158mm238mm

#  Non-invertible Symmetries of Class $\cS$ Theories

Vladimir Bashmakov  1 Michele Del Zotto  1,2
Azeem Hasan  2 and Justin Kaidi3,4
$1$ Department of Physics and Astronomy Uppsala University
Box 516 SE-75120 Uppsala  Sweden
$2$ Mathematics Institute Uppsala University
Box 480 SE-75106 Uppsala  Sweden
$3$ Department of Physics
University of Washington Seattle  WA  98195  USA
$4$ Kavli Institute for the Physics and Mathematics of the Universe
University of Tokyo Kashiwa  Chiba 277-8583  Japan

###### Abstract

We study the non-invertible symmetries of class ${\cal S}$ theories obtained
by compactifying the type $\mathfrak{a}_{p-1}$ 6d (2,0) theory on a genus $g$
Riemann surface with no punctures. After setting up the general framework, we
describe how such symmetries can be classified up to genus 5. Of central
interest to us is the question of whether a non-invertible symmetry is
intrinsic, i.e. whether it can be related to an invertible symmetry by
discrete gauging. We then describe the higher-dimensional origin of our
results, and explain how the Anomaly and Symmetry TFTs, as well as $N$-ality
defects, of class ${\cal S}$ theories can be obtained from compactification of
a 7d Chern-Simons theory. Interestingly, we find that the Symmetry TFT for
theories with intrinsically non-invertible symmetries can only be obtained by
coupling the 7d Chern-Simons theory to topological gravity.

#####  Abstract

————————–

November 2022

PREPRINT UUITP 50/22

###### Contents

  1. 1 Introduction
    1. 1.1 Global variants of $\mathfrak{su}(p)$ SYM
    2. 1.2 Modular orbits of global variants
    3. 1.3 Invariant global forms
    4. 1.4 Higher-dimensional point of view
    5. 1.5 Goals and organization
  2. 2 Non-invertible symmetries for class ${\cal S}$ theories
    1. 2.1 A review of class ${\cal S}$
    2. 2.2 Non-invertible symmetries and the modular group
    3. 2.3 Global variants of theories of class ${\cal S}$
    4. 2.4 Modular orbits of global variants
  3. 3 Non-invertible symmetries for genus-2 class ${\cal S}$
    1. 3.1 Genus 2 and the modular group $Sp(4,{\mathbb{Z}}_{p})$
    2. 3.2 Fixed points of $Sp(4,{\mathbb{Z}})$
    3. 3.3 Invariant global forms
  4. 4 Intrinsic versus non-intrinsic at higher genus
    1. 4.1 Characteristic polynomials of modular matrices
    2. 4.2 Invariant subspaces of modular matrices
    3. 4.3 Invariant global forms
  5. 5 A higher-dimensional perspective
    1. 5.1 The 6d (2,0) theory as a relative theory
    2. 5.2 Compactification on $\Sigma_{g,n}$
    3. 5.3 From the Anomaly TFT to the Symmetry TFT
    4. 5.4 Surgery defects
  6. A Symmetry generators at genus $2$
  7. B Some algebra
    1. B.1 Roots of unity over finite fields
    2. B.2 Integer matrices of finite order
    3. B.3 Intersections and the Jordan blocks
    4. B.4 Factorization $\phi_{15}(x)$ of over finite fields
    5. B.5 Composite characteristic polynomial
  8. C Basics of handlebodies
    1. C.1 Solid tori
    2. C.2 Higher genus handlebodies
  9. D Alternative expression for ${\cal C}_{F}(X_{4})$

##  1 Introduction

The study of generalized global symmetries [1, 2, 3, 4, 5, 6] is undergoing a
surprising evolution. Recent progress has taken us beyond the paradigm of
group-like symmetry to “categorical” symmetry, the structure of which is
encoded in a fusion (higher-)category. A particularly interesting class of
such symmetries are those which are “non-invertible.” Non-invertible
symmetries have long been known in $(1+1)$-dimensions [7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], but it was not until
fairly recently that they were realized in higher-dimensional theories, with
several different constructions now known [26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
54, 55, 56, 57, 58, 59, 60, 61, 62, 63].111 For further constructions of
generalised symmetries in higher-dimensional theories see e.g. [64, 65, 66,
67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85,
86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103,
104, 105]. This paper continues the study of non-invertible symmetries in a
particular class of supersymmetric $(3+1)$d theories, namely those of class
${\cal S}$ [106, 107].

A particular way of realising non-invertible symmetries, which we will heavily
use in this paper, is by half-space gauging [27]. Let us illustrate the
technique in the specific case of $SU(2)$ ${\cal N}=4$ super-Yang Mills (SYM)
theory. We begin by defining an operation $\sigma$ which gauges the
${\mathbb{Z}}_{2}^{(1)}$ one-form symmetry of the $SU(2)$ theory in half of
space-time, with Dirichlet boundary conditions at the interface. As discussed
in [26, 27], this gives rise to an interface between the $SU(2)$ and
$SO(3)_{+}$ theories, both at the same value of the Yang-Mills coupling
$\tau_{\mathrm{YM}}$. Because the theories on the left and the right of the
interface are in general distinct, $\sigma$ itself does not correspond to a
defect in a single theory. However, ${\cal N}=4$ SYM is known to enjoy an
$SL(2,{\mathbb{Z}})$ duality, which in particular contains an $S$-duality
operation denoted by ${\mathsf{S}}$. This operation acts on the global form
$SU(2)$ in the same way as $\sigma$, and also acts on the coupling. Composing
the two operations gives the configuration in Figure 1. From this we see that
the combined transformation ${\cal N}_{\mathsf{S}}:={\mathsf{S}}\sigma$ gives
rise to a symmetry in the $SU(2)$ theory, as long as we choose the couplings
on both sides to match, i.e.

$\displaystyle\tau_{\mathrm{YM}}=-1/\tau_{\mathrm{YM}}\hskip
36.135pt\Rightarrow\hskip 36.135pt\tau_{\mathrm{YM}}=i\leavevmode\nobreak\ .$
(1.1)

Since it involves a half-space gauging, this symmetry is non-invertible by
construction.

For this particular non-invertible symmetry, there is known to be an
alternative construction: namely, we can gauge an invertible symmetry with
mixed anomaly in the $SO(3)_{-}$ theory at $\tau_{\mathrm{YM}}=i$ [26]. In
general, if one begins with a theory with invertible, anomalous symmetries and
then does appropriate topological manipulations (in practice, discrete
gaugings) to obtain a non-invertible symmetry in a different global variant of
the theory [26], the resulting non-invertible symmetry is referred to as non-
intrinsic [35]: one can always find a different global variant of the theory
in which the consequences of the non-invertible symmetry can be reinterpreted
in terms of those of an invertible one. In the context of class ${\cal S}$, a
particular family of such non-intrinsic non-invertible symmetries was
identified in [39].

In contrast, non-invertible symmetries which cannot be related to anomalous
invertible symmetries via topological manipulations are referred to as
intrinsically non-invertible symmetries. One of the main goals of this work is
to classify non-invertible symmetries of general theories of class ${\cal S}$,
and to establish criteria for when they are intrinsic. Perhaps unsurprisingly,
we will find that intrinsic non-invertibility is the generic situation.

Since all of the relevant tools for the study of general class ${\cal S}$ can
be developed already at the level of 4d ${\cal N}=4$ theories, we will now
describe this case in full detail. The reader who has studied this
introduction should have no trouble following the more technical discussions
of the main text.

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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ Figure 1: At
$\tau_{\mathrm{YM}}=i$, the $SU(2)$ theory has a non-invertible defect ${\cal
N}_{\mathsf{S}}$, which can be understood as the composition of a defect
$\sigma$ implementing gauging of the ${\mathbb{Z}}_{2}^{(1)}$ one-form
symmetry, together with an invertible ${\mathsf{S}}$ defect.

###  1.1 Global variants of $\mathfrak{su}(p)$ SYM

In a general 4d ${\cal N}=4$ theory, one can try to identify non-invertible
symmetries descending from the ${\mathsf{S}}$ and ${\mathsf{S}}{\mathsf{T}}$
transformations at $\tau=i$ and $e^{2\pi i\over 3}$. This exercise was carried
out in [35] for all gauge algebras, including the exceptional and non-simply-
laced cases, with one family of exceptions: the theories $\mathfrak{su}(N)$
for $N>4$. The reason that this set of theories is the most difficult is that
in this case the one-form symmetry, and likewise the number of global
variants, grows with $N$. Thus the straightforward technique of enumerating
global variants and identifying combinations of
$\\{{\mathsf{S}},{\mathsf{T}},\sigma,\tau\\}$ (with $\sigma$ introduced
previously and $\tau$ corresponding to stacking with appropriate SPT phase)
leaving a particular variant unchanged is no longer tractable. However, if we
restrict to $N=p$ prime, then it turns out that significant progress can be
made. This was hinted at in [35], and will be discussed from a completely
different perspective now.

From now on we restrict to $N=p$ prime. It will prove useful to first
understand how the number of global variants grows with $p$. To specify a
global variant we must specify the charge lattice, as well as the invertible
phases with which we stack. Let us begin with the former. In order to specify
the charge lattice, we begin by choosing a single non-trivial point
$(e,m)\in{\mathbb{Z}}_{p}\times{\mathbb{Z}}_{p}$ (all physically realized
charge lattices must contain the trivial point $(0,0)$), and then allow all
other points $(e^{\prime},m^{\prime})$ which satisfy the mutual locality
condition [4]

$\displaystyle
em^{\prime}-me^{\prime}=0\,\,\,\mathrm{mod}\,\,p\leavevmode\nobreak\ .$ (1.2)

Note that it suffices to specify only a single point. One might have naively
expected that, after specifying the initial point $(e,m)$, there could be
multiple choices $(e^{\prime}_{1},m^{\prime}_{1})$ and
$(e_{2}^{\prime},m_{2}^{\prime})$ for the second point, which would give
different charge lattices. But for $p$ prime this is not the case: if
$(e^{\prime}_{1},m^{\prime}_{1})$ and $(e_{2}^{\prime},m_{2}^{\prime})$
satisfy the mutual locality condition with $(e,m)$, then they are also
mutually local to one another, and hence the charge lattice is independent of
the choice of the second point.222Concretely, we begin by assuming that $e\neq
0$ mod $p$ and consider the mutual locality conditions  $\displaystyle
em^{\prime}_{1}-me^{\prime}_{1}=0\,\,\,\mathrm{mod}\,\,p\leavevmode\nobreak\
,\hskip
43.36243ptem^{\prime}_{2}-me^{\prime}_{2}=0\,\,\,\mathrm{mod}\,\,p\leavevmode\nobreak\
.$ (1.3) Multiplying the first equation by $e^{\prime}_{2}$ and the second
equation by $e^{\prime}_{1}$, and then subtracting the two gives
$\displaystyle
e(e^{\prime}_{2}m^{\prime}_{1}-e^{\prime}_{1}m^{\prime}_{2})=0\,\,\,\mathrm{mod}\,\,p\leavevmode\nobreak\
.$ (1.4) Thus $(e^{\prime}_{1},m^{\prime}_{1})$ and
$(e_{2}^{\prime},m_{2}^{\prime})$ are mutually local as well. If on the other
hand $e=0\,\,\,\mathrm{mod}\,\,p$, then $m\neq 0\,\,\,\mathrm{mod}\,\,p$
(since we must specify a non-trivial point) and we may repeat the argument by
first multiplying by $m^{\prime}_{2}$ and $m^{\prime}_{1}$ and then
subtracting.

Furthermore, to specify the lattice we may restrict to pairs $(e,m)$ such that
$e$ and $m$ are coprime, i.e. $\mathrm{gcd}(e,m)=1$. Indeed, say that we
instead try to specify the lattice by a point $(\widetilde{e},\widetilde{m})$
such that $\mathrm{gcd}(\widetilde{e},\widetilde{m})=d$. Then there exist
coprime integers $(e,m)$ such that $(\widetilde{e},\widetilde{m})=d(e,m)$, and
we see that $(\widetilde{e},\widetilde{m})$ and $(e,m)$ are mutually local,

$\displaystyle\widetilde{e}m-m\widetilde{e}=d(em-me)=0\leavevmode\nobreak\ .$
(1.5)

Thus the lattice specified by $(\widetilde{e},\widetilde{m})$ contains
$(e,m)$, and is in fact identical to the one specified by $(e,m)$.

Of course, since we are working modulo $p$ we must be careful about what we
mean by coprime. For example, though the pairs $(1,2)$ and $(2,1)$ both
involve coprime integers, as elements of ${\mathbb{Z}}_{3}$ the two can be
related by $(1,2)=2\cdot(2,1)$. To avoid overcounting, we must therefore mod
out by the multiplicative group ${\mathbb{Z}}_{p}^{\times}$, which is of order
$p-1$ (it does not contain the zero element).

We may now count the number of allowed charge lattices, which is the number of
coprime pairs $(e,m)$ modulo ${\mathbb{Z}}_{p}^{\times}$. In general, the
number of $k$-plets $(n_{1},\dots,n_{k})$ such that $n_{i}$ are all mutually
coprime and coprime to $N$ is counted by the Jordan totient function

$\displaystyle J_{k}(N)=N^{k}\prod_{p|N}\left(1-{1\over
p^{k}}\right)\leavevmode\nobreak\ .$ (1.6)

In the current case we have $k=2$ and $N=p$ prime, so the number of coprime
pairs is simply $J_{2}(p)=p^{2}-1$. Dividing by ${\mathbb{Z}}_{p}^{\times}$
then gives the number of allowed charge lattices, namely ${J_{2}(p)\over
p-1}=p+1$.

For each charge lattice, we may further stack with $k$ copies of the
invertible phase333For $p=2$ this should be replaced by ${\pi k\over
2}\int{\cal P}(B)$ with ${\cal P}(B)$ the Pontryagin square of $B$.

$\displaystyle{2\pi k\over p}\int{p+1\over 2}B\cup B\leavevmode\nobreak\
,\hskip 36.135ptk=0,\dots,p-1$ (1.7)

where $B$ is the background gauge field for the ${\mathbb{Z}}_{p}^{(1)}$ one-
form symmetry. The theories with different $k$ count as different global
variants. We thus arrive at the final count for the number of global variants
$d(p)$ for $\mathfrak{su}(p)$ SYM, namely

$\displaystyle d(p)=p\times{J_{2}(p)\over p-1}=p(p+1)\leavevmode\nobreak\ .$
(1.8)

###  1.2 Modular orbits of global variants

We now notice a small “coincidence,” namely that444This formula can also be
found in [5]. We will give a geometric interpretation of it later on.

$\displaystyle
d(p)=p(p+1)={|SL(2,{\mathbb{Z}}_{p})|\over|{\mathbb{Z}}_{p}^{\times}|}\leavevmode\nobreak\
.$ (1.9)

What this means is that each global variant of $\mathfrak{su}(p)$ SYM can be
labelled by a ray-matrix $M\in SL(2,{\mathbb{Z}}_{p})$. Given one such label,
we may act on it with a modular transformation $\mathsf{S},\mathsf{T}\in
SL(2,{\mathbb{Z}})$ to obtain a unique new ray-matrix $M^{\prime}$. In fact,
transitivity of the group $SL(2,{\mathbb{Z}}_{p})$ assures us that every
global variant can be obtained in this way—in other words, there are no
disconnected modular orbits for $\mathfrak{su}(p)$ SYM.555See [108] for
similar, more refined statements. With this in mind, we choose our labels as
follows. We first choose one global variant to assign the identity matrix; for
concreteness we take this to be $PSU(p)_{0,0}$ (which for $p=2$ is
$SO(3)_{+,0}$).666The first subscript denotes the discrete theta angle, while
the second subscript denotes the number of copies of the invertible phase
(1.7) that are stacked with the theory. We then label each other global
variant by the particular element of the modular group that is needed to map
from $PSU(p)_{0,0}$ to that global variant,

$\displaystyle F({\mathsf{S}},{\mathsf{T}}):\hskip 14.45377ptM\mapsto
F({\mathsf{S}},{\mathsf{T}})^{T}M\leavevmode\nobreak\ ,\hskip
43.36243ptF({\mathsf{S}},{\mathsf{T}})\in
SL(2,{\mathbb{Z}})\leavevmode\nobreak\ .$ (1.10)

in the conventions

$\displaystyle\mathsf{S}=\left(\begin{matrix}0&-1\\\
1&0\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
36.135pt\mathsf{T}=\left(\begin{matrix}1&1\\\
0&1\end{matrix}\right)\leavevmode\nobreak$ (1.11)

For example, for $p=2$ we obtain the labelling shown in Figure 2.

