11institutetext: Université Paris-Saclay, CNRS, CEA, Institut de physique
théorique,
91191, Gif-sur-Yvette, France11institutetext: Laboratoire de Physique de
l’École Normale Supérieure, PSL University,
24 rue Lhomond, 75005 Paris, France22institutetext: Service de Physique
Théorique et Mathématique, Université Libre de Bruxelles and
International Solvay Institutes, Campus Plaine C.P. 231, B-1050 Bruxelles,
Belgium33institutetext: Mathematical Institute, University of Oxford,
Woodstock Road, Oxford, OX2 6GG, United Kingdom

# Generalized Toric Polygons, T-branes, and 5d SCFTs

Antoine Bourget 2 Andrés Collinucci 3 Sakura Schäfer-Nameki

###### Abstract

5d Superconformal Field Theories (SCFTs) are intrinsically strongly-coupled UV
fixed points, whose realization hinges on string theoretic methods: they can
be constructed by compactifying M-theory on local Calabi-Yau threefold
singularities or alternatively from the world-volume of 5-brane-webs in type
IIB string theory. There is a correspondence between 5-brane-webs and toric
Calabi-Yau threefolds, however this breaks down when multiple 5-branes are
allowed to end on a single 7-brane. In this paper, we extend this connection
and provide a geometric realization of brane configurations including
7-branes. A web with 7-branes defines a so-called generalized toric polygon
(GTP), which corresponds to combinatorial data that is obtained by removing
vertices along external edges of a toric polygon. We identify the geometries
associated to GTPs as non-toric deformations of toric Calabi-Yau threefolds
and provide a precise, algebraic description of the geometry, when 7-branes
are introduced along a single edge. The key ingredients in our analysis are
T-branes in a type IIA frame, which includes D6-branes. We show that
performing Hanany-Witten moves for the 7-branes on the type IIB side
corresponds to switching on semisimple vacuum expectation values on the
worldvolume of D6-branes, which in turn uplifts to complex structure
deformations of the Calabi-Yau geometries. We test the proposal by computing
the crepant resolutions of the deformed geometries, thereby checking
consistency with the expected properties of the SCFTs.

††preprint:

##  1 Introduction and Summary

The existence and characterization of interacting superconformal field
theories in five spacetime dimensions (5d SCFTs) is a remarkable prediction of
string theory Seiberg:1996bd . Two approaches have emerged that allow the
construction of 5d SCFTs within the framework of string theory: the low energy
limit of M-theory on $\mathbb{R}^{1,4}$ times a local Calabi-Yau threefold
Morrison:1996xf , and the world-volume of a brane-web in type IIB string
theory Aharony:1997ju ; Aharony:1997bh ; DeWolfe:1999hj ; Bergman:2015dpa ;
Zafrir:2015rga ; Zafrir:2015ftn ; Ohmori:2015tka ; Hayashi:2017btw ;
Hayashi:2018bkd ; Hayashi:2018lyv . The properties of the theory are encoded
in the geometry of the threefold in the first case, and in the charges of the
external $(p,q)$-5-branes in the second case. A middle ground is the type IIA
realization, which involves both geometry and branes Closset:2018bjz .

When the CY is toric, there is a precise dictionary between the brane-web and
M-theory realization: the charges of the external $(p,q)$-5-branes can be
encoded into an integral polygon, which in turn can be seen as the
intersection of a toric three-dimensional fan in $\mathbb{R}^{3}_{x,y,z}$ with
the plane $\\{z=1\\}$. The toric threefold $\mathbf{X}$ constructed from this
fan is such that the 5d SCFT obtained from M-theory on $\mathbf{X}$ coincides
with that on the world-volume of the brane-web Leung:1997tw . An example is
shown in figure 1.

A systematic geometric exploration and classification of 5d SCFTs was started
in Intriligator:1997pq ; Benini:2009gi ; Cherkis:2014vfa ; Xie:2017pfl ;
DelZotto:2017pti ; Alexandrov:2017mgi ; Jefferson:2017ahm ; Jefferson:2018irk
; Bhardwaj:2018yhy ; Bhardwaj:2018vuu ; Apruzzi:2018nre ; Apruzzi:2019vpe ;
Apruzzi:2019opn ; Apruzzi:2019enx ; Bhardwaj:2019fzv ; Bhardwaj:2019xeg ;
Apruzzi:2019kgb ; Closset:2020scj ; Closset:2020afy ; Closset:2021lwy ;
Bhardwaj:2020gyu ; Bhardwaj:2020kim ; Bhardwaj:2020ruf ; Bhardwaj:2020avz .
These studies reveal many detailed properties of the 5d SCFTs, such as their
UV enhanced flavor symmetry, their Coulomb branch (modeled in terms of the
crepant resolutions of the Calabi-Yau singularities), but also refined
information such as their generalized symmetries Morrison:2020ool ;
Albertini:2020mdx ; Bhardwaj:2020phs ; Apruzzi:2021vcu ; DelZotto:2022fnw ;
DelZotto:2022joo . What remains somewhat obscure in this framework is the
derivation of the full quantum corrected Higgs branch – though some progress
in the context of isolated hypersurface Calabi-Yau singularities can be made
Closset:2020scj ; Closset:2020afy ; Closset:2021lwy ; Collinucci:2020jqd ;
Collinucci:2021ofd .

Not surprisingly, these explorations reveal that only a small class of 5d
SCFTs have a realization in terms of toric Calabi-Yau threefolds. If such a
toric realization exists, then the geometry of the moduli space of
supersymmetric vacua, in particular the Higgs branch (but also Coulomb branch
and mixed branches) can be computed exactly – irrespective of whether the
singularity is isolated or not. The key tool is the connection between the
toric geometries and brane-webs, where in the latter these moduli space
questions have been determined in Bergman:2013aca ; Hayashi:2015fsa ;
Cremonesi:2015lsa ; Yonekura:2015ksa ; Gaiotto:2015una ; Ferlito:2017xdq ;
Cabrera:2018jxt . In this paper, we report progress in generalizing these
methods to a larger class of 5d SCFTs, which have not necessarily a toric
description.

$\longleftrightarrow$ Figure 1: Example of toric polygon (left) and dual
brane-web (right) in which lines denote 5-branes and circles denote 7-branes.
The 7-branes on which stacks of 5-branes are spaced to emphasize how the
5-brane end, here exactly one 5-brane ends on each 7-brane. This geometry
encodes a 5d SCFT of rank 3.

In Benini:2009gi , a generalization of toric polygons was introduced: a toric
Calabi-Yau threefold can be described in terms of a convex polygon in a square
integral lattice embedded into a 2-plane. The polygon associated to a toric
geometry has the property that all lattice points along the edges are part of
the toric data (corresponding to vertices). We will refer to these as _black
dots_. Generalizing this, Benini:2009gi  proposed to also allow some vertices
along the edges of the polygon to be unoccupied, which we will refer to as
_white dots_. In the dual description, allowing such white dots corresponds in
the web to several 5-branes that end on the same 7-brane. For a stack of $n$
5-branes, the boundary condition is encoded in an integer partition $\lambda$
of $n$. Figure 2 gives an example of a configuration that translates to the
$[3,2,2,1]$ partition of $8$. This combinatorial data will be referred to as
_Generalized Toric Polygons_ (GTP), and generalizes the standard toric
description. In this framework, the standard toric case considered in the
previous paragraph corresponds to a boundary condition where for each charge
$(p,q)$, there is an equal number of $(p,q)$-5-branes and $(p,q)$-7-branes,
with one 5-brane ending on one 7-brane, i.e. the partition is
$\lambda=[1^{n}]$. This generalization and its implications for characterizing
the moduli space of supersymmetric vacua using magnetic quivers were explored
in great detail in Cabrera:2018jxt ; Bourget:2019rtl ; Collinucci:2020jqd ;
vanBeest:2020kou ; VanBeest:2020kxw ; Bourget:2020asf ; Closset:2020scj ;
Giacomelli:2020gee ; Bourget:2020mez ; Akhond:2021ffo ; vanBeest:2021xyt ;
Bourget:2021csg ; Bourget:2021jwo . Note that O5 orientifold planes can also
be included in the webs Zafrir:2015ftn ; Bourget:2020gzi ; Akhond:2020vhc ;
Sperling:2021fcf , but we do not consider this possibility here.

Another way of interpreting GTPs is as non-convex would-be toric polygons.
These make sense when certain parameters, which map out the extended Coulomb
branch (i.e. gauge couplings and masses of hypers), are turned on, but the
non-convexity prevents one from considering the SCFT limit (i.e. from passing
to the origin of the extended Coulomb branch). From the dual brane-web point
of view, this is resolved using a combination of Hanany-Witten moves and
7-brane monodromies, and this plays a prominent role in the brane-web
manipulations of Bergman:2014kza ; Bergman:2015dpa ; Zafrir:2015ftn ;
Hayashi:2015zka ; Ohmori:2015tka ; Hayashi:2018bkd ; Hayashi:2019yxj ;
Bourget:2020gzi ; Uhlemann:2020bek ; Bergman:2020myx . Importantly, not all
non-convex polygons can be transformed into GTPs in this way, and it is in
general a hard question to decide whether a given polygon can be transformed
in this way or not. For this reason, it is simpler to take the GTPs as our
starting point.

GTPs can be thought of as generalizations of toric geometries. However, unlike
the precise dictionary between the combinatorial data of a toric polygon and
the algebraic geometry of the corresponding Calabi-Yau, no such dictionary
exists thus far for GTPs. The main purpose of this paper is to develop initial
steps in order to close this gap. See figure 3. In particular, in the
following, we will determine the algebraic geometric description of GTPs,
which have white dots along a single edge.

Figure 2: Correspondence between white dots on GTPs (left) and boundary
conditions of $(p,q)$ 5-branes on $(p,q)$ 7-branes (middle). Here we have
$(p,q)=(1,0)$ and the partition $\lambda=[3,2,2,1]$ of $n=8$. When we draw
brane-webs we usually ignore the detached 7-branes and separate the 7-branes
on a stack of 5-branes to show the boundary conditions (right).

Convex Integral PolygonToric $\mathrm{CY}_{3}$Brane-web5d SCFT

Toric Geometry

Dual

M-theory

Worldvolume

GTP
---
Non-Toric CY3 Brane-web with
---
7-brane b.c.
5d SCFT

Dual

M-theory

Worldvolume

Figure 3: Summary of the main question addressed in this paper. On the left
hand side, one starts from a convex polygon with integral vertices. It defines
a 5d SCFT in two distinct ways: from M-theory on the associated toric CY, and
from the dual brane-web. If on the contrary, as shown on the right hand side,
the polygon is not convex, i.e. is a generalized toric polygon (GTP), the
toric description is lost. However there still exists a dual brane-web, which
has non-trivial boundary conditions on 7-branes, and thus a 5d SCFT. The
central goal of this paper is to develop a map (the dashed line in the
diagram) from GTPs to (non-toric) geometry.

#### Summary.

We now give a schematic summary of the main ideas involved in our proposal.
Consider a length $n$ edge of a toric polygon, which we can assume to have
vertical orientation. In the brane-web, this corresponds to $n$ parallel semi-
infinite D5-branes. In the associated toric threefold, there is an asymptotic
region that approaches $\mathbb{C}^{2}/\mathbb{Z}_{n}\times\mathbb{C}$. Indeed
after a transverse T-duality, the D5-branes become D6-branes, which uplift to
$n$-centered Taub-NUT spaces. At strong string coupling
$g_{s}\rightarrow\infty$, this becomes $\mathbb{C}^{2}/\mathbb{Z}_{n}$.
Denoting two longitudinal directions as the $w$-complex plane, we arrive at a
local $\mathbb{C}^{2}/\mathbb{Z}_{n}\times\mathbb{C}$ patch. M-theory on this
geometry gives us $\mathcal{N}=1$ 7d SYM with $\mathrm{SU}(n)$ gauge group,
and we can represent it as the singular hypersurface in $\mathbb{C}^{4}$ given
by

$uv=z^{n}$ (1)

with the $w$-coordinate tagging along. The three adjoint scalars
$\phi_{i=1,2,3}$ on the worldvolume of the D6-branes can be grouped into a
complex scalar $\Phi=\phi_{1}+i\phi_{2}$, and the remaining real one
$\varphi=\phi_{3}$. In the M-theory uplift, $\Phi$ encodes algebraic
deformations to the hypersurface, and $\varphi$ encodes Kähler volumes of
resolutions. This grouping is of course arbitrary, and correlates with the
arbitrariness of choosing a complex structure on the noncompact K3 in
M-theory.

Having seen this, we can recast the hypersurface as a spectral equation for
the complexified adjoint Higgs field

$uv=\det(\textbf{1}_{n}\,z-\Phi(w))\,.$ (2)

Switching on constant vevs along the Cartan subalgebra of $\mathfrak{su}(n)$
will deform the equation and unfold the singularity into a deformed K3 times
the $w$-plane. However, switching on $w$-dependent vevs will turn this into a
_bona fide_ noncompact CY threefold. The geometry will be more or less
desingularized, depending on the Casimir invariants of $\Phi$ that are
switched on. _The claim of this paper, is that white dots correspond to
nilpotent elements in $\Phi$_. Note, that switching on a nilpotent $\Phi$
means that the spectral equation remains unchanged. In other words, the
D6-branes do not actually move, and the uplifted geometry underlying the
M-theory construction remains undeformed. This phenomenon is known as a _T-
brane_ Gomez:2000zm ; Donagi:2011jy ; Cecotti:2010bp . It is a non-Abelian
bound state of branes, whereby the worldvolume gauge group is (partially)
Higgsed, but the geometry of the branes is unaltered. However, the physics is
of course impacted by this T-brane, as we shall see momentarily.

