.

LMU-ASC 16/21

aainstitutetext: Lebedev Institute of Physics, Leninsky ave. 53, 119991
Moscow, Russia bbinstitutetext: Institute for Theoretical and Mathematical
Physics,
Lomonosov Moscow State University, 119991 Moscow, Russiaccinstitutetext:
Department of Natural Sciences, The Open University of Israel,
PO Box 808, Ra’anana 43537, Israelddinstitutetext: Arnold-Sommerfeld-Center
for Theoretical Physics, Ludwig-Maximilians-Universität, München,
Theresienstr. 37, D-80333 München, Germany

# A toy model for background independent string field theory

Maxim Grigoriev,111Corresponding author. c Adiel Meyer d and Ivo Sachs
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study gauge theories of background fields associated to BRST quantized
spinning particle models and identify background-independent algebraic
structures which allow to systematically reduce the spectrum of fields and
subject some of them to dynamical equations of motion. More specifically, we
construct a manifestly background-independent extension of the model based on
$N=2$ spinning particle. The extended system describes the on-shell spin-1
field coupled to off-shell background fields including metric and dilaton.
Tensoring with a given Lie algebra results in a non-abelian extension of the
model.

##  1 Introduction

Covariant string field theory Witten:1986cc ; Zwiebach:1993ie  is, by
construction, a Batalin-Vilkovisky (BV) formulation of some gauge system.
While the origins of the underlying gauge redundancies can easily be traced to
(super) reparameterization invariance on the world sheet, their space-time
interpretation are still to a large extent obscure. This is in contrast with
the BV formulation of the usual gauge theories where the starting point is a
dynamical system whose redundancies and equations of motion have a manifest
space-time interpretation.

On the other hand, the construction of string field theory usually requires
some choice of background metric222See however, Witten:1992qy ;
Shatashvili:1993ps  for a proposal of a manifestly background independent
formulation of open string field theory. in order to quantize the world-sheet
theory, although space-time itself is not a background independent concept in
string theory. While a perturbative construction around a given background is
quite sufficient to address questions like mass renormalization and background
shifts Sen:2015uoa  a global description becomes important to understand off-
shell symmetries or to look for disconnected critical points describing
classical solutions of string theory. Still, in some situations the
perturbative formulation may already be sufficient to find non-trivial vacua.
This is the case, for instance in Witten’s open bosonic string field theory
whose equation of motion describe the tachyon vacuum Schnabl:2005gv  which is
not continuously connected to the trivial vacuum. Open string field theory
also contains information about closed string equations of motion. Indeed
considering open string field theory, on a continuous deformation of the
closed string background one finds that classical consistency of open string
field theory implies the perturbative equations of motion for the closed
string Moeller:2010mh . The global structure of closed string field theory, on
the other hand, does not appear to be described by the available formulations
of open or closed string field theory.

The aim of this work is to analyze background independent structures of a
gauge field theory arising from the quantized spinning particle seen as a toy
model of the quantized string. More precisely, the point particle limit of
string field theory, where the world sheet reduces to a world line. In this
model it is possible to obtain a background independent parametrization of the
BV-configuration space starting from the graded Lie operator (super)algebra
arising from the quantization of the (spinning) world line. In so doing the
BRST operator of the model is understood as a generating function for
background fields while the nilpotency condition and the natural equivalence
are understood as equations of motion and gauge transformations respectively
Horowitz:1986dta ; Grigoriev:2006tt ; Dai:2008bh ; Bekaert:2013zya .

In fact, it turns out that even this smaller algebra is typically too rigid to
admit nilpotent elements other than that corresponding to the off-shell
background fields describing the underlying geometry and hence does not result
in dynamical gauge theories.333These background field can be made dynamical by
supplementing them with the induced action Segal:2002gd ; Tseytlin:2002gz ;
Bekaert:2010ky  or reinterpreting them as the boundary values of the on-shell
fields in the bulk in the context of AdS/CFT correspondence Bekaert:2013zya ;
Bekaert:2017bpy ; Grigoriev:2018wrx  However, by relaxing the nilpotency
condition to hold only on some subspace of the representation space the
spectrum naturally extends by genuine on-shell fields. Furthermore, the
equations imposed by nilpotency on these new fields have a familiar physical
interpretation of the fully non-linear Yang-Mills equations Dai:2008bh  for
$N=2$ spinning particle, Einstein equations Bonezzi:2018box  for $N=4$
spinning particle or Supergravity equations of motion Bonezzi:2010jr .
Equations of motion for background fields in the ambitwistor string arise in
an algebraically analogous way Adamo:2014wea .

These subspaces of the representation space have a simple interpretation in
the world line theory. They correspond to gauging subgroups of the R-symmetry
group that come with any extended world line supersymmetry. Some of these
gaugings also have an interpretation in string theory. They correspond to
level matching, level truncation and partity projection on the world sheet. In
more algebraic terms the theory is determined by a subalgebra of the graded
operator algebra that preserves the subspace and factorized over the left-
right ideal of operators that act trivially on the subspace. Although this is
performed with the graded algebra involving BFV-BRST ghosts, algebraically
this is just a usual constraint reduction or a version of quantum Hamiltonian
reduction.

The complete characterization of the spectrum of the gauge theory associated
to the spinning particle amounts to working out the operator BRST cohomology
in the reduced algebra. This is a tall order, even for the world line, since
one expects to have an infinite number of cohomology classes, corresponding to
background fields of all spins. Here we will reduce this problem by
introducing a filtration with an additional degree that is related to the
dimension of elements in the operator algebra. This allows us to disregard the
background of higher spins in an invariant way and work out the operator BRST
cohomology for some given filtrations for the spinning particle. In
particular, for $\mathcal{N}=2$ at filtration $0$ the theory is completely
characterised by an on-shell Maxwell field coupled to an unconstrained dilaton
field. Tensoring with the gauge Lie algebra results in the non-abelian
extension of this model. Gravity arises at filtration $1$ but equations of
motion for the latter, as well as for the dilaton only arise in the case of
$\mathcal{N}=4$ worldline SUSY.

Despite that we have succeeded to completely describe the spectrum only in
certain filtrations the system itself is perfectly well-defined and possesses
an important property of backgrond- independence. More specifically, to define
the BV-extended field space and the BRST differential (which in turn
determines equations of motion, gauge generators etc.) we do not need to
assume any fixed background fields to be predefined on the space-time. At
least locally, the operator algebra as well as the constraint determining the
reduction require for their construction only the canonical symplectic form on
the contangent bundle and the ghost grading. The subtle point, however, is
that gravity background contained in the degree 1 component of the algebra is
off-shell i.e. the nilpotency does not impose dynamical equations on the
background metric. This can be cured in the $\mathcal{N}=4$ generalization of
the model along the lines of Bonezzi:2018box .

Whether a dynamical system obtained in this way admits an action is a non-
trivial question. In order to construct an action one needs to endow the
operator algebra $\mathcal{A}$ with an (invariant) trace with negative degree,
which does not naturally arise for this type of algebras. Under certain
conditions, which usually amount to an operator-state correspondence between
elements of $\mathcal{A}$ and states of some module of $\mathcal{A}$, this
trace can be realised as an inner product on the latter but turns out to be
degenerate. We use this trace to construct a linearized action for which the
linearized nilpotency derives from a variational principle and then discuss
possible non-linear generalisations.

There is an alternative approach to the quantization of the relativistic
particle, which starts with Brink-Schwarz superparticle Brink:1981nb , where
space-time rather than world-line supersymmetry is manifest. Quantization of
the latter Sorokin:1988nj , after a suitable gauge fixing, gives rise to a
BRST operator $\widetilde{\Omega}$ which is a derivation of the standard
product of graded functions on the representations space Berkovits:2001rb .
The latter property allows to construct gauge-invariant polynomial actions on
the space of functions with an action of $\widetilde{\Omega}$. In particular,
non-abelian super-Yang-Mills theory in 10 dimensions arises as a cubic BV-
action in this way Berkovits:2001rb ; Berkovits:2002zk .

We begin in section 2 by reviewing how a general graded Lie super algebra
gives rise to a BV-BRST gauge system. In section 3 we focus on graded Lie
algebras arising from the world-line with ($\mathcal{N}$-extended) world-line
supersymmetry. However, these algebra do not generically determine on-shell
fields. In section 3.3 we then describe the reduction of $\mathcal{A}$, first
in general and then applied to the world line with $\mathcal{N}=2$ and
$\mathcal{N}=4$ and give a complete description of the theory of background
fields for $\mathcal{N}=2$ at filtration one, thus involving background fields
of spin no larger that two. In section 5 we employ the operator state
correspondence of Dai:2008bh  to construct a degenerate trace on $\mathcal{A}$
which is invariant at the linearized level. Nevertheless this trace gives rise
to a correct linearized action on $\mathcal{A}$. We then propose an ad-hoc
non-linear extension of which reproduces Yang-Mills theory up to higher
derivative terms.

##  2 Gauge theories associated to graded Lie ($L_{\infty}$-) algebras

For a given graded Lie super algebra $\mathcal{A}$ with a
$\mathbb{Z}$-grading, called ghost degree, there is a natural gauge system
associated to $\mathcal{A}$. To see this, consider a generic element $\Phi$ of
degree $1$, Grassmann parity 1, and subject it to the master equation

$[\Phi{,}\,\Phi]=0\,.$ (1)

In case there is a non-degenerate invariant inner product of ghost degree $-3$
on $\mathcal{A}$ this equation follows from a Lagrangian

$L=\frac{1}{6}\langle\Phi{,}\,[\Phi,\Phi]\rangle\,,$ (2)

suggested originally in Horowitz:1986dta . We should emphasise that
$\langle\cdot{,}\,\cdot\rangle$ does not directly originate form the standard
trace since it carries ghost degree $-3$.

In order to describe the gauge symmetries of this system we note that if $\Xi$
is an element of ghost degree $0$ it determines an infinitesimal gauge
transformation

$\delta\Phi=[\Phi{,}\,\Xi]\,.$ (3)

In particular, it maps solutions of (1) to solutions. Finite transformations
are obtained by exponentiation. In a similar way one defines gauge for gauge
symmetries. We refer to this gauge theory as the theory associated to
$\mathcal{A}$.

