch parameters, rather than an instanton expansion. A naive attempt
for refining topological vertex with an O5-plane seems more challenging
because those branes which are reflected over an O5-plane are of the same
$\Omega$ background parameters along the same edge. Hence, it is an
interesting direction to pursue to generalize the refined topological vertex
so that it is applicable to 5-brane system with an O5-plane. These would shed
some light on the partition functions from newly obtained 5-brane webs which
involve $G_{2}$ gauge theories, SO gauge theories with spinor matter, and Sp
or SU gauge theories of hypermultiplets in the rank-3 antisymmetric
representations.

It would be also interesting to study the relation between our proposal for a
$D$-type quiver theory and algebraic constructions based on Ding-Iohara-Miki
algebra where the presence of an ON-plane is discussed either from the point
of view of the reflection states Bourgine:2017rik  or as the introduction of a
new twisted vertex Kimura:2019gon .

###### Acknowledgements.

We thank Jean-Emile Bourgine, Shi Cheng, Hirotaka Hayashi, Songling He, Hee-
Cheol Kim, Minsung Kim, Kimyeong Lee, Xiaobin Li, Satoshi Nawata, Yuji
Sugimoto, Ryo Suzuki, Futoshi Yagi, and Ruidong Zhu for useful discussions. We
are also grateful to Shi Cheng and Songling He for providing useful
Mathematica codes and Futoshi Yagi and Yuji Sugimoto for their comments on the
draft. SSK thanks APCTP, Fundan university, KIAS, POSTECH and YMSC Tsinghua
university for the hospitality during his visit where part of this work is
done. SSK is partially supported by the Fundamental Research Funds for the
Central Universities 2682021ZTPY043. XYW is partially supported by NSFC grant
No. 11950410490.

##  Appendix A Characters

The characters for the fundamental weights of SO(8) $\chi_{\bf dim}^{\rm
SO(8)}$ are given as follows:

$\displaystyle\chi_{\bf 8v}^{\text{SO(8)}}$
$\displaystyle=\sum_{i=1}^{8}X_{i}^{2},$ $\displaystyle\chi_{\bf
28}^{\text{SO(8)}}$
$\displaystyle=\sum_{i=1}^{8}\sum_{j>i}^{8}X_{i}^{2}X_{j}^{2},$
$\displaystyle\chi_{\text{\bf 8s}}^{\text{SO(8)}}$
$\displaystyle=\frac{1}{2}\left(\prod_{i=1}^{4}\left(X_{i}+X_{i+4}\right)+\prod_{i=1}^{4}\left(X_{i}-X_{i+4}\right)\right),$
$\displaystyle\chi_{\text{\bf 8c}}^{\text{SO(8)}}$
$\displaystyle=\frac{1}{2}\left(\prod_{i=1}^{4}\left(X_{i}+X_{i+4}\right)-\prod_{i=1}^{4}\left(X_{i}-X_{i+4}\right)\right),$
(174)

where
$X_{i}\in\big{\\{}M_{1}^{1/2},\dots,M_{4}^{1/2},M_{1}^{-1/2},\dots,M_{4}^{-1/2}\big{\\}}$
with the flavor mass fagacities $M_{i}=e^{-\beta m_{i}}$.

The characters for the fundamental weights of SO(16) $\chi_{\bf dim}$ are
given as follows:

$\displaystyle\chi_{\bf 16}\,$ $\displaystyle=\sum_{i=1}^{16}Y_{i}^{2},$
$\displaystyle\chi_{\bf 120}\,$
$\displaystyle=\sum_{i=1}^{16}\sum_{j>i}^{16}Y_{i}^{2}Y_{j}^{2},$
$\displaystyle\chi_{\bf 560}\,$
$\displaystyle=\sum_{i=1}^{16}\sum_{j>i}^{16}\sum_{k>j}^{16}Y_{i}^{2}Y_{j}^{2}Y_{k}^{2},$
$\displaystyle\chi_{\bf 1820}\,$
$\displaystyle=\sum_{i=1}^{16}\sum_{j>i}^{16}\sum_{k>j}^{16}\sum_{l>k}^{16}Y_{i}^{2}Y_{j}^{2}Y_{k}^{2}Y_{l}^{2},$
$\displaystyle\chi_{\bf 4368}\,$
$\displaystyle=\sum_{i=1}^{16}\sum_{j>i}^{16}\sum_{k>j}^{16}\sum_{l>k}^{16}\sum_{m>l}^{16}Y_{i}^{2}Y_{j}^{2}Y_{k}^{2}Y_{l}^{2}Y_{m}^{2},$
$\displaystyle\chi_{\bf 8008}\,$
$\displaystyle=\sum_{i=1}^{16}\sum_{j>i}^{16}\sum_{k>j}^{16}\sum_{l>k}^{16}\sum_{m>l}^{16}\sum_{n>m}^{16}Y_{i}^{2}Y_{j}^{2}Y_{k}^{2}Y_{l}^{2}Y_{m}^{2}Y_{n}^{2},$
$\displaystyle\chi_{\bf 128}\,$
$\displaystyle=\frac{1}{2}\left(\prod_{i=1}^{8}\left(Y_{i}+Y_{i+8}\right)+\prod_{i=1}^{8}\left(Y_{i}-Y_{i+8}\right)\right),$
$\displaystyle\chi_{\overline{\bf 128}}\,$
$\displaystyle=\frac{1}{2}\left(\prod_{i=1}^{8}\left(Y_{i}+Y_{i+8}\right)-\prod_{i=1}^{8}\left(Y_{i}-Y_{i+8}\right)\right)\
,$ (175)

