# Defects via factorization algebras

Ivan Contreras and Chris Elliott and Owen Gwilliam

###### Abstract.

We provide a mathematical formulation of the idea of a defect for a field
theory, in terms of the factorization algebra of observables and using the BV
formalism. Our approach follows a well-known ansatz identifying a defect as a
boundary condition along the boundary of a blow-up, but it uses recent work of
Butson–Yoo and Rabinovich on boundary conditions and their associated
factorization algebras to implement the ansatz. We describe how a range of
natural examples of defects fits into our framework.

##  1\. Introduction

The notion of a defect in a field theory has played an increasingly important
role in physics and, perhaps surprisingly, in mathematics. Our goal in this
short paper is to offer a mathematical formulation of defects in field theory
that builds upon recent progress with the Batalin-Vilkovisky (BV) formalism
and higher algebra. Let us start by outlining one interpretation of the idea
of a defect.

Suppose we are studying a field theory $\mathcal{T}$ on a manifold $M$.
Loosely speaking, a defect is a modification of the theory along a submanifold
$D\subset M$ in a way that produces a new field theory. We call $D$ the
support of the defect, and the dimension of the defect is the dimension of
$D$. Well-known examples include the Wilson and ‘t Hooft line defects in
Yang–Mills theory: these are supported along one-dimensional submanifolds of a
four-dimensional manifold. (In physics these defects provide a means to
characterize the phase of a gauge theory, such as confining or Coulombic.)

We thus view a defect as consisting of a pair of data: the submanifold on
which its supported and how it affects (or couples) to the ambient theory.

######  Remark 1.1.

Note that this approach diverges from the terminology in some communities,
which might prefer to view a defect (or extended operator) as defined for some
large class of submanifolds. For instance, people often refer to a
representation $V$ of a group $G$ as giving a Wilson line operator, since one
can produce a line defect from $V$ on a large class of embedded 1-manifolds.

There is an ansatz for how to produce defects that will guide our approach in
this paper; it works by reducing the problem to studying boundary conditions.

######  Ansatz.

Let $\mathcal{T}$ be a classical field theory on a manifold $M$, which
determines a system of partial differential equations. There are two steps to
making a defect along a submanifold $D\subset M$:

  1. (1)

Take a small tubular neighborhood $\widetilde{D}\supset D$ whose complement is
a manifold with boundary $M\setminus\widetilde{D}$.

  2. (2)

Specify a boundary condition for the theory $\mathcal{T}$ on this manifold
with boundary.

If one takes a small enough neighborhood (or works in some limit as the
neighborhood shrinks), then this new theory (with the imposed boundary
condition) looks like the original theory but modified along $D$, as the
boundary condition affects the solutions to the equations of motion. We call
this new theory the theory $\mathcal{T}$ with a defect along $D$.

If we specify a quantization of $\mathcal{T}$ on $M\setminus\widetilde{D}$
with this boundary condition, then we produce a defect for a quantization of
$\mathcal{T}$. See Remark 1.1 for some background about this ansatz.

######  Remark 1.2.

Note that after removing $D$ from $M$, there may be solutions to the equations
of motion that do not extend across $D$. More generally, one may even consider
theories $\mathcal{T}$ defined only on $M-D$. We will continue to use the term
“defect” for such situations, where the boundary condition for the theory
$\mathcal{T}$ on $M\setminus\widetilde{D}$ prescribes the allowed limiting
behavior of $\mathcal{T}$ as one approaches $D$. This situation is also
commonly referred to by the term _disorder operator_. (For a discussion of
order operators, see Remark 4.2 below.)

An appealing aspect of this ansatz is that it is constructive, at least once
one spells out how to produce boundary conditions for quantum theories. A
drawback is that it is not a definition: certainly given any defect along $D$
and any tubular neighborhood $\widetilde{D}$ we should be able to obtain a
boundary condition as described in the ansatz, but we will see that this
assignment generally does not need to be an equivalence, because of the
reliance within the ansatz of a specific choice of neighborhood
$\widetilde{D}$.

A slight shift of perspective, however, offers a useful setting to pose an
actual definition. We shift our focus to the algebra of observables of a
theory with defect, rather than focussing directly on the theory itself, much
as it is often fruitful to focus on the algebra of functions on a space rather
than directly on the space. In the setting of QFT (at least in the
perturbative regime), the observables form a factorization algebra on the
spacetime manifold [CG21]. Recall that a factorization algebra $\mathcal{A}$
on a manifold $M$ is a local-to-global object, akin to a sheaf. (For a
systematic treatment, see [CG17, AF15] and references therein.) In particular,
it determines a functor $\mathcal{A}\colon\mathrm{Open}(M)\to\mathrm{Ch}$
assigning a cochain complex $\mathcal{A}(U)$ to each open set $U\subset M$.
For a QFT, $\mathcal{A}(U)$ consists of the observables with support in $U$,
i.e., that depend on the behavior of the fields only in the region $U$. The
key result of [CG21] is that if one constructs the QFT in the BV formalism,
the complex of observables forms a factorization algebra.

In the language of factorization algebras, we proffer the following view on
defects.

######  Definition 1.3.

Let $M$ be a manifold and $D\subset M$ a submanifold. Let $\mathcal{A}$ be a
factorization algebra on a manifold $M-D$. A defect along $D$ for
$\mathcal{A}$ is a factorization algebra $\mathcal{B}$ on $M$ with an
isomorphism

$\phi\colon\mathcal{B}|_{M-D}\xrightarrow{\cong}\mathcal{A}$ (1)

on the complement of $D$.

We view $\mathcal{B}$ as an extension of $\mathcal{A}$ along $D$, with
specifically prescribed modified behavior. In other words, we view $M$ as
stratified $D\subset M$ and we ask for factorization algebras that agree with
$\mathcal{A}$ on the big stratum $M-D$. When $\mathcal{A}$ is the algebra of
observables for a field theory, this definition clearly matches the behavior
of the observables of a defect in its heuristic form.

Our paper explains how one can implement the ansatz carefully to produce such
factorization algebras modelling a defect. More precisely, given a classical
BV field theory $\mathcal{T}$ on a manifold $M$, we will produce defects for
$\mathrm{Obs}^{\mathrm{cl}}_{\mathcal{T}}$ in the sense of Definition 1.3. The
construction is outlined in Section 2. The factorization algebra modelling the
defect depends explicitly on the choice of tubular neighborhood of $D$.

When, however, the theory satisfies a useful property near the defect — it is
topological normal to $D$ — we can then use recent results of Rabinovich
[Rab20, Rab21] to construct defects that do not depend on the size of the
neighborhood. Moreover, Rabinovich’s results allow one to construct the
quantum observables for the defects, when a BV quantization exists. We discuss
these results in Section 3.

Finally, we describe a number of examples, organized by codimension of the
defect, such as magnetic monopoles and Wilson lines, among others.

###  1.1. Some history, context, and future directions

The ansatz we have discussed is well-known, at least among quantum field
theorists, and has probably been known for several decades. We are not
knowledgeable, however, of how this idea appeared and evolved within the
literature. We found an explicit and useful articulation in [Kap06] but it
begins by acknowledging the idea is well-known. (Insights and suggestions
about the history and literature are welcome.)

The essential idea about how to capture defects with factorization algebras
has also been floating around the community of factorization algebraists for
at least a decade, and hence some version was known by many people. What
prompted us to document and extend these ideas is the powerful work of
Rabinovich, which makes it possible to implement the ansatz precisely and
rigorously using factorization algebras in a broad class of examples.

A careful treatment of “defects for a factorization algebra” is already
available in the topological setting, thanks to the pioneering work of
Ayala–Francis–Tanaka. The reader is encouraged to explore Section 4.3 of
[AFT17] for the complete story, but here we will gloss the key results and
explain how they fit into the story of this paper. (We will suppress all
subtleties about framings. Moreover, that paper works in the setting of
$\infty$-categories; below we will simply use the term “category.” The
interested reader can find details in [AFT17].)

Recall that a factorization algebra $\mathcal{F}$ on an $n$-dimensional
manifold is locally constant if the structure map
$\mathcal{F}(U)\to\mathcal{F}(U^{\prime})$ is an equivalence whenever
$U\subset U^{\prime}$ are each open subsets diffeomorphic to $\mathbb{R}^{n}$.
In any open ball in the manifold, such a factorization algebra is described by
an $E_{n}$-algebra. Ayala–Francis–Tanaka offer a characterization of all the
ways to extend a locally constant factorization algebra $\mathcal{A}$ on $M-D$
to a factorization algebra $\mathcal{B}$ on $M$ that is locally constant when
restricted along $D$. That means that for any inclusion of disks $U\subset
U^{\prime}$ where $U\cap D$ and $U^{\prime}\cap D$ are nonempty, the map
$\mathcal{B}(U)\to\mathcal{B}(U^{\prime})$ is an equivalence; hence
$\mathcal{B}$ along $D$ is locally determined by an $E_{k}$-algebra if $k$ is
the dimension of $D$. Hence, the local situation is when $M=\mathbb{R}^{n}$
and $D=\mathbb{R}^{k}$ is a vector subspace. A key result (Proposition 4.8) of
[AFT17] is that given the “bulk” $E_{n}$-algebra $A$, which determines
$\mathcal{A}$ on $\mathbb{R}^{n}-\mathbb{R}^{k}$, we need to pick an
$E_{k}$-algebra $B$ with values in the category of left modules for
$\int_{S^{n-k-1}}A$, where we view this $n-k-1$-sphere as linking the defect.
This factorization homology $\int_{S^{n-k-1}}A$ encodes how $A$ has to act on
$B$. (Alternatively, we pick a Swiss cheese algebra of dimensions $(k+1,k)$ of
the form $(\int_{S^{n-k-1}}A,B)$.) Ayala–Francis–Tanaka explain how to extend
factorization homology for such pairs $(A,B)$ to obtain all factorization
algebras that are stratified as $D^{k}\subset M^{n}$. In other words, they
reduce the classification of such defective factorization algebras to
classifying Swiss cheese algebras.