To continue we wish to associate a BV-BRST homological vector field to this
gauge system. For the moment we will be interested in the description at the
level of equations of motion and their gauge symmetries so that we disregard
the BV master action and the odd symplectic structure, focussing on the
homological vector filed known as BRST differential.444We will reconsider the
existence of a Lagrangian in section 5. More details on the BV formulation at
the level of equations of motion can be found in Barnich:2004cr ;
Kazinski:2005eb . In order to construct BV-BRST description to each element
$e_{A}$ of the homogeneous basis in $\mathcal{A}$ one associates a variable
(coordinate) $\psi^{A}$ of ghost degree
$\mathrm{gh}(\psi^{A})=-\mathrm{gh}(e_{A})+1$ and Grassmann parity
$|\psi^{A}|=|e_{A}|+1\,\,mod\,\,2$. These can be interpreted as follows: a
generic element from $\mathcal{A}$ has the form $\phi^{A}e_{A}$, where
$\phi^{A}$ are real numbers. Then one promotes each $\phi^{A}$ to a coordinate
$\psi^{A}$ on a graded supermanifold $M_{\mathcal{A}}$, associated to
$\mathcal{A}$ seen as a graded superspace. The algebra of functions on
$M_{\mathcal{A}}$ is then a graded super-commutative algebra generated by
$\psi^{A}$.

$M_{\mathcal{A}}$ is naturally a $Q$-manifold, with the $Q$-structure
determined by the algebraic structure on $\mathcal{A}$. To describe the
latter, it is useful to define the following canonical linear function on
$M_{\mathcal{A}}$ with values in $\mathcal{A}$:

$\Psi=\psi^{A}e_{A},\qquad\mathrm{gh}(\Psi)=1,\qquad|\Psi|=1\,,$ (4)

which we will call a string field in what follows. The $Q$-structure is then a
vector field
$Q=Q^{A}(\psi)\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\psi^{A}}$}\kern
1.0pt}{\frac{\partial}{\partial\psi^{A}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\psi^{A}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\psi^{A}}$}\kern
1.0pt}$ on $M_{\mathcal{A}}$, defined as follows: the action of $Q$ on linear
functions is defined through

$Q\Psi=\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{1}{2}$}\kern
1.0pt}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}[\Psi{,}\,\Psi]$ (5)

and then it is extended to all functions by requiring graded Leibnitz rule.
$Q$ is nilpotent by construction and the $Q$-manifold $(M_{\mathcal{A}},Q)$ is
a graded version of the usual Chevalley-Eilenberg (CE) complex which reduces
to the latter if $\mathcal{A}$ is a usual Lie algebra. In terms of components
one has

$Q\psi^{A}=U_{BC}^{A}\psi^{B}\psi^{C}\,\quad\text{with}\quad[e_{A}{,}\,e_{B}]=U_{AB}^{C}e_{C}\,.$
(6)

This is the BV-BRST formulation of the background independent gauge system
determined by a graded Lie algebra ${\cal{A}}$. It is background independent
in the sense that the system is defined without a reference to any fixed
vacuum solution. Moreover, the equations of motion and gauge symmetries are at
least quadratic in fields and hence do not arise as a perturbative expansion
of a more general system.

If $\mathcal{A}$ can be represented as an algebra of functions on space-time,
with coordinates $x^{\mu}$, with values in some “internal” linear space, it is
convenient to associate fields $\psi^{A}(x)$ to basis elements of the internal
space so that the string field really becomes a field defined on space-time.
Note that in this case the Lie bracket is in general a bidifferential
operator. $Q$ is then extended to all local functions by requiring
$[D_{\mu}{,}\,Q]=0$, where $D_{\mu}$ is a total derivative.

The graded supermanifold $M_{\mathcal{A}}$ equipped with $Q$ gives a BV
formulation of the above gauge system. In particular, equations of motion (1)
and gauge symmetries (3) are determined by $Q$. Indeed, the equations of
motion encoded in $Q$ can be obtained as
$\left(Q\Psi^{(-1)}\right)|_{\Psi^{(k)}=0}=0$, $k\neq 0$, where $\Psi^{(k)}$
denote a component of a string field containing ghost-degree $k$-fields.
Explicitly one gets ($k\neq 0$)

$Q\Psi^{(-1)}|_{\Psi^{(k)}=0}=\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{1}{2}$}\kern
1.0pt}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}{[\Psi,\Psi]}|_{\Psi^{(k)}=0}=\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{1}{2}$}\kern
1.0pt}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}[\Psi^{(0)}{,}\,\Psi^{(0)}]\,,$
(7)

giving the original equations (1) upon identifying $\Psi^{(0)}$ and
$\Phi$.555This requires some care in the case if fermions are present among
ghost degree zero fields. Similarly, one finds that $Q$ encodes the gauge
symmetries (3). Further details can be found in Grigoriev:2006tt .

Although we won’t have a need for it here, the above construction can be
generalized to the case where $\mathcal{A}$ is replaced by an
$L_{\infty}$-algebra (it goes without saying that we are dealing with
$\mathbb{Z}$-graded superalgebras). Suppose we are given an
$L_{\infty}$-algebra determined by the following polylinear maps on the space
$\mathcal{A}$:

$\mathcal{D}:\mathcal{A}\to\mathcal{A}\,,\qquad[\cdot,\cdot]:\mathcal{A}\otimes\mathcal{A}\to\mathcal{A}\,,\qquad[\cdot,\cdot,\cdot]:\mathcal{A}\otimes\mathcal{A}\otimes\mathcal{A}\to\mathcal{A}\,,\qquad\ldots\,,$
(8)

which are super-skew-symmetric, and satisfy $L_{\infty}$ conditions (i.e.
$\mathcal{D}$ is nilpotent, $[\cdot,\cdot]$ satisfies Jacobi identity modulo
$\mathcal{D}$-exact terms, etc.). In addition the ghost degree of a $k$-linear
map is assumed to be $2-k$. We note, in passing, that the construction also
applies to $A_{\infty}$-algebras just because each $A_{\infty}$ is an
$L_{\infty}$ as well.

All these structures can also be encoded in the $Q$-structure on
$M_{\mathcal{A}}$ determined by666In various contexts this construction
appeared in Zwiebach:1993ie ; Alexandrov:1995kv ; Grigoriev:2006tt ;
Erler:2013xta  for instance.

$Q\Psi=\mathcal{D}\Psi+\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{1}{2}$}\kern
1.0pt}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}[\Psi,\Psi]+\frac{1}{6}[\Psi,\Psi,\Psi]+\ldots$
(9)

and whose nilpotency encodes the $L_{\infty}$-conditions. With our conventions
$Q$ (as a vector field on $M_{\mathcal{A}}$) also carries ghost degree $1$.
This $Q$ again determines a gauge theory (see e.g. Grigoriev:2006tt ). In
particular, the explicit form of equations of motion can be obtained from $Q$
as $0=Q\Psi^{(-1)}|_{\Psi^{(l)}=0}$, $l\neq 0$. Explicitly this gives with
$\Psi^{(0)}=\Phi$,

$\mathcal{D}\Phi+\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{1}{2}$}\kern
1.0pt}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}[\Phi{,}\,\Phi]+\frac{1}{6}[{\Phi},{\Phi},{\Phi}]+\ldots=0\,,$
(10)

where the brackets are those determining $L_{\infty}$-structure. Furthermore,
gauge symmetries are determined by $Q$ through

$\delta_{\Xi}\Phi=(Q\Psi^{(0)})|_{\Psi^{0}=\Phi,\Psi^{(1)}=\Xi,\Psi^{(k)}=0\,\,k\neq
0,1\,,}\;,\quad$ (11)

where $\Xi$ is a generic element of ghost degree $0$ (gauge parameter).
Explicitly, this gives

$\delta_{\Xi}\Phi=\mathcal{D}\Xi+[\Phi{,}\,\Xi]+\frac{1}{2}[{\Phi},{\Phi},{\Xi}]+\ldots\,.$
(12)

Conversely, a perturbative gauge theory gives rise to an $L_{\infty}$-algebra.
Indeed, for a perturbative gauge theory the BV supermanifold of fields,
antifields and ghosts can be taken to be a linear supermanifold and the theory
is determined by a BRST differential $Q$ which is a formal series in
coordinates. This data is equivalent to that of an $L_{\infty}$ algebra
through (9). More abstractly, this can be expressed as a known statement that
$L_{\infty}$ algebras are in 1:1 with formal $Q$-manifolds; the respective
categories are equivalent. Although for a perturbatove gaueg theory the
$L_{\infty}$ structure is 1:1 with BV formulation the formulation of gauge
theories in terms of $L_{\infty}$-algebras can be quite useful (see e.g.
Jurco:2018sby  and references therein). Let us however mention a few important
subtleties:

  * •

The above discussion is given at the level of equations of motion. To work
with Lagrangian theories one needs a compatible symplectic structure on the
Q-manifold or a suitable invariant inner product on the $L_{\infty}$-algebra.

  * •

The BV formulation is more general because it is well-defined not just for
formal manifolds (i.e. perturative theories) and hence is especially relevant
in the case of background-independent theories. More precisely, given a
$Q$-manifold and a point of the zero locus of $Q$ (vacuum solution), the
tangent space at any point of its zero locus is naturally equipped with
$L_{\infty}$ structure Alexandrov:1995kv . In more physical terms this
corresponds to perturbative expansion of the theory around a vacuum solution
(the point of zero locus).

  * •

Another subtlety is that if we are dealing with a local gauge field theory its
structure is not captured by just $L_{\infty}$. From the BV perspective the
relevant setting is given by the local BV formalism, defined on suitable jet-
bundles (see e.g. Barnich:2000zw  and references therein). A more invariant
setup for local gauge theories at the level of equations of motion is provided
by so-called gauge PDEs (also known as parent formulation) Barnich:2010sw ;
Grigoriev:2012xg ; Grigoriev:2019ojp .

##  3 The relativistic point particles and background fields

In the present exploration we will focus on the algebra $\mathcal{A}$ that
arises from the BRST quantization of the relativistic particle, where the
degree is given by the BRST ghost number. Considering $\mathcal{A}$ as a
graded Lie superalgebra and applying the above construction one ends up with
the gauge theory with BRST differential $Q$, which defines a theory of
background fields on which the particle described by the constraint system
could propagate consistently. This is tantamount to nilpotency of the world
line BRST operator $\Omega$.777We should stress that $Q$ is not to be confused
with the world-line BRST charge.

###  3.1 Scalar particle

Let us illustrate this with the example of scalar point particle with no world
line SUSY. For this we consider the associative algebra $\mathcal{A}$ of
quantum operators (quantized phase space) determined by

$[x^{a}{,}\,p_{b}]=i\delta^{a}_{b}\,,\qquad[c{,}\,b]=1\,,$ (13)

where $[\,{,}\,]$ denotes the graded commutator and the Grassmann degree and
the ghost degree are assigned as

$\mathrm{gh}(x)=\mathrm{gh}(p)=0\,,\qquad\mathrm{gh}(c)=-\mathrm{gh}(b)=1\,,$
(14) $|x|=|p|=0\,,\qquad|c|=|b|=1\,.$ (15)

For a particle propagating on flat Minkowski space-time with metric
$\eta_{\mu\nu}$ and no other background fields present the world line BRST
operator is given by

$\Omega^{init}=-c\eta^{\mu\nu}p_{\mu}p_{\nu}\,,\quad|\Omega^{init}|=1$ (16)

We denote the deformation of $\Omega$ by

$\Omega^{ext}=c(g^{a_{1}\cdots a_{k}}p_{a_{1}}\cdots p_{a_{k}}+D(x))\,,\qquad
k\,{\geqslant}\,1\,,$ (17)

which describes a (higher-spin) background. Gauge transformations (3) amount
to a Weyl rescaling of $g_{a_{1}\cdots a_{k}}$ in conjunction with a shift in
$D$ plus higher spin diffeomorphisms. Nilpotency of $\Omega$, or equivalently
the master equation (1) is trivially satisfied due to $c^{2}=0$, so that the
associated conformal higher spin gauge theory is off-shell. This off-shell
higher spin theory has been proposed in Segal:2002gd  without employing BRST
technique. The above BRST description was in Grigoriev:2006tt .

