158mm238mm

#  Four-manifolds and Symmetry Categories of 2d CFTs

Vladimir Bashmakov  † Michele Del Zotto  †,‡
and Azeem Hasan  ‡
† Department of Physics and Astronomy Uppsala University
Box 516 SE-75120 Uppsala  Sweden
‡ Mathematics Institute Uppsala University
Box 480 SE-75106 Uppsala  Sweden

###### Abstract

In this paper we study the geometric origin of non-invertible symmetries of 2d
theories arising from the reduction of 6d $(2,0)$ theories on four-manifolds.
This generalizes and extends our previous results in the context of class
$\mathcal{S}$ theories to a wider realm of models. In particular, we find that
relative 2d field theories, such as the chiral boson, have a higher
dimensional origin in four-manifolds that are not null cobordant. Moreover, we
see that for the 2d theories with a 6d origin, the non-invertible symmetries
have a geometric origin as a sum over topologies from the perspective of the
7d symmetry TFT. In particular, we show that the Tambara-Yamagami non-
invertible symmetries $TY(\mathbb{Z}_{N})$ can be given a geometric origin of
this kind. We focus on examples that do not depend on spin structures, but we
analyse the simplest of such cases, finding an interesting parallel between
the extra choices arising in that context and symmetry fractionalization in
Maxwell theories.

#####  Abstract

————————–

May 2023

###### Contents

  1. 1 Introduction
  2. 2 Symmetry TFT from dimensional reduction
    1. 2.1 Topological boundary conditions
    2. 2.2 Dimensional reduction and boundary conditions
    3. 2.3 Two dimensional theories
    4. 2.4 Exceptional Symmetries and Non Invertible Defects
    5. 2.5 Codensation Surgery Defects
  3. 3 The 7d view of 3d Chern Simons
    1. 3.1 3d Abelian Chern-Simons theories
    2. 3.2 Discriminant group and topological boundary conditions
    3. 3.3 Categorical structure of Chern-Simons
    4. 3.4 Topological boundary conditions from $6d$
    5. 3.5 The case of $S^{2}\times S^{2}$ and $TY(\mathbb{Z}_{N})$
    6. 3.6 Algebra and geometry of topological boundary conditions
    7. 3.7 Non-invertible symmetries from del Pezzo surfaces
  4. 4 Spin Chern-Simons and abelian theory
  5. 5 Conclusions and outlook
  6. A A non-invertible T-duality
  7. B More on Spin CS and the self-dual tensor
    1. B.1 $dP_{1}$
    2. B.2 $S^{2}\times S^{2}$

##  1 Introduction

Finding a framework to study higher symmetries from a unified perspective is
an important open question.111 See footnote 26 for further comments and
references. A recent proposal to characterize the symmetry of a quantum field
theory (QFT) in $D$ spacetime dimensions is to exploit a $D+1$ dimensional
topological field theory (TFT)[1, 2], the so-called symmetry TFT. The symmetry
TFT encodes the symmetry structure of the QFT in terms of boundary conditions,
and it generalizes to higher dimensions the well known correspondence between
symmetries of 2d theories and 3d TFTs [3, 4, 5, 6, 7, 8, 9, 10]. For finite
higher group global symmetries (see eg. [11, 12, 13]) the symmetry TFT are
well-defined mathematical objects, the so-called finite homotopy TFTs of [14].
It is interesting to exhibit concrete examples for symmetries that are more
general than finite higher group. The problem of determining the symmetry TFT
for intrinsic non-invertible symmetries was recently studied in [15], the main
outcome of that study is that for a symmetry to be intrinsically non-
invertible, the resulting symmetry TFT must differ from a (possibly twisted)
Dijkgraaf-Witten (DW) cocycle [16].222 Here we recall that a given QFT has an
intrinsic non-invertible symmetry if the latter does not have an origin as an
otherwise invertible symmetry upon some topological manipulation on the
theory, such as eg. gauging a discrete symmetry [17]. An extremely powerful
tool to explore and produce examples of intrinsic (as well as non-intrinsic)
non-invertible symmetries is provided by higher dimensional SCFTs. In this
paper we continue the exploration of the corresponding symmetry TFTs and non-
invertible symmetries arising from the study of the dimensional reduction of
higher dimensional SCFTs [18, 19, 20], building on the seminal works [21, 22].

While in [18, 19, 23] the focus have been four-dimensional theories obtained
from compactifications of 6d (2,0) theories on Riemann surfaces without
punctures (see eg. [24, 25]), in this project we focus instead on the case of
two dimensional theories and four manifolds (see eg. [26, 27, 28, 29]).
Extending the construction outlined in [19] we recover several known aspects
of symmetry categories of 2d/3d theories from the 6d/7d perspective via
dimensional reduction. In particular, the fact that for 2d models many
features of symmetry categories are well-known provides a plethora of
interesting consistency checks for the technique.

For the sake of simplicity in this paper we focus on the $\mathfrak{a}_{N-1}$
6d (2,0) theories $\mathfrak{X}_{N}$. We find that whenever a given 4-manifold
is null-bordant, the resulting 2d theory is an absolute CFT. If that is not
the case then there are more possibilities, generalizing the well-known
example of a chiral boson. The signature of the 4-manifold is related to the
chirality of the resulting 2d SCFT in a natural way: four manifolds with non-
vanisihing signature can give rise to relative 2d field theories (where the
details are encoded in $N$). This gives an interesting parallel to theories
with and without conformal blocks, revisiting and extending several well-known
related studies [30, 31, 32]. As a first application to non-invertible
symmetries in two dimensions [33, 34, 35, 36] we recover, building on the
analysis of [37, 38, 15], the symmetry TFT for the Tambara-Yamagami
$\mathbb{Z}_{N}$ 2d categorical symmetry from $\mathfrak{X}_{N}$, where we
denote $\mathfrak{X}_{N}$ the 6d (2,0) SCFT of $\mathfrak{a}_{N-1}$-type. The
latter arises by looking at one of the simplest 4-manifolds with trivial
signature $Y_{4}=S^{2}\times S^{2}$. Swapping the two $S^{2}$’s in this
construction is equivalent to an electromagnetic $\mathbb{Z}_{2}$ duality, but
at a point where the two $S^{2}$’s have the same volume, this becomes a
symmetry of the theory, and we can gauge it: by this procedure one obtains the
$TY(\mathbb{Z}_{N})$ symmetry TFT [15]. Gauging such $\mathbb{Z}_{2}$ from the
perspective of the 7d CS theory is equivalent to a sum over topologies. This
example is interesting because it demonstrates that also in the case of two
dimensional compactifications, the non-invertible symmetries which arise as a
gauging from the perspective of the symmetry TFT can be interpreted as a sum
over topologies, as remarked in the context of the reduction on Riemann
surfaces leading to 4d/5d setups [18, 19, 23]. We stress however that the
4-manifold examples is much richer and one can obtain more general situations
where 4-manifolds that are not null-bordant nevertheless give rise to gapped
boundary conditions. These examples can be easily extended to more
4-manifolds. We present the general theory and we analyze in detail the cases
where there is no explicit dependence on the spin structure. We give some more
applications in the context of del Pezzo surfaces, 4-manifolds obtained from
blow ups of $\mathbb{P}^{2}$ at generic points.333These examples are
interesting as the corresponding wrapped M5 branes control the worldvolume
theoires of monopole strings of certain well-known 5d SCFTs [39, 40, 41]. We
conclude this paper with a discussion of some interesting preliminary results
towards extending our construction incorporating the dependence on the Wu
structure of the four-manifold in the context of a toy theory with $N=1$,
arising from the reduction of a single chiral tensor field on a four-manifold.
We find an interesting parallel between parition functions of the chiral boson
and the possible parition functions of Maxwell theories which differ by
symmetry fractionalization by studying the theory on
$\mathbb{T}^{2}\times\mathbb{P}^{2}$.

This paper is organized as follows: in section 2 we quickly and schematically
review the dimensional reduction of the $\mathfrak{X}_{N}$ theory to lower
dimensional field theories to set up notations and conventions. In section 3
we discuss in details the case of 4 manifolds. In section 4 we comment on the
dependence of the overall construction on the choice of a Wu structure for the
4-manifold, and we study the case of the dimensional reduction of a single
self-dual tensor as an example to illustrate some of the salient features of
this type of systems. In section 5 we discuss some future directions and
outlook, in particular commenting on how to extend our construction to the 3d
cases.

##  2 Symmetry TFT from dimensional reduction

We begin with a brief review of how to read off the symmetry TFT of the lower
dimensional theories from the symmetry TFT of the $6d\,(2,0)$ theory. This has
been covered extensively in e.g. [21, 22] (see also [19]) to which we refer
for further details.

We will focus on $6d\,(2,0)$ theory $\mathfrak{X}_{N}$. The symmetry TFT of
this theory $\mathfrak{F}_{N}$ contains a 7d Chern-Simons term which is going
to be of utmost importance for our analysis:

$\displaystyle\mathfrak{F}_{N}$
$\displaystyle=\frac{N}{4\pi}\int_{W_{7}}c\wedge\differential
c+\cdots\leavevmode\nobreak\ ,$ (2.1)

where $W_{7}$ is a seven manifold444 For the sake of simplicity we will
require that the fourth Wu class $\nu_{4}$ vanishes for $W_{7}$ but we will
relax this condition later — for a detailed discussion on the dependence of
this construction from $\nu_{4}$ we refer our readers to appendix A of [22].
and we are using a differential form notation $c\in H^{3}(W_{7},U(1)$. The
basic observables of this topological quantum field theory are the Wilson
surfaces supported on 3-manifolds $\Sigma_{3}$ inside $W_{7}$,

$\displaystyle\Phi_{q}(\Sigma_{3})=\exp(iq\oint_{\Sigma_{3}}c)\leavevmode\nobreak\
,$ $\displaystyle q\in\mathbb{Z}_{N}\leavevmode\nobreak\ .$ (2.2)

If two manifold $\Sigma_{3}$ and $\Sigma_{3}^{\prime}$ form a Hopf link inside
$W_{7}$, then the correlation functions involving Wilson surfaces supported on
them satisfy:

$\displaystyle\expectationvalue{\Phi_{q}(\Sigma_{3})\Phi_{q^{\prime}}(\Sigma_{3}^{\prime})\,\dots}$
$\displaystyle=\exp(\frac{2\pi
i}{N}qq^{\prime}\mbox{link}(\Sigma_{3},\Sigma_{3}^{\prime}))\expectationvalue{\dots}\leavevmode\nobreak\
,$ (2.3)

where $\dots$ denote any number of insertions which do not link with
$\Sigma_{3}$ or $\Sigma_{3}^{\prime}$ and link denotes the linking number. If
$W_{7}$ has a $6d$ boundary $X_{6}$ (which we assume to be compact), we can
push $\Sigma_{3}$ and $\Sigma_{3}^{\prime}$ to the boundary to obtain equal
time commutation relations,

$\displaystyle\Phi_{q}(\Sigma_{3})\Phi_{q^{\prime}}(\Sigma_{3}^{\prime})$
$\displaystyle=\exp(\frac{2\pi
i}{N}qq^{\prime}\expectationvalue{\Sigma_{3},\Sigma_{3}^{\prime}})\Phi_{q^{\prime}}(\Sigma_{3}^{\prime})\Phi_{q^{\prime}}(\Sigma_{3})\leavevmode\nobreak\
.$ (2.4)

The bracket $\expectationvalue{\,,\,}$ denotes the intersection pairing inside
the boundary $X_{6}$ of $W_{7}$. Since for generic values of $N$ the Wilson
surfaces $\Phi(\Sigma_{3})$ realize a non-commutative algebra at the boundary
we cannot impose Dirichlet boundary condition for $c$, in what follows we will
assume this is the case, but this assumption can be relaxed when $N$ is a
perfect square – see section 2.1. As a topological quantum field theory, the
7d Chern-Simons theory associates to $X_{6}$ a Hilbert space
$\mathfrak{F}_{N}(X_{6})$ on which $\Phi(\Sigma_{3})$ with $\Sigma_{3}\in
H_{3}(X_{6},\mathbb{Z}_{n})$ act, thus furnishing a representation of the non-
commutative algebra given by (2.4).

Regarding $\Sigma_{3}$ as an element of $H_{3}(X_{6},\mathbb{Z}_{n})$,
$\Phi_{q}(\Sigma_{3})=\Phi_{1}(q\Sigma_{3})$ so we will define
$\Phi(\Sigma_{3})=\Phi_{1}(\Sigma_{3})$ and drop the subscript. Since the
algebra given by (2.4) is non-commutative, all of $\Phi(\Sigma_{3})$ can’t be
simultaneously diagonalized. We can simultaneously diagonalize
$\Phi(\Sigma_{3})$ and $\Phi(\Sigma_{3}^{\prime})$ only if
$\expectationvalue{\Sigma_{3},\Sigma_{3}^{\prime}}=0\mod N$. Moreover, if a
state $\ket{K}$ is a common eigenvector of $\Phi(\Sigma_{3})$ and
$\Phi(\Sigma_{3}^{\prime})$ it is also an eigenvector of
$\Phi(q\Sigma_{3}+q^{\prime}\Sigma_{3}^{\prime})$. As a result the cycles
$\Sigma_{3}$ that diagonalize $\ket{K}$ define a lattice which is isotropic
i.e. no two cycles in it intersect. To define a representation of the
Heisenberg algebra, the best we can do is demand that a state is
simultaneously fixed by $\Phi(\Sigma_{3})$ with $\Sigma_{3}$ forming a maximal
isotropic lattice $\mathcal{L}$ of $H_{3}(X_{6},\mathbb{Z}_{N})$, i.e. this
state which we label by $\ket{\mathcal{L}}$ is defined by,

$\displaystyle\Phi(\Sigma_{3})\ket{\mathcal{L}}=\ket{\mathcal{L}}\leavevmode\nobreak\
,$ $\displaystyle\forall\Sigma_{3}\in\mathcal{L}\leavevmode\nobreak\ .$ (2.5)

Starting from this state $\mathcal{L}$ we can obtain other states in the
Hilbert space by acting on $\mathcal{L}$ with other
$\Phi(\Sigma_{3}^{\prime})$ such that $\Sigma_{3}^{\prime}$ intersects some
element of $\mathcal{L}$ i.e. the Hilbert space $\mathfrak{F}_{N}(X_{6})$ is
spanned by

$\displaystyle\ket{\mathcal{L},\Sigma_{3}^{\prime}}=\Phi(\Sigma_{3}^{\prime})\ket{\mathcal{L}}\leavevmode\nobreak\
.$ (2.6)

Such states are eigenstates of $\Phi(\Sigma_{3})$ with
$\Sigma_{3}\in\mathcal{L}$ with eigenvalue being $\exp(\frac{2\pi
i}{N}\expectationvalue{\Sigma_{3},\Sigma_{3}^{\prime}})$ and two states
$\ket{\mathcal{L},\Sigma_{3}^{\prime}}$ and
$\ket{\mathcal{L},\Sigma_{3}^{\prime\prime}}$ are linearly independent if and
only if they belong to different coset in
$\mathcal{L}^{\perp}=H_{3}(X_{6},\mathbb{Z}_{N})/\mathcal{L}$. This procedure
provides us with a representation of the algebra in (2.4) with dimension
$\absolutevalue{\mathcal{L}^{\perp}}$.

###  2.1 Topological boundary conditions

Each vector in the Hilbert space as described above gives us a boundary
condition for the $7d$ TQFT. However, they do not determine a (gapped)
topological boundary condition in general. The reason is that $X_{6}$ can in
general have a non-trivial group of large diffeomorphism i.e diffeomorphism
which are not continuously connected to identity, which we refer to as the
mapping class group of $X_{6}$, and we denote

$\mathcal{M}(X_{6})=\pi_{0}\text{Diff}(X_{6})$ (2.7)

in what follows. Given a non-trivial $F\in\mathcal{M}(X_{6})$, it generally
rotates the sublattice $\mathcal{L}$ into a different sublattice
$F(\mathcal{L})$ and the cycle $\Sigma_{3}^{\prime}$ into a different one
$F(\Sigma_{3}^{\prime})$. This transformation of states due to large
diffeomorphisms is how the symmetry TFT encodes the relative nature of
$6d\,(2,0)$ theory. To construct an absolute theory we must assign to each
$X_{6}$ a ray in the Hilbert space constructed above which is invariant under
all of $\mathcal{M}(X_{6})$. If one ends up with a projective representation
of $\mathcal{M}(X_{6})$, such an invariant state does not exist.

The requirement of invariance under large diffeomorphisms of all $X_{6}$ is
very stringent and for generic $N$ no such boundary condition exists. The
exception is when $N=M^{2}$ is a perfect square [22, 21]. In this case we
choose the isotropic sublattice $\mathcal{L}$ to be
$MH_{3}(X_{6},\mathbb{Z}_{M^{2}})$ i.e all cycles which are multiples of $M$.
This lattice is invariant under all isomorphism of
$H_{3}(X_{6},\mathbb{Z}_{M^{2}})$ and hence all diffeomorphisms of $X_{6}$.
However for generic $N$ no such choice exists and the $(2,0)$ theory is
relative.

