aainstitutetext: Department of Physics, POSTECH, Pohang 790-784,
Koreabbinstitutetext: Asia Pacific Center for Theoretical Physics, Postech,
Pohang 37673, Koreaccinstitutetext: School of Physics, University of
Electronic Science and Technology of China,
Chengdu, Sichuan 611731, China

# Bootstrapping BPS spectra of 5d/6d field theories

Hee-Cheol Kim a Minsung Kim c Sung-Soo Kim b Ki-Hong Lee

###### Abstract

We propose a systematic approach to computing the BPS spectra of any 5d/6d
supersymmetric quantum field theory in Coulomb phases, which admits either
gauge theory descriptions or geometric descriptions, based on the Nakajima-
Yoshioka’s blowup equations. We provide a significant generalization of the
blowup equation approach in terms of both properly quantized magnetic fluxes
on the blowup $\hat{\mathbb{C}}^{2}$ and the effective prepotential for 5d/6d
field theories on the Omega background which is uniquely determined by the
Chern-Simons couplings on their Coulomb branches. We employ our method to
compute BPS spectra of all rank-1 and rank-2 5d Kaluza-Klein (KK) theories
descending from 6d $\mathcal{N}=(1,0)$ superconformal field theories (SCFTs)
compactified on a circle with/without twist. We also discuss various 5d SCFTs
and KK theories of higher ranks, which include a few exotic cases such as new
rank-1 and rank-2 5d SCFTs engineered with frozen singularity as well as the
5d $SU(3)_{8}$ gauge theory currently having neither a brane web nor a smooth
shrinkable geometric description. The results serve as non-trivial checks for
a large class of non-trivial dualities among 5d theories and also as
independent evidences for the existence of certain exotic theories.

##  1 Introduction

Supersymmetric theories with eight supercharges in five and six dimensions are
a very rich subject that has been investigated over the past decades. Lots of
recent progress in this subject have been made in the classification of 5d and
6d superconformal theories (SCFTs) Heckman:2013pva ; Heckman:2015bfa ;
Bhardwaj:2015xxa  and 6d little string theories (LSTs) Bhardwaj:2015oru . Such
classifications have been carried out based on geometric properties of
F-theory compactified on local elliptic Calabi-Yau (CY) three-folds. Also, a
large class of 5d SCFTs has been classified using gauge theoretic
constructions Seiberg:1996bd ; Intriligator:1997pq ; Jefferson:2017ahm  and
M-theory compactified on local CY 3-folds Douglas:1996xp ; Intriligator:1997pq
; DelZotto:2017pti ; Xie:2017pfl ; Jefferson:2018irk ; Bhardwaj:2018yhy ;
Bhardwaj:2018vuu ; Apruzzi:2018nre ; Apruzzi:2019vpe ; Apruzzi:2019opn ;
Bhardwaj:2019fzv ; Apruzzi:2019kgb ; Apruzzi:2019enx . These higher
dimensional theories turn out to exhibit several fascinating features of
quantum field theories (QFTs) such as the existence of tensionless strings,
dualities, and symmetry enhancements. Moreover, they have played a pivotal
role in constructing and studying lower dimensional QFTs via
compactifications.

Supersymmetric partition functions have provided unexpectedly powerful methods
for exploring such features of higher dimensional field theories. Of
particular interest are the partition functions on $\mathbb{R}^{4}\times
S^{1}$ in 5d and $\mathbb{R}^{4}\times T^{2}$ (or the elliptic genera) in 6d,
which are the Witten indices on the $\Omega$-deformed $\mathbb{R}^{4}$
computing the number of the BPS states weighted by their electric charges and
angular momenta. Throughout the paper, we refer to them as the BPS partition
functions, though it may be an abuse of notation. Protected by the
supersymmetry, the BPS spectra encoded in the partition function can be used
to confirm non-trivial dualities and symmetry enhancements at various points
on the moduli space Bao:2011rc ; Hayashi:2013qwa ; Taki:2014pba ;
Mitev:2014jza ; Hayashi:2014wfa ; Gaiotto:2015una ; Hayashi:2015xla ;
Hayashi:2016abm ; Yun:2016yzw ; Hayashi:2017jze ; Kim:2018gjo ; Chen:2020jla .
Also, the Seiberg-Witten prepotential in the Coulomb phase can be reproduced
from the BPS partition function by taking the limit
$\epsilon_{1},\epsilon_{2}\rightarrow 0$ of two $\Omega$-deformation
parameters Nekrasov:2002qd ; Nekrasov:2003rj . In particular, other
supersymmetric observables such as the superconformal index
Bhattacharya:2008zy ; Kim:2012gu ; Kim:2013nva ; Iqbal:2012xm  and the $S^{5}$
partition function Kallen:2012va ; Kim:2012ava ; Kallen:2012zn ;
Lockhart:2012vp ; Imamura:2013xna ; Kim:2012qf  can be factorized into a
product of several BPS partition functions under localization, which signifies
importance of the partition function on the $\Omega$-background as a building
block for other observables in 5d and 6d.

Various computational tools for the 5d and 6d BPS partition functions have
been developed. First, the partition function on the $\Omega$-background for
5d $\mathcal{N}=1$ gauge theories is closely related to Nekrasov instanton
partition function counting the BPS bound states of instantons with other
charged particles on the Coulomb branch of the moduli space. The Nekrasov
instanton partition functions for gauge theories with the classical gauge
groups are computed through localization based on the ADHM constructions of
the instanton moduli space Atiyah:1978ri ; Nekrasov:2002qd ; Nekrasov:2003rj .
(See also Marino:2004cn ; Nekrasov:2004vw ; Fucito:2004gi ; Hwang:2014uwa  for
various generalizations). The ADHM constructions have also been used to
compute the elliptic genera of the self-dual strings in 6d SCFTs (in the
tensor branch) Haghighat:2013tka ; Kim:2014dza ; Haghighat:2014vxa ;
Gadde:2015tra . Even though the ADHM method is systematic and applicable for
many 5d and 6d gauge theories with classical gauge groups, ADHM construction
for the exceptional gauge groups is however still missing.111The ADHM-like
construction for the moduli space of instantons in the $G_{2}$ gauge theory
with fundamental matters was proposed in Kim:2016foj ; Kim:2018gjo . Moreover,
the ADHM method is not applicable when a gauge theory of classical group is
coupled to a large number of matter fields in generic representations. Hence,
there are still challenges and difficulties in the ADHM method when computing
the BPS partition functions for other more generic gauge theories in 5d and in
6d.

Topological vertex method Aganagic:2003db ; Iqbal:2007ii ; Awata:2008ed 222See
also the Ding-Iohara-Miki (DIM) algebra Ding:1996mq ; doi:10.1063/1.2823979
and its relation to topological vertex Aganagic:2012hs ; Awata_2012 ;
Bourgine:2017jsi . provides yet another systematic way of computing the BPS
partition functions when the 5d/6d theories are realized by Type IIB 5-brane
webs that are toric or toric-like Aharony:1997bh ; Benini:2009gi , by which we
mean those 5-brane webs which can be constructed from toric 5-brane webs
through Higgsing procedures Hayashi:2013qwa . Topological vertex has been
further developed so that it is also applicable for 5-brane webs with
$O5$-planes Kim:2017jqn ; Hayashi:2020hhb  or $ON$-planes Bourgine:2017rik ;
Kim-Wei2020 . Though fairly many 5-brane webs are known or discovered for 5d
theories and for 6d theories on a circle Kim:2015jba ; Hayashi:2015fsa ;
Bergman:2015dpa ; Zafrir:2015rga ; Hayashi:2015zka ; Hayashi:2015vhy ;
Zafrir:2015ftn ; Zafrir:2016jpu ; Hayashi:2018lyv ; Hayashi:2019yxj ;
Kim:2019dqn , there still remain challenges when the number of hypermultiplets
is large or the Chern-Simons (CS) level is high. For instance, 5-brane webs
for the 5d $SU(3)_{8}$ and $SU(4)_{8}$ gauge theories are unknown, and 5-brane
web realizations for gauge theories of exceptional groups are still far from
clear except for $G_{2}$ gauge theories Hayashi:2018bkd ; Kim:2019dqn .

In Nakajima:2003pg , Nakajima-Yoshioka formulated the so-called blowup
equations to compute the Nekrasov instanton partition functions for four-
dimensional $\mathcal{N}=2\ SU(N)$ gauge theories. The K-theoretic blowup
equations were established soon in Nakajima:2005fg ; Gottsche:2006bm  for
five-dimensional $\mathcal{N}=1\ SU(N)$ gauge theories. This blowup equation
approach has been further generalized to compute the instanton partition
functions for other simple gauge groups including exceptional gauge groups in
5d in Keller:2012da ; Kim:2019uqw . Moreover, a geometric generalization of
the blowup equation approach was formulated for certain local Calabi-Yau
3-folds Gu:2017ccq ; Huang:2017mis . This geometric formulation has been
extensively studied recently for computing refined BPS invariants of various
5d SCFTs and elliptic genera of 6d SCFTs admitting geometric constructions in
M-/F-theory on local (toric or elliptic) CY 3-folds Gu:2017ccq ; Huang:2017mis
; Gu:2018gmy ; Gu:2019dan ; Gu:2019pqj ; Gu:2020fem .

Despite the fact that the blowup method is a very powerful tool for computing
BPS spectra of a broad class of 5d/6d SCFTs even beyond the scope of other
computational methods, there is as yet no complete formulation of the blowup
equations that can be applicable for all supersymmetric theories in five and
six dimensions. For example, though the blowup equations obtained in
Kim:2019uqw  cover a large set of 5d gauge theories with simple gauge group
including some theories whose ADHM descriptions are not known, it still
requires further studies for plenty of other interesting theories such as 5d
quiver gauge theories, 5d theories with half-hypermultiplets, and 5d Kaluza-
Klein (KK) theories arising from 6d SCFTs compactified on a circle with outer-
automorphism twists. In particular, there exist some 5d/6d gauge theories, for
instance, the 5d $SU(3)_{8}$ gauge theory, that currently have neither ADHM
constructions nor shrinkable geometric descriptions nor associated 5-brane
webs, so that all the computational methods introduced above including the
blowup method cannot be used to compute their BPS partition functions.

The main aim of this paper is to devise a complete blowup formalism that
enables one to compute BPS spectra of all supersymmetric field theories having
UV completions in five or six dimensions. In this paper, we will generalize
the Nakajima-Yoshioka’s blowup equations in Nakajima:2003pg  to arbitrary
5d/6d gauge theories (including quiver gauge theories and twisted circle
compactifications of 6d theories) with matter fields in arbitrary
representations, and also extend the work of Gu:2017ccq ; Huang:2017mis  for a
geometric application of the blowup equations to any local Calabi-Yau 3-folds
(including elliptic and non-toric ones) based on novel geometric constructions
of 5d SCFTs/KK theories introduced in Jefferson:2018irk ; Bhardwaj:2019fzv .

The blowup equation is a functional equation identifying two partition
functions on different backgrounds, one is the $\Omega$-deformed
$\mathbb{C}^{2}$ and another one is the one point blow-up
$\hat{\mathbb{C}}^{2}$ of the $\mathbb{C}^{2}$, that are related to each other
by a smooth blow-up or blow-down transition. Each 5d/6d field theory can have
a number of blowup equations depending on background magnetic fluxes residing
on the blown-up $\mathbb{P}^{1}$ in $\hat{\mathbb{C}}^{2}$. As we will discuss
in section 3, such blowup equations can be solved recursively by expanding
them in terms of Kähler parameters of the theory.

The main input in our recursion process is the effective prepotential
$\mathcal{E}$ of a given 5d/6d SQFT evaluated on the $\Omega$-background. The
effective prepotential, as we will illustrate more precisely in section 2, is
fully determined by effective cubic and mixed Chern-Simons terms in the low
energy theory on the Coulomb branch which can be systematically calculated by
computing the induced Chern-Simons terms after integrating out charged
fermions Witten:1996qb  (and also by collecting classical Green-Schwarz
contributions for a 6d theory on a circle with/without a twist
Bhardwaj:2019fzv ). See Maldacena:1997de ; Katz:2020ewz  for the geometric
counterparts of the effective Chern-Simons terms. Therefore, the effective
prepotential, which is one of the main ingredients for our blowup formula, can
be systematically computed for every 5d/6d field theory having a UV
completion. Here we assume that every UV finite 5d/6d theory either has a
gauge theory description in 5d or in 6d on a circle with/without twist, or has
a geometric description as a local (elliptic) Calabi-Yau 3-fold, or can be
obtained by RG-flows thereof.

By seeding the effective prepotential as well as a consistent choice of
background magnetic fluxes on the blown-up $\mathbb{P}^{1}$ into the blowup
equations, we can bootstrap the spectrum of charged BPS states in a given
5d/6d theory. Since the effective prepotential can be easily prepared for any
arbitrary 5d/6d field theory, we now make a bold conjecture that we can
compute BPS spectra of all 5d/6d field theories by employing our bootstrap
method to formulate and solve blowup equations in those theories.

As concrete examples for our conjecture, we will apply our method to all
rank-1 and rank-2 5d supersymmetric field theories including also KK theories
to obtain their BPS spectra explicitly. This will involve the
$SU(3)_{15/2}+1{\bf F}$ theory (dual to the $\mathcal{N}=2$ $G_{2}$ gauge
theory), the $SU(3)_{8}$ theory, and new rank-1 and rank-2 5d SCFTs, which we
call the local $\mathbb{P}^{2}+1{\bf Adj}$ and the local
$\mathbb{P}^{2}\cup\mathbb{F}_{3}+1{\bf Sym}$, obtained by mass deformations
of the 5d $\mathcal{N}=2$ $Sp(N)_{\pi}$ gauge theories with $N=1,2$
respectively first introduced in Bhardwaj:2019jtr . We emphasize that the
partition functions of these theories cannot be obtained by other means since
they have none of ADHM constructions, conventional geometric constructions,
and also brane webs (but we will introduce brane webs for the new rank-1 and
rank-2 theories in this paper).

We will also compute BPS spectra of some higher rank theories. In particular,
we will show that the blowup equations for the $SU(5)_{8}$ theory can be
solved. The result shows that this theory may have no physical Coulomb branch
and thus be inconsistent in UV limit. This theory was ‘undetermined’ to exist
in Bhardwaj:2020gyu  because its existence was neither confirmed nor ruled out
with currently known techniques. Our computation provides a supporting
evidence that the $SU(5)_{8}$ theory has no UV completion. In this sense, our
bootstrap approach can be used to confirm or disprove the existence of certain
5d and 6d QFTs.

The organization of the paper is as follows. In section 2, we review salient
features of 5d/6d supersymmetric gauge theories and their geometry
engineering. In section 3, we explain the blowup equation as a tool for
bootstrapping BPS spectra of 5d/6d supersymmetric theories, and discuss our
main conjecture with instructive examples. Section 4 and section 5 are devoted
to cover all rank-1 and rank-2 5d theories including KK theories. We also
discuss some interesting higher rank theories, in section 6, including
$SU(4)_{8}$ and $SU(5)_{8}$. We then conclude with some subtle issues. In
Appendix A, we further discuss 1-loop partition functions of 6d SCFTs on a
circle with twists. In Appendix LABEL:sec:appendix2, some new 5-brane webs
associated with frozen singularity are presented.

##### Notation:

To avoid the cluttering of theories, we denote by
$G+N_{\mathbf{r}}\,\mathbf{r}$, the theory of gauge group $G$ with
$N_{\mathbf{r}}$ number of hypermultiplets in the representation $\mathbf{r}$.
Gauge group $G$ can be any classical groups, $SU(N)$, $SO(N)$, $Sp(N)$, and
exceptional groups $G_{2}$, $F_{4}$, $E_{6}$, $E_{7}$, $E_{8}$ as well as a
quiver gauge group. For hypermultiplets in the $\mathbf{r}$ representation of
$G$, we use the following shorthand notation: $\mathbf{F}$ for fundamental,
$\mathbf{bi\text{-}F}$ for bi-fundamental, $\mathbf{\Lambda}^{n}$ for rank-$n$
antisymmetric, $\mathbf{Sym}$ for symmetric, $\mathbf{Adj}$ for adjoint,
$\mathbf{S}$ for spinor, and $\mathbf{C}$ for conjugate spinor. For example,
$SU(2)+8\mathbf{F}$ means $SU(2)$ gauge theory with 8 hypermultiplets in the
fundamental representation. For the $\Omega$-deformation parameters
$\epsilon_{1},\epsilon_{2}$, we frequently use
$\epsilon_{+}=\frac{\epsilon_{1}+\epsilon_{2}}{2}$ and the fugacities
associated with them are denoted by $p_{1}=e^{-\epsilon_{1}}$ and
$p_{2}=e^{-\epsilon_{2}}$. We also denote the set of complex, real, rational,
integer numbers by $\mathbb{C}$, $\mathbb{R}$, $\mathbb{Q}$, $\mathbb{Z}$,
respectively.

##  2 5d theories on $\Omega$-background

In this section, we review some basic properties of 5d $\mathcal{N}=1$ QFTs
that have UV completions.

###  2.1 Gauge theories and effective prepotential

A large class of 5d SQFTs admits mass deformations that lead to non-Abelian
gauge theory descriptions at low energy. Consider a 5d $\mathcal{N}=1$ gauge
theory with a non-Abelian gauge group $G$. The theory consists of the vector
multiplet $\Phi$ for the gauge group $G$ and charged matter hypermultiplets.
The vector multiplet contains a real scalar field $\phi$ as well as the vector
field $A_{\mu}$. On the Coulomb branch of the moduli space, the scalar field
$\phi$ gets the expectation value in the Cartan subalgebra of the gauge group
$G$. This breaks the gauge group to its Abelian subgroup $U(1)^{r}$, with
$r={\rm rank}(G)$. Then the low-energy theory is described by an effective
theory with the Abelian groups. The scalar expectation values
$\phi^{i},i=1,\cdots,r$ for the Abelian gauge groups parameterize the Coulomb
branch of the moduli space.

The effective Abelian theory is characterized by a prepotential
$\mathcal{F}(\Phi)$ which is a cubic polynomial in the Abelian vector
multiplets $\Phi^{I}$ for both the dynamical gauge symmetry and the non-
dynamical flavor symmetry where the index $I$ labels both the dynamical and
the background vector multiplets. The exact prepotential can be computed by
1-loop calculations. For a general gauge group $G$ and matter hypermultiplets
in generic representations, the cubic prepotential in terms of the scalar
components is given by Witten:1996qb ; Intriligator:1997pq

$\mathcal{F}\\!=\\!\sum_{a}\\!\Big{(}\frac{m_{a}}{2}K^{a}_{ij}\phi^{a}_{i}\phi^{a}_{j}+\frac{\kappa_{a}}{6}d^{a}_{ijk}\phi^{a}_{i}\phi^{a}_{j}\phi^{a}_{k}\Big{)}+\frac{1}{12}\bigg{(}\sum_{e\in{\bf
R}}|e\cdot\phi|^{3}-\\!\sum_{f}\\!\sum_{w\in{\bf
w}_{f}}|w\cdot\phi+m_{f}|^{3}\\!\bigg{)},$ (1)

where $a$ runs over all non-Abelian subgroups $G_{a}\subset G$. Here,
$m_{a}=1/g_{a}^{2}$ is the inverse gauge couplings squared and $\kappa_{a}$ is
the classical Chern-Simons level, which is non-zero only for $G_{a}=SU(N)$
with $N\geq 3$, for the group $G_{a}$. $K^{a}_{ij}={\rm
Tr}(T^{a}_{i}T^{a}_{j})$ is the Killing form of $G_{a}$ and
$d^{a}_{ijk}=\frac{1}{2}{\rm Tr}T^{a}_{i}\\{T^{a}_{j},T^{a}_{k}\\}$ with the
generator $T^{a}_{i}$ in the fundamental representation of $G_{a}$. ${\bf R}$
and ${\bf w}_{f}$ are the roots and the weights for the $f$-th hypermultiplet
with mass $m_{f}$ of $G$, respectively. The mass parameters $m_{a}$ and
$m_{f}$ can be regarded as the scalar components in the background vector
multiplets for the topological symmetries and the flavor symmetries rotating
hypermultiplets respectively. We note from the prepotential that the Coulomb
branch is divided into distinct sub-chambers (or phases) distinguished by the
signs of masses inside absolute values in (1), and accordingly the
prepotential takes different values in the different sub-chambers.