$SU(2)_{0}$$\left(\begin{matrix}0&1\\\
1&0\end{matrix}\right)$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\tau}$$\color[rgb]{0,0.55,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.55,0}{{\mathsf{T}}}$$SU(2)_{1}$$\left(\begin{matrix}0&1\\\
1&1\end{matrix}\right)$$SO(3)_{+,0}$$\left(\begin{matrix}1&0\\\
0&1\end{matrix}\right)$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\tau}$$SO(3)_{+,1}$$\left(\begin{matrix}1&1\\\
0&1\end{matrix}\right)$$SO(3)_{-,0}$$\left(\begin{matrix}1&0\\\
1&1\end{matrix}\right)$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\tau}$$\color[rgb]{0,0.55,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.55,0}{{{\mathsf{S}}}}$$SO(3)_{-,1}$$\left(\begin{matrix}1&1\\\
1&0\end{matrix}\right)$$\color[rgb]{0,0.55,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.55,0}{\mathsf{S}}$$\color[rgb]{0,0.55,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.55,0}{\mathsf{T}}$$\color[rgb]{0,0.55,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.55,0}{\mathsf{S}}$$\color[rgb]{0,0.55,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.55,0}{\mathsf{T}}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sigma$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma}$
Figure 2: Labelling of global variants of $\mathfrak{su}(2)$ SYM by matrices
in $SL(2,{\mathbb{Z}}_{2})/{\mathbb{Z}}_{2}^{\times}=SL(2,{\mathbb{Z}}_{2})$.
They are defined such that modular transformations $\mathsf{S}$ and
$\mathsf{T}$ act from the left. The subscript $0,1$ denotes the number of
copies of the invertible phase ${\pi\over 2}\int{\cal P}(B)$ that we stack
with.

The topological operations $\sigma$ and $\tau$ generate
$SL(2,{\mathbb{Z}}_{p})$ and should likewise be realizable as a matrix action
on the global variants, but they should not act on the left, lest they fail to
commute with the modular transformations. Instead, the topological operations
act on the global variants from the right,

$\displaystyle G(\sigma,\tau):\hskip 14.45377ptM\mapsto
MG(\sigma,\tau)\leavevmode\nobreak\ ,\hskip 43.36243ptG(\sigma,\tau)\in
SL(2,{\mathbb{Z}}_{p})\leavevmode\nobreak\ ,$ (1.12)

with $\sigma$ and $\tau$ given by the same matrices as ${\mathsf{S}}$ and
${\mathsf{T}}$ respectively, as may be checked in the explicit example of
$\mathfrak{su}(2)$ in Figure 2.

There are two main takeaways from this discussion, both of which will
generalize to other class ${\cal S}$ theories:

  1. 1.

Every global variant can be mapped to every other by the action of both
${\mathsf{S}},{\mathsf{T}}\in SL(2,{\mathbb{Z}})$ and $\sigma,\tau\in
SL(2,{\mathbb{Z}}_{p})$.

  2. 2.

Every action of the modular group $F({\mathsf{S}},{\mathsf{T}})$ can be undone
by an appropriate topological manipulation $G(\sigma,\tau)$. This is true
independent of the global variant $M$ under consideration.

As we have mentioned before, the first of these implies that there are no
disconnected components in the orbit of ${\mathsf{S}},{\mathsf{T}}\in
SL(2,{\mathbb{Z}})$, nor in the orbit of $\sigma,\tau\in
SL(2,{\mathbb{Z}}_{p})$. On the other hand, the second of these means that,
for any $F({\mathsf{S}},{\mathsf{T}})$ leaving a certain value of the coupling
fixed (i.e. $F({\mathsf{S}},{\mathsf{T}})=\mathsf{S}$ or
${\mathsf{S}}{\mathsf{T}}$ in the current case), we will always get a symmetry
by dressing with the appropriate topological manipulation $G(\sigma,\tau)$.
This statement holds regardless of the global variant under consideration. The
topological manipulation will generically contain factors of $\sigma$ which
make the symmetry non-invertible, and so we see that non-invertible symmetries
are, in some sense, the norm.

From this point of view, instead of asking “For what global forms $M$ do we
have a non-invertible symmetry?” we should really ask “For what global forms
$M$ do we not have a non-invertible symmetry?” In other words, we ask if/when
the non-invertible symmetry becomes invertible. By point 1 above, we see that
if any single global variant has an invertible symmetry, then the potential
non-invertible symmetry in all other global variants is non-intrinsic.

###  1.3 Invariant global forms

We now focus on the identification of global forms with invertible symmetries.
A symmetry $F({\mathsf{S}},{\mathsf{T}})$ is invertible in the global form $M$
if it acts as777We could also allow for operations such as rescaling of the
background fields, but since we will set the background fields to zero in a
moment anyways we do not elaborate on these here.

$\displaystyle F({\mathsf{S}},{\mathsf{T}})^{T}M=\lambda
M\tau^{n}\leavevmode\nobreak\ ,\hskip
36.135pt\lambda\in{\mathbb{Z}}_{p}^{\times}\leavevmode\nobreak\
,\,\,\,\,n\in{\mathbb{Z}}_{p}\leavevmode\nobreak\ .$ (1.13)

The factor of $\lambda$ is allowed since $M$ is in the first place defined
only modulo ${\mathbb{Z}}_{p}^{\times}$. The factor of $\tau^{n}$ captures
potential anomalies of the invertible symmetry. If we are only concerned with
whether or not a symmetry is invertible, and not whether it is anomalous, then
it suffices to set all background gauge fields to zero, in which case the
factor of $\tau^{n}$ drops out, and we may redefine the label $M$ to be simply
the first column of the matrix $M$ we were using before (e.g.
$SU(2)\leftrightarrow\binom{0}{1},SO(3)_{+}\leftrightarrow\binom{1}{0},SO(3)_{-}\leftrightarrow\binom{1}{1}$).
We are then left with a simple eigenvalue equation

$\displaystyle F({\mathsf{S}},{\mathsf{T}})^{T}M=\lambda M\leavevmode\nobreak\
,\hskip 36.135pt\lambda\in{\mathbb{Z}}_{p}^{\times}\leavevmode\nobreak\ .$
(1.14)

The global form $M$ has an invertible symmetry if and only if this equation is
satisfied.

Let us begin by fixing to $F({\mathsf{S}},{\mathsf{T}})={\mathsf{S}}$. In this
case the possible eigenvalues $\lambda$ are given by the characteristic
polynomial

$\displaystyle\mathrm{det}({\mathsf{S}}-\lambda\mathds{1})=\lambda^{2}+1=0\,\,\,\mathrm{mod}\,\,p\leavevmode\nobreak\
.$ (1.15)

We must now ask if this equation admits a solution in
${\mathbb{Z}}_{p}^{\times}$. The answer to this question is a standard result
in the theory of quadratic residues—the number of solutions is given by

$\displaystyle\\#\,\,\mathrm{solutions}=\left\\{\begin{matrix}1&&p=2\\\
1+(-1)^{p-1\over 2}&&\mathrm{otherwise}\end{matrix}\right.$ (1.16)

Whenever a $\lambda$ in ${\mathbb{Z}}_{p}^{\times}$ exists, there
automatically exists a vector with coefficients in ${\mathbb{Z}}_{p}$
satisfying (1.14), and hence an invariant global form. For $p=2$ we see that
there is one solution, and hence one global form which is invariant under
${\mathsf{S}}$. This is none other than the $SO(3)_{-}$ theory, which is
indeed invariant under ${\mathsf{S}}$ [4].888Recall that we are turning off
the background fields for this analysis. In the presence of background fields
$SO(3)_{-}$ transforms under ${\mathsf{S}}$ by an invertible phase, indicating
a mixed ‘t Hooft anomaly between the invertible ${\mathsf{S}}$ symmetry and
the ${\mathbb{Z}}_{2}^{(1)}$ one-form symmetry. For $p\in 4{\mathbb{N}}+1$, we
see that there are two global forms left invariant, and otherwise there are no
global forms left invariant. Thus only for $p>2$ and $p\not\in
4{\mathbb{N}}+1$ is the ${\mathsf{S}}$ symmetry (dressed with the appropriate
topological manipulation $G(\sigma,\tau)$) intrinsically non-invertible.999A
similar statement can be found in Appendix C of [29].

We may carry out a similar analysis for
$F({\mathsf{S}},{\mathsf{T}})={\mathsf{S}}{\mathsf{T}}$. In this case the
possible eigenvalues $\lambda$ are given by the characteristic polynomial

$\displaystyle\mathrm{det}({\mathsf{S}}{\mathsf{T}}-\lambda\mathds{1})=\lambda^{2}-\lambda+1=0\,\,\,\mathrm{mod}\,\,p\leavevmode\nobreak\
.$ (1.17)

We again ask for which $p$ this admits a solution in
${\mathbb{Z}}_{p}^{\times}$. The answer is given by

$\displaystyle\\#\,\,\mathrm{solutions}=\left\\{\begin{matrix}0&&p=2\\\
1&&p=3\\\ 1+(-3|p)&&\mathrm{otherwise}\end{matrix}\right.$ (1.18)

where $(n|p)$ is the Legendre symbol. We conclude that there is no global form
left invariant for $p=2$, one global form left invariant for $p=3$ (identified
as the $PSU(3)_{1}$ theory in [35]), two global forms invariant for $p\in
3{\mathbb{N}}+1$, and none otherwise. Thus only for $p>3$ and $p\not\in
3{\mathbb{N}}+1$ is the ${\mathsf{S}}{\mathsf{T}}$ symmetry (dressed with the
appropriate topological manipulation $G(\sigma,\tau)$) intrinsically non-
invertible.

We have now understood the full spectrum of invertible modular symmetries in
the global variants of ${\cal N}=4$ $\mathfrak{su}(p)$ SYM for any prime $p$.
This in particular tells us whether non-invertible symmetries descending from
${\mathsf{S}}$ and ${\mathsf{S}}{\mathsf{T}}$ can be intrinsic. We summarize
the results for the first few primes in Table 1.

$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29
---|---|---|---|---|---|---|---|---|---|---
${\mathsf{S}}$ intrinsic? | ✗ | ✓ | ✗ | ✓ | ✓ | ✗ | ✗ | ✓ | ✓ | ✗
${\mathsf{S}}{\mathsf{T}}$ intrinsic? | ✓ | ✗ | ✓ | ✗ | ✓ | ✗ | ✓ | ✗ | ✓ | ✓
Table 1: For any $p$, there are global variants such that ${\mathsf{S}}$ and
${\mathsf{S}}{\mathsf{T}}$ give rise to non-invertible symmetries (upon
appropriate dressing with topological manipulations $G(\sigma,\tau)$). We may
then ask if these non-invertible symmetries are intrinsic or not. This may be
answered by asking if there exist global forms left invariant by
${\mathsf{S}}$ or ${\mathsf{S}}{\mathsf{T}}$. The results are shown for the
first few primes.

###  1.4 Higher-dimensional point of view

The 4d ${\cal N}=4$ theories that we have been discussing thus far are the
simplest examples of theories of class ${\cal S}$. In particular, they can be
obtained by compactifying the type $\mathfrak{a}_{p-1}$ 6d (2,0) theory on a
torus. An important point, to be elaborated on further in the main text, is
that the 6d (2,0) theory is not actually a well-defined theory, but rather a
“relative theory,” i.e. a theory living on the boundary of a non-trivial TFT
in 7d [109, 110, 111].

It may come as a surprise that the 4d ${\cal N}=4$ theories, which are well-
defined, can be obtained via torus compactification of the 6d (2,0) theory,
which is relative. In particular, when the coupled 6d-7d system is
compactified on a torus, one obtains a coupled 4d-5d system, and one might
worry that this indicates that the 4d theory is relative as well. This however
is not the case. To see this, note that the theory in the 7d bulk $W_{7}$ is
given schematically by

$\displaystyle S_{7d}={p\over 4\pi}\int_{W_{7}}c\wedge dc\leavevmode\nobreak\
,$ (1.19)

which upon compactification on $T^{2}$ with $H_{1}(T^{2},{\mathbb{Z}}_{p})$
generated by the usual ${\mathsf{A}},{\mathsf{B}}$ cycles gives

$\displaystyle S_{5d}={p\over 2\pi}\int_{W_{5}}b\wedge
d\widehat{b}\leavevmode\nobreak$ (1.20)

for $b:=\oint_{\mathsf{A}}c$ and $\widehat{b}:=\oint_{\mathsf{B}}c$. In
Section 5, this BF theory will be seen to be closely related to—though not in
general identical to—the Symmetry TFT (SymTFT) of the 4d ${\cal N}=4$ theory.

Crucially, unlike the 7d CS theory, the 5d BF theory admits topological
boundary conditions. It is this fact which allows the 4d ${\cal N}=4$ theory
to be well-defined—indeed, one may place the 5d BF theory on an interval with
a topological boundary condition on one end and the dimensionally-reduced 6d
(2,0) theory on the other end, and then shrink the slab to obtain a well-
defined 4d ${\cal N}=4$ theory. In this picture, the choice of global
structure of the 4d ${\cal N}=4$ theory (e.g. $SU(2)$ versus $SO(3)_{+}$
versus $SO(3)_{-}$) is captured by the choice of topological boundary.

The way that these topological boundary conditions are realized in 7d is by
taking the 7d manifold to be of the form $W_{7}=V_{3}\times X_{4}$, where
$\partial V_{3}=T^{2}$ and the product is not necessarily trivial. A
particularly simple class of three-manifolds $V_{3}$ with torus boundary are
solid tori, i.e. genus-1 handlebodies. As will be discussed in Section 5, the
number of distinct genus-1 handlebodies (at the level of homology over
${\mathbb{Z}}_{p}$) is given by $p+1$, matching precisely with the number of
distinct charge lattices of the 4d ${\cal N}=4$ theory. We may further
consider inserting charge $q\in{\mathbb{Z}}_{p}$ Wilson surfaces along the
non-contractible cycle of the solid torus, which capture the background gauge
fields for the one-form symmetry as well as the possible stackings with SPT
phases. We thus obtain a picture in which the choice of handlebody and
longitudinal Wilson surfaces determines completely the 5d topological boundary
conditions, and hence the global variant of the 4d ${\cal N}=4$ theory.

In this higher-dimensional language, the question of intrinsic versus non-
intrinsic non-invertible symmetry becomes a question of whether the isometries
of $T^{2}$ extend to isometries of the solid torus $V_{3}$. When the
isometries do extend, then all non-invertible symmetries are non-intrinsic,
and the BF theory given in (1.20) is the full SymTFT for the boundary theory
(at least, the full SymTFT capturing the one-form symmetry). On the other
hand, when the isometries do not extend, the non-invertible symmetries are
intrinsic, and the SymTFT is not (1.20), but rather a gauging of this theory
by an outer autormorphism symmetry, as discussed in [59]. Since this
automorphism symmetry shuffles the fields $b$ and $\widehat{b}$, the
7-dimensional realization of it should shuffle the ${\mathsf{A}}$\- and
${\mathsf{B}}$-cycles of the torus. In other words, this additional discrete
gauging in 5d corresponds to a sum over geometries in 7d. Obtaining the full
SymTFT for theories with intrinsically non-invertible symmetries thus involves
a (topological) quantum gravity computation in higher dimensions.

Let us close by saying that the theory obtained by gauging the outer
automorphism symmetry of the BF theory is not in general a gauge theory, even
for a higher-group symmetry [112]. In general, it is a difficult question when
exactly the result is a gauge theory, or more precisely a Dijkgraaf-Witten
(DW) theory. By recalling that a symmetry is non-intrinsically non-invertible
if and only if the SymTFT is a DW theory [59], our previous results on
intrinsic non-invertiblity (summarized for example in Table 1) immediately
allow us to answer this question:

######  Theorem 1.

Gauging the ${\mathbb{Z}}_{2}^{\mathrm{EM}}$ electro-magnetic duality symmetry
acting as $F={\mathsf{S}}$ in $(4+1)$d ${\mathbb{Z}}_{p}^{(2)}$ gauge theory
gives a spin Dijkgraaf-Witten theory if and only if $p\in 4{\mathbb{N}}+1$.

and likewise

######  Theorem 2.

Gauging the ${\mathbb{Z}}_{3}$ traility symmetry acting as $F=\mathsf{ST}$ in
$(4+1)$d ${\mathbb{Z}}_{p}^{(2)}$ gauge theory gives a spin Dijkgraaf-Witten
theory if and only if $p=3$ or $p\in 3{\mathbb{N}}+1$.

The reader who is mainly interested in this higher-dimensional perspective may
skip to Section 5, which is largely self-contained.

