# Hypertoric 2-categories $\mathcal{O}$ and symplectic duality

Benjamin Gammage and Justin Hilburn

###### Abstract.

We define 2-categories of microlocal perverse (resp. coherent) sheaves of
categories on the skeleton of a hypertoric variety and show that the
generators of these 2-categories lift the projectives (resp. simples) in
hypertoric category $\mathcal{O}.$ We then establish equivalences of
2-categories categorifying the Koszul duality between Gale dual hypertoric
categories $\mathcal{O}.$ These constructions give a prototype for
understanding symplectic duality via the fully extended 3d mirror symmetry
conjecture.

###### Contents

  1. 0 Introduction
    1. 0.1 Overview
    2. 0.2 Dual toric stacks
    3. 0.3 Hypertoric varieties and Lagrangian skeleta
    4. 0.4 Category $\mathcal{O}$ and decategorification
    5. 0.5 Predictions for Fueter theory
    6. 0.6 Notation and conventions
  2. 1 Hyperplanes and hyperkähler manifolds
    1. 1.1 Hyperplane arrangements
    2. 1.2 Gale duality
    3. 1.3 Toric hyperkähler manifolds
    4. 1.4 Toric holomorphic Lagrangians
  3. 2 Hypertoric categories $\mathcal{O}$
    1. 2.1 de Rham category $\mathcal{O}$
    2. 2.2 Betti category $\mathcal{O}$ and Riemann-Hilbert
  4. 3 Microlocal perverse schobers
    1. 3.1 Spherical functors and perverse schobers
    2. 3.2 2-category $\mathcal{O}$
  5. 4 Microlocal coherent schobers
    1. 4.1 Coherent sheaves of categories
    2. 4.2 Singular support
    3. 4.3 Monoidal structure
    4. 4.4 Microlocalization
    5. 4.5 The hypertoric 2-category
  6. 5 Proof of Theorems A & B
    1. 5.1 Notes from Justin
  7. 6 Justin
  8. 7 Category $\mathcal{O}$ stuff

##  0\. Introduction

This paper inaugurates a program to understand features of categories
$\mathcal{O}$ in terms of a pair of categorifications arising from symplectic
and algebraic geometry, realizing predictions from 3-dimensional
supersymmetric gauge theory.

###  0.1. Overview

The classical category $\mathcal{O}$ was introduced in [BGG] to study the
representation theory of a semisimple Lie algebra $\mathfrak{g}.$ It addresses
the difficulty that the category ${\sf Rep}(\mathfrak{g})$ of all
$\mathfrak{g}$-representations is too large to be manageable, whereas the
category ${\sf Rep}^{\sf f.d.}(\mathfrak{g})$ of finite-dimensional
representations is too constrained to be useful. The BGG category
$\mathcal{O}$ contains the finite-dimensional $\mathfrak{g}$-representations
and has enough projective objects but is still a finite-length abelian
category.

One of the many surprising properties of category $\mathcal{O}$ is the
phenomenon of Koszul duality. For each category $\mathcal{O}$ there is a dual
category $\mathcal{O}^{\vee}$ (which in this case is a category $\mathcal{O}$
for the Langlands dual Lie algebra $\mathfrak{g}^{\vee}$), such that the
endomorphism algebra of the projectives in $\mathcal{O}$ is equivalent to the
Ext algebra of the simples in $\mathcal{O}^{\vee}$. Moreover, within each
category $\mathcal{O},$ the (derived) endomorphism algebras of projectives and
simples are Koszul dual to each other. These relationships are summarized in
the following diagram, where the horizontal arrows are equivalences (with
gradings understood appropriately) and the vertical arrows are Koszul
dualities:

(0.1.1)
${\mathsf{end}_{\mathcal{O}}(P)}$${\mathsf{end}_{\mathcal{O}^{\vee}}(L^{\vee})}$${\mathsf{end}_{\mathcal{O}^{\vee}}(P^{\vee})}$${\mathsf{end}_{\mathcal{O}}(L).}$$\scriptstyle{\sim}$$\scriptstyle{\mathsf{KD}}$$\scriptstyle{\mathsf{KD}}$$\scriptstyle{\sim}$

Category $\mathcal{O}$ may be understood geometrically as a category of
perverse sheaves or regular D-modules on the Schubert-stratified flag variety.
In [BPW], the cotangent bundle to the flag variety (where the conormal to the
Schubert stratification lives) was reconceptualized as but one (quite
distinguished) example in the broader class of conical symplectic resolutions
(CSRs), including ADE surface singularities, Nakajima quiver varieties, and
slices in the affine Grassmannian. In [BLPW16] it was proposed that a category
$\mathcal{O}$, in analogy with the construction of [BGG], be associated to
each of these spaces. More specifically, a category $\mathcal{O}$ is
associated not to a CSR $X$, but to a sectorial CSR $(X\supset\mathbb{L}),$
containing the data of $X$ together with a Lagrangian subspace $\mathbb{L}$
determined as the stable set for a Hamiltonian vector field. The Koszul
duality of 0.1.1 now manifests as a relationship, called symplectic duality in
[BLPW16], relating invariants of seemingly unrelated pairs
$(X\supset\mathbb{L})$ and $(X^{\vee}\supset\mathbb{L}^{\vee}).$

Subsequently, symplectic duality was reframed as a phenomenon in
supersymmetric gauge theory. This process began with an observation of Gukov
and Witten: given a massive 3-dimensional $\mathcal{N}=4$ supersymmetric
quantum field theory, the Higgs and Coulomb branches of its moduli space of
vacua form a dual pair of sectorial CSRs. From this perspective, symplectic
duality is a consequence of 3d mirror symmetry, an involution on the space of
3d $\mathcal{N}=4$ theories first observed in [IS96]. As in the more familiar
2d mirror symmetry, there is a non-obvious correspondence between the
invariants of dual theories; in particular, the Higgs branch of one theory is
exchanged with the Coulomb branch of its dual.

The physical meaning of category $\mathcal{O}$ was clarified in [BDGH]: Each
3d $\mathcal{N}=4$ theory determines a pair of 3-dimensional topological field
theories (TFTs) — the 3d A and B-models111These theories are also known as 3d
Donaldson-Seiberg-Witten theory and Rozansky-Witten theory, respectively. —
which are exchanged by 3d mirror symmetry. After dimensionally reducing along
a circle in the $\Omega$-background,222This may be understood mathematically
as a periodic cyclic, i.e., Tate $S^{1}$-invariant, trace decategorification.
the 3d A and B-models become equivalent. The category of boundary conditions
for the resulting 2-dimensional TFT can be identified with the
$\mathbb{Z}/2$-graded category $\mathcal{O}$ for the Higgs branch — or, by 3d
mirror symmetry, for the Coulomb branch.333Other 2d reductions are discussed
in [BDGH]*Section 7, including one giving the graded lift of category
$\mathcal{O}$.

The process of dimensional reduction is lossy, and we propose instead to study
the 3d A- and B-models directly, in terms of their 2-categories of boundary
conditions. The basic objects in each of these 2-categories are given by
holomorphic Lagrangians, but otherwise the two 2-categories have a very
different flavor. Detailed expectations for the B-model 2-category have been
given in [KRS, KR, OR] and subsequently developed in many places; for our
purposes, the most important are [BZNP, Stefanich-QCoh, diFiore, Arinkin-
notes]. On the other hand, the A-model 2-category is less well-understood. It
should be a “Fukaya-type” 2-category, defined by counting $J$-holomorphic
2-disks and Fueter 3-disks, with Hom categories as studied in [Doan-
Rezchikov], but serious analytic and conceptual difficulties have obstructed a
full definition.

Inspired by similar theorems [NZ, GPS3] for Fukaya categories, we have
conjectured in [GMH] that the A-model 2-category is equivalent to one defined
topologically via the perverse schobers of [KS-schobers]. With 2-categories of
boundary conditions for both the A- and B-models in hand, and guided by
[BDGH], we are now able to state our prediction for the 2-categorical origins
of category $\mathcal{O}:$

######  Conjecture.

To a sectorial conical symplectic resolution $(X\supset\mathbb{L})$, there is
associated a pair of 2-categories

$2\mathcal{O}_{\mathsf{A}}:=\mu\mathsf{PervCat}(\mathbb{L}),\qquad
2\mathcal{O}_{\mathsf{B}}:=\mu\mathsf{CohCat}(\mathbb{L})$

of microlocal perverse (resp. coherent) schobers on $\mathbb{L},$ with
canonical generators $\mathcal{P}$ and $\mathcal{S},$ respectively. After
taking periodic cylic homology, these become the projectives and simples in a
$\mathbb{Z}/2$-graded version of category $\mathcal{O}$:

${\sf
HP}(\mathsf{end}_{2\mathcal{O}_{\mathsf{A}}}(\mathcal{P}))\simeq\mathsf{end}_{\mathcal{O}_{\mathbb{Z}/2}}(P),\qquad{\sf
HP}(\mathsf{end}_{2\mathcal{O}_{\mathsf{B}}}(\mathcal{S}))\simeq\mathsf{end}_{\mathcal{O}_{\mathbb{Z}/2}}(L).\qquad$

The 2-categories $2\mathcal{O}_{(\mathsf{A}/\mathsf{B})}$ are much richer than
the usual 1-category $\mathcal{O}$, which sees only the geometry of the
Lagrangian $\mathbb{L}.$ For instance, the $\mathbb{E}_{2}$ (i.e., Drinfeld)
center of $2\mathcal{O}_{\mathsf{B}}$ is equivalent to the whole category
$\mathsf{Coh}(X)$ of coherent sheaves on $X$; dually, the $\mathbb{E}_{3}$
center of $2\mathcal{O}_{\mathsf{A}}$ recovers the Coulomb branch construction
from [BFN1].

In terms of the above 2-categories, we can now interpret symplectic duality as
one consequence of a more fundamental statement about boundary conditions in
dual theories.

######  Conjecture (Fully extended 3d mirror symmetry).

Let $(X\supset\mathbb{L})$ and $(X^{\vee}\supset\mathbb{L}^{\vee})$ be a dual
pair of sectorial conical symplectic resolutions. Then there are equivalences
of 2-categories

$2\mathcal{O}_{\mathsf{A}}\simeq 2\mathcal{O}_{\mathsf{B}}^{\vee},\qquad
2\mathcal{O}_{\mathsf{B}}\simeq 2\mathcal{O}_{\mathsf{A}}^{\vee}$

exchanging the canonical generators.

As a result, application of periodic cylic homology to the diagram

(0.1.2)
${\mathsf{end}_{2\mathcal{O}_{\mathsf{A}}}(\mathcal{P})}$${\mathsf{end}_{2\mathcal{O}_{\mathsf{B}}^{\vee}}(\mathcal{S}^{\vee})}$${\mathsf{end}_{2\mathcal{O}_{\mathsf{A}}^{\vee}}(\mathcal{P})}$${\mathsf{end}_{2\mathcal{O}_{\mathsf{B}}}(\mathcal{S}).}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

recovers the horizontal equivalences in 0.1.1. These vertical Koszul dualities
thus appear only after decategorifying from $2\mathcal{O}$ to 1-category
$\mathcal{O}.$

In this paper, we prove the above conjectures in the case where $X,X^{\vee}$
are hypertoric varieties.

###  0.2. Dual toric stacks

Our story begins with the 3d mirror symmetry theorem formulated and proved in
[GMH]:

######  Theorem 0.1.

Let $\mathcal{S}$ be the stratification of $\mathbb{C}$ as
$\mathbb{C}=\mathbb{C}^{\times}\sqcup 0.$ Then there is an equivalence of
stable 2-categories444To simplify notation here, we index our 2-categories
here by the stratification $\mathcal{S}$ rather than the union of Lagrangian
conormals $\mathbb{L}$ to $\mathcal{S},$ as we do elsewhere in the paper. For
clarity, we also write “sheaves of categories” here for what elsewhere is
often denoted by the more economical word “schobers.” More broadly, our
notation throughout this paper differs from that in [GMH]: We write
$\mathsf{PervCat}$ (resp. $\mathsf{CohCat}$) for the 2-categories which were
written there as $\mathsf{Perv}^{(2)}$ (resp. $\mathsf{IndCoh}^{(2)}$).

(0.2.1)
$\mathsf{PervCat}_{\mathcal{S}}(\mathbb{C})\simeq\mathsf{CohCat}^{\mathbb{C}^{\times}}_{\mathcal{S}}(\mathbb{C})$

between the 2-category of “perverse sheaves of categories” [KS-schobers] on
$\mathbb{C}$ with a singularity at 0, and the 2-category of
$\mathbb{C}^{\times}$-equivariant “coherent sheaves of categories” [Ari-talk]
on $\mathbb{C}$ with a singularity at $0.$

More generally, let

(0.2.2) ${1}$${G}$${(\mathbb{C}^{\times})^{n}}$${F}$${1}$

be an exact sequence of tori, and let $\mathcal{S}$ denote the stratification
of $\mathbb{C}^{n}$ by coordinate hyperplanes. Then there is an equivalence of
stable 2-categories

(0.2.3)
$\mathsf{PervCat}^{G}_{\mathcal{S}}(\mathbb{C}^{n})\simeq\mathsf{CohCat}^{F^{\vee}}_{\mathcal{S}}(\mathbb{C}^{n}).$

The stratifications involved in the statements of Theorem 0.1 should be
understood as singular-support conditions inside the cotangent bundle of
$\mathrm{T}^{*}\mathbb{C}^{n}$ — or, more properly, inside the stacky
cotangent bundles $\mathrm{T}^{*}(\mathbb{C}^{n}/G)$ and
$\mathrm{T}^{*}(\mathbb{C}^{n}/F^{\vee}).$ A first goal of this paper is to
describe the equivalence 0.2.3 microlocally, so that we may restrict to stable
loci inside the stacky cotangent bundles.

###  0.3. Hypertoric varieties and Lagrangian skeleta

Throughout the paper, we fix the choice of exact sequence 0.2.2. The stack
$\mathrm{T}^{*}(\mathbb{C}^{n}/G)$ can also be written as a stacky Hamiltonian
reduction,

(0.3.1) $\mathrm{T}^{*}(\mathbb{C}^{n}/G)=\mu^{-1}(0)/G,$

where $\mu:\mathrm{T}^{*}\mathbb{C}^{n}\to\mathfrak{g}^{\vee}$ is the moment
map for the Hamiltonian $G$-action on $\mathrm{T}^{*}\mathbb{C}^{n}.$ We can
pass from the stack 0.3.1 to a smooth variety by GIT. Let
$t\in\mathfrak{g}^{\vee}_{\mathbb{Z}}$ be a character for the group $G$. The
character $t$ specifies a $G$-equivariant line bundle $\mathcal{O}(t)$ on
$\mathrm{T}^{*}(\mathbb{C}^{n}/G)$ and hence on $\mu^{-1}(0),$ which allows us
to perform a GIT quotient

(0.3.2) $\mathfrak{M}_{G}(t):=\mu^{-1}(0)/\\!/_{t}G=\mu^{-1}(0)^{t-ss}/G.$

######  Example 0.2.

We highlight two special cases of the above construction:

  1. (1)

For $G=\mathbb{C}^{\times}$ embedded diagonally in
$(\mathbb{C}^{\times})^{n},$ at generic parameter $t$ the variety
$\mathfrak{M}_{G}(t)$ is $\mathrm{T}^{*}\mathbb{P}^{n-1}.$

  2. (2)

If $G$ is the kernel of the multiplication map
$(\mathbb{C}^{\times})^{n}\to\mathbb{C}^{\times},$ then at generic parameter
$t,$ the the variety $\mathfrak{M}_{G}(t)$ is a resolution of the $A_{n-1}$
surface singularity $\mathbb{C}^{2}/\mathbb{Z}/(n-1).$

The spaces $\mathfrak{M}_{G}(t)$ were first introduced in [Goto], where they
were called toric hyperkähler manifolds, and studied further in [Konno] and in
[Biel-Dan, Haus-Sturm]; in [Harada-Proudfoot] they were called hypertoric
varieties, to avoid confusion with toric varieties. These spaces are the
simplest examples of symplectic resolutions, and often serve as a general
testing ground for aspects of the general theory, including categories
$\mathcal{O}$ [BLPW12] and Koszul duality [BLPW10], the quantum differential
equation [McBreen-Shenfeld], and K-theoretic quasimap counts [Smirnov-Zhou].
Following the program begun in [GMH], we seek to derive as much of this theory
as possible from an analysis of 2-categories. In this paper, we will recover
category $\mathcal{O},$ and the statement of Koszul duality for dual
categories $\mathcal{O}.$

In order to describe category $\mathcal{O}$ for these varieties, it will be
helpful to reframe the discussion in terms of a certain Lagrangian subspace.

######  Definition 0.3.

Let $\mathbb{L}\subset\mathrm{T}^{*}\mathbb{C}^{n}$ be the singular Lagrangian
given by the union of conormals to intersections of coordinate hyperplanes,
and let $\mathbb{L}_{G}$ be its image in $\mathrm{T}^{*}(\mathbb{C}^{n}/G).$

The Lagrangian $\mathbb{L}_{G}$ may be understood as a union of conormals to
stacky hyperplane intersections in the stack $\mathbb{C}^{n}/G$;
alternatively, one may note that $\mathbb{L}$ is contained in
$\mu^{-1}(0)\subset\mathrm{T}^{*}\mathbb{C}^{n},$ so that we can treat
$\mathbb{L}_{G}$ as its projection to $\mu^{-1}(0)/G.$ As a singular-support
condition, the conic Lagrangian $\mathbb{L}_{G}$ imposes the condition of
local constancy along toric strata of $\mathbb{C}^{n}/G$; we may therefore
rewrite 0.2.3 as the equivalence

(0.3.3)
$\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)\simeq\mathsf{CohCat}_{\mathbb{L}_{F^{\vee}}}(\mathbb{C}^{n}/F^{\vee})$

between perverse (resp. coherent) sheaves of categories with singular support
along the Lagrangian $\mathbb{L}_{G}$ (resp. $\mathbb{L}_{F^{\vee}}$).

Our goal in this paper is to move from the Lagrangian $\mathbb{L}_{G},$ inside
the stack $\mathrm{T}^{*}(\mathbb{C}^{n}/G),$ to a Lagrangian contained in the
stable locus $\mathfrak{M}_{G}(t).$ To this end, we will introduce a family of
Lagrangians $\mathbb{L}_{G}(t,m),$ depending on a pair of parameters. The
first of these is the stability condition
$t\in\mathfrak{g}^{\vee}_{\mathbb{Z}},$ and the latter is a cocharacter
$m\in\mathfrak{f}_{\mathbb{Z}}$, which determines an inclusion
$m:\mathbb{C}^{\times}\hookrightarrow F$ and therefore a Hamiltonian action of
$\mathbb{C}^{\times}$ on the symplectic stack
$\mathrm{T}^{*}(\mathbb{C}^{n}/G)$.

######  Definition 0.4.

For $(t,m)\in\mathfrak{g}^{\vee}_{\mathbb{Z}}\times\mathfrak{f}_{\mathbb{Z}},$
we write

$\mathbb{L}(t,m):=\\{x\in\mathbb{L}_{G}^{t-ss}\mid\lim_{\lambda\to\infty}m(\lambda)\cdot
x\text{ exists}\\}$

for the subspace of $\mathbb{L}_{G}$ consisting of those points which are
$t$-semistable and $m$-bounded.

When $t,m=0,$ we recover our original Lagrangian
$\mathbb{L}_{G}(0,0)=\mathbb{L}_{G}.$ In general, the Lagrangian
$\mathbb{L}_{G}(t,m)$ is a locally closed subset of $\mathbb{L}_{G}$: turning
on $t$ deletes the closed subset of $t$-unstable points, while turning on $m$
deletes the open subset of $m$-unbounded points.

######  Remark 0.5.

In the physics literature, $t$ is called the Fayet-Iliopolous (FI) parameter
and $m$ the mass parameter. We prefer the names stability parameter and
attraction parameter.

Observe that the parameters $t$ and $m$ for $\mathbb{L}_{G}$ play the opposite
roles for $\mathbb{L}_{F^{\vee}}.$ This is part of a general paradigm in
mirror symmetry that parameters exchange roles under mirror duality (which in
this case is implemented by dualizing the exact sequence 0.2.2). In order to
lift this duality to the level of 2-categories, we will have to understand how
the 2-categories in 0.3.3 change when we replace $\mathbb{L}_{G}$ (resp.
$\mathbb{L}_{F^{\vee}}$) with $\mathbb{L}_{G}(t,m)$ (resp.
$\mathbb{L}_{F^{\vee}}(m,t)$). In other words, we need a good theory of
microlocalization for perverse (resp. coherent) sheaves of categories.

