$\mathfrak{su}(4)^{2}$, while the actual symmetry is $\mathfrak{e}_{7}$. In
the resolution, see figure 12 on the right hand side, where we see the
representation

$({\bf 4},{\bf\overline{4}})\oplus({\bf\overline{4}},{\bf 4})\oplus
2\times({\bf 6},{\bf 1})\oplus 2\times({\bf 1},{\bf
6})\quad\longrightarrow\quad{\bf 56}\,.$ (158)

These indeed arise from the branching of the ${\bf 56}$ of $\mathfrak{e}_{7}$
to $\mathfrak{su}(4)^{2}$.

$T_{4}$$T_{4}$ with deformation $[1^{2},2]$$T_{4}$ with deformation $[2^{2}]$ Figure 12: CFDs: The intersection graph for $T_{4}$ (left hand side) and the deformation of $T_{4}$ described by the partition $[1^{2}2]$ (middle) and deformation of $T_{4}$ with partition $[2^{2}]$. The nodes are curves that are complete intersections between $\sum S_{i}$ (i.e. the sum of all compact divisors) and the non-compact divisors. The green nodes are $-2$ self-intersection curves, which correspond to roots of the flavor symmetry algebra, white are $-1$ curves, which can be thought of as bifundamental matter. On the left we see the $\mathfrak{su}(4)^{3}$ flavor symmetry and the matter in the $({\bf 4},{\bf 4},{\bf 1})$ etc representations. In the middle, we see the manifest flavor symmetry $\mathfrak{su}(4)^{2}\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)$, but the matter shown results in the enhancement to $\mathfrak{su}(8)\oplus\mathfrak{u}(1)$. On the right the flavor symmetry enhances to $\mathfrak{e}_{7}$.  Partitions | Rank | Symmetry | Equation | Fig.
---|---|---|---|---
$[1^{5}]$ | 6 | $\mathfrak{su}(5)^{3}$ | $W_{1}W_{2}W_{3}=Z^{5}$ | 13
$[2,1^{3}]$ | 5 | $\mathfrak{su}(5)^{2}\oplus\mathfrak{su}(3)\oplus\mathfrak{u}(1)$ | $W_{1}W_{2}W_{3}=Z^{3}(Z^{2}+W_{1})$ | 13
$[2^{2},1]$ | 4 | $\mathfrak{su}(5)^{2}\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)$ | $W_{1}W_{2}W_{3}=Z(Z^{2}+W_{1})^{2}$ | 15
$[3,1^{2}]$ | 3 | $\mathfrak{su}(10)\oplus\mathfrak{su}(2)$ | $W_{1}W_{2}W_{3}=Z^{2}(Z^{3}+W_{1})$ | 13
$[3,2]$ | 2 | $\mathfrak{su}(10)$ | $W_{1}W_{2}W_{3}=(Z^{2}+W_{1})(Z^{3}+W_{1})$ | 14
Table 3: Deformations of the $T_{5}$ model: the partition that defines the
GTP, the rank of the expected 5d SCFT, the flavor symmetry algebra, and the
deformed hypersurface equation.

#### Deformations of $T_{5}$.

Finally consider the $T_{5}$ model with deformations added along a single
edge. These are summarized in table 3.

First we recompute the resolution of the undeformed $T_{5}$ model, which is
given in figure 13. The flavor symmetry algebra and the representation of the
hypers is

$\mathfrak{g}_{UV}=\mathfrak{su}(5)^{3}:\qquad({\bf 5},{\bf\overline{5}},{\bf
1})\oplus({\bf 1},{\bf 5},{\bf\overline{5}})\oplus({\bf\overline{5}},{\bf
1},{\bf 5})\,.$ (159)

The CFD is shown in figure 13. Adding a single white dot results in a theory
with flavor algebra
$\mathfrak{su}(5)^{2}\oplus\mathfrak{su}(3)\oplus\mathfrak{u}(1)$, with the
CFD shown in the middle in figure 13, which has bifundamental matter
consistent with these flavor symmetry algebras.

$T_{5}$$T_{5}$ with deformation $[1^{3},2]$$T_{5}$ with deformation
$[1^{2},3]$ Figure 13: CFDs: The intersection graph of $-2$ (green) and $(-1)$
(white) curves, i.e. the CFD, for $T_{5}$ (LHS), the partition $[2,1^{3}]$ in
the middle, and $[3,1^{2}]$ on the right hand side.