###  1.5 Goals and organization

The main goal of this paper is to extend the analysis above to a more general
family of class ${\cal S}$ theories.101010Previous works studying global
variants and invertible symmetries of class ${\cal S}$ theories include [113,
114, 115, 94, 116]. As will be reviewed in Section 2, class ${\cal S}$
theories are obtained by reducing the 6d (2,0) theory of type
$\mathfrak{g}\in\\{\mathfrak{a},\mathfrak{d},\mathfrak{e}\\}$ on a Riemann
surface $\Sigma_{g,n}$ of genus $g$ with $n$ punctures. In this work we will
mainly focus on the subset of theories obtained by taking the 6d (2,0) theory
to be of type $\mathfrak{a}_{p-1}$ with $p$ prime, and with no punctures
$n=0$. We will denote the resulting theories by ${\cal
T}^{p,g,0}_{L}[\Omega]$, where $\Omega$ is the period matrix for the Riemann
surface and $L$ is a partial label of the global variant, to be discussed more
below. In the case of $g=1$, the theories ${\cal
T}^{p,1,0}_{L}[\tau_{\mathrm{YM}}]$ are precisely the ${\cal N}=4$
$\mathfrak{su}(p)$ theories discussed above, with coupling
$\tau_{\mathrm{YM}}$ and global variant labelled by $L$.

The first step in identifying (non-)invertible symmetries will be to identify
the points $\Omega$ such that the Riemann surface has an enhanced symmetry.
This is analogous to identifying the points $\tau_{\mathrm{YM}}=i$ or $e^{2\pi
i\over 3}$ at genus 1. Once we have obtained these points, together with the
generators $F\in Sp(2g,{\mathbb{Z}})$ of the enhanced symmetries, the second
step is to identify a topological manipulation $G$ which can undo the action
of this symmetry on the global form (but not on the period matrix).

At genus 1, we saw that for any $F$ and global variant $M$, there always
existed a topological manipulation $G$ which could undo the action of $F$ on
$M$, thereby turning $F$ into a (potentially non-invertible) symmetry of the
global variant $M$. In that sense, the second step just mentioned was actually
unnecessary. At higher genus, we will see that effectively the same is true.
There are several steps to proving this:

  1. 1.

Show that the full set of topological manipulations $G$ form
$Sp(2g,{\mathbb{Z}}_{p})$,

  2. 2.

Show that the global variants of genus $g$ class ${\cal S}$ are labelled by
elements in $Sp(2g,{\mathbb{Z}}_{p})$ (modulo some appropriate factor),

  3. 3.

Show that the global variants are acted on from the left by $F\in
Sp(2g,{\mathbb{Z}})$, and from the right by $G\in Sp(2g,{\mathbb{Z}}_{p})$.

These statements will be the subject of Section 2.

Having done this, we may then simply focus on the first task, namely
identification of the points of enhanced symmetry on a genus $g$ Riemann
surface. As one might imagine, this exercise has already been carried out in
the math literature up to genus 5 [117, 118, 119, 120, 121, 122], and we will
borrow results from there. The math results also include the characteristic
polynomials of the generators of the enhanced symmetries, which allow us to
evaluate the number of global forms in a way similar to that described above.

In Section 3 we carry out the analysis at genus 2 in significant detail, in
order to illustrate the many new features that arise at $g>1$. In Section 4 we
do a similar, somewhat abridged, analysis for genus $g=3,4,$ and $5$. Finally
in Section 5, we give a higher-dimensional perspective on all of the above
results, including a geometric origin of the topological boundary conditions
and condensation defects of the 5d SymTFT.

For the reader’s convenience, we also include four appendices. In Appendix A
we give explicit matrix representations for the generators of all enhanced
symmetries for genus 2 Riemann surfaces. In Appendix B we review various
arithmetic facts that are necessary for determining when a global form is
invariant under an enhanced symmetry. In Appendix C, we review some basic
facts about handlebodies necessary for understanding the passage from the
6d-7d system to the 4d-5d system. Finally in Appendix D we give an alternative
form for the condensation defects ${\cal C}_{F}(X_{4})$, first introduced in
Section 5.4, implementing automorphism symmetries of the 5d TFT.

##  2 Non-invertible symmetries for class ${\cal S}$ theories

The main goal of this work is to identify the spectrum of invertible versus
non-invertible symmetries in certain theories of class ${\cal S}$. To
introduce our notations we begin with a brief review of the class ${\cal S}$
construction.

###  2.1 A review of class ${\cal S}$

The class $\mathcal{S}$ construction begins with a 6d (2,0) theory of type
$\mathfrak{g}\in\\{\mathfrak{a},\mathfrak{d},\mathfrak{e}\\}$ on a closed,
compact six-manifold $X_{6}$, which is reduced on a genus $g$ Riemann surface
with $n$ punctures $\Sigma_{g,n}$ to obtain a four-dimensional theory on
$X_{4}\subset X_{6}$. Despite naive expectations, the four-dimensional
theories obtained in this way depend on more data than just the Riemann
surface $\Sigma_{g,n}$ and the algebra $\mathfrak{g}$. The origin of this
additional data lies in the fact that the 6d (2,0) theory is not itself a
well-defined theory (at least not for the cases of
$\mathfrak{g}=\mathfrak{a}_{p-1}$, which will be the focus of this work), but
rather a relative theory [109, 110]. This means that the 6d (2,0) theory
should be thought of as living on the boundary of a non-trivial bulk TQFT on a
seven-manifold $W_{7}$ with $\partial W_{7}=X_{6}$. This seven-dimensional
TQFT was identified as a certain Wu-Chern-Simons theory in [111]. The basic
setup is illustrated in Figure 3.

In this picture, the 6d (2,0) theory is identified with a state
$|\mathfrak{a}_{N-1}\rangle$ in the Hilbert space of the bulk TQFT. In order
to specify this state, one must first fix a basis for the Hilbert space, which
is specified by a maximal isotropic sublattice ${\cal L}\in
H_{3}(X_{6},{\mathbb{Z}}_{N})$ [113]; details will be given in Section 5.1.
For the present purposes it suffices to say that a maximal isotropic
sublattice is a maximal set of $M_{3}\in H_{3}(X_{6},{\mathbb{Z}}_{N})$ such
that

$\displaystyle\langle M_{3},M_{3}^{\prime}\rangle=0\leavevmode\nobreak\
,\hskip 36.135pt\,\,M_{3},M_{3}^{\prime}\in{\cal L}\leavevmode\nobreak\ ,$
(2.1)

where $\langle\cdot,\cdot\rangle$ is the standard intersection pairing of
3-cycles in 6d.

$|\mathfrak{a}_{N-1}\rangle$7d TQFT Figure 3: The 6d (2,0) theory of type
$\mathfrak{a}_{N-1}$ should be thought of as a state
$|\mathfrak{a}_{N-1}\rangle$ in the Hilbert space of a non-trivial 7d TQFT.
This TQFT does not admit topological boundary conditions.

In the context of class ${\cal S}$, we are interested in six-manifolds of the
form $X_{6}=\Sigma_{g,n}\times X_{4}$. In this case the group
$H_{3}(X_{6},{\mathbb{Z}}_{N})$ splits via the Künneth formula,

$\displaystyle H_{3}(X_{6},{\mathbb{Z}}_{N})\cong
H_{1}(\Sigma_{g,n},{\mathbb{Z}}_{N})\otimes
H_{2}(X_{4},{\mathbb{Z}}_{N})\leavevmode\nobreak\ ,$ (2.2)

subject to the additional assumption (which we will make throughout) that
$H_{1,3}(X_{4},{\mathbb{Z}})$ are trivial. There is an analogous splitting of
the maximal isotropic sublattice ${\cal L}$ as

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle L\otimes
H_{2}(X_{4},{\mathbb{Z}}_{N})\leavevmode\nobreak\ ,\hskip 36.135ptL\subset
H_{1}(\Sigma_{g,n},{\mathbb{Z}}_{N})\leavevmode\nobreak\ .$ (2.3)

We will likewise split $M_{3}\in H_{3}(X_{6},{\mathbb{Z}}_{N})$ into elements
$\gamma\in H_{1}(\Sigma_{g,n},{\mathbb{Z}}_{N})$ and $M_{2}\in
H_{2}(X_{4},{\mathbb{Z}}_{N})$ via $M_{3}=\gamma\otimes M_{2}$, on which the
6d intersection pairing decomposes as a product

$\displaystyle\langle
M_{3},M_{3}^{\prime}\rangle=\langle\gamma,\gamma^{\prime}\rangle\times(M_{2},M_{2}^{\prime})\leavevmode\nobreak\
.$ (2.4)

Here the pairing $\langle\gamma,\gamma^{\prime}\rangle$ is the usual
(antisymmetric) intersection pairing between 1-cocycles on $\Sigma_{g,n}$,
whereas $(M_{2},M_{2}^{\prime})$ is a symmetric pairing between 2-cocycles in
four dimensions.

As first discussed in [113], in order to specify the four-dimensional theory
we must specify not only the Riemann surface $\Sigma_{g,n}$ on which we
compactify, but also a lattice $L\in H_{1}(\Sigma_{g,n},{\mathbb{Z}}_{N})$.
This corresponds to the specification of the particular charge lattice of the
four-dimensional theory. Without specifying $L$, the four-dimensional theory
remains relative.

Furthermore, to fully specify the 4d theory we should also specify a
particular representative of the non-trivial classes of $L^{\perp}\otimes
H_{2}(X_{4},{\mathbb{Z}}_{N})$, with
$L^{\perp}:=H_{1}(\Sigma_{g,n},{\mathbb{Z}}_{N})/L$. The choice of
representatives in $L^{\perp}$ determines the SPT phases with which the 4d
theory is stacked, while the choice of elements in
$H_{2}(X_{4},{\mathbb{Z}}_{N})$ determines the background fields for the
corresponding one-form symmetries. In total, we will denote the representative
by ${\cal B}$. The geometric origin of this extra data will be explained in
Section 5.

The well-defined four-dimensional theory obtained by specifying the above
additional data will be denoted by ${\cal T}^{N,g,n}_{L}[\Omega,{\cal B}]$
where $\Omega$ is the period matrix of $\Sigma_{g,n}$, capturing the coupling
constants of the theory. The argument ${\cal B}$ collectively represents the
data about background fields for the one-form symmetries of the theory, and
will be suppressed unless important to the discussion. When we want to discuss
the class ${\cal S}$ theory without specifying the global variant (the analog
of ${\cal N}=4$ $\mathfrak{su}(N)$ SYM) then we will drop the $L$ subscript
and write ${\cal T}^{N,g,n}[\Omega]$.

Let us briefly review the example of the $\mathfrak{a}_{1}$ 6d (2,0) theory on
$\Sigma_{2,0}=T^{2}$. It is well-known that this gives 4d ${\cal N}=4$
$\mathfrak{a}_{1}$ super-Yang Mills. Taking the homology of the torus to be
spanned by the usual ${\mathsf{A}}$ and ${\mathsf{B}}$ cycles,

$\displaystyle
H_{1}(T^{2},{\mathbb{Z}}_{2})=\left\\{1,\,{\mathsf{A}},\,{\mathsf{B}},\,{\mathsf{A}}+{\mathsf{B}}\right\\}\leavevmode\nobreak\
,\hskip 36.135pt\langle{\mathsf{A}},{\mathsf{B}}\rangle=1$ (2.5)

we identify three maximal isotropic sublattices,

$\displaystyle L_{{\mathsf{A}}}=\\{1,\,{\mathsf{A}}\\}\leavevmode\nobreak\
,\hskip 36.135ptL_{{\mathsf{B}}}=\\{1,\,{\mathsf{B}}\\}\leavevmode\nobreak\
,\hskip
36.135ptL_{{\mathsf{A}}+{\mathsf{B}}}=\\{1,\,{\mathsf{A}}+{\mathsf{B}}\\}\leavevmode\nobreak\
.$ (2.6)

From this we expect three distinct charge lattices for the 4d theory, which we
denoted in the introduction by

$\displaystyle L_{{\mathsf{A}}}\leftrightarrow SU(2)\leavevmode\nobreak\
,\hskip 36.135ptL_{{\mathsf{B}}}\leftrightarrow SO(3)_{+}\leavevmode\nobreak\
,\hskip 36.135ptL_{{\mathsf{A}}+{\mathsf{B}}}\leftrightarrow
SO(3)_{-}\leavevmode\nobreak\ .$ (2.7)

The data about invertible phases is then given by specifying a representative
for the non-trivial class of $L^{\perp}$. For example, we have
$L^{\perp}_{\mathsf{A}}=\\{[1],[{\mathsf{B}}]\\}$, where the class
$[1]:=\\{1,{\mathsf{A}}\\}$ and
$[{\mathsf{B}}]:=\\{{\mathsf{B}},{\mathsf{B}}+{\mathsf{A}}\\}$, and the SPT
phase is specified by choosing the particular representative ${\mathsf{B}}$
(giving $SU(2)_{0}$) or ${\mathsf{B}}+{\mathsf{A}}$ (giving $SU(2)_{1}$).

One of the main utilities of the 6d point of view is that it is now easy to
see how the global variants map into one another under Montonen-Olive duality
transformations of the 4d ${\cal N}=4$ theory. Indeed, the Montonen-Olive
duality group is realized in the class ${\cal S}$ construction as the modular
group $SL(2,{\mathbb{Z}})$ of the Riemann surface $T^{2}$, which can be
generated by operations ${\mathsf{T}}$ and ${\mathsf{S}}$ acting on the
elements of $H_{1}(T^{2},{\mathbb{Z}}_{p})$ via,

$\displaystyle{\mathsf{T}}:$
$\displaystyle\mathsf{A}\rightarrow\mathsf{A}\leavevmode\nobreak\ ,\hskip
57.81621pt{\mathsf{S}}:\,\,\,\mathsf{A}\rightarrow\mathsf{B}\leavevmode\nobreak\
,$ (2.8) $\displaystyle\mathsf{B}\rightarrow\mathsf{A}+\mathsf{B}\hskip
63.59795pt\,\,\,\mathsf{B}\rightarrow-\mathsf{A}\leavevmode\nobreak\ .$

Alternatively, when realized on the vector $({\mathsf{B}},{\mathsf{A}})^{T}$,
these take the form of matrices

$\displaystyle{\mathsf{T}}=\left(\begin{matrix}1&1\\\
0&1\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
36.135pt{\mathsf{S}}=\left(\begin{matrix}0&-1\\\
1&0\end{matrix}\right)\leavevmode\nobreak\ .$ (2.9)

We thus see that the ${\mathsf{S}}$ operation exchanges $L_{{\mathsf{A}}}$ and
$L_{{\mathsf{B}}}$, and hence the $SU(2)$ and $SO(3)_{+}$ charge lattices,
whereas it leaves $L_{{\mathsf{A}}+{\mathsf{B}}}$, and hence the $SO(3)_{-}$
charge lattice, unchanged. On the other hand, ${\mathsf{T}}$ leaves the
$SU(2)$ charge lattice unchanged, whereas it exchanges the $SO(3)_{+}$ and
$SO(3)_{-}$ charge lattices. Tracking the action of ${\mathsf{S}}$ and
${\mathsf{T}}$ on the representatives of $L^{\perp}$ allows us to reproduce
the results in Figure 2.

###  2.2 Non-invertible symmetries and the modular group

We may now generalize the construction of non-invertible symmetries in 4d
${\cal N}=4$ theories, reviewed in the introduction and Figure 1, to arbitrary
class ${\cal S}$ theories ${\cal T}^{p,g,n}_{L}[\Omega]$. Our starting point
will be the application of an element $F$ of the relevant modular group
$\mathrm{Mod}(\Sigma_{g,n},p)$. For the case of no punctures, i.e. $n=0$, this
is $\mathrm{Mod}(\Sigma_{g,0},p)=Sp(2g,{\mathbb{Z}}_{p})$. An element of the
modular group will in general act on $L$ and $\Omega$, but will never act on
$p,g,$ or $n$. Having done a modular transformation, the next step is to do an
appropriate topological operation $G:=G(\sigma,\tau)$, to be explained below,
which acts on $L$ but not on $\Omega$. The situation is then as in Figure 4.

In order to have a non-invertible symmetry, we require the existence of a
solution to the following equations

$\displaystyle GF(L)=L\leavevmode\nobreak\ ,\hskip
57.81621ptF(\Omega)=\Omega\leavevmode\nobreak\ .$ (2.10)

The second equation is conceptually straightforward, and amounts to
identifying the fixed points of the modular group. The former instructs us to
look for a topological manipulation $G$ which can undo the action of $F$ on
the global variant $L$. We distinguish between three conceptually distinct
cases:

  * •

$F(L)=L$ and $G$ trivial: In this case the symmetry is invertible and non-
anomalous.111111Here we are referring to mixed anomalies between $F$ and the
one-form symmetry; there could still be self-anomalies of $F$, along the lines
of [123, 124, 125].

  * •

$F(L)=L$ and $G(L)=L$ with $G$ non-trivial: In this case $F$ again corresponds
to an invertible symmetry, but the partition function transforms by a
counterterm which is removed by $G$. In other words, $F$ is invertible and
anomalous.

