# Perverse schobers and 3d mirror symmetry

Benjamin Gammage, Justin Hilburn, and Aaron Mazel-Gee

###### Abstract

The proposed physical duality known as 3d mirror symmetry relates the
geometries of dual pairs of holomorphic symplectic stacks. It has served in
recent years as a guiding principle for developments in representation theory.
However, due to the lack of definitions, thus far only small pieces of the
subject have been mathematically accessible. In this paper, we formulate
abelian 3d mirror symmetry as an equivalence between a pair of 2-categories
constructed from the algebraic and symplectic geometry, respectively, of Gale
dual toric cotangent stacks.

In the simplest case, our theorem provides a spectral description of the
2-category of spherical functors – i.e., perverse schobers on the affine line
with singularities at the origin. We expect that our results can be extended
from toric cotangent stacks to hypertoric varieties, which would provide a
categorification of previous results on Koszul duality for hypertoric
categories $\mathcal{O}$. Our methods also suggest approaches to 2-categorical
3d mirror symmetry for more general classes of spaces of interest in geometric
representation theory.

Along the way, we establish two results that may be of independent interest:
(1) a version of the theory of Smith ideals in the setting of stable
$\infty$-categories; and (2) an ambidexterity result for co/limits of
presentable enriched $\infty$-categories over $\infty$-groupoids.

###### Contents

  1. 0 Introduction
    1. 0.1 Mathematical overview
    2. 0.2 Abelian gauge theories and hypertoric category $\mathcal{O}$

## 0 | Introduction

Mirror symmetry, which began life as a duality of 2-dimensional quantum field
theories, entered mathematics in [COGP] as a predicted equality between
Gromov–Witten invariants and period integrals associated to a dual pair of
Calabi–Yau manifolds. In [K94], this equality of numbers was upgraded to a
proposed equivalence of categories. These categories model the boundary
conditions in a pair of topological field theories – the A-model and B-model –
determined by the data of a symplectic or algebraic manifold. Kontsevich’s
proposal prefigured the result [Baez-Dolan, Costello-tcft, Lurie-cobordism]
that an $n$-dimensional topological field theory is determined by its
$(n-1)$-category of boundary conditions.

The program of 3d mirror symmetry [IS96] (also known as symplectic duality
[BLPW16]), which entails a duality between $3$-dimensional quantum field
theories, has so far been understood mathematically as expressing a relation
between invariants associated to dual pairs of symplectic resolutions.111The
literature on this subject is vast; some representative research directions
are described in [BFN1, BDGH, Okounkov-ICM, Kamnitzer-survey], and deeper
references to the field may be found there. As in (2d) mirror symmetry, most
of these invariants probe topological field theories – the 3d A- and
B-models.222The 3d B-model is also known as Rozansky–Witten theory [RW97]. The
$3$d A-model studied in this paper is essentially $3$d generalized Seiberg-
Witten theory. In general, this latter theory is different than that
considered in [Kap-Vyas, Kap-Set-Vyas]. Therefore the deepest formulation of
(topologically twisted) 3d mirror symmetry is as a conjectural equivalence of
2-categories.

Unfortunately, the mathematical existence of these 2-categories is also
conjectural. The B-type 2-category, first studied in [KRS, KR], is expected to
admit a formulation in the emerging mathematical language of coherent sheaves
of categories [Ga-shcat, Ari-talk], but the A-type 2-category, an expected
categorification of the Fukaya category, is much more mysterious. The
inspirational ICM address [Tel-ICM] first formulated the A-type 2-category
associated to pure gauge theory and described the equivalence of 2-categories
predicted by 3d mirror symmetry.

In this paper, we take the first steps beyond pure gauge theory, by proposing
2-categories associated to abelian gauge theories with matter. In mathematical
terms, we associate 2-categories to toric cotangent stacks – quotients of the
form ${{\rm T}^{*}}(\mathbb{C}^{n}/G)$ for $G\subset(\mathbb{C}^{\times})^{n}$
a torus – and prove equivalences between the 2-categories associated to dual
pairs of stacks. These equivalences, which can be understood in purely
mathematical terms, reveal heretofore hidden structure inside the world of
stable higher category theory.

In § 0.1, we provide a purely mathematical account of our results, focusing
on the basic duality between ${{\rm T}^{*}}\mathbb{C}$ and ${{\rm
T}^{*}}(\mathbb{C}/\mathbb{C}^{\times}).$ In this case, we understand 3d
mirror symmetry as an equivalence between a 2-category of perverse schobers on
$\mathbb{C}$ and a 2-category of coherent sheaves of categories on
$\mathbb{C}/\mathbb{C}^{\times}.$ Our main result is B, which gives a precise
statement of this equivalence. In § 0.2, which may be skipped by a reader
interested only in the mathematical results of this paper, we situate our
results in the context of abelian gauge theories with matter and give a
preview of our future applications to hypertoric categories $\mathcal{O}.$

### 0.1 | Mathematical overview

The notion of spherical functor [Anno-Logvinenko] has proven remarkably useful
as an organizing principle in contexts throughout algebraic and symplectic
geometry.

######  Definition 0.1.

An adjunction333The indices $\Phi,\Psi$ on the categories participating in
this adjunction are meant to recall the notation for the nearby and vanishing
cycles of a perverse sheaf.
$S:\mathcal{C}_{\Phi}\rightleftarrows\mathcal{C}_{\Psi}:S^{R}$ between stable
categories444Throughout this paper, we take the “implicit $\infty$” convention
(see LABEL:subsection:notn.and.conventions). is a spherical adjunction if it
satisfies the requirement that the endofunctors $T_{\Phi}:={\sf fib}({\sf
id}_{\mathcal{C}_{\Phi}}\xrightarrow{\eta}S^{R}S)$ and $T_{\Psi}:={\sf
cofib}(SS^{R}\xrightarrow{\varepsilon}{\sf id}_{\mathcal{C}_{\Psi}})$ are
invertible. The left adjoint $S$ is the underlying spherical functor of the
spherical adjunction.555We prefer the term “spherical adjunction” over the
more common term “spherical functor”, since it is more symmetric and in
particular adheres more closely to the intuition coming from symplectic
topology. On the other hand, the terminology “spherical functor” will also be
useful for us.

Examples include the pushforward of coherent sheaves on a divisor in algebraic
geometry, or the “Orlov/cup” functor in symplectic geometry; examples coming
from representation theory are also discussed in [KS-schobers]*§4 and [KS-
parabolic]*§8. It was recognized in [KS-schobers] that spherical functors
categorify the quiver presentation of perverse sheaves on $\mathbb{C}$ with
stratification $\mathcal{S}_{toric}$ given by
$\mathbb{C}=0\sqcup\mathbb{C}^{\times}.$

######  Theorem 0.2 ([Verdier]).

The category $\mathsf{Perv}(\mathbb{C},\mathcal{S}_{toric})$ is equivalent to
the category of diagrams of vector spaces $u:V_{\Phi}\rightleftarrows
V_{\Psi}:v$ satisfying the requirement that the endomorphisms
$T_{\Phi}:=1_{\Phi}-vu$ and $T_{\Psi}:=1_{\Psi}-uv$ are invertible.

For this reason, spherical functors are also referred to as perverse schobers
on $\mathbb{C}$ with stratification $\mathcal{S}_{toric}$ [BKS].

The collection of all spherical functors (not fixing the source or target
category) may be usefully organized into a 2-category $\mathsf{Sph}$, a full
sub-2-category of the 2-category ${\sf Fun}(\mathsf{Adj},{{\sf St}_{\Bbbk}})$
of stable adjunctions. As we shall see, this 2-category is remarkably rich:
the simple step of demanding that the twists be invertible imposes a great
deal of structure, including many symmetries not present for arbitrary stable
adjunctions.

The mathematical content of our paper is a spectral description of the
2-category of spherical functors. To give a concrete explanation of what we
mean by a “spectral” description, we consider the main precursor to our work,
the equivalences of 2-categories discussed in [Tel-ICM].

######  Definition 0.3.

Let $X$ be a stack which is 1-affine in the sense of [Ga-shcat]. We may
associate to $X$ the 2-categories $\mathsf{QCoh}^{(2)}(X):={\sf Mod}_{{\sf
Perf}(X)}({{\sf St}_{\Bbbk}})$ of quasicoherent sheaves of categories and
$\mathsf{Loc}^{(2)}(X):={\sf Fun}(X_{\sf{B}},{{\sf St}_{\Bbbk}})$ of local
systems of categories, where we write $X_{\sf{B}}$ for the homotopy type
(“Betti stack”) underlying $X$.

######  Example 0.4.

Fix a torus $T:=(\mathbb{C}^{\times})^{n}$, and let $T^{\vee}$ be its
Langlands dual. The (categorified) Fourier transform [Tel-ICM]*Remark 4.6 is
the equivalence of 2-categories

$\mathsf{QCoh}^{(2)}(BT)\simeq\mathsf{Loc}^{(2)}(T^{\vee})~{},$

which follows from the observation that $\mathsf{QCoh}^{(2)}(BT)$ and
$\mathsf{Loc}^{(2)}(T^{\vee})$ are both equivalent to the $2$-category of
categories equipped with $n$ commuting automorphisms.

There is also a dual Fourier transform [Tel-ICM]*Proposition 4.1

$\mathsf{Loc}^{(2)}(BT)\simeq\mathsf{QCoh}^{(2)}(T^{\vee})~{},$

where now the classifying stack appears on the automorphic side instead of the
spectral side. To see this, note that $\mathsf{Loc}^{(2)}(BT)$ is equivalent
to the 2-category of categories $\mathcal{C}$ equipped with a topological
action of the torus $T$. Such an action is equivalent to the data of an
$\mathbb{E}_{2}$ map $\mathbb{C}[T^{\vee}]\cong C_{*}(\Omega T)\rightarrow
HH^{*}(\mathcal{C}),$ as explained in [Tel-ICM]*Theorem 2.5.