Adding two white dots along a single edge in the partition $[3,1^{2}]$ we
obtain the right hand figure 13. The representations under
$\mathfrak{su}(5)^{2}\oplus\mathfrak{su}(2)$ are

$\qquad({\bf 5},{\bf\overline{5}},{\bf 1})\oplus({\bf 1},{\bf 5},{\bf
2})\oplus({\bf\overline{5}},{\bf 1},{\bf 2})\oplus({\bf 10},{\bf 1},{\bf
1})\oplus({\bf 1},{\bf\overline{10}},{\bf 1})\quad\longrightarrow\quad({\bf
45},{\bf 1})\oplus({\bf\overline{10}},{\bf 2})$ (160)

which is consistent with the flavor symmetry enhancement to

$\mathfrak{g}_{UV}=\mathfrak{su}(10)\oplus\mathfrak{su}(2)\,.$ (161)

Similarly for $[3,2]$ we computed the resolution and find the CFD shown in
figure 14, which shows the representations under $\mathfrak{su}(5)^{2}$ to be

$\qquad({\bf 5},{\bf\overline{5}})\oplus({\bf 10},{\bf 1})\oplus({\bf
1},{\bf\overline{10}})\quad\longrightarrow\quad{\bf 45}\,,$ (162)

which is consistent with the enhancement to $\mathfrak{su}(10)$. Finally
consider figure 15, where we manifestly see the $\mathfrak{su}(5)$ but the
$\mathfrak{su}(2)$ is decomposed into $\mathfrak{u}(1)$.

For partition $[2^{2},1]$, although the $\mathfrak{su}(2)$ factor of the
global symmetry is not directly visible in the resolution, an educated guess
allows to read the following representations of
$\mathfrak{su}(5)\oplus\mathfrak{su}(5)\oplus\mathfrak{su}(2)$ on figure 15:

$({\bf 5},{\bf 1},{\bf 1})\oplus({\bf 1},{\bf\overline{5}},{\bf 1})\oplus({\bf
10},{\bf 1},{\bf 2})\oplus({\bf 5},{\bf\overline{10}},{\bf
2})\oplus({\bf\overline{5}},{\bf 5},{\bf 1})\,.$ (163)

Again, the flavor symmetry algebra is consistent with the one predicted from
GTPs.

$T_{5}$ with deformation [2,3] Figure 14: CFDs: The intersection graph for
$T_{5}$ with the deformation labeled by the partition $[2,3]$. We see the
$\mathfrak{su}(5)^{2}\oplus\mathfrak{u}(1)$ and the matter in the $({\bf
5,{\bf 5}})\oplus({\bf 10},{\bf 1})\oplus({\bf 1},{\bf 10})$ which enhance
into ${\bf 45}$ of $\mathfrak{su}(10)$.  $T_{5}$ with deformation $[1,2^{2}]$
Figure 15: CFDs: The intersection graph for $T_{5}$ with the deformation
labeled by the partition $[1,2^{2}]$.

### Acknowledgments

We would like to thank Cyril Closset for collaboration in early stages of this
work. AB and SSN thank Hendrik Süß for discussions of related questions. This
work, in particular AC and SSN, was supported and enabled by the Fondation
Wiener-Anspach. This research is further supported by IISN-Belgium (convention
4.4503.15). AB is supported by the ERC Consolidator Grant
772408-Stringlandscape, and by the LabEx ENS-ICFP:
ANR-10-LABX-0010/ANR-10-IDEX-0001-02 PSL*. SSN is supported in part by the
“Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics”
and the EPSRC Open Fellowship EP/X01276X/1.

##  Appendix A Basic algebraic notions for $\mathfrak{sl}_{n}$

This appendix gathers a few elementary algebraic concepts needed in the bulk
of the paper. In the following, we take
$\mathfrak{g}=\mathfrak{sl}_{n}(\mathbb{C})$, which we represent as $n\times
n$ complex matrices with vanishing trace, and we pick the diagonal matrices
for the Cartan subalgebra $\mathfrak{h}$.

#### Nilpotent Orbits and Triples.

Let $e$ be a nilpotent element of $\mathfrak{g}$. By the Jacobson-Morozov
theorem, one can construct an $\mathfrak{sl}_{2}$ triple $(e,f,h)$, i.e. a
triple of elements of $\mathfrak{g}$ with $h\in\mathfrak{h}$ satisfying the
commutation relations

$[e,f]=h\,,\qquad[h,e]=2e\,,\qquad[h,f]=-2f\,.$ (164)

More precisely, there exists an embedding

$\rho:\mathfrak{sl}_{2}\longrightarrow\mathfrak{g}\,,$ (165)

such that $\rho(e)$ defines a nilpotent element of $\mathfrak{g}$. In our
case, nilpotent orbits are characterized by partitions of $n$. For the
partition $\lambda=(\lambda_{1},\dots,\lambda_{r})$, there is a canonical
triple, where $e$ is in the Jordan form specified by $\lambda$, $h$ is
diagonal and $f$ is lower diagonal, defined as follows. Consider first the
maximal orbit, $\lambda=(n)$. In this case we define the canonical triple
$e=J_{n}$, $h=H_{n}$ and $f=\tilde{J}_{n}$ where