The prepotential $\mathcal{F}$ determines the gauge kinetic terms in the
effective action with the gauge coupling

$\tau_{ij}^{\rm eff}=(g^{-2}_{\rm
eff})_{ij}=\partial_{i}\partial_{j}\mathcal{F}\ ,$ (2)

which sets the metric on the Coulomb branch and the cubic Chern-Simons terms
of the form

$S_{CS}=\frac{C_{IJK}}{24\pi^{2}}\int A^{I}\wedge F^{J}\wedge F^{K}\ ,\quad
C_{IJK}=\partial_{I}\partial_{J}\partial_{K}\mathcal{F}\ ,$ (3)

with the level $C_{IJK}$ quantized as $C_{IJK}\in\mathbb{Z}$ due to gauge
invariance of the Abelian symmetries Witten:1996qb .

Other topological terms in the effective action are also important in our
discussion later on the blowup equations. First, the effective action contains
the mixed gauge/gravitational Chern-Simons terms of the form, Bonetti:2011mw ;
Bonetti:2013ela ; Grimm:2015zea

$S_{\rm grav}=-\frac{1}{48}\int C^{G}_{i}A^{i}\wedge p_{1}(T)\ ,$ (4)

where $p_{1}(T)$ is the first Pontryagin class of the tangent bundle on the 5d
spacetime. Here $C_{i}^{G}$ is the level for the mixed Chern-Simons term and
it is quantized as $C_{i}^{G}\in\mathbb{Z}$ Chang:2019uag ; Katz:2020ewz . The
mixed Chern-Simons terms are induced at low energy by integrating out the
charged fermions. The induced level from the fermion 1-loop calculations is
Bonetti:2013ela ; Grimm:2015zea

$C_{i}^{G}=-\partial_{i}\bigg{(}\sum_{e\in{\bf
R}}|e\cdot\phi|-\sum_{f}\sum_{w\in{\bf w}_{f}}|w\cdot\phi+m_{f}|\bigg{)}\ .$
(5)

There also exists the mixed gauge/$SU(2)_{R}$ Chern-Simons terms of the form,

$S_{R}=\frac{1}{2}\int C_{i}^{R}A^{i}\wedge c_{2}(R)\ ,$ (6)

where $c_{2}(R)$ is the second Chern class of the $SU(2)_{R}$ R-symmetry
bundle. Due to the gauge invariance, the level $C^{R}_{i}$ is quantized as
$C^{R}_{i}\in 2\mathbb{Z}$. Note that the gauginos in the vector multiplets
are doublets, while the matter fermions are singlets under the $SU(2)_{R}$
R-symmetry. Thus this term receives 1-loop contributions only from the charged
gauginos and therefore it is independent of the number of hypermultiplets. The
mixed gauge/$SU(2)_{R}$ Chern-Simons level induced from the gaugino 1-loop
calculation is BenettiGenolini:2019zth

$\displaystyle C^{R}_{i}$ $\displaystyle=\frac{1}{2}\partial_{i}\sum_{e\in{\bf
R}}|e\cdot\phi|\ .$ (7)

We remark here that, in the Dynkin basis where the rows of the Cartan matrix
$A_{ij}$ for a gauge group $G$ are given by the simple roots, this level in
the low-energy effective theory is fixed to be $C^{R}_{i}=2$ for all $i$’s.

In this paper, we are interested in the partition functions of 5d
$\mathcal{N}=1$ theories on $\Omega$-deformed $\mathbb{R}^{4}\times S^{1}$.
This partition function is a Witten index counting BPS states in the 5d
theory, which is defined as333We can also define another Witten index as
$\hat{Z}(\phi,m;\epsilon_{1},\epsilon_{2})\equiv
Z(\phi,m;\epsilon_{1},\epsilon_{2})|_{(-1)^{F}\rightarrow(-1)^{2J_{R}}}$ by
replacing $(-1)^{F}$ in $Z$ by $(-1)^{2J_{R}}$. This index will be used later
when a 5d theory is put on the blowup $\hat{\mathbb{C}}^{2}$. Nekrasov:2002qd

$Z(\phi,m;\epsilon_{1},\epsilon_{2})={\rm
Tr}\left[(-1)^{F}e^{-\beta\\{Q,Q^{\dagger}\\}}e^{-\epsilon_{1}(J_{1}+J_{R})}e^{-\epsilon_{2}(J_{2}+J_{R})}e^{-\phi\cdot\Pi}e^{-m\cdot
H}\right]\ ,$ (8)

where $J_{1},J_{2}$ are the Cartan generators of the $SO(4)$ Lorentz rotation
and $J_{R}$ is the Cartan of the $SU(2)_{R}$ R-symmetry, and $\Pi$ and $H$ are
the gauge and the flavor charges respectively. $Q$ is a supercharge commuting
with $J_{1}+J_{R}$ and $J_{2}+J_{R}$, and $Q^{\dagger}$ is its conjugate.
$\beta$ is the radius of $S^{1}$ and $\epsilon_{1},\epsilon_{2}$ are the
$\Omega$-deformation parameters. We denote by $\phi$ and $m$ the chemical
potentials for the gauge and the flavor symmetries, respectively. The index
computes the BPS spectrum annihilated by the supercharges $Q$ and
$Q^{\dagger}$. So the index is independent of $\beta$.

The index can be represented by a path integral of the 5d theory on the
$\Omega$-background, which can be evaluated using localization Kim:2011mv ;
Kim:2012gu ; Hwang:2014uwa . We will call this path integral representation of
the Witten index the partition function on the $\Omega$-background or just BPS
partition function.

We compute the path integral on a vacuum on the Coulomb branch specified by
the expectation values $\phi$, which are now complexified by combining the
scalar vevs with the gauge holonomies around $S^{1}$ at infinity of
$\mathbb{R}^{4}$. The chemical potential $\phi$ in the above index is
identified with the complexified expectation value $\phi$ in the path
integral. Similarly, we identify the chemical potential $m$ with the
complexified background gauge field for a flavor symmetry. In the following
discussions, however, we shall take the chemical potentials $\phi$ and $m$ to
be pure real values.

In the localization, the BPS partition function receives perturbative and non-
perturbative instanton contributions which factorizes as

$Z=Z_{\rm pert}\cdot Z_{\rm inst}\ .$ (9)

The perturbative partition function $Z_{\rm pert}$ consists of the classical
action contribution and the 1-loop contribution. Actually, it depends on the
boundary condition at infinity of $\mathbb{R}^{4}$. We need to consider
boundary conditions preserving two supercharges $Q,Q^{\dagger}$ and being
compatible with the vacuum on the Coulomb branch. We shall choose the
following boundary condition at infinity: the vector multiplets associated
with the positive roots of gauge group $G$ survives, and the chiral halves of
hypermultiplets with positive masses, i.e., $w\cdot\phi+m_{f}>0$ survive. With
this boundary condition, the perturbative partition function can be written as

$\displaystyle Z_{\rm pert}$ $\displaystyle=Z_{\rm class}\cdot
Z_{\text{1-loop}}$ (10) $\displaystyle=e^{\mathcal{E}}\cdot{\rm
PE}\bigg{[}-\frac{1+p_{1}p_{2}}{(1-p_{1})(1-p_{2})}\\!\sum_{e\in{\bf
R}^{+}}\\!e^{-e\cdot\phi}+\frac{(p_{1}p_{2})^{1/2}}{(1-p_{1})(1-p_{2})}\sum_{f}\\!\sum_{w\in{\bf
w}_{f}}e^{-|w\cdot\phi+m_{f}|}\bigg{]},$

where $p_{1,2}\equiv e^{-\epsilon_{1,2}}$ and ${\bf R}^{+}$ denotes the
positive roots for the gauge group, and PE means the Plethystic exponential of
a letter index $f(\mu)$ with a chemical potential $\mu$ defined as

$\operatorname{PE}\left[f(\mu)\right]\equiv\exp\bigg(\sum_{n=1}^{\infty}\frac{1}{n}f(n\mu)\bigg{missing})\
.$ (11)

In (10), $\mathcal{E}=\mathcal{E}(\phi,m;\epsilon_{1},\epsilon_{2})$ in the
prefactor is a combination of the classical contribution and the Casimir
energy contribution coming from regularization of infinite products in the
1-loop part. In fact, $\mathcal{E}$ is effective prepotential, which includes
the cubic and mixed Chern-Simons terms (3), (4), (6), and their SUSY
completions, evaluated on the $\Omega$-background. We find, for the boundary
condition we chose above,

$\displaystyle\mathcal{E}(\phi,m;\epsilon_{1},\epsilon_{2})$
$\displaystyle=i\left(S_{CS}+S_{\rm
grav}+S_{R}+\cdots\right)|_{\phi,m,\epsilon_{1},\epsilon_{2}}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\left[\mathcal{F}+\frac{1}{48}C_{i}^{G}\phi^{i}(\epsilon_{1}^{2}+\epsilon_{2}^{2})+\frac{1}{2}C^{R}_{i}\phi^{i}\epsilon_{+}^{2}\right]\
,$ (12)

where $\cdots$ denotes the SUSY completions of the Chern-Simons terms and
$\epsilon_{+}\equiv\frac{\epsilon_{1}+\epsilon_{2}}{2}$. The first term
$\mathcal{F}$ in the bracket is the cubic prepotential (1) on the Coulomb
branch, and the other two terms are the contributions from the mixed
gauge/gravitational CS terms and the mixed gauge/$SU(2)_{R}$ CS terms,
respectively. This factor $\mathcal{E}$ can also be considered as an
equivariant integral of effective Chern-Simons terms by making the
replacements

$p_{1}(T)\rightarrow-(\epsilon_{1}^{2}+\epsilon_{2}^{2})\ ,\qquad
c_{2}(R)\rightarrow\epsilon_{+}^{2}\ ,$ (13)

with the equivariant parameters $\epsilon_{1,2}$ and $\phi,m$. A similar
interpretation for the Casimir energy of superconformal indices has been
proposed in Bobev:2015kza .

The instanton contribution $Z_{\rm inst}$ is in general given by a power
series expansion by the instanton numbers $k_{a}$ for each non-Abelian gauge
group factor. It can thus be written as

$Z_{\rm inst}=\sum_{k_{a}=0}^{\infty}\prod_{a}q_{a}^{k_{a}}Z_{\\{k_{a}\\}}\ ,$
(14)

where $q_{a}=e^{-m{}_{a}}$ is the instanton fugacity for the $a$-th gauge
group, and $Z_{\\{k_{a}\\}}$ denotes the path integral over instanton moduli
space with instanton numbers $\\{k_{a}\\}$. When a UV completion for the
instanton moduli space is known, for example by using ADHM construction (see
Nekrasov:2002qd ; Nekrasov:2003rj ; Nekrasov:2004vw ; Shadchin:2005mx  for
some early works), we can use it to compute the instanton partition function
$Z_{\\{k_{a}\\}}$ by localization. However, unfortunately, such ADHM
constructions for general gauge group $G$ and matter representations are yet
unknown.

Recently, there has been some progress on computation of the instanton
partition functions for more general gauge groups by using Nakajima-Yoshioka’s
blowup equations Nakajima:2003pg . See also Huang:2017mis ; Gu:2018gmy ;
Kim:2019uqw  and the references therein. Still, this method is applicable only
for very limited cases. In this paper, we will propose a new and simple
strategy to compute the instanton partition functions for arbitrary gauge
groups and matter representations based on the blowup formula. We expect that
our strategy can be applied to all the 5d gauge theories that have UV
completions as 5d SCFTs or 5d KK theories coming from 6d $\mathcal{N}=(1,0)$
theories on a circle with/without twists.

###  2.2 Geometric engineering

Many examples of 5d $\mathcal{N}=1$ theories have been engineered in M-theory
on local Calabi-Yau threefolds Morrison:1996xf ; Douglas:1996xp ;
Intriligator:1997pq . In this subsection we review some basic features of
M-theory compactification on a smooth non-compact 3-fold $X$ which gives rise
to a 5d SCFT or a 6d SCFT on $S^{1}$ (possibly with twists) in a singular
limit. See Jefferson:2018irk ; Bhardwaj:2019fzv  for more details.

A smooth 3-fold $X$ can be locally described as a neighborhood of a collection
of Kähler surfaces $S_{i}$. A Kähler surface $S_{i}$ inside $X$ is represented
by either a local $\mathbb{P}^{2}$ or a Hirzebruch surface with blowups
$\mathbb{F}_{n}^{b}$ where $n$ is the degree of the Hirzebruch surface and $b$
is the number of blowups. More explanation on the local $\mathbb{P}^{2}$ and
Hirzebruch surfaces can be found in Appendix A of Bhardwaj:2019fzv .

The volumes of complex $p$-cycles in $X$ are controlled by Kähler
deformations. There are normalizable Kähler deformations parameterized by
dynamical Kähler parameters $\phi_{i}$ assigned for each compact surface
$S_{i}$. The number $r$ of independent compact surfaces is the rank of the CY
3-fold $X$. Upon M-theory compactification, the Kähler moduli space of the
dynamical parameters $\phi_{i=1,\cdots,r}$ is identified with the Coulomb
branch moduli in the low-energy 5d theory, where $r$ is the rank of the gauge
group in the field theory.

There are also non-compact Kähler deformations parameterized by non-dynamical
Kähler parameters $m_{j=1,\cdots,r_{F}}$ where $r_{F}=h^{1,1}(X)-r$. These
non-dynamical parameters are identified with the mass parameters in the 5d
theory. For a given basis $S_{i},N_{j}\in H^{1,1}(X)$, one can then express
the Kähler form $J$ of $X$ as a linear sum of the Kähler parameters

$J=\sum_{I=1}^{h^{1,1}(X)}\phi_{I}D_{I}=\sum_{i=1}^{r}\phi_{i}S_{i}+\sum_{j=1}^{r_{F}}m_{j}N_{j}\
,$ (15)

where $D_{I=1,\cdots,r}=S_{i=1,\cdots,r}$ and
$D_{I=r+1,\cdots,h^{1,1}(X)}=N_{j=1,\cdots,r_{F}}$ are the divisors for the
compact and the non-compact 4-cycles inside $X$, respectively. In particular,
the Kähler parameters $\phi_{i}$ for elementary surfaces $S_{i}$ in this
geometric basis are directly mapped to the Coulomb branch parameters
$\phi_{i}$ in the Dynkin basis of the associated gauge group $G$. We will only
use the Dynkin basis for gauge groups in the following discussions.

The volumes of $p$-cycles in $X$ are measured with respect to the Kähler form
$J$. The total volume of the 3-fold $X$ is

${\rm vol}(X)=\frac{1}{3!}\int_{X}J^{3}\ .$ (16)

This is identified with the 5d cubic prepotential given in (1), i.e.
$\mathcal{F}={\rm vol}(X)$. Therefore the cubic Chern-Simons coefficients in
the 5d theory are geometrically determined by the triple intersections of
divisors in $X$ Cadavid:1995bk ; Ferrara:1996hh ; Witten:1996qb ;
Ferrara:1996wv  (See also Grimm:2011fx ; Bonetti:2011mw ; Bonetti:2013ela )

$C_{IJK}=D_{I}\cdot D_{J}\cdot D_{K}\equiv\int_{X}D_{I}\wedge D_{J}\wedge
D_{K}\ .$ (17)

The low-energy effective action in the 5d theory is also characterized by the
coefficients of the mixed Chern-Simons terms. The mixed gauge/gravitational
Chern-Simons level for a divisor $S_{i}$ is determined by its intersection
with $c_{2}(X)\in H^{4}(X,\mathbb{Z})$, the 2nd Chern-class of the 3-fold $X$,
as Cadavid:1995bk ; Ferrara:1996hh ; Bonetti:2011mw ; Bonetti:2013ela ;
Katz:2020ewz

$C_{i}^{G}=c_{2}(X)\cdot S_{i}\ .$ (18)

For a local $\mathbb{P}^{2}$ and a Hirzebruch surface $\mathbb{F}_{n}^{b}$
with $b$ blowups,

$\displaystyle c_{2}(X)\cdot\mathbb{P}^{2}=-6,\qquad
c_{2}(X)\cdot\mathbb{F}_{n}^{b}=-4+2b\,.$ (19)

In addition, we propose that the level $C^{R}_{i}$ of the mixed
gauge/$SU(2)_{R}$ Chern-Simons term is always

$C_{i}^{R}=2\ ,$ (20)

for all the basis surfaces $S_{i}$ represented by either a local
$\mathbb{P}^{2}$ or a Hirzebruch surface with blowups, in a Calabi-Yau 3-fold.
This is motivated by the field theory result in (7); the level $C^{R}_{i}$ is
always $2$ in the Dynkin basis of the gauge groups.

Therefore the cubic and the mixed Chern-Simons terms in the low-energy
effective field theory are fully expressed in terms of the topological data in
the CY $3$-fold $X$. The effective prepotential $\mathcal{E}$ in (2.1) for a
5d field theory can then be readily computed from the associated 3-fold $X$.
As we will see below, this effective prepotential $\mathcal{E}$ together with
few more information about primitive 2-cycles in a CY $3$-fold allows us to
compute the BPS partition function of the corresponding 5d theory by solving
the blowup equations.

The BPS spectrum in the 5d SQFT involves electric particles and (dual)
magnetic strings charged under the gauge groups. In the M-theory
compactification on $X$, these states arise from M2-branes and M5-branes
wrapping holomorphic curves and holomorphic surfaces respectively. Their
masses and tensions are determined by the volumes of the corresponding
$p$-cycles. The volume of a 2-cycle (or a curve) $C$ is

${\rm vol}(C)=-J\cdot C\ ,$ (21)

and the volume of a 4-cycle $S_{i}$ is given by

${\rm vol}(S_{i})=J\cdot J\cdot
S_{i}\equiv\partial_{i}\mathcal{F}=\frac{1}{2}\int_{X}J^{2}\wedge S_{i}\ .$
(22)

The Kähler surfaces are glued to each other by identifying a pair (or multiple
pairs) of holomorphic curves at the intersections as

$C_{ij}^{\alpha}\sim C_{ji}^{\alpha}\ ,$ (23)

where $C_{ij}^{\alpha}$ is a curve in $S_{i}$ and $C_{ji}^{\alpha}$ is a curve
in $S_{j}$ at the intersection of two adjacent surfaces $S_{i}\cap S_{j}$, and
$\alpha$ labels a pair of gluing curves. In order to be consistent with the
Calabi-Yau structure of $X$, a pair of gluing curves should satisfy the
condition

$(C^{\alpha}_{ij})^{2}+(C^{\alpha}_{ji})^{2}=2g-2\ ,$ (24)

where $g$ is the genus of curve $C_{ij}^{\alpha}$ and $C_{ji}^{\alpha}$.