Example 0.4 expressed a duality between the spaces $BT$ and $T^{\vee}$.
Similarly, our results will relate the space $\mathbb{C},$ equipped with
stratification $0\sqcup\mathbb{C}^{\times},$ to a dual geometry given by the
space $\mathbb{C}/\mathbb{C}^{\times}$ equipped with stratification
$0/\mathbb{C}^{\times}\sqcup\mathbb{C}^{\times}/\mathbb{C}^{\times}$. (In §
0.2, we will give context for the geometric duality relating both these
stratified spaces and the spaces described in Example 0.4.)

Consider the functor

$\mathsf{Coh}(\mathbb{C}/\mathbb{C}^{\times})\stackrel{{\scriptstyle
i^{*}}}{{\longrightarrow}}\mathsf{Coh}(B\mathbb{C}^{\times})$

of pullback along the inclusion
$i:0/\mathbb{C}^{\times}\hookrightarrow\mathbb{C}/\mathbb{C}^{\times}.$ This
functor is spherical, and we prove that it is the free spherical functor on a
categorified vanishing cycle, in the following sense.

######  Theorem A (LABEL:thm.gr.coreps.LHS).

The functor $\Phi:\mathsf{Sph}\to{{\sf St}_{\Bbbk}}$ taking a spherical
adjunction $\mathcal{C}_{\Phi}\rightleftarrows\mathcal{C}_{\Psi}$ to the
category $\mathcal{C}_{\Phi}$ is corepresented by the spherical functor
$\mathsf{Coh}(\mathbb{C}/\mathbb{C}^{\times})\xrightarrow{i^{*}}\mathsf{Coh}(B\mathbb{C}^{\times}).$

As a spherical functor, $i^{*}$ admits both adjoints $i_{!}\dashv i^{*}\dashv
i_{*},$ which are themselves spherical functors. Moreover, these adjoints
$i_{!}$ and $i_{*}$ are in fact equivalent (when considered as objects of
$\mathsf{Sph}$), and corepresent the functor $\Psi$ taking a spherical
adjunction $\mathcal{C}_{\Phi}\rightleftarrows\mathcal{C}_{\Psi}$ to the
category $\mathcal{C}_{\Psi}.$ Since the objects corepresenting the functors
$\Phi$ and $\Psi$ jointly generate $\mathsf{Sph},$ in the sense that the
functor corepresented by their direct sum is conservative, in the spirit of
the Barr-Beck theorem we may use their their monoidal endomorphism category to
give a presentation of the full 2-category $\mathsf{Sph}$.

######  Theorem B (LABEL:prop:spectral-description).

The 2-category of spherical functors admits a canonical equivalence
$\mathsf{Sph}\simeq{\sf Mod}_{\mathcal{A}}({{\sf St}_{\Bbbk}})$ with the
2-category of modules for the convolution monoidal category

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

As in Example 0.4, there is a dual form of B in which
$\mathbb{C}/\mathbb{C}^{\times}=0/\mathbb{C}^{\times}\sqcup\mathbb{C}^{\times}/\mathbb{C}^{\times}$
appears on the automorphic side and $\mathbb{C}=0\sqcup\mathbb{C}^{\times}$
appears on the spectral side.

######  Theorem C (LABEL:cor:ungauging-dim1).

The 2-category $\mathsf{Sph}$ admits a topological $S^{1}$-action, and the
invariant 2-category admits a canonical equivalence
$\mathsf{Sph}^{S^{1}}\simeq{\sf Mod}_{\mathcal{A}^{\prime}}({{\sf
St}_{\Bbbk}})$ with the 2-category of modules for the convolution monoidal
category

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

We deduce C from B by applying the following (twice categorified) Fourier
transform.

######  Theorem D (LABEL:thm:4d-fourier).

For any torus $T$, there is an equivalence of $3$-categories

$\mathsf{Loc}^{(3)}(BT)\simeq\mathsf{QCoh}^{(3)}(BT^{\vee})~{}.$

This equivalence is suitably functorial; in particular, it intertwines the
functor of $T$-invariants with the functor that forgets the $T^{\vee}$-action.

Towards establishing the above theorems, we prove two auxiliary results that
may be of independent interest, which we now describe.

Our first auxiliary result is an $\infty$-categorical version of the theory of
Smith ideals. In ordinary algebra, given a ring $R$, there is a bijective
correspondence between ideals of $R$ and surjective ring homomorphisms out of
$R$. Underlying this is the fact that a surjective homomorphism of abelian
groups is likewise determined by its kernel. By contrast, in higher algebra,
every homomorphism of derived abelian groups is determined by its fiber. Thus,
one may expect that every homomorphism of derived rings should be equivalent
data to its fiber equipped with suitable additional structure; and this is
what we establish.

######  Theorem E (LABEL:cor.ideals.are.algebras).

Fix a stably monoidal category $\mathcal{C}$. For each algebra object
$A\in{\sf Alg}(\mathcal{C})$, the fiber ${\sf
fib}(\operatorname{\mathbbm{1}}_{\mathcal{C}}\xrightarrow{\eta}A)\in\mathcal{C}$
admits a canonical enhancement to an ideal, i.e. a monoidal functor
$\mathbb{Z}_{\leq 0}\rightarrow\mathcal{C}$. Moreover, this construction
defines an equivalence

${\sf Alg}(\mathcal{C})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\sf
Idl}(\mathcal{C}):={\sf Fun}^{\sf mon}(\mathbb{Z}_{\leq 0},\mathcal{C})~{}.$

The theory of ideals developed in E is the main technical tool used in our
understanding of spherical functors and the proof of B. As we explain in
LABEL:rem:segal, this theory is the necessary ingredient to correct and
explain earlier work [Segal-autoequivalences] on spherical functors.

Our second auxiliary result is an ambidexterity theorem for limits and
colimits of presentable enriched categories.

######  Theorem F (LABEL:thm.prbl.ambidext).

Fix a presentably symmetric monoidal category $\mathcal{W}\in{\sf CAlg}({\sf
Pr}^{L})$. For any $\infty$-groupoid $\mathcal{A}$ and any functor
$\mathcal{A}\xrightarrow{F}{\sf Pr}^{L}_{\mathcal{W}}$ to the category of
presentably $\mathcal{W}$-enriched categories, there is a canonical
equivalence

${\sf colim}_{\mathcal{A}}(F)\simeq{\sf lim}_{\mathcal{A}}(F)~{}.$

### 0.2 | Abelian gauge theories and hypertoric category $\mathcal{O}$

It is expected that a symplectic stack $\mathfrak{X}$ admitting a
$\mathbb{C}^{\times}$ action for which the symplectic form has weight 2
determines a pair of 3-dimensional topological field theories, the 3d A-model
and B-model, respectively, relating to the symplectic and algebraic geometry
of $\mathfrak{X}.$ This situation encompasses all cotangent stacks
$\mathfrak{X}={{\rm T}^{*}}(M/G),$ for $M$ a smooth variety with action of a
reductive group $G$, and all examples we consider will be of this form.
Moreover, to a choice of Lagrangian $\mathbb{L}\subset\mathfrak{X},$ each of
these theories should associate a 2-category of boundary conditions supported
on $\mathbb{L}.$

The 3d mirror symmetry conjecture suggests that a 3d A-model associated to
some geometry $\mathfrak{X}$ may also be realized as a 3d B-model, and vice
versa. More precisely, in good cases, given a stack $\mathfrak{X},$ we can
find a “3d dual” space $\mathfrak{X}^{\vee}$ such that the A-model with target
$\mathfrak{X}$ is equivalent to the B-model with target
$\mathfrak{X}^{\vee}.$666Note that not every 3d A-model and B-model arises in
this way; thus, even a naïve version of the 3d mirror symmetry conjecture will
allow that the theory dual to a 3d A-model or B-model associated to
$\mathfrak{X}$ may not arise from the data of a symplectic stack
$\mathfrak{X}^{\vee}.$ This is analogous to the phenomenon, familiar from 2d
mirror symmetry, that a non-Calabi–Yau Kähler manifold will in general be
mirror to a Landau–Ginzburg model (rather than another Kähler manifold). (When
$\mathfrak{X}={{\rm T}^{*}}(N/G)$ for a linear $G$-representation $N$, the
ring of functions on the dual space is produced by the Coulomb branch
construction [N, BFN1].) In this case, we expect moreover that given a
Lagrangian $\mathbb{L}\subset\mathfrak{X},$ there ought to exist a Lagrangian
$\mathbb{L}^{\vee}\subset\mathfrak{X}^{\vee}$ such that there is an
equivalence between the respective 2-categories of boundary conditions
supported on $\mathbb{L}$ and $\mathbb{L}^{\vee}.$

######  Example 0.5.

Let $\mathfrak{X}={{\rm T}^{*}}({\sf pt}/T)$ for some $n$-torus
$T\simeq(\mathbb{C}^{\times})^{n}.$ The dual space in this case is the variety
$\mathfrak{X}^{\vee}={{\rm T}^{*}}(T^{\vee})$ given by the cotangent bundle to
the dual torus. Each of these spaces has a natural Lagrangian given by the
zero-section, $\mathbb{L}={\sf pt}/T\subset\mathfrak{X}$ and
$\mathbb{L}^{\vee}=T^{\vee}\subset\mathfrak{X}^{\vee}.$

To the geometric data $(\mathbb{L}\subset\mathfrak{X})$ and
$(\mathbb{L}^{\vee}\subset\mathfrak{X}^{\vee})$, the 3d A-model assigns the
2-categories $\mathsf{Loc}^{(2)}({\sf pt}/T)$ and
$\mathsf{Loc}^{(2)}(T^{\vee}),$ respectively, whereas the 3d B-model assigns
$\mathsf{QCoh}^{(2)}({\sf pt}/T)$ and $\mathsf{QCoh}^{(2)}(T^{\vee}).$ The
equivalences of 2-categories predicted by 3d mirror symmetry are described in
Example 0.4.