$(J_{n})_{i,j}=\delta_{i+1,j}\,,\qquad(H_{n})_{i,j}=\delta_{i,j}(n+1-2i)\,,\qquad(\tilde{J}_{n})_{i,j}=\delta_{i-1,j}j(n-j)\,.$
(166)

Then for any partition $\lambda$ the canonical triple is the block diagonal

$e=\mathrm{Diag}(J_{\lambda_{i}})\,,\qquad
f=\mathrm{Diag}(\tilde{J}_{\lambda_{i}})\,,\qquad
h=\mathrm{Diag}(H_{\lambda_{i}})\,.$ (167)

#### Slodowy Slices.

Given a triple $(e,f,h)$, one constructs the space

$\mathcal{S}_{e}=e+\mathfrak{g}^{f}\,,$ (168)

where $\mathfrak{g}^{f}$ is the centralizer of $f$ in $\mathfrak{g}$. This is
called the _Slodowy slice_ transverse to $e$. Note that this should not be
confused with the _nilpotent Slodowy slice_ , which is the intersection of the
Slodowy slice with the nilpotent cone. When $(e,f,h)$ is the canonical
nilpotent element (167) associated to partition $\lambda$, we call
$\mathcal{S}_{\lambda}$ the Slodowy slice.

In order to construct explicitly $\mathcal{S}_{\lambda}$, we first need the
set of matrices which commute with $\tilde{J}_{n}$. These are the matrices
$S(a_{1},\dots,a_{n})$ with $a_{1},\dots,a_{n}\in\mathbb{C}$, defined by888The
matrices $S(a_{1},\dots,a_{n})$ for low values of $n$ are:
$S(a_{1},a_{2})=\left(\begin{array}[]{cc}a_{1}&0\\\ a_{2}&a_{1}\\\
\end{array}\right),\quad
S(a_{1},a_{2},a_{3})=\left(\begin{array}[]{ccc}a_{1}&0&0\\\ 2a_{2}&a_{1}&0\\\
4a_{3}&2a_{2}&a_{1}\\\ \end{array}\right),\quad
S(a_{1},a_{2},a_{3},a_{4})=\left(\begin{array}[]{cccc}a_{1}&0&0&0\\\
3a_{2}&a_{1}&0&0\\\ 12a_{3}&4a_{2}&a_{1}&0\\\ 36a_{4}&12a_{3}&3a_{2}&a_{1}\\\
\end{array}\right).$ (169)

$(S(a_{1},\dots,a_{n}))_{ij}:=\sum\limits_{k=0}^{n-1}a_{k+1}\delta_{i-k,j}\prod\limits_{l=j}^{i-1}l(n-l)\,.$
(170)

Then the centralizer $\mathfrak{g}^{f}$ is a set of block matrices of sizes
$\lambda_{i}\times\lambda_{j}$, where the block $(i,j)$ depends on
$\mathrm{min}(\lambda_{i},\lambda_{j})$ parameters. We will not need its
explicit form here. We only need the form of the diagonal blocks ($i=j$),
which are of the form

$S(a_{1}^{(i)},\dots,a_{\lambda_{i}}^{(i)})\,.$ (171)

Using this, we can compute the dimension of the Slodowy slices, which is

$\mathrm{dim}\,\mathcal{S}_{\lambda}=-1+\sum\limits_{1\leq i,j,\leq
r}\mathrm{min}(\lambda_{i},\lambda_{j})=-(n+1)+2\sum\limits_{i=1}^{r}i\lambda_{i}$
(172)

and

$\mathrm{dim}\,\mathcal{S}_{\lambda}\cap\mathcal{N}=\mathrm{dim}\,\mathcal{S}_{\lambda}-(n-1)=-2n+2\sum\limits_{i=1}^{r}i\lambda_{i}\,.$
(173)

The dimension of the nilpotent cone $\mathcal{N}$ is $n(n-1)$, and therefore
we can deduce that the dimension of the nilpotent orbit of $e$ is

$\mathrm{dim}\,\mathcal{O}_{\lambda}=n(n+1)-2\sum\limits_{i=1}^{r}i\lambda_{i}\,,$
(174)

which matches known results.

