July 2023

HU-EP-23/22

Holography as Homotopy

Christoph Chiaffrino, Talha Ersoy and Olaf Hohm

Institute for Physics, Humboldt University Berlin,
Zum Großen Windkanal 6, D-12489 Berlin, Germany

<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

> We give an interpretation of holography in the form of the AdS/CFT
> correspondence in terms of homotopy algebras. A field theory such as a bulk
> gravity theory can be viewed as a homotopy Lie or $L_{\infty}$ algebra. We
> extend this dictionary to theories defined on manifolds with a boundary,
> including the conformal boundary of AdS, taking into account the cyclic
> structure needed to define an action with the correct boundary terms.
> Projecting fields to their boundary values then defines a homotopy retract,
> which in turn implies that the cyclic $L_{\infty}$ algebra of the bulk
> theory is equivalent, up to homotopy, to a cyclic $L_{\infty}$ algebra on
> the boundary. The resulting action is the ‘on-shell action’ conventionally
> computed via Witten diagrams that, according to AdS/CFT, yields the
> generating functional for the correlation functions of the dual CFT. These
> results are established with the help of new techniques regarding the
> homotopy transfer of cyclic $L_{\infty}$ algebras.

###### Contents

  1. 1 Introduction
  2. 2 Homotopy Transfer of Cyclic $L_{\infty}$ Algebras
    1. 2.1 Cyclic $L_{\infty}$ Algebras
    2. 2.2 Homotopy Transfer
  3. 3 Homotopy Retract to Boundary
    1. 3.1 Scalar Field Theory with Boundary
    2. 3.2 Scalar Green’s Function
    3. 3.3 Homotopy Retract
    4. 3.4 Example: The Harmonic Oscillator
  4. 4 Homotopy Retract for Gauge Theories
    1. 4.1 Hodge Decomposition on Manifolds with Boundary
    2. 4.2 Yang-Mills Theory as a cyclic $L_{\infty}$ Algebra
    3. 4.3 A Homotopy for Yang-Mills Theory
  5. 5 Homotopy Retract to Boundary of Anti-de-Sitter
    1. 5.1 Homotopy Retract for AdS
    2. 5.2 Homological Perturbation Lemma
    3. 5.3 Witten Diagrams and Correlation Functions
  6. 6 Summary and Outlook
  7. A Technical Details for Yang-Mills Theory with Boundary
    1. A.1 Cyclic Extension in Yang-Mills Theory
    2. A.2 Homotopy Retract in Yang-Mills Theory

##  1 Introduction

Holography in the form of the AdS/CFT correspondence is arguably one of the
most intriguing proposals that has emerged in quantum gravity. In its original
and most tested form, the AdS/CFT conjecture relates the large $N$ limit of
${\cal N}=4$ super Yang-Mills theory with gauge group $SU(N)$ to type IIB
supergravity on $AdS_{5}\times S^{5}$ [1, 2, 3]. Since then the AdS/CFT
correspondence has been conjectured to extend to many other examples as well
as to the full quantum regime, so that a theory of quantum gravity may be
fully equivalent to a more or less conventional quantum field theory in flat
space. While it remains to be seen whether this strongest form of the AdS/CFT
conjecture really can provide a fully-fledged definition of quantum gravity on
AdS, problems of quantum gravity are now routinely phrased in the language of
AdS/CFT.

Part of the psychological appeal of holography is undoubtedly due to the
mystery of having two theories defined in different dimensions that are
nevertheless fully equivalent. We are of course familiar with theories that on
the surface look quite different but that are secretly equivalent or dual to
each other. Examples include theories with a $p$-form that, for certain matter
couplings, are dual to theory with a $(D-p-2)$-form in $D$ dimensions. For
instance, in four dimensions a 2-form may be dual to a scalar, so that in
particular a theory with a gauge redundancy may be dual to a theory without
gauge redundancy. For such examples the duality is easily established by
writing a master action so that upon integrating out different fields one
obtains either the original theory or its dual, thereby proving their
equivalence. (See [4] for an example of such proof.) However, in such familiar
examples the dual theories are defined in the same dimension, hence leaving it
as somewhat of a mystery how holography could ever be proved from first
principles.

In this paper we will provide a framework for AdS/CFT in which the equivalence
of structures in different dimensions is built in. This is the framework of
homotopy algebras, which in pure mathematics arose in the context of algebraic
topology [5] and in theoretical physics in string field theory [6]. A
classical perturbative field theory is encoded in a homotopy Lie algebra or
$L_{\infty}$ algebra as follows [6, 7]: Denoting the fields collectively by
$\phi$ one writes the action as

$S[\phi]=\frac{1}{2}\omega(b_{1}(\phi),\phi)+\frac{1}{3!}\omega(b_{2}(\phi,\phi),\phi)+\frac{1}{4!}\omega(b_{3}(\phi,\phi,\phi),\phi)+\cdots\,,$
(1.1)

with a potentially infinite number of multi-linear maps $b_{n}$ (with $n$
inputs), subject to generalized Jacobi identities defining an $L_{\infty}$
structure, see [7, 8, 9, 10]. More precisely, the fields live in an integer
graded vector space $V$ (in degree zero), which also contains gauge parameters
(in degree $-1$), equations of motion or ‘anti-fields’ (in degree $1$), and so
on. Furthermore, in writing an action we actually assume a cyclic $L_{\infty}$
algebra, with inner product $\omega$ that can be viewed as a symplectic form
of odd degree. Such ingredients are familiar from the BV-BRST formalism [11,
12], whose relation to cyclic $L_{\infty}$ algebras we will further clarify in
this paper.

The usefulness of homotopy algebras is due to them behaving well under
operations known as homotopy retract and homotopy transfer [13, 14]. These are
defined by projections from the original graded vector space $V$ to a
potentially much smaller space. Since such projections are generally not
invertible, familiar algebraic structures such as Lie algebras are not
preserved. In the larger category of homotopy Lie algebras, however, they are
preserved provided the projection is invertible up to homotopy. Two homotopy
algebras so related are equivalent in the context of homotopy Lie algebras,
even if their underlying vector spaces have different dimensions. In
particular, the cohomology, which contains invariant physical information, is
preserved, so that the resulting homotopy algebra on the smaller space may
encode the same physical observables. It has been shown that homotopy transfer
formalizes the physical notions of integrating out fields, see e.g. [15, 16,
17], and passing over to gauge invariant variables [18] (in both cases
projecting from a larger to a smaller field space). Here we will show,
inspired by earlier results in [19, 20, 21, 22], that passing from a bulk
theory on a manifold with boundary to its boundary can be interpreted as a
homotopy transfer. In a slight generalization of this result we show in
particular that passing from AdS to its conformal boundary also provides a
homotopy transfer, hence proving that a theory on AdS, formulated as a
homotopy algebra, may be equivalent to another homotopy algebra associated to
the boundary.

In order to establish this result we have to refine the dictionary between
field theory and cyclic $L_{\infty}$ algebras to take the boundary into
account. Specifically, in order for (1.1) to reproduce the correct action
including boundary terms, the vector space $V$ has to be enlarged by ‘boundary
components’. For instance, for a scalar field theory on a Riemannian manifold
$M$ with boundary $\partial M$, the space of fields in degree zero is the
space of smooth functions $C^{\infty}(M)$, but the space in degree one
contains in addition the space of smooth functions $C^{\infty}(\partial M)$ on
the boundary. These spaces form the chain complex

${0}$${V^{0}=C^{\infty}(M)}$${V^{1}=C^{\infty}(M)\oplus C^{\infty}(\partial
M)}$${0\,,}$$\scriptstyle{b_{1}}$ (1.2)

where the only non-trivial differential $b_{1}$ is defined by

$b_{1}(\phi)=\left((-\Delta+m^{2})\phi\,,\;\partial_{n}\phi|_{\partial
M}\right)\,.$ (1.3)

Here, $\Delta$ is the Laplace operator on $M$ (we are working in Euclidean
signature), and $\partial_{n}$ is the derivative normal to the boundary.
Furthermore, we will show how to extend the complete cyclic $L_{\infty}$
algebra for scalar field theories and Yang-Mills theory, to this setting with
boundary, which requires adding an appropriate boundary term to the symplectic
form $\omega$.

Starting then from an $L_{\infty}$ algebra with boundary data included, we
will define a homotopy retract by introducing the projection map $p$ from
functions in the bulk to functions on the boundary. For a manifold $M$
carrying a boundary in the strict topological sense this projection is just
the restriction to the boundary: $p(\phi)=\phi\big{|}_{\partial M}$. The
prescription for AdS is slightly more subtle since here we have only a
conformal boundary. Working in Poincaré coordinates where the metric for
AdSd+1 reads $ds^{2}=\frac{1}{z^{2}}(dz^{2}+dx^{i}dx^{i})$, with
$d$-dimensional flat coordinates $x^{i}$, the projection of a scalar
$\phi(z,{x})$ includes a rescaling as follows:

$p(\phi):=\lim_{z\rightarrow 0}z^{-\Delta_{-}}\phi\;,$ (1.4)

where $\Delta_{-}=\frac{d}{2}-\sqrt{\frac{d^{2}}{4}+m^{2}}$. In both cases
there is also an inclusion map $i$ in the other direction, i.e., from boundary
functions to bulk functions. A given function $\phi_{0}({x})$ on the boundary
is mapped by $i$ to the solution of the bulk equation $(-\Delta+m^{2})\phi=0$
with boundary condition $\phi\big{|}_{\partial M}=\phi_{0}$ (or
$\lim_{z\rightarrow 0}z^{-\Delta_{-}}\phi=\phi_{0}$ for AdS). The projection
and inclusion maps are not inverse to each other since, for instance, the
inclusion maps only to on-shell fields, while the space $V^{0}$ contains all
off-shell fields. However, $i$ and $p$ are inverse ‘up to homotopy’, which
means that there is a degree $-1$ map $h$ so that $i\circ p={\bf 1}-b_{1}\circ
h-h\circ b_{1}$. This is sufficient to show that the $L_{\infty}$ algebra of
the bulk theory is transported to an equivalent $L_{\infty}$ algebra on the
boundary.

Let us relate the above construction to the conventional AdS/CFT dictionary.
Here one uses that according to the field/operator correspondence each
operator ${\cal O}$ of the CFT corresponds to a field $\phi$ in the bulk
theory whose boundary value $\phi_{0}$ can be thought of as a ‘source’ for
${\cal O}$ in that the generating functional or partition function is given by

$Z[\phi_{0}]=\Big{\langle}\exp\int d^{4}x\,\phi_{0}(x){\cal
O}(x)\Big{\rangle}_{\rm CFT}\;.$ (1.5)

This function encodes arbitrary correlation functions involving ${\cal O}$,
which can be obtained by taking suitable functional derivatives with respect
to $\phi_{0}$. The AdS/CFT conjecture states that this partition function is
computed by the on-shell action of the bulk theory. Specifically, given the
boundary value $\phi_{0}$ one may find, using perturbation theory, the
solution $\phi(\phi_{0})$ of the full non-linear bulk equations of motion that
approaches $\phi_{0}$ at the boundary. The on-shell action is obtained by
inserting this solution into the bulk action, subject to a procedure of
‘holographic renormalization’ to render the result finite [23]. This on-shell
action is conjectured to give the partition function (1.5):

$Z[\phi_{0}]=\exp\big{[}-S[\phi(\phi_{0})]\,\big{]}\;.$ (1.6)

The on-shell action defines, using (1.1), a boundary cyclic $L_{\infty}$
algebra. It is this $L_{\infty}$ algebra that is computed by the above
construction using homotopy transfer. Therefore, if AdS/CFT is correct, this
$L_{\infty}$ algebra encodes all correlation functions of the CFT. This result
follows from the detailed construction of the homotopy transported
$L_{\infty}$ brackets, which using the so-called perturbation lemma precisely
implements the standard technique of Witten diagrams [3], thereby giving a
homotopy algebra interpretation of the latter.

The reader may well wonder what the point of the homotopy algebra formulation
is, given that in the end one obtains the known method of Witten diagrams.
Apart from being of conceptual interest and giving, at least in our view, a
more systematic introduction of these techniques, the perhaps most important
point is that this formulation provides a conceptual framework for holography
(of structures in different dimensions that are nevertheless equivalent),
which hence may be a first baby step toward a first-principle derivation of
AdS/CFT. (The methods presented here also appear to be related to the use of
Hamilton-Jacobi theory for AdS/CFT [24, 25]).

Let us then try to imagine how a proof of the original AdS/CFT correspondence
between ${\cal N}=4$ super-Yang-Mills theory in four dimensions and type IIB
supergravity on $AdS_{5}\times S^{5}$ could look like. We start with the
cyclic $L_{\infty}$ algebra of IIB supergravity on $AdS_{5}\times S^{5}$,
which one could write down quite explicitly, thanks to techniques that were
developed during the last decade within a framework known as exceptional field
theory [26, 27].111These techniques allow one not only to embed the truncation
to five-dimensional gauged supergravity on $AdS_{5}$ into the full ten-
dimensional theory [28, 29], but also to encode all massive Kaluza-Klein modes
corresponding to the higher harmonics on $S^{5}$ [30], which are dual to
operators on the CFT side and hence need to be included. The results of this
paper establish a homotopy transfer from this $L_{\infty}$ algebra to the
$L_{\infty}$ algebra of the generating functional on the boundary
$\mathbb{R}^{4}$, and hence an $L_{\infty}$ morphism between them. This is
indicated in the diagram as a solid line.

Master Algebra? $L_{\infty}$ Algebra of Boundary CFT $L_{\infty}$ Algebra of
Bulk Theory $L_{\infty}$ Algebra of Boundary Correlators  homotopy transfer

In addition, one requires a map from the $L_{\infty}$ algebra of ${\cal N}=4$
super-Yang-Mills theory to the generating functional of its correlation
functions in the large $N$ limit. In this case we do not yet know of a
homotopy algebra formulation of this map, hence being indicated by a dashed
line in the diagram, but given results in [31, 32, 33, 34, 35, 36, 37]
concerning the closely related computation of scattering amplitudes and
quantum expectation values more generally [21], it is plausible that there is
an $L_{\infty}$ morphism in this direction. Supposing this to be the case, and
using that $L_{\infty}$ morphisms are invertible up to homotopy, one would
have an explicit $L_{\infty}$ morphism between ${\cal N}=4$ super-Yang-Mills
theory and type IIB supergravity on $AdS_{5}\times S^{5}$ provided the AdS/CFT
conjecture is correct. In order to actually prove the AdS/CFT conjecture one
would presumably have to define a master homotopy algebra, indicated by a
question mark in the diagram, from which both the ${\cal N}=4$ super-Yang-
Mills and the type IIB supergravity $L_{\infty}$ algebras can be obtained by
homotopy transfer. Finding this master homotopy algebra, which could be quite
abstract and may not be an $L_{\infty}$ algebra itself, and hence proving
AdS/CFT is of course a tall order. But we take it as progress that one can now
at least imagine how a proof could look like.

In the remainder of this introduction we outline the content of each section
and emphasize our new technical results:

In section 2 we review the formulation of $L_{\infty}$ algebras based on the
symmetric algebra $S^{c}V=\bigoplus_{n\geq 1}V^{\wedge n}$ of the underlying
vector space $V$, on which one defines the coderivation
$B=b_{1}+b_{2}+b_{3}+\cdots$ whose squaring to zero, $B^{2}=0$, encodes the
generalized Jacobi identities. This formulation is well-known, but we here
provide some important additions for the case of cyclic $L_{\infty}$ algebras
with symplectic form $\omega$ and their homotopy transfer. This allows us in
particular to clarify the relation to the BV formalism, which is
conventionally presented as an odd generalization of symplectic geometry, with
a homological vector field or derivation squaring to zero that is dual to the
coderivation. We show that the multilinear map given by
$S=\omega(\pi_{1}B\otimes{\bf 1})$, with $\pi_{1}$ the projection operator
from $S^{c}V$ to $V$, gives the classical action (1.1) when evaluated on
fields in degree zero, but also gives the complete BV master action when
evaluated on the full vector space $V$. Finally, with these results, we prove
that the homotopy transfer of the cyclic $L_{\infty}$ algebra yields an action
that equals the ‘on-shell action’ $\bar{S}$, i.e., $\bar{S}=S\circ i$, where
$i$ is the perturbed inclusion map from initial or boundary conditions to full
solutions of the ‘bulk’ equations. We imagine these results to be useful
beyond AdS/CFT.

In section 3 we discuss scalar field theories on Riemannian manifolds with
boundary and establish homotopy transfer to the boundary. To this end we
refine the $L_{\infty}$ description for theories in presence of a boundary
including the symplectic form, as outlined above, and show that the projector
to the boundary, together with the trivial inclusion and a homotopy map given
by the Green’s function, satisfies the homotopy relation. We illustrate this
for the one-dimensional harmonic oscillator, where the boundary is given by
the endpoints of the finite time interval $[t_{i},t_{f}]$.

In section 4 we generalize this construction to Yang-Mills theory, which we
take to be a toy model for bulk supergravity. We introduce and review Hodge
theory for manifolds with boundary and then specialize to Yang-Mills theory
whose cyclic $L_{\infty}$ algebra is generalized to take the boundary into
account, and we provide the homotopy transfer to the boundary. In particular,
using the multilinear map $S=\omega(\pi_{1}B\otimes{\bf 1})$ we then
automatically get the full BV action including boundary terms.

Finally, in section 5 we apply, and slightly generalize, these results by
considering scalar field theories on AdS. We establish the homotopy to its
conformal boundary, and compute the transported $L_{\infty}$ brackets on the
boundary using the perturbation lemma. In this way we recover the method of
Witten diagrams.

##  2 Homotopy Transfer of Cyclic $L_{\infty}$ Algebras

###  2.1 Cyclic $L_{\infty}$ Algebras

Generalities:
We give a short review of the definition of an $L_{\infty}$ algebra and how
they arise in classical field theory, for more details see for example [8, 7].
An $L_{\infty}$ algebra on a graded vector space $V$ consists of a collection
of graded symmetric maps

$b_{n}:V^{\otimes n}\rightarrow V$ (2.1)

of degree one satisfying various relations. The first three of them are

  1. 1.

$b_{1}^{2}=0$.

  2. 2.

$b_{1}b_{2}(x,y)=-b_{2}(b_{1}x,y)-(-)^{x}b_{2}(x,b_{1}y)$.

  3. 3.

$b_{1}b_{3}(x,y,z)+b_{3}(b_{1}x,y,z)+(-)^{x}b_{3}(x,b_{1}y,z)+(-)^{x+y}b_{3}(x,y,b_{1}z)=\\\
-b_{2}(b_{2}(x,y),z)-(-)^{(y+z)x}b_{2}(b_{2}(y,z),x)-(-)^{(x+y)z}b_{2}(b_{2}(z,x),y)$.

The first relation means that $b_{1}$ turns $V$ into a cochain complex, which
is just a vector space with a square zero map of degree one. The second
relation tells us that $b_{1}$ acts as a derivation on the product $b_{2}$.
The last relation implies that $b_{2}$ fails to satisfy Jacobi, and this
failure is measured by $b_{1}$ and $b_{3}$.

The infinite tower of relations among the $b_{n}$ can be encoded in a single
equation in the following way. For this we need to introduce the symmetric
graded coalgebra $S^{c}V$ over the graded vector space $V$. As a vector space,
we have

$S^{c}V=\bigoplus_{n\geq 1}V^{\wedge n}\,,$ (2.2)

i.e. it consists of polynomials over the elements of $V$ without constant
term. In particular, the space $V^{\wedge n}$ consists of order $n$ monomials
$x_{1}\cdots x_{n}$, where $x_{i}\in V$, and we have graded symmetry among the
$x_{i}$, i.e.

$x_{1}\cdots x_{i}x_{i+1}\cdots x_{n}=(-)^{x_{i}x_{i+1}}x_{1}\cdots
x_{i+1}x_{i}\cdots x_{n}\,.$ (2.3)

We then introduce a coproduct $\Delta:S^{c}V\rightarrow S^{c}V\otimes S^{c}V$.
On order $n$ monomial it is defined as

$\Delta(x_{1}\cdots
x_{n})=\sum_{i=1}^{n}\sum_{\sigma\in\bar{\text{Sh}}(i,n)}(-)^{\sigma(x_{1},...,x_{n})}x_{\sigma(1)}\cdots
x_{\sigma(i)}\otimes x_{\sigma(i+1)}\cdots x_{\sigma(n)}\,.$ (2.4)

Here, the second sum runs over all $n$-element permutations $\sigma$, such
that $\sigma(1)<...<\sigma(i)$ and $\sigma(i+1)<...<\sigma(n)$. The sign
$\sigma(x_{1},...,x_{n})$ is then the natural sign one gets from reordering
the $x_{i}$ with $\sigma$. The coproduct $\Delta$ is coassociative, i.e.