On the A-side, we lack even a general non-microlocal theory for perverse
sheaves of categories, so a full theory of microlocalization is out of reach.
Nevertheless, inspired by the microlocal behavior of the 1-category of
perverse sheaves, we can define directly our expectation for microlocal
perverse sheaves of categories on the Lagrangian $\mathbb{L}_{G}$. The key
fact is categorical Kirwan surjectivity, which has been established for DQ-
modules in [BLPW16] and will be proven, in the hypertoric setting, for
microlocal perverse sheaves in [CGH]. This theorem implies that the category
of microlocal perverse sheaves on $\mathbb{L}_{G}(t,m)$ admits a description
purely in terms of (non-microlocal) perverse sheaves.

We therefore imitate the construction of microlocal perverse sheaves on
$\mathbb{L}_{G},$ which we recall in § 2.2. For each component
$\mathbb{L}_{G}^{\alpha}$ of $\mathbb{L}_{G},$ we can define a projective
object $\mathcal{P}_{\alpha}$ in the 2-category
$\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)$ which corepresents the
functor taking a perverse sheaf of categories to its microstalk at a smooth
point of $\mathbb{L}_{G}^{\alpha}.$ In addition, if
$\mathbb{L}^{\prime}_{G}\subset\mathbb{L}_{G}$ is the closed embedding of a
union of the components of $\mathbb{L}_{G},$ we have an embedding
$\mathsf{PervCat}_{\mathbb{L}^{\prime}_{G}}(\mathbb{C}^{n}/G)\hookrightarrow\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G).$

######  Definition 0.6 (Definition 3.12).

The 2-category $\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))$ of microlocal
perverse sheaves of categories on the Lagrangian $\mathbb{L}_{G}(t,m)$ is
defined as the quotient

(0.3.4)
$\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m)):=\frac{\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)}{\mathsf{PervCat}_{\mathbb{L}_{G}^{t\text{-unstab}}}(\mathbb{C}^{n}/G),\langle\mathcal{P}_{\alpha}\rangle_{\alpha\in
m\text{-unbdd}}}$

of $\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)$ by the sub-2-category
generated by the perverse sheaves of categories with $t$-unstable microsupport
and the projective objects $\mathcal{P}_{\alpha}$ corresponding to
$m$-unbounded components $\mathbb{L}_{G}^{\alpha}$ of $\mathbb{L}_{G}.$

######  Remark 0.7.

The 2-category $\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))$ defined in 0.3.4
depends not only on the Lagrangian $\mathbb{L}_{G}(t,m)$ but also on the
ambient cotangent stack $\mathrm{T}^{*}(\mathbb{C}^{n}/G).$ See § 0.5 for a
conjectural justification for this dependence in terms of generalized Seiberg-
Witten theory.

On the B-side, more is known, thanks to works in progress of Dima Arinkin and
German Stefanich. Suppose the Lagrangian $\mathbb{L}_{F^{\vee}}$ is the image
in $\mathrm{T}^{*}Y$ of the conormal $\mathrm{N}^{*}_{X}Y$ of a map $X\to Y.$
Then we can present
$\mathsf{CohCat}_{\mathbb{L}_{F^{\vee}}}(\mathbb{C}^{n}/F^{\vee})$ as the
2-category with a single object, whose endomorphisms are given by the monoidal
category $\mathsf{Coh}(X\times_{Y}X).$ In other words, we have a presentation

$\mathsf{CohCat}_{\mathbb{L}_{F^{\vee}}}(\mathbb{C}^{n}/F^{\vee})\simeq{\sf
Mod}_{\mathsf{Coh}(X\times_{Y}X)}({\sf St})$

as the 2-category of module categories over $\mathsf{Coh}(X\times_{Y}X).$

Given a conic open subset $U\subset\mathbb{L}_{F^{\vee}},$ using Definition
4.14 we produce a singular-support condition $\tilde{U},$666This is
$F^{-1}(U)$ in the notation of § 4. in the sense of [AG], for coherent
sheaves on $X\times_{Y}X.$

######  Definition 0.8 (Definition 4.19).

With $X\to Y$ as above, we define the 2-category of microlocal coherent
sheaves of categories on $U$ as the 2-category

$\mu\mathsf{CohCat}(U):={\sf
Mod}_{\mathsf{Coh}_{\tilde{U}}(X\times_{Y}X)}({\sf St})$

of module categories for the monoidal category
$\mathsf{Coh}_{\tilde{U}}(X\times_{Y}X)$ of coherent sheaves on the fiber
product with singular support contained in $U$.

To justify our notation, we prove that the 2-category so defined is invariant
of the choice of map $X\to Y.$

######  Proposition 0.9 (Proposition 4.18 and Proposition 4.20).

Let $X\to Y$ and $X^{\prime}\to Y$ two maps whose conormals
$\mathrm{N}^{*}_{X}Y$ and $\mathrm{N}^{*}_{X^{\prime}}Y$ have the same image
in $\mathrm{T}^{*}Y,$ and let $U\subset\mathrm{T}^{*}Y$ be a conic open
subset. Then the monoidal categories $\mathsf{Coh}_{\tilde{U}}(X\times_{Y}X)$
and $\mathsf{Coh}_{\tilde{U}}(X^{\prime}\times_{Y}X^{\prime})$ are Morita
equivalent.

######  Remark 0.10.

As on the A-side, the above definition of microlocalization is very much
provisional. Ultimately, microlocal coherent sheaves should be global sections
of a sheaf of categories defined by stipulating that its stalks are calculated
by the procedure of Definition 0.8. That Definition 0.8 recovers the true
microlocalization in the case of hypertoric varieties would then follow from a
“2-categorical spectral Kirwan surjectivity” theorem. We leave the full
development of this coherent microlocal theory to future work.

Our first main theorem establishes our predicted mirror relationship between
the two 2-categories we have defined.

######  Theorem A.

There is an equivalence of stable 2-categories

(0.3.5)
$\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))\simeq\mu\mathsf{CohCat}(\mathbb{L}_{F^{\vee}}(m,t))$

between the 2-categories of microlocal perverse (resp. coherent) sheaves of
categories on $\mathbb{L}_{G}(t,m)$ (resp. $\mathbb{L}_{F^{\vee}}(m,t)$),
identifying the canonical generators of these 2-categories.

######  Remark 0.11.

Although in the introduction we work over $\mathbb{C}$ for the sake of
symmetry, A remains true for an arbitrary choice of coefficients. As we
explain in § 1, the spaces on the right-hand side of 0.3.5 can be defined
for a general coefficient ring $\Bbbk,$ which should then be taken as the
coefficient ring for the microlocal perverse schobers we consider (which will
still be defined on the complex algebraic stack
$\mathrm{T}^{*}(\mathbb{C}^{n}/G)$) on the left-hand side. This is part of the
usual paradigm in mirror symmetry and the Langlands program; cf. the
discussion in [Dev-Satake] regarding the geometric Satake equivalence. Later,
when we discuss category $\mathcal{O},$ we will specialize to coefficients
$\Bbbk=\mathbb{C}$ to make contact with the original calculations of
hypertoric category $\mathcal{O}.$

######  Example 0.12 (Example 3.15 and Example 4.32).

Let $G\simeq\mathbb{C}^{\times}\hookrightarrow(\mathbb{C}^{\times})^{2},$
embedded as the kernel of the multiplication map
$(\mathbb{C}^{\times})^{2}\xrightarrow{m}\mathbb{C}^{\times}\simeq F.$ Then
the 2-categories in 0.3.5 can be described as follows:

  * •

An object of $\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))$ is given by a
spherical adjunction

${\mathcal{C}}$${\mathcal{D}}$$\scriptstyle{F}$$\scriptstyle{\bot}$$\scriptstyle{F^{R}}$

equipped with the following extra data:

    * –

An identification of the cotwist ${\sf cofib}(FF^{R}\to{\sf
id}_{\mathcal{D}})$ with the 2-shift automorphism $[2]$;

    * –

An extra monad $M$ on $\mathcal{C}$;

    * –

An identification of twists ${\sf fib}({\sf id}_{\mathcal{C}}\to M)\simeq{\sf
fib}({\sf id}_{\mathcal{C}}\to F^{R}F).$

The Hom category between two such objects is the category of functors
commuting with all the above structure.

  * •

$\mu\mathsf{CohCat}(\mathbb{L}_{F^{\vee}}(m,t))$ is given by the 2-category
${\sf Mod}_{\mathcal{A}}({\sf St})$ of stable module categories over the
monoidal category

$\mathcal{A}:=\mathsf{Coh}\left((\mathbb{P}^{1}\sqcup
0)\times_{\mathbb{P}^{1}}(\mathbb{P}^{1}\sqcup 0)\right).$

Using Koszul duality to write
$\mathsf{Coh}(0\times_{\mathbb{P}}^{1}0)\simeq{\sf Perf}_{\Bbbk[\beta]}$ where
$\beta$ is a generator of degree 2, and Beilinson’s exceptional collection to
identify $\mathsf{Coh}(\mathbb{P}^{1})$ with modules over the Kronecker
quiver, we can rewrite $\mathcal{A}$ as the matrix monoidal category

(0.3.6) $\mathcal{A}\simeq\left(\begin{array}[]{ll}{\sf
Perf}_{\bullet\rightrightarrows\bullet}&{\sf Perf}_{\Bbbk}\\\ {\sf
Perf}_{\Bbbk}&{\sf Perf}_{\Bbbk[\beta]}\end{array}\right),$

giving an explicit presentation of the generating objects, 1-morphisms, and
2-morphisms in the 2-category ${\sf Mod}_{\mathcal{A}}({\sf St})$.

Under the equivalence 0.3.5, the generators of the rings whose perfect modules
form the components of $\mathcal{A}$ in 0.3.6 correspond on the A-side to the
structural morphisms among the spherical adjunctions and monads on the
categories $\mathcal{C}$ and $\mathcal{D}$: namely, the arrows in the
Kronecker quiver $\bullet\rightrightarrows\bullet$ correspond to the maps
$T\rightrightarrows{\sf id}_{\mathcal{C}}$ whose respective cofibers are the
monads $M$ and $F^{R}F$ on $\mathcal{C},$ and the degree-2 map $\beta$
corresponds to the map ${\sf id}_{\mathcal{D}}\to{\sf cofib}(FF^{R}\to{\sf
id}_{\mathcal{D}})\simeq[2].$

###  0.4. Category $\mathcal{O}$ and decategorification

As mentioned in A, each of our 2-categories comes equipped with a canonical
generator. Considered as an object in the automorphic (resp. spectral)
2-category, we will denote this generator by $\mathcal{P}$ (resp.
$\mathcal{S}$). By taking their endomorphisms, we obtain monoidal categories
$\mathsf{end}_{\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))}(\mathcal{P})$ and
$\mathsf{end}_{\mu\mathsf{CohCat}(\mathbb{L}_{F^{\vee}}(m,t))}(\mathcal{S}).$

For a dualizable category $\mathcal{C},$ we write ${\sf HH}(\mathcal{C})$ for
the Hochschild homology of $\mathcal{C},$ which may be understood as the trace
of the identity functor ${\sf id}_{\mathcal{C}}.$ The Hochschild homology
comes equipped with an $S^{1}$-action, for which we may take invariants ${\sf
HH}^{S^{1}}(\mathcal{C})$ or coinvariants ${\sf HH}_{S^{1}}(\mathcal{C})$ to
obtain cyclic homologies, or Tate invariants ${\sf HH}^{tS^{1}}(\mathcal{C})$
to obtain the periodic cyclic homology

${\sf HP}(\mathcal{C}):={\sf HH}^{tS^{1}}(\mathcal{C})\simeq{\sf cofib}({\sf
HH}_{S^{1}}(\mathcal{C})\to{\sf HH}^{S^{1}}(\mathcal{C})).$

The functor ${\sf HP}(-)$ (like ${\sf HH}(-)$) is symmetric monoidal, so that
for $\mathcal{C}$ a monoidal category, ${\sf HP}(\mathcal{C})$ is an algebra.
We will be interested in applying the functor ${\sf HP}(-)$ to the
endomorphism categories of the generators $\mathcal{P},\mathcal{S}$ discussed
above, and relating the resulting algebras to hypertoric categories
$\mathcal{O}.$

######  Remark 0.13.

For $\mathcal{C}$ a monoidal category, the Hochschild homology algebra ${\sf
HH}(\mathcal{C})$ (referred to in [BZCHN] as the “naïve trace”) is a first
approximation to a more sophisticated “categorical trace,” in a sense which is
made precise by [GKRV]*Theorem 3.8.5. The naive trace is sufficient for our
purposes here, but we will return to the calculation of categorical traces in
future work, extending the categorical trace computation in [GH-Betti] for the
2-categories 0.2.1.

As we have mentioned in § 0.1, categories $\mathcal{O}$ are supposed to
arise physically from reductions of 3d $\mathcal{N}=4$ gauge theories. From
this perspective, one class of input data from which one may produce a
category $\mathcal{O}$ is a stack of the form $\mathrm{T}^{*}(V/G),$ where $V$
is a representation of the reductive group $G$, together with a pair of
parameters specifying, respectively, a GIT stable locus
$\mathfrak{M}\subset\mathrm{T}^{*}(V/G)$ and a Hamiltonian
$\mathbb{C}^{\times}$ action on this stable locus. The stable set for this
$\mathbb{C}^{\times}$ action, inside the GIT-stable locus, is a Lagrangian
subspace $\mathbb{L}.$

Given the data of $(\mathfrak{M},\mathbb{L}),$ one may define a pair of
categories:

  * •

The Betti category $\mathcal{O}$ is the category $\mathcal{O}^{\sf
Bet}(\mathfrak{M},\mathbb{L}):=\mu\mathsf{Perv}_{\mathbb{L}}(\mathbb{L})$ of
microlocal perverse sheaves on $\mathbb{L}.$

  * •

The de Rham category $\mathcal{O}$ is a category $\mathcal{O}^{{\sf
dR}}(\mathfrak{M},\mathbb{L}):={\sf DQ}^{{\sf reg}}(\mathbb{L})$ is a category
of regular holonomic DQ-modules on $\mathbb{L}.$

Each of these categories $\mathcal{O}^{(*)}(\mathfrak{M},\mathbb{L})$ has a
collection of simple objects $L_{\alpha},$ with projective covers
$P_{\alpha}.$ We write $L:=\bigoplus L_{\alpha}$ and $P:=\bigoplus P_{\alpha}$
for the direct sum of the simples (resp. projectives). We will use periodic
cyclic homology to recover their endomorphism algebras, up to collapsing the
cohomological grading modulo 2. In the following theorem, we write
$\mathcal{O}^{(*)}_{\mathbb{Z}/2}(\mathfrak{M},\mathbb{L}):=\mathcal{O}^{(*)}(\mathfrak{M},\mathbb{L})\otimes_{{\sf
Mod}_{\Bbbk}}{\sf Mod}_{\Bbbk((u))}$ for the 2-periodization of category
$\mathcal{O}.$

######  Theorem B.

Let $\Bbbk=\mathbb{C}.$ There are equivalences of algebras

(0.4.1) $\displaystyle{\sf
HP}(\mathsf{end}_{\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))}(\mathcal{P}))$
$\displaystyle\simeq\mathsf{end}_{\mathcal{O}^{\sf
Bet}_{\mathbb{Z}/2}(\mathfrak{M}_{G}(t),\mathbb{L}(t,m))}(P),$ (0.4.2)
$\displaystyle{\sf
HP}(\mathsf{end}_{\mu\mathsf{CohCat}(\mathbb{L}_{F^{\vee}}(m,t))}(\mathcal{S}))$
$\displaystyle\simeq\mathsf{end}_{\mathcal{O}^{{\sf
dR}}_{\mathbb{Z}/2}(\mathfrak{M}_{F^{\vee}}(m),\mathbb{L}_{F^{\vee}}(m,t))}(L).$

######  Example 0.14.

We return to the 2-categories discussed in Example 0.12. In this case:

  * •

The category 0.4.1 is the (2-periodized) category of diagrams of vector spaces

${C}$${D}$$\scriptstyle{x}$$\scriptstyle{y}$

with the following relations:

    * –

The sphericality condition on $F$ imposes that the linear maps $1_{D}-xy$ and
$1_{C}-yx$ are invertible.

    * –

The exact triangle $FF^{R}\to{\sf id}_{\mathcal{D}}\to[2]$ decategorifies to
the relation $xy=0.$ (This implies the invertibility conditions above.)

(The extra monad $M$ decategorifies to $yx$ and thus does not contain any new
information.) We thus recover the usual quiver description for the category
$\mathcal{O}$ associated to $\widetilde{\mathbb{C}^{2}/\mathbb{Z}/2}$ (which
coincides with the classical BGG category $\mathcal{O}$ for $SL(2)$).

  * •

The category 0.4.2 is a 2-category of modules over the ring ${\sf
HP}(\mathcal{A})$, where $\mathcal{A}$ is as in Example 0.12. Using the
identification [Preygel-loopspaces]*Theorem 1.1.2

${\sf HP}(\mathcal{A})={\sf HP}\mathsf{Coh}\left((\mathbb{P}^{1}\sqcup
0)\times_{\mathbb{P}^{1}}(\mathbb{P}^{1}\sqcup 0)\right)\simeq
C_{*}^{BM}\left((\mathbb{P}^{1}\sqcup
0)\times_{\mathbb{P}^{1}}(\mathbb{P}^{1}\sqcup 0)\right)_{\mathbb{Z}/2},$

we find that this category is the subcategory of (2-periodized) D-modules on
$\mathbb{P}^{1}$ generated by the D-modules $\mathcal{O}_{\mathbb{P}^{1}}$ and
$\delta_{0}.$ These are the simple objects in category $\mathcal{O}$ for
$\mathrm{T}^{*}\mathbb{P}^{1},$ which once again agrees with the classical
category $\mathcal{O}$ for $SL(2).$

######  Remark 0.15.

The decategorification in B only recovers 2-periodized versions of category
$\mathcal{O}$. One might hope for an enhanced version of B which recovers
categories $\mathcal{O}$ without collapsing to a $\mathbb{Z}/2$-grading. Such
a result could be achieved by constructing a graded lift of the 2-categories
$\mathcal{O}$ we define in this paper. We leave this question to future work.

For a fixed hypertoric variety $\mathfrak{M}_{G}(t),$ the two flavors of
2-category $\mathcal{O}$ we define, namely
$\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))$ and
$\mu\mathsf{CohCat}(\mathbb{L}_{G}(t,m))$ are very different, and certainly
very far from being equivalent. However, one surprising feature of B is that
they agree after ${\sf HP}$-decategorification:

######  Corollary 0.16.

There is an equivalence of categories

$\displaystyle{\sf Perf}_{{\sf
HP}(\mathsf{end}_{\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))}(\mathcal{P}))}$
$\displaystyle\simeq\mathcal{O}^{\sf
Bet}_{\mathbb{Z}/2}(\mathfrak{M}_{G}(t),\mathbb{L}_{G}(t,m))$
$\displaystyle\simeq\mathcal{O}^{{\sf
dR}}_{\mathbb{Z}/2}(\mathfrak{M}_{G}(t),\mathbb{L}_{G}(t,m))\simeq{\sf
Perf}_{{\sf
HP}(\mathsf{end}_{\mu\mathsf{CohCat}(\mathbb{L}_{G}(t,m))}(\mathcal{S}))}.$

The above identification, given by the Riemann-Hilbert correspondence
$\mathcal{O}^{\sf Bet}\simeq\mathcal{O}^{{\sf dR}}$, is a categorification of
the fact that both matrix factorizations $\mathsf{MF}(X,f)$ and the Fukaya-
Seidel category $\mathsf{FS}(X,f)$ categorify the vanishing cohomology
$H^{*}(X,\varphi_{f})$ of a function $f.$

######  Remark 0.17.

Corollary 0.16 shows that the Koszul dualities among categories $\mathcal{O}$
appear only after decategorification: after applying ${\sf HP},$ the
equivalence of Corollary 0.16 supplies diagonal arrows to the diagram 0.1.2.
In other words, the miraculous fact underlying Koszul duality between dual
categories $\mathcal{O}$ is that the A- and B-type 2-categories $2\mathcal{O}$
for the same space, though a priori unrelated, have the same
decategorification — the 1-category $\mathcal{O}$ — with canonical generators
going to projectives (resp. simples) of this 1-category. We expect that
essentially all the instances of Koszul duality appearing in geometric
representation theory arise via decategorifications of this form (at least
after passing to a $\mathbb{Z}/2$ grading).