It is also possible that two curves in a single surface are glued together
while satisfying the Calabi-Yau condition (24). Such gluing is often called
self-gluing Jefferson:2018irk . A surface can have multiple self-glued curves.
With $s$ self-gluings, the canonical class $K_{S}$ of a surface $S$ changes to

$K_{S}^{\prime}=K_{S}+\sum_{i=1}^{s}(x_{i}+y_{i})\ ,$ (25)

where $(x_{i},y_{i})$ is $i$-th pair of self-glued curves.

The volume of a 2-cycle can in fact be written in terms of its intersection
with the canonical classes $K^{\prime}_{S}$ as

${\rm vol}(C)=-J\cdot C=-\sum_{I}\phi_{I}D_{I}\cdot
C=-\sum_{I}\phi_{I}K_{D_{I}}^{\prime}\cdot C\ .$ (26)

Also, the genus $g$ of a curve $C$ in a surface $S$ can be determined by the
modified adjunction formula Bhardwaj:2019fzv ; Bhardwaj:2020gyu :

$C\cdot(K_{S}+C)+\sum_{i=1}^{s}{\rm min}(C\cdot x_{i},C\cdot y_{i})=2g-2\ ,$
(27)

where $K_{S}$ is the original canonical divisor class of $S$.

We can now easily compute the triple intersection product $S_{i}\cdot
S_{j}\cdot S_{k}$ among three surfaces in a 3-fold $X$. For three distinct
surfaces, their intersection product is given by

$S_{i}\cdot S_{j}\cdot S_{k}=C_{ij}\cdot C_{ik}=C_{ji}\cdot C_{jk}=C_{ki}\cdot
C_{kj}\ \quad\ {\rm for}\ i\neq j\neq k\ .$ (28)

The triple intersection product of two distinct surfaces is

$S_{i}\cdot S_{i}\cdot S_{j}=K^{\prime}_{i}\cdot C_{ij}\ \quad\ {\rm for}\
i\neq j\ .$ (29)

Lastly, the triple intersection of a single surface $S_{i}$ is given by

$S_{i}^{3}=K_{i}^{\prime 2}\ .$ (30)

It is now straightforward to obtain the full effective prepotential
$\mathcal{E}$ on the $\Omega$-background for the low-energy theory from a
geometric construction.

###  2.3 6d SCFTs on $S^{1}$ with/without twists

Compactification of 6d SCFTs on a circle with/without outer automorphism
twists provides concrete UV completions of a large class of 5d field theories.
These 5d theories are often called 5d Kaluza-Klein (KK) theories. The
effective prepotential of such a 5d theory can be easily obtained from the 6d
classical action on the tensor branch and the action of outer automorphism as
well as matter content. The detailed procedure has been introduced in
Bhardwaj:2019fzv . We will now generalize this and propose full effective
prepotentials for 5d KK theories on the $\Omega$-background including the
contributions from mixed gauge/gravitational and gauge/$SU(2)_{R}$ Chern-
Simons terms.

##### 5d reductions without twist

Let us first consider the compactification of a 6d SCFT without a twist on its
tensor branch. We shall also consider a generic point on the Coulomb branch of
the resulting 5d theory where both tensor scalar fields and gauge holonomies
are turned on. The effective prepotential of the 5d theory can be written in
terms of the Chern-Simons coefficients which can be exactly calculable from
the classical action of the original 6d SCFT and 1-loop computations for
charged fermions.

The 6d tree-level action on the tensor branch is given by

$S_{\rm
tree}=\int-\frac{1}{4}\Omega^{\alpha\beta}G_{\alpha}\wedge*G_{\beta}-\Omega^{\alpha\beta}B_{\alpha}\wedge
X_{4\beta}\ ,$ (31)

and supersymmetric completions, where $\Omega^{\alpha\beta}$ is the negative-
definite, symmetric bilinear form of $T$ tensor fields and $G_{\alpha}$ is the
gauge-invariant field strength for the 2-form tensor field $B_{\alpha}$. The
second term in (31) is the Green-Schwarz term with the 4-form $X_{4\beta}$
defined as

$dG_{\alpha}=X_{4\alpha}\ ,\qquad
X_{4\alpha}=-\frac{1}{4}a_{\alpha}p_{1}(T)+\frac{1}{4}\sum_{a}b_{a,\alpha}{\rm
Tr}F_{a}^{2}+c_{\alpha}c_{2}(R)\ ,$ (32)

where $a_{\alpha},b_{a,\alpha},c_{\alpha}$ are fixed to cancel 1-loop gauge
anomalies via the Green-Schwarz-Sagnotti mechanism Sagnotti:1992qw , the
summation index $a$ runs over all the gauge and the flavor groups, and $F_{a}$
is the field strength for the $a$-th symmetry group. The circle reduction of
the classical action then gives rise to the following terms

$S_{\rm
tree}=\int-\frac{\tau}{4}\Omega^{\alpha\beta}F_{\alpha}\wedge*F_{\beta}-\Omega^{\alpha\beta}A_{0,\alpha}\wedge
X_{4\beta}+\cdots\ ,$ (33)

where $\tau\equiv 1/R$ is the inverse radius of the 6d circle and $F_{\alpha}$
denotes the $U(1)_{\alpha}$ field strength for the gauge field $A_{0,\alpha}$
obtained by reducing the tensor field $B_{\alpha}$ on the circle.

In order to compute the matter contribution to the effective Chern-Simons
terms, we need to perform fermion 1-loop computations including Kaluza-Klein
momentum states Bonetti:2011mw ; Bonetti:2013ela ; Grimm:2015zea . The cubic
Chern-Simons terms are captured by the 1-loop prepotential which is given by

$\mathcal{F}_{\text{1-loop}}=\frac{1}{12}\sum_{n\in\mathbb{Z}}\left(\sum_{e\in{\bf
R}}|n\tau+e\cdot\phi|^{3}-\sum_{f}\sum_{w\in{\bf
w}_{f}}|n\tau+w\cdot\phi+m_{f}|^{3}\right)\ ,$ (34)

where the sum for an integer $n$ is performed over all KK charges $n$, ${\bf
R}$ means collectively the roots of the 6d gauge groups, ${\bf w}_{f}$ is the
weight and $m_{f}$ is the mass parameter for the $f$-th hypermultiplet in 6d.
$\phi$’s are the gauge holonomies that become the scalar fields in the 5d
vector multiplets. The infinite sums in the prepotential can be regularized
using the zeta function regularization. In the 5d limit where
$\tau\gg\phi_{i},m_{f}$, the regularized cubic prepotential is given by
Bonetti:2013ela ; Grimm:2015zea

$\displaystyle\mathcal{F}_{\text{1-loop}}$
$\displaystyle=\frac{1}{12}\left(\sum_{e\in{\bf
R}}|e\cdot\phi|^{3}-\sum_{f}\sum_{w\in{\bf
w}_{f}}|w\cdot\phi+m_{f}|^{3}\right)$ $\displaystyle\ \ \ \
-\frac{\tau}{24}\left(\sum_{e\in{\bf
R}}(e\cdot\phi)^{2}-\sum_{f}\sum_{w\in{\bf
w}_{f}}(w\cdot\phi+m_{f})^{2}\right)\ .$ (35)

The first line comes from the zero KK momentum modes and the second line is
the contribution from the KK momentum states after the regularization. We
omitted the terms independent of the dynamical parameters $\phi$.

The mixed gauge/gravitational and gauge/$SU(2)_{R}$ Chern-Simons terms can be
computed in a similar manner. Note here that the contributions from the
positive and the negative KK momentum states cancel each other. So these
Chern-Simons terms receive the contribution only from the zero modes on a
circle. Therefore, the coefficients for the mixed Chern-Simons terms are,
respectively, Bonetti:2013ela ; Grimm:2015zea

$\displaystyle C^{G}_{i}=$ $\displaystyle-\partial_{i}\left(\sum_{e\in{\bf
R}}|e\cdot\phi|-\sum_{f}\sum_{w\in{\bf w}_{f}}|w\cdot\phi+m_{f}|\right)\ ,$
(36) $\displaystyle C^{R}_{i}=$
$\displaystyle~{}\frac{1}{2}\partial_{i}\sum_{e\in{\bf R}}|e\cdot\phi|\ .$
(37)

Here, $e$ and $w$ run over all the zero modes of the vectors and the hypers,
respectively, from the 6d theory.

The effective action of a 5d KK theory on the Coulomb branch can be obtained
by collecting the above tree-level and the 1-loop Chern-Simons terms. On the
$\Omega$-background, the full effective action is then given by

$\displaystyle\mathcal{E}_{6d}$ $\displaystyle=i(S_{\rm
tree}+S_{\text{1-loop}}+S_{\rm grav}+S_{R})|_{\epsilon_{1,2},\phi,m,\tau}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\left(\mathcal{E}_{\rm
tree}+\mathcal{F}_{\text{1-loop}}+\frac{1}{48}C^{G}_{i}\phi^{i}(\epsilon_{1}^{2}+\epsilon_{2}^{2})+\frac{1}{2}C_{i}^{R}\phi^{i}\epsilon_{+}^{2}\right)\
,$ (38) $\displaystyle\mathcal{E}_{\rm tree}$ $\displaystyle\equiv iS_{\rm
tree}|_{\epsilon_{1,2},\phi,m,\tau}$ (39)
$\displaystyle=-\frac{\tau}{2}\Omega^{\alpha\beta}\phi_{\alpha,0}\phi_{\beta,0}-\Omega^{\alpha\beta}\phi_{\alpha,0}\left(\frac{a_{\beta}}{4}(\epsilon_{1}^{2}+\epsilon_{2}^{2})+\frac{b_{a,\beta}}{2}K_{a,ij}\phi_{a,i}\phi_{a,j}+c_{\beta}\epsilon_{+}^{2}\right)\
,$

where $K_{a,ij}$ is the Killing form for the $a$-th symmetry group $G_{a}$. If
$G_{a}$ is a gauge group, then $\phi_{a,i}$ is the Coulomb branch parameter
for it, otherwise, it is the mass parameter for the flavor symmetry.
Consequently, we can compute the effective prepotentials on the Coulomb branch
evaluated on the $\Omega$-background directly from the knowledge of the 6d
SCFTs.

As discussed in Bhardwaj:2019fzv , when we compare this effective prepotential
of the 5d KK theory with geometry, in which $b_{a,\beta}=\delta_{a,\beta}$ for
gauge groups $G_{a}$, we need to shift the Coulomb branch parameters
$\phi_{\alpha,i}$ as

$\phi_{\alpha,i}\rightarrow\phi_{\alpha,i}-d_{i}^{\vee}\phi_{\alpha,0}\ ,$
(40)

for all $1\leq i\leq r_{\alpha}$ as well as for all $\alpha$, where
$d_{i}^{\vee}$ is the dual Coxeter label for the gauge group $G_{\alpha}$.
After this shift, the tensor parameter $\phi_{\alpha,0}$ becomes the Kähler
parameter for the affine node of the associated affine gauge algebra
$\hat{\mathfrak{g}}_{\alpha}$.

We claim that the function $\mathcal{E}_{6d}$ in (2.3) with the shift (40) is
the full effective prepotential on the Coulomb branch in the 5d reduction of
the 6d SCFT without a twist. This contains all the terms of the dynamical
Coulomb branch parameters that do not vanish on the $\Omega$-background.

As an example, consider the 6d minimal $SU(3)$ gauge theory consists of a
tensor multiplet (so $\alpha=1$) coupled to an $SU(3)$ vector multiplet. The
1-loop anomalies are cancelled by the Green-Schwarz-Sagnotti mechanism with
the data

$\Omega^{11}=-3\ ,\quad a_{1}=-\frac{1}{3}\ ,\quad b_{1}=1\ ,\quad c_{1}=1\ .$
(41)

Thus, the effective prepotential on the $\Omega$-background before the shift
(40) is given by

$\displaystyle\epsilon_{1}\epsilon_{2}\,\mathcal{E}=$
$\displaystyle~{}\frac{3}{2}\,\tau\phi_{0}^{2}+3\phi_{0}\Big{(}-\frac{1}{12}(\epsilon_{1}^{2}+\epsilon_{2}^{2})+\frac{1}{2}K_{ij}\phi_{i}\phi_{j}+\epsilon_{+}^{2}\Big{)}$
(42)
$\displaystyle+\frac{1}{6}\Big{(}8\phi_{1}^{3}+8\phi_{2}^{3}-3\phi_{1}\phi_{2}(\phi_{1}+\phi_{2})\Big{)}-\frac{\tau}{4}K_{ij}\phi_{i}\phi_{j}$
(43)
$\displaystyle-\frac{1}{12}(\phi_{1}+\phi_{2})(\epsilon_{1}^{2}+\epsilon_{2}^{2})+(\phi_{1}+\phi_{2})\epsilon_{+}^{2}\
,$ (44)

where $K_{ij}$ is the Killing form of the $SU(3)$ group. To compare with its
geometric construction, we first shift $\phi_{0}\rightarrow\phi_{0}+\tau/6$
and then perform the shift (40), $\phi_{1,2}\rightarrow\phi_{1,2}-\phi_{0}$.
The result is then

$\displaystyle\epsilon_{1}\epsilon_{2}\,\mathcal{E}^{\prime}=$
$\displaystyle~{}\frac{1}{6}\bigg{(}9\tau\phi_{0}^{2}+3\tau^{2}\phi_{0}+8(\phi_{0}^{3}+\phi_{1}^{3}+\phi_{2}^{3})$
(45)
$\displaystyle\quad-3\phi_{0}^{2}(\phi_{1}+\phi_{2})-3\phi_{1}^{2}(\phi_{0}+\phi_{2})-3\phi_{2}^{2}(\phi_{0}+\phi_{1})-6\phi_{0}\phi_{1}\phi_{2}\bigg{)}$
$\displaystyle-\frac{1}{12}(\phi_{0}+\phi_{1}+\phi_{2})(\epsilon_{1}^{2}+\epsilon_{2}^{2})+(\phi_{0}+\phi_{1}+\phi_{2})\epsilon_{+}^{2}\
.$ (46)

This is in perfect agreement with the effective prepotential for the geometry
for the minimal $SU(3)$ gauge theory that consists of three $\mathbb{F}_{1}$
surfaces glued along their $-1$ curves and one non-compact surface with Kähler
parameter ‘$-\tau$’ glued to the $\mathbb{F}_{1}$ surface for $\phi_{0}$ along
the base $+1$ curve as constructed in DelZotto:2017pti .

##### 5d reductions with twist

When a 6d SCFT has a discrete global symmetry $\Gamma$, we can compactify the
theory on a circle with a discrete holonomy $\gamma$ for the background gauge
field of the symmetry $\Gamma$. This is often called automorphism twist on
$\Gamma$ around the circle. Such twists for 6d SCFTs are well described in
section 3 of Bhardwaj:2019fzv . We will employ this prescription to compute
the effective prepotentials and the partition functions of 6d SCFTs
compactified on a circle with twists.

There are two kinds of discrete symmetries in 6d SCFTs. The first kind is the
symmetry arising from outer automorphism of gauge algebra $\mathfrak{g}$ which
permutes matter representations of the gauge algebra. The second one is the
symmetry from permutations of tensor fields. A general discrete symmetry is
generated by a combination of these two kinds of discrete symmetries.

$\mathfrak{\hat{g}}$ | $A_{2\ell}^{(2)}$ | $A_{2\ell-1}^{(2)}$ | $D^{(2)}_{\ell+1}$ | $E_{6}^{(2)}$ | $D_{4}^{(3)}$
---|---|---|---|---|---
$\mathfrak{h}$ | $C_{\ell}$ | $C_{\ell}$ | $B_{\ell}$ | $F_{4}$ | $G_{2}$
Table 1: Twisted affine Lie algebra $\mathfrak{\hat{g}}$ and invariant
subalgebra $\mathfrak{h}$ under outer automorphism. The superscript (p)
denotes order $p=2,3$ of twist.

The twist of the first kind on a gauge algebra $\mathfrak{g}$ splits
representations of $\mathfrak{g}$ into representations of the invariant
subalgebra $\mathfrak{h}$, listed in Table 1, under the outer automorphism.
(If the twist does not act on a gauge algebra, then the invariant subalgebra
is the same as the original gauge algebra, i.e. $\mathfrak{h}=\mathfrak{g}$.)
A 6d state in a representation of $\mathfrak{g}$ is then decomposed into
several representations of the subalgebra $\mathfrak{h}$. The decomposed
states will carry shifted KK momentum as $n\rightarrow n+r$ depending on their
charge under the discrete symmetry, where $n\in\mathbb{Z}$ and $r$ is a
fractional KK charge. See Appendix A for more details.

For example, a 5d KK state from twisting a 6d state in the adjoint
representation of the original gauge algebra will carry the following charges
Kac:1990gs

$\displaystyle A_{2\ell}^{(2)}\ \ $ $\displaystyle:\ \ {\bf Adj}\ {\rm of}\
A_{2\ell}\ \rightarrow\ {\bf Adj}_{0}\oplus{\bf F}_{1/4}\oplus{\bf
F}_{3/4}\oplus{\bf\Lambda}^{2}_{1/2}\oplus{\bf 1}_{1/2}\ \ {\rm of}\ C_{\ell}\
,$ $\displaystyle A_{2\ell-1}^{(2)}\ \ $ $\displaystyle:\ \ {\bf Adj}\ {\rm
of}\ A_{2\ell-1}\ \rightarrow\ {\bf Adj}_{0}\oplus{\bf\Lambda}^{2}_{1/2}\ \
{\rm of}\ C_{\ell}\ ,$ $\displaystyle D_{\ell+1}^{(2)}\ \ $ $\displaystyle:\ \
{\bf Adj}\ {\rm of}\ D_{\ell+1}\ \rightarrow\ {\bf Adj}_{0}\oplus{\bf
F}_{1/2}\ \ {\rm of}\ B_{\ell}\ ,$ $\displaystyle E_{6}^{(2)}\ \ $
$\displaystyle:\ \ {\bf Adj}\ {\rm of}\ E_{6}\ \rightarrow\ {\bf
Adj}_{0}\oplus{\bf F}_{1/2}\ \ {\rm of}\ F_{4}\ ,$ $\displaystyle D_{4}^{(3)}\
\ $ $\displaystyle:\ \ {\bf Adj}\ {\rm of}\ D_{4}\ \rightarrow\ {\bf
Adj}_{0}\oplus{\bf F}_{1/3}\oplus{\bf F}_{2/3}\ \ {\rm of}\ G_{2}\ ,$ (47)

where $\mathbf{Adj}_{r}$, $\mathbf{F}_{r}$, and $\mathbf{\Lambda}^{2}_{r}$
refer to the adjoint, fundamental, and 2nd rank antisymmetric representation
of subalgebra $\mathfrak{h}$, carrying shifted KK charge $r$ for the KK state.

The second kind of discrete symmetries is generated by a permutation $S$
acting on tensor nodes as $\alpha\rightarrow S(\alpha)$. The twist with $S$
then identifies the tensor nodes $\alpha$ and $S(\alpha)$. This brings the
intersection form $\Omega^{\alpha\beta}$ into another matrix
$\Omega_{S}^{\alpha^{\prime}\beta^{\prime}}$ where
$\alpha^{\prime},\beta^{\prime}$ parametrize orbits of tensor nodes permuted
by the action of $S$. The intersection matrix after the twist is determined by

$\Omega_{S}^{\alpha^{\prime}\beta^{\prime}}=\sum_{\beta\in\beta^{\prime}}\Omega^{\alpha\beta}\
,$ (48)

where $\alpha$ is any node in a given orbit $\alpha^{\prime}$. See
Bhardwaj:2019fzv  for more details.