The results of this paper take place in the situation where the space
$\mathfrak{X}$ is of the form ${{\rm T}^{*}}(N/G),$ where $N$ is a vector
space with linear action of the torus $G.$ Such a stack may be specified by
the data of an exact sequence of tori

${1}$${(\mathbb{C}^{\times})^{n}}$${1,}$$\scriptstyle{i}$$\scriptstyle{p}$ (1)

where we take $N=\mathbb{C}^{n}$ equipped with the action of $G$ induced from
the inclusion $i.$

######  Definition 0.6.

The toric cotangent stack determined by the exact sequence 1 is

$\mathfrak{X}(N,G):={{\rm T}^{*}}(N/G).$

If $\mathfrak{X}=\mathfrak{X}(N,G)$ is specified as above, then the 3d dual
$\mathfrak{X}^{\vee}$ is known [dBHOO, BLPW10, BFN1] to be a space of the same
form; namely, it is the toric cotangent stack associated to the dual exact
sequence of tori

${1}$∨${(\mathbb{C}^{\times})^{n}}$∨${1.}$$\scriptstyle{p^{\vee}}$$\scriptstyle{i^{\vee}}$
(2)

In other words, we identify the 3d dual as
$\mathfrak{X}(N,G)^{\vee}=\mathfrak{X}(N^{\vee},F^{\vee}).$

Now, a toric cotangent stack $\mathfrak{X}(N,G)$ admits a distinguished
Lagrangian subspace.

######  Definition 0.7.

We write $\mathbb{L}_{toric}\subset\mathfrak{X}(N,G)$ for the Lagrangian

$\mathbb{L}_{toric}=\bigcup_{\alpha}{\rm
T}^{*}_{\overline{\mathcal{O}}_{\alpha}}(N/G)$

given by the union of conormals to closures $\overline{\mathcal{O}}_{\alpha}$
of $(\mathbb{C}^{\times})^{n}$-orbits $\mathcal{O}_{\alpha}$ in $N/G.$ When we
discuss the dual cotangent stack
$\mathfrak{X}^{\vee}=\mathfrak{X}(N^{\vee},F^{\vee}),$ we will write
$\mathbb{L}_{toric}^{\vee}$ for the analogously defined Lagrangian in
$\mathfrak{X}^{\vee}$.

######  Example 0.8.

If $n=1$ and $G=\\{1\\},$ then the space $\mathfrak{X}(N,G)={{\rm
T}^{*}}\mathbb{C}$ with the Lagrangian $\mathbb{L}_{toric}=\mathbb{C}\cup{\rm
T}^{*}_{0}\mathbb{C}$ whose irreducible components are the zero-section and
the conormal to $0\in\mathbb{C}.$ The dual space $\mathfrak{X}^{\vee}$ is the
stack $\mathfrak{X}(N^{\vee},F^{\vee})={{\rm
T}^{*}}(\mathbb{C}/\mathbb{C}^{\times}),$ equipped with Lagrangian
$\mathbb{L}^{\vee}_{toric}=\mathbb{C}/\mathbb{C}^{\times}\cup{\rm
T}^{*}_{0/\mathbb{C}^{\times}}(\mathbb{C}/\mathbb{C}^{\times}).$

In general, it is not understood how to associate a 2-category of boundary
conditions to the data of a symplectic stack $\mathfrak{X}$ and a Lagrangian
$\mathbb{L}\subset\mathfrak{X}$: the assignment described in Example 0.5 is
one of the only cases where it has been possible to formulate 3d mirror
symmetry as an equivalence of 2-categories. The main difficulty is in
understanding the symplectic (A-type) 2-category; the B-type 2-category has
been explored in several works [KRS, KR, Ari-talk].

In this paper, we propose the following definitions for the 2-categories of
boundary conditions associated to the data of a toric cotangent stack
$\mathfrak{X}(N,G)$ equipped with the Lagrangian
$\mathbb{L}_{toric}\subset\mathfrak{X}(N,G).$

######  Proposal 0.9.

We model boundary conditions for the 3d A-model with target ${{\rm
T}^{*}}(N/G)$ supported on $\mathbb{L}_{toric}$ by the 2-category

$\mathsf{Perv}^{(2)}(N/G,\mathcal{S}_{toric}):=\mathsf{Perv}^{(2)}(N,\mathcal{S}_{toric})^{G}$

of $G$-invariant perverse schobers on $N$ equipped with stratification
$\mathcal{S}_{toric}$ by $(\mathbb{C}^{\times})^{n}$-orbits (incarnated as
$\mathsf{Sph}_{n}^{G}$ in LABEL:subsec:hypertoric). We model boundary
conditions for the 3d B-model with target ${{\rm T}^{*}}(N/G)$ supported on
$\mathbb{L}_{toric}$ by the 2-category

$\mathsf{IndCoh}^{(2)}(N/G,\mathcal{S}_{toric})$

of $G$-invariant Ind-coherent sheaves of categories with singular support in
the union of conormals to strata in $\mathcal{S}_{toric}.$ This latter
2-category may be defined [Ari-talk] as the 2-category ${\sf
Mod}_{\mathcal{A}}({{\sf St}_{\Bbbk}})$ of modules for the monoidal category
of coherent sheaves

$\mathcal{A}:=\mathsf{Coh}\left(\left(\bigsqcup_{\alpha}\overline{\mathcal{O}}_{\alpha}\right)\times_{N/G}\left(\bigsqcup_{\alpha}\overline{\mathcal{O}}_{\alpha}\right)\right),$

where we write $\bigsqcup_{\alpha}\overline{\mathcal{O}}_{\alpha}$ for the
disjoint union of closures of $(\mathbb{C}^{\times})^{n}$-orbits in $N/G.$

######  Remark 0.10.

Physically, the ${\sf hom}$-categories in $3$d A- and B-models should be
Fukaya–Seidel and matrix factorization categories, respectively, on path
spaces, as can be deduced by compactifying the 3d theory on an interval.777By
instead compactifying on an $S^{1},$ one obtains the categories of line
operators studied in [DGGH, HR, BN-voa]. The resulting 2d theories are twists
of the gauged Landau–Ginzburg model described in [BDGH]*Appendix A, whose
superpotential is a complexification of the equivariant symplectic action
functional on path spaces [Cieliebak, Frauenfelder]. On the B-side, the
resulting matrix factorization categories were computed in [KRS, KR] and shown
to match a $(\mathbb{Z}/2$-graded) version of the ${\sf hom}$-categories in
$\mathsf{IndCoh}^{(2)}.$ On the A-side, the “algebra of the infrared” [GMW,
KS-aoi, KSS] presents the resulting Fukaya–Seidel categories as
categorifications of Floer groups.888More precisely, these are
categorifications of the moment Floer groups of [Frauenfelder]. This justifies
our proposal to formalize the 2-category of boundary conditions in the 3d
A-model as a categorification of the Fukaya category. (For the relation
between perverse sheaves and the Fukaya category, see [GPS3, Jin].)

######  Remark 0.11.

In 0.9, we have expressed the 2-categories which 3d mirror symmetry associates
to a cotangent bundle ${{\rm T}^{*}}(N/G)$ in terms of the geometry of the
base $N/G.$ We expect that there will eventually be more “microlocal”
definitions of these 2-categories which do not privilege a particular
polarization.

The prediction that (topologically twisted) dual abelian 3d theories have
equivalent boundary conditions may now be formulated precisely as an
equivalence between the 2-categories described above. This is what we prove:

######  Theorem G (LABEL:thm:higherdim-mainthm).

Let $N=\mathbb{C}^{n}$ be a vector space equipped with an action of torus
$G\subset(\mathbb{C}^{\times})^{n},$ and let $F=(\mathbb{C}^{\times})^{n}/G$
be the quotient torus as in 1. Then there is an equivalence

$\mathsf{Perv}^{(2)}(N/G,\mathcal{S}_{toric})\simeq\mathsf{IndCoh}^{(2)}(N^{\vee}/F^{\vee},\mathcal{S}_{toric})$

between the A-model 2-category associated to
$\mathbb{L}_{toric}\subset\mathfrak{X}(N,G)$ and the B-model 2-category
associated to
$\mathbb{L}_{toric}^{\vee}\subset\mathfrak{X}(N^{\vee},F^{\vee}).$

As a corollary of G, we are able to rederive the prescription, from [BFN1],
for the algebra of functions on the 3d mirror dual to $\mathfrak{X}(N,G)$,
directly from our 2-category of A-type boundary conditions. As in their
approach, we follow the expectation that this algebra should be computed by
endomorphisms of the trivial line defect inside the category of line
operators. In our formulation, this is the endomorphism algebra of the unit
object $\operatorname{\mathbbm{1}}$ in the categorical center of the
2-category of boundary conditions. (By definition, for a 2-category of the
form $\mathcal{C}={\sf Mod}_{\mathcal{A}}(\mathcal{D}),$ for some algebra
$\mathcal{A}\in{\sf Alg}(\mathcal{D}),$ the categorical center
$Z_{cat}(\mathcal{C})$ is defined to be the center $Z(\mathcal{A})\in{\sf
Alg}_{\mathbb{E}_{2}}(\mathcal{D})$ of the algebra $\mathcal{A}.$)

######  Theorem H (LABEL:thm:e3-functions).

There is an equivalence

$\mathsf{end}_{Z_{cat}(\mathsf{Perv}^{(2)}(N/G,\mathcal{S}_{toric})}(\operatorname{\mathbbm{1}})\simeq\mathcal{O}(\mathfrak{X}(N^{\vee},F^{\vee}))_{regraded}$

between the algebra of endomorphisms of the unit in the categorical center of
the 2-category of A-type boundary conditions for $\mathfrak{X}(N,G)$ and a
regraded version of the algebra of functions on
$\mathfrak{X}(N^{\vee},F^{\vee})$ where functions linear in the cotangent
direction are placed into homological degree 2.