To give a simple example, take a vertical edge with $n+1$ dots, and replace
the second black dot from the top with a white dot as shown in figure 4. In
terms of the 5-branes, this corresponds to forming a bound state between the
two uppermost branes, and sending a suspended 5-brane segment to infinity.
From the D6-brane viewpoint, the bound state is understood as switching on a
vev for $\Phi$ along the minimal nilpotent orbit of $\mathfrak{su}(n)$

$\langle\Phi\rangle=\begin{pmatrix}0&1\\\ 0&0\\\ &&\ddots\end{pmatrix}\,.$ (3)

This binds the first two D6-branes, and partially Higgses

$\mathfrak{su}(n)\to\mathfrak{s}\left(\mathfrak{u}(1)\oplus\mathfrak{u}(n-2)\right)\,.$
(4)

More generally, we said above that a distribution of white dots on the edge is
encoded in a partition $\lambda$ of $n$. Our claim is that each such partition
$\lambda$ of $n$ translates into a vev for $\Phi$ along an element in the
nilpotent orbit $\mathcal{O}_{\lambda}$ of $\mathfrak{su}(n)$ that is uniquely
characterized by $\lambda$. The unbroken 7d gauge group on the D6-branes,
which will correspond to a subgroup of the total 5d flavor group, is then
broken to the commutant of this nilpotent element.

Figure 4: White dot and brane transition for a length 2 edge of a toric
polygon.

So far, our discussion parallels the picture developed several years ago in
Heckman:2016ssk ; DelZotto:2014hpa , in the 6d SCFT context, whereby geometric
data (about elliptic fibrations) was supplemented by nilpotent orbits, which
would partially Higgs an original theory and trigger various RG flows. At this
point, the reader might object that simply claiming that a white dot
translates to a nilpotent vev is not very interesting or verifiable, since
that data will be invisible to the geometry. While this is true, from the
5-brane-web perspective we know that a white dot opens up the possibility to
perform Hanany-Witten type transitions that were not possible in the presence
of black dots only. For instance, the following GTPs are related by such a
transition where one of the three leftmost 7-branes is moved to the right:

$\leftrightarrow$ (5)

Such HW type transitions provide non-trivial tests of our proposal: HW moves
correspond to changing the positions of branes, which in turn will impact the
dual geometry. We identify the subset of complex structure deformations that
are associated to these nilpotent vevs of $\Phi$. They are realized in terms
of vevs of $\Phi$ along a slice transverse to the nilpotent vev (inside the
full Lie algebra, not the nilpotent cone), known as the _Slodowy slice_. By
switching on such a vev, a subset of possible Casimir invariants will become
non-zero, leading to a deformation of the geometry, which is given by the
spectral equation of the the Higgs field.

For instance, in the simple case of $\mathfrak{su}(2)$, we take the initial
nilpotent vev along the minimal orbit

$\Phi_{0}=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}\,.$ (6)

The M-theory uplifted geometry corresponds to $\mathbb{C}^{2}/\mathbb{Z}_{2}$.
The Slodowy slice is given by matrices of the form

$\Phi=\begin{pmatrix}0&1\\\ a&0\end{pmatrix}\,,\quad{\rm with}\quad
a\in\mathbb{C}\,.$ (7)

The characteristic polynomial of this Higgs field is now non-trivial, and the
M-theory geometry deforms as follows

$uv=z^{2}\quad\longrightarrow\quad uv=z^{2}+a\,.$ (8)

$a_{3}$$a_{1}$$a_{7}$$e_{6}$$e_{7}$$\alpha\neq
0$$a_{1}$$a_{7}$$e_{6}$$e_{7}$$\alpha\neq 0$$\beta\neq 0$$a_{7}$$e_{6}$$e_{7}$

$W_{1}W_{2}W_{3}=Z^{4}$

$W_{1}W_{2}W_{3}=Z^{2}(Z^{2}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\alpha}W_{1})$

$W_{1}W_{2}W_{3}=(Z^{2}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\alpha}W_{1})(Z^{2}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\beta}W_{1})$

Figure 5:  Three GTPs are shown on the first line, and below the algebraic
equations characterizing the associated Calabi-Yau threefold geometry. The
model on the left is a toric threefold. The other two, non-toric GTPs, are
characterized in terms of deformations. Each of these geometries defines a 5d
SCFT. The Hasse diagrams of symplectic singularities for the Higgs branch of
these 5d SCFTs are shown below. The vertices represent symplectic leaves. For
transverse slices we use a standard notation where the closure of the minimal
nilpotent orbit of a simple Lie algebra is denoted using the lowercase form of
the name of the algebra, e.g. $e_{7}$ for algebra $E_{7}$. In red are drawn
the effects of the deformations.

The present paper elucidates this for all GTPs, which allow for a IIA-
description, i.e. whose white dots are along a single edge of the GTP. Dually,
all 7-branes are parallel, i.e. mutually local. The toy-model where Slodowy
slices appear is generalized in the following way. Higgs branches are
symplectic singularities beauville1999symplectic , to which one can associate
a Hasse diagram of symplectic leaves kaledin2006symplectic ; Bourget:2019aer ;
Eckhard:2020jyr . _In terms of this diagram, the T-brane data select a new,
lowest leaf of the foliation, and the transverse slice to that leaf is the
total space of a fibration over the complex structure moduli space of the
deformed Calabi-Yau threefold._ See for instance figure 5, where the
deformations of the $T_{4}$ 5d SCFT, realized on the threefold
$W_{1}W_{2}W_{3}=Z^{4}$, are displayed, along with the effect on the Higgs
branches.

In future work we will aim to generalize this to arbitrary GTPs, with mutually
non-local 7-branes. We conjecture that the above picture of transverse slices
in the Higgs branch extends to this situation. Eventually we hope to develop a
succinct description of the algebraic geometry of GTPs, as they exist for
toric polygons: A precise map between the combinatorial data and the basic
algebraic geometry, such as the set of divisors, curves, intersection numbers.

#### Plan.

In the rest of the paper, we spell out the details of the construction. An
essential tool is the T-brane, which is reviewed in section 2. The bulk of the
construction is then carried out explicitly in a representative example – that
of the $T_{n}$ SCFTs – in section 3, before generalizing to any GTP with white
dots on a single edge in section 4. As a first check, we reproduce there the
transition (5). Finally, in section 5 we provide consistency checks, by
computing the resolutions of the deformed threefold geometries. This shows
agreement of the UV flavor symmetry of the SCFT with the one expected from the
brane-web (and resulting Higgs branch).

##  2 T-branes and Kraft-Procesi Transitions

###  2.1 T-brane Basics

Consider $n$ parallel D7-branes in type IIB string theory on flat space. The
transverse space is the complex plane with coordinate $z$, which has
coordinate ring $R=\mathbb{C}[z]$. We call $z_{1},\dots,z_{n}$ the positions
of the $n$ branes. The stack of branes can be described as a
$\mathrm{D9}/\overline{\mathrm{D9}}$-brane tachyon condensate, which is
defined mathematically as the cokernel of the tachyon map

${R^{\oplus n}}$${R^{\oplus n}\,,}$$\scriptstyle{T}$ (9)

where $T=\mathrm{Diag}(z-z_{1},\dots,z-z_{n})$. This means that the D7-branes
correspond to the sheaf111We will use the formulation of branes modulo tachyon
condensation in terms of the derived category of coherent sheaves throughout
this paper. Some introduction to this topic can be found in Douglas:2000gi ;
Sharpe:2003dr ; Collinucci:2014qfa . $\mathcal{S}$ in the short exact sequence

${0}$${R^{\oplus n}}$${R^{\oplus n}}$${\mathcal{S}}$${0\,.}$$\scriptstyle{T}$
(10)

The matter on the system of D7-branes is described by fluctuations of the
tachyon, $\delta T$, which are defined up to linearized gauge transformations.
This corresponds to a self-Ext1 computation for the complex (9), i.e.
morphisms between the complex and the shifted version of that same complex,

${R^{\oplus n}}$${R^{\oplus n}}$${R^{\oplus n}}$${R^{\oplus
n}}$$\scriptstyle{\alpha_{L}}$$\scriptstyle{\delta
T}$$\scriptstyle{T}$$\scriptstyle{\hskip
14.22636pt\alpha_{R}}$$\scriptstyle{T}$ (11)

up to homotopies,

$\delta T\sim\delta T-T\cdot\alpha_{L}+\alpha_{R}\cdot T\,.$ (12)

Concretely, this means $\delta T$ is valued in the quotient ring of $n\times
n$ matrices $\text{Mat}_{n}(R)$ modulo the two matrix ideals in $R$ generated
by left and right multiplication by $T$,

$\delta T\in\text{Mat}_{n}(R)/(T\cdot,\cdot T)\,.$ (13)

Note also that the tachyon map $T$ can be expressed as a matrix given a choice
of basis for the D9 gauge bundle and the $\overline{\mathrm{D9}}$ gauge
bundle. These choices are independent, which means algebraically that only the
equivalence class of $T$ under the equivalence relation

$T\sim G_{L}\cdot T\cdot G_{R}^{-1}\,\qquad
G_{L},G_{R}\in\mathrm{GL}(n,\mathbb{C}[z])$ (14)

matters. A canonical representative of each such equivalence class is given by
the Smith Normal Form (SNF) computed in the ring $R=\mathbb{C}[z]$, i.e. any
matrix $T$ with entries in polynomials in $z$ is equivalent under (14) to a
unique diagonal matrix

$\mathrm{SNF}(T):=\mathrm{diag}(p_{1},\dots,p_{r},0,\dots,0)\,,$ (15)

where the $p_{i}$ are monic polynomials222Unicity is guaranteed up to
multiplication by units in the ring; demanding that the polynomials be monic,
i.e. have coefficient 1 for the term of highest degree, fixes this redundancy.
in $z$ such that $p_{1}|p_{2}|\dots|p_{r}$.

#### Example.

To illustrate the discussion of the previous paragraph, consider the case
$n=2$. The tachyon matrix is

$T=\begin{pmatrix}z-z_{1}&0\\\ 0&z-z_{2}\end{pmatrix}$ (16)

and the fluctuations belong to

$\delta T\in\begin{pmatrix}\frac{R}{(z-z_{1})}&\frac{R}{(z-z_{1},z-z_{2})}\\\
\frac{R}{(z-z_{1},z-z_{2})}&\frac{R}{(z-z_{2})}\end{pmatrix}\simeq\begin{cases}\begin{pmatrix}\mathbb{C}&0\\\
0&\mathbb{C}\end{pmatrix}&\qquad z_{1}\neq z_{2}\\\
\begin{pmatrix}\mathbb{C}&\mathbb{C}\\\
\mathbb{C}&\mathbb{C}\end{pmatrix}&\qquad z_{1}=z_{2}\\\ \end{cases}$ (17)

For any value of $z_{1},z_{2}$ there is $\mathrm{U}(1)$ adjoint matter on each
D7-brane, and for $z_{1}=z_{2}$ the $\mathrm{U}(1)^{2}$ gauge symmetry
enhances to $\mathrm{U}(2)$, and one can have fluctuations in the adjoint of
$\mathrm{U}(2)$. From the $\mathrm{U}(1)^{2}$ perspective this is simply
bifundamental matter.

Consider the case $z_{1}=z_{2}=0$. We can then activate an off-diagonal term:

$T_{[1,1]}=\begin{pmatrix}z&0\\\ 0&z\end{pmatrix}\quad\mapsto\quad
T_{[2]}=\begin{pmatrix}z&1\\\ 0&z\end{pmatrix}\,.$ (18)

After this activation, the fluctuations are reduced to

$\delta T_{[2]}\in\begin{pmatrix}0&0\\\
1&0\end{pmatrix}\mathbb{C}\oplus\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\mathbb{C}\,.$ (19)

The SNF reveals the same structure in a slightly different guise. Indeed

$\mathrm{SNF}\begin{pmatrix}z-z_{1}&0\\\
0&z-z_{2}\end{pmatrix}=\begin{cases}\begin{pmatrix}1&0\\\
0&(z-z_{1})(z-z_{2})\end{pmatrix}&\qquad z_{1}\neq z_{2}\\\
\begin{pmatrix}z-z_{1}&0\\\ 0&z-z_{1}\end{pmatrix}&\qquad z_{1}=z_{2}\\\
\end{cases}$ (20)

so the fluctuations are valued in

$\delta T\in\begin{cases}\begin{pmatrix}0&0\\\ 0&\mathbb{C}\oplus
z\mathbb{C}\end{pmatrix}&\qquad z_{1}\neq z_{2}\\\
\begin{pmatrix}\mathbb{C}&\mathbb{C}\\\
\mathbb{C}&\mathbb{C}\end{pmatrix}&\qquad z_{1}=z_{2}\\\ \end{cases}\,.$ (21)

Activating the off-diagonal term translates using the SNF into

$\mathrm{SNF}\begin{pmatrix}z-z_{1}&1\\\
0&z-z_{2}\end{pmatrix}=\begin{pmatrix}1&0\\\
0&(z-z_{1})(z-z_{2})\end{pmatrix}\,,$ (22)

valid for $z_{1}=z_{2}$ and $z_{1}\neq z_{2}$ alike. In particular, putting
both branes at the origin,

$\mathrm{SNF}\begin{pmatrix}z&1\\\ 0&z\end{pmatrix}=\begin{pmatrix}1&0\\\
0&z^{2}\end{pmatrix}\,,$ (23)

where we see the appearance of an infrared trivial complex

${R}$${R\quad\cong 0}$$\scriptstyle{1}$ (24)

and a so-called ’thick brane’

${R}$${R\,.}$$\scriptstyle{z^{2}}$ (25)

#### Multivariable polynomials.

The ring of polynomials in one variable $\mathbb{C}[z]$ has the property that
every ideal is principal. In particular, it is a _Bézout ring_ , which means
by definition that any ideal generated by finitely many generators is
principal. The ring $\mathbb{C}[x_{1},\dots,x_{n}]$ for $n>1$ on the other
hand is not a Bézout ring : it has non-principal finitely generated ideals
(for example, the ideal generated by $x_{1}$ and $x_{2}$). It turns out the
SNF is best defined in Bézout rings. By (stanley2016smith, , Theorem 2.1) an
SNF does not exist for matrices with coefficients in
$\mathbb{C}[x_{1},\dots,x_{n}]$. Thus, at face value it seems not possible to
use the SNF to describe intersecting branes.