###  2.2 Dimensional reduction and boundary conditions

The relative nature of the $6d$ theory described above can be exploited to
construct theories in lower dimensions which differ only in their global
structure i.e. these theories have the same spectrum of local operators but
they differ in their global properties such as the corresponding spectra of
extended operators, or their parition functions on compact manifolds with non-
trivial topology. The general procedure is to start with an $X_{6}=Y_{k}\times
M_{d}$, where $k=6-d$ and then dimensionally reduce the $6d$ theory along
$Y_{k}$ to obtain a $d$ dimensional theory.555 With a topological twist to
preserve some supersymmetry but the precise details twisting are not important
for us here. We denote the resulting theory

$D_{Y_{k}}\mathfrak{X}_{N}=\mathcal{T}_{N}[Y_{k}]$ (2.8)

This will in general give us another relative theory, defined as an edge mode
for the topological field theory $D_{Y_{k}}\mathfrak{F}_{N}$, obtained from
dimensionally reducing $\mathfrak{F}_{N}$ along $Y_{k}$, however if we can
choose a lattice $\mathcal{L}\subset H_{3}(Y_{k}\times M_{d},\mathbb{Z}_{N})$
such that it is invariant under large diffeomorphisms of $Y_{k}$ in a way
which allows to factor out the choice of $M_{d}$, then the theory
$D_{Y_{k}}\mathfrak{F}_{N}$ can admit a gapped boundary condition which allows
us to obtain an absolute theory.666 This choice is referred to as a choice of
polarization in section 3 of [22] We denote the latter interchangeably

$\langle\mathcal{L}|D_{Y_{k}}\mathfrak{X}_{N}\rangle\qquad\text{or}\quad
D_{Y_{k}}^{\mathcal{L}}\mathfrak{X}_{N}\quad\text{or}\quad\mathcal{T}_{N}[Y_{k}]_{\mathcal{L}}\in\text{(S)QFT}_{d}\,.$
(2.9)

Since the starting $6d$ theory is relative this means that
$H_{3}(Y_{k},\mathbb{Z}_{N})$ is typically not invariant under at least some
of the large diffeomorphism of $Y_{k}$. When the dimensionally reduced
$d$-dimensional theory is absolute this non-trivial action typically
translates to a statement about dualities of the theories thus obtained:

$\mathcal{T}_{N}[Y_{k}]_{\mathcal{L}}\longleftrightarrow\mathcal{T}_{N}[F(Y_{k})]_{F(\mathcal{L})}$
(2.10)

where $F\in\mathcal{M}(Y_{k}\times M_{d})$, but by construction ends up acting
non-trivially only as its induced action on $Y_{k}$,
$\pi_{Y_{k}}F\in\mathcal{M}(Y_{k})$. In what follows often we abuse notation,
leaving the above remark as understood. This is a well-known fact: we refer
our readers to [22, 21] for a review and further references. Since in the
current work we focus on the $2d$ theories let us briefly remind our readers
about this specific case [26, 27, 28].

$\mathfrak{F}_{N}$$|\mathfrak{X}_{N}\rangle$DimensionalReduction$D_{Y_{k}}\mathfrak{F}_{N}$$|D_{Y_{k}}\mathfrak{X}_{N}\rangle$$\langle\mathcal{L}|$
Figure 1: Dimensional reduction of the 7d/6d relative field theory. Provided a
maximally isotropic sublattice $\mathcal{L}$ exists in a way which is
independent of the choice of $M_{d}$ in the splitting $X_{6}=Y_{k}\times
M_{d}$ we obtain a corresponding absolute theory associated to a gapped
topological boundary condition for $D_{Y_{k}}\mathfrak{F}_{N}$.

###  2.3 Two dimensional theories

Consider the case of $2d$ theories obtained by reducing the $6d$ theory on a
four manifold $Y_{4}$ which we assume to be compact and simply connected. This
ensures that the $2d$ theory we obtain is conformal and in particular for
absolute theories this means that the partition functions on a torus
$\mathbb{T}^{2}$ are modular invariant.777 One can consider more general four
manifolds with non-trivial one cycles. In that case, we expect the
corresponding $2d$ theories to contain gauge fields such that their gauge
couplings scale with the size of the cycles. As a result in the limit where we
send the volume of $Y_{4}$ to zero and obtain a genuine $2d$ theory, the gauge
fields decouple. Hence these gauge fields are not important for the low energy
limit, but capture some KK modes. Moreover restricting to simply connected
four manifolds ensures that the $2d$ theory only depends on the conformal
class of the metric on $Y_{4}$. The mapping class group of diffeomorphism
preserving the conformal structure is bigger than that of large diffeomorphism
preserving the metric. Although it is not in general a tractable object for
internal manifolds $Y_{k}$ with $k>2$, in some simple examples we can
determine some of its relevant subgroups. That is typically enough to
constrain the symmetries. The aim of this paper is to exploit this geometric
control when it is available to construct interesting topological defects in
these 2d theories. Since the defects in two dimensional conformal field
theories are much better understood than their higher dimensional
counterparts, this allows us to make contact with existing literature.

We begin by constructing the symmetry TFT on a three manifold $W_{3}$ that
bounds $M_{2}$. This comes from reducing the 7d abelian Chern-Simons theory on
$Y_{4}\times W_{3}$. Choosing a basis $A_{i}$ of $2$-cycles on $Y_{4}$ we can
write $c$ as $c|_{W_{3}}=\sum_{i}a_{i}$ with,

$\displaystyle a_{i}$ $\displaystyle=\oint_{A_{i}}c\leavevmode\nobreak\ .$
(2.11)

Integrating over the $4$-manifold gives us an abelian Chern-Simons theory in
3-dimensions with the action,

$\displaystyle
S_{3d}=\frac{N}{4\pi}\int_{W_{3}}Q^{ij}\,a_{i}\wedge\differential a_{j}$
(2.12)

where $Q$ is the intersection form of $Y_{4}$. $Q$ is always unimodular and
symmetric [42].

For simply connected $Y_{4}$,

$\displaystyle H_{3}(Y_{4}\times M_{2},\mathbb{Z}_{N})\cong
H_{2}(Y_{4},\mathbb{Z}_{N})\otimes
H_{1}(M_{2},\mathbb{Z}_{N})\leavevmode\nobreak\ .$ (2.13)

For any Riemann surface the only subgroups of integral first homology that are
invariant under the mapping class group are the trivial group and the whole
homology itself. As a result a maximal isotropic sublattice of
$H_{3}(Y_{4}\times M_{2},\mathbb{Z}_{N})$ for generic $N$ must take the form
$L\otimes H_{1}(M_{2},\mathbb{Z}_{N})$ where $L$ is an sublattice of
$H_{2}(Y_{4},\mathbb{Z}_{N})$ isotropic with respect to intersection form $Q$
and with number of elements given by
$\sqrt{\absolutevalue{H_{2}(Y_{4},\mathbb{Z}_{N})}}$. From (2.12) it can be
seen that such an isotropic lattice also determines a topological boundary
condition for the symmetry TFT. However, since the intersection form $Q$ is
symmetric, it is much harder to find such a sublattice.

To illustrate this point, for any bilinear form $Q$ over any field
$\mathbb{F}$, the dimension of maximal isotropic sublattice of $Q$ is an
invariant of $Q$ called Witt index. For $\mathbb{F}$ the field of rational
numbers and $Q$ symmetric, the Witt index is given by the minimum of the
number of positive eigenvalues and negative eigenvalues. Hence
$H_{2}(Y_{4},\mathbb{Q})$ has an isotropic subspace of dimension
$\frac{1}{2}\dim(H_{2}(Y_{4},\mathbb{Q}))$ only if signature of $Q$ vanishes.
This should be contrasted with antisymmetric bilinear forms for which
isotropic subspaces of this dimension always exist. The bound on the dimension
of an isotropic subspace of homology with rational coefficients also limits
the rank of an integral isotropic sublattice.

For our purposes we will mostly consider $N=p$ a prime and hence
$\mathbb{F}=\mathbb{F}_{p}$ is a finite field. For this case isotropic
subspaces of dimension $\frac{1}{2}\dim(H_{2}(Y_{4},\mathbb{F}_{p}))$ can
exist even when the signature of the intersection form on
$H_{2}(Y_{4},\mathbb{Z})$ is not zero. These isotropic subspaces do not lift
to the isotropic subspaces of $H_{2}(Y_{4},\mathbb{Z})$ instead such a lift
contains cycles with intersections which are non-zero multiples of $p$. This
is to be contrasted with the situation familiar in the construction of class
$\mathcal{S}$ theories: isotropic integrals lifts always exist for the maximal
isotropic subspaces of the antisymmetric intersection forms of one-cycles on a
Riemann surfaces, and this was a crucial ingredient for the geometrization of
gapped topological boundary conditions in that context. The problem of
determining gapped topological boundary conditions becomes much more involved
in the case of dimensional reduction on four manifolds, a fact which is
reflected in the richness of both four manifolds and two dimensional CFT.
Exploring this interplay is among the main results of the current work.

Lastly we note that if we relax the condition that $Y_{4}$ is simply
connected, then the resulting theory has 1-form symmetries which lead to the
phenomenon of decomposition. The operators which implement these operations
are “topological points.” These 1-form symmetries were studied in [36] to
understand deconfinement in 2-dimensional adjoint QCD. It would be interesting
to geometrize the results obtained there to understand the low energy behavior
of a wider range gauge theories in two dimensions: we leave this study for
future work.

###  2.4 Exceptional Symmetries and Non Invertible Defects

In the preceding section we have described how to obtain the symmetry TFT of
the 2d theories which arise as the dimensional reduction of $6d\,(2,0)$
theory. However this description is incomplete: while the result is accurate
for theories at generic values of the moduli associated to the internal
manifold $Y_{k}$, it can be refined for certain special theories which we now
describe. The symmetry TFT we have obtained can have automorphisms: for the
case of dimensional reduction on a Riemann surface $\Sigma_{g}$ relevent for
class $\mathcal{S}$ theories these automorphism correspond to group of
$2g\times 2g$ matrices preserving the symplectic form i.e. the intersection
form of one cycles on the Riemann surface. Similarly, for the case of Chern-
Simons theory obtained by reduction on a $4$-manifold, this is the group of
matrices preserving the intersection form $Q$.

An element $F$ of the mapping class group of the manifold $Y_{k}$ on which we
reduce acts on the TFT through such a automorphism which leaves the symmetry
TFT invariant. However its action on the boundary state is typically non-
trivial, thus changing the theory on boundary to a dual theory. For the
theories of class $\mathcal{S}$ this actions changes various gauge couplings
to a dual value and at the same time it changes the maximal isotropic
sublattice $L$ of $H_{1}(\Sigma_{g},\mathbb{Z}_{N})$ (which determines the
isotropic sublattice and hence the global structure of the model) to a
different one. For the case of two dimensional theories the action of $F$ on
the boundary corresponds eg. to an operations that generalize changing the
radii of various compact bosons to a dual value, and just like the 4d case, in
order to give the complete statement of duality we must supply this change
with a change in the isotropic sublattice of $H_{2}(Y_{4},\mathbb{Z}_{N})$
which determines the global structure.

As a result the mapping class group element $F$ generally gives rise to an
operator in the symmetry TFT which when pushed to the boundary becomes a
topological interface between two dual theories. However things are different
if the continuous conformal moduli888e.g. the gauge couplings for $4d$
theories and the radii for $2d$ theories are stabilized by a subgroup
$\mathcal{G}$ of the mapping class group. In this case the defects
corresponding to a mapping class group elements $G\in\mathcal{G}$ implement a
duality symmetry. We can divide these symmetries into two broad classes:

  * •

If $\mathcal{G}$ fixes the global structure then the corresponding interface
is a defect in the theory implementing a (possibly anomalous) $\mathcal{G}$
symmetry. Two such defects fuse according to the multiplication in
$\mathcal{G}$;

  * •

If $\mathcal{G}$ changes the global structure, then we can undo this change by
half space gauging [43] to again obtain a topological defect in the theory.
Due to half space gauging the resulting defect becomes non-invertible.

Because of these additional symmetries, which are only present at these
special points, the symmetry TFT has to be modified. This is done by gauging a
suitable subgroup of $\mathcal{G}$, as explained in [15]. The topological
defects corresponding to these mapping class elements are constructed as
condensates of specific 3d Wilson lines for $\mathfrak{X}_{N}$, following the
same procedure already outlined in [19]. Since these are crucial for our
construction we review them quickly in the following section.

###  2.5 Codensation Surgery Defects

$\langle\mathcal{L}|$$|D_{Y_{k}}\mathfrak{X}_{N}\rangle$${\cal C}_{F}$Expand /
Shrink$\langle\mathcal{L}|D_{Y_{k}}\mathfrak{X}_{N}\rangle$$\langle
F(\mathcal{L})|D_{F(Y_{k})}\mathfrak{X}_{N}\rangle$${\cal N}_{F}$ Figure 2:
The surgery defect ${\cal C}_{F}$ is given a topological boundary and one
obtains a twist defect. Shrinking the slab, this twist defect becomes a
duality interface ${\cal N}_{F}$.

Obtaining a non-invertible symmetry along the lines described above requires
combining the action of a mapping class with a discrete gauging. This discrete
gauging operation acts only on the global structure i.e. the isotropic
sublattice $L$ and not on the continuous coupling and can be encoded as right
action of a matrix $F$ which preserves the intersection form. The resulting
defect $\mathcal{C}_{F}$ can be implemented as a condensation defect [44, 45]
following the procedure outlined in [15, 19]. The construction of
$\mathcal{C}_{F}$ can be streamlined using the $6d$ picture. The basic idea is
to construct these defect out of the operators $\Phi(\Sigma_{3})$ through the
isomorphism999We assume as before that $Y_{4}$ is simply connected and hence
in particular the homology is torsion free.

$\displaystyle H_{3}(Y_{4}\times M_{2},\mathbb{Z}_{N})\cong
H_{1}(M_{2},H_{2}(Y_{4},\mathbb{Z}_{N}))\leavevmode\nobreak\ .$ (2.14)

This allows to regard $\Phi(\Sigma_{3})$ as flux through a one-cycle of
$M_{2}$ with values in $H_{2}(Y_{4},\mathbb{Z}_{N})$. Hence the geometry
determines automatically the value of all the allowed fluxes. These include
both background fluxes and gauge fluxes and the precise description of which
correspond to which is dictated by the choice of the global structure. The
intersection pairing on $H_{3}(Y_{4}\times M_{2},\mathbb{Z}_{N})$ becomes a
combination of the antisymmetric pairing between the elements of
$H_{1}(M_{2},\mathbb{Z}_{N})$ and the symmetric pairing between the winding
and momentum modes of $2d$ compact bosons.101010This pairing is the analogue
of Dirac pairing for general dyonic lines in the 4d case.

The condensation defect $\mathcal{C}_{F}$ that implements the action of
mapping class $F$ is defined by,

$\displaystyle\Phi(\Sigma_{1})\times\mathcal{C}_{F}(M_{2})=\mathcal{C}_{F}(M_{2})\times\Phi(F\Sigma_{1})\leavevmode\nobreak\
,$ $\displaystyle\forall\Sigma_{1}\in
H_{1}(M_{2},H_{2}(Y_{4},\mathbb{Z}_{N}))\leavevmode\nobreak\ .$ (2.15)

Using the fact that $F$ respects the intersection form of $Y_{4}$ i.e.
$F^{T}QF=Q$ and assuming for simplicity that $N$ is a prime the
$\mathcal{C}_{F}$ can be written as,

$\displaystyle\mathcal{C}_{F}$
$\displaystyle=\frac{1}{\sqrt{\absolutevalue{H_{2}(Y_{4},\mathbb{Z}_{N})}}}\sum_{\Sigma_{1}\in
H_{1}(M_{2},H_{2}(Y_{4},\mathbb{Z}_{N}))}\exp(\frac{2\pi
i}{2N}\expectationvalue{F\Sigma_{1},\Sigma_{1}})\Phi((1-F)\Sigma_{1})\leavevmode\nobreak\
.$ (2.16)

The proof that this is indeed the case and that $\mathcal{C}_{F}$ as written
above implements the action in (2.15) is a simple modification of the
discussion in Appendix D of [19] where it is carried out with the roles of
$Y_{4}$ and $M_{2}$ reversed. It can be checked that the computation rests
solely on the fact that $F$ preserves the intersection form of the manifold
along which we carry out dimensional reduction.111111 In the special case of
$N=2$ if $F=\left(\begin{smallmatrix}0&1\\\ 1&0\end{smallmatrix}\right)$ i.e
the case of T-duality for $\mathfrak{a}_{1}$ theory described in appendix A
the expression for the condensate is different. The treatment for that case
can be found in [15] — see also section 3.5.

Now for such elements $F\in\mathcal{M}(Y_{k})$ corresponding to dualities
according to the discussion above, we can construct codimension-1 defects
implementing the associated symmetry from the bulk TFT. These condensation
defects can now be used to construct various defects in the boundary
$d$-dimensional theories by a standard trick [15]. Indeed, each of the
condensation defects above always admits at least one topological boundary
condition and this allows us to end them in the bulk $d+1$ dimensional TFT.
The resulting defects are codimension-2 and are better thought of as non-
genuine defects attached to a manifold that is one-dimension higher — see
Figure 2. Such non-genuine operators were referred to as “twist defects” in
[46, 15]. By moving the twist defects to the boundary as shown in Figure 2,
one obtains an interface between the $d$-dimensional theories that we denote
${\cal N}_{F}$. The fusion rule of these interfaces can be determined but the
result depends on the specifics of the given geometry.

##  3 The 7d view of 3d Chern Simons

As explained in the last section, the $7d$ abelian Chern-Simons theory (2.1)
when considered on a manifold $Y_{4}\times W_{3}$ and integrated along $Y_{4}$
gives $3d$ Chern-Simons theory with $K$-matrix given by the intersection form
of $Y_{4}$. In this section we explain in detail the relationship of the $7d$
Chern-Simons theory and the associated Heisenberg algebra (2.4) with various
aspects of the $3d$ Chern-Simons theory, especially its topological boundary
conditions and the structure of modular tensor category that its observables
encode. We illustrate these connections with the help of some simple examples
corresponding to $Y_{4}$ some del Pezzo surface.