###  0.5. Predictions for Fueter theory

Conjecturally, $\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))$ calculates the
2-category of boundary conditions for gauged Fueter theory (i.e., generalized
3d Seiberg-Witten theory [Taubes-generalizations]) associated to
$\mathrm{T}^{*}(\mathbb{C}^{n}/G)$, with generic real FI and mass parameters
$t,m$. Note that this theory is not the same as the A-twisted sigma model
(i.e., ungauged Fueter theory) on the stable locus
$\mathfrak{M}(t)\subset\mathrm{T}^{*}(\mathbb{C}^{n}/G)$: the gauge theory
(even with generic FI parameter) receives contributions from the unstable
locus. This explains the dependence of the 2-category
$\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))$ on the stack
$\mathrm{T}^{*}(\mathbb{C}^{n}/G)$ and not just the variety $\mathfrak{M}(t).$
Indeed, the factorization homology of this 2-category on a closed 2-manifold
is predicted in [BFK-surface] to recover information about the moduli space of
quasimaps, which depends on the stacky presentation of $\mathfrak{M}_{G}(t).$

So far, the Fueter 2-category associated to a symplectic variety or stack
$\mathfrak{M},$ equipped with Lagrangian skeleton $\mathbb{L},$ is not well-
understood or even defined (although preliminary work in the ungauged case is
available in [Doan-Rezchikov]). Heuristically, the objects of this 2-category
should be holomorphic Lagrangians in $\mathfrak{M}$; 1-morphisms should be
intersection points between these (after a displacement by a wrapping
Hamiltonian); and 2-morphisms should be $J_{\theta}$-holomorphic curves
between these.

Our 3d mirror symmetry comparison A thus gives us detailed predictions for the
behavior of the holomorphic Lagrangians in the gauged Fueter 2-category of
$\mathrm{T}^{*}(\mathbb{C}^{n}/G)$. The canonical generators of the 2-category
$\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))$ are expected to correspond to
Lagrangian cocores or linking disks to the components of the skeleton
$\mathbb{L}_{G}(t,m)),$ and A computes the Hom categories (and compositions)
among these.

######  Example 0.18.

We return to the situation of Example 0.12, where

$\mathbb{L}_{G}(t,m)=\mathbb{P}^{1}\cup\mathrm{T}^{*}_{\infty}\mathbb{P}^{1}\subset\mathrm{T}^{*}\mathbb{P}^{1}\subset\mathrm{T}^{*}(\mathbb{C}^{2}/\mathbb{C}^{\times}).$

The 2-category $\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))$ has two canonical
generators, corepresenting microstalk functors at the two components of
$\mathbb{L}_{G}(t,m),$ and we computed their endomorphism category to be

$\mathcal{A}\simeq\left(\begin{array}[]{ll}{\sf
Perf}_{\bullet\rightrightarrows\bullet}&{\sf Perf}_{\Bbbk}\\\ {\sf
Perf}_{\Bbbk}&{\sf Perf}_{\Bbbk[\beta]}\end{array}\right).$

Up to remembering the degree shift in the bottom-right corner, we can write
the matrix $\mathcal{A}$ entirely in terms of representations over quivers:
$\mathcal{A}\simeq{\sf Perf}_{\mathcal{Q}},$ where

$\mathcal{Q}\simeq\left(\begin{array}[]{cc}\bullet\rightrightarrows\bullet&\bullet\\\
\bullet&\bullet\circlearrowleft\end{array}\right).$

This presentation makes evident that each of the four Hom categories in
$\mathcal{A}$ has distinguished generating objects (the vertices) and
generating 1-morphisms (the arrows). It is natural to conjecture that these
correspond to the (wrapped) intersection points and the $J_{s}$-holomorphic
disks, respectively, among Lagrangian cocores in the Fueter 2-category.

######  Remark 0.19.

Note that Example 0.18 gives a conjectural calculation not of the Fueter
theory of $T^{*}\mathbb{P}^{1}$ but rather of the gauged Fueter theory of
$T^{*}\mathbb{C}^{2}.$ In Example 0.12, the embedding of $T^{*}\mathbb{P}^{1}$
into $T^{*}\mathbb{C}^{2}$ contributed the extra monad $M$, arising
geometrically from the interaction of $T^{*}_{\infty}\mathbb{P}^{1}$ with the
unstable Lagrangian
$T^{*}_{0/\mathbb{C}^{\times}}(\mathbb{C}^{2}/\mathbb{C}^{\times}).$ That
monad contributed one of the arrows in the Kronecker quiver appearing in the
upper-left corner of $\mathcal{Q}$ above. We therefore predict that the
endomorphism category of the generators of the Fueter 2-category for
$T^{*}\mathbb{P}^{1}$ itself should be given by

$\mathcal{A}^{\prime}:=\left(\begin{array}[]{cc}{\sf
Perf}_{\bullet\to\bullet}&{\sf Perf}_{\Bbbk}\\\ {\sf Perf}_{\Bbbk}&{\sf
Perf}_{\Bbbk[\beta]}\end{array}\right).$

###  0.6. Notation and conventions

Our categorical conventions are largely the same as in [GMH], and we refer to
§A there for more details. We use the “implicit $\infty$” convention: all
constructions are understood in the homotopy-coherent sense; in particular,
$(n-)$category always means $(\infty,n)$-category. Stable categories are
always assumed $\Bbbk$-linear for a fixed coefficient ring $\Bbbk.$ We write
${\sf St}$ for the stable 2-category of small ($\Bbbk$-linear) stable
categories.

By default, the categories we discuss are derived categories rather than the
hearts of their t-structure. If $\mathcal{C}$ is a category with t-structure,
we write $\mathcal{C}^{\heartsuit}$ for its abelian heart. So for instance,
for $X$ an algebraic stack, we write $\mathsf{Coh}(X)$ for the derived
category of coherent sheaves on $X$. Throughout the paper, “stack” will always
mean a derived stack which is quasi-compact, almost of finite presentation,
and with affine diagonal.

For $\Lambda\subset\mathrm{T}^{*}X$ a Lagrangian, we write
$\mathsf{Perv}_{\Lambda}(X)$ for the category of compact objects inside the
derived category of the heart of the perverse t-structure on the presentable
stable category of sheaves on $X$ with singular support in $\Lambda.$ (Thus,
our usage of $\mathsf{Perv}_{\Lambda}(X)$ differs from the usual one in two
ways: first, we treat this as a derived category rather than the heart of its
t-structure, and second, we allow some objects with infinite-dimensional
microstalks.) We retain the analogous convention for the category
$\mu\mathsf{Perv}_{\Lambda}(\Lambda)$ of microlocal perverse sheaves.

Ideals in a monoidal category will always be assumed to be idempotent-complete
full subcategories, closed under finite limits and colimits.

For a torus $G,$ we write $\mathfrak{g}$ for its Lie algebra (with dual
$\mathfrak{g}^{\vee})$, $\mathfrak{g}^{\vee}_{\mathbb{Z}}$ for the character
lattice (and $\mathfrak{g}_{\mathbb{Z}}$ for the cocharacter lattice), and
$\mathfrak{g}^{\vee}_{\mathbb{R}}:=\mathfrak{g}^{\vee}_{\mathbb{Z}}\otimes\mathbb{R},$
$\mathfrak{g}_{\mathbb{R}}:=\mathfrak{g}\otimes\mathbb{R}.$ When $G$ has an
action on a symplectic variety, the moment map is naturally valued in
$\mathfrak{g}^{\vee}$. If $G$ is defined over $\mathbb{C},$ we write $G_{c}$
for the maximal compact subgroup; if $G_{c}$ has a Hamiltonian action on a
(real) symplectic manifold, the moment map is valued in
$\mathfrak{g}^{\vee}_{\mathbb{R}}.$

Notational index: In Table 1 we collect some notation used frequently to
describe the geometry of hypertoric varieties. In the following list, we write
$\alpha\in 2^{[n]}$ for a sign vector, and $I\subset 2^{[n]}$ for a collection
of sign vectors.

Notation | Definition
---|---
$X_{G}^{\alpha}$ | $\\{z_{i}=0\mid i\notin\alpha\\}/G\subset\mathbb{A}^{n}/G$
$X_{G}$ | $\bigsqcup_{\alpha\in 2^{[n]}}X_{G}^{\alpha}$
$X_{G}^{I}$ | $\bigsqcup_{\alpha\in I}X_{G}^{\alpha}$
$\mathbb{L}_{G}^{\alpha}$ | $\mathrm{N}^{*}_{X_{G}^{\alpha}}(\mathbb{A}^{n}/G)\subset\mathrm{T}^{*}(\mathbb{A}^{n}/G)$
$\mathbb{L}_{G}$ | $\bigcup_{\alpha\in 2^{[n]}}\mathbb{L}_{G}^{\alpha}$
$\mathbb{L}_{G}^{I}$ | $\bigcup_{\alpha\in I}\mathbb{L}_{G}^{\alpha}$
Table 1. Index of notation for the basic subvarieties and conormal Lagrangians
considered in this paper

Acknowledgements The authors are grateful to Aaron Mazel-Gee for collaborating
on early stages of this project. BG would like to thank Elden Elmanto and
Adeel Khan for suggestions about periodic cyclic homology and Laurent Côté for
many long discussions about microlocal perverse sheaves and category
$\mathcal{O},$ and for collaboration on the companion paper [CGH]. JH would
like to thank Mat Bullimore, Tudor Dimofte, and Davide Gaiotto for their
collaborations on [BDGH]. He would also like to thank Kevin Costelo, Niklas
Garner, Wenjun Niu, and Philsang Yoo for many interesting discussions about
$3$d $\mathcal{N}=4$ theories and thank Aleksander Doan, Ahsan Khan, and Semon
Rezchikov for interesting discussions about Fueter maps.

BG acknowledges the support of an NSF Postdoctoral Research Fellowship,
DMS-2001897.

JH is part of the Simons Collaboration on Homological Mirror Symmetry
supported by Simons Grant 390287. This research was supported in part by
Perimeter Institute for Theoretical Physics. Research at Perimeter Institute
is supported by the Government of Canada through the Department of Innovation,
Science and Economic Development Canada and by the Province of Ontario through
the Ministry of Research, Innovation and Science.

##  1\. Hyperplanes and hyperkähler manifolds

Throughout this paper, we fix an exact sequence of tori

(1.0.1) ${1}$${G}$${D}$${F}$${1,}$$\scriptstyle{i}$$\scriptstyle{p}$

as well as an equivalence $D\simeq(\mathbb{G}_{m})^{n}.$

###  1.1. Hyperplane arrangements

From the exact sequence 1.0.1 and a choice of parameters
$(t,m)\in\mathfrak{g}^{\vee}_{\mathbb{R}}\times\mathfrak{f}_{\mathbb{R}},$ we
will define a polarized hyperplane arrangment in
$\mathfrak{f}^{\vee}_{\mathbb{R}}$ — that is, a hyperplane arrangement in
$\mathfrak{f}^{\vee}_{\mathbb{R}}$ together with an affine-linear functional
on $\mathfrak{f}^{\vee}_{\mathbb{R}}.$ We begin with the basic case where the
parameters are 0 and $G=\\{e\\}$ is trivial:

######  Definition 1.1.

We write $\mathcal{H}$ be the cooriented hyperplane arrangement in
$\mathfrak{d}^{\vee}_{\mathbb{R}}\simeq\mathbb{R}^{n}$ given by the $n$
coordinate hyperplanes, with their natural coorientations.

Observe that the exact sequence 1.0.1 induces an exact sequence

${0}$${\mathfrak{g}^{\vee}_{\mathbb{Z}}}$${\mathfrak{d}^{\vee}_{\mathbb{Z}}}$${\mathfrak{f}^{\vee}_{\mathbb{Z}}}$${0.}$$\scriptstyle{i^{\vee}_{\mathbb{Z}}}$$\scriptstyle{p^{\vee}_{\mathbb{Z}}}$

Given $t\in\mathfrak{g}^{\vee}_{\mathbb{R}},$ we write

$\mathfrak{f}^{\vee}_{\mathbb{R}}(t):=(i_{\mathbb{R}}^{\vee})^{-1}(t)\subset\mathfrak{d}^{\vee}_{\mathbb{R}}$

for the affine-linear subspace of $\mathfrak{d}^{\vee}_{\mathbb{R}}$ obtained
as a translation of the subspace $\mathfrak{f}^{\vee}_{\mathbb{R}}$ by $t.$ We
will assume that $\mathfrak{f}^{\vee}_{\mathbb{R}}(t)$ is not contained in any
of the hyperplanes in the arrangement $\mathcal{H}$.

######  Definition 1.2.

Given
$(t,m)\in\mathfrak{g}^{\vee}_{\mathbb{R}}\times\mathfrak{f}_{\mathbb{R}},$ we
define a polarized hyperplane arrangement $(\mathcal{H}_{G}(t),m)$, where
$\mathcal{H}_{G}(t)$ is the hyperplane arrangement on
$\mathfrak{f}_{\mathbb{R}}^{\vee}(t)$ obtained by intersecting with the
coordinate hyperplane arrangement $\mathcal{H},$ and the polarization $m$ is
the affine-linear functional on $\mathfrak{f}_{\mathbb{R}}^{\vee}(t)$ given by
$m\in\mathfrak{f}_{\mathbb{R}}$.

We will frequently need to refer to the chambers in these hyperplane
arrangements. Observe that one of the $2^{n}$ chambers in the cooriented
hyperplane arrangement $\mathcal{H}$ may be specified by labeling which of the
$n$ coordinates on $\mathbb{R}^{n}$ is positive and which is negative.

######  Definition 1.3.

A sign vector is an element of the set $2^{[n]}.$ Writing $2=\\{+,-\\},$ we
can think of a sign vector as a length $n$ word in $\\{+,-\\},$ hence the
name. We will also think of sign vectors as subsets of $[n],$ with
$(+,\ldots,+)$ corresponding to the full set $[n]=\\{1,...,n\\}$ and
$(-,\ldots,-)$ corresponding to the empty set.

For a sign vector $\alpha,$ we write $\mathcal{H}^{\alpha}$ for the
corresponding chamber of $\mathcal{H},$ and we write
$\mathcal{H}_{G}^{\alpha}(t)$ for the restriction of this chamber to
$\mathfrak{f}_{\mathbb{R}}^{\vee}(t).$

######  Definition 1.4.

We say that a sign vector $\alpha\in 2^{[n]}$ is $t$-unstable if the chamber
$\mathcal{H}_{G}^{\alpha}(t)$ does not appear in $\mathcal{H}_{G}(t).$777In
[BLPW10], these sign vectors are instead called infeasible, using the language
of linear programming. We prefer throughout to use language referencing the
geometry of the hypertoric variety.

We say that a sign vector $\alpha\in 2^{[n]}$ is $m$-bounded if the
restriction of $m$ to the chamber $\mathcal{H}_{G}^{\alpha}(t)$ (or
equivalently, to the chamber $\mathcal{H}_{G}^{\alpha}(0)$) is bounded.

Recall the following features of hyerplane arrangements:

######  Definition 1.5.

A hyperplane arrangement in $\mathbb{R}^{k}$ is simple if any $k$ hyperplanes
intersect in a single point and if the intersection of any $k+1$ hyperplanes
is empty.

The arrangement $\mathcal{H}_{G}(t)$ is unimodular if there is some basis for
$\mathfrak{g}_{\mathbb{Z}}$ where $p^{\vee}_{\mathbb{Z}}$ is given by a
totally unimodular integer matrix.

######  Definition 1.6.

We say that the parameters
$(t,m)\in\mathfrak{g}^{\vee}_{\mathbb{R}}\times\mathfrak{f}_{\mathbb{R}}$ are
generic if they satisfy the following conditions:

  * •

$t$ is chosen so that the hyperplane arrangement $\mathcal{H}_{G}(t)$ is
simple.

  * •

$m$ is chosen so that it is not constant on any 1-dimensional flat of
$\mathcal{H}_{G}(t)$ (or equivalently, on any 1-dimensional flat of
$\mathcal{H}_{G}(0)$).

###  1.2. Gale duality

Consider the exact sqeuence of tori obtained from 1.0.1 by taking duals:

(1.2.1)
${1}$${F^{\vee}}$${D^{\vee}}$${G^{\vee}}$${1.}$$\scriptstyle{p^{\vee}}$$\scriptstyle{i^{\vee}}$

We may repeat the constructions of § 1 with $G$ replaced by $F^{\vee}.$ Now
we use $m\in\mathfrak{f}$ as the parameter determining a translation of the
subspace
$\mathfrak{g}_{\mathbb{R}}\subset\mathfrak{d}_{\mathbb{R}}\simeq\mathbb{R}^{n},$
and by intersecting with the coordinate hyperplane arrangement in
$\mathbb{R}^{n}$ we obtain a hyperplane arrangement
$\mathcal{H}_{F^{\vee}}(m)$ in $\mathfrak{g}_{\mathbb{R}}.$

######  Definition 1.7.

We say that the polarized hyperplane arrangements $(\mathcal{H}_{G}(t),m)$ and
$(\mathcal{H}_{F^{\vee}}(m),t)$ are Gale dual to one another.

The basic theorem of linear programming is that the concepts of instability
and unboundedness are related to each other by this duality.

######  Theorem 1.8 ([BLPW10]*Theorem 2.4).

A sign vector $\alpha\in 2^{[n]}$ is $t$-unstable for $(\mathcal{H}_{G}(t),m)$
if and only if it is $t$-unbounded for $(\mathcal{H}_{F^{\vee}}(m),t).$

###  1.3. Toric hyperkähler manifolds

Before moving to hyperkähler manifolds, we begin with toric symplectic
algebraic stacks. To the exact sequence 1.0.1, we associate the stack

$\mathfrak{M}_{G}:=\mathrm{T}^{*}(\mathbb{A}^{n}/G),$

where $D$ is the torus of diagonal automorphisms of $\mathbb{A}^{n},$ and $G$
acts on $\mathbb{A}^{n}$ as a subtorus of $D$. This stack may be understood as
the global quotient

$\mathrm{T}^{*}(\mathbb{A}^{n}/G)=\mu_{G}^{-1}(0)/G,$

where $\mu_{G}:\mathrm{T}^{*}\mathbb{A}^{n}\to\mathfrak{g}^{\vee}$ is the
moment map for the Hamiltonian action of the torus $G.$ This moment map
factors as the composite

$\mathrm{T}^{*}\mathbb{A}^{n}\xrightarrow{\mu_{D}}\mathbb{A}^{n}\simeq\mathfrak{d}^{\vee}\xrightarrow{i^{\vee}}\mathfrak{g}^{\vee},$

with the first map given by

$\mu_{D}(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n})=(x_{1}y_{1},\ldots,x_{n}y_{n}).$

######  Definition 1.9.

Let $t\in\mathfrak{g}^{\vee}_{\mathbb{Z}}$. The hypertoric variety888If the
hyperplane arrangement $\mathcal{H}_{G}(t)$ is not unimodular, this is a
slight misnomer, as we will treat $\mathfrak{M}_{G}(t)$ as a DM stack in this
case. associated to the exact sequence 1.0.1 and the parameter $t$ is the GIT
quotient $\mathfrak{M}_{G}(t):=\mu_{G}^{-1}(0)/\\!\\!/_{t}G.$ Equivalently, we
may write
$\mathfrak{M}_{G}(t)=(\mathrm{T}^{*}(\mathbb{C}^{n}/G))^{t-ss}\subset\mathrm{T}^{*}(\mathbb{A}^{n}/G)$
as the $t$-semistable locus inside the stack $T^{*}(\mathbb{A}^{n}/G).$ When
$\Bbbk=\mathbb{C},$ the underlying complex-analyic space of
$\mathfrak{M}_{G}(t)$ is called a toric hyperkähler manifold.

######  Proposition 1.10 ([Biel-Dan]*Theorems 3.2 & 3.3).

If $t$ is generic, then the space $\mathfrak{M}_{G}(t)$ is a smooth Deligne-
Mumford stack. If moreover the hyperplane arrangement $\mathcal{H}_{G}(t)$ is
unimodular, then there are no strictly semistable points, and
$\mathfrak{M}_{G}(t)$ is a smooth variety.

