aainstitutetext: Center for High Energy Physics, Peking University,
Beijing 100871, Chinabbinstitutetext: School of Physics and Electronic
Science, East China Normal University,
Shanghai 200050, Chinaccinstitutetext: School of Physics, Korea Institute for
Advanced Study,
85 Hoegi-ro Dongdaemun-gu, Seoul 02455, Republic of Koreaddinstitutetext:
School of Physics,
Peking University, Beijing 100871, China

# 3d $\mathcal{N}=2$ theories from M-theory on CY4 and IIB brane box

Marwan Najjar b,c Jiahua Tian a,d Yi-Nan Wang<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study 3D $\mathcal{N}=2$ supersymmetric field theories geometrically
engineered from M-theory on non-compact Calabi-Yau fourfolds (CY4). We
establish a detailed dictionary between the geometry and topology of non-
compact CY4 and the physics of 3D $\mathcal{N}=2$ theories in three different
regimes. The first one is the Coulomb branch description when the CY4 is
smooth. The second one is non-abelian gauge theory when the CY4 has a
degenerate $\mathbb{P}^{1}$-fibration structure. The third one is the strongly
coupled SCFT from a CY4 singularity. We find interesting flavor symmetry
enhancements in the singular limit of CY4, as well as an interesting and
previously unexplored phenomenon in 3D, termed “flavor symmetry duality”. Many
examples are analyzed with an emphasis on toric CY4s and $\mathbb{C}^{4}$
orbifolds with crepant resolutions. We develop a new brane box method to study
the physics of Coulomb branch of 3D $\mathcal{N}=2$ theory that admits a toric
construction. Via IIB/M-theory duality we find that the brane box diagram
living in $\mathbb{R}^{3}$ can be physically realized as a configuration of
intersecting 4-branes which are extended objects in 8D maximal supersymmetric
theory, which is shown to be consistent via various chains of dualities. The
rank, effective gauge coupling and certain hints to flavor symmetry
enhancement of the 3D $\mathcal{N}=2$ theory are read off from the brane box
and cross-checked against the results obtained from geometric engineering. The
exotic branes in 8D maximal supersymmetric theory and the 4-string junctions
thereof are shown to play a crucial role in the construction of the brane box.

††preprint:

##  1 Introduction

In recent years, the classification and study of 5d superconformal field
theories (SCFTs) has been an active subject, from either M-theory on local
Calabi-Yau threefold geometries (canonical threefold singularities) or brane
web constructions in IIB superstring theory Seiberg:1996bd ; Morrison:1996xf ;
Intriligator:1997pq ; Aharony:1997ju ; Aharony:1997bh ; DeWolfe:1999hj ;
Benini:2009gi ; Kim:2012gu ; Bergman:2013aca ; Bergman:2013koa ;
Zafrir:2014ywa ; Bergman:2015dpa ; Hayashi:2015zka ; Zafrir:2015rga ;
Zafrir:2015uaa ; Zafrir:2015ftn ; Xie:2017pfl ; Ferlito:2017xdq ;
Jefferson:2017ahm ; Hayashi:2018bkd ; Hayashi:2018lyv ; Jefferson:2018irk ;
Bhardwaj:2018vuu ; Closset:2018bjz ; Cabrera:2018jxt ; Apruzzi:2018nre ;
Bhardwaj:2018yhy ; Apruzzi:2019vpe ; Apruzzi:2019opn ; Apruzzi:2019enx ;
Apruzzi:2019kgb ; Bhardwaj:2019ngx ; Bhardwaj:2019jtr ; Bhardwaj:2019fzv ;
Bhardwaj:2019xeg ; Hayashi:2019jvx ; Saxena:2020ltf ; Bhardwaj:2020gyu ;
Eckhard:2020jyr ; Morrison:2020ool ; Collinucci:2020jqd ; Closset:2020scj ;
vanBeest:2020kou ; Bhardwaj:2020ruf ; Hubner:2020uvb ; Bhardwaj:2020avz ;
VanBeest:2020kxw ; Kim:2020hhh ; Hayashi:2021pcj ; Apruzzi:2021vcu ;
Collinucci:2021ofd ; vanBeest:2021xyt ; Acharya:2021jsp ; Tian:2021cif ;
Closset:2021lwy ; Kim:2021fxx ; DelZotto:2022fnw ; Collinucci:2022rii ;
Xie:2022lcm ; DeMarco:2022dgh ; Bourget:2023wlb ; DeMarco:2023irn ; Mu:2023uws
. Physical information of the 5d SCFTs such as the properties of its Coulomb
branch (CB) and Higgs branch (HB), UV enhanced flavor symmetry, IR non-abelian
gauge descriptions and generalized symmetries have been extensively studied
via different approaches. Various partial classification results have been
achieved as well.

In contrast, very few progress has been made on the direct geometric
engineering of 3d $\mathcal{N}=2$ SUSY field theories and SCFTs
$\mathcal{T}^{\rm 3d}_{X_{4}}$ from M-theory on local Calabi-Yau fourfolds,
apart from the early works Leung:1997tw ; Diaconescu:1998ua ; Gukov:1999ya
and more recent developments Intriligator:2012ue ; Jockers:2016bwi ;
Chen:2022vvd . In this paper, we aim to establish a systematic framework of
this geometric approach, which represents an entirely new direction to study
the rich family of 3d $\mathcal{N}=2$ field theories Hanany:1996ie ;
Aharony:1997bx ; Aharony:1997gp ; Aharony:1997ju ; Kapustin:1999ha ;
Bergman:1999na ; Dorey:1999rb ; Tong:2000ky ; Borokhov:2002cg ; Gaiotto:2007qi
; Benini:2009qs ; Imamura:2011su ; Benini:2011cma ; Jafferis:2011ns ;
Dimofte:2011ju ; Benini:2011mf ; Cecotti:2011iy ; Dimofte:2011py ;
Closset:2012ep ; Closset:2012vg ; Closset:2012vp ; Intriligator:2013lca ;
Aharony:2013dha ; Aharony:2013kma ; Amariti:2015yea ; Closset:2019hyt ;
Eckhard:2019jgg ; Nii:2020ikd ; Sacchi:2021afk ; vanBeest:2022fss ;
Benvenuti:2023qtv .

We would like to first comment on the fundamental difference between the 5d
and 3d physics picture. We plot the different phases of 5d theories in figure
1. A 5d $\mathcal{N}=1$ SCFT $\mathcal{T}^{\rm 5d}_{X_{3}}$ is defined as
M-theory on a canonical threefold singularity, which is the full singular
limit of a non-compact CY3 $X_{3}$. The CB deformation of $\mathcal{T}^{\rm
5d}_{X_{3}}$ is given by M-theory on the crepantly resolved $X_{3}$, and one
obtains the IR CB effective theory after integrating out the massive modes and
RG flow to the deep IR. Finally, for some cases of $X_{3}$, one can define a
$\mathbb{P}^{1}$-fibration (ruling) structure and let all $\mathbb{P}^{1}$
fibers shrink to zero volume. From the field theory perspective, this
corresponds to an IR 5d non-abelian SUSY gauge theory description of the UV
SCFT $\mathcal{T}^{\rm 5d}_{X_{3}}$, which can be obtained after adding a
relevant deformation to $\mathcal{T}^{\rm 5d}_{X_{3}}$ and triggering an RG
flow. We have omitted the Higgs branch from the picture.

Figure 1: The relation between 5d SCFTs, non-abelian gauge theory description
and Coulomb branch theories.

In the 3d case, the general physical picture is different since for a generic
3d $\mathcal{N}=2$ SUSY field theory, one expect a (possibly free or gapped)
fixed point in the IR. We plot the different geometric limits that correspond
to different physical phases in figure 2. In the picture we assume that there
is no free $G_{4}$ flux being turned on.

Figure 2: Three geometric limits of $X_{4}$ and the physics of M-theory on
different limits. Note that there are three different IR limits.

Before getting into the details of $X_{4}$ or $\mathcal{T}^{\rm 3d}_{X_{4}}$
we comment on the different physical scales involved in this work. There is a
deep UV where every state in $\mathcal{T}^{\rm 3d}_{X_{4}}$ is effectively
massless though the physics at this scale is relatively unattractive to us, as
by construction we would expect to recover M-theory itself in the deep UV. The
physics that is interesting to us happens below the compactification scale,
which is much lower than the 11D Planck scale due to that our space $X_{4}$
has infinite volume (analogous to Cordova:2009fg ). At this scale where
gravity has been already decoupled while the gauge dynamics is still weakly
coupled, $\mathcal{T}^{\rm 3d}_{X_{4}}$ can be studied semi-classically
without caring too much about the quantum corrections. Therefore we expect
that there exists a very precise match between the geometric data of $X_{4}$
and the physical data of $\mathcal{T}^{\rm 3d}_{X_{4}}$. Finally, there is a
deep IR where all the massive states are integrated out in a Wilsonian manner.
At this scale, the gauge dynamics becomes strongly coupled such that any semi-
classical analysis fails. More importantly, the non-perturbative
superpotential that was suppressed by weak gauge coupling in the UV can no
longer be ignored at this scale and it does significantly modify the moduli
space in the IR, sometimes completely lifts it. We will remind the reader of
the scale we are at whenever it may not be clear from the context, and when it
is necessary we will distinguish between different scales via denoting by
$\mathcal{T}^{UV}_{X_{4}}$ the physical theory just below the compactification
scale where gravity has been decoupled while the gauge dynamics is still
weakly coupled, and by $\mathcal{T}^{IR}_{X_{4}}$ the physical theory in the
deep IR where all the massive states have been integrated out.

Now we briefly discuss the different geometric limits in figure 2, and we
present a more complete discussion in section 2.5. First, the case of M-theory
on fully resolved CY4 $X_{4}$ correspond to a CB description
$\mathcal{T}^{UV}_{X_{4}}$ that is a $U(1)^{r}$ gauge theory coupled to
charged matter fields (here $r$ is the rank of the 3d $\mathcal{N}=2$ theory).
From this point, it can RG flow to a deep IR CB effective theory
$\mathcal{T}^{IR}_{X_{4}}$. Secondly, if $X_{4}$ has a
$\mathbb{P}^{1}$-fibration structure, one can consider a geometric limit
$\widehat{X}_{4}$ where all the $\mathbb{P}^{1}$ fibers are shrunk to zero
volume. In this case, the W-bosons in the CB description become massless and
we obtain a 3d $\mathcal{N}=2$ non-abelian gauge theory
$\mathcal{T}^{UV}_{\widehat{X}_{4}}$. Such a theory would flow to another deep
IR theory $\mathcal{T}^{IR}_{\widehat{X}_{4}}$111One should keep in mind that
generically the theories $\mathcal{T}^{UV}_{X_{4}}$ and
$\mathcal{T}^{UV}_{\widehat{X}_{4}}$ may have a non-zero superpotential $W$,
which lead to no SUSY vacua (the CB is lifted). This issue is further
discussed in section 2.3. Nonetheless we would neither compute the detailed
dependence of the superpotential on all the geometric moduli fields nor
compute too much details about the deep IR effective field theories. Also note
that even if the CB is lifted, we still use the notion of CB to describe the
deformed 3d field theory from a resolved CY4.. Finally, if all the compact
cycles in $X_{4}$ are shrunk to zero volume, we arrive at the singular limit
$X_{\text{4,sing}}$ of $X_{4}$, with no geometric scale. M-theory on
$X_{\text{4,sing}}$ would give rise to a 3d $\mathcal{N}=2$ SCFT
$\mathcal{T}^{\rm 3d}_{X_{\text{4,sing}}}$ with $W=0$, since the geometric
picture contains no scale parameter at all. In section 2.5.3 we present a
detailed shrinkability criteria of $X_{4}$ for the existence of
$\mathcal{T}^{\rm 3d}_{X_{\text{4,sing}}}$.

In section 2, we establish a dictionary between the geometrical and
topological data of resolved $X_{4}$ and the physical properties of
$\mathcal{T}^{\rm 3d}_{X_{4}}$, including the geometric origins of gauge
fields, gauge coupling constants, CB parameters and FI parameters, background
gauge fields for flavor symmetries and charged particles. We discuss the BPS
particle spectrum from M2-brane wrapping 2-cycles $C$ on $X_{4}$, whose normal
bundle in $X_{4}$ is either
$\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$ or
$\mathcal{O}\oplus\mathcal{O}(-1)\oplus\mathcal{O}(-1)$, from the twisted
compactification of either 7D or 5D SYM theory, analogous to Beasley:2008dc ;
Intriligator:2012ue . We further discuss non-BPS higher modes as well in
appendix A. We also briefly discuss the schematics of superpotential and the
physical effects of turning on a non-zero $G_{4}$ flux. In particular, after
such $G_{4}$ flux is included, the singular limit of $X_{4}$ would be
topologically obstructed, since the 4-cycles supporting $G_{4}$ flux cannot
shrink to zero volume222We thank Hong Lu, Ruben Minasian and Dan Xie for
illuminating discussions about this point.. In the IR limit of the CB
effective field theory, a non-zero $G_{4}$ flux will induce a non-zero Chern-
Simons level from the dimensional reduction of 11d SUGRA action.

In section 3, we give a more detailed physical dictionary for the cases of
toric CY4 $X_{4}$, which is the main class of examples covered in this paper.
In section 3.1, we discuss the identification of ruling structures of $X_{4}$
and non-abelian gauge theory descriptions of $\mathcal{T}^{\rm 3d}_{X_{4}}$.
In section 3.2, we discuss in detail the non-abelian flavor symmetry
enhancement of the 3d $\mathcal{N}=2$ SCFT $\mathcal{T}^{\rm
3d}_{X_{\text{4,sing}}}$ at the singular limit $X_{\text{4,sing}}$. It can be
read off from the construction the flavor Cartan subalgebra from certain
linear combinations of non-compact divisors and flavor W-bosons from M2-brane
wrapping certain 2-cycles (which can be non-effective). We find a generic
flavor symmetry enhancement for the cases of local
$\mathbb{P}^{1}\times\mathcal{S}$. More interestingly, in section 3.3 we find
cases with two different possible flavor symmetry enhancements at the singular
limit and we will call this phenomenon “flavor symmetry duality”. In section
3.4 we review the calculation of 1-form symmetry and the SymTFT from geometry,
following the general recipe in vanBeest:2022fss .

In section 4.1, we discuss many geometric and physical aspects of toric CY4
examples. In particular, we studied the case of local
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ in detail. Such a
geometry has three $\mathbb{P}^{1}$-fibration structures, which lead to
triality among three different $SU(2)$ gauge theory descriptions. We have also
observed an interesting $\mathfrak{su}(3)$ flavor symmetry enhancement at the
singular limit via identifying flavor Cartan and flavor W-bosons on local
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ using the dictionary
developed in section 2 and 3. Furthermore, for this example we also discuss
its 1-form symmetry, the corresponding SymTFT and the consequences of turning
on $G_{4}$ flux. In this section we also present a number of other rank-1
examples, as well as higher-rank examples which admit $SU(N)$ gauge theory
limits. In section 4.2 we initiate the study of a rich class of non-compact
Calabi-Yau geometries which can be constructed by orbifolding $\mathbb{C}^{4}$
by a finite subgroup $\Gamma$ of $SU(4)$, i.e. $X_{4}=\mathbb{C}_{4}/\Gamma$.
While these finite groups have been already classified in HananyHe_SU4 ,
certain properties that are particularly relevant for geometrically
engineering $\mathcal{T}^{\rm 3d}_{X_{4}}$ have not been studied thoroughly in
literature. Via applying higher dimensional McKay correspondence ItoReid ;
ito2011mckay , we calculate and tabulate those group theoretical quantities
that correspond to certain discrete data of $\mathcal{T}^{\rm 3d}_{X_{4}}$,
e.g. its gauge rank and flavor rank. In section 4.3 we present a 3d
$\mathcal{N}=4$ example of local
$T^{2}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$. This case is technically
excluded by the shrinkability criteria we proposed in section 2.5.3, but it is
of physical interest in its own right. We observe an $\mathfrak{su}(2)$
geometric flavor symmetry enhancement in this case.

Besides geometric engineering of 3D $\mathcal{N}=2$ theory from M-theory on
non-compact CY4, we develop a new _brane box_ method to study 3D
$\mathcal{N}=2$ theories in section 5. We find that the branes in a brane box
diagram are the $(p,q,r)$ 4-branes labelled by a $(p,q,r)$ charge triplet in
8D maximal supersymmetric theory obtained from type IIB on $T^{2}$ studied in
Leung:1997tw ; Lu:1998sx ; Lu:1998mr . We emphasize that the brane boxes in
this work are fundamentally different from the brane webs that were used to
construct 3d $\mathcal{N}=2$ theories in, e.g. Hanany:1996ie ; Aharony:1997ju
; Bergman:1999na ; Benvenuti:2016wet ; Amariti:2019rhc ; Cheng:2021vtq .

In section 5.1, starting from a system of Kaluza-Klein (KK) monopoles in
M-theory, we establish the duality between a toric Calabi-Yau 4-fold (CY4) and
an intersecting brane/plane configuration in $\mathbb{R}^{3}$ following the
method in Leung:1997tw . The KK-monopole system has been studied via various
string dualities in the type II dual picture and in the language of 8D
supersymmetric theory in detail Lu:1998mr ; Lu:1998sx ; Eyras:1998hn ;
ortin2004gravity ; Elitzur:1997zn ; deBoer:2012ma . Following Leung:1997tw  we
show in section 5.1.1 that the M-theory KK-monopole configuration is
equivalent to a toric CY4 and it can be visually represented as a
configuration of intersecting planes in $\mathbb{R}^{3}$. Motivated by the
study of 5-brane webs Aharony:1997ju ; Aharony:1997bh  we find in section
5.1.2 that the intersecting planes are physically realized as intersecting
branes in type IIB via the IIB/M-theory duality. We then find in section 5.1.3
that these branes are best understood in the framework of 8D maximal
supersymmetric field theory obtained from IIB on $T^{2}$. In this 8D framework
certain symmetry properties, e.g. an $SL(3,\mathbb{Z})$ modular symmetry that
were manifest in the M-theory description but was obscured in the 10D IIB
description, become manifest again. The branes in the brane box are shown to
be the $(p,q,r)$ 4-branes studied in Leung:1997tw ; Lu:1998sx ; Lu:1998mr .
Various physical properties of a consistent intersecting $(p,q,r)$ 4-brane
configuration are then studied, including the $(p,q,r)$ charge conservation
law at the intersections of 4-branes or of 4-strings, brane bending when a
$(p,q,r)$ 4-brane meets a $(p^{\prime},q^{\prime},r^{\prime})$ 4-brane, etc.
We are then able to establish a dictionary between the diagrammatical
properties of the brane box diagram and the geometric properties of the dual
toric CY4 in section 5.1.4.

Having established the duality between a brane box diagram and a toric CY4 in
section 5.1, we devote section 5.2 to the study of the physics of a 3D
$\mathcal{N}=2$ theory on its CB when it is constructed as an intersecting
$(p,q,r)$ 4-brane system. We are able to read off the rank of CB, the
effective coupling and the limits of non-abelian gauge theory enhancement and
SCFT from the diagrammatical properties of the brane box without referring to
the dual toric geometry. The key is to relate the properties of the theory on
CB to (local) deformations of the branes in the brane box in a way similar to
that in Aharony:1997ju ; Aharony:1997bh . We cross-check the results obtained
from the brane box against the results obtained from toric geometry in section
4 and find perfect match.

Motivated by the introduction of 7-branes into a 5-brane web DeWolfe:1999hj ,
in section 5.3 we extend the construction of brane boxes to include the
codimension-2 branes in 8D, which are 8D 5-branes, that can be introduced into
the original brane box without further breaking SUSY. We review the relevant
facts about the relation between these 8D 5-branes and the exotic branes that
has been extensively studied in Hull:1994ys ; ortin2004gravity ;
Elitzur:1997zn ; Eyras:1999at ; Obers:1998fb ; deBoer:2010ud ; deBoer:2012ma ;
Bergshoeff:2011qk ; Bergshoeff:2011se ; Kleinschmidt:2011vu ; Kimura:2016yqa ;
Fernandez-Melgarejo:2018yxq . We find that these 8D 5-branes also carry
$(p,q,r)$ charges allowing the 4-branes and effective strings in 8D carrying
the same charge to end on them properly without breaking the law of charge
conservation. Furthermore, we show that a brane creation/annihilation process
similar to Hanany-Witten effects Hanany:1996ie  are realized in this extended
brane box system via studying brane movements in different type IIB
descriptions of the same brane box.