In order to disentangle background fields of different spins it turns out to
be useful to introduce an additional degree for $\mathcal{A}$, given by
Grigoriev:2012xg

$\begin{gathered}\deg{p_{a}}=1\,,\quad\deg{x^{a}}=0\,,\quad\deg{c}=-1\,,\quad\deg{b}=2\,.\end{gathered}$
(18)

A reader familiar with string theory will recognize the correspondence with
the conformal weights in the world sheet theory. The associated filtration is
given by
$\ldots\mathcal{A}_{0}\subset\mathcal{A}_{1}\subset\ldots\subset\mathcal{A}$,
where we denote by $\mathcal{A}_{r}\subset\mathcal{A}$ a subspace of elements
of degree $r$ or less

${\mathcal{A}_{r}}{\mathcal{A}_{s}}\subset\mathcal{A}_{r+s}\,,\qquad[\mathcal{A}_{r}{,}\,\mathcal{A}_{s}]\subset\mathcal{A}_{r+s-1}$
(19)

In particular, $\mathcal{A}_{r}$ is a subalgebra in $\mathcal{A}$ for
$r\,{\leqslant}\,0$ and $\mathcal{A}_{r}$ is a Lie subalgebra in $\mathcal{A}$
for $r\,{\leqslant}\,1$. An element $F\in\mathcal{A}_{r}$ of definite ghost
number parametrizes only a finite number of fields. In particular,
$\Omega\in\mathcal{A}_{1}$ describes a finite number of low spin fields. For
the bosonic world line we then have the result that the BRST operator
cohomology classes, i.e. non-exact deformations described by $\Omega^{ext}$,
are non-empty for all degrees $\,{\geqslant}\,-1$.

###  3.2 Spinning particles

A way to generate extensions of the above algebra $\mathcal{A}$ is to consider
models with world-line supersymmetry. In this section we describe
$\mathcal{N}=2$ and $\mathcal{N}=4$ spinning particle in some detail. We note
in passing that the $\mathcal{N}=1$ case is quite similar to $\mathcal{N}=0$
and was discussed in some detail in Grigoriev:2018wrx  mostly from the higher
spin theory point of view.

For $\mathcal{N}=2$, $\mathcal{A}$ is the associative algebra generated by the
operators arising from the quantisation of the world line theory which now
includes the world line fermions
$\theta^{\mu}=\frac{1}{\sqrt{2}}(\theta_{1}^{\mu}+i\theta_{2}^{\mu})$,
$\bar{\theta}^{\mu}=\frac{1}{\sqrt{2}}(\theta_{1}^{\mu}-i\theta_{2}^{\mu})$
together with the super reparametrisation ghosts
$c,b,\gamma,\bar{\gamma},\beta,\bar{\beta}$, subject to the commutation
relations

$[\theta^{a}{,}\,\bar{\theta}_{b}]=\delta^{a}_{b}\,,\quad[\gamma{,}\,\bar{\beta}]=1\,,\quad[\bar{\gamma}{,}\,\beta]=1$
(20)

where $[{,}\,]$ denotes the graded commutator with Grassmann parity

$|\gamma|=|\bar{\gamma}|=|\beta|=|\bar{\beta}|=0\,,\qquad|\theta|=|\bar{\theta}|=1\,.$
(21)

The ghost degree is

$\mathrm{gh}(\theta)=\mathrm{gh}(\bar{\theta})=0\,,\quad\mathrm{gh}(\gamma)=\mathrm{gh}(\bar{\gamma})=1\,,\quad\mathrm{gh}(\beta)=\mathrm{gh}(\bar{\beta})=-1\,,$
(22)

while the additional degree determining the filtration is assigned according
to

$\quad\deg{\theta}=\deg{\bar{\theta}}=\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{1}{2}$}\kern
1.0pt}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}\,\quad\deg{\gamma}=\deg{\bar{\gamma}}=-\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{1}{2}$}\kern
1.0pt}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}\,,\quad\deg{\beta}=\deg{\bar{\beta}}=\frac{3}{2}\,.$
(23)

It has been known for a long time that, like for the bosonic world line (16),
the constraints implied by (1) do not restrict the gravitational background
beyond the condition that the connection is torsion free Howe:1988ft . Thus,
like for $\mathcal{N}=0$, the gravitational background remains un-constraint
for $\mathcal{N}=2$. Furthermore, as we will see in section 4.3 symmetries of
$\Omega^{init}$ include conformal Killings in $4$ space-time dimensions. Let
us now turn to filtration zero. In filtration zero, the spin 1 background
fields are set to zero by nilpotency for $\mathcal{N}=2$ Dai:2008bh . However,
there is an off-shell scalar background which can be identified with the
dilaton. To see this, let us recall the structure of $\Omega$

$\displaystyle\Omega=c\mathcal{D}+\bar{\gamma}q+\gamma\bar{q}+\gamma\bar{\gamma}b\,.$
(24)

We then make the following Ansatz for the supercharges:

$q:=\theta^{\mu}(p_{\mu}-i\partial_{\mu}\varphi)\;,\quad\bar{q}:=\bar{\theta}^{\mu}(p_{\mu}+i\partial_{\mu}\varphi)\,.$
(25)

The algebra of the supercharges then takes the form

$\\{q,\bar{q}\\}=p^{2}-\,\partial^{2}\varphi+\,(\partial\varphi)^{2}+2\theta^{\mu}\bar{\theta}^{\nu}\,\partial_{\mu}\partial_{\nu}\varphi\;,$
(26)

This suggests to define the complete BRST operator as

$\displaystyle\Omega$
$\displaystyle=-c\,(p^{2}-\,\partial^{2}\varphi+\,(\partial\varphi)^{2}+2\theta^{\mu}\bar{\theta}^{\nu}\,\partial_{\mu}\partial_{\nu}\varphi)+\gamma\bar{q}+\bar{\gamma}q+\bar{\gamma}\gamma\,b\,$
$\displaystyle\equiv
c\mathcal{D}\;+\;\mathcal{G}\;+\;\bar{\gamma}\gamma\,b\,.$ (27)

Then $\Omega^{2}=0$ implies the two equations

$\displaystyle\mathcal{G}^{2}+\bar{\gamma}\gamma\,\mathcal{D}=0$
$\displaystyle[\mathcal{D},\mathcal{G}]=0\,.$ (28)

The vanishing of the first equation is guaranteed by the choice of non-minimal
coupling in $\mathcal{D}$. A short calculation then shows that the second
equation is also satisfied without imposing any condition on $\varphi$. This
calculation also shows that for $\mathcal{N}=2$, the filtration -1 background
field $D$, which was un-constraint for $\mathcal{N}=0$, is, for
$\mathcal{N}=2$, determined by an un-constraint filtration 0 dilaton
$\varphi$.

For $\mathcal{N}=4$, the world line fermions and super ghost are doubled with
commutation relations

$[\theta^{\mu}_{i}{,}\,\bar{\theta}^{j}_{\nu}]=\delta^{j}_{i}\delta^{\mu}_{\nu}\,,\qquad[\gamma_{i}{,}\,\bar{\beta}^{j}]=\delta^{j}_{i}\,,\qquad[\bar{\gamma}^{i}{,}\,\beta_{j}]=\delta^{i}_{j}\,.$
(29)

It has been known for many years that the $\mathcal{N}=4$ spinning particle
cannot be coupled to gravity. In addition the analysis in Bonezzi:2020jjq
shows that a non-constant dilaton is incompatible with nilpotency of $\Omega$
as well. Thus, although we don’t know the operator BRST at filtration $0$ and
$1$, we do know that gravity and the dilaton is not contained in it.

###  3.3 Constraint reduction of operator algebras

What we learned from the world line models with extended supersymmetry,
discussed in the last section, is that neither Yang-Mills nor Einstein Gravity
feature among the theories of background fields arising from these models.
This may seem odd in view of the fact that the world line should, in some
sense, be the low energy limit of the string from which world-sheet conformal
invariance does imply Yang-Mills - and Einstein equations. The solution to
this puzzle lies in the observation that the super Lie algebras that arise
from spinning world lines admits a consistent reduction. To see this we may
restrict to the case where $\mathcal{A}$ is an associative algebra, since this
is the case for all $\mathcal{A}$ that we have encountered so far.

Suppose now $\mathcal{A}$ comes together with its module $\mathcal{H}$ and
suppose we are interested in a subspace $\mathcal{H}_{0}\subset\mathcal{H}$.
Then one can consider a reduced algebra which is constructed in two steps.
First one considers a subalgebra $\mathcal{A}^{\prime\prime}$ of elements
preserving $\mathcal{H}_{0}$

$\mathcal{A}^{\prime\prime}=\\{a\in\mathcal{A}:a\mathcal{H}_{0}\subset\mathcal{H}_{0}\\}$
(30)

$\mathcal{A}^{\prime\prime}$ is clearly a subalgebra in $\mathcal{A}$.

In the next step one considers a quotient by the ideal in
$\mathcal{A}^{\prime\prime}$ that act trivially on $\mathcal{H}_{0}$:

$\mathcal{I}=\\{a\in\mathcal{A}^{\prime\prime}:a\mathcal{H}_{0}=0\\}$ (31)

It is clear that $\mathcal{I}$ is a left-right ideal in
$\mathcal{A}^{\prime\prime}$ and hence
$\mathcal{A}^{\prime}=\mathcal{A}^{\prime\prime}/\mathcal{I}$ is an
associative algebra. As we are going to see, starting from graded Heisenberg
algebra and its standard module one can build a variety of associative
algebras whose associated gauge field theories are known field theories such
as Yang-Mills or gravity.