  * •

$F(L)\neq L$: In this case the symmetry is non-invertible. There are two
subcases:

    * –

When $F(L)\neq L$, but there exists at least one $L^{\prime}\in
H_{1}(\Sigma_{g,n},{\mathbb{Z}}_{N})$ such that $F(L^{\prime})=L^{\prime}$,
then the symmetry is non-intrinsically non-invertible.

    * –

When $F(L)\neq L$, and there is no $L^{\prime}\in
H_{1}(\Sigma_{g,n},{\mathbb{Z}}_{N})$ such that $F(L^{\prime})=L^{\prime}$,
then the symmetry is intrinsically non-invertible.

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T}^{N,g,n}_{F(L)}[F(\Omega)]$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ Figure 4: When
$GF(L)=L$ and $F(\Omega)=\Omega$ admit solutions, then the composite ${\cal
N}_{F}:=FG$ gives a non-invertible symmetry of the class ${\cal S}$ theory.

The constraints in (2.10) were already discussed in detail in the case of
$(g,n)=(1,0)$ in the introduction. In that case, we observed that for $N=p$
prime, given an $F$ and $\Omega$ satisfying $F(\Omega)=\Omega$, there always
existed a $G$ which could undo the action of $F$ on any global variant $L$.
Hence any such $F$ automatically gave a symmetry, regardless of the global
form, with the (non-)invertibility stemming from whether $G$ contained
discrete gaugings $\sigma$. We will now see that this is again the case for
class ${\cal S}$ theories of type ${\cal T}^{p,g,0}_{L}[\Omega]$. The first
step in showing this is to understand the spectrum of global variants of
theories of class ${\cal S}$.

###  2.3 Global variants of theories of class ${\cal S}$

We restrict to the case of $N=p$ prime and no punctures $n=0$, and denote the
generators of $H_{1}(\Sigma_{g,0},{\mathbb{Z}}_{p})$ as ${\mathsf{A}}_{I}$ and
${\mathsf{B}}_{I}$ for $I=1,\dots,g$, with intersection pairing

$\displaystyle\langle{\mathsf{A}}_{I},{\mathsf{A}}_{J}\rangle=\langle{\mathsf{B}}_{I},{\mathsf{B}}_{J}\rangle=0\leavevmode\nobreak\
,\hskip
43.36243pt\langle{\mathsf{A}}_{I},{\mathsf{B}}_{J}\rangle=\delta_{IJ}\leavevmode\nobreak\
.$ (2.11)

This intersection pairing is preserved by matrices $F$ acting as

$\displaystyle\binom{\vec{{\mathsf{B}}}}{\vec{{\mathsf{A}}}}\rightarrow
F\binom{\vec{{\mathsf{B}}}}{\vec{{\mathsf{A}}}}$ (2.12)

as long as the matrices satisfy

$\displaystyle F^{T}\mathfrak{I}F=\mathfrak{I}\leavevmode\nobreak\ ,\hskip
36.135pt\mathfrak{I}=\left(\begin{matrix}0&-\mathds{1}_{g\times g}\\\
\mathds{1}_{g\times g}&0\end{matrix}\right)\leavevmode\nobreak\ .$ (2.13)

The matrices $F$ should have integer entries so that they map properly
normalized cycles amongst themselves, and hence the $F$ are matrices in
$Sp(2g,{\mathbb{Z}})$.

It will be useful for us to assign to each element ${\mathsf{A}}_{I}$ and
${\mathsf{B}}_{I}$ a $2g$-vector,

$\displaystyle v_{{\mathsf{B}}_{1}}=\left(\begin{matrix}1\\\
0_{2g-1}\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
21.68121ptv_{{\mathsf{B}}_{2}}=\left(\begin{matrix}0\\\ 1\\\
0_{2g-2}\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
21.68121pt\dots\leavevmode\nobreak\ ,\hskip
21.68121ptv_{{\mathsf{B}}_{g}}=\left(\begin{matrix}0_{g-1}\\\ 1\\\
0_{g}\end{matrix}\right)\leavevmode\nobreak\ ,$ $\displaystyle
v_{{\mathsf{A}}_{1}}=\left(\begin{matrix}0_{g}\\\ 1\\\
0_{g-1}\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
21.68121ptv_{{\mathsf{A}}_{2}}=\left(\begin{matrix}0_{g+1}\\\ 1\\\
0_{g-2}\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
21.68121pt\dots\leavevmode\nobreak\ ,\hskip
21.68121ptv_{{\mathsf{A}}_{g}}=\left(\begin{matrix}0_{2g-1}\\\
1\end{matrix}\right)\leavevmode\nobreak\ ,$

where $0_{n}$ represents an array of $n$ copies of $0$. Likewise
$v_{m{\mathsf{A}}_{I}+n{\mathsf{B}}_{J}}=m\,v_{{\mathsf{A}}_{I}}+n\,v_{{\mathsf{B}}_{J}}$.
In terms of these vectors the intersection pairing may then be represented as

$\displaystyle\langle{\mathsf{A}}_{I},{\mathsf{B}}_{J}\rangle=v_{{\mathsf{A}}_{I}}^{T}\,\mathfrak{I}\,v_{{\mathsf{B}}_{J}}\leavevmode\nobreak\
.$ (2.14)

A maximal isotropic sublattice $L$ is spanned by precisely half of the above
generators, i.e. $g$ distinct elements $\gamma_{i}$ for $i=1,\dots,g$. We may
then unambiguously label such an $L$ by a $2g\times g$ matrix whose $g$
columns are the $2g$-vectors corresponding to each of the generators of the
lattice. We denote this $2g\times g$ matrix by
$K_{L}=(v_{\gamma_{1}},\dots,v_{\gamma_{g}})$. Similarly, the background
gauge-field is specified by a representative ${\cal B}$ of $L^{\perp}\otimes
H^{2}(X_{4},{\mathbb{Z}}_{p})$, which can be labelled by a $2g\times g$ matrix
$K_{\cal B}$, where now the coefficients of the vector take values in
$H^{2}(X_{4},{\mathbb{Z}}_{p})$. Given a particular global variant, specified
by $L$ and the representative of $L^{\perp}\otimes
H^{2}(X_{4},{\mathbb{Z}}_{p})$, we may assign to it a $2g\times 2g$ matrix,

$\displaystyle M_{L,{\cal B}}:=(K_{L}\,\,K_{\cal B})\leavevmode\nobreak\ .$
(2.15)

A useful fact is that the matrices obtained in this way are elements of
$Sp(2g,{\mathbb{Z}}_{p})$. Indeed, denoting the generators of $L^{\perp}$ by
$\gamma_{i+g}$ for $i=1,\dots,g$, we have

$\displaystyle M_{L,{\cal B}}^{T}\,\mathfrak{I}\,M_{L,{\cal
B}}=\binom{K_{L}^{T}}{K_{\cal B}^{T}}\,\mathfrak{I}\,(K_{L}\,\,K_{\cal
B})=\left(\begin{matrix}v_{\gamma_{1}}^{T}\mathfrak{I}v_{\gamma_{1}}&\dots&v_{\gamma_{1}}^{T}\mathfrak{I}v_{\gamma_{2g}}\\\
&\ddots&\\\
v_{\gamma_{2g}}^{T}\mathfrak{I}v_{\gamma_{1}}&\dots&v_{\gamma_{2g}}^{T}\mathfrak{I}v_{\gamma_{2g}}\end{matrix}\right)=\mathfrak{I}\leavevmode\nobreak\
,$ (2.16)

where in the last step we have used the presentation of the intersection
pairing in (2.14), together with isotropicity and maximality of $L$.

Conversely, every element of $Sp(2g,{\mathbb{Z}}_{p})$ defines a global
variant. However, not all matrices denote distinct global variants. Given a
lattice $L$ spanned by $g$ elements $\gamma_{i}$ for $i=1,\dots,g$, it is
clear that linear transformations of these elements give rise to equivalent
bases for $L$. Similar statements hold for $L^{\perp}$, which is spanned by
$\gamma_{i+g}$ for $i=1,\dots,g$. In contrast, transformations which mix the
generators of $L$ and $L^{\perp}$ change the global form, and do not give
redundancies. The total set of redundancies is then given by block diagonal
matrices $\left(\begin{smallmatrix}Q&0\\\
0&Q^{\prime}\end{smallmatrix}\right)$ acting on
$(\gamma_{1},\dots,\gamma_{2g})^{T}$. By linearity of the labels $v_{\gamma}$,
these matrices act in the same way on $v_{\gamma}$, and hence we have the
identification under right-multiplication

$\displaystyle M_{L,{\cal B}}\cong M_{L,{\cal B}}\left(\begin{matrix}Q&0\\\
0&Q^{\prime}\end{matrix}\right)\leavevmode\nobreak\ .$ (2.17)

Importantly, the matrices $Q$ and $Q^{\prime}$ appearing here are actually not
independent—instead, $Q$ and $Q^{\prime}$ are constrained to satisfy
$Q^{\prime}Q^{T}=\mathds{1}_{g\times g}$ such that the resulting matrix is
symplectic. We conclude that each redundancy is given by a single element in
$GL(g,{\mathbb{Z}}_{p})$.

In summary, we expect that the total number of global variants of type
$\mathfrak{a}_{p-1}$ class ${\cal S}$ on $\Sigma_{g,0}$ is

$\displaystyle
d(g,p):={|Sp(2g,{\mathbb{Z}}_{p})|\over|GL(g,{\mathbb{Z}}_{p})|}\leavevmode\nobreak\
.$ (2.18)

In the particular case of $g=1$, this reduces to
$d(g,1)=|SL(2,{\mathbb{Z}}_{p})|/|{\mathbb{Z}}_{p}^{\times}|=p(p+1)$,
reproducing the result in (1.9). More generally, we have

$\displaystyle d(g,p)=p^{g^{2}}\prod_{m=1}^{g}{p^{2m}-1\over
p^{g}-p^{m-1}}=p^{{1\over
2}g(g+1)}\prod_{m=1}^{g}(p^{m}+1)\leavevmode\nobreak\ ,$ (2.19)

where the first equality follows from the known formulas for the orders of the
finite groups, and the second equality follows from some elementary algebra.
This formula admits a simple physical interpretation: the factor
$\prod_{m=1}^{g}(p^{m}+1)$ counts the number of maximal isotropic sublattices
of $H_{1}(\Sigma_{g,0},{\mathbb{Z}}_{N})$, while the factor $p^{{1\over
2}g(g+1)}$ counts the number of invertible phases that can be stacked with the
four-dimensional theory.121212The latter follows from noting that all SPT
phases are of the type $\int B_{i}\cup B_{j}$, with $i,j=1,\dots,g$. There are
${{1\over 2}g(g+1)}$ distinct combinations, and we can allow for between 0 and
$p-1$ copies of each.

####  2.3.1 An example

Let us give a concrete example before moving on. Consider the case of
$(g,p)=(2,2)$. We may begin by considering the maximally isotropic lattice
$L=\mathrm{span}\\{{\mathsf{A}}_{1},{\mathsf{A}}_{2}\\}$. The corresponding
$4\times 2$ matrix $K_{L}$ is then

$\displaystyle
K_{L}=(v_{{\mathsf{A}}_{1}}\,\,v_{{\mathsf{A}}_{2}})=\left(\begin{matrix}0&0&1&0\\\
0&0&0&1\end{matrix}\right)^{T}\leavevmode\nobreak\ .$ (2.20)

The orthogonal lattice $L^{\perp}=H_{1}(\Sigma_{2,0},{\mathbb{Z}}_{p})/L$
admits four explicit choices of basis vectors,
$\\{{\mathsf{B}}_{1},{\mathsf{B}}_{2}\\}$,
$\\{{\mathsf{B}}_{1}+{\mathsf{A}}_{1},{\mathsf{B}}_{2}\\}$,
$\\{{\mathsf{B}}_{1},{\mathsf{B}}_{2}+{\mathsf{A}}_{2}\\}$, and
$\\{{\mathsf{B}}_{1}+{\mathsf{A}}_{1},{\mathsf{B}}_{2}+{\mathsf{A}}_{2}\\}$,
with respective matrices $K_{\cal B}$ given by

$\displaystyle K_{\cal B}=\left(\begin{matrix}1&0&0&0\\\
0&1&0&0\end{matrix}\right)^{T}\leavevmode\nobreak\
,\,\,\,\left(\begin{matrix}1&0&1&0\\\
0&1&0&0\end{matrix}\right)^{T}\leavevmode\nobreak\
,\,\,\,\left(\begin{matrix}1&0&0&0\\\
0&1&0&1\end{matrix}\right)^{T}\leavevmode\nobreak\
,\,\,\,\left(\begin{matrix}1&0&1&0\\\
0&1&0&1\end{matrix}\right)^{T}\leavevmode\nobreak\ .$ (2.21)

The four corresponding global variants are thus labelled by the following
$4\times 4$ matrices,

$\displaystyle M_{L,{\cal B}}=\left(\begin{matrix}0&0&1&0\\\ 0&0&0&1\\\
1&0&0&0\\\ 0&1&0&0\end{matrix}\right)\leavevmode\nobreak\
,\,\,\,\left(\begin{matrix}0&0&1&0\\\ 0&0&0&1\\\ 1&0&1&0\\\
0&1&0&0\end{matrix}\right)\leavevmode\nobreak\
,\,\,\,\left(\begin{matrix}0&0&1&0\\\ 0&0&0&1\\\ 1&0&0&0\\\
0&1&0&1\end{matrix}\right)\leavevmode\nobreak\
,\,\,\,\left(\begin{matrix}0&0&1&0\\\ 0&0&0&1\\\ 1&0&1&0\\\
0&1&0&1\end{matrix}\right)\leavevmode\nobreak\ ,$ (2.22)

each of which is easily confirmed to be an element of
$Sp(4,{\mathbb{Z}}_{2})$. A similar exercise can be done for the remaining
choices of $L$.

Of course, we are always free to change our choice of bases such that e.g.
$L=\mathrm{span}\\{{\mathsf{A}}_{1}+{\mathsf{A}}_{2},{\mathsf{A}}_{2}\\}$
instead of $\mathrm{span}\\{{\mathsf{A}}_{1},{\mathsf{A}}_{2}\\}$, and this
should not change the global form. On the other hand, mixing between elements
of $L$ and $L^{\perp}$ will change the global form. Thus the redundancy in the
$4\times 4$ matrices given above is by right-multiplication

$\displaystyle M_{L,{\cal B}}\cong M_{L,{\cal B}}\left(\begin{matrix}Q&0\\\
0&Q^{\prime}\end{matrix}\right)$ (2.23)

with $Q$ and $Q^{\prime}$ both $2\times 2$ matrices in
$GL(2,{\mathbb{Z}}_{2})$. Requiring that the matrix is always symplectic fixes
$Q^{\prime}$ in terms of $Q$, as discussed above.

We note in closing that in our notation, for all $(g,p)$ the global form with
$L=\mathrm{span}\\{{\mathsf{B}}_{1},\dots,{\mathsf{B}}_{g}\\}$ and
$L^{\perp}=\mathrm{span}\\{{\mathsf{A}}_{1},\dots,{\mathsf{A}}_{g}\\}$ is
assigned the identity matrix in $Sp(2g,{\mathbb{Z}}_{p})$.

###  2.4 Modular orbits of global variants

In the previous subsection, we showed how to assign to each ${\cal
T}^{p,g,0}[\Omega]$ a distinct matrix in
$Sp(2g,{\mathbb{Z}}_{p})/GL(g,{\mathbb{Z}}_{p})$. The action of the modular
group $Sp(2g,{\mathbb{Z}}_{p})$ on the global variants follows from this
definition. Indeed, recall that any $F\in Sp(2g,{\mathbb{Z}}_{p})$ acts as in
(2.12), namely via left-multiplication

$\displaystyle F:\hskip
36.135pt\binom{\vec{{\mathsf{B}}}}{\vec{{\mathsf{A}}}}\mapsto
F\binom{\vec{{\mathsf{B}}}}{\vec{{\mathsf{A}}}}\leavevmode\nobreak\ .$ (2.24)

On the other hand, the vectors $v_{{\mathsf{A}}_{I}}$, $v_{{\mathsf{B}}_{I}}$
transform like a basis of charge vectors for the one-cycles, that is131313For
example, at genus 1 we have ${\mathsf{T}}:{\mathsf{A}}\mapsto{\mathsf{A}}$ and
$v_{\mathsf{A}}=\binom{0}{1}$. In order to obtain the appropriate
${\mathsf{T}}$ transform of $v_{\mathsf{A}}$, we must take
${\mathsf{T}}^{T}v_{\mathsf{A}}=\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)^{T}\binom{0}{1}=v_{\mathsf{A}}$, as opposed to
${\mathsf{T}}v_{\mathsf{A}}=v_{\mathsf{A}}+v_{\mathsf{B}}$.

