where both $P(x)$ and $Q(x)$ have degree $\frac{1}{2}\varphi(n)$. Over the
splitting field of $\phi_{n}(x)$, both $P(x)$ and $Q(x)$ split as well, and
isotropy requires that if $\lambda$ is a root of $P(x)$, then $\lambda^{-1}$
is a root of $Q(x)$. As a result, there are far fewer invariant isotropic
subspaces than invariant subspaces in general. For example, in the case when
$p\in n\mathbb{N}+1$ where $\phi_{n}(x)$ splits over $\mathbb{F}_{p}$, there
are $\binom{2g}{g}$ invariant subspaces of dimension $g$, but only $2^{g}$ of
them are isotropic.

Next, if $p$ is a factor of $n$, then for each eigenvalue $\lambda$ there is a
single Jordan block of size $\varphi(p^{k})$, where $p^{k}$ is the highest
power of $p$ dividing $n$. The eigenvalues themselves are roots of
$\phi_{\frac{n}{p^{k}}}(x)$. This Jordan decomposition means that for each
eigenvalue $\lambda$, we can find $\varphi(p^{k})$ generalized eigenvectors
$v^{(a)}_{\lambda}$ such that,

$\displaystyle Dv^{(0)}_{\lambda}$ $\displaystyle=\lambda
v^{(0)}_{\lambda}\leavevmode\nobreak\ ,$ $\displaystyle Dv^{(a)}_{\lambda}$
$\displaystyle=\lambda
v^{(a)}_{\lambda}+v^{(a-1)}_{\lambda}\leavevmode\nobreak\ .$ (4.7)

It can be shown that
$\expectationvalue{v^{(a)}_{\lambda_{1}},v^{(b)}_{\lambda_{2}}}$ is zero
unless $\lambda_{1}\lambda_{2}=1$ mod $p$ and $a+b\geq\varphi(p^{k})$; see
Appendix B.3. Unpacking this a bit, if $n=p^{k}$ or $n=2p^{k}$, then we obtain
a single Jordan block of size $\varphi(n)$ and there is a single invariant
isotropic subspace of dimension $d$ if $d\leq\frac{\varphi(n)}{2}=g$. This
subspace is the span of $v^{(0)}_{\pm 1},v^{(1)}_{\pm 1},\dots,v^{(d-1)}_{\pm
1}$ where $1$ is the eigenvalue when $n=p^{k}$ and $-1$ is the eigenvalue when
$n=2p^{k}$. There are no invariant isotropic subspaces of dimension greater
than $g=\frac{\varphi(n)}{2}$.

If $\frac{n}{p^{k}}>2$, each Jordan block for a given eigenvalue is isotropic
and the vector $v_{\lambda}^{(a)}$ can intersect only with
$v_{\lambda^{-1}}^{(b)}$ if $a+b\geq g$.

###  4.3 Invariant global forms

After the rather technical discussion above, we now determine the global forms
left invariant by matrices $F$ generating isometries of Riemann surfaces. In
this subsection we discuss all possible examples at genus 3, 4, 5 such that
the characteristic polynomial is a cyclotomic polynomial. See Appendix B.5 for
a discussion of the more general case in which the characteristic polynomial
is a product of cyclotomic polynomials.

####  4.3.1 Genus 3

We start with the case of genus 3, for which we are interested in cyclotomic
polynomials of degree $2g=6$. There are 4 of them, namely
$\phi_{7}(x),\phi_{9}(x),\phi_{14}(x),$ and $\phi_{18}(x)$. Since $18$ is
larger than the upper limit $4g+2=14$ for the order of cyclic subgroups of
$Sp(2g,{\mathbb{Z}})$, no $F$ can have $\phi_{18}(x)$ as its characteristic
polynomial. On the other hand, if a matrix $F$ has characteristic polynomial
$\phi_{14}(x)$, then the matrix $-F$ has characteristic polynomial
$\phi_{7}(x)$, and the invariant subspaces of $F$ and $-F$ are the same. Hence
we may focus only on the cases of $\phi_{7}(x)$ and $\phi_{9}(x)$.

#####  $\phi_{7}(x)$ :

According to [128], there are two fixed points at genus $3$ for which the
stabilizer subgroup has order $7$. Defining the matrices

$\displaystyle A=\left(\begin{matrix}0&1&0\\\ 1&0&1\\\
0&1&-1\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
21.68121ptB=\left(\begin{matrix}{\alpha+\alpha^{2}\over
1+3\alpha}&{\beta+\beta^{2}\over 1+3\beta}&{\gamma+\gamma^{2}\over
1+3\gamma}\\\ {1+2\alpha\over 1+3\alpha}&{1+2\beta\over
1+3\beta}&{1+2\gamma\over 1+3\gamma}\\\ {\alpha^{2}\over
1+3\alpha}&{\beta^{2}\over 1+3\beta}&{\gamma^{2}\over
1+3\gamma}\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
21.68121ptC=\mathrm{diag}\left(1,1,-1\right)$ (4.8)

for $\alpha=2\cos{2\pi\over 7}$, $\beta=2\cos{4\pi\over 7}$, and
$\gamma=2\cos{6\pi\over 7}$, the two isolated fixed points are given by

$\displaystyle\Omega_{7,1}$ $\displaystyle:=$ $\displaystyle-{1\over
2}A+iB\left(\mathds{1}-{1\over 4}B^{T}A^{2}B\right)^{1\over
2}B^{T}\leavevmode\nobreak\ ,$ $\displaystyle\Omega_{7,2}$ $\displaystyle:=$
$\displaystyle-{1\over 2}A(A+\mathds{1})+iB\left[\left(\mathds{1}-{1\over
4}B^{T}A^{2}B\right)^{1\over 2}B^{T}(A+\mathds{1})BC\right]B^{T}$ (4.9)

All of the non-identity elements for the symmetry groups at these points have
$\phi_{7}(x)$ as their characteristic polynomials.

We begin by considering the generic case for which $p$ is not a factor of 7.
This means that we consider $p\in 7{\mathbb{N}}+k$ for $k=1,\dots,6$.
Beginning with $p\in 7\mathbb{N}+1$, by the sufficient condition at the end of
Section 4.2 we see that $\mathbb{F}_{p}$ is the splitting field of
$\phi_{7}(x)$ and hence that 8 invariant subspaces exist. The non-invertible
symmetries are non-intrinsic.

Moving on to $p\in 7\mathbb{N}+2$ or $7\mathbb{N}+4$, we note that the
elements $2$ and $4$ have degree $3$ in $\mathbb{Z}^{\times}_{7}$, so the
polynomial $\phi_{7}(x)$ splits into two factors of degree $3$, c.f. (4.4).
The only possible factorization that can make the corresponding kernels
isotropic is $\phi_{7}(x)=P(x)Q(x)$ with

$\displaystyle
P(x)=(x-\lambda)(x-\lambda^{2})(x-\lambda^{4})\leavevmode\nobreak\ ,$
$\displaystyle
Q(x)=(x-\lambda^{3})(x-\lambda^{5})(x-\lambda^{6})\leavevmode\nobreak\ .$
(4.10)

Here $\lambda$ is a primitive seventh root of unity that exists in the
splitting field. We want to determine if $P(x)$ and $Q(x)$ are polynomials
over $\mathbb{F}_{p}$. This is the case if
$\lambda+\lambda^{2}+\lambda^{4}\in\mathbb{F}_{p}$, which can be shown to
happen if there exists a square root of $-7$ mod $p$. This is indeed the case
for $p\in 7\mathbb{N}+2$ or $7\mathbb{N}+4$. Hence in these cases there are
two invariant global forms given by $\ker(P(F))$ and $\ker(Q(F))$.

In the cases of $p\in 7{\mathbb{N}}+3$ and $7{\mathbb{N}}+5$ we note that the
elements $3$ and $5$ in $\mathbb{Z}_{7}$ both generate
$\mathbb{Z}^{\times}_{7}$, and hence there are no invariant global forms and
the symmetry is intrinsically non-invertible. Finally for the case of $p\in
7{\mathbb{N}}+6$ we note that the element $6$ has order $2$ in
$\mathbb{Z}_{7}^{\times}$, so the polynomial $\phi_{7}(x)$ splits into three
factors of degree 2. In this case all invariant subspaces are even-
dimensional, and in particular not equal to $g=3$. Hence there is no maximal
invariant subspace, and hence no invariant global form.

Having discussed the generic case of $p$ not dividing 7, we now discuss the
exceptional case of $p=7$. This is part of the family discussed in [39], where
it was seen to have a single invariant global form. This can be recovered from
our analysis in Section 4.2.1, which shows that there is a single Jordan block
with eigenvalue $1$ in this case. The unique invariant global form is the span
of first three vectors in the Jordan block.

The results so far are summarized in the first line of Table 7.

$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29
---|---|---|---|---|---|---|---|---|---|---
$\phi_{7}(x)$ | 2 | 0 | 0 | 1 | 2 | 0 | 0 | 0 | 2 | 8
$\phi_{9}(x)$ | 0 | 1 | 0 | 2 | 0 | 2 | 0 | 8 | 0 | 0
Table 7: The number of invariant global forms for enhanced symmetries $F$ with
characteristic polynomials given by $\phi_{7}(x)$ and $\phi_{9}(x)$, relevant
for genus 3. The $F$ give rise to intrinsically non-invertible symmetries if
and only if the entry above is $0$.

#####  $\phi_{9}(x)$ :

The reference [128] also contains one fixed point with stabilizer subgroup of
order 9. Defining the matrices

$\displaystyle A$ $\displaystyle=$ $\displaystyle\left(\begin{matrix}0&1&0\\\
1&-1&1\\\ 0&1&1\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
7.22743ptB={1\over
3}\left(\begin{matrix}-3+\alpha+\alpha^{2}&-3+\beta+\beta^{2}&-3+\gamma+\gamma^{2}\\\
-1+\alpha^{2}&-1+\beta^{2}&-1+\gamma^{2}\\\
1+\alpha&1+\beta&1+\gamma\end{matrix}\right)$ (4.11)

for $\alpha=2\cos{2\pi\over 9}$, $\beta+2\cos{4\pi\over 9}$, and
$\gamma=2\cos{8\pi\over 9}$, the isolated fixed point occurs at the following
value of the period matrix,

$\displaystyle\Omega_{9}:=-{1\over 2}A+iB\left(\mathds{1}-{1\over
4}B^{T}A^{2}B\right)^{1\over 2}B^{T}\leavevmode\nobreak\ .$ (4.12)

Any generator of the subgroup preserving this value has $\phi_{9}(x)$ as its
characteristic polynomial.

We begin with the generic case of $p$ not dividing 9. We would naively have to
consider the cases of $p\in 9{\mathbb{N}}+k$ for $k=1,\dots,8$, but since $p$
is prime we may skip the cases of $k=3,6$. Using the same arguments as for
$\phi_{7}(x)$ above, we find that the symmetry is intrinsically non-invertible
unless $p$ has order $3$ in $\mathbb{Z}_{9}^{\times}$. The latter occurs
whenever $p\in 9\mathbb{N}+1,9\mathbb{N}+4,$ or $9\mathbb{N}+7$, i.e. when
$p\in 3\mathbb{N}+1$. Let $\lambda$ be the primitive $9$-th root of unity in
the splitting field. For $p\in 3\mathbb{N}+1$, a primitive third root of unity
exists in $\mathbb{F}_{p}$ and must be $\lambda^{3}$. It can be shown that

$\displaystyle P(x)$ $\displaystyle=$
$\displaystyle(x-\lambda)(x-\lambda^{4})(x-\lambda^{7})=x^{3}-\lambda^{3}\leavevmode\nobreak\
,$ $\displaystyle Q(x)$ $\displaystyle=$
$\displaystyle(x-\lambda^{2})(x-\lambda^{5})(x-\lambda^{7})=x^{3}-\lambda^{6}\leavevmode\nobreak\
.$ (4.13)

Hence $P(x)$ and $Q(x)$ are both polynomials with coefficients in
$\mathbb{F}_{p}$, and they satisfy the criterion for the corresponding kernels
to be isotropic. If $p\in 9\mathbb{N}+4$ or $9\mathbb{N}+7$, then these are
the only two invariant global forms. If $p\in 9\mathbb{N}+1$, then
$\lambda\in\mathbb{F}_{p}$ and there are $8$ invariant global forms.

We finally discuss the exceptional case of $p=3$. In this case we obtain a
single Jordan block of size 6 and eigenvalue $1$, and hence there is a single
invariant global form consisting of the span of the first three vectors in the
Jordan block.

The results of the above discussion are given in the second line of Table 7.

####  4.3.2 Genus 4

For genus 4, we are interested in cyclotomic polynomials $\phi_{n}(x)$ with
degree $2g=8$ and $n\leq 4g+2=18$. There are again only two physically
distinct possibilities, namely $\phi_{15}(x)$ and $\phi_{16}(x)$. The
stabilizer subgroups at genus 4 were classified in [121] and contain cyclic
groups of order $15$ and $16$. The characteristic polynomial of any generator
$F$ of these groups was indeed found to be $\phi_{15}(x)$ and $\phi_{16}(x)$.

##### $\phi_{15}(x):$

We begin with the case of $\phi_{15}(x)$. One can again split the discussion
into two cases: $p$ not being a factor of 15, and $p$ being a factor of 15. We
will begin by analyzing the latter here, i.e. the cases of $p=3$ and $5$.

For $p=3$, the cyclotomic polynomial partially factorizes as

$\displaystyle\phi_{15}(x)=\phi_{5}(x)^{2}=(x^{4}+x^{3}+x^{2}+x+1)^{2}\leavevmode\nobreak\
.$ (4.14)

Since $\phi_{5}(x)$ does not split over $\mathbb{F}_{3}$, the only maximally
invariant subspace (i.e. invariant subspace of dimension 4) is
$\ker(\phi_{5}(F))$. This subspace is isotropic. This can be seen by passing
to the splitting field of $\phi_{5}(x)$ and using the discussion in Section
4.2.1, which tells us that the non-zero intersections are

$\displaystyle\expectationvalue{v_{\lambda}^{(0)},v_{\lambda^{4}}^{(1)}}$
$\displaystyle=\expectationvalue{v_{\lambda}^{(1)},v_{\lambda^{4}}^{(0)}}=\expectationvalue{v_{\lambda^{2}}^{(0)},v_{\lambda^{3}}^{(1)}}=\expectationvalue{v_{\lambda^{2}}^{(1)},v_{\lambda^{2}}^{(0)}}=1\leavevmode\nobreak\
$ (4.15)

where $\lambda$ is any the root of $\phi_{5}(x)$. On the other hand
$\ker{(\phi_{5}(M))}=\mbox{span}\\{v_{\lambda}^{(0)},v_{\lambda^{2}}^{(0)},v_{\lambda^{3}}^{(0)},v_{\lambda^{4}}^{(0)}\\}$
and is hence isotropic. Thus there is a single invariant global form.

Next we consider $p=5$, for which

$\displaystyle\phi_{15}(x)=(\phi_{3}(x))^{4}=(x^{2}+x+1)^{4}\leavevmode\nobreak\
.$ (4.16)

Once again, there is a single candidate for an invariant global form, namely
$\ker(\phi_{3}(F)^{2})$. It can be seen to be isotropic using the same
argument as for $p=3$.

The generic case when $p\neq 3,5$, requires some more involved algebra, which
we relegate to Appendix B.4. Here we only summarize the results:

  * •

The only case with intrinsic non-invertible symmetry is $p\in
15\mathbb{N}+14$.

  * •

If $p\in 15\mathbb{N}+2,15\mathbb{N}+7,15\mathbb{N}+8$ or $15\mathbb{N}+13$
there are two invariant global forms.

  * •

If $p\in 15\mathbb{N}+4$ or $15\mathbb{N}+11$ there are four invariant global
forms.

  * •

If $p\in 15\mathbb{N}+1$ there are eight invariant global forms.

$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29
---|---|---|---|---|---|---|---|---|---|---
$\phi_{15}(x)$ | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 2 | 0
$\phi_{16}(x)$ | 1 | 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0
Table 8: The number of invariant global forms for symmetry generators with
characteristic polynomials given by $\phi_{15}(x)$ and $\phi_{16}(x)$,
relevant for genus 4. There exist intrinsically non-invertible symmetries if
and only if the entry above is $0$.

#####  $\phi_{16}(x)$ :

We next consider the case of $\phi_{16}(x)$. The discussion again splits into
the generic case for which $p$ is not a factor of 16, and the exceptional case
when $p=2$. In the latter case there is a single Jordan block with eigenvalue
$1$, and hence a single invariant subspace spanned by
$v_{1}^{(0)},v_{1}^{(1)},v_{1}^{(2)}$, and $v_{1}^{(3)}$.

For $p>2$, the cyclotomic polynomial always factorizes to some degree. To
carry out the analysis, we start in the splitting field which has a primitive
sixteenth root of unity $\lambda$. In the splitting field we may write the
cyclotomic polynomial as

$\displaystyle\phi_{16}(x)=(x-\lambda)(x-\lambda^{3})(x-\lambda^{5})(x-\lambda^{7})(x-\lambda^{9})(x-\lambda^{11})(x-\lambda^{13})(x-\lambda^{15})\leavevmode\nobreak\
.$ (4.17)

When $p\in 16\mathbb{N}+1$, the root $\lambda$ is in $\mathbb{F}_{p}$ itself,
and there are 16 invariant global forms, c.f. the necessary condition in
Section 4.2.