$(\Delta\otimes 1)\Delta=(1\otimes\Delta)\Delta.$ (2.5)

This is simply the dual relation to the associativity property of products
$m:V\otimes V\rightarrow V$, which reads $m(m\otimes 1)=m(1\otimes m)$. We
also need the concept dual to a derivation. We say that $B$ is a coderivation
with respect to $\Delta$ if it satisfies

$(B\otimes 1+1\otimes B)\Delta=\Delta B\,.$ (2.6)

An $L_{\infty}$ algebra on $V$ can now be encoded in a degree one coderivation
$B$ on $S^{c}V$, such that $B^{2}=0$. The data $(S^{c}V,\Delta,B)$ is an
example of a differential graded coalgebra, which is the concept dual to that
of a differential graded algebra. The $L_{\infty}$ products
$\\{b_{n}\\}_{n\geq 1}$ are now given by

$\sum_{n\geq 1}b_{n}:=\pi_{1}B\,,$ (2.7)

where $\pi_{1}:S^{c}V\rightarrow V$ is the projection onto $V\subseteq
S^{c}V$, i.e. it extracts the linear part of a polynomial in $S^{c}V$. Also,
$b_{n}$ in (2.7) is the part of $\pi_{1}B:S^{c}V\rightarrow V$ that acts on
$V^{\wedge n}$. The infinite number of relations among the $b_{n}$ is now
encoded in the single equation $B^{2}=0$.

At this point we should mention that $B^{2}=0$ is equivalent to
$\pi_{1}B^{2}=0$, so the square zero condition does not encode further
information over those imposed on the $b_{n}$. The equivalence holds because
$B$ is a coderivation and not some general linear map. Coderivations on
$(S^{c}V,\Delta)$ have the property that there is a one-to-one correspondence
between linear maps $b:S^{c}V\rightarrow V$ and coderivations
$B:S^{c}V\rightarrow S^{c}V$. In other words, for any such $b$ (which
satisfies no additional property apart from being linear), there is a unique
coderivation $B$, such that $\pi_{1}B=b$. Also coderivations are closed under
taking graded commutators, i.e. for any two coderivations $A$ and $B$,
$[A,B]:=AB-(-)^{AB}BA$ is also a coderivation. For an odd $B$, we have
$B^{2}=\frac{1}{2}[B,B]$, so $B^{2}$ is also a coderivation independent of
whether it is zero or not, and as such $B^{2}=0$ is equivalent to
$\pi_{1}B^{2}=0$ (the coderivation $0:S^{c}V\rightarrow S^{c}V$ obviously is a
lift of the linear map $0:S^{c}V\rightarrow V$).

We can use $L_{\infty}$ algebras to encode the equations of motions and the
gauge structure of a quantum field theory. A field $\phi$ in $V$ of degree
zero satisfies the equations of motion if it is a solution to the Maurer-
Cartan equation

$\sum_{n\geq 1}\frac{1}{n!}b_{n}(\phi,...,\phi)=0\,.$ (2.8)

The gauge transformations are given by

$\delta\phi=\sum_{n\geq 0}\frac{1}{n!}b_{n+1}(\lambda,\phi,...,\phi)\,,$ (2.9)

where $\lambda$ has degree $-1$ and serves as the gauge parameter. The
$L_{\infty}$ relations then assure that the equations of motion are gauge
covariant under these gauge transformations.

For $L_{\infty}$ algebras one defines two notions of equivalence between two
different algebras. Suppose we have $L_{\infty}$ algebras
$(V,\\{b_{n}\\}_{n\geq 1})$ and $(W,\\{c_{n}\\}_{n\geq 1})$. Let $B$ be a
coderivation on $SV$ and $C$ a coderivation on $SW$ encoding the respective
$L_{\infty}$ products. We say that $(V,\\{b_{n}\\}_{n\geq 1})$ is isomorphic
to $(W,\\{c_{n}\\}_{n\geq 1})$, if there is a degree zero invertible coalgebra
morphism $\Phi:(SV,\Delta_{SV})\rightarrow(SW,\Delta_{SW})$, such that $\Phi
B=C\Phi$. $\Phi$ being a coalgebra morphism means that

$\Delta_{SW}\Phi=(\Phi\otimes\Phi)\Delta_{SV}\,.$ (2.10)

One can prove that $\Phi$ is invertible if and only if
$\varphi=\pi_{1}\circ\Phi|_{V}:V\rightarrow W$ is invertible. The second
notion of equivalence is weaker. We say that a degree zero coalgebra morphism
$\Phi:SV\rightarrow SW$ is a quasi-isomorphism, if
$\varphi=\pi_{1}\circ\Phi|_{V}:(V,b_{1})\rightarrow(W,c_{1})$ is a quasi-
isomorphism. This means that $\varphi$ is a (co)chain map, i.e. $\varphi
b_{1}=c_{1}\varphi$, and $\varphi$ descends to an isomorphism on cohomologies.
The most important result in the theory of $L_{\infty}$ algebras is that given
a quasi-isomorphism $i:(W,\\{c_{n}\\}_{n\geq
1})\rightarrow(V,\\{b_{n}\\}_{n\geq 1})$, there is always a quasi-isomorphism
$p:(V,\\{b_{n}\\}_{n\geq 1})\rightarrow(W,\\{c_{n}\\}_{n\geq 1})$, such that
the linear pieces of $p$ and $i$ are inverse to each other on cohomology of
$(V,b_{1})$ and $(W,c_{1})$.

General coalgebra morphisms $S^{c}V\rightarrow S^{c}W$ satisfy a property we
already encountered for coderivations. Namely, given any degree zero linear
map $f:S^{c}V\rightarrow W$, there is a unique coalgebra morphism
$F:S^{c}V\rightarrow S^{c}W$ such that $\pi_{1}\circ F=f$. Explicitly, given
$f:S^{c}V\rightarrow W$, the lift $F$ is given by

$F=\sum_{n\geq 1}\frac{1}{n!}f^{\otimes n}\Delta^{n}\,,$ (2.11)

where $\Delta^{1}=1$ and $\Delta^{n}=(1\otimes\Delta^{n-1})\Delta$. The $n$th
term in the sum is the part of $F$ that maps into $W^{\wedge n}\subseteq
S^{c}W$.

The map $f:SV\rightarrow W$ can be viewed as a non-linear map $f:V\rightarrow
W$ [38]. First of all in the ungraded case, given a point $x\in V$, we can
associate the non-linear map

$\begin{split}V\longrightarrow W\,,\qquad x\longmapsto\sum_{n\geq
1}\frac{1}{n!}f_{n}(x,\ldots,x)\,,\end{split}$ (2.12)

with $f_{n}=f\circ i_{n}$ and $i_{n}:V^{\wedge n}\hookrightarrow S^{c}V$ the
inclusion. Here, the $f_{n}$ are the Taylor coefficients of the non-linear map
$f$. One can further show that given $F:S^{c}U\rightarrow S^{c}V$ and
$G:S^{c}V\rightarrow S^{c}W$ with corresponding non-linear maps
$f:U\rightarrow V$ and $g:V\rightarrow W$, the non-linear map derived from
$G\circ F$ is $g\circ f$. In other words, the association $F\mapsto f$ is an
isomorphism of associative algebras, where the associative algebra structure
in both cases is given by composition of maps. This non-trivial fact is of
course necessary for the interpretation of coalgebra morphisms in terms of
non-linear maps to actually make sense.

In the graded case, it does not make sense to consider expressions like
$f_{n}(x,\ldots,x)$, since functions vanish on odd variables when $n\geq 2$.
Nevertheless, in this case $f$ still defines a map of graded spaces
$V\rightarrow W$. In the literature (especially in string field theory), one
still finds expressions like $f_{n}(x,\ldots,x)$ even in the graded case. This
makes sense if there is a basis $e^{i}$ and a dual basis $e_{i}^{*}$, in which
case one defines a “super field” $\Phi=\sum_{i}e^{i}e_{i}^{*}$. We can then
write

$f(\Phi)=\sum_{n\geq 1}\frac{1}{n!}f_{n}(\Phi,\ldots,\Phi)\,.$ (2.13)

However, we actually have $\frac{1}{n!}f_{n}(\Phi,...,\Phi)=f_{n}$, hence
$f(\Phi)=f$. The symmetry factor accounts for the fact that products of dual
vectors act as

$e_{1}^{*}\cdots e_{n}^{*}(a_{1},\ldots,a_{n})=\sum_{\sigma\in S_{n}}\pm
e_{1}^{*}(a_{\sigma(1)})\cdots e_{n}^{*}(a_{\sigma(n)})\,,$ (2.14)

where the sum runs over all permutations of $n$ elements and the sign is the
natural one coming from the grading. Its symmetrization ensures that these
products are well defined as functions on $V^{\wedge n}$. In this work we do
not want to assume the existence of a basis, and we therefore refrain from the
use of super fields.

From a physics perspective, coalgebra morphisms define maps between field
spaces. Among these maps, quasi-isomorphisms preserve the relevant physical
information. First of all, an isomorphism is just an invertible field
redefinition. A quasi-isomorphism on the other hand also includes the
possibility that some gauge degrees of freedom are fixed or that some fields
are evaluated on-shell. Of course, also the inverse is possible, i.e. that new
gauge degrees of freedom are added or that auxiliary fields are introduced.

Cyclic $L_{\infty}$ Algebras:
A cyclic $L_{\infty}$ algebra is an $L_{\infty}$ algebra equipped with a non-
degenerate inner product $\omega:V\otimes V\rightarrow\mathbb{C}$ of degree
$-1$, which is anti-symmetric in its entries, i.e.

$\omega(a,b)=-(-)^{(a-1)(b-1)}\omega(b,a)=-\omega(b,a)\,.$ (2.15)

The last equality follows by degree reasons: since the output of $\omega$ is
of degree zero (it is a number), it is only non-zero for $|a|+|b|=1$. In
addition, we require that the products $b_{n}$ are cyclic with respect to
$\omega$, meaning that

$\omega(a_{0},b_{n}(a_{1},...,a_{n}))=(-)^{a_{n}(a_{0}+\cdots+a_{n-1})}\omega(a_{n},b_{n}(a_{0},...,a_{n-1}))\,.$
(2.16)

Since the $b_{n}$ are graded symmetric, it follows that

$\omega(b_{n}\otimes 1):V^{\otimes(n+1)}\longrightarrow\mathbb{C}$ (2.17)

is graded symmetric in all entries.

Given an element $\phi\in V$ of degree zero, the inner product allows us to
define the action

$S[\phi]=\sum_{n\geq 1}\frac{1}{(n+1)!}\omega(b_{n}(\phi,...,\phi),\phi)\,.$
(2.18)

The variation of this action is given by

$\delta S[\phi]=\sum_{n\geq
1}\frac{1}{n!}\omega(b_{n}(\phi,...,\phi),\delta\phi)\,,$ (2.19)

where we used cyclicity. Since $\omega$ is non-degenerate, we deduce that the
equations of motion are indeed the Maurer-Cartan equations (2.8).

The action $S$ as a function on $V^{0}\subseteq V$ derives from the
multilinear function $\omega(\pi_{1}B\otimes 1)$, which is well defined as a
function on $S^{c}V$ by cyclicity of $B$, in the same fashion as general maps
$V\rightarrow W$ derive from coalgebra morphisms in the ungraded case.
Therefore, it makes sense to extend the action $S$ to a graded function on all
of $V$ by defining

$S:=\omega(\pi_{1}B\otimes 1)\,.$ (2.20)

Interestingly, this action defines a BV action, where the extension to non-
zero degrees then also contains terms involving anti-fields and ghosts. (One
can also derive an $L_{\infty}$ algebra from a BV action, which is perhaps
more familiar, see e.g. [39]). The anti-bracket can be defined via $\omega$.
For this reason, $\omega$ is often called a linear symplectic form on $V$.

For cyclic $L_{\infty}$ algebras, we also need to refine the notion of a
morphism. Given cyclic $L_{\infty}$ algebras $(V,\\{b_{n}\\}_{n\geq
1},\omega)$ and $(W,\\{b_{n}^{\prime}\\}_{n\geq 1},\omega^{\prime})$, a
morphism between them is a map $f:SV\rightarrow V$, such that

$\omega^{\prime}(a,b)=\omega(f_{1}(a),f_{1}(b))\
\,,\quad\omega(f_{n}(a_{1},...,a_{n}),g_{k}(b_{1},...,b_{k}))=0\ \,,$ (2.21)

for all $n+k>2$.

Below we need an alternative characterization of cyclicity. We claim that $B$
is cyclic if and only if there is a function $S:S^{c}V\rightarrow\mathbb{C}$,
such that

$S\circ Y=\omega(\pi_{1}B\otimes\pi_{1}Y)\circ\Delta$ (2.22)

for all coderivations $Y$. We consider this form advantageous in proofs, since
in this characterization we do not rely on an evaluation on elements $\phi\in
V$.

To check the claim we first note that if $B$ is cyclic in the sense of (2.16),
then

$S=\omega(\pi_{1}B\otimes 1)$ (2.23)

defines a function on $S^{c}V$. By the lifting property of coderivations, this
in turn means that any coderivation $Y$ acts on $S$ as

$Y=(1\otimes\pi_{1}Y)\circ\Delta\ .$ (2.24)

Therefore,

$S\circ Y=\omega(\pi_{1}B\otimes\pi_{1}Y)\circ\Delta\ .$ (2.25)

The other direction is a little bit more involved. We first need the
following. Let $F,G$ be maps on $S^{c}V\otimes V\rightarrow\mathbb{R}$ (for
example, $S=\omega(1\otimes\pi_{1}B)$ has this form independent of whether $B$
is cyclic or not). Then $F=G$ if and only if

$F(1\otimes\pi_{1}Y)\Delta=G(1\otimes\pi_{1}Y)\Delta$ (2.26)

for all coderivations $Y$. Obviously, if $F=G$, then the above equation holds.
On the other hand, suppose (2.26) holds. Without loss of generality, we can
assume that $F$ and $G$ are of the form $F,G:V^{\wedge n}\otimes
V\rightarrow\mathbb{R}$. Now we choose $Y$ to only have a linear piece
$\lambda:V\rightarrow V$. Equation (2.26) then says that

$\sum_{\sigma\in\mathbb{Z}_{n}}\pm(F-G)(x_{\sigma(0)},...,x_{\sigma(n-1)}|\lambda(x_{\sigma(n)}))=0\
,\quad x_{i}\in V\,,\;i=0,...,n\ .$ (2.27)

The vertical line $|$ separates the $V$ input from the $V^{\wedge n}$ inputs.
Let $E:=\\{e_{i}\\}_{i\in I}$ be an arbitrary basis of $V$.222We use that
every vector space, be it finite or infinite-dimensional, has a basis in the
sense that every vector can be expressed as a finite linear combination of
basis vectors. In this, and in life in general, we assume the axiom of choice.
We take $x_{i}\in E$ for all $i$. Let $k$ be the multiplicity of $x_{n}$ among
the $x_{i}$, i.e. $k$ is the number of times in which we have $x_{n}=x_{i}$
for $i=0,...,n$. Without loss of generality, we can assume that $k=1$ when
$x_{0}$ is odd, since otherwise at most one term in (2.27) survives and
equality holds for this choice of basis elements. We define $\lambda$ so that
$\lambda(e_{j})=e_{j}$, where $j\in I$ is such that $x_{n}=e_{j}$. We also
define $\lambda(e_{i})=0$ for all $i\neq j$, so $\lambda$ extends to a unique
function on all of $V$. With all these choices, equation (2.27) reduces to

$k(F-G)(x_{0},...,x_{n-1}|x_{n})=0\ .$ (2.28)

From this it follows that $F=G$ on basis elements. Therefore, $F=G$
everywhere.

Coming back to the original goal to show that (2.22) implies cyclicity in the
sense (2.16), assume that

$S\circ Y=(-)^{Y}\omega(\pi_{1}B\otimes\pi_{1}Y)\circ\Delta\ .$ (2.29)

Now since $S$ always takes at least two elements, we can think of it as a
function on $S^{c}V\otimes V$. In this case $Y$ acts as

$S\circ Y=S(1\otimes\pi_{1}Y)\circ\Delta\ ,$ (2.30)

where $S$ on the right hand side is thought of as a function of $V\otimes
S^{c}V$. We just proved that this then implies that

$S=\omega(\pi_{1}B\otimes 1)$ (2.31)

as functions on $S^{c}V\otimes V$. But since $S$ is graded symmetric, so is
$\omega(\pi_{1}B\otimes 1)$. Therefore, $B$ is cyclic in the sense given in
(2.16).

###  2.2 Homotopy Transfer

An important result in the theory of $L_{\infty}$ algebras is the homotopy
transfer theorem. In physical theories, it allows us to gauge fix and to
integrate out fields.

Suppose we have an $L_{\infty}$ algebra $(V,\\{b_{n}\\}_{n\geq 1})$ as well as
a chain complex $(W,c_{1})$ homotopy equivalent to $(V,b_{1})$. This means
that there are quasi-isomorphisms $p:(V,b_{1})\rightarrow(W,c_{1})$ and
$i:(W,c_{1})\rightarrow(V,b_{1})$, as well as a degree $-1$ map
$h:V\rightarrow V$ (the homotopy), such that

$1-ip=\\{b_{1},h\\}\,.$ (2.32)

The homotopy transfer theorem then states that the chain complex structure on
$(W,c_{1})$ extends to an $L_{\infty}$ structure $(W,\\{c_{n}\\}_{n\geq 1})$,
with the property that $i$ also has an extension to an $L_{\infty}$ morphism
$i^{\prime}:(W,\\{c_{n}\\}_{n\geq 1})\rightarrow(V,\\{b_{n}\\}_{n\geq 1})$,
such that the linear piece of $i^{\prime}:S^{c}W\rightarrow V$ is $i$. In
other words, $i^{\prime}|_{W}=i$.

From a physics perspective, a homotopy equivalence of chain complexes
$(V,b_{1})$ and $(W,c_{1})$ means that we have two equivalent _free_ theories.
The transfer theorem then says then once we turn on interactions on $V$, we
can find interactions for the theory on $W$ so that this interacting theory is
still equivalent to the now interacting theory on $V$.

Note that the homotopy transfer theorem does not guarantee us an extension for
$p$. Although we mentioned above that a $L_{\infty}$ quasi-isomorphism always
exists in the other direction, it is not necessarily an extension of $p$. In
this work, we make some more assumptions, which then also give an extension
$p^{\prime}$ of $p$. These assumptions will be needed when we talk about
homotopy transfer of cyclic algebras.

To state the result using these stronger assumptions, we first recall the
homological perturbation lemma and a version of it called the algebra
perturbation lemma. The latter gives the result we need as a special case.

The homological perturbation lemma assumes that we are given chain complexes
$(X,\text{d}_{X})$ and $(Y,\text{d}_{Y})$, as well as quasi-isomorphisms
$p:(X,\text{d}_{X})\rightarrow(Y,\text{d}_{Y})$ and
$i:(Y,\text{d}_{Y})\rightarrow(X,\text{d}_{X})$ together with a homotopy
$h:X\rightarrow X$, such that

$1-ip=\\{\text{d}_{X},h\\}\,.$ (2.33)

Assume further that we have a perturbation $\delta:X\rightarrow X$ of the
differential $\text{d}_{X}$, which means that we demand
$(\text{d}+\delta)^{2}=0$. We further demand that $\delta$ is small in the
sense that $(1+\delta h)$ is invertible. Then we have a differential
$\text{d}_{X}^{\prime}$ on $X$, quasi-isomorphisms
$i^{\prime}:(X,\text{d}_{X}+\delta)\rightarrow(Y,\text{d}_{Y}^{\prime})$ and
$p^{\prime}:(Y,\text{d}_{Y}^{\prime})\rightarrow(X,\text{d}_{X}+\delta)$, as
well as a homotopy $h^{\prime}:X\rightarrow X$ satisfying

$1-i^{\prime}p^{\prime}=\\{\text{d}_{V}+\delta,h^{\prime}\\}\,.$ (2.34)

Further, the perturbation lemma provides us with explicit formulae,

$i^{\prime}=1+hAi\,,\quad p^{\prime}=p+pAh\,,\quad
h^{\prime}=h+hAh\,,\quad\text{d}_{W}^{\prime}=\text{d}_{W}+pAi\,,$ (2.35)

where $A=(1+\delta h)^{-1}\delta$. In our case, we assume that we can always
write $A$ as a geometric series, i.e. $A=\sum_{n\geq 1}(-\delta h)^{n}\delta$,
which always works for our applications for $L_{\infty}$ algebras, since in
that case the sum will actually be finite when applied to an element in the
symmetric tensor coalgebra.