Now consider a 5d KK theory obtained by a twisted compactification of a 6d
SCFT. The 5d gauge group and matter content as well as their KK charges are
now fully fixed by the action of the twists discussed above. Knowing this, we
can compute the effective prepotential for any twisted KK theory. The tree-
level part $\tilde{\mathcal{E}}_{\rm tree}$ after a certain twist is given by

$\displaystyle\tilde{\mathcal{E}}_{\rm tree}$ $\displaystyle\equiv
i\tilde{S}_{\rm tree}|_{\epsilon_{1,2},\phi,m,\tau}$
$\displaystyle=-\frac{\tau}{2}K^{\alpha^{\prime}\beta^{\prime}}_{S}\phi_{\alpha^{\prime},0}\phi_{\beta^{\prime},0}-\Omega^{\alpha^{\prime}\beta^{\prime}}_{S}\phi_{\alpha^{\prime},0}\left(\frac{a_{\beta^{\prime}}}{4}(\epsilon_{1}^{2}+\epsilon_{2}^{2})+\frac{b_{a,\beta^{\prime}}}{2}\tilde{K}_{a,ij}\phi_{a,i}\phi_{a,j}+c_{\beta^{\prime}}\epsilon_{+}^{2}\right)\
.$

Here
$K_{S}^{\alpha^{\prime}\beta^{\prime}}\equiv\sum_{\alpha\in\alpha^{\prime},\beta\in\beta^{\prime}}\Omega^{\alpha\beta}$
and $\tilde{K}_{a,ij}$ is the Killing form for an invariant subalgebra
$\mathfrak{h}_{a}$ and $a$ runs over the gauge and the flavor symmetry groups.

The cubic prepotential receives contributions from the KK momentum states. We
compute, in the parameter regime $\tau\gg\phi,m_{f}$,

$\displaystyle\tilde{\mathcal{F}}_{\text{1-loop}}$
$\displaystyle=\frac{1}{12}\sum_{n\in\mathbb{Z}}\bigg{(}\sum_{e\in\oplus{\bf
R}_{r}}\big{|}(n+r)\tau+e\cdot\phi\big{|}^{3}-\sum_{f}\sum_{w\in\oplus{\bf
w}_{r,f}}\big{|}(n+r)\tau+w\cdot\phi+m_{f}\big{|}^{3}\bigg{)}$
$\displaystyle=\frac{1}{12}\bigg{(}\sum_{e\in{\bf
R}_{0}}|e\cdot\phi|^{3}-\sum_{f}\sum_{w\in{\bf
w}_{0,f}}|w\cdot\phi+m_{f}|^{3}\bigg{)}$ (49)
$\displaystyle\quad-\frac{\tau}{24}\bigg{(}\sum_{e\in\oplus{\bf
R}_{r}}\\!\big{(}6r(r\\!-\\!1)\\!+\\!1\big{)}(e\cdot\phi)^{2}-\sum_{f}\\!\\!\sum_{w\in\oplus{\bf
w}_{r,f}}\\!\\!\big{(}6r(r\\!-\\!1)\\!+\\!1\big{)}(w\cdot\phi+m_{f})^{2}\bigg{)}\
.$

Here in the first line, the first term comes from the 6d vector multiplets
decomposed into $\oplus{\bf R}_{r}$ representations of invariant subalgebras
$\otimes_{a}\mathfrak{h}_{a}$, and the second term corresponds to the 6d
hypermultiplets decomposed into $\oplus{\bf w}_{r,f}$ representations of
$\otimes_{a}\mathfrak{h}_{a}$. We used the zeta function regularization in the
second and third lines. Note that the cubic terms in the Coulomb branch
parameters $\phi$ receive contributions only from the zero KK momentum states
because the contributions from other higher KK towers are all cancelled each
other. Similarly, the mixed gauge/gravitational and the mixed
gauge/$SU(2)_{R}$ Chern-Simons coefficients receive contributions only from
zero KK momentum states. We compute

$\displaystyle\tilde{C}^{G}_{i}=~{}$
$\displaystyle-\partial_{i}\bigg{(}\sum_{e\in{\bf
R_{0}}}|e\cdot\phi|-\sum_{f}\sum_{w\in{\bf
w}_{0,f}}|w\cdot\phi+m_{f}|\bigg{)}\ ,$ $\displaystyle\tilde{C}^{R}_{i}=~{}$
$\displaystyle\frac{1}{2}\,\partial_{i}\sum_{e\in{\bf R}_{0}}|e\cdot\phi|\ .$
(50)

Then the effective prepotential on the $\Omega$-background for a 6d SCFT on a
circle with general outer automorphism twists is given by

$\displaystyle\mathcal{E}_{{\rm twist}}$ $\displaystyle=i(\tilde{S}_{\rm
tree}+\tilde{S}_{\text{1-loop}}+\tilde{S}_{\rm
grav}+\tilde{S}_{R})|_{\epsilon_{1,2},\phi,m,\tau}$
$\displaystyle=\frac{1}{\epsilon_{1}\epsilon_{2}}\left(\tilde{\mathcal{E}}_{\rm
tree}+\tilde{\mathcal{F}}_{\text{1-loop}}+\frac{1}{48}\tilde{C}^{G}_{i}\phi^{i}(\epsilon_{1}^{2}+\epsilon_{2}^{2})+\frac{1}{2}\tilde{C}_{i}^{R}\phi^{i}\epsilon_{+}^{2}\right)\
.$ (51)

Again, we need to perform the shift in (40) for the Coulomb branch parameters
when we compare this 6d result with the effective prepotential of a geometric
description or with the effective prepotentials of other 5d dual gauge
theories.

As an example, consider the 6d minimal $SU(3)$ theory and its twisted
compactification. This theory has discrete $\mathbb{Z}_{2}$ global symmetry
acting on $SU(3)$ representations. We can therefore compactify this theory on
a circle with twist of the $\mathbb{Z}_{2}$ symmetry. The invariant subalgebra
under the twist is $\mathfrak{h}=\mathfrak{su}(2)$ and the KK states in the 5d
theory take representations of $\mathfrak{su}(2)$. The vector multiplet of the
$\mathfrak{su}(3)$ algebra in 6d reduces to the following combination of KK
momentum states.

${\bf 8}\ {\rm of}\ \mathfrak{su}(3)\ \rightarrow\ {\bf 3}_{0}\oplus{\bf
2}_{1/4}\oplus{\bf 2}_{3/4}\oplus{\bf 1}_{1/2}\ {\rm of}\ \mathfrak{su}(2)\ .$
(52)

From this together with the data (41), we can compute the tree-level and the
loop contributions to the effective prepotential as

$\displaystyle\tilde{\mathcal{E}}_{\rm tree}=~{}$
$\displaystyle\frac{3}{2}\tau\phi_{0}^{2}+\phi_{0}\left(3\phi_{1}^{2}-\frac{1}{4}(\epsilon_{1}^{2}+\epsilon_{2}^{2})+3\epsilon_{+}^{2}\right)\
,$ $\displaystyle\tilde{\mathcal{F}}_{\text{1-loop}}=~{}$
$\displaystyle\frac{4}{3}\phi_{1}^{3}-\frac{5}{16}\tau\phi_{1}^{2}\ ,$ (53)

and the mixed Chern-Simons coefficients as

$\tilde{C}^{G}_{1}=-4\ ,\qquad\tilde{C}^{R}_{1}=2\ .$ (54)

Plugging all these terms into the formula (2.3), one obtains the effective
prepotential for the twisted minimal $SU(3)$ SCFT as

$\displaystyle\epsilon_{1}\epsilon_{2}\,\mathcal{E}=$
$\displaystyle~{}\frac{3}{2}\tau\phi_{0}^{2}+\phi_{0}\left(3\phi_{1}^{2}-\frac{1}{4}(\epsilon_{1}^{2}+\epsilon_{2}^{2})+3\epsilon_{+}^{2}\right)+\frac{4}{3}\phi_{1}^{3}-\frac{5}{16}\tau\phi_{1}^{2}$
$\displaystyle-\frac{1}{12}\phi_{1}(\epsilon_{1}^{2}+\epsilon_{2}^{2})+\phi_{1}\epsilon_{+}^{2}\
.$ (55)

As this theory is dual to the 5d pure $SU(3)$ gauge theory at the Chern-Simons
level $\kappa=9$, we can compare the effective prepotential $\mathcal{E}$ in
(2.3) with that for the dual 5d theory. For the comparison, we should shift
the Coulomb branch parameters as
$\phi_{0}\rightarrow\phi_{0}-\frac{1}{16}\tau$ and consecutively as
$\phi_{1}\rightarrow\phi_{1}-2\phi_{0}+\frac{\tau}{4}$ as guided by (40). Then
the shifted effective prepotential $\mathcal{E}^{\prime}$ becomes

$\displaystyle\epsilon_{1}\epsilon_{2}\,\mathcal{E}^{\prime}=$
$\displaystyle~{}\frac{\tau}{2}(\phi_{0}^{2}-\phi_{0}\phi_{1}+\phi_{1}^{2})+\frac{1}{6}\left(8\phi_{0}^{3}+8\phi_{1}^{3}+24\phi_{0}^{2}\phi_{1}-30\phi_{0}\phi_{1}^{2}\right)$
$\displaystyle-\Big{(}\frac{1}{12}(\epsilon_{1}^{2}+\epsilon_{2}^{2})-\epsilon_{+}^{2}\Big{)}(\phi_{0}+\phi_{1})\
,$ (56)

up to terms independent of $\phi_{i}$. The result is precisely the effective
prepotential for the pure $SU(3)_{9}$ theory with gauge coupling
$\frac{1}{g^{2}}=\tfrac{\tau}{2}$.

##  3 Blowup equations

In this section, we review and generalize the Nakajima-Yoshioka’s blowup
equations which are the main tool for computing the BPS partition functions of
5d/6d SQFTs on $\mathbb{C}^{2}\times S^{1}({\rm or~{}}\mathbb{C}^{2}\times
T^{2})$, and propose our main claim which enables one to compute the BPS
spectrum of any SQFT. We also present some instructive examples for computing
the partition functions.

###  3.1 Blowup equation review

To obtain the partition function $Z$ defined in (8), we first consider the
partition function $\hat{Z}$ on blowup $\hat{\mathbb{C}}^{2}$, where the
origin of the $\mathbb{C}^{2}$ is replaced by a compact 2-cycle
$\mathbb{P}^{1}$. The $\hat{\mathbb{C}}^{2}$ can be parametrized by the
projective coordinates $(z_{0},z_{1},z_{2})\sim(\lambda^{-1}z_{0},\lambda
z_{1},\lambda z_{2})$ for $\lambda\in\mathbb{C}\setminus\\{0\\}$. The Lorentz
generators $J_{1,2}$ act on the $\hat{\mathbb{C}}^{2}$ by

$\displaystyle(z_{0},z_{1},z_{2})\mapsto(z_{0},e^{\epsilon_{1}}z_{1},e^{\epsilon_{2}}z_{2})\
,$ (57)

with parameters $\epsilon_{1,2}$ for the Cartans of the $SO(4)$ rotations.
There are now two fixed points of the Lorentz rotations, the North pole and
the South pole of the coordinates $(0,1,0)$ and $(0,0,1)$ respectively on the
$\mathbb{P}^{1}$. Around these fixed points, the local coordinates are given
as $(z_{0}z_{1},z_{2}/z_{1})$ and $(z_{1}/z_{2},z_{0}z_{2})$, and thus their
weights under $J_{1,2}$ actions can be represented as
$(\epsilon_{1},\epsilon_{2}-\epsilon_{1})$ and
$(\epsilon_{1}-\epsilon_{2},\epsilon_{2})$ at the North and South poles
respectively.

By performing the localization the partition function will be given by a sum
over magnetic fluxes $\vec{n}$ on the $\mathbb{P}^{1}$, which is an
$r$-dimensional vector $\vec{n}=(n_{1},n_{2},\cdots,n_{r})\in\mathbb{Q}^{r}$
(running over the coweight lattices $\Gamma^{\vee}$ of gauge algebras), for
the maximal torus $U(1)^{r}$ of the gauge symmetry group $G$. In geometry,
such magnetic flux sum is performed for each compact 4-cycle. Also, background
magnetic fluxes $\vec{B}$ for the Abelian subgroup $U(1)^{r_{F}}$ of global
symmetries can be turned on, but we note that they are fixed and not summed
over. Quantization conditions for each set of magnetic fluxes
$(\vec{n},\vec{B})$ will be discussed shortly. In this paper, a flux set
$(\vec{n},\vec{B})$ represents a set of all allowed dynamical magnetic flux
vectors $\vec{n}_{i}$ and a fixed background flux vector $\vec{B}$.

At each flux background labelled by $\vec{n}$ and $\vec{B}$, the partition
function is localized at two fixed points on the $\mathbb{P}^{1}$ and the path
integral near each fixed point reduces to that of local $\Omega$-deformed
$\hat{\mathbb{C}}^{2}$ with shifted chemical potentials due to the magnetic
fluxes. As a consequence, the partition function $\hat{Z}$ can be written as
Nakajima:2003pg ; Nakajima:2005fg ; Gottsche:2006bm

$\displaystyle\Lambda(m_{j};\epsilon_{1},\epsilon_{2})\hat{Z}(\phi_{i},m_{j},B_{j};\epsilon_{1},\epsilon_{2})$
(58)
$\displaystyle=\sum_{\vec{n}}(-1)^{|\vec{n}|}\hat{Z}^{(N)}(\phi_{i}\\!+\\!n_{i}\epsilon_{1},m_{j}\\!+\\!B_{j}\epsilon_{1};\epsilon_{1},\epsilon_{2}\\!-\\!\epsilon_{1})\cdot\hat{Z}^{(S)}(\phi_{i}\\!+\\!n_{i}\epsilon_{2},m_{j}\\!+\\!B_{j}\epsilon_{2};\epsilon_{1}\\!-\\!\epsilon_{2},\epsilon_{2})\
,$

where $|\vec{n}|=\sum_{i}n_{i}$. Here $\hat{Z}^{(N)}$ and $\hat{Z}^{(S)}$ are
the partition function $\hat{Z}$ with shifted chemical potentials evaluated
near the North and South poles respectively. The shifts in the chemical
potentials $\phi_{i}$ and $m_{j}$ reflect the fact that magnetically charged
states experience angular momentum shifts under the flux background. The
prefactor $\Lambda(m_{j};\epsilon_{1},\epsilon_{2})$ does not depend on
dynamical parameters $\phi_{i}$, but depends only on mass parameters $m_{j}$
as well as $\epsilon_{1,2}$.

Now we will smoothly blow down the $\mathbb{P}^{1}$ at the origin. This is a
smooth transition bringing the blowup geometry $\hat{\mathbb{C}}^{2}$ back to
the flat $\mathbb{C}^{2}$ (with $\Omega$-deformation) without the
$\mathbb{P}^{1}$ at the origin. The claim in Nakajima:2003pg  was that for
certain theories, the partition function $\hat{Z}$ after the blowdown
procedure reduces to the usual BPS partition function $Z$ on $\mathbb{C}^{2}$.
In particular, the final partition function is independent of the background
fluxes $\vec{B}$ on the $\mathbb{P}^{1}$. The reason for this is the
following. The magnetic fluxes were supported on the $\mathbb{P}^{1}$ at the
origin, but the $\mathbb{P}^{1}$ has been blown down and disappeared. Then,
nothing remains to support these fluxes and moreover, there’s nowhere these
fluxes can flow on the flat $\mathbb{C}^{2}$. Therefore, we do not expect any
remnant of the fluxes after the transition.

We would like to make a remark on a subtle point in the presence of magnetic
fluxes about the fermion number operator and some modifications of $Z$
associated with it. Since the partition function $\hat{Z}$ was defined with
magnetic fluxes on $\hat{\mathbb{C}}^{2}$, the angular momentum for a state
with electric charge $\sf{e}$ is shifted by ${\sf e}\cdot n$ where $n$ is the
magnetic flux on $\mathbb{P}^{1}$ at the origin. Recall that the fermion
number operator $(-1)^{F}$ in the index in (8) can be also defined as
$(-1)^{2J_{1}}$. In the presence of the magnetic flux $n$, this should change
as $(-1)^{2J_{1}}\rightarrow(-1)^{2J_{1}+{\sf e}\cdot n}$. This is formally
equivalent to the following replacement in the index444Similar replacements
$(-1)^{F}\rightarrow(-1)^{2J_{R}}$ in the superconformal index and in the
holomorphic block for 3d SCFTs were discussed in Dimofte:2011py ; Beem:2012mb
.

$(-1)^{F}\ \ \rightarrow\ \ (-1)^{2J_{R}}\ ,$ (59)

with the Cartan $J_{R}$ of the $SU(2)_{R}$ charge. This indicates that the
partition function $\hat{Z}$ with magnetic fluxes on $\hat{\mathbb{C}}^{2}$ is
in fact defined with the operator $(-1)^{2J_{R}}$ instead of $(-1)^{F}$.
Moreover, since blowing down the $\mathbb{P}^{1}$ is a smooth transition, this
definition is still valid even after the transition. Thus, the partition
function (or the Witten index) in (58) before and after the transition is
defined with respect to the operator $(-1)^{2J_{R}}$.

One also finds that the replacement (59) can be implemented by a simple
redefinition of the angular momentum chemical potential as
$\epsilon_{1}\rightarrow\epsilon_{1}+2\pi i$. Therefore, after the transition
from $\hat{\mathbb{C}}^{2}$ to $\mathbb{C}^{2}$, the partition function in the
equation (58) can be written as

$\displaystyle\hat{Z}(\phi,m;\epsilon_{1},\epsilon_{2})$
$\displaystyle=e^{\mathcal{E}(\phi,m;\epsilon_{1},\epsilon_{2})}\cdot\hat{Z}_{GV}(\phi,m;\epsilon_{1},\epsilon_{2})\
,$ $\displaystyle\hat{Z}_{GV}(\phi,m;\epsilon_{1},\epsilon_{2})$
$\displaystyle\equiv Z_{GV}(\phi,m;\epsilon_{1}+2\pi i,\epsilon_{2})\ .$ (60)

Note that the prefactor $\mathcal{E}$ in the first equation is the same as the
prefactor before the replacement of $\epsilon_{1}$ because the redefinition
doesn’t affect the regularization factor in the path integral computation.
Here $Z_{GV}$ is the index part of the BPS partition function, which is
actually the refined Gopakumar-Vafa (GV) invariant Gopakumar:1998ii ;
Gopakumar:1998jq , defined as

$Z_{GV}(\phi,m;\epsilon_{1},\epsilon_{2})={\rm PE}\left[\sum_{j_{l},j_{r},{\bf
d}}(-1)^{2(j_{l}+j_{r})}N^{\bf
d}_{j_{l},j_{r}}\frac{\chi^{SU(2)}_{j_{l}}(p_{1}/p_{2})\,\chi^{SU(2)}_{j_{r}}(p_{1}p_{2})}{(p_{1}^{1/2}-p_{1}^{-1/2})(p_{2}^{1/2}-p_{2}^{-1/2})}e^{-{\bf
d}\cdot{\bf m}}\right]\ ,$ (61)

where ${\bf d}$ denotes the charge of a BPS state, ${\bf m}$ stands for the
chemical potentials (or Kähler parameters) $\phi,m$, and $N^{\bf
d}_{j_{l},j_{r}}$ is the degeneracy of a single-particle BPS state with spin
$(j_{l},j_{r})$ and charge ${\bf d}$, and $\chi_{j}^{SU(2)}$ is the $SU(2)$
character of spin $j$. Also, $j_{l}\equiv\tfrac{J_{1}-J_{2}}{2}$ and
$j_{r}\equiv\tfrac{J_{1}+J_{2}}{2}$. For example, the GV-invariants for a
hypermultiplet with Kähler parameter $\phi$ providing a BPS state with spin
$(0,0)$ are given by

$\displaystyle Z_{GV}^{\rm hyper}$ $\displaystyle={\rm
PE}\left[\frac{\sqrt{p_{1}p_{2}}}{(1-p_{1})(1-p_{2})}e^{-\phi}\right]=\prod_{i,j=0}^{\infty}\frac{1}{1-p_{1}^{i+1/2}p_{2}^{j+1/2}e^{-\phi}}\
,\quad$ $\displaystyle\hat{Z}_{GV}^{\rm hyper}$ $\displaystyle=Z^{\rm
hyper}_{GV}(\phi;\epsilon_{1}+2\pi
i,\epsilon_{2})=\prod_{i,j=0}^{\infty}\frac{1}{1+p_{1}^{i+1/2}p_{2}^{j+1/2}e^{-\phi}}\
.$ (62)

The equation (58) with the identification (3.1) is the celebrated blowup
equation for instanton partition functions on the $\Omega$-background
introduced in Nakajima:2003pg ; Nakajima:2005fg ; Gottsche:2006bm . See also
Huang:2017mis  for a geometric generalization of the blowup formula. The
blowup equation with non-trivial $\Lambda$ is called a unity blowup equation.
The prefactor $\Lambda$ can also be trivial, i.e. $\Lambda=0$, for certain
choices of fluxes, and the blowup equation in this case is called a vanishing
blowup equation Nakajima:2005fg ; Huang:2017mis .