######  Remark 0.12.

As we have mentioned previously, the proof of G exploits extra structure in
these 3d theories: their residual symmetries enhance them to boundary
conditions for 4-dimensional abelian gauge theories [GW]. The equivalence
described in D is a form of the Betti Geometric Langlands equivalence,
relating the A- and B-type twists of these 4-dimensional theories. This idea
was exploited in [BFN-ring, BFGT, HR] to compute rings/categories of
local/line operators and forms the underpinning for the relative Langlands
program of Ben-Zvi, Sakellaridis, and Venkatesh.

From the perspective of supersymmetric gauge theory or representation theory,
the support Lagrangian
$\mathbb{L}_{toric}\subset\mathfrak{X}=\mathfrak{X}(N,G)$ may seem unnatural;
usually, as in the theory of categories $\mathcal{O}$ for symplectic
resolutions [BLPW16], one studies a Lagrangian defined as the attracting set
for some Hamiltonian $\mathbb{C}^{\times}$-action on $\mathfrak{X}.$ Note that
a toric cotangent stack $\mathfrak{X}(N,G)$ defined via the exact sequence 1
admits a Hamiltonian action by the residual torus $F$; therefore, a
Hamiltonian $\mathbb{C}^{\times}$-action may be determined from a cocharacter
$m\in X_{*}(F)$ of the torus $F,$ specifying a 1-parameter subgroup acting on
$\mathfrak{X}(N,G).$

######  Definition 0.13.

Given a cocharacter $m\in X_{*}(F)$, the relative skeleton
$\mathbb{L}^{m}\subset\mathfrak{X}(N,G)$ is obtained by Hamiltonian reduction
from the Lagrangian $\overline{\mathbb{L}}^{m}\subset{{\rm T}^{*}}N$ which is
the union of attracting sets for all cocharacters of
$(\mathbb{C}^{\times})^{n}$ which map to $m$ under the projection $p_{*}$
[Web-gencat-O]*Definition 2.8. Following [GPS2], we refer to the process of
obtaining $\mathbb{L}^{m}$ by removing components of $\mathbb{L}_{toric}$ as
stop removal.

Our choice of Lagrangian $\mathbb{L}_{toric}$ is therefore not so unnatural:
it is a sort of “master Lagrangian” containing the relative skeleta
$\mathbb{L}^{m}$ for all possible choices of parameter $m.$

At this point it is reasonable to ask how the passage from
$\mathbb{L}_{toric}$ to the sub-Lagrangian $\mathbb{L}^{m}$ is reflected on
the 3d mirror dual. We first recall the parameter $t$ which is dual to $m$.

######  Definition 0.14.

Given a character $t\in X^{*}(G)$ for the torus $G$, we may treat $t$ as a GIT
stability parameter to define the hypertoric orbifold
$\mathfrak{X}(N,G)_{t}:={{\rm T}^{*}}N/\\!\\!/\\!\\!/_{t}G$ [Goto, Biel-Dan,
Haus-Sturm], which embeds as an open substack in $\mathfrak{X}(N,G).$ We write
$\mathbb{L}_{t}:=\mathbb{L}_{toric}\cap\mathfrak{X}(N,G)_{t}$ for the
restriction of the toric Lagrangian $\mathbb{L}_{toric}$ to this subset. We
refer to the passage from $\mathfrak{X}(N,G)$ to $\mathfrak{X}(N,G)_{t}$ as
microrestriction.

######  Definition 0.15.

Given $m\in X_{*}(F)$ and $t\in X^{*}(G),$ we may simultaneously impose the
GIT stability condition $t$ and pass to the relative skeleton
$\mathbb{L}_{t}^{m}:=\mathbb{L}^{m}\cap\mathfrak{X}(G,N)_{t}.$ This is the
category $\mathcal{O}$ Lagrangian associated to the exact sequence 1 and the
parameters $t$ and $m.$

Observe that when we dualize exact sequence 1 to 2, the roles of the
parameters $t$ and $m$ are exchanged: $m$ becomes a character of the subtorus
$F^{\vee},$ while $t$ becomes a cocharacter of the quotient torus $G^{\vee}.$
The prediction of 3d mirror symmetry is that $\mathfrak{X}(N,G)_{t}$ equipped
with Lagrangian $\mathbb{L}_{t}^{m}$ is dual to
$\mathfrak{X}(N^{\vee},F^{\vee})_{m}$ equipped with Lagrangian
$(\mathbb{L}^{\vee})_{m}^{t}.$ In slogan form: “Stop removal is dual to
microrestriction.”

In the sequel to this paper, we will explain the effects of stop removal and
microrestriction on both the 3d A-model and B-model 2-categories of boundary
conditions. We will then be able to make the above slogan into a rigorous
mathematical statement, through which we will establish 3d mirror symmetry as
an equivalence between “2-categories $\mathcal{O}$” associated to
$\mathbb{L}_{t}^{m}$ and $(\mathbb{L}^{\vee})_{m}^{t}.$


# Irreducible and site symmetry induced representations of single/double
ordinary/gray layer groups

B. Nikolića, I. Miloševića111Corresponding author<EMAIL_ADDRESS>T.
Vukovića, N. Lazića, S. Dmitrovića,
Z. Popovića and M. Damnjanovića,b
a
a NanoLab, Faculty of Physics, University of Belgrade,
Studentski trg 12, Belgrade, Serbia and
b Serbian Academy of Sciences and Arts,
Kneza Mihaila St. 35, Belgrade, Serbia

###### Abstract

Considered are eighty sets of layer groups, each set consisted of four groups:
ordinary single and double, and gray single and double layer group. Structural
properties of layer groups (factorization onto cyclic subgroups and existence
of grading according to the sequence of halving subgroups) enable efficient
symbolic computation (by POLSym code) of the relevant properties, real and
complex irreducible and allowed (half-)integer (co-)representations in
particular. This task includes, as the first step, classification of the
irreducible domains based on the group action in Brillouin zone combined with
torus topology. Also, the band (co-)representations induced from the
irreducible (co-)representations of Wyckoff position stabilizers (site
symmetry groups) are decomposed onto the irreducible components. These, and
other layer group symmetry related theoretical data relevant for physics,
layered materials in particular, are tabulated and made available through the
web site.

##  1 Introduction

Over last decade interest in layered structures has increased tremendously.
Graphene and related structures and then flatlands beyond graphene have been
discovered and systematically studied. Layered materials have been thoroughly
investigated and their fundamentally and technologically important properties
have been revealed. Recent research breakthrough relating topological phases
with symmetry, through the graph analysis of the band representations give
strong momentum to symmetry based research in solid state physics. While space
groups, symmetries of 3D crystals, are established knowledge, symmetry groups
of layered structures are not so exhaustively elaborated. Namely, first layer
groups (LGs) classification, their generators and Wyckoff positions are
published [1] while only forty years later, in International Tables for
Crystallography [2], tabulated are myriad of the relevant data including
various sets of generators, positions (Wyckoff symbol, multiplicity, site
symmetry, coordinates), asymmetric units, group generators, maximal subgroups
and minimal supergroups and other direct space related data. However,
reciprocal space and irreducible representations (IRs) are rarely discussed.
In fact, for each LG there are supergroups among the space groups, and data
for the supergroup (irreducible domains, IRs, etc.) can in principle be used
[3] to derive the corresponding data for the LG [4]. Being cumbersome, this
procedure is not straightforward and therefore, IRs of LGs have been derived
independently [5]. More than a decade ago Bilbao crystallographic server [6,
7] devoted a section to the LGs, which has recently been amended by the band
representations, being induced from the representations of the site symmetry
groups [4]. However, a bunch of information is still missing, and the present
paper is aimed at completing the field.

Therefore, we generate exhaustive data set, covering the following types of
LGs: single and double groups (relevant for spinless and spinful systems), as
well as ordinary and gray groups (systems without and with time reversal
symmetry, TRS). For all of these groups their IRs (for gray groups irreducible
co-representations, co-IRs), and band (co-)represenations are derived. It
should be stressed out that all derivations are performed by computer algebra,
meaning that a number of subroutines are developed (induction for general
points of Brillouin zone, reduction of various types of representations over
IRs parameterized by wave vector, including construction of band
representations, and their reduction) and gathered within POLSym code [8].
This means that beside the presented database, which are already directly
accessible, there are additional data (black-and-white LGs data, e.g.) which
at present can be obtained on request only (as the online platform is
currently under construction).

##  2 Layer groups: structure and generators

Gathering symmetries of quasi-2D crystals, LGs are generated by translations
(conventionally spanning $xy$-plane) and additional orthogonal
transformations. In a wider sense, when half-integer spin space is also
considered, orthogonal elements have twofold covering by ${\bm{S}U}(2)$
matrices, forming double groups altogether. Analogously, these (so called
ordinary single and double groups) are extended into the magnetic ones, when
time reversal is included.