In the section 5.4 we investigate various new diagrammatical properties of the
brane box diagram extended by adding codimension-2 branes, i.e. the $(p,q,r)$
5-branes in 8D. The notion of a global deformation of the brane box is then
introduced and used to calculate the rank of the flavor symmetry of the
corresponding 3D $\mathcal{N}=2$ theory. We then investigate the flavor
symmetry enhancement of a certain class of 3D $\mathcal{N}=2$ theories. Such a
theory is associated to a brane box diagram dual to a geometry in certain
special class of toric CY4’s which can be constructed via interlacing two
different toric CY3’s (see Appendix B for an introduction to the notion of
interlacing CY’s). The interlacing structure of geometry can be studied via
looking at the brane box at various decoupling limits in different type IIB
duality frames. At certain limits one recovers the ordinary 5-brane web
description with additional 7-branes and the flavor symmetries of the theories
associated to these brane webs can be read off using the standard methods in
DeWolfe:1998zf ; DeWolfe:1998bi ; DeWolfe:1999hj ; Distler:2019eky . Going
back to the brane box description via relaxing from the various limits, the
flavor symmetry of the 3D theory associated to the brane box can be viewed as
an interweaving of the symmetries that can be read off from the ordinary
5-brane webs obtained from looking at those limits.

##  2 Geometric dictionary

In this section, we consider several general aspects of 3D $\mathcal{N}=2$
theories $\mathcal{T}_{X_{4}}$ obtained from M-theory on a non-compact Calabi-
Yau fourfold $X_{4}$. We present the dictionary between the geometry and
topology of $X_{4}$ and the field theoretical quantities of 3d $\mathcal{N}=2$
theory. Certain subtle aspects of the dictionary will be explained by looking
at concrete examples where $X_{4}$ can be a (smooth) toric CY4, or a crepant
resolution of a CY4 singularity.

###  2.1 Uncharged states from supergravity multiplet and gauge coupling

We consider a non-compact resolved CY4 $X_{4}$ with a number of compact
divisors $D_{i}$ $(i=1,\dots,r)$, non-compact divisors $S_{\alpha}$
$(\alpha=1,\dots,f)$ and $2b$ 5-cycles. A non-compact divisor $S_{\alpha}$ in
$X_{4}$ cannot be written as a linear combination of $D_{i}$’s.

In the reduction of M-theory on $X_{4}$, the 3-form field $C_{3}$ in M-theory
can be decomposed as

$C_{3}=\sum_{i=1}^{r}A_{i}\wedge\omega_{i}^{(1,1)}+\sum_{k=1}^{b}(\mathcal{N}_{k}\omega_{k}^{(2,1)}+\widetilde{\mathcal{N}}_{k}\bar{\omega}_{k}^{(2,1)})+\sum_{\alpha=1}^{f}B_{\alpha}\wedge\omega_{\alpha}^{(1,1),F}$
(1)

There is a dynamical $U(1)$ gauge field $A_{i}$ associated with each compact
divisor $D_{i}\subset X_{4}$ where $\omega_{i}^{(1,1)}$ is the Poincaré dual
$(1,1)$-form of $D_{i}$. Following the notations in Grimm:2010ks , each
$\mathcal{N}_{k}$ is a real scalar associated to a (2,1)-form
$\omega_{k}^{(2,1)}$ which is the Poincaré dual of a compact 5-cycle in
$X_{4}$. In this paper we are mostly focusing on the either toric CY4 or
$\mathbb{C}^{4}$ orbifolds where there are no 5-cycles hence there will be no
$\mathcal{N}_{k}$’s in the corresponding 3d field theory. The decomposition of
other fields in the gravity multiplet can be fixed by SUSY.

Besides the dynamical gauge fields $A_{i}$, $C_{3}$ also gives rise to the
background $U(1)$ 1-form gauge fields $B_{\alpha}$ that generate 0-form flavor
symmetries. They correspond to non-compact divisors $S_{\alpha}$ Poincaré dual
to $\omega_{\alpha}^{(1,1),F}$. In some cases, one can gauge the flavor
symmetries by compactifying $S_{\alpha}$ to make $B_{\alpha}$ dynamical.

To compute the volume of various holomorphic cycles in $X_{4}$, we need to
introduce the Kähler $(1,1)$-form

$J(X_{4})=\sum_{i=1}^{r}a_{i}\omega_{i}^{(1,1)}+\sum_{\alpha=1}^{f}b_{\alpha}\omega_{\alpha}^{(1,1),F}\,,$
(2)

which is Poincaré dual to

$J^{c}(X_{4})=\sum_{i=1}^{r}a_{i}D_{i}+\sum_{\alpha=1}^{f}b_{\alpha}S_{\alpha}\,.$
(3)

Then the volume of each compact complex curve (2-cycle) $C$ can be computed as

$\displaystyle V_{C}$ $\displaystyle=\int_{C}J$ (4) $\displaystyle=C\cdot
J^{c}|_{X_{4}}\,.$

The volume of each compact complex surface (4-cycle) $\mathcal{S}$ can be
computed as

$\displaystyle V_{\mathcal{S}}$
$\displaystyle=\frac{1}{2}\int_{\mathcal{S}}J\wedge J$ (5)
$\displaystyle=\frac{1}{2}\mathcal{S}\cdot J^{c}\cdot J^{c}|_{X_{4}}\,.$

The volume of each compact divisor (6-cycle) $D$ can be computed as

$\displaystyle V_{D}$ $\displaystyle=\frac{1}{6}\int_{D}J\wedge J\wedge J$ (6)
$\displaystyle=\frac{1}{6}D\cdot J^{c}\cdot J^{c}\cdot J^{c}|_{X_{4}}\,.$

The 3d gauge coupling $g_{i}$ of dynamical gauge field $A_{i}$ is computed
from the dimensional reduction of the $G_{4}$ kinetic term in the 11D
supergravity action (we only keep the terms with the field strength $F_{i}$ of
$A_{i}$)333Here we omitted the higher-derivative corrections and other quantum
corrections of the supergravity.:

$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{1,2}\times X_{4}}G_{4}\wedge\star
G_{4}$ $\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{1,2}}F_{i}\wedge\star
F_{i}\int_{X_{4}}\omega_{i}^{(1,1)}\wedge\star\omega_{i}^{(1,1)}+(\dots)$ (7)
$\displaystyle=\frac{1}{2g_{i}^{2}}\int_{\mathbb{R}^{1,2}}F_{i}\wedge\star
F_{i}+(\dots)$

where

$\displaystyle\frac{1}{g_{i}^{2}}$
$\displaystyle=\int_{X_{4}}\omega_{i}^{(1,1)}\wedge\star\omega_{i}^{(1,1)}$
(8)
$\displaystyle=-\frac{1}{2}\int_{X_{4}}\omega_{i}^{(1,1)}\wedge\omega_{i}^{(1,1)}\wedge
J\wedge J$ $\displaystyle=-\frac{1}{2}\int_{D_{i}\cdot D_{i}}J\wedge J$
$\displaystyle=-\frac{1}{2}D_{i}\cdot D_{i}\cdot J^{c}\cdot J^{c}$
$\displaystyle=-\text{Vol}(D_{i}\cdot D_{i})\,.$

in terms of the 11D Planck unit. Here we used the expression for Hodge star
operator in CY4 Weissenbacher:2019mef , and that
Vol$(X_{4})\rightarrow\infty$. Note that in the toric cases, which will be
elucidated in section 3, (8) is equal to the sum of the volume of all toric
divisors on $D_{i}$.

Now let us discuss the two real parameters. The first one is real CB parameter
$\sigma_{i}$ which equals to the vev of the real scalar field in the dynamical
(gauge) $U(1)$ vector multiplet. The second one is the FI parameter
$\xi_{\alpha}$ which equal to the VEV of the scalar field in the background
(flavor) $U(1)$ vector multiplet. Analogous to the 5d case Morrison:1996xf ,
they come from the coefficients in the Kähler form:

$J(X_{4})=\sum_{i=1}^{r}a_{i}\omega_{i}^{(1,1)}+\sum_{\alpha=1}^{f}b_{\alpha}\omega_{\alpha}^{(1,1),F}\,.$
(9) $\sigma_{i}={a_{i}}\ ,\ \xi_{\alpha}={b_{\alpha}}\,.$ (10)

At a generic point on the CB, these parameters are non-vanishing.

###  2.2 Charged states from M2-brane wrapping modes

The charged BPS states444It was mentioned in Aharony:1997bx  that the total
energy of a 3d particle charged under $U(1)$ gauge symmetry diverges.
Nonetheless here our discussions are in the UV where the presence of light
W-bosons provide a screening effect of the otherwise logarithmically decayed
Coulomb interaction. of the 3d $\mathcal{N}=2$ theory come from M2-branes
wrapping various 2-cycles $C\subset X_{4}$. The charge of these BPS states
under the $U(1)$ gauge (flavor) symmetry associated with $D_{i}$
($S_{\alpha}$) can be calculated via the intersection numbers:

$q_{i}=C\cdot D_{i}\quad(\mathrm{or}\ C\cdot S_{\alpha})\,.$ (11)

In this paper we only consider the case of a single M2 (anti-M2) brane on a
rational curve $C=\mathbb{P}^{1}$, analogous to the discussions in the CY3
cases Witten:1996qb ; Kachru:2018nck ; Tian:2018icz . Moreover in this section
we assume that there is no $G_{4}$-flux on $X_{4}$.

The normal bundle of $C$ in $X_{4}$, $N_{C|X_{4}}$ is a rank-3 vector bundle
over $C$, which can be decomposed as

$N_{C|X_{4}}=\mathcal{O}(d_{1})\oplus\mathcal{O}(d_{2})\oplus\mathcal{O}(d_{3})\,.$
(12)

Applying Riemann-Roch formula on $C$ given the Ricci-flatness of $X_{4}$, we
have

$d_{1}+d_{2}+d_{3}=-2\,.$ (13)

If $C$ is a complete intersection curve of three divisors $D_{1}$, $D_{2}$ and
$D_{3}$, i.e.

$C=D_{1}\cdot D_{2}\cdot D_{3}\,,$ (14)

the degrees $d_{i}$ can be computed as $d_{i}=D_{i}\cdot C$, i.e.

$\displaystyle d_{1}$ $\displaystyle=D_{1}^{2}\cdot D_{2}\cdot D_{3}\ ,\
d_{2}=D_{1}\cdot D_{2}^{2}\cdot D_{3}\ ,\ d_{3}=D_{1}\cdot D_{2}\cdot
D_{3}^{2}\,.$ (15)

Note that since there is at least one $d_{i}\geq 0$, which means that
$h_{0}(N_{C|X_{4}})>0$, a $\mathbb{P}^{1}$ curve inside a Calabi-Yau fourfold
is never rigid.

To identify the particle spectrum of $\mathcal{T}_{X_{4}}$ we need to
calculate the spacetime quantum numbers of the BPS states from M2-brane
wrapping $C$ with different $N_{C|X_{4}}$. These results can be compared with
those from various literature on M-theory or F-theory on CY4 Beasley:2008dc ;
Donagi:2008ca ; Jockers:2016bwi .

  1. 1.

$N_{C|X_{4}}=\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$.

In this case, the moduli space of $C$ is a complex surface $S$, extended in
the directions of $\mathcal{O}\oplus\mathcal{O}$. The resulting subsector of
$\mathcal{T}_{X_{4}}$ can be viewed as a twisted reduction of a 7D SYM on $S$.

A 7D SYM in flat spacetime preserves the global symmetry group $SO(1,6)\times
SU(2)_{R}$. Upon reduction on a Riemannian fourfold $S$ we have the original
global symmetry group branches as follows:

$\begin{split}SO(1,6)\times SU(2)_{R}&\rightarrow SO(1,2)\times SO(4)\times
SU(2)_{R}\\\ &\simeq SO(1,2)\times SU(2)\times SU(2)\times
SU(2)_{R}\end{split}$ (16)

and in order to have a 3D theory with 4 supercharges we require that 4 out of
the original 16 supercharges under $(\mathbf{8},\mathbf{2})$ of $SO(1,6)\times
SU(2)_{R}$ to be preserved after the topological twist. The branching rule
(16) leads to:

$(\mathbf{8},\mathbf{2})\rightarrow(\mathbf{2},(\mathbf{2},\mathbf{1}))_{\pm\frac{1}{2}}+(\mathbf{2},(\mathbf{1},\mathbf{2}))_{\pm\frac{1}{2}}$
(17)

where $\pm\frac{1}{2}$ is the charge of $\mathbf{2}$ under the $U(1)$ center
of $SU(2)_{R}$. We further require $S$ to be Kähler hence the $SO(4)$ holonomy
becomes $U(2)_{L}$ under which $(\mathbf{2},\mathbf{1})$ becomes
$\mathbf{2}_{0}$ and $(\mathbf{1},\mathbf{2})$ becomes
$\mathbf{1}_{+1}\oplus\mathbf{1}_{-1}$ Beasley:2008dc . Applying the Kähler
condition, the original branching rule becomes:

$(\mathbf{8},\mathbf{2})\rightarrow(\mathbf{2},\mathbf{2}_{0})_{\pm\frac{1}{2}}+(\mathbf{2},\mathbf{1}_{+1}\oplus\mathbf{1}_{-1})_{\pm\frac{1}{2}}.$
(18)

After the topological twist $J=J_{L}+2J_{R}$ we have:

$(\mathbf{8},\mathbf{2})\rightarrow(\mathbf{2},\mathbf{2}_{\pm
1})_{\pm\frac{1}{2}}+(\mathbf{2},\mathbf{1}_{\pm
2})_{\pm\frac{1}{2}}+(\mathbf{2},\mathbf{1}_{0})_{\pm\frac{1}{2}}.$ (19)

The last piece of the above equation that is neutral under the $U(2)$ holonomy
of $S$ represents the 4 supercharges of the 3D $\mathcal{N}=2$ theory. From
(19) we can also read off that the 3D fermions transform as:

$\chi\in\Omega^{2,0}_{S},\ \widetilde{\chi}\in\Omega^{0,2}_{S},\
\psi\in\Omega^{1,0}_{S},\ \widetilde{\psi}\in\Omega^{0,1}_{S},\
\lambda,\widetilde{\lambda}\in\Omega^{0,0}_{S}.$ (20)

Next we look at the twisted reduction of the vector field
$(\mathbf{7},\mathbf{1})$. We have:

$\begin{split}(\mathbf{7},\mathbf{1})&\rightarrow(\mathbf{3},\mathbf{1}_{0})_{0}+(\mathbf{1},\mathbf{2}_{\pm
1})_{0}\end{split}$ (21)

from which we obtain:

$A_{\mu}\in\Omega^{0,0}_{S},\ \phi\in\Omega^{1,0}_{S},\
\widetilde{\phi}\in\Omega^{0,1}_{S}.$ (22)

Finally we look at the twisted reduction of the scalar field
$(\mathbf{1},\mathbf{3})$. We have:

$\begin{split}(\mathbf{1},\mathbf{3})\rightarrow(\mathbf{1},\mathbf{1}_{\pm
2})_{\pm 2}+(\mathbf{1},\mathbf{1}_{0})_{0}.\end{split}$ (23)

from which we obtain:

$\chi\in\Omega^{2,0}_{S},\ \widetilde{\chi}\in\Omega^{0,2}_{S},\
\rho\in\Omega^{0,0}_{S}.$ (24)

The states in (20), 22 and 24 can be packaged into the following 3D
$\mathcal{N}=2$ multiplets:

$\begin{split}&V:(A_{\mu},\lambda,\widetilde{\lambda},\rho)\in\Omega^{0,0}_{S},\\\
&\Psi:(\psi,\phi)\in\Omega^{1,0}_{S},\
\widetilde{\Psi}:(\widetilde{\psi},\widetilde{\phi})\in\Omega^{0,1}_{S}\\\
&X:(\chi,\rho)\in\Omega^{2,0}_{S},\
\widetilde{X}:(\widetilde{\chi},\widetilde{\rho})\in\Omega^{0,2}_{S}.\end{split}$
(25)

Therefore besides one vector multiplet $V$, there are $h^{0,1}(S)$ vector-like
chiral multiplets $\Psi+\widetilde{\Psi}$ and $h^{0,2}(S)$ vector-like chiral
multiplets $X+\widetilde{X}$. This result should be compared with the “bulk”
spectrum in Beasley:2008dc ; Donagi:2008ca .

  2. 2.

$N_{C|X_{4}}=\mathcal{O}\oplus\mathcal{O}(-1)\oplus\mathcal{O}(-1)$.

In this case, the moduli space of $C$ is a Riemann surface $\Sigma$ extended
in the direction of $\mathcal{O}$. The resulting 3D theory can be viewed as a
twisted reduction of 5D matters on $\Sigma$ where the 5D theory is a defect
theory living within the previously discussed 7D SYM.

We consider the following branching rule of 5D theory compactified on a
Riemannian twofold $Sigma$:

$SO(1,4)\times SU(2)_{R}\rightarrow SO(1,2)\times SO(2)\times U(1)_{R}.$ (26)

We will focus on the matter sector, i.e. the twisted reduction of the 5D
hypermultiplets. Since $\Sigma$ is complex, the $SO(2)$ holonomy becomes
$U(1)_{L}$.

Let us first look at the fermion in $\mathbf{4}$. Under the twisted reduction
we have:

$(\mathbf{4},\mathbf{1})\rightarrow\left(\mathbf{2},+\frac{1}{2}\right)_{0}+\left(\mathbf{2},-\frac{1}{2}\right)_{0}.$
(27)

Clearly the fermions cannot be twisted due to vanishing $R$-charge. Therefore
we obtain:

$\xi\in K^{1/2}_{\Sigma},\ \widetilde{\xi}\in\overline{K}^{1/2}_{\Sigma}.$
(28)

We then look at the complex scalars $(\mathbf{1},\mathbf{2})$. Upon the twist
$J=J_{L}+J_{R}$ the scalars become:

$(\mathbf{1},\mathbf{2})\rightarrow\left(\mathbf{1},\pm\frac{1}{2}\right)_{\pm\frac{1}{2}}.$
(29)

Therefore we obtain complex scalars:

$\eta\in K^{1/2}_{\Sigma},\ \widetilde{\eta}\in\overline{K}^{1/2}_{\Sigma}.$
(30)

The states in (28) and (29) can then be packaged into the following 3D
$\mathcal{N}=2$ multiplets:

$\Xi:(\xi,\eta)\in K^{1/2}_{\Sigma},\
\widetilde{\Xi}:(\widetilde{\xi},\widetilde{\eta})\in\overline{K}^{1/2}_{\Sigma}.$
(31)

Note that since
$\widetilde{\Xi}\in\overline{H^{0}(K_{\Sigma}^{1/2})}=H^{0}(K_{\Sigma}^{1/2})^{*}$555See
Beasley:2008dc  for a similar calculation in the context of 4D F-theory
compactification., we have:

$\chi(\Xi)=h^{0}(K_{\Sigma}^{1/2})-h^{0}(K_{\Sigma}^{1/2})=0.$ (32)

Therefore in the absence of $G_{4}$-flux the 3D spectrum is composed of
$h^{0}(K_{\Sigma}^{1/2})$ chiral multiplets $\Xi$ and
$h^{0}(K_{\Sigma}^{1/2})$ anti-chiral multiplets $\widetilde{\Xi}$. In other
words, the reduction of the 5D hypermultiplet on $\Sigma$ without $G_{4}$-flux
leads to a vector-like matter spectrum in 3D.