The purpose of this paper is to further generalize the above blowup formula
such that it can cover all the 5d supersymmetric theories which have
consistent UV completions. In addition, we will provide a systematic way to
compute the BPS partition function $Z$ for any 5d supersymmetric theories
using the blowup formula. More precisely, we propose the following conjecture:

Conjecture: The partition function $Z$ on the $\Omega$-background in (8) for
any 5d $\mathcal{N}=1$ field theory can be computed by solving the blowup
equations (58) with (i) consistent magnetic fluxes $\vec{n}$ and $\vec{B}$, up
to (ii) flop transitions.

Based on this conjecture, we will present in this paper how to bootstrap BPS
spectra of 5d field theories by solving the blowup equations. The seeds for
this bootstrapping are the effective prepotential $\mathcal{E}$ on the
$\Omega$-background and a set (or multiple sets) of consistent magnetic fluxes
$\vec{n}$ and $\vec{B}$. We have already introduced how to compute the
effective prepotential for every 5d field theory having either a gauge theory
description in 5d or a 6d field theory origin or a geometric realization in a
local CY 3-fold. We will now discuss how to choose consistent magnetic fluxes
$\vec{n}$ and $\vec{B}$, basically the points (i), (ii) in the conjecture.

####  3.1.1 Solving blowup equations

Let us explain how to compute the partition function $Z$ using the blowup
formula. We first remark that the index part of the partition function $Z$ of
any 5d/6d SQFT must take the form of the GV-invariant $Z_{GV}$ in (61). Now
consider a power series expansion of the GV-invariant with respect to the
fugacities $e^{-{\bf d}\cdot{\bf m}}$. The BPS states captured by the GV-
invariant satisfy the BPS mass formula $|M|={\bf d}\cdot{\bf m}$ and at a
generic point on the Coulomb branch (with mass parameters for global
symmetries turned on) they have positive masses ${\bf d}\cdot{\bf m}>0$.
Therefore the series expansion of $Z_{GV}$ in terms of the fugacities
$e^{-{\bf d}\cdot{\bf m}}$ on the Coulomb branch is well-defined.

The blowup equation (58) can be expressed in terms of power series in the
fugacities and can be solved iteratively. Practically, we first recast the
blowup equation as

$\displaystyle\Lambda(m_{j};\epsilon_{1},\epsilon_{2})\hat{Z}_{GV}(\phi_{i},m_{j};\epsilon_{1},\epsilon_{2})=\sum_{\vec{n}}(-1)^{|\vec{n}|}e^{-V(\phi_{i},m_{j},\vec{n},\vec{B};\epsilon_{1},\epsilon_{2})}$
$\displaystyle\times\hat{Z}_{GV}^{(N)}(\phi_{i}\\!+\\!n_{i}\epsilon_{1},m_{j}\\!+\\!B_{j}\epsilon_{1};\epsilon_{1},\epsilon_{2}\\!-\\!\epsilon_{1})\cdot\hat{Z}_{GV}^{(S)}(\phi_{i}\\!+\\!n_{i}\epsilon_{2},m_{j}\\!+\\!B_{j}\epsilon_{2};\epsilon_{1}\\!-\\!\epsilon_{2},\epsilon_{2})\
,$ (63)

where

$\displaystyle
V(\phi_{i},m_{j},\vec{n},\vec{B};\epsilon_{1},\epsilon_{2})\equiv\
\mathcal{E}(\phi_{i},m_{j};\epsilon_{1},\epsilon_{2})$ (64)
$\displaystyle\qquad\quad-\mathcal{E}^{(N)}(\phi_{i}\\!+\\!n_{i}\epsilon_{1},m_{j}\\!+\\!B_{j}\epsilon_{1};\epsilon_{1},\epsilon_{2}-\epsilon_{1})-\mathcal{E}^{(S)}(\phi_{i}\\!+\\!n_{i}\epsilon_{2},m_{j}\\!+\\!B_{j}\epsilon_{2};\epsilon_{1}-\epsilon_{2},\epsilon_{2})\
.$

We expand both sides of the blowup equation (3.1.1) and then try to find an
iterative solution of $\hat{Z}_{GV}$.

Importantly, we can use the fact that the GV-invariant should take a special
form in (61). Also, spins of states at each order are bounded by the maximum
spin $(j_{l}^{\rm max},j_{r}^{\rm max})$ in the series expansion, and the
characters $\chi_{j}^{SU(2)}$ with different spins are all orthogonal to each
other. Plugging the ansatz of the GV-invariant with a finite number of trial
states for a given charge ${\bf d}$ into the blowup equation and expanding it,
we can iteratively solve the equations to evaluate multiplicities
$N_{j_{l},j_{r}}^{\bf d}$ of BPS states.

We conjecture that every 5d field theory enjoys enough number of independent
blowup equations, enabling one to compute full BPS spectrum. It appears that a
generic 5d/6d SQFT admits at least one unity blowup equation which suffices to
determine all BPS degeneracies as shown in Huang:2017mis . For instance, all
5d and 6d gauge theories with only full hypermultiplets (without any unpaired
half hypermultiplets) have a number of unity blowup equations. We expect that
at least one of those unity equations in this case is formulated with a set of
consistent magnetic fluxes, whose definition will be given shortly, and thus
the solution to such unity equation will produce the correct BPS spectrum of
the gauge theory.

There are some theories having only vanishing blowup equations, though. The 6d
theories involving unpaired half hypermultiplets on a circle with/without
twists are such theories555On the other hand, we note that 5d gauge theories
with half hypermultiplets can have unity blowup equations. Gu:2020fem . An
analysis of small $\epsilon_{1,2}$ expansion in Huang:2017mis  suggests that a
single vanishing blowup equation without other information may not be
sufficient to determine all BPS degeneracies. Nevertheless, we propose that
those theories in fact have enough number of vanishing blowup equations so
that we can compute their BPS spectra by solving all the vanishing equations
together with other supplementary consistency conditions, like the positivity
of BPS degeneracies $N_{j_{l},j_{r}}^{\bf d}\geq 0$, conformity to geometric
realizations and dualities, and the KK tower structure of KK theories, etc. As
a concrete example for this, in section 3.2, we will show that a single
vanishing blowup equation for the pure $SU(2)_{\theta}$ gauge theories can be
solved with the aid of additional consistency conditions so that BPS spectra
can be obtained, though the partition function for the $SU(2)_{\theta}$ theory
can be also obtained from unity blowup equations as well.

Though it seems trivial, the condition $N_{j_{l},j_{r}}^{\bf d}\geq 0$ turns
out to be quite powerful. While solving the blowup equations, it usually
happens that degeneracies of BPS states captured in higher orders in the
expansion are fixed by BPS degeneracies appearing in lower orders.
Accordingly, the non-negativity of degeneracies $N_{j_{l},j_{r}}^{\bf d}\geq
0$ for all the BPS states in higher orders puts constraints on the possible
lower order BPS degeneracies. When taking into account higher expansion
orders, one finds more and more additional constraints on the lower order BPS
degeneracies, which would strongly restrict the allowed lower order
degeneracies and hence the BPS degeneracies in a given order may be completely
fixed at a certain stage in the iteration procedure666For example, the BPS
spectrum of the 6d $E_{7}$ gauge theory with a half hypermultiplet in the
fundamental representation was computed in Table 24 in Gu:2020fem  by solving
vanishing blowup equations. Several undetermined BPS degeneracies (denoted by
symbol ‘?’) can actually be fixed by the condition $N_{j_{l},j_{r}}^{\bf
d}\geq 0$. For instance, from the BPS spectrum given in Table 24 in Gu:2020fem
, we could fix many degeneracies as $2(0,1)$ for $\beta=(2,1,1,0,0,1,2,1,0)$,
$(0,1)$ for $\beta=(2,0,1,0,0,1,2,1,0)$, $(0,0)\oplus(0,1)$ for
$\beta=(1,0,1,0,0,1,2,1,0)$, $2(0,0)\oplus(0,1)$ for
$\beta=(1,0,1,0,0,2,2,1,0)$, and etc. All other undetermined degeneracies, but
$\beta=(0,3,3,0,1,0,0,0,0)$, are strongly constrained and actually have only
few possibilities. It seems that some higher degree computations can fix all
the lower order degeneracies in the table..

Dualities and geometric realizations can also be useful for computation. When
a theory enjoys a geometric construction or dualities, we can extract from
them yet another supplementary information about the BPS states. In
particular, when blowup equations have more than one distinct solutions, one
can use a geometric construction or dualities to pick up the right solution
for a given theory, which we will see with concrete examples below.

Consequently, we conjecture that one can compute BPS spectra of all 5d/6d
field theories using the blowup equations formulated from their geometric
realizations or gauge theory descriptions, or RG-flows thereof,777We are
assuming that every 5d field theory admitting a UV completion has either a
geometric realization or gauge theory descriptions in 5d or in 6d on a circle
possibly with twists, or can be obtained by an RG-flow from a UV complete
theory. even for the cases equipped with only vanishing blowup equations. A
similar conjecture for refined BPS invariants of a local CY 3-fold was given
in Huang:2017mis . We will provide evidences for our conjecture by explicitly
solving the blowup equations for all rank-2 theories and some interesting
higher rank theories in sections 5 and 6.

####  3.1.2 Magnetic flux quantization

The magnetic fluxes on the $\mathbb{P}^{1}$ in the blowup
$\hat{\mathbb{C}}^{2}$ cannot be arbitrary. They should satisfy suitable
quantization conditions. Let us explain the quantization conditions for the
magnetic fluxes in three different perspectives: the geometric perspective,
the 5d gauge theoretic perspective, and the 6d gauge theoretic perspective.

##### Geometries

In geometry, the magnetic flux $n_{i}$ (or $B_{j}$) can be turned on for each
(non-)compact divisor $D_{I}$ in a 3-fold. The flux quantization depends on
charges and spins $(j_{l},j_{r})$ of wrapped M2-branes on holomorphic
2-cycles. Consider an M2-brane wrapping a primitive curve $C_{i}$. Here the
primitive curve is a Mori cone generator and every holomorphic 2-cycle can be
expressed as a linear combination of primitive curves $C_{i}$ as
$C=\sum_{i}p_{i}C_{i}$ with non-negative integers $p_{i}$. The curve $C_{i}$
can consistently couple to a magnetic flux ${\sf F}$ if the following
condition is satisfied Huang:2017mis

${\sf F}\cdot C_{i}\ \text{ is integral/half-integral when }C_{i}^{2}\ {\rm
is\ even/odd}\ ,$ (65)

where $C_{i}^{2}$ is the self-intersection number of $C_{i}$. The flux ${\sf
F}$ is defined in geometry as ${\sf
F}\equiv\sum_{i=1}^{r}n_{i}S_{i}+\sum_{j=1}^{r_{F}}B_{j}N_{j}$, where
$S_{i},N_{j}\in H^{1,1}(X)$ are the basis of the compact and the non-compact
surfaces inside a 3-fold $X$, respectively. The above condition (65) is
equivalent to the condition that the magnetic flux on a charged M2-brane state
of spin $(j_{l},j_{r})$ wrapping on $C_{i}$ satisfies

$\displaystyle{\sf F}\cdot C_{i}~{}~{}$ $\displaystyle\text{is integral/half-
integral, when $2(j_{l}+j_{r})$ is odd/even}.$ (66)

This is because the spin of a curve $C_{i}$ is related to the self-
intersection as $2(j_{l}+j_{r})=C_{i}^{2}+1$ mod 2. From this we claim that
the flux ${\sf F}$, or equivalently $(\vec{n},\vec{B})$, must be quantized
such that all the primitive curves $C_{i}$ in a 3-fold satisfy the condition
(65). In general the solution to this quantization condition is not unique.
The proper choices of magnetic fluxes associated to spectrum of unitary BPS
states will be discussed below.

##### 5d Gauge Theories

We can easily translate the above geometric quantization condition to physical
conditions in 5d gauge theories. W-bosons and hypermultiplet states in a 5d
gauge theory correspond to primitive curves with $C^{2}=0$ and $C^{2}=-1$,
respectively. The charges of these elementary fields in the classical
Lagrangian are all known. Based on the classical information, we can first
quantize the magnetic fluxes $\vec{n}$ and $\vec{B}$ coupled to the elementary
fields. The geometric quantization condition implies that the total fluxes on
the W-bosons of the gauge group should be integral and those on the
perturbative hypermultiplets should be half-integral. Namely,

$\displaystyle\vec{n}\cdot e\in\mathbb{Z}\ ,\qquad\vec{n}\cdot
w_{f}+B_{f}\in\mathbb{Z}+\frac{1}{2}\ ,$ (67)

for all roots $e$, the weights $w_{f}$ associated with all hypermultiplets
$f$, and the fluxes $B_{f}$ for flavor symmetries.

We will use the first condition of (67) to fix the quantization of fluxes
$\vec{n}$, and use the second condition to quantize the background fluxes
$B_{f}$.888There is one exception. A half hypermultiplet does not admit its
mass parameter, so there is no corresponding $B_{f}$. In this case, the second
condition of (67) associated with the half-hyper further constrains the
quantization of $\vec{n}$. Depending on the gauge algebra, there are several
possibilities of magnetic fluxes satisfying (67). Since a state in a given
representation is obtained by subtracting roots from its highest weight state,
the quantization of $\vec{n}\cdot w_{f}$ is the same for every $w_{f}$ in a
fixed representation.

To find all possible quantizations for $\vec{n}$, it is convenient to
introduce the fundamental weight $\mu_{i}$ which is a dual basis of the
coroots $\alpha_{i}^{\vee}$ of the gauge algebra.999For a root $\alpha$ in
Euclidean space $E$, the coroot $\alpha^{vee}$ is a map from $E$ to
$\mathbb{R}$, defined to be
$\left\langle{\alpha^{\vee}},{x}\right\rangle=2(x,\alpha)/(\alpha,\alpha)$,
where $(\cdot,\cdot)$ is an inner product in $E$. The fundamental weight
$\mu_{i}$ is dual basis of $\alpha_{i}^{\vee}$, i.e.,
$\left\langle{\alpha_{i}^{\vee}},{\mu_{j}}\right\rangle=\delta_{ij}$. The
general weight vector can be written as a linear combination of $\mu_{i}$. For
example, write a simple root $\alpha_{i}=\sum_{k}a_{ij}\mu_{j}$. Then
$\left\langle{\alpha_{j}^{\vee}},{\alpha_{i}}\right\rangle=a_{ij}$ so that
$a_{ij}$ is an element of the Cartan matrix. The details can be found in many
textbooks about Lie algebras, for example, humphreys_introduction_1972  or
bump_lie_2013 . Consider a lattice $\Lambda=\oplus\mathbb{Z}\,\mu_{i}$ and the
corresponding root lattice $\Lambda_{r}$ which is a sublattice of $\Lambda$.
The lattice $\Lambda$ contains not only root lattice but also all possible
weight vectors. Thus, the number of possible quantizations of $\vec{n}$ with
$\vec{n}\cdot e\in\mathbb{Z}$ is counted by the number of elements in
$\Lambda/\Lambda_{r}$. This group is isomorphic to the center of simply
connected Lie group corresponding to gauge algebra humphreys_introduction_1972
; bump_lie_2013 :

$\displaystyle\begin{array}[]{cccccccccc}A_{r}&\hskip 4.30554ptB_{\ell}&\hskip
4.30554ptC_{\ell}&\hskip 4.30554ptD_{\ell,\text{odd}}&\hskip
4.30554ptD_{\ell,\text{even}}&\hskip 4.30554ptE_{6}&\hskip
4.30554ptE_{7}&\hskip 4.30554ptE_{8}&\hskip 4.30554ptF_{4}&\hskip
4.30554ptG_{2}\\\ \mathbb{Z}_{r+1}&\hskip 4.30554pt\mathbb{Z}_{2}&\hskip
4.30554pt\mathbb{Z}_{2}&\hskip 4.30554pt\mathbb{Z}_{4}&\hskip
4.30554pt\mathbb{Z}_{2}\times\mathbb{Z}_{2}&\hskip
4.30554pt\mathbb{Z}_{3}&\hskip 4.30554pt\mathbb{Z}_{2}&\hskip
4.30554pt\\{1\\}&\hskip 4.30554pt\\{1\\}&\hskip 4.30554pt\\{1\\}\end{array}\
.$ (70)

More explicitly, for the gauge algebra of type $A_{\ell}$, the possible
quantization for $\vec{n}$ is given by

$\displaystyle n_{i}\in\mathbb{Z}+\frac{h}{\ell+1}i\quad(1\leq i\leq\ell)\ ,$
(71)

where $n_{i}\equiv\vec{n}\cdot\mu_{i}$ and $h$ is a fixed integer subject to
$0\leq h\leq\ell$. For the gauge algebra of type $B_{\ell}$, there are two
quantizations,

$\displaystyle n_{i}$ $\displaystyle\in\mathbb{Z}\quad(1\leq i\leq\ell-1)\
,\qquad n_{\ell}\in\mathbb{Z}+\frac{h}{2}\ ,$ (72)

where $h=0,1$. For type $C_{\ell}$,

$\displaystyle n_{i}\in\mathbb{Z}+\frac{h}{2}\quad\text{for $i$ odd}\ ,\qquad
n_{i}\in\mathbb{Z}\quad\text{for $i$ even},$ (73)

where $h=0,1$. The set of the quantizations of types $B_{\ell}$ and $C_{\ell}$
has a $\mathbb{Z}_{2}$ structure. For type $D_{\ell}$ with $\ell$ odd,

$\displaystyle n_{i}$
$\displaystyle\in\mathbb{Z}+\frac{h}{2}\qquad(i=1,3,\cdots,\ell-3)\ ,\quad$
(74) $\displaystyle n_{i}$
$\displaystyle\in\mathbb{Z}\qquad(i=2,4,\cdots,\ell-2)\ ,\quad$ (75)
$\displaystyle n_{\ell-1}$ $\displaystyle\in\mathbb{Z}+\frac{h}{4},\qquad
n_{\ell}\in\mathbb{Z}+\frac{h+2}{4}\ ,$

where $h=0,1,2,3$. The set of the quantization for $D_{\ell,\text{odd}}$ hence
has a $\mathbb{Z}_{4}$ structure. For type $D_{\ell}$ with $\ell$ even, there
are four quantizations:

$\displaystyle n_{i}$
$\displaystyle\in\mathbb{Z}+\frac{h_{1}+h_{2}}{2}\qquad(i=1,3,\cdots,\ell-3)\
,$ (76) $\displaystyle n_{i}$
$\displaystyle\in\mathbb{Z}\qquad(i=2,4,\cdots,\ell-2)\ ,$ (77) $\displaystyle
n_{\ell-1}$ $\displaystyle\in\mathbb{Z}+\frac{h_{1}}{2},\qquad
n_{\ell}\in\mathbb{Z}+\frac{h_{2}}{2}\ ,$

where $(h_{1},h_{2})\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. For type $E_{6}$,

$\displaystyle(n_{1},n_{2},n_{3},n_{4},n_{5},n_{6})\in\mathbb{Z}^{6}+\frac{h}{3}(2,1,0,2,1,0)\
,$ (78)

with $h=0,1,2$, which has a $\mathbb{Z}_{3}$ structure. For type $E_{7}$,

$\displaystyle(n_{1},n_{2},n_{3},n_{4},n_{5},n_{6},n_{7})\in\mathbb{Z}^{7}+\frac{h}{2}(0,0,0,1,0,1,1)\
,$ (79)

with $h=0,1$, which has a $\mathbb{Z}_{2}$ structure. The remaining
exceptional algebras $E_{8}$, $F_{4}$ and $G_{2}$ admit only canonical flux
quantization defined as

$n_{i}\in\mathbb{Z}\ ,\quad B_{f}\in\mathbb{Z}+\frac{1}{2}\ .$ (80)

This canonical flux quantization holds for any gauge and flavor symmetry
groups except for the cases where the theory contains unpaired half
hypermultiplets. Most of the examples we will discuss below use the canonical
flux quantization.