When $p\in 16\mathbb{N}+9$, there exists a primitive eighth root of unity
$\sigma$ in $\mathbb{F}_{p}$ with $\sigma=\lambda^{2}$ and $\sigma^{4}=-1$.
Hence the product of two factors $(x-\lambda^{2i+1})(x-\lambda^{2j+1})$ is a
polynomial over $\mathbb{F}_{p}$ if

$\displaystyle\lambda^{2i+1}+\lambda^{2j+1}\in\mathbb{F}_{p}\hskip
36.135pt\Rightarrow\hskip
36.135pt\lambda(\sigma^{i}+\sigma^{j})\in\mathbb{F}_{p}\leavevmode\nobreak\ .$
(4.18)

Since $\lambda\notin\mathbb{F}_{p}$, this happens whenever
$\sigma^{i}=-\sigma^{j}$, which in turn happens whenever $i=j+4$. Hence

$\displaystyle\phi_{16}(x)=(x^{2}-\sigma)(x^{2}+\sigma)(x^{2}-\sigma^{3})(x^{2}+\sigma^{3})\leavevmode\nobreak\
$ (4.19)

and there are $4$ basic invariant subspaces of dimension 2. All of them are
isotropic.

When $p\in 8\mathbb{N}+5$ a primitive fourth root of unity $\kappa$ exists and
$\kappa=\lambda^{4}$. The cyclotomic polynomial $\phi_{16}(x)$ then factorizes
over $\mathbb{F}_{p}$ into two factors of degree 4, i.e
$\phi_{16}(x)=P(x)Q(x)$. Isotropy requires that if $\lambda$ is a root of
$P(x)$, then $\lambda^{-1}$ is a root of $Q(x)$. Imposing this together with
requiring that the coefficient of the constant term is a power of $\kappa$
leaves four possibilities,

$\displaystyle P(x)$
$\displaystyle=(x-\lambda)(x-\lambda^{3})(x-\lambda^{5})(x-\lambda^{7})\leavevmode\nobreak\
,$ $\displaystyle Q(x)$
$\displaystyle=(x-\lambda^{9})(x-\lambda^{11})(x-\lambda^{13})(x-\lambda^{15})\leavevmode\nobreak\
,$ $\displaystyle P(x)$
$\displaystyle=(x-\lambda)(x-\lambda^{3})(x-\lambda^{9})(x-\lambda^{11})\leavevmode\nobreak\
,$ $\displaystyle Q(x)$
$\displaystyle=(x-\lambda^{5})(x-\lambda^{7})(x-\lambda^{13})(x-\lambda^{15})\leavevmode\nobreak\
,$ $\displaystyle P(x)$
$\displaystyle=(x-\lambda)(x-\lambda^{5})(x-\lambda^{7})(x-\lambda^{11})\leavevmode\nobreak\
,$ $\displaystyle Q(x)$
$\displaystyle=(x-\lambda^{3})(x-\lambda^{9})(x-\lambda^{13})(x-\lambda^{15})\leavevmode\nobreak\
,$ $\displaystyle P(x)$
$\displaystyle=(x-\lambda)(x-\lambda^{7})(x-\lambda^{11})(x-\lambda^{13})\leavevmode\nobreak\
,$ $\displaystyle Q(x)$
$\displaystyle=(x-\lambda^{3})(x-\lambda^{5})(x-\lambda^{9})(x-\lambda^{15})\leavevmode\nobreak\
.$ (4.20)

In all four cases the requirement that the coefficient of $x^{2}$ is an
element of $\mathbb{F}_{p}$ leads to the requirement that
$w^{2}\in\mathbb{F}_{p}$, which is impossible for $p\in 8\mathbb{N}+5$. Hence
there are no invariant global forms in this case.

For the remaining case of $p\in 4\mathbb{N}+3$, a similar analysis shows that
there are again no invariant global forms. The results are summarized in Table
8.

####  4.3.3 Genus 5

Finally for genus $5$ the relevant cyclotomic polynomials are $\phi_{11}(x)$
and $\phi_{22}(x)$. If a matrix $F$ has characteristic polynomial
$\phi_{22}(x)$ then $-F$ has characteristic polynomial $\phi_{11}(x)$ and both
have the same invariant subspaces, and hence we deal only with $\phi_{11}(x)$
here.

$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29
---|---|---|---|---|---|---|---|---|---|---
$\phi_{11}(x)$ | 0 | 0 | 2 | 2 | 1 | 0 | 0 | 2 | 32 | 2
Table 9: The number of invariant global forms for symmetry generators with
characteristic polynomial given by $\phi_{11}(x)$, relevant for genus 5. There
exist intrinsically non-invertible symmetries if and only if the entry above
is $0$.

For the generic case in which $p$ does not divide 11, the only possibility for
the factorization of $\phi_{11}(x)=P(x)Q(x)$ satisfying the isotropy criterion
is

$\displaystyle P(x)$ $\displaystyle=$
$\displaystyle(x-\lambda)(x-\lambda^{3})(x-\lambda^{4})(x-\lambda^{5})(x-\lambda^{9})\leavevmode\nobreak\
,$ $\displaystyle Q(x)$ $\displaystyle=$
$\displaystyle(x-\lambda^{2})(x-\lambda^{6})(x-\lambda^{7})(x-\lambda^{8})(x-\lambda^{10})\leavevmode\nobreak\
$ (4.21)

with $\lambda$ a primitive eleventh root of unity. It can be shown that $P(x)$
and $Q(x)$ are polynomials over $\mathbb{F}_{p}$ if
$\eta:=\lambda+\lambda^{3}+\lambda^{4}+\lambda^{5}+\lambda^{9}\in\mathbb{F}_{p}$.
Note that $\eta$ satisfies a quadratic equation $\eta^{2}+\eta+3=0$ which has
a solution in $\mathbb{F}_{p}$ if $-11$ has a square root in $\mathbb{F}_{p}$.
This happens for $p\in 11\mathbb{N}+k$ for $k=1,5,7,8,$ and $9$. For the last
four cases there is no further factorization and there are only two invariant
global forms. For $p\in 11\mathbb{N}+1$ the polynomial $\phi_{11}(x)$ splits
completely and there are $32$ invariant global forms.

For the remaining cases of $p\in 11\mathbb{N}+k$ for $k=2,3,4,6,$ and $10$ the
symmetry is intrinsically non-invertible. Finally for the exceptional case of
$p=11$ there is a single invariant global form. These results are summarized
in Table 9.

##  5 A higher-dimensional perspective

In this final section we give a higher-dimensional perspective on the non-
invertible symmetries discussed above. To make the discussion self-contained,
we begin with a review of the relative nature of the 6d (2,0) theory. For most
of this section we will work the (2,0) theory of type $\mathfrak{a}_{N-1}$
without assuming that $N$ is prime, though this assumption will reappear
towards the end.202020We thank Kantaro Ohmori, Gabi Zafrir, and Yunqin Zheng
for numerous crucial discussions about the content of this section.

###  5.1 The 6d (2,0) theory as a relative theory

As was already mentioned in the introduction and Section 2.1, for generic $N$
the type $\mathfrak{a}_{N-1}$ 6d (2,0) theory is a relative theory, which
means that it is best thought of as living on the boundary of a non-trivial 7d
TQFT. This situation was shown in Figure 3. In order to specify the state
$|\mathfrak{a}_{N-1}\rangle$ corresponding to the 6d (2,0) theory, we must
first specify a basis for the Hilbert space of the 7d TQFT. To do so, we might
naively aim to specify boundary conditions for all fields in the bulk
7-dimensional theory, in particular the dynamical three-form field $c$.
However, due to a term proportional to $\int_{W_{7}}c\wedge dc$ in the action
of the bulk TQFT, the field $c$ becomes its own canonically conjugate
momentum, and thus there are non-trivial commutation relations between $c$ on
distinct 3-cycles. This means that it is not actually possible to specify the
boundary value of $c$ on all 3-cycles of $X_{6}$.

Let us be more concrete. Our starting point is the 7d Abelian CS
theory212121For simplicity, we will assume the presence of a Wu 4-structure
(so that the CS term is well-defined for arbitrary $N$) and take $c$ to be
topologically trivial (so that we do not need to work with differential
cohomology).

$\displaystyle S_{7d}={N\over 4\pi}\int_{W_{7}}c\wedge dc\leavevmode\nobreak$
(5.1)

which we write in differential form notation $c\in H^{3}(W_{7},U(1))$. This
theory has the following Wilson surfaces,

$\displaystyle\Phi_{q}(M_{3}):=e^{iq\oint_{M_{3}}c}\leavevmode\nobreak\
,\hskip 43.36243ptq\in{\mathbb{Z}}_{N}\leavevmode\nobreak\ ,\hskip
14.45377ptM_{3}\in H_{3}(W_{7},{\mathbb{Z}})\leavevmode\nobreak\ .$ (5.2)

Let us consider a configuration of two Wilson surfaces $\Phi_{q}(M_{3})$ and
$\Phi_{q^{\prime}}(M^{\prime}_{3})$, with $M_{3}$ and $M^{\prime}_{3}$ forming
a Hopf link in $W_{7}$. To evaluate this configuration via the path integral,
we consider the action with insertions (see e.g. [59])

$\displaystyle S_{\mathrm{CS}}$ $\displaystyle=$ $\displaystyle{N\over
4\pi}\int_{W_{7}}c\wedge dc+q\int_{M_{3}}c+q^{\prime}\int_{M_{3}^{\prime}}c$
(5.3) $\displaystyle=$ $\displaystyle{N\over 4\pi}\int_{W_{7}}c\wedge
dc+\int_{W_{7}}(q\omega_{M_{3}}+q^{\prime}\omega_{M_{3}^{\prime}})\wedge
c\leavevmode\nobreak\ ,$ (5.4)

where $\omega_{M}$ is the Poincaré dual of $M$. Integrating out $c$ then
imposes

$\displaystyle dc=-{2\pi\over
N}(q\omega_{M_{3}}+q^{\prime}\omega_{M_{3}^{\prime}})\leavevmode\nobreak\ .$
(5.5)

If we define a manifold $V_{4}$ such that

$\displaystyle\partial
V_{4}=\left(qM_{3}+q^{\prime}M_{3}^{\prime}\right)\leavevmode\nobreak\ ,$
(5.6)

then we have $c=-{2\pi\over N}\mathrm{PD}(V_{4})$. Plugging this back into the
action gives a term

$\displaystyle{\pi\over
N}\int_{M_{7}}(q\omega_{M_{3}}+q^{\prime}\omega_{M_{3}^{\prime}})\wedge\mathrm{PD}(V_{4})$
$\displaystyle=$ $\displaystyle{\pi\over
N}\int_{M_{7}}\mathrm{PD}\left((qM_{3}+q^{\prime}M_{3}^{\prime})\cap
V_{4}\right)$ (5.7) $\displaystyle=$ $\displaystyle{\pi\over
N}\int_{M_{7}}\left[q\mathrm{PD}(M_{3}\cap
V_{4})+q^{\prime}\mathrm{PD}(M_{3}^{\prime}\cap V_{4})\right]$
$\displaystyle=$ $\displaystyle{2\pi\over
N}qq^{\prime}\mathrm{link}(M_{3},M_{3}^{\prime})\leavevmode\nobreak\ ,$

where $\mathrm{link}(M_{3},M_{3}^{\prime})$ is the linking number of $M_{3}$
and $M_{3}^{\prime}$ in $W_{7}$. This linking comes about since $V_{4}$ is the
Seifert surface for $qM_{3}+q^{\prime}M_{3}^{\prime}$, and hence each time
$M_{3}$ pierces $V_{4}$ it links $M_{3}^{\prime}$ once.

To summarize, we have found that the Hopf link of Wilson surfaces supported on
$M_{3}$ and $M^{\prime}_{3}$ in $W_{7}$ evaluates to222222This is the direct
analog of the S-matrix element $S_{ab}$ between charge $a$ and $b$ anyons in
$U(1)_{k}$ CS theory in 3d, which is given by $S_{ab}=e^{{2\pi i\over k}ab}$.

$\displaystyle\langle\Phi_{q}(M_{3})\Phi_{q^{\prime}}(M_{3}^{\prime})\dots\rangle=e^{{2\pi
i\over
N}qq^{\prime}\mathrm{link}(M_{3},M_{3}^{\prime})}\langle\dots\rangle\leavevmode\nobreak\
.$ (5.8)

We may now push the configuration above to a 6d plane $X_{6}\subset W_{7}$.
This gives a statement about the equal-time commutation relation

$\displaystyle\Phi_{q}(M_{3})\Phi_{q^{\prime}}(M_{3}^{\prime})=e^{{2\pi i\over
N}qq^{\prime}\langle
M_{3},M_{3}^{\prime}\rangle}\Phi_{q^{\prime}}(M_{3}^{\prime})\Phi_{q}(M_{3})\leavevmode\nobreak\
.$ (5.9)

We see that the linking number in 7d has reduced to the (signed) intersection
number in 6d. One should be careful to note that the parity properties of the
two bilinear pairings are opposite: whereas the linking pairing was symmetric,
the intersection pairing is anti-symmetric.

The Wilson surfaces also satisfy the quantum torus algebra on a 6d slice,

$\displaystyle\Phi_{q}(M_{3})\Phi_{q}(M_{3}^{\prime})=e^{{2\pi i\over
N}{q^{2}\over 2}\langle
M_{3},M_{3}^{\prime}\rangle}\Phi_{q}(M_{3}+M_{3}^{\prime})\leavevmode\nobreak\
.$ (5.10)

Note that it is not obvious that the factor $e^{{2\pi i\over N}{q^{2}\over
2}}$ appearing in the quantum torus algebra is well-defined. This is because
the charge $q$ was only defined modulo $N$. Under $q\rightarrow q+N$ the
factor $e^{{2\pi i\over N}{q^{2}}}$ is single-valued, but this can fail to be
true when we take the square root.

For $N$ odd, there is no issue: noting that $e^{{2\pi i\over
N}{q^{2}}}=e^{{2\pi i\over N}(N+1){q^{2}}}$, we can simply define

$\displaystyle e^{{2\pi i\over N}{q^{2}\over 2}}:=e^{{2\pi i\over
N}(N+1){q^{2}\over 2}}$ (5.11)

which is manifestly well-defined since $N+1\over 2$ is an integer. A simpler
way of saying this is that $2$ is invertible in ${\mathbb{Z}}_{N}$ for $N$
odd.

For $N$ even, the discussion is more subtle. To begin, let us return to
$e^{{2\pi i\over N}{q^{2}}}$, which as we said is well-defined. In fact, even
this is slightly non-obvious, since one has to check that $q^{2}$ is well-
defined modulo $N$ when $q\rightarrow q+N$. This is indeed the case, since

$\displaystyle(q+N)^{2}-q^{2}=N(2q+N)$ (5.12)

and hence $q^{2}=(q+N)^{2}$ mod $N$. But now we see something more: when $N$
is even, then we in fact have

$\displaystyle(q+N)^{2}-q^{2}=2N\left(q+{N\over 2}\right)$ (5.13)

and hence $q^{2}=(q+N)^{2}$ mod $2N$. This means that we can promote $q^{2}$
to an element in ${\mathbb{Z}}_{2N}$, as opposed to an element in
${\mathbb{Z}}_{N}$ like $q$ was. We denote this promoted quantity by ${\cal
P}(q)\in{\mathbb{Z}}_{2N}$, which is known as a quadratic refinement of the
original pairing in ${\mathbb{Z}}_{N}$. We may now define

$\displaystyle e^{{2\pi i\over N}{q^{2}\over 2}}:=e^{{2\pi i\over N}{{\cal
P}(q)\over 2}}\leavevmode\nobreak\ ,$ (5.14)

which is well-defined since we only need single-valuedness under ${\cal
P}(q)\rightarrow{\cal P}(q)+2N$.

We now return to the specification of the basis of the Hilbert space of the 7d
TQFT. Noting that the factor of $q$ can be absorbed by replacing $M_{3}$ with
$qM_{3}$, we may without loss of generality restrict to $\Phi_{1}(M_{3})$,
which for simplicity we will denote by $\Phi(M_{3})$, and $M_{3}\in
H_{3}(X_{6},{\mathbb{Z}}_{N})$. Because of the non-commutativity of
$\Phi(M_{3})$ for generic $M_{3}\in H_{3}(X_{6},{\mathbb{Z}}_{N})$, it is not
possible to write a simultaneous eigenvector for all $\Phi(M_{3})$—in other
words, it is not possible to specify the boundary value of $c$ on all three-
cycles of $X_{6}$. Instead, the best one can do is to choose a maximal
isotropic sublattice ${\cal L}\in H_{3}(X_{6},{\mathbb{Z}}_{N})$ and to
specify all of the boundary values associated with ${\cal L}$. Here
isotropicity means that we have

$\displaystyle\langle M_{3},M_{3}^{\prime}\rangle=0\leavevmode\nobreak\
,\hskip 57.81621pt\forall\,M_{3},M_{3}^{\prime}\in{\cal L}\leavevmode\nobreak\
.$ (5.15)

In terms of this isotropic sublattice we may define a state $|{\cal
L},0\rangle$ such that

$\displaystyle\Phi(M_{3})|{\cal L},0\rangle=|{\cal
L},0\rangle\leavevmode\nobreak\ ,\hskip 36.135pt\forall\,\,M_{3}\in{\cal
L}\leavevmode\nobreak\ .$ (5.16)

On the other hand, applying $\Phi(M_{3}^{\prime})$ for $M_{3}^{\prime}$ in the
complement ${\cal L}^{\perp}:=H^{3}(X_{6},{\mathbb{Z}}_{N})/{\cal L}$ gives a
new state,

$\displaystyle|{\cal L},M_{3}^{\prime}\rangle:=\Phi(M_{3}^{\prime})|{\cal
L},0\rangle$ (5.17)

which by the commutation relations (5.9) satisfies

$\displaystyle\Phi(M_{3})|{\cal L},M_{3}^{\prime}\rangle=e^{{2\pi i\over
N}\langle M_{3},M_{3}^{\prime}\rangle}|{\cal
L},M_{3}^{\prime}\rangle\leavevmode\nobreak\ ,\hskip
36.135pt\forall\,\,M_{3}\in{\cal L}\leavevmode\nobreak\ .$ (5.18)

These states form an irreducible representation of the algebra (5.9).