When $\Bbbk=\mathbb{C},$ the complex-analytic spaces $\mathfrak{M}_{G}(t)$
were first introduced in [Goto], and further studied in [Konno, Biel-Dan,
Haus-Sturm].As these spaces have traditionally been studied as complex
manifolds, they are often constructed not as GIT quotients but as hyperkähler
quotients, as we now recall.

In the complex setting, not only is the action of $G$ on $T^{*}\mathbb{C}^{n}$
Hamiltonian (with moment map $\mu_{G}$) for the holomorphic symplectic form on
$T^{*}\mathbb{C}^{n},$ but the action of the compact torus $G_{c}$ is also
Hamiltonian for the real Kähler form on $T^{*}\mathbb{C}^{n}=\mathbb{C}^{2n}.$
The real moment map for this action is the composite

$T^{*}\mathbb{C}^{n}\xrightarrow{\mu_{D_{c}}}\mathbb{R}^{n}\simeq\mathfrak{d}^{\vee}_{\mathbb{R}}\xrightarrow{i^{\vee}_{\mathbb{R}}}\mathfrak{g}^{\vee}_{\mathbb{R}},$

with the first map given by

$\mu_{D^{c}}(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n})=\left(|x_{1}|^{2}-|y_{1}|^{2},\ldots,|x_{n}|^{2}-|y_{n}|^{2}\right).$

The following is a standard application of the Kempf-Ness theorem: see for
instance [Nakajima-Hilbertschemes]*Ch. 3.

######  Lemma 1.11.

For $t$ generic, there is an isomorphism of complex manifolds (or orbifolds if
$\mathcal{H}_{G}(t)$ is not unimodular)

$\mathfrak{M}_{G}(t)\simeq\left(\mu_{G}^{-1}(0)\cap\mu_{G^{c}}^{-1}(t)\right)/G_{c}.$

One very useful feature of the spaces $\mathfrak{M}_{G}(t)$ is the presence of
a dilating $\mathbb{G}_{m}$-action.

######  Definition 1.12.

Let $\SS=\mathbb{G}_{m}$ be the 1-dimensional torus which acts on
$\mathrm{T}^{*}\mathbb{A}^{n}\simeq\mathbb{A}^{2n}$ by scaling each of the
coordinates with weight $1.$

######  Lemma 1.13.

The action of $\SS$ on $\mathrm{T}^{*}\mathbb{A}^{n}$ descends to an action on
$\mathfrak{M}_{G}(t)$ which scales the symplectic form with weight 2.

###  1.4. Toric holomorphic Lagrangians

We now discuss some interesting Lagrangian subspaces of $\mathfrak{M}.$ As
usual, our discussion begins with the basic case of
$\mathfrak{M}=\mathrm{T}^{*}\mathbb{C}^{n}.$

######  Definition 1.14.

We define the following collection of conormal Lagrangians in our hypertyric
spaces, indexed by sign vectors $\alpha.$

  * •

Let $\alpha\in 2^{[n]}$ be a sign vector. We write

$X^{\alpha}:=\\{(z_{1},\ldots,z_{n})\in\mathbb{A}^{n}\mid z_{i}=0\text{ for
}i\notin\alpha\\}\subset\mathbb{A}^{n}$

for the subspace of $\mathbb{A}^{n}$ given by the intersection of coordinate
hyperplanes. We denote its image in $\mathbb{A}^{n}/G$ by $X_{G}^{\alpha},$
and we write
$\mathbb{L}_{G}^{\alpha}:=\mathrm{N}^{*}_{X_{G}^{\alpha}}(\mathbb{A}^{n}/G)$
for its conormal.

  * •

For $I\subset 2^{[n]}$ a collection of sign vectors, we write
$X_{G}^{I}:=\bigsqcup_{\alpha\in I}X_{G}^{\alpha}$ and
$\mathbb{L}_{G}^{I}:=\bigcup_{\alpha\in I}\mathbb{L}_{G}^{\alpha}.$

  * •

We write $X_{G}:=\bigsqcup_{\alpha\in 2^{[n]}}X_{G}^{\alpha}$ and
$\mathbb{L}_{G}:=\bigcup_{\alpha\in 2^{[n]}}\mathbb{L}_{G}^{\alpha}.$

As $X_{G}^{\alpha}\hookrightarrow\mathbb{A}^{n}/G$ is a closed immersion, we
may understand $\mathbb{L}_{G}^{\alpha}$ as a subspace of
$\mathrm{T}^{*}(\mathbb{A}^{n}/G).$ Similarly, we treat the singular
Lagrangians $\mathbb{L}_{G}^{I}$ and $\mathbb{L}_{G}$ as subspaces of
$\mathrm{T}^{*}(\mathbb{A}^{n}/G)$ as well.

We will be interested in descending these Lagrangians to the hypertoric
varieties $\mathfrak{M}_{G}(t),$ for generic
$t\in\mathfrak{g}^{\vee}_{\mathbb{Z}}.$

######  Definition 1.15.

We write

$\mathbb{L}_{G}(t,0):=\mathbb{L}_{G}\cap(\mathrm{T}^{*}(\mathbb{A}^{n}/G))^{t-ss}\subset\mathfrak{M}_{G}(t)$

for the locus of $t$-semistable points in the Lagrangian $\mathbb{L}_{G}.$

######  Proposition 1.16.

The locus of $t$-unstable points

$\mathbb{L}_{G}^{t\text{-unstab}}:=\mathbb{L}_{G}\setminus\mathbb{L}_{G}(t,0)\hookrightarrow\mathbb{L}_{G}$

in the Lagrangian $\mathbb{L}_{G}$ is given by the union
$\mathbb{L}_{G}^{t\text{-unstab}}=\bigcup_{\alpha\in
t\text{-unstab}}\mathbb{L}_{G}^{\alpha},$ where the set of $t$-unstable sign
vectors $\alpha$ is defined in Definition 1.4.

###### Proof.

This follows from the stability criterion [Hoskins-
stratifications]*Proposition 2.7. ∎

The Lagrangian $\mathbb{L}_{G}(t,0)\subset\mathfrak{M}_{G}(t)$ admits a simple
description. Observe that when $G=\\{e\\}$ is trivial, the Lagrangian
$\mathbb{L}=\mathbb{L}_{G}\subset\mathrm{T}^{*}\mathbb{A}^{n}$ is a normal
crossings variety, and when $\Bbbk=\mathbb{C},$ the chambers in the hyperplane
arrangement $\mathcal{H}$ (all orthants) are actually the moment polytopes for
the toric (actually, all $\mathbb{C}^{n}$) components of $\mathbb{L}.$ This
fact persists after the GIT quotient:

######  Theorem 1.17 ([Biel-Dan]*Theorem 6.5).

The component $(\mathbb{L}_{G}^{\alpha})^{t-ss}$ of $\mathbb{L}_{G}(t,0)$ is a
toric variety corresponding to the polytope $\mathcal{H}_{G}^{\alpha},$ and
these components meet according to the incidences of chambers in
$\mathcal{H}_{G}.$ When $\Bbbk=\mathbb{C},$ the chambers
$\mathcal{H}_{G}^{\alpha}$ are literally the moment polytopes for the
restriction to $\mathbb{L}_{G}(t,0)$ of the real moment map $\mu_{G^{c}}.$

In order to discuss (2-)categories $\mathcal{O},$ we will need one further
choice of parameter: the attraction parameter $m\in\mathfrak{f}_{\mathbb{Z}},$
which we understand as a cocharacter $m:\mathbb{G}_{m}\to F$ of the torus $F$
which acts on $\mathfrak{M}_{G}(t),$ determining a Hamiltonian action of a
1-dimensional torus $\mathbb{G}_{m}$ on $\mathfrak{M}_{G}(t)$.

######  Definition 1.18.

A point $x\in\mathrm{T}^{*}(\mathbb{A}^{n}/G)$ is $m$-bounded if there exists
a $\mathbb{G}_{m}$-equivariant map
$\mathbb{A}^{1}_{\infty}\to\mathfrak{M}_{G}(t)$ whose image contains $x,$
where we write $\mathbb{A}^{1}_{\infty}$ for the weight-$(-1)$ representation
of $\mathbb{G}_{m}.$

When $\Bbbk=\mathbb{C},$ so the variety $\mathfrak{M}_{G}(t)$ is a complex
manifold and $\mathbb{G}_{m}=\mathbb{C}^{\times}$, then Definition 1.18 is
just the condition that the limit $\lim_{\lambda\to\infty}m(\lambda)\cdot x$
exists. In general, we refer to [Drinfeld-Gaitsgory-hyperbolic]*§1 for more
information about the geometry of $\mathbb{G}_{m}$-actions on algebraic
spaces.

######  Lemma 1.19.

The $m$-bounded locus $\mathbb{L}_{G}^{m\text{-bdd}}$ in the Lagrangian
$\mathbb{L}_{G}$ is the union of components $\mathbb{L}_{G}^{\alpha}$
corresponding to sign vectors $\alpha$ which are $m$-bounded in the sense of
Definition 1.4.

######  Definition 1.20.

Fix parameters
$t\in\mathfrak{g}^{\vee}_{\mathbb{Z}},m\in\mathfrak{f}_{\mathbb{Z}}.$ Then we
write
$\mathbb{L}_{G}(t,m):=\mathbb{L}_{G}(t,0)^{m\text{-bdd}}\subset\mathfrak{M}_{G}(t)$
for the subset of $m$-bounded points in $\mathbb{L}_{G}(t,0),$ and we call it
the category $\mathcal{O}$ skeleton of $\mathfrak{M}_{G}(t).$

######  Remark 1.21.

The space of parameters $(t,m)$ is divided into chambers by a finite number of
rational walls, and none of the quantities we study in this paper changes as
$(t,m)$ moves within a fixed chamber. This suggests that it is reasonable to
take $(t,m)\in\mathfrak{g}^{\vee}_{\mathbb{R}}\times\mathfrak{f}_{\mathbb{R}}$
to be continuous and real-valued, and indeed it is possible to make sense of
the above constructions for such parameters: $t$ still defines a closed subset
of unstable points, and $m$ defines a Hamiltonian vector field (which may not
integrate to a $\mathbb{C}^{\times}$-action) which can still be used to define
the bounded points. Nevertheless, for simplicity, throughout this paper we
restrict our parameters to have integral values.

##  2\. Hypertoric categories $\mathcal{O}$

Throughout this section, we fix a generic choice of parameters
$(t,m)\in\mathfrak{g}^{\vee}_{\mathbb{Z}}\times\mathfrak{f}_{\mathbb{Z}},$
determining a Lagrangian
$\mathbb{L}(t,m)\subset\mathrm{T}^{*}(\mathbb{C}^{n}/G).$ Hypertoric category
$\mathcal{O},$ first studied in [BLPW10, BLPW12], should be a category of
regular DQ-modules or microlocal perverse sheaves on this Lagrangian. We will
refer to these two flavors of category $\mathcal{O}$ as the de Rham and Betti
categories $\mathcal{O},$ respectively. Here we will give a brief review of
these categories. The proofs of the statements about de Rham category
$\mathcal{O}$ can be found in [BLPW10, BLPW12], and those about Betti category
$\mathcal{O}$ will appear in [CGH].

###  2.1. de Rham category $\mathcal{O}$

Let $\mathcal{E}$ be a deformation quantization of the variety
$\mathfrak{M}_{G}(t),$ which may be produced as in [Kashiwara-Rouquier] by
quantum Hamiltonian reduction of the sheaf of microdifferential operators on
$\mathrm{T}^{*}\mathbb{C}^{n},$ followed by restriction to
$\mathfrak{M}_{G}(t).$ We write
$\mathcal{W}:=\mathcal{E}[\hbar^{-\frac{1}{2}}],$ so that $\mathcal{W}$ is a
sheaf of $\mathbb{C}((\hbar^{\frac{1}{2}}))$-algebras on
$\mathfrak{M}_{G}(t),$ which is moreover equivariant for the action of the
torus $\SS$ defined in Definition 1.12, with $|\hbar|=2.$

######  Definition 2.1.

An $\mathcal{E}$-module $\mathcal{F}$ is coherent if $\mathcal{F}/\hbar$ is a
coherent sheaf on $\mathfrak{M}_{G}(t).$ A $\mathcal{W}$-module is good if it
admits a coherent $\mathcal{E}$-lattice. We write ${\sf
Good}_{\mathcal{W}}^{\SS}$ for the derived category of $\SS$-equivariant good
$\mathcal{W}$-modules.

We refer to [Kashiwara-Rouquier] for the theory of $\SS$-equivariant good
$\mathcal{W}$-modules.

######  Definition 2.2 ([BLPW12]*Definition 6.2).

The de Rham category $\mathcal{O}$ is the full subcategory of ${\sf
Good}_{\mathcal{W}}^{\SS}$ on those $\mathcal{F}$ such that

  * •

the support of $\mathcal{F}$ is contained (set-theoretically) in
$\mathbb{L}_{G}(t,m),$ and

  * •

$\mathcal{F}$ admits an $\mathcal{E}$-lattice that is preserved by the action
of $m$.

We denote this category by $\mathcal{O}^{{\sf
dR}}(\mathfrak{M}_{G}(t),\mathbb{L}_{G}(t,m)).$

This category is studied extensively in [BLPW12], where it is shown to be
equivalent to the combinatorially-defined category introduced in [BLPW10]. An
alternative approach to studying this category is given in [Web-gencat-O]*§3.
We will summarize some facts about this category.

######  Theorem 2.3 ([BLPW12]).

$\mathcal{O}^{{\sf dR}}(\mathfrak{M}_{G}(t),\mathbb{L}_{G}(t,m))$ is a highest
weight category, with simple objects $L^{\alpha}$ corresponding to components
of the skeleton $\mathbb{L}_{G}(t,m)$. Each $L^{\alpha}$ has a projective
cover $P^{\alpha}.$

######  Corollary 2.4.

$\mathcal{O}^{{\sf dR}}(\mathfrak{M}_{G}(t),\mathbb{L}_{G}(t,m))$ is generated
by the simple objects $L^{\alpha}.$

The category $\mathcal{O}^{{\sf dR}}(\mathfrak{M}_{G}(t),\mathbb{L}_{G}(t,m))$
may therefore be described explicitly by calculating the endomorphism algebra
of its simple objects. The main technical tool used in performing this
calculation is the categorical Kirwan surjectivity theorem [BPW]*Theorem 5.31,
which says that the functor induced on deformation quantizations by pullback
along the open inclusion $\mathfrak{M}_{G}(t)\hookrightarrow
T^{*}(\mathbb{C}^{n}/G)$ admits a fully faithful left adjoint. As a result, we
obtain the following description of the de Rham category:

######  Proposition 2.5 ([Web-gencat-O]*Proposition 2.6).

There is an equivalence

(2.1.1) $\mathcal{O}^{{\sf dR}}(\mathfrak{M}(t),\mathbb{L}_{G}(t,m))\simeq{\sf
Dmod}_{\mathbb{L}_{G}^{m\text{-bdd}}}(\mathbb{C}^{n}/G)/{\sf
Dmod}_{\mathbb{L}_{G}^{m\text{-bdd},t\text{-unstab}}}(\mathbb{C}^{n}/G)$

between the de Rham category $\mathcal{O}$ and the quotient of the category of
D-modules with $m$-bounded microsupport by those with purely unstable
microsupport.

In the quotient presentation 2.1.1, the simples in $\mathcal{O}^{{\sf
dR}}(\mathfrak{M}(t),\mathbb{L}_{G}(t,m))$ correspond to the pushforwards of
functions D-modules on the subvarieties
$X^{\alpha}_{G}\hookrightarrow\mathbb{C}^{n}/G$ for the $m$-bounded sign
vectors $\alpha.$ Before quotienting out the unstably-supported D-modules, the
endomorphism algebra of these objects is given by the convolution algebra
$C^{\sf BM}_{*}(X^{m\text{-bdd}}\times_{\mathbb{C}^{n}/G}X^{m\text{-bdd}})$ of
Borel-Moore chains on fiber products of the varieties $X_{G}^{\alpha}$. The
endomorphism algebra of simples in category $\mathcal{O}$ can therefore be
described as a quotient of this algebra:

######  Theorem 2.6 ([Web-gencat-O]*Corollary 3.3).

Let $A^{{\sf dR}}$ be the quotient of the algebra $C^{\sf
BM}_{*}(X^{m\text{-bdd}}\times_{\mathbb{C}^{n}/G}X^{m\text{-bdd}})$ by the
ideal generated by $C^{\sf
BM}_{*}(X^{m\text{-bdd},t\text{-unstab}}\times_{\mathbb{C}^{n}/G}X^{m\text{-bdd},t\text{-unstab}}).$
Then there is an equivalence $\mathcal{O}^{{\sf
dR}}(\mathfrak{M}(t),\mathbb{L}_{G}(t,m))\simeq{\sf Perf}_{A^{{\sf dR}}}$
between the de Rham category $\mathcal{O}$ and the category of perfect modules
over $A^{{\sf dR}},$ induced by an equivalence $A^{{\sf
dR}}\simeq\mathsf{end}_{\mathcal{O}^{{\sf dR}}}(\bigoplus L^{\alpha})$ between
$A^{{\sf dR}}$ and the endomorphism algebra of the simple objects in
$\mathcal{O}^{{\sf dR}}.$

###  2.2. Betti category $\mathcal{O}$ and Riemann-Hilbert

The Betti category $\mathcal{O}$ will be defined as a category of microlocal
perverse sheaves: see [NS20] for the definition of microlocal sheaves on a
general (stably polarized) Weinstein manifold (extending the original work of
[Kashiwara-Schapira] on cotangent bundles) and [perverse-microsheaves] for the
definition of the perverse t-structure on a stably holomorphically polarized
holomorphic Lagrangian.

######  Notation 2.7.

In this section, we take our hypertoric variety $\mathfrak{M}_{G}(t)$ to be
defined over $\mathbb{C},$ and we write $\Bbbk$ for the coefficients in which
our perverse sheaves are valued.

######  Warning 2.8.

As noted in § 0.6, our conventions for the category
$(\mu)\mathsf{Perv}_{\Lambda}$ of (microlocal) perverse sheaves along a
Lagrangian $\Lambda$ differ in two ways from the standard ones:

  1. (1)

We do not require objects to have finite-dimensional microstalks but instead
write $(\mu)\mathsf{Perv}_{\Lambda}$ for the category of compact objects in
the presentable category of perverse sheaves with possibly infinite-
dimensional microstalks. For instance,
$\mathsf{Perv}_{\mathbb{C}^{\times}}(\mathbb{C}^{\times})\simeq\mathsf{Loc}(\mathbb{C}^{\times})$
denotes not the category ${\sf Mod}^{\text{perf}/\Bbbk}_{\Bbbk[x^{\pm}]}$ of
finite-dimensional $\Bbbk[x^{\pm}]$-modules but rather the category ${\sf
Perf}_{\Bbbk[x^{\pm}]}$ of perfect $\Bbbk[x^{\pm}]$-modules — including the
free module $\Bbbk[x^{\pm}],$ which corresponds to the universal local system
on $\mathbb{C}^{\times}$ given by the pushforward of the constant sheaf along
the universal covering map $\mathbb{C}\to\mathbb{C}^{\times}.$

  2. (2)

We denote by $(\mu)\mathsf{Perv}_{\Lambda}$ not the abelian category of
(microlocal) perverse sheaves but rather its derived category.

######  Definition 2.9.

The Betti category $\mathcal{O}$ is the category

$\mathcal{O}_{{\sf
Bet}}(\mathfrak{M}_{G}(t),\mathbb{L}_{G}(t,m)):=\mu\mathsf{Perv}_{\mathbb{L}_{G}(t,m)}(\mathbb{L}_{G}(t,m))$

of microlocal perverse sheaves on the Lagrangian $\mathbb{L}_{G}(t,m),$ with
stable polarization induced from the fiber polarization of
$\mathrm{T}^{*}(\mathbb{C}^{n}/G)$ by [NS-higgs]*Corollary 4.5.

Recall that there is a locally closed embedding
$\mathbb{L}_{G}(t,m)\hookrightarrow\mathbb{L}_{G}=\mathbb{L}/G,$ where
$\mathbb{L}\subset\mathrm{T}^{*}\mathbb{C}^{n}$ is the union of conormals to
intersections of coordinate hyperplanes.

######  Theorem 2.10 ([GGM]).

There is an equivalence between $\mathsf{Perv}_{\mathbb{L}}(\mathbb{C}^{n})$
and (compact objects in) the category of $n$-hypercubes of vector spaces
$\\{V_{\alpha}\mid\alpha\in 2^{[n]}\\}$ equipped with two families of linear
maps

(2.2.1) $u_{i}:V_{\alpha}\rightleftarrows V_{\alpha\cup\\{i\\}}:v_{i}$

such that all the operators $u_{i}$ and $v_{j}$ commute, and such that
$1-u_{i}v_{i}$ and $1-v_{i}u_{i}$ are invertible.