In summary, the perturbation lemma states that a homotopy equivalence is
stable under perturbations of the differential of $\text{d}_{X}$. To apply
this to $L_{\infty}$ algebras, we set $X=S^{c}V$ and $Y=S^{c}W$ with
differentials $\text{d}_{X}=B_{1}$ and $\text{d}_{Y}=C_{1}$, where $B_{1}$ and
$C_{1}$ act on $S^{c}V$ and $S^{c}W$ as the coderivations so that they lift
the linear differentials $b_{1}:V\rightarrow V$ and $c_{1}:W\rightarrow W$. We
now assume that we have a homotopy equivalence $1-ip=\\{h,b_{1}\\}$ between
$V$ and $W$. From this we can construct a homotopy equivalence between
$(S^{c}V,B_{1})$ and $(S^{c}W,C_{1})$ by lifting the maps $p$ and $i$ to
cohomomorphisms $P$ and $I$, as well as defining

$H(x_{1}\cdots x_{n})=\sum_{\sigma\in
S_{n}}\sum_{i=1}^{n}\pm\frac{1}{n!}x_{\sigma(1)}\cdots
x_{\sigma(i-1)}h(x_{\sigma(i)})ip(x_{\sigma(i+1)})\cdots ip(x_{\sigma(n)})\,.$
(2.36)

As always, the sign in front is the natural one coming from permuting the
elements according to their degree. Also there are signs of the form $(-)^{x}$
coming from commuting elements $x$ past $h$. Having this set up, one can then
show that

$1-IP=\\{H,B_{1}\\}\,.$ (2.37)

We can now apply the perturbation lemma using the perturbation
$\delta=\sum_{n\geq 2}B_{n}$, where $B_{n}$ is the lift of the product
$b_{n}:V^{\wedge n}\rightarrow V$. This then provides us with a differential
$C^{\prime}$ on $S^{c}W$ as well as with a homotopy eqivalence

$1-I^{\prime}P^{\prime}=\\{h,B_{1}+\delta\\}\,.$ (2.38)

Unfortunately, the perturbation lemma does not guarantee that $C^{\prime}$ is
a coderivation on $S^{c}W$, so it is not guaranteed that $C^{\prime}$ actually
induces an $L_{\infty}$ structure. Further, it is not true in general that
$I^{\prime}$ and $P^{\prime}$ are coalgebra morphisms. However, there is a
variant of the homological perturbation lemma that guarantees these
properties. For this, we further have to assume that

$H^{2}=0\,,\quad HI=0\,,\quad PH=0\,,\quad PI=1\,,$ (2.39)

which are sometimes called _side conditions_. For us it is enough to assume
these properties for $i$, $p$ and $h$, since then the above relations hold. If
the side conditions hold, we indeed have that $I^{\prime}$ and $P^{\prime}$
are coalgebra morphisms and $C^{\prime}$ is a coderivation. In particular it
follows that $C^{\prime}$ is of the form $C^{\prime}=\sum_{n\geq 1}c_{n}$,
i.e. we have $L_{\infty}$ products $c_{n}:W^{\wedge n}\rightarrow W$. Another
important property is that the perturbed data still satisfies the side
conditions.

For cyclic $L_{\infty}$ algebras we further require that

$\omega(hx,y)=(-)^{x}\omega(x,hy)$ (2.40)

in addition to the side conditions. Note that this implies in particular that

$\omega(i\otimes h)=\omega(h\otimes i)=\omega(h\otimes h)=0\,.$ (2.41)

We first check that $I^{\prime}$ is a cyclic $L_{\infty}$-morphism to $W$ with
respect to the cyclic structure $\omega(i\otimes i)$. We have

$i^{\prime}:=\pi_{1}I^{\prime}=i-h\sum_{n\geq 1}(-\delta H)^{n}\delta I\,,$
(2.42)

where on the right hand side $i$ and $h$ act as a zero on all $V^{\wedge n}$
for $n\geq 2$. Recall that $\delta=\sum_{n\geq 2}B_{n}$. Using (2.41), it
follows that

$\bar{\omega}=\omega(i^{\prime}\otimes i^{\prime})=\omega(i\otimes i)\ .$
(2.43)

Therefore, $I^{\prime}$ is indeed a cyclic $L_{\infty}$ morphism.

The last thing we have to prove is that the induced products $c_{n}$ are
cyclic. For this we use our alternative characterization of cyclicity (2.22).
This means that we need to find a function $\bar{S}$, such that

$\bar{S}=\omega(\pi_{1}C^{\prime}\otimes\pi_{1}Y)\circ\Delta$ (2.44)

for all coderivations $Y$. We claim that $\bar{S}=S\circ I^{\prime}$ has this
property. We check this by computing $S\circ I^{\prime}\circ Y$ for a given
coderivation $Y$. For this, recall that $I^{\prime}$ is given by

$I^{\prime}=\sum_{n\geq 0}\frac{1}{n!}{i^{\prime}}^{\wedge n}\Delta^{n}\ .$
(2.45)

Let $S_{n}$ the $n+1$-linear piece of $S$. We find that

$\begin{split}S_{n}\circ I^{\prime}\circ
Y&=\frac{1}{n!}S_{n}\circ{i^{\prime}}^{\otimes n}\circ\Delta^{n}\circ
Y=\frac{1}{(n-1)!}S_{n}\circ{i^{\prime}}^{\otimes
n}\circ(1^{\otimes{n-1}}\otimes Y)\circ\Delta^{n}\\\
&=S_{n}\circ({i^{\prime}}^{\otimes(n-1)}\otimes
i^{\prime}Y)(1\otimes\Delta^{n-1})\circ\Delta=S_{n}\circ(I^{\prime}\otimes
i^{\prime}Y)\circ\Delta\ .\end{split}$ (2.46)

Here we used that $Y$ is a coderivation, which implies that

$\Delta^{n}\circ Y^{\prime}=\sum_{k+l=n-1}(1^{\otimes k}\otimes Y\otimes
1^{\otimes l})\circ\Delta^{n}\ ,$ (2.47)

as well as the fact that $Y$ acts on a symmetric function, i.e.

$F\circ\sum_{k+l=n-1}(1^{\otimes k}\otimes Y\otimes 1^{\otimes
l})=nF\circ(Y\otimes 1^{\otimes(n-1)})\ ,$ (2.48)

where $F$ is a function $F:(S^{c}V)^{\otimes}\rightarrow\mathbb{C}$ symmetric
in each entry.

Coming back to our derived expression for $S_{n}$, we now sum over all $n$,
which then tells us that

$S\circ I\circ Y=S\circ(I^{\prime}\otimes i^{\prime}Y)\circ\Delta\ .$ (2.49)

We can now use that $S=\omega(\pi_{1}B\otimes 1)$. We then conclude

$\begin{split}S\circ I\circ Y&=\omega(\pi_{1}BI^{\prime}\otimes
i^{\prime}\circ Y)\circ\Delta=\omega(\pi_{1}I^{\prime}C^{\prime}\otimes
i^{\prime}\circ Y)\circ\Delta\\\ &=\omega(i\circ C^{\prime}\otimes i\circ
Y)\circ\Delta=(-)^{Y}\bar{\omega}(\pi_{1}C^{\prime}\otimes\pi_{1}Y)\circ\Delta\
.\end{split}$ (2.50)

Here we used that $B^{\prime}I^{\prime}=I^{\prime}C^{\prime}$ and
$\omega(i^{\prime}\otimes i^{\prime})=\omega(i\otimes i)=\bar{\omega}$. This
proves that $C^{\prime}$ is cyclic.

Note that our proof also tells us that we do not have to compute the
$C^{\prime}$ directly. Instead, we can compute $\bar{\omega}$ and $\bar{S}$
and find $C^{\prime}$ from

$\bar{S}=\omega(\pi_{1}C^{\prime}\otimes 1)\ .$ (2.51)

##  3 Homotopy Retract to Boundary

###  3.1 Scalar Field Theory with Boundary

We first look at a euclidean scalar field theory on some compact manifold $M$
with metric $g$ and with boundary $\partial M$. The action is given by

$S=\int_{M}\sqrt{g}\left(\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+\frac{1}{2}m^{2}\phi^{2}+V(\phi)\right)\,.$
(3.1)

For the moment, we assume that there are no issues arising due to the presence
of a boundary. In that case, the $L_{\infty}$ algebra has the underlying chain
complex

${0}$${V^{0}}$${V^{1}}$${0\,,}$$\scriptstyle{b_{1}}$ (3.2)

where $V^{0}=V^{1}=C^{\infty}(M)$ and the differential is given by
$b_{1}(\phi)=-\Delta\phi+m^{2}\phi$, with $\Delta$ the Laplacian with respect
to the metric $g$. The products $b_{n}$ depend on the form of the potential.
For instance, for an $n$-point interaction $\frac{\lambda}{n!}\phi^{n}$ we
have

$b_{k}(\phi_{1},...,\phi_{k})(x)=\begin{cases}\lambda\phi_{1}(x)\cdots\phi_{n-1}(x)&\text{for}\
k=n-1\\\ 0&\text{otherwise}\,,\end{cases}$ (3.3)

so that the Maurer-Cartan equation

$\sum_{k\geq 1}\frac{1}{k!}b_{k}(\phi,...,\phi)=0$ (3.4)

gives the correct equations of motion.

Let us denote by $\phi$ the fields in $V^{0}$ and by $\phi^{*}\in V^{1}$ the
degree one objects, which we also call _anti-fields_ due to their relation to
the anti-fields in the BV formalism. If there is no boundary, the cyclic
structure in that case is

$\omega(\phi^{*},\phi):=-\omega(\phi,\phi^{*}):=\int_{M}\sqrt{g}\phi^{*}\phi\,.$
(3.5)

This is only correct without a boundary, because when one computes the action
via

$S^{\prime}=\sum_{k\geq 1}\frac{1}{k!}\omega(b_{k}(\phi,...,\phi),\phi)\;,$
(3.6)

one finds

$S^{\prime}=\int_{M}\sqrt{g}\left(\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+\frac{1}{2}m^{2}\phi^{2}+V(\phi)\right)-\frac{1}{2}\int_{\partial
M}\sqrt{h}\phi\partial_{n}\phi\,,$ (3.7)

where $\partial_{n}$ is the derivative in the direction normal to the
boundary, and $h$ is the induced metric on the boundary. For simplicity we
assume that the potential does not contain derivatives. Otherwise, there would
also be boundary terms coming from the potential.

Due to boundary terms, the action $S^{\prime}$ in (3.7) is not equal to the
action $S$ in (3.1). Furthermore, $b_{1}$ is not cyclic with respect to
$\omega$. We could drop the requirement that we want a cyclic $L_{\infty}$
algebra and just consider a general $L_{\infty}$ algebra, in which case all
the physical information is encoded in the $b_{k}$, which is sufficient for
the equations of motion, but not the action. In order to encode an action we
have to modify the $L_{\infty}$ structure and/or the cyclic structure so that
we obtain (3.1) from (3.6), which we do in the following.

To this end we consider the general variation of the action (3.1) but keeping
boundary terms:

$\delta S=\int\sqrt{g}\left[\delta\phi(-\Delta+m^{2})\phi+\delta\phi
V^{\prime}(\phi)\right]+\int_{\partial
M}\sqrt{h}\delta\phi\partial_{n}\phi\,.$ (3.8)

We interpret this as implying that apart from the ‘bulk’ equations of motion

$(-\Delta+m^{2})\phi+V^{\prime}(\phi)=0\,,$ (3.9)

there is also the ‘boundary equation of motion’

$\partial_{n}\phi\big{|}_{\partial M}=0\,.$ (3.10)

This suggests to enlarge the space of antifields to $V^{1}=C^{\infty}(M)\oplus
C^{\infty}(\partial M)$ and then to define

$b_{1}^{\prime}(\phi)=\left(-\Delta\phi+m^{2}\phi\,,\;\partial_{n}\phi|_{\partial
M}\right)\;.$ (3.11)

Next we should account for the enlarged space in the definition of $\omega$.
Denoting by $\pi^{*}$ the fields in $C^{\infty}(\partial M)$, we define

$\omega^{\prime}((\phi^{*},\pi^{*}),\phi):=-\omega(\phi,(\phi^{*},\pi^{*})):=\int_{M}\sqrt{g}\phi^{*}\phi+\int_{\partial
M}\sqrt{h}\pi^{*}\phi\,.$ (3.12)

With these choices it is now straightforward to show that $S$ in (3.1) is
given by

$S=\sum_{n\geq
1}\frac{1}{(n+1)!}\omega^{\prime}(b_{n}^{\prime}(\phi),\phi)\,,$ (3.13)

where the higher products $b_{n}$, $n\geq 2$, are extended from
$C^{\infty}(M)$ to $C^{\infty}(M)\oplus C^{\infty}(\partial M)$ trivially,
i.e.
$b_{n}^{\prime}(\phi_{1},...,\phi_{n})=(b_{n}(\phi_{1},...,\phi_{n}),0)$.333This
can be extended to the case when $V$ does have derivative interactions: To
this end note that for a general Lagrangian density
$\sqrt{g}\mathcal{L}(\phi,\partial\phi,x)$, the boundary term coming from the
variation is always given by  $\int_{\partial
M}\sqrt{h}\frac{\delta\mathcal{L}}{\delta\partial_{n}\phi}\delta\phi\,.$
(3.14) The ‘boundary equations of motion’ then tell us that the canonical
momentum $\pi=\frac{\delta\mathcal{L}}{\delta\partial_{n}\phi}$ should vanish
at the boundary. If the canonical momentum depends nonlinearly on $\phi$, we
can account for this by modifying the higher products $b_{n}$. We will
encounter this phenomenon later when we consider Yang-Mills theory.

###  3.2 Scalar Green’s Function

In this section we work out the properties of the scalar Green’s function,
which will define the homotopy of the bulk theory to the boundary theory. To
an anti-field $\phi^{*}$, the Green’s function associates a field
$h(\phi^{*})$, such that

$(-\Delta+m^{2})h(\phi^{*})=\phi^{*}\,.$ (3.15)

Lemma 3.4.7 in [40] tells us that $G$ is unique, once we choose Dirichlet
boundary conditions $h(\phi^{*})\big{|}_{\partial M}=0$.

We will now prove that $h$ defines a homotopy to the space of fields living on
the boundary. In addition, since $h$ should give rise to a cyclic $L_{\infty}$
morphism, we need to prove the side conditions as well as symmetry with
respect to $\omega$. To simplify notation, we define $L=-\Delta+m^{2}$ (we
will not call it $b_{1}$, since $b_{1}$ now contains also the boundary
equations of motion). We will frequently use Green’s identity, which in our
case is the failure of $L$ being cyclic with respect to $\omega$:

$\omega(L\phi_{1},\phi_{2})+\omega(\phi_{1},L\phi_{2})=\int_{\partial
M}\sqrt{h}\Big{\\{}\phi_{1}\partial_{n}\phi_{2}-\phi_{2}\partial_{n}\phi_{1}\Big{\\}}\,.$
(3.16)

This relation can be used to define a symplectic form on phase space. To this
end we define $\pi(\phi)=(\partial_{n}\phi|_{\partial M},\phi|_{\partial M})$,
the projection of a field $\phi$ to its value and canonical momentum on the
boundary. Further defining the Hamiltonian symplectic form

$\omega_{\partial}((\varphi_{1},\varpi_{1}),(\varphi_{2},\varpi_{2}))=\int_{\partial
M}\sqrt{h}(\varphi_{1}\varpi_{2}-\varphi_{2}\varpi_{1})\,,$ (3.17)

we can write Green’s identity as

$\omega(L\phi_{1},\phi_{2})+\omega(\phi_{1},L\phi_{2})=\omega_{\partial}(\pi(\phi_{1}),\pi(\phi_{2}))\,.$
(3.18)

A non-linear version of this identity appears in the BV-BFV formalism of
Cattaneo et. al. [22], where it characterizes the failure of the BV symplectic
form to be invariant under the cohomological vector field.

We begin by proving symmetry of $h$ with respect to $\omega$, computing

$\begin{split}\omega(h\phi_{1}^{*},\phi_{2}^{*})+\omega(\phi_{1}^{*},h\phi_{2}^{*})&=\omega(h\phi_{1}^{*},Lh\phi_{2}^{*})+\omega(Lh\phi_{1}^{*},h\phi_{2}^{*})\\\
&=\omega_{\partial}(\pi(h\phi_{1}^{*}),\pi(h\phi_{2}^{*}))\,.\end{split}$
(3.19)

The right hand side vanishes if and only if $\omega_{\partial}$ vanishes on
the image of $\pi h$. In the context of symplectic geometry, a subspace where
the symplectic form vanishes is called isotropic. We can therefore say that
$h$ is symmetric if and only if $h$ maps to an isotropic subspace on the
boundary. In our case we demand that $h$ satisfies Dirichlet boundary
conditions, so the fields in the image of $h$ will vanish on the boundary, but
their canonical momentum can take arbitrary values. Fields satisfying the
constraint $\phi|_{\partial M}=0$ obviously specify an isotropic subspace.
Hence, a propagator with Dirichlet boundary conditions is necessarily
symmetric.

We next want to show that $h$ generates solutions to the free bulk equations
of motion. To this end we define a map $\iota:C^{\infty}(\partial M)\oplus
C^{\infty}(\partial M)\rightarrow C^{\infty}(M)$ to be the adjoint of $\pi h$,
in the sense

$\omega(\phi^{*},\iota(\varphi,\varpi)):=\omega_{\partial}(\pi
h\phi^{*},(\varphi,\varpi))\,.$ (3.20)

We show that the image of $\iota$ lies in the space of solutions to $L\phi=0$.
Suppose we have a general field $\phi$. We then find that

$\begin{split}\omega(\phi^{*},\iota\pi(\phi))&=\omega_{\partial}(\pi
h(\phi^{*}),\pi\phi)=\omega(Lh(\phi^{*}),\phi)+\omega(h(\phi^{*}),L\phi)\\\
&=\omega(\phi^{*},\phi)-\omega(\phi^{*},hL(\phi))\,,\end{split}$ (3.21)

where we used that $Lh=1$ and that $h$ is symmetric with respect to $\omega$.
Rewriting (3.21), we obtain

$\omega(\phi^{*},hL(\phi))=\omega(\phi^{*},(1-\iota\pi)(\phi))\,.$ (3.22)

Since this has to hold for all $\phi^{*}$, we can deduce that

$hL=1-\iota\pi$ (3.23)

on fields $\phi\in C^{\infty}(M)$. Acting on both sides with $L$, we find that
$L\iota\pi=0$. But from this it follows that $L\iota=0$, since $\pi$ is
obviously surjective. Therefore, the image of $\iota$ are solutions to
$L\phi=0$, which we wanted to show. In fact, all solutions are in the image of
$\iota$, since when $\phi$ is a solution, we see from (3.22) that

$\phi=\iota\pi(\phi)\,.$ (3.24)

.