#### Levi subalgebras and Slodowy slices.

A Borel subalgebra $\mathfrak{b}$ of $\mathfrak{g}$ is a maximal solvable
subalgebra. The standard choice, which we make, is to pick for $\mathfrak{b}$
the upper triangular matrices. The simple roots $\alpha_{i}$ can be indexed by
$i\in I:=\\{1,\dots,n-1\\}$, and to every subset $\Theta$ of $I$ one can
associate a parabolic subalgebra $\mathfrak{p}_{\Theta}$ of $\mathfrak{g}$
containing $\mathfrak{b}$. This parabolic subalgebra decomposes as a direct
sum

$\mathfrak{p}_{\Theta}=\mathfrak{l}_{\Theta}\oplus\mathfrak{n}_{\Theta}$ (175)

where $\mathfrak{l}_{\Theta}$ is the Levi subalgebra and
$\mathfrak{n}_{\Theta}$ is the nilradical of $\mathfrak{p}_{\Theta}$.

Let $\lambda=(\lambda_{1},\dots,\lambda_{r})$ be a partition of $n$. We
associate to this partition the set

$\Theta_{\lambda}=I-\\{\lambda_{1},\lambda_{1}+\lambda_{2},\dots,\lambda_{1}+\dots+\lambda_{r-1}\\}\,.$
(176)

This way, one can construct the corresponding Levi subalgebra
$\mathfrak{l}_{\lambda}:=\mathfrak{l}_{\Theta_{\lambda}}$. We then introduce
the intersection

$\mathcal{S}_{\lambda}^{0}:=\mathcal{S}_{\lambda}\cap\mathfrak{l}_{\lambda}\,.$
(177)

With out choices, $\mathfrak{l}_{\lambda}$ is simply the algebra of block
diagonal matrices, with blocks of respective sizes
$\lambda_{1},\dots,\lambda_{r}$, and $\mathcal{S}_{\lambda}^{0}$ is the set of
traceless matrices which are block diagonal, with diagonal blocks
$J_{\lambda_{i}}+S(a_{1},a_{2},\dots,a_{\lambda_{i}})$. The dimension is
$\mathrm{dim}\,\mathcal{S}_{\lambda}^{0}=n-1$, independent of $\lambda$.

#### Companion matrix.

It is easy to check that the matrix $J_{n}+S(0,a_{2},\dots,a_{n})$ can be
conjugated to a matrix of the form

$C(b_{2},\dots,b_{n}):=\begin{pmatrix}0&1&&&&\\\ &0&1&&&\\\
&&\ddots&\ddots&\ddots&&\\\ &&&&&0&1\\\
(-)^{n-1}b_{n}&(-)^{n-2}b_{n-1}&&\cdots&&-b_{2}&0\end{pmatrix}$ (178)

where the $b_{i}$ are known functions of the $a_{i}$. Note that the change of
basis depends explicitly on the $a_{i}$. A matrix in the form (178) is called
a _companion matrix_. It has the convenient property that its characteristic
polynomial (evaluated at $-z$) is

$\det(z\mathbf{1}_{n}+C)=z^{n}+\sum\limits_{i=2}^{n}b_{i}z^{n-i}\,.$ (179)

Any matrix in $\mathcal{S}^{0}_{\lambda}$ can then be conjugated to a block
diagonal matrix with blocks of the form $C(b_{2},\dots,b_{\lambda_{i}})$.

#### Cokernel of $\mathrm{ad}(e)$.

Consider a finite dimensional representation $\rho:\mathfrak{sl}(2)\rightarrow
V$ of $\mathfrak{sl}(2)$. One can decompose $V$ into irreducible
representations $V_{i}$ of dimensions $n_{i}$. In each irrep, the image of
$\rho(e)$ has codimension 1, and the cokernel is spanned by the weight space
of lowest weight, which by definition is precisely the kernel of $\rho(f)$.
Therefore in the Lie algebra $\mathfrak{g}$ seen as an $\mathfrak{sl}(2)$
representation defined by a triple $(e,f,h)$, the cokernel of $\mathrm{ad}(e)$
is $\mathfrak{g}^{f}$.