The quantization of fluxes $B_{a}$ for topological symmetries of gauge groups
is more involved because in general we do not know spins of instanton states.
Let us first discuss the quantization condition on $B_{a}$ when fluxes for the
gauge group $G_{a}$ satisfy the canonical flux quantization. In this case, a
single instanton state carries a unit charge under $U(1)_{a}$ topological
symmetry for the $a$-th gauge group. The charges of the instanton state under
other global symmetries can be computed by summing over contributions from the
zero modes of charged hypermultiplets on the instanton background. For a
hypermultiplet in a representation ${\bf r}$ of a gauge group $G_{a}$, the
zero mode contribution to the flavor charge of a 1-instanton state on the
Coulomb branch is given by its Dynkin index, i.e. $-T_{a}({\bf r})$.
Therefore, it follows from (65) that the ground state of a single instanton
for gauge group $G_{a}$ should induce a quantization condition

$B_{a}-\sum_{f}B_{f}T_{a}({\bf r}_{f})\ \in\ \mathbb{Z}\ \ {\rm or}\ \
\mathbb{Z}+1/2\ ,$ (81)

depending on the spin of the state. This will quantize the flux $B_{a}$
provided that $B_{f}$’s have already been quantized by (80). Here, we denote
by $f$ all hypermultiplets in the representation ${\bf r}_{f}$ of the gauge
group $G_{a}$ collectively. Since the canonical gauge fluxes $n_{i}$ bring
about only integral shifts, the gauge fluxes do not affect the above
quantization condition.

When $n_{i}^{a}$’s for a gauge group $G_{a}$ are not canonical, we can fix
$B_{a}$ by requiring the corresponding blowup equation to be solvable. We
first expand the blowup equation in terms of the instanton fugacity
$e^{-m_{a}}$ for a gauge group $G_{a}$, assuming that it is a unity equation.
The terms in the leading order in this expansion are given by a set of
magnetic flux vectors $\vec{n}$ that minimize

$\partial_{m_{a}}V=\frac{1}{2}(\vec{n},\vec{n})^{a}\equiv\frac{1}{2}K^{a}_{ij}n^{a}_{i}n^{a}_{j}\
,$ (82)

for all gauge groups, where $V$ is defined in (64). We call this set ${\rm
Min}(\vec{n})$. Since the GV-invariant $\hat{Z}_{GV}$ starts with $1$ in the
fugacity expansion and the prefactor $\Lambda$ is independent of the Coulomb
branch parameters $\phi_{i}$, the right-hand side in the blowup equation given
in (3.1.1) must start with terms independent of $\phi_{i}$. This means that
for a given $B_{a}$, there must exist at least a magnetic flux vector
$\vec{n}\in{\rm Min}(\vec{n})$ such that

$\partial_{\phi^{a}_{i}}V=0\ \ {\rm at}\ (\vec{n},\vec{B})\ {\rm for}\ {\rm
all}\ i\ {\rm of}\ G_{a}\ .$ (83)

This condition then fixes the background flux $B_{a}$ when the gauge fluxes
$n^{a}_{i}$ are non-canonical ones. We note that when the fluxes $n^{a}_{i}$
are all canonical ones, this condition is trivially satisfied since
$n^{a}_{i}\in$ Min($\vec{n})$ are all zero and $V$ is independent of
$\phi^{a}_{i}$ when $n^{a}_{i}=0$.

On the other hand, a vanishing blowup equation can be solved if there exists
at least a pair of minimal flux vectors $\vec{n}_{1},\vec{n}_{2}\in{\rm
Min}(\vec{n})$ with $|\vec{n}_{1}|-|\vec{n}_{2}|=1$, which is a necessary
condition for that two leading terms cancel each other in the expansion,
satisfying

$\partial_{\phi^{a}_{i}}V\ {\rm at}\
(\vec{n}_{1},\vec{B})\quad=\quad\partial_{\phi^{a}_{i}}V\ {\rm at}\
(\vec{n}_{2},\vec{B})\ ,$ (84)

for all $\phi_{i}^{a}$.

The conditions (83) and (84) will then leave only a finite number of allowed
fluxes $B_{a}$ for the topological symmetry of $G_{a}$ in the unity and
vanishing blowup equations, respectively, for non-canonical magnetic fluxes
$n^{a}_{i}$.

##### 6d Gauge Theories with/without Twists

Lastly, we will discuss the flux quantization conditions for the 5d KK
theories in terms of the associated 6d field theory data. Upon a circle
compactification, the 6d multiplets reduce to 5d vector multiplets and
hypermultiplets taking representations of the 6d gauge algebra $\mathfrak{g}$,
or the invariant subalgebra $\mathfrak{h}$ when $\mathfrak{g}$ is twisted. In
the 5d reduction of a 6d gauge theory, the magnetic fluxes for the gauge group
and those for the flavor group acting on 5d gauge and matter fields satisfy
the quantization condition in (67). Also, the magnetic flux $B_{\tau}$ of the
KK $U(1)$ symmetry should be an integer because all KK states from a single 6d
state should have the same (half-)integrality of fluxes. So it is natural to
fix

$B_{\tau}=0\ .$ (85)

We then need to determine the quantization conditions on the tensor fluxes
$n_{\alpha}$ (or $n_{\alpha^{\prime}}$ when the $\alpha$-th tensor node is
twisted) for tensor symmetries. Self-dual strings are charged under the tensor
symmetries. Their tensor charges are determined by the classical Green-Schwarz
terms evaluated on self-dual string backgrounds. The ground state of a single
string associated to the $\beta^{\prime}$-th (or $\beta$-th without twist)
tensor field carries charges for the $\alpha^{\prime}$-th tensor symmetry as

$\Omega^{\alpha^{\prime}\beta^{\prime}}_{S}b_{a,\beta^{\prime}}\ ,$ (86)

when the string is an instantonic string of the $a$-th gauge group. If the
$\beta^{\prime}$-th tensor supports no gauge algebra, then
$b_{a,\beta^{\prime}}=1$.

Let us first consider the cases where magnetic fluxes for a gauge group
$G_{a}$ coupled to the $\alpha^{\prime}$-th tensor fields, so when
$b_{a,\alpha^{\prime}}\neq 0$, satisfy the canonical flux quantization
condition in (80). The flavor and the KK charges for a single string can be
computed by counting the zero mode contributions from KK states on the string
background, which is the same computation we did above for instanton states in
5d gauge theories. Collecting all the zero mode contributions, we claim that
the tensor flux, which we denote by
$n_{\alpha^{\prime}}\equiv\tilde{n}_{\alpha^{\prime}}+B_{\alpha^{\prime}}$
with $\tilde{n}_{\alpha^{\prime}}\in\mathbb{Z}$, should satisfy the following
quantization condition: given the spin $(j_{l},j_{r})$ of the ground state,

$\displaystyle-\Omega^{\alpha^{\prime}\beta^{\prime}}_{S}\\!B_{\alpha^{\prime}}b_{a,\beta^{\prime}}$
$\displaystyle\\!-\\!\sum_{f_{0}}B_{f_{0}}T_{a}({\bf
r}_{f_{0}})\\!+\\!\frac{1}{6}B_{\tau}\\!\sum_{r}(6r(r\\!-\\!1)\\!+\\!1)\Big{(}T_{a}({\bf
R}_{r})\\!-\\!\sum_{f}T_{a}({\bf w}_{r,f})\Big{)}$
$\displaystyle\in\left\\{\begin{array}[]{ll}\mathbb{Z}&~{}\text{for~{}}2(j_{l}+j_{r})~{}\text{odd}\
,\\\ \mathbb{Z}+\frac{1}{2}&~{}\text{for~{}}2(j_{l}+j_{r})~{}\text{even}\
.\end{array}\right.$ (89)

This is essentially the quantization condition for $B_{\alpha^{\prime}}$.
Here, $f_{0}$ runs over hypermultiplets with KK-charge $0$, and $r$ runs over
all fractional KK-charge shifts including $r=0$. ${\bf R}_{r}$ and ${\bf
w}_{r,f}$ denote representations of the vector multiplets and the
hypermultiplets with fractional shift $r$ of KK charges, respectively. Note
that since gauge and tensor charges of the ground state on a self-dual string
are quantized to be integers and the associated fluxes $n_{a}$ and
$\tilde{n}_{\alpha^{\prime}}$ are all integers as well, they have no effect on
this quantization condition.

Now consider non-canonical magnetic fluxes for a gauge group $G_{a}$. Let us
first expand the blowup equation in terms of the fugacities
$e^{-\phi_{\alpha^{\prime}}}$ of tensor symmetries. The leading power of these
fugacities in this expansion is determined by the minimum of

$\partial_{\phi_{\alpha^{\prime}}}V=\Omega_{S}^{\alpha^{\prime}\beta^{\prime}}b_{a,\beta^{\prime}}\tilde{K}_{a,ij}n^{a}_{i}n^{a}_{j}\
.$ (90)

This is non-zero with non-integral gauge fluxes $n^{a}_{i}$. This means that
when gauge fluxes $n^{a}_{i}$ do not satisfy the canonical flux quantization
condition for any $G_{a}$, then the blowup equation written in terms of a 6d
field theory description is always a vanishing-type, i.e. $\Lambda=0$.

As we discussed for 5d gauge theories above, the vanishing blowup equation can
be solved only if we can find at least a pair of minimal flux vectors
$\vec{n}_{1},\vec{n}_{2}\in{\rm Min}(\vec{n})$ and background fluxes $\vec{B}$
subject to the condition given in (84), where ${\rm Min}(\vec{n})$ is a set of
magnetic flux vectors minimizing $(\vec{n},\vec{n})^{a}$ for all $G_{a}$ and
$\vec{B}$ here involves the tensor fluxes $B_{\alpha^{\prime}}$ as well as
other flavor fluxes. We can use this fact to fix the background magnetic
fluxes $B_{\alpha^{\prime}}$ of tensor symmetries.

We remark that if a 6d gauge theory involves unpaired half hypermultiplets and
thus the background flavor flux acting on the hypermultiplet cannot be turned
on, then we cannot choose the canonical flux quantization (80) for the fluxes
of the gauge group $G_{a}$ coupled to the half hypermultiplet. Therefore, we
have only vanishing blowup equations in such cases, as discussed in Gu:2020fem
.

####  3.1.3 Consistent magnetic fluxes

The above quantization conditions are a necessary condition but not a
sufficient one for the BPS partition functions $Z$ of a 5d theory to satisfy
the blowup equation (58) with chosen magnetic fluxes. Among the magnetic
fluxes satisfying the quantization conditions, only few of them can provide
consistent blowup equations whose solution correctly produces BPS spectrum of
a 5d field theory. We call such magnetic fluxes leading to the consistent
blowup equations consistent magnetic fluxes. With a wrong set of fluxes, one
would find an inconsistent blowup equation: the blowup equation has no
solution or the solution to the blowup equation does not take the form of a
GV-invariant, or the solution involves unphysical states and thus differs from
the desired BPS spectrum for a 5d theory. Therefore, in order to correctly
compute BPS spectrum using the blowup equations, it is crucial to know how to
choose proper consistent magnetic fluxes. We now present a set of criteria for
the consistent magnetic fluxes.

As discussed, all the BPS particles on the Coulomb branch must satisfy the BPS
mass formula. It is expressed as $|M|=\sum_{i}{\sf{e}}_{i}\phi_{i}$, when we
turned off mass parameters for global symmetries in strong coupling limit near
the UV fixed point. Here ${\sf{e}}_{i}$ denotes the charge of a state under
the $i$-th $U(1)$ gauge group. In geometry, these masses are identified with
the volumes of holomorphic 2-cycles for the BPS states as
$\sum_{i}{\sf{e}}_{i}\phi_{i}={\rm vol}(C)$ measured with respect to the
normalizable Kähler parameters $\phi_{i}$. It then follows that the mass
$\sum_{i}{\sf{e}}_{i}\phi_{i}$ (or ${\rm vol}(C)$) must be non-negative on the
Coulomb branch, i.e. $\sum_{i}{\sf{e}}_{i}\phi_{i}\geq 0$ for all BPS states.
In fact, the Coulomb branch $\mathcal{C}$ is the space of the dynamical Kähler
moduli $\phi_{i}$ defined as Jefferson:2017ahm ; Jefferson:2018irk

$\mathcal{C}=\bigg{\\{}\phi_{i},i=1,\cdots,r\,|\,{\sf{e}}\cdot\phi\equiv\sum_{i}{\sf{e}}_{i}\phi_{i}\geq
0\bigg{\\}}\ ,$ (91)

with $\phi_{i}>0$ when all mass parameters for global symmetries are switched
off. The Coulomb branch $\mathcal{C}$ must exist for a UV finite 5d theory
equipped with dynamical Kähler parameters. Otherwise, unitarity of the theory
will be violated at some point on the Coulomb branch and the theory cannot
have a consistent UV completion.

The Coulomb branch $\mathcal{C}$ is a collection of sub-chambers
$\mathcal{C}_{i}$ that are connected by so-called flop transitions. It is
possible that a BPS hypermultiplet becomes massless $M=0$ at the boundary
between two sub-chambers. If then, two sub-chambers are connected by a flop
transition flipping the mass sign of the hypermultiplet as $M>0\rightarrow
M<0$. The effective Chern-Simons terms in the low energy theory take different
expressions in the two sub-chambers because the CS terms depend on mass signs
of hypermultiplets. More precisely, the prepotential and the
gauge/gravitational Chern-Simons coefficients alter their forms under a flop
transition related to a hypermultiplet as

$\mathcal{F}\ \ \rightarrow\ \ \mathcal{F}+\frac{1}{6}M^{3}\ ,\qquad
C^{G}_{i}\ \ \rightarrow\ \ C^{G}_{i}-\partial_{i}M\ ,\qquad C_{i}^{R}\ \
\rightarrow\ \ C_{i}^{R}\ ,$ (92)

where $M$ here is the mass for the hypermultiplet before the flop transition.
In geometry, the flop transition corresponds to a geometric transition
$X\rightarrow X^{\prime}$ between two CY 3-folds $X$ and $X^{\prime}$
described by blowing down a $-1$ curve $C\subset X$ and blowing up a different
$-1$ curve $C^{\prime}\subset X^{\prime}$.

The partition function $Z$ computes degeneracies of BPS states on the Coulomb
branch of a 5d theory. Suppose that we solve the blowup equations for $Z$ in
the fugacity expansion and the BPS states are identified up to a certain
order. One can easily read off the masses for the BPS states up to that order.
For a consistent 5d field theory having dynamical Kähler parameters, there
must exist a non-vanishing Coulomb branch $\mathcal{C}$ defined in (91),
possibly after a finite number of flop transitions, where the masses of the
BPS states are all non-negative, i.e. ${\sf e}\cdot\phi\geq 0$, when all non-
dynamical parameters are turned off. If one cannot find a non-trivial Coulomb
branch, then it implies that the 5d theory is inconsistent in UV.

Note however that the partition function $Z$ is defined and computed in a
particular sub-chamber on the (extended) Coulomb branch with mass parameters
$m_{j}$ for global symmetries turned on. In the limit $m_{j}\rightarrow 0$,
some of hypermultiplets captured in the partition function can be massless and
then flop transitions should take place before their masses become negative.
Since this can generically happen, we should carefully examine the existence
of Coulomb branch by testing ${\sf e}\cdot\phi\geq 0$ for all the states in
$Z$, except for hypermultiplets (or states with spin $(j_{l},j_{r})=(0,0)$).
We note that for hypermultiplets, either ${\sf e}\cdot\phi\geq 0$ or ${\sf
e}\cdot\phi<0$ is allowed, as flop transitions can happen. If one finds a non-
trivial Coulomb branch where ${\sf e}\cdot\phi\geq 0$ for all the states in
$Z$ except for a hypermultiplet which is of ${\sf e}\cdot\phi<0$, then this
means that there must be a flop transition associated to the hypermultiplet as
we approach the UV fixed point. In this case we should first perform flop
transitions for the hypermultiplets with ${\sf e}\cdot\phi<0$ and then test
again ${\sf e}\cdot\phi\geq 0$ for all states including the hypermultiplets.

If one finds a chamber with ${\sf e}\cdot\phi\geq 0$ for all the states
including hypermultiplets, after a sequence of such flop transitions, it is a
strong indication that the theory has a UV completion with non-trivial Coulomb
branch and the solution $Z$ (or $\hat{Z}$) of the blowup equations computes
the BPS spectrum on the Coulomb branch around the UV fixed point. In this
case, we will refer to the set of magnetic fluxes as consistent magnetic
fluxes for the 5d theory on the blowup $\hat{\mathbb{C}}^{2}$. We conjecture
that there exist enough sets of consistent magnetic fluxes for every UV finite
5d theory and therefore we can compute its BPS spectrum using the blowup
equations.

Examining ${\sf e}\cdot\phi\geq 0$ for all BPS states is a formidable task as
there are infinitely many BPS states but the partition function $Z$ is
computed up to a certain order in the fugacity expansion. Practically we can
check this non-negativity of masses of BPS states only up to a certain higher
order. This however would not be very harmful from the perspective of
geometry. Recall that all holomorphic 2-cycles ${C}=\sum_{i}n_{i}C_{i}$ can be
written as a linear sum of primitive curves $C_{i}$ with non-negative integers
$n_{i}$ and the number of the primitive curves is finite for a local CY
3-fold. The primitive curves (or associated BPS states) are usually captured
at some lower orders in the expansion. We then need to check non-negativity of
volumes only for such primitive curves in lower orders. We expect this holds
also for general 5d theories. Hence one can in principle identify consistent
magnetic fluxes by computing the BPS partition function up to certain leading
orders.