###  2.1 Single ordinary layer groups

Single ordinary LGs are diperiodic subgroups of the Euclidean group
$E(3)={\bm{T}}\wedge{\bm{O}}(3)$ (semidirect product). Since the lattice is
invariant under the groups’ transformations, each LG ${\bm{G}}$ has invariant
translational subgroup ${\bm{T}}={\bm{T}}({\bm{a}}_{1},{\bm{a}}_{2})$, where
the lattice periods (basis) $\\{{\bm{a}}_{1},{\bm{a}}_{2}\\}$ span $xy$-plane.
General element is (Koster–Seitz notation)
$g=(~{}\\!A~{}\\!|\\!~{}{\bm{a}}~{}\\!)$, where $A$ is three-dimensional
orthogonal transformation preserving $xy$-plane (and $z$-axis) and ${\bm{a}}$
is a translational vector (belonging to the $xy$-plane); the transformations
$(~{}\\!A~{}\\!|\\!~{}0~{}\\!)$ will be written as $A$. The introduced setup
implies that $A$ can be represented (in a basis of the two vectors lying in
the $xy$-plane, and the third one along $z$-axis) by a block-diagonal
${\bm{O}}(3)$ matrix, with the upper $2\times 2$ block from ${\bm{O}}(2)$,
while $A_{33}=\pm 1$. Then, all transformations have linear representation by
$4\times 4$ matrices of the block form
$E(g)=E(~{}\\!A~{}\\!|\\!~{}{\bm{a}}~{}\\!)=\left(\begin{smallmatrix}A&{\bm{a}}\\\
\vec{0}&1\end{smallmatrix}\right)$; ${\bm{a}}$ is 3D column of the
translational vector components, while $\vec{0}=(0,0,0)$. The representation
is faithful, and therefore the transformations are naturally identified with
these matrices.

The key structural feature of single LGs (either ordinary or magnetic) is
their factorization into weak direct product of cyclic groups [5]:

${\bm{G}}={\bm{C}}(g_{1})\circ\cdots\circ{\bm{C}}(g_{[{\bm{G}}]}),$ (1)

where ${\bm{C}}(g_{i})$ is a cyclic group generated by $g_{i}$, and
$[{\bm{G}}]$ is the number of cycles. The elements
$\\{g_{1},\cdots,g_{[{\bm{G}}]}\\}$ will be called _factorization generators_
(the choice of the cycles in (1), may not be unique; in addition, the
generators $g_{i}$ are not unique for cycles of the length greater than 2). A
useful consequence of such a structure is that each element is expressed
uniquely as a monomial over generators:

$g=g^{\alpha_{1}}_{1}\cdots
g^{\alpha_{[{\bm{G}}]}}_{[{\bm{G}}]},\quad\alpha_{1},\alpha_{2}=0,\pm
1,\dots,\quad\alpha_{3},\alpha_{4},\dots=0,\dots,|g_{i}|-1,$ (2)

providing thus the identification of an element with its monomial exponents:
$g\equiv(\alpha_{1},\dots,\alpha_{[{\bm{G}}]})$. The convention here
introduced should be noted: the first two generators are of infinite order
(translations, screw axes or glide planes), and thus their exponents can take
any integer value; otherwise, the exponents are positive integers less than
the order of an element $g_{i}$: $|g_{i}|=|{\bm{C}}(g_{i})|$.

_Graded generators_ are another set, useful in various structural analyses and
construction of IRs (Subsection 4.1). Namely, generators of single ordinary
LGs can be arranged into four grades. First grade groups, ${\bm{G}}^{(1)}$,
can be factorized as a semi-direct product, ${\bm{T}}\wedge{\bm{C}}_{n}$, with
pure translational generators
$\ell_{1}=(~{}\\!\mathds{1}~{}\\!|\\!~{}{\bm{a}}_{1}~{}\\!)$ and
$\ell_{2}=(~{}\\!\mathds{1}~{}\\!|\\!~{}{\bm{a}}_{2}~{}\\!)$, and rotation
$\ell_{3}=C_{n}$, as the third generator. Thus, the first grade groups are
LG1, LG3, LG65, LG49 and LG73, corresponding to the orders $n=1,2,3,4,6$ of
the principle axis of rotation (note that $\ell_{3}=C_{1}$ for $n=1$). Each LG
${\bm{G}}$ has one maximal subgroup ${\bm{G}}^{(1)}$ of the first grade.
Second grade groups can be built expanding the first grade ones:
${\bm{G}}^{(2)}={\bm{G}}^{(1)}+\ell_{4}{\bm{G}}^{(1)}$ by adding a generator,
$\ell_{4}$, with the property $\ell^{2}_{4}\in{\bm{G}}^{(1)}$. Further
expansions (by a generator, the square of which belongs to the second grade)
can be continued until all eighty LGs are constructed (single ordinary
groups). In this way series ${\bm{G}}^{(i)}$ of subgroups is formed for each
LG ${\bm{G}}$, where the last in the series is
${\bm{G}}={\bm{G}}^{({\bm{G}})}$ (number $({\bm{G}})$ will be called _grade of
the group_), and $i$-th group of the series, ${\bm{G}}^{(i)}$, halves the next
one, ${\bm{G}}^{(i+1)}$. In general, the graded generators are not the minimal
set of generators; thus instead of (2), the unique expansion over graded
generators

$g=\ell_{1}^{\beta_{1}}\dots\ell_{({\bm{G}})+2}^{\beta_{({\bm{G}})+2}},\ \
\beta_{1},\beta_{2}=0,\pm 1,\dots;\ \beta_{3}=0,\dots,n-1;\
\beta_{4},\dots,\beta_{({\bm{G}})+2}=0,1$ (3)

should be used.

Each generator carries one or two quantum numbers: translations are associated
to momentum, ${\bm{k}}$, rotation $C_{n}$ to the $z$-component of the angular
momentum $m$, while the involutions $U$ (rotation for $\pi$ around horizontal
axis), $\sigma_{v}$ and $\sigma_{h}$ (vertical and horizontal mirror planes)
give the parity quantum numbers $\Pi_{U}$, $\Pi_{v}$ and $\Pi_{h}$ (sometimes
shortened to $U$, $V$ and $H$); in addition, the graded generators with $i>3$
(with allowed exponents $0,1$) which are not involutions carry pairs of
quantum numbers: $k$ and $\Pi_{U}$ for helical axes, $k$ and $\Pi_{h}$ or
$\Pi_{v}$ for (horizontal and vertical) glide planes, and $m$ and $h$ for
$S_{2n}=C_{2n}\sigma_{h}$. Moreover, in the last case the square
$\ell^{2}_{i}=\ell^{\prime}$ is one of the precedent graded generators. If
$\ell^{\prime}$ is substituted by $\ell_{i}$ (allowing for $\beta_{i}$ all
values from the cycle of $\ell_{i}$), the complete group will be generated
again; actually, the factorization generators used (at the web site and in the
POLSym code) are obtained by such reduction of the set of graded generators.
For instance, graded set of generators of a nonsymmorphic LG7 are
$\\{\ell_{1}=(~{}\\!\mathds{1}~{}\\!|\\!~{}{\bm{a}}_{1}~{}\\!),\ell_{2}=(~{}\\!\mathds{1}~{}\\!|\\!~{}{\bm{a}}_{2}~{}\\!),\ell_{3}=C_{2},\ell_{4}=(~{}\\!\sigma_{h}~{}\\!|\\!~{}{\bm{a}}_{2}/2~{}\\!)\\}$.
The first three elements generate first grade (halving) subgroup LG3, while
the whole group LG7 is generated by
$\\{g_{1}=\ell_{1},g_{2}=\ell_{4},g_{3}=\ell_{3}\\}$.

###  2.2 Clusters of layer groups

As it is well known, inclusion of a half-integer spin degree of freedom urges
for rotational group ${\bm{S}O}(3)$ extension to its universally covering
group ${\bm{S}U}(2)$, which is further inherited by all groups containing
rotations. These double coverings are known as double groups. Also, when TRS
is considered, besides the ordinary groups (single and double) gray and black-
and-white magnetic groups can be associated to the same LG. Thus the eighty
sets of LGs include ordinary LG ${\bm{G}}$, and the corresponding double LG
$\widetilde{{\bm{G}}}{}$, gray LG ${\bm{G}}1^{\prime}$, double gray LG
$\widetilde{{\bm{G}}}{}1^{\prime}$, magnetic and double magnetic groups making
the _clusters_ associated to each of the eighty ordinary single LGs. Since 368
black-and-white groups are not analyzed here, for the time being presented are
only the four-membered (${\bm{G}}$, $\widetilde{{\bm{G}}}{}$,
${\bm{G}}1^{\prime}$, $\widetilde{{\bm{G}}}{}1^{\prime}$) parts of each of the
clusters.