Physically, we may interpret the state $|{\cal L},0\rangle$ as imposing
Dirichlet boundary conditions $c|_{\partial}=0$ on all $M_{3}\in{\cal L}$.
This means the bulk Wilson surface $\Phi(M_{3})=e^{i\oint_{M_{3}}c}$
trivializes when moved to the boundary, reproducing the statement in (5.16).
By the canonical commutation relations, Dirichlet boundary conditions on
$M_{3}\in{\cal L}$ require Neumann boundary conditions on
$M_{3}^{\prime}\in{\cal L}^{\perp}$, and hence $\Phi(M_{3}^{\prime})$ for
$M_{3}^{\prime}\in{\cal L}^{\perp}$ gives a new state, which is $|{\cal
L},0\rangle$ with the insertion of a Wilson surface on $M_{3}^{\prime}$. This
is the state $|{\cal L},M_{3}^{\prime}\rangle$.232323These are the analogues
of the non-identity characters $\chi_{a}$ in 2d RCFT. The Wilson surfaces
discussed here are the analogues of the Verlinde lines in that context.

In terms of the basis of states $|{\cal L},M_{3}^{\prime}\rangle$, we may then
write the state $|\mathfrak{a}_{N-1}\rangle$ corresponding to the relative 6d
(2,0) theory as

$\displaystyle|\mathfrak{a}_{N-1}\rangle=\sum_{M_{3}^{\prime}\in{\cal
L}^{\perp}}Z_{{\cal L}}[M_{3}^{\prime}]|{\cal
L},M_{3}^{\prime}\rangle\leavevmode\nobreak$ (5.19)

for some appropriate coefficients $Z_{{\cal L}}[M_{3}^{\prime}]$. This vector
is sometimes referred to as the “partition vector” of the 6d theory. We should
be careful to note that the states $|{\cal L},{\cal B}\rangle$ do not define
topological boundary conditions for the bulk 7d TQFT; indeed, there do not in
general exist topological boundary conditions for this theory. Thus when one
refers to the 6d (2,0) “theory,” what one is really referring to is the above
vector.

###  5.2 Compactification on $\Sigma_{g,n}$

We now take $X_{6}=\Sigma_{g,n}\times X_{4}$, as for the class ${\cal S}$
construction. The 7d bulk theory is likewise put on $W_{7}=\Sigma_{g,n}\times
W_{5}$, where $\partial M_{5}=X_{4}$.242424Despite our notation, the product
here is not necessarily trivial, as we will discuss shortly. Because the four-
dimensional spacetime $X_{4}$ is attached to a bulk five-dimensional spacetime
$W_{5}$, we may expect that the theory is again only relative. However, we
will now show that the theory in $W_{5}$ can be made invertible, and hence
that this procedure defines an absolute theory in 4d.

Given the 6d spacetime $X_{6}=\Sigma_{g,n}\times X_{4}$, there is a particular
class of 7-manifolds $W_{7}$ obtained by taking $W_{7}=V_{g,n}\times X_{4}$
with $V_{g,n}$ a three-manifold with boundary $\Sigma_{g,n}$. For a given
Riemann surface $\Sigma_{g,0}$, there are many inequivalent three-manifolds
with boundary $\Sigma_{g,0}$. A particularly simple class of such manifolds
are the so-called handlebodies, whose basic properties are reviewed in
Appendix C. As discussed there, in order to construct a handlebody, one must
choose a set of $g$ meridians, i.e. a set of $g$ generators of
$H_{1}(\Sigma_{g,0},{\mathbb{Z}})$ which become trivialized as elements of
$H_{1}(V_{g,0},{\mathbb{Z}})$. The remaining $g$ generators of
$H_{1}(\Sigma_{g,0},{\mathbb{Z}})$ lift to generators of
$H_{1}(V_{g,0},{\mathbb{Z}})$ and are referred to as “longitudes.” We will
denote the meridians by $\mu_{i}$ and the longitudes by $\lambda_{i}$ for
$i=1,\dots,g$. Note that not every choice of $g$ generators of
$H_{1}(\Sigma_{g,0},{\mathbb{Z}})$ gives a legitimate set of meridians.
Indeed, as explained in Appendix C, each legitimate choice of meridians
corresponds to a maximal isotropic sublattice $L\subset
H_{1}(\Sigma_{g,0},{\mathbb{Z}})$. We will denote the handlebody specified by
the choice of meridians $L$ by $V_{g,0}^{L}$.

Given a handlebody $V_{g,0}^{L}$, there are multiple choices of longitudes,
which differ by shifts by meridians. Indeed, given one choice of longitudes
$\lambda_{i}$, the choice of longitudes

$\displaystyle\lambda_{i}^{\prime}=\lambda_{i}+\sum_{j=1}^{g}k_{ij}\,\mu_{j}\leavevmode\nobreak\
,\hskip 36.135ptk_{ij}\in{\mathbb{Z}}$ (5.20)

is another legitimate choice. We may also allow for linear combinations of the
$\lambda_{i}$, but we will fix a basis such that the intersection pairing of
$\lambda_{i}$ with the meridians $\mu_{i}$ is

$\displaystyle\langle\mu_{i},\lambda_{j}\rangle=-\langle\lambda_{j},\mu_{i}\rangle=\delta_{ij}\leavevmode\nobreak\
.$ (5.21)

We now consider the 7d CS theory (5.1) on this handlebody geometry. First we
define the following fields

$\displaystyle b_{i}=\oint_{\mu_{i}}c\leavevmode\nobreak\ ,\hskip
36.135pt\widehat{b}_{i}=\oint_{\lambda_{i}}c\leavevmode\nobreak\ ,\hskip
36.135pti=1,\dots,g\leavevmode\nobreak\ .$ (5.22)

Because the meridians $\mu_{i}$ are naively contractible, we might assume that
all of the $b_{i}$ are trivial. This however is not necessarily true. The
origin of the discrepancy is that we can have Wilson surfaces supported on
longitudes $\lambda_{i}$ which link with $\mu_{i}$, and hence effectively make
$\mu_{i}$ non-contractible.

To understand this, let us denote the Wilson surfaces wrapping meridians and
longitudes by

$\displaystyle\Phi_{i}(M_{2})$ $\displaystyle:=$
$\displaystyle\Phi(M_{2}\times\mu_{i})=e^{{2\pi i\over
N}\oint_{M_{2}}b_{i}}\leavevmode\nobreak\ ,$
$\displaystyle\widehat{\Phi}_{i}(M_{2})$ $\displaystyle:=$
$\displaystyle\Phi(M_{2}\times\lambda_{i})=e^{{2\pi i\over
N}\oint_{M_{2}}\widehat{b}_{i}}\leavevmode\nobreak\ ,$

respectively, where we have converted to discrete cocycle notation
$(b_{i},\widehat{b}_{i})\rightarrow{2\pi i\over N}(b_{i},\widehat{b}_{i})$. By
(5.8), these surfaces may have non-trivial linking in 5d. Indeed, let us
insert a Wilson 3-surface on the longitude $\lambda_{j}$ in 7d, with the
remaining support on $M_{2}^{\prime}\subset X_{4}$. This amounts to inserting
$\widehat{\Phi}_{j}(M_{2}^{\prime})$ in 5d. Equation (5.8) then gives

$\displaystyle\langle\Phi_{i}(M_{2})\widehat{\Phi}_{j}(M_{2}^{\prime})\dots\rangle=e^{{2\pi
i\over
N}\mathrm{link}(M_{2}\times\mu_{i},M_{2}^{\prime}\times\lambda_{j})}\langle\dots\rangle\leavevmode\nobreak\
.$ (5.23)

Clearly this means that we cannot set $\Phi_{i}(M_{2})=1$, and likewise
$b_{i}$ is not simply zero.

Let us now compactify the theory on $\Sigma_{g,0}$ in the presence of a series
of Wilson 3-surfaces on $\lambda_{j}\times M_{2,j}^{\prime}$. Denote the
Poincaré duals of $M_{2,j}^{\prime}$ by $B_{j}\in
H^{2}(X_{4},{\mathbb{Z}}_{N})$. The linking equation above becomes an operator
equation determining $\Phi_{i}(M_{2})$,

$\displaystyle\Phi_{i}(M_{2})=e^{{2\pi i\over
N}\sum_{j=1}^{g}\langle\mu_{i},\lambda_{j}\rangle(M_{2},M_{2,j}^{\prime})}=e^{{2\pi
i\over
N}\sum_{j=1}^{g}\langle\mu_{i},\lambda_{j}\rangle\oint_{M_{2}}B_{j}}\leavevmode\nobreak\
.$ (5.24)

Another way to say this is that in any 4d computation involving $\Phi_{i}$ we
may commute it to the right of all $\widehat{\Phi}_{j}$ using (5.9), at which
point it can be shrunk away. Hence insertion of $\Phi_{i}$ is equivalent to
insertion of the phase appearing in (5.9).

We see that the insertion of the longitudinal Wilson surface
$\widehat{\Phi}_{j}(M_{2,j}^{\prime})$ effectively fixes

$\displaystyle b_{i}=\sum_{j=1}^{g}\langle\mu_{i},\lambda_{j}\rangle
B_{j}\leavevmode\nobreak\ .$ (5.25)

In particular, if we work with a basis of longitudes satisfying (5.21), we
have simply $b_{i}=B_{i}$. The insertion of Wilson lines on longitudes has
allowed us to make $b_{i}$ non-zero, though it is still a constant, i.e. a
background field.

$X_{4}$$W_{7}$$\small{\widehat{\Phi}_{i}(M_{2,i})}$$\Sigma_{g,0}$$V^{L}_{g,0}$Shrink
$\Sigma_{g,0}$${\cal T}_{L}^{N,g,0}[\Omega,B_{i}]$$X_{4}$$W_{5}$ Figure 6: We
begin with a handlebody configuration in 7d specified by $L$, together with a
set of Wilson surfaces $\widehat{\Phi}_{i}(M_{2,i})$ on the longitudes. This
configuration has a single boundary, located on the right. One then
compactifies on $\Sigma_{g,n}$ to get a 5d bulk with a single 4d boundary. The
5d theory can be shown to be invertible, and hence the boundary theory is a
well-defined 4d theory labelled by $L$, the longitudes $\lambda_{i}$
(capturing stacking with SPT phases), and the background fields
$B_{i}=\mathrm{PD}(M_{2,i})$.

Upon compactification on $\Sigma_{g,0}$, the bulk CS theory of (5.1) then
reduces to,

$\displaystyle S_{7d}={N\over 4\pi}\int_{W_{7}}c\wedge
dc\,\,\,\xrightarrow{\text{compactify\,\,on\,\,}\Sigma_{g,0}}\,\,\,S_{5d}$
$\displaystyle=$ $\displaystyle{2\pi\over
N}\sum_{i=1}^{g}\int_{W_{5}}B_{i}\cup\delta\widehat{b}_{i}\leavevmode\nobreak\
.$ (5.26)

Unlike usual BF theory, only one of the fields appearing above is dynamical—in
other words, this is the theory of $g$ free ${\mathbb{Z}}_{N}$ background
gauge fields $B_{i}$. This theory is obviously invertible, which means that
the boundary four-dimensional theory is well-defined as a standalone theory.
This is how one obtains an absolute 4d theory from the relative 6d $(2,0)$
theory.

Of course, the set of Wilson surfaces $\widehat{\Phi}_{j}(M_{2,i}^{\prime})$
that we inserted in 7d was just one choice. We could have modified this choice
by placing the Wilson surfaces on different 2-cycles $M_{2,i}^{\prime}$. From
the four-dimensional perspective, this changes the background fields $B_{i}$
to which the one-form symmetries are coupled.

Alternatively, we could have changed the longitudes $\lambda_{i}$ on which the
Wilson surfaces were inserted. Say that we change longitudes to
$\lambda_{i}^{\prime}$ as defined in (5.20), with the corresponding Wilson
surfaces denoted by $\widehat{\Phi}^{\prime}_{i}(M_{2,i})$. Using the
commutation relations (5.9) and the quantum torus algebra (5.10), we find that

$\displaystyle\prod_{i}\widehat{\Phi}_{i}^{\prime}(M_{2,i})$ $\displaystyle=$
$\displaystyle\prod_{i}\left(e^{{2\pi i\over
N}\sum_{j=1}^{g}k_{ij}\langle\mu_{i},\lambda_{j}\rangle\int_{X_{4}}{B_{i}\cup
B_{i}\over
2}}\,\widehat{\Phi}_{i}(M_{2,i})\prod_{j=1}^{g}\Phi_{j}^{k_{ij}}(M_{2,j})\right)$
$\displaystyle=$ $\displaystyle\mathrm{exp}\left[{2\pi i\over
N}\left(\sum_{i}k_{ii}\int_{X_{4}}{B_{i}\cup B_{i}\over
2}+\sum_{i<j}k_{ij}\int_{X_{4}}B_{i}\cup
B_{j}\right)\right]\prod_{i}\widehat{\Phi}_{i}(M_{2,i})$

where in the second line we have used (5.21), and after commuting all
$\Phi_{j}$ operators to the right of $\widehat{\Phi}_{i}$ operators we have
shrunk them away. The factor of ${B_{i}\cup B_{i}\over 2}$ appearing above
should be understood as the appropriate quadratic refinement when applicable.
We see that changing to a different longitude stacks the 4d theory with an SPT
phase.

The final picture is thus as in Figure 6. We begin in 7d by specifying a
maximal isotropic sublattice $L$, which specifies a way to fill in the Riemann
surface $\Sigma_{g,0}$ to get a handlebody $V_{g,0}^{L}$. We may also choose
to insert some set of Wilson surfaces $\widehat{\Phi}_{i}(M_{2,i})$ along the
longitudes $\lambda_{i}$. Upon compactification on $\Sigma_{g,0}$, we obtain a
well-defined 4d theory coupled to an invertible 5d bulk. The choice of
meridians $L$ in 7d determines the charge lattice of the 4d theory. On the
other hand, the choice of longitudinal Wilson surfaces determines both the
background fields $B_{i}$ for the 1-form symmetry (coming from the choice of
$M_{2,i}$) together with the possible invertible phases that can be stacked
with the 4d theory (coming from the choice of $\lambda_{i}$).

###  5.3 From the Anomaly TFT to the Symmetry TFT

We have now seen how the (trivial) Anomaly TFT of the 4d theory can be
obtained by reducing the 7d CS term on an appropriate handlebody. It is also
enlightening to see how the Symmetry TFT (SymTFT) [110, 129, 130, 30, 131,
132, 57, 59] emerges in this picture.

At any point $y$ along the handlebody direction ${\mathbb{R}}_{+}$, we may
choose to split the handlebody into an interior portion
$V_{g,0}^{L,\text{in}}$ with $y\geq y_{*}$ and an exterior portion
$V_{g,0}^{\text{ext}}:=\Sigma_{g,0}\times[0,y_{*}]$. If we focus on the latter
and forget about boundary conditions, the reduction on the Riemann surface
gives

$\displaystyle S_{7d}={N\over 4\pi}\int_{W_{7}}c\wedge
dc\,\,\,\xrightarrow{\text{compactify\,\,on\,\,}\Sigma_{g,0}}\,\,\,S_{5d}$
$\displaystyle=$ $\displaystyle{2\pi\over
N}\sum_{i=1}^{g}\int_{X_{4}\times[0,y_{*}]}b_{i}\cup\delta\widehat{b}_{i}\leavevmode\nobreak\
,$ (5.28)

where now both $b_{i}$ and $\widehat{b}_{i}$ are dynamical—indeed, both the
longitudes $\lambda_{i}$ and meridians $\mu_{i}$ are non-trivial in
$V_{g,0}^{\text{ext}}$.

$y=\infty$$y_{*}$$y=0$$\Sigma_{g,0}$$V^{\text{ext}}_{g,0}$$V^{L,\text{in}}_{g,0}$Shrink
$\Sigma_{g,0}$$|{\cal T}^{N,g,0}[\Omega]\rangle$$\langle
L|$$\sum_{i=1}^{g}\int b_{i}d\widehat{b}_{i}$$\sum_{i=1}^{g}\int
B_{i}d\widehat{b}_{i}$ Figure 7: Splitting the handlebody into an inner
handlebody $V^{L,\text{in}}_{g,0}$ and an outer shell $V^{\text{ext}}_{g,0}$
separates the (trivial) Anomaly TFT from the Symmetry TFT. The location
$y_{*}$ of this splitting is arbitrary in 7d, and hence the boundary $\langle
L|$ separating the Anomaly and Symmetry TFTs is topological.