However, this is not a weakness but a feature, as we now demonstrate. Consider
the case of two variables, $x$ and $z$. Physically, this means we are
considering branes that share an $\mathbb{R}^{1,5}$ and wrap complex curves in
the $(x,z)$-plane. Consider a stack of $n$ branes at $x=0$ and a stack of $m$
branes at $z=0$. This is described by the diagonal matrix
$\mathrm{diag}(x,\dots,x,z,\dots,z)$. We can activate non-diagonal terms, that
we call $Q$ and $\tilde{Q}$ as they correspond to strings that yield
hypermultiplets at low energy

$T=\left(\begin{array}[]{cc}x\mathbf{1}_{n}&\widetilde{Q}\\\
Q&z\mathbf{1}_{m}\end{array}\right)\,.$ (26)

In order to pick a canonical diagonal form for this matrix, we need to _make a
choice of main variable_. Let us pick $z$. This means we extend the non-Bézout
ring $\mathbb{C}[x,z]$ to the ring $\mathbb{C}(x)[z]$, which is Bézout as it
is a polynomial ring in _one variable_ over the field $\mathbb{C}(x)$. This is
simply telling us that poles in $x$ have to be included. We now describe the
SNF over this ring. Assume first that the eigenvalues of $Q\widetilde{Q}$ are
all distinct, call them $\lambda_{1},\dots,\lambda_{m}$. Then the SNF is

$\mathrm{diag}\left(x,\dots,x,1,\dots,1,\prod\limits_{i=1}^{m}\left(z-\frac{\lambda_{i}}{x}\right)\right)\,.$
(27)

If some of the eigenvalues coincide, the form of the SNF changes. We can
collect this information, which is insensitive to the detailed properties of
the eigenvalues, by simply stating that the SNF is

$\left(\begin{array}[]{cc}x\mathbf{1}_{n}&0\\\
0&z\mathbf{1}_{n}-\frac{Q\widetilde{Q}}{x}\end{array}\right)\qquad\lambda_{i}\not=\lambda_{j}\,.$
(28)

Thus, from the point of view of the stack of $m$ branes at $z=0$, the presence
of the other stack is felt as a pole for the complex adjoint-valued Higgs
field living on the brane at $z=0$, Beasley:2008dc ; Anderson:2013rka ;
Anderson:2017rpr . Note that the situation is symmetric and one could have
chosen the other stack as the base one. This is exactly the same arbitrariness
we made when writing the ring as a Bézout ring.

###  2.2 Kraft-Procesi Transitions

Nilpotent orbits for $\mathfrak{sl}_{n}$ are in one-to-one correspondence with
partitions of $n$, and are partially ordered by inclusion of their closure
collingwood2017nilpotent . The nilpotent orbit associated to a partition
$\lambda$ of $n$ is denoted $\mathcal{O}_{\lambda}$. The partial order
corresponds to the well-known _dominance ordering_ for partitions,333The
dominance ordering is defined as follows. If $\lambda$ and $\mu$ are two
partitions of $n$, we say that $\lambda\geq\mu$ if and only if for all $j$,
$\sum\limits_{i=1}^{j}\lambda_{j}\geq\sum\limits_{i=1}^{j}\mu_{j}$.  and it
can be represented by a Hasse diagram, which indicates the covering relation
associated to this partial order. The diagram thus obtained also corresponds
to the stratification of the nilpotent cone (the set of all nilpotent
matrices) into symplectic leaves kraft1980minimal ; beauville1999symplectic ;
kaledin2006symplectic . Elementary degenerations between adjacent nilpotent
orbits are called Kraft-Procesi transitions. In the case of
$\mathfrak{sl}_{n}$ nilpotent orbits, these can be either closures of minimal
$\mathfrak{sl}_{m}$ nilpotent orbits or Kleinian singularities
$\mathbb{C}^{2}/\mathbb{Z}_{m}$ for $m\leq n$. This can be implemented in
brane setups Cabrera:2016vvv ; Cabrera:2017njm , where nilpotent orbit
closures are realized as Higgs or Coulomb branches of 3d $\mathcal{N}=4$
quiver theories.

#### Slodowy Slices.

Consider the case where $T=z\cdot\mathbf{1}_{n}+M$ and $M$ is a nilpotent
matrix. Equation (12) shows that

$\delta T\sim\delta
T+(\alpha_{R}-\alpha_{L})z+(\alpha_{R}M-M\alpha_{L})\,,\qquad\delta
T\in\text{Mat}_{n}(R)\,.$ (29)

Define $\alpha_{+}:=\frac{1}{2}(\alpha_{R}+\alpha_{L})$ and
$\alpha_{-}:=\frac{1}{2}(\alpha_{R}-\alpha_{L})$ this gives

$\delta T\sim\delta
T+2\alpha_{-}z+[\alpha_{+},M]+\\{\alpha_{-},M\\}\,,\qquad\delta
T\in\text{Mat}_{n}(R)\,.$ (30)

The image of the map

$\displaystyle\text{Mat}_{n}(R)$ $\displaystyle\to\text{Mat}_{n}(R)$ (31)
$\displaystyle\alpha_{-}$ $\displaystyle\mapsto
2\alpha_{-}z+\\{\alpha_{-},M\\}$

contains all of $\text{Mat}_{n}(zR)$,444This is easily proved by a recursive
argument on the degree.  so we can use $\alpha_{-}$ to eliminate all
$z$-dependence in $\delta T$. We still have the freedom to use $\alpha_{+}$,
which defines an equivalence relation

$\delta T\sim\delta T+[\alpha_{+},M]\,,\qquad\delta
T\in\text{Mat}_{n}(\mathbb{C})\,.$ (32)

The cokernel of the adjoint action by $M$ has dimension

$d_{\lambda}:=\sum\limits_{i}(2i-1)\lambda_{i}\,,$ (33)

where $\lambda$ is the partition that specifies the nilpotent orbit of $M$. If
one considers only traceless matrices, $\delta T$ then depends only on
$d_{\lambda}-1$ parameters.

Note that being in the cokernel of $\mathrm{ad}(M)$ corresponds to commuting
with the other nilpotent element in the $\mathfrak{sl}_{2}$-triple generated
by $M$. Thus $M+\delta T$ parameterizes the Slodowy slice $\mathcal{S}_{M}$
transverse to $M$ (see Appendix A for definitions and a proof of this
statement). Note that indeed that

$\forall
M\in\mathcal{O}_{\lambda}(\mathfrak{sl}_{n})\,,\qquad\mathrm{dim}\,\mathcal{S}_{M}=d_{\lambda}-1\,.$
(34)

#### Kraft-Procesi Transitions.

Consider two partitions $\lambda$ and $\mu$, which are immediately adjacent in
the partial order – one says that $\lambda$ _covers_ $\mu$ if they are
adjacent and $\lambda>\mu$. Then $\lambda$ and $\mu$ differ only in two
entries, say with indices $i<j$, with $\lambda_{i}-1=\mu_{i}$ and
$\lambda_{i+1}+1=\mu_{i+1}$, and one of the two following transitions occurs:

Condition | Transition name | Transverse slice
---|---|---
$j=i+1$ | $A_{\lambda_{i}-\lambda_{j}-1}$ | $\mathbb{C}^{2}/\mathbb{Z}_{\lambda_{i}-\lambda_{j}}$
$\mu_{i}=\mu_{j}$ | $a_{i-j-1}$ | $\overline{\mathcal{O}}_{\mathrm{min}}(\mathfrak{sl}(i-j,\mathbb{C}))$
(35)

The first corresponds to a Kleinian singularity, whereas the second is the
closure of a minimal nilpotent orbit. The equivalence class of tachyon
matrices for a partition $\mu=[\mu_{1},\dots,\mu_{r}]$ (with
$\mu_{1}\geq\dots\geq\mu_{r}$) is characterized by a common SNF, i.e.

$\text{SNF}(T_{\mu})=\begin{pmatrix}1&&&&&\\\ &\ddots&&&&\\\ &&1&&&\\\
&&&z^{\mu_{r}}&&\\\ &&&&\ddots&\\\ &&&&&z^{\mu_{1}}\end{pmatrix}\,.$ (36)

Starting from the SNF of partition $\mu$, the Kraft-Procesi transition is
realized using

$\mathrm{SNF}\begin{pmatrix}z^{\mu_{j}}&\alpha z^{\mu_{j}-1}\\\
0&z^{\mu_{i}}\end{pmatrix}=\begin{pmatrix}z^{\mu_{j}-1}&0\\\
0&z^{\mu_{i}+1}\end{pmatrix}=\begin{pmatrix}z^{\lambda_{j}}&0\\\
0&z^{\lambda_{i}}\end{pmatrix}\,$ (37)

for $\alpha\neq 0$.

Note that this is precisely the tachyon matrix formalism analog of the way
Kraft-Procesi transitions are realized in Hanany-Witten brane systems for 3d
$\mathcal{N}=4$ quiver theories in Cabrera:2016vvv .

#### Examples.

The equality (23) corresponds to the covering of partition $[1,1]$ by $[2]$,
whereby two branes are combined into a thick brane. A less trivial case is the
covering of $[2,2]$ by $[3,1]$, where we do not simply have two branes being
combined. Rather, one of the two thick branes needs to be broken. This is
realized in our framework using (37) as follows:

$\mathrm{SNF}\begin{pmatrix}z^{2}&\alpha z\\\
0&z^{2}\end{pmatrix}=\begin{pmatrix}z^{1}&0\\\ 0&z^{3}\end{pmatrix}\,$ (38)

for $\alpha\neq 0$.

When the nilpotent orbit Hasse diagram is linear, one can build matrices that
encode all partitions at once. This is the case for $n\leq 5$, where the
matrices are given by

$\left(\begin{array}[]{cc}z&a_{1}\\\ 0&z\\\
\end{array}\right)\,,\qquad\left(\begin{array}[]{ccc}z&a_{1}&0\\\ 0&z&a_{2}\\\
0&0&z\\\ \end{array}\right)\,,\qquad\left(\begin{array}[]{cccc}z&a_{1}&0&0\\\
0&z&za_{3}+a_{4}&0\\\ 0&0&z&a_{2}\\\ 0&0&0&z\\\ \end{array}\right)\,,$ (39)
$\left(\begin{array}[]{ccccc}z&a_{1}&0&0&0\\\ 0&z&za_{3}&0&a_{2}a_{3}a_{4}\\\
0&0&z&a_{2}&0\\\ -z^{3}a_{6}&0&z^{2}a_{1}a_{3}a_{6}&z&-za_{4}\\\
-za_{2}a_{4}a_{5}\left(a_{1}a_{2}a_{3}a_{6}+1\right)&0&a_{1}^{2}a_{2}^{2}a_{3}^{2}a_{4}a_{5}a_{6}&0&z\\\
\end{array}\right)\,.$ (40)

This means that the SNF of a matrix above with $a_{1},\cdots,a_{r}$ non-zero
gives precisely the $r$-th partition of $n$, the partitions being totally
ordered.

##  3 Example: GTPs for $T_{n}$ and Related Models

In this section, we use an intermediate step in correspondence between
M-theory on a CY threefold and IIB 5-brane-webs: IIA on a resolved
$\mathbb{C}^{2}/\mathbb{Z}_{n}$ singularity with D6-branes. This discussion
follows the philosophy of Closset:2018bjz . We start in this section with the
instructive example of $T_{n}$, and consider its description as well as GTPs
obtained by adding white dots along a single edge.