Under suitable assumptions the above reduction can be performed without making
use of representation. Indeed, suppose that we are given with certain elements
$T_{i}\in\mathcal{A}$ such that $[T_{i}{,}\,T_{j}]=C_{ij}^{k}T_{k}$, i.e. they
are in involution. Then one can define

$\displaystyle\mathcal{A}^{\prime\prime}$
$\displaystyle=\\{a\in\mathcal{A}:[a{,}\,T_{i}]=A_{i}^{j}T_{j}\\}$ (32)
$\displaystyle\mathcal{I}$
$\displaystyle=\\{a\in\mathcal{A}^{\prime\prime}:a=a^{i}T_{i}\\}$

This definition of $\mathcal{A}^{\prime\prime}$ and $\mathcal{I}$ is
equivalent provided $T_{i}$ are chosen such that $T_{i}\phi=0$ single out
$\mathcal{H}_{0}\subset\mathcal{H}$ and satisfy certain regularity
assumptions. This reduction is a version of the quantum Hamiltonian reduction.
In particular, the definition (32) is used to define higher-spin algebras
Eastwood:2002su ; Vasiliev:2003ev .

##  4 Field theory of the reduced $\mathcal{N}=2$ spinning particle

###  4.1 Maxwell-dilaton system

Here we apply the constraint reduction described above to the case where
$\mathcal{A}$ is an algebra of quantum operators of the BRST-quantized
particle with $\mathcal{N}=2$ supersymmetry and work out explicitly the
equations on the background field imposed by nilpotency of $\Omega$. The
module $\mathcal{H}$ is given by wave functions
$\phi(x|\theta,c,\beta,\gamma)$ which can be viewed as functions of $x$ with
values in polynomials in $\theta,c,\beta,\gamma$. The generators of
$\mathcal{A}$ are represented on wave functions in $x,\theta,c,\beta,\gamma$
according to

$p_{a}=-i\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
x^{a}}$}\kern 1.0pt}{\frac{\partial}{\partial
x^{a}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
x^{a}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
x^{a}}$}\kern
1.0pt}\,,\qquad\bar{\theta}_{a}=\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\theta^{a}}$}\kern
1.0pt}{\frac{\partial}{\partial\theta^{a}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\theta^{a}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\theta^{a}}$}\kern
1.0pt}\,,\qquad
b=\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
c}$}\kern 1.0pt}{\frac{\partial}{\partial
c}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
c}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
c}$}\kern
1.0pt},\qquad\bar{\gamma}=\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\beta}$}\kern
1.0pt}{\frac{\partial}{\partial\beta}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\beta}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\beta}$}\kern
1.0pt},\qquad\bar{\beta}=-\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\gamma}$}\kern
1.0pt}{\frac{\partial}{\partial\gamma}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\gamma}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\gamma}$}\kern
1.0pt}\,.$ (33)

On $\mathcal{H}$ the above data determines a linear gauge system describing
the totally anti-symmetric gauge fields. The standard procedure to construct
it is based on an inner product on the representation space realized as

$\langle\,{,}\,\rangle=\int d^{d}xdc\;\langle\,{,}\,\rangle_{0}\,,$ (34)

where $\langle\,{,}\,\rangle_{0}$ is the Fock space inner product uniquely
determined by identifying $\theta^{\dagger}=\bar{\theta}$,
$\gamma^{\dagger}=\bar{\gamma}$, $\beta^{\dagger}=-\bar{\beta}$, so that e.g.
$\langle\gamma{,}\,\beta\rangle_{0}=1$. The quadratic action is then written
as

$S[\phi]=\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{1}{2}$}\kern
1.0pt}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}\langle\phi{,}\,\Omega^{init}\phi\rangle\,,\qquad\mathrm{gh}(\phi)=0\,.$
(35)

The corresponding BV master actions is obtained by simply replacing $\phi$
with the associated string field $\Psi$.888In the case of $\mathcal{N}>2$
supersymmetric particles the associated linear system describes tensor fields
of more general symmetry type, including e.g. linearized graviton for
$\mathcal{N}=4$.

In order to define $\mathcal{A}^{\prime}$ we introduce the
$\Omega^{init}$-invariant operator

$N=\theta\cdot\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\theta}$}\kern
1.0pt}{\frac{\partial}{\partial\theta}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\theta}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\theta}$}\kern
1.0pt}+\gamma\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\gamma}$}\kern
1.0pt}{\frac{\partial}{\partial\gamma}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\gamma}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\gamma}$}\kern
1.0pt}+\beta\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\beta}$}\kern
1.0pt}{\frac{\partial}{\partial\beta}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\beta}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\beta}$}\kern
1.0pt}\,,\qquad[\Omega^{init}{,}\,N]=0\,$ (36)

counting the $U(1)$ $R$-charge. The subspace $\mathcal{H}_{0}$ is then defined
by $N\phi=q\phi$ for some fixed $q$ which we will soon set to one in order to
single out an irreducible subsystem of spin-1 gauge field. Gauge parameters
$\chi$ satisfy $\mathrm{gh}(\chi)=-1$ and $N\chi=q\chi$. If we restrict $\phi$
to $\mathcal{H}_{0}$ the restricted model indeed describes spin $1$ field. In
particular, (35) is equivalent to Maxwell Lagrangian for a complex spin-1
field.

Given the above $\mathcal{H}_{0}\subset\mathcal{H}$ one defines the reduced
operator algebra $\mathcal{A}^{\prime}$. The filtration on $\mathcal{A}$
defines that on $\mathcal{A}^{\prime}$ and from now we concentrate on the Lie-
subalgebra $\mathcal{A}^{\prime}_{1}$ and its associated gauge field theory.
Let us stress that the construction of $\mathcal{A}^{\prime}$ and the
filtration therein does not require any background fields to be defined on the
space-time and hence the associated field theory is background independent.

However, as we are going to see, on-shell fields are found in
$\mathcal{A}^{\prime}_{0}$, while the metric background enters through
$\Omega^{init}$ and is unconstrained by nilpotence. Later on we discuss how
this asymmetry can be cured by increasing the amount of world line
supersymmetry. In any case it is convenient to decompose $\Omega$ as

$\Omega=\Omega^{init}+\Omega^{ext}\,,\quad\Omega^{init}\in\mathcal{A}^{\prime}_{1}\,,\quad\Omega^{ext}\in\mathcal{A}^{\prime}_{0}\,$
(37)

and analyze the dynamics of the gauge fields parametrerizing $\Omega^{ext}$
keeping $\Omega^{init}$ to be a fixed nilpotent operator. Nilpotency of
$\Omega$ in $\mathcal{A}^{\prime}$ is then equivalent to

$\Omega^{2}\psi\equiv\left([\Omega^{init}{,}\,\Omega^{ext}]+\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{1}{2}$}\kern
1.0pt}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}[\Omega^{ext}{,}\,\Omega^{ext}]\right)\psi=0\,,\qquad\forall\psi\in\mathcal{H}_{0}\,.$
(38)

At the same time the natural gauge transformations for $\Omega^{ext}$ are
described by

$\delta\Omega^{ext}=[\Omega^{init}{,}\,\Xi]\,,\qquad\Xi\in\mathcal{A}^{\prime}_{0},\,\,\,\mathrm{gh}(\Xi)=0\,.$
(39)

Taking into account ghost number, Grassmann parity, filtration, and
$[N{,}\,\Omega^{ext}]=0$ the most general Ansatz for $\Omega^{ext}$ in the
sense of equivalence classes in $\mathcal{A}^{\prime}$ is given by

$\Omega^{ext}=c(B^{a}(x)p_{a}+\mathbf{F}^{ab}(x)\theta_{a}\bar{\theta}_{b}+D(x)+\mathcal{E}(x)(\beta\bar{\gamma}+\gamma\bar{\beta}))+\gamma(\mathbf{A}^{a}(x)\bar{\theta}_{a})+\bar{\gamma}(\bar{\mathbf{A}}^{a}(x)\theta_{a})\,.$
(40)

Here, and in what follows operators in $\mathcal{A}^{\prime}$ are understood
as representatives of equivalence classes and equalities are understood as
equalities in $\mathcal{A}^{\prime}$. In particular, in (40) we have
suppressed terms proportional to $\beta\bar{\gamma}-\gamma\bar{\beta}$ as they
can be reabsorbed into the redefinition of the trace of $\mathbf{F}^{ab}$ by
the equivalence relation defining $\mathcal{A}^{\prime}$.

As we already discussed the metric enter through $\Omega^{init}$ and we may
set it to any background value. For instance, $\Omega^{init}$ associated to
flat Minkowski space is given by

$\Omega^{init}=-cp^{2}+\gamma
p\cdot\bar{\theta}+\bar{\gamma}p\cdot\theta+\gamma\bar{\gamma}b\,,\quad\mathrm{gh}(\Omega^{init})=1\,,\;|\Omega^{init}|=1\,,\;(\Omega^{init})^{\dagger}=\Omega^{init}\,$
(41)

and is represented on $\mathcal{H}$ by

$\Omega^{init}\phi=\left(c\Box-i\gamma\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
x}$}\kern 1.0pt}{\frac{\partial}{\partial
x}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
x}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
x}$}\kern
1.0pt}\cdot\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\theta}$}\kern
1.0pt}{\frac{\partial}{\partial\theta}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\theta}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\theta}$}\kern
1.0pt}-i\theta\cdot\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
x}$}\kern 1.0pt}{\frac{\partial}{\partial
x}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
x}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
x}$}\kern
1.0pt}\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\beta}$}\kern
1.0pt}{\frac{\partial}{\partial\beta}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\beta}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\beta}$}\kern
1.0pt}+\gamma\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\beta}$}\kern
1.0pt}{\frac{\partial}{\partial\beta}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\beta}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\beta}$}\kern
1.0pt}\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
c}$}\kern 1.0pt}{\frac{\partial}{\partial
c}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
c}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial
c}$}\kern 1.0pt}\right)\phi\,.$ (42)

Now we turn to the detailed study of the fields encoded in $\Omega^{ext}$ and
for simplicity set $\Omega^{init}$ to be given by (42). More general
background metrics will be discussed in Section 4.3. A tedious, but mechanical
calculation shows that with this choice of $\Omega^{init}$ (38) implies that

$\displaystyle 0$
$\displaystyle=c\bar{\gamma}\theta^{b}\left(2i\partial_{a}\bar{\mathbf{A}}_{b}+i\partial_{b}B_{a}+\mathbf{F}_{ba}-\mathcal{E}\eta_{ab}\right)p^{a}$
(43)
$\displaystyle+c\gamma\bar{\theta}^{b}\left(2i\partial_{a}\mathbf{A}_{b}+i\partial_{b}B_{a}-\mathbf{F}_{ab}-\mathcal{E}\eta_{ab}\right)p^{a}$
$\displaystyle+\gamma\bar{\gamma}\left(B^{a}+\mathbf{A}^{a}+\bar{\mathbf{A}}^{a}\right)p_{a}$
$\displaystyle-2\gamma\bar{\gamma}cb\;\mathcal{E}$
$\displaystyle+\gamma\bar{\gamma}\left(D-i\partial_{a}\bar{\mathbf{A}}^{a}+\bar{\mathbf{A}}_{a}{\mathbf{A}}^{a}+\mathcal{E}\right)$
$\displaystyle+c\gamma\bar{\theta}^{b}\left(i\partial^{a}\mathbf{F}_{ab}+\partial^{a}\partial_{a}\mathbf{A}_{b}+i\partial_{b}D-iB^{a}\partial_{a}\mathbf{A}_{b}-\mathbf{F}_{ab}\mathbf{A}^{a}-{\mathcal{E}}\mathbf{A}_{b}\right)$
$\displaystyle+c\bar{\gamma}\theta^{a}\left(\partial^{b}\partial_{b}\bar{\mathbf{A}}_{a}+i\partial_{a}\left(D+\mathcal{E}\right)-iB^{b}\partial_{b}\bar{\mathbf{A}}_{a}+\mathbf{F}_{ab}\bar{\mathbf{A}}^{b}-\mathcal{E}\bar{\mathbf{A}}_{a}\right)$

as an equivalence class in $\mathcal{A}_{1}^{\prime}$. This gives rise to
algebraic relations between the fields as well as equations of motion. Note
that each line in (43) leads to one independent constraint. The first four
conditions then imply