###  3.1 3d Abelian Chern-Simons theories

The abelian Chern-Simons theories were classified in [47] and we briefly
recall the results here121212We normalize the periods of field strength using
the common convention is physics literature i.e. $\frac{F}{2\pi}$ has integral
periods. This convention was also used in the last section. However, this is
different from [47], in which the convention is to define $F$ so that it has
integral periods. An abelian Chern-Simons theory in 3d dimensions is
characterized by an integral lattice $\Lambda$ of rank $r$ endowed with a
symmetric bilinear form $K:\Lambda\times\Lambda\to\mathbb{Z}$. By choosing a
basis of $\Lambda$ we can encode $K$ in a symmetric $r\times r$ matrix
$K^{ij}$. The action of the this Chern-Simons theory on a 3-manifold is
written by introducing $r$ $U(1)$ gauge fields $A_{i}$ and is given by,

$\displaystyle S_{3d}=\frac{1}{4\pi}\int_{W_{3}}K^{ij}A_{i}\wedge\differential
A_{j}\leavevmode\nobreak\ .$ (3.1)

We are interested in how the action written above transforms under the gauge
transformations. For closed $W_{3}$, one way to determine this is to choose a
bounding $4$-manifold $W_{4}$ and extend the gauge fields to it. Calling these
gauge field $\widetilde{A}_{i}$ and the corresponding field strengths
$\widetilde{F}_{i}$ we can write down the four dimensional action,

$\displaystyle
S_{4d}=\frac{1}{4\pi}\int_{W_{4}}K^{ij}\widetilde{F}_{i}\wedge\widetilde{F}_{j}\leavevmode\nobreak\
.$ (3.2)

This is a $4d$ gauge theory with just the topological $\theta$ terms. The
Chern Simons action (3.1) is the boundary term for $S_{4d}$, so we need to
determine to what extent this action depends on the details of the particular
extension we have chosen for the boundary gauge fields. Since,
$\int_{W_{4}}K^{ij}\widetilde{F}_{i}\wedge\widetilde{F}_{j}$ is always an
integer times $4\pi^{2}$, the dependence of the exponentiated action
$\exp(iS_{4d})$ on the extension gives us two classes of Chern-Simons
theories,

  * •

A bilinear form $K:\Lambda\times\Lambda\to\mathbb{Z}$ is called even if
$K(x,x)=0\mod 2$ for all $x\in\Lambda$. For even $K$, the action is
independent of the extension into the bulk and such bilinear forms define
ordinary Chern-Simons theories. In this work, we will only deal with these
theories.

  * •

If $K$ is not even it is called odd. To define Chern-Simons theory for these
cases, we can choose a spin structure on $W_{3}$ and demand that $W_{4}$ has a
compatible spin structure. This ensures that
$\int_{W_{4}}K^{ij}\widetilde{F}_{i}\wedge\widetilde{F}_{j}$ is $8\pi^{2}$
times an integer and hence the extension is dependent only on the spin
structure chosen. Such theories are called spin Chern-Simons theories. In
modern language this makes the spin Chern-Simons theory defined by an odd $K$
one of the simpler examples of a relative theory.131313 We can be more precise
about this statement: if we regard a TFT as a functor from oriented bordisms
to vector spaces, the theory is relative, but if we instead regard a TFT as
functor from spin bordism the theory is absolute. Whether a theory is relative
or absolute depends on how much structure we are willing to put on the
manifolds.

###  3.2 Discriminant group and topological boundary conditions

Like the $7d$ Chern-Simons discussed in Section 2, the observables of abelian
$3d$ Chern-Simons theories are Wilson loops. The dual lattice $\Lambda^{*}$ of
a lattice $\Lambda$ is the set of linear map $\Lambda\to\mathbb{Z}$. To write
down a Wilson loop we pick a closed loop $\Sigma_{1}$ in the spacetime $W_{3}$
and an element $X\in\Lambda^{*}$. This map has a unique extension to a map
$\mathbb{R}^{r}\to\mathbb{R}$ which we also call $X$. The Wilson loop on
$\Sigma_{1}$ with charge $X$ which we will denote $\Phi_{X}(\Sigma_{1})$ is
defined by,

$\displaystyle\Phi_{X}(\Sigma_{1})=\exp(i\oint_{\Sigma_{1}}X(A_{i}))\leavevmode\nobreak\
.$ (3.3)

The Wilson loops can be pushed to the boundary $M_{2}$ of $W_{3}$ and their
equal time correlation functions take the form of a Heisenberg algebra similar
to (2.4). This setup and the topological boundary conditions for Abelian
Chern-Simons theories were studied in [48] and we now review the results of
interest to the current work. The partition vector the TFT assigns to boundary
furnishes a representation of this non-commutative algebra.

We start by describing the discriminant group $\mathrm{D}$ of a lattice
$\Lambda$. By fixing one element we can regard $K(x,\cdot)$ as a map
$\Lambda\to\mathbb{Z}$, i.e for a fixed $x$, $K(x,\cdot)$ is an element of the
dual lattice. Hence $K$ itself can be regarded as a map
$K:\Lambda\to\Lambda^{*}$, and its image is generally a subset of
$\Lambda^{*}$. The discriminant group $\mathrm{D}$ is the quotient
$\Lambda^{*}/\,\mbox{Im}(K)$. It is a finite Abelian group with the number of
elements equal to $\absolutevalue{\det(K)}$.

As an example let us consider the simplest case: that of a $U(1)$ Chern-Simons
theory at level $N$. The lattice $\Lambda$ in this case is the group of
integers $\mathbb{Z}$ with the bilinear form given by $K(x,y)=Nxy$. The dual
lattice $\Lambda^{*}$ is also isomorphic to $\mathbb{Z}$ with $n$ representing
the map $x\mapsto nx$. The $K$ then corresponds to multiplication of these
maps by $N$ and hence the discriminant group is $\mathbb{Z}/N\mathbb{Z}$ i.e.
$\mathbb{Z}_{N}$. The Wilson loops for the $7d$ Chern-Simons theory (2.2)
correspond precisely to the those with charges in discriminant group.

Restricting to the discriminant subgroup greatly simplifies the study of the
Wilson loops since their is now only a finite number of them. It is also
enough to classify the topological boundary conditions. Indeed one of the main
results of [48] is that the boundary conditions of a 3d Chern-Simons theory
correspond to the Lagrangian subgroups of $\mathrm{D}$. A lagrangian subgroup
$L$ of $\mathrm{D}$ is a subgroup which is isotropic with respect to the
intersection form $K$ and has precisely $\sqrt{\absolutevalue{\mathrm{D}}}$
elements.

###  3.3 Categorical structure of Chern-Simons

Three dimensional quantum field theories correspond to algebraic structures
called modular tensor categories. Abelian CS theories are prototypical
examples of 3d TFT and the corresponding modular tensor categories are well
known (see e.g. [49, 50, 48]). The data needed to define the modular tensor
category $\mathcal{C}$ are,

  1. 1.

A group of anyons which for abelian Chern-Simons theories is the discriminant
group $\mathrm{D}$.

  2. 2.

For each anyon $a$, its topological spin $\theta(a)$. For the CS theories
described above, it is,

$\displaystyle\theta(a)=\exp(2\pi i\frac{1}{2}K^{-1}(a,a))\leavevmode\nobreak\
,$ (3.4)

This expressions is well defined on $\mathrm{D}$ if $K$ is even i.e. when the
CS theory is an ordinary CS theory and not a spin CS theory.

  3. 3.

The braiding phase for each pair of anyons $a$ and $b$. For CS theories it is
given by,

$\displaystyle B(a,b)=\exp(2\pi iK^{-1}(a,b))\leavevmode\nobreak\ .$ (3.5)

The spin and braiding phases of anyons are required to satisfy several
consistency conditions which we do not describe here but which can be found in
the references cited above. In terms of these data we can reconstruct the
F-symbols and the hexagon relations which specify the corresponding symmetry
category, as discussed in detail in [50] to which we refer our readers.
Therefore the information which we recover from geometry completely determines
the theories we are after provided $N$ is even or $Q$ has even diagonal
elements. Whenever that is not the case more care is needed: the resulting
theories are spin Chern Simons and the construction crucially depends on the
choice of a trivialization for the Wu structure on the 4-manifold of interest.
We discuss some details about this in section 4 for the case of a 6d theory of
rank one corresponding to a free chiral self-dual tensor in 6d.

Let us discuss in slightly more detail on the aforementioned symmetries of
Chern-Simons theories. The description in terms of $K-$matrix is not unique:
the transformations of the form

$K\,\rightarrow\,V\,K\,V^{T},\qquad a\,\rightarrow\,Va,$ (3.6)

with $V\in GL(N,\,\mathbb{Z})$ is an invertible matrix, does not change the
braiding or fusion structure. We will be especially intereste in those
transformations, also called automorphisms, which leave the $K-$matrix
invariant:

$K\,=\,U\,K\,U^{T}.$ (3.7)

These transformations correspond to anyonic symmetry transformations: they
permute lines with the same spins and leave invariant the fusion rules [51,
52]. We will denote the group of automorphisms by $\text{Aut}(K)$. Not all the
elements in $\text{Aut}(K)$ correspond to a non-trivial transformation on
anyons, though. Indeed, the transformations of the form

$a\,\rightarrow\,a\,+\,K\,b,$ (3.8)

which leave an anyon invariant up to a local (with integer spin and trivial
braiding) particle. Transformations of this kind are referred to as inner
automorphisms and form a normal subgroup of the automorphism group,
$\text{Inner}(K)\,\unlhd\,\text{Aut}(K)$. Our ultimate interest is in the
quotient of the automorphism group with respect to internal automorphisms,
which is known as the outer automorphism group:

$\text{Outer}(K)\,=\,\frac{\text{Aut}(K)}{\text{Inner}(K)}.$ (3.9)

###  3.4 Topological boundary conditions from $6d$

As explained in Section 2.3, the $2d$ theory obtained by reducing $6d\,(2,0)$
theory of type $\mathfrak{a}_{N}$ on a compact simply connected four manifold
$Y_{4}$ has as its symmetry TFT an abelian CS theory with lattice
$\Lambda=H_{2}(Y_{4},\mathbb{Z})$ and the K-matrix,

$\displaystyle K=NQ\leavevmode\nobreak\ .$ (3.10)

Where $Q$ is the intersection form on $Y_{4}$. The discriminant subgroup is
then homology with coefficients in $\mathbb{Z}_{N}$ i.e
$H_{2}(Y_{4},\mathbb{Z}_{N})$. For square-free $N$ for which there are no
topological boundary conditions in $7d$, this discriminant group immediately
allows us to match the topological boundary conditions we expected from $6d$
with the result for $3d$ Chern-Simons theory obtained above.

For this let us recall from the general discussion in Section 2, that the
global structures for the $6d\,(2,0)$ theory of type $\mathfrak{a}_{N-1}$ on
$Y_{6-d}\times Z_{d}$ dimensionally reduced along $Y_{6-d}$ correspond to
precisely those maximal isotropic sublattices $\mathcal{L}$ of
$H_{3}(Y_{6-d}\times Z_{d},\mathbb{Z}_{N})$ which for all $Z_{d}$ are
invariant under the mapping class group of $d$. By Kunneth formula,

$\displaystyle H_{3}(Y_{4}\times M_{2},\mathbb{Z}_{N})\cong
H_{2}(Y_{4},\mathbb{Z}_{N})\otimes
H_{1}(M_{2},\mathbb{Z}_{N})\leavevmode\nobreak\ .$ (3.11)

Since no non-trivial subgroup of $H_{1}(M_{2},\mathbb{Z}_{N})$ is invariant
under the mapping class group of a two dimensional manifold $M_{2}$, the
maximal isotropic sublattice must take the form,

$\displaystyle\mathcal{L}=L\otimes
H_{1}(M_{2},\mathbb{Z}_{N})\leavevmode\nobreak\ .$ (3.12)

The fact that $\mathcal{L}$ is an maximal isotropic sublattice then means that
$L$ must be a Lagrangian subgroup of $H_{2}(Y_{4},\mathbb{Z}_{N})$ i.e. the
discriminant group for the symmetry TFT. If the intersection form $Q$ of
$Y_{4}$ is even, this arguments works for all square free $N$, however for odd
$Q$ we must restrict to even square free $N$ to obtain a CS theory without
dependence on spin structure.

###  3.5 The case of $S^{2}\times S^{2}$ and $TY(\mathbb{Z}_{N})$

In this section we discuss the details of the dictionary in one explicit
example, namely $Y_{4}=S^{2}\times S^{2}$. Let us begin by considering the
case $N=2$ which is slightly different from the general one. This example will
also provide an illustration of the condensate formalism. The action of
$D_{S^{2}\times S^{2}}\mathfrak{F}_{2}$ is

$\frac{2}{2\pi}\int\,A_{1}\wedge dA_{2},$ (3.13)

and so the corresponding $K-$matrix is given by two times the intersection
pairing on $S^{2}\times S^{2}$:

$K\,=2Q=\,\left(\begin{matrix}0&&2\\\ 2&&0\end{matrix}\right).$ (3.14)

A simple inspection reveals that $\text{Outer}(K)\,\simeq\,\mathbb{Z}_{2}$,
and is generated by the matrix

$F\,=\,\left(\begin{matrix}0&&1\\\ 1&&0\end{matrix}\right).$ (3.15)

From the 6d perspective $F$ arises as the element of $\mathcal{M}(S^{2}\times
S^{2})$ which swaps the two 2-spheres in the four-manifold. $H_{2}(S^{2}\times
S^{2},\mathbb{Z}_{2})$ has two generators $C_{1}$ and $C_{2}$, and by
considering $\Sigma_{3}^{i}=\gamma\cup C_{i}$ as supports for the 7d Wilson
surfaces we obtain the following basic lines in this model:

$\displaystyle\Phi(C_{1}\cup\gamma)\to\mathit{e}\,=\,e^{i\oint_{\gamma}A_{1}},\qquad\Phi(C_{2}\cup\gamma)\to\mathit{m}\,=\,e^{i\oint_{\gamma}A_{2}}$
$\displaystyle\qquad\Phi((C_{1}+C_{2})\cup\gamma)\to\psi\,=\,e^{i\oint
A_{1}+i\oint A_{2}},$ (3.16)

These satisfy the obvious Abelian fusion rules which can be derived
straighforwardly from the Heisenberg algebra:

$\displaystyle\mathit{e}\,\times\,\mathit{e}\,=\,\mathit{m}\,\times\,\mathit{m}\,=\,\psi\,\times\,\psi\,=\,1,$
$\displaystyle\mathit{e}\,\times\,\psi\,=\,\mathit{m},\qquad\mathit{m}\,\times\,\psi\,=\,\mathit{e},$
(3.17) $\displaystyle\mathit{e}\,\times\,\mathit{m}\,=\,\psi.$

The lines $\mathit{e}$ and $\mathit{m}$ are sometimes called the electric line
and the magnetic line, respectively, and the anyonic $\mathbb{Z}_{2}$ symmetry
corresponding to $F$ exchanges them,
$\mathit{e}\,\leftrightarrow\,\mathit{m}$, therefore being dubbed electric-
magnetic symmetry. The fermion line $\psi$ is invariant under the action of
electric-magnetic $\mathbb{Z}_{2}$.

Now, there must exist a topological co-dimension one operator, realizing the
electric-magnetic $\mathbb{Z}_{2}$ action, and, following [15, 19], we can
represent this operator as a condensate of lines. The idea goes as follows. We
can drag an anyon $a_{\mathbf{n}}$ from the left to the right through a
surface defect, and along the way it gets transformed into the line
$a_{F\mathbf{n}}$. We now exploit the folding trick and reflect the trajectory
of the line past the defect, bringing it to the left, but with the form of the
charge conjugate line $a_{-F\mathbf{n}}$ instead. Now we do not have anything
on the right, and so the line $a_{(\mathds{1}-F)\mathbf{n}}$ gets absorbed
(see fig. 3).

${\cal
C}_{F}(\Sigma_{2})$$a_{\mathbf{n}}(\gamma)$$a_{F\mathbf{n}}(\gamma)$${\cal
C}_{F}(\Sigma_{2})$$a_{(\mathds{1}-F)\mathbf{n}}(\gamma)$ Figure 3: The
automorphism transformation operators ${\cal C}_{F}(\Sigma_{2})$ change the
anyon lables in a given way. The folding trick implies that this operators
should be able to absorb lines of the form
$a_{(\mathds{1}-F)\mathbf{n}}(\gamma)$, and therefore are condensates of them.

In the case at hands $\mathbf{n}_{e}\,=\,\begin{pmatrix}1\\\ 0\end{pmatrix}$,
$F\,=\,\begin{pmatrix}0&&1\\\ 1&&0\end{pmatrix}$, and so the absorbed line has
the label $(\mathds{1}-F)\mathbf{n}_{e}\,=\,\begin{pmatrix}1\\\
-1\end{pmatrix}\,=\,\begin{pmatrix}1\\\ 1\end{pmatrix}\,\text{mod}\,2$, and is
recognized as the $\psi$ line. The condensate supported on a subamnifold
$\Sigma_{2}$ is then,

${\cal
C}_{F}(\Sigma_{2})\,=\,\frac{1}{|H_{0}(\Sigma_{2},\mathbb{Z}_{2})|}\,\sum_{\gamma\in
H_{1}(\Sigma_{2},\mathbb{Z}_{2})}\,\psi(\gamma).$ (3.18)

The normalization is chosen to provide the following fusion rule for
condensates:

${\cal C}_{F}\,\times\,{\cal
C}_{F}\,=\,\frac{|H^{1}(\Sigma_{2},\mathbb{Z}_{2})|}{|H^{0}(\Sigma_{2},\mathbb{Z}_{2})|^{2}}\,=\,\chi(\Sigma_{2},\mathbb{Z}_{2})^{-1}.$
(3.19)

We recognise here the 2d version of the condensate $\mathcal{C}_{F}$ we have
constructed in section 2 above which we obtain from the $\mathbb{Z}_{2}$
action that swaps the two spheres in geometry. It is interesting to compare
this definition with the one in equation (2.16). This is the geometrical
origin of the EM duality in this model.