Let us come back to (3.20). We can actually make an improvement there. Since
$h$ satisfies Dirichlet boundary conditions, $\iota(\varphi,\varpi)$ will be
independent of $\varpi$. We can therefore define a new map
$i:C^{\infty}(\partial M)\rightarrow C^{\infty}(M)$ to be the restriction of
$\iota$ to position variables. Likewise, we define a projection

$\begin{split}p:C^{\infty}(M)\rightarrow C^{\infty}(\partial
M)\,,\qquad\phi\mapsto\phi|_{\partial M}\,.\end{split}$ (3.25)

The relation (3.23) becomes

$hL=1-ip\,.$ (3.26)

We prefer $i$ over $\iota$ because $i$ is injective. This can easily be
deduced from (3.26) by acting with $p$ from the right and using $ph=0$ (i.e.
$h$ satisfies Dirichlet boundary conditions). It then follows that $p=pip$,
from which we can deduce that $pi=1$, since $p$ is surjective. From $pi=1$ we
then deduce that $i$ is injective. Furthermore, for any boundary value
$\varphi\in C^{\infty}(\partial M)$, we have that $pi(\varphi)=\varphi$, which
means that $i$ associates to $\varphi$ the solution to $L\phi=0$ with boundary
value $\phi|_{\partial M}=\varphi$. In other words, $i(\varphi)$ solves the
Dirichlet problem

$L\phi=0\ ,\qquad\phi|_{\partial M}=\varphi\,.$ (3.27)

In the context of AdS/CFT, $i$ is called the _boundary-to-bulk_ propagator,
while $h$ is the _bulk-to-bulk_ propagator.

We can extend $p$ to anti-fields via $p(\phi^{*})=0$ and $h$ to fields via
$h(\phi)=0$. Also, we define $i$ to have zero image in anti-fields. The
identity (3.26) then reads

$\\{L,h\\}=1-ip\,.$ (3.28)

This means that we have a homotopy from the full complex defined by the
differential $L:C^{\infty}(M)\rightarrow C^{\infty}(M)$ to the cohomology of
$L$, which is represented by the boundary fields in $C^{\infty}(\partial M)$.
Diagrammatically, this reads

${0}$${C^{\infty}(M)}$${C^{\infty}(M)}$${0\ ,}$${0}$${C^{\infty}(\partial
M)}$${0}$${0\ .}$$\scriptstyle{L}$$\scriptstyle{p}$ (3.29)

In fact, $h$ is a strong deformation retract (it satisfies the side conditions
(2.39)). We trivially have $hi=0$ and $h^{2}=0$. Further, we already saw that
$pi=1$ and $ph=0$.

Summarizing, we have found a homotopy to solutions with respect to the
differential $L:C^{\infty}(M)\rightarrow C^{\infty}(M)$. In the next
subsection we turn to the problem of extending this homotopy to the
differential $b_{1}:C^{\infty}(M)\rightarrow C^{\infty}(M)\oplus
C^{\infty}(\partial M)$.

###  3.3 Homotopy Retract

In the previous subsection we constructed maps $p:C^{\infty}(M)\rightarrow
C^{\infty}(\partial M)$ and $i:C^{\infty}(\partial M)\rightarrow
C^{\infty}(M)$ from fields defined on $M$ (bulk) to fields defined on
$\partial M$ (boundary), together with a homotopy $h$ from $1$ to $ip$. We now
want to extend this to the cyclic theory.

Recall from section 3.1 that in order to have a cyclic differential in the
presence of a boundary, we extended the standard scalar field theory complex

${0}$${C^{\infty}(M)}$${C^{\infty}(M)}$${0}$$\scriptstyle{L}$ (3.30)

to the complex

${0}$${C^{\infty}(M)}$${C^{\infty}(M)\oplus C^{\infty}(\partial
M)}$${0\,,}$$\scriptstyle{b_{1}}$ (3.31)

where

$b_{1}(\phi)=\left((-\Delta+m^{2})\phi\,,\;\partial_{n}\phi|_{\partial
M}\right)=\left(L(\phi)\,,\;p(\partial_{n}\phi)\right)\,.$ (3.32)

Here, $p$ is the projection to the boundary introduced in the previous
section, and $\partial_{n}$ is the derivative normal to the boundary.

We would like to use the above homotopy with respect to the differential $L$
also in the extended case, where the differential is given by $b_{1}$. In
other words, we want a homotopy that solves the bulk equations of motion
$L\phi=0$, but leaves the boundary equations of motion
$p(\partial_{n}\phi)=0$. The complex in this case then looks like

${0}$${C^{\infty}(\partial M)}$${C^{\infty}(\partial
M)}$${0\,,}$$\scriptstyle{c_{1}}$ (3.33)

where the differential $c_{1}$ computes $p_{0}(\partial_{n}\phi)$ on solutions
to the bulk equation of motion. In the previous subsection, we defined the map
$i:C^{\infty}(\partial M)\rightarrow C^{\infty}(M)$ that associates a solution
to given boundary data. In terms of this boundary data, the differential
$c_{1}$ is given by

$c_{1}(\varphi)=p\partial_{n}i(\varphi)\,.$ (3.34)

In fact, due to the Hamilton-Jacobi equations this can also be written as
$c_{1}(\varphi)=\frac{\delta(S\circ i)}{\delta\varphi}$. If we manage to find
a cyclic quasi-isomorphism to the complex (3.33), this is of course what we
would expect.

The first step to give a homotopy is to define a chain map between the bulk
and boundary theory, which is a commuting diagram of the form

${0}$${C^{\infty}(M)}$${C^{\infty}(M)\oplus C^{\infty}(\partial
M)}$${0}$${0}$${C^{\infty}(\partial M)}$${C^{\infty}(\partial
M)}$${0\,.}$$\scriptstyle{\partial}$$\scriptstyle{p_{0}}$$\scriptstyle{p_{1}}$$\scriptstyle{\bar{\partial}}$
(3.35)

In degree zero, we use the restriction map $p_{0}:=p$ we are already familiar
with. In degree one, we define

$p_{1}(\phi^{*},\pi^{*})=\pi^{*}-p_{0}\circ\partial_{n}h(\phi^{*})\,,$ (3.36)

where $h$ is the Dirichlet Green’s function. For $p_{0}$ and $p_{1}$ to
combine into a chain map, we need $c_{1}\circ p_{0}=p_{1}\circ b_{1}$, which
we now show:

$\begin{split}p_{1}b_{1}(\phi)&=p_{1}(L(\phi),p_{0}\partial_{n}\phi)=p_{0}\partial_{n}\phi-
p_{0}\partial_{n}hL(\phi)\\\ &=p_{0}\partial_{n}\phi-
p_{0}(\partial_{n}-\partial_{n}i_{0}p_{0})(\phi)=c_{1}p_{0}(\phi)\,.\end{split}$
(3.37)

Therefore, $p_{0}$ and $p_{1}$ combine into a chain map

${0}$${C^{\infty}(M)}$${C^{\infty}(M)\oplus C^{\infty}(\partial M)}$${0\
,}$${0}$${C^{\infty}(\partial M)}$${C^{\infty}(\partial M)}$${0\
.}$$\scriptstyle{b_{1}}$$\scriptstyle{p_{0}}$$\scriptstyle{p_{1}}$$\scriptstyle{c_{1}}$
(3.38)

For a fully fledged homotopy, we also need a chain map in the other direction.
In degree zero, we take $i_{0}:C^{\infty}(\partial M)\rightarrow
C^{\infty}(M)$. In degree one, we choose $i_{1}:C^{\infty}(\partial
M)\rightarrow C^{\infty}(M)\oplus C^{\infty}(\partial M)$ to be the inclusion
into the second factor. We prove that $i_{0}$ and $i_{1}$ combine into a chain
map:

$b_{1}i_{0}(\varphi)=(0,p_{0}\partial_{n}i_{0}(\varphi))=i_{1}c_{1}(\varphi)\,.$
(3.39)

Here, we used that $Li_{0}=0$. We therefore have a chain map

${0}$${C^{\infty}(M)}$${C^{\infty}(M)\oplus C^{\infty}(\partial M)}$${0\
,}$${0}$${C^{\infty}(\partial M)}$${C^{\infty}(\partial M)}$${0\
.}$$\scriptstyle{b_{1}}$$\scriptstyle{i_{0}}$$\scriptstyle{c_{1}}$$\scriptstyle{i_{1}}$
(3.40)

Note also that

$p_{0}i_{0}=1\ ,\qquad p_{1}i_{1}=1\ .$ (3.41)

The last step is to give a homotopy from $i\circ p$ to the identity. This is a
map $h_{1}:C^{\infty}(M)\oplus C^{\infty}(\partial M)\rightarrow
C^{\infty}(M)$ satisfying

$h_{1}b_{1}=1-i_{0}p_{0}\,,$ (3.42)

and

$b_{1}h_{1}=1-i_{1}p_{1}\,.$ (3.43)

Equation (3.42) is a rewriting of eq. (3.28) above if we set
$h_{1}(\phi^{*},\pi^{*})=h(\phi^{*})$. It remains to check (3.43):

$\begin{split}(1-i_{1}p_{1})(\phi^{*},\pi^{*})&=(\phi^{*},p_{0}\partial_{n}h(\phi^{*}))\\\
&=(Lh(\phi^{*}),p_{0}\partial_{n}h(\phi^{*}))=\partial
h(\phi^{*},\pi^{*})\,.\end{split}$ (3.44)

The existence of the homotopy together with (3.41) tells us that the chain map
in (3.35) is a quasi-isomorphism, i.e. that $p_{0}$ and $p_{1}$ descend to an
isomorphism on cohomology, with the inverse being the descendants of $i_{0}$
and $i_{1}$. The bulk algebra and the boundary algebra are therefore
equivalent.

We already noted that the differential $\bar{\partial}$ can also be derived
from the action $\bar{S}=S\circ i_{0}$ due to the Hamilton-Jacobi equations.
This is due to the fact that $i=(i_{0},i_{1})$ is a cyclic $L_{\infty}$
morphism (since it is linear, it is trivially so) to the theory with
differential $c_{1}$ and symplectic form

$\bar{\omega}(\pi^{*},\varphi)=\omega(i_{0}(\varphi),i_{1}(\pi^{*}))=\int_{\partial
M}\text{d}A\,\pi^{*}(x)\varphi(x)\,.$ (3.45)

The boundary action $\bar{S}=S\circ i$ can then be written as

$\bar{S}(\varphi)=\frac{1}{2}\bar{\omega}(c_{1}(\varphi),\varphi)\,.$ (3.46)

###  3.4 Example: The Harmonic Oscillator

Let us illustrate the above machinery by the one-dimensional harmonic
oscillator defined on an interval $I=[t_{i},t_{f}]$ (for a closely related
homological perspective on the harmonic oscillator, see [21]). The action is
given by

$S[x]=\int_{I}\text{d}t\frac{1}{2}(\dot{x}^{2}-\omega^{2}x^{2})\,,$ (3.47)

and the differential reads

$\begin{split}b_{1}:C^{\infty}(I)&\rightarrow C^{\infty}(I)\oplus
C^{\infty}(\partial I)\ ,\\\
x&\mapsto(-\ddot{x}-\omega^{2}x,\dot{x}|_{\partial I})\,.\end{split}$ (3.48)

Note that $\partial I=\\{t_{i},t_{f}\\}$, so $C^{\infty}(\partial
I)\cong\mathbb{R}^{2}$, under which $x|_{\partial I}$ becomes
$(x(t_{i}),x(t_{f}))$. The extended symplectic structure is

$\omega((x^{*},\pi_{i}^{*},\pi_{f}^{*}),x)=\int_{I}\text{d}t\,x^{*}(t)x(t)+\pi_{f}^{*}x(t_{f})-\pi_{i}^{*}x(t_{i})\,,$
(3.49)

with which one correctly obtains

$S[x]=\frac{1}{2}\omega(b_{1}(x),x)\,.$ (3.50)

We will now give the bulk-to-bulk propagator, from which we will derive the
boundary-to-bulk propagator using the general recipe we developed. Given a
function $t\mapsto x^{*}(t)$, the solution to
$L(x)=-\ddot{x}-\omega^{2}x=x^{*}$ with $x(t_{i})=x(t_{f})=0$ is

$x(t)=\int_{I}(K(t,s)+K(s,t))x^{*}(s)\ ,$ (3.51)

where

$K(t,s)=-\theta(t-s)\frac{\sin(s-t_{i})\sin\omega(t-t_{f})}{\sin\omega(t_{f}-t_{i})}\
.$ (3.52)

The fact that the integral kernel in (3.51) is symmetric in $t$ and $s$
implies that $h$ is symmetric with respect to the bulk symplectic structure

$\omega_{I}(x^{*},x)=\int_{I}\text{d}t\,x^{*}(t)x(t)\ .$ (3.53)

The inclusion map $i$ was constructed using the Hamiltonian symplectic form
$\omega_{\partial}$, which derives from $\omega_{I}$ as the failure of $L$
being cyclic. We have

$\omega_{I}(Lx_{1},x_{2})+\omega_{I}(x_{1},Lx_{2})=(x_{1}\dot{x}_{2}-\dot{x}_{1}x_{2})|^{t_{f}}_{t_{i}}=\omega_{\partial}(\pi(x_{1}),\pi(x_{2}))\
,$ (3.54)

where

$\omega_{\partial}((x_{i},p_{i},x_{f},p_{f}),(y_{i},y_{f},q_{i},q_{f}))=x_{f}q_{f}-p_{f}y_{f}-x_{i}q_{i}+p_{i}y_{i}$
(3.55)

and

$\begin{split}\pi:C^{\infty}(I)&\rightarrow C^{\infty}(\partial I)\oplus
C^{\infty}(\partial I)\cong\mathbb{R}^{4}\ ,\\\
x&\mapsto(x(t_{f}),\dot{x}(t_{f}),x(t_{i}),\dot{x}(t_{i}))\ .\end{split}$
(3.56)

The inclusion $\iota$ is then defined by

$\omega_{I}(x^{*},\iota(x_{i},p_{i},x_{f},p_{f}))=\omega_{\partial}(\pi
h(x^{*}),(x_{i},p_{i},x_{f},p_{f}))\ .$ (3.57)

On the right hand side we have

$\pi
h(x^{*})=(0,-\int_{I}\text{d}s\frac{\sin\omega(s-t_{i})}{\sin\omega(t_{f}-t_{i})}x^{*}(s),0,-\int_{I}\text{d}s\frac{\sin\omega(s-t_{f})}{\sin\omega(t_{f}-t_{i})}x^{*}(s))\,,$
(3.58)

and hence

$\begin{split}\omega_{\partial}(\pi
h(x^{*}),(x_{i},p_{i},x_{f},p_{f}))=&\int_{I}\text{d}s\,x^{*}(s)\frac{\sin\omega(s-t_{i})}{\sin\omega(t_{f}-t_{i})}x_{f}\\\
&+\int_{I}\text{d}s\,x^{*}(s)\frac{\sin\omega(t_{f}-s)}{\sin\omega(t_{f}-t_{i})}x_{i}\
.\end{split}$ (3.59)

From this we deduce that

$\iota(x_{i},p_{i},x_{f},p_{f})(t)=x_{f}\frac{\sin\omega(t-t_{i})}{\sin\omega(t_{f}-t_{i})}+x_{i}\frac{\sin\omega(t_{f}-t)}{\sin\omega(t_{f}-t_{i})}\
.$ (3.60)

As expected, this map is independent of the momenta $(p_{i},p_{f})$. In the
general discussion this observation suggested the definition of the improved
inclusion map

$i(x_{i},x_{f})=\iota(x_{i},0,x_{f},0)\ .$ (3.61)

In the other direction, we have the projector $p:x\mapsto x|_{\partial
I}=(x(t_{i}),x(t_{f}))$. It satisfies $pi=1$, which means that
$i(x_{i},x_{f})$ is indeed the solution to $Lx=0$ with boundary conditions
$x(t_{i})=x_{i}$ and $x(t_{f})=x_{f}$. It is also straightforward to check
that $1-ip=hL$.

The extended homotopy data that includes the boundary part is easy to read off
from our general discussion. We have

$\displaystyle p_{0}(x)=(x(t_{i}),x(t_{f}))$ $\displaystyle\ ,$ (3.62)
$\displaystyle
p_{1}(x^{*},\pi_{i}^{*},\pi_{f}^{*})=(\pi_{i}^{*}+\int_{I}\text{d}s\frac{\sin\omega(s-t_{i})}{\sin\omega(t_{f}-t_{i})}x^{*}(s),\pi_{f}^{*}+\int_{I}\text{d}s\frac{\sin\omega(s-t_{f})}{\sin\omega(t_{f}-t_{i})}x^{*}(s))$
$\displaystyle\ ,$ (3.63) $\displaystyle i_{0}(x_{i},x_{f})=i(x_{i},x_{f})\
,\quad i_{1}(\pi_{i}^{*},\pi_{f}^{*})=(0,\pi_{i}^{*},\pi_{f}^{*})\ .$ (3.64)

The boundary action then is

$S_{\partial}[x_{i},x_{f}]=\frac{\omega(x_{f}^{2}+x_{i}^{2})}{2}\frac{\cos\omega(t_{f}-t_{i})}{\sin\omega(t_{f}-t_{i})}-\frac{\omega
x_{i}x_{f}}{\sin\omega(t_{f}-t_{i})}\ .$ (3.65)

Just to make sure that this is correct, we compute $\partial_{x_{i}}S$ and
$\partial_{x_{f}}S$ to check the Hamilton-Jacobi equations:

$\begin{split}\partial_{x_{f}}S&=\frac{\omega
x_{f}\cos\omega(t_{f}-t_{i})-\omega
x_{i}}{\sin\omega(t_{f}-t_{i})}=\dot{x}(t_{f})\ ,\\\
\partial_{x_{i}}S&=\frac{\omega x_{i}\cos\omega(t_{f}-t_{i})-\omega
x_{f}}{\sin\omega(t_{f}-t_{i})}=-\dot{x}(t_{i})\ ,\end{split}$ (3.66)

which are indeed satisfied.

##  4 Homotopy Retract for Gauge Theories

In this section we apply the ideas we developed above for the scalar field to
Yang-Mills theory. To this end we review the Hodge decomposition for manifolds
with boundary, which is more involved compared to the familiar Hodge
decompositions of manifolds without boundary.

###  4.1 Hodge Decomposition on Manifolds with Boundary

The following review is based on [40]. Let $M$ be a $d$-dimensional compact
Riemannian manifold with metric $g$. We denote by $\Omega^{k}(M)$ the space of
$k$-forms on $M$. The metric allows us to define a Hodge star operator
$\star:\Omega^{k}(M)\rightarrow\Omega^{d-k}(M)$, satisfying
$\star^{2}=(-)^{k(d-k)}$ on $k$-forms, which defines an inner product on
$k$-forms for any $k=0,\ldots,d$:

$\begin{split}\Omega^{k}(M)\otimes\Omega^{k}(M)\longrightarrow\mathbb{R}\,,\qquad(a,b):=\int_{M}a\wedge\star
b\,.\end{split}$ (4.1)

Let $\text{d}:\Omega^{k}(M)\rightarrow\Omega^{k+1}(M)$ be the de-Rham
differential. We define its adjoint via
$\delta=-(-)^{n(n-k)}\star\text{d}\,\star$, with $k$ the degree of the form
$\delta$ acts on. This is used to define the Laplacian
$\Delta=\text{d}\delta+\delta\text{d}$. When $M$ does not have a boundary,
$\delta$ is the adjoint of d with respect to the inner product $(-,-)$, but in
presence of a boundary there are corrections in the form of boundary terms. In
order to determine these boundary terms let us introduce, for a given manifold
with boundary, a vector field $N$ normal to the boundary, which we normalize
such that $g(N,N)|_{\partial M}=1$. We denote by $\Gamma(TM)|_{\partial M}$
the space of sections of the tangent bundle $TM$ restricted to the boundary.
Note that this is not $\Gamma(T\partial M)$, since sections in
$\Gamma(TM)|_{\partial M}$ may not be tangent to $\partial M$. The normal
field $N$ then defines a splitting

$\Gamma(TM)|_{\partial M}=\Gamma(TM)|_{\partial
M}^{\parallel}\oplus\Gamma(TM)|_{\partial M}^{\perp}\,.$ (4.2)

Here, sections in $\Gamma(TM)|_{\partial M}^{\parallel}$ are those
proportional to $N$, while the sections in $\Gamma(TM)|_{\partial M}^{\perp}$
include those orthogonal to $N$. We have $\Gamma(TM)|_{\partial
M}^{\perp}\cong\Gamma(T\partial M)$.