$\displaystyle\mathbf{F}_{ab}=i\left(2\partial_{a}\mathbf{A}_{b}-\partial_{b}\mathbf{A}_{a}-\partial_{b}\bar{\mathbf{A}}_{a}\right)\
,\quad\mathbf{F}_{ab}=i\left(-2\partial_{b}\bar{\mathbf{A}}_{a}+\partial_{a}\mathbf{A}_{b}+\partial_{a}\bar{\mathbf{A}}_{b}\right)\
,$ $\displaystyle B^{a}=-\mathbf{A}^{a}-\bar{\mathbf{A}}^{a}\
,\quad\mathcal{E}=0\ ,$ (44)

and are equivalent to the condition that $[\Omega^{2},f(x,c)]=0$ for arbitrary
functions of $x$ and $c$. Note that $\mathbf{F}_{ab}$ has two different
expressions. In order for both expressions to be mutually consistent, we find,

$\bar{\mathbf{A}}_{a}=\mathbf{A}_{a}+\mathcal{C}_{a}\,,\qquad\partial_{a}\mathcal{C}_{b}-\partial_{b}\mathcal{C}_{a}=0,$
(45)

This allows to express $\mathcal{C}_{a}$ as $-2i\partial_{a}\Phi$ in terms of
a potential $\Phi$. Using the above relations we find that

$\mathbf{F}_{ab}=2i\left(\partial_{a}\mathbf{A}_{b}-\partial_{b}\mathbf{A}_{a}\right)-2\partial_{a}\partial_{b}\Phi\,.$
(46)

The last three conditions in (43) then imply the equations of motion

$\displaystyle\partial^{a}(\partial_{a}\mathbf{A}_{b}-\partial_{b}\mathbf{A}_{a})=2(\partial_{a}\mathbf{A}_{b}-\partial_{b}\mathbf{A}_{a})\partial^{a}\Phi\,.$
(47)

Thus, $\Omega^{2}=0$ encodes the complexified Maxwell equations coupled to an
off-shell complex scalar field. The reality condition
$\Omega=\Omega^{\dagger}$ with respect to the natural inner product in
$\mathcal{H}$ implies that $\bar{\mathbf{A}}=\mathbf{A}^{\dagger}$, that is
${\mathbf{A}}=A+id\varphi$ with $A$ and $\varphi$ real so that $\varphi$ can
be identified with the dilaton in (25). Then (47) becomes

$\displaystyle\partial_{a}F^{ab}=2F^{ab}\partial_{a}\varphi\,.$ (48)

where

$\displaystyle F_{ab}=\partial_{a}{A}_{b}-\partial_{b}{A}_{a}\,.$ (49)

A possible way to eliminate the dilaton from the spectrum is to impose reality
of $A_{a}$ as an additional condition. Further consistent restrictions of the
system could be obtained by imposing a reality condition in
$\mathcal{A}^{\prime}$ and $\mathcal{H}$ (which becomes pseudo-Euclidean then)
and requiring $\Omega$ to preserve the inner product in $\mathcal{H}$.

We already mentioned that natural gauge symmetries of the system are given by
(39). Taking into account ghost degree and filtration the general form of
$\Xi\subset\mathcal{A}^{\prime}_{0}$ is $\Xi=i\lambda$. One then obtains,

$\displaystyle[\Omega^{init}{,}\,\Xi]=c\left(i\partial^{a}\partial_{a}\lambda-2\partial_{a}\lambda
p^{a}\right)+\gamma\partial_{a}\lambda\bar{\theta}^{a}+\bar{\gamma}\partial_{a}\lambda\theta^{a}\,,$
(50)

giving

$\displaystyle\delta A_{b}=\partial_{b}\lambda\;,\qquad\delta\varphi=0\,.$
(51)

We have thus identified an operator algebra $\mathcal{A}^{\prime}_{1}$ such
that the most general nilpotent hermitean $\Omega$ of the structure
$\Omega=\Omega^{init}+\Omega^{ext}$ with
$\Omega^{ext}\in\mathcal{A}^{\prime}_{0}\subset\mathcal{A}^{\prime}_{1}$
describes Maxwell theory coupled to an off-shell dilaton.

###  4.2 Non-abelian extension and Yang-Mills fields

Here we consider the non-abelian generalization of (43). To this end as a
basic associative algebra $\widehat{\mathcal{A}}$ (replacing $\mathcal{A}$) we
take the tensor product of the previous $\mathcal{A}$ with a universal
enveloping algebra $\mathcal{U}^{\mathfrak{g}}$ of a given Lie algebra
$\mathfrak{g}$. Algebra $U^{\mathfrak{g}}$ is equipped with the standard
filtration $U^{\mathfrak{g}}_{0}\subset U^{\mathfrak{g}}_{1}\subset
U^{\mathfrak{g}}_{2}\ldots$, where $U^{\mathfrak{g}}_{k}$ contains elements of
order at most $k$ in the generators $T_{a}$ (basis elements of
$\mathfrak{g}$). In particular, $U^{\mathfrak{g}}_{0}$ is the 1-dimensional
subalgebra while $U^{\mathfrak{g}}_{1}/U^{\mathfrak{g}}_{0}$ seen as a Lie
algebra is $\mathfrak{g}$ itself. Furthermore, just like for $\mathcal{A}$ we
have

$\displaystyle U^{\mathfrak{g}}_{m}U^{\mathfrak{g}}_{n}\subset
U^{\mathfrak{g}}_{m+n}\,,\qquad[U^{\mathfrak{g}}_{m}{,}\,U^{\mathfrak{g}}_{n}]\subset
U^{\mathfrak{g}}_{m+n-1}\,.$ (52)

Now taking as $\widehat{\mathcal{A}}$ the tensor product of $\mathcal{A}$ and
$\mathcal{U}^{\mathfrak{g}}$ we obtain an associative algebra analogous to
$\mathcal{A}$. In particular, it is equipped with the total filtration
originating from filtration on $\mathcal{A}$ and $\mathcal{U}^{\mathfrak{g}}$
and moreover $\widehat{\mathcal{A}}^{\prime}_{1}$ is again a Lie subalgebra,
giving a consistent gauge theory determined by $\Omega^{2}=0$ for
$\Omega\in\widehat{\mathcal{A}}^{\prime}_{1}$. Again, construction of
$\widehat{\mathcal{A}}^{\prime}_{1}$ does not involve background fields and
hence the resulting theory is background-independent.

Now we decompose $\Omega$ with respect to the filtration in $\mathcal{A}$
factor as

$\Omega=\Omega^{init}+\Omega^{ext}$ (53)

so that $\Omega^{init}$ has $\mathcal{A}$-filtration $1$ and $\Omega^{ext}$
has $\mathcal{A}$-filtration $0$. If in addition $\Omega^{init}$ has
$\mathcal{A}$-degree $1$ it must be of the form
$\Omega^{init}\otimes\mathbf{1}$. So it automatically happens that
$\Omega^{init}$ is Abelian. At the same time because $\Omega^{ext}$ is of
total filtration $1$ and $\mathcal{A}$-filtration $0$ it can be seen as
$\mathcal{U}^{\mathfrak{g}}_{1}$ valued element of $\mathcal{A}$ and hence can
be decomposed as an Abelian element (i.e.
$\mathcal{U}^{\mathfrak{g}}{}_{0}$-valued element) and $\mathfrak{g}$-valued
one.

As far as the explicit computations are concerned, the nonabelian
generalization amounts to simply taking all the coefficient fields entering
$\Omega^{ext}$ to be matrix-valued and $\Omega^{init}$ proportional to the
unit matrix. To study the equations of motion encoded in
$[\Omega{,}\,\Omega]=0$ we can simply copy from the last subsection the terms
which are linear in the background fields since they are unchanged for matrix
valued fields. In particular the 3rd and 4th line in (43) imply that the
constraints

$\displaystyle
B^{a}=-\mathbf{A}^{a}-\bar{\mathbf{A}}^{a}\,,\quad\mathcal{E}=0\,,$ (54)

are unchanged. After substitution of (54) the generalization of the first
lines in (43) for matrix valued background fields is found to be

$\displaystyle 0$
$\displaystyle=c\bar{\gamma}\theta^{b}([\bar{\mathbf{\Pi}}_{b},\mathbf{\Pi}_{a}+\bar{\mathbf{\Pi}}_{a}]+\mathbf{F}_{ba})p^{a}$
(55)
$\displaystyle+c\gamma\bar{\theta}^{b}([\mathbf{\Pi}_{b},\mathbf{\Pi}_{a}+\bar{\mathbf{\Pi}}_{a}]-\mathbf{F}_{ab})p^{a}\,,$
(56)

where $\mathbf{\Pi}_{a}=p_{a}+\mathbf{A}_{a}$. This implies,

$\displaystyle 2\mathbf{F}_{[a,b]}$
$\displaystyle=[\mathbf{\Pi}_{b},\mathbf{\Pi}_{a}]+[\bar{\mathbf{\Pi}}_{b},\bar{\mathbf{\Pi}}_{a}]+[\mathbf{\Pi}_{b},\bar{\mathbf{\Pi}}_{a}]+[\bar{\mathbf{\Pi}}_{b},\mathbf{\Pi}_{a}]$
$\displaystyle 2\mathbf{F}_{(a,b)}$
$\displaystyle=[\mathbf{\Pi}_{b},\mathbf{\Pi}_{a}]-[\bar{\mathbf{\Pi}}_{b},\bar{\mathbf{\Pi}}_{a}]+[\mathbf{\Pi}_{b},\bar{\mathbf{\Pi}}_{a}]-[\bar{\mathbf{\Pi}}_{b},\mathbf{\Pi}_{a}]$
(57)

Symmetry in $(a,b)$ of the second equation implies, in particular, that

$\displaystyle[\mathbf{\Pi}_{b},\mathbf{\Pi}_{a}]=[\bar{\mathbf{\Pi}}_{b},\bar{\mathbf{\Pi}}_{a}]$
(58)

which is clearly satisfied if $\bar{\mathbf{A}}_{a}=\mathbf{A}_{a}$. Assuming
this in what follows, the forth line of (43), which is unchanged, imposes the
constraint

$\displaystyle
D=i\partial_{a}{\mathbf{A}}^{a}-\mathbf{A}^{a}{\mathbf{A}}_{a}\,,$ (59)

The non-abelian generalization of last two constraints in (43) then impose the
equations of motion for the remaining independent variable ${\mathbf{A}}^{a}$,

$\displaystyle\mathbf{\nabla}^{a}(\partial_{a}\mathbf{A}_{b}-\partial_{b}\mathbf{A}_{a}+i[\mathbf{A}_{a},\mathbf{A}_{b}])=0\,,$
(60)

so that we recover the Yang-Mills equation of motion as in Dai:2008bh .

