December 2020

HU-EP-20/43-RTG

Gauge Invariant Perturbation Theory via Homotopy Transfer

Christoph Chiaffrino, Olaf Hohm and Allison F. Pinto

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

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

> We show that the perturbative expansion of general gauge theories can be
> expressed in terms of gauge invariant variables to all orders in
> perturbations. In this we generalize techniques developed in gauge invariant
> cosmological perturbation theory, using Bardeen variables, by interpreting
> the passing over to gauge invariant fields as a homotopy transfer of the
> strongly homotopy Lie algebras encoding the gauge theory. This is
> illustrated for Yang-Mills theory, gravity on flat and cosmological
> backgrounds and for the massless sector of closed string theory. The
> perturbation lemma yields an algorithmic procedure to determine the higher
> corrections of the gauge invariant variables and the action in terms of
> these.

###### Contents

  1. 1 Introduction
  2. 2 Gauge invariant spin-1 and spin-2 theory
    1. 2.1 Maxwell’s theory
    2. 2.2 Gravity on flat space
    3. 2.3 Homotopy interpretation of linearized gravity
  3. 3 Gauge invariant closed string theory
    1. 3.1 Linearized double field theory on flat backgrounds
    2. 3.2 Chain complex of double field theory
    3. 3.3 Homotopy interpretation
  4. 4 Homotopy Transfer
    1. 4.1 General theory
    2. 4.2 Homotopy transfer and the homological perturbation lemma
    3. 4.3 Yang-Mills theory
  5. 5 Cosmological Perturbation Theory
    1. 5.1 Gravity around FLRW
    2. 5.2 Homotopy Interpretation
    3. 5.3 Quadratic action in terms of Bardeen variables
  6. 6 Discussion and Outlook

##  1 Introduction

One of the curious facts of modern physics is that its fundamental theories,
such as the standard model of particle physics and Einstein’s theory of
general relativity, are gauge theories. The gauge symmetries defining these
theories have a two-sided character: on the one hand, they uniquely determine
the interactions and are hence a powerful tool in controlling various features
of gauge field theories; on the other, they represent a mere redundancy of the
formulation, which complicates the physical interpretation because only gauge
invariant quantities may be observables. Logically, there are two ways of
eliminating this redundancy: one may impose conditions on the fundamental
fields as gauge fixing conditions (making sure that these conditions can
always be realized by legal gauge transformations) or one may aim to rewrite
the theory in terms of gauge invariant quantities. In this paper we will give
an effective procedure in perturbation theory to construct gauge invariant
variables to any order in fields and to write the action in terms of these
field variables, generalizing the methods of cosmological perturbation theory
[1, 2, 3]. While philosophically this seems to be quite different from simply
fixing a gauge, our results also establish an operational equivalence with
gauge fixing. In fact, when done properly, gauge fixing is just equivalent to
introducing gauge invariant variables, as the latter formulation may always be
reconstructed from the former. This will be explained and illustrated in more
detail in the main text and the conclusion section.

Let us begin with some observations on the two ways of eliminating a gauge
redundancy. The use of gauge fixing conditions is the standard text book
approach in (quantum) gauge theory, but while conceptually straightforward it
appears to have certain drawbacks. First, there are numerous possible gauge
fixing conditions, making it somewhat of an art to pick a convenient one (and
leaving one with the agonizing, but presumably irrational, worry that the
result in a different gauge may be different). Second, even if a gauge fixing
condition has been chosen, there are typically residual gauge transformations
and hence remaining redundancies that have to be dealt with. Most importantly,
since observables have to be gauge invariant, the physical interpretation of
results obtained in a particular gauge may be unclear. These issues seem to
have been particularly prominent in cosmological perturbation theory (see the
classic review [2] by Mukhanov et. al. for a survey of the issues that arose
historically). Maldacena, for instance, in his seminal computation of three-
point functions and non-Gaussianities, does the job twice, in two different
gauges, just to make sure that the result is the same [4]. Clearly, it would
be desirable to have a procedure that does not require arbitrary gauge
choices.

In first order cosmological perturbation theory, corresponding to the
quadratic action around an FLRW background (as needed for the computation of
two-point functions), there is a procedure introduced by Bardeen to formulate
the theory directly in terms of gauge invariant field variables [1], thereby
removing the need to pick a gauge fixing condition. These techniques are based
on a space/time split of fields and a decomposition into irreducible
components, such as scalar, vector or higher tensor modes, and so do not
preserve manifest Lorentz invariance. It is common lore that gauge
redundancies are only needed in order to make Lorentz invariance and locality
manifest. The results of this paper provide an explicit procedure to eliminate
any gauge redundancy, and in the process we indeed abandon manifest Lorentz
invariance and introduce a certain non-locality (but only in space, not in
time). While a loss of manifest Lorentz invariance may be too much to ask for
in particle physics applications, in cosmology the situation is different:
there is a preferred (cosmic) time, so that the space/time split needed for
the definition of Bardeen variables respects the symmetries of the FLRW
backgrounds, hence making these variables ideally suited for cosmological
perturbation theory.

Given this situation it seems desirable to extend gauge invariant cosmological
perturbation theory beyond first order, as would be needed, for instance, in
order to compute the three-point functions of [4] in a manifestly gauge
invariant manner. General aspects of non-linear perturbation theory can be
found in [5, 6, 7], and in our approach we will recover some of their results.
Specific results for second order perturbation theory have been obtained in
[8, 9, 10, 11, 12, 13, 14, 15, 16]. However, a cursory scan of these
references shows that the computations become quickly unwieldy and
nontransparent. In this paper we will completely systematize gauge invariant
perturbation theory to any order in perturbations by applying a powerful
mathematical result known as homotopy transfer theorem or perturbation lemma.
This is based on a formulation of gauge theories in terms of strongly homotopy
Lie or $L_{\infty}$ algebras [17, 18], which are generalizations of Lie
algebras by differentials and higher brackets, see [19, 20] for self-contained
reviews. The $L_{\infty}$ algebras encode the gauge transformations, field
equations, Bianchi identities, etc., of the field theory. (See [21] for the
general dictionary and the introduction of [22] for a concise summary. The
$L_{\infty}$ formulation is also closely related to the classical limit of the
Batalin-Vilkovisky (BV) formulation of field theory [23, 24]; more precisely,
the tangent space to the BV manifold is the vector space of the $L_{\infty}$
algebra, and the $L_{\infty}$ brackets are recovered as the Taylor
coefficients of the BV differential, which acts on the dual space of functions
[25].) Homotopy transfer refers to a map from an $L_{\infty}$ algebra to
another, generally ‘smaller’ $L_{\infty}$ algebra that is homotopy equivalent
(see [26, 22] for pedagogical introductions and [27, 28, 29, 32] for recent
applications). We will show that passing over to gauge invariant Bardeen-type
variables can be interpreted as homotopy transfer. The perturbation lemma then
provides an effective and algorithmic procedure to determine gauge invariant
variables to any order in perturbations.

In the remainder of the introduction we explain some of our results in more
technical detail. Let us consider Yang-Mills theory with gauge potential
$A_{\mu}$, which takes values in the Lie algebra of the gauge group. One
starts by performing a space/time split $A_{\mu}=(A_{0},A_{i})$ and then uses
the Helmholtz theorem familiar from 3d vector calculus to decompose the
3-vector into a divergenceless vector and the gradient of a scalar,
$A_{i}=\widehat{A}_{i}+\partial_{i}\psi$, where
$\partial_{i}\widehat{A}^{i}=0$. At the free linearized level, where the gauge
transformations read $\delta A_{\mu}=\partial_{\mu}\Lambda$, one infers that
the scalar $\psi$ transforms as $\delta\psi=\Lambda$, and so defining

$\bar{A}_{\mu}=(\bar{A}_{0},\bar{A}_{i})\equiv(A_{0}-\dot{\psi},A_{i}-\partial_{i}\psi)\,$
(1.1)

one quickly sees that these are gauge invariant field variables. It is
illuminating to rewrite this as

$A_{\mu}=\bar{A}_{\mu}+\partial_{\mu}s(A)\;,$ (1.2)

where $s(A)\equiv\psi$ is a function of $A$ (that is non-local in space by
picking out the scalar component of a vector-scalar decomposition). This
function will be interpreted as the homotopy map that is needed for the
application of the perturbation lemma. With the definition (1.1) this relation
is just an identity, but it is a useful one because it takes a general
unconstrained gauge field and decomposes it into a gauge invariant part plus a
pure gauge part, with the ‘effective’ gauge parameter being a (non-local)
function of the field $A$ itself. Then replacing in the quadratic action every
$A_{\mu}$ by the right-hand side of (1.2) it follows by (linearized) gauge
invariance that the $s(A)$ terms drop out, leaving the same action, but now in
terms of $\bar{A}_{\mu}$, where we are allowed to use
$\partial_{i}\bar{A}^{i}=0$, which follows from the definition of
$\bar{A}_{\mu}$. We thus infer that passing over to gauge invariant variables
is operationally equivalent to imposing Coulomb gauge $\partial_{i}A^{i}=0$
and working out the action in this gauge. As we will see, analogous remarks
apply to gravity on flat and cosmological backgrounds. While in the case of
Maxwell’s theory and even gravity linearized about flat space it is easy
enough to rearrange the action in terms of gauge invariant variables, in the
case of cosmological perturbation theory it simplifies the computation
significantly already to first order to use the above procedure for
determining the action in terms of Bardeen variables (which, of course, gives
the same result as the direct computation, which we also did).

As the main results of this paper we show, first, how the perturbation lemma
can be used to define gauge invariant field variables to arbitrary order in
perturbations and, second, that the action in terms of these gauge invariant
variables is just obtained by replacing the original fields by the gauge
invariant ones. We illustrate this for Yang-Mills theory by explicitly working
out the gauge invariant field to next order. Denoting by $p$ the projection
onto the part of the field that is gauge invariant to lowest order, i.e.,
$\bar{A}_{\mu}=p(A_{\mu})$, and by $\big{[}\cdot,\cdot\big{]}$ the Lie bracket
of the Yang-Mills gauge algebra, the perturbation lemma yields for the gauge
invariant variable to next order

$\widehat{A}_{\mu}=p\left(A_{\mu}+\tfrac{1}{2}\big{[}s(A),A_{\mu}+p(A_{\mu})\big{]}\right)\,.$
(1.3)

Gauge invariance (as opposed to gauge covariance) may be verified by an
explicit computation, which we also display in the main text. For the second
result we establish an identity, generalizing (1.2), of the form

$A_{\mu}\ =\ e^{\Delta_{\phi(A)}}\,\widehat{A}_{\mu}\;,$ (1.4)

where $\Delta$ is the operator defining the infinitesimal Yang-Mills gauge
transformation, i.e.,
$\Delta_{\Lambda}\widehat{A}_{\mu}=\partial_{\mu}\Lambda+\big{[}\widehat{A}_{\mu},\Lambda\big{]}$,
and $\phi$ is a (non-local) function of $A$. This relation is again
identically satisfied: we will give perturbative definitions for
$\widehat{A}_{\mu}(A)$ and $\phi(A)$ so that writing out the right-hand side
just reproduces $A_{\mu}$. Note that $\Delta_{\Lambda}\widehat{A}_{\mu}$ is
not actually the variation under infinitesimal gauge transformations, for this
would vanish due to $\widehat{A}_{\mu}$ being gauge invariant, which is the
reason we denoted it by $\Delta_{\Lambda}$. Nevertheless, since (1.4) takes
the form of a finite gauge transformation it follows by gauge invariance that
in the non-linear action we may simply replace $A_{\mu}$ by the gauge
invariant $\widehat{A}_{\mu}$. This result is so enticingly simple to be
almost disappointing, for it means that, to all orders in perturbations,
writing the action in terms of gauge invariant variables is operationally
equivalent to imposing a gauge fixing condition: Coulomb gauge in the present
example, for $\partial_{i}\widehat{A}^{i}=0$ holds to all orders, or
transverse gauge for the Bardeen variables of cosmological perturbation
theory. (As we will outline in the conclusion section, in principle one may
also define gauge invariant variables that mimic other gauge conditions,
although for cosmology the Bardeen variables seem to be preferred.)
Nevertheless, our results also show how to reconstruct from a computation in a
particular gauge the fully gauge invariant result written in gauge invariant
variables, thereby hopefully resolving some of the interpretational issues
alluded to above. More generally, we hope that our interpretation of gauge
invariant variables in terms of homotopy transfer and the corresponding
availability of powerful mathematical techniques will help systematize the
perturbative treatment of various problems, particularly in gravitational
physics.

The rest of this paper is organized as follows. In sec. 2 we illustrate gauge
invariant variables for the simple toy models of abelian spin-1 and spin-2
gauge theories (Maxwell’s theory and Fierz-Pauli theory) and explain how this
can be formulated in terms of chain complexes and homotopy transfer. While the
writing of these theories in terms of gauge invariant variables is completely
straightforward, we have not been able to find them in the literature before.
Then, in sec. 3, we work out the example of double field theory on a flat
background, which shows some new features by having an extended chain complex,
which is necessary due to the gauge algebra not being a Lie algebra but a
genuine $L_{\infty}$ algebra. The general theory of $L_{\infty}$ algebras and
homotopy transfer is reviewed in sec. 4 in a hopefully self-contained manner.
We then show how the homotopy transfer theorem allows for the construction of
gauge invariant variables to all orders for arbitrary gauge theories and
illustrate this explicitly for the example of Yang-Mills theory. This is the
core technical result of this paper. In sec. 5 we illustrate the power of
these techniques by determining the complete quadratic action for cosmological
perturbation theory in terms of gauge invariant variables by the simple
procedure explained above. To the best of our knowledge, this general action
has not been given in the literature before. Finally, in sec. 6 we close with
a brief discussion and outlook.

##  2 Gauge invariant spin-1 and spin-2 theory

In this section we illustrate the general approach for two simple theories:
Maxwell’s theory and gravity linearized about flat space. We explain the
formulation of these theories in terms of chain complexes, introduce gauge
invariant variables and establish the notion of homotopy transfer.

###  2.1 Maxwell’s theory

We start with Maxwell’s theory for a gauge potential $A_{\mu}$ with action
functional

$S=-\frac{1}{4}\int{\rm d}^{4}x\,F^{\mu\nu}F_{\mu\nu}\;,$ (2.1)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field
strength. This theory is invariant under the abelian gauge transformations
$\delta A_{\mu}=\partial_{\mu}\Lambda$. We aim to rewrite the theory in terms
of gauge invariant variables. To this end we perform a space/time split of the
gauge field, writing $A_{\mu}=(A_{0},A_{i})$, and employ the Helmholtz theorem
according to which any 3-vector $A_{i}$ can be decomposed into a
divergenceless vector plus the gradient of a scalar:

$A_{i}=\widehat{A}_{i}+\partial_{i}\psi\,,\qquad\partial_{i}\widehat{A}^{i}=0\,.$
(2.2)

Effectively, this decomposition exists whenever the spatial Laplacian
$\Delta\equiv\partial^{i}\partial_{i}$ is invertible (say in terms of the
familiar integral expression using the Green’s function), for then we can set

$\psi=\Delta^{-1}(\partial_{i}A^{i})\,,$ (2.3)

as follows by taking the divergence of (2.2). This is the (space) non-locality
inherent in this approach. One may read off the gauge transformations of the
vector $\widehat{A}_{i}$ and scalar $\psi$:

$\delta
A_{0}=\dot{\Lambda}\,,\qquad\delta\widehat{A}_{i}=0\,,\qquad\delta\psi=\Lambda\,.$
(2.4)

Thus, $\widehat{A}_{i}$ is gauge invariant. Furthermore, we observe that the
combination

$\Phi\equiv A_{0}-\dot{\psi}\;$ (2.5)

is gauge invariant. It is now straightforward to verify that the Maxwell
equations $\partial_{\mu}F^{\mu\nu}=0$ can be written in the manifestly gauge
invariant form

$\begin{split}\Delta\Phi&=0\,,\\\
\Box\,\widehat{A}_{i}+\partial_{i}\dot{\Phi}&=0\,.\end{split}$ (2.6)

Under the same assumption of invertibility of $\Delta$ the first equation
implies $\Phi=0$, so that the second equation yields the equation
$\Box\,\widehat{A}_{i}=0$ for the two independent degrees of freedom encoded
in $\widehat{A}_{i}$. The Maxwell action (2.1) can be written in the
manifestly gauge invariant form

$S=\frac{1}{2}\int{\rm
d}^{4}x\big{(}\widehat{A}^{i}\,\Box\,\widehat{A}_{i}-\Phi\Delta\Phi\big{)}\,.$
(2.7)