######  Example 2.11.

In case $n=1,$ we recover the classical description of perverse sheaves on
$(\mathbb{C},0)$ in terms of the data

${\Phi}$${\Psi}$$\scriptstyle{var}$$\scriptstyle{can}$

of their vanishing and nearby cycles, with canonical and variation maps
between them. (The general case may be recovered from this one by taking a
tensor product.)

######  Definition 2.12.

For $\alpha\in 2^{[n]},$ we write $P_{\alpha}$ for the object of
$\mathsf{Perv}_{\mathbb{L}}(\mathbb{C}^{n})$ which corepresents the functor
$\mathsf{Perv}_{\mathbb{L}}(\mathbb{C}^{n})\to{\sf Mod}_{\Bbbk}$ taking a
diagram 2.2.1 to the vector space $V_{\alpha}.$

The object $P=\bigoplus_{\alpha\in 2^{[n]}}P_{\alpha}$ is a projective
generator for the category $\mathsf{Perv}_{\mathbb{L}}(\mathbb{C}^{n}).$

######  Example 2.13.

Let $n=1.$ Then, the objects $P_{\Phi},P_{\Psi}$ are given by the respective
diagrams

${\Bbbk[m^{\pm}]}$${\Bbbk[m^{\pm}],}$${\Bbbk[m^{\pm}]}$${\Bbbk[m^{\pm}].}$$\scriptstyle{\sim}$$\scriptstyle{1-m}$$\scriptstyle{1-m}$$\scriptstyle{\sim}$

######  Lemma 2.14.

Let $I\subset 2^{[n]},$ and let
$\mathbb{L}^{I}=\bigcup\mathbb{L}^{\alpha}\subset\mathbb{L}$ be the union of
components $\mathbb{L}^{\alpha}$ with $\alpha\in I.$ Then there is an
equivalence

(2.2.2)
$\mathsf{Perv}_{\mathbb{L}^{I}}(\mathbb{C}^{n})\simeq\mathsf{Perv}_{\mathbb{L}}(\mathbb{C}^{n})/\langle
P_{\alpha}\mid\alpha\notin I\rangle.$

The right adjoint to the quotient functor gives a fully faithful embedding

(2.2.3)
$\mathsf{Perv}_{\mathbb{L}^{I}}(\mathbb{C}^{n})\hookrightarrow\mathsf{Perv}_{\mathbb{L}}(\mathbb{C}^{n})$

whose essential image is the diagrams $(V_{\bullet},u,v)$ with $V_{\alpha}=0$
for each $\alpha\notin I.$

###### Proof.

The equivalence 2.2.2 is described in [GPS3]*Theorem 4.13. On the presentable
categories ${\sf Ind}\mathsf{Perv},$ the quotient functor is left adjoint to
the natural inclusion

(2.2.4) ${\sf
Ind}\mathsf{Perv}_{\mathbb{L}^{I}}(\mathbb{C}^{n})\hookrightarrow{\sf
Ind}\mathsf{Perv}_{\mathbb{L}}(\mathbb{C}^{n}).$

To see that 2.2.4 preserves compact objects, observe that we can understand
this functor as a functor ${\sf Mod}_{A}\to{\sf Mod}_{B},$ where $A$ and $B$
are the endomorphism algebras of the projective generators in the respective
perverse sheaf categories, and in particular, $B$ is the path algebra of the
quiver with relations described in Theorem 2.10. In these terms, $A$ is the
quotient of $B$ by the idempotents at vertices not in $I$, and the functor
${\sf Mod}_{A}\to{\sf Mod}_{B}$ is restriction of scalars along the quotient
map $B\to A.$ Preservation of compact objects now follows from the fact that,
since $A$ is the quotient of $B$ by an idempotent, it is a projective
$B$-module. ∎

The foregoing calculations have been within the cotangent bundle
$T^{*}(\mathbb{C}^{n}),$ but to study the Betti category $\mathcal{O}$ we will
need to microlocalize away from the unstable locus. As on the de Rham side,
the result which will give us access to this microlocalization is a version of
the categorical Kirwan surjectivity theorem, whose proof will appear in [CGH]:

######  Theorem 2.15 ([CGH]).

Let $I^{\prime}\subset I\subset 2^{[n]}.$ Then there is an equivalence

$\mu\mathsf{Perv}(\mathbb{L}^{I}\setminus\mathbb{L}^{I^{\prime}})\simeq\mathsf{Perv}_{\mathbb{L}^{I}}(\mathbb{C}^{n})/\mathsf{Perv}_{\mathbb{L}^{I^{\prime}}}(\mathbb{C}^{n})$

presenting the category of microlocal sheaves on the locally closed subset
$\mathbb{L}^{I}\setminus\mathbb{L}^{I^{\prime}}\subset\mathbb{L}$ as a
quotient of categories of microlocal perverse sheaves on $\mathbb{L}^{I}$ by
those on $\mathbb{L}^{I^{\prime}}.$ The left adjoint to the quotient functor
gives a fully faithful embedding

(2.2.5)
$\mu\mathsf{Perv}(\mathbb{L}^{I}\setminus\mathbb{L}^{I^{\prime}})\hookrightarrow\mathsf{Perv}_{\mathbb{L}_{I}}(\mathbb{C}^{n})$

as the subcategory $of\mathsf{Perv}_{\mathbb{L}_{I}}(\mathbb{C}^{n})$
generated by the objects $P_{\alpha}$ for $\alpha\notin I^{\prime}.$

######  Example 2.16.

Consider the case where $n=1,$ and
$\mathbb{L}^{I^{\prime}}=\mathrm{T}^{*}_{0}\mathbb{C}$ is the cotangent fiber
over 0. In this case, we have

(2.2.6)
$\mu\mathsf{Perv}(\mathbb{L}\setminus\mathbb{L}^{I^{\prime}})\simeq\mu\mathsf{Perv}(\mathbb{C}^{\times})\simeq\mathsf{Loc}(\mathbb{C}^{\times}),$

since microlocal perverse sheaves on the Lagrangian
$\mathbb{C}^{\times}\subset\mathbb{C}\subset\mathrm{T}^{*}\mathbb{C}$ are just
local systems on $\mathbb{C}^{\times}.$ In these terms, the left adjoint to
the quotient functor
$\mathsf{Perv}_{\mathbb{L}}(\mathbb{C})\to\mu\mathsf{Perv}(\mathbb{L}\setminus\mathrm{T}^{*}_{0}\mathbb{C})$
is given by the proper pushforward

${\mathsf{Loc}(\mathbb{C}^{\times})}$${\mathsf{Perv}_{\mathbb{L}}(\mathbb{C})}$$\scriptstyle{j_{!}}$

along the open embedding $\mathbb{C}^{\times}\xrightarrow{j}\mathbb{C}.$ The
right orthogonal to this embedding is the subcategory
$\mathsf{Perv}_{T^{*}_{0}\mathbb{C}}(\mathbb{C})=\mathsf{Perv}_{0}(\mathbb{C})\hookrightarrow\mathsf{Perv}_{\mathbb{L}}(\mathbb{C})$
of perverse sheaves supported at $0\in\mathbb{C}.$

Finally, after our study of microlocal perverse sheaves inside of
$T^{*}\mathbb{C}^{n},$ we must pass to the $G$-quotient. Observe that each
vector space $V_{\alpha}$ in the hypercubical diagram described in Theorem
2.10 is equipped with $n$ commuting automorphisms (from the $1-u_{i}v_{i}$ or
$1-v_{i}u_{i}$), making it a representation of the group $\mathbb{Z}^{n}.$ The
inclusion $G\hookrightarrow(\mathbb{C}^{\times})^{n}$ induces an inclusion on
cocharacter lattices $\pi_{1}G\hookrightarrow\mathbb{Z}^{n},$ so that each
$V_{\alpha}$ is in particular a representation of the free abelian group
$\pi_{1}G.$

######  Lemma 2.17 ([CGH]).

The category $\mathsf{Perv}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)$ is equivalent
to the data of diagrams of vector spaces $(V_{\alpha})$ as in Theorem 2.10
such that the $\pi_{1}G$ action on each vector space $V_{\alpha}$ is trivial.

By combining the above results, we can give an explicit presentation of the
Betti category $\mathcal{O}.$

######  Definition 2.18.

Let $Q$ be the quiver whose vertices are indexed by $\alpha\in 2^{[n]}$ which
are $t$-stable, and whose arrows come in pairs

(2.2.7) $u_{i}:\alpha\rightleftarrows\alpha\cup\\{i\\}:v_{i}$

for each pair of vertices of the form $(\alpha,\alpha\cup\\{i\\}).$

Let $A^{\sf Bet}$ be the algebra $\Bbbk Q\otimes\Bbbk[\pi_{1}F]/I,$ where $I$
is the ideal generated by the following relations:

  1. (1)

$e_{\alpha}=0$ for each $\alpha$ which is $m$-unbounded, where we write
$e_{\alpha}$ for the idempotent (length-0 path) at $\alpha.$

  2. (2)

For each square in $Q$ of the form

${\alpha}$${\alpha^{\prime}}$${\alpha^{\prime\prime}}$${\alpha^{\prime\prime\prime},}$

the two length-2 paths from a vertex to its diagonal opposite commute.

  3. (3)

For each pair of vertices as in 2.2.7, we have
$1-u_{i}v_{i}=\overline{m}_{i}e_{\alpha\cup\\{i\\}},$
$1-v_{i}u_{i}=\overline{m}_{i}e_{\alpha},$ where $\overline{m}_{i}$ is the
image in $\Bbbk[\pi_{1}F]$ of the $i$th generator in
$\Bbbk[\pi_{1}D]\simeq\Bbbk[\mathbb{Z}^{n}]\simeq\Bbbk[m_{1}^{\pm},\ldots,m_{n}^{\pm}].$

######  Theorem 2.19 ([CGH]).

There is an equivalence

$\mathcal{O}^{{\sf Bet}}(\mathfrak{M}_{G}(t),\mathbb{L}_{G}(t,m))\simeq{\sf
Perf}_{A^{\sf Bet}}$

between the Betti category $\mathcal{O}$ and the category of perfect modules
over the algebra $A^{\sf Bet}$ defined in Definition 2.18.

######  Remark 2.20.

The algebra of Definition 2.18 may be most efficiently described in the
language of the hyperplane arrangement $\mathcal{H}_{G}(t)$ of Definition 1.2.
The vertices $\alpha$ which are $t$-stable correspond to chambers in
$\mathcal{H}_{G}(t),$ and pairs of $t$-stable vertices
$\alpha,\alpha\cup\\{i\\}$ are separated by the $i$th hyperplane in the
arrangement. Paths in the quiver $Q$ now correspond to paths between chambers
in $\mathcal{H}_{G}(t),$ and the relation (2) identifies the various shortest
paths between any pair of chambers.

######  Example 2.21.

Let $n=2$ and $G=\mathbb{C}^{\times}\hookrightarrow(\mathbb{C}^{\times})^{2}$
be the kernel of the multiplication map

${(\mathbb{C}^{\times})^{2}}$${\mathbb{C}^{\times}.}$$\scriptstyle{m}$

Then $\mathsf{Perv}_{\mathbb{L}_{G}}(\mathbb{C}^{2}/G)$ is the category of
diagrams

${V^{\\{1\\}}}$${V^{\\{1,2\\}}}$${V^{\emptyset}}$${V^{\\{2\\}},}$$\scriptstyle{v_{1}}$$\scriptstyle{u_{2}}$$\scriptstyle{v_{1}}$$\scriptstyle{v_{2}}$$\scriptstyle{u_{1}}$$\scriptstyle{u_{2}}$$\scriptstyle{u_{1}}$$\scriptstyle{v_{2}}$

where each vector space $V^{\alpha}$ is equipped with an automorphism
$M_{\alpha},$ with the relations

$\displaystyle 1-u_{1}v_{1}$ $\displaystyle=M,$ $\displaystyle 1-u_{2}v_{2}$
$\displaystyle=M^{-1},$ $\displaystyle 1-v_{1}u_{1}$ $\displaystyle=M,$
$\displaystyle 1-v_{2}u_{2}$ $\displaystyle=M^{-1}.$

For an appropriate choice of parameters $t,m,$ the component
$\mathbb{L}_{G}^{\\{2\\}}$ becomes $t$-unstable, while the smooth points of
$\mathbb{L}_{G}^{\emptyset}$ become $m$-unbounded. Now
$\mu\mathsf{Perv}(\mathbb{L}_{G}(t,m))$ is the quotient of
$\mathsf{Perv}_{\mathbb{L}_{G}}(\mathbb{C}^{2}/G)$ by the simple object at
$V^{\\{2\\}}$ and the projective object at $V^{\emptyset}.$ This has the
effect of forgetting $V^{\\{2\\}}$ and setting $V^{\emptyset}$ to 0, so that
$\mu\mathsf{Perv}(\mathbb{L}_{G}(t,m))$ is the category of diagrams

${V^{\\{1\\}}}$${V^{\\{1,2\\}}}$${0,}$$\scriptstyle{u_{2}}$$\scriptstyle{v_{1}}$$\scriptstyle{v_{2}}$$\scriptstyle{u_{1}}$

with monodromies at $V^{\\{1\\}}$ and $V^{\\{1,2\\}}$ and the same relations
as above (except those involving the bottom-right corner). Setting the bottom-
left corner to 0, together with the two relations at the upper-left vertex,
forces $v_{2}u_{2}=0,$ which also forces the final relation (invertibility of
$u_{2}v_{2}$), and we conclude that $\mu\mathsf{Perv}(\mathbb{L}_{G}(t,m))$ is
the category of diagrams of vector spaces

${V^{\\{1\\}}}$${V^{\\{1,2\\}}}$$\scriptstyle{u_{2}}$$\scriptstyle{v_{2}}$

with the single relation $v_{2}u_{2}=0.$ This is the usual description of the
BGG category $\mathcal{O}$ for $SL(2).$

We may reframe Theorem 2.19 as giving a presentation of Betti category
$\mathcal{O}$ as a subquotient of the category
$\mathsf{Perv}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)$:

######  Corollary 2.22.

There is an equivalence

$\mu\mathsf{Perv}(\mathbb{L}_{G}(t,m))\simeq\frac{\langle
P_{G}^{\alpha}\mid\alpha\text{ $t$-semistable}\rangle}{\langle
P_{G}^{\beta}\mid\beta\text{ $m$-unbounded}\rangle}.$

Finally, we note that although in § 2.1 we described $\mathcal{O}^{{\sf
dR}}$ in terms of the endomorphism algebra of simples, it may also be
described in terms of the endomorphism algebra of projectives. The resulting
algebra, described in [BLPW10]*§3, is a quiver algebra of the same form as
$A^{\sf Bet}.$ The following is a microlocal form of the Riemann-Hilbert
correspondence.

######  Proposition 2.23 ([CGH]).

Let $\Bbbk=\mathbb{C}.$ Then there is an equivalence

(2.2.8) $\mathcal{O}^{\sf
dR}(\mathfrak{M}_{G}(t),\mathbb{L}_{G}(t,m))\simeq\mathcal{O}^{\sf
Bet}(\mathfrak{M}_{G}(t),\mathbb{L}_{G}(t,m))$

between the Betti and de Rham categories $\mathcal{O}.$

We refer to [CGH] for more details and an explicit form of this equivalence.

##  3\. Microlocal perverse schobers

Our A-side 2-category $\mathcal{O}$ will be a category of microlocal perverse
schobers on the skeleton $\mathbb{L}(t,m),$ categorifying the presentation of
$\mathcal{O}^{\sf Bet}$ as a category of microlocal perverse sheaves.
Unfortunately, microlocal perverse schobers do not yet have a definition.
However, Theorem 2.15 tells us that microlocal perverse sheaves on
$\mathbb{L}$ can be defined purely in terms of categories of perverse sheaves
on $\mathbb{C}^{n}/G$ with singular supports in closed subsets of
$\mathbb{L}.$ We will therefore define microlocal perverse sheaves of
categories on $\mathbb{L}$ in the analogous way.

###  3.1. Spherical functors and perverse schobers

We begin by recalling that 2-category from [GMH]. We first need the notion of
a spherical adjunction.

######  Definition 3.1 ([RV-adj]).

The free adjunction is the (discrete, i.e., $(2,2)$ rather than $(\infty,2)$)
2-category $\mathsf{Adj}$ generated by 1-morphisms

(3.1.1) ${\Phi}$${\Psi}$$\scriptstyle{L}$$\scriptstyle{R}$

and the 2-morphisms ${\sf id}_{\Phi}\to RL,LR\to{\sf id}_{\Psi}$, with the
relations that the compositions $R\to RLR\to R$ and $L\to LRL\to L$ are the
identity maps ${\sf id}_{R}$ and ${\sf id}_{L},$ respectively.

######  Theorem 3.2 ([RV-adj]).

The 2-category $\mathsf{Adj}$ corepresents adjunctions in 2-categories.

######  Definition 3.3.

Let $\overline{\mathsf{Adj}}:=\Sigma^{(\infty,2)}\mathsf{Adj}$ be the
$\Bbbk$-linear stabilization of the 2-category $\mathsf{Adj}.$ The universal
(co)twist is the 1-morphism

$T_{\Phi}:={\sf fib}({\sf id}_{\Phi}\to RL)\qquad(\text{resp., }T_{\Psi}:={\sf
cofib}(LR\to{\sf id}_{\Psi}))$

in $\overline{\mathsf{Adj}}$ given by the (co)fiber of the (co)unit of the
universal adjunction 3.1.1.

The free spherical adjunction is the stable 2-category

$\mathsf{Sph}:=\overline{\mathsf{Adj}}[T_{\Phi}^{-1},T_{\Psi}^{-1}]$

obtained from $\mathsf{Adj}$ by localizing at the 1-morphisms
$T_{\Phi},T_{\Psi}.$999This differs from the usage in [GMH], where
$\mathsf{Sph}$ was synonymous with the 2-category
$\mathsf{PervCat}_{\mathbb{L}}(\mathbb{C}).$

Let $\mathcal{C}\rightleftarrows\mathcal{D}$ be an adjunction in stable
categories, which (by Theorem 3.2 and the stability of $\mathcal{C}$)
corresponds to a functor $F:\overline{\mathsf{Adj}}\to{\sf Cat}.$ If $F$
admits a factorization $\overline{\mathsf{Adj}}\to\mathsf{Sph}\to{\sf Cat},$
then we say that the adjunction $\mathcal{C}\rightleftarrows\mathcal{D}$ is
spherical, and that each of the adjoint functors is a spherical functor.

As usual, let
$\mathbb{L}=\bigcup_{\alpha}\mathrm{T}^{*}_{X^{\alpha}}\mathbb{C}^{n}\subset
T^{*}\mathbb{C}^{n}$ be the union of conormals to intersections of coordinate
hyperplanes.

######  Definition 3.4.

The 2-category of perverse schobers on $\mathbb{C}^{n}$ with singular support
in $\mathbb{L}$ (and coefficients in the 2-category ${\sf St}$ of stable
categories) is the 2-category
$\mathsf{PervCat}_{\mathbb{L}}(\mathbb{C}^{n}):={\sf
Fun}(\mathsf{Sph}^{\otimes n},{\sf St})$ of $\mathsf{Sph}^{\otimes
n}$-diagrams in stable categories.

######  Notation 3.5.

An object of $\mathsf{PervCat}_{\mathbb{L}}(\mathbb{C}^{n})$ is specified by
the data of a spherical categorical n-cube in the sense of [CDW]*Definition
5.4.3, namely a functor $2^{[n]}\to{\sf St}$ (where we treat $2^{[n]}$ as a
partially ordered set) such that each edge of the cube is a spherical functor,
and each face of the cube satisfies the Beck-Chevalley condition of
[CDW]*Definition 4.5.7. Such an object consists of a collection of categories
$(\mathcal{C}^{\alpha})_{\alpha\in 2^{[n]}},$ together with spherical
adjunctions

$u_{i}:\mathcal{C}_{\alpha}\rightleftarrows\mathcal{C}_{\alpha\cup\\{i\\}}:v_{i}$

satisfying the appropriate commutativity conditions. We will denote this
object by $(\mathcal{C}^{\bullet},u,v).$

The $D=(\mathbb{C}^{\times})^{n}$ action on $\mathbb{C}^{n}$ should make
$\mathsf{PervCat}_{\mathbb{L}}(\mathbb{C}^{n})$ into a topological
$D$-2-category (i.e., the underlying 2-category of a functor $BD\to 2{\sf
Pr}^{L,{\sf st}},$ where the target is the 3-category of presentable stable
2-categories), and indeed this structure was described in [GMH].

