Higher spins and Finsler geometry

Alessandro Tomasiello

Dipartimento di Matematica, Università degli Studi di Milano–Bicocca,
Via Cozzi 55, 20126 Milano, Italy
and
INFN, sezione di Milano–Bicocca

<EMAIL_ADDRESS>

Finsler geometry is a natural generalization of (pseudo-)Riemannian geometry,
where the line element is not the square root of a quadratic form but a more
general homogeneous function. Parameterizing this in terms of symmetric
tensors suggests a possible interpretation in terms of higher-spin fields. We
will see here that, at linear level in these fields, the Finsler version of
the Ricci tensor leads to the curved-space Fronsdal equation for all spins,
plus a Stueckelberg-like coupling. Nonlinear terms can also be systematically
analyzed, suggesting a possible interacting structure. No particular choice of
spacetime dimension is needed. The Stueckelberg mechanism breaks gauge
transformations to a redundancy that does not change the geometry. This is
however not enough to eliminate non-transverse modes, at least for some
versions of Finsler dynamics.

###### Contents

  1. 1 Introduction
  2. 2 Fronsdal equation
    1. 2.1 Spin two
    2. 2.2 Symmetric products
    3. 2.3 Differential operators
    4. 2.4 Flat space
    5. 2.5 Curved space
  3. 3 Finsler geometry
    1. 3.1 Distance
    2. 3.2 Connection
    3. 3.3 Curvature
  4. 4 Linearized analysis
    1. 4.1 Deformations
    2. 4.2 Parametrization
    3. 4.3 Sum over spins
  5. 5 Nonlinearities
    1. 5.1 General expansion
    2. 5.2 Spin-six nonlinear equation
  6. 6 Challenges for Finsler dynamics
    1. 6.1 Equations of motion
    2. 6.2 Broken Fronsdal gauge transformations
    3. 6.3 Perturbations
    4. 6.4 Discussion
  7. A More details on the symmetrized derivative

##  1 Introduction

The dynamics of fields with spin higher than two has an intricate structure.
Their free equations of motion are already quite complicated when they have a
mass [1], but simplifies in the massless case, where their equation of motion
mimics that of spin two, using the so-called Fronsdal operator ${\mathcal{F}}$
[2]. Their interactions have a highly constrained dynamics; for reviews see
for example [3, 4, 5, 6, 7]. It has long been surmised that they are important
for theories of quantum gravity. String theory does have a tower of such
fields, with masses related to the string tension. On the other hand, theories
of Vasiliev type describe massless higher spins in (A)dS [8, 9, 10, 11, 12].
Their status as quantum theories is bolstered by their holographic
interpretation [13, 14]. Higher spin fields and symmetries are expected more
generally in holography, although most often in a broken version [15]. Some
versions of these theories can indeed be even obtained as limits of string
theory [16, 17].

Finsler geometry is a natural generalization of Riemannian and pseudo-
Riemannian geometry; for some recent introductions see [18, 19, 20]. In the
former case, instead of writing the length of a curve as
$\int\mathrm{d}\tau\sqrt{2g_{2}}$, where
$g_{2}\equiv\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}$, one considers a
more general $\int\mathrm{d}\tau F(x,\dot{x})$. The homogeneity rule
$F(x,\lambda\dot{x})=\lambda F(x,\dot{x})$ is needed to ensure that such a
length does not depend on how the curve is parameterized, but this still
leaves many possibilities. A homogeneous analogue of a power series leads to a
fairly general expression:111Odd $s$ can also be introduced, as we will
discuss later.

$F^{2}=2g_{2}+\frac{\phi_{4}}{g_{2}}+\frac{\phi_{6}}{g_{2}^{2}}\ldots\,,\qquad\phi_{s}\equiv\frac{1}{s!}\phi_{\mu_{1}\ldots\mu_{s}}\dot{x}^{\mu_{1}}\ldots\dot{x}^{\mu_{s}}\,.$
(1.1)

So the first term in this expansion leads to the usual notion of distance; the
others are new. One can also think of this as a line element
$\mathrm{d}s^{2}=\mathrm{d}s^{2}_{0}$
$+(2/\mathrm{d}s^{2}_{0})\phi_{\mu\nu\rho\sigma}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}\mathrm{d}x^{\rho}\mathrm{d}x^{\sigma}$
$+\ldots$, where
$\mathrm{d}s^{2}_{0}=g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}$ is of the
customary (pseudo-)Riemannian type.

Analogues of the usual notions of connection and curvature have been studied
for Finsler geometry for a long time. These notions are most natural when $F$
is considered as a function on the tangent bundle $TM$,222More precisely,
because of the homogeneity constraint on $F$, one has to work either with the
_slit_ tangent bundle $TM-$(zero section), or with the _sphere_ bundle $SM$
inside it, where each fiber is quotiented by overall rescalings. with
$\dot{x}^{\mu}$ now promoted to a coordinate $y^{\mu}$ along the fiber.

Mathematically it looks natural to build a Finsler modification of general
relativity (GR); this has indeed been explored at length in the literature
(see for example [21, 22] for reviews). But from a physical point of view, no
model of gravity can be considered to be well-motivated unless it improves on
Einstein gravity on the crucial issue of its quantum behavior. This important
issue appears to have been relatively unexplored.

In any case, the appearance of the symmetric tensors
$\phi_{\mu_{1}\ldots\mu_{s}}$ in (1.1) suggests a relation to higher-spin
theories. Given the promising status of the latter as quantum theories, such a
link would also make Finsler geometry a lot more interesting physically. The
$\int\mathrm{d}\tau F$ would then be interpreted as a natural coupling of the
$\phi_{\mu_{1}\ldots\mu_{s}}$ to a particle.

A version of (1.1) with $s=2$ and one $s>2$ was considered in this context
[23, (5.8)], and an approximate gauge transformation for the particle action
was noticed. In [24] an analogue of Finsler geometry (without the homogeneity
constraint) was used for W-gravity in low dimensions. Most relevant for us,
(1.1) was found in [25] for $s=4$ and $g_{2}=\eta_{2}$ (flat space) to be
related to the spin-4 Fronsdal equation [2].

In this paper I will explore (1.1) more systematically, in the spirit of
taking the $\phi_{s}$ to be Finsler deformations around an ordinary
(pseudo-)Riemannian geometry described by
$g_{2}=\frac{1}{2}g_{\mu\nu}y^{\mu}y^{\nu}$. The first intriguing result is
that the Fronsdal kinetic operator ${\mathcal{F}}(\phi_{s})$ appears for all
spins $s>2$ and around any $g_{2}$. Among the curvature tensors, one that we
will call $\rho$ is the analogue of the ordinary Ricci tensor. We will see
that

$\rho\sim
R^{0}_{\alpha\beta}y^{\alpha}y^{\beta}+\sum_{s>2}g_{2}^{1-s/2}\left(-\frac{1}{2}{\mathcal{F}}(\phi_{s})+\alpha_{s}\mathrm{d}^{2}\phi_{s-2}\right)\,,$
(1.2)

to linear order in the $\phi_{s}$. The first term contains the ordinary Ricci
tensor of $g_{2}$; the rest is a series similar to (1.1). In a condensed
notation to be fully explain below, $\mathrm{d}$ represents the symmetrized
derivative $\nabla_{(\mu_{1}}\phi^{s-2}_{\mu_{2}\ldots\mu_{s})}$. The
coefficient $\alpha_{s}=\frac{1}{8}(s-4)(s+D-4)$, where $D$ is the spacetime
dimension, which we will leave unspecified throughout. A perhaps deeper
expression is

$\rho\sim
R^{0}_{\alpha\beta}y^{\alpha}y^{\beta}+\frac{1}{2}{\mathcal{F}}(\delta F^{2})$
(1.3)

with an appropriate understanding of the action of ${\mathcal{F}}$ on
arbitrary functions of $y$. This is valid for Finsler deformations $\delta
F^{2}$ around any (pseudo-)Riemannian geometry, without even using the
expansion (1.1).