In 2D spin-half space translations and spatial inversion act trivially and
thus only orthogonal transformations’ part $R=\det(A)A$ of
$g=(~{}\\!A~{}\\!|\\!~{}{\bm{a}}~{}\\!)$ is relevant. However, since
${\bm{S}U}(2)$ is a double covering group of the ${\bm{S}O}(3)$ rotational
group, two antipodal matrices $u(g)$ and $-u(g)$ are associated to each
Euclidean transformation $g$. Double LG $\widetilde{{\bm{G}}}{}$ is double
extension of the single LG ${\bm{G}}$, so that its action is expanded beyond
the Euclidean action in $\mathds{R}^{3}$: it acts also in the spin space
$\mathds{C}^{2}$. Therefore, two _antipodal_ elements $\tilde{g}{}=(g,u(g))$
and $\tilde{g}{}^{\prime}=(g,-u(g))$ are associated to each element
$g\in{\bm{G}}$. Note that either of the two opposite-sign matrices, mapped by
two-to-one homomorphism of ${\bm{S}U}(2)$ onto ${\bm{S}O}(3)$ into $R$, can be
taken as $u(g)$. Consequently, elements of double group are faithfully
represented by 6D matrices
$\tilde{E}{}(\tilde{g}{})=E(~{}\\!A~{}\\!|\\!~{}{\bm{a}}~{}\\!)\oplus u(A)$
[9]. For double groups pairwise intersections of cycles in (1) may contain
besides identity $\tilde{e}{}$, also its antipodal $\tilde{e}{}^{\prime}$,
hence this is not a weak direct product, which prevents uniqueness of the
exponents in (2). Fortunately, this can be fixed by adding antipodal to
identity $\tilde{e}{}^{\prime}$ of order two as the last generator [9]:

$\tilde{g}{}=\tilde{g}{}^{\alpha_{1}}_{1}\tilde{g}{}^{\alpha_{2}}_{2}\cdots\tilde{g}{}^{\alpha_{[{\bm{G}}]}}_{[{\bm{G}}]}\tilde{e}{}^{\prime\varepsilon},\quad\alpha_{1},\alpha_{2}=0,\pm
1,\dots,\ \ \alpha_{3},\alpha_{4},\dots=0,\dots,|g_{i}|-1,\varepsilon=0,1.$
(4)

Finally, in magnetic (single or double) group
${\bm{G}}t^{\prime}={\bm{G}}+t\theta{\bm{G}}$ there is additional generator
$t\theta$ such that $(t\theta)^{2}=t^{2}$ is an element of ${\bm{G}}$. Here we
consider only the case of gray groups, when $t$ is the identity; otherwise,
black-and-white groups are obtained. Clearly, this introduces additional
generator $\theta$.

Note that in gray groups $\theta$ is placed as the last generator (by
convention), while $\tilde{e}{}^{\prime}$ (double groups) is put right after
the geometrical transformation generators.

##  3 Brillouin zone and irreducible domain

Irreducible representations of group ${\bm{T}}$ are classified by quasi-
momentum ${\bm{k}}$ running over Brillouin zone (BZ) which has a form of a
torus $T^{2}$. The group is abelian, and IRs are 1D:
$\Delta^{({\bm{k}})}(~{}\\!\mathds{1}~{}\\!|\\!~{}{\bm{a}}_{i}~{}\\!)=\mathrm{e}^{\mathrm{i}{\bm{k}}_{i}{\bm{a}}_{i}}$
($i=1,2$). They are starting point for construction of IRs of a LG ${\bm{G}}$.
In the induction procedure, the first step is to determine orbits of
conjugation of $\Delta^{({\bm{k}})}({\bm{T}})$ by cosets of ${\bm{T}}$, which
effectively reduces to the ${\bm{k}}^{*}$-stars (orbits) of the polar vector
action of ${\bm{G}}$ in BZ. Set of star-representatives forms an irreducible
domain (ID). The latter consists of the generic (dense) stratum (with minimal
stabilizer ${\bm{G}}_{\bm{k}}$), which can be fully or partially bounded. The
special strata (having stabilizers which are nontrivial supergroups of the
generic stratum stabilizer ${\bm{G}}_{\bm{k}}$) are on the boundary of the
generic one. Here, two significant characteristics should be highlighted.
Firstly, as translations in BZ act trivially, they are subgroups of the all
stabilizers, and thus the stabilizers are LGs themselves. Consequently, only
isogonal point groups ${\bm{P}}_{I}$ are relevant, and thus stabilizing
elements pertaining to ${\bm{P}}_{I}$ are to be extended by the translations
to get the full little group. Secondly, isogonal transformations effectively
act as $2\times 2$ upper block of the usual polar ${\bm{O}}(3)$ action, i.e.
they form a subgroup of ${\bm{O}}(2)={\bm{S}O}(2)+U\,{\bm{S}O}(2)$ matrices,
and transformations differing by action in $z$-direction only are mutually
identical. Hence, there remain ten algebraically different point groups, which
by stratification of BZ form IDs: ${\bm{C}}_{1}$, ${\bm{C}}_{2}$,
${\bm{C}}_{3}$, ${\bm{C}}_{4}$, ${\bm{C}}_{6}$ and ${\bm{D}}_{1}$,
${\bm{D}}_{2}$, ${\bm{D}}_{3}$, ${\bm{D}}_{4}$, ${\bm{D}}_{6}$.

Further, a nontrivial topological characteristic related to the action of the
two-fold horizontal rotations $U\in{\bm{D}}_{n}$ on BZ-torus should be pointed
out. Namely, the $U$-axis can be either along ${\bm{k}}_{1}$ or ${\bm{k}}_{2}$
or bisector ${\bm{k}}_{0}={\bm{k}}_{1}+{\bm{k}}_{2}$ and these $U_{i}$-axes
($i=0,1,2$) are closed paths on torus. However, the axes are not homotopy
equivalent. Namely, as ${\bm{D}}_{n}$ has $n$ different $U$-axes, in the cases
$n=2$ and $n=3$, existence of the $U_{1}$-axis entails also presence of the
$U_{2}$-axis (i.e these two axes are equivalent), while in the cases $n=4$ and
$n=6$, all the U-axes are equivalent. To summarize, there are altogether nine
topologically (and algebraically) different dihedral groups:
${\bm{D}}^{1}_{1}$, ${\bm{D}}^{2}_{1}$, ${\bm{D}}^{0}_{1}$,
${\bm{D}}^{1}_{2}$, ${\bm{D}}^{0}_{2}$, ${\bm{D}}^{1}_{3}$,
${\bm{D}}^{0}_{3}$, (the superscripts indicate the above described position of
the $U$-axis within BZ) ${\bm{D}}_{4}$, and ${\bm{D}}_{6}$; these groups,
together with aforementioned ${\bm{C}}_{n}$ groups, make fourteen different
isogonal group actions on torus, ${\bm{P}}^{T}_{I}$, and thus BZ-
stratification results in fourteen topologically different IDs, Fig. 1. Hence,
having all the ID types of LGs at disposal, one can easily associate LGs to
IDs, with ${\bm{P}}^{T}_{I}$ being ${\bm{O}}(2)$-part of the isogonal group
${\bm{P}}_{I}$.

Figure 1: Irreducible domains of layer groups. Gray - generic stratum (G),
with black boundaries belonging to G. Color denotes stratum relative order
$|{\bm{G}}|/|{\bm{G}}_{K}|$: 1 (red) ,2 (green), 3 (blue), 4 (violet) and 6
(orange). Groups corresponding to IDs are indicated by colors (explained in
the central panel).

Time reversal transformation $\theta$ reverses ${\bm{k}}$ vectors, which is
equivalent to $\pi$-rotation around $z$-axis. Hence, inclusion of TRS
manifests as the inclusion of $C_{2}$ into ${\bm{P}}^{T}_{I}$, enlarging
${\bm{C}}_{1}$ to ${\bm{C}}_{2}$, ${\bm{C}}_{3}$ to ${\bm{C}}_{6}$, both
${\bm{D}}^{1}_{1}$ and ${\bm{D}}^{2}_{1}$ to ${\bm{D}}^{1}_{2}$,
${\bm{D}}^{0}_{1}$ to ${\bm{D}}^{0}_{2}$ and both ${\bm{D}}^{1}_{3}$ and
${\bm{D}}^{0}_{3}$ to ${\bm{D}}_{6}$, while the remaining groups stay the same
(as $C_{2}$ is their element). In this way seven ID-types of gray groups
${\bm{G}}1^{\prime}$ are identified. Black-and-white groups are of the form
${\bm{G}}\ell^{\prime}={\bm{G}}+\ell\theta{\bm{G}}$, where $\ell$ is a
Euclidean transformation preserving $xy$-plane, such that
$\ell^{2}\in{\bm{G}}$. For these groups ID is found in an analogous way as for
the gray groups (for which $\ell=\mathds{1}$, or equivalently
$\ell\in{\bm{G}}$): if ${\bm{P}}^{T}_{I}$ corresponds to ${\bm{G}}$, then
addition of $C_{2}\ell$ to ${\bm{P}}^{T}_{I}$ gives the toroidal isogonal
group of a black-and-white group.

Clearly, ${\bm{G}}$ (ordinary or magnetic) and its double group
$\widetilde{{\bm{G}}}{}$ have the same ID, as the extension by spin space does
not affect the translational group (and its IRs parameterized by the BZ wave
vector).

In conclusion, the LGs action yield fourteen ID types, each ID decomposes onto
strata, counted by $K$. Only generic stratum is 2D and dense. Lower-D strata,
special lines and special points make (partial) boundary of the generic one.
The sets of LGs associated to the specific IDs are listed in Table 1.