######  Definition 3.6.

The universal twist is the central element $\mathbf{T}\in\mathsf{aut}({\sf
id}_{\mathsf{Sph}})$ which acts on a spherical adjunction

${\mathcal{C}_{\Phi}}$${\mathcal{C}_{\Psi}~{}}$$\scriptstyle{F}$$\scriptstyle{\bot}$$\scriptstyle{F^{R}}$

by the automorphism

${\mathcal{C}_{\Phi}}$${\mathcal{C}_{\Psi}}$${\mathcal{C}_{\Phi}}$${\mathcal{C}_{\Psi}~{}.}$$\scriptstyle{F}$$\scriptstyle{\bot}$$\scriptstyle{F^{R}}$$\scriptstyle{T_{\Phi}}$$\scriptstyle{T_{\Psi}[-2]}$$\scriptstyle{F}$$\scriptstyle{\bot}$$\scriptstyle{F^{R}}$

######  Lemma 3.7 ([GMH], Proposition 3.16).

The universal twist is the generator of an $\mathbb{E}_{2}$-map
$\mathbb{Z}\to\mathsf{end}({\sf id}_{\mathsf{Sph}}),$ defining a topological
$B\mathbb{Z}$-action on $\mathsf{Sph}.$

By taking products, we obtain a topological $B(\mathbb{Z}^{n})\simeq D$-action
on $\mathsf{Sph}^{n},$ which we can restrict along the inclusion $G\to D$ to
obtain a $G$-action.

######  Definition 3.8.

We write $\mathsf{Sph}_{G}^{n}$ for the coinvariants of the $G$-action on
$\mathsf{Sph}.$ (See [GMH]*Lemma A.4 for the construction of these
coinvariants.) The 2-category of $G$-equivariant perverse schobers is defined
as

$\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G):={\sf
Fun}(\mathsf{Sph}^{n}_{G},{\sf St}).$

The 2-category $\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)$ is
therefore obtained as the invariants
$\mathsf{PervCat}_{\mathbb{L}}(\mathbb{C}^{n})^{G}$ for the $G$-action on
$\mathsf{PervCat}_{\mathbb{L}}(\mathbb{C}^{n}).$ An object of the resulting
2-category may be specified by the data of a spherical $n$-cube equiped with
trivializations of the appropriate compositions of (co)twist automorphisms on
each category $\mathcal{C}^{\alpha}.$ We will continue to denote such an
object by $(\mathcal{C}^{\bullet},u,v),$ leaving the trivializations implicit
in our notation.

Certain objects in $\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)$ will
play a distinguished role. Recall that an object of $\mathsf{Sph}^{n}$ may be
specified by a choice of $\alpha\in 2^{[n]}$; we continue to write $\alpha$ to
denote the image of this object in $\mathsf{Sph}^{n}_{G}.$

######  Definition 3.9.

The object $\alpha$ in $\mathsf{Sph}^{n}_{G}$ determines a corepresentable
functor ${\sf Hom}_{\mathsf{Sph}^{n}_{G}}(\alpha,-)$; we will refer to this
object of ${\sf Fun}(\mathsf{Sph}^{n}_{G},{\sf
St})=\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)$ by
$\mathcal{P}^{\alpha}_{G}.$

Tautologically, the object $\mathcal{P}^{\alpha}_{G}$ corepresents the
functional on $\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)$ which
takes a $G$-invariant spherical $n$-cube $(\mathcal{C}^{\bullet},u,v)$ to the
category $\mathcal{C}^{\alpha}.$ The objects $\mathcal{P}^{\alpha}_{G}$ are
therefore generators of the 2-category
$\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathsf{Sph}^{n}_{G},{\sf St})$; the main
theorem of [GMH], which we have stated as Theorem 0.1 above, is a calculation
of their monoidal category of endomorphisms
$\mathsf{end}\left(\bigoplus\mathcal{P}_{\alpha}\right).$

We can use the $\mathcal{P}^{\alpha}_{G}$ to describe perverse schobers on
$\mathbb{C}^{n}/G$ with singular support in closed subsets of
$\mathbb{L}_{G}.$

######  Definition 3.10.

Let $I\subset 2^{[n]},$ and $\mathbb{L}^{I}_{G}:=\bigcup_{\alpha\in
I}\mathbb{L}_{G}.$ Then the 2-category of perverse schobers on
$\mathbb{C}^{n}/G$ with singular support in $\mathbb{L}_{G}^{I}$ is the
quotient 2-category

(3.1.2)
$\mathsf{PervCat}_{\mathbb{L}_{G}^{I}}(\mathbb{C}^{n}/G):=\frac{\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)}{\langle\mathcal{P}_{\alpha}\rangle_{\alpha\notin
I}}.$

######  Lemma 3.11.

The right adjoint to the quotient functor 3.1.2 gives a fully faithful
embedding
$\mathsf{PervCat}_{\mathbb{L}_{G}^{I}}(\mathbb{C}^{n}/G)\hookrightarrow\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)$
as the sub-2-category of diagrams $(\mathcal{C}^{\bullet},u,v)$ where
$\mathcal{C}^{\alpha}\simeq 0$ for each $\alpha\notin I$.

###  3.2. 2-category $\mathcal{O}$

So far we have understood the 2-category
$\mathsf{PervCat}_{\mathbb{L}^{G}}(\mathbb{C}^{n})$ as a categorification of
equivariant perverse sheaves on $\mathbb{C}^{n}.$ We will now treat it
microlocally, imitating the description of microlocal perverse sheaves in §
2.2.

######  Definition 3.12.

For a stability parameter $t$ and an attraction parameter $m,$ the 2-category
of microlocal perverse schobers on $\mathbb{L}_{G}(t,m)$ is defined as the
quotient

(3.2.1)
$\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m)):=\frac{\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)}{\mathsf{PervCat}_{\mathbb{L}_{G}^{t\text{-unstab}}}(\mathbb{C}^{n}/G),\langle\mathcal{P}_{G}^{\alpha}\mid\alpha\text{
$m$-unbounded}\rangle}.$

We can also present 3.2.1 as a subquotient of
$\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)$, matching the
presentation of Betti category $\mathcal{O}$ given in Corollary 2.22:

######  Lemma 3.13.

There is an equivalence
$\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))\simeq\frac{\langle\mathcal{P}_{G}^{\alpha}\mid\text{$\alpha$
$t$-semistable}\rangle}{\langle\mathcal{P}_{G}^{\beta}\mid\text{$\beta$
$m$-unbounded}\rangle}.$

###### Proof.

As in the 1-categorical case, this follows from the fact that the left adjoint
to the quotient functor

$\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)\to\frac{\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)}{\mathsf{PervCat}_{\mathbb{L}_{G}^{t\text{-unstab}}}(\mathbb{C}^{n}/G)}$

embeds the quotient as the full sub-2-category of
$\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{n}/G)$ generated by the objects
$\mathcal{P}^{\alpha}_{G}$ for $t$-stable $\alpha.$ ∎

Categorifying Definition 2.18, we can give an explicit description of objects
in the 2-category $\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m)).$

######  Definition 3.14.

Quiver data for $\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))$ is the following
collection of data:

  * •

For each $\alpha$ which is both $t$-stable and $m$-bounded, a category
$\mathcal{C}^{\alpha}$.

  * •

For each $\mathcal{C}^{\alpha},$ a map
$\pi_{1}(F)\to\mathsf{aut}(\mathcal{C}^{\alpha})$, which we understand as a
map $\pi_{1}(D)\to\mathsf{aut}(\mathcal{C}^{\alpha})$ with a trivialization of
the composite $\pi_{1}(G)\to\pi_{1}(D)\to\mathsf{aut}(\mathcal{C}^{\alpha}).$
We write $T^{\alpha}_{i}\in\mathsf{aut}(\mathcal{C}^{\alpha})$ for the images
of the canonical generators of $\pi_{1}(D)\simeq\mathbb{Z}^{n}.$

  * •

On each $\mathcal{C}^{\alpha},$ a monad $M^{\alpha}_{i}$ for each
$i\in\alpha,$ and a comonad $C^{\alpha}_{i}$ for each $i\notin\alpha.$

  * •

For each $t$-stable pair $(\alpha,\alpha\cup\\{i\\}),$ a spherical adjunction

${\mathcal{C}^{\alpha}}$${\mathcal{C}^{\alpha\cup\\{i\\}}}$$\scriptstyle{u_{i}}$$\scriptstyle{v_{i}}$

together with equivalences $M^{\alpha\cup\\{i\\}}_{i}\simeq
u_{i}v_{i},C^{\alpha}_{i}\simeq v_{i}u_{i}$ of monads (resp. comonads), where
we take the convention that $\mathcal{C}^{\alpha}\simeq 0$ if $\alpha$ is not
$m$-bounded

  * •

For each $\mathcal{C}^{\alpha},$ an equivalence $T_{i}^{\alpha}\simeq{\sf
fib}({\sf id}_{\mathcal{C}^{\alpha}}\xrightarrow{\eta}M)$ for each
$i\in\alpha,$ and an equivalence $T_{i}^{\alpha}\simeq{\sf cofib}({\sf
id}_{\mathcal{C}^{\alpha}}\xrightarrow{\eta}M)[-2]$ for each $i\notin\alpha.$

We moreover require these data to satisfy the conditions that the $u_{i}$’s
and $v_{i}$’s all commute with each other and with the monads $M_{i}^{\alpha}$
and comonads $C_{i}^{\alpha}.$

It can be checked that quiver data as above determines an object in the
2-category $\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m)).$ We expect that it is
possible to define a stable 2-category directly from this quiver data,
although we have not done this here. If done correctly, this combinatorially
defined 2-category will agree with the 2-category
$\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))$ defined in Definition 3.12.

######  Example 3.15.

Here we consider the categorification of Example 2.21. Once again, let

$G\simeq\mathbb{C}^{\times}\hookrightarrow(\mathbb{C}^{\times})^{2}$

be the kernel of the multiplication map. Before imposing $G$-equivariance, we
have that $\mathsf{PervCat}_{\mathbb{L}}(\mathbb{C}^{2})$ is the 2-category of
Beck-Chevalley squares of spherical functors

(3.2.2)
${\mathcal{C}^{\\{1\\}}}$${\mathcal{C}^{\\{1,2\\}}}$${\mathcal{C}^{\emptyset}}$${\mathcal{C}^{\\{2\\}},}$$\scriptstyle{v_{1}}$$\scriptstyle{v_{2}}$$\scriptstyle{v_{1}}$$\scriptstyle{v_{2}}$

Sphericality equips each category $\mathcal{C}^{\alpha}$ with two
automorphisms, the relevant two of

$\displaystyle{\sf fib}({\sf id}_{\mathcal{C}^{\alpha}}\to$ $\displaystyle
v_{2}^{R}v_{2}),$ $\displaystyle{\sf cofib}(v_{2}v_{2}^{R}\to$
$\displaystyle{\sf id}_{\mathcal{C}^{\alpha}})[-2],$ $\displaystyle{\sf
fib}({\sf id}_{\mathcal{C}^{\alpha}}\to$ $\displaystyle v_{1}^{R}v_{1}),$
$\displaystyle{\sf cofib}(v_{1}v_{1}^{R}\to$ $\displaystyle{\sf
id}_{\mathcal{C}^{\alpha}})[-2].$

We obtain $\mathsf{PervCat}_{\mathbb{L}_{G}}(\mathbb{C}^{2}/G)$ from this
2-category by imposing additionally an equivalence between the two
automorphisms at each category $\mathcal{C}^{\alpha}.$

Finally, to produce the 2-category $\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m)),$
we must quotient by the simple object supported on $\mathcal{C}^{\\{2\\}}$ and
by the projective $\mathcal{P}_{G}^{\emptyset}.$ The first of these operations
has the effect of forgetting the lower-right vertex from the diagram 3.2.2
(and the functors ingoing and outgoing from it — but not the associated monad
$v_{1}^{R}v_{1}$ on $\mathcal{C}^{\\{1,2\\}}$). The latter operation sets to 0
the category $\mathcal{C}^{\emptyset}$, along with the monad $v_{1}^{R}v_{1}$
on $\mathcal{C}^{\\{1\\}},$ so that the automorphism at
$\mathcal{C}^{\\{1\\}}$ (which remains identified with ${\sf
cofib}(v_{2}v_{2}^{R}\to{\sf id}_{\mathcal{C}^{\\{1\\}}})[-2]$) is identified
with ${\sf id}_{\mathcal{C}^{\\{1\\}}}.$

In summary, the 2-category $\mu\mathsf{PervCat}(\mathbb{L}_{G}(t,m))$ is the
2-category of spherical functors

${\mathcal{C}^{\\{1\\}}}$${\mathcal{C}^{\\{1,2\\}}}$$\scriptstyle{v_{2}}$

together with an extra monad $``v_{1}^{R}v_{1}"$ on $\mathcal{C}^{\\{1,2\\}},$
and identifications

${\sf fib}({\sf
id}_{\mathcal{C}^{\\{1,2\\}}}\xrightarrow{\eta}``v_{1}^{R}v_{1}")\simeq{\sf
fib}({\sf
id}_{\mathcal{C}^{\\{1,2\\}}}\xrightarrow{\eta}v_{2}^{R}v_{2}),\qquad{\sf
cofib}(v_{2}v_{2}^{R}\to{\sf id}_{\mathcal{C}^{\\{1\\}}})\simeq[2].$

##  4\. Microlocal coherent schobers

###  4.1. Coherent sheaves of categories

Beginning with a symplectic variety $\mathfrak{M},$ a proposal is given in
[KRS, KR] for a an algebraic 2-category whose objects are supported on
Lagrangian subvarieties in $\mathfrak{M},$ and it is further explained that
when $\mathfrak{M}=\mathrm{T}^{*}Y$ is a cotangent bundle, the part of the
category supported on a conical Lagrangian $\mathbb{L}\subset\mathrm{T}^{*}Y$
may be described in terms of coherent-sheaf categories of spaces proper over
$Y.$ (This may motivate the notation in [Tel-ICM] for this 2-category as
$\sqrt{\mathfrak{Coh}}(\mathrm{T}^{*}Y).$)

Given a conic Lagrangian $\mathbb{L}\subset\mathrm{T}^{*}Y$ in the cotangent
bundle of an algebraic stack $Y$, we are therefore motivated to study the
2-category of coherent sheaves of categories over $Y$ with singular support in
$\mathbb{L}.$ A full mathematical account of this 2-category has not yet
appeared in the literature, though it is expected in forthcoming works of
Stefanich and Arinkin.101010See [Ari-talk] for motivation for the construction
of this 2-category, and [Stefanich-QCoh] for the simpler case of quasicoherent
sheaves of ($n$-)categories.) Nevertheless, enough is known about this
2-category for our present purposes.

We will need only the following two facts: first, that if $X$ is a smooth
stack with a proper map $X\to Y,$ then $\mathsf{Coh}(X)$ defines an object in
this 2-category, and second, that the Hom category between two such objects
$\mathsf{Coh}(X)$ and $\mathsf{Coh}(X^{\prime})$ is the category ${\sf
Fun}_{{\sf Perf}(Y)}^{ex}(\mathsf{Coh}(X),\mathsf{Coh}(X^{\prime}))$ of exact
${\sf Perf}(Y)$-linear functors between the coherent-sheaf categories, with
compositions among these Hom categories given by composition of functors.
These Hom categories have been computed directly in [BZNP-kernels].

######  Theorem 4.1 ([BZNP-kernels]*Theorem 1.1.3).

Let $X,X^{\prime},Y$ be smooth perfect stacks, and let $X\to Y\leftarrow
X^{\prime}$ be proper maps. Then there is an equivalence

${\sf Fun}_{{\sf
Perf}(Y)}(\mathsf{Coh}(X),\mathsf{Coh}(X^{\prime}))\simeq\mathsf{Coh}(X\times_{Y}X^{\prime})$

between the category of exact ${\sf Perf}(Y)$-linear functors and the category
of coherent sheaves on the fiber product $X\times_{Y}X^{\prime},$ associating
to a coherent sheaf $K$ on $\mathsf{Coh}(X\times_{Y}X^{\prime})$ the integral
transform functor $p_{X^{\prime},*}(p_{X}^{*}(-)\otimes K).$ Moreover, in the
case $X=X^{\prime},$ this is naturally a monoidal equivalence.

Theorem 4.1 justifies the following definition, due to Arinkin.

######  Definition 4.2.

Let $X,Y$ be smooth perfect stacks and $f:X\to Y$ a proper map, and write
$\overline{\mathrm{N}^{*}_{X}Y}$ for the image in $T^{*}Y$ of the conormal
Lagrangian $\mathrm{N}^{*}_{X}Y.$ Then we say the 2-category of coherent
sheaves of categories on $Y$ with singular support
$\overline{\mathrm{N}^{*}_{X}Y}$ is the 2-category

(4.1.1) $\mathsf{CohCat}_{\overline{\mathrm{N}^{*}_{X}Y}}(Y):={\sf
Mod}_{\mathsf{Coh}(X\times_{Y}X)}({\sf St})$

of module categories for the monoidal category $\mathsf{Coh}(X\times_{Y}X).$

######  Remark 4.3.

The notation in Definition 4.2 may seem misleading, as a priori it is unclear
that the 2-category defined by 4.1.1 depends only on the subset
$\overline{\mathrm{N}^{*}_{X}Y}\subset\mathrm{T}^{*}Y$, rather than on the
choice of map $X\to Y$. This invariance statement is promised as [Arinkin-
notes]*Theorem 10, but so far a proof has not appeared in the literature. We
provide a proof below as Proposition 4.18.

###  4.2. Singular support

We would like to introduce a modification of Definition 4.2 to account for
non-closed singular support conditions. For this, we will need the theory of
coherent singular support developed in [AG]. Recall that $Z$ a quasi-smooth
Artin stack, the stack of singularities of $Z$ is the classical Artin stack

$\mathsf{Sing}(Z):=(\mathrm{T}^{*}[-1]Z)^{cl}$

obtained by taking the underlying classical stack of the $(-1)$-shifted
cotangent bundle. If $Z$ is affine, there is a map

$\Gamma(\mathsf{Sing}(Z),\mathcal{O}_{\mathsf{Sing}(Z)})\to{\sf
HH}^{\text{even}}(Z)$

from functions on $\mathsf{Sing}(Z)$ to the even Hochschild cohomology of $Z$.
As a result, in general, the category $\mathsf{Coh}(Z)$ localizes over
$\mathsf{Sing}(Z)$ (in the conic topology).

######  Definition 4.4.

Let $\Lambda\subset\mathsf{Sing}(Z)$ be a conic closed subset, with open
complement $U$. Then we write
$\mathsf{Coh}_{\Lambda}(Z)\subset\mathsf{Coh}(Z)$ for the full subcategory of
coherent sheaves supported on $\Lambda,$ and
$\mathsf{Coh}_{U}(Z):=\mathsf{Coh}(Z)/\mathsf{Coh}_{\Lambda}(Z)$ for the
localization away from sheaves supported on $\Lambda.$

######  Definition 4.5.

Given a map $g:Z\to W$ of quasi-smooth stacks, consider the correspondence

(4.2.1)
${\mathsf{Sing}(Z)}$${\mathsf{Sing}(W)\times_{W}Z}$${\mathsf{Sing}(W),}$$\scriptstyle{dg^{*}}$$\scriptstyle{\tilde{g}}$

which we will use to define a pushforward and pullback of singular support
conditions along $g$:

  1. (1)

For a conic subset $U\subset\mathsf{Sing}(Z),$ we define

$g_{*}U:=\tilde{g}((dg^{*})^{-1}(U))\subset\mathsf{Sing}(W).$

  2. (2)

For a conic subset $V\subset\mathsf{Sing}(W),$ we define

$g^{!}V:=df^{*}(Z\times_{W}V)\subset\mathsf{Sing}(Z).$

######  Lemma 4.6 ([AG]*Proposition 7.1.3).

The pushforward and pullback functors preserve singular supports, inducing
functors

$f_{*}:\mathsf{Coh}_{\Lambda_{Z}}(Z)\to\mathsf{Coh}_{f_{*}\Lambda_{Z}}(W),\qquad
f^{!}:\mathsf{Coh}_{\Lambda_{W}}(W)\to\mathsf{Coh}_{f^{!}\Lambda_{Z}}(Z).$

We refer to [AG] for more information about coherent singular support
conditions.