## References

  * (1) N. Seiberg, _Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics_ , _Phys. Lett. B_ 388 (1996) 753–760, [hep-th/9608111].
  * (2) D. R. Morrison and N. Seiberg, _Extremal transitions and five-dimensional supersymmetric field theories_ , _Nucl. Phys. B_ 483 (1997) 229–247, [hep-th/9609070].
  * (3) O. Aharony and A. Hanany, _Branes, superpotentials and superconformal fixed points_ , _Nucl. Phys. B_ 504 (1997) 239–271, [hep-th/9704170].
  * (4) O. Aharony, A. Hanany and B. Kol, _Webs of (p,q) five-branes, five-dimensional field theories and grid diagrams_ , _JHEP_ 01 (1998) 002, [hep-th/9710116].
  * (5) O. DeWolfe, A. Hanany, A. Iqbal and E. Katz, _Five-branes, seven-branes and five-dimensional E(n) field theories_ , _JHEP_ 03 (1999) 006, [hep-th/9902179].
  * (6) O. Bergman and G. Zafrir, _5d fixed points from brane webs and O7-planes_ , _JHEP_ 12 (2015) 163, [1507.03860].
  * (7) G. Zafrir, _Brane webs, $5d$ gauge theories and $6d$ $\mathcal{N}=(1,0)$ SCFT’s_, _JHEP_ 12 (2015) 157, [1509.02016].
  * (8) G. Zafrir, _Brane webs and $O5$-planes_, _JHEP_ 03 (2016) 109, [1512.08114].
  * (9) K. Ohmori and H. Shimizu, _$S^{1}/T^{2}$ compactifications of 6d $\mathcal{N}=\left(1,\;0\right)$ theories and brane webs_, _JHEP_ 03 (2016) 024, [1509.03195].
  * (10) H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, _Discrete theta angle from an O5-plane_ , _JHEP_ 11 (2017) 041, [1707.07181].
  * (11) H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, _5-brane webs for 5d $\mathcal{N}$ = 1 G2 gauge theories_, _JHEP_ 03 (2018) 125, [1801.03916].
  * (12) H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, _Dualities and 5-brane webs for 5d rank 2 SCFTs_ , _JHEP_ 12 (2018) 016, [1806.10569].
  * (13) C. Closset, M. Del Zotto and V. Saxena, _Five-dimensional SCFTs and gauge theory phases: an M-theory/type IIA perspective_ , _SciPost Phys._ 6 (2019) 052, [1812.10451].
  * (14) N. C. Leung and C. Vafa, _Branes and toric geometry_ , _Adv. Theor. Math. Phys._ 2 (1998) 91–118, [hep-th/9711013].
  * (15) K. A. Intriligator, D. R. Morrison and N. Seiberg, _Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces_ , _Nucl. Phys. B_ 497 (1997) 56–100, [hep-th/9702198].
  * (16) F. Benini, S. Benvenuti and Y. Tachikawa, _Webs of five-branes and N=2 superconformal field theories_ , _JHEP_ 09 (2009) 052, [0906.0359].
  * (17) S. A. Cherkis, _Phases of Five-dimensional Theories, Monopole Walls, and Melting Crystals_ , _JHEP_ 06 (2014) 027, [1402.7117].
  * (18) D. Xie and S.-T. Yau, _Three dimensional canonical singularity and five dimensional $\mathcal{N}$ = 1 SCFT_, _JHEP_ 06 (2017) 134, [1704.00799].
  * (19) M. Del Zotto, J. J. Heckman and D. R. Morrison, _6D SCFTs and Phases of 5D Theories_ , _JHEP_ 09 (2017) 147, [1703.02981].
  * (20) S. Alexandrov, S. Banerjee and P. Longhi, _Rigid limit for hypermultiplets and five-dimensional gauge theories_ , _JHEP_ 01 (2018) 156, [1710.10665].
  * (21) P. Jefferson, H.-C. Kim, C. Vafa and G. Zafrir, _Towards Classification of 5d SCFTs: Single Gauge Node_ , 1705.05836.
  * (22) P. Jefferson, S. Katz, H.-C. Kim and C. Vafa, _On Geometric Classification of 5d SCFTs_ , _JHEP_ 04 (2018) 103, [1801.04036].
  * (23) L. Bhardwaj and P. Jefferson, _Classifying $5d$ SCFTs via $6d$ SCFTs: Rank one_, _JHEP_ 07 (2019) 178, [1809.01650].
  * (24) L. Bhardwaj and P. Jefferson, _Classifying 5d SCFTs via 6d SCFTs: Arbitrary rank_ , _JHEP_ 10 (2019) 282, [1811.10616].