Table 1: Irreducible domains of layer groups. Columns: counter (ID), topological class of the 2D isogonal group ( ${\bm{P}}^{T}_{I}$), ordinals of associated ordinary single and double groups (${\bm{G}}$, $\widetilde{{\bm{G}}}{}$), IDs of the corresponding gray groups (ID of ${\bm{G}}1^{\prime}$ & $\widetilde{{\bm{G}}}{}1^{\prime}$), bolded when different from the IDs of the counterparts (without TRS, ID-column). ID | ${\bm{P}}^{T}_{I}$ |  ${\bm{G}}$, $\widetilde{{\bm{G}}}{}$ | ID of ${\bm{G}}1^{\prime}$ & $\widetilde{{\bm{G}}}{}1^{\prime}$
---|---|---|---
1 | ${\bm{C}}_{1}$ | 1,4,5 | 2
2 | ${\bm{C}}_{2}$ | 2,3,6,7 | 2
3 | ${\bm{D}}^{2}_{1}$ | 9,12,28,29,30 | 8
4 | ${\bm{D}}^{1}_{1}$ | 8,11,27,31,32,33,34 | 8
5 | ${\bm{D}}^{0}_{1}$ | 10,13,35,36 | 9
6 | ${\bm{C}}_{3}$ | 65,74 | 10
7 | ${\bm{C}}_{4}$ | 49,50,51,52 | 7
8 | ${\bm{D}}^{1}_{2}$ | 14,15,16,17,19,20,21,23,24,25, | 8
|  | 37,38,39,40,41,42,43,44,45,46 |
9 | ${\bm{D}}^{0}_{2}$ | 18,22,26,47,48 | 9
10 | ${\bm{C}}_{6}$ | 66,73,75 | 10
11 | ${\bm{D}}^{1}_{3}$ | 67,69,78 | 14
12 | ${\bm{D}}^{0}_{3}$ | 68,70,79 | 14
13 | ${\bm{D}}_{4}$ | 53,54,55,56,57,58, | 13
|  | 59,60,61,62,63,64 |
14 | ${\bm{D}}_{6}$ | 71,72,76,77,80 | 14

##  4 Representations

Group representations are well defined on the generators only, as homomorphism
of the representation and unique monomial expansion (2) give:

$D(g)=D^{\alpha_{1}}(g_{1})D^{\alpha_{2}}(g_{2})\cdots
D^{\alpha_{[{\bm{G}}]}}(g_{[{\bm{G}}]}),$ (5)

for arbitrary group element $g$. Here, in addition to the above introduced
Euclidean representation $E(~{}\\!A~{}\\!|\\!~{}{\bm{a}}~{}\\!)$ and natural
spin representation $u({\tilde{g}{}})$, considered are IRs and allowed
representations. The latter are representations of stabilizers, which are also
LGs, i.e groups of infinite order. However, since pure translations are
represented as (multiple of) $\Delta^{({\bm{k}})}$, to avoid non-unique choice
of generators, they are given on the elements of the little co-group (which
forms transverzal of ${\bm{T}}$ in the little group).

###  4.1 Irreducible representations

Standard prescription for construction of IRs uses ID in a way that for each
representative ${\bm{k}}$ of a connected stratum $K$ its little group
(stabilizer) ${\bm{G}}_{K}$ is found. Then the allowed representations (by
definition, IRs of ${\bm{G}}_{K}$ subducing on ${\bm{T}}$ multiple of
$\Delta^{({\bm{k}})}({\bm{T}})$) are constructed. As the little group is a LG
itself, the little co-group ${\bm{G}}_{K}/{\bm{T}}$ is a crystallographic
axial point group, and its projective IR
$d^{(K\kappa)}({\bm{G}}_{K}/{\bm{T}})$ is one factor of the allowed
representation, while the other one is a phase related to
$\Delta^{({\bm{k}})}({\bm{T}})$ (compensating the factor system of the
aforementioned projective representation). Finally, each of these
representations $d^{(K{\bm{k}}\kappa)}({\bm{G}}_{\bm{K}})$ gives by induction
[10, 11, 12] one IR of the whole group:
$D^{(K{\bm{k}}\kappa)}({\bm{G}})=d^{(K{\bm{k}}\kappa)}({\bm{G}}_{K})\uparrow{\bm{G}}$;
gathering all of them for each $K$ in ID, the complete set of non-equivalent
IRs of ${\bm{G}}$ is obtained. IRs are grouped in the $K\kappa$-series of the
representations of the same form, differing in the value of ${\bm{k}}$, with
the same (projective) representation $d^{(K\kappa)}({\bm{G}}_{K}/{\bm{T}})$.

An alternative procedure encoded in POLSym utilizes graded generators, and
skips the allowed representations, except for the simplest, first grade LGs
${\bm{G}}^{(1)}$. These groups are symmorphic, with little co-groups
${\bm{C}}_{n_{K}}$, where $n_{K}$ divides $n$ (for generic stratum $n_{K}=1$,
while the remaining strata are special points for which $n_{K}>1$), having
only one-dimensional linear (trivial projective) IRs. Hence, allowed IRs of
the stabilizers are
$d^{(K{\bm{k}}\kappa)}(\ell_{i})=\Delta^{({\bm{k}})}(\ell_{i})$ ($i=1,2$) and
$d^{(K{\bm{k}}\kappa)}(\ell_{3}=C_{n})=\mathrm{e}^{\mathrm{i}2\kappa\pi/n_{K}}$,
where $\kappa$ is integer or (for double groups) half-integer within the
interval $(-n_{K}/2,n_{K}/2]$. The induction procedure gives
$n/n_{K}$-dimensional matrices. For arbitrary group ${\bm{G}}$ IRs are
obtained from IRs of its (unique) first grade subgroup ${\bm{G}}^{(1)}$.
Namely, as ${\bm{G}}^{(1)}$ is a halving subgroup in ${\bm{G}}^{(2)}$, IRs of
the latter are straightforwardly found from IRs of ${\bm{G}}^{(1)}$ using
induction procedure for index-two supergroups [13, 8]. The same method is
successively applied in the follow-up grade(s), until IRs of ${\bm{G}}$ are
determined. For magnetic groups, the final step is the so called $*$-induction
[13, 8] of co-IRs of a magnetic group from IRs of its halving subgroup. It is
important to find the strata of ${\bm{G}}$ first and then in each step IRs of
${\bm{G}}^{(i)}$ for each of the strata of ${\bm{G}}$ are constructed. The
additional symmetries of the higher-grade groups may refine ID, in a way that
special lines and special points of the preceding grade group are retained,
while the new ones may emerge.

###  4.2 Allowed representations

Allowed representations are of relevance for many physical applications and
since in the here applied induction-procedure their construction is skipped,
the method of extracting allowed representations from IRs is developed. At
first, for the representations associated to the $K$-stratum, set of IRs
$D^{(K{\bm{k}}\kappa)}({\bm{G}})$ is selected. Each element $f$ of the
stabilizer ${\bm{G}}_{K}$ can be expressed in terms of the group generators in
the form (2), and thus the matrices $D^{(K{\bm{k}}\kappa)}(f)$ are easily
found using (5). As has been previously emphasised, little group is a LG and
thus there are infinitely many such elements. However, the stabilizers are
characterized by the stabilizer co-group elements, i.e. the coset
representatives of the translational subgroup ${\bm{T}}$ in ${\bm{G}}_{K}$.
The orbit order (star ${\bm{k}}^{*}$ of ${\bm{k}}$ from $K$) is
$|{\bm{k}}^{*}|=|{\bm{G}}/{\bm{G}}_{K}|$ (equal to the ratio of the isogonal
groups of ${\bm{G}}$ and ${\bm{G}}_{K}$), and the above described induction-
procedure from the little group allowed representation enlightens that when
$D^{(K{\bm{k}}\kappa)}(f)$ is partitioned into
$|D|/|{\bm{k}}^{*}|=|D|/|{\bm{G}}/{\bm{G}}_{K}|$-dimensional blocks (here,
$|D|$ denotes the dimension of the representation $D$ and
$|d^{(K{\bm{k}}\kappa)}|=|D|/|{\bm{G}}/{\bm{G}}_{K}|$ is order of the coset
space ${\bm{G}}/{\bm{G}}_{K}$), the first block gives the allowed
representation: $d^{(K{\bm{k}}\kappa)}(f)=D^{(K{\bm{k}}\kappa)}_{11}(f)$.

##  5 The web site

All here described results (with a number of the relevant extensions) are
presented at the web-site www.nanolab.group/layer (except for the clusters 58
and 80 a free registration is needed).

The home page Layer groups contains menu (Fig. 2, left margin), with
Introduction gathering several notes on the purpose and main ideas of the
site, Notation giving the system of labeling (group transformations and
Brillouin zone strata, in particular) with comparison to other referent
sources, as well as definitions of the not so frequently used notions. Follows
page Groups overview with general data on the LGs: their ordinal and
international symbol [2], holohedry, factorization, isogonal group, grade
$({\bm{G}})$, i.e. the number of graded generators minus 2, ID, symmorphism,
the factorization generators that are used in POLSym. Generators are given in
the standard order: firstly the translational ones (including possible
fractional translations), then, depending on the group, $g_{3}$ is either
$C_{n}$ or $S_{2n}=C_{2n}\sigma_{\mathrm{h}}$, and the remaining generators
(at most two for ordinary single groups) follow. To remind, for double groups
the next one is the antipodal to identity, $\tilde{e}{}^{\prime}$. In gray
groups the last generator is time reversal $\theta$.

Generators are specified at page Generators where listed are all generators
(22 in total) used for LGs (including double and gray ones). The first eight
generators comprise lattice and fractional translations, and therefore contain
lattice parameters. Analogously, generators 9 and 15, $C_{n}$ and $S_{2n}$
depend on the principle axis order; all these parameters are to be substituted
when used in a particular group. Then matrices of Euclidean and natural spin
representations follow. Euler angles are the next entry. In fact, orthogonal
part $A$ of a generator is either $R$ or $JR$; while $J$ is spatial inversion,
$R$ is a rotation from ${\bm{S}O}(3)$ defined by three Euler angles (real
numbers) complemented with number 1 if $A$ is a rotation, and with $-1$,
otherwise. The orders of the generators in Euclidean and sum of the Euclidean
and spin space, as well as conventional order of the latter in the described
notation which includes $\tilde{e}{}^{\prime}$ follow. Finally, quantum
numbers introduced by the generators are given: ${\bm{k}}$ for translations,
$m$ for rotations, and different parities. Glide planes and screw axes, in
addition to ${\bm{k}}$, introduce one more quantum number, associated to the
orthogonal element. Analogously, $S_{2n}$ has rotational part yielding quantum
number $m$ and reflectional part yielding parity quantum number $H$.