This gives us a picture in which a dynamical BF theory inhabits $[0,y_{*}]$,
with a boundary condition at $y=y_{*}$ specified by $L$. In particular, $L$
specifies which directions of the Riemann surface become trivial in the
handlebody (i.e. meridians), and hence which of the dynamical fields of the BF
theory become background. In other words, each choice of $L$ is a choice of
Dirichlet boundary conditions for different sets of fields of the BF theory.
The setup is shown in Figure 7. Because the 7d geometry is completely
independent of the location of $y_{*}$, the 5d picture should likewise be
independent of $y_{*}$, meaning that the interface between the BF theory and
the anomaly theory is topological.

While it is tempting to refer to the dynamical BF theory we have obtained as
the SymTFT for the boundary 4d theories, one should be careful with this
language. In general, the SymTFT is defined such that it captures the full
symmetry (higher-)category of the boundary theory, and it is not always true
that the bulk BF theory does so.252525In fact, if we also account for zero-
form symmetries such as the ${\mathbb{Z}}_{2N}$ chiral symmetry of the
boundary, then the BF theory is never the full SymTFT; at best it only
captures the one-form symmetry sector. We will focus only on this one-form
symmetry sector here, defining the “full SymTFT” as such.  Indeed, the BF
theory above is a simple example of a Dijkgraaf-Witten (DW) theory, and as
explained in [59] the SymTFT is a DW theory if and only if the boundary fusion
category has no intrinsically non-invertible symmetries. Hence straightforward
reduction of the 7d CS theory on a Riemann surface can only capture the full
SymTFT in cases in which all potential non-invertible symmetries are non-
intrinsic. To obtain the SymTFT for intrinisically non-invertible symmetries,
it turns out that the 7d CS theory must be coupled to topological gravity, as
we will now explain.262626Of course, it is not inconsistent to have the 4d
theory coupled to the 5d BF theory alone, so the sum over geometries is
ultimately optional. It is only necessary if we want to obtain the 4d theory
coupled to the full SymTFT for the one-form symmetry sector.  This gives a
physical relevance for the notion of intrinsic non-invertibility.

To begin our discussion, we define two families of operations on the
handlebody configuration. The first are modular transformations of the Riemann
surface, which act throughout the entire handlebody and generically change the
period matrix of the Riemann surface at each cross section. Note that because
the theory in the bulk of the handlebody is topological, the only sensitivity
to the change in the period matrix is on the boundary Riemann surface, where
the 6d $(2,0)$ theory lives.

The second family of operations involve excising from the handlebody an inner
handlebody, like in the case of the splitting into $V_{g,0}^{L,\text{in}}$ and
$V_{g,0}^{\text{ext}}$ from before, but now gluing them back together with a
non-trvial element of the modular group $Sp(2g,{\mathbb{Z}}_{N})$—in other
words, we perform “surgery” on the three-manifold [133]. Because this
operation happens in the interior of the handlebody, it does not affect the
boundary Riemann surface, and in particular it does not change the period
matrix in the region in which the theory is sensitive to it. This operation
does however change the global form $L$, because it changes which cycles in
the handlebody are contractible (fixing the conventions for the cycles on the
boundary Riemann surface).

The handlebody configuration we have been describing thus far has two
“boundaries”: one is the obvious boundary where the 6d (2,0) theory lives,
while the other is off at infinity, where the longitudinal Wilson surfaces
fixing the background fields live. We can now perform surgery by an element
$F$ of the modular group, as described above. Since this happens in the
interior of the handlebody, it does not affect the boundary where the 6d (2,0)
theory lives, but it does change the “inner boundary”, i.e. it changes the
global form. This operation may then be combined with a modular transformation
$F$ acting on the entire handlebody. The modular transformation reverts the
interior of the handlebody back to its original form, but changes the value of
the period matrix of the boundary Riemann surface. If we fix ourselves to a
value of the period matrix that is invariant under the action of $F$, then the
geometry with the combined surgery and modular transformations has the same
boundaries as the original geometry. Importantly though, the full geometry is
not identical to the original one, due to the internal twist.

In quantum gravity (and quantum field theory in general), one is instructed to
sum over all configurations with the same asymptotic boundary conditions. If
we were to treat the 7d CS theory in a (topological) quantum gravity context,
then we would likewise sum over handlebody configurations with all such
internal twists. Furthermore, we should allow for different twists at each
point of $X_{4}$, captured by a discrete metric connection.272727Without the
sum over geometries, this setup is similar to that of theories of “class
$\mathcal{F}$ ” [134], though in the current case we still take the boundary
$X_{6}=\Sigma_{g,n}\times X_{4}$ to be a trivial product.

How do we interpret this topological gravity computation from the point of
view of the compactified 5d theory? As we will explain in the next subsection,
the surgery defects described above descend to condensation defects
implementing outer automorphism symmetries of the 5d BF theory. Promoting
background gravity to dynamical gravity in 7d then corresponds to gauging of
these automorphisms, with the role of the discrete gauge field being played by
the dimensional reduction of the discrete metric connection. The result is not
a BF theory in general.

We may write the action for the resulting 5d theory in terms of twisted
cocycles as follows [59]. First, we write (5.28) in the form of a $K$-matrix
theory,

$\displaystyle S_{5d}={2\pi\over 2N}\int_{W_{5}}\mathbf{b}^{T}\cup
K\delta\mathbf{b}\leavevmode\nobreak\ ,\hskip
36.135ptK=\left(\begin{matrix}\widetilde{K}&\mathds{1}_{g\times g}\\\
-\mathds{1}_{g\times g}&0\end{matrix}\right)\leavevmode\nobreak\ ,$ (5.29)

where we have defined
$\mathbf{b}=(b_{1},\dots,b_{g},\widehat{b}_{1},\dots,\widehat{b}_{g})$. The
matrix $\widetilde{K}$ has elements $(\widetilde{K})_{ij}=k_{ij}$, with
$k_{ij}$ as defined in (5.20). This theory has an $Sp(2g,{\mathbb{Z}}_{N})$
outer automorphism symmetry which acts on $\mathbf{b}$ in the usual matrix
representation, and is a zero-form symmetry.

Let us assume that the enhanced symmetry is cyclic and is generated by a
single element $F\in Sp(2g,{\mathbb{Z}}_{N})$. Gauging this symmetry amounts
to promoting $\mathbf{b}$ to a twisted cocycle. If $F$ is of order $|F|$, the
action after gauging is given by

$\displaystyle S_{5d}={2\pi\over
2N}\int_{W_{5}}\mathbf{b}^{T}\cup_{\eta}K\delta_{\eta}\mathbf{b}+{2\pi\over|F|}\int_{W_{5}}x\cup\delta\eta$
(5.30)

where $\eta$ is the background gauge field for $F$, descending in an obvious
manner from the discrete metric connection in 7d, and $x$ is a Lagrange
multiplier field enforcing that $\eta$ is a ${\mathbb{Z}}_{|F|}$-valued
cocycle. The twisted cup products and differentials are defined by

$\displaystyle(\mathbf{a}\cup\mathbf{b})_{ijk\ell
m}:=\mathbf{a}_{ijk}F^{\eta_{ik}}\mathbf{b}_{k\ell m}\leavevmode\nobreak\
,\hskip
21.68121pt(\delta_{\eta}\mathbf{b})_{ijk\ell}:=F^{\eta_{ij}}\mathbf{b}_{jk\ell}-\mathbf{b}_{ik\ell}+\mathbf{b}_{ij\ell}-\mathbf{b}_{ijk}\leavevmode\nobreak\
,$ (5.31)

so that the term appearing in the integrand in components is

$\displaystyle(\mathbf{b}^{T}\cup_{\eta}K\delta_{\eta}\mathbf{b})_{ijk\ell
pq}=\mathbf{b}^{T}_{ijk}F^{\eta_{ik}}K\left(F^{\eta_{k\ell}}\mathbf{b}_{\ell
pq}-\mathbf{b}_{kpq}+\mathbf{b}_{k\ell q}-\mathbf{b}_{k\ell
p}\right)\leavevmode\nobreak\ .$ (5.32)

Here we take $F$ to be given in the appropriate matrix representation. Note
that similar expressions can be written down in the non-cyclic (and even non-
Abelian) cases, though the expressions are more complicated and we do not
write them down here. This gives a complete description of the SymTFT for the
4d theory.

What we have just described is the generic scenario. However, in some cases
something special happens: it can sometimes be the case that there is a global
form $L$ (i.e. a choice of handlebody) which is left invariant by modular
transformations, and hence also by surgery.282828For the current discussion we
set the background gauge fields in 4d to zero—i.e. we do not insert
longitudinal Wilson lines inside the handlebody. In this case, the geometry
with the combined surgery and modular transformation not only has the same
boundary as the original geometry, but is exactly identical to it (at fixed
points of the period matrix). In this case there are not distinct geometries
to sum over, and upon compactification we obtain (5.30) with the field $\eta$
now only a background field. The same holds for global forms which are related
to an invariant global form by a surgery operation. This is exactly as
expected: when there is an invariant global form $L$, the symmetries are non-
intriniscally non-invertible, and this is only possible if the bulk theory is
Dijkgraaf-Witten (and in this case BF).

Incidentally, note that this gives an alternative, though obviously
equivalent, approach to searching for invariant global forms: we ask when an
isometry of the Riemann surface $\Sigma_{g,n}$ can be extended to an isometry
of $V_{g,n}^{L}$, with coefficients valued in ${\mathbb{Z}}_{N}$. It is known
that this is possible if and only if the isometry maps meridians of
$V_{g,n}^{L}$ to meridians [133] modulo ${\mathbb{Z}}_{N}$. It is easy to
convince oneself that for $N=p$ prime this gives the same results as
identified using 4d techniques before. We may also understand the
transformation law given in (2.26) from the current perspective: in the
current language, $F$ correspond to modular transformations while $G$
corresponds to surgery operations. Whereas the former acts by changing the
basis in which the global form $M_{L,{\cal B}}$ is expressed, the latter fixes
the basis but changes the “charges” labelling the global variant.

Finally, let us note that for $(g,n)=(1,0)$ and $F=\mathsf{S}$, the theory
obtained by compactifying the 7d CS + topological gravity theory on
$V_{1,0}^{L}$ is precisely the SymTFT for $(3+1)$d theories with duality
defects, studied in detail in [59]. One question which was left unanswered in
that work was when the SymTFT can be recast as a Dijkgraaf-Witten theory. Our
current results allow us to answer this question when $N=p$ is prime, as was
already discussed in the introduction in Theorems 1 and 2. We may use the
results in Sections 3 and 4 to make analogous statements about discrete
gaugings of $({\mathbb{Z}}_{p}^{(2)})^{\otimes g}$ gauge theory in $(4+1)$d,
with $g\leq 5$.

###  5.4 Surgery defects

We close this section by providing more detail on the surgery defects
discussed above. In particular, let us see how they descend to condensation
defects in 5d. First we define the following surface operators,

$\displaystyle\Phi_{\mathbf{n}}(M_{2})=e^{{2\pi i\over N}{1\over 2}e\cdot
m(M_{2},M_{2})}\prod_{i=1}^{g}\widehat{\Phi}_{i}^{e_{i}}(M_{2})\prod_{i=1}^{g}\Phi_{i}^{m_{i}}(M_{2})\leavevmode\nobreak\
,\hskip
36.135pt\mathbf{n}=(e_{1},\dots,e_{g};m_{1},\dots,m_{g})\leavevmode\nobreak\
,$ (5.33)

where $e\cdot m:=\sum_{j=1}^{g}e_{j}m_{j}$. These are the most general Wilson
surfaces that can be constructed in the 5d TQFT. Each of these surface
operators descends from a Wilson three-surface in 7d, and is associated with a
cycle $\gamma_{\mathbf{n}}$ on $\Sigma_{g,n}$ via

$\displaystyle\Phi_{\mathbf{n}}(M_{2})=\Phi(M_{2}\times\gamma_{\mathbf{n}})\leavevmode\nobreak\
,\hskip
36.135pt\gamma_{\mathbf{n}}:=\sum_{j=1}^{g}e_{j}\lambda_{j}+\sum_{j=1}^{g}m_{j}\mu_{j}\leavevmode\nobreak\
.$ (5.34)

The surgery defects in 7d correspond to loci across which $F$ acts on
$\gamma_{\mathbf{n}}$ to produce
$F(\gamma_{\mathbf{n}})=\gamma_{F^{T}\mathbf{n}}$. We thus expect that surgery
defects in 7d should reduce to defects ${\cal C}_{F}(X_{4})$ in 5d
implementing

$\displaystyle{\cal
C}_{F}(X_{4}):\,\,\,\Phi_{\mathbf{n}}(M_{2})\mapsto\Phi_{F^{T}\mathbf{n}}(M_{2})\leavevmode\nobreak\
\hskip 57.81621ptM_{2}\subset X_{4}\leavevmode\nobreak\ .$ (5.35)

It is easy to see that such defects are condensation defects. Indeed, consider
Figure 8. On the left-hand side of the figure we have a configuration in which
$\Phi_{\mathbf{n}}(M_{2})$ enters ${\cal C}_{F}(X_{4})$ from the left and
$\Phi_{F^{T}\mathbf{n}}(M_{2})$ exits from the right. We may now fold
$\Phi_{F^{T}\mathbf{n}}(M_{2})$ to the left to obtain the figure on the right.
This configuration tells us that ${\cal C}_{F}(X_{4})$ must be able to absorb
$\Phi_{(\mathds{1}-F^{T})\mathbf{n}}(M_{2})$, for any $\mathbf{n}$. Let us
assume that $\mathbf{n}=(1,0,\dots,0)$. Then
$\Phi_{(\mathds{1}-F^{T})\mathbf{n}}=\Phi_{(\mathds{1}-F^{T})_{1}}$, where we
use the notation that ${M_{i}}$ represents the $i$-th column of a matrix $M$.
We conclude that ${\cal C}_{F}(X_{4})$ can absorb
$\Phi_{(\mathds{1}-F^{T})_{1}}$, and hence ${\cal C}_{F}(X_{4})$ should be a
condensate of $\Phi_{(\mathds{1}-F^{T})_{1}}$. The same argument can be used
for $\mathbf{n}=(0,\dots,0,1,0,\dots,0)$ with the $1$ in the $i$-th slot,
which tells us that ${\cal C}_{F}(X_{4})$ must be a simultaneous condensate of
all $\Phi_{(\mathds{1}-F^{T})_{i}}$ for $i=1,\dots,2g$.

One subtlety is that, for certain values of $N$, two separate vectors
$(\mathds{1}-F^{T})_{i}$ and $(\mathds{1}-F^{T})_{j}$ may be equal modulo $N$.
In such cases the full one-form symmetry is not quite ${\mathbb{Z}}_{N}^{2g}$,
but a quotient thereof. This happens for example in the case of
$F={\mathsf{S}}$ at genus 1, where the two vectors $(1-{\mathsf{S}})_{i}$ are
$(1,-1)$ and $(1,1)$, which are equivalent for $N=2$; a treatment of the
subtlety in this case can be found in [59]. More generally, some of the
$(\mathds{1}-F^{T})_{i}$ or multiples thereof may be linear combinations of
the others. For simplicity here we will only focus on the case for which this
does not happen, i.e. $\mathrm{ker}(\mathds{1}-F^{T})=0$ mod $N$, and will
furthermore assume that $N=p$ is prime. The more general case is discussed in
Appendix D.

${\cal
C}_{F}(X_{4})$$\Phi_{\mathbf{n}}(M_{2})$$\Phi_{F^{T}\mathbf{n}}(M_{2})$${\cal
C}_{F}(X_{4})$$\Phi_{(\mathds{1}-F^{T})\mathbf{n}}(M_{2})$ Figure 8: The
defects ${\cal C}_{F}(X_{4})$ can absorb Wilson surfaces
$\Phi_{(\mathds{1}-F^{T})\mathbf{n}}(M_{2})$, and hence should be condensates
of them.