It is clear that there are generalizations of the solution just presented. We
are not aware of the general solution999See Mostow:1980yb  for a thourough
discussion of solutions to (58). of (58) compatible with the remaining
constraints. Let us just point that (58) is solved by

$\displaystyle\bar{\mathbf{\Pi}}_{a}=g^{-1}\mathbf{\Pi}_{a}g$ (61)

provided $g$ commutes with $\mathfrak{g}$. With this Ansatz the equation (60)
becomes instead

$\displaystyle\mathbf{\nabla}^{a}(\partial_{a}\mathbf{A}_{b}-\partial_{b}\mathbf{A}_{a}+i[\mathbf{A}_{a},\mathbf{A}_{b}])=i(\partial_{a}\mathbf{A}_{b}-\partial_{b}\mathbf{A}_{a}+i[\mathbf{A}_{a},\mathbf{A}_{b}])(\bar{\mathbf{A}}_{a}-\mathbf{A}_{a})\,,$
(62)

This is a non-abelian generalization of (47) and describes complexified Yang-
Mills theory coupled to a non-abelian, but $\mathfrak{g}$-invariant Dilaton
parametrised by $\bar{\mathbf{A}}_{a}-\mathbf{A}_{a}$.

###  4.3 Off-Shell Gravity

In this subsection we generalize the derivation in section 4 to allow for a
generic background metric. For this take $\Omega_{g}^{init}\in\mathcal{A}_{1}$
to be of the form

$\displaystyle\Omega_{g}^{init}=c\mathcal{D}+\bar{\gamma}\theta^{\mu}\Pi_{\mu}+\gamma\bar{\theta}^{\mu}\Pi_{\mu}+\gamma\bar{\gamma}b$
(63)

where $\Pi_{\mu}=p_{\mu}-i\omega_{\mu ab}\theta^{a}\bar{\theta}^{b}$ and

$\displaystyle\mathcal{D}$
$\displaystyle=\Pi^{2}+R_{\mu\nu\lambda\sigma}\theta^{\mu}\bar{\theta}^{\nu}\theta^{\lambda}\bar{\theta}^{\sigma}\,,\quad\Pi^{2}=-g^{\mu\nu}\Pi_{\mu}\Pi_{\nu}+ig^{\mu\nu}\Gamma^{\lambda}_{\mu\nu}\Pi_{\lambda}.$

Here, $x^{\mu}$ and $\theta^{a}$ are regarded as independent variables while
$\theta^{\mu}=e^{\mu}_{a}\theta^{a}$. Note that now we work in general space-
time coordinates $x^{\mu}$ and keep Latin indices to denote the orthonormal
basis in the flat tangent space. One finds Howe:1988ft  (see also
Getzler:2016fek  for a recent discussion of the spinning particles on curved
backgrounds) that $\Omega^{init}_{g}$ is nilpotent, provided the connection is
torsionless.

As before, we deform $\Omega$ as

$\displaystyle\Omega=\Omega_{g}^{init}+\Omega^{ext},$ (64)

with

$\displaystyle\Omega^{ext}=$ $\displaystyle
c\left(-2A^{\mu}\Pi_{\mu}+e^{\mu}_{a}e^{\nu}_{b}\left(F_{\mu\nu}-2D_{\mu}\partial_{\nu}\varphi\right)\theta^{a}\theta^{b}+iD_{\mu}A^{\mu}-A^{2}+D^{\mu}\partial_{\mu}\varphi-(\partial_{\mu}\varphi)^{2}\right)$
$\displaystyle+\gamma
e^{\mu}_{a}\left(A_{\mu}+i\partial_{\mu}\varphi\right)\bar{\theta}^{a}+\bar{\gamma}e^{\mu}_{a}\left(A_{\mu}-i\partial_{\mu}\varphi\right)\theta^{a},$
(65)

where $A_{\mu}$ and $\varphi$ are assumed real and $D_{\mu}$ is the Levi-
Civita covariant derivative acting on tensor fields. From nilpotency of the
BRST operator restricted to $\mathcal{A}^{\prime}_{1}$, we find,

$\displaystyle\Omega^{2}=c\bar{\gamma}\theta^{\sigma}\left(D^{\nu}F_{\nu\sigma}-2F_{\nu\sigma}\partial^{\nu}\varphi\right)-c\gamma\bar{\theta}^{\sigma}\left(D^{\nu}F_{\nu\sigma}-2F_{\nu\sigma}\partial^{\nu}\varphi\right)$
(66)

From which one can read off the background field equations,

$\displaystyle D^{\nu}F_{\nu\sigma}=2F_{\nu\sigma}\partial^{\nu}\varphi\,,$
(67)

which we recognise as the Maxwell equations coupled to off-shell metric and
Dilaton fields. Note that as expected we do not find the gravitational
background field equations. Those do arise when considering the
$\mathcal{N}=4$ relativistic particle Bonezzi:2018box ; Bonezzi:2020jjq .

It is well known that in four dimensions Maxwell theory is conformally
invariant. An efficient way to identify global symmetries in the present
approach is to look for the ”zero modes” of the gauge transformation (3) that
correspond to symmetries of $\Omega^{init}$. For simplicity, we consider
$\Omega^{init}$ with Minkowski metric. After some trial and error we come up
with Ansatz

$\displaystyle\Xi_{c}=i{\xi_{a}}{p^{a}}+\frac{1}{2}\partial_{a}\xi^{a}+{t_{ab}}\left({{\theta^{a}}{\bar{\theta}^{b}}-{{\theta}^{b}}{{\bar{\theta}}^{a}}}\right)-\lambda\left({bc-
cb}\right)-\widehat{f}\left({\gamma\bar{\beta}+\beta\bar{\gamma}}\right)+ic{\widehat{A}_{a}}\left({\beta{{\bar{\theta}}^{a}}+\bar{\beta}{\theta^{a}}}\right)$
(68)

for $\Xi_{c}\in{\cal{A}}_{1}$. This leads to

$\displaystyle[\Omega^{init}{,}\,\Xi_{c}]=$
$\displaystyle-c\left({2{\partial_{b}}{\xi_{a}}{p^{a}}{p^{b}}-i{\partial^{e}}\left({\partial_{e}}{\xi_{d}}+\partial_{d}\xi_{e}\right){p^{d}}}-\frac{1}{2}\partial_{e}\partial^{e}\partial_{d}\xi^{d}\right)$
$\displaystyle+{\partial_{a}}{\xi_{d}}\left({\gamma{{\bar{\theta}}^{a}}+\bar{\gamma}{\theta^{a}}}\right){p^{d}}{-\frac{i}{2}\left(\gamma\bar{\theta}^{a}+\bar{\gamma}\theta^{a}\right)\partial_{a}\partial_{d}\xi^{d}}$
$\displaystyle+\left({c\left({2i{\partial_{a}}{t_{de}}{p^{a}}+{\partial^{a}}{\partial_{a}}{t_{de}}}\right)-i\left({\gamma{{\bar{\theta}}^{a}}+\bar{\gamma}{\theta^{a}}}\right){\partial_{a}}{t_{de}}}\right)\left({{\theta^{d}}{{\bar{\theta}}^{e}}-{\theta^{e}}{{\bar{\theta}}^{d}}}\right)$
$\displaystyle+2{t_{ae}}\left({\gamma{{\bar{\theta}}^{e}}+\bar{\gamma}{\theta^{e}}}\right){p^{a}}$
(69)
$\displaystyle+c\left({2\lambda{p^{2}}-{\partial^{a}}{\partial_{a}}\lambda-2i{\partial_{a}}\lambda{p^{a}}+i{{\widehat{A}}_{d}}\left({{\theta^{d}}{{\bar{\theta}}^{a}}-{\theta^{a}}{{\bar{\theta}}^{d}}}\right){p_{a}}-{\partial_{a}}{{\widehat{A}}_{d}}{\theta^{a}}{{\bar{\theta}}^{d}}}\right)$
$\displaystyle-c\gamma\bar{\beta}{\partial_{a}}{\widehat{A}^{a}}-c\left({{\partial^{a}}{\partial_{a}}\widehat{f}+i\left({2{\partial_{a}}\widehat{f}+{{\widehat{A}}_{a}}}\right){p^{a}}}\right)\left({\gamma\bar{\beta}+\beta\bar{\gamma}}\right)$
$\displaystyle+\frac{i}{2}\left({2{\partial_{a}}\lambda+{{\widehat{A}}_{a}}}\right)\left({\gamma{{\bar{\theta}}^{a}}+\bar{\gamma}{\theta^{a}}}\right)\left({bc-
cb}\right)+2\gamma\bar{\gamma}b\left({\lambda-\widehat{f}}\right)$
$\displaystyle{+\frac{i}{2}\bar{\gamma}{\theta^{a}}\left({2{\partial_{a}}\widehat{f}-{{\widehat{A}}_{a}}}\right)+\frac{i}{2}\gamma{\bar{\theta}^{d}}{\widehat{A}_{d}}}-\left({\gamma{{\bar{\theta}}^{a}}+\bar{\gamma}{\theta^{a}}}\right)\widehat{f}{p_{a}}$

For $\lambda=\widehat{f}=\widehat{A}_{a}=0$ and
$t_{ab}=\frac{1}{4}\left(\partial_{a}\xi_{b}-\partial_{b}\xi_{a}\right)$ the
commutator $[\Omega^{init}{,}\,\Xi_{c}]$ then vanishes on the solutions of the
Killing equation $\partial_{a}\xi_{b}+\partial_{b}\xi_{a}=0$, in agreement
with Poincaré invariance. For $d=4$ with

$\displaystyle\lambda=\widehat{f}=\frac{1}{4}\partial_{a}\xi^{a},\qquad\
\widehat{A}_{a}=-2\partial_{a}\lambda,\qquad\
t_{ab}=\frac{1}{4}\left(\partial_{a}\xi_{b}-\partial_{b}\xi_{a}\right),$ (70)

the vanishing of (69) implies the conformal Killing equation
$\partial_{a}\xi_{b}+\partial_{b}\xi_{a}=\frac{1}{2}\partial_{c}\xi^{c}\eta_{ab}$
in agreement with conformal invariance the Maxwell action in four dimensions.