A second interesting point is that the Finsler Ricci tensor is not linear in
the $\phi_{s}$; going to higher orders gives rise to expressions that are more
complicated but still manageable, especially for low $s$. It is natural to
interpret these as higher spin interactions, similar to the graviton
interactions appearing in the perturbative expansion of GR around Minkowski
space.

Unfortunately there is also bad news. What makes the Fronsdal operator
especially important is that it admits gauge transformations: under
$\delta\phi_{s}=\mathrm{d}\lambda_{s-1}$,
$\delta{\mathcal{F}}(\phi_{s})=\frac{1}{2}\mathrm{d}^{3}\lambda^{\prime}_{s-1}$,
with $\lambda^{\prime}_{s-1}$ representing the trace
$\lambda^{\mu}{}_{\mu\mu_{2}\ldots\mu_{s}}$. Thus the traceless
$\lambda_{s-1}$ give a large set of gauge transformations.333This traceless
condition is often viewed with suspicion; there is a way to get rid of it
[26]. In the context of Finsler geometry we will see that in a sense this
condition is rather natural. On the other hand, the double symmetrized
derivative $\mathrm{d}^{2}$ in (1.2) is not invariant. The two terms combine
to transform as
$\delta({\mathcal{F}}(\phi_{s})-2\alpha_{s}\mathrm{d}^{2}\phi_{s-2})=\mathrm{d}^{3}(-\frac{1}{4}\lambda^{\prime}_{s-1}+\alpha_{s}\lambda_{s-3})$.
The gauge transformation of each field is “eaten” by that of a higher spin, in
a sort of Stückelberg mechanism. Unfortunately, the surviving transformations
can be shown to be a trivial rearrangement of fields that leaves the geometry
unchanged. (An equivalent, sleeker analysis is possible from (1.3).)

This essential lack of gauge transformations (beyond the usual
diffeomorphisms) is worrisome. We will see that the equations of motion for a
Finsler analogue of GR, the simplest of which just reads $\rho=0$, have many
perturbative solutions that are not transverse to the momentum, and thus would
presumably create trouble upon quantization. These solutions also have many
free parameters; this might give hope that a gauge transformation can indeed
be defined for this system. Unfortunately I have found no such candidate, but
I have not excluded it either.

There are of course many other possible problems with a theory of higher spins
[3]. The Vasiliev theory manages to avoid them in a rather sophisticated
fashion. In contrast, the idea of a Finsler higher spin theory should be
regarded as moving here one of its first steps. Nevertheless, the present
results make it worth exploring further in the future.

In Sec. 2 some aspects of higher spin field theory are reviewed, setting up
notation. Sec. 3 is a lightning introduction to Finsler geometry. In Sec. 4 I
connect the two, showing that a linearization of the Finsler analogue $\rho$
of the Ricci tensor contains infinitely many Fronsdal operators, as promised
above. In Sec. 5 a partial analysis is given beyond the linear order in the
$\phi_{s}$. Sec. 6 considers the physical issues of gauge transformations and
of perturbative solutions, with some details in App. A.

##  2 Fronsdal equation

The Fronsdal equation [2] describes a massless free field in a totally
symmetric representation. We will give here a very basic review of some its
features. There are many deeper discussions in the literature [3, 4, 5].

###  2.1 Spin two

As a warm-up, let us recall the equation for a spin-two field. This can be
obtained by applying the vacuum equations of general relativity,
$R_{\mu\nu}=0$, to an infinitesimal perturbation $g_{\mu\nu}+h_{\mu\nu}$ of a
background metric $g_{\mu\nu}$. The connection is deformed by

$\delta\Gamma^{\mu}_{\nu\rho}=g^{\mu\sigma}\left(\nabla_{(\nu}h_{\rho)\sigma}-\frac{1}{2}\nabla_{\sigma}h_{\nu\rho}\right)$
(2.1)

and the Riemann tensor by

$\delta
R_{\nu\rho}=2\nabla_{[\mu}\delta\Gamma^{\mu}_{\nu]\rho}=-\frac{1}{2}\nabla^{2}h_{\nu\rho}+\nabla^{\mu}\nabla_{(\nu}h_{\rho)\mu}-\frac{1}{2}\nabla_{\nu}\nabla_{\rho}h^{\prime}\,,$
(2.2)

where $h^{\prime}=g^{\mu\nu}h_{\mu\nu}$. So setting the right-hand side to
zero is the equation of motion for $h_{\mu\nu}$.

Sometimes it can be useful to rewrite (2.2) in terms of the _Lichnerowicz
operator_ $\Delta$:

$\displaystyle\delta R_{\nu\rho}=-\frac{1}{2}\Delta\delta
g_{\nu\rho}+\nabla_{(\nu}\nabla^{\mu}\delta
g_{\rho)\mu}-\frac{1}{2}\nabla_{\rho}\nabla_{\nu}\delta g^{\prime}\,,$ (2.3a)
$\displaystyle\Delta\delta g_{\nu\rho}\equiv\nabla^{2}\delta
g_{\nu\rho}-2R_{(\nu}{}^{\sigma}\delta
g_{\rho)\sigma}+2R_{\mu\nu\rho\sigma}h^{\mu\sigma}\,.$ (2.3b)

The gauge transformations of general relativity are infinitesimal coordinate
changes; if we take

$\delta h_{\mu\nu}=2\nabla_{(\mu}\xi_{\nu)}\,,$ (2.4)

(2.2) gives

$\delta_{\xi}R_{\mu\nu}=\xi^{\rho}\nabla_{\rho}R_{\mu\nu}+2\nabla_{(\mu}\xi^{\rho}R_{\nu)\rho}=L_{\xi}R_{\mu\nu}$
(2.5)

after using the familiar

$[\nabla_{\mu},\nabla_{\nu}]v^{\rho}=R^{\rho}{}_{\sigma\mu\nu}v^{\sigma}$
(2.6)

and its generalization on tensors. If the original metric is a solution of the
vacuum equation, $R_{\mu\nu}=0$, and so (2.2) is invariant.

###  2.2 Symmetric products

To set the stage for a generalization to higher spins, let us consider now a
condensed notation. A version of this formalism is used one way or another in
much of the higher-spin literature; see for example [3, (2.10)]. My
definitions are adapted to Finsler geometry, so some normalizations might be a
bit unfamiliar.