We now have all the ingredients for solving the blowup equations and thus are
ready for bootstrapping BPS spectrum of any 5d field theory. In the followings
we will explicitly illustrate the bootstrap procedure with a large number of
interesting examples.

###  3.2 Instructive examples

In this section, we present instructive examples which illustrate our proposal
in section 3.1 for bootstrapping the BPS partition functions.

####  3.2.1 5d pure $SU(2)_{\theta}$ with $\theta=0,\pi$

As the simplest example, consider the 5d $\mathcal{N}=1$ pure $SU(2)_{\theta}$
gauge theory with a discrete theta angle $\theta=0$ or $\pi$. As the
perturbative descriptions of two theories are the same, they have the same
prepotential. It follows from (1) that the cubic prepotential is given by

$\displaystyle 6\mathcal{F}=6m\phi^{2}+8\phi^{3}\ ,$ (93)

where $m=1/g^{2}$ is the gauge coupling and $\phi$ is the Coulomb branch
parameter for the $SU(2)$ gauge symmetry. The effective prepotential defined
in (2.1) evaluated on the $\Omega$-background for both theories takes the form
of

$\displaystyle\mathcal{E}=\frac{1}{\epsilon_{1}\epsilon_{2}}\quantity(\mathcal{F}-\frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}}{12}\phi+\epsilon_{+}^{2}\phi)\
,$ (94)

where the second term comes from the mixed gauge/gravitational CS term with
$C^{G}_{\phi}=-4$ in (18), and the third term comes from the mixed
gauge/$SU(2)_{R}$ CS term with $C^{R}_{\phi}=2$. The mixed CS terms are also
insensitive to the discrete theta angle.

We now use the effective prepotential $\mathcal{E}$ as an initial input for
the blowup equation to obtain BPS spectrum of this theory. Recall that though
$\mathcal{E}$ for both $SU(2)_{0}$ and $SU(2)_{\pi}$ theories are the same,
they have different UV fixed points and thus they have different BPS spectra.
This is reflected in the blowup equation by two distinct choices of background
magnetic flux $B_{m}$ for the topological $U(1)$ symmetry:

$B_{m}\in\left\\{\begin{array}[]{ll}\mathbb{Z}&\ {\rm for}\ SU(2)_{0}\\\
\mathbb{Z}+1/2&\ {\rm for}\ SU(2)_{\pi}\end{array}\right.\ .$ (95)

To choose the flux $B_{m}$ properly, it is convenient to consider geometric
construction of two theories. The $SU(2)_{0}$ theory is geometrically
engineered by a CY 3-fold containing a single Hirzebruch surface
$\mathbb{F}_{0}$, while the $SU(2)_{\pi}$ theory is engineered by another
3-fold containing a Hirzebruch surface $\mathbb{F}_{1}$.

fe (a) $\mathbb{F}_{0}$

feh (b) $\mathbb{F}_{1}$

Figure 1: Geometric constructions of (a) $SU(2)_{0}$ and (b) $SU(2)_{\pi}$.
The fiber $\mathbb{P}^{1}$ in a Hirzebruch surface $\mathbb{F}_{n}$ is denoted
by $f$, the base $\mathbb{P}^{1}$ is denoted by $e$, and $h=e+f$ for
$\mathbb{F}_{1}$.

The primitive 2-cycles (or Mori cone generators) in a Hirzebruch surface are
the fiber $f$ and the base $e$. The volumes of these 2-cycles are given by

$\displaystyle\left\\{\begin{array}[]{ll}\mathbb{F}_{0}&:\quad\operatorname{vol}(f)=2\phi,\quad\operatorname{vol}(e)=2\phi+m\,,\\\
\mathbb{F}_{1}&:\quad\operatorname{vol}(f)=2\phi,\quad\operatorname{vol}(e)=\phi+m\,.\end{array}\right.$
(98)

Here $m$ is a Kähler parameter for a non-compact 4-cycle that is identified
with the gauge coupling in the gauge theory. This implies that the curve $e$
in each geometry carries charge $+1$ for the $U(1)$ topological symmetry.

Let us consider a canonically quantized magnetic flux $n\in\mathbb{Z}$ for the
$SU(2)$ gauge symmetry in both $\mathbb{F}_{0}$ and $\mathbb{F}_{1}$ theories.
This magnetic flux can suitably couple to the state coming from a wrapped
M2-brane on the fiber curve $f$. This state is a W-boson with gauge charge
$+2$ and it feels an integer magnetic flux $2n$, which is compatible with the
quantization condition in (65).

Two theories have different quantization conditions on the background flux
$B_{m}$ for the $U(1)$ topological symmetry. Consider first the theory of
$\mathbb{F}_{0}$. The $e$ curve in this geometry gives rise to a vector
multiplet with charge $+2$ for the gauge symmetry and charge $+1$ for the
topological symmetry. From the quantization condition (65) with
$n\in\mathbb{Z}$, the background flux should be quantized as
$B_{m}\in\mathbb{Z}$. On the other hand, the state associated to the $e$ curve
in the theory of $\mathbb{F}_{1}$ is a hypermultiplet with gauge charge $+1$
and topological charge $+1$. Then the quantization condition (65) requires the
background flux to be half-integrally quantized, $B_{m}\in\mathbb{Z}+1/2$.
Hence we find the following quantizations for the magnetic fluxes:

$\displaystyle\left\\{\begin{array}[]{ll}\mathbb{F}_{0}&:\quad n\in\mathbb{Z}\
,\quad B_{m}\in\mathbb{Z}\\\ \mathbb{F}_{1}&:\quad n\in\mathbb{Z}\ ,\quad
B_{m}\in\mathbb{Z}+1/2\,.\end{array}\right.$ (101)

We note however that not all integer/half-integer fluxes $B_{m}$ are allowed
in the blowup equations; the blowup equations only with the consistent
magnetic fluxes can be solved to produce the correct BPS spectrum, as
discussed in the previous subsection. If one uses other sets of magnetic
fluxes, then the solution to the blowup equation will contain some BPS states
with negative volumes ${\rm vol}(C)<0$ on the Coulomb branch, which is surely
inconsistent. So we first need to identify the consistent magnetic fluxes
$B_{m}$.

Let us try to solve the blowup equation with magnetic fluxes $n\in\mathbb{Z}$
and some $B_{m}$. The magnetic fluxes result in the shifts of parameters in
the North and the South poles as $\phi\to\phi+n\,\epsilon_{1,2}$, $m\to
m+B_{m}\,\epsilon_{1,2}$. The GV-invariant (or the Witten index) can be
decomposed into two parts, the perturbative and the instanton parts, as

$\displaystyle Z_{GV}(\phi,m;\epsilon_{1},\epsilon_{2})$
$\displaystyle=\mathcal{Z}_{\rm
pert}(\phi;\epsilon_{1},\epsilon_{2})\cdot\mathcal{Z}_{\rm
inst}(\phi,m;\epsilon_{1},\epsilon_{2})\ ,$ $\displaystyle\mathcal{Z}_{\rm
pert}(\phi;\epsilon_{1},\epsilon_{2})$ $\displaystyle={\rm
PE}\left[-\frac{1+p_{1}p_{2}}{(1-p_{1})(1-p_{2})}e^{-2\phi}\right]\ ,$
$\displaystyle\mathcal{Z}_{\rm inst}(\phi,m;\epsilon_{1},\epsilon_{2})$
$\displaystyle=\sum_{k=0}^{\infty}q^{k}Z_{k}(\phi;\epsilon_{1},\epsilon_{2})\
,$ (102)

with the instanton fugacity $q\equiv e^{-m}$ and $Z_{0}=1$. The perturbative
part comes from the W-boson (or the M2-brane state on $f$) on the Coulomb
branch. Plugging the GV-invariant into (3.1.1), one finds a unity blowup
equation that can be expanded in terms of the instanton fugacity as

$\displaystyle\sum_{k,k^{\prime}=0}^{\infty}q^{k+k^{\prime}}\Lambda_{k^{\prime}}\hat{Z}_{k}(\phi;\epsilon_{1},\epsilon_{2})=$
$\displaystyle\
\sum_{n\in\mathbb{Z}}\sum_{k_{1},k_{2}=0}^{\infty}(-1)^{n}e^{-V}\frac{\hat{\mathcal{Z}}_{\text{pert}}^{(N)}\hat{\mathcal{Z}}_{\text{pert}}^{(S)}}{\hat{\mathcal{Z}}_{\text{pert}}}(q\,p_{1}^{B_{m}})^{k_{1}}(q\,p_{2}^{B_{m}})^{k_{2}}$
(103) $\displaystyle\
\times\hat{Z}_{k_{1}}^{(N)}(\phi+n\epsilon_{1};\epsilon_{1},\epsilon_{2}-\epsilon_{1})\cdot\hat{Z}^{(S)}_{k_{2}}(\phi+n\epsilon_{2};\epsilon_{1}-\epsilon_{2},\epsilon_{2})\,,$

with $V$ defined in (64), where $\Lambda_{0}=1$ which is fixed by the zeroth
order equation in the expansion. Here, the hatted functions are defined with
the shift in the parameter $\epsilon_{1}$ as

$\hat{f}(\phi,m;\epsilon_{1},\epsilon_{2})\equiv f(\phi,m;\epsilon_{1}+2\pi
i,\epsilon_{2})\ .$ (104)

One can extract the $k$-th order of the instanton expansion of this equation,
which can be schematically written as

$\displaystyle\Lambda_{k}(\epsilon_{1},\epsilon_{2})+\hat{Z}_{k}(\phi;\epsilon_{1},\epsilon_{2})=$
$\displaystyle~{}p_{1}^{kB_{m}}\,\hat{Z}_{k}(\phi;\epsilon_{1},\epsilon_{2}-\epsilon_{1})+p_{2}^{kB_{m}}\,\hat{Z}_{k}(\phi;\epsilon_{1}-\epsilon_{2},\epsilon_{2})$
(105) $\displaystyle+(\text{terms with }\hat{Z}_{r<k}\ \text{and
}\Lambda_{r<k})\ .$ (106)

Here the $k$-instanton partition function $\hat{Z}_{k}$ at the $k$-th order in
the expansion is independent of the background magnetic flux $B_{m}$ and
appears only with a trivial gauge flux $n=0$. All $\phi$ independent terms in
the second line (first by expanding it in $e^{-\phi}$) will be absorbed into
$\Lambda_{k}$. This equation can be solved once we know the solutions $Z_{r}$
at the lower orders $r<k$. In particular, since $\hat{Z}_{k}$ is independent
of $B_{m}$, when there exist 3 (or more) distinct consistent fluxes $B_{m}$,
we can exactly solve three linearly independent equations for 3 unknown
functions, $\hat{Z}_{k},\hat{Z}_{k}^{(N)},$ and $\hat{Z}^{(S)}_{k}$, and hence
find a closed expression for the $k$-instanton partition function
$\hat{Z}_{k}$ at each $k$.

The first order of the blowup equation can be written explicitly as

$\displaystyle\Lambda_{1}+\hat{Z}_{1}=$ $\displaystyle\
p_{1}^{B_{m}}\hat{Z}_{1}^{(N)}+p_{2}^{B_{m}}\hat{Z}_{1}^{(S)}-\frac{(p_{1}p_{2})^{B_{m}+1}e^{-2(2+B_{m})\phi_{1}}}{(1-e^{-2\phi})(1-p_{1}e^{-2\phi})(1-p_{2}e^{-2\phi})(1-p_{1}p_{2}e^{-2\phi})}$
$\displaystyle-\frac{(p_{1}p_{2})^{B_{m}-1}e^{-2(2-B_{m})\phi_{1}}}{(1-e^{-2\phi_{1}})(1-p_{1}^{-1}e^{-2\phi_{1}})(1-p_{2}^{-1}e^{-2\phi_{1}})(1-(p_{1}p_{2})^{-1}e^{-2\phi_{1}})}\
.$ (107)

The last two terms on the right-hand side come from the perturbative parts
with the gauge fluxes $n=1$ and $n=-1$ respectively. One can then easily see
that if $B_{m}>2$ or $B_{m}<-2$, then the last two terms will start with a
negative power of $e^{-\phi}$ in the expansion. This means that the solution
$\hat{Z}_{1}$ to the above equation will involve a BPS state of mass
$|M|=m+a\phi$ with a negative coefficient $a$. This is problematic because
this state violates the unitarity condition on its mass, i.e. $a\phi\geq 0$,
on the Coulomb branch parametrized by $\phi>0$ in the UV limit where
$m\rightarrow 0$. In addition, one can check that this state is not a
hypermultiplet and thus there is no associated flop transition. Thus the
solution is inconsistent due to the states with negative masses (or volumes)
in the UV limit. This tells us that solving the blowup equations with flux
$|B_{m}|>2$ cannot give the correct BPS partition function for the pure
$SU(2)_{\theta}$ gauge theory.

On the other hand, if $-2\leq B_{m}\leq 2$, then the solution will not contain
any states with ${\sf{e}}\cdot\phi<0$. Therefore we conclude that the
consistent magnetic fluxes for the $SU(2)_{0}$ and $SU(2)_{\pi}$ theories are

$\displaystyle\left\\{\begin{array}[]{ll}\mathbb{F}_{0}:~{}~{}SU(2)_{0}&\quad
n\in\mathbb{Z}\ ,\ \ B_{m}=-2,\ -1,\ 0,\ 1,2\\\
\mathbb{F}_{1}:~{}~{}SU(2)_{\pi}&\quad n\in\mathbb{Z}\ ,\ \
B_{m}=-\frac{3}{2},\ -\frac{1}{2},\ \frac{1}{2},\ \frac{3}{2}\
.\end{array}\right.$ (110)

Since we have more than three sets of consistent fluxes $(n,B_{m})$ for both
cases, we can compute a closed expression of the partition function at each
instanton order. For example, the 1-instanton partition functions are given as
follows:

$\displaystyle Z_{1}^{SU(2)_{0}}(\phi;\epsilon_{1},\epsilon_{2})$
$\displaystyle=\frac{p_{1}p_{2}(1+p_{1}p_{2})e^{-2\phi}}{(1-p_{1})(1-p_{2})(1-p_{1}p_{2}e^{-2\phi})(e^{-2\phi}-p_{1}p_{2})}\
,$ (111) $\displaystyle Z_{1}^{SU(2)_{\pi}}(\phi;\epsilon_{1},\epsilon_{2})$
$\displaystyle=-\frac{p_{1}^{3/2}p_{2}^{3/2}(1+e^{-2\phi})e^{-\phi}}{(1-p_{1})(1-p_{2})(1-p_{1}p_{2}e^{-2\phi})(e^{-2\phi}-p_{1}p_{2})}\
.$ (112)

By repeating this procedure for the blowup equations to higher orders in $q$,
it is straightforward to obtain the $k$-instanton partition functions $Z_{k}$.

We note that though there are many theories which possess three different sets
of consistent magnetic fluxes, many of which are discussed in Kim:2019uqw , a
much larger class of theories do not allow such three distinct sets. However,
as discussed in the previous subsection and also in Huang:2017mis ; Gu:2019pqj
, a single unity blowup equation with the consistent magnetic fluxes
$(\vec{n},\vec{B})$ is enough to compute the BPS partition function. We will
now illustrate this by computing the BPS spectra of the pure $SU(2)_{\theta}$
theories, using only a single background flux $B_{m}$ giving a unity blowup
equation.

##### Solving a unity blowup equation

Let us start with the following ansatz for the GV-invariant.

$\displaystyle Z_{GV}(\phi,m;\epsilon_{1,2})$ $\displaystyle={\rm
PE}\left[\sum_{j_{l},j_{r}}\sum_{d_{1},d_{2}=0}^{\infty}(-1)^{2(j_{l}+j_{r})}N_{j_{l},j_{r}}^{(d_{1},d_{2})}A_{j_{l},j_{r}}(\epsilon_{1},\epsilon_{2})e^{-d_{1}{\rm
vol}(e)-d_{2}{\rm vol}(f)}\right]$ $\displaystyle
A_{j_{l},j_{r}}(\epsilon_{1},\epsilon_{2})$
$\displaystyle\equiv\frac{\chi_{j_{l}}^{SU(2)}(p_{1}/p_{2})\chi_{j_{r}}^{SU(2)}(p_{1}p_{2})}{(p_{1}^{1/2}-p_{1}^{-1/2})(p_{2}^{1/2}-p_{2}^{-1/2})}\
.$ (113)

Here ${\bf d}=(d_{1},d_{2})$ denotes the degrees $d_{1}$ and $d_{2}$ of
2-cycles $e$ and $f$ respectively. The BPS states with $d_{1}=0$ at
0-instanton sector are all captured by the perturbative spectrum of the
$SU(2)$ theory. The perturbative spectrum then fixes
$N_{0,\frac{1}{2}}^{(0,1)}=1$ and $N_{j_{l},j_{r}}^{(0,d_{2})}=0$ for
$d_{2}>1$.

We shall now put the ansatz to the blowup equation and solve it order by order
in $d_{1}$ expansion. Let us first consider the $SU(2)_{0}$ theory with the
background flux $B_{m}=0$. The equation at $d_{1}=1$ order is given in
(3.2.1). To solve this equation, we further expand the equation in $d_{2}$ as

$\displaystyle\Lambda_{1}=$
$\displaystyle~{}\hat{Z}^{(N)}_{1}+\hat{Z}^{(S)}_{1}-\hat{Z}_{1}-\big{(}p_{1}p_{2}+(p_{1}p_{2})^{-1}\big{)}e^{-4\phi}$
(114)
$\displaystyle-\frac{(1+p_{1})(1+p_{2})(1+p_{1}^{3}p_{2}^{3})}{p_{1}^{2}p_{2}^{2}}e^{-6\phi}+\mathcal{O}(e^{-8\phi})\
.$ (115)

This equation can be easily solved order by order in $d_{2}$ or in the powers
of $e^{-\phi}$.

It turns out that this equation uniquely determines all the degeneracies
$N_{j_{l},j_{r}}^{(1,d_{2})}$, except for the degeneracies
$N_{0,\frac{1}{2}}^{(1,d_{2})}$ for all $d_{2}\geq 0$. The result is listed in
Table 2.

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,0)$ |  $N_{0,\frac{1}{2}}^{(1,0)}(0,\frac{1}{2})$ | $(1,1)$ |  $N_{0,\frac{1}{2}}^{(1,1)}(0,\frac{1}{2})\oplus(0,\frac{3}{2})$
$(1,2)$ |  $N_{0,\frac{1}{2}}^{(1,2)}(0,\frac{1}{2})\oplus(0,\frac{5}{2})$ | $(1,3)$ |  $N_{0,\frac{1}{2}}^{(1,3)}(0,\frac{1}{2})\oplus(0,\frac{7}{2})$
Table 2: The result of solving the $SU(2)_{0}$ blowup equation at $d_{1}=1$
order for the background flux $B_{m}=0$.

The degeneracies $N_{0,\frac{1}{2}}^{(1,d_{2})}$ are not determined in the 1st
order computation because they vanish in the combination
$\hat{Z}^{(N)}_{1}+\hat{Z}^{(S)}_{1}-\hat{Z}_{1}$ and thus do not appear in
the above equation. These undetermined degeneracies at $d_{1}=1$ order are all
fixed by solving the blowup equation in the next order or higher orders:
$N_{0,\frac{1}{2}}^{(1,0)}=1$ and $N_{0,\frac{1}{2}}^{(1,d_{2})}=0$ for
$d_{2}>0$.