Having constructed the transformation operator on manifolds without boundary,
we can consider a situation when the operator ends on a curve $\gamma$. This
is achieved by imposing Dirichlet boundary conditions on the boundary: in our
case $(A_{1}+A_{2})|_{\gamma}=0$. This is a non-genuine co-dimension 2 defect,
which we will denote $\sigma_{0}(\gamma)$; it is non-genuine, because it comes
with a surface attached. More explicitly, we can define the defect line in the
following way:

$\sigma_{0}(\gamma)\,=\,\frac{1}{|H^{0}(\Sigma_{2},\gamma,\mathbb{Z}_{2})|}\,\sum_{\gamma\in
H_{1}(\Sigma_{2},\mathbb{Z}_{2})}\,\psi(\gamma),$ (3.20)

where the normalization is introduced, following [15]. We can also locate an
electric line $\mathit{(}e)$ along the boundary $\gamma$: this gives rise to a
new defect $\sigma_{1}(\gamma)$. Can we get to more defects by putting along
the boundary the other to lines? $\psi$ trivialises along the boundary due to
the boundary conditions. For the magnetic line we have
$\int_{\gamma}\,A_{2}\,=\,\int_{\gamma}\,(A_{1}\,+\,2A_{2})\,=\,\int_{\gamma}\,A_{1}$.
In the first equality we used that $(A_{1}+A_{2})|_{\gamma}\,=\,0$, and the
second equality is valid mod $2$. Thus, the magnetic line is equivalent to the
electric line on the defect boundary, and so again leads to $\sigma_{1}$.
These statements can be summarized by the following fusion rules:

$\displaystyle\
\mathit{e}\,\times\,\sigma_{\lambda}\,=\,\mathit{m}\,\times\,\sigma_{\lambda}\,=\sigma_{\lambda+1},$
$\displaystyle\psi\,\times\,\sigma_{\lambda}\,=\,\sigma_{\lambda},$ (3.21)

where $\lambda$ is a mod $2$ index. With slightly more work (see [15] for the
details), one can also obtain the defect fusion rules:

$\displaystyle\sigma_{0}\,\times\,\sigma_{0}\,=\,\sigma_{1}\,\times\,\sigma_{1}\,=\,\chi(M_{2},\mathbb{Z}_{N})^{-1}\,(1\,+\,\psi),$
$\displaystyle\sigma_{0}\,\times\,\sigma_{1}\,=\,\chi(M_{2},\mathbb{Z}_{N})^{-1}\,(\mathit{e}\,+\mathit{m}).$
(3.22)

Here the overall coefficient, depending on the two-manifold $M_{2}$, attached
to the defect, can be removed by an appropriate counterterm. We are not
specifying the curves for line operators in the equations above, assuming that
they all have the same support. The fusions of the defects with the same
labels gives a higher gauging of $\psi$ line along a curve. The fusion of two
diffects with different labels can be obtain by fusing the condensate with the
$\mathit{e}$ line (or equivalently with the m line).

This discussion generalizes straightforwardly to the $\mathbb{Z}_{N}$ theories
with arbitrary $N>2$. The action for $D_{S^{2}\times S^{2}}\mathfrak{F}_{N}$
that arises from dimensional reduction is

$\frac{N}{2\pi}\int\,A_{1}\wedge dA_{2}\,.$ (3.23)

Again we see this system has an anyonic symmetry given by $\mathbb{Z}_{2}$
that exchanges $A_{1}$ with $A_{2}$. Sometimes we can also refer to it as the
electric-magnetic symmetry. This symmetry is clearly geometrized by swapping
the two $S^{2}$’s. For this example we have $N$ electric, $N$ magnetic and in
total $N^{2}$ dyonic lines, that have a 6d origin from operators
$\Phi(\Sigma_{3})$ of the following forms:

$\Phi((nC_{1}+mC_{2})\cup\gamma)\to\mathit{\ell}_{n,m}\,=\,e^{(in\oint_{\gamma}A_{1}+im\oint_{\gamma}A_{2})}.$
(3.24)

In analogy with the preceding discussion, we consider the surface defects,
implementing the electric-magnetic symmetry. The folding trick implies that
the line $\mathit{\ell}_{1,-1}$ must be absorbed by this defect (see [15] for
a thorough discussion), which is therefore given by the condensate

${\cal
C}_{F}(\Sigma_{2})\,=\,\frac{1}{|H_{0}(\Sigma_{2},\mathbb{Z}_{N})|}\,\sum_{\gamma\in
H_{1}(\Sigma_{2},\mathbb{Z}_{H})}\,\mathit{\ell}_{1,-1}(\gamma).$ (3.25)

This is indeed a surgery defect arising from condensation of the type we
discussed in section 2. Using the dictionary between the Wilson surfaces of
the 7d theory and the 3d Wilson lines in equation (3.24), it can be checked
that this indeed reproduces the expression in (2.16). For this we note that in
this case the element of $\mathcal{M}(S^{2}\times S^{2})$ that we are
representing is

$F=\left(\begin{matrix}0&1\\\ 1&0\end{matrix}\right)$ (3.26)

that acts on homology by exchanging $C_{1}$ and $C_{2}$. The image of $1-F$ is
generated by $C_{1}-C_{2}$ and hence the lines that participate in the
condensate are $\Phi((C_{1}-C_{2})\cup\gamma)$ with $\gamma\in
H_{1}(\Sigma_{2},\mathbb{Z}_{N})$ i.e. $l_{1,-1}(\gamma)$. Since all the
phases in (2.16) vanish for this example, that expression reduces to (3.25).
One can check that the fusion of these surface defects is still governed by
$\mathbb{Z}_{2}$ group law.

As before, we can put the defect ${\cal C}_{F}$ on a two-manifold with
boundary, imposing Dirichlet boundary conditions (see figure 4): this will
give rise to a non-genuine defect operator, which we dub ${\cal N}_{F}$.
Following the derivation in [15] we see that also in this case the defect in
(2.16) gives rise to the EM duality defect, and that from such a condensate
insertion with an appropriate twist, we can obtain the non-invertible
generator for the non-invertible Tambara-Yamagami symmetry
$TY(\mathbb{Z}_{N})$.

$\langle\mathcal{L}|$$|D_{S^{2}\times S^{2}}\mathfrak{X}_{N}\rangle$${\cal
C}_{F}$Expand / Shrink$\langle\mathcal{L}|D_{S^{2}\times
S^{2}}\mathfrak{X}_{N}\rangle$$\langle F(\mathcal{L})|D_{F(S^{2}\times
S^{2})}\mathfrak{X}_{N}\rangle$${\cal N}_{F}$ Figure 4: The surgery defect
${\cal C}_{F}$ is given a topological boundary and one obtains a twist defect.
Shrinking the slab, this twist defect becomes a duality interface ${\cal
N}_{F}$.

###  3.6 Algebra and geometry of topological boundary conditions

Finding a Lagrangian subgroup of a finite abelian group is a purely algebraic
problem. The same problem for an antisymmetric bilinear form determines the
global structures of $4d$ theories of class $\mathcal{S}$. In [19, 22] it was
realized that for class $\mathcal{S}$ theories, this algebraic problem can be
geometrized in an elegant manner: the boundary conditions are engineered by a
null bordism of Riemann surface defining the class $\mathcal{S}$ theory. In
this section we explore whether such a geometrization is possible for $2d$
theories too. This turns out not to be the case: the bordism of $Y_{4}$ when
they exist determine a boundary condition for the $2d$ theory. However such
geometric boundary conditions are rather special and (depending on $N$)
boundary conditions exist even for those manifolds which are not null bordant.
Before expanding on this we recall some basic facts about $4$ manifolds and
their intersection forms.

For smooth simply connected manifolds the intersections forms are rather
special.141414see e.g. [53] for an elementary exposition. They are unimodular
i.e. the determinant of $Q$ is $\pm 1$. For topological four manifolds,
Freedman showed that every symmetric unimodular bilinear form is realized as
the intersection form of a simply connected four manifold. However situation
is much more complicated for smooth four manifolds and we illustrate the
complexity with the help of some families of four manifold.

#####  $\\#^{n}\mathbb{P}^{2}$:

We start with the example of connected sum of $n$ copies of complex projective
plane $\mathbb{P}^{2}$. The second homology has rank $n$ and the intersection
form is the $n\times n$ identity matrix. This intersection form is also
positive definite i.e. $Q(x,x)>0$ for all 2 cycles $x$. We can also consider
the orientation reversal of these manifolds, which gives a negative definite
intersection form i.e $Q(x,x)<0$. Identity and minus identity are in fact the
only definite intersection forms possible. Other definite intersection forms
such as $E_{8}$ Cartan matrix matrix are not realized as the intersection form
of any smooth manifold.

$\mathbb{P}^{2}$ is not null bordant and in fact generates the oriented
bordism group in four dimensions i.e. every $4$ manifold is bordant to one of
$\\#^{n}\mathbb{P}^{2}$. However as we will see Lagrangian subgroups of
$H_{2}(\\#^{n}\mathbb{P}^{2},\mathbb{Z}_{N})$ exist for some values of $N$.

#####  $\\#^{n}(S^{2}\times S^{2})$:

Since intersection form is symmetric its eigenvalues are real. An important
invariant of a symmetric bilinear form is the signature defined by
$b_{+}-b_{-}$, where $b_{\pm}$ are the number of positive and negative
eigenvalues. A four manifold is null bordant iff the signature of its
intersection form is $0$. The simplest example of such a manifold is
$S^{2}\times S^{2}$ which was considered in Section 2.3. It is also the
simplest manifold with an even intersection form which is,

$\displaystyle H=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\leavevmode\nobreak\ .$
(3.27)

A null bordism of $S^{2}\times S^{2}$ can be constructed by filling one of the
two spheres into a ball. Thus we obtain two null bordisms which correspond to
the two boundary conditions (which exist for all $N$) described in Section
2.3. For $N=2$ however there is an extra global structure which is not
engineered by a bordism.

The connected sum of $n$ copies of $S^{2}\times S^{2}$ have as intersection
form $n$ copies of $H$. $2^{n}$ bordisms for it can be constructed by filling
a sphere to get a ball into each of $n$ connected components and these
correspond to topological boundary conditions which are natural generalization
of ones considered for $S^{2}\times S^{2}$. However more topological boundary
conditions exist and it would be interesting to figure out whether the extra
boundary conditions also correspond to bordisms. We can ask this question in a
slightly general setting: let us suppose that there is a Lagrangian subgroup
$L$ of $H_{2}(Y_{4},\mathbb{Z}_{N})$ along with a Lagrangian subgroup
$\widetilde{L}$ of $H_{2}(Y_{4},\mathbb{Z})$151515By Lagrangian subgroup in
this context we mean a free abelian of half the rank which is isotropic w.r.t.
the intersection form. such that $L$ is the mod $N$ reduction of
$\widetilde{L}$. In this case we can hope to engineer a bordism geometrizing
the boundary condition specified by $L$ by filling in the collection of two
cycles in $\widetilde{L}$. This procedure is in direct analogy with how
handlebodies are constructed, but there might be obstructions to filling in
the all the cycles in $\widetilde{L}$ smoothly. The question is whether such
obstructions exist and if they do in what circumstances?

#####  $K3$ surfaces:

Topologically all compact Calabi-Yau two fold are K3 surfaces. Their
intersection form contain 2 copies of the Cartan matrix of the $E_{8}$ Lie
group and 3 copies of $H$. The two copies of $E_{8}$ contribute $8$ each to
the signature. Since $H$ has signature $0$, the signature of $K3$ surfaces is
$16$. $16$ is also the minimum positive value of the signature for a spin $4$
manifold, and indeed K3 is the generator of the spin bordism group in four
dimensions, i.e. any spin manifold is bordant to the connected sum of some
number of K3s.

The lattice $H_{2}(Y_{4},\mathbb{Z})$ with the quadratic form induced by the
intersection pairing on $Y_{4}$ admits Lagrangian sublattices only when the
signature of $Y_{4}$ is $0$. For this reason the topological boundary
conditions for the $6d\,(2,0)$ theory of type $\mathfrak{a}_{N-1}$ are not
simply classified by a mod $N$ reduction of the Lagrangian subgroups of
$H_{2}(K3,\mathbb{Z})$. However, as explained in the following section, there
are some Lagrangian sublattices when $N$ is a prime or when $N$ is a product
of two primes, and it seems natural to conjecture that they exist for all
$N$.161616 Since an M5 brane wrapping a K3 is dual to a Heterotic string, it
is natural to expect to be able to reproduce multiple heterotic strings from
multiple M5 branes. However, the theory on a stack of $N$ M5s is, decoupling
the center of mass, a 6d (2,0) $A_{N-1}$ SCFT. The choices that we see
realized here then might be interesting for studying dualities between
M-theory and heterotic strings. It would be interesting to study this in
greater detail in the future.

####  3.6.1 Finding Lagrangian subgroups

We now describe when topological boundary conditions exist by exploiting and
extending the treatment of Section 4.2 of [54].171717 We thank Iñaki García
Etxebarria for pointing us to [54]. The treatment in this section improved
considerably thanks to this. The background material for this section can be
found in e.g. [55, 56], here we quickly summarize the results we need. The
salient point is that many bilinear forms which are different over integers
become equivalent over $\mathbb{Z}_{N}$. We will restrict ourselves to
bilinear forms which are of relevance to four manifolds i.e. we assume that
the bilinear form $Q$ is the $\mod N$ reduction of an integral unimodular
form.

To start with let us consider a Lagrangian subgroup $L$ of
$H_{2}(Y_{4},\mathbb{Z}_{N})$ with $N=N_{1}N_{2}$ with $N_{1}$ and $N_{2}$
coprime. All elements of $L$ whose order is coprime to $N_{2}$ form a subgroup
of $L$ which we denote $L_{1}$. Similarly all elements with order coprime to
$N_{1}$ form a subgroup of $L$ which we denote $L_{2}$. Any element $g\in
L_{1}$ can be written as $N_{2}g_{1}$ and it can be shown that all such
$g_{1}$ form a Lagrangian subgroup $\widetilde{L}_{1}$of
$H_{2}(Y_{4},\mathbb{Z}_{N_{1}})$. Similarly any element of $g\in L_{2}$ can
be written as $N_{1}g_{2}$ and such $g_{2}$ form a Lagrangian subgroup
$\widetilde{L}_{2}$ of $H_{2}(Y_{4},\mathbb{Z}_{N_{2}})$. Hence $L$ can be
decomposed into a Lagrangian subgroup of $\widetilde{L}_{1}$ of
$H_{2}(Y_{4},\mathbb{Z}_{N_{1}})$ and a Lagrangian subgroup
$\widetilde{L}_{2}$ of $H_{2}(Y_{4},\mathbb{Z}_{N_{2}})$. The converse is also
true, given Lagrangian subgroups $\widetilde{L}_{1}$ and $\widetilde{L}_{2}$
we can assemble them into a Lagrangian subgroup $L$ of
$H_{2}(Y_{4},\mathbb{Z}_{N})$. Applying this argument repeatedly to
$N=p_{1}^{k_{1}}\dots p_{i}^{k_{i}}$, we obtain that Lagrangian subgroups of
$H_{2}(Y_{4},\mathbb{Z}_{N})$ exist if and only if Lagrangian subgroups exist
for each of $H_{2}(Y_{4},\mathbb{Z}_{p^{k_{i}}_{i}})$. Hence we can restrict
ourselves to finding Lagrangian subgroups for
$H_{2}(Y_{4},\mathbb{Z}_{p^{k}})$ with $p$ a prime.

Let us start with the case of $N=p^{k}$ with $p$ an odd prime. In this case
there are at most two inequivalent classes of bilinear forms for any given
dimension $n$ (which in our case is $b_{2}(Y_{4})$). When $n=2m+1$ is odd, the
two classes are represented by,

$\displaystyle Q^{(n)}_{+}=\mathds{1}\oplus\underbrace{H\oplus\cdots\oplus
H}_{m\text{ times}}\leavevmode\nobreak\ ,$ $\displaystyle
Q^{(n)}_{-}=(-\mathds{1})\oplus\underbrace{H\oplus\cdots\oplus H}_{m\text{
times}}\leavevmode\nobreak\ ,$ (3.28)

here $H$ is as in (3.27) and $\mathds{1}$ represents the $1\times 1$ identity
matrix.

When $n=2m$ is even, the two classes are,

$\displaystyle Q_{+}^{(n)}=\underbrace{H\oplus\cdots\oplus H}_{m\text{
times}}\leavevmode\nobreak\ ,$ $\displaystyle
Q^{(n)}_{-}=\mathds{1}_{2}\oplus\underbrace{H\oplus\cdots\oplus H}_{m-1\text{
times}}\leavevmode\nobreak\ .$ (3.29)

A bilinear form with determinant $\pm 1$ belongs to the class of $Q^{(n)}_{+}$
if $\det(Q)=(-1)^{\lfloor\frac{n}{2}\rfloor}$ and $Q^{(n)}_{-}$ if
$\det(Q)=-(-1)^{\lfloor\frac{n}{2}\rfloor}$. However $Q^{(n)}_{\pm}$ are not
always inequivalent. When $p=1\mod 4$, $-1$ is a quadratic residue and as a
result $Q^{(n)}_{+}$ can be rescaled by a square of $-1$ to get a bilinear
form equivalent to $Q^{(n)}_{-}$ and vice versa. Hence for $p=1\mod 4$, the
intersection form on $H_{2}(Y_{4},\mathbb{Z}_{p^{k}})$ is completely
determined the second Betti number $b_{2}(Y_{4})$. While for $p=3\mod 4$, $-1$
is not a quadratic residue and $Q^{(n)}_{\pm}$ are genuinely inequivalent.

Next let us discuss the case of $N=2^{k}$ which is slightly more involved. A
complete set of invariants for a bilinear form $Q$ in this case is given by
the dimension $n$, the evenness,181818 Unlike the case of odd primes $p^{k}$,
evenness in $\mathbb{Z}_{2^{k}}$ is well defined, hence one can distinguish
between even and odd bilinear forms. and the signature modulo $8$. Even forms
exist only when $n=2m$ and there is a unique even form,
$E^{(m)}=Q_{+}^{(2m)}$, for each $m$.