Next, we briefly discuss the inclusion of matter, in which case the non-
dynamical $\Phi$ is determined by the charge density and hence non-vanishing.
Specifically, we can couple a current
$j^{\mu}=(\rho,\widehat{j}^{i}+\partial^{i}\sigma)$, with
$\partial_{i}\widehat{j}^{i}=0$, for which the conservation law
$\partial_{\mu}A^{\mu}=0$ implies $\dot{\rho}+\Delta\sigma=0$. Using this one
may verify that adding the matter coupling $A_{\mu}j^{\mu}$ to (2.1) results
in the manifestly gauge invariant action

$S=\int{\rm
d}^{4}x\Big{(}\frac{1}{2}\widehat{A}^{i}\,\Box\,\widehat{A}_{i}-\frac{1}{2}\Phi\Delta\Phi-\Phi\rho-\widehat{A}_{i}\widehat{j}^{i}\Big{)}\,.$
(2.8)

#### Homotopy interpretation

We now give an algebraic interpretation of the passing over to gauge invariant
variables in terms of homotopy transfer. We first explain that the ‘data’ of a
free gauge theory, such as gauge transformations, equations of motion, and
Bianchi identities, can be encoded in a so-called chain complex. This is a
sequence of vector spaces $X_{i}$ with maps (abstract differentials)
$\partial_{i}:X_{i}\rightarrow X_{i-1}$ that square to zero in the sense that
$\partial_{i-1}\circ\partial_{i}=0$. (We note that this encodes the same
information as the BV-BRST differential, see, e.g., [30, 31].) For Maxwell’s
theory the chain complex reads

$\begin{array}[]{ccccccccccc}X_{1}&\xlongrightarrow{\partial_{1}}&X_{0}{}&\xlongrightarrow{\partial_{0}}&X_{-1}&\xlongrightarrow{\partial_{-1}}&X_{-2}\\\\[6.45831pt]
\\{\Lambda\\}&&\\{A_{\mu}\\}&&\\{E^{\mu}\\}&&\\{f\\}\end{array}$ (2.9)

where we have indicated in the second line the interpretation of each space:
$X_{1}$ is the space of gauge parameters $\Lambda$, $X_{0}$ is the space of
gauge fields $A_{\mu}$, $X_{-1}$ is the space of field equations $E^{\mu}$,
and $X_{-2}$ a space of scalars encoding the Bianchi identity. The
differentials are defined as

$\begin{split}\partial_{1}(\Lambda)_{\mu}\ &=\ \partial_{\mu}\Lambda\,,\\\
\partial_{0}(A)_{\mu}\ &=\
\Box\,A_{\mu}-\partial_{\mu}(\partial^{\nu}A_{\nu})\,,\\\ \partial_{-1}(E)\
&=\ \partial_{\mu}E^{\mu}\;.\end{split}$ (2.10)

The gauge transformations and field equations are then encoded in $\delta
A=\partial_{1}(\Lambda)$ and $\partial_{0}(A)=0$, respectively. The nilpotency
of the differential amounts to $\partial_{0}\circ\partial_{1}=0$ and
$\partial_{-1}\circ\partial_{0}=0$, which encode (linearized) gauge invariance
and the Bianchi identity, respectively. (In the following we will often skip
the subscript on $\partial$ and similar maps as the source and target spaces
are usually clear from the context.)

Next we define projection maps $p$ from this chain complex to the complex of
gauge invariant variables:

$\begin{array}[]{ccccccccccc}X_{1}&\xlongrightarrow{\partial}&X_{0}{}&\xlongrightarrow{\partial}&X_{-1}&\xlongrightarrow{\partial}&X_{-2}\\\\[6.45831pt]
\Big{\downarrow}{p}&&\Big{\downarrow}{p}&&\Big{\downarrow}{p}&&\Big{\downarrow}{p}\\\\[6.45831pt]
\\{0\\}&\xlongrightarrow{\bar{\partial}=0}&\bar{X}_{0}&\xlongrightarrow{\bar{\partial}}&\bar{X}_{-1}&\xlongrightarrow{\bar{\partial}=0}&\\{0\\}\end{array}$
(2.11)

Note, in particular, that the space of gauge parameters is projected to zero,
indicating that there are no gauge redundancies left, in agreement with the
interpretation of $\bar{X}_{0}$ as the space of gauge invariant fields. For
the fields the projector is defined by

$\bar{A}_{\mu}=p(A_{\mu})=(\bar{A}_{0},\bar{A}_{i})\equiv(\Phi,\widehat{A}_{i})\,,$
(2.12)

and consists of the gauge invariant variables defined above. For the space
$X_{-1}$ of field equations the projector is

$\bar{E}^{\mu}=p(E^{\mu})=E^{\mu}-(0,\partial^{i}\Delta^{-1}(\partial_{\mu}E^{\mu}))\,.$
(2.13)

From this definition it follows

$\partial_{\mu}\bar{E}^{\mu}\equiv 0\,,$ (2.14)

i.e., the projected space $\bar{X}_{-1}$ consists of 4-vectors with zero
divergence. All other projection maps in (2.11) are trivial. We also have to
introduce inclusion maps going in the other direction,
$\iota:\bar{X}\rightarrow X$, that simply view a projected object as an
element of the original space, i.e., $\iota(\bar{A}_{\mu})=\bar{A}_{\mu}$ and
$\iota(\bar{E}^{\mu})=\bar{E}^{\mu}$.

We are now ready to explain the homotopy relation that is the basis for all
our subsequent applications. We first note that, by definition, the
composition $p\circ\iota$ equals the identity on $\bar{X}$, but in general the
composition $\iota\circ p$ going the other direction does not equal the
identity on $X$ since the projection of course shrinks the space. However, in
the case at hand $\iota\circ p$ is equal to the identity up to homotopy, which
means that there are degree $+1$ maps $s_{i}:X_{i}\rightarrow X_{i+1}$ (going
in the opposite direction of the differential $\partial$) so that

$(\iota\circ p)_{i}={\rm id}_{X_{i}}-\partial_{i+1}\circ
s_{i}-s_{i-1}\circ\partial_{i}\;.$ (2.15)

The significance of this relation, to be explained in sec. 4, is that an
$L_{\infty}$ structure on $X$ (encoding non-linear gauge symmetries and
dynamics) can then be transported to an $L_{\infty}$ structure on $\bar{X}$
(for instance encoding non-linear dynamics without any gauge symmetries left).

Let us now define the maps $s_{i}:X_{i}\rightarrow X_{i+1}$ so that (2.15)
holds. Evaluating the homotopy relation on fields $A_{\mu}\in X_{0}$ one
obtains

$\iota p(A_{\mu})=\bar{A}_{\mu}=A_{\mu}-\partial_{\mu}(s_{0}(A))\,,$ (2.16)

where we assumed

$s_{-1}\equiv 0\,,$ (2.17)

whose consistency will be confirmed momentarily. Recalling (2.12) and the
definition (2.5) of gauge invariant variables we have

$\bar{A}_{\mu}=(\Phi,\widehat{A}_{i})=(A_{0}-\dot{\psi},A_{i}-\partial_{i}\psi)=A_{\mu}-\partial_{\mu}\psi\,.$
(2.18)

Therefore, the homotopy relation (2.16), which we also write as

$A_{\mu}=\bar{A}_{\mu}+\partial_{\mu}(s_{0}(A))\;,$ (2.19)

is obeyed for

$s_{0}(A)=\psi=\Delta^{-1}(\partial_{i}A^{i})\;,$ (2.20)

where we recalled the definition (2.3) of $\psi$ in terms of $A_{\mu}$. Next,
we evaluate the homotopy relation on $E^{\mu}\in X_{-1}$,

$\iota p(E^{\mu})=\bar{E}^{\mu}=E^{\mu}-s^{\mu}(\partial\cdot E)\;,$ (2.21)

where we used (2.17) and the definition (2.10) of the differential
$\partial(E)\equiv\partial\cdot E\equiv\partial_{\mu}E^{\mu}$. Recalling the
projection (2.13) of $E^{\mu}$ one infers that the homotopy relation is obeyed
upon setting for arbitrary $f\in X_{-2}$

$s^{\mu}(f)=(0,\partial^{i}\Delta^{-1}f)\;.$ (2.22)

Finally, having fixed the homotopy maps, we have to verify the homotopy
relation on $X_{1}$ and $X_{-2}$. For $\Lambda\in X_{1}$ we compute

$\iota p(\Lambda)=0=\Lambda-s(\partial\Lambda)\,,$ (2.23)

which is obeyed thanks to
$s(\partial\Lambda)=\Delta^{-1}\partial_{i}\partial^{i}\Lambda=\Lambda$, upon
using (2.20). On $f\in X_{-2}$ one finds

$\iota p(f)=0=f-\partial_{\mu}(s^{\mu}(f))\;,$ (2.24)

which indeed is identically satisfied for (2.22). Summarizing, we have shown
that for homotopy maps $s_{i}$ defined by (2.17), (2.20) and (2.22) the
homotopy relation (2.15) is satisfied. This proves that passing over to gauge
invariant variables can be interpreted as homotopy transfer.

###  2.2 Gravity on flat space

We now redo the above analysis for gravity linearized about flat Minkowski
space, as a preparation for the technically more involved case of cosmological
perturbation theory. Expanding the spacetime metric as
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, with $\eta_{\mu\nu}$ the Minkowski
metric, to second order in fluctuations, the Einstein-Hilbert action reduces
to the Fierz-Pauli action

$S_{\rm FP}=-\frac{1}{2}\int{\rm d}^{4}x\,h^{\mu\nu}G_{\mu\nu}(h)\,,$ (2.25)

where $G_{\mu\nu}$ denotes the linearized Einstein tensor

$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}R\,\eta_{\mu\nu}\;,$ (2.26)

with the linearized Ricci tensor

$R_{\mu\nu}\equiv-\frac{1}{2}\big{(}\Box\,h_{\mu\nu}-2\,\partial^{\rho}\partial_{(\mu}h_{\nu)\rho}+\partial_{\mu}\partial_{\nu}h\big{)}\;,$
(2.27)

and the linearized scalar curvature

$R\equiv\eta^{\mu\nu}R_{\mu\nu}=\partial^{\mu}\partial^{\nu}h_{\mu\nu}-\Box\,h\;.$
(2.28)

The Fierz-Pauli action is invariant under linearized diffeomorphisms

$\delta h_{\mu\nu}=\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu}\,,$ (2.29)

a fact that is also expressed in the Bianchi identity
$\partial^{\mu}G_{\mu\nu}\equiv 0$.

As for Maxwell’s theory, we will next rewrite linearized gravity in terms of
gauge invariant variables, illustrating in a simpler setting the techniques of
cosmological perturbation theory. In the same way that we decomposed a
3-vector into a divergence-free vector plus the gradient of a scalar one
begins with a so-called scalar-vector-tensor (SVT) decomposition. Here, a
symmetric tensor is decomposed into a traceless and divergence-free tensor
plus the trace part plus the gradients of vectors and scalars. Specifically,
we write

$\begin{split}h_{00}&=-2\phi\,,\\\ h_{0i}&={B}_{i}+\partial_{i}B\,,\\\
h_{ij}&=\widehat{h}_{ij}+2C\delta_{ij}+\partial_{i}{E}_{j}+\partial_{j}{E}_{i}+2(\partial_{i}\partial_{j}E-\tfrac{1}{3}\delta_{ij}\Delta
E)\;,\end{split}$ (2.30)

subject to the constraints

$\partial^{i}\widehat{h}_{ij}=\widehat{h}^{i}{}_{i}=\partial^{i}{E}_{i}=\partial^{i}B_{i}=0\,.$
(2.31)

We next determine the gauge transformations of the components fields, where we
subject the gauge parameter to the same decomposition
$\xi_{\mu}=(\xi_{0},\xi_{i})$, where

$\xi_{i}=\zeta_{i}+\partial_{i}\chi\,,\qquad\partial_{i}\zeta^{i}=0\,.$ (2.32)

One finds with (2.29)

$\begin{split}\delta h_{00}&=2\,\dot{\xi}_{0}\,,\\\ \delta
h_{0i}&=\dot{\zeta}_{i}+\partial_{i}(\dot{\chi}+\xi_{0})\,,\\\ \delta
h_{ij}&=\partial_{i}\zeta_{j}+\partial_{j}\zeta_{i}+2\,\partial_{i}\partial_{j}\chi\,.\end{split}$
(2.33)

From the first equation in (2.30) we then immediately infer

$\delta\phi=-\dot{\xi}_{0}\,.$ (2.34)

Next, varying the second equation in (2.30),

$\delta h_{0i}=\delta{B}_{i}+\partial_{i}\delta
B=\dot{\zeta}_{i}+\partial_{i}(\dot{\chi}+\xi_{0})\;,$ (2.35)

and comparing the divergence-free and gradient parts on both sides of the
equation we obtain

$\delta{B}_{i}=\dot{\zeta}_{i}\,,\qquad\delta B=\dot{\chi}+\xi_{0}\,.$ (2.36)

A similar analysis gives the following gauge transformations for the
components of $h_{ij}$:

$\begin{split}\delta{E}_{i}&=\zeta_{i}\,,\qquad\,\,\delta
C=\frac{1}{3}\,\Delta\chi\,,\\\ \delta
E&=\chi\,,\qquad\delta\widehat{h}_{ij}=0\,.\end{split}$ (2.37)

We learn that $\widehat{h}_{ij}$ is gauge invariant. In addition, it is easy
to see with (2.34), (2.36) and (2.37) that we can build three more gauge
invariant combinations:

$\begin{split}\Sigma_{i}\equiv\dot{{E}}_{i}-B_{i}\,,\qquad\Psi\equiv-C+\frac{1}{3}\Delta
E\,,\qquad\Phi\equiv\phi+\dot{B}-\ddot{E}\,.\end{split}$ (2.38)

It should be noted that these quantities are not unique in that any function
of gauge invariant objects is also gauge invariant. It is instructive to count
the number of gauge invariant quantities (say in four dimensions):
$\widehat{h}_{ij}$ has $6-3-1=2$ components (subtracting the trace and
divergence constraints from the six components of a symmetric 2-tensor in
three-dimensional space), $\Sigma_{i}$ has $2$ components (since it is
divergence-free), while $\Phi$ and $\Psi$ each have one degree of freedom,
making a total of six degrees of freedom. This is what one should expect,
since we started with the ten components of $h_{\mu\nu}$ which are subject to
gauge redundancies with parameters $\xi_{\mu}$ having four components.

We can now write the vacuum Einstein equations in terms of the gauge invariant
quantities $\widehat{h}_{ij},\Sigma_{i},\Phi$ and $\Psi$. After some algebra
one finds for the components of the Einstein tensor (2.26):

$\begin{split}G_{00}&=2\Delta\Psi\,,\\\
G_{0i}&=\frac{1}{2}\Delta\Sigma_{i}+2\partial_{i}\dot{\Psi}\,,\\\
G_{ij}&=-\frac{1}{2}\Box\,\widehat{h}_{ij}-\partial_{i}\partial_{j}\big{(}\Phi-\Psi\big{)}+\delta_{ij}\big{(}\Delta\Phi-\Delta\Psi+2\ddot{\Psi}\big{)}+\partial_{(i}\dot{\Sigma}_{j)}\,.\end{split}$
(2.39)

It is also instructive to analyze the gauge invariant content of linearized
gravity using these equations. From the equation $G_{00}=0$ we infer
$\Delta\Psi=0$, which by the assumed invertibility of $\Delta$ tells us
$\Psi=0$. Using this in the second equation $G_{0i}=0$ then tells us
$\Delta\Sigma_{i}=0$ and hence that $\Sigma_{i}=0$. The final equation
$G_{ij}=0$ then reduces to

$-\frac{1}{2}\Box\,\widehat{h}_{ij}-\partial_{i}\partial_{j}\Phi+\delta_{ij}\Delta\Phi=0\,.$
(2.40)

Taking the trace of this equation (recalling that $\widehat{h}_{ij}$ is
traceless) we learn $\Delta\Phi=0$ and hence $\Phi=0$, so that the equations
finally reduce to

$\Box\,\widehat{h}_{ij}=0\,.$ (2.41)

We thus find, as expected, that the gravitational field $h_{\mu\nu}$ carries
the two propagating degrees of freedom encoded in the divergence-free and
traceless $\widehat{h}_{ij}$.