We introduce a formal variable $y^{\mu}$ (which will later acquire an
interpretation as a velocity vector). To a completely symmetric tensor
$\phi_{\mu_{1}\ldots\mu_{s}}$ we associate a polynomial

$\phi_{s}\equiv\frac{1}{s!}\phi_{\mu_{1}\ldots\mu_{s}}y^{\mu_{1}}\ldots
y^{\mu_{s}}\,;$ (2.7)

this is of course similar to the notation in (1.1). It is convenient to also
define

$\begin{split}\phi^{s}_{\mu}&\equiv\partial_{y^{\mu}}\phi_{s}=\frac{1}{(s-1)!}\phi_{\mu_{2}\ldots\mu_{s}}y^{\mu_{2}}\ldots
y^{\mu_{s}}\,,\\\
\phi^{s}_{\mu\nu}&\equiv\partial_{y^{\mu}}\partial_{y^{\nu}}\phi_{s}=\frac{1}{(s-2)!}\phi_{\mu_{3}\ldots\mu_{s}}y^{\mu_{3}}\ldots
y^{\mu_{s}}\,,\end{split}$ (2.8)

and so on. The $s$ is a label and can be written up or down as convenient, and
will sometimes be omitted when clear from the context. Notice that

$y^{\mu}\phi^{s}_{\mu}=s\phi_{s}\,.$ (2.9)

The trace is defined as

$\phi^{\prime}_{s}\equiv\phi^{\mu}_{s\,\mu}=g^{\mu\nu}\partial_{y^{\mu}}\partial_{y^{\nu}}\phi_{s}\,.$
(2.10)

A product of two polynomials $\phi_{s}\psi_{s^{\prime}}$ represents in this
language the symmetric product
$\frac{(s+s^{\prime})!}{s!s^{\prime}!}\phi_{(\mu_{1}\ldots\mu_{s}}\psi_{\mu_{s+1}\ldots\mu_{s+s^{\prime}})}$.
We will often encounter products $g_{2}\phi_{s}$ or more generally
$g_{2}^{k}\phi_{s}$, with $g_{2}=\frac{1}{2}g_{\mu\nu}y^{\mu}y^{\nu}$ and
$g_{\mu\nu}$ a metric. It is useful to compute the trace of such objects:

$(g_{2}\phi_{s})^{\prime}=g^{\mu\nu}\partial_{y^{\mu}}(y_{\nu}\phi_{s}+g_{2}\phi^{s}_{\nu})=D\phi_{s}+2y^{\mu}\phi_{\mu}^{s}+g_{2}\phi_{s}^{\prime}=(D+2s)\phi_{s}+g_{2}\phi^{\prime}_{s}\,.$
(2.11)

One can rewrite this more abstractly by introducing the operators

$t\equiv
g^{\mu\nu}\partial_{y^{\mu}}\partial_{y^{\nu}}=\partial_{y^{\mu}}\partial_{y_{\mu}}\,,\qquad\mathrm{deg}\equiv
y^{\mu}\partial_{y^{\mu}}\,,$ (2.12)

so that $t\phi_{s}=\phi_{s}^{\prime}$ and $\mathrm{deg}\phi_{s}=s\phi_{s}$,
respectively. With the usual commutator rules, it is easy to work out

$[t,g_{2}\cdot]=\partial_{y^{\mu}}y^{\mu}+y^{\mu}\partial_{y^{\mu}}=D+2\deg\,,$
(2.13)

where $g_{2}\cdot$ is multiplication by $g_{2}$. This reproduces (2.11).

Any $\phi_{s}$ can be written in terms of traceless tensors:

$\phi_{s}=\phi_{s,0}+g_{2}\phi_{s,1}+\ldots
g_{2}^{s/2}\phi_{s,s}\,,\qquad\phi_{s,k}^{\prime}=0\,.$ (2.14)

The degree $(s-2k)$ fields $\phi_{s,k}$ can also be written in terms of the
iterated traces $\phi^{(j)}$ (in a notation where $\phi^{(1)}=\phi^{\prime}$,
$\phi^{(2)}=\phi^{\prime\prime}$ and so on):

$\phi_{s,0}=\phi_{s}+t_{s1}g_{2}\phi_{s}^{\prime}+t_{s2}\frac{g_{2}^{2}}{2}\phi_{s}^{\prime\prime}+\ldots\,,\qquad\phi_{s,j}=(-)^{j}t_{s-j+1,j}(\phi^{(j)})_{0}\,,$
(2.15)

where

$t_{s1}\equiv-\frac{1}{D+2s-4}\,,\qquad
t_{s,j}\equiv-\frac{t_{s,j-1}}{D+2s-2j-2}\,.$ (2.16)

###  2.3 Differential operators

The placeholder variable $y$ is inert under derivatives. So
$\nabla_{\mu}\phi_{s}=\frac{1}{s!}\nabla_{\mu}\phi_{\mu_{1}\ldots\mu_{s}}y^{\mu_{1}}\ldots
y^{\mu_{s}}$. We will often encounter the operator

$\mathrm{d}\equiv y^{\mu}\nabla_{\mu}\,.$ (2.17)

This name is inspired by
$\mathrm{d}=\mathrm{d}x^{\mu}\wedge\partial_{\mu}=\mathrm{d}x^{\mu}\wedge\nabla_{\mu}$
in exterior algebra, which represents an antisymmetrized derivative, similar
to how (2.17) represents a _symmetrized_ derivative:

$\mathrm{d}\phi_{s}=\frac{1}{s!}\nabla_{(\mu}\phi_{\mu_{1}\ldots\mu_{s})}y^{\mu}y^{\mu_{1}}\ldots
y^{\mu_{s}}\,.$ (2.18)

(Writing the symmetrizer on the indices is here of course optional, as the
product of $y$’s enforces it anyway.) A crucial difference is that the
$\mathrm{d}$ in exterior algebra squares to zero, while (2.17) does not.

In this language, the equation of motion $R_{\mu\nu}=0$ from (2.2), namely
$\frac{1}{2}\nabla^{2}h_{\nu\rho}-\nabla^{\mu}\nabla_{(\nu}h_{\rho)\mu}+\frac{1}{2}\nabla_{\nu}\nabla_{\rho}h^{\prime}=0$,
can be rewritten by multiplying by $y^{\nu}y^{\rho}$ as

$\nabla^{2}h_{2}-\nabla_{\mu}\mathrm{d}h_{2}^{\mu}+\frac{1}{2}\mathrm{d}^{2}h^{\prime}=0\,.$
(2.19)

The gauge transformation (2.4) reads

$\delta h_{2}=\mathrm{d}\xi_{1}\,.$ (2.20)

We seem not to have gained much by this rewriting. However, the nice
properties of (2.19) now suggest to replace $h_{2}$ in (2.19) by a polynomial
$\phi$ in $y$, obtaining the equation of motion