When $n=2m$, the odd forms have even signatures, while when $n=2m+1$ is odd,
the odd forms have odd signatures. For $n=2m$ even, we have odd forms with
signatures (0,2,4,6): the odd form of signature $0$ with lowest dimension is
$\mbox{diag}(1,-1)$. For signature $2$ it is $\mathds{1}_{2}$, for signature
$6=-2\mod 8$, it is $-\mathds{1}_{2}$. Lastly, odd forms with signature $4$
exist only in dimension four and higher with the four dimensional one being
$\mathds{1}_{4}$. All these forms can be combined with suitable factors of
$E^{(m^{\prime})}$ to give odd forms with the same signature in higher
dimensions, the distinct classes of bilinear forms are represented by,

$\displaystyle Q_{0}^{(2m+2)}$ $\displaystyle=\mbox{diag}(1,-1)\oplus
E^{(m)}\leavevmode\nobreak\ ,$ $\displaystyle Q_{2}^{(2m+2)}$
$\displaystyle=\mathds{1}_{2}\oplus E^{(m)}\leavevmode\nobreak\ ,$
$\displaystyle Q_{4}^{(2m+4)}$ $\displaystyle=\mathds{1}_{4}\oplus
E^{(m)}\leavevmode\nobreak\ ,$ $\displaystyle Q_{6}^{(2m+2)}$
$\displaystyle=(-\mathds{1}_{2})\oplus E^{(m)}\leavevmode\nobreak\ .$ (3.30)

Similarly odd forms with odd signature exist when $n=2m+1$ is odd. A
representative for each class is,

$\displaystyle Q_{1}^{(2m+1)}$ $\displaystyle=\mathds{1}\oplus
E^{(m)}\leavevmode\nobreak\ ,$ $\displaystyle
Q_{7}^{(2m+1)}=(-\mathds{1})\oplus E^{(m)}\leavevmode\nobreak\ ,$
$\displaystyle Q_{3}^{(2m+3)}$ $\displaystyle=\mathds{1}_{3}\oplus
E^{(m)}\leavevmode\nobreak\ ,$ $\displaystyle
Q_{5}^{(2m+3)}=(-\mathds{1}_{3})\oplus E^{(m)}\leavevmode\nobreak\ .$

However since $1=-1$ for $\mathbb{Z}_{2}$, for $k=1$ most of these forms are
not distinct. This is because for $k=1$ the signature is only well defined
$\mod 2$ and not $\mod 8$ as in the general case. It can be seen that the
distinct ones are represented by one of $E^{(m)},Q_{0}^{(2m+2)}$ and
$Q_{1}^{(2m+1)}$.

Having described the equivalence classes let us now turn to the problem of
finding Lagrangian subspaces. Once again we start with $N=p^{k}$ with $p$ an
odd prime. Instead of trying to classify all the Lagrangian subspaces we will
restrict to those that have a basis. 191919 More technically we consider only
those Lagrangian subgroups which are free submodules of
$H_{2}(Y_{4},\mathbb{Z}_{p^{k}})$. If $Y_{4}$ is simply connected
$H_{2}(Y_{4},\mathbb{Z}_{p^{k}})$ itself is always a free module. This
assumption does have some physical relevance: only in this case the global
symmetry is $\mathbb{Z}_{p^{k}}^{\frac{1}{2}b_{2}(Y_{4})}.$ This is a
technical assumption but allows us to deal with the problem of finding
Lagrangian subgroups using tools of linear algebra. For instance, consider the
case $k=1$. Since $\mathbb{Z}_{p}$ is a field all subspaces and hence all
Lagrangian subspaces admit a basis. However, this is not necessarily true for
$k>1$, e.g. for $k=2l$ even, $p^{l}H_{2}(Y_{4},\mathbb{Z}_{p^{2l}})$ is a
Lagrangian subspace, but it does not have a basis.

Let us start with the case of $n=b_{2}(Y_{4})=2m$. Suppose that a Lagrangian
subspace $L$ with a basis exists and we extend this basis to a basis of
$H_{2}(Y_{4},\mathbb{Z}_{p^{k}})$. In this basis the the matrix representing
$Q$ takes the form,

$\displaystyle Q$ $\displaystyle=\begin{pmatrix}0&A^{T}\\\
A&B\end{pmatrix}\leavevmode\nobreak\ ,$ (3.32)

with $A,B$ $m\times m$ matrices. $B$ must be symmetric and since $Q$ is non-
degenerate $A$ must be invertible. The Lagrangian subspace $L$ is the row span
of the first $m$ rows in this basis. Let us perform a change of basis $Q\to
SQS^{T}$ with $S=\left(\begin{smallmatrix}A^{-1}&0\\\
0&\mathds{1}\end{smallmatrix}\right)$. This basis change preserves $L$ but $Q$
becomes,

$\displaystyle Q=\begin{pmatrix}0&\mathds{1}\\\
\mathds{1}&B\end{pmatrix}\leavevmode\nobreak\ .$ (3.33)

Hence for any Lagrangian subspace $L$ we can choose a basis of
$H_{2}(Y_{4},\mathbb{Z}_{p^{k}})$ such that $L$ is the span of first $m$ rows
and the intersection matrix takes the form above. This hold even for for
$p=2$. For odd primes, we can do a further change of basis which also
preserves $L$, $Q\to SQS^{T}$ with $S=\left(\begin{smallmatrix}\mathds{1}&0\\\
-\frac{1}{2}(B+B^{T})&\mathds{1}\end{smallmatrix}\right)$. This changes $Q$ to

$\displaystyle Q=\begin{pmatrix}0&\mathds{1}\\\
\mathds{1}&0\end{pmatrix}\leavevmode\nobreak\ ,$ (3.34)

which is related $Q^{(n)}_{+}$ as defined in (3.29) by a permutation of basis
elements. Hence if such an $L$ exists $Q$ must be equivalent to $Q^{(m)}_{+}$
and any other Lagrangian subspace with a basis is related to $L$ by some
element of $SO(Q^{(n)}_{+},\mathbb{Z}_{p^{k}})$ i.e. the group of matrices
preserving $Q^{(n)}_{+}$.

We have shown that when $p=3\mod 4$, $Q^{(n)}_{-}$ has no Lagrangian subgroup
with a basis, we now show it has no Lagrangian subgroups at all when $k$ is
odd. To show this let us suppose that $x=\left(\begin{smallmatrix}a\\\ b\\\
\vec{c}\end{smallmatrix}\right)\in L$ which is an isotropic subspace w.r.t
$Q^{(n)}_{-}$. Then $Q^{(n)}_{-}(x,x)=a^{2}+b^{2}=0$. Since for $p=3\mod 4$, a
square root of $-1$ doesn’t exist, we must have $a^{2}=b^{2}=0$. For $k$ odd
the only solution to $a^{2}=0$ is $a=0$. Since all the $\vec{c}$ form a
Lagrangian subgroup of $Q^{(n-2)}_{+}$, an isotropic subspace for
$Q^{(n)}_{-}$ can have at most the size of Lagrangian subspace of
$Q^{(n-2)}_{-}$ but that is smaller than the size of a Lagrangian subgroup of
$Q^{(n)}_{-}$. For $k=2l$ even, Lagrangian subgroups exist and it can be seen
that any such subgroup takes the form
$L\cong(p^{l}\mathbb{Z}_{p^{2l}}^{2}\oplus L^{\prime})$ where $L^{\prime}$ is
a Lagrangian subgroup for $Q^{(n-2)}_{+}$.

When $n=2m+1$ is odd a Lagrangian subspace can exist only when $k=2l$ is even.
In that case a Lagrangian subgroup takes the form $L\cong
p^{l}\mathbb{Z}_{p^{2l}}\oplus L^{\prime}$ where $L$ is a Lagrangian subgroup
for $Q^{(n-1)}_{+}$.

Next we quickly deal with the case of $N=2^{k}$. In this case use methods
similar to those employed above it can be checked that for $k$ odd, Lagrangian
subspaces exist only for signature $0$ forms i.e. $E^{(m)}$ and
$Q^{(2m+2)}_{0}$. For $k$ even the Lagrangian subgroups exist for all $l$.
Lastly we draw attention to the case of $k=1$. Over $\mathbb{F}_{2}$ both
forms of even dimensions $E^{(m)}$ and $Q^{(2m+2)}_{0}$ admit Lagrangian
subgroups while the odd dimensional form $Q^{(2m+1)}_{1}$ does not.

Lastly let us summarize the topological boundary conditions the discussion
above allows to construct,

  * •

If the intersection form $Q$ of $Y_{4}$ is even, it is equivalent to $E^{(m)}$
over $\mathbb{Z}_{2^{k}}$ and $Q^{(2m)}_{+}$ over $\mathbb{Z}_{p^{k}}$ where
$m=\frac{1}{2}b_{2}(Y_{4})$. In this case Lagrangian subgroups exist for all
$N$. The intersection form for the Chern-Simons theory given by $NQ$ is even
for all $N$ and hence we obtain topological boundary conditions for all $N$.
This includes the case of $\\#^{n}(S^{2}\times S^{2})$ and K3 surfaces and
more generally all spin manifolds.

  * •

If the intersection form of $Y_{4}$ is odd we must choose $N$ to be even to
make the intersection form $NQ$ of CS theory even. When $b_{2}(Y_{4})=2m$ is
even there are following case to consider,

    * –

If $N$ is an even perfect square, the topological boundary conditions always
exist.

    * –

If the signature of $Q$ is $0\mod 8$ then $\det(Q)=(-1)^{m}$, and as a result
the topological boundary conditions exist for all even $N$. The manifolds in
this class include the del Pezzo $dP_{1}$ which will be described later and
which has the intersection form $\mbox{diag}(-1,1)$. $\\#^{8n}\mathbb{P}^{2}$
are further examples.

    * –

If the signature of $Q$ is not $0\mod 8$ but $\det(Q)=(-1)^{m}$, the
topological boundary conditions exist for all $N=2M$ when $M$ is odd. Examples
of manifolds in this category are $\\#^{8n+4}\mathbb{P}^{2}$.

    * –

If the signature of $Q$ is not $0\mod 8$ and $\det(Q)=-(-1)^{m}$, the
topological boundary conditions exist when $N=2M$ where $M$ is odd and all
primes in the prime decomposition of $M$ are congruent to $1\mod 4$.
$\\#^{4n+2}\mathbb{P}^{2}$ belong to this class of manifolds.

  * •

If $b_{2}(Y_{4})$ is odd, the topological boundary conditions exist if and
only if $N$ is an even perfect square.

###  3.7 Non-invertible symmetries from del Pezzo surfaces

After describing the topological boundary conditions and how to determine
their existence from both the $6d$ perspective and the perspective of the $3d$
symmetry TFT for generic $Y_{4}$, we discuss the special $Y_{4}$’s which are
stabilized by some subgroup of mapping class group $Y_{4}$.

By definition an element $F$ of the mapping class group acts on
$H_{2}(Y_{4},\mathbb{Z}_{N})$ as a matrix which we will also denote by $F$
that preserves the intersection form i.e. $F^{T}QF=Q$. However not every
matrix $F$ can be realized as the action of a global diffeomorphism on
$H_{2}(Y_{4},\mathbb{Z}_{N})$ and determining which $F$ lift to a mapping
class requires great control over geometry. As an easy example of topological
obstruction to such a lift let us consider the topology of such a two cycle
$C$. Every two cycle $C$ can be realized as a Riemann surface $\Sigma_{g}$ in
$Y_{4}$ and the genus $g$ of this Riemann surface can vary with the cycle. A
diffeomorphism will always preserve the topology of a submanifold and hence
will map a cycle to another cycle realized as a submanifold of the same genus.
However this genus is invisible to the abstract homology groups.

This and other difficulties of studying four manifolds and the moduli spaces
of conformal structures associated to them, make studying the theories with
exceptional symmetries more complicated compared to class $\mathcal{S}$
theories. For class $\mathcal{S}$ theories, the moduli space (i.e. space of
gauge couplings) is understood and at least for low genus can be used to give
an exhaustive account of theories with exceptional symmetries [19]. As a
result instead of attempting a general treatment we study a class of examples
where (a subgroup) of mapping class group and its action on
$H_{2}(Y_{4},\mathbb{Z}_{N})$ is known and can be used to construct $2d$ CFT
with the corresponding exceptional symmetries which according to the general
discussion of Section 2 are non-invertible.

####  3.7.1 del Pezzo surfaces

The del Pezzo surfaces are complex surfaces (thus of real dimension four) with
positive first Chern class. Up to homeomorphism there are exactly 9 of them. 8
of these can be obtained from $\mathbb{P}^{2}$ by blowing up points in general
position.202020General position here means that no three points lie on a line,
no four points on a plane and so on. We will denote by $dP_{k}$ the manifold
obtain by blowing up $k$ points on $\mathbb{P}^{2}$. Topologically it is the
same as $\mathbb{P}^{2}\\#^{k}\overline{\mathbb{P}^{2}}$ where
$\overline{\mathbb{P}^{2}}$ means $\mathbb{P}^{2}$ with the opposite of
standard orientation. The remaining del Pezzo surface is topologically
$S^{2}\times S^{2}$ and corresponds to the $\mathbb{F}_{0}$ Hirzebruch
surface, which we have already described above (see also Appendix A).
Consequently in this section we will focus on the $dP_{k}$s.

Let us denote the non-trivial cycles of the $dP_{k}$ $E_{i}$ for $1\leq i\leq
k$ (each $E_{i}$ is the exceptional cycle corresponding to the $i^{th}$ blown
up point) and $H$ (which is the pullback of the hyperplane section of
$\mathbb{P}^{2}$ under the projection which shrinks all the cycles $E_{i}$ to
points).212121ie. the blown down of all $E_{i}$ The intersection form is,

$\displaystyle Q(H,H)=1\leavevmode\nobreak\ ,$ $\displaystyle
Q(H,E_{i})=0\leavevmode\nobreak\ ,$ $\displaystyle
Q(E_{i},E_{j})=-\delta_{ij}\leavevmode\nobreak\ .$ (3.35)

This gives us a Lorentzian version of the intersection form on
$\\#^{n}\mathbb{P}^{2}$. The group preserving it is thus $SO(1,k;\mathbb{Z})$.
This intersection form is odd and to simplify our analysis we restrict to the
case of $6d\,(2,0)$ of type $\mathfrak{a}_{2N-1}$. From the discussion in
Section 3.6.1, the topological boundary conditions always exist when
$N=2M^{2}$. In addition,

  * •

For $dP_{1}$ there are topological boundary conditions when
$N=2^{l}p_{1}^{k_{1}}\cdots p_{i}^{k}$ and $p_{i}=1\mod 4$ for all $i$.

  * •

For $dP_{5}$ there are topological boundary conditions when
$N=p_{1}^{k_{1}}\cdots p_{i}^{k}$ and $p_{i}=1\mod 4$ for all $i$.

  * •

For $dP_{3}$ and $dP_{7}$, there are topological boundary conditions for all
$N$.

This exhausts the list of $N$ for which we can find topological boundary
conditions. In particular, for all the cases not listed above the resulting 2d
theories will be relative field theories by construction.

Now in the cases above, we want to identify those cases that lead to non-
invertible symmetries. The quantities giving rise to the parameters of
interest for the $2d$ CFTs out of the dimensional reduction of the $6d\,(2,0)$
theory on $dP_{k}$ are the relative volumes of the $E_{i}$ cycles, for $1\leq
i\leq k$,

$\displaystyle R_{i}$
$\displaystyle=\operatorname{Vol}(E_{i})\leavevmode\nobreak\ ,$
$\displaystyle\sum_{i=1}^{k}R_{i}=1\leavevmode\nobreak\ .$ (3.36)

where the constraint on the RHS above comes by requiring that the resulting 2d
theory is conformally invariant (and hence depend only on the ratios of the
$R_{i}$ and hence one can always rescale them so that the RHS holds).

The case of $dP_{1}$ is somewhat special and we deal with it separately. In
this case there are no free parameters. The group $SO(1,1;\mathbb{Z})$ is
$\mathbb{Z}_{2}$ generated by $-\mathsf{1}$. This transformation necessarily
preserves every lattice and as a result all topological boundary conditions
are in separate orbits of the mapping class group.

####  3.7.2 Exceptional symmetries from del Pezzo surfaces

From now on we consider the case of $dP_{3},dP_{5}$ and $dP_{7}$. For them
there are equivalence classes of maps up to isotopy which arise from the
exchange of two blowups points $i$ and $j$. The action of this mapping class
on $H_{2}(dP_{k},\mathbb{Z})$ is to exchange $E_{i}$ and $E_{j}$. The action
of this diffeomorphism is simply $R_{i}\leftrightarrow R_{j}$ and when
$R_{i}=R_{j}$, we obtain a theory with an extra $\mathbb{Z}_{2}$ symmetry.
Similarly when more $l$ of $R_{i}$ coincide we obtain an exceptional $S_{l}$
symmetry corresponding to a permutations of the corresponding blowups.

However, following [57] we can construct a bigger group of exceptional
symmetries corresponding to the Weyl group of lie algebras $E_{k}$. The first
Chern class of $dP_{k}$ is given by,

$\displaystyle c_{1}(dP_{k})$
$\displaystyle=3H-\sum_{l=1}^{k}E_{l}\leavevmode\nobreak\ .$ (3.37)

Its orthogonal complement $K_{k}$ is the sublattice of
$H_{2}(dP_{k},\mathbb{Z})$ consisting of two-cycles $x$ such that
$Q(c_{1}(dP_{k}),x)=0$. It is an integral lattice of rank $k$ with a basis
$C_{i}$ given by,

$\displaystyle C_{i}$ $\displaystyle=E_{i}-E_{i+1}\leavevmode\nobreak\ ,$
$\displaystyle 1\leq i\leq k-1\leavevmode\nobreak\ ,$ $\displaystyle C_{k}$
$\displaystyle=H-E_{1}-E_{2}-E_{3}\leavevmode\nobreak\ .$ (3.38)

The intersection form induced by $Q$ on $K_{k}$ is negative definite and in
the basis written above equals to minus the Cartan matrix of the
$\mathfrak{e}_{k}$ Lie algebra.222222 Where in the (by now) standard
conventions of [57]: $\mathfrak{e}_{1}=\mathfrak{a}_{1}$,
$\mathfrak{e}_{2}=\mathfrak{a}_{1}\times\mathfrak{a}_{1}$,
$\mathfrak{e}_{3}=\mathfrak{a}_{1}\times\mathfrak{a}_{2}$,
$\mathfrak{e}_{4}=\mathfrak{a}_{4}$, $\mathfrak{e}_{5}=\mathfrak{so}_{10}$,
$\mathfrak{e}_{6}$, $\mathfrak{e}_{7}$, $\mathfrak{e}_{8}$. Therefore $K_{k}$
as a lattice is isomorphic to the root lattice of the exceptional Lie algebra
$\mathfrak{e}_{k}$, but with a negative definite bilinear form instead of
positive definite.232323We choose have chosen to define $dP_{k}$ with opposite
orientation and this would have made the intersection form on $K_{k}$ positive
definite. However we follow the convention of [57] which is also more found
commonly in string theory literature. The group preserving this intersection
form on $K_{k}$ is then $\text{Weyl}(\mathfrak{e}_{k})$, the Weyl group of
$\mathfrak{e}_{k}$, namely the group generated by the reflections $s_{i}$,

$\displaystyle s_{i}(x)=x+Q(C_{i},x)C_{i}\leavevmode\nobreak\ .$ (3.39)

Since $Q(C_{i},c_{1}(dP_{k}))=0$, this action preserves the first Chern class.
As a result, the Weyl group of $\mathfrak{e}_{k}$ is the subgroup of
$SO(1,k;\mathbb{Z}_{k})$ that in addition to the intersection form also
preserves the first Chern class. It can also be realized as a subgroup of
diffeomorphism of $dP_{k}$. From (3.7.2) it can be seen that for $i\neq k$,
$s_{i}$ corresponds precisely to exchanging $E_{i}$ and $E_{i+1}$. These are
the transpositions which generate the $S_{k}$ subgroup of the Weyl group
described above. The diffeomorphism that implements $s_{k}$ is less intuitive
and we refer to [57] for its details.