Let us now analyze the action in terms of gauge invariant quantities.
Inserting (2.39) into the Fierz-Pauli action (2.25) it is straightforward to
bring it into the manifestly gauge invariant form

${\cal L}_{\rm
FP}=\frac{1}{4}\widehat{h}^{ij}\,\Box\,\widehat{h}_{ij}-\frac{1}{2}\Sigma^{i}\Delta\Sigma_{i}+(4\Phi-2\Psi)\Delta\Psi+6\Psi\ddot{\Psi}\,.$
(2.42)

This action is not diagonal, but it can be diagonalized by a suitable (space
non-local) field redefinition. It is convenient to do this computation right
away in the more general setting where one includes matter sources via an
energy-momentum tensor $T^{\mu\nu}$. The energy-momentum tensor may also be
subjected to an SVT decomposition:

$\begin{split}T^{00}&=\rho\,,\\\
T^{0i}&=q^{i}+\partial^{i}q\,,\quad\partial_{i}q^{i}=0\,,\\\
T^{ij}&=\Pi^{ij}+\partial^{i}\Pi^{j}+\partial^{j}\Pi^{i}+\partial^{i}\partial^{j}\Pi-\frac{1}{3}\delta^{ij}\Delta\Pi+p\delta^{ij}\,,\end{split}$
(2.43)

where

$\Pi^{i}{}_{i}=\partial_{i}\Pi^{ij}=\partial_{i}\Pi^{i}=0\,.$ (2.44)

The energy-momentum conservation law $\partial_{\mu}T^{\mu\nu}=0$ then implies

$\begin{split}\dot{\rho}+\Delta q&=0\,,\\\ \dot{q}^{j}+\Delta\Pi^{j}&=0\,,\\\
p+\dot{q}+\frac{2}{3}\Delta\Pi&=0\,.\end{split}$ (2.45)

Using these one finds for the coupling to gravity:

$\frac{1}{2}h_{\mu\nu}T^{\mu\nu}=\frac{1}{2}\widehat{h}_{ij}\Pi^{ij}-\Sigma_{i}q^{i}-\Phi\rho+\frac{3}{2}\Psi\,p\,,$
(2.46)

and hence for the complete Lagrangian:

$\begin{split}{\cal L}_{\rm
FP,\,matter}=&\,\frac{1}{4}\widehat{h}^{ij}\,\Box\,\widehat{h}_{ij}-\frac{1}{2}\Sigma^{i}\Delta\Sigma_{i}+(4\Phi-2\Psi)\Delta\Psi+6\Psi\ddot{\Psi}\\\
&+\frac{1}{2}\widehat{h}_{ij}\Pi^{ij}-\Sigma_{i}q^{i}-\Phi\rho-3\Psi\,p\,.\end{split}$
(2.47)

We note that the components of the energy-momentum tensor $q$, $\Pi^{i}$, and
$\Pi$ have dropped out, i.e., we lost four degrees of freedom, which matches
the number of conservation constraints.

Our next task is to diagonalize this action so that it expresses the physical
content directly. We can perform the following field redefinition of $\Phi$
(leaving $\Psi$ unchanged):

$\Phi\rightarrow\Phi^{\prime}=\Phi-\frac{1}{2}\Psi+\frac{3}{2}\Delta^{-1}\ddot{\Psi}\,.$
(2.48)

This implies

$(4\Phi-2\Psi)\Delta\Psi=4\Phi^{\prime}\Delta\Psi-6\Delta^{-1}\ddot{\Psi}\Delta\Psi\,,$
(2.49)

so that after cancelling the $\Delta$’s this cancels the $\Psi\ddot{\Psi}$
term. We obtain for the action in the new variables:

$\begin{split}{\cal L}_{\rm
FP}=&\,\frac{1}{4}\widehat{h}^{ij}\,\Box\,\widehat{h}_{ij}-\frac{1}{2}\Sigma^{i}\Delta\Sigma_{i}+4\Phi^{\prime}\Delta\Psi\\\
&+\frac{1}{2}\widehat{h}_{ij}\Pi^{ij}-\Sigma_{i}q^{i}-\Phi^{\prime}\rho-\frac{1}{2}\Psi\rho+\frac{3}{2}\dot{\Psi}q-3\Psi\,p\,,\end{split}$
(2.50)

where we integrated by parts and used the conservation law
$\Delta^{-1}\dot{\rho}=-q$. Integrating by parts again and using the last of
(2.45) this can also be rewritten as

$\begin{split}{\cal L}_{\rm
FP}=&\,\frac{1}{4}\widehat{h}^{ij}\,\Box\,\widehat{h}_{ij}-\frac{1}{2}\Sigma^{i}\Delta\Sigma_{i}+4\Phi\Delta\Psi\\\
&+\frac{1}{2}\widehat{h}_{ij}\Pi^{ij}-\Sigma_{i}q^{i}-\Phi\rho-\frac{1}{2}\Psi(\rho+3p-2\Delta\Pi)\,,\end{split}$
(2.51)

where we dropped the prime on $\Phi$. We could further diagonalize the terms
in the first line by defining

$\Phi_{\pm}:=2\Psi\pm\Phi\,,$ (2.52)

for which

$\begin{split}{\cal L}_{\rm
FP}=&\,\frac{1}{4}\widehat{h}^{ij}\,\Box\,\widehat{h}_{ij}-\frac{1}{2}\Sigma^{i}\Delta\Sigma_{i}+\frac{1}{2}\Phi_{+}\Delta\Phi_{+}-\frac{1}{2}\Phi_{-}\Delta\Phi_{-}\\\
&+\frac{1}{2}\widehat{h}_{ij}\Pi^{ij}-\Sigma_{i}q^{i}-\frac{1}{2}(\Phi_{+}-\Phi_{-})\rho-\frac{1}{8}(\Phi_{+}+\Phi_{-})(\rho+3p-2\Delta\Pi)\,.\end{split}$
(2.53)

This is linearized gravity expressed in purely physical (gauge invariant)
quantities. Let us finally note that we may integrate out $\Sigma_{i}$ and
$\Phi_{\pm}$ by solving their own equations of motion, using the invertibility
of $\Delta$.

###  2.3 Homotopy interpretation of linearized gravity

We now interpret the gauge invariant variables for gravity in terms of
homotopy transfer in the same way as for Maxwell’s theory, focusing for
simplicity on the vacuum case. We start from the chain complex defining
linearized gravity

$\begin{array}[]{ccccccccccc}X_{1}&\xlongrightarrow{\partial}&X_{0}{}&\xlongrightarrow{\partial}&X_{-1}&\xlongrightarrow{\partial}&X_{-2}\\\\[6.45831pt]
\\{\xi_{\mu}\\}&&\\{h_{\mu\nu}\\}&&\\{E_{\mu\nu}\\}&&\\{F_{\mu}\\}\end{array}$
(2.54)

where the spaces $X_{0}$ and $X_{-1}$ consist of symmetric but otherwise
unconstrained Lorentz tensors (so that, for instance, $E_{\mu\nu}\in X_{-1}$
in general has a non-vanishing divergence). The differentials are defined by

$\begin{split}\partial(\xi)_{\mu\nu}&=\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu}\;,\\\
\partial(h)_{\mu\nu}&=G_{\mu\nu}(h)\,,\\\
\partial({E})_{\mu}&=\partial^{\nu}{E}_{\nu\mu}\,,\end{split}$ (2.55)

where $G_{\mu\nu}(h)$ is the linearized Einstein tensor (2.26). As before, the
relation $\partial^{2}=0$ amounts to gauge invariance of $G_{\mu\nu}$ (when
evaluated on $\xi\in X_{1}$) and to the Bianchi identity
$\partial^{\mu}G_{\mu\nu}=0$ (when evaluated on $h\in X_{0}$).

In the same way as indicated in the diagram (2.11) we want to map with $p$
down to the space of gauge invariant fields $\bar{X}_{0}$, and
$\bar{X}_{1}=\\{0\\}$ since there is then no gauge redundancy left. We write

$p(h_{\mu\nu})=\bar{h}_{\mu\nu}=\begin{pmatrix}\bar{h}_{ij}&\bar{h}_{0i}\\\
\bar{h}_{i0}&\bar{h}_{00}\end{pmatrix}\,,$ (2.56)

where in terms of the gauge invariant variables

$\bar{h}_{ij}=\widehat{h}_{ij}-2\Psi\delta_{ij}\;,\qquad\bar{h}_{0i}=-\Sigma_{i}\;,\qquad\bar{h}_{00}=-2\Phi\;.$
(2.57)

Note that from these definitions it follows that

$\partial^{i}\bar{h}_{ij}-\tfrac{1}{3}\partial_{j}(\bar{h}^{i}{}_{i})=0\,,\qquad\partial^{i}\bar{h}_{i0}=0\;.$
(2.58)

This is formally identical to the transverse gauge condition (but conceptually
different because no gauge has been chosen). In order to identify the homotopy
map so that the homotopy relation (2.15) holds it is convenient to write the
original field $h_{\mu\nu}$ in terms of the projected (gauge invariant) field
plus a pure gauge term,

$h_{\mu\nu}=\bar{h}_{\mu\nu}+\partial_{\mu}A_{\nu}+\partial_{\nu}A_{\mu}\;,$
(2.59)

where

$A_{\mu}=(A_{0},A_{i})=(B-\dot{E},E_{i}+\partial_{i}E)\;,$ (2.60)

as follows quickly with the formulas of the previous section. We now want to
define a homotopy map $s:X_{0}\rightarrow X_{1}$ so that (2.15) holds, for
which we compute

$(\iota p-{\rm
id})(h_{\mu\nu})=\bar{h}_{\mu\nu}-h_{\mu\nu}=-\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}=-\partial(A)_{\mu\nu}\;,$
(2.61)

where we recalled that the inclusion map $\iota:\bar{X}_{0}\rightarrow X_{0}$
is simply $\iota(\bar{h}_{\mu\nu})=\bar{h}_{\mu\nu}$. Assuming that the second
term on the right-hand side of the homotopy relation (2.15) vanishes due to
$s_{-1}=0$ we infer

$s_{0}(h)_{\mu}=A_{\mu}\in X_{1}\,,$ (2.62)

where we mean that $A_{\mu}$ is defined in terms of the SVT-decomposed
components of $h_{\mu\nu}$ as in (2.60). Next, we have to check the homotopy
relation on $\xi\in X_{1}$, which reads

$(\iota p-{\rm id})(\xi)=-\xi=-s_{0}(\partial\xi)\,,$ (2.63)

since $p(\xi)=0$. This relation is satisfied since $\partial(\xi)$ by (2.55)
is already of the pure gauge form and hence $s(\partial\xi)=\xi$.

Let us now determine the homotopy map on $X_{-2}$. We first need to specify
how the projection map is extended to $X_{-1}$ and $X_{-2}$. For $E\in X_{-1}$
we set

$\bar{E}_{\mu\nu}\equiv
p({E})_{\mu\nu}=\begin{pmatrix}{E}_{00}&{E}_{0j}-\partial_{j}\Delta^{-1}(\partial^{\nu}{E}_{\nu
0})\\\ {E}_{i0}-\partial_{i}\Delta^{-1}(\partial^{\nu}{E}_{\nu
0})&{E}_{ij}-2\partial_{(i}\Delta^{-1}(\partial^{\nu}{E}_{j)\nu})+\partial_{i}\partial_{j}\Delta^{-2}(\partial^{\mu}\partial^{\nu}{E}_{\mu\nu})\end{pmatrix}.$
(2.64)

This implements the projection to divergence-free Lorentz tensors:

$\partial^{\mu}\bar{E}_{\mu\nu}\equiv 0\;,$ (2.65)

and so $\bar{X}_{-1}$ consistent of symmetric tensors with vanishing
divergence. Consequently, there is no non-trivial differential to
$\bar{X}_{-2}$ and hence it is natural to assume that this space trivializes.
Thus, we set for $F\in X_{-2}$

$p(F)=0\;.$ (2.66)

We now claim that the homotopy map from $X_{-2}$ to $X_{-1}$ acting on
$F_{\mu}=(F_{0},F_{i})$ is given by

$s(F)_{\mu\nu}=\begin{pmatrix}s(F)_{00}&s(F)_{0j}\\\
s(F)_{i0}&s(F)_{ij}\end{pmatrix}=\begin{pmatrix}0&\partial_{j}\Delta^{-1}F_{0}\\\
\partial_{i}\Delta^{-1}F_{0}&2\partial_{(i}\Delta^{-1}F_{j)}-\partial_{i}\partial_{j}\Delta^{-2}(\partial_{\mu}F^{\mu})\end{pmatrix}.$
(2.67)

Let us verify the homotopy relation on ${E}\in X_{-1}$. We compute

$\begin{split}(\iota p-{\rm
id})({E})_{\mu\nu}&=\begin{pmatrix}0&-\partial_{i}\Delta^{-1}(\partial^{\nu}{E}_{\nu
0})\\\ -\partial_{i}\Delta^{-1}(\partial^{\nu}{E}_{\nu
0})&-2\partial_{(i}\Delta^{-1}(\partial^{\nu}{E}_{j)\nu})+\partial_{i}\partial_{j}\Delta^{-2}(\partial^{\mu}\partial^{\nu}{E}_{\mu\nu})\end{pmatrix}\\\
&=-s(\partial{E})_{\mu\nu}\;,\end{split}$ (2.68)

where the second equality follows with (2.67). Since ${s}({E})=0$ this
establishes the homotopy relation. Finally, for the homotopy relation on $F\in
X_{-2}$ we have with (2.66)

$(\iota p-{\rm id})(F)_{\mu}=-F_{\mu}\,,$ (2.69)

while we compute for the right-hand side, using $S(\partial(F))=0$,

$\begin{split}-\partial(s(F))_{\mu}=-\partial^{\nu}s(F)_{\nu\mu}&=-\begin{pmatrix}\partial^{0}s(F)_{00}+\partial^{i}s(F)_{i0}\\\
\partial^{0}s(F)_{0i}+\partial^{j}s(F)_{ji}\end{pmatrix}\\\
&=\begin{pmatrix}-F_{0}\\\
-\partial_{i}\Delta^{-1}(\partial^{0}F_{0})-\partial_{i}\Delta^{-1}(\partial^{j}F_{j})-F_{i}+\partial_{i}\Delta^{-1}(\partial_{\mu}F^{\mu})\end{pmatrix}\\\
&=\begin{pmatrix}-F_{0}\\\ -F_{i}\end{pmatrix}\,.\end{split}$ (2.70)

This establishes the homotopy relation and completes our discussion of the
homotopy interpretation of gauge invariant variables in linearized gravity.

##  3 Gauge invariant closed string theory

In this section we analyze the free action for the massless fields of closed
bosonic string theory on a constant background in terms of gauge invariant
field variables and show again that it can be interpreted as homotopy
transfer. We employ the formalism of double field theory, subjected to a
space/time split. This theory shows new features due to the chain complex
being extended by a space of trivial parameters and a space of Bianchi
identities for Bianchi identities (which in turn become important for the non-
linear gauge algebra which is a genuine $L_{\infty}$ algebra).

###  3.1 Linearized double field theory on flat backgrounds

Our starting point is double field theory, which formulates the universal
sector of the target space action of closed string theory, consisting of
metric, B-field and dilaton, in terms of fields living on a doubled space [38,
39, 40, 41]. The doubling of coordinates is a feature of string theory on
toroidal backgrounds, where they encode momentum and winding modes [42], but
as far as the free quadratic theory is concerned the doubling of coordinates
is consistent for any background with constant metric and B-field. Since the
free theory is all we need for the following discussion we will not elaborate
on the physical meaning of the doubled coordinates any further.