With this motivation, we may now define the following condensation defect

$\displaystyle{\cal
C}_{F}(X_{4}):={|H^{0}(X_{4},{\mathbb{Z}}_{p})|^{2g}\over|H^{1}(X_{4},{\mathbb{Z}}_{p})|^{2g}}\sum_{M_{2}^{1},\dots,M_{2}^{2g}\in
H_{2}(X_{4},{\mathbb{Z}}_{p})}e^{{2\pi i\over p}{1\over
2}\sum_{i,j=1}^{2g}R^{F}_{ij}(M_{2}^{i},M_{2}^{j})}\prod_{i=1}^{2g}\Phi_{(\mathds{1}-F^{T})_{i}}(M_{2}^{i})\leavevmode\nobreak\
.$ (5.36)

The $R^{F}_{ij}$ are $F$-dependent coefficients, whose explicit form will be
given shortly. In practice, it is actually more useful to rewrite the
operators $\Phi_{(\mathds{1}-F^{T})_{i}}(M_{2})$ appearing above in terms of
products of the elementary operators $\widehat{\Phi}_{j}(M_{2})$ and
$\Phi_{j}(M_{2})$ introduced previously. To do so, we write

$\displaystyle\Phi_{(\mathds{1}-F^{T})_{i}}(M_{2}^{i})=\Phi(M_{2}^{i}\times\Gamma_{i})\leavevmode\nobreak\
,\hskip
36.135pt\Gamma_{i}:=\sum_{j=1}^{g}\alpha^{i}_{j}\lambda_{j}+\sum_{j=1}^{g}\beta_{j}^{i}\mu_{j}\leavevmode\nobreak\
.$ (5.37)

The coefficients $\alpha^{i}_{j}$ and $\beta^{i}_{j}$ are fixed in terms of
the matrix elements of $F$ straightforwardly,

$\displaystyle
F=\mathds{1}-\left(\begin{matrix}\alpha^{1}_{1}&\dots&\alpha_{1}^{2g}\\\
\vdots&&\vdots\\\ \alpha^{1}_{g}&\dots&\alpha_{g}^{2g}\\\
\beta^{1}_{1}&\dots&\beta_{1}^{2g}\\\ \vdots&&\vdots\\\
\beta^{1}_{g}&\dots&\beta_{g}^{2g}\end{matrix}\right)\leavevmode\nobreak\ .$
(5.38)

Using (5.33) and the commutation relations (5.9), we may then expand

$\displaystyle\prod_{i=1}^{2g}\Phi_{(\mathds{1}-F^{T})_{i}}(M_{2}^{i})$
$\displaystyle=$ $\displaystyle\mathrm{exp}\left\\{{2\pi i\over
p}\left[{1\over
2}\sum_{i=1}^{2g}\alpha^{i}\cdot\beta^{i}(M_{2}^{i},M_{2}^{i})+\sum_{i<j}^{2g}\beta^{i}\cdot\alpha^{j}(M_{2}^{i},M_{2}^{j})\right]\right\\}$
(5.39) $\displaystyle\hskip
36.135pt\times\prod_{j=1}^{g}\widehat{\Phi}_{j}\left(\sum_{i=1}^{2g}\alpha_{j}^{i}M_{2}^{i}\right)\prod_{j=1}^{g}\Phi_{j}\left(\sum_{i=1}^{2g}\beta_{j}^{i}M_{2}^{i}\right)$

with $\alpha^{i}\cdot\beta^{i}:=\sum_{j=1}^{2g}\alpha^{i}_{j}\beta^{i}_{j}$,
whence the surgery defect may be rewritten as

$\displaystyle{\cal C}_{F}(X_{4})$ $\displaystyle=$
$\displaystyle{|H^{0}(X_{4},{\mathbb{Z}}_{p})|^{2g}\over|H^{1}(X_{4},{\mathbb{Z}}_{p})|^{2g}}\sum_{M_{2}^{1},\dots,M_{2}^{2g}\in
H_{2}(X_{4},{\mathbb{Z}}_{p})}e^{{2\pi i\over p}{1\over
2}\sum_{i,j=1}^{2g}(\beta^{i}\cdot\alpha^{j}+R^{F}_{ij})(M_{2}^{i},M_{2}^{j})}$
(5.40) $\displaystyle\hskip
57.81621pt\times\,\prod_{j=1}^{g}\widehat{\Phi}_{j}\left(\sum_{i=1}^{2g}\alpha_{j}^{i}M_{2}^{i}\right)\prod_{j=1}^{g}\Phi_{j}\left(\sum_{i=1}^{2g}\beta_{j}^{i}M_{2}^{i}\right)\leavevmode\nobreak\
.$

Having introduced this notation, we may now give the expression for the
coefficients $R^{F}_{ij}$ in terms of $\alpha_{i}^{j}$ and $\beta_{i}^{j}$,

$\displaystyle R^{F}_{ij}=\left\\{\begin{matrix}-\beta^{i}_{j}&&j\leq g\\\
\alpha_{j-g}^{i}&&j>g\end{matrix}\right.$ (5.41)

which completes the definition of the surgery defects ${\cal C}_{F}(X_{4})$.

Let us also note that this defect can be written in a more streamlined fashion
by defining it in terms of a sum over a single manifold $M_{2}$ in
$H_{2}(X_{4},{\mathbb{Z}}_{p}^{2g})$, as opposed to $2g$ manifolds $M_{2}^{i}$
in $H_{2}(X_{4},{\mathbb{Z}}_{p})$,

$\displaystyle{\cal
C}_{F}(X_{4})={|H^{0}(X_{4},{\mathbb{Z}}_{p}^{2g})|\over|H^{1}(X_{4},{\mathbb{Z}}_{p}^{2g})|}\sum_{M_{2}\in
H_{2}(X_{4},\mathbb{Z}_{p}^{2g})}\exp(\frac{2\pi
i}{2p}\expectationvalue{FM_{2},M_{2}})\Phi((\mathds{1}-F)M_{2})\leavevmode\nobreak\
.$ (5.42)

The notation here is such that $F$ now acts on the coefficient system of the
manifold, instead of on the manifolds themselves. We discuss this expression
in more detail in Appendix D.

We may similarly define the conjugate defect $\overline{{\cal
C}_{F}}(X_{4}):=\chi(X_{4},{\mathbb{Z}}_{p})^{-2g}{\cal
C}_{F}(X_{4})^{\dagger}$, where we have used the freedom to normalize this
operator with an Euler counterterm. The dagger acts as complex conjugation,
and also transposes the order of operators. With these definitions, one finds
the following fusion rules,

$\displaystyle{\cal C}_{F}(X_{4})\times\overline{{\cal C}_{F}}(X_{4})$
$\displaystyle=$ $\displaystyle\overline{{\cal C}_{F}}(X_{4})\times{\cal
C}_{F}(X_{4})\,\,=\,\,\mathds{1}\leavevmode\nobreak\ ,$
$\displaystyle\Phi_{\mathbf{n}}(M_{2})\times{\cal C}_{F}(X_{4})$
$\displaystyle=$ $\displaystyle{\cal
C}_{F}(X_{4})\times\Phi_{F^{T}\mathbf{n}}(M_{2})\leavevmode\nobreak\ ,$
$\displaystyle\Phi_{F^{T}\mathbf{n}}(M_{2})\times\overline{{\cal
C}_{F}}(X_{4})$ $\displaystyle=$ $\displaystyle\overline{{\cal
C}_{F}}(X_{4})\times\Phi_{\mathbf{n}}(M_{2})\leavevmode\nobreak\ .$ (5.43)

In particular, we see that the fusions are generically non-Abelian.

An important point is that the fusion rules above are valid only subject to
certain constraints on the matrix $F$. In particular, explicit computation
leads one to conclude that the following constraints on $\alpha^{i}_{j}$,
$\beta^{i}_{j}$ must hold,

$\displaystyle\alpha^{i}\cdot\beta^{j}-\beta^{i}\cdot\alpha^{j}$
$\displaystyle=$ $\displaystyle\beta_{i}^{j}-\beta_{j}^{i}$
$\displaystyle\alpha^{i}\cdot\beta^{j+g}-\beta^{i}\cdot\alpha^{j+g}$
$\displaystyle=$ $\displaystyle\beta_{i}^{j+g}+\alpha_{j}^{i}$
$\displaystyle\alpha^{i+g}\cdot\beta^{j+g}-\beta^{i+g}\cdot\alpha^{j+g}$
$\displaystyle=$ $\displaystyle\alpha_{j}^{i+g}-\alpha_{i}^{j+g}$ (5.44)

Remarkably, these conditions are satisfied exactly when $F$ is symplectic.
Indeed, if $F$ is symplectic then $F^{T}\mathfrak{I}F=\mathfrak{I}$, and hence

$\displaystyle(\mathds{1}-F)^{T}\mathfrak{I}(\mathds{1}-F)=\mathfrak{I}(\mathds{1}-F)+(\mathds{1}-F)^{T}\mathfrak{I}\leavevmode\nobreak\
.$ (5.45)

This gives a series of equations which are quadratic in $\alpha^{i}_{j}$,
$\beta^{i}_{j}$ on the left and linear in $\alpha^{i}_{j}$, $\beta^{i}_{j}$ on
the right, which are precisely the equations in (5.4).

${N\over 2\pi}\sum_{i=1}^{g}\int b_{i}d\widehat{b_{i}}$$\langle L|$$|{\cal
T}^{N,g,n}[\Omega]\rangle$${\cal C}_{F}$Expand / Shrink${\cal
T}^{N,g,n}_{L}[\Omega]$${\cal T}^{N,g,n}_{F(L)}[F\Omega]/\langle
F\rangle$${\cal N}_{F}$ Figure 9: When the surgery defect ${\cal C}_{F}$ is
given a topological boundary, one obtains a twist defect. Shrinking the slab,
this twist defect becomes an $|F|$-ality interface ${\cal N}_{F}$ in $(3+1)$d.

For every element of $F\in Sp(2g,{\mathbb{Z}}_{p})$, we have thus constructed
codimension-1 defects implementing the symmetry in the bulk 5d TQFT. These
condensation defects can now be used to construct $|F|$-ality defects in the
boundary 4d gauge theories. Indeed, each of the condensation defects above
admits at least one topological boundary condition (namely, Dirichlet boundary
conditions for the condensate) which allows us to terminate them in the bulk.
The result can be thought of as a non-genuine 3-manifold defect attached to a
4-manifold. Such non-genuine operators were referred to as “twist defects” in
[135, 136, 59], and we denote them by $T_{F}(M_{3},M_{4})$. Concretely, we
have

$\displaystyle
T_{F}(M_{3},M_{4}):={|H^{0}(M_{4},M_{3},{\mathbb{Z}}_{p})|^{2g}\over|H^{1}(M_{4},M_{3},{\mathbb{Z}}_{p})|^{2g}}\sum_{M_{2}\in
H_{2}(M_{4},\mathbb{Z}_{p}^{2g})}\exp(\frac{2\pi
i}{2p}\expectationvalue{FM_{2},M_{2}})\Phi((1-F)M_{2})\leavevmode\nobreak\ ,$
(5.46)

where the only difference with the expression for the condensation defect
${\cal C}_{F}(M_{4})$ is in the overall normalization factor, which is now in
relative cohomology.292929Note however that the sum is still over absolute
homology, since the Lefschetz dual of relative cohomology is absolute
homology. The fusion rules follow simply from those above. By moving the twist
defects to the boundary as shown in Figure 9, one obtains an interface between
the four-dimensional theory,

$\displaystyle{\cal N}_{F}(M_{3}):=T_{F}(M_{3},M_{4})|_{y\rightarrow
0}\leavevmode\nobreak\ ,$ (5.47)

which fuse as

$\displaystyle{\cal N}_{F}(M_{3})\times\overline{{\cal
N}_{F}}(M_{3})={1\over|H^{0}(M_{3},{\mathbb{Z}}_{p})|^{2g}}\sum_{M_{2}^{1},\dots,M_{2}^{2g}\in
H_{2}(M_{3},{\mathbb{Z}}_{p})}\prod_{j=1}^{g}\widehat{\Phi}_{j}\left(\sum_{i=1}^{2g}\alpha_{j}^{i}M_{2}^{i}\right)\prod_{j=1}^{g}\Phi_{j}\left(\sum_{i=1}^{2g}\beta_{j}^{i}M_{2}^{i}\right)\leavevmode\nobreak\
.$ (5.48)

One can further obtain the fusion rules of ${\cal N}_{F}(M_{3})$ with ${\cal
N}_{F}(M_{3})$ or ${\cal N}_{G}(M_{3})$ for generic $G\in
Sp(2g,{\mathbb{Z}}_{p})$, thereby proving that they are $|F|$-ality defects as
claimed. Since these fusion rules will appear in an upcoming work [137] we do
not expand on them here.

## Acknowledgements

We would like to thank Christian Copetti, Luis Diogo, Alice Hedenlund, Kantaro
Ohmori, Jian Qiu, Gabi Zafrir, and Yunqin Zheng for many useful conversations.
We also thank Yunqin Zheng for a careful reading of the manuscript. The work
of MDZ and AH is supported by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program (grant agreement
No. 851931). MDZ also acknowledges support from the Simons Foundation Grant
$\\#$888984 (Simons Collaboration on Global Categorical Symmetries). VB is
supported from the Knut and Alice Wallenberg Foundation under grant KAW
2021.0170, VR grant 2018-04438 and the Olle Engkvists Stiftelse grant n.
2180108.

##  Appendix A Symmetry generators at genus $2$

In this appendix we give explicit matrix representation of the generators of
isometries for genus 2 Riemann surfaces. These matrices were also given in
[126]. First, the matrices appearing in Table 2 are given by,

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}0&-1&-1&-1\\\ 0&0&-1&0\\\ 0&0&0&-1\\\
1&0&0&1\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
36.135ptM_{1}=\left(\begin{matrix}0&0&0&1\\\ 0&0&1&1\\\ 1&-1&0&0\\\
-1&0&0&0\end{matrix}\right)\leavevmode\nobreak\ ,$ $\displaystyle M_{2}$
$\displaystyle=$ $\displaystyle\left(\begin{matrix}0&0&1&1\\\ 0&0&1&0\\\
0&-1&0&0\\\ -1&1&0&0\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
36.135ptM_{3}=\left(\begin{matrix}0&0&0&1\\\ 0&0&-1&0\\\ 0&-1&0&0\\\
1&0&0&0\end{matrix}\right)\leavevmode\nobreak\ ,$ $\displaystyle M_{4}$
$\displaystyle=$ $\displaystyle\left(\begin{matrix}0&0&1&0\\\ 0&0&0&1\\\
-1&0&-1&0\\\ 0&-1&0&0\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
36.135ptM_{5}=\left(\begin{matrix}0&0&1&0\\\ 0&-1&0&0\\\ -1&0&0&0\\\
0&0&0&-1\end{matrix}\right)\leavevmode\nobreak\ ,$ $\displaystyle M_{6}$
$\displaystyle=$ $\displaystyle\left(\begin{matrix}0&0&-1&0\\\ 0&0&0&1\\\
1&0&0&0\\\ 0&-1&0&0\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
36.135ptM_{7}=\left(\begin{matrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{matrix}\right)\leavevmode\nobreak\ ,$ $\displaystyle M_{8}$
$\displaystyle=$ $\displaystyle\left(\begin{matrix}-1&1&1&0\\\ 1&0&0&1\\\
-1&0&0&0\\\ 1&-1&0&1\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
36.135ptM_{9}=\left(\begin{matrix}0&0&0&-1\\\ 1&0&1&0\\\ 0&1&0&1\\\
-1&0&0&0\end{matrix}\right)\leavevmode\nobreak\ ,$ $\displaystyle M_{10}$
$\displaystyle=$ $\displaystyle\left(\begin{matrix}0&0&1&0\\\ 0&0&0&-1\\\
-1&0&-1&0\\\ 0&1&0&1\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
36.135ptC=\left(\begin{matrix}-1&0&0&0\\\ 0&-1&0&0\\\ 0&0&-1&0\\\
0&0&0&-1\end{matrix}\right)\leavevmode\nobreak\ .$ (A.1)

In addition, the remaining matrices appearing in Tables 3 and 4 are given by

$\displaystyle N_{1}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}0&0&1&0\\\ 0&1&0&0\\\ -1&0&0&0\\\
0&0&0&1\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
36.135ptN_{2}=\left(\begin{matrix}0&1&-1&0\\\ -1&0&0&1\\\ 0&0&0&1\\\
0&0&-1&0\end{matrix}\right)\leavevmode\nobreak\ ,$ $\displaystyle N_{3}$
$\displaystyle=$ $\displaystyle\left(\begin{matrix}0&-1&0&0\\\ 1&0&0&0\\\
0&0&0&-1\\\ 0&0&1&0\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
36.135ptN_{4}=\left(\begin{matrix}1&0&1&0\\\ 0&1&0&0\\\ -1&0&0&0\\\
0&0&0&1\end{matrix}\right)\leavevmode\nobreak\ ,$ $\displaystyle N_{5}$
$\displaystyle=$ $\displaystyle\left(\begin{matrix}-1&1&0&0\\\ 0&1&0&0\\\
0&0&-1&0\\\ 0&0&1&1\end{matrix}\right)\leavevmode\nobreak\ ,\hskip
36.135ptN_{6}=\left(\begin{matrix}1&0&0&0\\\ 1&-1&0&0\\\ 0&0&1&1\\\
0&0&0&-1\end{matrix}\right)\leavevmode\nobreak\ ,$ $\displaystyle P_{1}$
$\displaystyle=$ $\displaystyle\left(\begin{matrix}1&0&0&0\\\ 0&-1&0&0\\\
0&0&1&0\\\ 0&0&0&-1\end{matrix}\right)\leavevmode\nobreak\ .$ (A.2)

##  Appendix B Some algebra

In this appendix we collect and give details on various arithmetic statements
that are used in the main text.

###  B.1 Roots of unity over finite fields

Over the complex numbers the polynomial $x^{n}-1$ has $n$ roots, each with
multiplicity one. We can ask if the same is true over an extension of
$\mathbb{F}_{p}$ which splits $x^{n}-1$.

Over any field, if a polynomial $P(x)$ has a root $\lambda$ with multiplicity
greater than $1$, then $\lambda$ is also a root of $\derivative{x}P(x)$. For
$x^{n}-1$ this means that there are roots with multiplicity greater than $1$
if and only if $n\lambda^{n-1}$ = 0. Since $\lambda^{n-1}$ can’t be zero, this
is possible only if $n=0\mod p$. Hence if $p$ is not a factor of $n$, there
are $n$ distinct roots of unity.