Brillouin zones presents IDs of LGs.

Lastly, there is a table with links to eighty clusters of LG Layer group $N$,
$N=1,\dots,80$. For the time being, only data for four-membered parts of
clusters is presented (ordinary single and double, and gray single and double
groups, while the data on black and white groups is under construction); these
groups will be denoted here as ”LG $N$ SPIN MAG” where SPIN may be single or
double, and MAG ordinary or gray. For each of them, at a separate linked page,
given are the following data:

LG $N$ SPIN MAG: Generators Data on generators of a particular group:
Euclidean and natural spin representation, Euler angles, quantum numbers,
order of the element used in (2), and for double groups given is also the
order of the element applied in (4).

LG $N$ SPIN MAG: Wyckoff Stratification of the Euclidean space obtained by the
group action, i.e. description of the fundamental domain. For each stratum its
label, coordinates of the general Wyckof position, stabilizer (site-symmetry
group), elements of the stabilizer (exponents of (2) or (4)), order of the
stabilizer. Topology: adjacency graph of strata. Maximal Wyckoff positions are
singled out.

LG $N$ SPIN MAG - Brillouin zone Layer group action onto Brillouin zone: each
stratum (generic, special lines and points) is labeled by a symbol, general
${\bm{k}}$ vector with its star, a typical numeric point of the stratum, order
of little co-group and its elements (exponents of the monomial in generators),
associated IRs’ ordinals, with the IR dimension and allowed representations.
Depicted are Brillouin zone strata and IDs.

LG $N$ SPIN MAG - IRs Tables of (co)-IRs. Beside ordinals (used in POLSym) the
matrices (${\bm{k}}$-dependent in general) of $K\kappa$-series are given for
the group generators. Also some data are presented: corresponding quantum
numbers (the first one is label giving the momenta values stratum), symbol of
IR based on the quantum numbers (bar denotes negative sign, underline denotes
division by two) and dimension of the allowed representation.

LG $N$ SPIN MAG - Allowed representations Allowed representations of the
little groups. For each stratum the elements of the little co-group are given
(monomial exponents in generators) together with their matrix representation.

LG $N$ SPIN MAG - Site symmetry induced band representations Band
representations induced from IRs of the site stabilizers. For each stratum its
label is given together with the stabilizer (site symmetry group) ${\bm{F}}$
and the standard point group generators; then IRs $D^{(\sigma)}({\bm{F}})$ of
the stabilizer are enumerated and explicated by matrix representation of the
mentioned generators, accompanied by standard quantum numbers. Finally,
frequencies of (co)-IRs of the whole group in the induced (co-)representation
$D^{(\sigma)}({\bm{F}})\uparrow{\bm{G}}$ are tabulated.

Figure 2: Illustration of the web site information: co-IRs of single gray LG10
and RIRs (if the $\theta$-columns are disregarded) of single ordinary LG10.

##  6 Discussions

Most of the calculations, including construction of IRs and allowed
representations, are performed by symbolic programming with code POLSym. The
key ingredients are the afore described structural properties, LG
factorization and graded hierarchy of index-two subgroups, suitable for the
application of the modified group projector technique [14], utilized by POLSym
and being also efficient for antilinear operators.

The two sets of generators are defined; one set is suited to the factorization
(1), and the other for the graded LG series. The first two generators are
translations (generalized in (1)). The third generator in (1) is $C_{n}$ or
$S_{2n}$, while in the grade-suited convention it is $C_{n}$ solely. There are
many groups with $n=1$, and thus the third generator is identity. To establish
universality of the notation (and subsequent derivation of the results by
computer algebra), the tables at the web site (as well as the code POLSym)
refer to this generator (and use it) in a standard manner, at the cost of
possible redundancy, i.e. by giving a piece of a trivial information. The
remaining generators are the elements which are squared into the subgroup
generated by the preceding generators. Elements $\tilde{e}{}^{\prime}$ and
$\theta$, when present, stand at the position $[G]+1$ and at the last
position, respectively.

From (1) it stems that each LG can be factorized onto the _generalized 2D
translational group_ and axial point group, ${\bm{G}}=\hat{{\bm{T}}}{\bm{P}}$
[5], where the group of generalized translations $\hat{{\bm{T}}}$ comprises
the infinite part, the product $\hat{{\bm{T}}}={\bm{C}}(g_{1}){\bm{C}}(g_{2})$
of 1D generalized translations leaving the $xy$-plane invariant. The latter
are depicted at the page Factorization (the link can be found at Groups
overview legend).

Note that the LGs obtained by scaling of the periods (when scaling is either
independent on the present orthogonal symmetries, or orthogonal symmetry
imposes certain restrictions onto the scaling factors) are isomorphic, and
thus they have the same IRs. Consequently, IRs can be given in the
translational period independent form, i.e. ${\bm{k}}$-components take values
from the interval $(-\pi,\pi]$, while translations are dimensionless integers;
alternatively, one may assume that the periods are length units along the
translational axes.

In many physical applications, especially for the sake of interpretation of
the obtained physical results, real representations are more convenient than
the complex ones. Therefore, instead of complex IRs, the so called physical
(or real) irreducible representations (RIRs) are used. They are of importance
not only when there is time reversal invariance, but also, in the estimation
of the electron-phonon coupling in layered materials [15] where it is RIRs-
based analysis which explains Jahn-Teller (in)stability in 2D, as well as the
absence of coupling between flexural phonons and electrons. By definition,
RIRs are representations of ordinary (single or double) group ${\bm{G}}$, with
real matrix elements, which cannot be decomposed onto real representations.
Technically, RIRs coincide with subduced to ${\bm{G}}$ IRs of the gray group
${\bm{G}}1^{\prime}$ and thus they can easily be read out from the given
irreducible matrix representations of the gray groups, Fig. 2. Therefore they
are not listed separately.

POLSym subroutines BZStabilizers and FundamentalDomain are subroutines which
output the afore presented information. The relevant procedures are suitable
also for black-and-white groups, and the results and related set of data are
available on request, for the time being.

Finally, let us note that POLSym program has broad scope of applications, far
beyond the here presented ones, and it is expected in the near future that the
POLSym platform will be made available to the scientific community.

## References

  * [1] E. A. Wood.  80 diperiodic groups in three dimensions.  Bell System Monograph, 4680, 1964.
  * [2] V. Kopsky and D. B. Litvin.  Subperiodic groups, volume E of International Tables for Crystallography.  Kluwer, Dordrecht, 1 edition, 2002.
  * [3] R. Evarestov and V. P. Smirnov.  Site Symmetry in Crystals: Theory and Applications, volume 106 of Springer Series in Solid-State Sciencesz.  Springer, 2012.
  * [4] G. de la Flor, D. Orobengoa, R. A. Evarestov, Y. E. Kitaev, E. Tascie, and M. I. Aroyo.  The site-symmetry induced representations of layer groups on the bilbao crystallographic server.  J. Appl. Cryst., 52:1214 -1221, 2019.
  * [5] Ivanka Milošević, Božidar Nikolić, Milan Damnjanović, and Maja Krčmar.  Irreducible representations of diperiodic groups.  Journal of Physics A: Mathematical and General, 31(15):3625, 1998\.
  * [6] M. I. Aroyo, J. M. Perez-Mato, C. Capillas, E. Kroumova, S. Ivantchev, G. Madariaga, A. Kirov, and H. Wondratschek.  Bilbao crystallographic server i: Databases and crystallographic computing programs.  Z. Krist., 221(1):15–27, 2006.
  * [7] M. I. Aroyo, A. Kirov, C. Capillas, J. M. Perez-Mato, and H. Wondratschek.  Bilbao crystallographic server ii: Representations of crystallographic point groups and space groups.  Acta Cryst. A, 62:115–128, 2006.
  * [8] Milan Damnjanović and Ivanka Milošević.  Line Groups in Physics: Theory and Applications to Nanotubes and Polymers, volume 801 of Lecture Notes in Physics.  Springer, Berlin, 2010.
  * [9] N Lazić, M Milivojević, T Vuković, and M Damnjanović.  Double line groups: structure, irreducible representations and spin splitting of the bands.  Journal of Physics A: Mathematical and Theoretical, 51(22):225203, 2018.
  * [10] G. W. Mackey.  Induced representations of locally compact groups. i.  Ann. of Math., 55:101 139, 1952.
  * [11] E. P. Wigner.  Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra.  New York: Academic Press, 1959.
  * [12] T. Janssen.  Crystallographic groups.  North-Holland, Amsterdam, 1973.
  * [13] L. Jansen and M. Boon.  Theory of Finite Groups. Applications in Physics.  North-Holland, Amsterdam, 1967.
  * [14] Milan Damnjanović and Ivanka Milošević.  Full symmetry implementation in condensed matter and molecular physics — modified group projector technique.  Physics Reports, 581:1, 2015.
  * [15] Božidar Nikolić, Ivanka Milošević, and Milan Damnjanović.  Electron-phonon (de)coupling in 2d.  Physica E: Low-dimensional Systems and Nanostructures, 126:114468, 2021.

#### Synopsis

For single and double, ordinary and gray layer groups ${\bm{G}}$, all
irreducible (half-)integer (co-)representations $D^{(\mu)}({\bm{G}})$ are
derived as well as the allowed (co-)representations of little groups. Band
representations for all types of layer groups are induced from the irreducible
representations of the site symmetry groups and decomposed onto the
irreducible components. Together with many other important characteristics of
the layer groups, these results are presented at the new web-site.