We use double field theory in a space/time split, as worked out by Naseer in
[43], and expand the action to quadratic order in fluctuations. The details of
this computation will be detailed in a separate publication for the more
general case of time-dependent backgrounds, so here we only display the
result. The field content is given by

$h_{a\bar{b}}\;,\quad{\cal A}_{a}\;,\quad{\cal
A}_{\bar{a}}\;,\quad\phi\;,\quad\varphi\;,$ (3.1)

where $h_{a\bar{b}}$ encodes the spatial components of the metric and
$B$-field fluctuations, the vectors ${\cal A}_{a}$ and ${\cal A}_{\bar{a}}$
encode (linear combinations of) the space-time component of the metric and
$B$-field fluctuations, respectively, while $\varphi$ encodes the fluctuations
of the dilaton and $\phi$ encodes the fluctuation of the time-time component
of the metric (the lapse function). All fields depend on time $t$ and doubled
spatial coordinates $x^{M}=(\tilde{x}_{i},x^{i})$, where $i=1,...,d$ and
$M=1,...,2d$, with dual derivatives $\partial_{M}$. The dependence of the
fields on doubled coordinates is subject to the level-matching constraint

$\eta^{MN}\partial_{M}\partial_{N}f=2\tilde{\partial}^{i}\partial_{i}f=0\,,\qquad\eta_{MN}\
=\ \begin{pmatrix}0&\delta^{i}{}_{j}\\\ \delta_{i}{}^{j}&0\end{pmatrix}\;,$
(3.2)

where $f$ stands for any of the fields, and $\eta_{MN}$ is the $O(d,d)$
invariant metric that is used to raise and lower indices $M,N,\ldots$. The
action is written in terms of the above fields and the background metric and
$B$-field that are conveniently encoded in a constant background frame
$E_{A}{}^{M}$, with flat index $A=(a,\bar{a})$, which is subject to the
constraint that the following metric is block-diagonal:

${\cal G}_{AB}\ \equiv\ E_{A}{}^{M}E_{B}{}^{N}\eta_{MN}\ =\
\begin{pmatrix}-\delta_{ab}&0\\\ 0&\delta_{\bar{a}\bar{b}}\end{pmatrix}\;.$
(3.3)

This metric is used to raise and lower flat indices $a,b,\ldots$ and
$\bar{a},\bar{b},\ldots$. Furthermore, we define the derivative operators

$D_{A}\equiv E_{A}{}^{M}\partial_{M}\equiv(D_{a},D_{\bar{a}})\,,$ (3.4)

in terms of which the constraint (3.2) reads

$\Delta\equiv-{2}D^{a}D_{a}=2D^{\bar{a}}D_{\bar{a}}\;.$ (3.5)

Thus, despite the doubling of spatial coordinates there is a unique Laplacian.

We are now ready the write the quadratic double field theory action:

$\begin{split}{\cal L}_{\rm
DFT}=&-4(D_{t}\varphi)^{2}-D_{t}h^{a\bar{b}}D_{t}h_{a\bar{b}}\\\ &+8D^{a}\phi
D_{a}\varphi-8D^{a}\varphi D_{a}\varphi-8D_{a}\varphi
D_{\bar{b}}h^{a\bar{b}}+4D_{a}\phi D_{\bar{b}}h^{a\bar{b}}\\\
&-2D^{a}h^{b\bar{c}}D_{a}h_{b\bar{c}}+2D^{c}h^{a\bar{b}}D_{a}h_{c\bar{b}}-2D^{\bar{c}}h^{a\bar{b}}D_{\bar{b}}h_{a\bar{c}}\;,\end{split}$
(3.6)

where we defined the covariant time derivative operators

$\begin{split}D_{t}\varphi&=\partial_{t}\varphi+\frac{1}{2}D_{a}{\cal
A}^{a}+\frac{1}{2}D_{\bar{a}}{\cal A}^{\bar{a}}\,,\\\
D_{t}h_{a\bar{b}}&=\partial_{t}h_{a\bar{b}}-D_{a}{\cal
A}_{\bar{b}}+D_{\bar{b}}{\cal A}_{a}\,.\end{split}$ (3.7)

As unbarred and barred indices are consistently contracted the above action
has a manifest $SO(d)_{L}\times SO(d)_{R}$ invariance, a double copy structure
inherited from the full string theory. There is also an $O(d,d)$ duality,
i.e., an invariance under a simultaneous action on both the fields and the
background. We note that the action is the complete quadratic action for the
massless string fields as there is a field basis for which the higher-
derivative $\alpha^{\prime}$ corrections become only relevant at cubic level.
The action is invariant under the gauge symmetries with parameters
$(\xi_{a},\xi_{\bar{a}},\xi^{0},\xi_{0})$, given by

$\begin{split}\delta h_{a\bar{b}}&=D_{a}\xi_{\bar{b}}-D_{\bar{b}}\xi_{a}\,,\\\
\delta\phi&=\dot{\xi}^{0}\,,\\\
\delta\varphi&=-\frac{1}{2}D_{a}\xi^{a}-\frac{1}{2}D_{\bar{a}}\xi^{\bar{a}}\,,\\\
\delta{\cal A}_{a}&=\dot{\xi}_{a}+D_{a}({\xi}_{0}+\xi^{0})\,,\\\ \delta{\cal
A}_{\bar{a}}&=\dot{\xi}_{\bar{a}}+D_{\bar{a}}({\xi}_{0}-\xi^{0})\,.\end{split}$
(3.8)

There is a gauge symmetry for gauge symmetries since gauge parameters of the
form

$\xi_{a}=D_{a}\chi\;,\quad\xi_{\bar{a}}=D_{\bar{a}}\chi\;,\quad{\xi}_{0}=-\dot{\chi}\;,$
(3.9)

where $\chi$ is an arbitrary scalar, do not generate a transformation of
fields. Note that the derivatives (3.7) are invariant under gauge
transformations w.r.t. $(\xi_{a},\xi_{\bar{a}})$. They are also invariant
under $\xi_{0}$ transformations thanks to the constraint (3.5). This leaves
$\xi^{0}$ transformations as the only symmetry linking the terms in the first
line of (3.6) to the rest of the action. The total invariance, subject to the
level-matching constraint (3.5), is easy to verify. It should be emphasized
that at the level of the free theory one does not have to worry about how to
implement the constraint (3.5) on products of fields, since under an integral
it is sufficient that all fields and gauge parameters satisfy the constraint
[39].

Our goal in the following is to rewrite linearized double field theory in
terms of gauge invariant variables. We start by subjecting the fields to a
scalar-vector-tensor (SVT) decomposition as for conventional gravity, writing

$\begin{split}h_{a\bar{b}}&=\widehat{h}_{a\bar{b}}+D_{a}B_{\bar{b}}-D_{\bar{b}}B_{a}+D_{a}D_{\bar{b}}E\,,\\\
{\cal A}_{a}&=A_{a}+D_{a}A\,,\\\ {\cal
A}_{\bar{a}}&=A_{\bar{a}}+D_{\bar{a}}\bar{A}\,,\end{split}$ (3.10)

where

$D^{a}\,\widehat{h}_{a\bar{b}}=D^{\bar{b}}\,\widehat{h}_{a\bar{b}}=0\;,\qquad
D^{a}B_{a}=D^{\bar{a}}B_{\bar{a}}=0\;,\qquad
D^{a}A_{a}=D^{\bar{a}}A_{\bar{a}}=0\,.$ (3.11)

Note that, in contrast to conventional gravity, we have not taken out the
trace part of $\widehat{h}_{a\bar{b}}$ since there is no $SO(d)_{L}\times
SO(d)_{R}$ invariant way to do so. Similarly, we write for the gauge
parameter:

$\xi_{a}=\zeta_{a}+D_{a}\lambda\,,\qquad\xi_{\bar{a}}=\zeta_{\bar{a}}+D_{\bar{a}}\bar{\lambda}\,,$
(3.12)

where

$D^{a}\zeta_{a}=D^{\bar{a}}\zeta_{\bar{a}}=0\,.$ (3.13)

It is straightforward to work out the gauge transformations of the SVT
components:

$\begin{split}\delta\widehat{h}_{a\bar{b}}&=0\,,\qquad\delta
B_{a}=\zeta_{a}\,,\qquad\delta
B_{\bar{a}}=\zeta_{\bar{a}}\,,\qquad\qquad\qquad\delta
E=\bar{\lambda}-\lambda\,,\\\ \delta A_{a}&=\dot{\zeta}_{a}\,,\quad\;\;\delta
A_{\bar{a}}=\dot{\zeta}_{\bar{a}}\,,\qquad\;\;\delta
A=\dot{\lambda}+\tilde{\xi}_{0}+\xi^{0}\,,\qquad\delta\bar{A}=\dot{\bar{\lambda}}+\tilde{\xi}_{0}-\xi^{0}\,,\\\
\delta\varphi&=\frac{1}{4}\Delta(\lambda-\bar{\lambda})\,,\qquad\qquad\quad\delta\phi=\dot{\xi}^{0}\,.\end{split}$
(3.14)

We observe that $\widehat{h}_{a\bar{b}}$ is gauge invariant. In addition, it
is easy to see that one may define two gauge invariant vector modes and two
gauge invariant scalar modes:

$\begin{split}\widehat{A}_{a}&=A_{a}-\dot{B}_{a}\,,\\\
\widehat{A}_{\bar{a}}&=A_{\bar{a}}-\dot{B}_{\bar{a}}\,,\\\
\Psi&=\phi-\frac{1}{2}\frac{d}{dt}(A-\bar{A})-\frac{1}{2}\ddot{E}\,,\\\
\Phi&=\varphi+\frac{1}{4}\Delta E\,.\end{split}$ (3.15)

Thus, compared to pure gravity (c.f. (2.38)) we have one more vector mode,
which corresponds to the vector component of the B-field. Somewhat
surprisingly, however, we do not obtain an additional scalar mode
corresponding to the dilaton. In order to clarify this point let us count the
total number of off-shell field components and of gauge invariant quantities,
taking for definiteness $d=3$. There are then 17 off-shell field components
distributed as

$h_{a\bar{b}}\,:\;\;9\,\quad{\cal A}_{a}\,:\;\;3\,\quad{\cal
A}_{\bar{a}}\,:\;\;3\,\quad\phi\,:\;\;1\,\quad\varphi\,:\;\;1\,.$ (3.16)

Naively there are eight gauge redundancies ($3+3$ for $\xi_{a}$ and
$\xi_{\bar{a}}$ and $1+1$ for ${\xi}_{0}$ and $\xi^{0}$), but there is the
one-parameter gauge symmetry of gauge symmetries (3.9), leaving seven gauge
redundancies of fields. We thus expect ten gauge invariant combinations, and
indeed the counting works with the above gauge invariant variables:

$10\ =\ 4(\widehat{h}_{a\bar{b}})\ +\ 2(\widehat{A}_{a})\ +\
2(\widehat{A}_{\bar{a}})\ +\ 1(\Psi)\ +\ 1(\Phi)\,.$ (3.17)

Note, in particular, that $\widehat{h}_{a\bar{b}}$ encodes in a gauge
invariant way all four propagating degrees of freedom, i.e., the spin-2 tensor
modes with two degrees of freedom, the scalar mode given by the B-field and
the scalar mode given by the dilaton.

We now return to the quadratic double field theory Lagrangian (3.6) and
express it in terms of the gauge invariant variables. It is a straightforward
computation to verify that the Lagrangian can be written as

${\cal
L}=-4\dot{\Phi}^{2}-\widehat{h}^{a\bar{b}}\,\Box\,\widehat{h}_{a\bar{b}}+\frac{1}{2}\widehat{A}^{a}\Delta\widehat{A}_{a}-\frac{1}{2}\widehat{A}^{\bar{a}}\Delta\widehat{A}_{\bar{a}}-4\Phi\Delta\Phi+4\Psi\Delta\Phi\,,$
(3.18)

where $\Box=-\frac{\partial^{2}}{\partial t^{2}}+\Delta$ is the d’Alembert
operator. As encountered before for conventional gravity, the action is not
diagonal and includes a term with time derivatives for $\Phi$. This does not
mean that $\Phi$ is propagating, however, since we may perform the following
field redefinition

$\Psi\rightarrow\Psi^{\prime}=\Psi-\Phi+\Delta^{-1}\ddot{\Phi}\,,$ (3.19)

which removes the $\dot{\Phi}^{2}$ term, after which the Lagrangian reads

${\cal
L}=-\widehat{h}^{a\bar{b}}\,\Box\,\widehat{h}_{a\bar{b}}+\frac{1}{2}\widehat{A}^{a}\Delta\widehat{A}_{a}-\frac{1}{2}\widehat{A}^{\bar{a}}\Delta\widehat{A}_{\bar{a}}+4\Psi^{\prime}\Delta\Phi\;.$
(3.20)

This form makes it manifest that only $\widehat{h}^{a\bar{b}}$ propagates,
while the other modes could be integrated out to eliminate them, i.e., in the
vacuum case the Lagrangian reduces to ${\cal
L}=-\widehat{h}^{a\bar{b}}\,\Box\,\widehat{h}_{a\bar{b}}$.

###  3.2 Chain complex of double field theory

In this subsection we set the stage for the interpretation of gauge invariant
variables in double field theory in terms of homotopy transfer by defining the
chain complex encoding the free theory. This brings in some conceptually new
features due to the chain complex being extended compared to Yang-Mills and
gravity theories. The chain complex reads

$\begin{array}[]{ccccccccccc}X_{2}&\xlongrightarrow{\partial}&X_{1}&\xlongrightarrow{\partial}&X_{0}{}&\xlongrightarrow{\partial}&X_{-1}&\xlongrightarrow{\partial}&X_{-2}&\xlongrightarrow{\partial}&X_{-3}\\\\[6.45831pt]
\\{\chi\\}&&\\{\xi\\}&&\\{\Psi\\}&&\\{{\cal E}\\}&&\\{{\cal
G}\\}&&\\{\rho\\}\end{array}$ (3.21)

We have to define each vector space and each differential map: $X_{2}$ is the
space of trivial gauge parameters, which here are scalars $\chi$. $X_{1}$ is
the space of gauge parameters $\xi$, and $X_{0}$ is the space of fields
$\Psi$, which read in terms of field components

$\xi=\begin{pmatrix}\xi^{a}\\\ \xi^{\bar{a}}\\\ \xi_{0}\\\
\xi^{0}\end{pmatrix}\in X_{1}\;,\qquad\Psi=\begin{pmatrix}h_{a\bar{b}}\\\
\phi\\\ \varphi\\\ {\cal A}_{a}\\\ {\cal A}_{\bar{a}}\end{pmatrix}\in
X_{0}\;.$ (3.22)

Next, $X_{-1}$ is the space of field equations ${\cal E}$, which have the same
index structure as fields, and $X_{-2}$ is the space of Bianchi identities,
which have the same index structure as the gauge parameters. Finally, $X_{-3}$
is the space of Bianchi identities for Bianchi identities, which have the same
index structure as the trivial gauge parameters and are thus scalars. More
abstractly, we have the following equivalence between vector spaces and their
duals:

$X_{-1}\cong(X_{0})^{\star}\,,\quad X_{-2}\cong(X_{1})^{\star}\,,\quad
X_{-3}\cong(X_{2})^{\star}\,.$ (3.23)

Let us now give the differential maps $\partial$. From the form of the trivial
gauge parameters (3.9) we read off for $\partial_{2}:X_{2}\rightarrow X_{1}$,

$\partial(\chi)=\begin{pmatrix}D_{a}\chi\\\ D_{\bar{a}}\chi\\\ -\dot{\chi}\\\
0\end{pmatrix}\in X_{1}\;.$ (3.24)

Next, the gauge transformations (3.8) determine the differential
$\partial_{1}:X_{1}\rightarrow X_{0}$ via $\delta_{\xi}\Psi=\partial(\xi)$:

$\begin{split}\partial(\xi)=\begin{pmatrix}D_{a}\xi_{\bar{b}}-D_{\bar{b}}\xi_{a}\\\
\dot{\xi}^{0}\\\
-\frac{1}{2}D_{a}\xi^{a}-\frac{1}{2}D_{\bar{a}}\xi^{\bar{a}}\\\
\dot{\xi}_{a}+D_{a}({\xi}_{0}+\xi^{0})\\\
\dot{\xi}_{\bar{a}}+D_{\bar{a}}({\xi}_{0}-\xi^{0})\end{pmatrix}\in
X_{0}\,.\end{split}$ (3.25)

The field equations ${\cal E}=0$ are encoded in the differential
$\partial_{0}:X_{0}\rightarrow X_{-1}$ via

$\begin{pmatrix}{\cal E}_{a\bar{b}}\\\ {\cal E}_{\phi}\\\ {\cal
E}_{\varphi}\\\ {\cal E}_{a}\\\ {\cal E}_{\bar{a}}\end{pmatrix}\ =\
\partial\begin{pmatrix}h_{a\bar{b}}\\\ \phi\\\ \varphi\\\ {\cal A}_{a}\\\
{\cal A}_{\bar{a}}\end{pmatrix}\in X_{-1}\,,$ (3.26)

for which one obtains upon variation of (3.6)