###  3.1 The Setup

Consider the toric local Calabi-Yau defined by the toric diagram with vertices
at coordinates $(0,0)$, $(n,0)$ and $(n,n)$, drawn here for $n=5$:

$W_{1}$$W_{3}$$W_{2}$ (41)

The generators of the dual cone are

$\displaystyle(-1,0,0)$ $\displaystyle\leftrightarrow$ $\displaystyle W_{1}$
(42) $\displaystyle(1,-1,n)$ $\displaystyle\leftrightarrow$ $\displaystyle
W_{2}$ (43) $\displaystyle(0,1,0)$ $\displaystyle\leftrightarrow$
$\displaystyle W_{3}$ (44) $\displaystyle(0,0,1)$
$\displaystyle\leftrightarrow$ $\displaystyle Z\,.$ (45)

The first three generators are vectors normal to the 2-dimensional facets on
the fan, drawn in blue arrows when projected on the CY plane in (41). As an
algebraic variety, the toric threefold is simply a hypersurface in
$\mathbb{C}^{4}$:

$W_{1}W_{2}W_{3}=Z^{n}\quad\subset\quad\mathbb{C}^{4}\langle
W_{1},W_{2},W_{3},Z\rangle\,.$ (46)

This space has non-isolated singularities, specified by the following
intersecting ideals:

$I_{\rm
sing}=\left(W_{1},W_{2},Z\right)\cap\left(W_{1},W_{3},Z\right)\cap\left(W_{2},W_{3},Z\right)\,.$
(47)

Along each such ideal, there is a family of $A_{n-1}$-singularities. These are
in one-to-one correspondence with the three edges of the toric graph.

The singular threefold admits three different (albeit linearly dependent)
$\mathbb{C}^{*}$-actions which act on the coordinates with the following
weights:

$\begin{array}[]{c|cccc}&W_{1}&W_{2}&W_{3}&Z\\\
\hline\cr\mathbb{C}^{*}_{1}&0&1&-1&0\\\ \mathbb{C}^{*}_{2}&1&0&-1&0\\\
\mathbb{C}^{*}_{3}&1&-1&0&0\end{array}$ (48)

As explained in toric language in Closset:2018bjz , we can define projections
$\pi_{i}$, for $i=1,2,3$, with respect to these actions, and this will bring
us down to IIA. Let $(i,j,k)$ be a permutation of $(1,2,3)$. The way to reduce
along a particular $\mathbb{C}^{*}$-action $\mathbb{C}^{*}_{i}$ is to pick the
pair of ‘charged’ coordinates $W_{j}$ and $W_{k}$, and setup a
$\mathbb{C}^{*}$-fibration over an new complex coordinate $V_{jk}$ as follows:

$\mathbb{C}^{*}_{i}:\mathbb{C}[W_{1},W_{2},W_{3},Z]\cong\frac{\mathbb{C}[W_{1},W_{2},W_{3},Z,V_{jk}]}{(W_{j}W_{k}-V_{jk})}\,.$
(49)

Now we can rewrite the threefold in the following presentation:

$\frac{\mathbb{C}[W_{1},W_{2},W_{3},Z]}{(W_{1}W_{2}W_{3}-Z^{n})}\cong\frac{\mathbb{C}[W_{1},W_{2},W_{3},Z,V_{jk}]}{(W_{j}W_{k}-V_{jk};\,W_{i}V_{jk}-Z^{n})}\,.$
(50)

The IIA reduction is achieved by reducing over the
$S^{1}\subset\mathbb{C}^{*}$ action in each case. The non-compact part
$\mathbb{R}\subset\mathbb{C}^{*}$ becomes a transverse direction to the
D6-branes. The projection is simply defined as dropping the pair of
coordinates $(W_{j},W_{k})$, leaving us with a local K3 with an $A_{n-1}$
Klein singularity

$\frac{\mathbb{C}[W_{i},V_{jk},Z]}{(W_{i}V_{jk}-Z^{n})}\,.$ (51)

To simplify the notations, we switch to the more standard

$X:=W_{i}\qquad Y:=V_{jk}$ (52)

so that the $A_{n-1}$ singularity (a local K3) is described by

$XY=Z^{n}\,.$ (53)

There are D6-branes on the locus defined by the ideal

$I_{\rm D6}=(Y,Z^{n})\,,$ (54)

which we call the D6 ideal. It is, first of all a stack of $n$ non-compact
D6-branes. However, since this passes through the singularity, more is at play
here, and we need to resolve the local K3 to refine our understanding.

In order to describe the resolution of the $A_{n-1}$ orbifold (53), we
introduce homogeneous coordinates $(z_{1},e_{1},\dots,e_{n-1},z_{2})$ with
$n-1$ $\mathbb{C}^{*}$-actions

$\begin{array}[]{|c| c c c c c c c c
c|}\hline\cr&z_{1}&e_{1}&e_{2}&e_{3}&\ldots&e_{n-3}&e_{n-2}&e_{n-1}&z_{2}\\\
\hline\cr\mathbb{C}^{*}_{1}&1&-2&1&0&\ldots&0&0&0&0\\\
\mathbb{C}^{*}_{2}&0&1&-2&1&\ldots&0&0&0&0\\\ \vdots&&&&&&&&&\\\
\mathbb{C}^{*}_{n-2}&0&0&0&0&\ldots&1&-2&1&0\\\
\mathbb{C}^{*}_{n-1}&0&0&0&0&\ldots&0&1&-2&1\\\ \hline\cr\end{array}$ (55)

The coordinates are homogeneous with respect to the $n-1$ projective actions,
by which the space is quotiented. Each row gives the list of weights of the
coordinate with respect to each such action. Just as one must excise
particular loci when creating standard projective space, so must one excise a
number of loci here.

$\cdots$$e_{1}=0$$e_{2}=0$$e_{n-2}=0$$e_{n-1}=0$$z_{1}=0$$z_{2}=0$ Figure 6:
Geometry of the resolved $A_{n-1}$ singularity. Each curved line is a
$\mathbb{P}^{1}$ while the straight lines are the non compact divisors
$z_{1,2}=0$.

Specifically, if a coordinate is set to zero, then only one its two
‘neighbors’ are allowed to vanish. See figure 6. For example, we can have
$e_{2}=0$, and $e_{3}=0$, or $e_{2}=0$ and $e_{1}=0$, but the pair
$(e_{1},e_{3})$ does not form a valid ideal for a vanishing locus. In terms of
the coordinates $(X,Y,Z)$, we have

$X=\prod\limits_{i=0}^{n}e_{i}^{n-i}\,,\quad
Y=\prod\limits_{i=0}^{n}e_{i}^{i}\,,\quad Z=\prod\limits_{i=0}^{n}e_{i}$ (56)

where we have introduced $e_{0}:=z_{1}$ and $e_{n}:=z_{2}$. The locus
$e_{i}=0$ corresponds to the $i$-th exceptional $\mathbb{P}^{1}$. The loci
$z_{1}=0$ and $z_{2}=0$ correspond to noncompact holomorphic curves
intersecting the first and last $\mathbb{P}^{1}$, respectively. Line bundles
over this space are characterized by their first Chern class, which is encoded
as $\mathcal{O}(k_{1},\ldots,k_{n-1})$. A section of this bundle is a
polynomial of homogeneous multi-degree $(k_{1},\ldots,k_{n-1})$.

The brane locus (54) is now given by

$I_{\rm
D6}=\left(\prod\limits_{i=0}^{n}e_{i}^{i},\prod\limits_{i=0}^{n}e_{i}^{n}\right)=\left(\prod\limits_{i=0}^{n}e_{i}^{i}\right)\,.$
(57)

The interpretation is as follows: there are $i$ D6-brane wrapping the $i$-th
$\mathbb{P}^{1}$, and $n$ non-compact D6-branes on the curve $z_{2}=0$, which
intersects the $(n-1)$-th sphere at one point. At the SCFT point, all the
$\mathbb{P}^{1}$’s shrink to zero size. On the Coulomb branch, where the
Kähler volumes are non zero, the effective theory can be read from the ideal
(57). It is described by the quiver:

$\mathrm{U}(1)$$\mathrm{U}(2)$$\cdots$$\mathrm{U}(n-2)$$\mathrm{U}(n-1)$$n$
(58)

###  3.2 Tachyon Condensation Picture

The theory (58) is encoded by via the tachyon condensation/coherent sheaf
language as follows. First recall that in terms of complexes, one can describe
the branes as follows:

  * •

A brane $B_{i}$ wrapped on the $i$-th $\mathbb{P}^{1}$ given by $e_{i}=0$ can
be described as the cokernel of the complex of line bundles:

${B_{i}:\qquad\mathcal{O}(-e_{i})}$${\mathcal{O}\,,}$$\scriptstyle{e_{i}}$
(59)

where $\mathcal{O}$ is the structure sheaf over the local K3, and
$\mathcal{O}(-e_{i})$ is the dual of the line bundle $\mathcal{O}(e_{i})$, of
which $e_{i}$ is a section. For instance,
$\mathcal{O}(-e_{1})=\mathcal{O}(2,-1,0,\ldots,0)$.

  * •

A noncompact ‘flavor brane’ $B_{F}$ at the locus $z_{2}=0$, intersecting the
rightmost $\mathbb{P}^{1}$ (given by $e_{n-1}=0$), is given by the following
complex:

${B_{n}:\qquad\mathcal{O}(-z_{2})}$${\mathcal{O}\,.}$$\scriptstyle{z_{2}}$
(60)

  * •

More generally, a noncompact D6-brane intersecting the $i$-th exceptional
$\mathbb{P}^{1}$ will be given by the zero-locus of a section of
$\mathcal{O}(0,\ldots,0,1,0,\ldots,0)$, where the ‘1’ is the $i$-th entry.

Define $N=\frac{1}{2}n(n+1)$ D8-branes with gauge bundle
$\mathcal{F}_{\mathrm{D8}}:=\mathcal{O}^{\oplus N}$ and $N$ anti-D8-branes
with gauge bundle

$\mathcal{F}_{\overline{\mathrm{D8}}}:=\mathcal{O}(2,-1,0,\cdots,0)\oplus\mathcal{O}(-1,2,-1,\cdots,0)^{\oplus
2}\oplus\cdots\oplus\mathcal{O}(0,\cdots,0,-1)^{\oplus n}\,.$ (61)

We then define the tachyon map

$T:\mathcal{F}_{\overline{\mathrm{D8}}}\rightarrow\mathcal{F}_{\mathrm{D8}}$
(62)

as a diagonal matrix

$T=\mathrm{Diag}(e_{1}\cdot\mathbf{1}_{1},e_{2}\cdot\mathbf{1}_{2},\dots,e_{n-1}\cdot\mathbf{1}_{n-1},z_{2}\cdot\mathbf{1}_{n})\,.$
(63)

Note that $\det T=Y$, so that the equations of the threefold are

$W_{j}W_{k}=\det T=Y\,,\qquad XY=Z^{n}\,.$ (64)

from which one recover the original equation $W_{i}W_{j}W_{k}=Z^{n}$. The
resulting D6-brane system is defined as the cokernel $\mathcal{S}:={\rm
cok}(T)$ of this map, which is the locus where $T$ fails to be invertible. So
$\mathcal{S}$ is reducible:

$\mathcal{S}=\bigoplus\limits_{i=1}^{n}\mathcal{O}_{e_{i}}^{\oplus i}\,,$ (65)

where $\mathcal{O}_{p}$ means the structure sheaf with support over $p=0$.
There is an exact sequence

${\mathcal{F}_{\overline{\mathrm{D8}}}}$${\mathcal{F}_{\mathrm{D8}}}$${\mathcal{S}}$${0\,.}$$\scriptstyle{T}$
(66)

#### Fluctuations.

The fluctuations around this background are computed as self-extensions, in
the same way as in section 2. This means that the fluctuations $\delta T$ of
the tachyon $T$ belong to the self-extension group,

$\delta T\in{\rm Ext}^{1}(\mathcal{S},\mathcal{S})={\rm
Hom}_{\mathcal{D}(K3)}(\mathcal{S},\mathcal{S}[1])\,,$ (67)

where we consider the Homs in the derived category of coherent sheaves on K3
$\mathcal{D}(K3)$, and the $[1]$ means that we shift the complex one step to
the left. The matrix $\delta T$ can be decomposed in blocks, and there will be
non-zero fluctuations just above and below the diagonal.

In practice, the fluctuations are subjected to three conditions, that we will
be using repeatedly in the following sections:

  1. $(i)$

The $\mathbb{C}^{\ast}$ weights of the columns of $T$ and $\delta T$ need to
be compatible with (63) ;

  2. $(ii)$

The fluctuations are regular sections of the relevant sheaves (no pole on the
support locus) ;

  3. $(iii)$

The components of $\delta T$ are subject to the identifications (13).