where
$Y_{i}\in\big{\\{}M_{1}^{1/2},\dots,M_{8}^{1/2},M_{1}^{-1/2},\dots,M_{8}^{-1/2}\big{\\}}$.

The characters for higher dimensional irreducible representations can be
expressed in terms of the product of the characters of these fundamental
weights. For instance, among those that appear in (90),

$\displaystyle\chi_{\overline{\bf 1920}}+\chi_{\overline{\bf 128}}=\chi_{\bf
16}\cdot\chi_{\bf 128}\ ,$ (176) $\displaystyle\chi_{\bf 1920}+\chi_{\bf
128}=\chi_{\bf 16}\cdot\chi_{\overline{\bf 128}}\ ,$ (177)
$\displaystyle\chi_{\bf 13312}+\chi_{\bf 1920}+\chi_{\bf 128}=\chi_{\bf
120}\cdot\chi_{\bf 128}\ .$ (178)

##  Appendix B Reflected 5-brane web with $C^{\text{R}}$ and
$\tilde{f}^{\text{R}}$.

In the main text, we have introduced new vertex factor $C^{\text{R}}$ and
framing factor $\tilde{f}^{\text{R}}$ which account for vertex and framing
factors that are reflected over an ON-plane. For a 5-brane web with an ON-
plane, one needs to choose the fundamental region, as there is reflected
5-brane configuration over the ON-plane. Because of the equivalence between
the original 5-brane web and the reflected image, the partition function based
on either the original 5-brane web or the reflected one should be equivalent.
In other words, even though one performs the topological vertex computation
based on reflected web diagram with $C^{\text{R}}$ and $\tilde{f}^{\text{R}}$
factors, the resulting partition function should not make any difference. The
new factors $C^{\text{R}},\tilde{f}^{\text{R}}$ and conventional $C,\tilde{f}$
are on an equal footing.

As a concrete example, let us consider a $D_{4}$-type quiver with SU(3)1 at
the center node connected to tree SU(2) nodes, which is discussed in
subsection 3.3. The corresponding 5-brane configuration is presented in Figure
17. The corresponding reflected 5-brane configuration can be easily depicted
as given in Figure 19, where the role of $C,\tilde{f}$ and
$C^{\text{R}},\tilde{f}^{\text{R}}$ exchanged, compared to the original
5-brane web in Figure 17. In particular, $C,\tilde{f}$ only appear on the
strip next to an ON-plane on the left. Note that the preferred direction
framing factor $f$ in the reflected web diagram is the same as that in the
unreflected web diagram as is mentioned in the main text. One can check that
the partition function based on this reflected 5-brane web is the same as the
one discussed in subsection 3.3.

Figure 19: Reflected 5-brane configuration for a $D_{4}$-type quiver theory
discussed in Figure 17.  Figure 20: Web diagram of SU$(2)+1\mathbf{F}$ denoted
by $C^{\text{R}}$ and $\tilde{f}^{\text{R}}$.

It naturally follows that even for a 5-brane web without an ON-plane, one can
use the new factors $C^{\text{R}}$ and $\tilde{f}^{\text{R}}$, instead of
using the conventional factors $C$ and $\tilde{f}$. For instance, consider a
5-brane web for SU(2)+1F given in Figure 20. Here in the figure, we have
assigned the new vertex and framing factors
$C^{\text{R}},\tilde{f}^{\text{R}}$. Note that four vertex factors out of the
five $C^{\text{R}}$’s in fact reduce to the usual vertex factor $C$ because
one of their legs in non-preferred direction is external, and hence
effectively we have only one $C^{\text{R}}$ factor presented. One can check
that the partition function computation with this $C^{\text{R}}$ and
$\tilde{f}^{\text{R}}$ is straightforward and the result is the same as the
partition function obtained with $C$ and $\tilde{f}$.