$\begin{split}{\cal
E}_{a\bar{b}}&=\partial_{t}(D_{t}h_{a\bar{b}})-2D_{a}D_{\bar{b}}\phi+{\cal
R}_{a\bar{b}}\,,\\\ {\cal E}_{\phi}&={\cal R}\;,\\\ {\cal
E}_{\varphi}&=\partial_{t}(D_{t}\varphi)+\frac{1}{2}\Delta\phi+{\cal R}\,,\\\
{\cal
E}_{a}&=D_{a}(D_{t}\varphi)+\frac{1}{2}D^{\bar{b}}(D_{t}h_{a\bar{b}})\,,\\\
{\cal
E}_{\bar{a}}&=D_{\bar{a}}(D_{t}\varphi)-\frac{1}{2}D^{b}(D_{t}h_{b\bar{a}})\,,\end{split}$
(3.27)

with the spatial curvatures

$\begin{split}{\cal R}&=-\Delta\varphi+D_{a}D_{\bar{b}}h^{a\bar{b}}\,,\\\
{\cal R}_{a\bar{b}}&=2\left(-\frac{1}{2}\Delta
h_{a\bar{b}}-D_{a}D^{c}h_{c\bar{b}}+D_{\bar{b}}D^{\bar{c}}h_{a\bar{c}}+2D_{a}D_{\bar{b}}\varphi\right)\;.\end{split}$
(3.28)

The gauge invariance of (3.27), and thus the nilpotency
$\partial_{0}\circ\partial_{1}=0$, may be easily verified, using in particular

$\delta(D_{t}h_{a\bar{b}})=2D_{a}D_{\bar{b}}\xi^{0}\;,\qquad\delta(D_{t}\varphi)=-\frac{1}{2}\Delta\xi^{0}\;.$
(3.29)

We next observe that the curvatures (3.28) obey the Bianchi identities

$\begin{split}&D^{a}{\cal R}_{a\bar{b}}-2D_{\bar{b}}{\cal R}=0\,,\\\
&D^{\bar{b}}{\cal R}_{a\bar{b}}+2D_{a}{\cal R}=0\,,\end{split}$ (3.30)

from which in turn one may derive Bianchi identities for the ${\cal E}$. These
identities are encoded in the next differential
$\partial_{-1}:X_{-1}\rightarrow X_{-2}$:

$\partial{\cal E}\equiv\begin{pmatrix}(\partial{\cal E})^{a}\\\ (\partial{\cal
E})^{\bar{a}}\\\ (\partial{\cal E})_{0}\\\ (\partial{\cal
E})^{0}\end{pmatrix}\equiv\partial\begin{pmatrix}{\cal E}_{a\bar{b}}\\\ {\cal
E}_{\phi}\\\ {\cal E}_{\varphi}\\\ {\cal E}_{a}\\\ {\cal
E}_{\bar{a}}\end{pmatrix}=\begin{pmatrix}D_{\bar{b}}{\cal
E}^{a\bar{b}}+2D^{a}{\cal E}_{\varphi}-2\partial_{t}{\cal E}^{a}\\\ D_{b}{\cal
E}^{b\bar{a}}-2D^{\bar{a}}{\cal E}_{\varphi}+2\partial_{t}{\cal
E}^{\bar{a}}\\\ D_{a}{\cal E}^{a}+D_{\bar{a}}{\cal E}^{\bar{a}}\\\ D_{a}{\cal
E}^{a}-D_{\bar{a}}{\cal E}^{\bar{a}}-\partial_{t}{\cal
E}_{\phi}\,\end{pmatrix}\,.$ (3.31)

Indeed, for ${\cal E}$ given by (3.27) these combinations vanish identically,
which proves $\partial_{-1}\circ\partial_{0}=0$. Finally, the differential
$\partial_{-2}:X_{-1}\rightarrow X_{-3}$, defined by

$\partial\begin{pmatrix}{\cal G}^{a}\\\ {\cal G}^{\bar{a}}\\\ {\cal G}_{0}\\\
{\cal G}^{0}\end{pmatrix}\equiv\dot{\cal G}_{0}+\frac{1}{2}D_{a}{\cal
G}^{a}-\frac{1}{2}D_{\bar{a}}{\cal G}^{\bar{a}}\;,$ (3.32)

encodes Bianchi identities of Bianchi identities. Indeed, for ${\cal G}$’s
given by the right-hand side of (3.31) this vanishes identically, irrespective
of what the ${\cal E}$’s are, establishing
$\partial_{-2}\circ\partial_{{-1}}=0$. This completes our discussion of the
chain complex encoding the gauge structure and dynamics of linearized double
field theory on flat space.

###  3.3 Homotopy interpretation

We now turn to the homotopy transfer from the chain complex (3.21) to a
projected complex in which in particular the fields are gauge invariant and so
the gauge symmetries trivialize. Since we now also have gauge for gauge
symmetries it is in principle conceivable that the space of gauge parameters
is projected to something non-zero but trivial, but we will see in the
following that the natural homotopy relations work with

$p_{2}=p_{1}=0\;,$ (3.33)

and so we indeed take $\bar{X}_{2}=\\{0\\}=\bar{X}_{1}$. In order to define
the homotopy map $s_{1}:X_{1}\rightarrow X_{2}$ we follow a similar strategy
as for the fields in Maxwell’s theory and standard gravity. We first decompose
the gauge parameters into combinations that are invariant under the gauge for
gauge transformations $\delta_{\chi}\xi=\partial(\chi)$. Using the SVT
decomposition (3.12) for the gauge parameters and (3.24) one finds

$\begin{split}\delta_{\chi}\zeta^{a}&=0\;,\qquad\delta_{\chi}\zeta^{\bar{a}}=0\;,\\\
\delta_{\chi}\lambda&=\chi\;,\qquad\delta_{\chi}\bar{\lambda}=\chi\;,\\\
\delta_{\chi}\xi_{0}&=-\dot{\chi}\;,\quad\delta_{\chi}\xi^{0}=0\;.\end{split}$
(3.34)

We can then define the particular combination of gauge for gauge invariant
parameters

$\begin{split}\bar{\xi}^{a}&\equiv\zeta^{a}+\frac{1}{2}D^{a}(\lambda-\bar{\lambda})\;,\\\
\bar{\xi}^{\bar{a}}&\equiv\zeta^{\bar{a}}-\frac{1}{2}D^{\bar{a}}(\lambda-\bar{\lambda})\;,\\\
\bar{\xi}_{0}&\equiv\xi_{0}+\frac{1}{2}(\dot{\lambda}+\dot{\bar{\lambda}})\;,\\\
\bar{\xi}^{0}&\equiv\xi^{0}\;.\end{split}$ (3.35)

With these we can write a general decomposition of a gauge parameter into a
gauge for gauge invariant parameter plus the trivial or ‘pure gauge for gauge’
part,

$\xi=\begin{pmatrix}\xi^{a}\\\ \xi^{\bar{a}}\\\ \xi_{0}\\\
\xi^{0}\end{pmatrix}=\begin{pmatrix}\bar{\xi}^{a}+D^{a}f\\\
\bar{\xi}^{\bar{a}}+D^{\bar{a}}f\\\ \bar{\xi}_{0}-\dot{f}\\\
\bar{\xi}^{0}\end{pmatrix}\;.$ (3.36)

A quick computation confirms that this holds identically for

$f\equiv\frac{1}{2}(\lambda+\bar{\lambda})\;.$ (3.37)

It is convenient to write (3.36) more briefly as

$\xi=\bar{\xi}+\partial(f)\;.$ (3.38)

In analogy to previous cases it is natural to define the homotopy map $s$
acting on gauge parameters as

$s(\xi)=f\;.$ (3.39)

The homotopy relation on $X_{2}$ then follows quickly, using $s(\chi)=0$:

$({\rm id}-\iota p)(\chi)=\chi=s(\partial\chi)\;.$ (3.40)

Next, we define the homotopy $s_{0}:X_{0}\rightarrow X_{1}$ from fields to
gauge parameters, using again the strategy of decomposing a field into a gauge
invariant part plus a pure gauge part. Denoting the gauge invariant quantities
with a bar for consistency of notation, we rename (3.15) into

$\bar{h}_{a\bar{b}}=\widehat{h}_{a\bar{b}}\;,\quad\bar{\varphi}=\Phi\;,\quad\bar{\phi}=\Psi\;,\quad\bar{{\cal
A}}_{a}=\widehat{A}_{a}\;,\quad\bar{{\cal
A}}_{\bar{a}}=\widehat{A}_{\bar{a}}\;.$ (3.41)

These define the projection map $p_{0}:X_{0}\rightarrow\bar{X}_{0}$,

$p_{0}(\Psi)=\bar{\Psi}\,,$ (3.42)

using the notation (3.22) (and its obvious extension to $\bar{X}_{0}$). It is
then a straightforward computation to verify that the original fields are
related to the gauge invariant fields via

$\begin{split}h_{a\bar{b}}&=\bar{h}_{a\bar{b}}+D_{a}F_{\bar{b}}-D_{\bar{b}}F_{a}\;,\\\
\phi&=\bar{\phi}+\dot{F}^{0}\;,\\\
\varphi&=\bar{\varphi}-\frac{1}{2}D_{a}F^{a}-\frac{1}{2}D_{\bar{a}}F^{\bar{a}}\;,\\\
{\cal A}_{a}&=\bar{\cal
A}_{a}+\dot{F}_{a}+D_{a}(\widetilde{F}_{0}+F^{0})\,,\\\ {\cal
A}_{a}&=\bar{\cal
A}_{\bar{a}}+\dot{F}_{\bar{a}}+D_{\bar{a}}(\widetilde{F}_{0}-F^{0})\;,\end{split}$
(3.43)

where the ‘pure gauge’ $F$ terms are given in terms of the SVT components of
the fundamental fields by

$\begin{split}F_{a}&=B_{a}-\frac{1}{2}D_{a}E\;,\\\
F_{\bar{a}}&=B_{\bar{a}}+\frac{1}{2}D_{\bar{a}}E\;,\\\
F^{0}&=\frac{1}{2}(\dot{E}+A-\bar{A})\;,\\\
\widetilde{F}_{0}&=\frac{1}{2}(A+\bar{A})\;.\end{split}$ (3.44)

We can also write this relation compactly as $\Psi=\bar{\Psi}+\partial F$.
These define the homotopy map $s$ from the space of fields to the space of
gauge parameters,

$s(\Psi)=\begin{pmatrix}F^{a}\\\ F^{\bar{a}}\\\ \widetilde{F}_{0}\\\
F^{0}\end{pmatrix}\;.$ (3.45)

From these formulas it follows by a direct computation that $s$ acting on a
field that is pure gauge gives back the gauge for gauge invariant part (3.35)
of the parameter:

$s(\partial\xi)=\bar{\xi}\;.$ (3.46)

The homotopy relation evaluated on $\xi\in X_{1}$ then follows with (3.38)

$({\rm id}-\iota p)(\xi)=\xi=\partial(s(\xi))+s(\partial\xi)=\partial
f+\bar{\xi}\,,$ (3.47)

where we used (3.39). Finally, we can inspect the homotopy relation for
$\Psi\in X_{0}$,

$({\rm id}-\iota p)(\Psi)=\Psi-\bar{\Psi}=\partial(F)=\partial(s(\Psi))\,,$
(3.48)

where we used (3.43) and (3.45). This shows that the homotopy relation is
satisfied if we take the homotopy to be trivial on the space of field
equations,

$s_{-1}=0\,.$ (3.49)

Summarizing, on the sub-complex $X_{2}\rightarrow X_{1}\rightarrow X_{0}$ the
maps (3.39) and (3.45) define a consistent homotopy transfer.

In the remainder of this section we show that the homotopy transfer can be
extended to the entire chain complex, which requires two new non-trivial
homotopy maps. As in the Yang-Mills and gravity examples, however, the spaces
of Bianchi (and Bianchi for Bianchi) identities will be projected to zero:

$\bar{X}_{-2}=\bar{X}_{-3}=\\{0\\}\;,$ (3.50)

i..e, we set $p_{-2}=p_{-3}=0$. The projector
$p_{-1}:X_{-1}\rightarrow\bar{X}_{-1}$ is non-trivial, however, and projects
onto the space of tensors of the equations-of-motion type that satisfy the
Bianchi identities identically. Writing for the projected tensors $\bar{\cal
E}=p({\cal E})$ one finds

$\begin{split}\bar{\cal E}_{a\bar{b}}&={\cal
E}_{a\bar{b}}+2D_{a}\Delta^{-1}(\partial{\cal
E})_{\bar{b}}-2D_{\bar{b}}\Delta^{-1}(\partial{\cal E})_{a}\\\
&\qquad\;\;\,+4D_{a}D_{\bar{b}}\Delta^{-2}\left((\dot{\partial{\cal
E}})^{0}-\frac{1}{2}D^{c}(\partial{\cal
E})_{c}-\frac{1}{2}D^{\bar{c}}(\partial{\cal E})_{\bar{c}}\right)\,,\\\
\bar{\cal E}_{\phi}&={\cal E}_{\phi}\,,\\\ \bar{\cal E}_{\varphi}&={\cal
E}_{\varphi}\,,\\\ \bar{\cal E}_{a}&={\cal
E}_{a}+D_{a}\Delta^{-1}\left(({\partial{\cal E}})_{0}+({\partial{\cal
E}})^{0}\right)\,,\\\ \bar{\cal E}_{\bar{a}}&={\cal
E}_{\bar{a}}-D_{\bar{a}}\Delta^{-1}\left(({\partial{\cal
E}})_{0}-({\partial{\cal E}})^{0}\right)\,,\end{split}$ (3.51)

where the components of $(\partial{\cal E})$ are defined in (3.31). This is
the appropriate projector since one may now verify that the following
relations hold identically, just as a consequence of the definition of the
bar:

$\begin{split}0&\equiv D_{\bar{b}}\bar{\cal E}^{a\bar{b}}+2D^{a}\bar{\cal
E}_{\varphi}-2\partial_{t}\bar{\cal E}^{a}\,,\\\ 0&\equiv D_{b}\bar{\cal
E}^{b\bar{a}}-2D^{\bar{a}}\bar{\cal E}_{\varphi}+2\partial_{t}\bar{\cal
E}^{\bar{a}}\,,\\\ 0&\equiv D_{a}\bar{\cal E}^{a}+D_{\bar{a}}\bar{\cal
E}^{\bar{a}}\,,\\\ 0&\equiv D_{a}\bar{\cal E}^{a}-D_{\bar{a}}\bar{\cal
E}^{\bar{a}}-\partial_{t}\bar{\cal E}_{\phi}\,.\end{split}$ (3.52)

We have now completed the definition of the projection of the entire chain
complex. It remains to identify the homotopy maps. We continue to take the
homotopy from the space of field equations to the space of fields to be
trivial, $s({\cal E})=0$. In order to find the homotopy map on the space of
Bianchi identities we use this to evaluate the homotopy relation on ${\cal
E}\in X_{-1}$:

$({\rm id}-\iota p)({\cal E})={\cal E}-\bar{\cal E}=s(\partial{\cal E})\;.$
(3.53)

Since we expressed the projector (3.51) already in terms of $\partial{\cal E}$
we can immediately read off the homotopy map so that the homotopy relation
holds:

$\begin{split}s({\cal G})_{a\bar{b}}&=-2D_{a}\Delta^{-1}{\cal
G}_{\bar{b}}+2D_{\bar{b}}\Delta^{-1}{\cal
G}_{a}-4D_{a}D_{\bar{b}}\Delta^{-2}\left(\dot{{\cal
G}}^{0}-\frac{1}{2}D^{c}{\cal G}_{c}-\frac{1}{2}D^{\bar{c}}{\cal
G}_{\bar{c}}\right)\,,\\\ s({\cal G})_{\phi}&=0\,,\\\ s({\cal
G})_{\varphi}&=0\,,\\\ s({\cal G})_{a}&=-D_{a}\Delta^{-1}\left({\cal
G}_{0}+{\cal G}^{0}\right)\,,\\\ s({\cal
G})_{\bar{a}}&=D_{\bar{a}}\Delta^{-1}\left({\cal G}_{0}-{\cal
G}^{0}\right)\,.\end{split}$ (3.54)

Let us next turn to the homotopy relation on ${\cal G}\in X_{-2}$. Recalling
$p({\cal G})=0$, it reads

$({\rm id}-\iota p)({\cal G})={\cal G}=\partial(s({\cal G}))+s(\partial{\cal
G})\;.$ (3.55)

The first term on the right-hand side can be computed directly with (3.31) and
(3.54), for which one finds

$\partial(s({\cal G}))={\cal
G}+2\begin{pmatrix}D_{a}\Delta^{-1}\big{(}\dot{\cal
G}_{0}+\tfrac{1}{2}D^{c}{\cal G}_{c}-\tfrac{1}{2}D^{\bar{c}}{\cal
G}_{\bar{c}}\big{)}\\\ D_{\bar{a}}\Delta^{-1}\big{(}\dot{\cal
G}_{0}+\tfrac{1}{2}D^{c}{\cal G}_{c}-\tfrac{1}{2}D^{\bar{c}}{\cal
G}_{\bar{c}}\big{)}\\\ 0\\\ 0\end{pmatrix}\,.$ (3.56)

Looking back at (3.32) we infer that the failure of $\partial(s({\cal G}))$ to
give back ${\cal G}$ involves $\partial{\cal G}$. This implies that (3.55) is
satisfied if we define a non-trivial homotopy map $s:X_{-3}\rightarrow X_{-2}$
as follows

$s(\rho)=-2\begin{pmatrix}D_{a}\Delta^{-1}\rho\\\
D_{\bar{a}}\Delta^{-1}\rho\\\ 0\\\ 0\end{pmatrix}\;.$ (3.57)

Finally, it then remains to check that the homotopy relation holds on $\rho\in
X_{-3}$:

$({\rm id}-\iota p)(\rho)=\rho=\partial(s(\rho))\;,$ (3.58)

using $\partial\rho=0$. This identity is quickly verified with (3.32) and
(3.57). With this we have completed the proof that the entire chain complex
(3.21) of double field theory can be homotopy transferred to the complex
$\bar{X}_{0}\rightarrow\bar{X}_{-1}$ of gauge invariant fields and their field
equations, where all redundancies encoded in gauge symmetries and Bianchi
identities have been eliminated.