This result is powerful and satisfying, and it tells one how to use
computational methods and results from topology and algebra to classify
defects (for factorization algebras). It is wholly complementary to the
physical point of view. On the other hand, it is only applicable to
topological field theories and topological defects therein; it would be
nontrivial to formulate a version that works in more geometric settings. (It
does proffer guidance, however, and it matches the behavior of, say, vertex
modules corresponding to modules over a kind of “algebra of modes” for a
vertex algebra.) Their classification does not capture all physical
expectations and requirements, however, as it merely asks for ways to extend a
factorization algebra. Physics often suggests some additional properties or
features of the extension. Compare, for instance, with the problem of
identifying which modules over a deformation quantization of a Poisson algebra
should be seen as physically relevant: not all $D$-modules seem to have a
place in quantum mechanics (or at least an obvious place).

Looking to the future, we expect the approach to defects that we describe here
to generalize to several more sophisticated contexts. Here are two natural
extensions.

  1. (1)

The definitions we introduce here make sense even when $D\subseteq M$ is not a
submanifold. All we will need is that $D$ admits an open neighborhood $U$ in
$M$ with a continuous contracting map $U\to D$. For example, one could
consider situations where the defect space $D$ is singular, like a nice
immersion.

  2. (2)

Suppose $D$ is a stratified manifold (possibly singular, as in the example
above). One could form an extension of the approach explained here to model
theories admitting defects within defects. We hope that a careful analysis of
this idea would lead to an instantiation of higher categorical structures
associated to defects, as explained in Kapustin’s ICM address [Kap10].

Pursuing these directions would allow for many more physical ideas to be
translated into the language of factorization algebras.

######  Remark 1.4.

The approach to defects that we take here is related to Costello and Li’s
conjectures [CL16] about holography for twisted supersymmetric theories, which
uses Koszul duality to understand and characterize defects. In their work on
this approach, Paquette and Williams [PW21] develop language for the
consideration of line defects, and in particular of a _universal_ defect.
Their definition provides examples of line defects in the sense of this paper.
More specifically, their line defects can be thought of as “order” type, and
they work under the assumption that the theory is topological in the direction
spanned by the line. We will describe some examples of this type in Section 4.

### Acknowledgments

We have benefited from discussions with many people about defects,
factorization algebras, and the BV formalism. We would like to thank Iván
Burbano, Dylan Butson, Alberto Cattaneo, Kevin Costello, John Francis, Ben
Heidenreich, Rune Haugseng, John Huerta, Theo Johnson-Freyd, Pavel Mnev,
Eugene Rabinovich, Ingmar Saberi, Pavel Safronov, Claudia Scheimbauer, Michele
Schiavina, Christoph Schweigert, Stephan Stolz, Matt Szczesny, Peter Teichner,
Alessandro Valentino, Konstantin Wernli, Brian Williams, and Philsang Yoo;
undoubtedly more should be listed, as this is a frequent topic of
conversation. The National Science Foundation supported O.G. through DMS
Grants No. 1812049 and 2042052. I.C. thanks the Amherst College Provost and
Dean of the Faculty’s Research Fellowship (2021-2022).

##  2\. The construction

Let $M$ be a smooth manifold and let $\mathcal{T}$ denote a classical BV
theory on $M$, following Costello’s definition in Chapter 5 of [Cos11]. The
associated factorization algebra $\mathrm{Obs}^{\mathrm{cl}}_{\mathcal{T}}$ is
described in Chapter 5 of [CG21]. We will now explain how to produce something
analogous for theories with defects.

Let $D\subset M$ be a submanifold. Equip the restriction of the tangent bundle
$TM|_{D}$ with a fiberwise metric, and let $n\colon N\to D$ denote the normal
bundle to $D$, which inherits a fiberwise metric. Let $B_{r}(D)\subset N$
denote the disk bundle of radius less than $r$, and suppose that the
exponential map provides a diffeomorphism

$\rho\colon B_{3}(D)\xrightarrow{\cong}U$

with a tubular neighborhood $U$ of $D$. (We can always adjust the metric on
$TM|_{D}$ to accomplish this.) We then have a family of tubular neighborhoods

$U_{t}=\rho(B_{t}(D))$

around $D$ parametrized by $t\in(0,3)$. Note that we will only work with
values of $t$ up to $1$.

Let $M_{t}=M-U_{t}$ denote the complement of the tubular neighborhood $U_{t}$.
It is a manifold with boundary, and let

$\partial_{D}M_{t}=\partial U_{t}\subset M_{t}$

denote its boundary along $D$. (If $M$ has boundary itself, we will ignore
that region and focus only on the boundary introduced by excising a
neighborhood of $D$.) Suppose we have a theory $\mathcal{T}$ defined on $M-D$,
so that the theory $\mathcal{T}$ restricts to a theory on $M_{t}$, and,
following the ansatz, we wish to impose a boundary condition $\mathcal{L}_{t}$
along $\partial_{D}M_{t}$ and construct the factorization algebra of classical
observables given that boundary condition. We will use
$\mathrm{Obs}^{\mathrm{cl}}_{\mathcal{L}_{t}}$ to denote this factorization
algebra on $M_{t}$. (In a moment we will describe what we mean by a boundary
condition and how it determines a factorization algebra.)

Suppose we have produced this factorization algebra
$\mathrm{Obs}^{\mathrm{cl}}_{\mathcal{L}_{t}}$ and want to produce the
associated factorization algebra modelling the observables in the theory on
$M$ with defect. The essential idea is simple: provide a map $\pi_{t}\colon
M_{t}\to M$ collapsing $\partial_{D}M_{t}$ to $D$ and push
$\mathrm{Obs}^{\mathrm{cl}}_{\mathcal{L}}$ forward along this map. This
pushforward $(\pi_{t})_{*}\mathrm{Obs}^{\mathrm{cl}}_{\mathcal{L}_{t}}$ ought
to model a defect for $\mathrm{Obs}^{\mathrm{cl}}_{\mathcal{T}}$ on $M$. We
now make more precise what we want to do.

For a specified value of $t$ in $(0,1)$, let $f_{t}\colon[0,3]\to[0,3]$ be a
smooth non-decreasing function such that

$\displaystyle\begin{split}&f_{t}\text{ is a diffeomorphism preserving the
boundary},\\\ &f_{t}(s)=0\text{ for }s\in[0,t]\text{ and,}\\\
&f_{t}(s)=s\text{ for }s\in[2t,3].\end{split}$ (2)

Such a function exists, and the space of these functions is contractible. We
now use it to define a map $F_{t}:B_{3}(D)\to B_{3}(D)$ as follows. For each
point $x\in D$, let $B_{3}(D)_{x}$ denote the fiber over $x$ in the disk
bundle, and let $|v|$ denote the length of a point $v\in B_{3}(D)_{x}$. Then
set $F_{t}(x,v)=(x,f_{t}(|v|)v/|v|)$ for any point $(x,v)\in B_{3}(D)$. This
map rescales each fiber so that vectors within distance $t$ of $x$ get
collapsed to $x$, those with distance at least $2t$ are left unchanged, and
those in between get stretched.

Now define

$\pi_{t}(p)=\begin{cases}F_{t}\circ\rho^{-1}(x),&p\in U\\\ p,&p\in
M-U_{2t}=M_{2t}\end{cases}.$

Note that outside $U_{2t}$, the map $\pi_{t}$ leaves $M$ unchanged, while
inside it collapses the boundary $\partial_{D}M_{t}$ onto $D$ and “stretches”
the region between $\partial_{D}M_{t}$ and $\partial_{D}M_{2t}$ to fill the
neighborhood of $D$.

Hence, given any factorization algebra $\mathcal{A}$ on $M$, we find that

$\mathcal{A}|_{M_{2t}}\cong\left((\pi_{t})_{*}(\mathcal{A}|_{M_{t}})\right)|_{M_{2t}},$

i.e., this construction leaves a factorization algebra unchanged outside
$U_{2t}$. Pursuing this idea, we can define a factorization algebra that
models observables on a theory with defect well outside the neighborhood
$U_{2t}$, by the following procedure.