In particular, if the moduli space $\Sigma$ is a $\mathbb{P}^{1}$, since we
know that

$h^{0}(K_{\mathbb{P}^{1}}^{1/2})=h^{0}(\mathcal{O}(-1))=0\,,$ (33)

there would not be any BPS matter chiral multiplet at all in absence of
$G_{4}$ flux. This is consistent with the fact that Dirac operator on
$\mathbb{P}^{1}\equiv S^{2}$ has no zero mode. For all the cases of toric
$\mathbb{P}^{1}$ curves $C$ in a toric CY4 $X_{4}$ with
$N_{C|X_{4}}=\mathcal{O}\oplus\mathcal{O}(-1)\oplus\mathcal{O}(-1)$, the
previous statement applies and the matter field is absent. For the higher
Eigenstates of the Dirac operator on $S^{2}$, we present a detailed discussion
in appendix A.

For the other cases, such as
$N_{C|X_{4}}=\mathcal{O}(1)\oplus\mathcal{O}(-1)\oplus\mathcal{O}(-2)$, the
moduli space $\mathcal{M}_{C}$ of $C$ is more complicated, and the resulting
BPS states are typically higher-spin multiplets. In the case of curves on CY4,
different representatives of a given homology class $[C]\in
H_{2}(X_{4},\mathbb{Z})$ can have different normal bundles. We will not
discuss these complicated cases in the current paper.

###  2.3 Superpotential

The superpotential $W$ plays an important role in theories with four
supercharges. In this section we discuss a number of possible sources of $W$:

  1. 1.

Euclidean M5-brane over a divisor $D$

This type of superpotential was introduced in Witten:1996bn . If we have

$h^{0,1}(D)=h^{0,2}(D)=h^{0,3}(D)=0\,,$ (34)

there is a non-perturbative superpotential generated from Euclidean M5-brane
wrapping $D$, of the form

$W_{EM5}=T(m_{a})e^{-V_{D}+i\phi_{D}}\,.$ (35)

$V_{D}$ is the volume of $D$ and $\phi_{D}$ is an axion field and $T(m_{a})$
is a function depending on all other moduli. In fact,
$(\phi_{D},\eta_{D},V_{D})$ form a 3d $\mathcal{N}=2$ linear multiplet. Note
that the $V_{D}$ is not directly related to the $U(1)$ gauge coupling in (8).

  2. 2.

Polynomial superpotentials among chiral multiplets

This is analogous to the Yukawa superpotential in 4d F-theory
Blumenhagen:2007zk ; Beasley:2008dc . When we have $n$ curves $C_{i}$, and
over a common point $p$ in their moduli spaces $\mathcal{M}_{C_{i}}$ they
satisfy

$\sum_{i=1}^{n}(C_{i})|_{p}=0\,,$ (36)

it induces a superpotential with degree $n$:

$W_{Pol}=AX_{1}X_{2}\dots X_{n}$ (37)

where $X_{i}$ is the chiral matter field from $C_{i}$.

  3. 3.

Euclidean M2-brane over a 3-cycle

When $X_{4}$ has rigid 3-cycles $\gamma$, Euclidean M2-brane over it generates
non-perturbative superpotentials of the form Harvey:1999as

$W_{EM2}=Ae^{-V_{\gamma}+i\phi_{\gamma}}\,.$ (38)

Again $A$ would be a function of other moduli fields.

We comment on the lifting of 3d $\mathcal{N}=2$ Coulomb branch due to a non-
trivial superpotential. In general when there are rigid divisors satisfying
(34) in the resolved $X_{4}$, we would expect the non-perturbative
superpotential in (35) to appear. This would always happen for the cases of
local toric CY4, which are the main examples in this paper. Hence for generic
values of other moduli fields, we would expect that the Coulomb branch is
lifted (becomes massive) by the non-zero superpotential, and there is no SUSY
vacuum. Nonetheless, it is certainly possible that the prefactor $T(m_{a})$ in
(35) would vanish for special values of moduli fields $m_{a}$, and hence an
actual SUSY vacuum exists. We would not discuss the complications of SUSY
breaking in this paper, since there is no known way to compute the prefactor
$T(m_{a})$ in the general geometric setups.

In general, in the absence of $G_{4}$-flux there will be no obstruction to
achieving the geometric limit where all compact divisors along with the
subvarieties thereof shrink. In fact this is guaranteed by our construction of
$X_{4}$ such that all of its compact divisors are shrinkable, see section
2.5.3. At the end point of this limiting process there is no length scale in
the geometry hence there must be no energy scales in the corresponding field
theory in the UV either. Therefore the physical theory associated to the
geometry at the geometric limit is indeed a CFT.

Generally a CFT living in an infinite flat spacetime has zero vacuum energy.
The existence of a zero energy vacuum implies no SUSY breaking, assuming we
have started with supersymmetry. Therefore we expect the CFT associated with
$X_{4}$ at the geometric limit to be a SCFT. This in turn suggests no
superpotential be present. Hence $W$ must vanish at the geometric limit in the
absence of $G_{4}$-flux.

###  2.4 $G_{4}$-flux

$G_{4}$-flux is an essential ingredient in compactification on Calabi-Yau
fourfolds. It satisfies the quantization condition Witten:1996md

$G_{4}+\frac{1}{2}c_{2}(X_{4})\in H^{4}(X_{4},\mathbb{Z})\,.$ (39)

We restrict ourselves to self-dual $G_{4}$ flux of the type (2,2) in order to
preserve supersymmetry. The inclusion of $G_{4}$-flux changes the matter
spectrum of the 3d $\mathcal{N}=2$ theory. In practice, we consider the
Poincaré dual of $G_{4}$:

$G_{4}^{c}\in H_{4}(X_{4},\mathbb{R})\,.$ (40)

In order to make the total energy of the $G_{4}$-flux finite, it is necessary
that $G_{4}$ is compactly supported Gukov:1999ya :

$G_{4}\in H^{4}_{cpct}(X_{4},\mathbb{R})\,$ (41)

which in turn means that its Poincaré dual $G_{4}^{c}$ should only have
compact components in $H_{4}(X_{4},\mathbb{R})$.

In terms of $G_{4}^{c}$ and intersection numbers, (39) can be written as

$(G_{4}^{c}+\frac{1}{2}c_{2}^{c}(X_{4}))\cdot\mathcal{S}\in\mathbb{Z}\quad\ ,\
\forall\mathcal{S}\in H_{4}^{cpct}(X_{4},\mathbb{Z})\,.$ (42)

Here $c_{2}^{c}(X_{4})$ is the Poincaré dual 4-cycle of $c_{2}(X_{4})$, and
$\mathcal{S}$ is an arbitrary compact 4-cycle of $X_{4}$. When $X_{4}$ is
toric we have

$c_{2}^{c}(X_{4})=\sum_{\sigma_{2}=v_{i}v_{j}}D_{i}\cdot D_{j}\,,$ (43)

where $\sigma_{2}$ are all the 2d cones of $X_{4}$ and $D_{i}$ is the toric
divisor corresponding to the ray $v_{i}$.

Now we briefly discuss the change in the spectrum of M2-brane wrapping modes
after introducing $G_{4}$-flux. Again we will focus on the case where the
normal bundle of curve wrapped by an M2-brane is either
$\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$ or
$\mathcal{O}\oplus\mathcal{O}(-1)\oplus\mathcal{O}(-1)$.

  1. 1.

For the bulk matter from M2-brane wrapping a curve $C\simeq\mathbb{P}^{1}$
with $N_{C|X_{4}}=\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$, we
suppose that the compact divisor $D$ is a $\mathbb{P}^{1}$-fibration over a
complex surface $S$. Then the intersection product

$D\cdot G_{4}^{c}\in CH_{1}(S)$ (44)

defines a divisor class in $S$, which further corresponds to a line bundle
$L\in\mathrm{Pic}(S)$ on $S$. Now the chiral and anti-chiral spectrum are

$\displaystyle\mathrm{chiral}:$ $\displaystyle h^{1}(S,L)+h^{0}(S,L\otimes
K_{S})$ (45) $\displaystyle\mathrm{anti-chiral}:$ $\displaystyle
h^{2}(S,L)+h^{1}(S,L\otimes K_{S})$

In particular, the net chirality is Beasley:2008dc :

$\chi=-\int_{S}c_{1}(S)c_{1}(L)\,.$ (46)

  2. 2.

For the chiral matter from M2-brane wrapping a curve $C\simeq\mathbb{P}^{1}$
with $N_{C|X_{4}}=\mathcal{O}\oplus\mathcal{O}(-1)\oplus\mathcal{O}(-1)$,
suppose that the moduli space of $C$ is $\mathcal{M}_{C}=\Sigma$ and we
further define a complex surface

$\mathcal{S}=C\rightarrow\Sigma\in D\,,$ (47)

as the fibration of $C$ over $\Sigma$. Such $\mathcal{S}$ is also known as the
_matter surface_ of $C$ in the context of 4D F-theory compactification
Donagi:2008ca ; Beasley:2008dc .

The intersection product

$\mathcal{S}\cdot G_{4}^{c}\in CH_{0}(\Sigma)$ (48)

defines a divisor class and line bundle $L\in\mathrm{Pic}(\Sigma)$ on
$\Sigma$. The chiral and anti-chiral spectrum are

$\displaystyle\mathrm{chiral}:$ $\displaystyle
h^{0}(\Sigma,L\otimes\sqrt{K_{\Sigma}})$ (49) $\displaystyle\mathrm{anti-
chiral}:$ $\displaystyle h^{1}(\Sigma,L\otimes\sqrt{K_{\Sigma}})\,.$

We can compute the net chirality

$\chi(C)=G_{4}^{c}\cdot\mathcal{S}\,.$ (50)

Another important consequence of non-trivial $G_{4}$-flux is the generation of
an effective Chern-Simons term in the action from the dimensional reduction of
classical 11d SUGRA action:

$S_{CS}=\frac{1}{4}\int\sum_{i,j=1}^{r}k_{ij}A_{i}\wedge F_{j}\,.$ (51)

Here $A_{i}$ $(i=1,\dots,r)$ are the $U(1)$ gauge fields associated to each
compact divisor $D_{i}$. Note that such a CS term is present in the UV theory
$\mathcal{T}^{UV}_{X_{4}}$ in figure 2. The Chern-Simons level $k_{ij}$ can be
computed from the $G_{4}$ flux as

$\displaystyle k_{ij}$
$\displaystyle=\int_{X_{4}}G_{4}\wedge\omega_{i}^{(1,1)}\wedge\omega_{j}^{(1,1)}$
(52) $\displaystyle=G_{4}^{c}\cdot D_{i}\cdot D_{j}\,.$

When there exists a 4D F-theory lift of our 3D $\mathcal{N}=2$ theory, it
should match the 1-loop corrections of _massless_ fermions Grimm:2011fx :

$k_{ij}=\frac{1}{2}\sum_{f}(q_{f})_{i}(q_{f})_{j}\mathrm{sign}\left(\sum_{k=1}^{r}(q_{f})_{k}\sigma^{k}+m_{f}\right)\,$
(53)

where $\sigma^{k}$ is the CB parameter of $U(1)_{k}$ and $m_{f}$ is the
classical mass of the fermion $f$ and one has to set $m_{f}\rightarrow 0$. In
absence of the parity anomaly, the Chern-Simons levels should be all
quantized, i.e. $k_{ij}\in\mathbb{Z}$. It is clear that $k_{ij}$’s vanish when
$G_{4}^{c}=0$. This matches $k_{ij}$ calculated via (53) when there exists a
4D lift due to the vanishing $m_{f}$ and the trivial chiral index Grimm:2011fx
.

Finally, turning on $G_{4}$-flux induces a contribution to the superpotential,
known as the Gukov-Vafa-Witten superpotential Gukov:1999ya

$W_{GVW}=\int_{X_{4}}G_{4}\wedge\Omega_{4}\,.$ (54)

Nonetheless, since we have already required that $G_{4}$ has no $(0,4)$
component, the above term vanishes.

In the cases of compact CY4, there is also the scalar potential term in 3d

$V(G_{4})=\pi\int_{X_{4}}G_{4}\wedge*G_{4}$ (55)

and the $D$-term superpotential

$D=\int_{X_{4}}G_{4}\wedge J\wedge J\,.$ (56)

Nonetheless, since our $X_{4}$ is always non-compact, such terms will not play
any role here either.

In studying various geometric limits of $X_{4}$, we need to be careful that
$G_{4}$-flux cannot pass through shrinking 4-cycles. More precisely, the limit
Vol$(\mathcal{S})\rightarrow 0$ cannot be achieved if

$\int_{\mathcal{S}}G_{4}=G_{4}^{c}\cdot\mathcal{S}\neq 0\,.$ (57)

for a compact 4-cycle $\mathcal{S}$.

###  2.5 Geometric limits and 3d $\mathcal{N}=2$ SCFTs

We take $X_{4}$ to be a smooth non-compact CY4, which has a singular limit
$X_{4,\text{sing}}$ when all the compact cycles in $X_{4}$ shrink to zero
volume. We discuss the three geometric limits and the corresponding physics
respectively, see Figure 2.

Note that the various limiting behaviours in Figure 2 are analogous to what
happens in 5D obtained from M-theory compactified on a local CY3
Apruzzi:2019vpe ; Apruzzi:2019opn . In 5D at a generic point on the Coulomb
branch the IR EFT is a free $U(1)^{r}$ theory where $r$ is the rank of the CB.
Along certain locus in the CB the theory can be enhanced to a non-Abelian
gauge theory. It happens typically when the fibral $\mathbb{P}^{1}$ of a ruled
surface in CY3 is shrunk to zero volume. In the limit where all compact cycles
in the CY3 are shrunk to zero volume the theory becomes a 5D SCFT.

The 3D theory is different from 5D theory in several aspects. One important
difference is that a Chern-Simons term generated after integrating out a
massive charged fermion generates a mass for the gauge field in the presence
of YM Lagrangian Dunne:1998qy ; DESER1982372 . Therefore the existence of
massive charged fermions in the UV will drastically change the IR physics.
Clearly, in the absence of massive charged fermions in UV no CS term will be
generated along the RG flow, hence the IR will be a strongly coupled SYM. In
the following subsections we will assume the existence of massive charged
fermions in UV, which is generally true for a theory obtained from geometric
engineering on a smooth variety.

####  2.5.1 Generic point on the Coulomb branch

We first consider M-theory on a smooth non-compact $X_{4}$, which leads to a
$U(1)^{r}$ gauge theory with non-zero gauge coupling and massive charged
matter fields. The rank $r$ equals to the number of irreducible compact
divisors $D_{i}$, i.e. $r=h^{1,1}(X_{4})$. We denote the $U(1)$ gauge fields
by $A_{1}$, $\dots$, $A_{r}$.

Besides the compact divisors, there are also $f=b_{2}(X_{4})-b_{6}(X_{4})$
non-compact divisors $S_{\alpha}$, which correspond to non-dynamical 1-form
gauge fields $B_{1}$, $\dots$, $B_{f}$. These can be interpreted as background
gauge fields for the flavor symmetries, with the field contents summarized in
the vector multiplet:

$(\sigma_{\alpha},B_{\alpha},D_{\alpha})\,.$ (58)

In particular, the VEV $\langle\sigma_{\alpha}\rangle:=m_{\alpha}$ is the FI
parameter for the flavor symmetry $B_{\alpha}$ Closset:2019hyt .

Even in the absence of $G_{4}$-flux, there will be massive BPS particles from
M2-brane wrapping 2-cycles $C\subset X_{4}$ are charged under both $U(1)^{r}$
and $U(1)^{f}$. Such states include

  * •

W-bosons of the broken gauge symmetries, whose charge under $U(1)^{r}$ takes
the form of a weight in the adjoint representation of certain enhanced non-
abelian Lie algebra. $C$ has normal bundle
$N_{C|X_{4}}=\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$.

  * •

Flavor W-bosons, whose charge under $U(1)^{f}$ takes the form of a weight in
the adjoint representation of the enhanced flavor algebra $G_{F}$. They are
also uncharged under all the gauge groups, which means $C\cdot D_{i}=0$,
$\forall D_{i}$. $C$ has normal bundle
$N_{C|X_{4}}=\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$.

  * •

Matter chiral multiplets from $C$ with normal bundle
$N_{C|X_{4}}=\mathcal{O}\oplus\mathcal{O}(-1)\oplus\mathcal{O}(-1)$. If the
moduli space of $C$ is $\mathbb{P}^{1}$ (e. g. in the toric cases), strictly
speaking there is no zero mode in absence of $G_{4}$ flux. Nonetheless there
are still massive particles from M2-brane wrapping $C$ which will be discussed
shortly.

  * •

Other states where $C$ has a different normal bundle than the previous cases.

Now we consider the RG flow to the IR. In the IR, the kinetic terms of
$U(1)^{r}$ become irrelevant. However, there will be effective Chern-Simons
terms from integrating out massive fermions with charge $q_{i}(i=1,\dots,r)$
under the gauge groups $U(1)^{r}$ and $q_{\alpha}^{F}(\alpha=1,\dots,f)$ under
the flavor Cartan $U(1)^{f}$ PhysRevLett.52.18 ; Dunne:1998qy ;
Intriligator:2013lca . The quantizations of these $U(1)$ charges are the same
from the intersection number calculations. There are three types of terms
arising from such integrating-out procedure:

  * •

The CS terms from integrating out massive fermions charged under gauge
symmetries:

$\left(\frac{1}{8\pi}\sum_{f}(q_{f})_{i}(q_{f})_{j}\mathrm{sign}\left(\sum_{k=1}^{r}(q_{f})_{k}\sigma^{k}+m_{f}\right)\right)A_{i}\wedge
dA_{j}\,,$ (59)

  * •

The mixed CS terms from integrating out massive fermions charged under both
gauge and flavor symmetries:

$\left(\frac{1}{8\pi}\sum_{f}(q_{f})_{i}(q_{f})_{\alpha}^{F}\mathrm{sign}\left(\sum_{k=1}^{r}(q_{f})_{k}\sigma^{k}+m_{f}\right)\right)A_{i}\wedge
dB_{\alpha}\,,$ (60)

  * •

The CS terms from integrating out massive fermions charged under flavor
symmetries:

$\left(\frac{1}{8\pi}\sum_{f}(q_{f})_{\alpha}^{F}(q_{f})_{\beta}^{F}\mathrm{sign}\left(\sum_{k=1}^{r}(q_{f})_{k}\sigma^{k}+m_{f}\right)\right)B_{\alpha}\wedge
dB_{\beta}\,.$ (61)

Here $m_{f}$ is the UV real mass of the particle and there is a shift term
related to the CB parameters $\sigma^{k}$. Note that one should sum over
fermions from both the M2-brane and anti-M2-brane. In terms of Lie algebra
representations, it means that we need to sum over the weights $w$ and $-w$
when they are both present Grimm:2011fx . When $G_{4}=0$, the 3d
$\mathcal{N}=2$ spectrum is generally expected to be non-chiral, which means
that the numbers of chiral fermions in the weight $w$ and $-w$ are identical,
and they would give rise to identical contributions to (59), (60), (61) but
with opposite signs. In this case, we expect that the CS terms are all
vanishing, which is consistent with the dimensional reduction of 11d SUGRA
action (51). For the non-BPS states from higher eigenmodes of the Dirac
operators on the moduli space of complex curves, we give a more detailed
discussion in appendix A, and the physical result of no CS terms still holds.

When there is a non-zero $G_{4}$-flux on $X_{4}$, the will be additional
chiral zero-modes from M2-brane wrapping curves, as well as contributions to
the CS terms depending on the $G_{4}$-flux:

$\left(\frac{1}{4\pi}G_{4}^{c}\cdot D_{i}\cdot D_{j}\right)A_{i}\wedge
dA_{j}\,,$ (62) $\left(\frac{1}{4\pi}G_{4}^{c}\cdot D_{i}\cdot
S_{\alpha}\right)A_{i}\wedge dB_{\alpha}\,,$ (63)
$\left(\frac{1}{4\pi}G_{4}^{c}\cdot S_{\alpha}\cdot
S_{\beta}\right)B_{\alpha}\wedge dB_{\beta}\,.$ (64)

Moreover, after adding $G_{4}$-flux, there can be additional flavor symmetry
rotating different new chiral modes. The charges and mixed CS term for these
new flavor symmetries can be computed in an analogous way, and we would not
discuss in more details.