  * (25) F. Apruzzi, L. Lin and C. Mayrhofer, _Phases of 5d SCFTs from M-/F-theory on Non-Flat Fibrations_ , _JHEP_ 05 (2019) 187, [1811.12400].
  * (26) F. Apruzzi, C. Lawrie, L. Lin, S. Schafer-Nameki and Y.-N. Wang, _5d Superconformal Field Theories and Graphs_ , _Phys. Lett. B_ 800 (2020) 135077, [1906.11820].
  * (27) F. Apruzzi, C. Lawrie, L. Lin, S. Schafer-Nameki and Y.-N. Wang, _Fibers add Flavor, Part I: Classification of 5d SCFTs, Flavor Symmetries and BPS States_ , _JHEP_ 11 (2019) 068, [1907.05404].
  * (28) F. Apruzzi, C. Lawrie, L. Lin, S. Schafer-Nameki and Y.-N. Wang, _Fibers add Flavor, Part II: 5d SCFTs, Gauge Theories, and Dualities_ , _JHEP_ 03 (2020) 052, [1909.09128].
  * (29) L. Bhardwaj, P. Jefferson, H.-C. Kim, H.-C. Tarazi and C. Vafa, _Twisted Circle Compactifications of 6d SCFTs_ , _JHEP_ 12 (2020) 151, [1909.11666].
  * (30) L. Bhardwaj, _Do all 5d SCFTs descend from 6d SCFTs?_ , _JHEP_ 04 (2021) 085, [1912.00025].
  * (31) F. Apruzzi, S. Schafer-Nameki and Y.-N. Wang, _5d SCFTs from Decoupling and Gluing_ , _JHEP_ 08 (2020) 153, [1912.04264].
  * (32) C. Closset, S. Schafer-Nameki and Y.-N. Wang, _Coulomb and Higgs Branches from Canonical Singularities: Part 0_ , _JHEP_ 02 (2021) 003, [2007.15600].
  * (33) C. Closset, S. Giacomelli, S. Schafer-Nameki and Y.-N. Wang, _5d and 4d SCFTs: Canonical Singularities, Trinions and S-Dualities_ , _JHEP_ 05 (2021) 274, [2012.12827].
  * (34) C. Closset, S. Schafer-Nameki and Y.-N. Wang, _Coulomb and Higgs branches from canonical singularities. Part I. Hypersurfaces with smooth Calabi-Yau resolutions_ , _JHEP_ 04 (2022) 061, [2111.13564].
  * (35) L. Bhardwaj and G. Zafrir, _Classification of 5d $\mathcal{N}$ = 1 gauge theories_, _JHEP_ 12 (2020) 099, [2003.04333].
  * (36) L. Bhardwaj, _More 5d KK theories_ , _JHEP_ 03 (2021) 054, [2005.01722].
  * (37) L. Bhardwaj, _Flavor symmetry of 5d SCFTs. Part I. General setup_ , _JHEP_ 09 (2021) 186, [2010.13230].
  * (38) L. Bhardwaj, _Flavor symmetry of 5 $d$ SCFTs. Part II. Applications_, _JHEP_ 04 (2021) 221, [2010.13235].
  * (39) D. R. Morrison, S. Schafer-Nameki and B. Willett, _Higher-Form Symmetries in 5d_ , _JHEP_ 09 (2020) 024, [2005.12296].
  * (40) F. Albertini, M. Del Zotto, I. n. García Etxebarria and S. S. Hosseini, _Higher Form Symmetries and M-theory_ , _JHEP_ 12 (2020) 203, [2005.12831].
  * (41) L. Bhardwaj and S. Schäfer-Nameki, _Higher-form symmetries of 6d and 5d theories_ , _JHEP_ 02 (2021) 159, [2008.09600].
  * (42) F. Apruzzi, L. Bhardwaj, J. Oh and S. Schafer-Nameki, _The Global Form of Flavor Symmetries and 2-Group Symmetries in 5d SCFTs_ , _SciPost Phys._ 13 (2022) 024, [2105.08724].
  * (43) M. Del Zotto, J. J. Heckman, S. N. Meynet, R. Moscrop and H. Y. Zhang, _Higher symmetries of 5D orbifold SCFTs_ , _Phys. Rev. D_ 106 (2022) 046010, [2201.08372].
  * (44) M. Del Zotto, I. n. García Etxebarria and S. Schafer-Nameki, _2-Group Symmetries and M-Theory_ , _SciPost Phys._ 13 (2022) 105, [2203.10097].
  * (45) A. Collinucci and R. Valandro, _The role of U(1)’s in 5d theories, Higgs branches, and geometry_ , _JHEP_ 10 (2020) 178, [2006.15464].
  * (46) A. Collinucci, M. De Marco, A. Sangiovanni and R. Valandro, _Higgs branches of 5d rank-zero theories from geometry_ , _JHEP_ 10 (2021) 018, [2105.12177].
  * (47) O. Bergman, D. Rodríguez-Gómez and G. Zafrir, _5-Brane Webs, Symmetry Enhancement, and Duality in 5d Supersymmetric Gauge Theory_ , _JHEP_ 03 (2014) 112, [1311.4199].