######  Definition 2.1.

We define a factorization algebra $\mathrm{Obs}^{\mathrm{cl}}_{t}$ — the
_observables with an effective defect_ — on a manifold $M$ with submanifold
$D\subset M$ using the following input data:

  1. (1)

a classical BV field theory $\mathcal{T}$ on the manifold $M-D$,

  2. (2)

a tubular neighborhood $U$ of $M-D$ with collar coordinate $\rho\colon
B_{3}(D)\cong U$,

  3. (3)

a choice of real number $t$ in $(0,1)$,

  4. (4)

a function $f_{t}$ satisfying conditions (2),

  5. (5)

a local boundary condition $\mathcal{L}_{t}$ along $\partial_{D}M_{t}$.

This data determines a factorization algebra
$\mathrm{Obs}^{\mathrm{cl}}_{\mathcal{L}_{t}}$ on $M_{t}$. Let
$\mathrm{Obs}^{\mathrm{cl}}_{t}$ denote the factorization algebra

$(\pi_{t})_{*}\mathrm{Obs}^{\mathrm{cl}}_{\mathcal{L}_{t}}$

on $M$.

This factorization algebra $\mathrm{Obs}^{\mathrm{cl}}_{t}$ is nearly, but not
exactly, a defect of $\mathrm{Obs}^{\mathrm{cl}}_{\mathcal{T}}$ in the sense
of Definition 1.3. It satisfies the weaker condition that

$\mathrm{Obs}^{\mathrm{cl}}_{t}|_{M_{2t}}=\mathrm{Obs}^{\mathrm{cl}}_{\mathcal{T}}|_{M_{2t}}.$
(3)

This behavior is, however, a good match with physical intuition: it says that
near $D$, the physical defect does change the behavior of the observables, but
at some distance away from $D$ (i.e., on $M_{2t}$), the local behavior of
observables are unchanged. This physical defect changes, of course, the
observables on any open set containing $D$. In particular, the global
observables with defect $\mathrm{Obs}^{\mathrm{cl}}_{t}(M)$ are (typically)
sensitive to the physical defect and do not agree with the global observables
$\mathrm{Obs}^{\mathrm{cl}}_{\mathcal{T}}(M)$ of the defect-free theory.

######  Take-away.

This construction realizes the ansatz at the level of factorization algebras,
provided one can construct the factorization algebra of observables for a
classical BV theory with a local boundary condition.

In particular, this construction produces a factorization algebra that is
“effectively” a defect for $\mathrm{Obs}^{\mathrm{cl}}_{\mathcal{T}}$ in the
sense that it satisfies the weaker condition (3) rather than the stronger
condition (1).

######  Remark 2.2.

We could ask for slightly more data. Instead of fixing the value of the radius
$t\in(0,1)$ and then choosing $f_{t},\mathcal{L}_{t}$ for this fixed radius,
we could choose a compatible family of such data for all values of $t$. First,
choose a smooth function $f(t,s)\colon(0,1]\times[0,3]\to[0,3]$ so that
$f_{t_{0}}(s)=f(t_{0},s)$ satisfies conditions (2) for each choice of
$t=t_{0}$, and $f(t,s_{0})$ is monotonic for each fixed $s_{0}$. Next, define
a smoothly varying family $\mathcal{L}_{t}$ of boundary conditions for each
$t\in(0,1]$. In this way, we produce a _family_ of factorization algebras
$\mathrm{Obs}^{\mathrm{cl}}_{t}$ parameterized by the interval $(0,1)_{t}$.
One can then investigate the limiting prefactorization algebra as $t\to 0$.

###  2.1. On boundary conditions

We now return to discussing what boundary conditions mean for a classical BV
field theory. A complete treatment would be lengthy and technical, so we
sketch the key ideas here and point the interested reader to [Rab20, Rab21]
for a special class of theories of high relevance to this paper. In addition,
we will describe a number of examples in Section 4 below that we hope will
give the reader a good sense of what we mean.

######  Remark 2.3.

We would like to highlight another powerful approach to boundary problems
involving the BV formalism: the BV–BFV formalism developed by Cattaneo, Mnev,
Reshetikhin, and others [CMR14, CMR18]. This approach has proven to be
successful in modifying BV field theory on manifolds with boundary and
corners. In particular, there has been work on the BV description of AKSZ
observables [Mne15], and more recently [MSW20], an interpretation of the WZW
action functional in arbitrary codimension, via Witten’s descent.

For a theory $\mathcal{T}$ on a manifold $N$ with boundary, let
$\mathcal{S}{\rm ol}_{\mathcal{T}}$ denote the sheaf of solutions to the
equations of motion for the theory. In the BV formalism for perturbative
theories, this is a sheaf of formal derived spaces with a local
$-1$-symplectic pairing, in the following sense.

######  Definition 2.4.

Let $L$ denote a local $L_{\infty}$-algebra on $N$, as defined in [CG21,
§3.1.3], and let $BL$ denote the associated sheaf of formal derived spaces. A
_local $k$-shifted symplectic structure_ on $BL$ is a fiberwise nondegenerate
density-valued graded skew-symmetric pairing

$\omega\colon L\otimes L\to\mathrm{Dens}_{N}[k],$

inducing an invariant pairing on the $L_{\infty}$ algebra of compactly
supported sections of $L$ under integration.

Near the boundary $\partial N$ (or along any hypersurface, really), we
anticipate the following structure to hold in good cases.

######  Hypothesis 2.5.

The formal derived space $\mathcal{S}{\rm ol}_{\mathcal{T}}^{\wedge}(\partial
N)$ of jets of solutions near the boundary $\partial N$ is represented by a
local $L_{\infty}$-algebra $L_{\partial}$ on $\partial N$ carrying a natural
$0$-shifted local symplectic structure

$\omega_{\partial}\colon L_{\partial}\otimes
L_{\partial}\to\mathrm{Dens}_{\partial N}.$

As we will see in Section 4, this hypothesis is automatically satisfied in
many natural examples, including topological theories such as BF and
Chern–Simons theory, as well as non-topological examples such as Yang–Mills
theory. It would be interesting to establish the technical assumptions on the
theory $\mathcal{T}$ to guarantee the validity of this hypothesis, but we will
not pursue this in the present paper. (In general some version of
presymplectic reduction may be necessary.)

######  Example 2.6.

We mention two quick examples to orient the reader.

  1. (1)

First, consider a one-dimensional field theory of maps from a line into a
target Riemannian manifold $X$. If one uses the standard action functional,
then for a hypersurface (i.e., point) $t$ in the line, the space of jets of
solutions is equivalent to $T^{*}X$.

  2. (2)

Second, consider abelian Chern-Simons theory on an oriented 3-manifold $N$, so
the sheaf of solutions is modeled by the shifted de Rham complex
$\Omega^{\bullet}_{N}[1]$, where the shift appears so that 1-forms (i.e.,
deformations of the flat connection ${\rm d}_{\mathrm{dR}}$) are in degree 0.
Pick a closed, oriented 2-dimensional hypersurface $S\subset N$. Then jets of
solutions near $S$ is equivalent to $\Omega^{\bullet}(S)[1]$, as the Poincaré
lemma tells us that the de Rham complex is insensitive (at the level of
cohomology) to the normal direction. Poincaré duality tells us that
$\Omega^{\bullet}(S)[1]$ has a natural symplectic structure by the wedge-and-
integrate pairing. As a sheaf on $S$, this de Rham complex has local
symplectic pairing given by the wedge product, so the local symplectic
structure is well-defined even when $S$ is noncompact.

We can now offer a structural formulation of a local boundary condition.

######  Definition 2.7.

A local boundary condition for $\mathcal{T}$ on $\partial N$ is a sheaf
$\mathcal{L}$ on $\partial N$ of formal derived spaces equipped with the
structure of a Lagrangian for the sheaf $\mathcal{S}{\rm
ol}_{\mathcal{T}}^{\wedge}$ of local $0$-shifted symplectic formal derived
spaces.

Let us briefly unpack exactly what sort of data we are specifying. A
Lagrangian structure is a map of sheaves of formal derived spaces

$f\colon\mathcal{L}\to\mathcal{S}{\rm ol}_{\mathcal{T}}^{\wedge}$

together with a trivialization of $f^{*}\omega$, the pullback of the
symplectic pairing, that is non-degenerate. Non-degeneracy here means that the
induced map

$T_{\mathcal{L}}\to\mathrm{Fib}(T^{*}_{\mathcal{L}}\overset{f^{*}\omega}{\to}f^{*}T_{\mathcal{S}{\rm
ol}_{\mathcal{T}}^{\wedge}})$

is an equivalence. The original definition in the derived setting is due to
Pantev, Toën, Vaquié and Vezzosi [Pan+13], and a more expository account is
provided the lectures [Cal14] of Calaque. (In the setting of field theory,
there are often serious functional analysis issues that appear. See [Rab21]
for a careful treatment in the most important situations for this paper.)

We call this data a boundary condition because we will use it to specify which
solutions we care about: we want solutions on $N$ that live in $\mathcal{L}$
near the boundary. We call this data a local boundary condition because it is
a sheaf on $\partial N$, and hence is local-to-global.

######  Example 2.8.

We return to our examples above.

  1. (1)

For a particle traveling through $X$, a boundary condition is precisely a
Lagrangian submanifold of $T^{*}X$.