###  4.3. Monoidal structure

######  Notation 4.7.

Throughout this section, we fix smooth perfect stacks $Y,X_{i}$ and proper
maps $f_{i}:X_{i}\to Y.$ We write $X_{ij}:=X_{i}\times_{Y}X_{j}$ for the fiber
products over $Y$, and similarly
$X_{ijk}:=X_{i}\times_{Y}X_{j}\times_{Y}X_{k}$ for the triple fiber products.

In Theorem 4.1 and Definition 4.2 we have made use of the functoriality of
$\mathsf{Coh}$ under correspondences: to a correspondence

(4.3.1) ${Z}$${C}$${Z^{\prime}}$$\scriptstyle{f}$$\scriptstyle{g}$

with $g$ proper, we associate the functor.

(4.3.2) $g_{*}f^{*}:\mathsf{Coh}(Z)\to\mathsf{Coh}(Z^{\prime}).$

In the case where the correspondence 4.3.1 is

(4.3.3) ${X_{ij}\times
X_{jk}}$${X_{ijk}}$${X_{ik},}$$\scriptstyle{(p_{ij},p_{jk})}$$\scriptstyle{p_{jk}}$

we recover the rule for composition of integral transforms:

######  Definition 4.8.

Using 4.3.2, the correspondence 4.3.3 (together with the higher fiber products
for associativity data) defines a monoidal structure on the category
$\mathsf{Coh}(X_{ii})$ together with a
$\mathsf{Coh}(X_{ii}),\mathsf{Coh}(X_{jj})$-bimodule structure on the category
$\mathsf{Coh}(X_{ij}).$ We call this the convolution monoidal structure.

We will need to understand how this algebraic structure on the categories
$\mathsf{Coh}(X_{ij})$ interacts with the the theory of coherent singular
support. This question has been previously studied in [BZNP, BZCHN].

######  Definition 4.9.

Let $\Lambda_{12}\subset\mathsf{Sing}(X_{12})$ and
$\Lambda_{23}\subset\mathsf{Sing}(X_{23}),$ and consider the diagram

(4.3.4) ${X_{12}\times
X_{23}}$${X_{123}}$${X_{13},}$$\scriptstyle{\delta_{2}}$$\scriptstyle{p_{13}}$

where we write $\delta_{2}$ as shorthand for $(p_{12},p_{23}).$ The
convolution of singular support conditions $\Lambda_{ij}$ is defined by

(4.3.5)
$\Lambda_{12}*\Lambda_{23}:=(p_{13})_{*}(\delta_{2})^{!}(\Lambda_{12}\boxtimes\Lambda_{23}).$

The singular support condition $\Lambda_{ij}$ is $X_{ii}$-stable if it is
preserved by convolution with $\mathsf{Sing}(X_{ii})$:

$\mathsf{Sing}(X_{ii})*\Lambda_{ij}\subset\Lambda_{ij}.$

######  Remark 4.10.

The operation of convolution may be more succinctly reframed in the language
of Lagrangian correspondences: given a correspondence 4.3.1, the shifted
conormal to $C$ defines a Lagrangian correspondence

(4.3.6) ${\mathrm{T}^{*}[-1]X}$${\mathrm{N}^{*}_{C}[-1](X\times
Y)}$${\mathrm{T}^{*}[-1]Y.}$

A Lagrangian $\Lambda\to\mathrm{T}^{*}[-1]X$ may be understood as a Lagrangian
correspondence

${{\sf pt}}$${\Lambda}$${\mathrm{T}^{*}[-1]X,}$

which we may compose with the Lagrangian correspondence
$\mathrm{N}^{*}_{C}[-1](X\times Y)$ to obtain a new Lagrangian

(4.3.7) $\Lambda\circ C:=\Lambda\times_{\mathrm{N}^{*}_{C}[-1](X\times
Y)}\mathrm{T}^{*}[-1]X\to\mathrm{T}^{*}[-1]Y.$

After taking classical part $(\Lambda\circ C)^{cl},$ we recover the singular-
support condition $g_{*}f^{!}\Lambda$ obtained by transferring the singular-
support condition $\Lambda^{cl}$ along the correspondence $C$.

Moreover, the Lagrangian correspondence 4.3.6 may be factored as the
composition of a pair of Lagrangian correspondences. Given a map $g:Z\to
Z^{\prime},$ the shifted conormal to the graph of $Z$ is equivalent to the
pullback of the shifted cotangent bundle,

$\mathrm{N}^{*}_{\Gamma_{g}}[-1](Z\times
Z^{\prime})\simeq\mathrm{T}^{*}[-1]Z^{\prime}\times_{Z^{\prime}}Z,$

and therefore it defines the Lagrangian correspondence 4.2.1, with the
pushforward and pullback $g_{*},g^{!}$ being the respective pushforward and
pullback with this Lagrangian correspondence with its reverse. Given a
correspondence 4.3.1, the convolution 4.3.7 is the composition of the
pushforward and pulback $g_{*}f^{!},$ implemented by the composed Lagrangian
correspondence

(4.3.8) $\mathrm{N}^{*}_{C}[-1](X\times
Y)\simeq\mathrm{N}^{*}_{\Gamma_{f}}[-1](X\times
C)\circ\mathrm{N}^{*}_{\Gamma_{g}}[-1](C\times X).$

In what follows, we will often be able to simplify computations involving
convolution using the language of Lagrangian correspondences, with the
understanding that the Lagrangians discussed in these terms are derived
schemes, and at the end of a computation we must pass to their classical parts
to recover a singular support condition in the singularity space
$\mathsf{Sing}.$

We would like to relate the convolution of Lagrangians to the categorical
convolution structures of Definition 4.8.

######  Definition 4.11.

We will write $L_{i}:=\mathrm{N}^{*}_{X_{i}}Y$ for the conormal to the map
$X_{i}\to Y,$ which comes equipped with a map $L_{i}\to\mathrm{T}^{*}Y$ We
write

$L_{ij}:=L_{i}\times_{\mathrm{T}^{*}Y}L_{j},\qquad
L_{ijk}:=L_{i}\times_{\mathrm{T}^{*}Y}L_{j}\times_{\mathrm{T}^{*}Y}L_{k}$

for their fiber products over $\mathrm{T}^{*}Y,$ and write $q_{i}$ (resp.
$q_{ij}$) for the projections to $L_{i}$ (resp. $L_{ij}$).

######  Lemma 4.12.

There is an equivalence $\mathrm{T}^{*}[-1]X_{ij}\simeq L_{ij}.$ Taking
classical parts, we obtain an equivalence $\mathsf{Sing}(X_{ij})\simeq
L_{ij}^{cl}.$

###### Proof.

Consider the cocartesian diagram

${\mathbf{L}_{Y}}$${\mathbf{L}_{X_{j}}}$${0}$${\mathbf{L}_{X_{i}}}$${\mathbf{L}_{X_{ij}}}$${\mathbf{L}_{f_{i}}}$${0}$${\mathbf{L}_{f_{j}}}$${\mathbf{L}_{Y}[1]}$

of sheaves on $X_{ij}$. Here the upper left cocartesian square is the standard
formula for the cotangent complex of a pullback, the left and upper
cocartesian rectangles define the relative cotangent complexes, and the outer
cocartesian square is the definition of the suspension.

The desired map $\mathrm{T}^{*}[-1]X_{ij}\to L_{ij}$ comes from noticing that
$L_{i}$ and $L_{j}$ are the total spaces of $\mathbf{L}_{f_{i}}[-1]$ and
$\mathbf{L}_{f_{j}}[-1]$ respectively. The fact that this is a
symplectomorphism is Remark 2.20 in [Calaque]. ∎

######  Lemma 4.13.

There is an equivalence of Lagrangian correspondences

${\mathrm{T}^{*}[-1]X_{12}\times\mathrm{T}^{*}[-1]X_{23}}$${\mathrm{N}^{*}_{X_{123}}[-1](X_{12}\times
X_{23}\times X_{13})}$${\mathrm{T}^{*}[-1]X_{13}}$${L_{12}\times
L_{23}}$${L_{123}}$${L_{13}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

between the correspondence underlying the operation of convolution on
Lagrangians and that given by the triple fiber product of conormals.

###### Proof.

There is a strongly cocartesian diagram

${\mathbf{L}_{X_{2}}}$${\mathbf{L}_{X_{23}}}$${\mathbf{L}_{Y}}$${\mathbf{L}_{X_{3}}}$${\mathbf{L}_{X_{12}}}$${\mathbf{L}_{X_{123}}}$${\mathbf{L}_{X_{1}}}$${\mathbf{L}_{X_{13}}}$

of sheaves on $X_{123}$. Using the fact that the total cofiber of any face of
the cube must vanish, we find

$\displaystyle{\sf
cofib}(\mathbf{L}_{X_{13}}\to\mathbf{L}_{X_{123}})\simeq{\sf
cofib}(\mathbf{L}_{X_{3}}\to\mathbf{L}_{X_{23}})\simeq{\sf
cofib}(\mathbf{L}_{Y}\to\mathbf{L}_{X_{2}})\simeq\mathbf{L}_{f_{2}}.$

The $(-1)$-shifted conormal bundle $\mathrm{T}^{*}_{X_{123}}[-1](X_{12}\times
X_{13}\times X_{23})$ is the total space of the complex

$\displaystyle{\sf
fib}(\mathbf{L}_{X_{12}}\oplus\mathbf{L}_{X_{13}}\oplus\mathbf{L}_{X_{23}}\to\mathbf{L}_{X_{123}})[-1]$
$\displaystyle\simeq{\sf
cofib}(\mathbf{L}_{X_{12}}\oplus\mathbf{L}_{X_{13}}\oplus\mathbf{L}_{X_{23}}\to\mathbf{L}_{X_{123}})[-2]$
$\displaystyle\simeq{\sf
cofib}(\mathbf{L}_{X_{12}}\oplus\mathbf{L}_{X_{23}}\to{\sf
cofib}(\mathbf{L}_{X_{13}}\to\mathbf{L}_{X_{123}}))[-2]$
$\displaystyle\simeq{\sf
cofib}(\mathbf{L}_{X_{12}}\oplus\mathbf{L}_{X_{23}}\to\mathbf{L}_{f_{2}})[-2]$
$\displaystyle\simeq{\sf
fib}(\mathbf{L}_{X_{12}}\oplus\mathbf{L}_{X_{23}}\to\mathbf{L}_{f_{2}})[-1].$

The total space of the last complex is $L_{12}\times_{L_{2}}L_{23}\simeq
L_{123},$ as desired. ∎

######  Definition 4.14.

We write

$F:\mathsf{Sing}(X_{ij})\to\mathrm{T}^{*}Y$

for the map induced on classical parts by the projection
$L_{ij}=L_{i}\times_{\mathrm{T}^{*}Y}L_{j}\to\mathrm{T}^{*}Y.$

The map $F$ just defined gives some justification for the relevance of
singular support conditions, and the relation to Definition 4.2.

######  Lemma 4.15.

The image of the map $F$ coincides with the intersection in $\mathrm{T}^{*}Y$
of the images of the conormals $\mathrm{N}^{*}_{X_{i}}Y\to\mathrm{T}^{*}Y.$ In
particular, when $X_{i}=X_{j}=X,$ the image of $F$ coincides with the image of
$\mathrm{N}^{*}_{X}Y\to\mathrm{T}^{*}Y.$

###### Proof.

This is immediate from the identification of $\mathsf{Sing}(X_{ij})$ with the
fiber product of conormals in $\mathrm{T}^{*}Y.$ ∎

We now specialize to the case where $X_{i}=X$ for all $i$, so that we are
studying the monoidal category $\mathsf{Coh}(X\times_{Y}X).$ We will use the
map $F$ to pull back conic subsets of $\mathrm{T}^{*}Y$ (or of the image of
$\mathrm{N}^{*}_{X}Y$ in $\mathrm{T}^{*}Y$) to singular support conditions on
$X\times_{Y}X.$ We must first check that such singular-support conditions are
respected by the monoidal structure on $\mathsf{Coh}(X\times_{Y}X)$:

######  Lemma 4.16.

Let $\Lambda\subset\mathrm{T}^{*}Y$ be a closed conic subset. Then the
pullback $F^{-1}(\Lambda)$ is stable under convolution with
$\mathsf{Sing}(X\times_{Y}X).$

###### Proof.

From Lemma 4.13, we know that the convolution of Lagrangians (or, in this
case, coisotropics) may be computed by composition with the Lagrangian
correspondence

${L_{12}\times L_{23}}$${L_{123}}$${L_{13},}$

so that $F^{-1}(\Lambda)*\mathsf{Sing}(X\times_{Y}X)$ is the coisotropic in
$L_{123}$ described by the fiber product

$(F^{-1}(\Lambda)\times L_{23})\times_{L_{12}\times L_{23}}L_{123}\to
L_{123}.$

This fiber product imposes the condition on points of $L_{123}$ that their
projection to $\mathrm{T}^{*}Y$ lives in $\Lambda,$ and its image in $L_{13}$
again consists of points satisfying the same condition. ∎

Our main tool in relating convolution singular supports to convolution of
coherent sheaf categories will be the following calculation:

######  Proposition 4.17 ([BZCHN]*Proposition 3.30).

Let $\Lambda_{12}\subset\mathsf{Sing}(Z_{12}),$
$\Lambda_{23}\subset\mathsf{Sing}(Z_{23})$ be $Z_{22}$-stable conic subsets.
Then there is an equivalence of categories

$\mathsf{Coh}_{\Lambda_{12}}(Z_{23})\otimes_{\mathsf{Coh}(Z_{22})}\mathsf{Coh}_{\Lambda_{23}}(Z_{23})\simeq\mathsf{Coh}_{\Lambda_{12}*\Lambda_{23}}(Z_{13}).$

Proposition 4.17 allows us to establish the following invariance result, which
justifies the notation of Definition 4.2.

######  Proposition 4.18.

Suppose that the images in $\mathrm{T}^{*}Y$ of the conormals
$\mathrm{N}^{*}_{X_{1}}Y$ and $\mathrm{N}^{*}_{X_{2}}Y$ coincide. Then there
is an equivalence of module 2-categories

${\sf Mod}_{\mathsf{Coh}(X_{1}\times_{Y}X_{1})}\simeq{\sf
Mod}_{\mathsf{Coh}(X_{2}\times_{Y}X_{2})},$

given by tensoring with the bimodule category
$\mathsf{Coh}(X_{1}\times_{Y}X_{2}).$

###### Proof.

To see that the bimodule $\mathsf{Coh}(X_{12})$ defines a Morita equivalence
between the monoidal categories $\mathsf{Coh}(X_{11})$ and
$\mathsf{Coh}(X_{22}),$ we will show that the bimodule $\mathsf{Coh}(X_{12})$
gives the inverse Morita equivalence, by exhibiting an equivalence

(4.3.9)
$\mathsf{Coh}(X_{1}\times_{Y}X_{2})\otimes_{\mathsf{Coh}(X_{2}\times_{Y}X_{2})}\mathsf{Coh}(X_{2}\times_{Y}X_{1})\simeq\mathsf{Coh}(X_{1}\times_{Y}X_{1}).$

By Proposition 4.17, the left-hand side of 4.3.9 is equivalent to the category
$\mathsf{Coh}_{\Lambda^{cl}}(X_{1}\times_{Y}X_{1}),$ where
$\Lambda^{cl}=\mathsf{Sing}(X_{12})*\mathsf{Sing}(X_{21})$ is given by
convolution of the coisotropic full singular-support conditions for $X_{12}$
and $X_{21}.$

$\Lambda^{cl}$ is the underlying classical (and reduced) space of the
coisotropic $\Lambda$ given by applying the Lagrangian correspondence
$L_{121}$ to $L_{12}\times L_{23}$:

$\Lambda\simeq(L_{12}\times L_{21})\times_{L_{12}\times L_{21}}L_{121}\simeq
L_{121}\to L_{11}.$

By assumption, the images in $\mathrm{T}^{*}Y$ of the conormals to $X_{1}$ and
$X_{2}$ coincide, so that the map on sets induced by $L_{121}\to L_{11}$ is
surjective, and we conclude that $\Lambda^{cl}=\mathsf{Sing}(X_{11})$ is the
entire singular-support condition for $X_{11},$ so that the left-hand side of
4.3.9 is equivalent to the whole category of coherent sheaves on
$X_{1}\times_{Y}X_{1}.$ ∎

###  4.4. Microlocalization

We now return to the case of a single map $f:X\to Y$ (where $X$ may be
disconnected). Using the map
$F:\mathsf{Sing}(X\times_{Y}X)\simeq(\mathrm{N}^{*}_{X}Y\times_{\mathrm{T}^{*}Y}\mathrm{N}^{*}_{X}Y)^{cl}\to\mathrm{T}^{*}Y,$
we propose the following definition, allowing us to study coherent sheaves of
categories on $Y$ microlocally away from the zero-section in
$\mathrm{T}^{*}Y.$

######  Definition 4.19.

Let $X\to Y$ be as in Definition 4.2, and let
$U\subset\overline{\mathrm{N}^{*}_{X}Y}$ be an open subset of the image in
$\mathrm{T}^{*}Y$ of the conormal $\mathrm{N}^{*}_{X}Y,$ with closed
complement $\Lambda=\overline{\mathrm{N}^{*}_{X}Y}\setminus U$. Then we define
the 2-category of microlocal coherent sheaves of categories on $U$ to be the
2-category

$\mu\mathsf{CohCat}(U):={\sf
Mod}_{\mathsf{Coh}_{F^{-1}(U)}(X\times_{Y}X)}({\sf St})$

of module categories for the monoidal category
$\mathsf{Coh}_{U}(X\times_{Y}X)\simeq\frac{\mathsf{Coh}(X\times_{Y}X)}{\mathsf{Coh}_{F^{-1}(\Lambda)}(X\times_{Y}X)}$
of coherent sheaves with singular support in $U$.

The microlocal analogue of Proposition 4.18 remains true.

######  Proposition 4.20.

Suppose that the images in $T^{*}Y$ of $\mathrm{N}^{*}_{X_{1}}Y$ and
$\mathrm{N}^{*}_{X_{2}}Y$ coincide. Then there is an equivalence of
2-categories

$\mu\mathsf{CohCat}(U\cap\overline{\mathrm{N}^{*}_{X_{1}}Y})\simeq\mu\mathsf{CohCat}(U\cap\overline{\mathrm{N}^{*}_{X_{2}}Y}).$

###### Proof.

The monoidal category
$\overline{\mathcal{A}}_{i}:=\mathsf{Coh}_{F^{-1}}(U)(X_{i}\times_{Y}X_{i})$
is a quotient of the monoidal category
$\mathcal{A}_{i}:=\mathsf{Coh}(X_{i}\times_{Y}X_{i})$ be the ideal
$\mathcal{I}_{i}$ generated by
$\mathsf{Coh}_{F^{-1}(\Lambda)}(X_{i}\times_{Y}X_{i}).$

From Proposition 4.18, we know that the
$\mathcal{A}_{1},\mathcal{A}_{2}$-bimodule category
$\mathcal{M}:=\mathsf{Coh}(X_{2}\times_{Y}X_{1})$ gives a Morita equivalence
between $\mathcal{A}_{1}$ and $\mathcal{A}_{2}.$ To show that the quotient
$\mathcal{M}/\mathcal{M}\mathcal{I}_{1}$ induces a Morita equivalence between
the quotient algebras $\overline{\mathcal{A}}_{i},$ it is sufficient to show
that $\mathcal{I}_{2}\mathcal{M}$ and $\mathcal{M}\mathcal{I}_{1}$ are
equivalent subcategories of $\mathcal{M},$ or in other words that the two
convolutions

$\mathsf{Coh}_{F^{-1}(\Lambda)}(X_{2}\times_{Y}X_{2})\otimes_{\mathsf{Coh}(X_{2}\times_{Y}X_{2})}\mathsf{Coh}(X_{2}\times_{Y}X_{1}),\quad\mathsf{Coh}_{F^{-1}(\Lambda)}(X_{2}\times_{Y}X_{1})\otimes_{\mathsf{Coh}(X_{1}\times_{Y}X_{1})}\mathsf{Coh}(X_{1}\times_{Y}X_{1})$

agree. By Proposition 4.17 we are reduced to checking that the convolutions
$F^{-1}(\Lambda)*L_{21}$ and $L_{21}*F^{-1}(\Lambda)$ determine the same
singular support condition inside of $\mathsf{Sing}(X_{2}\times_{Y}X_{1}).$
Since both of these convolutions agree with
$F^{-1}(\Lambda)\subset\mathsf{Sing}(X_{2}\times_{Y}X_{1}),$ we are done. ∎

######  Remark 4.21.

By [BZNP]*Theorem 1.2.10, the monoidal center of the monoidal category
$\mathsf{Coh}(X\times_{Y}X)$ is given by the category
$\mathsf{Coh}_{prop/Y}(\mathcal{L}_{\sf Bet}Y)$ of coherent sheaves on the
loop space $\mathcal{L}_{\sf Bet}Y:={\sf Map}(S^{1},Y)$ whose pushforward to
$Y$ is coherent. If $Y$ is a scheme, then $\mathsf{Sing}(\mathcal{L}_{\sf
Bet}Y)\simeq\mathrm{T}^{*}Y,$ and localization over the monoidal center of
$\mathsf{Coh}(X\times_{Y}X)$ recovers the microlocalization defined above. For
$Y$ a stack, $\mathcal{L}_{\sf Bet}Y$ offers a refinement of this
microlocalization.