Given a $k$-form $\alpha$, we write $\mathbf{t}\alpha$ for its restriction to
$\Gamma(TM)|_{\partial M}^{\perp}$. Since the direct sum decomposition allows
us to define a projector $\Gamma(TM)|_{\partial
M}\rightarrow\Gamma(TM)|_{\partial M}^{\perp}$, we can think of $\mathbf{t}$
as an endomorphism on $\Gamma(TM)|_{\partial M}$. This in turn allows us to
define $\mathbf{n}\alpha=\alpha|_{\Gamma(M)|_{\partial M}}-\mathbf{t}\alpha$.
We then write $\Omega_{D}^{k}(M)=\ker\mathbf{t}$ and
$\Omega_{N}^{k}(M)=\ker\mathbf{n}$, where the subscripts stand for Dirichlet
and Neumann boundary conditions. To make contact with the previous section,
note that the scalar fields in $\Omega_{D}^{0}(M)$ are exactly the scalar
fields vanishing on the boundary.

We can use the maps $\mathbf{t}$ and $\mathbf{n}$ to give a version of Green’s
theorem for forms. The failure of $\delta$ to be the adjoint of d is a
boundary term,

$(\text{d}\alpha,\beta)-(\alpha,\delta\beta)=\int_{\partial
M}\mathbf{t}\alpha\wedge\star\mathbf{n}\beta\ .$ (4.3)

The failure obviously vanishes if either $\alpha$ satisfies Dirichlet or
$\beta$ satisfies Neumann boundary conditions.

We now come to the Hodge decomposition in the presence of a boundary. We
define the space of harmonic $k$-forms to be
$\mathcal{H}^{k}:=\ker\text{d}|_{\Omega^{k}(M)}\cap\ker\delta|_{\Omega^{k}(M)}$.
We then have a decomposition (theorem 2.4.2 in [40])

$\Omega^{k}(M)=\text{d}\Omega^{k-1}_{D}(M)\oplus\delta\Omega^{k+1}_{N}(M)\oplus\mathcal{H}^{k}(M)\
.$ (4.4)

This decomposition is orthogonal with respect to $(-,-)$. Also, d
(respectively $\delta$) preserves Dirichlet (respectively Neumann) boundary
conditions, so forms in $\text{d}\Omega^{k-1}_{D}(M)$ (respectively in
$\delta\Omega^{k+1}_{N}(M)$) satisfy the same boundary conditions.

Following theorem 2.4.8 in [40], the decomposition can be further improved in
two distinct ways. We define
$\mathcal{H}^{k}_{ex}(M):=\text{d}\Omega^{k-1}(M)\cap\mathcal{H}^{k}(M)$ and
$\mathcal{H}^{k}_{coex}(M):=\delta\Omega^{k+1}(M)\cap\mathcal{H}^{k}(M)$. We
further denote by $\mathcal{H}_{D}^{k}(M)$ and $\mathcal{H}_{N}^{k}(M)$ the
spaces of harmonic forms with boundary conditions indicated by the subscripts.
We then have

$\mathcal{H}^{k}(M)=\mathcal{H}_{D}^{k}(M)\oplus\mathcal{H}^{k}_{coex}(M)=\mathcal{H}_{N}^{k}(M)\oplus\mathcal{H}^{k}_{ex}(M)\
.$ (4.5)

Both decompositions are again orthogonal with respect to $(-,-)$.

We end the discussion on the Hodge decomposition by giving an interpretation
in terms of de-Rham cohomology. This will later help to understand the form of
topological solutions in Yang-Mills theory. However, it is not necessary to
follow the rest of the paper.

Without a boundary, the de-Rham cohomology can be identified with the harmonic
forms. This is no longer true here. To compute the cohomology, we decompose a
given $k$-form $\alpha$ using (4.4). We write

$\alpha=\text{d}a+\delta b+c\ ,$ (4.6)

where $a$ satisfies Dirichlet boundary conditions, $b$ satisfies Neumann
boundary conditions and $c$ is harmonic. We find that $\text{d}\alpha=0$ if
and only if $\delta b=0$, since $\text{d}\alpha=\text{d}\delta b$ and

$(\text{d}\delta b,b)=(\delta b,\delta b)\geq 0\ .$ (4.7)

Here we used Green’s theorem, where the boundary term is zero since $b$
satisfies Neumann boundary conditions. The inner product $(-,-)$ is positive
definite, so $\text{d}\alpha=\text{d}\delta b=0$ implies $\delta b=0$. Hence,
any closed form is given by

$\alpha=\text{d}a+c\ .$ (4.8)

We further write
$c=e+\text{d}f\in\mathcal{H}^{k}_{N}(M)\oplus\mathcal{H}_{ex}^{k}(M)$. Since
this decomposition is a direct sum, $e$ cannot be exact unless it is zero.
Therefore $\alpha=e$ up to exact elements. So we can identify the $k$th de-
Rham cohomology with $\mathcal{H}^{k}_{N}(M)$.

The space $\mathcal{H}_{D}^{k}(M)$ has a similar interpretation. Since the
pullback of any function commutes with the exterior derivative, we can
restrict d to the kernel $\ker f^{*}$ of any smooth map $f$. In particular, we
can restrict it to $\ker j^{*}$, where $j:\partial M\rightarrow M$ is the
boundary inclusion. As noted before, we have $\Omega_{D}^{\bullet}(M)=\ker
j^{*}$. The cohomology of $\ker j^{*}$ is often denoted by
$H^{\bullet}(M,\partial M)$ and called the relative de-Rham cohomology with
respect to $\partial M$. We will show that $H^{k}(M,\partial
M)\cong\mathcal{H}_{D}^{k}(M)$. For any $k$-form $\alpha$, we again use the
de-Rham composition

$\alpha=\text{d}a+\delta b+c$ (4.9)

as before and further decompose $c=e+\delta
f\in\mathcal{H}_{D}^{k}(M)\oplus\mathcal{H}_{coex}^{k}(M)$. We already learned
that $\text{d}\alpha=0$ implies $\delta b=0$. We now assume that $\alpha$
further satisfies Dirichlet boundary conditions. $\text{d}a$ and $e$ satisfy
this condition by assumption. On the other hand, $\delta f$ cannot satisfy
Dirichlet boundary conditions unless it is zero, since
$\mathcal{H}_{D}^{k}(M)\cap\mathcal{H}_{coex}^{k}(M)=\\{0\\}$ due to the fact
that the decomposition of $c$ is a direct sum. This proves that a closed form
$\alpha$ satisfying Dirichlet boundary conditions can be written as
$\alpha=e\in\mathcal{H}^{k}_{D}(M)$ up to exact terms. This proves the claim
$H^{k}(M,\partial M)\cong\mathcal{H}_{D}^{k}(M)$.

###  4.2 Yang-Mills Theory as a cyclic $L_{\infty}$ Algebra

Cyclic $L_{\infty}$ Algebra of Yang-Mills Theory without a Boundary

We now describe Yang-Mills as an $L_{\infty}$ algebra. We fix an
$n$-dimensional manifold $M$. Yang-Mills fields are then given by Lie algebra
valued one-forms on $M$, at least for the sake of perturbation theory. Under
the same assumption, the gauge field $A$ globally splits into a form component
and a Lie algebra component, i.e. $A\in\Omega^{1}(M)\otimes\mathfrak{g}$ for
some fixed Lie algebra $\mathfrak{g}$.

The curvature of the gauge field $A\in\Omega^{1}(M)\otimes\mathfrak{g}$ is
given by $F=\text{d}A+\frac{1}{2}[A,A]$, where $[-,-]$ combines the wedge
product of forms with the Lie bracket of the Lie algebra. The Yang-Mills
action reads

$S_{0}[A]=\frac{1}{2}\int_{M}{\rm tr}(F\wedge\star F)\,,$ (4.10)

and its Euler-Lagrange equations are given by

$0=\text{d}A+\frac{1}{2}\text{d}\star[A,A]+[A,\star\text{d}A]+\frac{1}{2}[A,\star[A,A]]\
.$ (4.11)

The chain complex describing the free part of Yang-Mills theory is given by

${\Omega^{0}(M)\otimes\mathfrak{g}}$${\Omega^{1}(M)\otimes\mathfrak{g}}$${\Omega^{n-1}(M)\otimes\mathfrak{g}}$${\Omega^{n}(M)\otimes\mathfrak{g}\
.}$d$\scriptstyle{\text{d}\star\text{d}}$d (4.12)

Here d is the usual de-Rham differential acting trivially on the Lie algebra
part. Since we want the physical fields to live in degree zero, the complex
starts in degree $-1$ and ends in degree $2$. As usual, we denote the
differential collectively by $b_{1}$. From now on, we will write
$\Omega^{\bullet}(M)$ for $\Omega^{\bullet}(M)\otimes\mathfrak{g}$.

Let us find the $L_{\infty}$ products. Below, any product not explicitly given
is zero. The $L_{\infty}$ products should be graded symmetric and such that

$b_{1}(A)+\frac{1}{2}b_{2}(A,A)+\frac{1}{6}b_{3}(A,A,A)=0$ (4.13)

is the equation of motion. The products on fields are then

$b_{2}(A_{1},A_{2})=\text{d}\star[A_{1},A_{2}]+[A_{1},\star\text{d}A_{2}]+[A_{2},\star\text{d}A_{1}]\
,$ (4.14)

and

$b_{3}(A_{1},A_{2},A_{3})=[A_{1},\star[A_{2},A_{3}]]+[A_{2},\star[A_{3},A_{1}]]+[A_{3},\star[A_{1},A_{2}]]\
.$ (4.15)

The products on gauge parameters $c\in\Omega^{0}(M)$ are such that

$\delta_{c}A=b_{1}(c)+b_{2}(c,A)$ (4.16)

are the gauge transformations of $A$. Since these are given by $\delta
A=\text{d}c+[A,c]$, we need to define

$b_{2}(c,A)=b_{2}(A,c)=-[c,A]$ (4.17)

for any gauge parameter $c$ and any gauge field $A$. For this bracket to
satisfy the Jacobi identity, we further need

$b_{2}(c_{1},c_{2})=-[c_{1},c_{2}]$ (4.18)

for any two gauge parameters $c_{1},c_{2}$. Gauge invariance of the theory is
ensured by gauge covariance of the equations of motion. Let
$A^{*}\in\Omega^{d-1}(M)$, i.e. $A^{*}$ lives in the space of equations of
motion. The gauge transformations of $A^{*}$ are encoded in

$b_{2}(c,A^{*})=-b_{2}(A^{*},c)=-[c,A^{*}]\ .$ (4.19)

One can show that the products defined above satisfy the $L_{\infty}$
relations if we ignore the space $\Omega^{d}(M)\otimes\mathfrak{g}$ in degree
2. This space is needed, however, in order to later define a cyclic structure.
We call the vector space $\Omega^{d}(M)\otimes\mathfrak{g}$ the space of
Noether identities, and denote its elements or fields by $c^{*}$. We can then
extend the products via

$b_{2}(A,A^{*})=b_{2}(A^{*},A)=[A,A^{*}]$ (4.20)

for any gauge field $A$ and any equation of motion $A^{*}$, as well as

$b_{2}(c,c^{*})=b_{2}(c^{*},c)=-[c,c^{*}]$ (4.21)

for any gauge parameter $c$ and any Noether identity $c^{*}$. These products
reflect the fact that

$\text{D}^{2}_{A}\star F=0\ .$ (4.22)

Without a boundary, the cyclic structure of Yang-Mills theory can be defined
by

$\omega(c,c^{*})=-\omega(c^{*},c)=\int_{M}c\wedge c^{*}\
,\qquad\omega(A^{*},A)=-\omega(A,A^{*})=\int_{M}A\wedge A^{*}\ .$ (4.23)

Recall from 2.1 that we can extend the classical action $S_{0}$ to an action
$S$, defined as a multilinear function

$S=\sum_{n\geq 1}\omega(b_{n}\otimes 1)\ .$ (4.24)

In the case of Yang-Mills theory, we find

$\begin{split}S&=\int_{M}\frac{1}{2}A\wedge\Big{(}\text{d}\star\text{d}A+\frac{1}{2}\text{d}\star[A,A]+[A,\star\text{d}A]+\frac{1}{4}[A,[A,A]]\Big{)}\\\
&\quad-\int_{M}(\text{d}c+[c,A])\wedge A^{*}+\frac{1}{2}c^{*}\wedge[c,c]\\\
&=\frac{1}{2}\int_{M}F\wedge\star F+A^{*}\wedge
D_{A}c+\frac{1}{2}c^{*}\wedge[c,c]\ .\end{split}$ (4.25)

This is the BV extended action of Yang-Mills theory. In our case, the above
expression is understood as a multilinear map as follows. The variables
$\\{c^{*},A^{*},A,c\\}$ are not the fields of the complex, but rather define
projection maps to the subspaces the fields live in. For example

$c^{*}:(\Omega^{0}(M)\oplus\Omega^{1}(M)\oplus\Omega^{d-1}(M)\oplus\Omega^{d}(M))\rightarrow\Omega^{d}(M)\
.$ (4.26)

A wedge product of $n$ field variables
$\alpha_{1},...,\alpha_{n}\in\\{c^{*},A^{*},A,c\\}$ on $n$ fields
$\mathcal{A}_{1},...,\mathcal{A}_{n}$ in the complex is then

$(\alpha_{1}\wedge...\wedge\alpha_{n})(\mathcal{A}_{1},...,\mathcal{A}_{n})=\sum_{\sigma\in\mathcal{S}_{n}}\pm\alpha_{1}(\mathcal{A}_{\sigma(1)})\wedge...\wedge\alpha_{n}(\mathcal{A}_{\sigma(n)})\
.$ (4.27)

The wedge product on the right is the wedge product on forms combined with the
product of Lie algebra elements. The sign is the natural one coming from the
degree of the fields (which, we recall, in general is not the form
degree).444Another way of thinking of the wedge product in the action is the
following. Suppose we have multilinear maps
$f,g:S^{c}(V^{\bullet})\rightarrow\Omega^{\bullet}(M)$, where $V^{\bullet}$ is
the underlying complex of the theory. The domain $S^{c}(V)$ has a coproduct
$\Delta$, while the image $\Omega^{\bullet}(M)$ has a product $\nabla$. We can
then define an associative product $f\wedge g:=\nabla(f\otimes g)\Delta$ on
the space of functions from $S^{c}(V)$ to $\Omega^{\bullet}(M)$. It is
straightforward to see that this agrees with the definition given in the text
and also makes it manifest that the product is associative. With this it is
straightforward to check that (4.25) is exactly $S=\sum_{n\geq
1}\omega(b_{n}\otimes 1)$.

Cyclic $L_{\infty}$ Algebra in Presence of Boundary
When the manifold has a boundary, cyclicity of the differential $b_{1}$ with
respect to $\omega$ fails due to boundary terms, just like in the case of the
scalar field. We find

$\omega(b_{1}A_{1},A_{2})+\omega(A_{1},b_{1}A_{2})=\int_{\partial
M}A_{1}\wedge\star\text{d}A_{2}-\int_{\partial
M}A_{2}\wedge\star\text{d}A_{1}=\omega_{\partial}(A_{1},A_{2})\ .$ (4.28)

The failure defines a symplectic structure, pairing the gauge field $A$ with
its canonical momentum $\star\text{d}A$. There is further an induced
symplectic structure due to the gauge symmetry:

$\omega(b_{1}c,A^{*})-\omega(c,b_{1}A^{*})=-\int_{\partial M}c\wedge
A^{*}=\omega_{\partial}(c,A^{*})\ .$ (4.29)

To define the symplectic structure independently of the bulk data, we
introduce the following spaces,

$X^{-1}=\Omega^{0}(\partial M)\ ,\qquad X^{0}=\Omega^{1}(\partial
M)\times\Omega^{n-2}(\partial M)\ ,\qquad X^{1}=\Omega^{n-1}(\partial M)\ .$
(4.30)

We can then define the symplectic structure using the obvious pairing.
Further, there exists a projection from the bulk data to the boundary data via

$\begin{split}\pi:V^{\bullet}&\rightarrow X^{\bullet}=\Omega^{0}(\partial
M)\oplus\Omega^{1}(\partial M)\times\Omega^{n-2}(\partial
M)\oplus\Omega^{n-1}(\partial M)\;,\\\
(c,A,A^{*},c^{*})&\mapsto(i^{*}c,i^{*}A,i^{*}\star\text{d}A,i^{*}A^{*})\;.\end{split}$
(4.31)

Here, $i:\partial M\rightarrow M$ is the inclusion and $i^{*}$ its pullback.
The failure of $b_{1}$ being cyclic can now be written as

$\omega(b_{1}\mathcal{A},\mathcal{B})+(-)^{\mathcal{A}}\omega(\mathcal{A},b_{1}\mathcal{B})=\omega_{\partial}(\pi\mathcal{A},\pi\mathcal{B})\
,$ (4.32)

which, like in the scalar field case, is the $L_{\infty}$ version of the BV-
BFV relation [22].

To ensure cyclicity of the theory, we introduce the following complex

${\Omega^{0}(M)}$${\Omega^{1}(M)}$${\Omega^{n-1}(M)\oplus\Omega^{n-2}(\partial
M)}$${\Omega^{n}(M)\oplus\Omega^{n-1}(\partial M)\
,}$$\scriptstyle{b_{1}}$$\scriptstyle{b_{1}}$$\scriptstyle{b_{1}}$ (4.33)

where now

$b_{1}(c)=\text{d}c\ ,\qquad
b_{1}(A)=(\text{d}\star\text{d}A,i^{*}\star\text{d}A)\ ,\qquad
b_{1}(A^{*},B^{*})=(\text{d}A^{*},i^{*}A^{*}-\text{d}B^{*})\ .$ (4.34)

We also extend the inner product via

$\begin{split}\omega(c,(c^{*},\eta^{*}))&=\int_{M}c\wedge c^{*}-\int_{\partial
M}i^{*}c\wedge\eta^{*}\;,\\\ \omega((A^{*},B^{*}),A)&=\int_{M}A\wedge
A^{*}+\int_{\partial M}i^{*}A\wedge B^{*}\ ,\end{split}$ (4.35)

which makes the free theory cyclic.

Since the cubic interaction $b_{2}$ has derivatives, it picks up boundary
terms. To ensure cyclicity, we need to extend it as

$\displaystyle
b_{2}^{\prime}(A_{1},A_{2})=(b_{2}(A_{1},A_{2}),i^{*}\star[A_{1},A_{2}])\ ,$
$\displaystyle
b_{2}^{\prime}(A,(A^{*},B^{*}))=(b_{2}(A,A^{*}),-[i^{*}A,B^{*}])\ ,$ (4.36)
$\displaystyle b_{2}^{\prime}((A^{*},B^{*}),c)=(-[A^{*},c],-[B^{*},i^{*}c])\
,$ $\displaystyle
b_{2}^{\prime}((c^{*},\eta^{*}),c)=(b_{2}(c^{*},c),[\eta^{*},i^{*}c])\ .$
(4.37)

For the three-bracket, it turns out that $b_{3}$ does not need any
corrections, i.e. it is still non-zero on fields only and it extends from
$\Omega^{n-1}(M)$ to $\Omega^{n-1}(M)\times\Omega^{n-2}(\partial M)$ by zero:

$b_{3}^{\prime}(A_{1},A_{2},A_{3})=(b_{3}(A_{1},A_{2},A_{3}),0)\ .$ (4.38)

###  4.3 A Homotopy for Yang-Mills Theory

We will now give a homotopy, first for the non-cyclic variant of Yang-Mills
theory. We give a more in-depth derivation of the homotopy in appendix A.2. As
we did with the scalar field, we then extend it to the cyclic theory.

We first give the propagator, i.e. the homotopy on equations of motion
$A^{*}\in\Omega^{n-1}(M)$. Let
$P_{1}:\Omega^{n-1}(M)\rightarrow\mathcal{H}_{N}^{n-1}$ be the orthogonal
projection to $\mathcal{H}_{N}^{n-1}$ and
$I_{1}:\mathcal{H}_{N}^{n-1}\rightarrow\Omega^{n-1}(M)$ be the canonical
inclusion. Then, let $G_{1}$ be the unique Green’s function solving

$\Delta G_{1}(A^{*})=\star(1-I_{1}P_{1})A^{*}\;,$ (4.39)

such that $G_{1}(A^{*})\in(\mathcal{H}_{D}^{1}(M))^{\perp}$. We also introduce
the orthogonal projection
$P_{2}:\Omega^{n-1}(M)\rightarrow\delta\Omega^{n}_{N}(M)$ and the canonical
inclusion $I_{2}:\delta\Omega^{n}\rightarrow\Omega^{n-1}(M)$. We define the
homotopy on $A^{*}$ to be

$h_{1}(A^{*})=G_{1}(1-I_{2}P_{2})(A^{*})\;.$ (4.40)

We show in the appendix that $h$ satisfies Lorenz gauge, i.e. $\delta h=0$.