$\displaystyle F:\hskip 36.135pt(v_{\vec{{\mathsf{B}}}}\,,\leavevmode\nobreak\
v_{\vec{{\mathsf{A}}}})\,\mapsto\,F^{T}(v_{\vec{{\mathsf{B}}}}\,,\leavevmode\nobreak\
v_{\vec{{\mathsf{A}}}})\leavevmode\nobreak\ .$ (2.25)

This means that by construction modular transformations act via left-
multiplication by $F^{T}$ on the matrix $M_{L,{\cal B}}$ labelling the global
form. By transitivity of $Sp(2g,{\mathbb{Z}}_{p})$, we see that every global
variant must be connected to every other via modular transformations—i.e.
there are no disconnected modular orbits.

On the other hand, the topological manipulations $G\in
Sp(2g,{\mathbb{Z}}_{p})$ commute with modular transformations, and act on the
global forms from the right. This fact, together with the fact that the
transformations $G$ span $Sp(2g,{\mathbb{Z}}_{p})$, will be made more apparent
in Section 5. Thus in general we have the transformations

$\displaystyle M_{L,{\cal B}}\rightarrow F^{T}M_{L,{\cal
B}}G\leavevmode\nobreak\ .$ (2.26)

From this we may draw two important conclusions:

  1. 1.

Every global variant can be mapped to every other by the action of both the
modular group and the topological manipulations.

  2. 2.

Every action $F$ of the modular group can be undone by an appropriate
topological manipulation $G$. This is true independent of the global variant
$M_{{\cal L},{\cal B}}$ under consideration.

The problem of identifying (non-)invertible symmetries is thus reduced to the
identification of points of enhanced symmetry for $\Sigma_{g,0}$.

##  3 Non-invertible symmetries for genus-2 class ${\cal S}$

We now begin our analysis of points of enhanced symmetry for the case of genus
2. For pedagogy we will work with explicit matrix representations of all group
elements here. For higher genus somewhat more abstract techniques will be
required.

###  3.1 Genus 2 and the modular group $Sp(4,{\mathbb{Z}}_{p})$

We work with the basis of $H_{1}(\Sigma_{g,0},{\mathbb{Z}}_{p})$ given above,
namely $\\{{\mathsf{A}}_{I},{\mathsf{B}}_{I}\\}$ for $I=1,\dots,g$ satisfying
(2.11). For general $g$ the first cohomology group of $\Sigma_{g,0}$ is given
by $H^{1}(\Sigma_{g,0},{\mathbb{R}})={\mathbb{R}}^{2g}$, and admits a
splitting into Dolbeault cohomology groups
$H^{1}(\Sigma_{g,0},{\mathbb{R}})=H^{(1,0)}(\Sigma_{g,0})\oplus
H^{(0,1)}(\Sigma_{g,0})$. A canonical basis of
$H^{(1,0)}(\Sigma_{g,0},{\mathbb{Z}})$ is given by holomorphic (1,0)-forms
$\omega_{I}$ for $I=1,\dots,g$, with periods normalized as

$\displaystyle\oint_{{\mathsf{A}}_{I}}\omega_{J}=\delta_{IJ}\leavevmode\nobreak\
.$ (3.1)

In terms of these forms we may define the period matrix $\Omega$ of
$\Sigma_{g,0}$,

$\displaystyle\oint_{{\mathsf{B}}_{I}}\omega_{J}=\Omega_{IJ}\leavevmode\nobreak\
.$ (3.2)

The matrices $F\in Sp(2g,{\mathbb{Z}})$ then act on $\Omega$ as

$\displaystyle F:\hskip
14.45377pt\Omega\mapsto(A\Omega+B)(C\Omega+D)^{-1}\leavevmode\nobreak\ ,\hskip
36.135ptF=\left(\begin{matrix}A&B\\\
C&D\end{matrix}\right)\leavevmode\nobreak\ .$ (3.3)

At genus 2, the period matrix is a $2\times 2$ matrix

$\displaystyle\Omega=\left(\begin{matrix}\tau_{1}&\tau_{2}\\\
\tau_{2}&\tau_{3}\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
36.135pt\mathrm{det}(\mathrm{Im}(\tau_{i}))>0\leavevmode\nobreak\ ,\hskip
14.45377pt\mathrm{Tr}(\mathrm{Im}(\tau_{i}))>0\leavevmode\nobreak\ .$ (3.4)

Note that the limit of $\tau_{2}\rightarrow 0$ is a degeneration limit in
which the genus-2 Riemann surface splits into two genus-1 Riemann surfaces
with complex structure moduli $\tau_{1}$ and $\tau_{3}$. Indeed, we have an
embedding of the corresponding modular groups $SL(2,{\mathbb{Z}})\times
SL(2,{\mathbb{Z}})\subset Sp(4,{\mathbb{Z}})$.

Each value of the period matrix $\Omega$ corresponds to a certain choice of
couplings for the corresponding four-dimensional class ${\cal S}$ theory, and
in certain limits one may have a Lagrangian description of the theory as a
(generalized) quiver theory, as shown in Figure 5. In the separating limit
mentioned in the previous paragraph, in which e.g. the tube in the top left
side of Figure 5 is stretched infinitesimally thin and infinitely long, one
retrieves two copies of the appropriate $\mathfrak{a}_{p-1}$ ${\cal N}=4$
theory, which have the non-invertible symmetries studied in [35]. More
generally, the points at which non-invertible symmetries arise will be
strongly-coupled and non-Lagrangian.

$p$$p$$p$

$p$$p$$p$

Figure 5: Two weakly-coupled limits for class ${\cal S}$ on $\Sigma_{2,0}$,
with the corresponding generalized quivers. The trivalent junctions represent
$T_{p}$ theories, while the circle represent $\mathfrak{a}_{p-1}$ gauge nodes.

For the purposes of this section it will be useful to have an explicit matrix
representation of the modular group $Sp(4,{\mathbb{Z}})$, which is given by

$\displaystyle Sp(4,{\mathbb{Z}})=\langle T_{1},\leavevmode\nobreak\
T_{2},\leavevmode\nobreak\ S_{12},\leavevmode\nobreak\
W\rangle\leavevmode\nobreak\ ,$ (3.5)

in terms of the following four generators,

$\displaystyle T_{1}=\left(\begin{matrix}1&0&1&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{matrix}\right),\leavevmode\nobreak\
T_{2}=\left(\begin{matrix}1&0&0&0\\\ 0&1&0&1\\\ 0&0&1&0\\\
0&0&0&1\end{matrix}\right),\leavevmode\nobreak\
S_{12}=\left(\begin{matrix}0&0&1&0\\\ 0&0&0&1\\\ -1&0&0&0\\\
0&-1&0&0\end{matrix}\right),\leavevmode\nobreak\
W=\left(\begin{matrix}1&0&0&1\\\ 0&1&1&0\\\ 0&0&1&0\\\
0&0&0&1\end{matrix}\right)\leavevmode\nobreak\ .$

These generators act on the basis of one-cycles as,

$\displaystyle T_{1}\left(\begin{matrix}{\mathsf{B}}_{1}\\\
{\mathsf{B}}_{2}\\\ {\mathsf{A}}_{1}\\\
{\mathsf{A}}_{2}\end{matrix}\right)=\left(\begin{matrix}{\mathsf{B}}_{1}+{\mathsf{A}}_{1}\\\
{\mathsf{B}}_{2}\\\ {\mathsf{A}}_{1}\\\
{\mathsf{A}}_{2}\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
14.45377ptT_{2}\left(\begin{matrix}{\mathsf{B}}_{1}\\\ {\mathsf{B}}_{2}\\\
{\mathsf{A}}_{1}\\\ {\mathsf{A}}_{2}\end{matrix}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}{\mathsf{B}}_{1}\\\
{\mathsf{B}}_{2}+{\mathsf{A}}_{2}\\\ {\mathsf{A}}_{1}\\\
{\mathsf{A}}_{2}\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
14.45377ptS_{12}\left(\begin{matrix}{\mathsf{B}}_{1}\\\ {\mathsf{B}}_{2}\\\
{\mathsf{A}}_{1}\\\
{\mathsf{A}}_{2}\end{matrix}\right)=\left(\begin{matrix}{\mathsf{A}}_{1}\\\
{\mathsf{A}}_{2}\\\ -{\mathsf{B}}_{1}\\\
-{\mathsf{B}}_{2}\end{matrix}\right)\leavevmode\nobreak\ ,$ $\displaystyle
W\left(\begin{matrix}{\mathsf{B}}_{1}\\\ {\mathsf{B}}_{2}\\\
{\mathsf{A}}_{1}\\\ {\mathsf{A}}_{2}\end{matrix}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}{\mathsf{B}}_{1}+{\mathsf{A}}_{2}\\\
{\mathsf{B}}_{2}+{\mathsf{A}}_{1}\\\ {\mathsf{A}}_{1}\\\
{\mathsf{A}}_{2}\end{matrix}\right)\leavevmode\nobreak\ .$ (3.6)

We see that $T_{1}$ and $T_{2}$ act as separate $T$ transformations on the two
sub-tori, whereas $S_{12}$ acts as a simultaneous $S$ transformation on both
tori. The $W$ transformation is an intrinsically genus-2 transformation.

###  3.2 Fixed points of $Sp(4,{\mathbb{Z}})$

With an explicit basis of generators for $Sp(4,{\mathbb{Z}})$, we may now
search for elements admitting fixed points in their action on $\Omega$. As one
might have anticipated, these results have already been obtained in the
mathematics literature [117, 118, 119]; here we simply quote them. There are
two qualitatively distinct classes of fixed loci:

Order | Subgroup | Generators | $\Omega$
---|---|---|---
10 | ${\mathbb{Z}}_{10}$ | $\phi$ | $\left(\begin{smallmatrix}\varepsilon&\varepsilon+\varepsilon^{-2}\\\ \varepsilon+\varepsilon^{-2}&-\varepsilon^{-1}\end{smallmatrix}\right)$
24 | $({\mathbb{Z}}_{2}\times{\mathbb{Z}}_{6})\rtimes{\mathbb{Z}}_{2}$ | $M_{1},M_{2},M_{3}$ | ${i\over\sqrt{3}}\left(\begin{smallmatrix}2&1\\\ 1&2\end{smallmatrix}\right)$
| ${\mathbb{Z}}_{12}\times{\mathbb{Z}}_{2}$ | $C,M_{4}$ | $\left(\begin{smallmatrix}\rho&0\\\ 0&i\end{smallmatrix}\right)$
32 | $({\mathbb{Z}}_{4}\times{\mathbb{Z}}_{4})\rtimes{\mathbb{Z}}_{2}$ | $M_{5},M_{6},M_{7}$ | $\left(\begin{smallmatrix}i&0\\\ 0&i\end{smallmatrix}\right)$
48 | $GL(2,3)$ | $M_{7},M_{8}$ | ${1\over 3}\left(\begin{smallmatrix}1+2i\sqrt{2}&-1+i\sqrt{2}\\\ -1+i\sqrt{2}&1+2i\sqrt{2}\end{smallmatrix}\right)$
72 | ${\mathbb{Z}}_{3}\times({\mathbb{Z}}_{6}\times{\mathbb{Z}}_{2})\rtimes{\mathbb{Z}}_{2}$ | $M_{7},M_{9},M_{10}$ | $\left(\begin{smallmatrix}\rho&0\\\ 0&\rho\end{smallmatrix}\right)$
Table 2: Singular moduli of $Sp(4,{\mathbb{Z}})$ occurring at isolated values
of $\Omega$. Here we have defined $\rho:=e^{2\pi i\over 3}$ and
$\varepsilon:=e^{2\pi i\over 5}$.

##### Isolated loci:

The first type of fixed loci are those of dimension zero, i.e. isolated fixed
points. This is the only case which occurred at genus 1, where the fixed loci
were the isolated points $\tau_{\mathrm{YM}}=i$ and $e^{2\pi i\over 3}$. The
list of all such isolated fixed points, together with the stabilizing subgroup
of $Sp(4,{\mathbb{Z}})$, are collected in Table 2. An explicit matrix
representation for the generators $\phi$, $C$, and $M_{i}$ for $i=1,\dots,10$
is given in Appendix A.141414Note that the same results can be found in [126].

Let us make some comments about these fixed points. First, the fixed points at
$\Omega=\left(\begin{smallmatrix}\rho&0\\\
0&i\end{smallmatrix}\right),\left(\begin{smallmatrix}i&0\\\
0&i\end{smallmatrix}\right)$, and $\left(\begin{smallmatrix}\rho&0\\\
0&\rho\end{smallmatrix}\right)$ correspond to degeneration limits of the
genus-2 Riemann surface. The (non-)invertible symmetries corresponding to
these points are the same as those discussed in the introduction up to an
additional symmetry corresponding to permutation of the two genus-1 Riemann
surfaces. In particular, the question of intrinsic versus non-intrinsic
invertibility as a function of the prime $p$ is the same as before. Thus we do
not discuss these cases here.

The remaining three fixed points are intrinsically genus 2. The first fixed
point is
$\Omega=\left(\begin{smallmatrix}\varepsilon&\varepsilon+\varepsilon^{-2}\\\
\varepsilon+\varepsilon^{-2}&-\varepsilon^{-1}\end{smallmatrix}\right)$ with
$\varepsilon=e^{2\pi i\over 5}$, which leads to an enhanced
${\mathbb{Z}}_{10}$ symmetry of the Riemann surface. This is the direct analog
of the ${\mathbb{Z}}_{6}$ symmetry at $\tau_{\mathrm{YM}}=e^{2\pi i\over 3}$
at genus-1.151515More generally, for every genus $g$ there is an isolated
point at which there is an enhanced ${\mathbb{Z}}_{4g+2}$ symmetry. Similarly
at $\Omega={i\over\sqrt{3}}\left(\begin{smallmatrix}2&1\\\
1&2\end{smallmatrix}\right)$ we have an enhanced
$({\mathbb{Z}}_{2}\times{\mathbb{Z}}_{6})\rtimes{\mathbb{Z}}_{2}$ symmetry,
and at $\Omega={1\over 3}\left(\begin{smallmatrix}1+2i\sqrt{2}&-1+i\sqrt{2}\\\
-1+i\sqrt{2}&1+2i\sqrt{2}\end{smallmatrix}\right)$ an enhanced $GL(2,3)$
symmetry. The latter is in fact the most symmetric (connected) genus-2 Riemann
surface, known as the Bolza surface.

We should point out here that unlike for 4d ${\cal N}=4$ SYM, the enhanced
symmetries can now be non-Abelian. This can give rise to non-Abelian fusion
rules involving non-invertible defects, as will be discussed further in
Section 5.4. Note that while non-Abelian non-invertible fusion rules were
identified previously for triality defects in [29], the non-Abelian fusion
rules here are of a slightly different nature: in particular, the fusion rules
continue to be non-Abelian even modulo condensation defects. In the language
of [34], the fusion rules are non-Abelian even at the level of local fusion.

##### Extended loci:

Unlike for genus 1, at higher genus there can also be entire loci of fixed
points. At genus 2, these can be of complex dimension 1 or 2. We list the
former in Table 3, and the latter in Table 4. Of these, the ones at
$\Omega=\left(\begin{smallmatrix}i&0\\\
0&\tau_{3}\end{smallmatrix}\right),\left(\begin{smallmatrix}\tau_{1}&0\\\
0&\tau_{1}\end{smallmatrix}\right),\left(\begin{smallmatrix}\rho&0\\\
0&\tau_{3}\end{smallmatrix}\right),$ and
$\left(\begin{smallmatrix}\tau_{1}&0\\\ 0&\tau_{3}\end{smallmatrix}\right)$
correspond to degeneration limits, and will not be discussed further. Let us
now comment on the remaining fixed loci.

First there are two loci of complex dimension one
$\Omega=\left(\begin{smallmatrix}\tau_{1}&{1\over 2}\\\ {1\over
2}&\tau_{1}\end{smallmatrix}\right)$ and
$\left(\begin{smallmatrix}\tau_{1}&{1\over 2}\tau_{1}\\\ {1\over
2}\tau_{1}&\tau_{1}\end{smallmatrix}\right)$ which have enhanced symmetries
that are respectively $D_{8}$ and $D_{12}$. Second, there is a complex
dimension two locus $\Omega=\left(\begin{smallmatrix}\tau_{1}&\tau_{2}\\\
\tau_{2}&\tau_{1}\end{smallmatrix}\right)$ with an enhanced
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ symmetry. Note that the existence of
such loci suggests the possibility of having continuous families of theories,
parameterized by either 1 or 2 complex couplings, all of which have non-
invertible symmetries. This is in contrast to the case of 4d ${\cal N}=4$, for
which the non-invertible symmetries only emerged at isolated points
$\tau_{\mathrm{YM}}=i$ and $e^{2\pi i\over 3}$.