Since the product of two $n$-th roots of unity is another $n$-th root of
unity, $n$-th roots of unity form a group. This group is always cyclic, just
like the group of complex $n$-th roots of unity. To see this let us assume the
contrary, namely that there exist two $n$-th roots of unity $\sigma$ and
$\kappa$ such that the intersection of their orbits is the identity, and such
that they generate a subgroup which is not cyclic. Let $k$ be the order of
$\sigma$ and $l$ the order of $\kappa$. Non-cyclicity means that
$\gcd(k,l)=m>1$. Hence, $\sigma^{\frac{k}{m}}$ and $\kappa^{\frac{l}{m}}$ both
generate a subgroup of order $m$. Unique factorization of $x^{m}-1$ then
demands that $\sigma^{\frac{k}{m}}=\kappa^{\frac{l}{m}}$, which contradicts
the assumption that the orbits do not intersect.

Hence if $p$ is not a factor of $n$, the group of $n$-th roots of unity is a
cyclic group of order $n$.

Let us next consider the case where $p$ is a factor of $n$, namely $n=p^{k}m$
with $p$ not dividing $m$. Since

$\displaystyle\binom{p^{k}}{\ell}=0\mod p\hskip 36.135pt\mbox{ if }\hskip
36.135pt\ell\neq 0,\,p^{k}\leavevmode\nobreak\ $ (B.1)

we have
$(x^{m}-1)^{p^{k}}=\sum_{\ell=0}^{p^{k}}\binom{p^{k}}{\ell}(-1)^{p^{k}-\ell}x^{m\ell}=x^{n}-1$
mod $p$. So the roots of $x^{n}-1$ are roots of $x^{m}-1$ with multiplicity
$p$. Since $p$ is not a factor of $m$, the group of $n=p^{k}m$ roots of unity
is a cyclic group of order $m$.

Noting further that

$\displaystyle x^{n}-1=\prod_{k|n}\phi_{k}(x)\leavevmode\nobreak\ $ (B.2)

with $\phi_{k}(x)$ the $k$-th cyclotomic polynomial, we can also determine the
factorization of $\phi_{n}(x)$ from this. When $p$ is not a factor of $n$, the
roots of $\phi_{n}(x)$ are primitive $n$-th roots of unity. More explicitly,
if $\lambda$ is the generator of $n$-th roots of unity then the roots of
$\phi_{n}$ are $\lambda^{r}$ with $\gcd(r,n)=1$.

When $n=p^{k}m$, (B.2) along with an inductive argument show that

$\displaystyle\phi_{n}(x)=(\phi_{m}(x))^{\varphi(p^{k})}\leavevmode\nobreak\
.$ (B.3)

In this case the roots of $\phi_{n}(x)$ are the primitive $m$-th roots of
unity, each with multiplicity $\varphi(p^{k})=p^{k}-p^{k-1}$.

###  B.2 Integer matrices of finite order

Let $F$ be a matrix of order $k$ with integer coefficients. Then as a complex
matrix it has a Jordan normal form; see the discussion around (3.9) for the
definition of Jordan normal form. Each Jordan block corresponds to an
eigenvalue $\lambda$ and contains an eigenvector $v$ with $\lambda$ as its
eigenvalue. The condition that $F$ has order $k$ then translates to,

$\displaystyle F^{k}v=\lambda^{k}v=v\leavevmode\nobreak\ .$ (B.4)

Hence $\lambda$ must be a $k$-th root of unity. Furthermore, every Jordan
block must be of size $1$, i.e. $F$ is diagonalizable as a complex matrix. To
see this, let us suppose that a Jordan block with eigenvalue $\lambda$ is not
of size 1. Then in addition to the eigenvector $v$, there is a vector $u$ such
that

$\displaystyle Fu=\lambda u+v\leavevmode\nobreak\ .$ (B.5)

Applying $F$ repeatedly, we deduce

$\displaystyle F^{k}u=\lambda^{k}u+k\lambda^{k-1}v\leavevmode\nobreak\ .$
(B.6)

Since $F^{k}=\mathds{1}$, the above condition tells us that
$k\lambda^{k-1}v=0$, which is impossible. Thus all integers matrices of finite
order are diagonalizable as complex matrices. Things will be different when we
consider matrices over finite fields.

Since $F$ is an integer matrix, its characteristic polynomial $P_{F}(x)$ is an
integer polynomial. Let $\lambda_{i}$ be the (complex) eigenvalues of $F$.
Then by definition

$\displaystyle P_{F}(x)=\prod_{i}(x-\lambda_{i})\leavevmode\nobreak\ .$ (B.7)

As explained above, all eigenvalues of $F$ are roots of unity. Hence the only
possible integer eigenvalues are $\pm 1$. All other eigenvalues are complex,
and their product must give an integer polynomial. Complex numbers which are
roots of monic polynomials with integer coefficients are known as “algebraic
integers”, and hence the eigenvalues are all algebraic integers.

Every algebraic integer $\lambda$ has a corresponding minimal polynomial
$\phi(x)$, which is the unique monic polynomial of the smallest degree such
that $\lambda$ is a root of $\phi(x)$. If $P(x)$ is any other integer
polynomial with root $\lambda$, then the minimal polynomial $\phi(x)$ of
$\lambda$ must be a factor of $P(x)$. Roots of unity are algebraic integers
since they are roots of $x^{n}-1$. The minimal polynomial for a primitive
$k$-th root of unity is the cyclotomic polynomial $\phi_{k}(x)$. As a result,
the characteristic polynomial $P_{F}(x)$ of a finite order integer matrix $F$
must be a product of cyclotomic polynomials. Hence if one primitive $k$-th
root of unity is a root of $P_{F}(x)$, then all other primitive $k$-th roots
of unity must also be roots, all with the same multiplicity.

###  B.3 Intersections and the Jordan blocks

Recall that the action of a modular transformation on the cycles in
$H_{1}(\Sigma_{g,0},\mathbb{F}_{p})$ is represented by a matrix $F\in
Sp(2g,\mathbb{F}_{p})$. This means that $F$ preserves the symplectic form
$\mathfrak{I}$,

$\displaystyle F^{T}\mathfrak{I}F=\mathfrak{I}\leavevmode\nobreak\ .$ (B.8)

If we work over the splitting field of the characteristic polynomial of $F$,
the cycles organize themselves into Jordan blocks which are (partially)
labelled by the eigenvalues $\lambda$ of $F$. A Jordan block with eigenvalue
$\lambda$ is spanned by vectors $v^{(i)}_{\lambda}$ such that

$\displaystyle Fv^{(0)}$ $\displaystyle=\lambda v^{(0)}\leavevmode\nobreak\ ,$
$\displaystyle Fv^{(i)}$ $\displaystyle=\lambda
v^{(i)}+v^{(i-1)}\leavevmode\nobreak\ .$ (B.9)

In particular $v^{(0)}_{\lambda}$ is an eigenvector with eigenvalue $\lambda$.
Since $F$ satisfies (B.8), we have

$\displaystyle(v^{(0)}_{\lambda_{1}})^{T}\mathfrak{I}v^{(0)}_{\lambda_{2}}=(v^{(0)}_{\lambda_{1}})^{T}F^{T}\mathfrak{I}Fv^{(0)}_{\lambda_{2}}=\lambda_{1}\lambda_{2}(v^{(0)}_{\lambda_{1}})^{T}\mathfrak{I}v^{(0)}_{\lambda_{2}}\leavevmode\nobreak\
,$

which then implies
$(\lambda_{1}\lambda_{2}-1)(v^{(0)}_{\lambda_{1}})^{T}\mathfrak{I}v^{(0)}_{\lambda_{2}}=0$
mod $p$. Hence
$\expectationvalue{v_{\lambda_{1}}^{(0)},v_{\lambda_{2}}^{(0)}}=0$ unless
$\lambda_{2}=\lambda_{1}^{-1}$ mod $p$. Following the same steps iteratively,
we can obtain that
$\expectationvalue{v_{\lambda_{1}}^{(i)},v_{\lambda_{2}}^{(i)}}=0$ unless
$\lambda_{2}=\lambda_{1}^{-1}$ mod $p$ for any $i$. This means in particular
that Jordan blocks themselves are isotropic unless the eigenvalue $\lambda=\pm
1$.

Now let us apply this to integer matrices of finite order as above. As
explained, the characteristic polynomial of such a matrix is a product of
cyclotomic polynomials,

$\displaystyle
P_{F}(x)=(\phi_{n_{1}}(x))^{k_{1}}(\phi_{n_{2}}(x))^{k_{2}}\dots(\phi_{n_{l}}(x))^{k_{l}}$
(B.10)

The module $\mathbb{Z}^{2g}$ on which $F$ acts splits as

$\displaystyle\mathbb{Z}^{2g}\cong\Lambda_{1}\oplus\Lambda_{2}\oplus\dots\oplus\Lambda_{l}\leavevmode\nobreak\
,$ $\displaystyle\Lambda_{i}:=\ker(\Phi_{n_{i}}(F))\leavevmode\nobreak\ .$
(B.11)

This splitting relies on the diagonalizability of $F$. The $\Lambda_{i}$ are
invariant under the action of $F$, and using the intersections in diagonalized
form we can deduce that the vectors in $\Lambda_{i}$ do not intersect vectors
in $\Lambda_{j}$ unless $i=j$. Furthermore, we can split each $\Lambda_{i}$
into $k_{i}$ parts,

$\displaystyle\Lambda_{i}=\Lambda_{i}^{(1)}\oplus\Lambda_{i}^{(2)}\oplus\dots\oplus\Lambda_{i}^{(k_{i})}\leavevmode\nobreak\
,$ (B.12)

where each $\Lambda_{i}^{(1)}$ is of dimension $\varphi(n_{i})$ and is
annihilated by $\phi_{n_{i}}(F)$. This means that regarded as a complex vector
space, the space $\Lambda_{i}^{(a)}$ contains one eigenvector for each
primitive $n$-th root of unity, and furthermore vectors in $\Lambda_{i}^{(a)}$
don’t intersect those in $\Lambda_{i}^{(b)}$ unless $a=b$.

This remains true on passage from integers to finite fields. As a result,
whenever $\phi_{n_{i}}(x)$ has $\varphi(n_{i})$ distinct roots over
$\mathbb{F}_{p}$, each $\Lambda_{i}^{(a)}$ will contain one eigenvector for
each of these roots. On the other hand, if $p$ is a factor of $n$ there can be
non-trivial Jordan blocks. In fact, in this case each $\Lambda^{(a)}_{i}$
contains one Jordan block of size $\varphi(n)$ for each root of
$\phi_{\frac{n_{i}}{p^{k}}}(x)$ where $p^{k}$ is the highest power of $p$ that
divides $n_{i}$. This can be seen as a statement about the order of
$\widetilde{F}$, defined to be $F$ restricted to $\Lambda_{i}^{(a)}$. Over
integers $\widetilde{F}$ has order $n=mp^{k}$, while over $\mathbb{F}_{p}$ it
has this order iff there is a single Jordan block for each root of
$\phi_{m}(x)$. To see this, let us suppose that the order of $\widetilde{F}$
is $mp^{\ell}$ with $\ell\leq k$. In that case we must have that over
integers,

$\displaystyle\widetilde{F}^{mp^{\ell}}$ $\displaystyle=\mathds{1}+pN$ (B.13)

where $N$ is some non-zero matrix. Let us recall that over its splitting field
the matrix $\widetilde{F}$ is diagonalizable, with eigenvalues that are
primitive $n$-th roots of unity. The eigenvalues of
$\widetilde{F}^{mp^{\ell}}$ are then primitive $p^{k-\ell}$-th roots of unity,
each with multiplicity $\frac{\varphi(n)}{\varphi(p^{k-\ell})}$. Hence we
obtain

$\displaystyle\det(x-M^{mp^{\ell}})=\left(\phi_{p^{k-\ell}}(x)\right)^{\frac{\varphi(n)}{\varphi(p^{k-\ell})}}\Rightarrow\det(\mathds{1}-M^{mp^{\ell}})=p^{\frac{\varphi(n)}{\varphi(p^{k-\ell})}}$
(B.14)

where we have used the fact that $\phi_{p^{k}}(1)=p$ for $k>1$ and $p>2$. Now
going back to (B.13), we see that this means,

$\displaystyle p^{\varphi(n)}\det(N)$
$\displaystyle=p^{\frac{\varphi(n)}{\varphi(p^{k-\ell})}}$ (B.15)

Hence $N$ cannot be an integer matrix unless $k=\ell$.

We can now ask about the intersection number of a vector $v$ in a Jordan block
of eigenvalue $\lambda$ with another vector $u$ in a different Jordan block.
First we note that since $\mathfrak{I}$ is non-degenerate there must exist at
least one vector which intersects $v$, and from the discussion above it must
lie in a Jordan block of eigenvalue $\lambda^{-1}$. Now we can check that

$\displaystyle(v^{(0)}_{\lambda^{-1}})^{T}F^{T}\mathfrak{I}Fv^{(1)}_{\lambda}=(v^{(0)}_{\lambda^{-1}})^{T}\mathfrak{I}v^{(1)}_{\lambda}\hskip
21.68121pt\Rightarrow\hskip
21.68121pt(v^{(0)}_{\lambda^{-1}})^{T}v^{(0)}_{\lambda}=0\leavevmode\nobreak\
,$ (B.16)

i.e. if there is a non-trivial trivial Jordan block for $\lambda$, then the
eigenvector $v^{(0)}_{\lambda}$ cannot intersect $v^{(0)}_{\lambda^{-1}}$. The
argument above obviously does not work if $v^{(1)}_{\lambda}$ does not exist.
We can then iterate this computation and deduce that
$\expectationvalue{v_{\lambda}^{(0)},v_{\lambda^{-1}}^{(a)}}=0$, unless
$v_{\lambda^{-1}}^{(a+1)}$ doesn’t exist. In that case we need to compute
$(v^{(a+1)}_{\lambda^{-1}})^{T}F^{T}\mathfrak{I}Fv^{(0)}_{\lambda}$.
Similarly, by an appropriate sequence of computations of
$(v^{(a)}_{\lambda^{-1}})^{T}F^{T}\mathfrak{I}Fv^{(b)}_{\lambda}$, we can
conclude that $\expectationvalue{v^{(a)}_{\lambda},v^{(b)}_{\lambda^{-1}}}=0$
unless $a+b-1$ is greater than or equal to the size of the Jordan blocks, i.e
$\varphi(p^{k})$. In fact, this can be simplified even further: the
$v^{(a)}_{\lambda}$ are not uniquely determined by (B.3). In fact we can
change $v^{(a)}_{\lambda}$ by any arbitrary linear combination of
$v^{(b)}_{\lambda}$ with $b<a$, and (B.3) will still hold. By using this
freedom we can choose $v^{(b)}_{\lambda}$ in such a way that the non-zero
intersection numbers are
$\expectationvalue{v_{\lambda}^{(a)},v_{\lambda^{-1}}^{(\varphi(p^{k})-1-a)}}$.

###  B.4 Factorization $\phi_{15}(x)$ of over finite fields

We now focus on the study of invariant subspaces of $\phi_{15}(x)$, which is
relevant to the study of global forms of genus 4 class ${\cal S}$ theories
invariant under a certain order 15 symmetries, discussed in Section 4.3.2.

Passing to the splitting field of $\phi_{15}(x)$, we obtain a primitive
fifteenth root of unity $\lambda$. The roots of $\phi_{15}(x)$ are then:
$\lambda,\lambda^{2},\lambda^{4},\lambda^{7},\lambda^{8},\lambda^{11},\lambda^{13},$
and $\lambda^{14}$. Since $\lambda^{3}$ is a primitive fifth root of unity and
$\lambda^{5}$ is a primitive third root of unity, we obtain two relations,

$\displaystyle\phi_{3}(\lambda^{5})$
$\displaystyle\,\,=\,\,\lambda^{10}+\lambda^{5}+1=0\leavevmode\nobreak\ ,$
$\displaystyle\phi_{5}(\lambda^{3})$
$\displaystyle\,\,=\,\,\lambda^{12}+\lambda^{9}+\lambda^{6}+\lambda^{3}+1=0\leavevmode\nobreak\
.$ (B.17)

The multiplicative groups ${\mathbb{Z}}_{15}^{\times}$ is not cyclic, and as a
result there is no element of order $8$ in it. Thus over any $\mathbb{F}_{p}$
we can write $\phi_{15}(x)=P(x)Q(x)$ where $P(x),Q(x)$ are polynomials over
$\mathbb{F}_{p}$ that can potentially factorize further. We want to determine
when they determine isotropic invariant global form. Here we need to divide
into three cases,

  1. 1.