Let $G_{0}:\Omega^{0}(M)\rightarrow\Omega^{0}_{D}(M)$ be the scalar Green’s
function we are familiar with from our discussion of the scalar field. On
gauge fields $A$, we use it to define the homotopy

$h_{0}(A)=G_{0}\delta A\,.$ (4.41)

We then find

$h_{0}b_{1}c=G_{0}\Delta c=(1-i_{-1}p_{-1})(c)\;,$ (4.42)

where $p_{-1}=i^{*}$, i.e. it restricts the gauge parameter to the boundary
and $i_{-1}$ computes the harmonic $c$ with given boundary value.

With the homotopy on fields and equations of motion at hand, we can now
compute the commutator $\\{b_{1},h\\}$. We have

$(1-i_{0}p_{0})(A)=(b_{1}h_{0}+h_{1}b_{1})(A)\;,$ (4.43)

where
$p_{0}:\Omega^{1}(M)\rightarrow\mathcal{H}_{D}^{1}(M)\oplus\Omega^{1}(\partial
M)$ is the projection defined by $p_{0}(A)=(P_{3}(A),i^{*}A)$, with
$P_{3}:\Omega^{1}(M)\rightarrow\mathcal{H}_{D}^{1}(M)$ the orthogonal
projection. In the other direction, let $u:\Omega^{1}(\partial
M)\rightarrow\Omega^{1}(M)$ be the map that associates to a one form
$a_{0}\in\Omega^{1}(\partial M)$ the unique solution
$u(\alpha)\in\Omega^{1}(M)$ satisfying $\Delta u(a_{0})=0$, such that
$i^{*}r(a_{0})=a_{0},i^{*}\delta
u(a_{0})=0,u(a_{0})\in(\mathcal{H}_{D}^{1})^{\perp}(M)$. Further, let
$I_{3}:\mathcal{H}^{1}_{D}(M)\rightarrow\Omega^{1}(M)$ be the canonical
inclusion. We then define
$i_{0}:\mathcal{H}^{1}_{D}(M)\oplus\Omega^{1}(\partial
M)\rightarrow\Omega^{1}(M)$ as the sum $i_{0}=I_{3}+r$.

The last piece of the homotopy is given by
$h_{2}:\Omega^{n}(M)\rightarrow\Omega^{n-1}(M)$, where

$h_{2}=(-)^{n}\delta\star G_{0}\star\,.$ (4.44)

One can show that $b_{1}h_{2}=1$. This is consistent with the fact that any
connected compact manifold with boundary has no cohomology in top degree. On
the other hand, $h_{2}b_{1}+b_{1}h_{1}=1-i_{1}p_{1}$, where $p_{1}=P_{1}$ and
$i_{1}=I_{1}$.

The homotopy $h_{\bullet}$, together with the chain maps $i_{\bullet}$ and
$b_{\bullet}$ give rise to a homotopy

${X^{\bullet}:0}$${\Omega^{0}(M)}$${\Omega^{1}(M)}$${\Omega^{n-1}(M)}$${\Omega^{n}(M)}$${0}$${Y^{\bullet}:0}$${\Omega^{0}(\partial
M)}$${\mathcal{H}_{D}^{1}(M)\oplus\Omega^{1}(\partial
M)}$${\mathcal{H}_{N}^{n-1}(M)}$${0}$${0\,.}$d$\scriptstyle{p_{-1}}$$\scriptstyle{\text{d}\star\text{d}}$$\scriptstyle{p_{0}}$d$\scriptstyle{p_{1}}$$\scriptstyle{0}$$\scriptstyle{\text{(}0{,}\text{d})}$$\scriptstyle{0}$$\scriptstyle{0}$
(4.45)

As in the scalar field case, the complex $Y^{\bullet}$ of the boundary theory
is not symmetric, in the sense that $Y^{i}$ is not isomorphic to $Y^{1-i}$.
This just reflects the fact that the theory is not cyclic. We will now extend
the homotopy to the cyclic complex, which we introduced in the previous
section.

Recall that in order to obtain a cyclic theory, we extended both the space of
equations of motion and the space of Noether identities in the following way:

$\Omega^{n-1}(M)\mapsto\Omega^{n-1}(M)\times\Omega^{n-2}(\partial M)\
,\qquad\Omega^{n}(M)\mapsto\Omega^{n}(M)\times\Omega^{n-1}(\partial M)\ .$
(4.46)

For the boundary complex, we make the ansatz

${\Omega^{0}(\partial M)}$${\Omega^{1}(\partial
M)\oplus\mathcal{H}_{D}^{1}(M)}$${\Omega^{n-2}(\partial
M)\oplus\mathcal{H}_{N}^{n-1}(M)}$${\Omega^{n-1}(\partial M)\
.}$$\scriptstyle{c_{1}}$$\scriptstyle{c_{1}}$$\scriptstyle{c_{1}}$ (4.47)

We denote the boundary field by $\eta\in\Omega^{0}(\partial M)$,
$(a_{0},\mathcal{A})\in\Omega^{1}(\partial M)\times\mathcal{H}_{D}^{1}(M)$,
$(B^{*},\mathcal{A}^{*})$ and $\eta^{*}\in\Omega^{d-1}(\partial M)$. Here, we
define

$c_{1}(c)=(\text{d}c,0)\ ,\qquad
c_{1}(a_{0},\mathcal{A})=(i^{*}\star\text{d}i_{0}(a_{0},\mathcal{A}),0)\
,\qquad
c_{1}(B^{*},\mathcal{A}^{*})=\text{d}B^{*}-i^{*}i_{1}(\mathcal{A}^{*})\ .$
(4.48)

Note that, among other reasons, $c_{1}^{2}=0$ follows from
$\text{d}\star\text{d}i_{0}(B,\mathcal{A})=0$, since the image of $i_{0}$ are
solutions to the equations of motion. Also, we have been particularly pedantic
when using the inclusion
$i_{1}:\mathcal{H}_{N}^{n-1}(M)\rightarrow\Omega^{n-1}(M)$ before applying the
pullback $i^{*}$.

As an ansatz for the homotopy, we will extend $h_{1}$ trivially to a map

$h_{1}^{\prime}:\Omega^{n-1}(M)\oplus\Omega^{n-2}(\partial
M)\longrightarrow\Omega^{1}(M)\ ,$ (4.49)

i.e. we set $h_{1}^{\prime}(A^{*},B^{*})=h_{1}(A)$, where on the right hand
side we denote by $h_{1}$ the original homotopy. In this way, we do not affect
the action of the homotopy on the gauge fields. On the other hand, for an
equation of motion $(A^{*},B^{*})\in X^{1}$, we find

$b_{1}^{\prime}h_{1}^{\prime}(A^{*},B^{*})=(b_{1}h_{1}(A^{*}),i^{*}\star\text{d}h_{1}(A^{*}))\
.$ (4.50)

The second component is the canonical momentum of $h_{1}(A^{*})$. We also need
an extension $h_{2}^{\prime}$ of $h_{2}$, which can be obtained by defining

$\begin{split}h_{2}^{\prime}:\Omega^{n}(M)\times\Omega^{n-1}(\partial
M)&\longrightarrow\Omega^{n-1}(M)\times\Omega^{n-2}(\partial M)\\\
(c^{*},\eta^{*})&\longmapsto(h_{2}(c^{*}),0)\ .\end{split}$ (4.51)

We then find

$(b_{1}^{\prime}h_{1}^{\prime}+h_{2}^{\prime}b_{1}^{\prime})(A^{*},B^{*})=(A^{*},B^{*})-(i_{1}p_{1}(A^{*}),B^{*}-i^{*}\star\text{d}h_{1}(A^{*}))\
.$ (4.52)

Furthermore,

$b_{1}^{\prime}h_{2}^{\prime}(c^{*},\eta^{*})=(c^{*},\eta^{*})-(0,\eta^{*}-i^{*}h_{2}(c^{*}))\
.$ (4.53)

This suggests that we define the following projections:

$\begin{split}p_{1}^{\prime}:\Omega^{n-1}(M)\times\Omega^{n-2}(\partial
M)&\longrightarrow\Omega^{n-2}(\partial M)\times\mathcal{H}_{N}^{n-1}(M)\ ,\\\
(A^{*},B^{*})&\longmapsto(B^{*}-i^{*}\star\text{d}h_{1}(A^{*}),p_{1}(A^{*}))\
,\\\ p_{2}^{\prime}:\Omega^{n}(M)\times\Omega^{n-1}(\partial
M)&\longrightarrow\Omega^{n-2}(\partial M)\ ,\\\
(c^{*},\eta^{*})&\longmapsto\eta^{*}-i^{*}h_{2}(c^{*})\ .\end{split}$ (4.54)

We also set $p_{0}^{\prime}=p_{0}$ and $p_{-1}^{\prime}=p_{-1}$. We need to
check that these satisfy the chain map property, i.e.

$c_{1}p_{-1}^{\prime}=p_{0}^{\prime}b_{1}^{\prime}\ ,\qquad
c_{1}p_{0}^{\prime}=p_{1}^{\prime}b_{1}^{\prime}\ ,\qquad
c_{1}p_{1}^{\prime}=p_{2}^{\prime}b_{1}^{\prime}\ .$ (4.55)

From the previous section we know that the first equation holds, since the
maps involved did not receive a cyclic extension. Now let $A\in\Omega^{1}(M)$.
We then find

$\begin{split}p_{1}^{\prime}b_{1}^{\prime}(A)&=p_{1}^{\prime}(i^{*}\star\text{d}A,b_{1}A)=(i^{*}\star\text{d}A-i^{*}\star\text{d}h_{1}b_{1}(A),p_{1}b_{1}(A))\\\
&=(i^{*}\star\text{d}(1-h_{1}b_{1})(A),0)=(i^{*}\star\text{d}(b_{1}h_{0}+i_{0}p_{0})(A),0)=(i^{*}\star\text{d}i_{0}p_{0}(A),0)\\\
&=c_{1}p_{0}^{\prime}(A)\ .\end{split}$ (4.56)

This proves the first chain map relation. Here, we used the homotopy relation
$h_{1}b_{1}+b_{1}h_{0}=1-i_{0}p_{0}$ as well as the fact that $b_{1}$ on gauge
parameters $h_{0}(A)$ acts as $\text{d}h_{0}(A)$.

Next, we consider $(A^{*},B^{*})\in\Omega^{n-1}(M)\times\Omega^{n-2}(\partial
M)$ and compute

$\begin{split}c_{1}p_{1}^{\prime}(A^{*},B^{*})&=c_{1}(B^{*}-i^{*}\star\text{d}h_{1}(A^{*}),p_{1}(A^{*}))=\text{d}(B^{*})-\text{d}i^{*}\star\text{d}h_{1}(A^{*})-i^{*}i_{1}p_{1}(A^{*})\\\
&=\text{d}B^{*}-i^{*}(b_{1}h_{1}+i_{1}p_{1})(A^{*})=\text{d}B^{*}-i^{*}(1-h_{2}b_{1})(A^{*})\\\
&=p_{2}^{\prime}(\text{d}A^{*},B^{*}-i^{*}A^{*})=p_{2}^{\prime}b_{1}^{\prime}(A^{*},B^{*})\
.\end{split}$ (4.57)

This completes the prove that the $p_{j}^{\prime}$ combine into a chain map.

We also need the inclusion maps, which themselves should give rise to chain
maps. In positive degree, we define them via the trivial inclusions, which are

$\begin{split}i_{1}^{\prime}:\mathcal{H}_{N}^{n-1}(M)\times\Omega^{n-2}(\partial
M)&\rightarrow\Omega^{n-1}(M)\times\Omega^{n-2}(\partial M)\ ,\\\
i_{2}^{\prime}:\Omega^{n-1}(\partial
M)&\rightarrow\Omega^{n}(M)\times\Omega^{n-1}(\partial M)\ .\end{split}$
(4.58)

Here, $i_{1}$ identifies $\mathcal{H}_{N}^{n-1}(M)$ as a subspace of
$\Omega^{n-1}(M)$ and acts on $\Omega^{n-2}(\partial M)$ as the identity,
while $i_{2}$ is just the inclusion into the second factor.

We further define $i_{0}^{\prime}:=i_{0}$ and $i_{-1}^{\prime}:=i_{-1}$. To
prove the chain map property, we need to show that

$c_{1}i_{-1}^{\prime}=i_{0}^{\prime}b_{1}^{\prime}\ ,\qquad
i_{1}^{\prime}c_{1}=b_{1}^{\prime}i_{0}^{\prime}\ ,\qquad
i_{2}^{\prime}c_{1}=b_{1}^{\prime}i_{1}^{\prime}\ .$ (4.59)

Again, we proved the first equation already in the previous section, since all
the maps involved did not need a cyclic extension. We further have

$i_{1}^{\prime}c_{1}(\mathcal{A},B)=i_{1}^{\prime}(0,i^{*}\star\text{d}i_{0}B)=b_{1}i_{0}(\mathcal{A},B)\
,$ (4.60)

as well as

$i_{2}^{\prime}c_{1}(\mathcal{A}^{*},B^{*})=i_{2}^{\prime}(\text{d}B^{*}-i^{*}i_{1}\mathcal{A}^{*})=(0,\text{d}B^{*}-i^{*}i_{1}\mathcal{A}^{*})=b_{1}^{\prime}i_{1}^{\prime}(\mathcal{A}^{*},B^{*})\
.$ (4.61)

This proves that the $i_{j}^{\prime}$ combine into a chain map.

It is straightforward to see that
$1-i^{\prime}_{i}p^{\prime}_{i}=b_{1}^{\prime}h_{i}+h_{i+1}b_{1}^{\prime}$,
when we recall that we already computed the right hand side of that equation.
We also have $p_{i}i_{i}=1$, as well as $p_{i}h_{i+1}=0$ and $h_{i}i_{i}=0$.

##  5 Homotopy Retract to Boundary of Anti-de-Sitter

In this section we apply the above general framework of homotopy algebras for
field theories defined on manifolds with a boundary to AdS/CFT. This case
entails further subtleties as AdS does not have a boundary in the strict
topological sense, but only a ‘conformal boundary’ [3]. This in turn implies
that the projection operator involves a rescaling by the $z$ coordinate in the
Poincaré patch. We establish the homotopy retract for this case and show that
determining the homotopy transferred cyclic $L_{\infty}$ algebra, and hence
the on-shell action, by means of the homological perturbation lemma, amounts
to the familiar method of Witten diagrams, hence providing a homological
interpretation of the latter.

###  5.1 Homotopy Retract for AdS

We begin by collecting some generalities about Anti-de-Sitter (AdS) space,
which we here take to be of Euclidean signature. We work in the Poincaré
patch, for which the metric $g$ of $(d+1)$-dimensional Euclidean $AdS$ is
given by

$ds^{2}=\frac{1}{z^{2}}\Big{(}dz^{2}+\sum_{i=1}^{d}dx_{i}^{2}\Big{)}\;.$ (5.1)

In these coordinates the conformal boundary is given by $d$-dimensional flat
(Euclidean) space with metric $\sum_{i=1}^{d}dx_{i}^{2}$, which we can think
of as being obtained from (5.1) by taking the limit $z\to 0$. [For instance,
we can rescale $z\rightarrow\lambda z$ and then send $\lambda\rightarrow 0$,
for which the second term in (5.1) dominates over the first term.]

Next, we inspect a simple field theory defined on AdS, with a free scalar
field $\phi$, whose coordinate dependence we display, in line with (5.1), as
$\phi(x,z)$. The free action for a massive scalar then reads

$\begin{split}S&=\frac{1}{2}\int_{AdS}\sqrt{g}\left(g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+m^{2}\phi^{2}\right)\\\
&=\frac{1}{2}\int\text{d}^{d}x\text{d}z\left({z^{-d+1}}\left((\partial_{z}\phi)^{2}+(\partial_{i}\phi)^{2}\right)+{z^{-d-1}}m^{2}\phi^{2}\right)\;,\end{split}$
(5.2)

where the second line follows with (5.1). The equations of motion following
from this action are

$(-z^{2}\partial_{i}\partial_{i}+m^{2})\phi+(d-1)z\partial_{z}\phi-z^{2}\partial_{z}^{2}\phi=0\;,$
(5.3)

where $\partial_{i}\partial_{i}$ is the flat space (or boundary) Laplacian,
i.e., with indices contracted with the flat metric. In order to inspect some
simple solutions we make the ansatz $\phi=z^{\Delta}\phi_{0}$, where
$\phi_{0}$ and $\Delta$ are constant (the latter being referred to as the
conformal dimension). Insertion into (5.3) then yields a quadratic equation
for $\Delta$ whose solutions are
$\Delta_{\pm}=\frac{d}{2}\pm\sqrt{\frac{d^{2}}{4}+m^{2}}$, where $\Delta_{+}$
is positive and $\Delta_{-}$ is negative for $m>0$. For more general solutions
of (5.3) that depend on $x$ we impose the boundary condition that they
asymptote for $z\rightarrow 0$ the solution with $\Delta_{-}$, i.e.,

$\phi(x,z)=z^{\Delta_{-}}(\phi_{0}(x)+{\cal O}(z))\;.$ (5.4)

In other words, we parametrize solutions using $\phi_{0}(x)$. Note that we
cannot just evaluate these solutions at $z=0$ since, owing to $\Delta_{-}$
being negative, they become singular.

In order to now describe the free scalar field on AdS as a chain complex, we
define an $L_{\infty}$ algebra with a degree $0$ space of fields $X^{0}$. We
assume that these fields are of the form

$\begin{split}\phi(x,z)&=\phi^{-}(x)z^{\Delta_{-}}(1+a_{1}z+\cdots)+\phi^{+}(x)z^{\Delta_{+}}(1+b_{1}z+\cdots)\\\
&\equiv\phi^{-}(x,z)+\phi^{+}(x,z)\,\end{split}.$ (5.5)

This is the space of off-shell fields, for which we can define the following
projection to the boundary:

$p(\phi)(x):=\lim_{z\rightarrow 0}\,z^{-\Delta_{-}}\phi(x,z)\,.$ (5.6)

To make the action finite, one has to apply _holographic renormalization_ ,
see e.g. [23]. In this work, we will ignore this issue, since here it is only
important that the boundary term obtained when integrating by parts in the
kinetic term is given by

$\begin{split}\lim_{z\rightarrow
0}\int_{\mathbb{R}^{d}}\text{d}^{d}x\,z^{-d+1}\phi(x,z)\partial_{z}\phi(x,z)&=\lim_{z\rightarrow
0}\int_{\mathbb{R}^{d}}\text{d}^{d}x\,z^{-\Delta_{-}}\phi(x,z)z^{-\Delta_{+}}z\partial_{z}\phi(x,z)\\\
&=\int_{\mathbb{R}^{d}}\text{d}^{d}x\,\Delta_{+}\phi^{-}(x)\phi^{+}(x)\,.\end{split}$
(5.7)

The first line makes sense for finite values of $z$. In the limit
$z\rightarrow 0$, we then make the substitution

$z^{-\Delta_{+}}z\partial_{z}\phi(x,z)\mapsto
z^{-\Delta_{+}}z\partial_{z}\phi_{+}(x,z)\,,$ (5.8)

which can be justified using holographic renormalization.