Order | Subgroup | Generators | $\Omega$
---|---|---|---
8 | ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{4}$ | $C,N_{1}$ | $\left(\begin{smallmatrix}i&0\\\ 0&\tau_{3}\end{smallmatrix}\right)$
| $D_{8}$ | $M_{7},N_{2}$ | $\left(\begin{smallmatrix}\tau_{1}&{1\over 2}\\\ {1\over 2}&\tau_{1}\end{smallmatrix}\right)$
| $D_{8}$ | $M_{7},N_{3}$ | $\left(\begin{smallmatrix}\tau_{1}&0\\\ 0&\tau_{1}\end{smallmatrix}\right)$
12 | ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{6}$ | $C,N_{4}$ | $\left(\begin{smallmatrix}\rho&0\\\ 0&\tau_{3}\end{smallmatrix}\right)$
| $D_{12}$ | $N_{5},N_{6}$ | $\left(\begin{smallmatrix}\tau_{1}&{1\over 2}\tau_{1}\\\ {1\over 2}\tau_{1}&\tau_{1}\end{smallmatrix}\right)$
Table 3: Singular moduli of $Sp(4,{\mathbb{Z}})$ occurring along complex one-dimensional loci. Order | Subgroup | Generators | $\Omega$
---|---|---|---
4 | ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ | $C,P_{1}$ | $\left(\begin{smallmatrix}\tau_{1}&0\\\ 0&\tau_{3}\end{smallmatrix}\right)$
| ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ | $C,M_{7}$ | $\left(\begin{smallmatrix}\tau_{1}&\tau_{2}\\\ \tau_{2}&\tau_{1}\end{smallmatrix}\right)$
Table 4: Singular moduli of $Sp(4,{\mathbb{Z}})$ occurring along complex two-
dimensional loci.

###  3.3 Invariant global forms

By the results of Section 2.3, we know that each geometric symmetry $F$ gives
rise to a symmetry of ${\cal T}^{p,2,0}_{L}[\Omega]$, regardless of the choice
of global variant $L$. Depending on $L$, the topological manipulation $G$
which is required to undo the action of $F$ on $L$ may involve a discrete
gauging, and hence may give rise to a non-invertible symmetry. Indeed, this is
the generic situation. What remains is only the question of whether the non-
invertible symmetries are intrinsic or not. We now answer this question. We
warn that the following analysis is somewhat technical; the reader only
interested in the answer can refer to Tables 5 and 6.

To answer the question of intrinsic non-invertibility, we search for global
forms which are invariant under $F$. The existence of such global forms will
depend delicately on the value of $p$. For an invariant global form $M_{{\cal
L},{\cal B}}$ to exist, we must have

$\displaystyle F^{T}M_{L,{\cal B}}=M_{L,{\cal
B}}\,P\,\chi(\tau)\leavevmode\nobreak\ ,\hskip
36.135ptP=\left(\begin{matrix}Q&0\\\
0&Q^{\prime}\end{matrix}\right)\leavevmode\nobreak\ ,$ (3.7)

where $Q$ and $Q^{\prime}$ are elements of $GL(2,{\mathbb{Z}}_{p})$ and the
matrix $Q^{\prime}$ is completely determined in terms of $Q$ by the equation
$Q^{\prime}Q^{T}=\mathds{1}_{g\times g}$, as explained around (2.17). The
reason for allowing the matrix $P$ on the right-hand side is that $M_{L,{\cal
B}}$ is, in the first place, defined only up to such transformations. The term
$\chi(\tau)$ is a topological manipulation involving only invertible $\tau$
operations (and potentially automorphisms of the one-form symmetry), which
captures ‘t Hooft anomalies of the invertible symmetry. As in the genus-1
analysis in the introduction, we will set the background fields to zero, which
will allow us to drop the $\chi(\tau)$ operation and reduce $M_{L,{\cal B}}$
to the $2g\times g$ matrix $K_{L}$. We thus aim to solve

$\displaystyle F^{T}K_{L}=K_{L}Q\leavevmode\nobreak\ ,\hskip 36.135ptQ\in
GL(g,{\mathbb{Z}}_{p})\leavevmode\nobreak\ .$ (3.8)

The question is now whether any solution to this equation exists. For genus-1,
the corresponding equation was a standard eigenvalue equation (1.14), and we
could understand the existence of solutions by simply trying to solve the
characteristic polynomial modulo $p$. The current case is somewhat more
involved.

Given a matrix $Q$ over an algebraically closed field (such as ${\mathbb{C}}$)
we may always do a similarity transformation to put $Q$ in Jordan normal form,
i.e. $Q=B^{-1}JB$. A matrix $J$ in Jordan normal form is a block diagonal
matrix $\bigoplus_{i}J_{i}$ where each block $J_{i}$ has the form

$\displaystyle J_{i}=\left(\begin{matrix}\lambda_{i}&1&0&\dots&0\\\
0&\lambda_{i}&1&\dots&0\\\ 0&0&\lambda_{i}&\dots&0\\\ &&\vdots&&\\\
0&0&\dots&0&\lambda_{i}\end{matrix}\right)\leavevmode\nobreak\ ,$ (3.9)

i.e. it is an upper-triangular matrix with a single number $\lambda_{i}$ on
the diagonal and ones on the superdiagonal. For the case of $Q$ being a
$2\times 2$ matrix, there are two possible Jordan decompositions,

$\displaystyle J=\left(\begin{matrix}\lambda_{1}&0\\\
0&\lambda_{2}\end{matrix}\right)\leavevmode\nobreak\
,\,\,\,\,\left(\begin{matrix}\lambda_{1}&1\\\
0&\lambda_{1}\end{matrix}\right)\leavevmode\nobreak\ .$ (3.10)

In the current case we are concerned with $Q$ defined over the finite field
${\mathbb{Z}}_{p}$, which in general is not algebraically closed. However, by
lifting to the splitting field ${\mathbb{Z}}_{p}[\lambda_{i}]$, it is again
always possible to put $Q$ in Jordan normal form (of course, if the
$\lambda_{i}$ are all valued in ${\mathbb{Z}}_{p}$, then the splitting is
possible in ${\mathbb{Z}}_{p}$ itself). We may thus write

$\displaystyle Q=B^{-1}JB\leavevmode\nobreak\ ,\hskip 36.135ptQ\in
GL(g,{\mathbb{Z}}_{p})\leavevmode\nobreak\ ,\,\,\,\,B,J\in
GL(g,{\mathbb{Z}}_{p}[\lambda_{i}])\leavevmode\nobreak\ .$ (3.11)

In order for $Q$ to be an element of $GL(g,{\mathbb{Z}}_{p})$, we require
$\mathrm{det}(Q)=\mathrm{det}(B^{-1}JB)=\mathrm{det}(J)\in{\mathbb{Z}}_{p}^{\times}$.
Thus even though the entries of $J$ may not in general be valued in
${\mathbb{Z}}_{p}^{\times}$, the product of the diagonal entries must be
valued in ${\mathbb{Z}}_{p}^{\times}$.

The quantities $\lambda_{i}$ above are related to the modular transformation
$F$ as follows. We begin by rewriting (3.8) as

$\displaystyle F^{T}\widetilde{K}_{L}=\widetilde{K}_{L}J\leavevmode\nobreak\
,\hskip 36.135pt\widetilde{K}_{L}:=K_{L}B^{-1}\leavevmode\nobreak\ .$ (3.12)

Writing $\widetilde{K}_{L}=(\widetilde{v}_{1},\widetilde{v}_{2})$ at genus 2,
we then have

$\displaystyle
F^{T}\,(\widetilde{v}_{1},\widetilde{v}_{2})=(\lambda_{1}\widetilde{v}_{1},\lambda_{2}\widetilde{v}_{2})\leavevmode\nobreak\
,\hskip
36.135ptF^{T}\,(\widetilde{v}_{1},\widetilde{v}_{2})=(\lambda_{1}\widetilde{v}_{1},\widetilde{v}_{1}+\lambda_{1}\widetilde{v}_{2})\leavevmode\nobreak$
(3.13)

for the two choices of $J$ in (3.10). Either way, this tells us that all
$\lambda_{i}$ must be eigenvalues of $F$.

A further simplifying feature is that $F$ is always the generator of a finite
group, meaning that $F^{k}=\mathds{1}$ for appropriate $k$. In the case that
$J$ is diagonal, this tells us that

$\displaystyle(\widetilde{v}_{1},\widetilde{v}_{2})=(F^{T})^{k}\,(\widetilde{v}_{1},\widetilde{v}_{2})=(\lambda_{1}^{k}\widetilde{v}_{1},\lambda_{2}^{k}\widetilde{v}_{2})$
(3.14)

and hence that the $\lambda_{i}$ must be $k$-th roots of unity. On the other
hand, if $J$ is a non-trivial Jordan block, this tells us that

$\displaystyle(\widetilde{v}_{1},\widetilde{v}_{2})=(F^{T})^{k}\,(\widetilde{v}_{1},\widetilde{v}_{2})=(\lambda_{1}^{k}\widetilde{v}_{1},\,\lambda_{1}^{k}\widetilde{v}_{2}+k\lambda_{1}^{k-1}v_{1})$
(3.15)

The first equation in this pair would imply $\lambda_{1}^{k}=1$, upon which
the second would imply that $k\lambda_{1}^{k-1}v_{1}=0$ mod $p$. Since neither
$\lambda_{1}$ or $v_{1}$ can be trivial, we conclude that having a non-trivial
Jordan block is only possible if $p|k$.

Once we have identified a $J$ satisfying the above properties, it is
guaranteed that there will exist a matrix $K_{L}$ with coefficients in
${\mathbb{Z}}_{p}$ satisfying (3.8)—this follows since the entries of $K_{L}$
satisfy a set of linear equations with coefficients in ${\mathbb{Z}}_{p}$,
which always admit solutions in ${\mathbb{Z}}_{p}$. However, it is not clear
that the matrices obtained in this way will have columns spanning an isotropic
sublattice. For $K_{L}$ to correspond to a legitimate global form, we must
verify this. Here it is useful to note two facts:

  1. 1.

If we write $K_{L}=(v_{1},v_{2})$ and
$\widetilde{K}_{L}=(\widetilde{v}_{1},\widetilde{v}_{2})$, then the vectors
$v_{1},v_{2}$ span an isotropic subspace if and only if
$\widetilde{v}_{1},\widetilde{v}_{2}$ do. To prove e.g. the forward direction,
we assume that $K_{L}$ is isotropic, i.e.

$\displaystyle K_{L}^{T}\mathfrak{I}K_{L}=0_{2\times 2}\leavevmode\nobreak\ ,$
(3.16)

which then implies that

$\displaystyle\widetilde{K}_{L}^{T}\mathfrak{I}\widetilde{K}_{L}=(K_{L}B^{-1})^{T}\mathfrak{I}K_{L}B^{-1}=(B^{-1})^{T}K_{L}^{T}\mathfrak{I}K_{L}B^{-1}=0_{2\times
2}\leavevmode\nobreak\ .$ (3.17)

The reverse direction follows analogously.

  2. 2.

Let $v_{1},v_{2}$ be two eigenvectors of $F$ with eigenvalues
$\lambda_{1},\lambda_{2}$. Since $F$ is symplectic we have

$\displaystyle
v_{2}^{T}\mathfrak{I}v_{1}=v_{2}^{T}F^{T}\mathfrak{I}Fv_{1}=(Fv_{2})^{T}\mathfrak{I}Fv_{1}=\lambda_{1}\lambda_{2}v_{2}^{T}\mathfrak{I}v_{1}\leavevmode\nobreak\
,$ (3.18)

or in other words $(\lambda_{2}\lambda_{1}-1)v_{2}^{T}\mathfrak{I}v_{1}=0$ mod
$p$. This means that the span of $v_{1}$ and $v_{2}$ is necessarily isotropic
unless $\lambda_{1}\lambda_{2}=1$ mod $p$.

Using these two facts, we will be able to identify the solutions $K_{L}$
corresponding to legitimate global forms.

To summarize the discussion above, in order to identify invariant global forms
we begin by identifying the eigenvalues of $F$, which is done by solving the
characteristic equation. Not all of the eigenvalues will be elements of
${\mathbb{Z}}_{p}^{\times}$. We choose pairs of eigenvalues which are
conjugates (i.e. such that
$\lambda_{1}\lambda_{2}\in{\mathbb{Z}}_{p}^{\times}$ or
$\lambda_{1}^{2}\in{\mathbb{Z}}_{p}^{\times}$) and construct all possible $J$,
subject to the constraint that $J$ can only have non-trivial Jordan blocks if
$p$ divides the order of $F$. Each such $J$ will give rise to a generalized
eigenvalue equation as in (3.8). There will always exist a matrix $K_{L}$ with
coefficients in ${\mathbb{Z}}_{p}$ satisfying this equation, but to ensure
that this matrix corresponds to a legitimate global form we must check that
its columns span an isotropic sublattices. This may equivalently be done by
checking isotropicity of the solutions $\widetilde{K}_{L}$ to (3.12).

We now apply this procedure for each of the symmetries mentioned above.

#####  ${\mathbb{Z}}_{10}$:

We begin by asking if there exist global forms invariant under the generator
$\phi$ of the ${\mathbb{Z}}_{10}$ symmetry. An explicit matrix form for $\phi$
is given in Appendix A, from which we obtain the characteristic equation,

$\displaystyle\mathrm{det}(\phi-\lambda\mathds{1}_{4\times
4})=\lambda^{4}-\lambda^{3}+\lambda^{2}-\lambda+1=0\,\,\,\mathrm{mod}\,\,p\leavevmode\nobreak\
.$ (3.19)

We begin by giving two representative examples before presenting the general
solution.

We first consider the case of $p=5$, for which the equation above can be
rewritten as

$\displaystyle\lambda^{4}+4\lambda^{3}+6\lambda^{2}+4\lambda+1=(\lambda+1)^{4}=0\,\,\,\mathrm{mod}\,\,p\leavevmode\nobreak\
.$ (3.20)

Thus there exists a single solution $\lambda=-1$ mod $5$ of multiplicity 4.
Since $5$ divides the order $|\phi|=10$, we are allowed to have a non-trivial
Jordan block, and hence the possible choices for $J$ are

$\displaystyle J=\left(\begin{matrix}-1&0\\\
0&-1\end{matrix}\right)\leavevmode\nobreak\ ,\left(\begin{matrix}-1&1\\\
0&-1\end{matrix}\right)\leavevmode\nobreak\ .$ (3.21)

The determinant of $J$ is automatically in ${\mathbb{Z}}_{5}^{\times}$ since
the diagonal entries are. The most general matrices $\widetilde{K}_{L}$
satisfying (3.12) for these $J$ are then found to be (up to linear
transformations on the columns),

$\displaystyle\widetilde{K}_{L}=\left(\begin{matrix}3&3\\\ 1&1\\\ 1&1\\\
1&1\end{matrix}\right)\leavevmode\nobreak\ ,\,\,\left(\begin{matrix}3&3\\\
1&1\\\ 1&0\\\ 1&4\end{matrix}\right)\leavevmode\nobreak\ ,$ (3.22)

respectively. In the first case, we see that the two columns are identical,
and will not span a maximal sublattice. Hence this does not correspond to a
legitimate label for a global form. In the second case, the two columns are
linearly independent, but we must check that they span an isotropic
sublattice, i.e. $\widetilde{K}_{L}\mathfrak{I}\widetilde{K}_{L}=0$. One
easily confirms that this is the case. We thus conclude that for $p=5$ there
is a single global form with an invertible $\phi$ symmetry, labelled by the
second matrix in (3.22).

We next consider $p=11$. In this case all four solutions of (3.19) are in
${\mathbb{Z}}_{11}$, and are given by

$\displaystyle\lambda=2,\,6,\,7,\,8\leavevmode\nobreak\ .$ (3.23)

Because $11\nmid 10$, no non-trivial Jordan blocks are allowed, so there are 6
possible $J$,161616We neglect cases where $J$ is proportional to the identity,
since in such cases the columns of $\widetilde{K}_{L}$ will not span an
isotropic sublattice.