  2. (2)

For abelian Chern-Simons theory on an oriented 3-manifold $N$ with boundary,
one can specify a boundary condition on $S=\partial N$ by fixing a complex
structure. The local boundary condition is then the map of sheaves

$\Omega^{1,\bullet}_{S}\hookrightarrow\Omega^{\bullet}_{S},$

which is manifestly isotropic because any two $(1,*)$-forms have trivial wedge
product.

The data of a local boundary condition determines a new sheaf on $N$ of formal
derived spaces, as follows. Let $i\colon\partial N\hookrightarrow N$ be the
inclusion of the boundary. By definition, we have a diagram of sheaves

${\mathcal{S}{\rm
ol}_{\mathcal{T}}}$${i_{*}\mathcal{L}}$${i_{*}\mathcal{S}{\rm
ol}_{\mathcal{T}}^{\wedge}}$$\scriptstyle{r}$$\scriptstyle{i_{*}f}$

where $r$ is the map that restricts a solution to its jet of a solution near
the boundary. The homotopy fiber product $\mathcal{F}$ of this diagram is the
sheaf encoding solutions to the equations of motion that satisfy the boundary
condition. As $\mathcal{F}$ is a sheaf of formal derived spaces, its algebra
$\mathcal{O}(\mathcal{F})$ of functions determines a cosheaf of dg commutative
algebras (and hence a commutative factorization algebra) on $N$.

##  3\. A theorem

We now describe a condition under which we obtain a defect in the sense of
Definition 1.3 because the theory has nice behavior near $D$. We begin by
reviewing some relevant recent results of Rabinovich, starting with classical
theories before discussing quantum theories. Boundary conditions in the
classical BV formalism were previously discussed in a paper of Butson and Yoo
[BY16], which also offers a bounty of examples.

For a manifold $N$ with boundary, Rabinovich [Rab20] defines a classical BV
theory that is topological normal to its boundary in Definition 2.4 of loc.
cit. (after Butson–Yoo [BY16, Definition 3.8]). Definition 2.21 then offers a
clean characterization of a local boundary condition for such a theory, and in
Definition 2.25 he characterizes the sheaf of solutions satisfying the
boundary condition. The totality of such data he dubs a classical bulk-
boundary system. A key result of Rabinovich’s paper, explained and proved in
Section 4 of loc. cit., is that the observables of this bulk-boundary system
form a factorization algebra with a defect along the boundary (in the sense of
our Definition 1.3).

Let us briefly sketch Rabinovich’s definition of a theory that is topological
normal to its boundary. We refer to the original paper for a full account.

######  Definition 3.1 (See [Rab20, Definition 2.4]).

A classical BV theory $\mathcal{T}$ on a manifold $M$ with boundary $\partial
M$ is _topological normal to its boundary_ if there exists a tubular
neighborhood $U$ of $\partial M$, a collar coordinate $\phi\colon\partial
M\times[0,\varepsilon)\to U$, and an isomorphism for the graded vector bundle
$E$ of fields

$\phi^{*}E|_{U}\cong
E_{\partial}\boxtimes\Lambda^{\bullet}T^{*}([0,\varepsilon)),$

where $E_{\partial}$ is a graded vector bundle on the boundary such that the
solutions $\mathcal{S}{\rm ol}_{\mathcal{T}}$ to the equations of motion for
$\mathcal{T}$, when viewed as a sheaf on $[0,\varepsilon)$ (i.e., in terms of
the collar parameter), is locally constant and takes values in solutions
$\mathcal{S}{\rm ol}_{\mathcal{T}_{\partial}}$ for a “boundary theory”
$\mathcal{T}_{\partial}$. (We refer to [Rab20] for the full definition of the
boundary theory.)

We will apply Rabinovich’s results in the following setting. Let
$p\colon\mathrm{Bl}_{D}(M)\to M$ denote the blow-up of $M$ along $D$
associated to the tubular neighborhood $U$ and collar coordinate $\rho$. Let’s
describe it in explicit terms. Let $D$ have codimension $k$ as a submanifold.
We have fixed a disk bundle $B_{3}(D)$ inside the normal bundle $N\to D$,
equipped with a metric. Let $S_{1}(D)$ denote the unit sphere bundle inside
$N\to D$. Observe that there is a natural diffeomorphism

$\phi:B_{3}(D)-D\xrightarrow{\cong}S_{1}(D)\times(0,3)$

by using the fiberwise metric; this map produces a diffeomorphism $U-D\cong
S_{1}(D)\times(0,3)$. Then

$\mathrm{Bl}_{D}(M)=(M-D)\cup_{U-D}\left(S_{1}(D)\times[0,3)\right),$

which simply attaches a copy of $D$ to the “end of the cylinder”
$S_{1}(D)\times(0,3)$. The map $p\colon\mathrm{Bl}_{D}(M)\to M$ is the
identity away from the boundary $\partial\mathrm{Bl}_{D}(M)\cong
S_{1}(D)\times\\{0\\}$, and it collapses the sphere bundle down to $D$. Let
$\mathring{\mathrm{Bl}}_{D}(M)$ denote
$\mathrm{Bl}_{D}(M)-\partial\mathrm{Bl}_{D}(M)$, and let
$\mathring{p}\colon\mathring{\mathrm{Bl}}_{D}(M)\to(M-D)$ denote the
restriction of $p$ to the complement of the boundary.

######  Definition 3.2.

For a smooth manifold $M$ with smooth submanifold $D$, let $\mathcal{T}$ be a
classical BV theory on $M-D$. We say $\mathcal{T}$ is _topological normal to
$D$_ if the pullback of $\mathcal{T}$ along the blow-up map $\mathring{p}$
extends to a theory on $\mathrm{Bl}_{D}(M)$ that is topological normal to its
boundary.

Note a key property of such a theory. For any choice of radius $0<r<R<3$, let
$A_{(r,R)}$ denote the annular bundle over $D$ (i.e., the points in the normal
bundle of distance between the radii). For any choices of radii
$0<r<r^{\prime}<R^{\prime}<R<3$, the restriction of solutions from the bigger
annular bundle $A_{(r,R)}$ to the smaller annular bundle
$A_{(r^{\prime},R^{\prime})}$ is an equivalence:

$\mathcal{S}{\rm ol}_{\mathcal{T}}(A_{(r,R)})\to\mathcal{S}{\rm
ol}_{\mathcal{T}}(A_{(r^{\prime},R^{\prime})}).$

Thus the observables are likewise equivalent along the extension map

$\mathrm{Obs}^{\mathrm{cl}}_{\mathcal{T}}(A_{(r^{\prime},R^{\prime})})\to\mathrm{Obs}^{\mathrm{cl}}_{\mathcal{T}}(A_{(r,R)})$

determined by those radii. This claim follows immediately from the
topological-normal-to-the-boundary condition, because the pullback map
$\Omega^{\bullet}((r,R))\to\Omega^{\bullet}((r^{\prime},R^{\prime}))$ along
the inclusion of intervals is a quasi-isomorphism.

Moreover, a choice of local boundary condition $\mathcal{L}$ on
$\mathrm{Bl}_{D}(M)$ determines a local boundary condition $\mathcal{L}_{t}$
for every $t$, as solutions are locally constant with respect to the collar
coordinate.

As an immediate corollary of Rabinovich’s work, we thus obtain the following
result.

######  Theorem 3.3.

If the classical BV theory $\mathcal{T}$ is topological normal to $D\subset
M$, then a local boundary condition $\mathcal{L}_{t}$ determines a
factorization algebra $\mathrm{Obs}^{\mathrm{cl}}_{t}$ on $M$ that is a defect
for $\mathrm{Obs}^{\mathrm{cl}}_{\mathcal{T}}$, i.e.,

$\mathrm{Obs}^{\mathrm{cl}}_{t}|_{M-D}\cong\mathrm{Obs}^{\mathrm{cl}}_{\mathcal{T}}|_{M-D}.$

The point is that the observables for $\mathcal{T}$ do not care about the
“width” of a collar, so that when we push forward along $\pi_{t}$, the
stretching of the annular neighborhood is irrelevant.

######  Remark 3.4.

It is important to notice that this theorem applies for a _fixed_ value of the
radius $t$ around the defect. This is particular to theories that are
topological normal to the boundary. For theories without this condition, in
order to obtain a genuine defect one would need to define a family of theories
for all $t$ in the interval $(0,1)$, and take an appropriate limit as $t\to
0$, as discussed briefly in Remark 2.2.

######  Remark 3.5.

We expect that the hypothesis can be weakened from “topological normal to $D$”
to “rescaling-equivariant normal to $D$.” Compare with [Kap06], where Kapustin
requires the theory with defect to have a symmetry by the group of conformal
transformations preserving the support of the defect.

Another important feature of Rabinovich’s work is that he explains how to
quantize classical BV theories that are topological normal to the boundary.
That is, he offers a rigorous renormalization method (building upon Costello’s
approach in [Cos11] and the work of Albert [Alb16]) and formulates a version
of the quantum master equation. His central result is that, when a BV
quantization exists (i.e., the master equation is satisfied), it has a
factorization algebra of quantum observables. Hence, as another corollary of
Rabinovich’s work, we have the following.