  * (48) H. Hayashi, S.-S. Kim, K. Lee, M. Taki and F. Yagi, _A new 5d description of 6d D-type minimal conformal matter_ , _JHEP_ 08 (2015) 097, [1505.04439].
  * (49) S. Cremonesi, G. Ferlito, A. Hanany and N. Mekareeya, _Instanton Operators and the Higgs Branch at Infinite Coupling_ , _JHEP_ 04 (2017) 042, [1505.06302].
  * (50) K. Yonekura, _Instanton operators and symmetry enhancement in 5d supersymmetric quiver gauge theories_ , _JHEP_ 07 (2015) 167, [1505.04743].
  * (51) D. Gaiotto and H.-C. Kim, _Duality walls and defects in 5d $\mathcal{N}=1$ theories_, _JHEP_ 01 (2017) 019, [1506.03871].
  * (52) G. Ferlito, A. Hanany, N. Mekareeya and G. Zafrir, _3d Coulomb branch and 5d Higgs branch at infinite coupling_ , _JHEP_ 07 (2018) 061, [1712.06604].
  * (53) S. Cabrera, A. Hanany and F. Yagi, _Tropical Geometry and Five Dimensional Higgs Branches at Infinite Coupling_ , _JHEP_ 01 (2019) 068, [1810.01379].
  * (54) A. Bourget, S. Cabrera, J. F. Grimminger, A. Hanany and Z. Zhong, _Brane Webs and Magnetic Quivers for SQCD_ , _JHEP_ 03 (2020) 176, [1909.00667].
  * (55) M. van Beest, A. Bourget, J. Eckhard and S. Schafer-Nameki, _(Symplectic) Leaves and (5d Higgs) Branches in the Poly(go)nesian Tropical Rain Forest_ , _JHEP_ 11 (2020) 124, [2008.05577].
  * (56) M. Van Beest, A. Bourget, J. Eckhard and S. Schafer-Nameki, _(5d RG-flow) Trees in the Tropical Rain Forest_ , _JHEP_ 03 (2021) 241, [2011.07033].
  * (57) A. Bourget, J. F. Grimminger, A. Hanany, M. Sperling, G. Zafrir and Z. Zhong, _Magnetic quivers for rank 1 theories_ , _JHEP_ 09 (2020) 189, [2006.16994].
  * (58) S. Giacomelli, M. Martone, Y. Tachikawa and G. Zafrir, _More on $\mathcal{N}=2$ S-folds_, _JHEP_ 01 (2021) 054, [2010.03943].
  * (59) A. Bourget, S. Giacomelli, J. F. Grimminger, A. Hanany, M. Sperling and Z. Zhong, _S-fold magnetic quivers_ , _JHEP_ 02 (2021) 054, [2010.05889].
  * (60) M. Akhond and F. Carta, _Magnetic quivers from brane webs with O7 +-planes_, _JHEP_ 10 (2021) 014, [2107.09077].
  * (61) M. van Beest and S. Giacomelli, _Connecting 5d Higgs branches via Fayet-Iliopoulos deformations_ , _JHEP_ 12 (2021) 202, [2110.02872].
  * (62) A. Bourget, J. F. Grimminger, M. Martone and G. Zafrir, _Magnetic quivers for rank 2 theories_ , _JHEP_ 03 (2022) 208, [2110.11365].
  * (63) A. Bourget, J. F. Grimminger, A. Hanany, R. Kalveks and Z. Zhong, _Higgs branches of U/SU quivers via brane locking_ , _JHEP_ 08 (2022) 061, [2111.04745].
  * (64) A. Bourget, J. F. Grimminger, A. Hanany, M. Sperling and Z. Zhong, _Magnetic Quivers from Brane Webs with O5 Planes_ , _JHEP_ 07 (2020) 204, [2004.04082].
  * (65) M. Akhond, F. Carta, S. Dwivedi, H. Hayashi, S.-S. Kim and F. Yagi, _Five-brane webs, Higgs branches and unitary/orthosymplectic magnetic quivers_ , _JHEP_ 12 (2020) 164, [2008.01027].
  * (66) M. Sperling and Z. Zhong, _Balanced B and D-type orthosymplectic quivers \- magnetic quivers for product theories_ , _JHEP_ 04 (2022) 145, [2111.00026].
  * (67) O. Bergman and G. Zafrir, _Lifting 4d dualities to 5d_ , _JHEP_ 04 (2015) 141, [1410.2806].
  * (68) H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, _6d SCFTs, 5d Dualities and Tao Web Diagrams_ , _JHEP_ 05 (2019) 203, [1509.03300].
  * (69) H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, _Rank-3 antisymmetric matter on 5-brane webs_ , _JHEP_ 05 (2019) 133, [1902.04754].
  * (70) C. F. Uhlemann, _Wilson loops in 5d long quiver gauge theories_ , _JHEP_ 09 (2020) 145, [2006.01142].