${\mathcal{F}}(\phi)\equiv\nabla^{2}\phi-\nabla_{\mu}\mathrm{d}\phi^{\mu}+\frac{1}{2}\mathrm{d}^{2}\phi^{\prime}=0\,$
(2.21)

irrespectively of the degree of $\phi$. This is the _Fronsdal equation_ for
massless totally symmetric fields. It is much simpler than its explicit
expression: for $\phi=\phi_{s}$ as in (2.7),

$\nabla^{2}\phi_{\mu_{1}\ldots\mu_{s}}-s\nabla_{\mu}\nabla_{(\mu_{1}}\phi_{\mu_{2}\ldots\mu_{s})}{}^{\mu}+\frac{s(s-1)}{2}\nabla_{(\mu_{1}}\nabla_{\mu_{2}}\phi_{\mu_{3}\ldots\mu_{s})\mu}{}^{\mu}=0\,.$
(2.22)

We can also rewrite

${\mathcal{F}}=\nabla^{2}-\nabla^{\mu}\mathrm{d}\partial_{y^{\mu}}+\frac{1}{2}\mathrm{d}^{2}t=\Delta-\mathrm{d}\mathrm{d}^{\dagger}+\frac{1}{2}\mathrm{d}^{2}t\,,$
(2.23)

with
$\Delta=\nabla^{2}-R_{\mu\nu}y^{\mu}\partial_{y_{\nu}}+R_{\mu\rho\nu\sigma}y^{\mu}y^{\nu}\partial_{y_{\rho}}\partial_{y_{\sigma}}$
as in (2.3b), recalling (2.12), and with the definition

$\mathrm{d}^{\dagger}\equiv
g^{\mu\nu}\partial_{y^{\mu}}\nabla_{\nu}=\partial_{y_{\mu}}\nabla_{\mu}\,.$
(2.24)

We note for later use:

$[t,\mathrm{d}]=2\mathrm{d}^{\dagger}\,,\qquad[t,\mathrm{d}^{\dagger}]\,,\qquad[\mathrm{d}^{\dagger},g_{2}\cdot]=\mathrm{d}\,,\qquad[\mathrm{d},g_{2}\cdot]=0\,,[\mathrm{d},\mathrm{d}^{\dagger}]=-2\nabla^{2}+\Delta\,.$
(2.25)

###  2.4 Flat space

We will now see that (2.21) inherits some of the nice properties of (2.19). In
this subsection we work in flat space, $g_{\mu\nu}=\eta_{\mu\nu}$.

In flat space, a $\phi_{s}$ give a representation of the Lorentz group; it is
irreducible if $\phi^{\prime}_{s}=0$. Otherwise, (2.14) can be viewed as a
decomposition into $s$ irreducible representations.

Generalizing the gauge transformation (2.20), we take that of a field
$\phi=\phi_{s}$ to be

$\delta\phi_{s}=\mathrm{d}\lambda_{s-1}\,.$ (2.26)

In components this reads
$\delta\phi_{\mu_{1}\ldots\mu_{s}}=s\partial_{(\mu_{1}}\lambda_{\mu_{2}\ldots\mu_{s})}$.
We find

$\delta\phi^{\mu}=\partial_{y_{\mu}}(y^{\alpha}\partial_{\alpha}\lambda)=\partial^{\mu}\lambda+\mathrm{d}\lambda^{\mu}\,,\qquad\delta\phi^{\mu\nu}=2\partial^{(\mu}\lambda^{\nu)}+y^{\alpha}\partial_{\alpha}\lambda^{\mu\nu}\,.$
(2.27)

(As in (2.21), when the equations don’t depend on the degree/spin we sometimes
omit it, to get more readable expressions.) So for the Fronsdal equation
(2.21):

$\begin{split}\delta{\mathcal{F}}(\phi)&={\mathcal{F}}(\mathrm{d}\lambda)=\partial^{2}\mathrm{d}\lambda-\mathrm{d}\partial_{\mu}(\partial^{\mu}\lambda+\mathrm{d}\lambda^{\mu})+\frac{1}{2}\mathrm{d}^{2}(2\partial_{\mu}\lambda^{\mu}+\mathrm{d}\lambda^{\prime})\\\
&=\frac{1}{2}\mathrm{d}^{3}\lambda^{\prime}\,.\end{split}$ (2.28)

In particular, ${\mathcal{F}}(\phi_{s})$ is invariant under $\lambda_{s-1}$
that are traceless:

$\lambda^{\prime}=\lambda_{\alpha}^{\alpha}=0\,.$ (2.29)

Recall that more explicitly this means
$\lambda^{\alpha}{}_{\alpha\mu_{3}\ldots\mu_{s-1}}=0$.

Trying to write an action for the Fronsdal equation leads to a second
constraint. Up to integration by parts,

$S=\int\mathrm{d}^{D}x\,\phi\,{\mathcal{G}}\,,\qquad{\mathcal{G}}\equiv{\mathcal{F}}(\phi)-\frac{1}{2}\eta_{2}{\mathcal{F}}^{\prime}(\phi)\,.$
(2.30)

This action is in general not gauge invariant: $\delta
S=\int\mathrm{d}^{D}x\,\lambda\,\partial_{\mu}{\mathcal{G}}^{\mu}\neq 0$.
Fortunately, a few more cancellations yield

$\partial_{\mu}{\mathcal{G}}^{\mu}=-\frac{1}{4}\mathrm{d}^{3}\phi^{\prime\prime}\,.$
(2.31)

We see that the action is invariant if we also impose that the double trace of
the field vanishes:

$\phi^{\prime\prime}=\phi^{\alpha\beta}{}_{\alpha\beta}=0\,.$ (2.32)

(Recall that more explicitly this means
$\phi^{\alpha\beta}{}_{\alpha\beta\mu_{5}\ldots\mu_{s}}=0$.) In this case
$\phi$ consists of two irreducible representations, its trace $\phi^{\prime}$
and its traceless part.

With this constraint, the variation $\delta
S=\int\mathrm{d}^{D}x\delta\phi{\mathcal{G}}=0$ now only imposes that the
double-traceless part of ${\mathcal{G}}$ vanishes. Fortunately, it is simple
to see (just by counting the number of available indices) that (2.32) in fact
directly implies
${\mathcal{F}}^{\prime\prime}={\mathcal{G}}^{\prime\prime}=0$. So the action
does imply ${\mathcal{G}}=0$. This in turn sets to zero
${\mathcal{G}}^{\prime}$, which is proportional to ${\mathcal{F}}^{\prime}$.
So the equation of motion is indeed ${\mathcal{F}}=0$.

For spin two, the first constraint (2.29) was automatic because in that case
$\lambda_{1}$ was in fact $\xi_{\mu}$, which only has one index. (2.31)
becomes the linearized flat-space version of the spin-two identity
$\nabla^{\mu}(R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu})=0$, a consequence of the
Bianchi identity. The double-trace constraint (2.32) is also automatic, since
the field only has two indices.