####  2.5.2 Non-abelian enhancement locus on the Coulomb branch

In the cases where $X_{4}$ has a $\mathbb{P}^{1}$-fibration structure,
$U(1)^{r}$ can be enhanced to certain non-abelian gauge groups $G$ when the
fibral $\mathbb{P}^{1}$s shrink to zero volume. More precisely we require each
compact divisor of $X_{4}$ is a $\mathbb{P}^{1}$-ruled subvariety and all the
compact divisors can be glued in an uniform way (see section 3 for a detailed
explanation of this point). In this geometry the M2-brane wrapping modes over
the fibral $\mathbb{P}^{1}$s become the massless W-bosons whose charges are
determined by its intersection numbers with the compact divisors associated to
$r$ Cartan $U(1)$’s. This is a non-Abelian gauge theory limit of the
$U(1)^{r}$ theory on the Coulomb branch.

Besides the massless W-bosons, there are also massive particles from M2-brane
wrapping curves in the base directions, which are charged under both the gauge
symmetry and $U(1)^{f}$ flavor symmetry. We will see in section 3 and section
4 that these states should be interpreted as disorder operators in 3D.

Therefore in this case, along the RG flow one first integrates out the massive
fermion modes charged under both $U(1)$ and non-Abelian symmetries and gets a
3D CS theory with both $G$ and $U(1)^{f}$ as the gauge algebra.

Here we comment on the precise correspondence between the non-abelian gauge
coupling and the geometric data of $X_{4}$. For this recall that from (8) we
have:

$\frac{1}{g_{i}^{2}}=-\text{Vol}(D_{i}\cdot D_{i}).$ (65)

For simplicity let us focus on one such compact divisor $D$ and study the
corresponding $U(1)$ gauge coupling:

$\frac{1}{g^{2}}=-\text{Vol}(D\cdot D)=-\text{Vol}(K_{D})=-K_{D}\cdot
J^{c}\cdot J^{c}.$ (66)

Recall that the current situation where the UV theory admits a gauge theory
description, $D$ is a $\mathbb{P}^{1}$-ruled variety which is topologically
$C\times B$, $C\simeq\mathbb{P}^{1}$. Alternatively $D$ can be viewed as the
projective bundle associated to a rank-2 vector bundle $E$ over $B$. Hence we
have:

$-K_{D}=2\xi-\pi^{*}(K_{B}+\det(E))$ (67)

where $\xi$ is the natural line bundle $O_{D}(1)$ on $D$ (i.e. a section of
fibral $C\simeq\mathbb{P}^{1}$) and $\pi$ is the projection of $D$ to $B$.
Since $\pi^{*}(K_{B}+\det(E))$ is a vertical divisor of $D$, its volume will
be suppressed by the hierarchy needed for a valid gauge theory description.
Therefore the gauge coupling will be set by:

$\frac{1}{g^{2}}=-K_{D}\cdot J^{c}\cdot J^{c}\simeq 2\xi\cdot J^{c}\cdot
J^{c}.$ (68)

Since $[\xi]=[B]$, we arrive at:

$\frac{1}{g^{2}}\simeq 2\text{Vol}(B).$ (69)

Therefore, given the hierarchy $\text{Vol}(B)\gg\text{Vol}(C)$ the gauge
coupling $1/g^{2}$ is essentially set by the volume of the base $B\subset D$
666A very similar calculation along the same line of logic was done in
Halverson:2022mtc  in a slightly different context..

####  2.5.3 3d $\mathcal{N}=2$ SCFT at the origin of the Coulomb branch

In the singular limit where all the compact divisors $D_{i}$ are shrunk to
zero volume such that the theory is at the origin of CB, since all $D_{i}\cdot
D_{i}$’s shrink with $D_{i}$, all $U(1)$ gauge theories discussed in section
2.5.1 will become strongly coupled assuming the description (8) in a smooth
geometry can be extrapolated to the singular limit. Even without looking into
the details of the limiting process it is natural to expect that the theory
will be conformal due to the absence of all the scales in the singular
geometry $X_{4,\text{sing}}$.

Now we discuss the conditions for reaching the singular limit of the smooth
local CY4 $X_{4}$, such that the 3d $\mathcal{N}=2$ SCFT $\mathcal{T}^{\rm
3d}_{X_{\text{4,sing}}}$ exists.

The first condition is the shrinkability of $X_{4}$. Before presenting the
proposal, we provide a streamlined set of shrinkability conditions for the
local CY3 case $X_{3}$, stated differently from Jefferson:2018irk .

$\bullet$ Shrinkability for local CY3

  1. 1.

For any compact curve $C=D\cdot S\subset X_{3}$, where $D$ is a compact
divisor and $S$ is a non-compact divisor, we require that
$N_{\mathcal{C}|D}-K_{C|D}\geq 0$. Equivalently, it means that $C\cdot D\leq
0$. The condition can exclude non-examples such as local $\mathbb{F}_{3}$.

  2. 2.

Consider a general point in the Kähler cone such that all compact curves
Vol$(C)\geq 0$ and compact divisors vol($D)\geq 0$, it is required that for
any compact divisor $D\subset X_{3}$, we have

$\text{Vol}(D\cdot D)\equiv\text{Vol}(-K_{D})=0$ (70)

if and only if all curves $C$ on $D$ have zero volume. Physically, the
condition (70) means that the gauge coupling $g$ of $U(1)$ associated to $D$
satisfies $1/g^{2}\rightarrow 0$ Morrison:1996xf  only when $D$ shrinks
exactly to a point. The condition can exclude non-examples such as local dP9
or local $T^{2}\times\mathbb{P}^{1}$.

Now let us present the analogous shrinkability criterion for local CY4
$X_{4}$.

$\bullet$ Shrinkability for local CY4

  1. 1.

For any compact 4-cycle $\mathcal{S}=D\cdot S$, where $D$ is a compact divisor
and $S$ is a non-compact divisor, we require that
$N_{\mathcal{S}|D}-K_{\mathcal{S}|D}\geq 0$, or equivalently
$N_{\mathcal{S}|D}-K_{\mathcal{S}|D}$ is effective on $D$.

  2. 2.

Moreover, for any compact curve $C=D\cdot S_{1}\cdot S_{2}$ where $D$ is a
compact divisor and $S_{1}$, $S_{2}$ are non-compact divisors, we require that
$C\cdot D\leq 0$.

  3. 3.

Consider a general point in the Kähler cone such that all compact curves
Vol$(C)\geq 0$, 4-cycles Vol$(\mathcal{S})\geq 0$ and compact divisors
vol($D)\geq 0$, it is required that for any compact divisor $D\subset X_{4}$,
we have

$\text{Vol}(D\cdot D)\equiv\text{Vol}(-K_{D})=0$ (71)

if and only if all 4-cycles $\mathcal{S}$ of $D$ have zero volume. It means
that the gauge coupling $g$ of $U(1)$ associated to $D$ satisfies
$1/g^{2}\rightarrow 0$ (8) only when $D$ shrinks to a point or a line.

We list a number of cases that are excluded by the above criteria, where
$X_{4}$ has a $T^{2}$ fibration structure in which case the resulting 3D SCFT
is not $\mathcal{N}=2$.

(a) If $X_{4}$ is a non-trivial $T^{2}$-fibration, then the geometry $X_{4}$
has a 4d F-theory uplift when the $T^{2}$-fibration shrink to zero volume.
This corresponds to the geometric construction of a 4d $\mathcal{N}=1$ theory.

(b) If $X_{4}$ is a direct product of $T^{2}$ and a local CY3 $X_{3}$, then
the geometry describes a 3d $\mathcal{N}=4$ theory with the volume of $T^{2}$
is finite. It uplifts to 4d $\mathcal{N}=2$ when the Vol$(T^{2})\rightarrow
0$, where the theory is IIB on $X_{3}$.

(c) If $X_{4}$ is a direct product of (local) K3 and (local) K3, the theory is
also 3d $\mathcal{N}=4$.

(d) If $X_{4}$ is a direct product of $T^{4}$ and a local K3 $X_{2}$, the
geometry describes a 3d $\mathcal{N}=8$ theory.

Specifically, in the cases of toric CY4, the shrinkability conditions are
equivalent to requiring the toric polytope $\mathcal{P}_{3}$ to be convex.

There is also additional criterion in the CY4 case about the $G_{4}$ flux. It
is required that no compact 4-cycle $\mathcal{S}$ in $X_{4}$ can support any
$G_{4}$-flux, that is for any $\mathcal{S}$,

$\int_{\mathcal{S}}G_{4}=0\,.$ (72)

This means that we cannot turn on $G_{4}$ flux in the resolved $X_{4}$ as its
existence obstructs the singular limit.

In the singular limit, the M2-brane wrapping modes all become massless, which
include the W-bosons of the $U(1)^{f}$ geometric flavor symmetry as well.
Hence $U(1)^{f}$ can be enhanced to a non-abelian flavor symmetry group
(algebra) $G_{F}$.

##  3 3d $\mathcal{N}=2$ theories from toric CY4

In this section, we consider a smooth toric CY4 $X_{4}$, which has a singular
limit $X_{4,\text{sing}}$ when all the compact cycles in $X_{4}$ shrink to
zero volume. There exists a $GL(4,\mathbb{Z})$ transformation, such that all
rays of $X_{4}$ take the form of $\tilde{v}_{i}=(v_{i},1)$. We define the
convex hull of $v_{i}$s to be the 3d polytope $\mathcal{P}_{3}$. One can
compute the quadruple intersection numbers of divisors using the standard
toric methods.

###  3.1 Fibration structures and non-abelian gauge theory

When the local CY4 has a $\mathbb{P}^{1}$-fibration structure (ruling), one
can generally consider the non-abelian gauge theory description when the
$\mathbb{P}^{1}$ fiber direction is shrunk to zero volume. If there are
multiple $\mathbb{P}^{1}$-fibration structures, it would imply dualities
between different non-abelian gauge theories in the UV.

In the toric CY4, the $\mathbb{P}^{1}$-fibration structures can be read off
from the structure of (triangulated) $\mathcal{P}_{3}$. Namely, whenever there
is a set of straight parallel lines $L_{i}$ in $\mathcal{P}_{3}$ that pass
through all the lattice points in $\mathcal{P}_{3}$ while not intersecting the
boundary of $\mathcal{P}_{3}$ over non-lattice point, then one can define a
$\mathbb{P}^{1}$ fibration structure where the base are along the $L_{i}$
direction in the toric diagram and the $\mathbb{P}^{1}$ fibers are transversal
to the $L_{i}$ direction.

To determine the non-abelian gauge group and flavor symmetry group in the
geometric limit of a particular $\mathbb{P}^{1}$-fibration, one count the
number $n_{i}$ of internal lattice points on the parallel lines $L_{i}$. If
these internal lattice points are in the interior of $\mathcal{P}_{3}$, they
give rise to a $SU(n_{i}+1)$ gauge group factor. If these points are on the
boundary of $\mathcal{P}_{3}$, they give rise to a $\mathfrak{su}(n_{i}+1)$
flavor symmetry factor.

When there are multiple $SU(n_{i}+1)$ gauge group factors, one may ask whether
there are bifundamental matter fields charged between them. In general, if
there is no $G_{4}$ flux, the absence of BPS states from M2-brane wrapping
$\mathcal{O}\oplus\mathcal{O}(-1)\oplus\mathcal{O}(-1)$ curves in the toric
CY4 would imply that there are no bifundamental matter, unlike the 5d cases.

For example, in the case of local
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$, from the toric
diagram in figure 6 we see that there are three set of lines that satisfy the
criteria above, and they represents three different $\mathbb{P}^{1}$-fibration
structures. For instance if one choose the line $L_{1}=(0,0,1)\sim(0,0,-1)$
colored in red, then the base direction is given by $S_{3}$ and the fiber is
given by $D\cdot S_{1}\cdot S_{2}$. There is an $SU(2)$ gauge theory
description since the line $L_{1}$ has one interior point that corresponds to
a compact divisor. For the other two $\mathbb{P}^{1}$-fibration structures
colored in green and blue, they give $SU(2)$ gauge theory descriptions as
well.

The case in figure 5 also has three distinct $\mathbb{P}^{1}$-fibration
structures from the lines in $x$, $y$ and $z$-axes. For the
$\mathbb{P}^{1}$-fibration structure giving by parallel lines in red, one can
read off an $SU(3)$ gauge group and $SU(3)^{2}$ non-abelian flavor symmetry.
For the $\mathbb{P}^{1}$-fibration structure given by the lines in green, one
read off $SU(2)^{2}$ gauge group and $SU(2)^{4}$ flavor symmetry. For the
$\mathbb{P}^{1}$-fibration structure in blue, one read off $SU(2)^{2}$ gauge
group and $SU(2)^{6}$ flavor symmetry.

Note that we have not discussed the full flavor symmetry enhancement, which is
elaborated in section 3.2.

###  3.2 Flavor symmetry from toric CY4

We focus on the geometric flavor symmetry algebra $G_{F}$ associated to the
non-compact divisors in $X_{4}$. The rank of $G_{F}$ equals to the total
number of linearly independent non-compact divisors, which can also be
expressed as

$f:=\text{rk}(G_{F})=b_{2}(X_{4})-b_{6}(X_{4})\,.$ (73)

In the case of smooth $X_{4}$, the 3d $\mathcal{N}=2$ theory is on its
extended Coulomb branch such that only the maximal abelian subgroup
$U(1)^{f}\subset G_{F}$ can be seen. In the singular limit
$X_{4,\text{sing}}$, $U(1)^{f}$ can be enhanced to certain larger flavor
symmetry algebra $G_{F}$.

In order to see the enhancement, one should first identify the linear
combinations of non-compact divisors $S_{1}$, $\dots$, $S_{f}$ that generate
the Cartan subalgebra of $G_{F}$ (the flavor Cartans). Then for the parts that
enhance to non-abelian factors $S_{k}$, there exist flavor W-bosons wrapping
curves $C_{k}=S_{k}\cdot\mathcal{D}$ where $\mathcal{D}$ is a compact 4-cycle.
The flavor W-boson $C_{k}$ satisfy the following conditions

  1. 1.

$C_{k}$ is neutral under all $U(1)$ gauge groups generated by compact
divisors.

  2. 2.

$C_{k}$ has the correct normal bundle in $X_{4}$:
$N_{C_{k}|X_{4}}=\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$.

  3. 3.

The intersection matrix $C_{k}\cdot S_{j}$ correctly corresponds to (the
negative of) the Cartan matrix of $G_{F}$.

Such conditions provide constraints on the choices of $S_{k}$ and $C_{k}$,
which may not be unique.

For a toric $X_{4}$, the flavor symmetry enhancement can be partially read off
from the toric data. For $X_{4}$ with a 3d polytope $\mathcal{P}_{3}$, there
are two ways to get an enhanced non-abelian flavor symmetry factor:

  1. 1.

Each edge $\theta=v_{i}v_{j}$ on a facet of $\mathcal{P}_{3}$ contributes to a
non-abelian flavor symmetry factor $\mathfrak{su}(l^{*}(\theta)+1)$, where
$l^{*}(\theta)$ is the number of interior points on the edge $\theta$.

To see this, we construct a new 3d polytope $\widetilde{\mathcal{P}}_{3}$ by
adding a new vertex $v$, such that the edge $\theta$ becomes an internal edge
in $\widetilde{\mathcal{P}}_{3}$. In this case, let us take the
$l^{*}(\theta)$ internal points and its adjacent cones, which form a sub-
polytope $\mathcal{P}_{f}\subset\widetilde{\mathcal{P}}_{3}$.
$\mathcal{P}_{f}$ precisely describes a string of $l^{*}(\theta)$
$\mathbb{P}^{1}$’s fibered over a complex surface (see Figure 3).

Figure 3: A new polyhedron $\widetilde{\mathcal{P}}_{3}$ is made by adding an
extra vertex at the intersection of the three green edges to the original
tetrahedron $\mathcal{P}_{3}$ bounded by the red and the blue edges. The red
edge is the one that is made internal, i.e. $\theta$ by adding the extra
vertex.

Hence the internal points of $\theta$ generate the Cartan subalgebra
$U(1)^{l^{*}(\theta)}$ of a non-abelian gauge group
$\mathfrak{su}(l^{*}(\theta)+1)$, and we restore the full non-abelian gauge
group when all the $\mathbb{P}^{1}$ fibers are shrunk to zero volume.

Finally, let us decouple this $\mathfrak{su}(l^{*}(\theta)+1)$ gauge group by
removing the vertex $v$, and $\mathfrak{su}(l^{*}(\theta)+1)$ becomes a flavor
symmetry in the original 3d $\mathcal{N}=2$ theory from $X_{4}$.

  2. 2.

For the internal points of a 2d face $F\subset\mathcal{P}_{3}$, each of them
generates a $\mathfrak{u}(1)$ flavor symmetry, though further enhancement to a
non-abelian flavor algebra may exist given a suitable triangulation of
$\mathcal{P}_{3}$.

As an example of the second case, consider the configuration of the following
2d facet $\mathcal{F}$:

$S_{1}$$S$$S_{2}$$S_{3}$$S_{4}$ (74)

We denote by $D$ the compact divisor that connects all points in
$\mathcal{F}$. To see the enhanced $\mathfrak{su}(3)$ flavor symmetry, we
write down the flavor Cartan subalgebra generators

$F_{1}=S_{1}+S\ ,\ F_{2}=S_{3}\,.$ (75)

The flavor W-boson associated to $F_{1}$ and $F_{2}$ are

$C_{1}=D\cdot F_{1}\cdot S_{1}=D\cdot S\cdot S_{1}$ (76)

and

$C_{2}=D\cdot F_{2}\cdot S=D\cdot S\cdot S_{3}\,.$ (77)

We can check the intersection matrix

$\begin{array}[]{c|cc}&F_{1}&F_{2}\\\ \hline\cr C_{1}&-2&1\\\
C_{2}&1&-2\end{array}$ (78)

which is indeed the Cartan matrix of $\mathfrak{su}(3)$.

Furthermore, we argue that there can be a flavor symmetry enhancement in the
general cases of local $\mathbb{P}^{1}\times\mathcal{S}$, where $\mathcal{S}$
is a weak-Fano complex surface that is not $\mathbb{P}^{2}$, see the detailed
examples in Section 4. In the resolved geometry, we have a single compact
divisor $D$ and a number of non-compact divisors $S$, $S_{1}$, $\dots$,
$S_{n}$. Note that $D\cdot S\cong\mathcal{S}$, and $D\cdot S\cdot S_{i}$
corresponds to the curves $C_{i}$ on $\mathcal{S}$. In general, we have the
following intersection numbers on $X_{4}$:

$\displaystyle S^{2}=0\ ,\ D\cdot S\cdot S_{i}\cdot S_{j}=(C_{i}\cdot
C_{j})|_{\mathcal{S}}\ ,\ D\cdot D\cdot S\cdot S_{i}=(K_{D}\cdot
C_{i})|_{\mathcal{S}}\ ,\ $ (79) $\displaystyle D\cdot D\cdot S_{i}\cdot
S_{j}=-2(C_{i}\cdot C_{j})|_{\mathcal{S}}\,.$

In the toric language, $D$ has the toric coordinate $v_{D}=(0,0,0)$, $S$
corresponds to the toric coordinates $(0,0,1)$ and $(0,0,-1)$ and $S_{i}$s has
toric coordinates $(v_{i,x},v_{i,y},0)$.