We note in passing that if we consider gauge transformations
$[\Omega^{init}{,}\,\Xi_{c}]$ with the parameter $\Xi_{c}$ determined through
(70) by $\xi^{a}$ that is not necessarily a conformal Killing, but still
satisfying $\Box\partial_{a}\xi^{a}=0$, one finds that the gauge variation of
$\Omega$ is equivalent to

$\displaystyle\delta
g_{ab}=-\partial_{a}\xi_{b}-\partial_{b}\xi_{a}+\frac{1}{4}\partial_{c}\xi^{c}\eta_{ab}\,.$
(71)

If we consider generic $\Xi^{a}$ the respective gauge transformation do not
amount to the transformation of the metric so that further background fields
are to be introduce in $\Omega^{init}$ in order to realize such gauge
transformations. The explicit identification of a a complete set of background
fields remains an open problem.

###  4.4 $\mathcal{N}=4$ extension

As another illustration of the general reduction described at the beginning of
this section we revisit the relativistic particle with $\mathcal{N}=4$ world
line supersymmetry briefly described in section 3. For $\mathcal{N}=4$ there
is more freedom in choosing the subspace $\mathcal{H}_{0}$, corresponding to
different choices of generators of the $SO(4)$ R-symmetry present in this
model. For the maximal reduction we impose that Bonezzi:2018box

$\begin{split}N_{i}&:=\theta_{i}\cdot\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\theta_{i}}$}\kern
1.0pt}{\frac{\partial}{\partial\theta_{i}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\theta_{i}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\theta_{i}}$}\kern
1.0pt}+\gamma_{i}\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\gamma_{i}}$}\kern
1.0pt}{\frac{\partial}{\partial\gamma_{i}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\gamma_{i}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\gamma_{i}}$}\kern
1.0pt}+\beta_{i}\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\beta_{i}}$}\kern
1.0pt}{\frac{\partial}{\partial\beta_{i}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\beta_{i}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\beta_{i}}$}\kern
1.0pt}\;,\quad i=1,2\\\\[5.69054pt]
Y&:=\theta^{1}\cdot\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\theta_{2}}$}\kern
1.0pt}{\frac{\partial}{\partial\theta_{2}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\theta_{2}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\theta_{2}}$}\kern
1.0pt}+\gamma^{1}\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\gamma_{2}}$}\kern
1.0pt}{\frac{\partial}{\partial\gamma_{2}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\gamma_{2}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\gamma_{2}}$}\kern
1.0pt}+\beta^{1}\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\beta_{2}}$}\kern
1.0pt}{\frac{\partial}{\partial\beta_{2}}}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\beta_{2}}$}\kern
1.0pt}{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{\partial}{\partial\beta_{2}}$}\kern
1.0pt}\;,\\\\[5.69054pt]
T&:=\frac{\partial^{2}}{\partial\theta_{1}\cdot\partial\theta_{1}}+\frac{\partial^{2}}{\partial\gamma_{1}\partial\beta_{2}}-\frac{\partial^{2}}{\partial\gamma_{2}\partial\beta_{2}}\;,\end{split}$
(72)

commute with $\Omega$. Furthermore $\mathcal{H}_{0}\subset\mathcal{H}$ is
defined by $N_{i}\phi=\phi$, $i=1,2$ and $Y\phi=T\phi=0$ for
$\phi\in\mathcal{H}_{0}$. This defines the reduced operator algebra
$\mathcal{A}^{\prime}$ along the same lines as in section 3.3. The filtration
on $\mathcal{A}$ is introduced in a similar way, giving us a Lie subalgebra
$\mathcal{A}^{\prime}_{1}$ and its associated gauge field theory. Let us
stress that the construction of $\mathcal{A}^{\prime}_{1}$ again does not
require specification of background fields and hence the associated field
theory is background independent. While we do not have a full characterization
of the spectrum in filtration 1 we know that the metric is constrained by
nilpotency of $\Omega\in\mathcal{A}^{\prime}_{1}$ to be Ricci flat
Bonezzi:2018box .

Another consistent choice of a subspace
$\widetilde{\mathcal{H}}_{0}\supset\mathcal{H}_{0}$ and the corresponding
$\mathcal{A}^{\prime}$, is defined by $N_{i}\phi=\phi$, $i=1,2$ and $Y\phi=0$
Bonezzi:2018box . For this choice of $\mathcal{A}^{\prime}$, nilpotency of
$\Omega$ implies the coupled equations Bonezzi:2020jjq

$R_{\mu\nu}-\Lambda
g_{\mu\nu}+2\nabla_{\mu}\nabla_{\nu}\varphi=0\;,\quad\nabla^{2}\varphi-2\,\nabla^{\mu}\varphi\,\nabla_{\mu}\varphi+2\Lambda\varphi=0\;,$
(73)

for the metric in degree 1, the dilaton in degree 0 and the cosmological
constant $\Lambda$ in degree -1. Note that the Maxwell field is set to zero by
nilpotency for all choices of $\mathcal{H}_{0}$. This can be seen by adding a
vector potential to $\Omega$ describing a Minkowski background, that is

$\displaystyle q_{i}$
$\displaystyle=\theta_{i}^{a}\left(p_{a}+A_{a}\right)\quad\text{and}\quad\bar{q}^{j}=\bar{\theta}^{i\,a}\left(p_{a}+A_{a}\right)\,$
(74)

with commutator,

$\displaystyle\bar{\gamma}^{i}\gamma_{j}\left\\{q_{i},\bar{q}^{j}\right\\}=\bar{\gamma}^{i}\gamma_{i}H+\frac{i}{2}\bar{\gamma}^{i}\gamma_{j}\theta_{i}^{a}\bar{\theta}^{j\,b}F_{ab},$
(75)

where $H=p^{2}+2A^{a}p_{a}-i\partial_{a}A^{a}+A^{2}$ and
$F_{ab}=2i\left(\partial_{a}A_{b}-\partial_{b}A_{a}\right)$. Nilpotency of
$\Omega$ in $\mathcal{A}^{\prime}$ then implies $F_{ab}=0$.

##  5 Operator-state correspondence, physical states and action

Having obtained the Yang-Mills equations of motion as a consequence of the
nilpotency of $\Omega$ we will now address the question of constructing an
action from which these equations, or equivalently, (1) derive. The field
content encoded in $\Omega^{ext}\in\mathcal{A}^{\prime}_{0}$ does not seem to
admit an invariant inner product required for the action principle so our
strategy is to split the component equations encoded in (43) into off-shell
algebraic constraints and the remaining equations. More specifically, the off-
shell constraints are encoded in

$[\Omega^{2}{,}\,f(x,c)]=0$ (76)

for any $f(x,c)$. These encode101010Strictly speaking they are not algebraic
because one of the equations is $d(\bar{\mathbf{A}}-\mathbf{A})=0$. However,
in terms of our parameterization $\bar{\mathbf{A}}-\mathbf{A}=d\Phi$ the
differential constraint is not present. the first four lines of (43) and
sufficient to eliminate all the fields save for $\mathbf{A},\Phi$ and $D$.

###  5.1 Inner product and quadratic Lagrangians

Then we look for an action functional for $\Omega^{ext}$ such that upon
eliminating all the fields save for $\bar{\mathbf{A}},\Phi,D$ via (76) its
Euler-Lagrange equations reproduce the remaining equations from (43), i.e.
Maxwell equations on the dilaton background. For this purpose we need an
invariant inner product on $\mathcal{A}^{\prime}_{0}$ which thanks to the
algebraic constraints could well be degenerate. More precisely, it should
become nondegenerate only after imposing these constraints and disregarding
the dilaton $\Phi$ because it is off-shell.

In order to construct such an inner product, following Dai:2008bh , we first
observe that there exists a map $\mu$ from $\mathcal{A}^{\prime}$ to
$\mathcal{H}_{0}$ that can be used to pull the standard inner product on
$\mathcal{H}_{0}$ back to $\mathcal{A}^{\prime}$. To construct this map we
pick an element in $\mathcal{H}_{0}$ represented by the wave function $\beta$
(as before here we use the identification of the Fock space vectors and
functions in creation operators $\beta,\gamma,\theta,\bar{\theta}$). Thanks to
$\Omega^{init}\beta=0$ it is in the cohomology of $\Omega^{init}$ of ghost
degree -1. Then for a generic element $O\in\mathcal{A}^{\prime\prime}$ the map
$\mu$ is defined as

$\mu(O)=O\beta$ (77)

and it induces a well-defined map $\mathcal{A}^{\prime}\to\mathcal{H}_{0}$
because $\beta\in\mathcal{H}_{0}$ so that $\mu$ acts trivially on
$\mathcal{I}\subset\mathcal{A}^{\prime\prime}$. Finally the degenerate inner
product on $\mathcal{A}^{\prime}$ is defined as:

$\langle a{,}\,b\rangle_{\boldsymbol{\beta}}:=\langle
a\boldsymbol{\beta}{,}\,b\boldsymbol{\beta}\rangle\,.$ (78)

It turns out this inner product is nondegenerate on a subspace of
$\mathcal{A}^{\prime}_{0}$ singled out by the constraints (76) and with the
dilaton set to a fixed background value. Indeed, generic
$\Omega^{ext}\in\mathcal{A}^{\prime}_{0}$ that belongs to the subspace is
parameterized by the unconstrained complex $\bar{\mathbf{A}}_{a},D$ and is
mapped by $\mu$ to

$\displaystyle\psi=\mu(\Omega^{ext})=\Omega^{ext}\beta=\bar{\mathbf{A}}_{b}\theta^{b}+Dc\beta\,,$
(79)

which is a generic ghost degree $0$ element of $\mathcal{H}_{0}$ so the map is
indeed 1:1. Analogous considerations apply to other ghost degrees so that the
inner product is indeed nondegenerate.