$\displaystyle
J=\mathrm{diag}(2,6),\,\mathrm{diag}(2,7),\,\mathrm{diag}(2,8),\,\mathrm{diag}(6,7),\,\mathrm{diag}(6,8),\,\mathrm{diag}(7,8)\leavevmode\nobreak\
.$ (3.24)

The determinant of $J$ is automatically in ${\mathbb{Z}}_{11}^{\times}$ since
the diagonal entries are. The corresponding matrices $\widetilde{K}_{L}$
satisfying (3.12) are found to be

$\displaystyle\widetilde{K}_{L}=\left(\begin{matrix}1&5\\\ 3&4\\\ 5&9\\\
1&1\end{matrix}\right)\leavevmode\nobreak\ ,\,\,\left(\begin{matrix}1&6\\\
3&9\\\ 5&3\\\ 1&1\end{matrix}\right)\leavevmode\nobreak\
,\,\,\left(\begin{matrix}1&7\\\ 3&5\\\ 5&4\\\
1&1\end{matrix}\right)\leavevmode\nobreak\ ,\,\,\left(\begin{matrix}5&6\\\
4&9\\\ 9&3\\\ 1&1\end{matrix}\right)\leavevmode\nobreak\
,\,\,\left(\begin{matrix}5&7\\\ 4&5\\\ 9&4\\\
1&1\end{matrix}\right)\leavevmode\nobreak\ ,\,\,\left(\begin{matrix}6&7\\\
9&5\\\ 3&4\\\ 1&1\end{matrix}\right)\leavevmode\nobreak\ .$ (3.25)

We must now check which of these has columns spanning an isotropic sublattice,
i.e. $\widetilde{K}_{L}^{T}\mathfrak{I}\widetilde{K}_{L}=0$. As mentioned
above, isotropicity is automatic if the eigenvalues of $J$ are not inverses,
and in the current case we have $6=2^{-1}$ mod $11$ and $8=7^{-1}$ mod $11$.
So we need only check the first and last matrices above, in which case we find
$\widetilde{K}_{L}^{T}\mathfrak{I}\widetilde{K}_{L}=6,4$ mod $11$
respectively. We see that these cases are not isotropic, and do not correspond
to legitimate global forms. We conclude that for $p=11$ there are four global
forms with an invertible $\phi$ symmetry.

Having given two explicit examples, we now turn to generic $p$. We first look
for the number of solutions in ${\mathbb{Z}}_{p}^{\times}$ to the
characteristic equation. The number of such solutions turns out to be

$\displaystyle\\#\,\,\mathrm{solutions}=\left\\{\begin{matrix}1&&p=5\\\
4&&p\in 10{\mathbb{N}}+1\\\
0&&\mathrm{otherwise}\end{matrix}\right.\leavevmode\nobreak\ .$ (3.26)

For generic $p\in 10{\mathbb{N}}+1$, we see that all four solutions
$\\{\lambda_{1},\dots,\lambda_{4}\\}$ are in ${\mathbb{Z}}_{p}^{\times}$. One
can furthermore show that these come in inverse pairs, say
$\lambda_{2}=\lambda_{1}^{-1}$ and $\lambda_{4}=\lambda_{3}^{-1}$. Because
$(10{\mathbb{N}}+1)\nmid 10$, it is not possible to have non-trivial Jordan
blocks, so there are six choices for $J$, namely

$\displaystyle J=\mathrm{diag}(\lambda_{i},\lambda_{j})\leavevmode\nobreak\
,\hskip 36.135pt1\leq i<j\leq 4\leavevmode\nobreak\ ,$ (3.27)

all of which have determinant in ${\mathbb{Z}}_{p}^{\times}$ since the
diagonal entries are in ${\mathbb{Z}}_{p}^{\times}$. The corresponding
matrices $\widetilde{K}_{L}$ satisfying (3.12) are

$\displaystyle\widetilde{K}_{L}=\left(\begin{matrix}p+\lambda_{i}-1&p+\lambda_{j}-1\\\
(p-1)\lambda_{i}^{3}&(p-1)\lambda_{j}^{3}\\\
p-1+\lambda_{i}+(p-1)\lambda_{i}^{2}+\lambda_{i}^{3}&p-1+\lambda_{j}+(p-1)\lambda_{j}^{2}+\lambda_{j}^{3}\\\
1&1\end{matrix}\right)\leavevmode\nobreak\ .$ (3.28)

What remains is to check that the columns span an isotropic sublattice. This
is automatically the case if $\lambda_{i}$ and $\lambda_{j}$ are not modular
inverses (which is four of the six cases), but must be checked otherwise.
Taking $\lambda_{j}=\lambda_{i}^{-1}$, we find

$\displaystyle\widetilde{v}_{i}^{T}\mathfrak{I}\widetilde{v}_{j}=\lambda_{i}(2\lambda_{i}-1)-\lambda_{j}(2\lambda_{j}-1)\,\,\,\mathrm{mod}\,\,p\leavevmode\nobreak\
.$ (3.29)

It is easy to show that the right-hand side can never vanish for
$\lambda_{j}\neq\lambda_{i}$, and hence these cases do not correspond to
legitimate global forms. The remaining four cases do correspond to legitimate
global forms though, and so we conclude that for any $p\in 10{\mathbb{Z}}+1$
there exist four global forms with invertible $\phi$ symmetry.

We finally turn to the case of $p>5$ and $p\not\in 10{\mathbb{N}}+1$. In this
case there do not exist any solutions to the characteristic equation in
${\mathbb{Z}}_{p}^{\times}$. Furthermore, it can be shown that the product of
any two elements which are not inverses of each other does not return an
element of ${\mathbb{Z}}_{p}^{\times}$. Thus there are no $J$ that we can
write down with the requisite properties, and hence no global forms with an
invertible $\phi$ symmetry. The results so far are summarized in the first
line of Table 5.

$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29
---|---|---|---|---|---|---|---|---|---|---
$\phi$ intrinsic? | ✓ | ✓ | ✗ | ✓ | ✗ | ✓ | ✓ | ✓ | ✓ | ✓
$M_{1}$ intrinsic? | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
$M_{2}$ intrinsic? | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
$M_{3}$ intrinsic? | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
$M_{7}$ intrinsic? | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
$M_{8}$ intrinsic? | ✗ | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✓ | ✗
$N_{2}$ intrinsic? | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
$N_{5}$ intrinsic? | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
$N_{6}$ intrinsic? | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
Table 5: For any $p$, there are global variants such that the modular
transformations in Tables 2, 3, 4 give rise to non-invertible symmetries (upon
appropriate dressing with topological manipulations $G$). We may then ask if
these non-invertible symmetries are intrinsic or not. This is answered by
looking for global forms left invariant by the modular transformations. The
results are shown for the first few primes.

#####  $GL(2,3)$:

We next consider the case of $GL(2,3)$. This group is generated by matrices
$M_{7}$ and $M_{8}$, which are of respective order 2 and 8. The explicit form
of the matrices is given in Appendix A, which allows us to compute the
relevant characteristic equations

$\displaystyle M_{7}:\hskip
14.45377pt\lambda^{4}-2\lambda^{2}+1=0\,\,\,\mathrm{mod}\,\,p\leavevmode\nobreak\
,\hskip 36.135ptM_{8}:\hskip
14.45377pt\lambda^{4}+1=0\,\,\,\mathrm{mod}\,\,p\leavevmode\nobreak\ .$ (3.30)

These admit the following number of solutions in ${\mathbb{Z}}_{p}^{\times}$,

$\displaystyle M_{7}:\hskip 14.45377pt\\#\,\,\mathrm{solutions}$
$\displaystyle=$ $\displaystyle\left\\{\begin{matrix}1&&p=2\\\
2&&\mathrm{otherwise}\end{matrix}\right.$ $\displaystyle M_{8}:\hskip
14.45377pt\\#\,\,\mathrm{solutions}$ $\displaystyle=$
$\displaystyle\left\\{\begin{matrix}1&&p=2\\\ 4&&p\in 8{\mathbb{N}}+1\\\
0&&\mathrm{otherwise}\end{matrix}\right.$

We begin by analyzing $M_{7}$. Starting with $p=2$, the characterstic equation
simplifies to $(\lambda+1)^{4}=0$ mod $2$ and the only solution is
$\lambda=1$. The possibilities for $J$ are $J=\left(\begin{smallmatrix}1&0\\\
0&1\end{smallmatrix}\right),\,\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)$; in the first case we have
$\widetilde{K}_{L}=\left(\begin{smallmatrix}1&1&1&1\\\
0&0&1&1\end{smallmatrix}\right)^{T}$, while in the second case we can have
either $\widetilde{K}_{L}=\left(\begin{smallmatrix}1&1&1&1\\\
0&1&0&1\end{smallmatrix}\right)^{T},\,\left(\begin{smallmatrix}1&1&0&0\\\
0&1&0&0\end{smallmatrix}\right)^{T}$. It is straightforward to check that
$\widetilde{K}_{L}^{T}\mathfrak{I}\widetilde{K}_{L}=0$ mod $2$ in all cases,
and hence all correspond to legitimate global forms. Thus for $p=2$ there are
three global forms with an invertible $M_{7}$ symmetry.

Moving on to more general $p$, there are two solutions $\lambda=\pm 1$ mod $p$
in ${\mathbb{Z}}_{p}^{\times}$. The product of the remaining two solutions
does not give an element in ${\mathbb{Z}}_{p}^{\times}$, so we need not
consider them. Since in general $p\nmid 2$, there can be no non-trivial Jordan
blocks, and the only possible $J$ are

$\displaystyle
J=\mathrm{diag}(1,1),\,\mathrm{diag}(1,-1),\,\mathrm{diag}(-1,1),\,\mathrm{diag}(-1,-1)\leavevmode\nobreak\
.$ (3.31)

The corresponding $\widetilde{K}_{L}$ take the form

$\displaystyle\widetilde{K}_{L}=\left(\begin{matrix}a&b\\\ a&b\\\ c&d\\\
c&d\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
36.135pta,b,c,d\in{\mathbb{Z}}_{p}\leavevmode\nobreak\ .$ (3.32)

Since the eigenvalues are not modular inverses, all such $\widetilde{K}_{L}$
are isotropic and label legitimate global forms. In total there are
$(p+1)^{2}$ distinct such $\widetilde{K}_{L}$, and hence for generic $p$ there
are a total of $(p+1)^{2}$ global forms with an invertible $M_{7}$ symmetry.

We next turn to $M_{8}$. The exercise here proceeds in much the same way as
before, and one begins by showing that there is a single global form with
invariant $M_{8}$ symmetry for $p=2$ and four global forms with invariant
$M_{8}$ symmetry for $p\in 8{\mathbb{N}}+1$. However, we now encounter a
feature that we have not seen in previous examples. In particular, though
there do not exist solutions to the characteristic equation in
${\mathbb{Z}}_{p}^{\times}$ for $p\not\in 8{\mathbb{N}}+1$, products of two of
them can give elements of ${\mathbb{Z}}_{p}^{\times}$, and we may use these
conjugate pairs to construct $J$. Indeed, over ${\mathbb{C}}$ the solutions to
the characteristic polynomial are simply $\lambda_{n}=e^{2\pi i(2n-1)\over 8}$
for $n=1,\dots,4$, and we see in particular that
$\lambda_{1}\lambda_{4}=\lambda_{2}\lambda_{3}=1$, which is an element in
${\mathbb{Z}}_{p}^{\times}$. We must thus allow for the following matrices

$\displaystyle J=\mathrm{diag}(e^{2\pi i\over 8},\,e^{14\pi i\over
8})\leavevmode\nobreak\ ,\,\,\mathrm{diag}(e^{6\pi i\over 8},\,e^{10\pi i\over
8})\leavevmode\nobreak\ .$ (3.33)

Focusing on the first case for instance, we find that the corresponding
$\widetilde{K}_{L}$ is of the form

$\displaystyle\widetilde{K}_{L}=\left(\begin{matrix}1&1\\\
1+\sqrt{2}&1+\sqrt{2}\\\ -{1\over\sqrt{2}}(1-i)&-{1\over\sqrt{2}}(1+i)\\\
{1\over\sqrt{2}}(1+i(1+\sqrt{2}))&{1\over\sqrt{2}}(1-i(1+\sqrt{2}))\end{matrix}\right)\leavevmode\nobreak\
,$ (3.34)

whose columns satisfy
$\widetilde{v}_{1}^{T}\mathfrak{I}\widetilde{v}_{2}=4i(1+\sqrt{2})$. We would
like this quantity to be zero mod $p$ so that the columns span an isotropic
sublattice, but before that we must first make sense of what $i$ and
$i\sqrt{2}$ even mean in ${\mathbb{Z}}_{p}$. There are various inequivalent
ways to make sense of these quantities: for example, we may demand the
existence of elements $x$ and $y$ in ${\mathbb{Z}}_{p}$ satisfying either

$\displaystyle x^{2}=-1\,\,\,\mathrm{mod}\,\,p\leavevmode\nobreak\ ,\hskip
36.135pty^{2}=2\,\,\,\mathrm{mod}\,\,p\leavevmode\nobreak\ ,$ (3.35)

or

$\displaystyle x^{2}=-1\,\,\,\mathrm{mod}\,\,p\leavevmode\nobreak\ ,\hskip
36.135pty^{2}=-2\,\,\,\mathrm{mod}\,\,p\leavevmode\nobreak\ .$ (3.36)

Considering all of the possibilities and demanding that
$\widetilde{v}_{1}^{T}\mathfrak{I}\widetilde{v}_{2}=0$ mod $p$, we find that
there are no invariant global forms if and only if $p\in 8{\mathbb{N}}+7$.
These results are collected in Table 5.

Because the full $GL(2,3)$ symmetry is generated by both $M_{7}$ and $M_{8}$,
we can now ask a somewhat more refined question: are there global forms which
are left invariant under the full $GL(2,3)$, i.e. under both $M_{7}$ and
$M_{8}$? We will call the full non-Abelian symmetry $GL(2,3)$ intrinsic if
there is no global form such that both $M_{7}$ and $M_{8}$ are invertible,
even if there are global forms such that one or the other is individually
invertible. In order to understand, for a given $p$, whether $GL(2,3)$ is
intrinsic or not, we must know more than just the characteristic equation.
Here we turn to a computerized search, which gives the results in Table 6. We
see that for e.g. $p=2,3$ there exists a global variant such that the full
$GL(2,3)$ becomes invertible,171717More concretely, for $p=2$ there is one
global variant fixed under both $M_{7}$ and $M_{8}$,  $\displaystyle K_{{\cal
L}}=\left(\begin{matrix}1&1&0&1\\\
0&0&1&1\end{matrix}\right)^{T}\leavevmode\nobreak\ ,$ (3.37) while for $p=3$
there are two global forms fixed by both,  $\displaystyle K_{{\cal
L}}=\left(\begin{matrix}1&0&0&1\\\
0&1&1&0\end{matrix}\right)^{T}\leavevmode\nobreak\ ,\hskip 36.135ptK_{{\cal
L}}=\left(\begin{matrix}1&0&1&1\\\
0&1&1&1\end{matrix}\right)^{T}\leavevmode\nobreak\ .$ (3.38) whereas for
$p=5,7$ no such global form exists.

$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29
---|---|---|---|---|---|---|---|---|---|---
${GL(2,3)}$ intrinsic? | ✗ | ✗ | ✓ | ✓ | ✗ | ✓ | ✗ | ✗ | ✓ | ✓
$({\mathbb{Z}}_{2}\times{\mathbb{Z}}_{6})\rtimes{\mathbb{Z}}_{2}$ intrinsic? | ✗ | ✗ | ✓ | ✗ | ✓ | ✗ | ✓ | ✗ | ✓ | ✓
$D_{8}$ intrinsic? | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
$D_{12}$ intrinsic? | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ intrinsic? | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
Table 6: We call a rank $r>1$ group intrinsic if there is no global form such
that all generators are simultaneously invertible. We do allow some of the
generators to become invertible, though.

#####  $({\mathbb{Z}}_{2}\times{\mathbb{Z}}_{6})\rtimes{\mathbb{Z}}_{2}$:

We finally mention the case of
$({\mathbb{Z}}_{2}\times{\mathbb{Z}}_{6})\rtimes{\mathbb{Z}}_{2}$, which is
generated by the matrices $M_{1},M_{2},$ and $M_{3}$. The characteristic
equations of $M_{1}$ and $M_{2}$ are identical, and the characteristic
equation for $M_{3}$ coincides with that for $M_{7}$ above,

$\displaystyle M_{1},M_{2}:\hskip
36.135pt\lambda^{4}+2\lambda^{2}+1=0\,\,\,\mathrm{mod}\,\,p\leavevmode\nobreak\
,$ (3.39) $\displaystyle M_{3}:\hskip
36.135pt\lambda^{4}-2\lambda^{2}+1=0\,\,\,\mathrm{mod}\,\,p\leavevmode\nobreak\
.$

The number of invariant global forms are to be the following

$\displaystyle M_{1},M_{2}:\hskip
14.45377pt\\#\,\,\mathrm{invariant\,\,forms}$ $\displaystyle=$
$\displaystyle\left\\{\begin{matrix}p+3&&p\in 4{\mathbb{N}}+1\\\
p+1&&\mathrm{otherwise}\end{matrix}\right.$ (3.40) $\displaystyle M_{3}:\hskip
36.135pt\\#\,\,\mathrm{invariant\,\,forms}$ $\displaystyle=$
$\displaystyle\left\\{\begin{matrix}3&&p=2\\\
(p+1)^{2}&&\mathrm{other}\,\,p\end{matrix}\right.$

By an analysis similar to the previous ones, one finds that for all primes $p$
the symmetries $M_{1}$, $M_{2}$, and $M_{3}$ are non-intrinsic. However, the
full group $({\mathbb{Z}}_{2}\times{\mathbb{Z}}_{6})\rtimes{\mathbb{Z}}_{2}$
can still be intrinsic: that is, there exist some primes for which there does
not exist a global form simultaneously invariant under all three of $M_{1}$,
$M_{2}$, and $M_{3}$. This is summarized in Table 6.