Now first for all the $(-2)$-curves $C_{i}$ on $\mathcal{S}$, they also give
rise to flavor W-bosons $C_{i}=D\cdot S\cdot S_{i}$ in $X_{4}$ corresponding
to flavor Cartan $F_{i}=S_{i}$ in the Dynkin diagram of $G_{F,nA}$. One can
check that

$D\cdot C_{i}=D\cdot D\cdot S\cdot S_{i}=(K_{D}\cdot
C_{i})|_{\mathcal{S}}=0\,.$ (80)

Furthermore, whenever $\mathcal{S}\neq\mathbb{P}^{2},\mathbb{F}_{1}$, there
exists another flavor Cartan $F=S_{\alpha}-S$, where $S_{\alpha}$ corresponds
to a rational 0-curve $C_{\alpha}$ on $\mathcal{S}$
($(C_{\alpha}^{2})|_{\mathcal{S}}=0$, $g(C_{\alpha})=0$). The flavor W-boson
for $F$ is given by $C=D\cdot(S_{\alpha}-S)\cdot(S_{\alpha}+F_{j})$. Here
$F_{j}$ is a flavor Cartan satisfying $F_{j}\cdot C_{\alpha}=1$, which
corresponds to a $(-2)$-curve $C_{j}$ on $\mathcal{S}$. One can compute the
charges

$\displaystyle C\cdot(S_{\alpha}-S)$ $\displaystyle=-2$ (81) $\displaystyle
C_{i}\cdot(S_{\alpha}-S)$ $\displaystyle=(C_{\alpha}\cdot
C_{i})|_{\mathcal{S}}$ $\displaystyle C\cdot F_{i}$
$\displaystyle=(-C_{\alpha}\cdot C_{i}-C_{i}\cdot C_{j})|_{\mathcal{S}}\,.$

Now let us see whether the flavor Cartans $F$ and $F_{i}$’s can be combined
into a larger non-abelian subalgebra with the correct (negative of) Cartan
matrix. First, one can compute that

$C_{j}\cdot(S_{\alpha}-S)=1\ ,\ C\cdot F_{j}=1\,,$ (82)

which it seems that one can attach the node $F$ to $F_{j}$ in the Dynkin
diagram.

Here comes a subtlety: if there are other flavor Cartans $F_{k}$ intersecting
$F_{j}$ $(k\neq j)$, there would be a negative charge

$C\cdot F_{k}=(-C_{k}\cdot C_{j})|_{\mathcal{S}}\neq 0\,.$ (83)

Nonetheless, the problem can be remove by redefining the flavor Cartan
$F_{k}\rightarrow F_{k}^{\prime}=S_{k}+S$, and we get $C\cdot
F_{k}^{\prime}=0$ and there is no issue.

Note that in the construction for the additional flavor Cartan
$F=S_{\alpha}-S$ and its flavor W-boson
$D\cdot(S_{\alpha}-S)\cdot(S_{\alpha}+F_{j})$, they are not effective in
$X_{4}$. Nonetheless, as one reaches the full singular limit
$X_{\text{4,sing}}$ of $X_{4}$, the flavor W-bosons all become massless and
there is no obstruction to having such BPS states from non-effective curves.
From another perspective, one can consider a complex structure deformation
$X_{4}\rightarrow X_{4}^{\prime}$ where $X_{4}^{\prime}$ has a larger
effective cone of divisors and curves such that $S_{\alpha}-S$ becomes
effective.

As an example, we draw the toric diagram of the case of
$\mathcal{S}=\mathbb{F}_{2}$ in figure 4. As one can see, in the deformed
$X_{4}^{\prime}$, the red lines and points lie on the same plane, where the
red points form the enhanced $\mathfrak{su}(3)$ flavor algebra. The ray
$v=(1,0,0)$ exactly corresponds to the flavor Cartan $F$ in the original
geometry $X_{4}$. There is a subtlety that after the deformation
$X_{4}^{\prime}$ only has a single $\mathbb{P}^{1}$-fibration structure (along
the $y$-axis), while $X_{4}$ has two $\mathbb{P}^{1}$-fibration structures.
Precisely speaking, the deformation $X_{4}\rightarrow X_{4}^{\prime}$ changes
the physical description along the Coulomb branch, but they still give rise to
the identical enhanced flavor symmetry at the singular limit
$X_{\rm{X_{4},sing}}$.

Figure 4: The toric diagram of $X_{4}=$ local
$\mathbb{P}^{1}\times\mathbb{F}_{2}$ and its deformation $X_{4}^{\prime}$. On
$X_{4}^{\prime}$, one explicitly see the flavor W-bosons of $\mathfrak{su}(3)$
flavor symmetry, which are not effective on $X_{4}$.

In summary, whether $\mathcal{S}$ has a 0-curve $C_{\alpha}$ that intersects
some $(-2)$-curves, it can be assigned node in the Dynkin diagram of the
enhanced flavor algebra $G_{F,nA}$. Note that the analysis works for non-toric
$\mathcal{S}$ as well.

###  3.3 Flavor symmetry dualities

In the cases of 3d $\mathcal{N}=2$ theories from M-theory on local CY4
singularities, we observe a new phenomenon of “flavor symmetry duality”, that
follows from the existence of multiple $\mathbb{P}^{1}$-fibration structures
mentioned in section 3.1.

Let us consider a 2d facet $\mathcal{F}$ in the toric diagram with two
possible $\mathbb{P}^{1}$-fibration structures, such as

(84)

If one considers a 5d theory from M-theory on the toric threefold
$X_{\mathcal{F}}$ in the figure (84), it would have IR dual descriptions
$SU(3)_{0}+2\mathbf{F}$ and $SU(2)_{\pi}-SU(2)_{\pi}$ Jefferson:2018irk . In
the description $SU(3)_{0}+2\mathbf{F}$, the vertical $\mathbb{P}^{1}$ fibers
$S_{2}\cdot S_{5}$, $S_{3}\cdot S_{5}$, $S_{3}\cdot S_{6}$ are shrunk to zero
volume. While in the description $SU(2)-SU(2)$, the horizontal
$\mathbb{P}^{1}$ fibers $S_{2}\cdot S_{3}$, $S_{3}\cdot S_{5}$, $S_{3}\cdot
S_{7}$ are shrunk to zero volume. In the singular CY3 limit, the massless
states can be expressed as either $SU(3)$ or $SU(2)^{2}$ representations.

Now let us consider the 3d $\mathcal{N}=2$ case where figure 5 is a 2d facet,
and all divisors $S_{i}$ are non-compact. We assume that the non-compact CY4
$Y_{\mathcal{F}}$ also has two $\mathbb{P}^{1}$-fibration structures. Now in
the limit where the vertical fibers are shrunk to zero volume, the 3d theory
would have enhanced flavor symmetry $\mathfrak{su}(3)$. On the other hand, in
the limit where the horizontal fibers are shrunk to zero volume, the enhanced
flavor symmetry is $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$. Finally, in the
full singular limit where all curves are shrunk to zero size, one can either
choose the $\mathfrak{su}(3)$ or $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$
flavor symmetry enhancements. These two choices represent two different ways
to arrange the physical states as non-abelian flavor Lie algebra
representations, and either of them makes sence. We hence call this “flavor
symmetry duality”.

For example let us consider the resolved non-compact CY4 in figure 5.

Figure 5: A rank-2 example with three $\mathbb{P}^{1}$-fibration structures,
and a 2d facet that demonstrates the flavor symmetry duality. The total flavor
rank of the corresponding 3D theory is 18.

This describes a rank-2 theory with compact divisors $D_{1}$ and $D_{2}$, and
there are two copies of the facet (84). It has three
$\mathbb{P}^{1}$-fibration structures:

  1. 1.

when the vertical $\mathbb{P}^{1}$ fibers (in red) shrinks to zero volume, we
have a $SU(3)$ gauge theory with $\mathfrak{su}(3)$ flavor symmetry
enhancement from the facet $\mathcal{F}$. Counting the other non-compact
divisors, the enhanced flavor symmetry from toric geometry is
$\mathfrak{su}(3)^{\oplus 2}\oplus\mathfrak{u}(1)^{\oplus 14}$.

  2. 2.

When the horizontal $\mathbb{P}^{1}$ fibers (in green) shrink to zero volume,
we have a $SU(2)^{2}$ gauge theory with
$\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ flavor symmetry enhancement from the
facet $\mathcal{F}$. Counting the other non-compact divisors, the enhanced
flavor symmetry from toric geometry is $\mathfrak{su}(2)^{\oplus
4}\oplus\mathfrak{u}(1)^{\oplus 14}$.

  3. 3.

In the third limit where the $\mathbb{P}^{1}$ fibers along the $D_{1}\cdot
S_{2}$ direction (in blue) shrink to zero volume, we have a $SU(2)^{2}$ gauge
theory with no flavor symmetry enhancement from the facet $\mathcal{F}$.
Counting the other non-compact divisors, the enhanced flavor symmetry from
toric geometry is $\mathfrak{su}(2)^{\oplus 6}\oplus\mathfrak{u}(1)^{\oplus
12}$.

Finally in the full singular limit of $X_{4}$, we can choose to use either of
the enhanced flavor symmetry $\mathfrak{su}(3)$ or
$\mathfrak{su}(2)\oplus\mathfrak{su}(2)$, as there is a flavor symmetry
duality between them. After we count all the contributions of the edges and
faces, the flavor symmetry duality is between $G_{F}=\mathfrak{su}(3)^{\oplus
2}\oplus\mathfrak{su}(2)^{\oplus 6}\oplus\mathfrak{u}(1)^{\oplus 8}$ and
$G_{F}=\mathfrak{su}(2)^{\oplus 10}\oplus\mathfrak{u}(1)^{\oplus 8}$.

Such flavor duality appears in a variety of 3D theories which can be seen as
follows. For a 2D facet $\mathcal{F}$ we consider both its associated local
CY3 $X_{\mathcal{F}}$ and a toric CY4 $Y_{\mathcal{F}}$ whose associated 3D
polyhedron $\mathcal{P}_{3}$ has $\mathcal{F}$ one of its 2D facets. If the 5D
SCFT obtained from M-theory on $X_{\mathcal{F}}$ admits multiple IR gauge
theory descriptions, then the 3D SCFT obtained from M-theory on
$Y_{\mathcal{F}}$ admits an IR flavor duality. From its construction it is
clear that this duality depends only on the property of $\mathcal{F}$ and is
independent from the way it is embedded in $\mathcal{P}_{3}$.

###  3.4 Higher-form symmetries and SymTFT from toric CY4

The computation of higher-form symmetries and SymTFT action from M-theory on a
CY4 was developed in vanBeest:2022fss  , which we briefly recap here.

We denote by $\mathcal{T}_{X_{4}}$ the 3D theory obtained from M-theory
compactification on CY4 $X_{4}$. To compute the 1-form symmetry of
$\mathcal{T}_{X_{4}}$, we need to compute the Smith normal form of the
intersection matrix $\\{C_{i}\cdot D_{j}\\}$, where $C_{i}$
$(i=1,\dots,b_{2}(X_{4}))$ are compact 2-cycles on $X_{4}$ and $D_{j}$
$(j=1,\dots,r=b_{6}(X_{4}))$ are the compact divisors.

Taking the _Smith normal form_ of the matrix $\\{C_{i}\cdot D_{j}\\}$:

$SNF(\\{C_{i}\cdot D_{j}\\})=\begin{pmatrix}l_{1}&0&\dots&0\\\
0&l_{2}&\dots&0\\\ \vdots&\vdots&\ddots&0\\\ 0&0&\dots&l_{r}\\\ 0&0&\dots&0\\\
\vdots&\vdots&\ddots&0\\\ 0&0&\dots&0\\\ \end{pmatrix}=A\cdot\\{C_{i}\cdot
D_{j}\\}\cdot B\,,$ (85)

one can read off the 1-form symmetry

$\Gamma=\bigoplus_{i=1}^{r}(\mathbb{Z}/\Gamma_{i}\mathbb{Z})=\bigoplus_{\alpha}\mathbb{Z}_{l_{\alpha}}\,.$
(86)

In the toric cases a simpler algorithm can be applied Morrison:2020ool ;
Luo:2023ive . One take the $n\times 4$ matrix where each row is the 4d toric
ray $\tilde{v}_{i}=(v_{i},1)$. It is required that $v_{i}$ locates on the
boundary of the polytope $\mathcal{P}$. Such a $n\times 4$ matrix has the same
Smith normal form as the one in (88), which gives rise to the same 1-form
symmetry.

We write down the generators of $\mathbb{Z}_{l_{\alpha}}$ 1-form symmetry
using a linear combination of compact divisors:

$Z_{6}^{\alpha}=B_{\alpha j}D_{j}\,.$ (87)

Besides the 1-form symmetries, there are also torsional $(-1)$-form
symmetries, which one can compute from the intersection matrix of compact
4-cycles $\\{\mathcal{S}_{i}\cdot\mathcal{S}_{j}\\}$
$(i=1,\dots,b_{4}(X_{4}))$. We can use the Smith decomposition

$SNF(\\{\mathcal{S}_{i}\cdot\mathcal{S}_{j}\\})=\begin{pmatrix}m_{1}&0&\dots&0\\\
0&m_{2}&\dots&0\\\ \vdots&\vdots&\ddots&0\\\ 0&0&\dots&m_{r}\\\ 0&0&\dots&0\\\
\vdots&\vdots&\ddots&0\\\ 0&0&\dots&0\\\
\end{pmatrix}=\mathcal{A}\cdot\\{\mathcal{S}_{i}\cdot\mathcal{S}_{j}\\}\cdot\mathcal{B}\,,$
(88)

to read off such symmetry $\bigoplus_{i}(\mathbb{Z}/m_{i}\mathbb{Z})$ and the
generators $Z_{4}^{\beta}=\mathcal{B}_{\beta j}\mathcal{S}_{j}$ for each
$\mathbb{Z}_{m_{\beta}}$ factor.

The SymTFT for the 3d $\mathcal{N}=2$ theory can be computed following the
general formula in vanBeest:2022fss :

$\frac{S_{\rm
SymTFT}}{2\pi}=-\sum_{j\alpha\beta}\Lambda_{22}^{j\alpha\beta}b^{\alpha}\int_{\mathcal{M}_{4}}F_{2}^{j}\cup
B_{2}^{\beta}-\sum_{\alpha\beta\gamma}\frac{\Lambda_{24}^{\alpha\beta\gamma}}{2}b^{\alpha}\int_{\mathcal{M}_{4}}B_{2}^{\beta}\cup
B_{2}^{\gamma}\,.$ (89)

Here $F_{2}^{j}$ corresponds to the field strength of 0-form flavor symmetry,
where $j$ labels each non-compact divisors $S_{j}$. $b^{\alpha}$ denotes the
amount of torsional background flux from $G_{4}$ over
$\mathbb{Z}_{m_{\alpha}}$ torsional 4-cocycles (which is dual to the 4-cycle
$Z_{4}^{\alpha}$). $B_{2}^{\beta}$ corresponds to the background gauge field
of $\mathbb{Z}_{l_{\beta}}$ 1-form symmetry, which arise from $G_{4}$ over
torsional 2-cocycles (which is dual to the 6-cycle $Z_{6}^{\beta}$). The
coefficients can be computed from the intersection number calculation

$\displaystyle\Lambda_{22}^{j\alpha\beta}$
$\displaystyle=\left[\frac{S_{j}\cdot Z_{4}^{\alpha}\cdot
Z_{6}^{\beta}}{m_{\alpha}l_{\beta}}\right]_{\rm mod\ 1}\,,$ (90)
$\displaystyle\Lambda_{24}^{\alpha\beta\gamma}$
$\displaystyle=\left[\frac{Z_{4}^{\alpha}\cdot Z_{6}^{\beta}\cdot
Z_{6}^{\gamma}}{m_{\alpha}l_{\beta}l_{\gamma}}\right]_{\rm mod\ 1}\,.$

Note that for the SymTFT to be non-trivial, one has to introduce torsional
flux $b^{\alpha}$ that generates discrete $(-1)$-form symmetries.

##  4 Examples from non-compact geometries

In this section we present concrete examples of non-compact CY4 and analyze
the physics of the corresponding 3D theories.

###  4.1 Toric Calabi-Yau fourfolds

In this section we study non-compact CY4’s that can be explicitly constructed
as the canonical bundle of a complex 3D compact variety, whose Ricci-flatness
trivially follows from $c_{1}=0$ and Yau’s theorem. Many of the examples that
fall into this class can be constructed torically which allows us to analyze
their geometry with standard toric geometry tools. Using the dictionary
developed in section 2 we compute both the naive and the enhanced flavor
symmetry group, the gauge group and various other important physical
quantities. We will also present an analysis of the possible $G_{4}$-flux
profile on such CY4 and the resulting change in the particle spectrum in field
theory. In particular, we analyze the cases of local $\mathbb{P}^{3}$, local
$\mathbb{P}^{1}\times\mathbb{P}^{2}$,
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$, local
$\mathbb{P}^{1}\times\mathbb{F}_{1}$ and local
$\mathbb{P}^{1}\times\text{dP}_{2}$ in full details.

####  4.1.1 Local $\mathbb{P}^{3}$

We first consider the simple case of $X_{4}=$ local $D$, where
$D=\mathbb{P}^{3}$. It gives rise to a $U(1)$ gauge theory on the CB. There
are no curves with normal bundle
$\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$ or
$\mathcal{O}\oplus\mathcal{O}(-1)\oplus\mathcal{O}(-1)$ on $\mathbb{P}^{3}$,
and there is no $\mathbb{P}^{1}$-fibration structure either. Thus there is no
geometric limit such that the gauge group is enhanced to $G=SU(2)$.

We define a non-compact divisor $S$ such that $D\cdot S|_{D}=H$ is the
hyperplane class of $D$. The compact curves on $X_{4}$ are in the homology
class $C_{a}=aD\cdot S^{2}=aH^{2}|_{D}$. We have

$K_{D}=-4H=-4D\cdot S\,$ (91)

and the intersection numbers

$D\cdot S^{3}=1\ ,\ D^{2}\cdot S^{2}=-4\ ,\ D^{3}\cdot S=16\ ,\ D^{4}=-64\,.$
(92)

##### Kähler form and $U(1)$ gauge coupling

We denote the (Poincaré dual) of Kähler form to be

$\displaystyle J^{c}$ $\displaystyle=xS$ (93)
$\displaystyle=-\frac{1}{4}xD\,.$

We can compute

$\text{Vol}(C_{a})=ax\ ,\ \text{Vol}(D\cdot S)=\text{Vol}(H)=\frac{1}{2}x^{2}\
,\ \text{Vol}(D)=\frac{1}{6}x^{3}\,.$ (94)

At a generic point in the Kähler moduli space, $x>0$, $D$ and any complex
curve $C\subset D$ have a positive volume. Hence the 3d $\mathcal{N}=2$ theory
$\mathcal{T}^{UV}_{X_{4}}$ is a $U(1)$ gauge theory coupled to an infinite
tower of massive particle states, which are generated by M2-branes wrapping
every $C\in H_{2}(D,\mathbb{Z})$.

The gauge coupling is (8)

$\frac{1}{g^{2}}=\text{Vol}(-D\cdot D)=2x^{2}\,.$ (95)

In the strong coupling limit $x\rightarrow 0$, these massive BPS states become
massless, and one recovers the geometric limit of local $\mathbb{P}^{3}$
singularity $X_{\text{4,sing}}$, leading to a 3d $\mathcal{N}=2$ SCFT
$\mathcal{T}^{\rm 3d}_{X_{\text{4,sing}}}$.

##### M2-brane wrapping states, 1-form symmetry and SymTFT

For the massive BPS states in this case, let us consider M2 brane wrapping
curves $C_{a}$ on $D=\mathbb{P}^{3}$. Its charge under the $U(1)$ gauge group
is

$C_{a}\cdot D|_{X_{4}}=C_{a}\cdot K_{D}|_{D}=-4aH^{3}|_{D}=-4a\,.$ (96)

Hence every massive particle BPS state has charge $-4a$ $(a\in\mathbb{Z})$
under the $U(1)$. We therefore expect a $\mathbb{Z}_{4}$ 1-form symmetry in
both $\mathcal{T}^{UV}_{X_{4}}$ and $\mathcal{T}^{\rm 3d}_{X_{\text{4,sing}}}$
from the Coulomb branch analysis.

For the intersection between 4-cycles, we have

$(D\cdot S)\cdot(D\cdot S)=-4\,,$ (97)

hence there is a $\mathbb{Z}_{4}$ $(-1)$-form symmetry as well with the
generator

$Z_{4}=D\cdot S\,.$ (98)

Using (89) we can compute the SymTFT action

$\frac{S_{\text{SymTFT}}}{2\pi}=-\frac{1}{8}b\int_{\mathcal{M}_{4}}B_{2}\cup
B_{2}\,,$ (99)

where $b$ is the torsional $G_{4}$ flux over $\mathbb{Z}_{4}$ torsional
4-cocycles and $B_{2}$ is the background gauge field for $\mathbb{Z}_{4}$
1-form symmetry.