###  3.3 Example: $T_{2}$

For concreteness we work out in detail the case $n=2$ for $T_{n}$ before
treating the general case. The tachyon matrix is

$\leavevmode\hbox to119.98pt{\vbox
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-59.98888pt}{-9.56194pt}\pgfsys@invoke{
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36.21667pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathcal{O}(2)\oplus\mathcal{O}(-1)^{\oplus 2}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
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}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\mathcal{O}^{\oplus
3}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
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}\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{12.64447pt}{-7.06194pt}\pgfsys@lineto{35.84448pt}{-7.06194pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{36.04446pt}{-7.06194pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.91304pt}{-4.70917pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{T}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,,\qquad\qquad
T=\left(\begin{array}[]{cc}e_{1}&0\\\
0&z_{2}\cdot\mathbf{1}_{2}\end{array}\right)$ (68)

and we give names to the blocks in the fluctuation $\delta T$ in
correspondence with the quiver

$\delta T=\begin{pmatrix}\Phi_{1}&\widetilde{Q}_{1}\\\
Q_{1}&\Phi_{2}\end{pmatrix}\,\qquad\qquad\raisebox{-0.5pt}{ \leavevmode\hbox
to118.64pt{\vbox
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.96593}{0.25882}{-0.25882}{-0.96593}{289.42221pt}{-1.3115pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ (69)

The three conditions listed above give

$\delta
T\in\begin{pmatrix}\Gamma(\mathcal{O}(-2)_{(e_{1})})&\Gamma(\mathcal{O}(1)_{(e_{1},z_{2})})\cdot\mathbb{C}^{1\times
2}\\\ \Gamma(\mathcal{O}(-2)_{(e_{1},z_{2})})\cdot\mathbb{C}^{2\times
1}&\Gamma(\mathcal{O}(1)_{(z_{2})})\cdot\mathbb{C}^{2\times
2}\end{pmatrix}=\begin{pmatrix}0&\mathbb{C}^{1\times 2}\cdot z_{1}\\\
\mathbb{C}^{2\times 1}\cdot\frac{1}{z_{1}^{2}}&\mathbb{C}^{2\times 2}\cdot
z_{1}\end{pmatrix}\,.$ (70)

Here, $\Gamma$ indicates that we take sections of the bundles, and
$\mathbb{C}^{n\times m}$ is the set of $n\times m$ matrix of complex numbers.
The last equality is easily checked, e.g. for the lower-left entry, the
regular sections are generated by rational functions in $z_{1}$ alone, having
$\mathbb{C}^{\ast}$-weight $-2$, and poles are allowed as the support is the
intersection between two curves where $z_{1}\neq 0$. In terms of Ext groups
between $B_{1}$ (see (59)) and $B_{2}$ (see (60)), we can write (using e.g. a
spectral sequence argument)

$Q_{1}\in\mathrm{Ext}^{1}(B_{1},B_{2})=H^{0}(\mathcal{O}(e_{1})_{e_{1},z_{2}})=\left\langle\frac{1}{z_{1}^{2}}\right\rangle\,.$
(71)
$\widetilde{Q}_{1}\in\mathrm{Ext}^{1}(B_{2},B_{1})=H^{0}(\mathcal{O}(z_{2})_{e_{1},z_{2}})=\langle
z_{1}\rangle\,.$ (72)

To summarize, the background tachyon plus fluctuation is given by

$T+\delta T=\left(\begin{array}[]{cccc}e_{1}&&&\widetilde{q}_{1}z_{1}\\\
\frac{q_{1}}{z_{1}^{2}}&&&z_{2}\cdot\mathbf{1}_{2}+z_{1}\varphi_{2}\end{array}\right)\,,$
(73)

where we have pulled out all dependencies in $z_{1}$, $e_{1}$ and $z_{2}$, so
that $\widetilde{q}_{1}\in\mathbb{C}^{1\times 2}$,
$q_{1}\in\mathbb{C}^{2\times 1}$ and $\varphi_{2}\in\mathbb{C}^{2\times 2}$
are pure constants:

$Q_{1}=\frac{q_{1}}{z_{1}^{2}}\,,\qquad\widetilde{Q}_{1}=\widetilde{q}_{1}z_{1}\,\,,\qquad\Phi_{2}=\varphi_{2}z_{1}\,.$
(74)

Note that the term $\frac{q_{1}}{z_{1}^{2}}$ ensures that (73) is defined on
the locus $z_{1}\neq 0$

We can perform basis changes from the left and right, using (14), as follows:

$\left(\begin{array}[]{cc}1&0\\\
-\frac{q_{1}}{z_{1}^{2}e_{1}}&\mathbf{1}_{2}\end{array}\right)\cdot\left(\begin{array}[]{cc}e_{1}&\widetilde{q}_{1}z_{1}\\\
\frac{q_{1}}{z_{1}^{2}}&z_{2}\cdot\mathbf{1}_{2}\end{array}\right)\cdot\left(\begin{array}[]{cc}1&-\frac{\widetilde{q}_{1}z_{1}}{e_{1}}\\\
0&\mathbf{1}_{2}\end{array}\right)=\left(\begin{array}[]{cc}e_{1}&0\\\
0&z_{2}\cdot\mathbf{1}_{2}-\frac{M}{z_{1}e_{1}}\end{array}\right)\,$ (75)

$e_{1}=0$$z_{1}=0$$z_{2}=0$$\Phi$$\varphi_{1}$${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}Q_{1},\tilde{Q}_{1}}$

$e_{1}=0$$z_{1}=0$$z_{2}=0$$\varphi_{2}=-\frac{M}{X}$$\varphi_{1}$

Figure 7: Intersecting branes before and after the transformation that maps
the off-diagonal Higgs-field entries $Q_{1},\tilde{Q}_{1}$ to diagonal ones
with a pole. The orange dot signals the pole in the Higgs field at $X=0$.

In the last step we have introduced the meson matrix

$M:=q_{1}\widetilde{q}_{1}\,.$ (76)

In the 5d effective field theory, F-term conditions impose that $M$ be
nilpotent. This can also be demonstrated mathematically via the so-called
_cone construction_ in the derived category of coherent sheaves. See
Collinucci:2014qfa  for examples of this mechanism.

This diagonalization shows that giving a vev to the meson field can be
subsumed into a shift of vev of the adjoint field $\phi$ on the flavor branes,
with a pole. Using the coordinate $X=z_{1}^{2}e_{1}$ (see (56)) on the
$z_{2}=0$ plane, we can write this as

$T+\delta T\sim\left(\begin{array}[]{cc}e_{1}&0\\\
0&z_{2}\cdot\mathbf{1}_{2}+z_{1}\cdot\varphi_{2}\end{array}\right)\qquad\textrm{with}\qquad\varphi_{2}=-\frac{M}{X}\,.$
(77)

The transformation from (73) to (77) means that we can regard in an
appropriate regime the intersecting branes in figure 7 as a stack of two
branes on $z_{2}=0$ with a complex codimension-one defect on its world-volume
at $e_{1}=0$. This is in agreement with the findings of Beasley:2008dc ;
Anderson:2017rpr , where the authors find that a vev of bifundamental fields
at the intersection of two branes can be subsumed into a pole for the adjoint
of one of the two branes. In the picture, we trade the description in figure 7
on the left with the right.

Up to a change of basis we can take $M$ in canonical Jordan form. There are
two possibilities, corresponding to the two partitions of $n=2$:

$M_{[1^{2}]}=\begin{pmatrix}0&0\\\ 0&0\end{pmatrix}\qquad\textrm{and}\qquad
M_{[2]}=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}\,.$ (78)

Consider the latter case,

$T_{[2]}:=\begin{pmatrix}e_{1}&\\\
&t_{[2]}\end{pmatrix}:=\left(\begin{array}[]{ccc}e_{1}&0&0\\\
0&z_{2}&-\frac{1}{z_{1}e_{1}}\\\ 0&0&z_{2}\end{array}\right)\,.$ (79)

We still have $\det T_{[2]}=ez_{2}^{2}=Y$. So the brane configuration has not
changed geometrically. Accordingly, the M-theory uplift is still given by
(64). However, the tachyon matrix shows us that the flavor brane no longer
carries an $\mathrm{SU}(2)$ group. This is the hallmark of a _T-brane_ : A
non-abelian bound state of branes that does not realize the gauge group that
it would naively have given its geometry. This T-brane effect is the IIA
counterpart of the change in boundary conditions in the dual type IIB brane-
webs

(80)

In this particular case, once the D5-segment has been sent away, a Hanany-
Witten move where the 7-brane detaches completely becomes possible:

(81)

How does this translate into the IIA language, i.e. in terms of the tachyon
field? Let us see what deformations are available, starting from the new
vacuum defined by (79). Using the results of section 2, the fluctuations of
$t_{[2]}$ are

$\delta t_{[2]}=z_{1}\cdot\begin{pmatrix}0&0\\\
\alpha&0\end{pmatrix}\,,\qquad\alpha\in\mathbb{C}\,.$ (82)

Now in order to see how this affects the geometry, we add this perturbation to
$T_{[2]}$

$T_{[2]}+\delta t_{[2]}=\begin{pmatrix}e&0&0\\\ 0&z_{2}&-\frac{1}{z_{1}e}\\\
0&\alpha z_{1}&z_{2}\end{pmatrix}\,.$ (83)

Now the geometry (64) is deformed to

$W_{j}W_{k}=\det T=Y+\alpha\,,\qquad XY=Z^{2}\,.$ (84)

which reduces to the hypersurface

$W_{1}W_{2}W_{3}=Z^{2}+\alpha W_{i}\,.$ (85)

So the CY threefold is fully desingularized. From the IIA perspective, we see
that two flavor branes have recombined with one gauge brane, to give rise to a
noncompact brane that can escape the singular locus. This is in full agreement
with the 5-brane-web picture, whereby the 7-brane moves off to the left,
becomes fully detached from the NS5-branes, and can escape to infinity, see
(81). This corresponds precisely to removing the $A_{1}$ singularity.

To summarize, the white dot is represented in the IIA picture as a nilpotent
vev with poles on the flavor D6-stack. The further Hanany-Witten move that
actually deforms the M-theory geometry is implemented by switching on a
further vev on the flavor stack along the _Slodowy slice_ with respect to the
initial singular nilpotent vev.

###  3.4 General Case: $T_{n}$

The general $T_{n}$ case is very similar to the $T_{2}$ example treated above.
The fluctuations around the tachyon background are denoted by fields as
follows:

$1$$2$$\dots$$n-2$$n-1$$n$$Q_{1}$$Q_{n-2}$$Q_{n-1}$$\widetilde{Q}_{1}$$\widetilde{Q}_{n-2}$$\widetilde{Q}_{n-1}$$\Phi_{1}$$\Phi_{2}$$\Phi_{n-2}$$\Phi_{n-1}$
(86)

Generalizing (71) and (72), we find that the 5d hypermultiplets that reside at
the intersection of the curves $e_{i}=0$ and $e_{i+1}=0$ are described by

$Q_{i}\in{\rm
Ext}^{1}(B_{i},B_{i+1})=H^{0}(\mathcal{O}(e_{i})_{(e_{i},e_{i+1})})=\left\langle\frac{Y}{Z^{i+1}}e_{i}\right\rangle=\left\langle\prod\limits_{j\neq
i}e_{j}^{j-i-1}\right\rangle\,.$ (87)

and

$\tilde{Q_{i}}\in{\rm
Ext}^{1}(B_{i+1},B_{i})=H^{0}(\mathcal{O}(e_{i+1})_{(e_{i},e_{i+1})})=\left\langle\frac{Z^{i}}{Y}e_{i+1}\right\rangle=\left\langle\prod\limits_{j\neq
i+1}e_{j}^{i-j}\right\rangle\,.$ (88)

On the other hand, one can check that ${\rm Ext}^{1}(B_{i},B_{j})=0$ for
$|i-j|>1$. Therefore the tachyon matrix $T$, plus the fluctuations $\delta T$,
fit schematically (we will provide the explicit form below) into the matrix

$T+\delta
T=\left(\begin{array}[]{ccccccc}e_{1}\cdot\mathbf{1}_{1}&\widetilde{Q}_{1}&&&&\\\
Q_{1}&e_{2}\cdot\mathbf{1}_{2}&\ddots&&&\\\ &&\ddots&\ddots&\ddots&&\\\
&&&&&e_{n-1}\cdot\mathbf{1}_{n-1}&\widetilde{Q}_{n-1}\\\
&&&&&Q_{n-1}&z_{2}\cdot\mathbf{1}_{n}\end{array}\right)\,.$ (89)

As in (74), we introduce the notation

$Q_{i}=q_{i}\cdot\frac{Y}{Z^{i+1}}e_{i}\qquad\tilde{Q}_{i}=\tilde{q}_{i}\cdot\frac{Z^{i}}{Y}e_{i+1}\,,$
(90)

such that $q_{i}$ and $\tilde{q}_{i}$ are constants. We can also have
fluctuations on the diagonal, which we write as

$\Phi_{i}=\varphi_{i}\cdot\frac{Y}{Z^{i+1}}e_{i}\,.$ (91)

With this notation the F-terms at the $i$th node can be written

$\widetilde{q}_{1}q_{1}=0,\quad{\rm and}\quad
q_{i}\widetilde{q}_{i}=\widetilde{q}_{i+1}q_{i+1}\quad i=1,\ldots,n-2\,.$ (92)

We now come back to the reason why (89) is only a schematic form. The $i$-the
hyper $(Q_{i},\widetilde{Q}_{i})$ is only well defined at the
$(e_{i},e_{i+1})$ intersection, but has poles in the nearby patches. Hence,
(89) is not well-defined over the whole target space. This is due to the
projective nature of the resolved K3. Therefore, we must study it patch by
patch. A good local affine coordinate on the locus $e_{i}=0$ for the
hemisphere where $e_{i+1}$ (respectively $e_{i-1}$) does not vanish is
$\frac{Y}{Z^{i}}$ (respectively $\frac{Z^{i}}{Y}$):

$e_{i}=0$$e_{i+1}=0$Coordinate $\frac{Y}{Z^{i}}$Coordinate $\frac{Z^{i+1}}{Y}$
(93)

Note in particular that for $i=N$ (respectively $i=0$), this is compatible
with the affine coordinate on the locus $z_{2}=0$ (resp. $z_{1}=0$) being
simply $X$ (resp. $Y$), as chosen in (77).