##  4 Homotopy Transfer

In this section we introduce the general framework of $L_{\infty}$-algebras
and homotopy transfer and explain how it determines gauge invariant variables.
We begin by giving a brief but hopefully self-contained introduction to the
coalgebra formulation that is most useful for our purposes. In the second
subsection we give the details of the perturbation lemma, and in the third
subsection we illustrate the resulting techniques for the example of Yang-
Mills theory.

###  4.1 General theory

#### The symmetric coalgebra

A convenient formulation of $L_{\infty}$-algebras makes use of the notions of
coalgebras and coderivations. This construction can be found for example in
[18]. The very similar construction for $A_{\infty}$-algebras is nicely
described in [37]. We only need one type of coalgebra. Given a graded vector
space $X_{\bullet}$ (the total vector space of the chain complex), we can
define the symmetric graded coalgebra $S^{c}(X_{\bullet})$. As a vector space,
we can think of it as the Fock space made from elements in $X_{\bullet}$,
where elements of odd degree are fermionic (anti-commuting), and elements of
even degree are bosonic (commuting). For our purposes, we can exclude the zero
particle state. We therefore define

$S^{c}(X_{\bullet})=\bigoplus_{n\geq 1}X_{\bullet}^{\wedge n}.$ (4.1)

The wedge denotes the graded symmetric product of vector spaces.

The structure turning $S^{c}(X_{\bullet})$ into a coalgebra is the coproduct

$\Delta:S^{c}(X_{\bullet})\rightarrow S^{c}(X_{\bullet})\otimes
S^{c}(X_{\bullet}).$ (4.2)

Notice that the direction of the map $\Delta$ is opposite from what we want
from an ordinary product. It takes one input and gives two outputs. The
coproduct is the sum of all possible ways to split an element in
$S^{c}(X_{\bullet})$ into two in a symmetric fashion. We give a few examples
before defining it in general. When $\Delta$ acts on a single element $x\in
X_{\bullet}\subseteq S^{c}(X_{\bullet})$ there is no sensible way to split it
into two pieces, and we therefore define

$\Delta(x)=0.$ (4.3)

Next we can consider $x_{1}\wedge x_{2}\in X_{\bullet}\wedge
X_{\bullet}\subseteq S^{c}(X_{\bullet})$. In that case, the obvious split
reads

$\Delta(x_{1}\wedge x_{2})=x_{1}\otimes x_{2}+(-)^{x_{1}x_{2}}x_{2}\otimes
x_{1}.$ (4.4)

The sum makes sure that $\Delta$ is well defined on $S^{c}(X_{\bullet})$, i.e.
that it is invariant under interchanging $x_{1}$ and $x_{2}$, up to a sign. As
a final example, we consider a cubic element $x_{1}\wedge x_{2}\wedge x_{3}\in
S^{c}(X_{\bullet})$. In that case,

$\displaystyle\Delta(x_{1}\wedge x_{2}\wedge x_{3})$
$\displaystyle=(x_{1}\wedge x_{2})\otimes x_{3}+(-)^{x_{2}x_{3}}(x_{1}\wedge
x_{3})\otimes x_{2}+(-)^{x_{1}(x_{2}+x_{3})}(x_{2}\wedge x_{3})\otimes x_{1}$
(4.5) $\displaystyle+x_{1}\otimes(x_{2}\wedge
x_{3})+(-)^{x_{1}x_{2}}x_{2}\otimes(x_{1}\wedge
x_{3})+(-)^{x_{3}(x_{1}+x_{2})}x_{3}\otimes(x_{1}\wedge x_{2}).$ (4.6)

The signs in front of each term are just obtained by permuting the three
elements to a particular order. The definition of the coproduct to higher
powers continues along the same lines. The general formula reads

$\Delta(x_{1}\wedge\cdots\wedge x_{n})=\sum_{l=1}^{n-1}\sum_{\sigma\in
S(l,n-l)}\pm(x_{\sigma(1)}\wedge\cdots\wedge
x_{\sigma(l)})\otimes(x_{\sigma(l+1)}\wedge\cdots\wedge x_{\sigma(n)}).$ (4.7)

Here, the second sum runs over a subset $S(p,q)$ of the permutation group of
$p+q$ elements. An element $\sigma$ in there is called a _(p,q)-unshuffle_. It
is a permutation with the condition that $\sigma(i)\leq\sigma(j)$ whenever
$1\leq i\leq j\leq p$ or $p+1\leq i\leq j\leq p+q$. The permutation $\sigma$
then determines the sign, which is just obtained by a graded permutation of
the elements by $\sigma$.

We now establish some properties of this coalgebra. The coproduct is
coassociative, which means that

$(\Delta\otimes{\mathbb{I}})\circ\Delta=({\mathbb{I}}\otimes\Delta)\circ\Delta.$
(4.8)

This property allows one to uniquely define higher coproducts

$\Delta_{n}:=(\Delta_{n-1}\otimes{\mathbb{I}})\circ\Delta=({\mathbb{I}}\otimes\Delta_{n-1})\circ\Delta,\quad\Delta_{1}:={\mathbb{I}}.$
(4.9)

We now turn to the definition of cohomomorphisms, where the higher coproducts
will be useful. A cohomomorphism $\phi:S^{c}(X_{\bullet})\rightarrow
S^{c}(Y_{\bullet})$ between symmetric tensor coalgebras is a map of degree
zero that preserves the coproduct. This means that

$(\phi\otimes\phi)\circ\Delta=\Delta\circ\phi.$ (4.10)

A useful fact is that such a cohomomorphism is uniquely determined by its
image in $Y_{\bullet}\subseteq S^{c}(Y_{\bullet})$. In other words, defining
$\pi_{1}:S^{c}(Y_{\bullet})\rightarrow Y_{\bullet}$ as the projection onto
$Y_{\bullet}$, the association $\phi\mapsto\pi_{1}\circ\phi$ is bijective as a
map from cohomomorphisms to linear maps of degree zero. The inverse is given
by

$f\mapsto\exp(f):=\sum_{n\geq 1}\frac{1}{n!}f^{\wedge n}\circ\Delta_{n}\,,$
(4.11)

where $f^{\wedge n}:S^{c}(X_{\bullet})^{\otimes n}\rightarrow
S^{c}(Y_{\bullet})$ is given by

$f^{\wedge n}(l_{1}\otimes\cdots\otimes l_{n}):=f(l_{1})\wedge\cdots\wedge
f(l_{n}).$ (4.12)

Homomorphisms of coalgebras may seem very strange to readers not familiar with
the topic. A more familiar way to think about cohomomorphisms is as non-linear
maps between vector spaces. This can be found for example in [36]. On the one
hand, given a map $f:S^{c}(X_{\bullet})\rightarrow Y_{\bullet}$, we can
construct a formal Taylor series

$f(x):=\sum_{n\geq 1}\frac{1}{n!}f_{n}(x,...,x)\,,$ (4.13)

where $f_{n}$ is the $n$-linear piece of $f$. We say that this is formal,
since we do not care about convergence of this Taylor series. On the other
hand, we can always obtain a cohomomorphism from a formal Taylor series by
reading off the $n$-linear piece. Associating a cohomomorphism to a formal
Taylor series is an isomorphism in the sense that composing cohomorphisms is
the same as composing the Taylor series. In other words, given two
cohomomorphisms $\phi_{1}:S^{c}(X_{\bullet})\rightarrow S^{c}(Y_{\bullet})$
and $\phi_{2}:S^{c}(Y_{\bullet})\rightarrow S^{c}(Z_{\bullet})$, we have that

$(\pi_{1}\circ\phi_{2}\circ\phi_{1})(x)=\pi_{1}\circ\phi_{2}(\pi_{1}\circ\phi_{1}(x)).$
(4.14)

This ensures that the two pictures are indeed equivalent.

The last fact about cohomomorphisms we need for later applications is that a
cohomorphism $f:S^{c}(X_{\bullet})\rightarrow Y_{\bullet}$ is invertible if
and only if its linear piece is. We do not give a full proof of this fact, but
only the basic argument. It is based on the Taylor series picture. We write

$y=f(x)=f_{1}(x)+\sum_{n\geq 2}\frac{1}{n!}f_{n}(x,...,x).$ (4.15)

We want to solve for $y$ perturbatively. Assuming that $f_{1}$ is invertible,
we can write

$x=f_{1}^{-1}(y)-\sum_{n\geq 2}\frac{1}{n!}f_{1}^{-1}\circ f_{n}(x,...,x).$
(4.16)

We can now plug in the left hand side into the right hand side over and over
again. Notice that the $p+1$-linear term in $y$ no longer changes after the
$p$th iteration of this process. In this way, we are able to perturbatively
determine the $p$-linear piece of the inverse to any order.

Having discussed cohomomorphisms, we now turn to coderivations. A linear map
$Q:S^{c}(X_{\bullet})\rightarrow S^{c}(X_{\bullet})$ is called a coderivation,
if it satisfies the co-Leibniz rule

$\Delta\circ Q=(Q\otimes{\mathbb{I}}+{\mathbb{I}}\otimes Q)\circ\Delta.$
(4.17)

Similar to cohomomorphisms, a coderivation is determined by its image in
$X^{\bullet}\subseteq S^{c}(X^{\bullet})$. The lift is of course different in
this case. Let $b_{k}:X_{\bullet}^{\wedge k}\rightarrow X_{\bullet}$ be an
$k$-linear map. We obtain a coderivation by defining

$b_{k}(x_{1}\wedge\cdots\wedge x_{n}):=\sum_{\sigma\in S(k,n-k)}\pm
b_{k}(x_{\sigma(1)},...,x_{\sigma(k)})\wedge x_{\sigma(k+1)}\wedge\cdots\wedge
x_{\sigma(n)}.$ (4.18)

The sign is again obtained by permuting the graded vectors with the
permutation $\sigma$. Also, the above is defined to be zero when $k>n$. From
this perspective, the data of a coderivation is equivalent to a collection of
multilinear maps $\\{b_{k}\\}_{k\geq 1}$.

####  $L_{\infty}$-algebras

We now have gathered all the tools to define an $L_{\infty}$-algebra. Given a
graded vector space $X_{\bullet}$, an $L_{\infty}$-algebra is given by a
coderivation $Q$ of degree $-1$ on $S^{c}(X_{\bullet})$, such that $Q^{2}=0$.
A morphism of $L_{\infty}$-algebras
$\phi:(X_{\bullet},Q_{1})\rightarrow(Y_{\bullet},Q_{2})$ is a cohomomorphism,
such that $\phi\circ Q_{1}=Q_{2}\circ\phi$. The simplicity of the square zero
condition and the definition of morphism is the main reason to use the
language of coalgebras.

In the previous section we explained that a coderivation is equivalent to a
collection of graded symmetric maps $b_{k}:X^{\wedge k}_{\bullet}\rightarrow
X_{\bullet}$. The condition $Q^{2}=0$ defines relations among the $b_{k}$.
There are infinitely many of them, and we explicitly state the first few. In
order to do so, let us first note the following. Coderivations, just as
derivations, form a graded Lie algebra with respect to the commutator bracket
$[Q_{1},Q_{2}]:=Q_{1}\circ Q_{2}-(-)^{Q_{1}Q_{2}}Q_{2}\circ Q_{1}$. This in
particular means that $Q^{2}=\frac{1}{2}[Q,Q]$ is a coderivation, independent
of whether it is zero or not. This implies in particular that it is
identically zero, if and only if it is zero when its output is projected to
$X_{\bullet}$. Hence, all relations can be stated in terms of projecting to
one output. One finds the following conditions

$\displaystyle 0=$ $\displaystyle\ b_{1}^{2},$ (4.19) $\displaystyle 0=$
$\displaystyle\ b_{1}(b_{2}(x,y))+b_{2}(b_{1}(x),y)+(-)^{x}b_{2}(x,b_{1}(y)),$
(4.20) $\displaystyle 0=$ $\displaystyle\
b_{1}b_{3}(x,y,z)+b_{3}(b_{1}(x),y,z)+(-)^{x}b_{3}(x,b_{1}(y),z)+(-)^{x+y}b_{3}(x,y,b_{1}(z)),$
$\displaystyle+b_{2}(b_{2}(x,y),z)+(-)^{z}b_{2}(b_{2}(z,x),y)+(-)^{y+z}b_{2}(b_{2}(y,z),x),$
(4.21) $\displaystyle\vdots$

Equation (4.19) tells us that $(X_{\bullet},b_{1})$ defines a chain complex.
Condition (4.20) means that the differential $b_{1}$ acts as a derivation with
respect to the product $b_{2}$. Finally, (4.21) says that $b_{2}$ satisfies
some Jacobi identity, up to terms that involve $b_{1}$ and $b_{3}$.

Let us look at some examples. We first consider a differential graded Lie
algebra, which is a very special type of $L_{\infty}$-algebra. In that case,
all $b_{n}$ are zero for $n\geq 3$. They occur for example in the theory of
Lie algebra valued forms. Given some manifold $M$ and a Lie algebra
$\mathfrak{g}$, we can consider the complex
$\Omega^{\bullet}(M)\otimes\mathfrak{g}$. We have a linear map
$\tilde{b}_{1}=\text{d}$ given by the de Rham differential and a product
$\tilde{b}_{2}=[-,-]$ from the Lie bracket. With the usual conventions, the de
Rham differential is of degree one and the Lie bracket is graded anti-
symmetric of degree zero. To match our conventions, we define a complex
$X_{\bullet}$ such that

$X_{n}=\Omega^{1-n}(M)\otimes\mathfrak{g}.$ (4.22)

In this convention, both $\tilde{b}_{1}$ and $\tilde{b}_{2}$ have degree $-1$.
In order to match the signs in the conditions (4.19 \- 4.21), we set

$b_{1}(x)=\tilde{b}_{1}(x),\quad b_{2}(x,y)=(-)^{x}\tilde{b}_{2}(x,y).$ (4.23)

The above example shows how $L_{\infty}$-algebras generalize ordinary Lie
algebras. In the Lie algebra example, the Jacobi identity in (4.21) is
strictly satisfied, so there is no need to have a non-trivial $b_{3}$. On the
other hand, whenever there is a $b_{3}$, one occasionally says that $b_{2}$
satisfies the Jacobi identity up to homotopy $b_{3}$. This is one reason why
$L_{\infty}$-algebras are also known as homotopy Lie algebras.

In physics, $L_{\infty}$-algebras arise as a formulation of classical field
theories. We do not go into detail in how this connection is established
precisely, but merely state the idea. In the case of a simple gauge theory,
one often introduces four vector spaces ranging from degree one to minus two.
In the previous sections, we already saw that the free theory is then
described by a chain complex

${0}$${X_{1}}$${X_{0}}$${X_{-1}}$${X_{-2}}$${0.}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\partial_{0}}$$\scriptstyle{\partial_{-1}}$
(4.24)

Here, $X_{0}$ is the space of fields. To the left we have the space $X_{1}$ of
gauge parameters. Given $\Lambda\in X_{1}$ and $\phi\in X_{0}$, the
differential $b_{1}$ defines a gauge transformation
$\delta_{\Lambda}\phi=\partial_{1}\Lambda$. The free equations of motion are
encoded in the condition $\partial_{0}\phi=0$. Gauge independence of the
equations of motion reads $\partial_{0}\circ\partial_{1}=0$. The physical
content of the theory is therefore encoded in the homology
$H_{0}(X_{\bullet})=\frac{\ker\partial_{0}}{\text{im}\,\partial_{1}}$. Since
$\partial_{0}\phi$ gives the equation of motion, we think of $X_{-1}$ as the
space of equations of motion. Finally, we call $X_{-2}$ the space of
generalized Bianchi identities. The Bianchi identities are encoded in
$\partial_{-1}\circ\partial_{0}=0$. The complex can continue to other degrees
if the theory has reducible gauge symmetries. Higher degrees then encode gauge
symmetries among gauge symmetries, and gauge symmetries among gauge symmetries
of gauge symmetries, etc. This in turn leads to higher Bianchi identities in
negative degrees.