####  4.1.2 Local $\mathbb{P}^{1}\times\mathbb{P}^{2}$

Now let us consider the case with an $SU(2)$ gauge theory description. To
construct such a theory one starts with a compact complex threefold $D$ which
is a $\mathbb{P}^{1}$-fibration over a (weak-Fano) complex surface
$\mathcal{S}$:

$D=\mathbb{P}^{1}\hookrightarrow\mathcal{S}$ (100)

Let us denote the fibral $\mathbb{P}^{1}$ by $F$. The normal bundle of $F$ is

$\displaystyle N_{F|D}$ $\displaystyle=\mathcal{O}\oplus\mathcal{O}\,,$ (101)
$\displaystyle N_{F|X_{4}}$
$\displaystyle=\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)\,.$

If $\mathcal{S}$ satisfies $h^{0,1}(\mathcal{S})=h^{0,2}(\mathcal{S})=0$, the
M2-brane wrapping mode over $F$ is a single vector multiplet with charge

$F\cdot D=-2$ (102)

under the $U(1)$ gauge group associated to $D$. The anti-M2-brane wrapping
mode over $F$ is also a vector multiplet, whose $U(1)$ charge is +2. Hence
they form the W-bosons of the $SU(2)$ gauge group. In the limit of
Vol$(F)\rightarrow 0$, the W-bosons become massless and we restore the non-
abelian $SU(2)$ gauge group. On the other hand, there are also M2 brane
wrapping modes over curves $C\subset\mathcal{S}$, which correspond to massive
BPS states that are different for each choice of $\mathcal{S}$.

In this section, we consider the case of a local
$\mathbb{P}^{1}\times\mathbb{P}^{2}$, i.e. $\mathcal{S}=\mathbb{P}^{2}$. In
this case, we write down the rays in the toric fan of CY4
$\tilde{v}_{i}=(v_{i},1)$:

$v=(0,0,0)\ ,\ v_{1}=(1,0,0)\ ,\ v_{2}=(0,1,0)\ ,\ v_{3}=(-1,-1,0)\ ,\
v_{4}=(0,0,1)\ ,\ v_{5}=(0,0,-1)\,.$ (103)

The 4d cones are

$\\{\tilde{v}\tilde{v}_{1}\tilde{v}_{2}\tilde{v}_{4}\ ,\
\tilde{v}\tilde{v}_{1}\tilde{v}_{3}\tilde{v}_{4}\ ,\
\tilde{v}\tilde{v}_{2}\tilde{v}_{3}\tilde{v}_{4}\ ,\
\tilde{v}\tilde{v}_{1}\tilde{v}_{2}\tilde{v}_{5}\ ,\
\tilde{v}\tilde{v}_{1}\tilde{v}_{3}\tilde{v}_{5}\ ,\
\tilde{v}\tilde{v}_{2}\tilde{v}_{3}\tilde{v}_{5}\\}\,.$ (104)

The Picard group of $X_{4}$ is generated by the compact divisor $D$ associated
to $\tilde{v}$, the non-compact divisor $S$ associated to $\tilde{v}_{4}$ and
$H$ associated to $\tilde{v}_{1}$, see the following figure of
$\mathcal{P}_{3}$:

The canonical class of $D$ is:

$K_{D}=(-2S-3H)|_{D}\,.$ (105)

The triple intersection numbers on $D$ are

$S\cdot H^{2}|_{D}=1\ ,\ S^{3}|_{D}=S^{2}\cdot H|_{D}=H^{3}|_{D}=0\,.$ (106)

The (Poincaré dual) of Kähler form is taken as

$\displaystyle J^{c}$ $\displaystyle=xS+yH$ (107)
$\displaystyle=-\frac{y}{3}D+\left(x-\frac{2y}{3}\right)S\,.$

We discuss the following physical aspects.

##### Kähler moduli and $U(1)$ gauge coupling

From (107) we can compute the volume of curves and cycles

$\displaystyle\text{Vol}(D\cdot S\cdot H)=y\ ,\ \text{Vol}(D\cdot H\cdot
H)=x\,,$ (108) $\displaystyle\text{Vol}(D\cdot S)=\frac{1}{2}y^{2}\ ,\
\text{Vol}(D\cdot H)=xy\,,$ $\displaystyle\text{Vol}(D)=\frac{1}{2}xy^{2}\,.$

Hence if all of these volume are non-nagetive, it is required that $x>0$,
$y>0$.

The $U(1)$ gauge coupling (8) is

$\frac{1}{g^{2}}=\text{Vol}(-D\cdot D)=y^{2}+3xy\,.$ (109)

##### Charged particles from M2-brane wrapping modes and 1-form symmetry

For an M2-brane wrapping a curve in the homology class

$C_{ab}=D\cdot(aS\cdot H+bH^{2})\,,$ (110)

its charge under the $U(1)$ gauge group is

$C_{ab}\cdot D=(C\cdot K_{D})|_{D}=-2b-3a\,.$ (111)

For example the curve $C_{10}=D\cdot S\cdot H$ has charge 3, which breaks the
$\mathbb{Z}_{2}$ 1-form symmetry of the $SU(2)$ gauge group.

On the other hand, $C_{01}=D\cdot H\cdot H$ is the $\mathbb{P}^{1}$ fiber with
normal bundle
$N_{C_{01}|X_{4}}=\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$.

#####  $SU(2)$ gauge theory limit

M2-brane wrapping $C_{01}$ acts as the W-boson for the $SU(2)$ gauge group,
which become massless in the limit

$\displaystyle\text{Vol}(C_{01})$ $\displaystyle=x=0\,.$ (112)

In this limit if $y>0$, we get an $SU(2)$ gauge theory description with
$SU(2)$ gauge coupling

$\frac{1}{g_{YM}^{2}}=y^{2}\,.$ (113)

Furthermore, in the limit $y\rightarrow 0$, we get a strongly coupled SCFT
$\mathcal{T}^{\rm 3d}_{X_{\text{4,sing}}}$ where all the volume of compact
cycles (LABEL:P1P2-volumes) shrink to zero.

##### Flavor symmetry $G_{F}$ for $\mathcal{T}^{\rm 3d}_{X_{\text{4,sing}}}$

In this case, there is no flavor W-boson from M2-brane wrapping a curve with
normal bundle $\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$, and the
flavor symmetry would be $G_{F}=\mathfrak{u}(1)$ for $\mathcal{T}^{\rm
3d}_{X_{\text{4,sing}}}$ at the singular limit. Let us choose the non-compact
divisor generating $G_{F}$ to be $S$, the charges of BPS states from M2-brane
wrapping the generators of Mori cone are in table 1. $Q_{e}$ is the gauge
$U(1)$ charge and $Q_{A}$ is the flavor charge.

$U(1)$ Charge | $Q_{e}$ | $Q_{A}$
---|---|---
Divisor | D | S
$C_{10}$ | $-3$ | 0
$C_{01}$ | $-2$ | 1
Table 1: Gauge and flavor charges of M2-brane wrappinP1P2-chargesg states on
local $\mathbb{P}^{1}\times\mathbb{P}^{2}$.

#####  $c_{2}$ and additional $G_{4}$ flux on the resolved $X_{4}$

Now we can compute the Poincaré dual of $c_{2}(X_{4})$, which is

$c_{2}^{c}(X_{4})=2D\cdot S+3D\cdot H+3H^{2}+6S\cdot H\,.$ (114)

From

$\frac{1}{2}c_{2}^{c}(X_{4})\cdot D\cdot H\in\mathbb{Z}\ ,\
\frac{1}{2}c_{2}^{c}(X_{4})\cdot D\cdot S\in\mathbb{Z}\,,$ (115)

we see that a non-zero $G_{4}$ flux is not required.

On the other hand, if one chooses to turn on $G_{4}$ flux, the most general
$G_{4}^{c}$ dual to compact 4-cycles satisfying (42) is

$G_{4}^{c}=mD\cdot S+nD\cdot H(m,n\in\mathbb{Z})\,.$ (116)

We can compute the chirality for the adjoint matter charged under $SU(2)$,
which comes from M2-brane wrapping the $\mathbb{P}^{1}$ fiber $D\cdot H\cdot
H$. The curve $D\cdot H\cdot H$ has normal bundle $N_{D\cdot H\cdot
H}=\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$

$\displaystyle\chi$ $\displaystyle=-\int_{D\cdot S}c_{1}(D\cdot S)c_{1}(L)$
(117) $\displaystyle=D\cdot G_{4}^{c}\cdot(-3H)$ $\displaystyle=(mS+nH)\cdot
K_{D}\cdot(-3H)$ $\displaystyle=9m+6n\,.$

From the $G_{4}$ flux there are induced CS terms from 11d SUGRA action
(51)(52)

$\displaystyle S_{CS}$ $\displaystyle=\frac{1}{4\pi}(G_{4}^{c}\cdot D\cdot
D)\int A\wedge dA+\frac{1}{2\pi}(G_{4}^{c}\cdot D\cdot S)\int A\wedge
dB+\frac{1}{4\pi}(G_{4}^{c}\cdot S\cdot S)\int B\wedge dB$ (118)
$\displaystyle=\frac{1}{4\pi}(9m+12n)\int A\wedge dA-\frac{3n}{2\pi}\int
A\wedge dB\,.$

Here $A$ is the $U(1)$ gauge field and $B$ is the $G_{F}=\mathfrak{u}(1)$
background gauge field.

####  4.1.3 Local $\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$

The case of $X_{4}=$ local
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ is particularly
interesting because it admits multiple $\mathbb{P}^{1}$-fibration structures
and a non-trivial flavor symmetry enhancement $G_{F}=\mathfrak{su}(3)$. We
write down the rays in the toric fan of $X_{4}$ $\tilde{v}_{i}=(v_{i},1)$:

$\displaystyle v=(0,0,0)\ ,\ v_{1}=(1,0,0)\ ,\ v_{2}=(0,1,0)\ ,\
v_{3}=(0,0,1)\ ,\ v_{4}=(-1,0,0)\ ,\ $ (119) $\displaystyle v_{5}=(0,-1,0)\ ,\
v_{6}=(0,0,-1)\,.$

The cones are

$\\{\tilde{v}\tilde{v}_{1}\tilde{v}_{2}\tilde{v}_{3}\ ,\
\tilde{v}\tilde{v}_{1}\tilde{v}_{2}\tilde{v}_{6}\ ,\
\tilde{v}\tilde{v}_{1}\tilde{v}_{3}\tilde{v}_{5}\ ,\
\tilde{v}\tilde{v}_{1}\tilde{v}_{5}\tilde{v}_{6}\ ,\
\tilde{v}\tilde{v}_{2}\tilde{v}_{3}\tilde{v}_{4}\ ,\
\tilde{v}\tilde{v}_{2}\tilde{v}_{4}\tilde{v}_{6}\ ,\
\tilde{v}\tilde{v}_{3}\tilde{v}_{4}\tilde{v}_{5}\ ,\
\tilde{v}\tilde{v}_{4}\tilde{v}_{5}\tilde{v}_{6}\\}\,.$ (120)

The Picard group of $X_{4}$ is generated by the divisors $D$, $S_{1}$, $S_{2}$
and $S_{3}$ which corresponds to the rays $\tilde{v}$, $\tilde{v}_{1}$,
$\tilde{v}_{2}$ and $\tilde{v}_{3}$ respectively. The corresponding 3D
polytope $\mathcal{P}_{3}$ is drawn in Figure 6.

Figure 6: $\mathcal{P}_{3}$ of local
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$.

The canonical class of $D$ is:

$K_{D}=(-2S_{1}-2S_{2}-2S_{3})|_{D}\,.$ (121)

The intersection numbers on $D$ are

$S_{1}\cdot S_{2}\cdot S_{3}|_{D}=1\ ,\
S_{1}^{2}|_{D}=S_{2}^{2}|_{D}=S_{3}^{2}|_{D}=0\,.$ (122)

We discuss the following physical aspects of the model.

##### Kähler form and $U(1)$ gauge coupling

We take the (Poincaré dual of) the Kähler class to be

$\displaystyle J^{c}$ $\displaystyle=xS_{1}+yS_{2}+zS_{3}$ (123)
$\displaystyle=-\frac{1}{6}(x+y+z)D+\frac{1}{3}(2x-y-z)(S_{1}-S_{2})+\frac{1}{3}(x+y-2z)(S_{2}-S_{3})\,.$

The volume of curves are

$\text{Vol}(D\cdot S_{1}\cdot S_{2})=z\ ,\ \text{Vol}(D\cdot S_{2}\cdot
S_{3})=x\ ,\ \text{Vol}(D\cdot S_{1}\cdot S_{3})=y\,.$ (124)

The criteria for all the compact cycles to have non-negative volume is $x\geq
0$, $y\geq 0$, $z\geq 0$.

The gauge coupling of $U(1)$ CB gauge group is (8)

$\displaystyle\frac{1}{g^{2}}$ $\displaystyle=-\frac{1}{2}D\cdot D\cdot
J^{c}\cdot J^{c}$ (125) $\displaystyle=2(xy+xz+yz)\,.$

We can see that it is equal to the sum of the volume of all the toric divisors
of $D$

$\frac{1}{g^{2}}=2\text{Vol}(D\cdot S_{1})+2\text{Vol}(D\cdot
S_{2})+2\text{Vol}(D\cdot S_{3})\,.$ (126)

##### Charged particles from M2-brane wrapping modes and 1-form symmetry

For M2-brane wrapping a curve in the homology class

$C_{abc}=D\cdot(aS_{1}\cdot S_{2}+bS_{2}\cdot S_{3}+cS_{1}\cdot S_{3})\,,$
(127)

its charge under the $U(1)$ gauge group is

$C_{abc}\cdot D=-2a-2b-2c\,,$ (128)

which is always a multiple of 2, hence the $\mathbb{Z}_{2}$ 1-form symmetry is
preserved on the Coulomb branch.

The Mori cone of $X_{4}$ is generated by $C_{100}$, $C_{010}$ and $C_{001}$,
note that these curves all have normal bundle
$N_{C_{100}|X_{4}}=N_{C_{010}|X_{4}}=N_{C_{001}|X_{4}}=\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$.
Their charges under the $U(1)$ gauge group are

$C_{100}\cdot D=C_{010}\cdot D=C_{001}\cdot D=-2\,.$ (129)

Hence M2-brane wrapping one of these curves would give rise to a vector
multiplet, which can act as the W-boson for an $SU(2)$ gauge group in a
certain $\mathbb{P}^{1}$ fibration limit, which will be discussed later.

##### Non-abelian flavor symmetry enhancement $G_{F}$

Now let us discuss the non-abelian 0-form symmetry enhancement at the singular
point $X_{\text{4,sing}}$. We first identify the generators of the flavor
Cartan subalgebra $\mathfrak{u}(1)^{2}$, which are

$F_{1}=S_{1}-S_{2}\ ,\ F_{2}=S_{2}-S_{3}\,.$ (130)

We identify the flavor W-bosons of $F_{1}$ and $F_{2}$ to be M2-branes
wrapping

$C_{1}=D\cdot S_{3}\cdot(S_{1}-S_{2})$ (131)

and

$C_{2}=D\cdot S_{1}\cdot(S_{2}-S_{3})\,.$ (132)

One can check the following requirements:

  1. 1.

The flavor W-bosons $C_{1}$ and $C_{2}$ are neutral under the $U(1)$ gauge
symmetry generated by $D$, as one can check that $C_{1}\cdot D=C_{2}\cdot
D=0$.

  2. 2.

The flavor W-bosons $C_{1}$ and $C_{2}$ has the correct normal bundle
$N_{C_{1}|X_{4}}=N_{C_{2}|X_{4}}=\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$.

Note that the curves $C_{1}$ and $C_{2}$ are not effective on the particular
resolved geometry, they represent degrees of freedom that are decoupled to the
3d $\mathcal{N}=2$ SCFT in the singular limit777The story is analogous to the
$SU(2)$ flavor symmetry enhancement of the 5d $SU(2)_{0}$ SCFT from M-theory
on local $\mathbb{P}^{1}\times\mathbb{P}^{1}$, where the flavor Cartan and
flavor W-boson are not effective..

Now let us compute the charge of $C_{1}$ and $C_{2}$ under the flavor Cartan
generators:

$\begin{array}[]{c|cc}&F_{1}&F_{2}\\\ \hline\cr C_{1}&-2&1\\\
C_{2}&1&-2\end{array}$ (133)

As one can see, they form the Cartan matrix of an $\mathfrak{su}(3)$ flavor
algebra. Hence the 0-form flavor symmetry of the local
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ theory is actually
enhanced to $G_{F}=\mathfrak{su}(3)$ at the singular limit!

One can also check that the curves $C_{abc}$ with the same normal bundles form
$\mathfrak{su}(3)$ representations. Given a triple $(a,b,c)$, the curves

$C_{abc}\ ,\ C_{acb}\ ,\ C_{bac}\ ,\ C_{bca}\ ,\ C_{cab}\ ,\ C_{cba}$ (134)

all have the same normal bundles, which can be combined into a larger
representation.

For example, let us consider the case of $(a,b,c)=(1,0,0)$. The curves
$C_{100}$, $C_{010}$, $C_{001}$ have the charges $Q_{e}$ under the gauge
Cartan and $Q_{A}$, $Q_{B}$ under the flavor Cartans as shown in table 2.
Hence one can see that they form a $\mathbf{3}$ representation of the enhanced
flavor symmetry $\mathfrak{su}(3)$.

For the case of $(a,b,c)=(1,1,0)$. The curves $C_{110}$, $C_{011}$, $C_{101}$
has the following charges under the flavor Cartan

$\begin{array}[]{c|cc}&F_{1}&F_{2}\\\ \hline\cr C_{110}&1&-1\\\ C_{011}&0&1\\\
C_{101}&-1&0\end{array}$ (135)

They form a $\bar{\mathbf{3}}$ representation of the enhanced flavor symmetry
$\mathfrak{su}(3)$.

$U(1)$ Charge | $Q_{e}$ | $Q_{A}$ | $Q_{B}$
---|---|---|---
Divisor | $D$ | $F_{1}$ | $F_{2}$
$C_{100}$ | $-2$ | 0 | $-1$
$C_{010}$ | $-2$ | 1 | 0
$C_{001}$ | $-2$ | $-1$ | 1
Table 2: U(1) gauge charge and flavor charges of BPS states from M2-brane
wrapping the Mori cone generators of local
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$.

#####  $c_{2}$ and additional $G_{4}$ flux on the resolved $X_{4}$

Now let us discuss the $G_{4}$ flux. As $c_{2}(X_{4})$ is an even element in
$H^{4}(X_{4},\mathbb{Z})$, a non-zero $G_{4}$ flux is not required.
Nonetheless, one can still choose to add $G_{4}$ flux of the general form

$G_{4}^{c}=D\cdot(mS_{1}+nS_{2}+kS_{3})\,.$ (136)

The flux will induce chiral matter from M2-brane wrapping $C_{100}$, $C_{010}$
and $C_{001}$.