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##  1 Introduction

Despite its success in describing the elementary particles and their
interactions, the Standard Model (SM) is still incomplete, e.g., it does not
account for neutrino masses, baryon asymmetry or dark matter. Thus, the
discovery of physics beyond the Standard Model (BSM) is one of the main goals
of particle physics. In particular, it is a core component of the physics
program of the two multipurpose experiments, ATLAS [atlas] and CMS [cms] at
the Large Hadron Collider (LHC) at CERN. So far hundreds of searches for BSM
physics have been conducted, but no significant deviation from the SM
predictions has been found. With only a few exceptions (e.g., [D0:2000vuh,
H1:2008aak, gensearchCDF, gensearch1, gensearch2, gensearch3]), most of these
searches were conducted following the blind-analysis paradigm according to
which the data is only looked at in the last step of the analysis, after most
of the time and efforts were invested. Moreover, since the data are looked at
in the end, these analyses were typically designed to inspect a specific
region of the _space of observables_ \- the space spanned by all observables
of the recorded data. As a result, despite thousands of person-years invested,
a large portion of the observable space has yet to be fully exploited (see
also Ref. [Craig:2016rqv, Kim:2019rhy]). The risk of missing a discovery by
studying only a limited number of final states could be mitigated by
prioritizing the searches and focusing the efforts on high priority ones.
Traditionally, this is mostly done based on theoretical considerations.
However, by now, the searches with the strongest theoretical motivation have
mostly been conducted and to a large extent, none of the many remaining ones
are, a priori, more motivated than the others. This calls for investigating
additional search paradigms.

Complementary to the blind searches, we propose extending the discovery
potential of the LHC with a data-directed paradigm (DDP). Similarly to
[gensearch1, gensearch3, gensearchCDF], its principal objective is to
efficiently scan large portions of the observable-space for hints of new
physics (NP), but unlike [gensearch1, gensearch3, gensearchCDF] without using
any Monte-Carlo (MC) simulation. We look directly at the data, in an attempt
to identify regions in the space of observables that exhibit deviations from a
theoretically well established property of the SM. Such regions should be
considered as data-directed BSM hypotheses, as opposed to theoretically-
motivated ones, and could be studied using traditional data-analysis methods.
As detailed in [ddp-bumphunt], a search in the DDP can be implemented with two
key ingredients: a) a theoretically well established property of the SM and b)
an efficient algorithm to search for deviations from this property.

In this work, we show that any symmetry of the SM can be exploited in such a
data-directed search. Symmetries can be used to split the data into two
mutually exclusive samples which should only differ by statistical
fluctuations. By comparing them, we become sensitive to any potential BSM
process which breaks this symmetry. In some cases, systematic detector effects
could also affect the symmetry. There are methods to account for these effects
in principle and so we do not consider them further in this proof-of-principle
study. In an experimental realization of the symmetry-based DDP search, such
systematic effects must be taken into account.

The concept of exploiting symmetries of the SM for data-driven BSM searches
was previously proposed in [emmePheno] and [chris3]. It is also implemented in
the ATLAS search for lepton flavor violating (LFV) decays of the Higgs (H)
Boson [emmeRun1], and the search for asymmetry between $e^{+}\mu^{-}$ and
$e^{-}\mu^{+}$ events [epmum]. In the former, the SM background is estimated
from the data using the electron-muon ($e/\mu$) symmetry method, based on the
premise that kinematic properties of SM processes are, to a good
approximation, symmetric under the exchange of prompt electrons and prompt
muons111This approximate symmetry derives from the _lepton flavor
universality_ of the electroweak force. Phase-space effects and Higgs
interactions only violate it at negligible levels within the energy range of
LHC collisions, since the mass of the two leptons is negligible.. In this
case, the sample of all recorded data events with one electron and one muon in
the final state is split into the $e\mu$ and $\mu e$ samples, which differ
only by the $p_{T}$ ordering of the two leptons. The Higgs LFV signal is
expected to contribute only to one of these two samples, while the other is
used as the background estimate. In [emmeRun1], it was shown that systematic
effects which violate the expected SM symmetry, e.g., the different detection
efficiencies of electrons and muons, can be accounted for and the symmetry can
be restored. However, the implementation of the search still follows the
blind-analysis paradigm where only a specific signal is searched for, in a
small theoretically-motivated subset of the observable-space.