In the patch that contains the intersection
$\\{e_{i}=0\\}\cap\\{e_{i+1}=0\\}$, the tachyon fluctuation is expressed as

$\delta T=\begin{pmatrix}\Phi_{i}&\widetilde{Q}_{i}\\\
Q_{i}&\Phi_{i+1}\end{pmatrix}\in\begin{pmatrix}\mathbb{C}^{i\times
i}\cdot\frac{Y}{Z^{i+1}}e_{i}&\mathbb{C}^{i\times(i+1)}\cdot\frac{Z^{i}}{Y}e_{i+1}\\\
\mathbb{C}^{(i+1)\times
i}\cdot\frac{Y}{Z^{i+1}}e_{i}&\mathbb{C}^{(i+1)\times(i+1)}\cdot\frac{Z^{i}}{Y}e_{i+1}\end{pmatrix}$
(94)

Using line and row transformations, one finds

$\displaystyle\begin{pmatrix}\mathbf{1}_{i}&\widetilde{q}_{i}\frac{Z^{i}}{Y}\\\
-q_{i}\frac{Y}{Z^{i+1}}&\mathbf{1}_{i+1}-\frac{q_{i}\widetilde{q}_{i}}{Z}\end{pmatrix}\begin{pmatrix}e_{i}\left(\mathbf{1}_{i}-\frac{\widetilde{q}_{i}q_{i}}{Z}\right)&0\\\
0&e_{i+1}\cdot\mathbf{1}_{i+1}\end{pmatrix}\begin{pmatrix}\mathbf{1}_{i}&-\widetilde{q}_{i}\frac{Z^{i}e_{i+1}}{Ye_{i}}\\\
q_{i}\frac{Ye_{i}}{Z^{i+1}e_{i+1}}&\mathbf{1}_{i+1}-\frac{q_{i}\widetilde{q}_{i}}{Z}\end{pmatrix}$
(95) $\displaystyle=\begin{pmatrix}e_{i}\cdot\mathbf{1}_{i}&0\\\
0&e_{i+1}\left(\mathbf{1}_{i+1}-\frac{q_{i}\widetilde{q}_{i}}{Z}\right)\end{pmatrix}\,.$

Effectively, this transforms a pole of the form

$\begin{pmatrix}e_{i}\cdot\mathbf{1}_{i}+\frac{Y}{Z^{i+1}}e_{i}\varphi_{i}&0\\\
0&e_{i+1}\cdot\mathbf{1}_{i+1}\end{pmatrix}\,,\qquad\qquad\varphi_{i}=\frac{-\widetilde{q}_{i}q_{i}}{(Y/Z^{i})}$
(96)

into a pole of the form

$\begin{pmatrix}e_{i}\cdot\mathbf{1}_{i}&0\\\
0&e_{i+1}\mathbf{1}_{i+1}+\frac{Z^{i}}{Y}e_{i+1}\varphi_{i+1}\end{pmatrix}\,,\qquad\qquad\varphi_{i+1}=\frac{-q_{i}\widetilde{q}_{i}}{(Z^{i+1}/Y)}\,.$
(97)

Let us introduce the $n\times n$ meson matrix

$M=q_{n-1}\widetilde{q}_{n-1}\,.$ (98)

As argued for the $T_{2}$ previously, this meson matrix must be nilpotent. We
can characterize the tachyon fluctuation entirely by the last pole (97) for
$i=n-1$, which gives

$\varphi_{n}=-\frac{M}{X}\,,$ (99)

consistently with what we found for the $n=2$ case in (77). In summary, the
bifundamental matter between the various branes can be subsumed under a shift
of the 7d $\mathrm{SU}(n)$-adjoint Higgs $\varphi_{n}$, as a simple pole with
residue equal to a nilpotent element $M$. Since $M^{n}=0$, we still have

$\det T=\prod_{i=1}^{n}e_{i}^{i}=Y\,.$ (100)

Hence, the brane locus remains unscathed despite the activation of the matter
fields.

###  3.5 Interpretation in terms of Generalized Toric Polygons

We saw in the last subsection that the geometry of the threefold can be
affected by the presence of a pole of the form (99). This pole can be encoded
in two ways: in the nilpotent meson matrix $M$, or in the list of
hypermultiplet deformations $q_{i}$ and $\widetilde{q}_{i}$ satisfying the
relations (92). Let us call $\mathcal{Q}$ the set

$\mathcal{Q}=\\{\mathbf{q}=(q_{1},\widetilde{q}_{1},\dots,q_{n-1},\widetilde{q}_{n-1})\in\mathbb{C}^{2\times
1}\times\dots\mathbb{C}^{(n-1)\times n}|\eqref{eq:Fterms}\\}\,.$ (101)

We also call $\mathcal{N}$ the set of nilpotent $n\times n$ complex matrices.
The map

$\displaystyle m:\mathcal{Q}$ $\displaystyle\rightarrow$
$\displaystyle\mathcal{N}$ (102) $\displaystyle\mathbf{q}$
$\displaystyle\mapsto$ $\displaystyle M=q_{n-1}\widetilde{q}_{n-1}$

is certainly not injective, as given a nilpotent matrix $M$, there are
infinitely many families $\\{q_{i},\widetilde{q}_{i}\\}_{i=1,\dots,n-1}$
mapping to it. Let us define

$\displaystyle r:\mathcal{Q}$ $\displaystyle\rightarrow$
$\displaystyle\mathbb{N}^{n-1}$ (103) $\displaystyle\mathbf{q}$
$\displaystyle\mapsto$
$\displaystyle(r_{i}=\mathrm{rank}(q_{i}\widetilde{q}_{i}))_{i=1,\dots,n-1}\,.$

Figure 8: On top, we depict a part of the resolved GTP for the $T_{9}$ theory,
with white dots on the right edge, along with an internal triangulation
consistent with the white dots. The presence of white dots propagates into the
interior of the GTP, thereby limiting the possible resolutions. Each column in
the GTP corresponds to a boundary condition for D5-branes ending on D7-branes.
This information is encoded in the ranks $r_{i}$ of the matrices appearing in
$\delta T$. On the bottom, we show the associated D5-brane boundary conditions
(where each circle denotes a D7-brane).

For a given $M\in\mathcal{N}$, the ranks of the bilinears in
$q_{i}\widetilde{q}_{i}$ that map to $M$ are not fixed. In other words,
$r(m^{-1}(M))$ contains more than one element. However, there is a unique
element of $\mathbf{r}^{\mathrm{min}}\in r(m^{-1}(M))$ that minimizes the sum
of the entries. If $M\in\mathcal{O}_{\lambda}$, then $\mathbf{r}_{0}$ is given
by the partial sums of the transpose of $\lambda$,

$\mathbf{r}^{\mathrm{min}}=\left(\sum\limits_{j=n+1-i}^{n}\lambda^{T}_{j}\right)_{i=1,\dots,n-1}\,.$
(104)

Note that this coincides with the ranks of the linear quiver

$\mathbf{r}^{\mathrm{min}}_{1}$$\mathbf{r}^{\mathrm{min}}_{2}$$\cdots$$\mathbf{r}^{\mathrm{min}}_{n-1}$$n$
(105)

whose 3d $\mathcal{N}=4$ Coulomb branch is the corresponding nilpotent orbit
closure. The ranks in (104) represent the minimal deformations needed in
$\delta T$ to produce the pole (99). In simple cases, the quivers (105) appear
as embedded in the magnetic quiver describing the Higgs branch of the 5d
theory. For a more precise statement about the embedding, the reader is
referred to vanBeest:2020kou .

In the GTP, these ranks can be encoded with white dots _inside_ the polygon,
using again the notation introduced in Benini:2009gi . For instance, for $n=9$
and partition $\lambda=[4,3,2]$, we can draw the configuration shown in figure
8. The minimal ranks correspond to the number of white dots in each column:
here we get $\mathbf{r}^{\mathrm{min}}=(0,0,0,0,0,1,3,6)$. These numbers give
the brane configuration which saturates the s-rule, as illustrated in the
lower part of figure 8.

$\vdots$$\vdots$$\mathcal{H}$$\mathcal{H}^{\prime}$Pole Figure 9: Schematic
form of a Hasse diagram of the Higgs branch $\mathcal{H}$ viewed as a
symplectic singularity. The effect of the pole prescription corresponds
geometrically to imposing a higher dimensional base leaf (the transverse slice
is shown in red – here it is minimal, but it does not have to be in general).
The remaining Higgs branch $\mathcal{H}^{\prime}$ is the transverse slice.

#### Impact on the Hasse diagram.

The vevs of Higgs branch operators in the 5d theory can be projected on the
space of complex structure deformations of the geometry. The fact that the
pole (99) freezes some of these deformations means geometrically that the
resulting pole-deformed Higgs branch is the slice in the initial Higgs branch
transverse to these imposed deformations. This can be represented as follows.

The Higgs branch $\mathcal{H}$ with no pole is a symplectic singularity, which
can be depicted using its Hasse diagram of singularities. The effect of the
pole is to freeze certain deformations, represented here as a forced choice of
a higher dimensional bottom symplectic leaf. The resulting Higgs branch
$\mathcal{H}^{\prime}$ is the transverse slice to that leaf. See figure 9 for
a schematic depiction.

The analysis above applies to the theory in any phase. In the case of the IIA
reduction on the resolved $A_{n-1}$ singularity shown in figure 6, the Higgs
branch of the $T_{n}$ theory becomes the nilpotent cone of
$\mathfrak{sl}_{n}$. The transverse slices are then identified with the
Slodowy slices. The example $n=4$ is illustrated in figure 10. If
$M\in\mathcal{O}_{\lambda}$ for $\lambda$ some partition of $n$, then the
corresponding Higgs branch is the Slodowy slice
$\mathcal{S}_{\lambda}\cap\mathcal{N}$. Moving on to the SCFT phase, the Higgs
branch is no longer a nilpotent orbit closure, but the general picture stays
the same: the Higgs branch is restricted to a transverse slice. This is what
we already mentioned in the introduction, see figure 5. The Hasse diagrams
drawn in figure 5 can be reproduced independently using the quiver subtraction
algorithm Cabrera:2018ann ; Bourget:2021siw  on the magnetic quivers extracted
from the GTPs.

$0$$3$$4$$5$$6$$a_{3}$$a_{1}$$A_{1}$$A_{3}$$M\in\mathcal{O}_{[1^{4}]}$$M\in\mathcal{O}_{[2,1^{2}]}$$M\in\mathcal{O}_{[2^{2}]}$$M\in\mathcal{O}_{[3,1]}$
Figure 10: Diagram for the nilpotent cone of $\mathfrak{sl}_{4}$. The dots are
nilpotent orbits. When $M$ belongs to a given orbit, the Higgs branch of the
resulting theory is the transverse Slodowy slice, represented by a bracket on
the right.

###  3.6 Hanany-Witten Moves

In the previous section, we defined the IIA counterpart of a ‘white dot’ as a
nilpotent residue for the adjoint complex scalar on the flavor D6-branes. The
nilpotency implies that there will be no repercussions on the geometry of the
branes, and hence, the M-theory CY will not be deformed. This is the hallmark
of a ‘T-brane’ Cecotti:2010bp .

We would now like to determine how the T-brane configuration impacts the
physics of the 5d theory. This is done using the brane-web language, with
Hanany-Witten moves, as we have explained in detail in section 3.3. The key
point is that the nilpotent orbit of $M$ determines which Hanany-Witten move
can be performed in order to detach any of the 7-branes.

In the IIA setup, flavor D6-branes are now allowed to move across exceptional
$\mathbb{P}^{1}$s, thereby changing the quiver structure. The way this is seen
at the level of the Higgs field, is that by activating vevs along the Slodowy
(transverse) slice to the nilpotent vev with poles, the characteristic
polynomial of the tachyon matrix actually becomes deformed.

This can be seen by computing the self-Ext group
$\textrm{Ext}^{1}\left(B_{F}^{(\text{nil})},B_{F}^{(\text{nil})}\right)$ of
the nilpotent configuration on the flavor brane. Using the computation from
section 2, we are looking for $\delta T_{F}$ such that

$\delta T_{F}\sim\delta T_{F}+\frac{z_{2}}{Z}\,[M,g]\,.$ (106)

This is equivalent to requiring that $\delta T_{F}$ be on a transverse slice
to $M$, gauge equivalent to the Slodowy slice. We will choose a gauge such
that $\delta T_{F}$ be the companion matrix to $M$ as in Gaiotto:2008sa ,
which is referred to as a ‘reconstructible Higgs’ in Cecotti:2010bp . The
details on how this is done are given in Appendix A. Say $T_{\rm nil}$ is in
the maximal nilpotent orbit, then the ‘Hanany-Witten’ tachyon will take the
form

$T_{\rm HW}=T_{\rm nil}+\delta T_{F}=\frac{z_{2}}{Z}\begin{pmatrix}Z&1&&&&\\\
&Z&1&&&\\\ &&\ddots&\ddots&\ddots&&\\\ &&&&&Z&1\\\
(-)^{n-1}a_{n}X&(-)^{n-2}a_{n-1}X&&&&-a_{2}X&Z\end{pmatrix}\,,$ (107)

where the $a_{i}$ are constants (the fluctuation $\delta T_{F}$ is
proportional to $X$ as a consequence of (94) with $i=N-1$). This matrix has
determinant

$\det(T_{\rm
HW})=\left(\frac{z_{2}}{Z}\right)^{n}\,\left(Z^{n}+a_{2}XZ^{n-2}+\ldots+a_{n}X\right)\,.$
(108)

More generally, for $M$ in the $[\lambda_{1},\lambda_{2},\ldots,\lambda_{r}]$
partition, we take a block diagonal matrix where each block will take the
form:

$(T_{\rm HW})_{i}=\frac{z_{2}}{Z}\begin{pmatrix}Z&1&&&&\\\ &Z&1&&&\\\
&&\ddots&\ddots&\ddots&&\\\ &&&&&Z&1\\\
(-)^{\lambda_{i}-1}a_{\lambda_{i}}^{(i)}X&(-)^{\lambda_{i}-2}a^{(i)}_{\lambda_{i}-1}X&&&&-a^{(i)}_{2}X&Z\end{pmatrix}\,.$
(109)

Note that by doing so, we are not using the full Slodowy slice
$\mathcal{S}_{\lambda}$ (which contains non-block-diagonal matrices) but
instead restrict to the intersection
$\mathcal{S}_{\lambda}^{0}=\mathcal{S}_{\lambda}\cap\mathfrak{l}_{\lambda}$
with the Levi subalgebra $\mathfrak{l}_{\lambda}$, see Appendix A. This is
physically justified, as it guarantees that the flavor symmetry will not be
further broken by the Hanany-Witten moves than it already has been by the
white dots. The missing parameters, in
$\mathcal{S}_{\lambda}-\mathcal{S}_{\lambda}^{0}$, are associated to the non
splitting of the flavor branes, illustrated on an example below in (117).