We can solve the blowup equation iteratively and compute the degeneracies of
higher degree curves. The resulting BPS spectra for $d_{1}\leq 2$, $d_{2}\leq
3$ are summarized in Table 3. One can readily see that this result agrees with
the above result obtained by solving three blowup equations in (111).

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,0)$ |  $(0,\frac{1}{2})$ | $(1,1)$ |  $(0,\frac{3}{2})$
$(1,2)$ |  $(0,\frac{5}{2})$ | $(1,3)$ |  $(0,\frac{7}{2})$
$(2,1)$ |  $(0,\frac{5}{2})$ | $(2,2)$ |  $(0,\frac{5}{2})\oplus(0,\frac{7}{2})\oplus(\frac{1}{2},4)$
$(2,3)$ | $(0,\frac{5}{2})\oplus(0,\frac{7}{2})\oplus 2(0,\frac{9}{2})\oplus(\frac{1}{2},4)\oplus(\frac{1}{2},5)\oplus(1,\frac{11}{2})$
Table 3: BPS spectrum of the $SU(2)_{0}$ theory for $d_{1}\leq 2$, $d_{2}\leq
3$, where $\mathbf{d}=(d_{1},d_{2})$ labels the states from an M2-brane
wrapping $d_{1}e+d_{2}f$ curve in $\mathbb{F}_{0}$.

Next, consider the $SU(2)_{\pi}$ theory with a background magnetic flux
$B_{m}=\frac{1}{2}$. The computation is basically the same as the previous
case with $\theta=0$. The blowup equation at $d_{1}=1$ order is expanded as

$\displaystyle\Lambda_{1}=$
$\displaystyle~{}p_{1}^{1/2}\hat{Z}^{(N)}_{1}+p_{2}^{1/2}\hat{Z}^{(S)}_{1}-\hat{Z}_{1}-(p_{1}p_{2})^{1/2}e^{-3\phi}$
(116)
$\displaystyle-\frac{1+(1+p_{1})(1+p_{2})p_{1}^{2}p_{2}^{2}}{(p_{1}p_{2})^{3/2}}e^{-5\phi}+\mathcal{O}(e^{-7\phi})\
.$ (117)

All the degeneracies $N_{j_{l},j_{r}}^{(1,d_{2})}$ are uniquely determined by
solving this equation, except for the degeneracies $N_{0,0}^{(1,d_{2})}$ for
all $d_{2}\geq 0$. The undetermined degeneracies $N_{0,0}^{(1,d_{2})}$ are
also fixed uniquely by solving the equations in higher orders. We list the
result for $d_{1}\leq 2,d_{2}\leq 3$ in Table 4. Like the $SU(2)_{0}$ case,
this result agrees with the expansion of (112).

$\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$ | $\mathbf{d}$ |  $\oplus N_{j_{l},j_{r}}^{\mathbf{d}}(j_{l},j_{r})$
---|---|---|---
$(1,0)$ |  $(0,0)$ | $(1,1)$ |  $(0,1)$
$(1,2)$ |  $(0,2)$ | $(1,3)$ |  $(0,3)$
$(2,2)$ |  $(0,\frac{5}{2})$ | $(2,3)$ |  $(0,\frac{5}{2})\oplus(0,\frac{7}{2})\oplus(\frac{1}{2},4)$
Table 4:  BPS spectrum of the $SU(2)_{\pi}$ theory for $d_{1}\leq 2$,
$d_{2}\leq 3$, where $\mathbf{d}=(d_{1},d_{2})$ labels the states from an
M2-brane wrapping $d_{1}e+d_{2}f$ curve in $\mathbb{F}_{1}$.

##### Solving a vanishing blowup equation

In the $SU(2)$ gauge theories, it is possible to turn on non-canonically
quantized magnetic flux as $n\in\mathbb{Z}+1/2$ because the W-boson carries
charge +2 for the gauge symmetry. Then we can choose $B_{m}\in\mathbb{Z}$
which is a consistent quantization for the instanton state corresponding to
the curves $e$ in both $\mathbb{F}_{0}$ and $\mathbb{F}_{1}$. So the magnetic
fluxes

$\displaystyle n\in\mathbb{Z}+1/2\ ,\quad B_{m}=0\ ,$ (118)

satisfy the quantization conditions for both $\theta=0$ and $\pi$. These
fluxes with the effective prepotential for the $SU(2)$ theories lead to a
vanishing blowup equation. We will now show that this vanishing blowup
equation alone can be solved with additional geometric data. In fact, we find
two independent solutions to the blowup equation and they correspond to
respectively the BPS spectrum of the $SU(2)_{0}$ theory and that of the
$SU(2)_{\pi}$ theory.

Solving the vanishing blowup equation is much harder than solving unity
equations because of huge cancellations in the expansion and also non-
uniqueness of solutions. Let us start with an ansatz for the GV-invariant as

$\displaystyle
Z_{\mathrm{GV}}=\operatorname{PE}\quantity[\,\sum_{j_{l},j_{r}}\sum_{d_{1},d_{2}=0}^{\infty}(-1)^{2(j_{l}+j_{r})}N_{j_{l},j_{r}}^{(d_{1},d_{2})}A_{j_{l},j_{r}}(\epsilon_{1},\epsilon_{2})e^{-(d_{1}m+d_{2}\phi)}]\,.$
(119)

As discussed in Gu:2017ccq , the spins of a degree $(d_{1},d_{2})$ state are
bounded as

$\displaystyle j_{l}$ $\displaystyle\leq\frac{C^{2}\\!+\\!K_{S}\cdot
C}{2}\\!+\\!1=\frac{d_{1}d_{2}\\!-\\!2d_{1}^{2}\\!-\\!d_{2}\\!+\\!2}{2}\,,$
(120) $\displaystyle j_{r}$ $\displaystyle\leq\frac{C^{2}\\!-\\!K_{S}\cdot
C}{2}=\frac{d_{1}d_{2}\\!-\\!2d_{1}^{2}\\!+\\!d_{2}}{2}\,,$ (121)

where $C$ is a curve with volume $d_{1}m+d_{2}\phi$ and $K_{S}$ is the
canonical class in $\mathbb{F}_{0}$ or $\mathbb{F}_{1}$.

The zeroth-order in the expansion of the blowup equation is

$\displaystyle\sum_{n\in\\{\pm
1/2\\}}(-1)^{n}e^{-V}\frac{\hat{\mathcal{Z}}_{\mathrm{pert}}^{(N)}\hat{\mathcal{Z}}_{\mathrm{pert}}^{(S)}}{\hat{\mathcal{Z}}_{\mathrm{pert}}}=0\,,$
(122)

which is automatically satisfied by the perturbative spectrum
$N_{0,1/2}^{(0,2)}=1$. The 1st order blowup equation with $d_{1}=1$ is then
given by

$\displaystyle\sum_{n\in\\{\pm
1/2\\}}(-1)^{n}e^{-V}\frac{\hat{\mathcal{Z}}_{\mathrm{pert}}^{(N)}\hat{\mathcal{Z}}_{\mathrm{pert}}^{(S)}}{\hat{\mathcal{Z}}_{\mathrm{pert}}}\quantity(\hat{Z}_{1}^{(N)}+\hat{Z}_{1}^{(S)})=0\,.$
(123)

This equation for each $d_{2}$ can be rephrased in terms of the multiplicities
$N_{j_{l},j_{r}}^{(1,d_{2})}$ as

$\displaystyle\sum_{j_{l},j_{r}}(-1)^{2(j_{l}+j_{r})}N_{j_{l},j_{r}}^{(1,d_{2})}\quantity(A_{j_{l},j_{r}}(\epsilon_{1},\epsilon_{2}/\epsilon_{1})p_{1}^{-d_{2}/2}+A_{j_{l},j_{r}}(\epsilon_{1}/\epsilon_{2},\epsilon_{2})p_{2}^{-d_{2}/2}-A_{j_{l},j_{r}}(\epsilon_{1},\epsilon_{2}))$
$\displaystyle=\sum_{j_{l},j_{r}}(-1)^{2(j_{l}+j_{r})}N_{j_{l},j_{r}}^{(1,d_{2})}\quantity(A_{j_{l},j_{r}}(\epsilon_{1},\epsilon_{2}/\epsilon_{1})p_{1}^{d_{2}/2}+A_{j_{l},j_{r}}(\epsilon_{1}/\epsilon_{2},\epsilon_{2})p_{2}^{d_{2}/2}-A_{j_{l},j_{r}}(\epsilon_{1},\epsilon_{2}))\,.$
(124)

It turns out that this equation is not enough to fix the multiplicities
$N_{j_{l},j_{r}}^{(1,d_{2})}$, and in fact it has infinitely many solutions.
For example, if $N_{j_{l},j_{r}}^{(1,d_{2})}=c^{(d_{2})}$ with an integer
number $c^{(d_{2})}$ is a solution, then
$N_{j_{l},j_{r}}^{(1,d_{2})}=2c^{(d_{2})}$ is also another solution. We thus
need to solve higher order equations to determine them.

The 2nd order of the blowup equation can be written as

$\displaystyle\sum_{n\in\\{\pm
1/2\\}}(-1)^{n}e^{-V}\frac{\hat{\mathcal{Z}}_{\mathrm{pert}}^{(N)}\hat{\mathcal{Z}}_{\mathrm{pert}}^{(S)}}{\hat{\mathcal{Z}}_{\mathrm{pert}}}$
$\displaystyle\quantity(\hat{Z}_{2}^{(N)}+\hat{Z}_{2}^{(S)}+\hat{Z}_{1}^{(N)}\hat{Z}_{1}^{(S)})$
$\displaystyle+\sum_{n\in\\{\pm
3/2\\}}(-1)^{n}e^{-V}\frac{\hat{\mathcal{Z}}_{\mathrm{pert}}^{(N)}\hat{\mathcal{Z}}_{\mathrm{pert}}^{(S)}}{\hat{\mathcal{Z}}_{\mathrm{pert}}}=0\
.$ (125)

Unfortunately, this equation does not help us to fix the degeneracies
$N_{j_{l},j_{r}}^{(1,d_{2})}$ because they do not appear in this order. We
need the blowup equation at higher orders to fully fix
$N_{j_{l},j_{r}}^{(1,d_{2})}$. We find that at the order of the blowup
equation involving $N_{j_{l},j_{r}}^{(4,10)}$, the degeneracy
$N_{j_{l},j_{r}}^{(1,1)}$ is first fixed to be

$\displaystyle N_{j_{l},j_{r}}^{(1,1)}=0\ ,\quad\text{except
for}~{}~{}N_{0,0}^{(1,1)}=\Bigg{\\{}\begin{array}[]{ll}1&\quad\text{when~{}}\theta=\pi\\\
0&\quad\text{when~{}}\theta=0\end{array}\ .$ (128)

Here we fixed the discrete theta angles from the geometric information:
$N_{j_{l},j_{r}}^{(1,1)}=0$ for the $SU(2)_{0}$ theory, while
$N_{0,0}^{(1,1)}=1$ for the $SU(2)_{\pi}$ theory.

Now consider the $SU(2)_{0}$ theory. Solving the blowup equation at the order
having terms of $N_{j_{l},j_{r}}^{(4,12)}$ again gives two sets of solutions:
one with $N_{j_{l},j_{r}}^{(1,2)}=0$ for all $(j_{l},j_{r})$ and the other
with $N_{j_{l},j_{r}}^{(1,2)}=0$ but $N_{0,\frac{1}{2}}^{(1,2)}=1$. However,
the geometric realization suggests that there exists a BPS state with charge
$m+2\phi$, so only the second solution is acceptable. The next order blowup
equation which contains $N_{j_{l},j_{r}}^{(4,13)}$ gives
$N_{j_{l},j_{r}}^{(1,3)}=0$ for all $(j_{l},j_{r})$. Now to fix
$N_{j_{l},j_{r}}^{(1,4)}$, instead of calculating higher order equations, one
can use other consistency conditions. First, we know from geometry that there
are effective curves with volume $m+4\phi$. In addition, the blowup equation
yields that $N_{j_{l},j_{r}}^{(1,4)}=0$ for $(j_{l},j_{r})\neq(0,\frac{3}{2})$
and $N_{1,\frac{15}{2}}^{(3,12)}=1-N_{0,\frac{3}{2}}^{(1,4)}$. Notably,
$N_{1,\frac{15}{2}}^{(3,12)}$ must be a non-negative integer and
$N_{0,\frac{3}{2}}^{(1,4)}\neq 0$ from geometry. Therefore the only possible
solution is $N_{0,\frac{3}{2}}^{(1,4)}=1$. This result reproduces all the BPS
states of the $SU(2)_{0}$ theory up to $(d_{1},d_{2})=(1,4)$. In this way, a
vanishing blowup equation together with other consistency conditions can
determine the BPS spectrum in this theory. We expect other higher degree
states can also be captured by solving higher order blowup equations in this
manner.

Next, consider the $SU(2)_{\pi}$ theory. The blowup equation at the order
containing $N_{j_{l},j_{r}}^{(4,11)}$ fixes $N_{j_{l},j_{r}}^{(1,2)}=0$ for
all $(j_{l},j_{r})$. The higher order equations containing
$N_{j_{l},j_{r}}^{(4,14)}$ has two sets of solutions: one with
$N_{j_{l},j_{r}}^{(1,3)}=0$ for all $(j_{l},j_{r})$, and the other one with
$N_{j_{l},j_{r}}^{(1,3)}=0$ but $N_{0,1}^{(1,3)}=1$. Also, the geometry of
$\mathbb{F}_{1}$ says that there is a state with charge $m+3\phi$, so only the
second one is acceptable. The next order equation containing
$N_{j_{l},j_{r}}^{(4,15)}$ fixes $N_{j_{l},j_{r}}^{(1,4)}=0$. It is also
possible to fix $N_{j_{l},j_{r}}^{(1,4)}$ by consistency conditions instead of
calculating higher order equations. The blowup equation fixes
$N_{j_{l},j_{r}}^{(1,4)}=0$ for $(j_{l},j_{r})\neq(0,\frac{3}{2})$, and
$N_{1,\frac{15}{2}}^{(3,12)}=-N_{0,\frac{3}{2}}^{(1,4)}$, from which the only
possible solution is $N_{0,\frac{3}{2}}^{(1,4)}=0$. This result agrees with
the spectrum of the $SU(2)_{\pi}$ theory that we computed above using unity
bloup equations.

These computations show that, although the BPS spectrum is not completely
determined only by solving vanishing blowup equations, we may be able to
uniquely determine the BPS degeneracies by requiring the solution of the
vanishing blowup equations to be physically or geometrically consistent.

####  3.2.2 5d pure $SU(3)_{\kappa\leq 7}$

Let us now discuss rank-2 theories. A non-trivial example is the pure $SU(3)$
gauge theory at the Chern-Simons level $\kappa$. When $\kappa$ is even, it is
geometrically engineered as

(129)

Whereas, if $\kappa$ is odd, it is engineered by

(130)

Each geometry contains three primitive curves: the fiber $f_{1}$ in
$\mathbb{F}_{n_{1}}|_{1}$, the fiber $f_{2}$ and the base $e_{2}$ in
$\mathbb{F}_{n_{2}}|_{2}$. The volumes of the primitive 2-cycles are

$\displaystyle\operatorname{vol}(f_{1})=2\phi_{1}-\phi_{2},\quad\operatorname{vol}(f_{2})=-\phi_{1}+2\phi_{2},$
$\displaystyle\operatorname{vol}(e_{2})=\left\\{\begin{array}[]{ll}\left(1-\tfrac{\kappa}{2}\right)\phi_{1}+\phi_{2}+m&\
\ {\rm for\ even}\ \kappa\\\ \frac{1-\kappa}{2}\phi_{1}+2\phi_{2}+m&\ \ {\rm
for\ odd}\ \kappa\end{array}\right.\ .$ (133)

These theories with CS level $|\kappa|\leq 9$ have UV completions
Jefferson:2018irk ; Bhardwaj:2019jtr . The 5-brane webs for $|\kappa|\leq 7$
and $|\kappa|=9$ have been constructed in Hayashi:2018lyv .

Surprisingly the BPS spectra of the $SU(3)_{\kappa}$ theories for all
$|\kappa|\leq 9$ can be computed by employing our bootstrapping approach
explained above. The $SU(3)_{8}$ theory is of particular interest. Unlike
other $\kappa$, this theory at $\kappa=8$ has no consistent geometric
realization Jefferson:2018irk ; Bhardwaj:2019jtr ; Bhardwaj:2020gyu  and also
its 5-brane web is unknown yet. Moreover the usual ADHM construction for
instantons in this theory does not work. So it was a quite challenging task to
compute its BPS partition function. The $SU(3)_{8}$ theory and its blowup
equations will be discussed separately in subsection 5.2.5. The $SU(3)_{9}$
theory, which is a KK theory, will be discussed in subsection 3.2.3.

In this subsection, we will illustrate how to bootstrap the BPS spectra of
$|\kappa|\leq 7$ theories and their dual theories using the blowup equations
as well as their geometric descriptions.

Consider the $SU(3)_{\kappa}$ theory put on the blowup $\hat{\mathbb{C}}_{2}$.
We turn on the magnetic flux $\textsf{F}=(n_{1},n_{2},B_{m})$ for two
Hirzebruch surfaces and a non-compact divisor of the non-normalizable Kähler
parameter $m$. The quantization conditions for the fluxes are given by their
intersections with the primitive 2-cycles as

$\displaystyle\textsf{F}\cdot f_{1}=2n_{1}-n_{2}\ \in\ \mathbb{Z}\
,\qquad\textsf{F}\cdot f_{2}=2n_{2}-n_{1}\ \in\ \mathbb{Z}\ ,$
$\displaystyle\textsf{F}\cdot
e_{2}=\left\\{\begin{array}[]{ll}\left(1-\tfrac{\kappa}{2}\right)n_{1}+n_{2}+B_{m}\
\in\ \mathbb{Z}+\frac{1}{2}&\ \ {\rm for\ even}\ \kappa\\\
\frac{1-\kappa}{2}n_{1}+2n_{2}+B_{m}\ \in\ \mathbb{Z}&\ \ {\rm for\ odd}\
\kappa\end{array}\right.\ ,$ (136)

which follow from the geometric quantization condition (65) together with the
self-intersections of the curves, $f_{1}^{2}=f_{2}^{2}=0$ and $e_{2}^{2}=-1$
for even $\kappa$ and $e_{2}^{2}=0$ for odd $\kappa$. Each theory has 3 or
more sets of magnetic fluxes satisfying the quantization conditions. We can
use, for example, the following fluxes for solving the blowup equations:

$n_{1},n_{2}\in\mathbb{Z}\quad{\rm and}\quad
B_{m}=\left\\{\begin{array}[]{ll}-\frac{1}{2},\frac{1}{2},\frac{3}{2}&\quad\kappa=0,2,4,6\\\
-1,0,1&\quad\kappa=1,3,5,7\ .\end{array}\right.$ (137)

Let us now consider the Coulomb branch of the theory where
$2\phi_{1}-\phi_{2}\geq 0$ and $2\phi_{2}-\phi_{1}\geq 0$ with
$\phi_{1},\phi_{2}>0$. The effective prepotential $\mathcal{E}$ on the Coulomb
branch takes the following form

$\displaystyle\mathcal{E}$