We first assume that the constant term of $P(x)$ and $Q(x)$ is $1$, since that
is only power of $\lambda$ that is in $\mathbb{F}_{p}$ for all $p$, and hence
the only value allowed without further assumptions on $p$. The only possible
factorization satisfying this and the isotropy criterion is:

$\displaystyle P(x)$ $\displaystyle=$
$\displaystyle(x-\lambda)(x-\lambda^{2})(x-\lambda^{4})(x-\lambda^{8})\leavevmode\nobreak\
,$ $\displaystyle Q(x)$ $\displaystyle=$
$\displaystyle(x-\lambda^{7})(x-\lambda^{11})(x-\lambda^{13})(x-\lambda^{14})\leavevmode\nobreak\
.$ (B.18)

These are polynomials over $\mathbb{F}_{p}$ if
$\eta:=\lambda+\lambda^{2}+\lambda^{4}+\lambda^{8}\in\mathbb{F}_{p}$. Using
the relations (B.4), it can be checked that $\eta$ satisfies,

$\displaystyle\eta^{2}-\eta+4=0\hskip 21.68121pt\Rightarrow\hskip
21.68121pt\eta=\frac{1\pm\sqrt{-15}}{2}\leavevmode\nobreak\ .$ (B.19)

Hence $P(x)$ and $Q(x)$ are polynomials over $\mathbb{F}_{p}$ whenever $-15$
has a square root in $\mathbb{F}_{p}$. Using the Legendre symbol, this can be
seen to happen whenever $p\in 15\mathbb{N}+k$ for $k=1,2,4,$ and $8$. Notice
that when $k=1,4$ there are also other values available for the constant term
of $P(x)$ and $Q(x)$. This means that $P(x)$ and $Q(x)$ can split further for
these cases, and as we will see below that they do. For the other cases
$k=2,8$ there are exactly 2 invariant global forms.

  2. 2.

If $p\in 3\mathbb{N}+1$ then $\lambda^{5}\in\mathbb{F}_{p}$ and hence $P(x)$
and $Q(x)$ can have a constant term which is a power of $\lambda^{5}$. As a
result, in addition to the possibility given in (1), there is another
possibility,

$\displaystyle P(x)$ $\displaystyle=$
$\displaystyle(x-\lambda)(x-\lambda^{4})(x-\lambda^{7})(x-\lambda^{13})\leavevmode\nobreak\
,$ $\displaystyle Q(x)$ $\displaystyle=$
$\displaystyle(x-\lambda^{2})(x-\lambda^{8})(x-\lambda^{11})(x-\lambda^{14})\leavevmode\nobreak\
.$ (B.20)

Using (B.4), it can be verified that

$\displaystyle(-\lambda-\lambda^{4}-\lambda^{7}-\lambda^{13})^{3}=(-\lambda^{2}-\lambda^{8}-\lambda^{11}-\lambda^{14})^{3}=1\leavevmode\nobreak\
.$ (B.21)

This together with (B.4) can then be used to show that $P(x)$ and $Q(x)$ are
polynomials over $\mathbb{F}_{p}$. This means that when $p\in 15\mathbb{N}+7$
or $15\mathbb{N}+13$, there are 2 invariant global forms. For $p\in
15\mathbb{N}+4$ there are 4.

  3. 3.

Finally if $p\in 5\mathbb{N}+1$, then $\lambda^{3}\in\mathbb{F}_{p}$. Moreover
using (B.4) we can show that,

$\displaystyle(-\lambda-\lambda^{11})^{5}=(-\lambda^{2}-\lambda^{7})^{5}=(-\lambda^{4}-\lambda^{14})^{5}=(-\lambda^{8}-\lambda^{13})^{5}=1\leavevmode\nobreak\
.$ (B.22)

Hence all of the four sums are fifth roots of unity, and thus in
$\mathbb{F}_{p}$. This gives four factors of $\phi_{15}$ which are polynomials
over $\mathbb{F}_{p}$, namely $(x-\lambda)(x-\lambda^{11})$,
$(x-\lambda^{2})(x-\lambda^{7})$, $(x-\lambda^{4})(x-\lambda^{14})$, and
$(x-\lambda^{8})(x-\lambda^{13})$. In all 4 cases, the coefficient of both the
linear and constant term is a fifth root of unity. Hence there are $4$
invariant global forms when $p\in 15\mathbb{N}+6$ or $p\in 15\mathbb{N}+11$.

When $p\in 15\mathbb{N}+1$, all 3 cases above coincide and there are 8
invariant global forms. In the remaining case, i.e. when $p\in
15\mathbb{N}+14$, the factorization gives non-isotropic invariant forms and
hence in that case there are no invariant global forms.

###  B.5 Composite characteristic polynomial

Finally, in this appendix we make some comments on the case in which the
characteristic polynomial of $F$ is not a cyclotomic polynomial, but a product
thereof. Concretely, take $P_{F}(x)=(\phi_{n}(x))^{k}$ with $n>2$. In this
case, over the complex numbers each of the $n$ eigenvalues of $P_{F}(x)$ has
multiplicity $k$. For the generic case when $p$ is not a factor of $n$, this
fact carries over to the field extension of $\mathbb{F}_{p}$ which splits
$P_{F}(x)$. Hence each eigenvalue $\lambda$ has an eigenspace of degeneracy
$k$. We can always choose a basis $v_{\lambda}^{(i)},1\leq i\leq k$ of these
spaces such that the non-zero intersection numbers are,303030The ordering in
the expression is arbitrary and its only purpose is to fix the sign of the
intersection pairing.

$\displaystyle\expectationvalue{v_{\lambda}^{(i)},v_{\lambda^{-1}}^{(i)}}=1\leavevmode\nobreak\
,$ $\displaystyle\hskip 36.135pt\mbox{ if
}\,\,\,\lambda>\lambda^{-1}\leavevmode\nobreak\ .$ (B.23)

This degeneracy means that often a large number of invariant subspaces exist.
In fact, if $k=2l$ is even then $F$ has invariant spaces for every prime field
$\mathbb{F}_{p}$. This is because any subspace spanned by $l$ eigenvectors of
each eigenvalue is annihilated by $(\phi_{n}(F))^{l}$ and hence is an
invariant subspace, and because of the intersections given in (B.31) many of
these are isotropic.

As an illustration let us consider the case of $k=2$. If $\phi_{n}(x)$ is
irreducible over $\mathbb{F}_{p}$, then an invariant global form consists of
precisely one vector from each of the eigenvalues $\lambda^{i}$, where $1\leq
i<n,\gcd(i,n)=1$ and $\lambda^{n}=1$. There are $p+1$ ways of picking an
eigenvector (up to scalar multiplication) with the eigenvalue $\lambda^{i}$
from the 2 dimensional eigenspace. Once we pick an eigenvector $v_{\lambda}$
for $\lambda^{i}$, then isotropy forces us to pick the unique (again up to
scalar multiplication) eigenvector $u_{\lambda^{-1}}$ for
$\lambda^{-i}=\lambda^{n-i}$ which does not intersect $u$. As a result, there
are $(p+1)^{\frac{\varphi(n)}{2}}$ invariant isotropic subspaces. If
$\phi_{n}(x)$ factorizes then more invariant global forms exist and can be
counted using similar arguments, we will refrain from a general exposition
here.

If $k$ is odd, then invariant global forms require factorization of
$\phi_{n}(x)$ over $\mathbb{F}_{p}$. The counting of the number of invariant
forms is tedious, but is a straightforward application of the arguments above
combined with those used in the preceding sections. The non-generic case when
$p$ is a factor of $n$ is yet more involved because there are multiple non-
trivial Jordan blocks.

Finally we discuss the most general case,

$\displaystyle P_{F}(x)=\prod_{n}(\phi_{n}(x))^{k_{n}}\leavevmode\nobreak\ .$
(B.24)

Over the complex numbers each $\ker(\phi_{n}(F)^{k_{n}})$ is an invariant
subspace and, moreover, a vector in such a subspace only intersects with other
vectors in the same subspace.

For the generic prime fields $\mathbb{F}_{p}$313131Generic means that if
$k_{n}\neq 0$ then $p$ is not a factor of $n$. This is the generic case
because for a given $F$, there are only finitely many primes not satisfying
this. this means that an invariant global form consists of invariant maximal
isotropic lattice from each of $\ker(\phi_{n}(F)^{k_{n}})$. As a result the
number of invariant global forms is a product: If we denote by
$N^{(p)}_{n_{1},k_{1};n_{2},k_{2};\dots;n_{a},k_{a}}$, the number of global
forms for,

$\displaystyle
P_{F}(x)=(\phi_{n_{1}}(x))^{k_{1}}\dots(\phi_{n_{a}}(x))^{k_{a}}\leavevmode\nobreak\
,$ (B.25)

then for generic $p$,

$\displaystyle
N^{(p)}_{n_{1},k_{1};n_{2},k_{2};\dots;n_{a},k_{a}}=N^{(p)}_{n_{1},k_{1}}N^{(p)}_{n_{2},k_{2}}\dots
N^{(p)}_{n_{a},k_{a}}\leavevmode\nobreak\ .$ (B.26)

This factorization essentially holds for non-generic $p$ too, but
multiplicities need to be adjusted to take into account the factorization
(4.4) and the more complicated structure of the Jordan blocks.

####  B.5.1 An all genus example

As an example with composite characteristic polynomials, we consider an
infinite family of fixed point indexed by genus $g$. This fixed point is a
genus $g$ surface obtained by identifying the diagonally opposite sides of a
regular $(4g+2)$-sided polygon in hyperbolic space. This surface has a
$\mathbb{Z}_{4g+2}$ isometry corresponding to rotation by $\frac{2\pi}{4g+2}$.
The non-zero entries for the matrix $F$ for the corresponding transformation
take the form (in a basis with non-standard intersection form, c.f. [39]),

$\displaystyle F_{i+1,i}=1\leavevmode\nobreak\ ,\hskip
21.68121ptF_{2g,i}=(-1)^{i}\leavevmode\nobreak\ ,\hskip
36.135pt0<i<2g\leavevmode\nobreak\ .$ (B.27)

The characteristic polynomial of $F$ can be evaluated using,

$\displaystyle\det(X)=\sum_{\pi\in
S_{n}}\operatorname{sign}(\pi)\prod_{i=1}^{n}X_{\pi(i),i}\leavevmode\nobreak\
$ (B.28)

where $X$ is an $n\times n$ matrix and $S_{n}$ is the group of permutations of
$n$ objects. Since $F$ has few non-zero entries, it is possible to enumerate
all the permutations $\pi$ that contribute to $\det(x-F)$. We do so by
identifying permutations contributing to a particular power of $x$. For
$x^{2n}$, the permutations that contribute to its coefficient are those that
fix precisely $2n$ of the $2g$ objects. The rest of the $2n-2g$ objects are
permuted among themselves. For such a permutation to have a non-zero
contribution we must have $\pi(i)=i+1$ for all those $2n-2g$ objects, except
for $i=2g$. Completing the cycle then requires $\pi(2g)=2g-2n+1$. This is an
odd permutation, and thus the coefficient is $1$.

Next consider the odd powers of $x$. For $x^{2n-1}$, the only permutation that
contributes is the identity permutation since the diagonal element
$F_{2g,2g}=1$, and hence the coefficient is $-1$. For other odd powers
$x^{2n+1}$ the only permutation that contributes is again the one consisting
of the cycle $\pi(i)=i+1$ for $i\geq 2g-2n$ and now $\pi(2g)=2g-2n$. This is
an even permutation and the coefficient is $-1$. Hence

$\displaystyle
P_{F}(x)=x^{2g}-x^{2g-1}+x^{2g-2}+\dots+x^{2}-x+1\leavevmode\nobreak\ .$
(B.29)

We can mod out by $\mathbb{Z}_{2}$ by considering $F^{2}$ as the generator of
$\mathbb{Z}_{2g+1}$. Since we know the eigenvalues from the characteristic
polynomial, it is straightforward to obtain that,

$\displaystyle
P_{F^{2}}(x)=x^{2g}+x^{2g-1}+\dots+1=\frac{x^{2g+1}-1}{x-1}\leavevmode\nobreak\
.$ (B.30)

Hence it is easy to obtain the expansion of $P_{F^{2}(x)}$ in terms of
cyclotomic polynomials,

$\displaystyle P_{F^{2}}(x)=\prod_{m\neq
1,m|2g+1}\phi_{m}(x)\leavevmode\nobreak\ .$ (B.31)

We can now easily determine when the polynomial $P_{F^{2}(x)}$ splits
completely over $\mathbb{F}_{p}$ when $p$ is not a divisor of $2g+1$. This
requires that for any divisor $m$ of $2g+1$, $p=2m+1$. Requiring that this
happens simultaneously for all the divisors means that
$p\in(2g+1)\mathbb{N}+1$ and in this case there are $2^{2g}$ invariant global
forms. We can also easily find a sufficient criteria for the existence of
intrinsic non-invertible symmetry. The group $\mathbb{Z}_{n}^{\times}$ is
cyclic if $n$ is a prime and there are $\varphi(n-1)$ generators in this
group. Hence if $p\in(2g+1)\mathbb{N}+k$ where $k$ is one of the $\varphi(2g)$
generators of $\mathbb{Z}_{n}^{\times}$ then the symmetry generated by $M$ is
intrinsically non-invertible. However whenever $p$ is not in one of these
cases, i.e. when $\phi_{2g+1}(x)$ is not irreducible or split over
$\mathbb{F}_{p}$, then we need the details of how it factors to decide whether
the symmetry is invertible or not.

The next simplest case is when $2g+1=q^{k}$ with $q$ a prime. Here the
sufficient condition for the invariant global forms to exist is that $p\in
q^{k}\mathbb{N}+1$. Since $\phi_{q}(x)$ is a factor of $P_{F^{2}}(x)$ we have
intrinsic non-invertible symmetry whenever $p\in(2g+1)\mathbb{N}+k$ where $k$
is one of $\varphi(q-1)$ generators of $\mathbb{Z}_{q}^{\times}$.

For the most general case we can write down a similar pair of sufficient
conditions for complete splitting and irreducibility of $P_{F}(x^{2})$ when
$2g+1=q_{1}^{k_{1}}\dots q_{m}^{k_{m}}$. When $p=2l(2g+1)+1$, then $2^{2g}$
invariant forms always exist. Similarly, if any of the cyclotomic polynomials
appearing in (B.31) are irreducible over $\mathbb{F}_{p}$ then the symmetry is
intrinsically non-invertible. So if $p\in(2g+1)\mathbb{N}+k$ where $k$
generates one of $\mathbb{Z}_{q_{i}}^{\times}$, then the symmetry is
intrinsically non-invertible.

##  Appendix C Basics of handlebodies

In this appendix we review some basic facts about handlebodies which will be
used in Section 5. These facts can be found in many standard math texts, such
as [133].

###  C.1 Solid tori

Handlebodies are a class of three manifolds obtained by “filling in” a genus
$g$ Riemann surface (for simplicity we will assume no punctures in this
appendix). The simplest case is filling in a torus to obtain a solid torus,
homeomorphic to $S^{1}\times D^{2}$.323232Not every three-manifold with torus
boundary is a solid torus. Indeed, for any knot $K$ we could consider the knot
exterior in $S^{3}$. For any $K$ besides the unknot this will not be a solid
torus, but rather a “cube with knotted hole”. More concretely, given a torus
$\Sigma_{1,0}$, we may construct a solid torus $V_{1,0}$ by choosing any curve
$\mu\in H_{1}(\Sigma_{1,0},{\mathbb{Z}})$ on $\Sigma_{1,0}$ to be the
“meridian”, i.e. the curve which becomes contractible in the handlebody. The
remaining generator of $H_{1}(\Sigma_{1,0},{\mathbb{Z}})$, which remains as a
non-trivial generator of $H_{1}(V_{1,0},{\mathbb{Z}})$, is referred to as the
“longitude”, and will be denoted by $\lambda$.

Meridians and longitudes are not on equal footing: all meridians of $V_{1,0}$
are ambiently isotopic, and hence it makes sense to discuss the meridian of
$V_{1,0}$, but there are infinitely many ambient isotopy classes of
longitudes.333333Any two longitudes are related by homeomorphism, though.
These statements can be better understood by first introducing the following
non-trivial theorem:

######  Theorem 3.

The set of isotopy classes $[\gamma]$ of non-trivial closed curves $\gamma$ on
$\Sigma_{1,0}$ is in bijection with ordered pairs of coprime integers $(m,n)$,
such that $[\gamma]$ corresponds to $(m,n)$ if and only if $\gamma$ is
homologous to $m\mu+n\lambda$.

Since $\mu$ is trivial in $\pi_{1}(V_{1,0})$, whereas $\lambda$ generates
$\pi_{1}(V_{1,0})$, a curve $\gamma$ is homotopically trivial in $V_{1,0}$ if
and only if it is labelled by $(m,n)=(\pm 1,0)$. Thus all homotopically
trivial curves in $V_{1,0}$, i.e. all meridians, are in a single isotopy class
(modulo change of orientation).

On the other hand, a curve $\gamma$ labelled by $(m,n)$ represents a generator
of $H_{1}(V_{1,0})$, and hence a longitude, if and only if $n=\pm 1$. This
means that any curve labelled by $(m,\pm 1)$ is a longitude. There are thus an
infinite number of them, labelled by integers $m$. Note that each $(m,\pm 1)$
can be obtained from the curve $(m,\pm 1)$ and doing $m$ Dehn twists around
the meridian.