We conclude with a quick review of gauge fixing. First we notice

$\delta\left(\partial_{\mu}\phi^{\mu}-\frac{1}{2}\mathrm{d}\phi^{\prime}\right)=\partial^{2}\lambda\,,\qquad\left(\partial_{\mu}\phi^{\mu}-\frac{1}{2}\mathrm{d}\phi^{\prime}\right)^{\prime}=-\frac{1}{2}\mathrm{d}\phi^{\prime\prime}=0\,.$
(2.33)

So we can always find a $\lambda$ such that
$\partial_{\mu}\phi^{\mu}-\frac{1}{2}\mathrm{d}\phi^{\prime}=0$, by inverting
the operator $\partial^{2}$. This simplifies the Fronsdal equation to
$\partial^{2}\phi=0$. There is still a residual gauge invariance, consisting
of $\lambda$ such that $\partial^{2}\lambda=0$ (and which are traceless).

We now go to momentum space, where we have $p^{2}\phi(p)=0$,
$p_{\mu}\phi^{\mu}=\frac{1}{2}p_{1}\phi^{\prime}$, with $p_{1}=y^{\mu}p_{\mu}$
as by now familiar. We pick a vector $u^{\mu}$ such that $u^{\mu}p_{\mu}=1$.
Using (2.27),

$\delta(u^{\mu}\phi_{\mu})=\lambda+u_{\mu}y^{\alpha}p_{\alpha}\lambda^{\mu}=M\lambda\,,\qquad
M\equiv 1+p_{1}u^{\mu}\partial_{y^{\mu}}\,.$ (2.34)

It can be shown that $M$ is invertible, with inverse
$M^{-1}=\sum_{l}(-)^{l}(p_{1}^{l}/l!)\partial_{y^{\mu_{1}}}\ldots\partial_{y^{\mu_{l}}}$.
Moreover
$M^{-1}(-u^{\mu}\phi_{\mu})=\sum_{l}(-)^{l}(p_{1}^{l}/l!)u_{\mu_{1}}\ldots
u_{\mu_{l}}\phi^{\mu_{1}\ldots\mu_{l}}$ can be shown to be traceless. So by
picking this $\lambda$ we can set $u^{\mu}\phi_{\mu}$ to zero. Since $M$ is
invertible, it has no kernel, and so there is no residual gauge
transformation. Finally, $p_{\mu}\phi^{\mu}=\frac{1}{2}p_{1}\phi^{\prime}$
implies $2p^{\nu}\phi_{\mu\nu}=p_{\mu}\phi^{\prime}+p_{1}\phi^{\prime}_{\mu}$;
contracting this with $u_{\mu}$ we find now that $\phi^{\prime}=0$, and so
also that $p_{\mu}\phi^{\mu}=0$. This is the analogue of the transverse
traceless gauge for spin-two fields.

###  2.5 Curved space

In curved space, the gauge transformation of $\phi_{s}$ is still (2.26), but
it now represents
$\delta\phi_{\mu_{1}\ldots\mu_{s}}=s\nabla_{(\mu_{1}}\lambda_{\mu_{2}\ldots\mu_{s})}$,
with covariant derivatives. The transformation of (2.21) is more complicated:
some terms that canceled in (2.28) no longer do so, because of the non-
commutativity of covariant derivatives recalled in (2.6).

Besides the explicit indices, one also needs to take care of the tensor
indices inside $\phi_{s}$, made explicit in (2.7). In this respect, the
following identity is useful:

$[\nabla_{\mu},\nabla_{\nu}]\phi=-R^{\rho}{}_{\sigma\mu\nu}y^{\sigma}\partial_{y^{\rho}}\phi\,.$
(2.35)

Using also the second Bianchi identity we obtain

$\delta{\mathcal{F}}(\phi)=\frac{1}{2}\mathrm{d}^{3}\lambda^{\prime}-y^{\alpha}y^{\beta}(\lambda^{\nu}\nabla_{\nu}R_{\alpha\beta}+2\nabla_{(\alpha}\lambda^{\nu}R_{\beta)\nu})+\mathrm{d}\rho_{\mu\nu}\lambda^{\mu\nu}+2\rho_{\mu\nu}\mathrm{d}\lambda^{\mu\nu}\,.$
(2.36)

We have introduced

$\rho_{\mu\nu}\equiv R_{\mu\alpha\nu\beta}y^{\alpha}y^{\beta}\,$ (2.37)

which will be useful later on. The parenthesis with the Ricci terms in (2.36)
is reminiscent of the Lie derivative
$L_{\xi}R_{\alpha\beta}=\xi^{\mu}\nabla_{\mu}R_{\alpha\beta}+2\nabla_{(\alpha}\xi^{\nu}R_{\beta)\nu}$,
except that
$\lambda^{\mu}=\lambda^{\mu}_{s-1}=\frac{1}{(s-1)!}\lambda^{\mu}{}_{\mu_{2}\ldots\mu_{s-1}}$
has $(s-2)$ hidden indices. For $s=2$ this expression does reduce to the Lie
derivative in (2.5).

The appearance of the Riemann tensor in (2.36) is famously problematic [27].
We can still take $\lambda^{\prime}=0$ to make the first term disappear. The
Ricci tensor appears in the ordinary gravity equations, and in particular it
simply vanishes for vacuum GR, so those terms can also be reasonably set to
zero. However, setting the Riemann tensor to any particular value appears
unjustified. So the gauge invariance of the Fronsdal equation is broken in
general.

If one simply postulates that the background geometry is maximally symmetric,
as in AdSD or dSD, then

$R_{\mu\nu\rho\sigma}=\frac{\Lambda}{D-1}(g_{\mu\rho}g_{\nu\sigma}-g_{\mu\sigma}g_{\nu\rho})\,.$
(2.38)

The constant is chosen so that $R_{\mu\nu}=\Lambda g_{\mu\nu}$. (2.36) now
becomes the simpler

$\delta{\mathcal{F}}(\phi_{s})=\frac{1}{2}\mathrm{d}^{3}\lambda^{\prime}_{s-1}+\frac{4\Lambda}{D-1}g_{2}\mathrm{d}\lambda^{\prime}_{s-1}-\Lambda\mu_{s}^{2}\mathrm{d}\lambda_{s-1}\,,\qquad\mu_{s}^{2}\equiv\frac{2(s-1)(D+s-3)}{D-1}\,.$
(2.39)

(We reinstated the spin labels because now the equation does explicitly depend
on $s$.) Recall that $g_{2}=\frac{1}{2}g_{\mu\nu}y^{\mu}y^{\nu}$, per our
notation (2.7). This is still non-zero even if $\lambda^{\prime}=0$. But the
modified equation

${\mathcal{F}}(\phi_{s})+\Lambda\mu_{s}^{2}\phi_{s}=0$ (2.40)

does have a gauge invariance.

##  3 Finsler geometry

We will give here a relatively quick introduction to Finsler geometry. Some
relatively recent accounts are [18, 19, 20], in increasing order of detail.