In particular, consider the case of the unique maximally symmetric point for
$dP_{k}$ (for which all $R_{i}$ are equal, in our normalization
$R_{i}=\frac{1}{k}$ for $1\leq i\leq k$). Since no $\frac{k+1}{2}$ dimensional
sublattice of $H_{2}(dP_{k},\mathbb{F}_{p})$ can be invariant under all the
Weyl reflections, $s_{i}$, following the procedure outlined in Section 2.4 we
obtain non-invertible symmetries corresponding to
$\text{Weyl}(\mathfrak{e}_{k})$ elements realized at this point.

##  4 Spin Chern-Simons and abelian theory

So far we have avoided dealing with spin Chern-Simons theories and their
analogue in $7d$ i.e. Chern-Simons theory with odd level which depends on a
Wu4 structure. In this section we illustrate some of the subtleties which
arise in this case. Our main focus will be theory of an abelian self dual
2-form gauge field in $6$ dimensions. This theory is a higher dimensional
analogue of a chiral boson is two dimensions. It is also relevant for the
discussion of $6d\,(2,0)$ theories since the tensor multiplet contains self
dual gauge fields and it is the global properties of these self dual gauge
fields which are controlled by the Chern-Simons term of symmetry TFT, much in
the same way that 3d Chern-Simons controls the global properties of Wess-
Zumino-Witten models.242424To be precise the analogy should be with WZW model
at level 1, for which the 3d bulk theory is $SU(N)$ CS theory which is related
by a level rank duality (see e.g. [58]) to the abelian CS theory at level $N$.
We will return on the dependence on the Wu structure for the more general case
of the 6d (2,0) $\mathfrak{X}_{\mathfrak{g}}$ theories in future work.

The connection of the global properties of the theory of self dual gauge
fields with Chern-Simons theory was realized in [31] and expanded upon in [59,
60, 61, 62] among other works to elucidate aspects of $S$ duality in four
dimensional gauge theories. We start by recapping some of the salient results.
The starting point is the fact that for general $7$ manifolds, the periods of
field strength $c$ field in (2.1) obey a shifted quantization condition, i.e.

$\displaystyle\oint_{Y_{4}}\differential c$
$\displaystyle=\frac{1}{2}\oint_{Y_{4}}v_{4}\mod 1\leavevmode\nobreak\ ,$
(4.1)

with $v_{4}$ being the fourth (integral) Wu class. If the level $k$ is even
then this subtlety goes away since the periods of $v_{4}$ are integral.
However for odd $k$, this makes $\frac{1}{2}v_{4}$ an ever present background
field that cannot be turned off i.e. unlike the more familiar case where
background gauge fields form a group, in this case they have merely the
structure of an affine space. This means in particular that to define the
Chern-Simons theory on a 7 manifold with boundary $X_{6}$, we must choose a
trivialization of $v_{4}$ on the boundary. We will now specialize to level 1
which is the relevant level for the abelian theory. The result of the shifted
quantization is that the boundary theory is a relative theory whose partition
vector forms a representation of a $\mathbb{Z}_{2}$ Heisenberg algebra i.e.
the algebra in (2.4) with $N=2$.

Up till now, we have considered the partition vector as an abstract object.
The action of Heisenberg algebra on it gives us a handle on many important
properties of it. In the case of abelian theory however the different
components of the partition vector can be calculated following the recipe of
[61], which we now recount briefly.

The self dual $2$ form field $B^{+}$ satisfies the self duality condition,
$H=\star\,H$ with $H=\differential B^{+}$ the field strength and $\star$
represents the Hodge star. We want to compute the Euclidean partition function
so that after the Wick rotation, the self duality constraint becomes $H=i\star
H$. The difficulty in writing a partition function comes from the fact that
the self duality constraint is in tension with the quantization of periods of
$H$ i.e. we cannot simultaneously impose $H=i\star H$ and that
$\oint_{\Sigma_{3}}H\in 2\pi\mathbb{Z}$ for every $\Sigma_{3}\in
H_{3}(X_{6},\mathbb{Z})$. The best that can be done is to impose it on a
maximal isotropic sublattice of $H_{3}(X_{6},\mathbb{Z})$. In view of (4.1) it
is actually enough that this lattice is isotropic modulo $2$ i.e. an element
of the partition vector is indexed by a half rank sublattice $\mathcal{L}$ of
$H_{3}(X_{6},\mathbb{Z})$ such that its projection to
$H_{3}(X_{6},\mathbb{Z}_{2})$ is isotropic. This choice refines the possible
choices of global structures for the resulting theories.

In the explicit examples we will always choose $\mathcal{L}$ to be an integral
isotropic lattice. The background fields then correspond to the (modulo 2
reduction of) $H_{3}(X_{6},\mathbb{Z})/\mathcal{L}$ and we can represent them
by choosing an isotropic lattice $\mathcal{L}^{\perp}$ such that
$H_{3}(X_{6},\mathbb{Z})=\mathcal{L}\oplus\mathcal{L}^{\perp}$. To write down
an element of the partition vector, we choose $\mathcal{A}_{i}$-cycles which
form a basis of $\mathcal{L}$ and $\mathcal{B}_{i}$ cycles which form a basis
of $\mathcal{L}^{\perp}$ such that the intersection form on
$H_{3}(X_{6},\mathbb{Z})$ is,

$\displaystyle\expectationvalue{\mathcal{A}_{i},\mathcal{B}_{j}}=\delta_{ij}\leavevmode\nobreak\
.$ (4.2)

Next we choose to restrict to self dual 3-forms which have integral periods
over elements of $\mathcal{L}$. In particular we choose a basis of such forms
$\lambda_{j}$ such that,

$\displaystyle\oint_{\mathcal{A}_{i}}\lambda_{j}$
$\displaystyle=\delta_{ij}\leavevmode\nobreak\ .$ (4.3)

The periods of the $\lambda_{j}$ over $\mathcal{B}_{i}$ can be arbitrary
complex numbers and define the period matrix $\tau_{ij}$ i.e,

$\displaystyle\tau_{ij}=\oint_{\mathcal{B}_{j}}\lambda_{i}\leavevmode\nobreak\
.$ (4.4)

The period matrix $\tau$ can be shown to be symmetric i.e.
$\tau_{ij}=\tau_{ji}$. With period matrix in hand, the elements of partition
vector we are after are,

$\displaystyle Z\begin{bmatrix}\theta\\\ \phi\end{bmatrix}$
$\displaystyle=\frac{1}{\Delta}\mathcal{I}_{\mathcal{L}}(\theta,\phi;\tau)\leavevmode\nobreak\
,$ $\displaystyle\mathcal{I}_{\mathcal{L}}(\theta,\phi;\tau)$
$\displaystyle=\sum_{n\in\mathcal{L}+\theta}\exp(i\pi\sum_{i,j}\tau_{ij}n^{i}n^{j}+2\pi
i\sum_{i}n^{i}\phi_{i})\leavevmode\nobreak\ .$ (4.5)

Where $\theta\in(\frac{1}{2}\mathcal{L})/\mathcal{L}$ and
$\phi\in(\frac{1}{2}\mathcal{L}^{\perp})/\mathcal{L}^{\perp}$. Both $\theta$
and $\phi$ change under the action of the mapping class group of $X_{6}$ and
the relative nature of the theory is exhibited in the change of
$\mathcal{I}_{L}(\theta,\phi;\tau)$ under this variation.

The factor $\mathcal{I}_{L}(\theta,\phi;\tau)$ is the analogue of the
instantonic contributions to the path integral of $4d$ Maxwell theory. The
factor $\Delta$ is the contribution of the non-zero modes to the path integral
and as a result if two lattices $\mathcal{L}$ and $\mathcal{L}^{\prime}$ are
related by the action of some element of mapping class group, $\Delta$ is same
for both of them. As a result we do not attempt to compute it but we emphasize
that if $\mathcal{L}$ and $\mathcal{L}^{\prime}$ are in the disconnected
orbits under the action of mapping class group there is no reason to expect
$\Delta$ to be same for both of them.

We will compute $Z$ above in the examples of the form $\mathbb{T}^{2}\times
Y_{4}$ where $Y_{4}$ is a compact simply connected four manifold. In the
applications to the $4d$ S-duality, it is more natural (see the discussion in
Section 2) to choose $\mathcal{L}$ to be (an integral lift) of the form
$L\otimes H_{2}(Y_{4},\mathbb{Z}_{2})$ where $L$ is a maximal isotropic
sublattice of $H_{1}(\mathbb{T}^{2},\mathbb{Z}_{2})$. The $4d$ partition
functions thus obtained also turn out to have well known $2d$ interpretations.
However we can also consider $\mathcal{L}$ to be of the form
$H_{1}(\mathbb{T}^{2},\mathbb{Z}_{2})\otimes L$ where $L$ is a Lagrangian
subgroup of $H_{2}(Y_{4},\mathbb{Z}_{2})$. These partition functions turn out
to be different from the first kind but must be recognized as partition
functions of the $6d$ theory dimensionally reduced along $Y_{4}$.

As an explicit example let us consider the case of $6d$ abelian theory on
$\mathbb{T}^{2}\times\mathbb{P}^{2}$ and reinterpret the result of [61] in the
language of higher form symmetries. Since
$H_{2}(\mathbb{P}^{2},\mathbb{Z}_{2})=\mathbb{Z}_{2}$ has no Lagrangian
subgroup, any maximal isotropic subgroup of
$H_{2}(\mathbb{P}^{2},\mathbb{Z}_{2})$ and consequently any maximal isotropic
sublattice $\mathcal{L}$ is given by $\mathcal{L}=L\otimes
H_{2}(\mathbb{P}^{2},\mathbb{Z}_{2})$ where $L$ is a maximal isotropic
sublattice of $H_{1}(\mathbb{T}^{2},\mathbb{Z}_{2})$. Since the intersection
form of $\mathbb{P}^{2}$ is odd the $3d$ symmetry TFT is a level 1 U(1) CS
theory. Then we expect to obtain a relative $2d$ theory, or, equivalently, a
$4d$ theory with multiple global structures. We will take $L$ to be generated
by the usual $A$ cycle on $\mathbb{T}^{2}$. The lone generator $S$ of
$H_{1}(\mathbb{P}^{2},\mathbb{Z})$ has self intersection $1$. A basis of three
forms with integral periods on $A\cup S$ is then,

$\displaystyle\lambda=\differential\overline{z}\wedge\kappa\leavevmode\nobreak\
,$ (4.6)

where $\kappa$ is the generator of integral harmonic two forms on
$\mathbb{P}^{2}$ and $\overline{z}$ is the anti-holomorphic coordinate on
$\mathbb{T}^{2}$. The period matrix can be seen to have a single entry equal
to $\overline{\tau}$. The partition function given by (4) is then,

$\displaystyle\mathcal{I}(\theta,\phi;\overline{\tau})=\sum_{n\in\mathbb{Z}+\theta}\exp(i\pi\overline{\tau}n^{2}+2\pi
i\phi\,n)$ (4.7)

The choice of Wu structure is encoded in the values of $\theta$ and $\phi$.
Either can be $0$ or $\frac{1}{2}$, and thus there four distinct choices of
$\mathcal{I}$. Interestingly these are precisely the four Jacobi theta
functions. From the $2d$ perspective, reducing a self dual two form on
$\mathbb{P}^{2}$ we obtain a chiral (i.e. self dual) boson in $2d$. This boson
lives on the $\mathbb{T}^{2}$ boundary of the spin Chern-Simons theory at
level $1$, and hence to properly define the theory we need to choose a spin
structure on $\mathbb{T}^{2}$ and a compatible spin structure on $3d$ bulk.
The familiar four choices of spin structure correspond to the four possible
conformal blocks. Defining $\overline{q}=e^{i\pi\overline{\tau}}$ we see that

  * •

the spin structure given by $0\in H_{1}(\mathbb{T}^{2},\mathbb{Z}_{2})$
correspond to $\mathcal{I}(0,0;\overline{\tau})=\theta_{3}(\overline{q})$;

  * •

the spin structure given by $A\in H_{1}(\mathbb{T}^{2},\mathbb{Z}_{2})$
corresponds to
$\mathcal{I}(\frac{1}{2},0;\overline{\tau})=\theta_{2}(\overline{q})$;

  * •

the spin structure given by $B\in H_{1}(\mathbb{T}^{2},\mathbb{Z}_{2})$
corresponds to
$\mathcal{I}(0,\frac{1}{2};\overline{\tau})=\theta_{4}(\overline{q})$;

  * •

the spin structure given by $A+B\in H_{1}(\mathbb{T}^{2},\mathbb{Z}_{2})$
corresponds to
$\mathcal{I}(\frac{1}{2},\frac{1}{2};\overline{\tau})=\theta_{1}(\overline{q})$.

Now let us take a look at how the choices of spin structures become refined
choices of global structure of the $4d$ theory on $\mathbb{P}^{2}$ — we will
see that this is ultimately related to the fractionalization of the Poincaré
rotation symmetry, and we indeed produce all the fractionalization classes for
Maxwell theory [63, 64]. Let us recall that the Maxwell theory has
$U(1)_{e}\times U(1)_{m}$ one-form symmetry and the object charged under it
are Wilson and ’t Hooft lines respectively. In the $6d$ picture the Wilson
lines and the ’t Hooft lines supported on a closed loop $\gamma_{1}$252525
These curves are not topological so $\gamma_{1}$ need not be a non-trivial
cycle. Indeed for $\mathbb{P}^{2}$ there are no such cycles. arise from the
following integrals of the self-dual 2-forms:

$\displaystyle W^{(e)}(\gamma_{1})=\text{exp}(i\oint_{A\cup\gamma_{1}}B^{+})$
$\displaystyle
W^{(m)}(\gamma_{1})=\text{exp}(i\oint_{B\cup\gamma_{1}}B^{+})\leavevmode\nobreak\
.$ (4.8)

From the 7d perspective, we are after the corresponding charge operators. The
latter are realised by $\Phi(\Sigma_{3})$ such that they have a surface
$\Sigma_{2}$ in $\mathbb{P}^{2}$ that links with $\gamma_{1}$ and wrap a
1-cycle in $\mathbb{T}^{2}$, corresponding to whether these are electric or
magnetic generators. However here we see that to properly define these
surfaces we need to keep track of the dependence of the $7d$ Chern-Simons on
the choice of a trivialization of $v_{4}$ on the boundary $X_{6}$. Different
trivializations correspond to choices parametrized by classes in
$H^{3}(X_{6},\mathbb{Z}_{2})$. For the space at hand there are four choices,
namely $0,\widetilde{A}\wedge w_{2}(\mathbb{P}^{2}),\widetilde{B}\wedge
w_{2}(\mathbb{P}^{2})$ and $(\widetilde{A}+\widetilde{B})\wedge
w_{2}(\mathbb{P}_{2})$ where $\widetilde{A}$ and $\widetilde{B}$ are Poincaré
duals to $A$ and $B$ cycles on $\mathbb{T}^{2}$. Now to each of these choices
correspond a different representative for the electric and the magnetic fluxes
associated to $\Phi(\Sigma_{3})$. This is how charge fractionalization is
encoded in the choice of a trivialization of $v_{4}$ on the boundary. If we
chose the trivial one, then all the fluxes will be integer and in particular
no flux will depend on $w_{2}(\mathbb{P}^{2})$. Then we can choose all lines
to be bosons. Instead in the other three choices, we are forced to give
$w_{2}(\mathbb{P}^{2})$ flux for respectively the electric, the magnetic, and
both electric and magnetic one-form symmetries in the $4d$ theory. Linking
with such a flux changes the statistics of the corresponding line. Since the
one form multiplying $w_{2}(\mathbb{P}_{2})$ determines a spin structure on
$\mathbb{T}^{2}$, we obtain the following dictionary between the spin
structure of $\mathbb{T}^{2}$ and the statistics of Wilson and ’t Hooft lines
for the Maxwell theory:

  * •

If the spin structure is $0$: both Wilson and ’t Hooft lines are bosonic;

  * •

If the spin structure is $A$: the ’t Hooft line can link with the
corresponding one-form symmetry flux and become fermionic, while the Wilson
lines stay bosonic;

  * •

If the spin structure is $B$: the Wilson lines become fermionic while the ’t
Hooft lines are bosonic.

  * •

If the spin structure is $A+B$: both Wilson and ’t Hooft lines are fermionic.

The last of these four cases is a version of Maxwell theory called “all
fermion electrodynamics” and it is known to have an anomaly [65]. This fact
can be recovered from the $6d$ picture by noticing that the partition function
for it is given by $\theta_{1}(\overline{q})$ which is notoriously identically
zero, thus signaling an anomaly.

One can consider more examples of this effect: the 6d partition function, even
for the simplest case of the free self-dual tensor field, is richer than just
the Maxwell theory, and that is encoded in the further choices that arise from
the trivialization of $v_{4}$ on the boundary. We discuss the examples of
$dP_{1}$ and $S^{2}\times S^{2}$ in detail in appendix B for the interested
reader.