To capture interactions in a theory, we need to extend the data of a chain
complex to that of an $L_{\infty}$-algebra. Given a field $\phi\in X_{0}$, we
would like to write the non-linear equations of motion in a power series as

$0=\sum_{n\geq 1}\frac{1}{n!}b_{n}(\phi,...,\phi).$ (4.25)

In this context, the equations of motion are also known as Maurer-Cartan
equations. On the other hand, gauge symmetries in an interacting theory may be
non-linear. Given a gauge parameter $\Lambda\in X_{1}$, we write the gauge
transformation it generates on a field $\phi$ as

$\delta_{\Lambda}\phi=\sum_{n\geq
1}\frac{1}{(n-1)!}b_{n}(\phi,...,\phi,\Lambda).$ (4.26)

Here, the $L_{\infty}$-relations become important. The theory is gauge
invariant precisely when the relations are satisfied. For example, in Chern-
Simons theory one precisely uses the fact that we have a Lie bracket in order
to prove gauge invariance. In other degrees, there are also relations about
non-linear Bianchi identities and higher gauge structure.

####  $L_{\infty}$-morphisms and gauge transformations

We now aim to explain in a little more detail how to translate from the
coalgebra language to more familiar formulas for the gauge transformations. We
first recall that according to (4.13) any coalgebra morphism
$f:S^{c}(X_{\bullet})\rightarrow Y_{\bullet}$ defines a formal Taylor series

$f(x)=\sum_{n\geq 1}\frac{1}{n!}f_{n}(x,...,x).$ (4.27)

We defined $L_{\infty}$-morphisms
$f:(X_{\bullet},Q_{1})\rightarrow(Y_{\bullet},Q_{2})$ to be coalgebra
morphisms that commute with the coderivations, i.e. $Q_{2}\circ f=f\circ
Q_{1}$. We will now show that $f$, when viewed as a function, is covariant
under gauge transformations. We need to introduce some conventions. For now,
we denote by $b:=\pi_{1}\circ Q_{1}$ the part of $Q_{1}$ with one output.
Also, we introduce the exponential as the map $X_{0}\rightarrow
S^{c}(X_{\bullet})$, defined for an element $x$ of degree zero by

$\exp(x):=\sum_{n\geq 1}\frac{1}{n!}x^{\wedge n}\,.$ (4.28)

This map has the characteristic property

$\Delta(\exp(x))=\exp(x)\otimes\exp(x)\,.$ (4.29)

From this it follows that for a coalgebra morphism $f$ we have

$f(\exp(x))=\exp(f(x)).$ (4.30)

By a slight abuse of notation, on the left-hand side we view $f$ as a
coalgebra morphism and on the right-hand side we view $f$ as a map
$X_{\bullet}\rightarrow Y_{\bullet}$. We can now write the gauge
transformation (4.26) as

$\delta_{\Lambda}\phi=\frac{\partial}{\partial\varepsilon}b(\exp(\phi+\varepsilon\Lambda)),$
(4.31)

where we introduced a formal parameter $\varepsilon$ of degree minus one, and
we take the odd $\varepsilon$ to be nilpotent, $\varepsilon^{2}=0$.

For the following to work, we need to assume that the part of $f$ that takes
values in the field space $Y_{0}$ only depends on fields in $X_{0}$. In this
case, we can effectively assume that

$b(\exp(\phi+\varepsilon\Lambda))=b(\exp(\phi+\varepsilon\Lambda))\exp(\phi+\varepsilon\Lambda)\simeq
b(\exp(\phi+\varepsilon\Lambda))\exp(\phi)\,,$ (4.32)

which holds as arguments of $f$, and we used $\simeq$ instead of the equality
sign to indicate that the above assumption on $f$ is needed. We now apply the
$L_{\infty}$-morphism $f$ to both sides of (4.31) and use (4.30) as well as
(4.32). We find

$\begin{split}f(\delta_{\Lambda}\phi)&=f(\delta_{\Lambda}\exp(\phi))=\frac{\partial}{\partial\varepsilon}f(b(\exp(\phi+\varepsilon\Lambda))\exp(\phi))=\frac{\partial}{\partial\varepsilon}f(b(\exp(\phi+\varepsilon\Lambda)))\\\
&=\frac{\partial}{\partial\varepsilon}b(\exp(f(\phi+\varepsilon\Lambda)))=\frac{\partial}{\partial\varepsilon}b(\exp(f(\phi)+\varepsilon
f^{(1)}(\phi,\Lambda)))=\delta_{f^{(1)}(\phi,\Lambda)}f(\phi)\,,\end{split}$
(4.33)

where we used

$f(\phi+\varepsilon\Lambda)=f(\phi)+\varepsilon f^{(1)}(\phi,\Lambda)\,,$
(4.34)

with

$f^{(1)}(\phi,\Lambda)=\sum_{n\geq
1}\frac{1}{(n-1)!}f_{n}(\phi,...,\phi,\Lambda).$ (4.35)

Equation (4.33) shows that $L_{\infty}$-morphisms, when thought of as
functions, are covariant with respect to gauge transformations. In particular,
if the target space has no gauge transformations, $f$ is gauge invariant.

One may wonder what happens when we do not make the assumption on $f$. In that
case additional terms proportional to the equations of motion enter. In that
case, $f$ is only gauge invariant on shell (assuming that $Y_{\bullet}$ has no
gauge transformations, i.e. $Y_{1}=0$). One would obtain such $f$ when
applying the procedure explained below to theories with open gauge algebras.

###  4.2 Homotopy transfer and the homological perturbation lemma

A central concept of $L_{\infty}$-algebras is that of homotopy transfer, see
[26].111We note that homotopy transfer is also closely related to the
elimination of generalized auxiliary fields in the BV-BRST formulation of
gauge theories, see [44, 31]. One asks the following question: Suppose that we
have a pair of chain complexes $(X_{\bullet},b_{1})$ and
$(Y_{\bullet},c_{1})$, and suppose that $X_{\bullet}$ is equipped with an
$L_{\infty}$-structure, given by an infinite tower of products
$\\{b_{i}\\}_{i\geq 1}$ and such that $b_{1}$ is the differential of the chain
complex $X_{\bullet}$, under which conditions can we then transfer the
$L_{\infty}$-structure of $X_{\bullet}$ to $Y_{\bullet}$, such that the two
structures are equivalent? This is possible provided $Y_{\bullet}$ is a strong
deformation retract of $X_{\bullet}$,222A strong deformation retract is
actually not required, but it will be necessary for applying the homological
perturbation lemma introduced below. which means that there are chain maps
$p_{0}:X_{\bullet}\rightarrow Y_{\bullet}$ and
${\iota}_{0}:Y_{\bullet}\rightarrow X_{\bullet}$, together with a degree one
map $s:X_{\bullet}\rightarrow X_{\bullet}$, such that

$1-{\iota}_{0}\circ p_{0}=b_{1}s+sb_{1},\quad p_{0}\circ{\iota}_{0}=1,\quad
s\circ{\iota}_{0}=0,\quad p_{0}\circ s=0,\quad s^{2}=0.$ (4.36)

Here we use the subscript 0 to distinguish these maps from deformations to be
provided by the deformation lemma. The first of the above conditions implies
that ${\iota}_{0}$ and $p_{0}$ are inverse on homology. Then there exists an
$L_{\infty}$-structure $\\{c_{n}\\}_{n\geq 1}$ on $Y_{\bullet}$, together with
$L_{\infty}$-morphisms $p:(X_{\bullet},\\{b_{n}\\}_{n\geq
1})\rightarrow(Y_{\bullet},\\{c_{n}\\}_{n\geq 1})$ and
${\iota}:(Y_{\bullet},\\{c_{n}\\}_{n\geq
1})\rightarrow(X_{\bullet},\\{b_{n}\\}_{n\geq 1})$ such that
$p\circ{\iota}=1$, and their linear pieces are given by $p_{0}$ and
${\iota}_{0}$, respectively. More generally, we say that two
$L_{\infty}$-structures are equivalent, if there exists a morphism such that
its linear part is an isomorphism on homology. So the $L_{\infty}$-structures
on $X_{\bullet}$ and $Y_{\bullet}$ are equivalent, precisely since
${\iota}_{0}$ (or $p_{0}$) is an isomorphism on homology.

The formulas for the morphisms and the induced $L_{\infty}$-structure are
given perturbatively as cohomomorphisms and coderivation respectively. The
fact that $p_{0}$ and ${\iota}_{0}$ are linear simplifies their lift
drastically. We have

$p_{0}(x_{1}\wedge\cdots\wedge x_{n})=p_{0}(x_{1})\wedge\cdots\wedge
p_{0}(x_{n}),\quad{\iota}_{0}(y_{1}\wedge\cdots\wedge
y_{n})={\iota}_{0}(y_{1})\wedge\cdots\wedge{\iota}_{0}(y_{n}).$ (4.37)

We also have to construct a lift of the homotopy $s$ to coalgebras. We first
define it on tensor powers

$\displaystyle s_{n}(x_{1}\otimes\cdots\otimes x_{n})=\,$ $\displaystyle
s(x_{1})\wedge x_{2}\cdots\wedge x_{n}+(-)^{x_{1}}{\iota}\circ p(x_{1})\wedge
s(x_{2})\wedge x_{3}\cdots\wedge x_{n}$ (4.38)
$\displaystyle+...+(-)^{x_{1}+...x_{n-1}}{\iota}\circ
p(x_{1})\wedge\cdots\wedge{\iota}\circ p(x_{n-1})\wedge s(x_{n}).$ (4.39)

We also define

$q_{n}(x_{1}\wedge\cdots\wedge x_{n})=\frac{1}{n!}\sum_{\sigma\in S_{n}}\pm
x_{\sigma(1)}\otimes\cdots\otimes x_{\sigma(n)},$ (4.40)

where the sum runs over all permutations $\sigma$ of $n$ elements, and the
sign is the natural one coming from commuting graded objects. We can now
define a map $s_{n}\circ q_{n}:X_{\bullet}^{\wedge n}\rightarrow X^{\wedge
n}_{\bullet}$. Finally, we obtain the map $s:=\sum_{n\geq 1}s_{n}\circ q_{n}$
from the coalgebra $S^{c}(X_{\bullet})$ to itself.

The perturbed $L_{\infty}$-morphisms, as well as the induced products, can be
obtained by applying the homological perturbation lemma, see [33] for a nice
discussion. They read

${\iota}=(1+s\circ\delta)^{-1}\circ{\iota}_{0},\quad
p=p_{0}\circ(1+\delta\circ s)^{-1},\quad
Q=c_{1}+\delta\circ(1+s\circ\delta)^{-1}.$ (4.41)

Here, $\delta:=\sum_{n\geq 2}b_{n}$ is the coderivation corresponding to the
products defining the $L_{\infty}$-structure on $X_{\bullet}$, but without the
linear piece. Also, we define the inverses appearing in the above formulas as
a geometric series, i.e. $(1+x)^{-1}=\sum_{n\geq 0}(-x)^{n}$. The fact that
${\iota}$ and $p$ are coalgebra morphisms is guaranteed by the fact that we
considered $s$ to be a strong deformation retract [35, 34].

Let us recapture the essence of the homotopy transfer theorem from the point
of view of field theory. We noted earlier that free field theories are
equivalent if the associated complexes are homotopic to each other (in
particular, the fields are equivalent on-shell and up to gauge
transformations). In general, such a homotopy arises either by gauge fixing
some of the fields or by solving equations of motion. One then obtains an
effective theory in terms of a smaller set of fields. The homotopy transfer
then allows us to carry this over to an interacting theory. Applying the
procedure allows us to extract from any field theory an effective theory of a
smaller set of fields. The projection map $p:(X_{\bullet},\\{b_{n}\\}_{n\geq
1})\rightarrow(Y_{\bullet},\\{c_{n}\\}_{n\geq 1})$ gives rise to a non-linear
relation between the old (larger) set of fields in $X_{\bullet}$ and the new
smaller one in $Y_{\bullet}$. The fact that $p$ is an $L_{\infty}$-morphism
ensures that the relation is covariant with respect to gauge transformations
and that equations of motion are correctly translated from the original theory
to the effective theory. More precisely, if $x$ solves the Maurer-Cartan
equation on $X_{\bullet}$, then $p(x)$, where $p(x)$ is given by a formal
Taylor series as in (4.13), solves the Maurer-Cartan equation on
$Y_{\bullet}$. The equivalence property (i.e. that the linear part of $p$ is
an isomorphism on homology) ensures that sets of solutions modulo gauge
transformations of both theories are actually isomorphic [45].

Below we will use the homotopy transfer as a tool to remove gauge degrees of
freedom. The main advantage of this approach is that the map $p$ produced by
the transfer theorem automatically gives a relation between the original gauge
redundant fields, and the new gauge invariant fields. Moreover, the fact that
$p$ is an $L_{\infty}$-morphism implies that the relation is invariant under
the full non-linear gauge transformations of the original space. We define the
gauge invariant variable by $\widehat{\phi}\equiv p(\phi)$. Its gauge
invariance follows from (4.33):

$\delta_{\Lambda}\widehat{\phi}=p(\delta_{\Lambda}\phi)=\delta_{p^{(1)}(\phi,\Lambda)}p(\phi)=0\,,$
(4.42)

which vanishes since $p^{(1)}(\phi,\Lambda)\in Y_{1}=0$ for the complex of
gauge invariant variables. Therefore, $\widehat{\phi}=p(\phi)$ is gauge
invariant.

###  4.3 Yang-Mills theory

####  $L_{\infty}$-algebra description

The free theory of Yang-Mills has already been covered in sec. 2. We now give
the description of the full interacting theory as an $L_{\infty}$-algebra,
beginning with the gauge transformations. The non-linear gauge transformation
of Yang-Mills theory reads

$\delta_{\Lambda}A_{\mu}=\partial_{\mu}\Lambda+[A_{\mu},\Lambda].$ (4.43)

When we compare this to (4.26), we find that we have to set

$b_{2}^{\mu}(A,\Lambda)=b_{2}(\Lambda,A^{\mu})=[A^{\mu},\Lambda].$ (4.44)

There are no higher products as long as a gauge parameter is involved, i.e.
$b_{k}(\Lambda,...)=0$ for all $k\geq 3$. The equations of motion read

$0=\Box A_{\mu}-\partial_{\mu}\partial\cdot
A+[A^{\nu},\partial_{\nu}A_{\mu}-\partial_{\mu}A_{\nu}]+\partial^{\nu}[A_{\nu},A_{\mu}]+[A^{\nu},[A_{\nu},A_{\mu}]].$
(4.45)

When we compare this to the general form (4.25), we find that,

$b_{2}^{\mu}(A,B)=[A_{\nu},\partial^{\nu}B^{\mu}-\partial^{\mu}B^{\nu}]+(A\leftrightarrow
B)$ (4.46)

and

$b_{3}^{\mu}(A,B,C)=[A_{\nu},[B^{\nu},C^{\mu}]]+\text{permutations},$ (4.47)

where $A_{\mu},B_{\nu},C_{\rho}$ are gauge fields. Higher brackets are zero.