  * (71) O. Bergman and D. Rodríguez-Gómez, _The Cat’s Cradle: deforming the higher rank E 1 and ${\tilde{E}}_{1}$ theories_, _JHEP_ 02 (2021) 122, [2011.05125].
  * (72) T. Gomez and E. R. Sharpe, _D-branes and scheme theory_ , hep-th/0008150.
  * (73) R. Donagi and M. Wijnholt, _Gluing Branes, I_ , _JHEP_ 05 (2013) 068, [1104.2610].
  * (74) S. Cecotti, C. Cordova, J. J. Heckman and C. Vafa, _T-Branes and Monodromy_ , _JHEP_ 07 (2011) 030, [1010.5780].
  * (75) J. J. Heckman, T. Rudelius and A. Tomasiello, _6D RG Flows and Nilpotent Hierarchies_ , _JHEP_ 07 (2016) 082, [1601.04078].
  * (76) M. Del Zotto, J. J. Heckman, A. Tomasiello and C. Vafa, _6d Conformal Matter_ , _JHEP_ 02 (2015) 054, [1407.6359].
  * (77) A. Beauville, _Symplectic singularities_ , _arXiv preprint math/9903070_ (1999) .
  * (78) D. Kaledin, _Symplectic singularities from the poisson point of view_ , .
  * (79) A. Bourget, S. Cabrera, J. F. Grimminger, A. Hanany, M. Sperling, A. Zajac and Z. Zhong, _The Higgs mechanism — Hasse diagrams for symplectic singularities_ , _JHEP_ 01 (2020) 157, [1908.04245].
  * (80) J. Eckhard, S. Schafer-Nameki and Y.-N. Wang, _Trifectas for T N in 5d_, _JHEP_ 07 (2020) 199, [2004.15007].
  * (81) M. R. Douglas, _D-branes, categories and N=1 supersymmetry_ , _J. Math. Phys._ 42 (2001) 2818–2843, [hep-th/0011017].
  * (82) E. Sharpe, _Lectures on D-branes and sheaves_ , 7, 2003.  hep-th/0307245.
  * (83) A. Collinucci and R. Savelli, _T-branes as branes within branes_ , _JHEP_ 09 (2015) 161, [1410.4178].
  * (84) R. P. Stanley, _Smith normal form in combinatorics_ , _Journal of Combinatorial Theory, Series A_ 144 (2016) 476–495.
  * (85) C. Beasley, J. J. Heckman and C. Vafa, _GUTs and Exceptional Branes in F-theory - I_ , _JHEP_ 01 (2009) 058, [0802.3391].
  * (86) L. B. Anderson, J. J. Heckman and S. Katz, _T-Branes and Geometry_ , _JHEP_ 05 (2014) 080, [1310.1931].
  * (87) L. B. Anderson, J. J. Heckman, S. Katz and L. P. Schaposnik, _T-Branes at the Limits of Geometry_ , _JHEP_ 10 (2017) 058, [1702.06137].
  * (88) D. H. Collingwood and W. M. McGovern, _Nilpotent orbits in semisimple Lie algebras_.  Routledge, 2017.
  * (89) H. Kraft and C. Procesi, _Minimal singularities ingl n_ , _Inventiones mathematicae_ 62 (1980) 503–515.
  * (90) S. Cabrera and A. Hanany, _Branes and the Kraft-Procesi Transition_ , _JHEP_ 11 (2016) 175, [1609.07798].
  * (91) S. Cabrera and A. Hanany, _Branes and the Kraft-Procesi transition: classical case_ , _JHEP_ 04 (2018) 127, [1711.02378].
  * (92) S. Cabrera and A. Hanany, _Quiver Subtractions_ , _JHEP_ 09 (2018) 008, [1803.11205].
  * (93) A. Bourget, J. F. Grimminger, A. Hanany, M. Sperling and Z. Zhong, _Branes, Quivers, and the Affine Grassmannian_ , 2102.06190.
  * (94) D. Gaiotto and E. Witten, _Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory_ , _J. Statist. Phys._ 135 (2009) 789–855, [0804.2902].
  * (95) A. Hanany and R. Kalveks, _Quiver Theories for Moduli Spaces of Classical Group Nilpotent Orbits_ , _JHEP_ 06 (2016) 130, [1601.04020].
  * (96) K. Altmann, _One-parameter families containing three-dimensional toric gorenstein singularities_ , _Explicit birational geometry of_ (2000) 21–50.
  * (97) C. Lawrie and S. Schafer-Nameki, _The Tate Form on Steroids: Resolution and Higher Codimension Fibers_ , _JHEP_ 04 (2013) 061, [1212.2949].
  * (98) L. Bhardwaj, M. Bullimore, A. E. V. Ferrari and S. Schafer-Nameki, _Generalized Symmetries and Anomalies of 3d N=4 SCFTs_ , 2301.02249.