######  Theorem 3.6.

If the classical BV theory $\mathcal{T}$ is topological normal to $D\subset M$
with a local boundary condition $\mathcal{L}_{t}$ that admits a BV
quantization, then the quantum bulk-boundary system determines a factorization
algebra $\mathrm{Obs}^{\mathrm{q}}_{t}$ on $M$ that is a defect for
$\mathrm{Obs}^{\mathrm{q}}_{\mathcal{T}}$, i.e.,

$\mathrm{Obs}^{\mathrm{q}}_{t}|_{M-D}\cong\mathrm{Obs}^{\mathrm{q}}_{\mathcal{T}}|_{M-D}.$

##  4\. Examples

We organize our examples by codimension of the defect. In each example we will
focus on how to formulate the field theory and boundary condition along the
blowup of a submanifold; we do not analyze the associated factorization
algebra. At the classical level, most statements about the factorization
algebra boil down to statements about the behavior of solutions with the
boundary condition. We postpone such analysis of more interesting examples,
including quantizations, to future work.

######  Remark 4.1.

Another rich source of examples arises by applying our construction to
examples from [GRW20], which constructs the quantizations and factorization
algebras for free bulk-boundary systems (i.e., where the bulk theory is free
and the boundary condition is linear). Thus, the reader can produce defects
for abelian Chern-Simons theory (including higher dimensional Chern-Simons
theories) or the Poisson sigma-model into a Poisson vector space. The relevant
factorization algebras are analyzed, to some extend, in [GRW20]. See [BY16]
for a number of interacting bulk-boundary systems of interest to
mathematicians and physicists. We also mention that [CY19] contains a wealth
of examples and insights about defects that would be well-suited to treatment
by the approach advocated by this paper.

###  4.1. Codimension 1

When the defect’s support is a dividing hypersurface, the defect is often
called a domain wall. We will offer a few examples, building from point
defects in mechanical systems towards domain walls for BF theories.

####  4.1.1. Topological Mechanics

Following [GRW20, Rab20], the data of topological mechanics can be encoded in
a symplectic vector space $V$; in physical terms, we are studying maps from a
real line (or worldline) $\mathbb{R}$ into a target $V$. The factorization
algebra of classical observables for this system encodes the Poisson algebra
$\mathcal{O}(V)=\operatorname{Sym}(V^{*})$ of functions on $V$. A choice of
Lagrangian vector subspace $L\subset V$ provides a boundary condition for the
half-line $[0,\infty)$; in physical terms, it means the path must start in
$L$. In [GRW20], it is proven that the factorization algebra of classical
bulk-boundary observables encodes $\mathcal{O}(V)=\operatorname{Sym}(V^{*})$
as the bulk observables together with the module

$\mathcal{O}(L)=\operatorname{Sym}(L^{*})\cong\operatorname{Sym}((V/L)^{*})$

as the boundary observables. In addition, this system can be quantized, and
[GRW20] shows the factorization algebra of quantum bulk-boundary observables
encodes the Weyl algebra ${\rm Weyl}(V^{*})$ as the bulk observables together
with the Fock module ${\rm Fock}(L^{*})$ for the boundary observables. (As a
vector space, ${\rm Fock}(L^{*})\cong\mathcal{O}(L)$.) The Fock space is a
right module for ${\rm Weyl}(V^{*})$ as the boundary is on the left end of the
half-line.

We can leverage those results to construct point defects. Take the point
defect for topological mechanics to be supported at the origin, so the
complement of the origin has two boundary points. Pick Lagrangian subspaces
$L_{-}$ and $L_{+}$ for boundary conditions on $(-\infty,0]$ and on
$[0,\infty)$, respectively. Our main construction gives a factorization
algebra of quantum observables with a defect where on $\mathbb{R}-\\{0\\}$, it
encodes ${\rm Weyl}(V^{*})$ but for an interval containing the origin, the
observables encode ${\rm Fock}(L_{-}^{*})\otimes{\rm Fock}(L_{+}^{*})$. Note
that this tensor product of Fock modules is a bimodule for the Weyl algebra.

####  4.1.2. Domain walls in BF theories

Let $\mathfrak{g}$ denote a Lie algebra for a Lie group $G$. Consider BF
theory on an oriented $n$-manifold $M$ in the BV formalism: the graded vector
space of fields is

$\Omega^{\sharp}(M,\mathfrak{g})[1]\oplus\Omega^{\sharp}(M,\mathfrak{g}^{*})[2-n]$

where an element of the first summand is called the $A$-field and where an
element of the second summand is called the $B$-field. (We mean here the
graded vector space of differential forms, without its differential, and use
$\sharp$ to indicate this.) The action functional is

$S_{\mathrm{BF}}(A,B)=\int_{M}(B\wedge F_{A})$

where $F_{A}={\rm d}A+\frac{1}{2}[A,A]$ and the notation $(-\wedge-)$
indicates that we use the evaluation pairing between $\mathfrak{g}$ and its
dual $\mathfrak{g}^{*}$ but wedge the form component. The equations of motion
are $F_{A}=0$ and $\nabla_{A}B=0$. In other words, this theory picks out a
flat connection and a horizontal section of the coadjoint bundle.

Take $M$ to be a connected manifold and $D$ a dividing hypersurface, so that

$M-D=M_{0}\sqcup M_{1}$

is a disjoint union of two manifolds. Let $\overline{M}_{0}=M-M_{1}$, and let
$\overline{M}_{1}=M-M_{0}$, so that $\overline{M}_{i}$ is a manifold with
boundary isomorphic to $D$. Then

$M\cong\overline{M}_{0}\cup_{D}\overline{M}_{1}$

by construction.

As the equations of motion do not care about the geometry of the manifold,
just its underlying topology, we see that $\mathcal{S}{\rm
ol}_{\mathrm{BF}}^{\wedge}(D)$, the formal space of solutions near the
boundary of $\overline{M}_{i}$, is modeled by

$\Omega^{\bullet}(D,\mathfrak{g})[1]\oplus\Omega^{\bullet}(D,\mathfrak{g}^{*})[2-n],$

since the de Rham complex in the collar direction is quasi-isomorphic to
$\mathbb{R}$ by the Poincaré lemma.

There are many possible choices of local boundary conditions here. Two obvious
options are the summands: one can take

$\Omega^{\bullet}(D,\mathfrak{g})[1],$

which corresponds to taking the $B$-field to be zero, or one can take

$\Omega^{\bullet}(D,\mathfrak{g}^{*})[2-n],$

which corresponds to taking the $A$-field to be zero. Other options arise from
taking shifted conormal spaces to a “subspace” of
$\Omega^{\bullet}(D,\mathfrak{g})[1]$, which models the formal neighborhood of
the trivial connection among all flat $G$-bundles. (We will mention other
possibilities in a moment, to produce “order” operators.)

Fix local boundary conditions $\mathcal{L}_{0}$ for BF theory on
$\overline{M}_{0}$ and $\mathcal{L}_{1}$ for BF theory on $\overline{M}_{1}$.
Then our main construction produces a factorization algebra on $M$ that agrees
with $\mathrm{Obs}^{\mathrm{cl}}_{\mathrm{BF}}$ on $M-D$ but, for an open set
$U$ intersecting $D$, is given by

$\mathrm{Obs}^{\mathrm{cl}}_{\mathcal{L}_{0}}(U\cap\overline{M}_{0})\otimes\mathrm{Obs}^{\mathrm{cl}}_{\mathcal{L}_{1}}(U\cap\overline{M}_{1}).$

Note the similarity with the construction in topological mechanics; by
compactifying along $D$ (at least in a collar neighborhood), the two
constructions can be identified.

######  Remark 4.2.

We want to remark on another useful class of domain walls, which provide
examples of order operators. The essential idea is simple: put a field theory
$\mathcal{T}^{\prime}$ on the hypersurface $D$ that couples to the theory
$\mathcal{T}$ on $M$. Away from $D$, the fields of $\mathcal{T}$ should
satisfy the usual equations of motion, but on an open intersecting $D$, there
is an interesting system of equations involving the fields of both
$\mathcal{T}$ and $\mathcal{T}^{\prime}$. The observables should thus provide
a factorization algebra with a defect along $D$. In terms of this paper’s
point of view, note that if one writes $M$ as the union $(M-U)\cup_{\partial
U}U$ for some tubular neighborhood $U$ of $D$, then one obtains a local
boundary condition along $\partial U$ by taking the boundary values of the
coupled field theory on $U$. (We will not show such boundary conditions are
always Lagrangian, but in practice it is typically the case.) We will discuss
examples of this nature in higher codimension in Sections 4.2.3 and 4.3.2
below. An extensive discussion of the order operator case can be found in
[PW21], which provides motivation and many useful examples, as well as
references to the pertinent literature.

###  4.2. Codimension 2

In the setting of BF gauge theories, we exhibit a class of codimension 2
defects that depend upon monodromy of flat connections. We will then examine
point defects for 2-dimensional free scalar theory; our approach works for
point defects of scalar theory in any dimension. Finally, we explain how to
produce a Wilson line defect for Chern-Simons theory.