##  5 Conclusions and outlook

The study of higher symmetries is undergoing a rapid evolution in recent
years.262626 For a (partial) list of references accounting for the recent
progress on this topic we refer our readers to eg. [66, 67, 11, 12, 68, 69,
70, 71, 72, 13, 73, 74, 37, 54, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85,
86, 87, 88, 89, 90, 91, 92, 93, 22, 94, 95, 96, 36, 97, 98, 99, 100, 101, 102,
103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 38, 113, 114, 115, 116, 117,
118, 119, 120, 121, 43, 122, 123, 124, 125, 2, 126, 127, 128, 129, 130, 131,
132, 133, 134, 135, 17, 45, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145,
146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 15, 158, 159, 23,
160, 19, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174,
18, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189,
190, 191, 192, 193, 194, 195, 196, 197, 198], as well as to [199] for a recent
review. In this paper we have investigated a technique to study the known
topological symmetries of 2d theories from a higher dimensional perspective,
thus recovering many known results about these theories from geometry. This
analysis confirms the idea outlined in [19] that indeed null bordisms of the
compactification manifold are a tool to induce gapped boundary conditions, and
that summing over topologies in that context is an operation equivalent to
gauging a discrete symmetry. In particular, we have seen that the Tambara-
Yamagami $\mathbb{Z}_{N}$ symmetries can be obtained in this way. It would be
interesting in the future to have a deeper exploration of the 2d symmetry
categories that have a 4d origin. This analysis is based on establishing a
dictionary between 4-manifolds and fusion 1-category data along the lines of
the lattice approach [84]. In particular, it would be interesting to
understand if the Haagerup symmetry category (see eg. [200] and references
therein) can be given a higher dimensional origin, as it was recently shown
that the latter cannot have a simple 3d gauge theory origin as a boundary of a
Chern Simons gauge theory [201].

The methods discussed in this paper can be extended straightforwardly to other
$\mathfrak{g}$ and manifolds with different dimensions. More precisely, we
exploit the fact that 6d (2,0) theories $\mathfrak{X}_{\mathfrak{g}}$ for
$\mathfrak{g}$ generic are relative field theories [202, 203]:
$\mathfrak{X}_{\mathfrak{g}}$ exists as a boundary condition for a 7d TFT,
$\mathfrak{F}_{\mathfrak{g}}$, and for $\mathfrak{g}$ generic
$\mathfrak{F}_{\mathfrak{g}}$ does not have gapped boundary conditions. For
this reason $\mathfrak{X}_{\mathfrak{g}}$ does not assign to six manifolds
$X_{6}$ a partition function, but rather a partition vector in the Hilbert
space $\mathfrak{F}_{\mathfrak{g}}(X_{6})$, a collection of conformal blocks.
Upon reduction on a $k$ dimensional manifold $Y_{k}$, we obtain a new system
$D_{Y_{k}}\mathfrak{X}_{\mathfrak{g}}$ which is also a quantum field theory in
$d=6-k$ dimensions. This theory also comes with a symmetry TFT which is
obtained via dimensional reduction of $\mathfrak{F}_{\mathfrak{g}}$,
$D_{Y_{k}}\mathfrak{F}_{\mathfrak{g}}$. Interestingly
$D_{Y_{k}}\mathfrak{F}_{\mathfrak{g}}$ can have gapped boundary conditions
even if $\mathfrak{F}_{\mathfrak{g}}$ does not, and the latter arise from
null-bordisms of $Y_{k}$, $k+1$ dimensional manifolds that have boundary
$Y_{k}$, which we can use to smoothly interpolate between $Y_{k}$ and a point.
In the context of the 4d class $\mathcal{S}$ theories that we considered in
[18, 19], $Y_{2}$ is a genus $g$ riemann surface without punctures, which
always admit (several) such null-bordisms, realized via three-dimensional
handlebodies. For two dimensional field theories instead, only those four
manifolds that have vanishing signature $\sigma(Y_{4})=b_{2}^{+}-b_{2}^{-}$
can be null-bordant. If one considers the theory
$D_{Y_{4}}\mathfrak{X}_{\mathfrak{g}}$ for $\mathfrak{g}=\mathfrak{a}_{N-1}$
with $N$ prime and $\sigma(Y_{4})=0$, the resulting 2d theory is always an
absolute theory. It would be interesting to study these correspondences more
in detail in the future for other values of $N$ and general $\mathfrak{g}$.
Also here we expect then to be able to formulate the symmetry structure of the
theories $D_{Y_{k}}\mathfrak{X}_{\mathfrak{g}}$ in terms of the boundary
conditions of $D_{Y_{k}}\mathfrak{F}_{\mathfrak{g}}$ and the condensates of
defects of $\mathfrak{F}_{\mathfrak{g}}$ to give rise to interesting
generalizations of duality defects in various dimensions. In particular, it
would be interesting to start establishing dictionaries between $k$-manifolds
and $(5-k)$-fusion category data.

## Acknowledgements

We would like to thank Justin Kaidi for initial collaboration on this project
as well as for many illuminating discussions that helped us shaping these
ideas and refining these techniques. We especially thank Iñaki García
Etxebarria for comments on the previous version of this manuscrip which lead
to a substantial improvement of the results in Section 3.6. We also thank
Christian Copetti, Luis Diogo, Babak Haghighat, Tzu-Chen Huang, Alice
Hedenlund, Kantaro Ohmori, and Jian Qiu for many useful conversations. 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). The work of 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 A non-invertible T-duality

As the simplest example of this general framework let us consider the example
of theory obtained by dimensional reduction of $6d\,(2,0)$ theory of type
$\mathfrak{a}_{1}$ on $S^{2}\times S^{2}$. In this case the conformal
structure of $S^{2}\times S^{2}$ is fixed by the sole dimensionless parameter
$R^{2}$ which is the ratio of the areas of two spheres. The two spheres
intersect each other so the intersection form is the Pauli matrix
$\sigma_{1}$.

The symmetry TFT is a $U(1)^{2}$ Chern-Simons theory with only off diagonal
couplings,

$\displaystyle S_{3d}=\frac{1}{4\pi}\int_{W_{3}}(a_{1}\wedge\,\differential
a_{2}+a_{2}\wedge\,\differential a_{1})$ (A.1)

The mapping class group is $\mathbb{Z}_{2}$ corresponding to exchange of two
spheres which we call $S_{1}$ and $S_{2}$. Hence it changes $R\to\frac{1}{R}$
and corresponds to a T-duality which for abelian case considered from the six
dimensional perspective in [31].

The global structures of this $2d$ theory correspond to isotropic sublattices
of $H_{2}(S^{2}\times S^{2},\mathbb{Z}_{2})$. Notice that modulo $2$, the
intersection form of $S^{2}\times S^{2}$ is same as the intersection form of
$\mathbb{T}^{2}$. So in precise analogy with $4d\,\mathcal{N}=4$ SYM with
gauge algebra $\mathfrak{su}(2)$ there are three global structures, generated
by $S_{1},S_{2}$ and $S_{1}+S_{2}$ respectively.

The first two global structures, determined by $S_{1}$ and $S_{2}$
respectively are exchanged by the T-duality and as a result they have non-
invertible T-duality at the self dual point $R=1$. While the third one
generated by $S_{1}+S_{2}$ is fixed by the T-duality and at $R=1$ T-duality
becomes an anomalous symmetry. This is very similar to the treatment of
$\mathcal{N}=4\,\mathfrak{su}(2)$ SYM under S-transformation of
$SL(2,\mathbb{Z})$ mapping class group at $\tau=i$. However there is a crucial
difference, in $4d$ case the non-invertible symmetries can be related to the
anomalous symmetry using $T$-transformation, this however is not the case for
the 2d theory since mapping class group has two disconnected orbits.

We can also consider the dimensional reduction of $\mathfrak{a}_{p-1}$ theory
on $S^{2}\times S^{2}$. For $p>2$ there are only two global structures
generated by $S_{1}$ and $S_{2}$ respectively. The two global structures are
interchanged under $T$-duality and at $R=1$ we obtain a non-invertible
symmetry.

##  Appendix B More on Spin CS and the self-dual tensor

###  B.1 $dP_{1}$

As the next case let us consider the abelian theory on $\mathbb{T}^{2}\times
dP_{1}$. We again start with a lattice $\mathcal{L}$ which is relevant for
$4d$ S-duality and is generated by $A\cup H$ and $A\cup E_{1}$ in the notation
of Section 3.7.1. A basis of self dual form with integral periods over this
sublattice is,

$\displaystyle F_{1}$
$\displaystyle=\differential\overline{z}\wedge\omega_{+}\leavevmode\nobreak\
,$ $\displaystyle F_{2}=\differential z\wedge\omega_{-}\leavevmode\nobreak\ .$
(B.1)

Where $\omega_{+}$ and $\omega_{-}$ are 2-forms Poincaré dual to $H$ and
$E_{1}$ respectively. Choosing $B\cup H$ and $B\cup E_{1}$ for the basis for
$\mathcal{L}^{\perp}$ we obtain, the period matrix,

$\displaystyle\begin{pmatrix}-\overline{\tau}&&0\\\
0&&\tau\end{pmatrix}\leavevmode\nobreak\ .$ (B.2)

As a result $\mathcal{I}(\theta,\phi;\tau,\overline{\tau})$ is given by,

$\displaystyle\mathcal{I}(\theta,\phi;\tau,\overline{\tau})=\sum_{n_{1}\in\mathbb{Z}+\theta_{1},n_{2}\in\mathbb{Z}+\theta_{2}}\exp(\pi
i(n_{1}^{2}\tau-n_{2}^{2}\overline{\tau})+2\pi
i(n_{1}\phi_{1}+n_{2}\phi_{2}))\leavevmode\nobreak\ .$ (B.3)

As expected [204], since the intersection form of $dP_{1}$ is odd, this
partition function is not invariant under $\tau\to\tau+1$ but only under
$\tau\to\tau+2$.

However it is possible to write another partition function for $dP_{1}$ too
which is not related to (B.3) by any duality. $E+H$ has self intersection $0$
and as a result the lattice generated by $A\cup(H+E_{1})$ and $B\cup(H+E_{1})$
is isotropic. This is a lattice of the form
$\mathcal{L}=H_{1}(\mathbb{T}^{2},\mathbb{Z})\otimes L$ with $L$ a Lagrangian
subgroup of $H_{2}(dP_{1},\mathbb{Z})$. However unlike the cases we have
considered above no such basis which is of the form
$H_{1}(\mathbb{T}^{2},\mathbb{Z})\otimes L^{\perp}$ can be written for
$\mathcal{L}^{\perp}$. So we choose a basis for $\mathcal{L}^{\perp}$ which is
$B\cup H$ and $A\cup E_{1}$. The resulting period matrix is,

$\displaystyle\frac{1}{2\tau_{2}}\begin{pmatrix}i\tau\overline{\tau}&i\overline{\tau}\\\
i\overline{\tau}&i\end{pmatrix}\leavevmode\nobreak\ ,$ (B.4)

where $\tau=\tau_{1}+i\tau_{2}$. The resulting
$\mathcal{I}(\theta,\phi;\tau,\overline{\tau})$ is,

$\displaystyle\mathcal{I}(\theta,\phi,\tau,\overline{\tau})$
$\displaystyle=\sum_{n_{1}\in\mathbb{Z}+\theta_{1},n_{2}\in\mathbb{Z}+\theta_{2}}\exp(-\left(\frac{\tau\overline{\tau}}{2\tau_{2}}n_{1}^{2}+\frac{1}{2\tau_{2}}n_{2}^{2}\right)-\frac{\overline{\tau}}{\tau_{2}}n_{1}n_{2})\leavevmode\nobreak\
,$
$\displaystyle=\sum_{n_{1}\in\mathbb{Z}+\theta_{1},n_{2}\in\mathbb{Z}+\theta_{2}}\exp(-\pi\frac{\left(\tau_{1}n_{1}-n_{2}\right)^{2}+\tau_{2}^{2}n_{1}^{2}}{2\tau_{2}}+\pi
i(n_{1}n_{2}+2n_{1}\phi_{1}+2n_{2}\phi_{2}))\leavevmode\nobreak\ .$ (B.5)

Due to the term involving $n_{1}n_{2}$, this partition function is not
invariant under $\tau\to\tau+1$ even when $\phi=0$ i.e when background fields
have been switched off. This illustrates the point that for a spin Chern-
Simons theory a Lagrangian subgroup of the discriminant group does not
necessarily give a topological boundary condition. Just like the expression in
(B.3) it is invariant under $\tau\to\tau+2$. However the two have a different
behavior for $\tau\to-\frac{1}{\tau}$. The expression above is invariant while
that in (B.3) transforms as a modular form.

###  B.2 $S^{2}\times S^{2}$

As a last example let us consider case $6d$ theory on $\mathbb{T}^{2}\times
S^{2}\times S^{2}$. The two dimensional theory obtained by dimensionally
reducing along $S^{2}\times S^{2}$ has a symmetry TFT which is an ordinary
Chern-Simons and hence we should expect an modular invariant partition
function. Similarly the Maxwell theory on $S^{2}\times S^{2}$ is invariant
under $\tau\to\tau+1$. However the partition function the $6d$ theory gives on
$\mathbb{T}^{2}\times S^{2}\times S^{2}$ is not unique. This is because
$H_{3}(\mathbb{T}^{2}\times S^{2}\times S^{2})$ has multiple maximal isotropic
sublattices with at least two distinct orbits under
$SL(2,\mathbb{Z})\times\mathbb{Z}_{2}$ mapping class group. As in the previous
example the first class consists of $\mathcal{L}$ of the form $L\times
H_{2}(S^{2}\times S^{2},\mathbb{Z})$. Such an $\mathcal{L}$ transforms into
another lattice of this same class under the action of $SL(2,\mathbb{Z})$
mapping class group of $\mathbb{T}^{2}$ and is invariant under the
$\mathbb{Z}_{2}$ mapping class group of $S^{2}\times S^{2}$. The second class
consists of $\mathcal{L}$ of the form $H_{1}(\mathbb{T}^{2},\mathbb{Z})\times
L^{\prime}$ and the lattices in this class are invariant under the
$SL(2,\mathbb{Z})$ but can transform under the $\mathbb{Z}_{2}$ mapping
classes of $S^{2}\times S^{2}$. There are no mapping classes that transform a
lattice in one class to the other and as we will see the partition functions
obtained from the two classes are different.272727The multitude of the
partition functions is a consequence of the fact that the manifolds we have
considered are not symmetric enough that all maximally isotropic sublattices
of $H_{3}(X_{6},\mathbb{Z})$ lies in the same orbit of mapping class group.
For the case of $\mathbb{T}^{6}$ which has a mapping class group
$SL(6,\mathbb{Z})$ big enough for this, the partition function for theory of
two form is known [205] and it is $SL(6,\mathbb{Z})$ invariant and hence
unique.

Let us consider a example of the first class with a basis of $\mathcal{L}$
given by $A\cup C_{1}$ and $A\cup C_{2}$ where $C_{1}$ and $C_{2}$ are the two
spheres in $S^{2}\times S^{2}$ regarded as 2-cycles. A basis for
$\mathcal{L}^{\perp}$ is given by $B\cup C_{2}$ and $A\cup C_{1}$. The Hodge
star on $S^{2}\times S^{2}$ depends on $R^{2}$ the relative volume of two
spheres and is given by,

$\displaystyle\star C_{1}=R^{2}\,C_{2}\leavevmode\nobreak\ ,$
$\displaystyle\star C_{2}=\frac{1}{R^{2}}\,C_{1}\leavevmode\nobreak\ .$ (B.6)

A basis of self dual forms with integral periods on $\mathcal{L}$ is given by,

$\displaystyle F_{1}=\differential x\wedge\omega_{1}+iR^{2}\,\differential
y\wedge\omega_{2}\leavevmode\nobreak\ ,$ $\displaystyle F_{2}=\differential
x\wedge\omega_{2}+\frac{i}{R^{2}}\,\differential
y\wedge\omega_{1}\leavevmode\nobreak\ ,$ (B.7)

with $\omega_{1}$ and $\omega_{2}$ being the Poincare duals of $C_{1}$ and
$C_{2}$. The corresponding period matrix is,

$\displaystyle\begin{pmatrix}\frac{i\tau_{2}}{R^{2}}&&\tau_{1}\\\
\tau_{1}&&iR^{2}\tau_{2}\end{pmatrix}\leavevmode\nobreak\ .$ (B.8)

Which results in,282828Since $\theta$ and $\phi$ do not influence the modular
properties we have set them to zero here in order to emphasize the modular
properties.

$\displaystyle\mathcal{I}(\tau,\overline{\tau},R)=\sum_{n_{1},n_{2}}\exp(-\pi\left(\frac{n_{1}^{2}}{R^{2}}+n_{2}^{2}R^{2}\right)\tau_{2}-2\pi
in_{1}n_{2}\tau_{1})\leavevmode\nobreak\ .$ (B.9)

This is explicitly invariant under $R\to\frac{1}{R}$ and under $\tau\to\tau+1$
and transforms as modular form under $\tau\to\frac{1}{\tau}$ as expected for
the partition function of $4d$ Maxwell theory.292929Correctly matching the
weight requires inclusion of the factor of $\Delta$ which we are ignoring for
now.

Although writing this partition function requires choosing a maximal isotropic
sublattice $L$ of $H_{1}(\mathbb{T}^{2},\mathbb{Z})$ the result itself is
independent of $L$ and hence gives a well defined $2d$ theory which can
recognized as usual compact boson of radius $R^{2}$.