The moduli space of $C_{100}$ is $D\cdot S_{3}$, and we can compute the
chirality of $C_{100}$ as

$\displaystyle\chi(C_{100})$ $\displaystyle=G_{4}^{c}\cdot
D\cdot(-2S_{1}-2S_{2})$ (137) $\displaystyle=4m+4n+8k\,.$

Similarly, the moduli space of $C_{010}$ and $C_{001}$ are $S_{1}$ and $S_{2}$
respectively, and we have

$\displaystyle\chi(C_{010})$ $\displaystyle=8m+4n+4k$ (138)
$\displaystyle\chi(C_{001})$ $\displaystyle=4m+8n+4k\,.$

Note that the chiralities $\chi(C_{100})$, $\chi(C_{010})$ and $\chi(C_{001})$
are different for generic choices of $m$, $n$ and $k$ which in turn seems to
imply that the $SU(3)$ flavor symmetry is broken for a generic choice of
$G_{4}$-flux since we have shown in table 2 that M2-branes wrapping
$(C_{100},C_{010},C_{001})$ form a $\mathbf{3}$ under $SU(3)$ without
$G_{4}$-flux. This is not surprising if we look solely at the curves $C_{abc}$
since a generic choice of $G_{4}^{c}$ clearly breaks the symmetric group
$S_{3}$ symmetry of (127), which one can also see from Figure 6 that there is
an apparent symmetry among $S_{1}$, $S_{3}$ and $S_{3}$ before introducing
$G_{4}$-flux. But this is a bit puzzling if we look at M2-brane wrapping modes
on $C_{1}$ and $C_{2}$ from which we obtain the 6 flavor W-bosons of the
enhanced $SU(3)$ flavor symmetry. The $G_{4}$-flux does not give rise to non-
trivial chiral indices for the states obtained from M2-brane wrapping $C_{1}$
and $C_{2}$ therefore the $SU(3)$ flavor symmetry does not seem to be broken
at all from this point of view.

For this, recall that in Section 2.4 we have commented that a careless choice
of $G_{4}$-flux may prevent certain compact 4-cycle from shrinking therefore
the SCFT limit cannot be achieved. Consider for example
$S=D\cdot(S_{1}+S_{2})\subset D$, we have $G_{4}^{c}\cdot S\neq 0$ for generic
$G_{4}^{c}$. Therefore $S$, hence $D$ cannot be shrunk. But since we have
shown that the flavor W-bosons are obtained from M2-brane wrapping $D\cdot
S_{1}\cdot(S_{2}-S_{3})$ and $D\cdot S_{3}\cdot(S_{1}-S_{2})$ respectively,
the fact that $D$ being unshrinkable implies that the enhancement to $SU(3)$
can never be realized with generic $G_{4}^{c}$. In this sense the breaking of
$SU(3)$ by $G_{4}^{c}$ becomes almost tautological.

From the $G_{4}$ flux there are induced CS terms

$\displaystyle S_{CS}$ $\displaystyle=\frac{1}{4\pi}(8m+8n+8k)\int A\wedge
dA+\frac{1}{2\pi}(2m-2n)\int A\wedge dB_{1}+\frac{1}{2\pi}(2n-2k)\int A\wedge
dB_{2}$ (139) $\displaystyle+\frac{1}{2\pi}(k+m-n)\int B_{1}\wedge
dB_{2}-\frac{2k}{4\pi}\int B_{1}\wedge dB_{1}-\frac{2m}{4\pi}\int B_{2}\wedge
dB_{2}\,.$

Here $A$ is the $U(1)$ gauge field, $B_{1}$ and $B_{2}$ are the background
gauge fields of the flavor Cartan $F_{1}$ and $F_{2}$ of
$G_{F}=\mathfrak{su}(3)$.

##### SymTFT from geometry

Although adding a free $G_{4}$ flux would destroy the validity of geometric
singularity description, one can still discuss the addition of torsional
$G_{4}$ flux and the resulting SymTFT action for the 3d $\mathcal{N}=2$
theory, as in section 3.4.

Let us first compute the intersection matrix between 4-cycles

$\\{\mathcal{S}_{i}\cdot\mathcal{S}_{j}\\}=\begin{array}[]{c|ccc}&D\cdot
S_{1}&D\cdot S_{2}&D\cdot S_{3}\\\ \hline\cr D\cdot S_{1}&0&-2&-2\\\ D\cdot
S_{2}&-2&0&-2\\\ D\cdot S_{3}&-2&-2&0\end{array}$ (140)

Its Smith normal form is
$SNF(\\{\mathcal{S}_{i}\cdot\mathcal{S}_{j}\\})=$diag$(2,2,4)$. The generators
of $\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{4}$ $(-1)$-form
symmetries are

$Z_{4}^{1}=D\cdot S_{1}\ ,\ Z_{4}^{2}=D\cdot S_{2}\ ,\
Z_{4}^{3}=D\cdot(-S_{1}-S_{2}+S_{3})\,.$ (141)

We can thus plug in (89) to compute the SymTFT action

$\displaystyle\frac{S_{\rm SymTFT}}{2\pi}$
$\displaystyle=\frac{1}{2}\int_{\mathcal{M}_{4}}[(b_{2}-b_{1})F_{2}^{1}+(b_{3}-b_{2})F_{2}^{2}]\cup
B_{2}+\frac{1}{4}(2b_{1}+2b_{2}-b_{3})\int_{\mathcal{M}_{4}}B_{2}\cup
B_{2}\,.$ (142)

Here $b_{i}$ are the torsional flux associated to $Z_{4}^{i}$, $F_{2}^{i}$
$(i=1,2)$ are field strength of $U(1)$ background gauge fields corresponding
to the flavor Cartan generators in (130) and $B_{2}$ is the background gauge
field for the $\mathbb{Z}_{2}$ 1-form symmetry.

#####  $\mathbb{P}^{1}$-fibration structures and the triality

Now let us discuss the different $\mathbb{P}^{1}$-fibration structures,
following section 3.1. We have three different $\mathbb{P}^{1}$-fibration
structures along the red, green and blue lines in figure 6. In each of these
cases when the $\mathbb{P}^{1}$-fiber shrinks to zero volume, we have an
effective UV $SU(2)$ gauge theory description.

For instance let us consider the $\mathbb{P}^{1}$-fibration structure in red,
where $D\cdot S_{3}$ is the base while $C_{100}\equiv D\cdot S_{1}\cdot S_{2}$
is the $\mathbb{P}^{1}$ fiber. When $z=0$, the $\mathbb{P}^{1}$ fiber
$C_{100}$ shrinks to zero volume. We require that in this geometric limit,
$x,y>0$. In this case, the gauge coupling (125) becomes the $SU(2)$ gauge
coupling

$\frac{1}{g_{YM}^{2}}=2xy\,.$ (143)

For the BPS states in table 2, $C_{100}$ becomes the massless W-boson for
$SU(2)$ in this case, while $C_{010}$ and $C_{001}$ are two massive vector
multiplets. For the physics interpretation, we conjecture that $C_{010}$ can
be thought as the twisted dimensional reduction of the dyonic instanton in 5d
$SU(2)_{0}$ theory Jia:2021ikh  on $\mathbb{P}^{1}=C_{001}=D\cdot S_{1}\cdot
S_{3}$. For the other state $C_{001}$, it is regarded as the twisted
dimensional reduction of the dyonic instanton in 5d $SU(2)_{0}$ theory on
$C_{010}$ as well. These BPS states are all $(0+1)$-dimensional disorder
operators in the 3d $\mathcal{N}=2$ $SU(2)$ field theory.

Alternatively, we can argue that $C_{010}$ and $C_{001}$ cannot be charged
local operators charged under $SU(2)$ gauge group, because they cannot be
completed into a full $SU(2)$ adjoint representation (the single uncharged
$U(1)$ gauge boson already formed the $SU(2)$ gauge group with the W-boson
$C_{100}$). Thus in the $SU(2)$ gauge theory description, such operators can
only be disorder operators.

Now let us discuss the flavor symmetry in this limit. The volume of the flavor
W-bosons (131), (132) are

$\text{Vol}(F_{1})=y-x\ ,\ \text{Vol}(F_{2})=z-y\,.$ (144)

Now we find an interesting phenomenon: when $x=y$, the volume of the flavor
W-boson $F_{1}$ vanishes, and we have an enhanced flavor symmetry
$\mathfrak{su}(2)\oplus\mathfrak{u}(1)$. When $x\neq y$, the flavor symmetry
is still $\mathfrak{u}(1)^{\oplus 2}$.

When going to the $\mathbb{P}^{1}$-fibration structure in green, we take
Vol$(C_{010})=x=0$ and $y,z>0$ in (124). the disorder operator from M2-brane
wrapping $C_{010}$ becomes the massless W-boson in this limit. On the other
hand, the old W-boson $C_{100}$ becomes a disorder operator. The $SU(2)$ gauge
coupling is now

$\frac{1}{g_{YM}^{2}}=2yz\,.$ (145)

Finally, if we take the $\mathbb{P}^{1}$-fibration structure in blue, which
means that Vol$(C_{001})=y=0$ and $x,z>0$. Now M-brane wrapping the curve
$C_{001}$ becomes the massless W-boson while the other $C_{ab0}$ corresponds
massive disorder operators. The $SU(2)$ gauge coupling in this limit is

$\frac{1}{g_{YM}^{2}}=2xz\,.$ (146)

In summary, there are three $\mathbb{P}^{1}$-fibration structures, and we have
a triality among the three different limits, see figure 7.

Figure 7: Three different geometric limits of the local
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ geometry, where one
of the $\mathbb{P}^{1}$ fibers shrink to zero volume. They represents three
different $SU(2)$ gauge theory descriptions.

####  4.1.4 Local $\mathbb{P}^{1}\times\mathbb{F}_{1}$

We write down the rays in the toric fan of CY4 $\tilde{v}_{i}=(v_{i},1)$:

$\displaystyle v=(0,0,0)\ ,\ v_{1}=(1,0,0)\ ,\ v_{2}=(0,1,0)\ ,\
v_{3}=(0,0,1)\ ,\ v_{4}=(-1,-1,0)\ ,\ $ (147) $\displaystyle v_{5}=(0,-1,0)\
,\ v_{6}=(0,0,-1)\,.$

The cones are the same as the local
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ case (120). The
Picard group of $X_{4}$ is generated by the divisors $D$, $S_{1}$, $S_{3}$ and
$S_{5}$ which corresponds to the rays $\tilde{v}$, $\tilde{v}_{1}$,
$\tilde{v}_{3}$ and $\tilde{v}_{5}$ respectively, see figure 8.

Figure 8: $\mathcal{P}_{3}$ of local $\mathbb{P}^{1}\times\mathbb{F}_{1}$.

The canonical class of $D$ is:

$K_{D}=(-3S_{1}-2S_{3}-2S_{5})|_{D}\,.$ (148)

The intersection numbers on $D$ are

$S_{1}\cdot S_{3}\cdot S_{5}|_{D}=1\ ,\ S_{1}^{2}|_{D}=S_{3}^{2}|_{D}=0\ ,\
S_{1}\cdot S_{5}^{2}|_{D}=0\ ,\ S_{3}\cdot S_{5}^{2}|_{D}=-1\,.$ (149)

We discuss the following physical aspects of the model.

##### Charged particles from M2-brane wrapping modes and 1-form symmetry

For M2-brane wrapping a curve in the homology class

$C_{abc}=D\cdot(aS_{1}\cdot S_{3}+bS_{1}\cdot S_{5}+cS_{3}\cdot S_{5})\,,$
(150)

its charge under the $U(1)$ gauge group is

$C_{abc}\cdot D=-2a-2b+c\,.$ (151)

The 1-form symmetry is trivial.

The Mori cone generators of $X_{4}$ are

  * •

$C_{100}=D\cdot S_{1}\cdot S_{3}$ with normal bundle
$N_{C_{100}|X_{4}}=\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$.
M2-brane wrapping $C_{100}$ gives a vector multiplet.

  * •

$C_{010}=D\cdot S_{1}\cdot S_{5}$ with normal bundle
$N_{C_{100}|X_{4}}=\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$. In
fact, $C_{010}$ is the $\mathbb{P}^{1}$ fiber in
$D=\mathbb{P}^{1}\times\mathbb{F}_{1}$. M2-brane wrapping $C_{100}$ gives a
vector multiplet as well.

  * •

$C_{001}=D\cdot S_{3}\cdot S_{5}$ with
$N_{C_{001}|X_{4}}=\mathcal{O}\oplus\mathcal{O}(-1)\oplus\mathcal{O}(-1)$ and
moduli space $\mathbb{P}^{1}$. Following the discussions in section 2.2, when
there is no $G_{4}$ flux, there is no BPS state (zero mode) from M2-brane
wrapping $C_{001}$.

Despite that $C_{001}=D\cdot S_{3}\cdot S_{5}$ gives no zero mode, one can
consider the curve $C_{101}=D\cdot S_{3}\cdot S_{2}$ with normal bundle
$\mathcal{O}\oplus\mathcal{O}(1)\oplus\mathcal{O}(-3)$. M2-brane wrapping such
curve would give a rich set of higher-spin massive particles on the CB.

##### Non-abelian flavor symmetry enhancement $G_{F}$

The non-compact divisor generating flavor Cartan of $\mathfrak{su}(2)$ is
$F_{1}=S_{5}-S_{3}+S_{1}$, with flavor W-boson

$\displaystyle C$ $\displaystyle=D\cdot(S_{5}-S_{3}+S_{1})\cdot S_{1}$ (152)
$\displaystyle=D\cdot(S_{5}-S_{3})\cdot S_{1}\,,$

one can compute that
$N_{C|X_{4}}=\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$ and $C\cdot
D=0$. The non-compact divisor for the other flavor Cartan can be chosen as
$F_{2}=S_{1}$888Note that there are freedom to choose $F_{1}$ and $F_{2}$. One
can choose $F_{1}=S_{5}-S_{3}+aS_{1}$ $(a\in\mathbb{Z})$ and $F_{2}$ to be an
arbitrary linear combination of non-compact divisors.. Hence the enhanced
flavor symmetry is $G_{F}=\mathfrak{su}(2)\oplus\mathfrak{u}(1)$.

We list the gauge charges and flavor symmetry charges under $F_{1}$, $F_{2}$
for M2-branes wrapping Mori cone generators $C_{100}$, $C_{010}$, $C_{001}$ in
table 3. $Q_{e}$ is the gauge charge and $Q_{A}$, $Q_{B}$ are flavor charges.

$U(1)$ Charge | $Q_{e}$ | $Q_{A}$ | $Q_{B}$
---|---|---|---
Divisor | $D$ | $F_{1}$ | $F_{2}$
$C_{100}$ | $-2$ | 1 | 0
$C_{010}$ | $-2$ | $-1$ | 0
$C_{001}$ | $-1$ | 0 | 1
Table 3: U(1) gauge charge and flavor charges of BPS states from M2-brane
wrapping the Mori cone generators of local
$\mathbb{P}^{1}\times\mathbb{F}_{1}$.

#####  $c_{2}$ and additional $G_{4}$ flux on resolved $X_{4}$

Now we consider $G_{4}$ flux. First we compute

$c_{2}^{c}(X_{4})=2D\cdot S_{3}+3D\cdot S_{1}+6S_{1}\cdot S_{3}+4S_{2}\cdot
S_{3}\,,$ (153)

which has even intersection number with any 4-cycle on $X_{4}$. Hence no non-
zero $G_{4}$-flux is required. Nonetheless we can add $G_{4}$-flux whose most
general form is:

$G_{4}^{c}=mD\cdot S_{1}+nD\cdot S_{3}+kD\cdot
S_{5}\quad(m,n,k\in\mathbb{Z}).$ (154)

Now we compute the chirality of certain charged matter fields from M2-brane
wrapping 2-cycles. For an M2-brane wrapping $C_{100}=D\cdot S_{1}\cdot S_{3}$
with normal bundle
$N_{C_{100}|X_{4}}=\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$, its
moduli space is $D\cdot S_{5}\sim\mathbb{P}^{1}\times\mathbb{P}^{1}$. The
chirality is

$\chi(C_{100})=-\int_{D\cdot S_{5}}c_{1}(D\cdot S_{5})c_{1}(L)=D\cdot
G_{4}^{c}\cdot(-2S_{1}-2S_{3})=2k+4m\,.$ (155)

For an M2-brane wrapping $C_{010}=D\cdot S_{1}\cdot S_{5}$ with normal bundle
$N_{C_{010}|X_{4}}=\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$, its
moduli space is $D\cdot S_{3}\sim\mathbb{F}_{1}$. The chirality is

$\chi(C_{010})=-\int_{D\cdot S_{2}}c_{1}(D\cdot S_{3})c_{1}(L)=D\cdot
G_{4}^{c}\cdot(-3S_{1}-2S_{5})=2k+4m+4n\,.$ (156)

For an M2-brane wrapping $C_{001}=D\cdot S_{3}\cdot S_{5}$ with normal bundle
$N_{C_{001}|X_{4}}=\mathcal{O}\oplus\mathcal{O}(-1)\oplus\mathcal{O}(-1)$, its
moduli space is $D\cdot S_{1}\cdot S_{5}\sim\mathbb{P}^{1}$. To compute the
chirality we need to integrate $G_{4}$ over the matter surface $D\cdot S_{5}$.
The chirality is

$\chi(C_{001})=D\cdot S_{5}\cdot G_{4}^{c}=2k-2m+n\,.$ (157)

Finally, we compute the effective Chern-Simons term from 11d SUGRA action,
which is free of parity anomaly

$\displaystyle S_{CS}$ $\displaystyle=\frac{1}{4\pi}(8m+8n+4k)\int A\wedge
dA+\frac{1}{2\pi}(k-3n)\int A\wedge dB_{1}-\frac{1}{2\pi}(2k+2n)\int A\wedge
dB_{2}$ (158) $\displaystyle+\frac{1}{2\pi}(k-n)\int B_{1}\wedge
dB_{2}+\frac{1}{4\pi}(n-2m)\int B_{1}\wedge dB_{1}\,.$

Here $A$ is the $U(1)$ gauge field, $B_{1}$ and $B_{2}$ are background gauge
fields of the flavor Cartans $F_{1}$ and $F_{2}$.

#####  $\mathbb{P}^{1}$-fibration structures and duality

Let us write the Poincaré dual of the Kähler form in terms of

$J^{c}=xS_{1}+yS_{3}+zS_{5}\,.$ (159)

The $U(1)$ gauge coupling on CB is (8)

$\displaystyle\frac{1}{g^{2}}$ $\displaystyle=-\frac{1}{2}D\cdot D\cdot
J^{c}\cdot J^{c}$ (160) $\displaystyle=2(xy+xz+yz)-z^{2}\,.$

The volume of various curves are

$\text{Vol}(C_{100})=z\ ,\ \text{Vol}(C_{010})=y\ ,\
\text{Vol}(C_{001})=x-z\,.$ (161)

For all the compact cycles to have a non-negative volume, we require that
$x\geq z\geq 0$, $y\geq 0$.

Local $\mathbb{P}^{1}\times\mathbb{F}_{1}$ has two distinct $\mathbb{P}^{1}$
fibration structures.

If one shrink the $\mathbb{P}^{1}$ fiber $C_{010}=D\cdot S_{1}\cdot S_{5}$
along the red direction in figure 8, the gauge theory description is $SU(2)$
with massive matter fields from the curves $C_{a0c}$ in the base $D\cdot
S_{3}\sim\mathbb{F}_{1}$. In this limit $y=0$, $x>z>0$, and the $SU(2)$ gauge
coupling is

$\frac{1}{g_{YM}^{2}}=2xz-z^{2}\,.$ (162)

On the other hand, if one shrink the $\mathbb{P}^{1}$ fiber $C_{100}=D\cdot
S_{1}\cdot S_{3}$ along the green direction, the gauge theory description is
$SU(2)$ with massive matter fields from the curves $C_{0bc}$ in the base
$D\cdot S_{5}\sim\mathbb{F}_{0}$. In this limit $z=0$, $x,y>0$, and the
$SU(2)$ gauge coupling is

$\frac{1}{g_{YM}^{2}}=2xy\,.$ (163)

####  4.1.5 Local $\mathbb{P}^{1}\times\mathbb{F}_{2}$

We write down the rays in the toric fan of CY4 $\tilde{v}_{i}=(v_{i},1)$:

$\displaystyle v=(0,0,0)\ ,\ v_{1}=(1,0,0)\ ,\ v_{2}=(0,1,0)\ ,\
v_{3}=(0,0,1)\ ,\ v_{4}=(-1,-2,0)\ ,\ $ (164) $\displaystyle v_{5}=(0,-1,0)\
,\ v_{6}=(0,0,-1)\,.$

The cones are the same as the local
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ case (120). The
Picard group of $X_{4}$ is generated by the divisors $D$, $S_{1}$, $S_{3}$ and
$S_{5}$ which corresponds to the rays $\tilde{v}$, $\tilde{v}_{1}$,
$\tilde{v}_{3}$ and $\tilde{v}_{5}$ respectively, see figure 9.