The above definitions correctly reproduce gauge transformations and equations
of motion. However, we also want to include the space $X_{-2}$ of Bianchi
identities. Moreover, even without it, one can check that the above products
do not satisfy the $L_{\infty}$-relations. There are two reasons. At the non-
linear level, gauge transformations are not abelian, but are again
proportional to a gauge transformation. We can define

$b_{2}(\Lambda_{1},\Lambda_{2})=-b_{2}(\Lambda_{2},\Lambda_{1})=-[\Lambda_{1},\Lambda_{2}].$
(4.48)

Another issue is that a gauge transformation of the equations of motion does
not give zero, but is again proportional to the equations of motion. The
equation of motion $D_{\nu}F^{\nu\mu}$ transforms in the adjoint
representation of the gauge algebra. To account for this fact, we should
define a product between gauge parameters $\Lambda\in X_{1}$ and equations of
motion $E^{\mu}\in X_{-1}$. We set

$b_{2}^{\mu}(E,\Lambda)=-b_{2}^{\mu}(\Lambda,E)=[E^{\mu},\Lambda].$ (4.49)

To obtain the correct Bianchi identities, we need to introduce brackets
between scalars $B\in X_{-2}$ and gauge parameters. By degree reasons, this is
the only non-vanishing combination. We can determine the bracket via the rule

$\partial b_{2}(E,\Lambda)=-b_{2}(\partial E,\Lambda)-b_{2}(E,\Lambda).$
(4.50)

The left hand side is

$\partial_{\mu}[E^{\mu},\Lambda]=[\partial_{\mu}E^{\mu},\Lambda]+[E^{\mu},\partial_{\mu}\Lambda].$
(4.51)

From this it follows that

$b_{2}(B,\Lambda)=m_{2}(\Lambda,B)=-[B,\Lambda],\qquad
b_{2}(E,A)=b_{2}(A,E)=-[E^{\mu},A_{\mu}],$ (4.52)

where $\Lambda\in X_{1},A_{\mu}\in X_{0},E^{\mu}\in X_{-1},B\in X_{-2}$.

#### Homotopy transfer

We recall the homotopy from the complex $X_{\bullet}$ to the gauge invariant
complex $\bar{X}_{\bullet}$. We denote this by

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}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{102.42761pt}{-22.26276pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ .$ (4.53)

Since $p_{0}{\iota}_{0}=1$ and therefore
$\bar{\partial}=\bar{\partial}p_{0}{\iota}_{0}=p_{0}\partial{\iota}_{0}$,
$\bar{\partial}$ is obtained by restricting $\partial$ to $\bar{X}_{\bullet}$.
On gauge fields $A_{\mu}$, $s$ is given by

$s(A)=\Delta^{-1}\partial_{i}A^{i}.$ (4.54)

This allows us to write

$A_{\mu}=\bar{A}_{\mu}+\partial_{\mu}s(A),$ (4.55)

where $\bar{A}_{\mu}$, as a function of $A_{\mu}$, is invariant under gauge
transformations. In the notation of section 2, we would write

$\bar{A}_{\mu}=p_{0}(A_{\mu})=A_{\mu}-\partial_{\mu}\Delta^{-1}\partial^{i}A_{i}.$
(4.56)

We can now apply the homotopy transfer to obtain a relation

$\widehat{A}_{\mu}=p(A_{\mu})$ (4.57)

that is invariant under non-linear gauge transformations. Recall that $p$ was
given by

$p=p_{0}(1+\delta s)^{-1},$ (4.58)

where $\delta=\sum_{k\geq 2}b_{k}$. All maps involved are lifted so that they
act on coalgebras. To get an idea what $p$ looks like, we expand it to
quadratic order. Its linear part is just $p_{0}$, so we directly turn to its
quadratic part. Given two gauge fields $A,B\in X_{0}$, it is

$\displaystyle-p_{0}b_{2}s(A\wedge B)$
$\displaystyle=-p_{0}b_{2}\frac{1}{2}(s(A)\wedge B+A\wedge
s(B)+s(A)\wedge{\iota}_{0}p_{0}(B)+{\iota}_{0}p_{0}(A)\wedge s(B))$ (4.59)
$\displaystyle=-\frac{1}{2}p_{0}([(1+{\iota}_{0}p_{0})(A_{\mu}),s(B)]-[s(A),(1+{\iota}_{0}p_{0})(B)]).$
(4.60)

We now turn this into a non-linear relation between $\widehat{A}_{\mu}$ and
$A_{\mu}$. In order to do this, we need to insert the appropriate symmetry
factors.

$\displaystyle\widehat{A}_{\mu}=p(A_{\mu})$
$\displaystyle=p_{0}(A_{\mu})-\frac{1}{4}p_{0}([(1+{\iota}p)(A_{\mu}),s(A)]-[s(A),(1+{\iota}p)(A_{\mu})])$
(4.61)
$\displaystyle=p_{0}(A_{\mu})+\frac{1}{2}p_{0}[s(A),(1+{\iota}p)(A_{\mu})]\,,$
(4.62)

which is the relation (1.3) quoted in the introduction.

As a consistency check, we now want to proof that $\widehat{A}_{\mu}$ is
indeed gauge invariant to second order. We first compute the linear variation
of the quadratic term. Using
$s(\partial_{\mu}\Lambda)=\Lambda,\partial_{\mu}s(A_{\mu})=(1-{\iota}p)(A),p(\partial_{\mu}...)=0$,
we find that

$\displaystyle\frac{1}{2}p_{0}[s(\partial_{\mu}\Lambda),(1+{\iota}p)(A_{\mu})]+\frac{1}{2}p_{0}[s(A),(1+{\iota}p)(\partial_{\mu}\Lambda)]$
(4.63)
$\displaystyle=\frac{1}{2}p_{0}([\Lambda,(1+{\iota}p)(A_{\mu})]+[s(A),\partial_{\mu}\Lambda])$
(4.64)
$\displaystyle=\frac{1}{2}p_{0}([\Lambda,(1+{\iota}p)(A_{\mu})]-[(1-{\iota}p)(A_{\mu}),\Lambda])$
(4.65) $\displaystyle=-p_{0}([A_{\mu},\Lambda]).$ (4.66)

On the other hand, the variation of the linear part reads

$p_{0}(\partial_{\mu}\Lambda+[A_{\mu},\Lambda])=p_{0}([A_{\mu},\Lambda]),$
(4.67)

which exactly cancels (4.66). This proves gauge invariance of
$\widehat{A}^{\mu}$ to second order.

We now come to the question what the effective equations of motion are. Recall
that the general formula for the associated coderivation is

$\bar{Q}=\bar{\partial}+p_{0}\delta(1+s\delta)^{-1}{\iota}_{0}.$ (4.68)

$\bar{Q}$ simplifies drastically in our case. First of all, since
$\bar{X}_{\bullet}$ only contains fields in degree zero and one, $\bar{Q}$ can
only be nonzero when acting on degree zero elements, i.e. on gauge fields
$\widehat{A}_{\mu}$ only. Next we note that

$(1+s\delta)^{-1}=\sum_{n\geq 0}(-s\delta)^{n}$ (4.69)

acts as the identity on these fields. The reason is that $\delta$, when
restricted to fields in $X_{0}$, produces an element in $X_{-1}$ . But $s$ is
defined to be zero on these elements. The sum in (4.69) breaks after the
zeroth term. This implies that

$\bar{Q}=\bar{\partial}+p_{0}\delta i_{0}=p_{0}(\partial+\delta){\iota}_{0}.$
(4.70)

Therefore, the full equations of motion can be obtained by restriction to
$\partial_{i}A^{i}=0$.

#### Gauge invariant action

We now want to explain why the equations of motion (and the action) can be
obtained by simply replacing $A_{\mu}$ by $\widehat{A}_{\mu}$. We will argue,
within the $L_{\infty}$ framework, that there is a map $\phi(A)$ to the space
of gauge parameters so that together with the map $\widehat{A}(A)$ constructed
above the following identity, c.f. (1.4) in the introduction, holds:

$A_{\mu}\ =\ e^{\Delta_{\phi(A)}}\,\widehat{A}_{\mu}\;.$ (4.71)

Here $\Delta_{\phi}$ is the operator defining infinitesimal gauge
transformations,
$\Delta_{\phi}(A_{\mu})=\partial_{\mu}\phi+\big{[}A_{\mu},\phi\big{]}$. Since
this takes the form of a finite gauge transformation, it follows from gauge
invariance of the action that we may simply replace $A_{\mu}$ by
$\widehat{A}_{\mu}$.

In the last section we constructed a homotopy from $X_{\bullet}$ to
$\bar{X}_{\bullet}$. The homotopy forgets about some of the physically
irrelevant data (in this case the gauge degrees of freedom). We now find it
convenient to still pass from $A_{\mu}$ to $\bar{A}_{\mu}$, but to not forget
about the gauge degrees of freedom. Rather, we represent the gauge degrees of
freedom by another field $\phi$, denoting the space encoding $\phi$ by
$Y_{0}$. Let us now define the map

$\begin{split}F:\,\bar{X}_{0}\oplus Y_{0}&\rightarrow X_{0},\\\
(\bar{A}_{\mu},\phi)&\mapsto{\iota}_{0}(\bar{A}_{\mu})+j_{0}(\bar{A}_{\mu},\phi),\end{split}$
(4.72)

where $j_{0}(\bar{A}_{\mu},\phi)=\partial_{\mu}\phi$. The inverse map is given
by

$\begin{split}F^{-1}:X_{0}&\rightarrow\bar{X}_{0}\oplus Y_{0}\\\
A_{\mu}&\mapsto(p_{0}(A_{\mu}),q_{0}(A_{\mu}))\,,\end{split}$ (4.73)

where $q_{0}$ is defined like the homotopy,
$q_{0}=\Delta^{-1}\partial_{i}A^{i}$, but viewed as a degree 0 map. With this
it is straightforward to verify that these maps are indeed inverses to each
other. These maps formalize the intuitively clear fact that, at least at the
linearized level, the space of all gauge fields $X_{0}$ is isomorphic (of the
same ‘size’) to the sum of the space of gauge invariant fields plus the space
of pure gauge fields.

Our goal is now to extend the above maps to the non-linear or interacting
level by constructing non-linear extensions $q$ and $j$ of $q_{0}$ and
$j_{0}$. This mimics the way the perturbation lemma provides non-linear
corrections to $p_{0}$. In view of the desired identity (4.71), we define the
map ${\cal F}:\bar{X}_{0}\oplus Y_{0}\rightarrow X_{0}$ given by

${\cal F}(\widehat{A}_{\mu},\phi)=e^{\Delta_{\phi}}\widehat{A}_{\mu}\;,$
(4.74)

where $\widehat{A}_{\mu}\in\bar{X}_{0}$ is denoted by a hat to indicate that
this will be the fully gauge invariant field. The map ${\cal F}$ serves as a
nonlinear extension of the maps ${\iota}_{0}$ and $j_{0}$, since to linear
order we have with (4.72)

${\cal F}={\iota}_{0}+j_{0}+\cdots\,.$ (4.75)

Recall that, in the $L_{\infty}$-setup, maps are invertible perturbatively if
and only if their linear part is. Because of (4.75), this is true for ${\cal
F}$. In this way, we can in principle determine the non-linear extension
${\cal F}^{-1}$ of (4.73) and hence the non-linear extension $q$ of $q_{0}$.
This in turn determines the function $\phi(A)$ perturbatively from the ansatz

$\big{(}\widehat{A}_{\mu},\phi(A)\big{)}\equiv\big{(}\widehat{A}_{\mu},q(A)\big{)}\equiv{\cal
F}^{-1}(A_{\mu})\;.$ (4.76)

Let us illustrate this to quadratic order, by expanding ${\cal F}$ to this
order:

$A_{\mu}={\cal
F}(\widehat{A}_{\mu},\phi)=\widehat{A}_{\mu}+\partial_{\mu}\phi+\big{[}\widehat{A}_{\mu},\phi\big{]}+\frac{1}{2}\big{[}\partial_{\mu}\phi,\phi\big{]}+\cdots\
.$ (4.77)

We now apply $q_{0}$ to both sides. After bringing $A_{\mu}$ to the right and
$\phi$ to the left, and recalling $q_{0}(\partial_{\mu}\phi)=\phi$ we find

$\phi=q_{0}(A_{\mu})-q_{0}\big{[}\widehat{A}_{\mu},\phi\big{]}-\frac{1}{2}q_{0}\big{[}\partial_{\mu}\phi,\phi\big{]}\,.$
(4.78)

We then write $\widehat{A}_{\mu}$ and $\phi$ to linear order in $A_{\mu}$,
recalling (4.62) and that to leading order $\phi(A)=s(A)$,

$\displaystyle\phi$
$\displaystyle=q_{0}(A_{\mu})-q_{0}\big{[}{\iota}_{0}p_{0}(A_{\mu}),s(A)\big{]}-\frac{1}{2}q_{0}\big{[}(1-{\iota}_{0}p_{0})(A_{\mu}),s(A)\big{]}$
$\displaystyle=q_{0}(A_{\mu})-\frac{1}{2}q_{0}\big{[}(1+{\iota}_{0}p_{0})(A_{\mu}),s(A)\big{]}\,.$
(4.79)

This is $q$ to first non-trivial order. It should now be clear that, in
principle, we could continue this procedure to determine $\phi$ as a function
of $A_{\mu}$ to arbitrary order.

As a consistency check, we now want to see whether ${\cal F}$ really inverts
$A_{\mu}\mapsto(p(A_{\mu}),q(A_{\mu}))$. We do this using (4.77) to express
$\widehat{A}_{\mu}$ as a function of $A_{\mu}$, and then check whether it
matches what we found through the perturbation lemma. Applying $p_{0}$ to both
sides of (4.77) tells us that

$\widehat{A}_{\mu}=p_{0}(A_{\mu})-p_{0}\big{[}\widehat{A}_{\mu},\phi\big{]}-\frac{1}{2}p_{0}\big{[}\partial_{\mu}\phi,\phi\big{]}\,.$
(4.80)

Expressing $\widehat{A}_{\mu}$ and $\phi$ in terms of $A_{\mu}$ to linear
order gives

$\displaystyle\widehat{A}_{\mu}$
$\displaystyle=p_{0}(A_{\mu})-p_{0}\big{[}{\iota}_{0}p_{0}(A_{\mu}),s(A)\big{]}-\frac{1}{2}p_{0}\big{[}(1-{\iota}_{0}p_{0})(A_{\mu}),s(A)\big{]}$
$\displaystyle=p_{0}(A_{\mu})-\frac{1}{2}p_{0}\big{[}(1+{\iota}_{0}p_{0})(A_{\mu}),s(A)\big{]}\,.$
(4.81)

This is exactly what we found in (4.62).

##  5 Cosmological Perturbation Theory

We now turn to the main motivation of this paper, the perturbation theory
about cosmological or FLRW backgrounds. As an illustration for the techniques
developed in previous sections we determine the general quadratic action of
Einstein-Hilbert coupled to a scalar expanded around FLRW in terms of gauge
invariant variables. We do this both by a brute-force computation and using
the methods of homotopy transfer, arriving at the same result (that to the
best of our knowledge has not appeared before).

###  5.1 Gravity around FLRW

We consider quadratic perturbations in an FLRW universe with minimally-coupled
scalar matter. The theory is described by the action,

$S\ =\ \int d^{4}x\ \sqrt{-g}\
\bigg{\\{}R-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\mathcal{X}\partial_{\nu}\mathcal{X}-V(\mathcal{X})\bigg{\\}}\,,$
(5.1)

where $\mathcal{X}$ denotes the scalar matter field and $V(\mathcal{X})$ is
its potential. The expansion of the fields around the purely time-dependent
background is as follows:

$\begin{split}g_{\mu\nu}(\eta,x)&\ =\
a^{2}(\eta)(\eta_{\mu\nu}+h_{\mu\nu}(\eta,x))\,,\\\ \mathcal{X}(\eta,x)&\ =\
{\mathcal{X}}^{(0)}(\eta)+\varphi(\eta,x)\,,\end{split}$ (5.2)

where $\eta$ is conformal time. The background dynamics are governed by the
Friedmann equations and the equation of motion of $\mathcal{X}^{(0)}$:

$\displaystyle H^{2}\ =\ \frac{1}{6}a^{2}{\rho}\,,$ (5.3)
$\displaystyle\dot{H}+H^{2}\ =\ \frac{1}{12}a^{2}({\rho}-3{p})\,,$ (5.4)
$\displaystyle\ddot{\mathcal{X}}^{(0)}+2H\dot{\mathcal{X}}^{(0)}+a^{2}V^{\prime}\big{(}{\mathcal{X}}^{(0)}\big{)}\
=\ 0\,,$ (5.5)

where the dot denotes the derivative with respect to conformal time,
$H\equiv\frac{\dot{a}}{a}$ is the Hubble parameter, and the prime indicates a
derivative with respect to $\mathcal{X}^{(0)}$. ${\rho}$ and ${p}$ are the
background density and pressure, respectively:

$\displaystyle{\rho}\ =\
\frac{1}{2}a^{-2}\dot{\mathcal{X}}^{(0)2}+V\big{(}\mathcal{X}^{(0)}\big{)}\,,$
(5.6) $\displaystyle{p}\ =\
\frac{1}{2}a^{-2}\dot{\mathcal{X}}^{(0)2}-V\big{(}\mathcal{X}^{(0)}\big{)}\,,$
(5.7)