####  4.2.1. A point defect in 2-dimensional BF theory: monodromy

Take $M=\mathbb{R}^{2}$ and the origin $D=0$ as a submanifold. Let
$\mathfrak{g}$ be the Lie algebra of a Lie group $G$. If we consider BF theory
for $\mathfrak{g}$ on $\mathring{M}=\mathbb{R}^{2}-\\{0\\}$, then a solution
to the equations of motion provides a flat connection $\nabla={\rm d}+A$. Such
a flat connection has monodromy ${\rm Mon}(\nabla)$ for a loop $\gamma$ that
winds once around the origin, and this monodromy is an element of $G$, only up
to conjugation. (And every element of $G$ can appear as monodromy. ) In other
words, we have ${\rm Mon}(\nabla)\in G/G$, the adjoint quotient space, and, in
fact, two flat connections with the same monodromy are gauge-equivalent (for a
textbook account, see [Tau11, §13.2]).

In a BF theory, a solution also involves a choice of $\nabla$-closed
$\mathfrak{g}^{*}$-valued function $B$, i.e., a $\mathfrak{g}^{*}$-valued
function that is horizontal for $\nabla$. Such a function is completely
determined by its value at one point $p$ in $\mathring{M}$, i.e., an element
of $\mathfrak{g}^{*}$. Thus, the quotient space $(G\times\mathfrak{g}^{*})/G$,
with respect to the adjoint-coadjoint action, parametrizes solutions to the BF
theory on $\mathring{M}$, and hence perturbative BF theories. Given a pair
$(m,b)\in G\times\mathfrak{g}^{*}$, let $\mathcal{T}_{(m,b)}$ denote the
perturbative BF theory around a pair $(\nabla,B)$ lying in the corresponding
$G$-orbit, i.e.

$[{\rm Mon}(\nabla),B(p)]=[m,b].$

Let $\mathrm{Obs}^{\mathrm{cl}}_{(m,b)}$ denote the factorization algebra on
$\mathring{M}$ of classical observables of this theory.

We note that such a theory is topological. In particular, if we choose any
closed disk $D_{t}$ of radius $t<|p|$, the formal space of jets of solutions
$\mathcal{S}{\rm ol}_{\mathrm{BF}}^{\wedge}(\partial D_{t})$ at the boundary
circle $\partial D_{t}=\partial(\mathbb{R}^{2}-\mathring{D}_{t})$ is
canonically independent of $t$.

Suppose for simplicity that $[m,b]=[e,0]$. Similarly to the previous example,
this space of formal jets of solutions is modeled by the shifted de Rham
complex

$\Omega^{\bullet}(\partial D_{t},\mathfrak{g}\oplus\mathfrak{g}^{*})[1].$

If we choose a non-trivial value of $[m,b]$ then we must instead use a twisted
de Rham complex with differential modified by the chosen basepoint. We can
choose natural boundary conditions by taking forms valued in any Lagrangian
linear subspace of $\mathfrak{g}\oplus\mathfrak{g}^{*}=T^{*}\mathfrak{g}$.
There are several natural classes of example.

  1. (1)

If $\mathfrak{g}$ admits an inner product
$\kappa\colon\operatorname{Sym}^{2}(\mathfrak{g})\to\mathbb{R}$, there is an
induced linear isomorphism $f_{\kappa}\colon\mathfrak{g}\to\mathfrak{g}^{*}$,
and there are boundary conditions $\mathcal{L}_{s}$ associated to each real
number $s$, given by the image of the map
$(1,sf_{\kappa})\colon\mathfrak{g}\to\mathfrak{g}\oplus\mathfrak{g}^{*}$. Such
boundary conditions impose the condition $B=sf_{\kappa}(A)$ at the boundary.

  2. (2)

If we choose a Lie subalgebra $\mathfrak{l}\subseteq\mathfrak{g}$, there is an
associated Lagrangian subspace
$\mathfrak{l}\oplus(\mathfrak{l}^{*})^{\perp}\subseteq\mathfrak{g}\oplus\mathfrak{g}^{*}$,
inducing associated boundary conditions $\mathcal{L}_{\mathfrak{l}}$. Such
boundary conditions restrict the monodromy of $\nabla$ to lie in the
subalgebra $\mathfrak{l}$.

######  Remark 4.3.

There was nothing particular to $M=\mathbb{R}^{2}$ in the calculation we just
outlined. If we take $M=\mathbb{R}^{n}$ and take $D$ to be a choice of
embedding $\mathbb{R}^{n-2}\to M$, then we can perform an identical
calculation. For example, this provides a description of a class of line
defects in 3-dimensional BF theory, and of a class of surface defects in
4-dimensional BF theory. Small and straightforward modifications allow one to
construct such “monodromy defects” for a codimension 2 submanifold inside any
oriented manifold, although characterizing the allowable monodromies can be
elaborate.

######  Remark 4.4.

As we discussed in Section 1.1, each choice of boundary condition will lead to
a module for the algebra of classical observables on an annulus. One can
describe this algebra fairly concretely starting from the dg Lie algebra
$\mathcal{S}{\rm ol}_{\mathrm{BF}}^{\wedge}(\partial D_{t})[-1]$, by forming
the Chevalley–Eilenberg cochains.

####  4.2.2. Point defects in scalar field theory

We will now move on to discussing another example of point defects in
$M=\mathbb{R}^{2}$, but now in a non-topological theory (so the results of
Section 3 will not be applicable). We will consider the example of a free
classical scalar field theory on $\mathbb{R}^{2}$ equipped with its flat
metric, with a defect at the origin, so $D=\\{0\\}\subseteq\mathbb{R}^{2}$.
The blow-up is easy to describe here: it is given by excising a disk
$D_{R}(0)$, so

$\mathbb{R}^{2}-D_{R}(0)=\\{z\in\mathbb{R}^{2}\colon|z|\geq R\\}.$

We will trace the dependence on radius $R$ below.

The free scalar field theory has graded space of fields

$C^{\infty}_{\mathbb{C}}(\mathbb{R}^{2})\oplus
C^{\infty}_{\mathbb{C}}(\mathbb{R}^{2})[-1],$

where $C^{\infty}_{\mathbb{C}}(\mathbb{R}^{2})$ denotes the space of smooth
complex-valued functions. The action functional is

$S(\phi)=\int\phi\Delta\phi$

and depends only on the field in cohomological degree zero. Here $\Delta$ is
the usual Laplacian operator. The formal space of solutions near $\phi=0$ is
modelled by the complex

$C^{\infty}_{\mathbb{C}}(\mathbb{R}^{2})\overset{\Delta}{\to}C^{\infty}_{\mathbb{C}}(\mathbb{R}^{2})$

concentrated in degrees zero and one. The space of jets of solutions along the
boundary $S^{1}_{R}$ of the blowup can also easily be described. Let us use
polar coordinates $(r,\theta)$ on $\mathbb{R}^{2}$, and let $t=r-R$ be a
collar coordinate near our boundary. Then jets of fields near the origin are
modelled by

$C^{\infty}_{\mathbb{C}}(\mathbb{R}^{2})^{\wedge}=C^{\infty}_{\mathbb{C}}(S^{1})[[t]],$

and the differential becomes

$\displaystyle\Delta^{\wedge}$
$\displaystyle=\partial^{2}_{t}+\frac{1}{t+R}\partial_{t}+\frac{1}{(t+R)^{2}}\partial_{\theta}^{2}$
$\displaystyle=\partial^{2}_{t}+\frac{1}{R}\sum_{m\geq
0}\left(\frac{t}{R}\right)^{m}\partial_{t}+\frac{1}{R^{2}}\left(\sum_{m\geq
0}\left(\frac{t}{R}\right)^{m}\right)^{2}\partial_{\theta}^{2}.$

We will use the more attractive $\mathcal{D}$ to denote this operator
$\Delta^{\wedge}$ obtained by restricting to jets along the boundary
$S^{1}_{R}$, and we write it succinctly as

$\mathcal{D}=\partial_{t}^{2}+g_{R}(t)\partial_{t}+g_{R}(t)^{2}\partial_{\theta}^{2},$

where

$g_{R}(t)=\frac{1}{R}\sum_{m\geq 0}\left(\frac{t}{R}\right)^{m}$

denotes the geometric series. It is routine to verify the following properties
of the operator $\mathcal{D}$:

  1. (1)

$\mathcal{D}$ is surjective and thus $\operatorname{coker}(\mathcal{D})=0$.

  2. (2)

$\ker(\mathcal{D})\cong C^{\infty}_{\mathbb{C}}(S^{1})$.

(This second fact boils down to showing that any solution is determined by its
$t^{0}$-term.) These facts imply that there is a quasi-isomorphism of the form

$C^{\infty}_{\mathbb{C}}(S^{1})\hookrightarrow\mathcal{S}{\rm
ol}^{\wedge}_{\partial},$

where the injective map is defined by the inclusion of the kernel of the
operator $\mathcal{D}$. This inclusion is a section of a quasi-isomorphism
$\mathcal{S}{\rm ol}^{\wedge}_{\partial}\to C^{\infty}_{\mathbb{C}}(S^{1})$
given by projection onto the $t^{0}$ component in the degree zero part of
$\mathcal{S}{\rm ol}^{\wedge}$. We thus have a concrete description of
boundary conditions for the free scalar field on $\mathbb{R}^{2}$.