Figure 9: $\mathcal{P}_{3}$ of local $\mathbb{P}^{1}\times\mathbb{F}_{2}$.

In this case, the geometry and physics are almost exactly the same as the
local $\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ case, with the
linear relation

$(S_{5})|_{\mathbb{P}^{1}\times\mathbb{F}_{2}}=(S_{2}-S_{1})_{\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}}\,.$
(165)

Hence in the case of local $\mathbb{P}^{1}\times\mathbb{F}_{2}$, $S_{5}$ is a
flavor Cartan by itself, and $D\cdot S_{5}\cdot S_{1}$ is its flavor W-boson.
In this case note that $D\cdot S_{5}\cdot S_{1}$ is an effective curve,
nonetheless the M2-brane wrapping mode over it is neutral under the $U(1)$
gauge symmetry, hence it can be decoupled from the theory.

The main difference here is that there are only two $\mathbb{P}^{1}$ fibration
structures, in red and green. The physical discussions of duality is similar
to case of local $\mathbb{P}^{1}\times\mathbb{F}_{2}$, and we would not repeat
here.

####  4.1.6 Local $\mathbb{P}^{1}\times dP_{2}$

The rays in the toric fan of $X_{4}$ are $\tilde{v}_{i}=(v_{i},1)$

$\displaystyle v=(0,0,0)\ ,\ v_{1}=(1,0,0)\ ,\ v_{2}=(0,1,0)\ ,\
v_{3}=(0,0,1)\ ,\ v_{4}=(-1,0,0)\ ,\ $ (166) $\displaystyle v_{5}=(-1,-1,0)\
,\ v_{6}=(0,-1,0)\ ,\ v_{7}=(0,0,-1)\,.$

The Picard group of $X_{4}$ is generated by divisors $D$, $S_{1}$, $S_{2}$,
$S_{3}$ and $S_{5}$ which corresponds to the rays $v$, $v_{1}$, $v_{2}$,
$v_{3}$ and $v_{5}$ respectively, see figure 10.

Figure 10: $\mathcal{P}_{3}$ of local $\mathbb{P}^{1}\times dP_{2}$.

$K_{D}=(-2S_{1}-2S_{2}-2S_{3}+S_{5})|_{D}\,.$ (167)

The non-zero intersection numbers on $D$ are

$S_{3}\cdot S_{1}\cdot S_{2}|_{D}=1\ ,\ S_{3}\cdot S_{5}^{2}|_{D}=-1\,.$ (168)

##### Charged particles from M2-brane wrapping modes and 1-form symmetry

For the M2-brane wrapping a curve in the homology class

$C_{abcd}=D\cdot(aS_{3}\cdot S_{1}+bS_{3}\cdot S_{2}+cS_{3}\cdot
S_{5}+dS_{1}\cdot S_{2})\,,$ (169)

its charge under the $U(1)$ gauge group is

$C_{abcde}\cdot D=-2a-2b-c-2d\,.$ (170)

Hence there is no 1-form symmetry.

In this case, the Mori cone generators $D\cdot S_{3}\cdot S_{4}$,
$D_{\cdot}S_{3}\cdot S_{5}$ and $D\cdot S_{3}\cdot S_{6}$ all have normal
bundle $\mathcal{O}\oplus\mathcal{O}(-1)\oplus\mathcal{O}(-1)$, and there is
no M2-brane zero mode. Nonetheless, the curves $C_{1000}=D\cdot S_{3}\cdot
S_{1}$ and $C_{0100}=D\cdot S_{3}\cdot S_{2}$ have normal bundle
$\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$ and still give vector
multiplets. Together wtih $C_{0001}=D\cdot S_{1}\cdot S_{2}$, they form the
three possible $\mathbb{P}^{1}$ fibers in the three $\mathbb{P}^{1}$-fibration
structures colored in red, green and blue in figure 10.

##### Non-abelian flavor symmetry enhancement $G_{F}$

The flavor Cartan subalgebra generators are

$F_{1}=S_{1}-S_{2}\ ,\ F_{2}=S_{2}-S_{3}\ ,\ F_{3}=S_{5}\,.$ (171)

The flavor W-boson for $F_{1}$ is M2-brane wrapping

$C_{1}=D\cdot S_{3}\cdot(S_{1}-S_{2})\,.$ (172)

The flavor W-boson for $F_{2}$ is M2-brane wrapping

$C_{2}=D\cdot S_{1}\cdot(S_{2}-S_{3})\,.$ (173)

There is no flavor W-boson for $F_{3}$. Moreover, we can compute the
intersection matrix

$\begin{array}[]{c|cc}&F_{1}&F_{2}\\\ \hline\cr C_{1}&-2&1\\\
C_{2}&1&-2\end{array}$ (174)

Hence there is a non-trivial flavor symmetry enhancement to
$G_{F}=\mathfrak{su}(3)\oplus\mathfrak{u}(1)$.

The charges of BPS states under $U(1)$ gauge group and flavor Cartans are
given in table 4. $Q_{e}$ is the gauge charge and $Q_{A}$, $Q_{B}$, $Q_{C}$
are the flavor charges.

$U(1)$ Charge | $Q_{e}$ | $Q_{A}$ | $Q_{B}$ | $Q_{C}$
---|---|---|---|---
Divisor | $D$ | $F_{1}$ | $F_{2}$ | $F_{3}$
$C_{1000}$ | $-2$ | $-1$ | 1 | 0
$C_{0100}$ | $-2$ | 1 | 0 | 0
$C_{0010}$ | $-1$ | 0 | 0 | $-1$
$C_{0001}$ | $-2$ | 0 | $-1$ | 0
Table 4: U(1) gauge charge and flavor charges of BPS states from M2-brane
wrapping the curves on local $\mathbb{P}^{1}\times dP_{2}$.

One can see that $C_{1000}$, $C_{0100}$ and $C_{0001}$ form a $\mathbf{3}_{0}$
representation of $G_{F}=\mathfrak{su}(3)\oplus\mathfrak{u}(1)$.

#####  $c_{2}$ and additional $G_{4}$ flux on the resolved $X_{4}$

Now let us consider the $G_{4}$ flux. First since

$c_{2}^{c}(X_{4})=2S_{3}\cdot(2S_{1}+2S_{2}-S_{5})+D\cdot(2S_{1}+2S_{2}+2S_{3}-S_{5})+3S_{1}\cdot
S_{2}-2S_{5}^{2}\,,$ (175)

one can compute that

$\frac{1}{2}c_{2}^{c}(X_{4})\cdot D\cdot S_{1}\in\mathbb{Z}\ ,\
\frac{1}{2}c_{2}^{c}(X_{4})\cdot D\cdot S_{2}\in\mathbb{Z}\ ,\
\frac{1}{2}c_{2}^{c}(X_{4})\cdot D\cdot S_{3}\in\mathbb{Z}\ ,\
\frac{1}{2}c_{2}^{c}(X_{4})\cdot D\cdot S_{5}\in\mathbb{Z}\,.$ (176)

Therefore a non-vanishing $G_{4}$ is not required.

Nonetheless, the most general form of $G_{4}^{c}$ is

$\displaystyle G_{4}^{c}=D\cdot(nS_{5}+pS_{1}+qS_{2}+kS_{3})\,.$ (177)

Now let us compute the chirality of charged matter fields from M2-brane
wrapping specific 2-cycles.

First for M2-brane wrapping $C_{1000}=D\cdot S_{1}\cdot S_{3}$, its normal
bundle is $N_{C_{1000}}=\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$.
The moduli space of of $C_{1000}$ is $D\cdot
S_{2}\sim\mathbb{P}^{1}\times\mathbb{P}^{1}$. The chirality of matter field
with charge $-2$ under the $U(1)$ gauge group is

$\displaystyle\chi(C_{1000})$ $\displaystyle=-\int_{D\cdot S_{2}}c_{1}(D\cdot
S_{2})c_{1}(L)$ (178) $\displaystyle=D\cdot
G_{4}^{c}\cdot(-2S_{3}-S_{1}-S_{4})$ $\displaystyle=4n+4p+8q+3k\,.$

For M2-brane wrapping $C_{0100}=D\cdot S_{2}\cdot S_{3}$, its normal bundle is
$N_{C_{0100}}=\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$. The moduli
space of $C_{0100}$ is $D\cdot S_{1}\sim\mathbb{P}^{1}\times\mathbb{P}^{1}$.
The chirality of matter field with charge $-2$ under the $U(1)$ gauge group is

$\displaystyle\chi(C_{0100})$ $\displaystyle=-\int_{D\cdot S_{1}}c_{1}(D\cdot
S_{1})c_{1}(L)$ (179) $\displaystyle=D\cdot
G_{4}^{c}\cdot(-2S_{3}-S_{2}-S_{6})$ $\displaystyle=4n+8p+4q+3k\,.$

For M2-brane wrapping $C_{0001}=D\cdot S_{1}\cdot S_{2}$, its normal bundle is
$N_{C_{0001}}=\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}(-2)$. The moduli
space of $C_{0001}$ is $D\cdot S_{3}\sim dP_{2}$. The chirality of matter
field with charge $-2$ under the $U(1)$ gauge group is

$\displaystyle\chi(C_{0001})$ $\displaystyle=-\int_{D\cdot S_{3}}c_{1}(D\cdot
S_{3})c_{1}(L)$ (180) $\displaystyle=D\cdot
G_{4}^{c}\cdot(-2S_{1}-2S_{2}+S_{5})$ $\displaystyle=2n+4p+4q+7k\,.$

Finally, for M2-brane wrapping $C_{0010}=D\cdot S_{3}\cdot S_{5}$, its normal
bundle is
$N_{C_{0010}}=\mathcal{O}\oplus\mathcal{O}(-1)\oplus\mathcal{O}(-1)$. The
moduli space of $C_{0010}$ is $D\cdot S_{5}\cdot S_{6}\sim\mathbb{P}^{1}$. To
compute the chirality of such matter field with charge $-1$ under the $U(1)$
gauge group, we integrate $G_{4}$ over the matter surface $D\cdot S_{5}$:

$\chi(C_{0010})=D\cdot S_{5}\cdot G_{4}^{c}=2n-k\,.$ (181)

We compute the effective Chern-Simons term from 11d SUGRA action, which is
free of parity anomaly

$\displaystyle S_{CS}$ $\displaystyle=\frac{1}{4\pi}(7k+4n+8p+8q)\int A\wedge
dA+\frac{1}{2\pi}(2p-2q)\int A\wedge dB_{1}+\frac{1}{2\pi}(n-2k+2q)\int
A\wedge dB_{2}$ (182) $\displaystyle+\frac{1}{2\pi}(2n-k)\int A\wedge
dB_{3}-\frac{2k}{4\pi}\int B_{1}\wedge dB_{1}-\frac{2p}{4\pi}\int B_{2}\wedge
dB_{2}$ $\displaystyle-\frac{k}{4\pi}\int B_{3}\wedge
dB_{3}+\frac{1}{2\pi}(k+p-q)\int B_{1}\wedge dB_{2}+\frac{n}{4\pi}\int
B_{2}\wedge dB_{3}\,.$

Here $A$ is the $U(1)$ gauge field, and $B_{1}$, $B_{2}$, $B_{3}$ are
background gauge fields of flavor Cartans $F_{1}$, $F_{2}$, $F_{3}$ (171)
respectively.

####  4.1.7 Local $\mathbb{P}^{1}\times dP_{N}(gdP_{N})$ models

According to the general rules of flavor symmetry enhancement in Section 3.2,
for the general local $\mathbb{P}^{1}\times\mathcal{S}$ rank-1 models where
$\mathcal{S}=dP_{N}(gdP_{N})$, the $E_{N}$ flavor symmetry is enhanced to a
larger flavor symmetry algebra. Namely, one look at the curve configurations
on $gdP_{N}$ of type $E_{N}$, which geometrically realizes the largest flavor
symmetry algebra (in 5d) derenthal2014singular . The $(-2)$-curves on
$\mathcal{S}$ and the single 0-curve on $\mathcal{S}$ combine into a larger
Dynkin diagram of the enhanced flavor symmetry algebra. From this general
rule, we list the flavor enhancement at the singular limit in table 5.

$\mathcal{S}$ | Enhanced $G_{F}$
---|---
$\mathbb{P}^{2}$ | $\mathfrak{u}(1)$
$\mathbb{F}_{1}$ | $\mathfrak{su}(2)\oplus\mathfrak{u}(1)$
$\mathbb{F}_{2}$ | $\mathfrak{su}(3)$
$dP_{2}$ | $\mathfrak{su}(3)\oplus\mathfrak{u}(1)$
$dP_{3}$ | $\mathfrak{su}(4)\oplus\mathfrak{su}(2)$
$dP_{4}$ | $\mathfrak{su}(6)$
$dP_{5}$ | $\mathfrak{so}(12)$
$dP_{6}$ | $E_{7}$
$dP_{7}$ | $\hat{E}_{7}$
Table 5: Flavor enhancement for the 3d $\mathcal{N}=2$ SCFT $\mathcal{T}^{\rm
3d}_{X_{\text{4,sing}}}$ from M-theory on the local
$\mathbb{P}^{1}\times\mathcal{S}$ singularities.

As one can see, it is even possible to get affine flavor symmetry in this
class of models.

####  4.1.8 Local $\mathbb{P}^{1}\times T_{N}$ models

We discuss non-compact CY4 which is a local $\mathbb{P}^{1}\times T_{N}$,
where $T_{N}$ corresponds to the CY3 singularity
$\mathbb{C}^{3}/(\mathbb{Z}_{N}\times\mathbb{Z}_{N})$ that gives rise to the
5d $T_{N}$ theory Eckhard:2020jyr .

In the resolved geometry $\widetilde{X}_{4}$, the non-compact divisors
$S_{1},\dots,S_{3N+2}$ correspond to the rays

$\displaystyle v_{1}=(0,N,0)\ ,\ v_{2}=(1,N-1,0),\dots,v_{N}=(N-1,1,0)\ ,\ $
(183) $\displaystyle v_{N+1}=(N,0,0)\ ,\
v_{N+2}=(N-1,0,0),\dots,v_{2N}=(1,0,0)\ ,\ $ $\displaystyle v_{2N+1}=(0,0,0)\
,\ v_{2N+2}=(0,1,0),\dots,v_{3N}=(0,N-1,0)\ ,\ $ $\displaystyle
v_{3N+1}=(0,0,1)\ ,\ v_{3N+2}=(0,0,-1)\,.$

While the compact divisors are denoted as $D_{1},\dots,D_{(N-1)(N-2)/2}$.
Hence the rank $r=(N-1)(N-2)/2$.

For the flavor symmetry enhancement, we plot the configuration of flavor
curves on the resolution of
$\mathbb{C}^{3}/(\mathbb{Z}_{N}\times\mathbb{Z}_{N})$ (see Eckhard:2020jyr ):

(184)

Note that the curves in the red rectangle effectively form a 0-curve on the
compact divisors, which give rise to an additional node in the Dynkin diagram
of $G_{F,nA}$ following Sec.3.2.

Hence for $N\geq 4$, the enhanced flavor symmetry is $G_{F,nA}=SU(N)\times
SU(2N)$.

For $N=3$, there is a further enhancement $G_{F,nA}=E_{7}\supset SU(3)\times
SU(6)$, because of the identification with the case of local
$\mathbb{P}^{1}\times dP_{6}$.

####  4.1.9 Higher-rank fibration over $\mathbb{P}^{1}\times\mathbb{P}^{1}$

We can also consider toric geometries that give rise to $r>1$ theories, which
are $\mathbb{P}^{1}$s fiber over a base
$\mathcal{S}=\mathbb{P}^{1}\times\mathbb{P}^{1}$. First let us consider the
geometry

(185)

In this case the two compact divisors are $D_{1}$, $D_{2}$ and the non-compact
divisors are labeled by $S_{1}$, $S_{2}$, $S_{3}$, $S_{4}$. There is linear
equivalence relation

$S_{3}=D_{2}+2S_{4}\,.$ (186)

The gauge W-bosons are $C_{1}^{g}=D_{1}\cdot S_{1}\cdot S_{2}$,
$C_{2}^{g}=D_{2}\cdot S_{1}\cdot S_{2}$, which satisfies the intersection
matrix

$\begin{array}[]{c|cc}&D_{1}&D_{2}\\\ \hline\cr C_{1}^{g}&-2&1\\\
C_{2}^{g}&1&-2\end{array}$ (187)

Hence in the limit where the $\mathbb{P}^{1}$ fibers are all shrunk to zero
volume limit, there is an $SU(3)$ gauge theory description.

For the flavor symmetry enhancements, we can again write down the flavor
Cartans

$F_{1}=S_{1}-S_{2}\ ,\ F_{2}=S_{2}-S_{3}$ (188)

and their corresponding flavor W-bosons

$C_{1}=D_{1}\cdot(S_{1}-S_{2})\cdot S_{3}\ ,\
C_{2}=D_{1}\cdot(S_{2}-S_{3})\cdot S_{1}\,.$ (189)

From the intersection matrix

$\begin{array}[]{c|cc}&F_{1}&F_{2}\\\ \hline\cr C_{1}&-2&1\\\
C_{2}&1&-2\end{array}$ (190)

we see that the flavor symmetry enhancement to $G_{F}=\mathfrak{su}(3)$ at the
singular limit of $X_{4}$.

We can even consider higher-rank configurations by adding new rays below
$S_{4}$ in (185). The analysis is analogous to the $SU(3)$ case, in general we
will get a theory with $SU(r+1)$ gauge theory limit and flavor symmetry
enhancement $G_{F}=\mathfrak{su}(3)$ at the singular limit.

One can also consider non-trivial fibrations over
$\mathbb{P}^{1}\times\mathbb{P}^{1}$, which we would not discuss the details
here.

###  4.2 $\mathbb{C}^{4}$ orbifolds

In this section we study a rich class of geometries obtained from orbifolding
$\mathbb{C}^{4}$ by finite subgroups of $SU(4)$. We will denote such orbifolds
by $\mathbb{C}^{4}/\Gamma$ for a finite subgroup $\Gamma\subset SU(4)$ and by
$\widetilde{\mathbb{C}^{4}/\Gamma}$ its crepant resolution if there exists
one. Strictly speaking $\Gamma$ is a subgroup of $SL(4,\mathbb{C})$ but we
will not distinguish between the two and this subtlety is not important for
our discussion.

The significance of such orbifolds $\mathbb{C}^{n}/\Gamma$, where $\Gamma$ is
a finite subgroup of $SU(n)$ for general $n$, lies in the fact that
$\mathbb{C}^{n}/\Gamma$ possesses an $SU(n)$ holonomy. Therefore SUSY field
theories can be geometrically engineered from string/M-theory compactification
on $\mathbb{C}^{n}/\Gamma$. In contrast, from a physical point of view
assuming the existence of a crepant resolution for $\mathbb{C}^{n}/\Gamma$ is
somewhat less general in our setup. When $n=4$ this assumption is equivalent
to assuming the existence of a UV Coulomb branch hence making the physics more
tractable. Certainly some mathematical aspects of $\mathbb{C}^{n}/\Gamma$
become tricky if no crepant resolution exists, but those issues are beyond the
scope of the current paper.

It is well-known that a crepant resolution may not exist for any
$\mathbb{C}^{4}/\Gamma$ and an existence theorem is yet to be established
ItoReid ; ito1998mckay ; ito2011mckay . In this section we will avoid this
subtlety and discuss only general physical properties that a 3D theory
associated to $\mathbb{C}^{4}/\Gamma$ must possess whenever at least one
crepant resolution exists. We will leave a more detailed investigation of
physical aspects of these theories and its link to various mathematical
concepts to future work.

####  4.2.1 Group data of $\Gamma$ and M-theory on
$\widetilde{\mathbb{C}^{n}/\Gamma}$ or $\mathbb{C}^{n}/\Gamma$

In this subsection we discuss the general relation between the group
theoretical data of a finite subgroup $\Gamma$ of $SU(n)$ and the geometrical
and topological data of $\widetilde{\mathbb{C}^{n}/\Gamma}$, assuming the