###  3.1 Distance

In Riemannian geometry, the length $\gamma$ of a curve is given by the
integral
$\int_{\gamma}\mathrm{d}\tau\sqrt{g_{\mu\nu}(x)\dot{x}^{\mu}\dot{x}^{\nu}}$,
with $\tau$ a coordinate on $\gamma$ and
$\dot{x}^{\mu}\equiv\partial_{\tau}x^{\mu}$. The idea of Finsler geometry is
to generalize this to

$\int\mathrm{d}\tau F(x,\dot{x})\,$ (3.1)

with $F(x,\dot{x})$ any function of $x^{\mu}$, $\dot{x}^{\mu}$ that is
homogeneous of degree one in velocity:

$F(x,\lambda\dot{x})=\lambda F(x,\dot{x})\,.$ (3.2)

This requirement is needed so that the length (3.1) of $\gamma$ is invariant
under $\tau\to\tau^{\prime}(\tau)$ and does not depend on how we parameterize
it. Actually it is customary to call

$\dot{x}^{\mu}\equiv y^{\mu}$ (3.3)

the velocity variable, and we will do so. This can be viewed as a coordinate
along the fiber of the tangent bundle $TM$. In this sense, $F$ can be viewed
as a function on $TM$, a fact that will be useful later.

Clearly there are many new options opened by the generalization (3.1), (3.2).
Two simple possibilities that have been considered in the mathematical
literature are the _Randers_ choice $F=\phi_{1}+\sqrt{2g_{2}}$ and the _$s$
-th root_ choice $F=(\phi_{s})^{1/s}$, with
$\phi_{s}=\frac{1}{s!}\phi_{\mu_{1}\ldots\mu_{s}}y^{\mu_{1}}\ldots
y^{\mu_{s}}$ as in (2.7) and (1.1). That equation also shows infinitely many
new possibilities, which will be explored in Sec.4 below.

Most of the mathematical literature has been devoted to the case where $F$ is
positive, thus defining a notion of distance that generalizes Riemannian
geometry.444Not all metric spaces are of this form, as pointed out in [28].
There exist even more general notions of distance that cannot be obtained as
integrals of a function $F(x,\dot{x})$. In that case, $F$ is also taken to be
smooth, and the _fundamental tensor_

$g_{\mu\nu}=\frac{1}{2}\partial_{y^{\mu}}\partial_{y^{\nu}}F^{2}$ (3.4)

is required to be positive-definite for all $x$ and $y$. When there is danger
of confusion between (3.4) and a Riemannian metric, I will call the latter
$g^{0}_{\mu\nu}$. Notice that (3.4) depends on $y$, except if
$F^{2}=2g^{0}_{2}=g^{0}_{\mu\nu}(x)y^{\mu}y^{\nu}$, where
$g_{\mu\nu}=g^{0}_{\mu\nu}(x)$ is an ordinary Riemannian metric. (3.4) is used
to lower indices in Finsler geometry; given that it is positive-definite, its
inverse $g^{\mu\nu}$ is used to raise them.

For physics, we are more interested in generalizing pseudo-Riemannian
spacetimes; for this, we require instead (3.4) to have signature $(-+++)$. Now
in spite of its name $F^{2}$ is no longer assumed to be positive, similar to
how the symbol $\mathrm{d}s^{2}$ can have either sign in ordinary pseudo-
Riemannian geometry. We take $\int\mathrm{d}\tau\sqrt{F^{2}}$ to measure
distances, and $\int\mathrm{d}\tau\sqrt{-F^{2}}$ to measure proper time.

Even with the understanding that $F^{2}$ can have either sign, some choices
are not defined for all $y$. An example is the pseudo-Randers
$F^{2}=\phi_{1}+\sqrt{-2g_{2}}$, with $g_{\mu\nu}$ pseudo-Riemannian. Another
example is encountered when generalizing (1.1) to include odd $s$, with terms
$\phi_{3}/\sqrt{-g_{2}}+\phi_{5}/(-g_{2})^{3/2}+\ldots$. Physically this
appears to be fine as long as, for every $x$, $F^{2}$ is real above a cone in
the tangent space $T_{x}M$. Definitions in the mathematical literature vary as
to whether to impose $F^{2}$ to be defined for all $y$ or not [29]. The rest
of our discussion in this Section will be basically valid for either
signature.

Before we proceed further, we note that the homogeneity condition (3.2) has
several interesting consequences. Recall the _Euler theorem_ for a homogeneous
function of degree $k$:

$\phi(\lambda y)=\lambda^{k}\phi(y)\quad\Rightarrow\quad
y^{\mu}\partial_{y^{\mu}}\phi=k\phi\,.$ (3.5)

A monomial such as (2.7) is obviously homogeneous of degree $s$, as we saw in
(2.9). If $\phi=\phi(x,y)$ is analytic around $y=0$, it can be Taylor expanded
in such a basis:

$\phi=\phi_{0}+\phi_{\mu}y^{\mu}+\frac{1}{2}\phi_{\mu\nu}y^{\mu}y^{\nu}+\ldots\,.$
(3.6)

In this sense, such a function can be seen as a collection of completely
symmetric tensors $\phi_{\mu_{1}\ldots\mu_{s}}$.

$F^{2}$ is homogeneous of degree two; moreover, each $\partial_{y^{\mu}}$
lowers the homogeneity degree by one, so for example (3.4) has degree zero.
This gives the following useful identities:

$y^{\mu}\partial_{y^{\mu}}F^{2}=2F^{2}\,,\qquad
y^{\mu}g_{\mu\nu}=\frac{1}{2}\partial_{y^{\nu}}F^{2}\,,\qquad
y^{\mu}y^{\nu}g_{\mu\nu}=F^{2}\,,\qquad
y^{\mu}\partial_{y^{\mu}}g_{\nu\rho}=0\,.$ (3.7)

###  3.2 Connection

Just like in ordinary (pseudo-)Riemannian geometry, a connection is needed in
order to write derivatives that transform well under coordinate changes. A
first piece of information is obtained by varying (3.1): this gives the
familiar-looking geodesic equation
$\ddot{x}^{\mu}+\gamma^{\mu}_{\nu\rho}\dot{x}^{\nu}\dot{x}^{\rho}=0$ with

$\gamma^{\mu}_{\nu\rho}=\frac{1}{2}g^{\mu\sigma}(\partial_{\nu}g_{\rho\sigma}+\partial_{\rho}g_{\nu\sigma}-\partial_{\sigma}g_{\nu\rho})\,.$
(3.8)

Recall however that $g$ now also depends on $y^{\mu}=\dot{x}^{\mu}$. The
combination appearing in the geodesic equation appears often enough that we
give it a name, _spray coefficients_ :

$G_{\mu}\equiv\gamma_{\mu\nu\rho}y^{\nu}y^{\rho}=\frac{1}{2}(y^{\nu}\partial_{\nu}\partial_{y^{\mu}}-\partial_{\mu})F^{2}\,.$
(3.9)

The second equality uses (3.7).