Unfortunately such an inner product is not invariant, i.e. in general for an
antihermitean $a$

$\langle[a{,}\,c]{,}\,b\rangle_{\boldsymbol{\beta}}+\langle
c{,}\,[a{,}\,b]\rangle_{\boldsymbol{\beta}}\neq 0\,.$ (80)

However, if $a\boldsymbol{\beta}=0$ the above becomes an equality. This is the
property we employ to construct the linearized action. Indeed, consider the
quadratic approximation of
$\frac{1}{4}\langle\Omega{,}\,[\Omega{,}\,\Omega]\rangle_{{\boldsymbol{\beta}}}$,

$S[\Omega^{ext}]=\mathchoice{\raisebox{0.5pt}{\footnotesize$\displaystyle\frac{1}{2}$}\kern
1.0pt}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}\langle\Omega^{ext}{,}\,\Omega^{init}\Omega^{ext}\rangle_{{\boldsymbol{\beta}}}$
(81)

where for the moment we do not assume $\Omega^{ext}$ hermitean. This action is
by construction gauge invariant and using (79) and the algebraic constraints
can be rewritten as the standard $\langle\psi{,}\,\Omega^{init}\psi\rangle$
action (c.f. (35)) for the spin-1 field. Explicitly one gets

$\displaystyle S[\Omega^{ext}]$ $\displaystyle=-\frac{1}{2}\int
d^{d}x\;\left(\bar{\mathbf{A}}^{\dagger}_{b}\Box\bar{\mathbf{A}}_{b}+iD^{\dagger}(\partial\cdot\bar{\mathbf{A}})+i\bar{\mathbf{A}}^{\dagger}_{b}\partial^{b}D-D^{\dagger}D\right)$
(82)

which upon elimination of the auxiliaries $D,D^{\dagger}$ gives the standard
complexified Maxwell action. Note that the action does not depend on the
dilaton $\Phi$ and hence imposes no equations on it while the complexified
Maxwell equation it determines coincides with the linearization of (47) as
expected. Note that (82) holds irrespective whether we assume the algebraic
constraint (76) or not. In particular we can impose (76) before or after
obtaining the equations of motion from (82).

As we have seen hermiticity on $\Omega^{ext}$ together with (76) implies that
$\bar{\mathbf{A}}_{a}=A_{a}-i\partial_{a}\phi$ with $A_{a},\phi$ real so that
the above complexified Maxwell action reduces to the one for the real field
$A_{a}$ while the dependence on the dilaton drops out because $\phi$ does not
contribute to the Faraday tensor. In particular, the resulting equation of
motion do not involve dilaton and is a lineraization of the real Maxwell-
dilaton system (48).

###  5.2 Non-linear extension

In this last section we describe an attempt of a non-linear generalization of
(81). Here we will assume $\bar{\mathbf{A}}=\mathbf{A}$ and hermitean so that,
in particular, the dilaton is absent. In addition we assume the algebraic
constraint (76). Then $\Omega$ takes the simpler form

$\displaystyle\Omega=-c(\Pi^{2}+2[\Pi_{a},\Pi_{b}]\theta^{a}\bar{\theta}^{b}-\widetilde{D})+\bar{\gamma}\theta^{\mu}\Pi_{\mu}+\gamma\bar{\theta}^{\mu}\Pi_{\mu}+b\bar{\gamma}\gamma\,,$
(83)

where $\widetilde{D}=D-i\partial\cdot A+A^{2}$ is the shifted auxiliary field
which we assume complex in what follows. The natural extension111111The
pefactor $\frac{1}{4}$ instead of the expected $\frac{1}{6}$ from (2)
originates in the non-invariance of the inner product. of (81)

$S[\Omega]=\frac{1}{4}\langle\Omega{,}\,[\Omega{,}\,\Omega]\rangle_{{\boldsymbol{\beta}}}$
(84)

then evaluates to

$\frac{1}{4}\langle\Omega{,}\,[\Omega,\Omega]\rangle{}_{\boldsymbol{\beta}}=-\frac{1}{2}\int
d^{d}x\;\textbf{tr}_{\mathfrak{u}}\left((\Pi^{2}-\widetilde{D}^{\dagger})\Pi\cdot\Pi-(\Pi^{2}-\widetilde{D}^{\dagger})(\Pi^{2}-\widetilde{D})-\Pi^{\mu}(\Pi^{2}-\widetilde{D})\Pi_{\mu}\right.\\\
\left.\qquad\qquad\qquad+\Pi\cdot\Pi(\Pi^{2}-\widetilde{D})-2\Pi^{\mu}[\Pi_{\mu},\Pi_{\nu}]\Pi^{\nu}\right)$
(85)

and, after elimination of $\widetilde{D}$ and $\widetilde{D}^{\dagger}$,

$S[\Omega]=\frac{1}{2}\int
d^{d}x\;\textbf{tr}_{\mathfrak{u}}\left(\Pi^{\mu}\Pi^{2}\Pi_{\mu}-\Pi^{2}\Pi^{2}+2\Pi^{\mu}[\Pi_{\mu},\Pi_{\nu}]\Pi^{\nu}\right)=\\\
\frac{1}{2}\int d^{d}x\;\textbf{tr}_{\mathfrak{u}}\left(A^{\mu}(p^{2}+A\cdot
p+p\cdot A+A^{2})A_{\mu}-(\partial\cdot
A)^{2}+(A^{2})^{2}+[A^{\nu},A^{\mu}][\Pi_{\mu},\Pi_{\nu}]\right)$ (86)

In the commutative case this reproduces (82), but for a non-abelian gauge
group the relative factors don’t match to reproduce the Yang-Mills action.
This is a consequence of the lack of cyclicity of (80) that does not allow us
to bring the first line in (86) into the form
$[\Pi_{\mu},\Pi_{\nu}][\Pi^{\mu},\Pi^{\nu}]$. We may try to remedy this by
starting with an alternative inner product given by

$\langle a{,}\,b\rangle^{\prime}{}_{\boldsymbol{\beta}}:=\int
d^{d}xd^{d}y\langle x|a\boldsymbol{\beta}{,}\,b\boldsymbol{\beta}|y\rangle
K_{\epsilon}(x,y)\,,$ (87)

where $K_{\epsilon}(x,y)$ is some regulator function to be determined. For now
we just assume that it is compatible with the required cyclicity so that we
can cyclicly permute $\Pi^{\mu}$ and get

$\displaystyle{\rm Tr}(\Omega^{\dagger}[\Omega,\Omega])=\frac{2}{N}\int
d^{d}xd^{d}y\;\textbf{tr}_{\mathfrak{u}}<x|[\Pi^{\nu},\Pi^{\mu}][\Pi_{\mu},\Pi_{\nu}]+\widetilde{D}^{\dagger}\widetilde{D}-\Pi^{2}\widetilde{D}|y>K_{\epsilon}(x,y)$
(88)

where $N=\int d^{d}xK_{\epsilon}(x,0)$ is a normalisation factor. Variation
w.r.t. $\widetilde{D}^{\dagger}$ then gives $\widetilde{D}=0$ and thus
produces the expected Yang-Mills action in the absence of the dilaton.

This still leaves us with the task of finding a regulator function
$K_{\epsilon}(x,y)$ which commutes with $\Pi^{\mu}$ to ensure the assumed
cyclicity. A natural Ansatz for $K_{\epsilon}(x,y)$ that comes to mind is

$\displaystyle
K_{\epsilon}(x,y)=<x|e^{-\epsilon\Pi^{2}}|y>\stackrel{{\scriptstyle\epsilon\to
0}}{{\to}}\delta(x-y)\,.$ (89)

This regulator does not commute with $\Pi^{\mu}$, due to

$\displaystyle[\Pi^{\mu},\Pi^{2}]=-\frac{1}{2}\left(\Pi^{\nu}F_{\mu\nu}+F^{\mu}_{\;\;\nu}\Pi^{\mu}\right)$
(90)

However, the extra terms generated by this are suppressed by as
$\epsilon^{r+s}\nabla_{\lambda}^{r}(F_{\mu\nu})^{s}$. Thus, we may say that
the trace regulated in this way is cyclic (on $\Pi^{\mu}$) up to terms of
higher orders in derivatives or fields (or both).121212We may note that the
Ansatz $K_{\epsilon}(x,y)=<x|e^{-\epsilon p^{2}}|y>$ works equally well to
leading order in $\epsilon$.

##  6 Conclusions

The purpose of this work was to propose a precise algebraic framework allowing
to interpret the results in Dai:2008bh ; Bonezzi:2018box ; Bonezzi:2010jr  as
a toy model for a background independent formulation of string field theory.
More specifically, we employed the BV-BRST description of the background
fields associated to quantum constrained systems which allows to associate to
a given graded operator algebra $\mathcal{A}$ (and more generally
$L_{\infty}$-algbera) a gauge field theory in the BV-BRST formulation. If
$\mathcal{A}$ is an operator algebra of a quantized (spinning) particle, the
assocoiated field theory is usually off-shell in the sense that it only
implies algebraic constraints and gauge symmetries but not dynamical
equations.

For the spinning particle with $\mathcal{N}=2$ supersymmetry there is a
natural constraint reduction of $\mathcal{A}$ associated to the gauging of the
world line $R$-symmetry. The space of background fields associated to this
algebra is infinite reflecting, in particular, higher spin backgrounds. We
then introduced a suitable filtration on $\mathcal{A}$ which allows to
identify in an invariant way a finite subalgebra whose associated fields to
not involve infinite tower of higher spins. We explicitly analyzed gauge
fields associated to filtration 0 subalgebra and find that they are exhasuted
by Maxwell field supplemented by an off-shell dilaton. We also propose a
nonabelian extension of the model leading to a Yang-Mills generalization of
the system. We did not work out the complete operator BRST-cohomology for all
ghost numbers for this problem but instead focused on the physical fields.
Working out the later would be an interesting extension of the present note.

Increasing the amount of world-line symmetry leads to interesting
generalizations of $\mathcal{A}$. In particular, the $\mathcal{N}=4$ particle
is known to give rise to on-shell gravity and its extensions. The complete
description of the underlying gauge theory and its manifestly background
formulation appears to be possible but industrious. Finding a suitable
parametrization for this problem would be another interesting extension of the
the present work. We note, in passing, that background fields associated to
extended supersymmetric particles with generic even $\mathcal{N}$ have been
discussed in Bekaert:2009fg  from the higher spin theory perspective.

Finally we explored the possibility of constructing a Lagrangian formulation
of the gauge theory associated to $\mathcal{N}=2$ particle. Generically, this
requires an invarinat inner product of degree -3 on $\mathcal{A}$ as a new
ingredient to the formalism. In the quadratic approximation this turns out to
be rather straightforward and instructive. On the other hand, we have not
succeeded in constructing a satisfactory description of the latter at the non-
linear level due to the lack of a cyclic property of our construction. This is
in contrast to string theory or the pure spinor approach, where a cubic action
on the representation space is known Berkovits:2001rb ; Berkovits:2003ny  and
indeed reproduces super Yang-Mills theory in 10 dimensions. The reason that
this construction cannot be repeated for the world line is that in the pure
spinor approach the undeformed BRST operator (corresponding to
$\Omega^{init}$) above is a derivation of the product of functions on $H$.
This not the case in our formalism.

## Acknowledgments

We profited from discussions with Roberto Bonezzi. I.S. would like to Warren
Siegel for helpful discussions. This work was funded by the Excellence Cluster
Origins of the DFG under Germany’s Excellence Strategy EXC-2094 390783311. The
work of M.G. was supported in part by the Russian Science Foundation grant
18-72-10123.

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