In terms of the DDP proposed here, no specific signal is searched for.
Instead, the full $e\mu$ and $\mu e$ samples are compared in many different
sub-samples (corresponding to exclusive selections of the data), and any
significant deviation observed is considered a potential sign for NP, to be
further investigated. Thus, sensitivity to many more possible BSM processes
and scenarios is enabled, and this $e\mu/\mu e$ comparison becomes a general
test for lepton flavor universality in the final state containing one electron
and one muon. Similarly, different final states including a number of
electrons, muons and other objects can be probed ($ee$ vs $\mu\mu$, $e+$jet vs
$\mu+$jet, etc.), each potentially sensitive to different BSM manifestations.
In this context, the recent hints for non-universality in the RK measurements
from LHCb [rk] are in fact hints of an asymmetry between the $ee$ and $\mu\mu$
samples in the decay of $b$ hadrons to a Kaon and two same-flavor leptons.
Likewise, the comparison of $e^{-}\mu^{+}$ to $e^{+}\mu^{-}$ in [chris3] and
[epmum] is a test of CP symmetry in the lepton sector. Other symmetries could
be used in similar implementations, such as forward-backward or time-reversal
symmetries.

Given the large number of symmetries in the SM which can be violated in BSM
scenarios, the potential benefits of implementing such symmetry-based generic
searches are significant. However, interpreting the results must be done with
care. Indeed, a data-directed search will naturally be tuned to identify
regions including statistical fluctuations, or other measurement effects which
could induce asymmetries. If a detected signal originates from a statistical
fluctuation, it will disappear with more collected data. If it originates from
a detector or other systematic effect which are correctly modeled in MC
simulations then it can be ruled out. Any residual asymmetry can be considered
a data-directed BSM hypotheses, to be inspected using standard analysis
techniques. In this manner, the risk to claim a false discovery should not be
higher than when implementing hundreds of searches in the blind-paradigm,
since the trial factor is high in both cases [eilamLook].

The aim of this paper is to draw the attention on the potential for
discovering BSM physics when implementing searches in the DDP, and in
particular, data-directed searches based on symmetries of the SM. In this
context, we lay the groundwork for a generic method to compare two data
samples, and quantify the level of any discrepancy between them, if present.
As previously discussed, we do not address here the treatment of eventual
systematic effects which can deteriorate the expected SM symmetry between the
two samples. Nonetheless, as shown in [emmeRun1] and [epmum], in analyses that
were based on symmetry considerations, such effects can be accounted for.

Since the goal is to quickly scan multiple sub-regions of the space of
observables in a large number of final states, a fast method for identifying
asymmetries is needed. We develop this method based on a simplified framework
using MC simulated data. Different test statistics can be used to compare the
two samples (e.g. Kolmogorov-Smirnov [ks-test], student $t$-test [student-
test]). Here the samples are represented by 2D histograms222The generalization
to $n$-dimensional histograms is straightforward. of predetermined properties
of the data and compared using a simple test statistic referred to as the
$N_{\sigma}$ test. Since the method is fast, multiple 2D histograms of all the
existing properties and their combinations can be compared efficiently. We
leave to future work the generalization of this study for a more comprehensive
and optimized implementation.

When working with histograms, there is no a priori way to choose the bins,
which is particularly challenging in many dimensions. One solution to this
challenge is to make use of machine learning. Starting from [ano1, ano2] based
on [ano3], there have been a variety of proposals to perform anomaly detection
with machine learning by comparing two samples [ano1, ano2, DAgnolo:2018cun,
DAgnolo:2019vbw, Amram:2020ykb, gensearch2, Nachman:2020lpy,
Andreassen:2020nkr, Benkendorfer:2020gek, Hallin:2021wme, dAgnolo:2021aun]
(see Ref. [2010.14554, 2112.03769, 2102.02770, Kasieczka:2021xcg,
Aarrestad:2021oeb] for recent reviews). We demonstrate that asymmetries can be
identified using a similar approach as complementary to the binned DDP
(henceforth, simply ‘the DDP’). Nevertheless, for now such methods require
training at least one Neural Net (NN) for each event selection. This is time
consuming and restricts the number of selections that can be tested, which
could be limiting in the context of the DDP, depending on the available
computational resources.

The sensitivity of the proposed DDP search is compared to that of two
likelihood-based test statistics. While both assume exact knowledge of the
signal shape, one represents an ideal search in which also the distribution of
the symmetric background components are exactly known, and the other
represents the expected sensitivity of a traditional blind analysis search
employing a symmetry-based background estimation. According to the Neyman-
Pearson lemma [neyman1933ix] these are the most sensitive tests for the
respective scenarios they consider.

This paper is organized as follows. Section LABEL:sec:quant describes some of
the statistical properties of the DDP symmetry search. The simulated data used
for our numerical studies is presented in Sec. LABEL:sec:data. Results for the
DDP are given in Sec. LABEL:sec:results and a complementary approach using
neural networks is discussed in Sec. LABEL:sec:NN. The paper ends with
conclusions and outlook in Sec. LABEL:sec:discussion.