We observe that there is a natural presymplectic structure on
$\ker\mathcal{D}\cong\mathcal{C}^{\infty}_{\mathbb{C}}(S^{1})$ given by

$\omega_{D}(e^{ik\theta},e^{il\theta})=\begin{cases}1&\text{if }k>0,l=-k,\\\
-1&\text{if }k<0,l=-k,\\\ 0&\text{if }k\neq l.\end{cases}$

Note that this is well-defined as a pairing on smooth functions on the space
$S^{1}$ because the Fourier coefficients of any smooth function decay more
rapidly than any polynomial.

This presymplectic structure is degenerate due to the existence of the
constant Fourier coefficient. As a result, our example only satisfies a
weakened version of Hypothesis 2.5. We could resolve this issue by a mild
alteration, where we replace the complex controlling jets of solutions by its
quotient by the constant factor $\mathbb{C}$.

The following are examples of Lagrangians $L\Longrightarrow
C^{\infty}_{\mathbb{C}}(S^{1})/\mathbb{C}$ with respect to this symplectic
form.

  1. (a)

Take the positive modes

$L_{0}=\\{f\in C^{\infty}_{\mathbb{C}}(S^{1})\colon\hat{f}_{n}=0\text{ if
}n<0\\}/\mathbb{C}.$

Here $\hat{f}_{n}$ denote the $n^{\text{th}}$ Fourier coefficient of the
function $f$. This boundary condition corresponds to the inclusion of the
space of functions that extend across the interior disk, so it is the “trivial
defect” that recovers the scalar field theory on $\mathbb{R}^{2}$.

  2. (b)

More generally, we could consider the following Lagrangian:

$L_{0}=\\{f\in C^{\infty}_{\mathbb{C}}(S^{1})\colon\hat{f}_{n}=0\text{ if
}n\in S\\}/\mathbb{C}.$

where $S\subset\mathbb{Z}$ is a subset such that $0\notin S$ and $k\in
S\iff-k\notin S$, for all $k\neq 0$. We refer to such Lagrangians as
“spectral” boundary conditions as they are determined by spectral properties
of the function along the boundary and not pointwise behavior along the
boundary.

A variation of (a) is the following: if $\eta$ denotes the choice of a
conformal class of a metric on $D_{R}(0)$, then

$L_{\eta}=\left\\{\eta\mbox{--harmonic functions on the disk}\right\\}$

is a Lagrangian. (This construction offers a useful view on the appearance of
the “restricted” or “semi-infinite” Grassmannian in conformal field
theory.)One can formulate more exotic spectral boundary conditions too.

######  Remark 4.5.

We point out that our approach here works in higher dimensions. For free
scalar theory on $\mathbb{R}^{n}$ with a point defect at the origin, one
simply replaces $S^{1}$ by $S^{n-1}$, and the Fourier modes by spherical
harmonics. The remaining analysis is parallel.

####  4.2.3. Chern–Simons theory coupled to a charged fermion: a Wilson line

Let us now take $M=\mathbb{R}^{3}$ and $D=\mathbb{R}\times(0,0)$, the
$x$-axis. We will describe an “order” type defect obtained by coupling a
background gauge theory on $M$ to a charged particle along the line $D$. Thus,
fix a Lie algebra $\mathfrak{g}$ with a non-degenerate pairing, and let $V$ be
a finite-dimensional representation of $\mathfrak{g}$ equipped with an
invariant inner product. We define our BV theory on $\mathbb{R}^{3}$ to be

$\mathcal{T}=\Omega^{\bullet}(\mathbb{R}^{3})\otimes\mathfrak{g}[1],$

with the BV bracket defined using the pairing on $\mathfrak{g}$. This is a
perturbative description of _Chern–Simons theory_ near the trivial flat
connection.

If we choose a tubular neighborhood $U$ of the embedded line $\mathbb{R}$,
with boundary $\partial U$, the formal space $\mathcal{S}{\rm
ol}^{\wedge}_{\mathrm{CS}}(\partial U)$ of jets of solutions near the boundary
is determined by the shifted de Rham complex

$\Omega^{\bullet}(\partial U)\otimes\mathfrak{g}[1],$

with cohomology concentrated in degrees $-1$ and $0$.

We can define a boundary condition in a trivial way.

######  Definition 4.6.

Let $i_{\partial}\colon\partial U\to U$ denote the inclusion of the boundary
of $U$. There is a canonical Lagrangian given by the restriction map

$i_{\partial}^{*}(\Omega^{\bullet}(U)\otimes\mathfrak{g}[1])\to\Omega^{\bullet}(\partial
U)\otimes\mathfrak{g}[1].$

It is the _trivial_ defect along $D$.

We can enhance this trivial defect by coupling to an additional field along
the line $D$, valued in the representation $V$. Define a topological _free
fermion_ theory on $\mathbb{R}$ by setting

$\mathcal{T}_{V}=\Omega^{\bullet}(\mathbb{R})\otimes\Pi V$

with BV pairing defined using the inner product on $V$ and wedge-and-
integration of the forms. Here $\Pi$ indicates that we place the
representation $V$ in odd degree for an auxiliary
$\mathbb{Z}/2\mathbb{Z}$-grading. The action functional is

$S_{V}(\psi)=\int_{\mathbb{R}}(\psi,{\rm d}\psi)$

for the associated BV field theory, with $\psi\in\mathcal{T}_{V}$.

We will now define a defect by coupling the topological free fermion
$\mathcal{T}_{V}$ along $D$ to the Chern-Simons theory in the bulk. Observe
that the $\mathfrak{g}$-module structure on $V$ makes
$\mathcal{T}_{V}|_{U}[-1]$ into a module for the sheaf of dg Lie algebras
$i^{*}_{D}\mathcal{T}[-1]$, where $i_{D}$ is the inclusion of the defect line
in $U$. In terms of action functionals, this means that we can view the gauge
field $A$ as modifying the fermion action by minimal coupling:

$S_{V,{\rm min}}(\psi;A)=\int_{\mathbb{R}}(\psi,({\rm d}+A)\psi).$

The equation of motion for this theory picks out sections $\psi$ that are
horizontal (or flat) for the connection ${\rm d}+A$; if we worked with a
circular defect rather than a line, we could ask about the holonomy of the
connection on this vector bundle. In this way, the fermionic theory allows one
to encodes the Wilson operator (i.e., trace of holonomy).

We can formulate a BV theory that involves both the gauge field and the
fermion in terms of the super dg Lie algebra

$\mathcal{E}_{\mathfrak{g},V}(U)=\left(\Omega^{\bullet}(U)\otimes\mathfrak{g}\right)\ltimes(i_{D})_{*}\left(\Omega^{\bullet}(\mathbb{R})\otimes\Pi
V[-1]\right)$

with bracket generated by the Lie bracket on $\mathfrak{g}$ and the action of
$\mathfrak{g}$ on $V$. The associated action functional is the sum of the
Chern-Simons action and minimally coupled action for the fermion.

######  Definition 4.7.

The charged line defect associated to a free fermion valued in the
representation $V$ of $\mathfrak{g}$ is the boundary condition

$i_{\partial}^{*}\mathcal{E}_{\mathfrak{g},V}(U)\to\Omega^{\bullet}(\partial
U)\otimes\mathfrak{g}[1].$

If we set $V=0$, we recover the trivial defect defined above.

This charged line defect is another example of an order operator, as discussed
in Remark 4.2.

######  Remark 4.8.

Note that once we introduce a non-trivial representation $V$, the boundary
condition is no longer associated to the inclusion of a subcomplex. In the
derived setting, as we have here, a map can admit a Lagrangian structure even
if it is not a degreewise inclusion.

######  Remark 4.9.

Let us conclude this section by mentioning an additional interesting
collection of codimension two defects that one could hope to describe in the
present formalism. Costello and Yamazaki [CY19] consider a large class of
surface defects of both order and disorder type in _four-dimensional
Chern–Simons theory_ : a gauge theory analogous to Chern–Simons theory, but
defined on $\mathbb{R}^{2}\times C$ where $C$ is a Riemann surface, in which
the solutions to the equations of motion are topological in the two real
directions and holomorphic in the complex direction. They consider defects
placed along planes of the form $\mathbb{R}^{2}\times\\{z\\}$ for points $z\in
C$, from which they are able to engineer a large number of interesting
integrable systems on the Riemann surface. We should note that these examples
will not be topological normal to the boundary, but only holomorphic in the
normal directions.

###  4.3. Codimension 3

We now turn to the most well-known examples from gauge theory: we describe the
magnetic monopole and the Wilson line in 4-dimensional Yang-Mills theory. The
results of Section 3 will not apply in this section, because Yang–Mills theory
is not topological (it may be possible to recover similar results using the
weaker condition of conformal invariance, but do not make any claims in this
direction at present).

####  4.3.1. The magnetic monopole

On four-dimensional manifolds, Yang–Mills theory admits a first-order
formulation that is convenient for producing boundary conditions, so we will
stick to dimension 4 in this paper. We will also only consider Yang–Mills
theory for the abelian group $\mathrm{U}(1)$. (In a companion paper [CEG22],
we will work out the monopole for abelian Yang–Mills theory in arbitrary
dimensions.)