Putting together all these blocks, and the non-flavor part of the tachyon
matrix, the end result of the full deformation is

$Y\mapsto\frac{1}{X}\cdot\prod\limits_{i=1}^{r}\left(Z^{\lambda_{i}}+X\left(\sum_{j=0}^{\lambda_{i}-2}a^{(i)}_{\lambda_{i}-j}Z^{j}\right)\right)\,,$
(110)

where $\sum_{i}\lambda_{i}=n$.

Actually it can be argued that only the coefficients $a_{\lambda_{i}}^{(i)}$
affect the physics near the singularity: coefficient $a^{(i)}_{\lambda_{i}-j}$
for $j\neq 0$ correspond to a shift of $\frac{Z^{\lambda_{i}-j}}{X}$, which is
using (93) the coordinate along a $\mathbb{P}^{1}$ with which the brane has
zero intersection. Therefore the equation simplifies to

$Y\mapsto\frac{1}{X}\cdot\prod\limits_{i=1}^{r}\left(Z^{\lambda_{i}}+Xa^{(i)}\right)\,,$
(111)

where we have renamed $a^{(i)}:=a^{(i)}_{\lambda_{i}}$. Reverting to the
original notations $(W_{1},W_{2},W_{3},Z)$ for the coordinates in
$\mathbb{C}^{4}$, see (52), one finally gets

$W_{1}W_{2}W_{3}=\prod\limits_{i=1}^{r}\left(Z^{\lambda_{i}}+a^{(i)}W_{1}\right)\,.$
(112)

#### Examples.

Let us work out a few examples. Equation (110) is worked out explicitly for
$T_{3}$ and $T_{4}$ in table 1. The factorization allows to read off the
corresponding quivers, which are the magnetic quivers for nilpotent orbits of
$\mathfrak{sl}(3)$ and $\mathfrak{sl}(4)$ Hanany:2016gbz , as expected.

Partition | $\det T$ | Factorization
---|---|---
$[1^{3}]$ | $Y$ | ${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}e_{1}e_{2}^{2}}(z_{2}^{3})$
$[2,1]$ | $Y+a_{2}Z$ | ${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}e_{1}e_{2}}(z_{2})(e_{2}z_{2}^{2}+a_{2}z_{1})$
$[3]$ | $Y+a_{2}Z+a_{3}$ | smooth

Partition | $\det T$ | Factorization
---|---|---
$[1^{4}]$ | $Y$ | ${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}e_{1}e_{2}^{2}e_{3}^{3}}(z_{2}^{4})$
$[2,1^{2}]$ | $Y+a_{2}Z^{2}$ | ${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}e_{1}e_{2}^{2}e_{3}^{2}}(e_{3}z_{2}^{2}+a_{2}z_{1}^{2}e_{1})(z_{2}^{2})$
$[2^{2}]$ | $Y+(a_{2}^{(1)}+a_{2}^{(2)})Z^{2}+a_{2}^{(1)}a_{2}^{(2)}X$ | ${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}e_{1}e_{2}^{2}e_{3}}(e_{3}z_{2}^{2}+a_{2}^{(1)}z_{1}^{2}e_{1})(e_{3}z_{2}^{2}+a_{2}^{(2)}z_{1}^{2}e_{1})$
$[3,1]$ | $Y+a_{2}Z^{2}+a_{3}Z$ | ${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}e_{1}e_{2}e_{3}}(e_{2}e_{3}^{2}z_{2}^{3}+a_{2}z_{1}^{2}e_{1}e_{2}e_{3}z_{2}+a_{3}z_{1})(z_{2})$
$[4]$ | $Y+a_{2}Z^{2}+a_{3}Z+a_{4}$ | smooth

Table 1: Equations defining the brane configurations in $T_{3}$ and $T_{4}$
with white dots on one edge after HW moves. In the last column, the equation
is rewritten in terms of the toric variables for the resolution, and maximally
factorized. The factor in orange give the ranks of the low energy quiver while
the terms within brackets give the flavor ranks.

We can illustrate the problem discussed in the previous paragraph with
partition $[2,1]$. The generic matrix in the Slodowy slice would give rise to
the final block for the tachyon matrix given by

$\left(\begin{array}[]{ccc}Z&1&0\\\ -a_{2}X&Z&\alpha X\\\ \beta X&0&Z+\gamma
X\end{array}\right)\,,$ (113)

The resulting, deformed, equation then reads

$\det
T=e_{1}e_{2}{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}{\left(a_{2}\gamma
e_{1}z_{1}^{3}+a_{2}z_{1}z_{2}+\gamma e_{1}e_{2}z_{1}^{2}z_{2}^{2}+\alpha\beta
e_{1}z_{1}^{3}+e_{2}z_{2}^{3}\right)}}\,,$ (114)

from which one reads off a quiver with two abelian nodes and a single flavor
brane _system_ :

$e_{1}=0$$e_{2}=0$$z_{1}=0$$z_{2}=0$ (115)

This system, drawn in teal, intersects both $\mathbb{P}^{1}$ divisors at
$e_{1}=0$ and $e_{2}=0$. This breaks the global isometry $\mathfrak{u}(1)$ of
the nilpotent Slodowy slice $\mathcal{S}_{[2,1]}\cap\mathcal{N}$. By
restricting to the Levi subalgebra, i.e. sending $\alpha$, $\beta$ and
$\gamma$ to zero, one gets one more factorization:

$\det
T=e_{1}e_{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{(z_{2})}}{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}{\left(a_{2}z_{1}+e_{2}z_{2}^{2}\right)}}\,.$
(116) $e_{1}=0$$e_{2}=0$$z_{1}=0$$z_{2}=0$ (117)

The flavor branes are now disjoint and can be moved independently:

$e_{1}=0$$e_{2}=0$$z_{1}=0$$z_{2}=0$ (118)

Each flavor brane intersects a single $\mathbb{P}^{1}$, and the global
symmetry of the resulting Higgs branch (still in the resolved phase) is read
to be $\mathfrak{s}(\mathfrak{u}(1)\oplus\mathfrak{u}(1))=\mathfrak{u}(1)$.

##  4 General Discussion

###  4.1 White Dots along a Single Edge

Having discussed the case of $T_{n}$ in detail, we move to a generic toric
polygon and its GTP deformations. Let $P$ be a convex polygon in
$\mathbb{R}^{2}$ with vertices in $\mathbb{Z}^{2}$. Pick an edge on $P$, and
denote by $n$ its length (i.e. the edge contains exactly $n+1$ points in
$\mathbb{Z}^{2}$). Using an $\mathrm{SL}(2,\mathbb{Z})$ transformation, one
can assume without loss of generality that this edge extends between the
vertices $(0,0)$ and $(0,n)$, and that all vertices of $P$ other than $(0,0)$
and $(0,n)$ have coordinates $(i,j)$ with $i<0$:

$\cdots$$\cdots$$(0,0)$$(0,n)$ (119)

In the toric threefold, the length $n$ edge is translated into the presence of
a dimension 1 singular stratum with transverse slice of type $A_{n-1}$. In the
brane-web picture, this means that $n$ D5-branes extend to infinity, and the
boundary condition, encoded in how they end on D7-branes, can again be encoded
in a T-brane datum. This can be done explicitly using the IIA reduction as in
section 3 for any given polygon (and we do so for the example of a rectangular
$P$ in the next section), even though it would be extremely cumbersome to try
to give general formulas that would apply to any polygon. The general idea,
however, is clear: in the $T_{n}$ example discussed in section 3, the
undeformed equation $W_{1}W_{2}W_{3}=Z^{n}$ yields the $A_{n-1}$ singularity
transverse to the line $W_{2}=W_{3}=Z=0$ by setting $W_{1}$ equal to a
constant, say $W_{1}=c$, giving

$cW_{2}W_{3}=Z^{n}\,.$ (120)

It is then this equation that is modified by $(i)$ adding a nilpotent pole of
the form (99) with $M$ in the prescribed nilpotent orbit, and $(ii)$
performing HW moves to deform the geometry, in effect replacing

$Z^{n}\rightarrow\prod\limits_{i}(Z^{\lambda_{i}}+a^{(i)}W_{1})\,.$ (121)

The general case is obtained by mimicking this procedure: among the equations
for the toric threefold corresponding to the polygon $P$, the line with
transverse singularity $A_{n-1}$ can be parametrized by a coordinate $W_{1}$,
and the transverse slice is again given by (120). This is one of the equations
defining the toric threefold, called a ”boundary equation” in altmann2000one .
This equation should then be replaced by

$cW_{2}W_{3}=\prod\limits_{i}(Z^{\lambda_{i}}+a^{(i)}W_{1})\,.$ (122)

This is in agreement with (altmann2000one, , Theorem 4.8). A useful mnemonic
is that in the toric diagram (41), the deformation of the edge labelled by
$W_{1}$ involves an $A_{n-1}$ equation involving the coordinates labelling the
_adjacent_ edges, here $W_{2}$ and $W_{3}$, with the degree $n$ polynomial in
$Z$ deformed by powers of $W_{1}$. In the following subsections, we give
examples to illustrate the scope of our results, and also show certain
limitations.

#### Successive deformations.

Before discussing the examples, we want to address a natural question: given a
GTP with white dots on an edge of length $n$ characterized by a partition
$\mu$, is it possible to deform it to the same GTP with partition $\lambda$
instead? One necessary condition is that $\lambda>\mu$, and it is also
sufficient if the polygon is large enough.555If the GTP is small, then certain
deformations could lead to violations of the S-rule.  In this case, one can
deform the nilpotent pole according to the general rules spelled out in
section 2.

For instance, if $n=4$, we can use the explicit form (39). Going from
partition $[2,1^{2}]$ to $[2^{2}]$ is straightforward as it just involves
merging two Jordan blocks, while going from $[2^{2}]$ to $[3,1]$ requires
using higher powers of $z$ (recall (37)):

$\left(\begin{array}[]{cccc}z&1&0&0\\\ 0&z&0&0\\\ 0&0&z&0\\\ 0&0&0&z\\\
\end{array}\right)$$\left(\begin{array}[]{cccc}z&1&0&0\\\ 0&z&0&0\\\
0&0&z&\alpha\\\ 0&0&0&z\\\
\end{array}\right)$$\left(\begin{array}[]{cccc}z&1&0&0\\\ 0&z&z\beta&0\\\
0&0&z&\alpha\\\ 0&0&0&z\\\ \end{array}\right)$$\alpha\neq 0$$\beta\neq 0$
(123)

However it is worth mentioning that this can be done on the T-brane _before_
HW deformations are added. Crucially, the two operations do not commute in
general. Here for instance the deformed equations for $T_{4}$ with partition
$[2,2]$ and $[3,1]$, given in table 1, cannot be deformed from one to the
other.

###  4.2 Rectangular Box

Consider a rectangular box of size $n\times m$,

$m$$n$$W_{1}$$W_{3}$$W_{2}$$W_{4}$ (124)

The 5d SCFT this describes admits several well-known low energy gauge theory
deformations. The toric threefold singularity is not a hypersurface, but it is
described by the pair of equations

$W_{1}W_{3}=Z^{m}\,,\qquad W_{2}W_{4}=Z^{n}\,.$ (125)

We can add white dots on the right segment, labeled by $W_{1}$, exactly as in
section 3. The deformed equations are

$W_{1}W_{3}=Z^{m}\,,\qquad
W_{2}W_{4}=\prod\limits_{i}(Z^{\lambda_{i}}+a^{(i)}W_{1})\,.$ (126)

Note that again, the coordinate $W_{1}$ is used to deform the $A_{n-1}$
equation involving the coordinates labeling the adjacent edges $W_{2}$ and
$W_{4}$.