We now specialize to the situation that $X=\bigsqcup_{\alpha}X_{\alpha}$ is a
disjoint union of components, so that the monoidal category
$\mathsf{Coh}(X\times_{Y}X)$ splits (non-monoidally) into its “matrix
entries”:

$\mathsf{Coh}(X\times_{Y}X)\simeq\bigoplus_{(\alpha,\alpha^{\prime})}\mathsf{Coh}(X_{\alpha}\times_{Y}X_{\alpha^{\prime}}).$

The diagonal entries

(4.4.1) $\mathsf{Coh}(X_{\alpha}\times_{Y}X_{\alpha})\simeq{\sf Fun}_{{\sf
Perf}(Y)}(\mathsf{Coh}(X_{\alpha}),\mathsf{Coh}(X_{\alpha}))$

of this matrix are themselves monoidal categories, with unit object
(corresponding to the identity functor under the equivalence 4.4.1) given by
the pushforward $(\Delta_{\alpha})_{*}\mathcal{O}_{X_{\alpha}}$ under the
diagonal map

$\Delta_{\alpha}:X_{\alpha}\to X_{\alpha}\times_{Y}X_{\alpha}.$

This object plays a distinguished role:

######  Definition 4.22.

With $X=\bigsqcup_{\alpha}X_{\alpha}$ as above, we write

$E_{\alpha}:=(\Delta_{\alpha})_{*}\mathcal{O}_{X_{\alpha}}$

for the matrix idempotent corresponding to $X_{\alpha}.$

######  Proposition 4.23.

Let $I=\\{\alpha_{1},\ldots,\alpha_{k}\\}.$ Write
$X_{I}:=\bigsqcup_{\alpha_{i}\in I}X_{\alpha}$ and let $\Lambda_{I}$ be the
image in $\mathrm{T}^{*}Y$ of the conormal $\mathrm{N}^{*}_{X_{I}}.$ Then the
smallest ideal in $\mathsf{Coh}(X\times_{Y}X)$ containing $\langle
E_{\alpha_{i}}\rangle_{\alpha_{i}\in I}$ is
$\mathsf{Coh}_{F^{-1}(\Lambda_{I})}(X\times_{Y}X).$

###### Proof.

Let $E_{I}=\bigoplus_{\alpha_{i}\in I}E_{\alpha_{i}}.$ Since $E_{I}$ is the
monoidal unit of $\mathsf{Coh}(X_{I}\times_{Y}X_{I}),$ the smallest ideal
containing $E_{I}$ may be written as the tensor product

(4.4.2)
$\mathsf{Coh}(X\times_{Y}X_{I})\otimes_{\mathsf{Coh}(X_{I}\times_{Y}X_{I})}\mathsf{Coh}(X_{I}\times_{Y}X).$

More By Proposition 4.17, 4.4.2 is equivalent to
$\mathsf{Coh}_{\Lambda}(X\times_{Y}X),$ where $\Lambda$ is the convolution of
$\mathsf{Sing}(X\times_{Y}X_{I})$ with $\mathsf{Sing}(X_{I}\times_{Y}X).$ In
the notation of the previous section, writing $X_{1}=X_{3}=X$ and
$X_{2}=X_{I},$ we have

$\Lambda\simeq L_{12}*L_{23}\simeq(L_{12}\times L_{23})\times_{L_{12}\times
L_{23}}L_{123}\simeq L_{123}\to L_{13}.$

The image of $L_{123}$ in $L_{13}^{cl}\simeq\mathsf{Sing}(X\times_{Y}X)$
agrees with the set of points whose image under
$F:\mathsf{Sing}(X\times_{Y}X)\to\mathrm{T}^{*}Y$ agrees with the image in
$\mathrm{T}^{*}Y$ of the conormal $\mathrm{N}^{*}_{X_{I}}Y.$ ∎

######  Definition 4.24.

Let $X=\bigsqcup_{\alpha}X^{\alpha}\to Y$ as above. Then we write
$\mathcal{S}^{\alpha}$ for the $\mathsf{Coh}(X\times_{Y}X)$-module category
$\mathsf{Coh}(X^{\alpha}\times_{Y}X).$ For
$U\subset\overline{\mathrm{N}^{*}_{X}Y}$ open, we continue to write
$\mathcal{S}^{\alpha}$ for the image of $\mathcal{S}^{\alpha}$ in
$\mu\mathsf{CohCat}(U).$

###  4.5. The hypertoric 2-category

We now specialize to the case of interest to us.

######  Notation 4.25.

As in Definition 1.14, for a sign vector $\alpha,$ we write
$X^{\alpha}_{G}:=\\{\prod_{i\in\alpha}z_{i}=0\\}/G$ for the $G$-equivariant
intersection of coordinate hyperplanes in $\mathbb{A}^{n},$ equipped with its
closed embedding $X^{\alpha}_{G}\hookrightarrow\mathbb{A}^{n}/G.$ We write
$X_{G}:=\bigsqcup_{\alpha\in 2^{[n]}}X_{G}^{\alpha},$ equipped with the map
$X_{G}\to\mathbb{A}^{n}/G$ restricting to the standard embedding on each
component. More generally, for $I\subset 2^{[n]}$ a collection of sign
vectors, we write $X_{G}^{I}:=\bigsqcup_{\alpha\in I}X_{G}^{\alpha}.$

######  Notation 4.26.

Let $\mathcal{A}_{G}:=\mathsf{Coh}(X_{G}\times_{\mathbb{A}^{n}/G}X_{G}),$
equipped with a monoidal structure by convolution. More generally, for
$I\subset 2^{[n]}$ a collection of sign vectors, we write
$\mathcal{A}_{G}^{I}:=\mathsf{Coh}(X_{G}^{I}\times_{\mathbb{A}^{n}/G}X_{G}^{I}).$

The relevance of $\mathcal{A}_{G}^{I}$ is immediate from Definition 4.2:

######  Corollary 4.27.

There is an equivalence of 2-categories

(4.5.1) $\mathsf{CohCat}_{\mathbb{L}_{G}^{I}}(\mathbb{A}^{n}/G)\simeq{\sf
Mod}_{\mathcal{A}_{G}^{I}}({\sf St})$

between $\mathsf{CohCat}_{\mathbb{L}_{G}^{I}}(\mathbb{A}^{n}/G)$ and the
2-category of module categories for $\mathcal{A}_{G}^{I}.$

######  Example 4.28.

Let $n=1$ and $G=\\{1\\}\subset(\mathbb{G}_{m})^{n}$ be the trivial torus.
Then $\mathsf{CohCat}_{\mathbb{L}_{G}}(\mathbb{A}^{1})$ is the 2-category of
module categories over the monoidal category

(4.5.2)
$\mathcal{A}_{G}=\left(\begin{array}[]{cc}\mathsf{Coh}(\mathbb{A}^{1})&\mathsf{Coh}(0)\\\
\mathsf{Coh}(0)&\mathsf{Coh}(0\times_{\mathbb{A}^{1}}0)\end{array}\right).$

Each entry in the matrix 4.5.2 admits a single categorical generator, with a
commutative ring of endomorphisms, so that we may rewrite the matrix as

$\left(\begin{array}[]{cc}{\sf Perf}_{\Bbbk[x]}&{\sf Perf}_{\Bbbk}\\\ {\sf
Perf}_{\Bbbk}&{\sf Perf}_{\Bbbk[\beta]}\end{array}\right),$

where $\beta$ is in cohomological degree $2.$ In other words, the 2-category
$\mathsf{CohCat}_{\mathbb{L}_{G}}(\mathbb{A}^{1})$ has two objects, four
generating 1-morphisms (namely, the rings $\Bbbk[x],\Bbbk,\Bbbk,$ and
$\Bbbk[\beta]$ above) and two generating 2-morphisms (namely, $x$ and
$\beta$). The relations among these can be made explicit, so that this
2-category thus admits a completely combinatorial description. Working in the
same fashion, such a description can be given for all the 2-categories
$\mathsf{CohCat}_{\mathbb{L}_{G}}(\mathbb{A}^{n}),$ for a general choice of
$G$. (However, if $G$ is nontrivial, then the set of generating 1-morphisms in
this description must be allowed to be infinite.)

######  Example 4.29.

For $m\in\mathfrak{f}_{\mathbb{Z}}$ a choice of attraction parameter, we may
let $I=m\text{-bdd}\subset 2^{[n]}$ be the collection of sign vectors which
are $m$-bounded. In this case,
$\mathbb{L}_{G}^{m\text{-bdd}}=\mathbb{L}_{G}(0,m),$ and we recover the
2-category

(4.5.3) $\mathsf{CohCat}_{\mathbb{L}_{G}(0,m)}(\mathbb{A}^{n}/G)\simeq{\sf
Mod}_{\mathcal{A}_{G}^{m\text{-bdd}}}({\sf St})$

associated to the $m$-bounded locus
$\mathbb{L}_{G}(0,m)=\mathbb{L}_{G}^{m\text{-bdd}}$ in the Lagrangian
$\mathbb{L}_{G}.$

We are now ready to study the 2-category
$\mu\mathsf{CohCat}(\mathbb{L}_{G}(t,m))$ of microlocal coherent sheaves of
categories on the category $\mathcal{O}$ Lagrangian. To apply Definition 4.19,
we need to present $\mathbb{L}_{G}(t,m))$ as an open subset of a conormal
Lagrangian. There is an obvious choice, namely the open inclusion

(4.5.4)
$\mathbb{L}_{G}(t,m)\hookrightarrow\mathbb{L}_{G}(0,m)=\mathbb{L}_{G}^{m\text{-bdd}}.$

Applying definition Definition 4.19, we may therefore write

$\mu\mathsf{CohCat}(\mathbb{L}_{G}(t,m))\simeq{\sf
Mod}_{\mathsf{Coh}_{F^{-1}(\mathbb{L}_{G}(t,m))}(X^{m\text{-bdd}}\times_{\mathbb{A}^{n}/G}X^{m\text{-bdd}})}({\sf
St}).$

######  Lemma 4.30.

Let $\Lambda:=\mathbb{L}_{G}(0,m)\setminus\mathbb{L}_{G}(t,m)$ be the closed
complement of the embedding 4.5.4. Then
$\mathsf{Coh}_{F^{-1}(\Lambda)}(X^{m\text{-bdd}}\times_{\mathbb{A}^{n}/G}X^{m\text{-bdd}})\subset\mathsf{Coh}(X^{m\text{-bdd}}\times_{\mathbb{A}^{n}/G}X^{m\text{-bdd}})$
is the ideal generated by
$\mathsf{Coh}(X^{m\text{-bdd},t\text{-unstab}}\times_{\mathbb{A}^{n}/G}X^{m\text{-bdd},t\text{-unstab}}).$

###### Proof.

By Proposition 4.23,
$\mathsf{Coh}_{F^{-1}(\Lambda)}(X^{m\text{-bdd}}\times_{\mathbb{A}^{n}/G}X^{m\text{-bdd}})$
is the smallest ideal in
$\mathsf{Coh}(X^{m\text{-bdd}}\times_{\mathbb{A}^{n}/G}X^{m\text{-bdd}})$
containing the matrix idempotents $E_{\alpha}$ for $\alpha\in
m\text{-bdd},t\text{-unstab}.$ But these matrix idempotents are monoidal
generators of the monoidal subcategory
$Coh(X^{m\text{-bdd},t\text{-unstab}}\times_{\mathbb{A}^{n}/G}X^{m\text{-bdd},t\text{-unstab}}).$
∎

######  Corollary 4.31.

There is an equivalence

$\mu\mathsf{CohCat}(\mathbb{L}_{G}(t,m))\simeq{\sf
Mod}_{\frac{\mathcal{A}_{G}^{m\text{-bdd}}}{\mathcal{A}_{G}^{m\text{-bdd},t\text{-unstab}}}}({\sf
St})$

presenting $\mu\mathsf{CohCat}(\mathbb{L}_{G}(t,m))$ as the 2-category of
module categories for the monoidal category obtained as the quotient of
$\mathcal{A}_{G}^{m\text{-bdd}}$ by the ideal generated by the subcategory
$\mathcal{A}_{G}^{m\text{-bdd},t\text{-unstab}}.$

###### Proof.

This follows immediately from Definition 4.19 and Lemma 4.30. ∎

######  Example 4.32.

We return to the situation of Example 0.12, where
$F^{\vee}\simeq\mathbb{G}_{m}\hookrightarrow(\mathbb{G}_{m})^{2}$ is the
diagonal torus.

Before turning on the stability or attraction parameters, the skeleton
$\mathbb{L}_{F^{\vee}}$ is the image in
$\mathrm{T}^{*}(\mathbb{A}^{2}/\mathbb{G}_{m})$ of the union of conormals to
the zero section, $x$-axis, $y$-axis, and origin. The coherent 2-category
admits a presentation

$\mathsf{CohCat}_{\mathbb{L}_{F^{\vee}}}(\mathbb{A}^{2}/F^{\vee})={\sf
Mod}_{\mathcal{A}_{F}^{\vee}}({\sf St}),$

as modules over the monoidal category
$\mathcal{A}:=\mathsf{Coh}^{\mathbb{G}_{m}}(X\times_{Y}X),$ where we write

$X=(\mathbb{A}^{2}\sqcup\mathbb{A}^{1}_{x}\sqcup\mathbb{A}^{1}_{y}\sqcup
0)\to\mathbb{A}^{2}=Y,$

with the maps given by embeddings of closures of toric strata. In other words,
presenting each component in the fiber product $X\times_{Y}X$ as the
appropriate shifted normal bundle, we have

(4.5.5)
$\mathcal{A}\simeq\mathsf{Coh}^{\mathbb{G}_{m}}\left(\begin{array}[]{llll}\mathbb{A}^{2}&\mathbb{A}^{1}_{x}&\mathbb{A}^{1}_{y}&0\\\
\mathbb{A}^{1}_{x}&\mathrm{N}_{\mathbb{A}^{1}_{x}}[-1]\mathbb{A}^{2}&0&\mathrm{N}_{0}[-1]\mathbb{A}^{1}_{y}\\\
\mathbb{A}^{1}_{y}&0&\mathrm{N}_{\mathbb{A}^{1}_{y}}[-1]\mathbb{A}^{2}&\mathrm{N}_{0}[-1]\mathbb{A}^{1}_{x}\\\
0&\mathrm{N}_{0}[-1]\mathbb{A}^{1}_{y}&\mathrm{N}_{0}[-1]\mathbb{A}^{1}_{x}&\mathrm{N}_{0}[-1]\mathbb{A}^{2}\end{array}\right).$

If we pick the stability parameter
$m\in\mathfrak{f}^{\vee}_{\mathbb{Z}}=\mathbb{Z}$ to be positive, then the
unstable locus in $\mathrm{T}^{*}(\mathbb{A}^{2}/\mathbb{G}_{m})$ is
$\mathrm{T}^{*}_{0}(\mathbb{A}^{2}),$ so that passage to the $m$-semistable
locus is restriction to the open subset
$\mathcal{U}:=\mathrm{T}^{*}\mathbb{P}^{1}=\mathrm{T}^{*}((\mathbb{A}^{2}\setminus\\{0\\})/\mathbb{G}_{m})$
inside $\mathrm{T}^{*}(\mathbb{A}^{2}/\mathbb{G}_{m}).$

Passing to the $m$-semistable locus is a localization of the monoidal category
$\mathcal{A},$ so that we can write
$\mu\mathsf{CohCat}(\mathbb{L}_{F^{\vee}}(m,0))\simeq{\sf
Mod}_{\mathcal{A}|_{\mathcal{U}}}({\sf St}),$ where

(4.5.6)
$\mathcal{A}|_{\mathcal{U}}\simeq\mathsf{Coh}^{\mathbb{G}_{m}}\left(\begin{array}[]{llll}\mathbb{A}^{2}\setminus
0&\mathbb{A}^{1}_{x}\setminus 0&\mathbb{A}^{1}_{y}\setminus 0&\emptyset\\\
\mathbb{A}^{1}_{x}\setminus 0&\mathrm{N}_{\mathbb{A}^{1}_{x}\setminus
0}[-1](\mathbb{A}^{2}\setminus 0)&\emptyset&\emptyset\\\
\mathbb{A}^{1}_{y}\setminus
0&\emptyset&\mathrm{N}_{\mathbb{A}^{1}_{y}\setminus
0}[-1](\mathbb{A}^{2}\setminus 0)&\emptyset\\\
\emptyset&\emptyset&\emptyset&\emptyset\end{array}\right).$

Using the fact that the $\mathbb{G}_{m}$-action on the spaces in 4.5.6 is
free, and removing the final row and column (whose values are all the zero
category $\mathsf{Coh}(\emptyset)\simeq 0$ — not to be confused with the
category $\mathsf{Coh}(0)\simeq{\sf Mod}_{\Bbbk}$), we may rewrite 4.5.6 as

(4.5.7)
$\mathcal{A}|_{\mathcal{U}}\simeq\mathsf{Coh}\left(\begin{array}[]{lll}\mathbb{P}^{1}&0&\infty\\\
0&T_{0}[-1]\mathbb{P}^{1}&\emptyset\\\
\infty&\emptyset&T_{\infty}[-1](\mathbb{P}^{1})\end{array}\right),$

where we have identified the images of the $x$\- and $y$-axes with the points
$0,\infty\in\mathbb{P}^{1}=(\mathbb{A}^{2}\setminus 0)/\mathbb{G}_{m}$. This
monoidal category evidently admits a convolution presentation as before, where
now

$\mathcal{A}|_{\mathcal{U}}\simeq\mathsf{Coh}\left((\mathbb{P}^{1}\sqcup
0\sqcup\infty)\times_{\mathbb{P}^{1}}(\mathbb{P}^{1}\sqcup
0\sqcup\infty)\right).$

Finally, we impose a nonzero attraction parameter $t,$ passing from the “TIE
fighter” Lagrangian

$\mathbb{L}_{F^{\vee}}(m,0)=\mathbb{P}^{1}\cup\mathrm{T}^{*}_{0}\mathbb{P}^{1}\cup\mathrm{T}^{*}_{\infty}\mathbb{P}^{1}$

to the closed subspace

$\mathbb{L}_{F^{\vee}}(m,t)=\mathbb{P}^{1}\cup\mathrm{T}^{*}_{0}\mathbb{P}^{1}.$

At the level of categories, this has the effect of deleting the third row and
column of 4.5.7, so that we have at last

$\mu\mathsf{CohCat}(\mathbb{L}_{F^{\vee}}(m,t))={\sf
Mod}_{\mathcal{A}^{\prime}}({\sf St})$

where

(4.5.8) $\mathcal{A}^{\prime}=\mathsf{Coh}\left((\mathbb{P}^{1}\sqcup
0)\times_{\mathbb{P}^{1}}(\mathbb{P}^{1}\sqcup
0)\right)\simeq\mathsf{Coh}\left(\begin{array}[]{ll}\mathbb{P}^{1}&0\\\
0&T_{0}[-1]\mathbb{P}^{1}\end{array}\right).$

######  Remark 4.33.

Example 4.32 is misleading in some ways, due to the simplicity of the stable
locus $\mathrm{T}^{*}((\mathbb{A}^{2}\setminus\\{0\\})/\mathbb{G}_{m}).$ As a
result, the theory of coherent singular support conditions was not really