We noticed earlier that $F$ is a function on $TM$. It proves useful to also
think of the derivatives $\partial_{\mu}$ and $\partial_{y^{\mu}}$ as vector
fields on that bundle. Under a coordinate change $x^{\mu}\to
x^{\prime}{}^{\mu}=x^{\prime}{}^{\mu}(x)$, the velocity variables should
transform as $y^{\mu}\to y^{\prime}{}^{\mu}=\frac{\partial
x^{\prime}{}^{\mu}}{\partial x^{\nu}}y^{\nu}$. If we now view both formulas
together as a coordinate change on $TM$, it follows that

$\displaystyle\partial_{\mu}$
$\displaystyle\to\partial^{\prime}_{\mu}=\frac{\partial
x^{\prime}{}^{\nu}}{\partial x^{\mu}}\frac{\partial}{\partial
x^{\prime}{}^{\nu}}+\frac{\partial y^{\prime}{}^{\nu}}{\partial
x^{\mu}}\frac{\partial}{\partial y^{\prime}{}^{\nu}}=\frac{\partial
x^{\prime}{}^{\nu}}{\partial
x^{\mu}}\partial^{\prime}_{\nu}+y^{\rho}\frac{\partial^{2}x^{\prime}{}^{\nu}}{\partial
x^{\mu}\partial x^{\rho}}\partial_{y^{\prime}{}^{\nu}}\,,$ (3.10a)
$\displaystyle\partial_{y^{\mu}}$
$\displaystyle\to\partial_{y^{\prime}{}^{\mu}}=\frac{\partial
y^{\prime}{}^{\nu}}{\partial y^{\mu}}\frac{\partial}{\partial
y^{\prime}{}^{\nu}}=\frac{\partial x^{\prime}{}^{\nu}}{\partial
x^{\mu}}\partial_{y^{\prime}{}^{\nu}}\,.$ (3.10b)

The $\partial_{\mu}$ transform in a complicated fashion, but it turns out that
the combination

$\delta_{\mu}\equiv\partial_{\mu}-N^{\nu}{}_{\mu}\partial_{y^{\nu}}\,,\qquad
N^{\nu}{}_{\mu}\equiv\frac{1}{2}\partial_{y^{\mu}}G^{\nu}$ (3.11)

transforms without the second term in (3.10a). A dual issue appears when
considering forms: on $TM$ the $\mathrm{d}y^{\mu}$ transform in a complicated
fashion, and this is solved by introducing

$\delta y^{\mu}\equiv\mathrm{d}y^{\mu}+N^{\mu}{}_{\nu}\mathrm{d}x^{\nu}\,.$
(3.12)

To see the importance of (3.11), consider the case of (pseudo-)Riemannian
geometry, $F^{2}=2g^{0}_{2}=g^{0}_{\mu\nu}y^{\mu}y^{\nu}$. The fundamental
tensor is $g_{\mu\nu}=g^{0}_{\mu\nu}$, and
$N^{\mu}{}_{\nu}=\gamma^{\mu}_{\nu\rho}y^{\rho}$.555This is not true in
general, because $\gamma^{\mu}_{\nu\rho}$ depends on $y$. By using (3.7) one
can show that $\partial_{y^{\nu}}G_{\mu}=2\gamma_{\mu\nu\rho}y^{\rho}$, but
$N^{\nu}{}_{\mu}$ contains the derivative of $G^{\mu}$ with an _upper_ index.
On each of the monomials
$T_{s}=\frac{1}{s!}T_{\mu_{1}\ldots\mu_{s}}y^{\mu_{1}}\ldots y^{\mu_{s}}$
appearing in (1.1), we obtain
$\delta_{\mu}T_{s}=\frac{1}{s!}\nabla_{\mu}T_{\mu_{1}\ldots\mu_{s}}y^{\mu_{1}}\ldots
y^{\mu_{s}}$, where $\nabla$ is the covariant derivative relative to the
(pseudo-)Riemannian metric $g^{0}_{\mu\nu}$. So for ordinary
(pseudo-)Riemannian geometry $\delta_{\mu}$ coincides with the usual covariant
derivative $\nabla_{\mu}$, even when acting on objects that depend on $y$.

Returning to the general discussion, the importance of (3.11) suggests
introducing the _Chern connection_

$\Gamma^{\mu}_{\nu\rho}=\frac{1}{2}g^{\mu\sigma}(\delta_{\nu}g_{\rho\sigma}+\delta_{\rho}g_{\nu\sigma}-\delta_{\sigma}g_{\nu\rho})\,.$
(3.13)

This has zero torsion, and satisfies

$D_{\mu}g_{\nu\rho}\equiv\delta_{\mu}g_{\nu\rho}-\Gamma^{\sigma}_{\mu\nu}g_{\sigma\rho}-\Gamma^{\sigma}_{\mu\rho}g_{\nu\sigma}=0\,.$
(3.14)

We define the operator $D_{\mu}$ on any tensor, with $\Gamma$ acting on the
indices in the way that is familiar in (pseudo-)Riemannian geometry, and the
partial derivative replaced by $\delta$:666In the literature this derivative
is usually denoted by a “slash” at the end:
$T^{\mu_{1}\ldots\mu_{k}}{}_{\nu_{1}\ldots\nu_{l}|\mu}$.

$\begin{split}D_{\mu}T^{\mu_{1}\ldots\mu_{k}}_{\nu_{1}\ldots\nu_{l}}=\delta_{\mu}T^{\mu_{1}\ldots\mu_{k}}_{\nu_{1}\ldots\nu_{l}}&+\Gamma^{\mu_{1}}_{\mu\rho}T^{\rho\mu_{2}\ldots\mu_{k}}_{\nu_{1}\ldots\nu_{l}}+\Gamma^{\mu_{2}}_{\mu\rho}T^{\mu_{1}\rho\mu_{3}\ldots\mu_{k}}_{\nu_{1}\ldots\nu_{l}}+\ldots+\Gamma^{\mu_{k}}_{\mu\rho}T^{\mu_{1}\ldots\mu_{k-1}\rho}_{\nu_{1}\ldots\nu_{l}}\\\
&-\Gamma^{\rho}_{\mu\nu_{1}}T^{\mu_{1}\ldots\mu_{k}}_{\rho\nu_{2}\ldots\nu_{l}}-\Gamma^{\rho}_{\mu\nu_{2}}T^{\mu_{1}\ldots\mu_{k}}_{\nu_{1}\rho\nu_{3}\ldots\nu_{l}}-\ldots-\Gamma^{\rho}_{\mu\nu_{l}}T^{\mu_{1}\ldots\mu_{k}}_{\nu_{1}\ldots\nu_{l-1}\rho}\,.\end{split}$
(3.15)

Once again, this reduces to $\nabla_{\mu}$ in (pseudo-)Riemannian geometry,
even if $T$ depends on $y$. In this sense, (3.14) is more natural than the
analogous equality containing $\gamma^{\mu}_{\nu\rho}$ and ordinary partial
derivatives.

At a more conceptual level, on $TM$ there is no connection that preserves the
metric and has zero torsion; this is unlike in (pseudo-)Riemannian geometry,
where (3.8) is the Levi-Civita connection and does have both properties. (3.8)
and (3.13) both have zero torsion, but both fail to respect the metric in the
$y$ directions of $TM$. If on $TM$ we write the exterior differential
$\mathrm{d}_{TM}\equiv\mathrm{d}x^{\mu}\partial_{\mu}+\mathrm{d}y^{\mu}\partial_{y^{\mu}}=\mathrm{d}x^{\mu}\delta_{\mu}+\delta
y^{\mu}\partial_{y^{\mu}}$, (3.14) implies

$\mathrm{d}_{TM}g_{\mu\nu}=\delta
y^{\rho}\partial_{y^{\rho}}g_{\mu\nu}=\frac{1}{2}\delta
y^{\rho}\partial_{y^{\mu}}\partial_{y^{\nu}}\partial_{y^{\rho}}F^{2}\,.$
(3.16)

This property is called _almost compatibility_ with the metric of the Chern
connection (3.13). Other connections exist with nice properties, the most
notable being the Cartan connection, which also has components along the $y$
coordinates; it is metric-compatible but has torsion [30, Thm. 1.4.2].