Nevertheless by choosing an $\mathcal{L}$ from the second class we can write a
different partition function. For this we choose $\mathcal{L}$ to be generated
by $A\cup C_{1}$ and $B\cup C_{2}$ with the basis for $\mathcal{L}^{\perp}$
given by $B\cup C_{2}$ and $A\cup C_{1}$. A basis of self dual three forms
with integral periods on $\mathcal{L}$ is given by,

$\displaystyle F_{1}$
$\displaystyle=\frac{1}{2i\tau_{2}}\left(\overline{\tau}\differential
z\wedge(\omega_{1}-R^{2}\omega_{2})-\tau\differential\overline{z}\wedge(\omega_{1}+R^{2}\omega_{2})\right)\leavevmode\nobreak\
,$ $\displaystyle F_{2}$ $\displaystyle=\frac{i}{2\tau_{2}}\left(\differential
z\wedge(\omega_{1}-R^{2}\omega_{2})-\differential\overline{z}\wedge(\omega_{1}+R^{2}\omega_{2})\right)\leavevmode\nobreak\
.$ (B.10)

This data results in the period matrix,

$\displaystyle\frac{1}{\tau_{2}}\begin{pmatrix}i\tau\overline{\tau}R^{2}&-i\tau_{1}R^{2}\\\
-i\tau_{1}R^{2}&iR^{2}\end{pmatrix}\leavevmode\nobreak\ ,$ (B.11)

and hence the partition function correspond to,

$\displaystyle\mathcal{I}(R,\tau,\overline{\tau})$
$\displaystyle=\sum_{n_{1},n_{2}}\exp(-\frac{((\tau_{1}n_{1}-n_{2})^{2}+\tau_{2}^{2}n_{1}^{2})R^{2}}{\tau_{2}})\leavevmode\nobreak\
.$ (B.12)

This expression can be verified to be invariant under $SL(2,\mathbb{Z})$
mapping class group of $\mathbb{T}^{2}$ and hence gives the partition function
of an absolute theory.

## References

  * [1] D. S. Freed, G. W. Moore, and C. Teleman, “Topological symmetry in quantum field theory,” arXiv:2209.07471 [hep-th].
  * [2] F. Apruzzi, F. Bonetti, I. n. G. Etxebarria, S. S. Hosseini, and S. Schafer-Nameki, “Symmetry TFTs from String Theory,” arXiv:2112.02092 [hep-th].
  * [3] G. W. Moore and N. Seiberg, “Taming the Conformal Zoo,” Phys. Lett. B 220 (1989) 422–430.
  * [4] N. Reshetikhin and V. G. Turaev, “Invariants of three manifolds via link polynomials and quantum groups,” Invent. Math. 103 (1991) 547–597.
  * [5] V. G. Turaev and O. Y. Viro, “State sum invariants of 3 manifolds and quantum 6j symbols,” Topology 31 (1992) 865–902.
  * [6] J. Fuchs, I. Runkel, and C. Schweigert, “TFT construction of RCFT correlators 1\. Partition functions,” Nucl. Phys. B 646 (2002) 353–497, arXiv:hep-th/0204148.
  * [7] J. Fuchs, I. Runkel, and C. Schweigert, “TFT construction of RCFT correlators. 2. Unoriented world sheets,” Nucl. Phys. B 678 (2004) 511–637, arXiv:hep-th/0306164.
  * [8] J. Fuchs, I. Runkel, and C. Schweigert, “TFT construction of RCFT correlators. 3. Simple currents,” Nucl. Phys. B 694 (2004) 277–353, arXiv:hep-th/0403157.
  * [9] J. Fuchs, I. Runkel, and C. Schweigert, “TFT construction of RCFT correlators IV: Structure constants and correlation functions,” Nucl. Phys. B 715 (2005) 539–638, arXiv:hep-th/0412290.
  * [10] J. Fjelstad, J. Fuchs, I. Runkel, and C. Schweigert, “TFT construction of RCFT correlators. V. Proof of modular invariance and factorisation,” Theor. Appl. Categor. 16 (2006) 342–433, arXiv:hep-th/0503194.
  * [11] A. Kapustin and R. Thorngren, “Topological Field Theory on a Lattice, Discrete Theta-Angles and Confinement,” Adv. Theor. Math. Phys. 18 no. 5, (2014) 1233–1247, arXiv:1308.2926 [hep-th].
  * [12] A. Kapustin and R. Thorngren, “Higher symmetry and gapped phases of gauge theories,” arXiv:1309.4721 [hep-th].
  * [13] F. Benini, C. Córdova, and P.-S. Hsin, “On 2-Group Global Symmetries and their Anomalies,” JHEP 03 (2019) 118, arXiv:1803.09336 [hep-th].
  * [14] D. S. Freed, M. J. Hopkins, J. Lurie, and C. Teleman, “Topological Quantum Field Theories from Compact Lie Groups,” in A Celebration of Raoul Bott’s Legacy in Mathematics.  5, 2009.  arXiv:0905.0731 [math.AT].
  * [15] J. Kaidi, K. Ohmori, and Y. Zheng, “Symmetry TFTs for Non-Invertible Defects,” arXiv:2209.11062 [hep-th].
  * [16] R. Dijkgraaf and E. Witten, “Topological Gauge Theories and Group Cohomology,” Commun. Math. Phys. 129 (1990) 393.
  * [17] J. Kaidi, G. Zafrir, and Y. Zheng, “Non-Invertible Symmetries of $\mathcal{N}=4$ SYM and Twisted Compactification,” arXiv:2205.01104 [hep-th].
  * [18] V. Bashmakov, M. Del Zotto, and A. Hasan, “On the 6d Origin of Non-invertible Symmetries in 4d,” arXiv:2206.07073 [hep-th].
  * [19] V. Bashmakov, M. Del Zotto, A. Hasan, and J. Kaidi, “Non-invertible Symmetries of Class $\mathcal{S}$ Theories,” arXiv:2211.05138 [hep-th].
  * [20] A. Antinucci, C. Copetti, G. Galati, and G. Rizi, “”Zoology” of non-invertible duality defects: the view from class $\mathcal{S}$,” arXiv:2212.09549 [hep-th].
  * [21] Y. Tachikawa, “On the 6d origin of discrete additional data of 4d gauge theories,” JHEP 05 (2014) 020, arXiv:1309.0697 [hep-th].
  * [22] S. Gukov, P.-S. Hsin, and D. Pei, “Generalized global symmetries of $T[M]$ theories. Part I,” JHEP 04 (2021) 232, arXiv:2010.15890 [hep-th].
  * [23] A. Antinucci, F. Benini, C. Copetti, G. Galati, and G. Rizi, “The holography of non-invertible self-duality symmetries,” arXiv:2210.09146 [hep-th].
  * [24] D. Gaiotto, “N=2 dualities,” JHEP 08 (2012) 034, arXiv:0904.2715 [hep-th].
  * [25] D. Gaiotto, G. W. Moore, and A. Neitzke, “Wall-crossing, Hitchin Systems, and the WKB Approximation,” arXiv:0907.3987 [hep-th].
  * [26] A. Gadde, S. Gukov, and P. Putrov, “Fivebranes and 4-manifolds,” Prog. Math. 319 (2016) 155–245, arXiv:1306.4320 [hep-th].
  * [27] P. Putrov, J. Song, and W. Yan, “(0,4) dualities,” JHEP 03 (2016) 185, arXiv:1505.07110 [hep-th].
  * [28] S. Gukov, D. Pei, P. Putrov, and C. Vafa, “4-manifolds and topological modular forms,” JHEP 05 (2021) 084, arXiv:1811.07884 [hep-th].
  * [29] M. Dedushenko, S. Gukov, and P. Putrov, “Vertex algebras and 4-manifold invariants,” in Nigel Hitchin’s 70th Birthday Conference, vol. 1, pp. 249–318.  5, 2017.  arXiv:1705.01645 [hep-th].
  * [30] J. A. Harvey, G. W. Moore, and A. Strominger, “Reducing S duality to T duality,” Phys. Rev. D 52 (1995) 7161–7167, arXiv:hep-th/9501022.
  * [31] E. P. Verlinde, “Global aspects of electric - magnetic duality,” Nucl. Phys. B 455 (1995) 211–228, arXiv:hep-th/9506011.
  * [32] N. Seiberg and W. Taylor, “Charge Lattices and Consistency of 6D Supergravity,” JHEP 06 (2011) 001, arXiv:1103.0019 [hep-th].
  * [33] J. Frohlich, J. Fuchs, I. Runkel, and C. Schweigert, “Kramers-Wannier duality from conformal defects,” Phys. Rev. Lett. 93 (2004) 070601, arXiv:cond-mat/0404051.
  * [34] D. S. Freed and C. Teleman, “Topological dualities in the Ising model,” arXiv:1806.00008 [math.AT].
  * [35] C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin, “Topological Defect Lines and Renormalization Group Flows in Two Dimensions,” JHEP 01 (2019) 026, arXiv:1802.04445 [hep-th].
  * [36] Z. Komargodski, K. Ohmori, K. Roumpedakis, and S. Seifnashri, “Symmetries and strings of adjoint QCD2,” JHEP 03 (2021) 103, arXiv:2008.07567 [hep-th].
  * [37] R. Thorngren and Y. Wang, “Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases,” arXiv:1912.02817 [hep-th].
  * [38] R. Thorngren and Y. Wang, “Fusion Category Symmetry II: Categoriosities at $c$ = 1 and Beyond,” arXiv:2106.12577 [hep-th].
  * [39] N. Seiberg, “Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics,” Phys. Lett. B 388 (1996) 753–760, arXiv:hep-th/9608111.
  * [40] M. R. Douglas, S. H. Katz, and C. Vafa, “Small instantons, Del Pezzo surfaces and type I-prime theory,” Nucl. Phys. B 497 (1997) 155–172, arXiv:hep-th/9609071.
  * [41] O. J. Ganor and A. Hanany, “Small E(8) instantons and tensionless noncritical strings,” Nucl. Phys. B 474 (1996) 122–140, arXiv:hep-th/9602120.
  * [42] J. Milnor, “On simply connected 4-manifolds,” vol. Proc. Int. Symp. Algebraic. Topology, Mexico.  1958\.
  * [43] Y. Choi, C. Cordova, P.-S. Hsin, H. T. Lam, and S.-H. Shao, “Noninvertible duality defects in 3+1 dimensions,” Phys. Rev. D 105 no. 12, (2022) 125016, arXiv:2111.01139 [hep-th].
  * [44] D. Gaiotto and T. Johnson-Freyd, “Condensations in higher categories,” arXiv:1905.09566 [math.CT].
  * [45] K. Roumpedakis, S. Seifnashri, and S.-H. Shao, “Higher Gauging and Non-invertible Condensation Defects,” arXiv:2204.02407 [hep-th].
  * [46] M. Barkeshli, P. Bonderson, M. Cheng, and Z. Wang, “Symmetry Fractionalization, Defects, and Gauging of Topological Phases,” Phys. Rev. B 100 no. 11, (2019) 115147, arXiv:1410.4540 [cond-mat.str-el].
  * [47] D. Belov and G. W. Moore, “Classification of Abelian spin Chern-Simons theories,” arXiv:hep-th/0505235.
  * [48] A. Kapustin and N. Saulina, “Topological boundary conditions in abelian Chern-Simons theory,” Nucl. Phys. B 845 (2011) 393–435, arXiv:1008.0654 [hep-th].
  * [49] S. D. Stirling, Abelian Chern-Simons theory with toral gauge group, modular tensor categories, and group categories.  PhD thesis, Texas U., Math Dept., 2008.  arXiv:0807.2857 [hep-th].
  * [50] Y. Lee and Y. Tachikawa, “A study of time reversal symmetry of abelian anyons,” JHEP 07 (2018) 090, arXiv:1805.02738 [hep-th].
  * [51] M. N. Khan, J. C. Y. Teo, and T. L. Hughes, “Anyonic Symmetries and Topological Defects in Abelian Topological Phases: an application to the $ADE$ Classification,” Phys. Rev. B 90 no. 23, (2014) 235149, arXiv:1403.6478 [cond-mat.str-el].
  * [52] J. C. Y. Teo, T. L. Hughes, and E. Fradkin, “Theory of Twist Liquids: Gauging an Anyonic Symmetry,” Annals Phys. 360 (2015) 349–445, arXiv:1503.06812 [cond-mat.str-el].
  * [53] R. J. Stern, Instantons and the Topology of 4-Manifolds, pp. 333–341.  Springer New York, New York, NY, 2001.
  * [54] I. n. Garcίa Etxebarria, B. Heidenreich, and D. Regalado, “IIB flux non-commutativity and the global structure of field theories,” JHEP 10 (2019) 169, arXiv:1908.08027 [hep-th].
  * [55] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, vol. 290 of Grundlehren der mathematischen Wissenschaften.  Springer, 1988.  https://doi.org/10.1007/978-1-4757-2016-7.
  * [56] R. Miranda and D. Morrison, “Embeddings of Integral Quadratic Forms,”. https://web.math.ucsb.edu/~drm/manuscripts/eiqf.pdf.
  * [57] A. Iqbal, A. Neitzke, and C. Vafa, “A Mysterious duality,” Adv. Theor. Math. Phys. 5 (2002) 769–808, arXiv:hep-th/0111068.
  * [58] P.-S. Hsin and N. Seiberg, “Level/rank Duality and Chern-Simons-Matter Theories,” JHEP 09 (2016) 095, arXiv:1607.07457 [hep-th].
  * [59] E. Witten, “On flux quantization in M theory and the effective action,” J. Geom. Phys. 22 (1997) 1–13, arXiv:hep-th/9609122.
  * [60] E. Witten, “Five-brane effective action in M theory,” J. Geom. Phys. 22 (1997) 103–133, arXiv:hep-th/9610234.
  * [61] E. Witten, “Duality relations among topological effects in string theory,” JHEP 05 (2000) 031, arXiv:hep-th/9912086.
  * [62] D. Belov and G. W. Moore, “Holographic Action for the Self-Dual Field,” arXiv:hep-th/0605038.
  * [63] T. D. Brennan, C. Cordova, and T. T. Dumitrescu, “Line Defect Quantum Numbers & Anomalies,” arXiv:2206.15401 [hep-th].
  * [64] D. Delmastro, J. Gomis, P.-S. Hsin, and Z. Komargodski, “Anomalies and Symmetry Fractionalization,” arXiv:2206.15118 [hep-th].
  * [65] J. Wang, X.-G. Wen, and E. Witten, “A New SU(2) Anomaly,” J. Math. Phys. 60 no. 5, (2019) 052301, arXiv:1810.00844 [hep-th].
  * [66] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, “Generalized Global Symmetries,” JHEP 02 (2015) 172, arXiv:1412.5148 [hep-th].
  * [67] D. Gaiotto, G. W. Moore, and A. Neitzke, “Framed BPS States,” Adv. Theor. Math. Phys. 17 no. 2, (2013) 241–397, arXiv:1006.0146 [hep-th].
  * [68] O. Aharony, N. Seiberg, and Y. Tachikawa, “Reading between the lines of four-dimensional gauge theories,” JHEP 08 (2013) 115, arXiv:1305.0318 [hep-th].
  * [69] M. Del Zotto, J. J. Heckman, D. S. Park, and T. Rudelius, “On the Defect Group of a 6D SCFT,” Lett. Math. Phys. 106 no. 6, (2016) 765–786, arXiv:1503.04806 [hep-th].
  * [70] E. Sharpe, “Notes on generalized global symmetries in QFT,” Fortsch. Phys. 63 (2015) 659–682, arXiv:1508.04770 [hep-th].
  * [71] Y. Tachikawa, “On gauging finite subgroups,” SciPost Phys. 8 no. 1, (2020) 015, arXiv:1712.09542 [hep-th].
  * [72] C. Córdova, T. T. Dumitrescu, and K. Intriligator, “Exploring 2-Group Global Symmetries,” JHEP 02 (2019) 184, arXiv:1802.04790 [hep-th].
  * [73] P.-S. Hsin, H. T. Lam, and N. Seiberg, “Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d,” SciPost Phys. 6 no. 3, (2019) 039, arXiv:1812.04716 [hep-th].
  * [74] Z. Wan and J. Wang, “Higher anomalies, higher symmetries, and cobordisms I: classification of higher-symmetry-protected topological states and their boundary fermionic/bosonic anomalies via a generalized cobordism theory,” Ann. Math. Sci. Appl. 4 no. 2, (2019) 107–311, arXiv:1812.11967 [hep-th].
  * [75] J. Eckhard, H. Kim, S. Schafer-Nameki, and B. Willett, “Higher-Form Symmetries, Bethe Vacua, and the 3d-3d Correspondence,” JHEP 01 (2020) 101, arXiv:1910.14086 [hep-th].
  * [76] Z. Wan, J. Wang, and Y. Zheng, “Higher anomalies, higher symmetries, and cobordisms II: Lorentz symmetry extension and enriched bosonic / fermionic quantum gauge theory,” Ann. Math. Sci. Appl. 05 no. 2, (2020) 171–257, arXiv:1912.13504 [hep-th].
  * [77] O. Bergman, Y. Tachikawa, and G. Zafrir, “Generalized symmetries and holography in ABJM-type theories,” JHEP 07 (2020) 077, arXiv:2004.05350 [hep-th].
  * [78] D. R. Morrison, S. Schafer-Nameki, and B. Willett, “Higher-Form Symmetries in 5d,” JHEP 09 (2020) 024, arXiv:2005.12296 [hep-th].
  * [79] F. Albertini, M. Del Zotto, I. n. Garcίa Etxebarria, and S. S. Hosseini, “Higher Form Symmetries and M-theory,” JHEP 12 (2020) 203, arXiv:2005.12831 [hep-th].
  * [80] P.-S. Hsin and H. T. Lam, “Discrete theta angles, symmetries and anomalies,” SciPost Phys. 10 no. 2, (2021) 032, arXiv:2007.05915 [hep-th].
  * [81] I. Bah, F. Bonetti, and R. Minasian, “Discrete and higher-form symmetries in SCFTs from wrapped M5-branes,” JHEP 03 (2021) 196, arXiv:2007.15003 [hep-th].
  * [82] M. Del Zotto, I. n. Garcίa Etxebarria, and S. S. Hosseini, “Higher form symmetries of Argyres-Douglas theories,” JHEP 10 (2020) 056, arXiv:2007.15603 [hep-th].
  * [83] I. Hason, Z. Komargodski, and R. Thorngren, “Anomaly Matching in the Symmetry Broken Phase: Domain Walls, CPT, and the Smith Isomorphism,” SciPost Phys. 8 no. 4, (2020) 062, arXiv:1910.14039 [hep-th].