#### Example.

The following example is a useful check of this proposal, as the resulting
model is related to $T_{3}$. Consider the case $m=3$, $n=2$. Placing a white
dot on the length 2 edge yields the equations

$W_{1}W_{3}=Z^{3}\,,\qquad W_{2}W_{4}=Z^{3}+aW_{1}\,.$ (127)

We can now use the second equation to solve for $W_{1}$, and we get the
hypersurface

$W_{2}W_{3}W_{4}=Z^{2}(aZ+W_{3})\,.$ (128)

In the limit where $a\rightarrow\infty$, the D7-brane, in the web
interpretation, has crossed the whole brane system from right to left. We are
left with

$W_{2}W_{3}W_{4}=Z^{3}\,,$ (129)

and we have reproduced the prediction (5)

$\leftrightarrow$ (130)

This is a consistency check on the validity of our proposal.

###  4.3 Generic Triangle

In this subsection we consider more examples which involve threefolds what are
not hypersurfaces in $\mathbb{C}^{4}$. Consider the case where the toric
polygon is a triangle, of arbitrary shape. Pick one edge of length $n$. Then
an $\mathrm{SL}(2,\mathbb{Z})$ transformation can bring the triangle to a
frame where its three vertices are:

$\\{(0,0);(0,n);(-a,b)\\}\,,\qquad a\in\mathbb{Z}_{>0},\,b\in\mathbb{Z}\,.$
(131)

This means the toric fan is generated by the three vectors $(0,0,1)$,
$(0,n,1)$ and $(-a,b,1)$. Among the generators of the dual cone we have

$\displaystyle(-1,0,0)$ $\displaystyle\leftrightarrow$ $\displaystyle W_{1}$
(132)
$\displaystyle\left(\frac{n-b}{d_{2}},-\frac{a}{d_{2}},\frac{an}{d_{2}}\right)$
$\displaystyle\leftrightarrow$ $\displaystyle W_{2}$ (133)
$\displaystyle\left(\frac{b}{d_{1}},\frac{a}{d_{1}},0\right)$
$\displaystyle\leftrightarrow$ $\displaystyle W_{3}$ (134)
$\displaystyle(0,0,1)$ $\displaystyle\leftrightarrow$ $\displaystyle Z\,.$
(135)

In order to write the equation of the threefold, we have introduced
$d_{1}=\mathrm{gcd}(a,b)$, $d_{2}=\mathrm{gcd}(a,n-b)$ and
$d_{3}=\mathrm{gcd}(a,b,n-b)$. One of the equations for the toric threefold is

$W_{1}^{n/d_{3}}W_{2}^{d_{2}/d_{3}}W_{3}^{d_{1}/d_{3}}=Z^{an/d_{3}}\,,$ (136)

but in general there are other equations, as there are other generators in the
dual cone. In the case of $T_{n}$ (41), equation (136) is sufficient and
reduces to (46), but this is a special case. To illustrate this phenomenon, we
consider an example.

#### Example.

We take the case $a=2$ and $b=0$.

$2$$2n$$W_{1}$$W_{2}$$W_{3}$ (137)

The dual cone is generated by 5 vectors:666There is an algorithmic way to see
that the fifth vector $(1,0,2)$ is needed, an no other. This is the
computation of the Hilbert basis of the semi-group $\sigma^{\vee}\cap M$,
using standard notation in toric geometry. This Hilbert basis is unique, and
in the present case it has 5 elements, given here.

$\displaystyle(-1,0,0)$ $\displaystyle\leftrightarrow$ $\displaystyle W_{1}$
(138) $\displaystyle(n,-1,2n)$ $\displaystyle\leftrightarrow$ $\displaystyle
W_{2}$ (139) $\displaystyle(0,1,0)$ $\displaystyle\leftrightarrow$
$\displaystyle W_{3}$ (140) $\displaystyle(0,0,1)$
$\displaystyle\leftrightarrow$ $\displaystyle Z$ (141) $\displaystyle(1,0,2)$
$\displaystyle\leftrightarrow$ $\displaystyle T$ (142)

and the threefold is described by two equations in $\mathbb{C}^{5}$:

$W_{1}^{n}W_{2}W_{3}=Z^{2n}\,,\qquad W_{1}T=Z^{2}\,.$ (143)

The first equation corresponds to (136). The singular locus associated to the
length $2n$ edge is $Z=W_{2}=W_{3}=T=0$, parametrized by $W_{1}$. Setting as
before $W_{1}=c$, we can eliminate $T$ and we find as expected an equation
$c^{n}W_{2}W_{3}=Z^{2n}$ that we can deform. Therefore, the system of
equations for the triangle (137) with one white dot on the long edge according
to our conjecture is

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}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{14.22638pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
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\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\qquad\leftrightarrow\qquad\begin{cases}W_{1}^{n}W_{2}W_{3}=Z^{2n-2}(Z^{2}+aW_{1})\\\
W_{1}T=Z^{2}\end{cases}$ (144)

##  5 Testing the Proposal: Resolution of Singularities

To test the proposal for the deformation of singularities, we compute the
resolutions for the deformed singularities. For the starting point, which we
take to be a toric variety, the resolution is easily obtained in a
combinatorial fashion, by a complete triangulation of the toric polygon.
However, no such computational simplification exists yet for the resolution of
GTPs. In view of this, we therefore need to revert to resolving the actual
algebraic varieties. We will carry this out for the $T_{n}$ theories.

###  5.1 Crepant Resolution of $T_{n}$

The simplest type of singularities are the hypersurfaces that realize the
$T_{n}$ theories. The flavor symmetry algebra is $\mathfrak{su}(n)^{3}$ except
for $n=3$, where we expect $\mathfrak{e}_{6}$. This can be easily detected
from the toric geometry by extracting the set of curves that are complete
intersections between compact and non-compact (i.e. flavor) divisors. The
resulting so-called combined fiber diagram (CFD) Apruzzi:2019vpe ;
Apruzzi:2019opn ; Apruzzi:2019enx  is easily read off from the toric polygon
Eckhard:2020jyr .

Here we will take the more laborious path of resolving the hypersurface
singularity, in preparation for resolving the deformed singularities. The
hypersurface in $\mathbb{C}^{4}$ is

$W_{1}W_{2}W_{3}=Z^{n}\,.$ (145)

The first resolution for all of these singularities is simply to remove the
locus $W_{1}=W_{2}=W_{3}=Z=0$, which is achieved by inserting a
$\mathbb{P}^{3}$ with projective coordinates $[W_{1},W_{2},W_{3},Z]$ (for a
detailed exposition of resolutions from a physicist’s perspective, see
Lawrie:2012gg ). Denoting the exceptional section of the blowup by
$\delta_{1}$ the equation, after proper transform, which ensures that the
resolution is crepant, becomes

$W_{1}W_{2}W_{3}=Z^{n}\delta_{1}^{n-3}\,.$ (146)

This can be further resolved by consecutively blowing up the loci
$W_{1}=W_{2}=W_{3}=\delta_{i}=0$ i.e. by inserting another projective space
with coordinates

$[W_{1},W_{2},W_{3},\delta_{i}]\,,\qquad
i=1,\cdots,\left\lfloor{n-3i}\right\rfloor\,,$ (147)

which implies that the coordinates cannot vanish at the same time (and thus
the above is a relation in the Stanley-Reissner ideal of the hypersurface).
This is iterated until we reach

$W_{1}W_{2}W_{3}=Z^{n}\prod_{i=1}^{\lfloor n-3i\rfloor}\delta_{i}^{n-3i}\,.$
(148)

The remaining blowups will resolve the local singularities along $W_{1}=0$,
$W_{2}=0$ and $W_{3}=0$ respectively, by small resolutions, i.e.

$[W_{i},Z]\,,\quad\text{or}\quad[W_{i},\delta_{i}]\,.$ (149)

Let us denote the compact divisors by $S_{i}$ and the non-compact divisors by
$D_{a}$. We then find from these resolutions the intersection matrix

$\mathcal{G}_{a}=\sum_{i}S_{i}\cdot D_{a}\cdot D_{a}$ (150)

to be precisely the adjacency matrix of the CFD for $T_{n}$: i.e. three
$\mathfrak{su}(n)$ Cartan matrices, pairwise connected by $-1$ curves. We have
shown examples in figures 11 and 12.

$T_{3}$$T_{3}$ with deformation $[1,2]$ Figure 11: CFDs: The intersection
graph for $T_{3}$ (left hand side) and the deformation of $T_{3}$ described by
the partition $[1,2]$ (right). 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 $\mathfrak{su}(3)^{3}$ flavor symmetry and the matter in the $({\bf
3},{\bf 3},{\bf 1})$ etc, which enhances to $\mathfrak{e}_{6}$. On the right
the theory is rank 0.

###  5.2 Resolution of Deformations of $T_{n}$

#### Deformations of $T_{3}$.

By including the deformations, some of the above resolutions get obstructed.
Lets consider the simplest case of $T_{3}$, which itself is the rank 1 Seiberg
theory with flavor symmetry $\mathfrak{e}_{6}$. After a single blowup
$[W_{1},W_{2},W_{3},Z]$ we get one compact divisor $\delta_{1}=0$, and three
$A_{2}$ singularities. Resolving yields the CFD shown in figure 11 on the left
hand side. The representations are $({\bf 3},{\bf\overline{3}},{\bf 1})$, and
cyclic permutations, and thus we get the known enhancement to
$\mathfrak{e}_{6}$ 777From the geometry we are also able to extract the flavor
symmetry group Apruzzi:2021vcu ; Bhardwaj:2023zix , which however will not
play a role in the present paper.,

$({\bf 3},{\bf\overline{3}},{\bf 1})\oplus({\bf 1},{\bf
3},{\bf\overline{3}})\oplus({\bf\overline{3}},{\bf 1},{\bf
3})\longrightarrow\quad{\bf 27}\,.$ (151)

Adding the deformation results in $W_{1}W_{2}W_{3}=Z(Z^{2}+W_{1})$, which does
not allow for a big resolution. There are small resolutions, along e.g.
$W_{1}=Z=0$, indicating that this is a rank 0 theory.

#### Deformations of $T_{4}$.

More interestingly we can start with $T_{4}$. All deformations involving a
single edge and yielding a positive rank theory are shown in table 2. The
deformation that is associated to the partition $[2,1^{2}]$ along one edge is
the hypersurface

$W_{1}W_{2}W_{3}=Z^{2}(Z^{2}+W_{1})\,.$ (152)

The first resolution (147) results in

$W_{1}W_{2}W_{3}=\delta_{2}Z^{4}+Z^{2}W_{1}\,.$ (153)

Continuing the blowup results in the intersection matrix between the sum of
compact divisors $\sum_{i}S_{i}$ and each of the non-compact ones

$\left(\sum_{i}S_{i}\right)\cdot D_{a}\cdot
D_{b}=\left(\setcounter{MaxMatrixCols}{12}\begin{smallmatrix}-1&1&0&0&0&0&0&0&1&0&0&0\\\
1&-1&0&0&0&0&0&0&0&0&0&1\\\ 0&0&-2&1&0&0&0&0&0&1&0&0\\\
0&0&1&-1&1&0&0&0&0&0&0&0\\\ 0&0&0&1&-2&0&0&0&1&0&0&0\\\
0&0&0&0&0&-2&0&1&1&0&0&0\\\ 0&0&0&0&0&0&-2&1&0&0&0&1\\\
0&0&0&0&0&1&1&-1&0&0&0&0\\\ 1&0&0&0&1&1&0&0&-2&0&0&0\\\
0&0&1&0&0&0&0&0&0&-1&1&0\\\ 0&0&0&0&0&0&0&0&0&1&-2&1\\\
0&1&0&0&0&0&1&0&0&0&1&-2\\\ \end{smallmatrix}\right)_{ab}\,.$ (154)

This is shown in figure 12, from which the manifest flavor symmetry algebra is
read off from the collection of $-2$ curves

$\mathfrak{su}(4)^{2}\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)\,.$ (155)

The additional $\mathfrak{u}(1)$ follows from the general rule on flavor
symmetry algebras from CFDs as laid out in Apruzzi:2019kgb . The matter is in
the representations, which combine naturally into the representations of the
enhanced symmetry algebra as follows

$({\bf 4},{\bf\overline{4}},{\bf 1})\oplus({\bf 6},{\bf 1},{\bf 1})\oplus({\bf
1},{\bf 6},{\bf 1})\oplus({\bf\overline{4}},{\bf 1},{\bf 2})\oplus({\bf
1},{\bf 4},{\bf 2})\quad\longrightarrow\quad({\bf 28},{\bf 1})\oplus({\bf
8},{\bf 2})\,,$ (156)

where we identify the enhanced flavor symmetry algebra as

$\mathfrak{g}_{UV}=\mathfrak{su}(8)\oplus\mathfrak{su}(2)\,.$ (157)

Partitions | Rank | Free | Symmetry | Equation | Fig.
---|---|---|---|---|---
$[1^{4}]$ | 3 | 0 | $\mathfrak{su}(4)^{3}$ | $W_{1}W_{2}W_{3}=Z^{4}$ | 12
$[2,1^{2}]$ | 2 | 0 | $\mathfrak{su}(8)\oplus\mathfrak{su}(2)$ | $W_{1}W_{2}W_{3}=Z^{2}(Z^{2}+W_{1})$ | 12
$[2^{2}]$ | 1 | 0 | $\mathfrak{e}_{7}$ | $W_{1}W_{2}W_{3}=(Z^{2}+aW_{1})(Z^{2}+bW_{1})$ | 12
Table 2: The partition that defines the distribution of white dots in the GTP,
the rank of the 5d SCFT, the flavor symmetry algebra, as well as hypersurface
equation for the deformations of $T_{4}$.

Similarly for the partition $[2^{2}]$, the manifest symmetry is