##### Invertible subspaces, extended loci cases.

Having exhaustively analyzed the isolated fixed loci, we now turn to the
extended loci. Since the analysis is similar to the one above, we will be
brief.

#####  $D_{8}$:

We begin with the case of $D_{8}$ generated by the matrices $M_{7}$ and
$N_{2}$, given in Appendix A. The generator $M_{7}$ was analyzed previously in
the context of the $GL(2,3)$ symmetric point, with the conclusion that it is
never intrinsic. As for $N_{2}$, the characteristic polynomial is given by

$N_{2}:\hskip
36.135pt\lambda^{4}+2\lambda^{2}+1=0\,\,\,\mathrm{mod}\,\,p\leavevmode\nobreak\
,$ (3.41)

which is the same as for $M_{1}$ and $M_{2}$.

The number of invariant global forms is found to be

$N_{2}:\hskip
14.45377pt\\#\,\,\mathrm{invariant\,\,forms}=\left\\{\begin{matrix}p+3&&p\in
4{\mathbb{N}}+1\\\ p+1&&\mathrm{otherwise}\end{matrix}\right.$ (3.42)

This is obtained roughly as follows. For $p\in 4{\mathbb{N}}+1$ the element
$-1$ is a quadratic residue, meaning that there is an element
$a\in\mathbb{Z}_{p}$ such that $a^{2}=-1$. The matrix $N_{2}$ then has two
distinct eigenvalues $\lambda_{1,2}=\pm a$. Let the $a$ eigenspace be spanned
by $v_{1},\,v_{2}$ and the $-a$ eigenspace be spanned by $u_{1},\,u_{2}$. It
is guaranteed that $v_{2}^{T}\mathfrak{I}v_{1}=u_{2}^{T}\mathfrak{I}u_{1}=0$,
and furthermore that the bases can be chosen to satisfy
$v_{1}^{T}\mathfrak{I}u_{2}=v_{2}^{T}\mathfrak{I}u_{1}=0$. We then have the
following isotropic lattices: $\langle v_{1},v_{2}\rangle$, $\langle
u_{1},u_{2}\rangle$, $\langle v_{1},u_{2}\rangle$, $\langle
v_{2},u_{1}\rangle$ and $\langle v_{1}+kv_{2},u_{1}+su_{2}\rangle$, where
$k\in\mathbb{Z}_{p}^{\times}$ and
$s=-k^{-1}(v_{2}^{T}\mathfrak{J}u_{2})/(v_{1}^{T}\mathfrak{J}u_{1})$. There
are thus a total of $4+(p-1)=p+3$ invariant global forms. A similar analysis
can be performed for $p\not\in 4{\mathbb{N}}+1$.

We can also ask if there are isotropic sublattices invariant under both
$M_{7}$ and $N_{2}$. The simplest way to check this is to start with the
sublattices invariant under $M_{7}$ described around (3.32), and then to check
which (if any) of them are invariant under $N_{2}$. A simple computation
demonstrates that for $p>2$, out of the $(p+1)^{2}$ global forms invariant
under $M_{7}$, there are $p-2$ global forms invariant under the whole $D_{8}$,
while for $p=2$ there are two such forms. Thus the full $D_{8}$ is never
intrinsic.

#####  $D_{12}$:

We next consider the case of $D_{12}$, generated by $N_{5}$ and $N_{6}$. Both
generators have the same characteristic polynomial:

$N_{5},\,N_{6}:\hskip
36.135pt\lambda^{2}-2\lambda^{2}+1=0\,\,\,\mathrm{mod}\,\,p\leavevmode\nobreak\
,$ (3.43)

which has one root for $p=2$ ($\lambda=1$) and two different roots for prime
$p>2$ ($\lambda=\pm 1$). The number of invariant isotropic sublattices (which
is also the same for both $N_{5}$ and $N_{6}$) is

$N_{5},\,N_{6}:\hskip
14.45377pt\\#\,\,\mathrm{invariant\,\,forms}=\left\\{\begin{matrix}3&&p=2\\\
(p+1)^{2}&&\mathrm{otherwise}\end{matrix}\right.$ (3.44)

We can again ask if there are global forms invariant with respect to $N_{5}$
and $N_{6}$ simultaneously. We find the following

$\\#\,\,\mathrm{simult.\,\,invariant\,\,forms}=\left\\{\begin{matrix}2&&p=2\\\
7&&p=3\\\ p+1&&\mathrm{otherwise}\end{matrix}\right.$ (3.45)

from which we conclude that $D_{12}$ is never intrinsically non-invertible.

#####  $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$:

Finally we consider the case of the ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$
symmetry spanned by $\\{C,M_{7}\\}$. The analysis in this case is particularly
simple: we have already seen that $M_{7}$ fail to be intrinsic for every value
of $p$, while the generator $C$ is obviously never intrinsic, since it leaves
invariant every isotropic sublattice. The combined symmetry is also obviously
non-intrinsic.

##  4 Intrinsic versus non-intrinsic at higher genus

In the previous section we understood in detail the spectrum of
(non-)invertible symmetries for class ${\cal S}$ theories of genus $2$. We now
turn to higher genus theories. The special points in moduli space have been
classified up to genus 5 [121, 122], and many examples are known at higher
genera. Our discussion here will not be as exhaustive as before, and will
instead focus on cyclic subgroups of enhanced symmetries.

In the references [121, 122], the stabilizer subgroups at the special points
of moduli space are realized as subgroups of $GL(g,\mathbb{C})$ acting on
holomorphic one-forms. Given such a matrix $U=A+iB$ with $A$ and $B$ real, we
can construct a $2g\times 2g$ matrix

$\displaystyle M=\begin{pmatrix}A&-B\\\ B&A\end{pmatrix}\leavevmode\nobreak\
.$ (4.1)

For stabilizer subgroups this matrix is real symplectic, and moreover is
similar to an integer symplectic matrix $F$. It is this integer matrix which
we will analyze.

We warn the reader that the content of this section is rather technical, and
the arithmetically-averse reader may wish to skip directly to Section 5. For
such readers, let us simply note here that the content of this section
translates to an answer to the following physical question. Given a
$({\mathbb{Z}}_{p}^{(2)})^{\otimes 2g}$ gauge theory in $(4+1)$d, there is an
$Sp(2g,{\mathbb{Z}}_{p})$ zero-form symmetry acting via outer automorphisms,
and we may gauge certain subgroups of this symmetry. In general, the result
will not be another gauge theory, even for a higher-group symmetry. The
results of this section tell us, for $g\leq 5$, when the result is a gauge
theory: namely, this happens whenever there is a non-zero entry in Tables 7,
8, or 9.

###  4.1 Characteristic polynomials of modular matrices

We start by introducing some relevant machinery. Consider elements $F\in
Sp(2g,{\mathbb{Z}})$ leaving some point in moduli space fixed. Such elements
are always of finite order. This implies that as complex matrices the $F$ are
diagonalizable, and moreover that their eigenvalues are $k$-th roots of unity.
These two facts together mean that the characteristic polynomials $P_{F}(x)$
are products of cyclotomic polynomials $\phi_{i}(x)$ with total degree $2g$;
see Appendices B.1 and B.2 for a detailed discussion. Since cyclotomic
polynomials of a given degree are limited, there are relatively few
characteristic polynomials at a given genus.

As an example, consider the case of genus 1, for which we are interested in
products of cyclotomic polynomials of total degree 2. There are two possible
reducible characteristic polynomials:

$\displaystyle\phi_{1}(x)^{2}=(x-1)^{2}=P_{\mathds{1}}(x)$
$\displaystyle\phi_{2}(x)^{2}=(x+1)^{2}=P_{-\mathds{1}}(x)\leavevmode\nobreak\
,$ (4.2)

which are clearly the characteristic polynomials of $\mathds{1}$ and
$-\mathds{1}$. There are also two possible irreducible characteristic
polynomials,181818 Note that we could have also considered $\phi_{3}(-x)$, but
in general for a $2g\times 2g$ diagonalizable matrix $M$ we have
$P_{-M}(x)=P_{M}(-x)$, and hence when considering the action of stabilizer
subgroups the matrices with characteristic polynomials $P(x)$ and $P(-x)$ are
identified in ${Sp}(2g,\mathbb{Z})/\mathbb{Z}_{2}$.

$\displaystyle\phi_{3}(x)=x^{2}+x+1=P_{{\mathsf{S}}{\mathsf{T}}}(x)\leavevmode\nobreak\
,$ $\displaystyle\phi_{4}(x)=x^{2}+1=P_{{\mathsf{S}}}(x)\leavevmode\nobreak\
,$ (4.3)

which we recognize as the characteristic polynomials of the
${\mathsf{S}}{\mathsf{T}}$ and ${\mathsf{S}}$ elements of
$SL(2,{\mathbb{Z}})$.

Of special interest to us are the cases in which $P_{F}(x)$ is a single
cyclotomic polynomial $\phi_{n}(x)$, as opposed to a product thereof. We have
just seen that $F={\mathsf{S}}$ and ${\mathsf{S}}{\mathsf{T}}$ are of this
type, for example. Such matrices $F$ have order $n$, and as will be explained
later serve as the building blocks for all other cases. Note that the degree
of the $n$-th cyclotomic polynomial $\phi_{n}(x)$ is Euler’s totient function
$\varphi(n)$, i.e the number of integers less than $n$ which are coprime to
$n$. However, not all $\phi_{n}(x)$ such that $\varphi(n)=2g$ need to be
considered: indeed, a matrix with characteristic polynomial $\phi_{n}(x)$
generates a cyclic subgroup of order $n$, and since the maximum order of
cyclic isometries is known to be $4g+2$ by the “$(4g+2)$ theorem” [127], only
$\phi_{n}(x)$ with $n\leq 4g+2$ are relevant.

####  4.1.1 Cyclotomic polynomials over finite fields

Over integers, the cyclotomic polynomials $\phi_{n}(x)$ are by definition
irreducible, i.e. they do not factorize. On the other hand, over complex
numbers they factorize completely, with the roots being primitive $n$-th roots
of unity $\exp(2\pi i\frac{m}{n})$ with $m$ and $n$ coprime. The situation
over finite fields is somewhat intermediate, in that $\phi_{n}(x)$ can
factorize partially, but not always completely. We are ultimately interested
in prime fields $\mathbb{F}_{p}$, but it is convenient to start with an
appropriate extension of $\mathbb{F}_{p}$ in which $\phi_{n}(x)$ does split
completely. Over this field $\phi_{n}(x)$ has $\varphi(n)$ roots. These roots
are not necessarily distinct: if $p^{k}$ is the highest power of $p$ that
divides $n$, the number of distinct roots is
$\varphi\left(\frac{n}{p^{k}}\right)$,191919Another way of saying this is that
the group of $n$-th roots of unity in the splitting field of $\phi_{n}(x)$
over $\mathbb{F}_{p}$ has order $\varphi\left(\frac{n}{p^{k}}\right)$. and
each of these roots occurs with multiplicity $\varphi(p^{k})$, i.e. we have

$\displaystyle\phi_{n}(x)$
$\displaystyle=\left(\phi_{\frac{n}{p^{k}}}(x)\right)^{\varphi(p^{k})}\leavevmode\nobreak\
;$ (4.4)

see Appendix B.1 for more details. As we will see below, this fact essentially
fixes the Jordan normal form of all $F$ of interest.

With the above factorization in hand, we now restrict to $\phi_{n}(x)$ with
$\gcd(n,p)=1$ and consider how $\phi_{n}(x)$ can factorize over
$\mathbb{F}_{p}$. The simplest case is when $n$ divides $p-1$. In this case
$\mathbb{F}_{p}$ contains a multiplicative element of order $n$, which we call
$\omega$, and $\phi_{n}(x)$ splits completely,

$\displaystyle\phi_{n}(x)=\prod_{\begin{subarray}{c}m<n\\\
\gcd(m,n)=1\end{subarray}}(x-\omega^{m})\leavevmode\nobreak\ .$ (4.5)

This is not the only possible factorization that can happen. Indeed, since
$\gcd(n,p)=1$, the element $p$ is an invertible element in $\mathbb{Z}_{n}$.
If $m$ is the order of $p$ in $\mathbb{Z}_{n}$, then $\phi_{n}(x)$ can also
split into $\frac{\varphi(n)}{m}$ factors over $\mathbb{F}_{p}$, all of which
have the same degree $m$.

###  4.2 Invariant subspaces of modular matrices

We have seen how the characteristic polynomials of $F$ are related to
cyclotomic polynomials. We now explain how to use this fact to identify
invariant global forms, i.e. maximal isotropic sublattices invariant under
$F$. To begin, let us temporarily forget the constraints of maximality and
isotropicity and just identify invariant subpaces in general. It is again
convenient to start by considering $F$ such that the characteristic polynomial
(over integers) of $P_{F}(x)$ is a single cyclotomic polynomial $\phi_{n}(x)$.

Let us first consider the generic case, in which we are working over a finite
field $\mathbb{F}_{p}$ such that $p$ is not a factor of $n$. As explained
above, in this case the eigenvalues of $F$ in a suitable extension of
$\mathbb{F}_{p}$ are all pairwise distinct and given by the roots of
$\phi_{n}(x)$ as in (4.5). As a result, over this field the homology of
$\Sigma_{g,n}$ splits into eigenspaces, each of which is one-dimensional, and
any invariant subspace can be written as a sum of these eigenspaces. From this
fact we can derive a sufficient condition and a necessary condition for the
existence of non-trivial invariant subspaces.

First for the sufficient condition, when $p-1\in n\mathbb{N}$ all the roots of
$\phi_{n}(x)$ are in $\mathbb{F}_{p}$ itself, and hence all the eigenspaces
described above give a decomposition of $H_{1}(\Sigma_{g,0},\mathbb{F}_{p})$.
There are $\binom{2g}{d}$ invariant subspaces of dimension $d$.

The necessary condition comes from the fact that all invariant subspaces are
annihilated by some factor of the characteristic polynomial. All factors of
$\phi_{n}(x)$ over $\mathbb{F}_{p}$ have degree $m$ where $m$ is the
multiplicative order of $p$ in $\mathbb{Z}_{n}$. Hence invariant subspaces of
dimension $d$ exist only if $d$ is a multiple of $m$, and in that case there
are $\binom{\varphi(n)/m}{d/m}$ such subspaces.

####  4.2.1 Invariant isotropic subspaces

The considerations above allow us to understand which subspaces are invariant
under the action of an enhanced symmetry $F$. But to identify invariant global
forms we need to identify which of those subspaces have dimension $g$ and are
isotropic. Naively there is no reason to expect that the Jordan decomposition
of a matrix $F$ can tell us which of its invariant subspaces are isotropic,
since all similar matrices have the same Jordan normal form and there is no
reason to expect that similarity transformations preserve isotropy. However,
the fact that $F$ preserves the intersection form, together with the
restrictions we have derived using the finite order of $F$, actually suffices
to fix which of the invariant subspaces are isotropic.

To see this, we begin not in $\mathbb{F}_{p}$ but in the field extension which
splits the characteristic polynomial $P_{F}(x)=\phi_{n}(x)$. We now consider
two cases:

First, we consider the generic case in which $p$ is not a factor of $n$ and we
take $n>2$ since $n=1,2$ are trivial. Then all of the eigenvalues of
$\phi_{n}(x)$ are distinct. We saw around (3.18) that two eigenvectors $v_{1}$
and $v_{2}$ with eigenvalues $\lambda_{1}$ and $\lambda_{2}$ can intersect
only if $\lambda_{1}\lambda_{2}=1$ mod $p$, i.e. the two eigenvalues are
modular inverses of each other. From (4.5), it can be seen that if $\lambda$
is an eigenvalue then $\lambda^{-1}$ is also an eigenvalue. Since the
eigenvector $v$ with eigenvalue $\lambda$ does not intersect the complement of
eigenvector $v^{\prime}$ with eigenvalue $\lambda^{-1}$, the eigenvectors $v$
and $v^{\prime}$ must have non-zero intersection, since the intersection form
$\mathfrak{I}$ of $\Sigma_{g,0}$ is non-degenerate. As a result, if an
invariant global form is to exist the polynomial $\phi_{n}(x)$ must factorize
over $\mathbb{F}_{p}$ as

$\displaystyle\phi_{n}(x)=P(x)Q(x)\leavevmode\nobreak\ ,$ (4.6)