###  C.2 Higher genus handlebodies

We may now proceed to the case of more general handlebodies $V_{g,0}$. In
order to specify the handlebody, we must again choose a set of $g$ generators
of $H_{1}(\Sigma_{g,0},{\mathbb{Z}})$ to be meridians, i.e. to be trivialized
in $H_{1}(V_{g,0},{\mathbb{Z}})$. Given a meridian $\mu_{1}$, note that any
$\gamma$ intersecting $\mu_{1}$ at a single point must be a longitude. Indeed,
we have the following,

######  Proposition 1.

Given two curves $\gamma_{1}$ and $\gamma_{2}$ with labels $(m_{1},n_{1})$ and
$(m_{2},n_{2})$, the algebraic intersection number is given by

$\displaystyle\langle\gamma_{1},\gamma_{2}\rangle=m_{1}n_{2}-n_{1}m_{2}\leavevmode\nobreak\
.$ (C.1)

This proposition implies that if $\langle\gamma_{1},\gamma_{2}\rangle=\pm 1$,
then we have $m_{1}n_{2}-n_{1}m_{2}=\pm 1$. Now if we assume that $\gamma_{1}$
is a meridian, we must have $(m_{1},n_{1})=(\pm 1,0)$, and hence we conclude
that $n_{2}=\pm 1$. This means that $\gamma_{2}$ is indeed a longitude.

In the current context, what this tells us is that when we choose our
meridians, we must choose them such that they have zero intersection pairing.
In other words, the choice of meridians is given by a choice of maximal
isotropic sublattice $L\subset H_{1}(\Sigma_{g,0},{\mathbb{Z}})$.

As we discussed above for solid tori, the set of meridians $L$ is a well-
defined property of the handlebody. On the other hand, there are many choices
for the set of longitudes. Each choice is related by shifts by the meridians.
The invariant quantity associated to the handlebody is thus not the set of
longitudes themselves, but the coset
$L^{\perp}=H_{1}(\Sigma_{g,0},{\mathbb{Z}})/L$.

##  Appendix D Alternative expression for ${\cal C}_{F}(X_{4})$

In this appendix we give details on the more streamlined expression for ${\cal
C}_{F}(X_{4})$ given in (D.5). The basic idea is to replace the $2g$ two-
cycles $M_{2}^{i}$ valued in $\mathbb{Z}_{p}$ with a single two-cycle $M_{2}$
valued in $\mathbb{Z}_{p}^{2g}$. Another way of saying the same thing is we
consider a single cycle in $H_{2}(X_{4},\mathbb{Z}_{p}^{2g})$ instead of $2g$
cycles in $H_{2}(X_{4},\mathbb{Z}_{p})$. This also makes direct contact with
the 6d picture since,343434As in the main text, we assume
$H_{1,3}(X_{4},{\mathbb{Z}})=0$.

$\displaystyle H_{3}(\Sigma_{g,0}\times X_{4},\mathbb{Z}_{p})\cong
H_{2}(X_{4},H_{1}(\Sigma_{g,0},\mathbb{Z}_{p}))\cong
H_{2}(X_{4},\mathbb{Z}_{p}^{2g})\leavevmode\nobreak\ .$ (D.1)

In this picture given $H_{2}(X_{4},\mathbb{Z}_{p}^{2g})$ we can define the
antisymmetric pairing $\expectationvalue{\cdot,\cdot}$ which is the
combination of the symmetric intersection pairing on homology with the
antisymmetric Dirac paring on $\mathbb{Z}_{p}^{2g}$. In class $\mathcal{S}$
theories, this is the same as the intersection pairing on
$H_{3}(\Sigma_{g,0},\mathbb{Z}_{p})$ by (D.1). As a result, the operators
$\Phi(M_{2})$ with $M_{2}\in H_{2}(X_{4},\mathbb{Z}_{p}^{2g})$ satisfy the
algebra in (5.9). There is also a symmetric pairing $(\cdot,\cdot)$ which is
the combination of symmetric intersection pairing with the canonical symmetric
pairing on the group $\mathbb{Z}_{p}^{2g}$ of coefficients. The two are
related by,

$\displaystyle\expectationvalue{M_{2},M^{\prime}_{2}}=(M_{2},\mathfrak{I}M^{\prime}_{2})\leavevmode\nobreak\
.$ (D.2)

We would like to construct a defect ${\cal C}_{F}(X_{4})$ that acts on
$\Phi(M_{2})$ with $M_{2}\in H_{2}(X_{4},\mathbb{Z}_{p}^{2g})$ as

$\displaystyle{\cal
C}_{F}(X_{4}):\,\,\,\Phi(M_{2})\mapsto\Phi(FM_{2})\leavevmode\nobreak\ ,\hskip
36.135ptM_{2}\subset X_{4}\leavevmode\nobreak\ .$ (D.3)

This is equivalent to the fusion rule,

$\displaystyle\Phi(M_{2})\times{\cal C}_{F}(X_{4})={\cal
C}_{F}(X_{4})\times\Phi(FM_{2})\leavevmode\nobreak\ ,\hskip
36.135ptM_{2}\subset X_{4}\leavevmode\nobreak\ .$ (D.4)

In the equations above $F$ is a $2g\times 2g$ matrix with coefficients in
$\mathbb{Z}_{p}$, and $FM_{2}$ represents an automorphism of the coefficient
system of $H_{2}(X_{4},\mathbb{Z}_{p}^{2g})$.

The most general form ${\cal C}_{F}(X_{4})$ can take is

$\displaystyle{\cal
C}_{F}(X_{4})={|H^{0}(X_{4},{\mathbb{Z}}_{p}^{2g})|\over|H^{1}(X_{4},{\mathbb{Z}}_{p}^{2g})|}\sum_{M_{2}\in
H_{2}(X_{4},\mathbb{Z}_{p}^{2g})}\exp(\frac{2\pi
i}{2p}\expectationvalue{M_{2},M_{2}}_{F})\Phi((\mathds{1}-F)M_{2})\leavevmode\nobreak\
,$ (D.5)

where $\expectationvalue{\cdot,\cdot}_{F}$ is a bilinear form which we will
determine by demanding that the fusion rules given by (D.4) are satisfied.
From (D.5), it is clear that only the symmetric part of it contributes. The
reason that we have included a factor of $(\mathds{1}-F)$ in
$\Phi((\mathds{1}-F)M_{2})$ is to implement folding trick, as discussed around
Figure 8. Another way to see that it should be there is to notice that if
$FM_{2}=M_{2}$ then the defect ${\cal C}_{F}(X_{4})$ should be transparent to
$M_{2}$, and hence $\Phi(M_{2})$ should drop out of the condensate.

We now fix the bilinear pairing $\expectationvalue{\cdot,\cdot}_{F}$ by
evaluating $\Phi(\widetilde{M_{2}})\times{\cal C}_{F}(X_{4})$, which gives

$\displaystyle{|H^{1}(X_{4},{\mathbb{Z}}_{p}^{2g})|\over|H^{0}(X_{4},{\mathbb{Z}}_{p}^{2g})|}$
$\displaystyle\Phi(\widetilde{M_{2}})\times{\cal C}_{F}(X_{4})$
$\displaystyle=$ $\displaystyle\sum_{M_{2}\in
H_{2}(X_{4},\mathbb{Z}_{p}^{2g})}\exp(\frac{2\pi
i}{2p}\expectationvalue{M_{2},M_{2}}_{F})\Phi(\widetilde{M_{2}})\times\Phi((\mathds{1}-F)M_{2})$
$\displaystyle=$ $\displaystyle\sum_{M_{2}\in
H_{2}(X_{4},\mathbb{Z}_{p}^{2g})}\exp(\frac{2\pi
i}{2p}\left(\expectationvalue{M_{2},M_{2}}_{F}+\expectationvalue{\widetilde{M_{2}},(\mathds{1}-F)M_{2}}\right))\Phi(\widetilde{M_{2}}+(\mathds{1}-F)M_{2})$
$\displaystyle=$ $\displaystyle\sum_{M_{2}\in
H_{2}(X_{4},\mathbb{Z}_{p}^{2g})}\exp(\frac{2\pi
i}{2p}\left(\expectationvalue{M_{2},M_{2}}_{F}+\expectationvalue{\widetilde{M_{2}},(\mathds{1}-F)M_{2}}\right))$
$\displaystyle\hskip
86.72377pt\times\Phi((\mathds{1}-F)\widetilde{M_{2}}+(\mathds{1}-F)M_{2}+F\widetilde{M_{2}})$
$\displaystyle=$ $\displaystyle\sum_{M_{2}\in
H_{2}(X_{4},\mathbb{Z}_{p}^{2g})}\mathrm{exp}\left(\frac{2\pi
i}{2p}\left(\expectationvalue{M_{2},M_{2}}_{F}+\expectationvalue{\widetilde{M_{2}},(\mathds{1}-F)M_{2}}-\expectationvalue{(\mathds{1}-F)(\widetilde{M_{2}}+M_{2}),F\widetilde{M_{2}}}\right)\right)$
$\displaystyle\hskip
86.72377pt\times\Phi((\mathds{1}-F)(\widetilde{M_{2}}+M_{2}))\times\Phi(F\widetilde{M_{2}})$
$\displaystyle=$ $\displaystyle\sum_{M_{2}\in
H_{2}(X_{4},\mathbb{Z}_{p}^{2g})}\exp\left(\frac{2\pi
i}{2p}\left(\expectationvalue{M_{2}-\widetilde{M_{2}},M_{2}-\widetilde{M_{2}}}_{F}+\expectationvalue{\widetilde{M_{2}},(\mathds{1}-F)(M_{2}-\widetilde{M_{2}})}\right.\right.$
$\displaystyle\hskip
50.58878pt-\expectationvalue{(\mathds{1}-F)M_{2},F\widetilde{M_{2}}}\Big{)}\bigg{)}\Phi((\mathds{1}-F)M_{2})\times\Phi(F\widetilde{M_{2}})\leavevmode\nobreak\
.$

We now demand that the left-hand side is proportional to ${\cal
C}_{F}(X_{4})\times\Phi(F\widetilde{M_{2}})$. If we assume that
$\mathrm{ker}(\mathds{1}-F)$ is trivial, then this simply amounts to the
requirement that

$\displaystyle\expectationvalue{M_{2}-\widetilde{M_{2}},M_{2}-\widetilde{M_{2}}}_{F}+\expectationvalue{\widetilde{M_{2}},(\mathds{1}-F)(M_{2}-\widetilde{M_{2}})}-\expectationvalue{(\mathds{1}-F)M_{2},F\widetilde{M_{2}}}=\expectationvalue{M_{2},M_{2}}_{F}\leavevmode\nobreak\
.$ (D.6)

To solve for $\expectationvalue{\cdot,\cdot}_{F}$ we first consider the case
$\widetilde{M_{2}}=M_{2}$. This gives

$\displaystyle\expectationvalue{\widetilde{M_{2}},\widetilde{M_{2}}}_{F}$
$\displaystyle=-\expectationvalue{\widetilde{M_{2}},F\widetilde{M_{2}}}=\expectationvalue{F\widetilde{M_{2}},\widetilde{M_{2}}}\leavevmode\nobreak\
,$ (D.7)

which in turn motivates

$\displaystyle\expectationvalue{M_{2},\widetilde{M_{2}}}_{F}:=\expectationvalue{FM_{2},\widetilde{M_{2}}}\leavevmode\nobreak\
.$ (D.8)

This pairing is not symmetric, but only the symmetric part contributes to
(D.5). By making use of (D.2) and (5.38), it can be shown that this definition
matches that appearing in (5.40) and (5.41).

Inserting this definition into the more general (D.6), we see that this is
satisfied only if

$\displaystyle\expectationvalue{FM_{2},F\widetilde{M_{2}}}$
$\displaystyle=\expectationvalue{M_{2},\widetilde{M_{2}}}\leavevmode\nobreak\
,$ (D.9)

i.e if $F$ is a symplectic matrix. It can also be checked that ${\cal
C}_{F}(X_{4})\times\overline{{\cal C}_{F}}(X_{4})=\Phi(0)=\mathds{1}$ if we
define $\overline{{\cal
C}_{F}}(X_{4})=\chi(X_{4},{\mathbb{Z}}_{p})^{-2g}\,{\cal
C}_{F}(X_{4})^{\dagger}$.

More care is required in the case that $\mathrm{ker}(\mathds{1}-F)$ is
nontrivial. There are two cases to be considered here. First, consider the
case in which $\mathds{1}-F$ has no non-trivial Jordan blocks with eigenvalue
1. In this case we have the splitting

$\displaystyle
H_{2}(X_{4},{\mathbb{Z}}_{p}^{2g})=\mathrm{ker}(\mathds{1}-F)\oplus\mathrm{im}(\mathds{1}-F)\leavevmode\nobreak\
,$ (D.10)

and we see that requiring $\Phi(\widetilde{M_{2}})\times{\cal
C}_{F}(X_{4})={\cal C}_{F}(X_{4})\times\Phi(F\widetilde{M_{2}})$ no longer
imposes (D.6), but rather the seemingly looser requirement

$\displaystyle\sum_{N_{2}\in\mathrm{ker}(1-F)}e^{{2\pi i\over 2p}\langle
M_{2}+N_{2},M_{2}+N_{2}\rangle_{F}}$ $\displaystyle=$
$\displaystyle\sum_{N_{2}\in\mathrm{ker}(\mathds{1}-F)}e^{{2\pi i\over
2p}\expectationvalue{M_{2}+N_{2}-\widetilde{M_{2}},M_{2}+N_{2}-\widetilde{M_{2}}}_{F}}$
$\displaystyle\hskip 57.81621pt\times\,e^{{2\pi i\over
2p}\left(\expectationvalue{\widetilde{M_{2}},(\mathds{1}-F)(M_{2}-\widetilde{M_{2}})}-\expectationvalue{(\mathds{1}-F)M_{2},F\widetilde{M_{2}}}\right)}\leavevmode\nobreak\
,$

obtained by comparing the coefficients of $\Phi((\mathds{1}-F)M_{2})$ on both
sides. Similar to before, in order to determine the pairing
$\langle\cdot,\cdot\rangle_{F}$ we first restrict to
$\widetilde{M_{2}}=M_{2}+N_{2}$, in which case we obtain

$\displaystyle\sum_{N_{2}\in\mathrm{ker}(\mathds{1}-F)}e^{{2\pi i\over
2p}\langle M_{2}+N_{2},M_{2}+N_{2}\rangle_{F}}$ $\displaystyle=$
$\displaystyle e^{{2\pi i\over 2p}\langle
FM_{2},M_{2}\rangle}\sum_{N_{2}\in\mathrm{ker}(\mathds{1}-F)}e^{{2\pi i\over
2p}\langle N_{2},(\mathds{1}-F)M_{2}\rangle}$ (D.12) $\displaystyle=$
$\displaystyle|\mathrm{ker}(\mathds{1}-F)|\,e^{{2\pi i\over 2p}\langle
FM_{2},M_{2}\rangle}$

where we have used the fact that any $M_{2}$ satisfies $\langle
N_{2},(\mathds{1}-F)M_{2}\rangle=0$. Thus in this case we obtain the form of
the defect

$\displaystyle{\cal
C}_{F}(X_{4})={|H^{0}(X_{4},{\mathbb{Z}}_{p}^{2g})|\over|H^{1}(X_{4},{\mathbb{Z}}_{p}^{2g})|}|\mathrm{ker}(\mathds{1}-F)|\sum_{M_{2}\in\mathrm{im}(\mathds{1}-F)}\exp(\frac{2\pi
i}{2p}\expectationvalue{FM_{2},M_{2}})\Phi((\mathds{1}-F)M_{2})\leavevmode\nobreak\
.$ (D.13)

On the other hand, if there exists a non-trivial Jordan blocks with eigenvalue
1, then the splitting in (D.10) is no longer possible, and the above
manipulations are no longer valid.353535We can see that the splitting fails in
the case of a non-trivial Jordan block of size $k>1$ by considering the vector
$v^{(k-1)}$ in the notation of (B.3), which cannot be written as a sum of an
element in the image and an element of the kernel.  In that case, an
alternative way to proceed (which works as long as $p\neq 2$) is to work in a
basis where $F$ is given by

$\displaystyle F=\begin{pmatrix}F^{\prime}&0\\\ 0&\widetilde{F}\end{pmatrix}$
(D.14)

where $F^{\prime}$ is a block containing all Jordan blocks of eigenvalue $1$
and $\widetilde{F}$ contains the rest of the blocks. Then instead of starting
with $F$, we start with the matrices,

$\displaystyle F^{-}:=\begin{pmatrix}-F^{\prime}&0\\\
0&\widetilde{F}\end{pmatrix}\leavevmode\nobreak\ ,$ $\displaystyle
I^{-}:=\begin{pmatrix}-\mathds{1}&0\\\
0&\mathds{1}\end{pmatrix}\leavevmode\nobreak\ .$ (D.15)

Since $F^{-}$ and $I^{-}$ both have no non-trivial Jordan blocks of eigenvalue
$1$, we can write down the defects for both of them. We then define the defect
for $F$ in a roundabout manner as363636When $p=2$ this trick no longer works.
For a discussion of this case in the example of $g=1$ and $F={\mathsf{S}}$,
see [59].

$\displaystyle\mathcal{C}_{F}(X_{4}):=\mathcal{C}_{F^{-}}(X_{4})\times\mathcal{C}_{I^{-}}(X_{4})\leavevmode\nobreak\
.$ (D.16)

We note that in practice this is not an irrelevant subtlety: non-trivial
Jordan blocks of eigenvalue $1$ appear in our discussion whenever the
characteristic polynomial of $F$ is $\phi_{p}(x)$ and the finite field we are
working over is $\mathbb{Z}_{p}$. A concrete example is the case of the
${\mathbb{Z}}_{5}\subset{\mathbb{Z}}_{10}$ symmetry appearing at genus 2,
discussed in Section 3.3, when $p=5$.

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