The last line in (5.7) suggests that we define $\pi(x)=\Delta_{+}\phi^{+}(x)$
to be the canonical momentum on the boundary. Since
$\partial_{n}=z\partial_{z}$ is a vector field normal to the boundary and
satisfying $g(\partial_{n},\partial_{n})=1$, this is very similar to what
happens in the scalar field case on ordinary manifolds. Explicitly, the
canonical momentum is obtained as the limit

$\pi(x):=\lim_{z\rightarrow 0}z^{-\Delta_{+}}\partial_{n}\phi^{+}(x,z)\;.$
(5.9)

The space of equations of motion $X^{1}$ then consists of fields
$(\phi^{*},\pi^{*})$, where $\phi^{*}$ is a bulk field and $\pi^{*}$ is a
boundary field. We then have a chain complex

${0}$${X^{0}}$${X^{1}}$${0\,,}$$\scriptstyle{b_{1}}$ (5.10)

where

$b_{1}(\phi):=((-\Delta+m^{2})\phi\,,\;\Delta_{+}\phi^{+})\,.$ (5.11)

We define the cyclic structure as

$\omega((\phi^{*},\pi^{*}),\phi):=\int_{AdS}\text{d}^{d}x\text{d}z\sqrt{g}\,\phi^{*}(x,z)\phi(x,z)+\int_{\mathbb{R}^{d}}\text{d}^{d}x\,\phi^{-}(x)\pi^{*}(x)\,.$
(5.12)

This correctly reproduces the free action, since

$\begin{split}\frac{1}{2}\omega(b_{1}\phi,\phi)&=\int_{AdS}\text{d}^{d}x\text{d}z\,\sqrt{g}\phi(-\Delta+m^{2})\phi+\int_{\mathbb{R}^{d}}\text{d}^{d}x\,\Delta_{+}\phi^{-}(x)\phi^{+}(x)\\\
&=\int_{AdS}\text{d}^{d}x\text{d}z\,\sqrt{g}(g^{\mu\nu}\partial_{\mu}\phi(x)\partial_{\nu}\phi(x)+m^{2}\phi(x))\,.\end{split}$
(5.13)

#### Green’s Function

The Green’s function $G$ of the kinetic operator $L=-\Delta+m^{2}$ is given by
a certain Gaussian hypergeometric function [41]. The details are not important
for us. In the following, we only need its behavior close to the boundary.
Explicitly, it satisfies

$G(x,z_{1};y,z_{2})\stackrel{{\scriptstyle z_{1}\rightarrow
0}}{{\longrightarrow}}c\frac{(z_{1}z_{2})^{\Delta_{+}}}{(z_{2}^{2}+(x-y)^{2})^{\Delta_{+}}}\,,$
(5.14)

where $c$ is some normalization constant that is irrelevant for our purposes.
The Green’s function $G$ satisfies Dirichlet boundary conditions in the sense
that its weighted boundary value is zero, i.e.

$ph(\phi^{*}):=\lim_{z_{1}\rightarrow
0}z_{1}^{-\Delta_{-}}\int\text{d}z_{2}\text{d}^{d}w\,G(x,z_{1},y,w)\phi^{*}(y,w)=0$
(5.15)

for all fields $\phi^{*}$.

The associated inclusion map $i:X^{0}\rightarrow X^{1}$ can be found as in the
case of a scalar field on an ordinary manifold. Explicitly, we first compute
the boundary symplectic structure via

$\omega(L\phi_{1},\phi_{2})-\omega(\phi_{1},L\phi_{2})=\int_{\mathbb{R}^{d}}\text{d}^{d}x\,\Delta_{+}(\phi_{1}^{-}(x)\phi_{2}^{+}(x)-\phi_{2}^{-}(x)\phi_{1}^{+}(x))=:\omega_{\partial}(\phi_{1},\phi_{2})\,.$
(5.16)

Now recall that, given a boundary field $\phi_{0}$, the solution $i(\phi_{0})$
can be found via

$\omega(i(\phi_{0}),\phi^{*})=\omega_{\partial}(\phi_{0},\pi h(\phi^{*}))\,,$
(5.17)

where

$\pi(\phi)=\lim_{\varepsilon\rightarrow
0}(\varepsilon^{-\Delta_{-}}\phi(x,\varepsilon),\varepsilon^{-\Delta_{+}}\varepsilon\partial_{\varepsilon}\phi(x,\varepsilon))=(\phi^{-}(x),\Delta_{+}\phi^{+}(x))$
(5.18)

is the projection onto phase space. We find

$\omega_{\partial}(\phi_{0},\pi
h(\phi^{*}))=\int_{\mathbb{R}^{d}}\text{d}^{d}x\,\phi_{0}(x)\int_{AdS}\text{d}^{d}y\,\text{d}z\,\lim_{\varepsilon\rightarrow
0}\varepsilon^{-\Delta_{+}}\varepsilon\partial_{\varepsilon}G(x,\varepsilon,y,z)\phi^{*}(y,z)\,.$
(5.19)

Therefore,

$\begin{split}i_{0}(\phi_{0})(x,z)&=\int_{R^{d}}\text{d}^{d}y\,\lim_{\varepsilon\rightarrow
0}\varepsilon^{-\Delta_{+}}\varepsilon\partial_{\varepsilon}G(x,z,y,\varepsilon)\phi_{0}(y)\\\
&=c\Delta_{+}\int_{R^{d}}\text{d}^{d}y\,\frac{z^{\Delta_{+}}}{(z^{2}+(x-y)^{2})^{\Delta_{+}}}\phi_{0}(y)\,.\end{split}$
(5.20)

This indeed satisfies $p_{0}i_{0}(\phi_{0})=\phi_{0}$ [3]. Further, by
construction it also satisfies $hL=1-i_{0}p_{0}$. Therefore, we have a
deformation retract

${0}$${X^{0}}$${X^{1}}$${0\ ,}$${0}$${\bar{X}^{0}}$${0}$${0\
.}$$\scriptstyle{p}$$\scriptstyle{-\Delta+m^{2}}$ (5.21)

The space $\bar{X}^{0}$ consists of fields defined on the conformal boundary
of AdS.

#### Cyclic Extension of the Homotopy

From our experience with the scalar field on a manifold with boundary we know
that the projection map in degree one is given by

$p_{1}(\phi^{*},\pi^{*})(x)=\pi^{*}(x)-\lim_{\varepsilon\rightarrow
0}\varepsilon^{-\Delta_{+}}\varepsilon\partial_{\varepsilon}h(\phi^{*})(x,\varepsilon)\,.$
(5.22)

The second term on the right hand side computes the canonical momentum of a
solution with source $\phi^{*}$. It projects to $\bar{X}^{1}$, which consists
only of the boundary anti-fields $\pi^{*}$. The inclusion in degree one is the
standard one, i.e. $i(\pi^{*})=(0,\pi^{*})$. From
$b_{1}(\phi)=(L\phi,\Delta_{+}\phi_{+})$ it follows that

$\begin{split}p_{1}b_{1}(\phi)(x)&=\Delta_{+}\phi_{+}(x)-\lim_{\varepsilon\rightarrow
0}\varepsilon^{-\Delta_{+}}\varepsilon\partial_{\varepsilon}hL(\phi)(x,\varepsilon)\\\
&=\Delta_{+}\phi_{+}(x)-\lim_{\varepsilon\rightarrow
0}\varepsilon^{-\Delta_{+}}\varepsilon\partial_{\varepsilon}(1-i_{0}p_{0})(\phi)(x,\varepsilon)=\lim_{\varepsilon\rightarrow
0}\varepsilon^{-\Delta_{+}}\varepsilon\partial_{\varepsilon}i_{0}p_{0}(\phi)(x,\varepsilon)\,,\end{split}$
(5.23)

where we used that $\lim_{\varepsilon\rightarrow
0}\varepsilon^{-\Delta_{+}}\varepsilon\partial_{\varepsilon}\phi(x,\varepsilon)=\Delta_{+}\phi_{+}(x)$.
The differential $c_{1}$ on the boundary fields should compute the canonical
momentum of a solution with boundary value $\phi_{0}$. Therefore, we define

$c_{1}(\phi_{0})(x)=\lim_{\varepsilon\rightarrow
0}\varepsilon^{-\Delta_{+}}\varepsilon\partial_{\varepsilon}i_{0}(\phi_{0})(x,\varepsilon)\,.$
(5.24)

Using (5.23), we then find that $p_{1}b_{1}=c_{1}p_{0}$, so $p_{0}$ and
$p_{1}$ define a chain map. In order to verify that $i_{0}$ and $i_{1}$ also
define a chain map we compute

$b_{1}i_{0}(\phi_{0})=(0,\lim_{\varepsilon\rightarrow
0}\varepsilon^{-\Delta_{+}}\varepsilon\partial_{\varepsilon}i_{0}(\phi_{0})(x,\varepsilon))=i_{1}c_{1}(\phi_{0})\,.$
(5.25)

Finally, we want to extend the homotopy. We extend $h$ so that it is
independent of $\pi^{*}$, i.e. $h(\phi^{*},\pi^{*})=h(\phi^{*})$. We then find

$\begin{split}b_{1}h(\phi^{*},\pi^{*})&=(\phi^{*},\lim_{\varepsilon\rightarrow
0}\varepsilon^{-\Delta_{+}}\varepsilon\partial_{\varepsilon}h(\phi^{*})(x,\varepsilon))=(1-i_{1}p_{1})(\phi^{*},\pi^{*})\,.\end{split}$
(5.26)

As before, we also have $hb_{1}=1-i_{0}p_{0}$. Therefore, we have in fact a
strong deformation retract, given by

${0}$${X^{0}}$${X^{1}}$${0\ ,}$${0}$${\bar{X}^{0}}$${\bar{X}^{1}}$${0\
.}$$\scriptstyle{p_{0}}$$\scriptstyle{b_{1}}$$\scriptstyle{p_{1}}$$\scriptstyle{c_{1}}$
(5.27)

###  5.2 Homological Perturbation Lemma

We can now apply the perturbation lemma by adding a cubic potential. This
corresponds to the 2-bracket:

$b_{2}(\phi_{1},\phi_{2})=\lambda\phi_{1}\phi_{2}\,.$ (5.28)

To get the boundary action by using the inclusion map we apply the
perturbation lemma, which gives us the perturbed inclusion map, which in
degree zero is given by

$i^{\prime}_{0}=(1+hb_{2})^{-1}i_{0}=\sum_{n\geq 0}(-hb_{2})^{n}i_{0}\,.$
(5.29)

Note that all maps have to be lifted to maps on the coalgebra. We will compute
this map perturbatively in the coupling constant $\lambda$. We denote this
expansion as

$i^{\prime}_{0}=i_{(0)}+i_{(1)}+i_{(2)}+...\,.$ (5.30)

In our case, $i_{(n)}$ is the $n$th term of the sum in the formula (5.29).

The homotopy $h$ is lifted by first defining it as a map
$\tilde{h}:(X^{\bullet})^{\otimes n}\rightarrow(X^{\bullet})^{\wedge n}$ as

$h(a_{1}\otimes\cdots\otimes
a_{n})=\sum_{k=1}^{n}(-)^{a_{1}+...+a_{k}}ip(a_{1})\wedge\cdots
ip(a_{k-1})\wedge h(a_{k})\wedge a_{k+1}\wedge\cdots\wedge a_{n}\,.$ (5.31)

We can then restrict it to a map acting on $(X^{\bullet})^{\wedge n}$ by using
the map $s:(X^{\bullet})^{\wedge n}\rightarrow(X^{\bullet})^{\otimes n}$,
given by

$s(a_{1}\wedge\cdots\wedge a_{n})=\frac{1}{n!}\sum_{\sigma\in P(n)}\pm
a_{\sigma(1)}\otimes\cdots\otimes a_{\sigma(n)}\,,$ (5.32)

and then setting $h:=\tilde{h}\circ s$. Here, $P(n)$ denotes the group of
permutations.

We now compute the different terms in the expansion (5.30). We already know
from (5.20) that

$i_{(0)}(\phi_{0})(x_{1},z_{1})=c\Delta_{+}\int\text{d}^{d}x_{2}\frac{z^{\Delta_{+}}\phi_{0}(x_{2})}{(z_{1}^{2}+|x_{1}-x_{2}|^{2})^{\Delta_{+}}}\,.$
(5.33)

To first order in $\lambda$, we find

$\begin{split}i_{(1)}(\phi_{0},\psi_{0})(x,z)=-\lambda c\Delta_{+}\int
d^{d}x_{1}dz_{1}&d^{d}y_{1}d^{d}y_{2}\sqrt{g}(x_{1},z_{1})G(x,z;x_{1},z_{1})\\\
&\cdot\frac{z_{1}^{2\Delta_{+}}\phi_{0}(y_{1})\psi_{0}(y_{2})}{\left((z_{1}^{2}+|x_{1}-y_{1}|^{2})(z_{1}^{2}+|x_{1}-y_{2}|^{2})\right)^{\Delta_{+}}}\,.\end{split}$
(5.34)

For the contribution quadratic in $\lambda$ we do not give the explicit
expressions but merely state that the perturbation lemma tells us that

$i_{(2)}=hb_{2}(hb_{2}(i_{0}(\phi_{0}),i_{0}(\psi_{0})),i_{0}(\rho_{0}))+\text{cyclic}\,.$
(5.35)

The three terms included in the cyclic sum are precisely analogous to $s,t$
and $u$ channels.

###  5.3 Witten Diagrams and Correlation Functions

We will now use the previous results for the inclusion map to get correction
terms for the action in dependence of boundary data. The free theory will give
us an action of boundary fields, from which we can read off the two point
function. The correction terms of orders $\lambda$ and $\lambda^{2}$ will give
us the first and second correction terms to the action: the three and four-
point functions.

The starting point for all calculations is the surface term of the action,
obtained by integration by parts of the action:

$\begin{split}S=&\;\frac{1}{2}\int_{AdS}\sqrt{g}\text{d}^{d}x\text{d}z\phi(x,z)\big{[}(-\Delta+m^{2})\phi(x,z)+\frac{\lambda}{2}\phi^{2}(x,z)\big{]}\\\
&\;-\frac{\lambda}{3}\int_{AdS}\sqrt{g}\text{d}^{d}x\text{d}z\phi^{3}(x,z)+\frac{\Delta_{+}}{2}\int_{\mathbb{R}^{d}}\text{d}^{d}x\phi^{-}(x)\phi^{+}(x)\,.\end{split}$
(5.36)

The first line vanishes when we consider fields satisfying the full (bulk)
equations of motion. The perturbation lemma is such that

$i^{\prime}(\phi_{0}):=\sum_{n\geq
0}\frac{1}{n!}i_{(n)}(\phi_{0},...,\phi_{0})$ (5.37)

satisfies the full non-linear equations of motion. Therefore, when computing
$\bar{S}=S\circ i^{\prime}(\phi_{0})$, we can drop the first line in (5.36).
Using this together with the computation of $i_{(n\leq 3)}$ in the previous
section, we can compute the boundary action $\bar{S}$ to quartic order. We can
further simplify the computation using the fact that $\phi^{-}=p(\phi)$ and
$ph=0$. But all the $i_{(n\geq 1)}$ end in a propagator, so we find that
$\phi^{-}(x)$ reduces to $\phi_{0}$ in all orders. Only the linear part of
$i^{\prime}$ contributes.

#### Boundary Action

We get the boundary action $\bar{S}(\phi_{0})=S\circ i^{\prime}(\phi_{0})$ to
zeroth order in $\lambda$ from (5.36) by noting that $\phi^{-}(x)=\phi_{0}(x)$
and

$\begin{split}\Delta_{+}\phi^{+}(x)&=\lim_{z\rightarrow
0}z^{-\Delta_{+}}\partial_{z}z\int\text{d}^{d}y\frac{c\Delta_{+}z^{\Delta_{+}}}{(z^{2}+|x-y|^{2})^{\Delta_{+}}}\phi_{0}(y)\\\
&=\int\text{d}^{d}y\frac{c\Delta^{2}_{+}}{|x-y|^{\Delta_{+}}}\phi_{0}(y)\,.\end{split}$
(5.38)

Therefore, to this order we find

$\bar{S}_{(0)}(\phi_{0})=\int\text{d}^{d}x\text{d}^{d}y\,\frac{c\Delta_{+}^{2}}{|x-y|^{2\Delta_{+}}}\,.$
(5.39)

By the AdS/CFT dictionary, this is the two-point function of an operator of
scaling dimension $\Delta_{+}$ of the CFT on the boundary. It is customary to
represent it as the following Witten diagram

$K$$AdS$$\partial AdS$

Next, we compute $\bar{S}$ to first order in $\lambda$. As noted before, on-
shell the action reduces to

$\begin{split}S=-\frac{\lambda}{3}\int_{AdS}\sqrt{g}\text{d}^{d}x\text{d}z\,\phi^{3}(x,z)+\frac{\Delta_{+}}{2}\int_{\mathbb{R}^{d}}\text{d}^{d}x\,\phi_{0}(x)\phi^{+}(x)\,.\end{split}$
(5.40)

The first term already is linear in the coupling constant, so only $i_{(0)}$
contributes. On the other hand, in the second term, we need $i_{(1)}$ to
compute $\Delta_{+}\phi^{+}$ to first order in $\lambda$. With this, we find

$\begin{split}\bar{S}_{(1)}(\phi_{0})=\int\sqrt{g}\text{d}^{d}x\text{d}z\text{d}^{d}x_{1}&\text{d}^{d}x_{2}\text{d}^{d}x_{3}\,\phi_{0}(x_{1})\phi_{0}(x_{2})\phi_{0}(x_{3})\\\
&\times\frac{\lambda
c^{3}\Delta_{+}^{3}z^{3\Delta_{+}}}{|(z^{2}+|x-x_{1}|^{2})|(z^{2}+|x-x_{2}|^{2})|(z^{2}+|x-x_{3}|^{2})|^{\Delta_{+}}}\;.\end{split}$
(5.41)

This generates the three-point function of the boundary CFT. The corresponding
Witten diagram is

$K$$K$$K$$AdS$$\partial AdS$

Finally, the term quadratic in the coupling constant is given by

$\begin{split}\bar{S}_{(2)}(\phi_{0})=\frac{1}{4!}&\int\sqrt{g}(z_{1})\sqrt{g}(z_{2})\text{d}^{d}y_{1}\text{d}z_{1}\text{d}^{d}y_{2}\text{d}z_{2}\prod_{i=1}^{4}\text{d}^{d}x_{i}\phi_{0}(x_{i})\\\
&\times\frac{3\lambda^{2}c^{4}\Delta_{+}^{4}z_{1}^{2\Delta_{+}}z_{2}^{2\Delta_{+}}G(y_{1},z_{1};y_{2},z_{2})}{|(z_{1}^{2}+|y_{1}-x_{1}|^{2})(z_{1}^{2}+|y_{1}-x_{2}|^{2})(z_{2}^{2}+|y_{2}-x_{3}|^{2})(z_{2}^{2}+|y_{2}-x_{4}|^{2})|^{\Delta_{+}}}\;,\end{split}\,.$
(5.42)

whose Witten diagram is

$K$$K$$G$$K$$K$$AdS$$\partial AdS$

##  6 Summary and Outlook

Our goal in this paper was to show that homotopy algebras are an ideal
framework to formalize holography or the AdS/CFT correspondence, the main
reason being that the equivalence of structures associated to spaces in
different dimensions is natural in the world of homotopy algebras. More
precisely, two spaces of different dimensions may be homotopy equivalent, in
which case homotopy algebras associated to these spaces are equivalent.

We showed that field theories defined on a ‘bulk’ spacetime with boundary (or
conformal boundary in the case of AdS) can be encoded in a cyclic $L_{\infty}$
algebra. To this end we refined the dictionary between field theories and
cyclic $L_{\infty}$ algebras by extending the underlying chain complex with
proper boundary components. We then showed, for the examples of scalar field
theory and Yang-Mills gauge theory, how the passing over to the boundary
defines a homotopy retract of the underlying chain complex. There is then a
homotopy transfer of the $L_{\infty}$ algebra encoding the bulk theory to a
‘boundary’ cyclic $L_{\infty}$ algebra, which in turn gives the on-shell
action conventionally computed by Witten diagrams. Strictly speaking, in order
to obtain finite results further boundary terms need to be included following
the procedure of holographic renormalization [23]. Such additional boundary
terms may be encoded, for instance, in a further extension of the cyclic
$L_{\infty}$ algebra. We leave a more detailed homotopy algebra formulation of
holographic renormalization to future work.

If the AdS/CFT conjecture is correct then the boundary cyclic $L_{\infty}$
algebra yields the generating functional of the correlation functions of the
dual field theory defined on the boundary. Thus, the results of this paper
formalize one arrow of the AdS/CFT dictionary, as depicted in the figure of
the introduction, thereby providing a promising first step toward a first-
principle understanding or perhaps even proof of AdS/CFT.