Let $M$ be a Riemannian 4-manifold with Hodge star operator $\star$. (In fact,
to describe a Yang–Mills theory, we only need the data of a Hodge star
operator—up to rescaling—not the full metric.) Fix a principal
$\mathrm{U}(1)$-bundle $L\to M$ with a non-degenerate inner product, and fix a
connection $\nabla$ on $L$. (The choice allows us to describe the space of
connections as sections of a vector bundle.) The first-order formulation of
$\mathrm{U}(1)$-Yang–Mills theory has graded space of fields

$\begin{array}[]{cccc}-1&0&1&2\\\
\hline\cr\Omega^{0}(M,L)&\Omega^{1}(M,L)&\Omega^{2}_{+}(M,L)&\\\
&\Omega^{2}_{+}(M,L)&\Omega^{3}(M,L)&\Omega^{4}(M,L)\end{array}$

where $\Omega^{2}_{+}(M,L)$ denotes the self-dual 2-forms,( i.e., $\alpha$
such that $\star\alpha=\alpha$, using the inner product to identify $L$\- and
$L^{*}$-valued 2-forms) and where the top row indicates the cohomological
degree. We call an element of the first row the $A$-field and an element of
the second row the $B$-field. The action functional is

$S_{\mathrm{YM}}(A,B)=\int_{M}B\wedge\nabla_{+}A-c(\star B)\wedge A$

where $c$ denotes a coupling constant and where $\nabla_{+}$ denotes the
covariant derivative followed by projection onto self-dual 2-forms. The
equations of motion are

$\nabla_{+}A=cB\qquad\text{and}\qquad\nabla B=0,$

which together imply the usual equations of motion. For instance if $c\neq 0$,
we deduce that $\nabla\ast\nabla A=0$. (When $c=0$, we recover self-dual
Yang–Mills theory.)

A solution to the equation of motion for $\mathrm{U}(1)$-Yang–Mills theory is
a connection $\widetilde{\nabla}=\nabla+A$ on the bundle $L$. The tangent
complex at $\widetilde{\nabla}$ to the derived space of solutions is modelled
by the cochain complex

$\mathcal{E}_{M}=\quad\vbox{\hbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
22.64305pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\Omega^{0}(M,L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
28.86526pt\raise 6.61111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\widetilde{\nabla}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 46.64305pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
46.64305pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\Omega^{1}(M,L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
96.59581pt\raise 7.36111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.86113pt\hbox{$\scriptstyle{\widetilde{\nabla}_{+}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 115.92915pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
115.92915pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\Omega^{2}_{+}(M,L)}$}}}}}}}{\hbox{\kern
204.8583pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-41.13776pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
46.64305pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\Omega^{2}_{+}(M,L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
98.15137pt\raise-34.52664pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\widetilde{\nabla}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
115.92915pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
90.53604pt\raise-15.9404pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.62848pt\hbox{$\scriptstyle{-c\cdot\star}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 129.30707pt\raise-5.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
115.92915pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\Omega^{3}(M,L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
167.43747pt\raise-34.52664pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\widetilde{\nabla}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
185.21526pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
185.21526pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\Omega^{4}(M,L)}$}}}}}}}\ignorespaces}}}}\ignorespaces}}$
(4)

concentrated in degrees $-1,0,1,2$. Note that there is a natural subcomplex
given by the top row, i.e., the subcomplex consisting only of $A$-fields. If
$M$ is a manifold with boundary, then jets of this subcomplex along $\partial
M$ yield a local boundary condition. Let $\mathcal{L}_{B=0}$ denote this
boundary condition.

######  Remark 4.10.

We can describe this in a little more detail, though we will not include a
detailed calculation in this paper. We can model the jets of solutions along
$\partial M$, i.e. the formal space $\mathcal{S}{\rm
ol}^{\wedge}_{\mathrm{YM}}(\partial M)$, by the cochain complex

$\mathcal{E}_{\partial M}=\quad\vbox{\hbox{\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
33.80571pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\Omega^{0}(\partial M,L|_{\partial
M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
36.4415pt\raise 7.48611pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.73611pt\hbox{$\scriptstyle{\widetilde{\nabla}|_{\partial
M}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 60.35086pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
60.35086pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\Omega^{1}(\partial M,L|_{\partial
M})}$}}}}}}}{\hbox{\kern 187.85829pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-41.13776pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
57.80571pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{z\Omega^{2}(\partial
M,L|_{\partial
M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
131.87065pt\raise-33.65164pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.73611pt\hbox{$\scriptstyle{\widetilde{\nabla}|_{\partial
M}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern
154.50743pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
154.50743pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{z\Omega^{3}(\partial
M,L|_{\partial M})}$}}}}}}}\ignorespaces}}}}\ignorespaces}}$ (5)

concentrated in degrees $-1,0$ and 1, where $z$ is a chosen formal coordinate
normal to $\partial M\subseteq M$. So $\mathcal{L}_{B=0}$ is equivalent to the
complex

$\textstyle{\Omega^{0}(\partial M,L|_{\partial
M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\nabla}|_{\partial
M}}$$\textstyle{\Omega^{1}(\partial M,L|_{\partial M})}$

in degrees $-1$ and 0.

We can now finally formulate magnetic monopoles, as follows. Let
$M=\mathbb{R}^{4}$ and fix a line $D\subset M$, so $D\cong\mathbb{R}$. We
require that $D$ “extend to infinity,” i.e., it is not contained in any
compact subset of $M$; for simplicity, we also require that $D$ is unknotted.
For example, when the metric is Euclidean, use an axis of the usual Cartesian
coordinate system. The manifold $M-D$ is then diffeomorphic to
$S^{2}\times\mathbb{R}^{2}$. In this situation, $\mathrm{U}(1)$-bundles up to
isomorphism are computed by
$\mathrm{H}^{1}(M-D,\mathrm{U}(1))\cong\mathrm{H}^{2}(M-D,\mathbb{Z})$, which
is isomorphic to $\mathbb{Z}$. For a line bundle $L$, its first Chern class
$c_{1}(L)$ is the identifying cohomology class. Any line bundle $L$ does admit
solutions to the equations of motion for $\mathrm{U}(1)$-Yang–Mills theory,
and a physicist would call $c_{1}(L)$ the magnetic charge of the solution.
When $c_{1}(L)\neq 0$, such a solution is called a magnetic monopole. (We can
view the case of $c_{1}(L)=0$ as a charge-free monopole.)

Thus, let $\nabla$ be a magnetic monopole with magnetic charge $m$. Then if we
impose the boundary condition $\mathcal{L}_{B=0}$ along the line $D$, we can
use our construction for any small value of $t$ to produce an effective
factorization algebra for the monopole.

######  Remark 4.11.

There is a qualitatively different story describing monopoles in non-abelian
Yang–Mills–Higgs theory (such as ’t Hooft–Polyakov monopoles), where there is
no locus on which the fields become singular. We will discuss such examples in
[CEG22].

####  4.3.2. Wilson line defects in Yang–Mills theory

As a final example, let us discuss Wilson lines in Yang–Mills theory. This
example will be defined very similarly to the order type operators described
above, particularly in Section 4.2.3.

To begin, recall that the irreducible representations of $\mathrm{U}(1)$ are
all one-dimensional, labelled by their weight $n\in\mathbb{Z}$. We will denote
this representation by $V_{n}$; we will view $n$ as the electric charge of a
charged particle.

Let $M=\mathbb{R}^{4}$, and let $D\subseteq M$ be an embedding of a line with
tubular neighborhood $U$. Write $i_{D}\colon D\to U$ for the inclusion. We can
describe a “coupled” theory along the line using the complex

$\mathcal{E}_{U,n}=\mathcal{E}_{U}\ltimes(i_{D})_{*}(\Omega^{\bullet}(D)\otimes
V_{n}),$

where $\mathcal{E}_{U}$ is the complex from Equation (4), and where
$\mathfrak{u}(1)\ltimes V_{n}$ is the Lie algebra with underlying vector space
$\mathbb{R}\oplus V_{n}$ whose only non-trivial bracket is generated by

$[(1,0),(0,v)]=(0,nv).$

This is an example of the representations $\mathfrak{g}\ltimes V$ described in
Section 4.2.3, but with the representation $V$ now placed in even rather than
odd degree.

We can define the Wilson line defect associated to the representation $V_{n}$.

######  Definition 4.12.

Choose a tubular neighborhood $U$ of $D$ in $M$, and let $\iota_{\partial}$ be
the embedding of the boundary of $U$ in $M$. The _Wilson line defect_ with
electric charge $n$ is the defect

$\iota_{\partial}^{*}\mathcal{E}_{U,n}\to\mathcal{E}_{\partial U},$

where $\mathcal{E}_{\partial U}$ is the complex modelling germs of solutions
to the equations of motion along $\partial U$ (e.g. as defined in Equation
(5)).

######  Remark 4.13.

If we _also_ chose a magnetic monopole with magnetic charge $m$, singular
along $D$, we could similarly define a Wilson line defect with electric charge
$n$ in this background: one can define a defect by taking the magnetic charge
$m$ monopole defect $\mathcal{L}_{B=0}$ and forming the tensor product with
the electric charge $n$ Lie algebra $\mathfrak{u}(1)\ltimes V_{n}$ defined
above. The result is called a _dyonic defect_ with charge $(m,n)$.

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