In order to achieve this most ambitious goal, the results of this paper need
to be extended in various directions. First, the computation of correlation
functions starting from a boundary field theory needs to be formalized in
terms of homotopy algebras so that one has an $L_{\infty}$ morphism from the
bulk to the boundary theory (and the other way around). Second, the
formulation of bulk theories on AdS in terms of cyclic $L_{\infty}$ algebras
needs to be extended to genuine gravity theories, ideally to theories such as
type IIB supergravity in ten dimensions including all massive Kaluza-Klein
modes on, say, $AdS_{5}\times S^{5}$. As alluded to in the introduction, using
techniques from exceptional field theory it is clear in principle how to write
down these algebras [26, 27, 28, 29, 30], but it remains to include the
boundary components. More importantly, as discussed in the introduction, in
order to actually prove the AdS/CFT correspondence one would presumably need
to write down some kind of ‘master’ homotopy algebra, say along the lines of
[42], from which both the boundary and bulk $L_{\infty}$ algebras are
derivable via some sort of homotopy transfer.

### Acknowledgements

We would like to thank Alex Arvanitakis, Roberto Bonezzi, Tomas Codina, Felipe
Diaz-Jaramillo, Allison Pinto, Ivo Sachs and Barton Zwiebach for useful
discussions and collaborations on related topics.

This work is funded by the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreement No
771862).

##  Appendix A Technical Details for Yang-Mills Theory with Boundary

###  A.1 Cyclic Extension in Yang-Mills Theory

In this appendix we discuss some technical details of the cyclic extensions of
the Yang-Mills brackets given in sec. 4.2.

We begin by noticing from (4.28) that

$\omega(b_{1}A_{1},A_{2})+\int_{\partial
M}A_{2}\wedge\star\text{d}A_{1}=\omega(b_{1}A_{2},A_{1})+\int_{\partial
M}A_{1}\wedge\star\text{d}A_{2}\;,$ (A.1)

so the combination on the left-hand side (or equivalently the right-hand side)
is symmetric. This is also clear from the fact that

$\frac{1}{2}\omega(b_{1}A,A)+\frac{1}{2}\int_{\partial
M}A\wedge\star\text{d}A$ (A.2)

is the kinetic term of the classical action $S_{0}$. Similarly, from (4.29) we
find that

$\omega(b_{1}A^{*},c)+\int_{\partial M}c\wedge A^{*}=-\omega(b_{1}c,A^{*})\ $
(A.3)

is graded symmetric.

In the scalar field case, we only considered interactions without derivatives.
On the other hand, the cubic vertex in Yang-Mills theory has a derivative.
This derivative also leads to a failure of $b_{2}$ being cyclic. We compute

$\begin{split}\omega(b_{2}(A_{1},A_{2}),A_{3})&=\int_{M}A_{3}\wedge(\text{d}\star[A_{1},A_{2}]+[A_{2},\star\text{d}A_{1}]+[A_{1},\star\text{d}A_{2}])\\\
&=\int_{M}A_{1}\wedge[A_{2},\star\text{d}A_{3}]-\int_{\partial
M}A_{3}\wedge\star[A_{1},A_{2}]\\\
&\quad+\int_{M}\text{A}_{1}\wedge\text{d}\star[A_{2},A_{3}]+\int_{\partial
M}A_{1}\wedge\star[A_{2},A_{3}]\\\
&\quad+\int_{M}A_{1}\wedge[A_{3},\star\text{d}A_{2}]\\\
&=\omega(b_{2}(A_{2},A_{3}),A_{1})-\int_{\partial
M}A_{3}\wedge\star[A_{1},A_{2}]+\int_{\partial M}\star[A_{2},A_{3}]\wedge
A_{1}\;.\end{split}$ (A.4)

Rewriting this equation, we find that

$\omega(b_{2}(A_{1},A_{2}),A_{3})+\int_{\partial
M}A_{3}\wedge\star[A_{1},A_{2}]=\omega(b_{2}(A_{2},A_{3}),A_{1})+\int_{\partial
M}A_{1}\wedge\star[A_{2},A_{3}]\;.$ (A.5)

As before, we find that the above defines a symmetric product. Again, this is
also obvious from the fact that the left-hand side is equal to the cubic part
of the classical action $S_{0}$.

We now have all the information we need to define an actual cyclic
$L_{\infty}$ algebra. We extend the space of equations of motion by including
a space of boundary fields $\Omega^{n-2}(\partial M)\otimes\mathfrak{g}$,
whose elements we call $B^{*}$. Further, we extend the space of Noether
identities by a boundary part $\Omega^{n-1}(\partial M)\otimes\mathfrak{g}$,
whose fields we call $\eta^{*}$. We then define the extended complex

${\Omega^{0}(M)}$${\Omega^{1}(M)}$${\Omega^{n-1}(M)\oplus\Omega^{n-2}(\partial
M)}$${\Omega^{n}(M)\oplus\Omega^{n-1}(\partial M)\
,}$$\scriptstyle{b_{1}^{\prime}}$$\scriptstyle{b_{1}^{\prime}}$$\scriptstyle{b_{1}^{\prime}}$
(A.6)

with

$b_{1}^{\prime}(c)=\text{d}c\ ,\qquad
b_{1}^{\prime}(A)=(\text{d}\star\text{d}A,i^{*}\star\text{d}A)\ ,\qquad
b_{1}^{\prime}(A^{*},B^{*})=(\text{d}A^{*},i^{*}A^{*}-\text{d}B^{*})\,,$ (A.7)

where $i^{*}$ is the pullback w.r.t. the inclusion map from the boundary to
the bulk. We also extend the inner product via

$\begin{split}\omega^{\prime}(c,(c^{*},\eta^{*}))&=\int_{M}c\wedge
c^{*}-\int_{\partial M}i^{*}c\wedge\eta^{*}\;,\\\
\omega^{\prime}((A^{*},B^{*}),A)&=\int_{M}A\wedge A^{*}+\int_{\partial
M}i^{*}A\wedge B^{*}\ ,\end{split}$ (A.8)

for which one finds

$\begin{split}\omega^{\prime}(b_{1}^{\prime}A_{1},A_{2})&=\omega(b_{1}^{\prime}A_{2},A_{1})=\int_{M}\text{d}A_{1}\wedge\star\text{d}A_{2}\
,\\\
\omega^{\prime}(b_{1}^{\prime}c,(A^{*},B^{*}))&=-\omega^{\prime}(b_{1}^{\prime}(A^{*},B^{*}),c)=-\int_{M}\text{d}c\wedge
A^{*}-\int_{\partial M}i^{*}\text{d}c\wedge B^{*}\ .\end{split}$ (A.9)

Therefore, we correctly reproduce the classical free action. The BV extended
action gets the additional term

$-\int_{\partial M}i^{*}\text{d}c\wedge B^{*}\ .$ (A.10)

For the cubic vertex, we found that the naive product is also not cyclic. We
therefore have to extend the product $b_{2}$ to

$b_{2}^{\prime}(A_{1},A_{2})=(b_{2}(A_{1},A_{2}),i^{*}\star[A_{1},A_{2}])\in(\Omega^{n-1}(M)\oplus\Omega^{n-2}(\partial
M))\otimes\mathfrak{g}\;.$ (A.11)

One then finds

$\omega^{\prime}(b_{2}^{\prime}(A_{1},A_{2}),A_{3})=\int_{M}\text{d}A_{1}\wedge\star[A_{2},A_{3}]+\text{cyclic}\;,$
(A.12)

which is manifestly symmetric. We are not done, however, since we still need
to make sure that $b_{1}^{\prime}$ is a derivation with respect to
$b_{2}^{\prime}$. We compute

$\begin{split}b_{1}^{\prime}b_{2}^{\prime}(A_{1},A_{2})&=(b_{1}b_{2}(A_{1},A_{2}),i^{*}b_{2}(A_{1},A_{2})-i^{*}\text{d}\star[A_{1},A_{2}])\\\
&=(-b_{2}(b_{1}A_{1},A_{2})-b_{2}(A_{1},b_{1}A_{2}),[i^{*}A_{1},i^{*}\star\text{d}A_{2}]+[i^{*}A_{2},i^{*}\star\text{d}A_{1}])\,.\end{split}$
(A.13)

We see that in the second component of the above expression,
$\star\text{d}A_{i}$ is the boundary contribution of $b_{1}^{\prime}(A_{i})$.
Therefore, we set

$b_{2}^{\prime}(A,(A^{*},B^{*}))=(b_{2}(A,A^{*}),-[i^{*}A,B^{*}])\ .$ (A.14)

A priori, it is not guaranteed that $b_{1}^{\prime}$ is a derivation of that
product. However, the output of $b_{2}^{\prime}$ in that case already has top
degree, and therefore $b_{1}$ acts trivially. So the derivation property is
trivially satisfied.

We extended $b_{2}^{\prime}$ so that it also pairs fields with boundary anti-
fields $B^{*}$. For this reason, it is no longer guaranteed that it is cyclic.
Therefore, we need to compute

$\begin{split}\omega^{\prime}(b_{2}^{\prime}(A,(A^{*},B^{*})),c)&=\omega^{\prime}(([A,A^{*}],-[i^{*}A,B^{*}]),c)\\\
&=-\int_{M}[A,A^{*}]\wedge c-\int_{\partial M}[i^{*}A,B^{*}]\wedge i^{*}c\\\
&=\int_{M}A\wedge[c,A^{*}]+\int_{\partial M}i^{*}A\wedge[i^{*}c,B^{*}]\\\
&=-\int_{M}A\wedge[A^{*},c]-\int_{\partial M}i^{*}A\wedge[B^{*},i^{*}c]\\\
&=\omega^{\prime}(b_{2}^{\prime}((A^{*},B^{*}),c),A)\ ,,\end{split}$ (A.15)

where the final equation is the statement of cyclicity for $b_{2}^{\prime}$.
This is hence satisfied if we define

$b_{2}^{\prime}((A^{*},B^{*}),c)=(-[A^{*},c],-[B^{*},i^{*}c])\ .$ (A.16)

Once more, we need to check the derivation property of $b_{1}^{\prime}$:

$\begin{split}b_{1}^{\prime}b_{2}^{\prime}((A^{*},B^{*}),c)&=(b_{1}b_{2}(A^{*},c),-i^{*}[A^{*},c]+\text{d}[B^{*},i^{*}c])\\\
&=(-b_{2}(b_{1}A^{*},c)+b_{2}(A^{*},b_{1}c),-[i^{*}A^{*}-\text{d}B^{*},i^{*}c]-[\text{i}^{*}dc,B^{*}])\\\
&=b_{2}^{\prime}((A^{*},B^{*}),b_{1}^{\prime}c)-(b_{2}(b_{1}A^{*},c),[i^{*}A^{*}-\text{d}B^{*},i^{*}c])\,.\end{split}$
(A.17)

From this we see that we need to define

$b_{2}^{\prime}((c^{*},\eta^{*}),c)=(b_{2}(c^{*},c),[\eta^{*},i^{*}c])\ .$
(A.18)

We now have everything we want. $b_{1}^{\prime}$ and $b_{2}^{\prime}$ are
cyclic and $b_{1}^{\prime}$ is a derivation of $b_{2}^{\prime}$. Further, one
can show that $b_{2}^{\prime}$ satisfies the Jacobi identity up to homotopy
$b_{3}^{\prime}$, if we set

$b_{3}^{\prime}(A_{1},A_{2},A_{3})=(b_{3}(A_{1},A_{2},A_{3}),0)\,.$ (A.19)

###  A.2 Homotopy Retract in Yang-Mills Theory

Here we give a more in-depth derivation of the homotopy given in section 4.3.
On free equations of motion, the homotopy should be the Green’s function. We
are looking for solutions to

$\text{d}\star\text{d}A=A^{*}\>.$ (A.20)

To solve this, we use Lorenz gauge $\delta A=0$. In this case, the equation of
motion for $A$ can be written as

$\Delta A=\star A^{*}\;.$ (A.21)

Note that this equation does not require $A^{*}$ to be conserved
($\text{d}A^{*}=0$), unlike the original Yang-Mills equation (A.20). However,
we claim that if we have a solution to (A.21) together with $i^{*}\delta A=0$,
then $A$ satisfies Lorenz gauge if and only if $A^{*}$ is conserved.

To prove this, suppose we have a solution to (A.21) with $i^{*}\delta A=0$. We
first assume that $A$ satisfies Lorenz gauge. Then, the equation (A.21)
reduces to the original Yang-Mills equation $\text{d}\star\text{d}A=A^{*}$.
From this it immediately follows that $\text{d}A^{*}=0$. On the other hand,
assume that $A^{*}$ is conserved. From (A.21), we then deduce that

$\delta\text{d}\delta A=0\;.$ (A.22)

We define $f=\delta A$. From (A.22), we then find

$\delta\text{d}f=\Delta f=0\ ,$ (A.23)

where we used the fact that $f$ is a zero form (i.e. a Lie algebra valued
function). Now $i^{*}f=i^{*}\delta A=0$ by assumption. But this means that $f$
should be a harmonic functions vanishing on the boundary. But from this it
follows that $f=0$ everywhere and therefore $0=\delta A$.

We now turn to existence of solutions, where we will solve (A.21) instead of
(A.20). Lemma 3.4.7 in [40] states that the equation $\Delta A=\star A^{*}$,
with boundary condition $i^{*}(A)=a_{0},i^{*}\delta A=0$, has a solution, if
and only if $A^{*}\in(\mathcal{H}_{N}^{d-1}(M))^{\perp}$, where $\perp$ is the
orthogonal with respect to the inner product $(-,-)$ defined in (4.1). The
solution is unique up to $A\in\mathcal{H}^{1}_{D}(M)$. In general, we can
project $A^{*}$ to $(\mathcal{H}_{N}^{d-1}(M))^{\perp}$ first using the Hodge
decomposition

$\Omega^{n-1}(M)=\text{d}\Omega^{n-2}_{D}(M)\oplus\delta\Omega^{n}(M)\oplus\mathcal{H}_{N}^{d-1}(M)\oplus\mathcal{H}_{ex}^{d-1}(M)\;,$
(A.24)

and then solving the problem such that $A\in(\mathcal{H}^{1}_{D}(M))^{\perp}$.
Taking the solution with $i^{*}A=0$, this defines a Green’s function
$G:\Omega^{n-1}(M)\rightarrow\Omega^{1}(M)$, such that

$\Delta G_{1}(A^{*})=\star(1-I_{1}P_{1})A^{*}\ ,$ (A.25)

where $P_{1}:\Omega^{n-1}(M)\rightarrow\mathcal{H}_{N}^{n-1}(M)$ is the
projector to $\mathcal{H}_{N}^{n-1}(M)$ using the Hodge decomposition and
$I_{1}$ is the inclusion of that space. Just like in the scalar field case, we
take the Green’s function satisfying Dirichlet boundary conditions. Note that
here $A$ being Dirichlet means that $i^{*}A=0$, and not $A|_{\partial M}=0$.
There is also a solution to the latter boundary condition under different
assumptions. But we do not want to solve for the normal component, since it
will be determined by the ‘constraint equation’.555This is a constraint
equation if we think of the direction normal to the boundary as “time”.

Apart from $G_{1}$, we will also use the scalar Green’s function $G_{0}$,
which we encountered in the section on scalar field theory. With this, we now
have all the tools to define the homotopy. First of all, on
$V^{1}=\Omega^{n-1}(M)\otimes\mathfrak{g}$, we define

$h(A^{*})=G_{1}(1-I_{2}P_{2})(A^{*})\ ,$ (A.26)

where $P_{2}$ and $I_{2}$ are the projection and inclusion onto and from
$\delta\Omega^{n}_{N}(M)$ coming from the Hodge decomposition. We use them,
since as we saw before, this ensures that $h(A^{*})$ satisfies Lorenz gauge.
We further define

$h(A)=G_{0}\delta A\ $ (A.27)

to find

$hb_{1}c=G_{0}\delta\text{d}c=G_{0}\Delta c=1-i_{-1}p_{-1}\;,$ (A.28)

where $p_{-1}=i^{*}$ and $i_{-1}$ associates a solution $\Delta c=0$ to a
given boundary value $c_{0}$. The relation (A.28) is exactly as in the scalar
field case. So the homotopy interpolates between a bulk gauge parameter to a
boundary gauge parameter (the word _gauge_ will be justified below). On the
other hand, to see what the homotopy does on gauge fields, we compute
$hb_{1}(A)$ and $b_{1}h(A)$ separately. First of all, we have
$b_{1}h(A)=\text{d}G_{0}\delta A$. We claim that this projects $A$ to
$\text{d}a\in\text{d}\Omega_{D}(M)$, i.e. to a ‘pure gauge’ component. To see
this, we use the Hodge decomposition (4.4) to find

$b_{1}h(A)=\text{d}G_{0}\delta\text{d}a\;.$ (A.29)

From the scalar field case, we know that $G_{0}\delta\text{d}=1-i_{-i}p_{-1}$.
But from $a\in\Omega^{0}_{D}(M)$, it follows that $p_{-1}(a)=0$. So we find
$G_{0}\delta\text{d}a=a$ and therefore $b_{1}h(A)=\text{d}a$. On the other
hand,

$hb_{1}(A)=G_{1}(1-I_{2}P_{2})\text{d}\star\text{d}A=G_{1}\text{d}\star\text{d}A\;,$
(A.30)

where we used that $P_{2}\text{d}=0$, which follows from the fact that the
image of d is orthogonal to $\delta\Omega_{N}(M)$. We now show that
$b_{1}(1-hb_{1})(A)=0$. To this end one computes

$\begin{split}b_{1}hb_{1}(A)&=\text{d}\star\text{d}G_{1}\text{d}\star\text{d}A=-(-)^{n}\star\Delta
G_{1}\text{d}\star\text{d}A=-(-)^{n}(1-I_{1}P_{1})\star\delta\text{d}A\\\
&=\text{d}\delta\text{d}A=b_{1}(A)\ .\end{split}$ (A.31)

Here we used that $P_{1}\delta=0$, which is true since the image of $\delta$
is orthogonal to $\mathcal{H}_{N}^{n-1}$. From the above it follows that
$b_{1}(1-hb_{1})=0$, so $hb_{1}$ projects to solutions to the equations of
motion.

Recall that our original goal is to find a homotopy to a boundary theory. This
will not be entirely possible, since there is a topological obstruction in
$(\mathcal{H}^{1}_{D})^{\perp}$, which is part of the space of solutions not
removeable by a choice of gauge. For this reason, we first define
$P_{3}:\Omega^{1}(M)\rightarrow\mathcal{H}^{1}_{D}(M)$ using the Hodge
decomposition and $I_{3}:\mathcal{H}^{1}_{D}(M)\rightarrow\Omega^{1}(M)$, the
natural inclusion. We project the gauge fields $A$ via
$p_{0}(A)=(i^{*}A,P_{3}(A))$. To define our inclusion, we associate to a
boundary configuration $a_{0}\in\Omega^{1}(\partial M)$ the unique solution to
$\Delta A=0$, such that $i^{*}A=a_{0},i^{*}\delta
A=0,A\in(\mathcal{H}_{D}^{1})^{\perp}(M)$. Since all boundary conditions are
linear, the map $r:\Omega^{1}(\partial M)\rightarrow\Omega^{1}(M)$ associating
to a boundary one-form $a_{0}$ a harmonic bulk form is linear. We can then
define the linear map $i_{0}:\Omega^{1}(\partial
M)\oplus\mathcal{H}^{1}_{D}(M)\rightarrow\Omega^{1}(M)$, where
$i_{0}=r+I_{3}$. We recall also that $i^{*}\delta A=0$ and $\delta\Delta A=0$
implies $\delta A=0$, so $A$ satisfies Lorenz gauge. Therefore $\Delta
A=\star\text{d}\delta\text{d}A=0$, so $A$ solves the linear equations of
motion of Yang-Mills theory. By construction, we have $p_{0}i_{0}=1$. To show
the homotopy property

$(1-i_{0}p_{0})(A)=(b_{1}h+hb_{1})(A)\ ,$ (A.32)

we prove that $(1-b_{1}h-hb_{1})(A)$ is the sum of the topological part of
$A$, i.e. $I_{3}P_{3}(A)\in\mathcal{H}_{D}^{1}(M)$, plus a solution
$\mathcal{A}$ with the boundary conditions

$i^{*}\mathcal{A}=\mathcal{A}\,,\qquad i^{*}\delta\mathcal{A}=0\
,\qquad\mathcal{A}\in(\mathcal{H}_{D}^{1}(M))^{\perp}\ .$ (A.33)

Since $i_{0}p_{0}(A)$ has the same property, it follow from uniqueness of
solutions that (A.32) holds.