The coefficients we introduced previously are related to the Chern connection:

$\Gamma^{\mu}_{\nu\rho}y^{\rho}=N^{\mu}{}_{\nu}\,,\qquad
N^{\mu}{}_{\nu}y^{\nu}=G^{\mu}\,.$ (3.17)

The second is a simple consequence of $G^{\mu}$ being homogeneous in $y$ with
degree two; the first is a little more involved.

It is also useful to notice that

$[\partial_{y^{\mu}},\delta_{\nu}]=-\partial_{y^{\mu}}N^{\rho}{}_{\nu}\partial_{y^{\rho}}=-(L^{\rho}_{\mu\nu}+\Gamma^{\rho}_{\mu\nu})\partial_{y^{\rho}}\,,\qquad
L^{\rho}_{\mu\nu}\equiv
y^{\sigma}\partial_{y^{\mu}}\Gamma^{\rho}_{\nu\sigma}\,.$ (3.18)

We introduced the _Landsberg tensor_ $L^{\rho}_{\mu\nu}$. As an application,

$\partial_{y^{\mu}}\delta_{\nu}T=D_{\nu}\partial_{y^{\mu}}T-L^{\rho}_{\mu\nu}\partial_{y^{\rho}}T\,$
(3.19)

on a function $T=T(x,y)$. One can show that

$y^{\nu}L^{\rho}_{\mu\nu}=0\,,$ (3.20)

so in particular (3.19) implies

$y^{\nu}\partial_{y^{\mu}}\delta_{\nu}T=y^{\nu}D_{\nu}\partial_{y^{\mu}}T=\mathrm{d}\partial_{y^{\mu}}T\,.$
(3.21)

In the last step we have defined

$\mathrm{d}\equiv y^{\mu}D_{\mu}\,$ (3.22)

by analogy with the $\mathrm{d}$ in (2.17), to which it reduces in the
(pseudo-)Riemannian case.

The contraction

$L^{\mu}=g^{\nu\rho}L^{\mu}_{\nu\rho}\,$ (3.23)

is called _mean_ Landsberg tensor. On the other hand,

$L^{\mu}_{\mu\nu}=y^{\rho}\partial_{y^{\nu}}\Gamma^{\mu}_{\mu\rho}=\frac{1}{2}y^{\rho}\partial_{y^{\nu}}\delta_{\rho}\log\det
g=\mathrm{d}I_{\nu}\,,$ (3.24)

where

$I_{\mu}=\frac{1}{2}\partial_{y^{\mu}}\log\det g\,,$ (3.25)

is called the _mean Cartan_ tensor.

###  3.3 Curvature

In view of (3.13), we now define

$\frac{1}{2}R^{\mu}{}_{\nu\rho\sigma}\equiv\delta_{[\rho}\Gamma^{\mu}_{\sigma]\nu}+\Gamma^{\mu}_{[\rho|\lambda}\Gamma^{\lambda}_{|\sigma]\nu}\,.$
(3.26)

It has many of the usual properties of the ordinary Riemann tensor:
$R^{\mu}{}_{\nu(\rho\sigma)}=0$, $R^{\mu}{}_{[\nu\rho\sigma]}=0$. However, it
is not antisymmetric in the first two indices: rather, it is possible to show

$R_{(\mu\nu)\rho\sigma}=-\frac{1}{2}\partial_{y^{\alpha}}g_{\mu\nu}R^{\alpha}{}_{\beta\rho\sigma}y^{\beta}\,.$
(3.27)

A particular contraction of (3.26) with the $y$ can be reexpressed in terms of
$N^{\mu}{}_{\nu}$ or of $G^{\mu}$:

$\displaystyle\rho^{\mu}{}_{\rho}$ $\displaystyle\equiv
R^{\mu}{}_{\alpha\rho\beta}y^{\alpha}y^{\beta}=2y^{\sigma}\delta_{[\rho}N^{\mu}{}_{\sigma]}$
(3.28a)
$\displaystyle=\partial_{\rho}G^{\mu}-\frac{1}{4}\partial_{y^{\rho}}G^{\nu}\partial_{y^{\nu}}G^{\mu}-\frac{1}{2}y^{\nu}\partial_{\nu}\partial_{y^{\rho}}G^{\mu}+\frac{1}{2}G^{\nu}\partial_{y^{\nu}}\partial_{y^{\rho}}G^{\mu}\,.$
(3.28b)

(3.27) implies that $\rho_{\mu\rho}=\rho_{\rho\mu}$, $\rho_{\mu\rho}\equiv
g_{\mu\nu}\rho^{\nu}{}_{\rho}$. We already encountered $\rho_{\mu\nu}$ in
(2.37) in the (pseudo-)Riemannian case.777In the literature on Finsler
geometry, $\rho^{\mu}{}_{\nu}$ is called $(1/F^{2})R^{\mu}{}_{\nu}$. I have
decided to change this notation in order to avoid confusion with the Ricci
tensor in (pseudo-)Riemannian geometry. Even more confusingly,
$\rho^{\mu}{}_{\mu}$ is often called “Ricci scalar”; we will not follow this
custom. The trace contains the Finsler extension of the Ricci tensor:

$\rho\equiv\rho^{\mu}{}_{\mu}=R^{0}_{\mu\nu}y^{\mu}y^{\nu}\,\qquad(\text{when
}F^{2}=g^{0}_{\mu\nu}y^{\mu}y^{\nu})\,.$ (3.29)

##  4 Linearized analysis

We would now like to obtain some formulas for infinitesimal deformations of
Finsler structures, inspired by (2.1), (2.2) in the ordinary
(pseudo-)Riemannian case. As anticipated, this will lead to the appearance of
the Fronsdal operator.

###  4.1 Deformations

We first take a small detour and show a formula for a _finite_ deformation.
Given a $F^{2}$, consider a second Finsler structure $\tilde{F}^{2}$. It
determines a new fundamental tensor, connection and curvature; we will denote
them all with a tilde. After recalling (3.11), one finds

$\tilde{G}^{\mu}=G^{\mu}+\frac{1}{2}\tilde{g}^{\mu\rho}(y^{\nu}\partial_{y^{\rho}}\delta_{\nu}-\delta_{\rho})\tilde{F}^{2}\,.$
(4.1)

It can be shown [18, Thm. 3.3.1] that the geodesics of $\tilde{F}^{2}$
coincide with those of $F^{2}$ iff
$\delta_{\nu}\tilde{F}=y^{\mu}\partial_{y^{\nu}}\delta_{\mu}\tilde{F}$; in
this case $\tilde{G}^{\mu}=G^{\mu}+Py^{\mu}$, with
$P=\tilde{F}^{-1}y^{\nu}\delta_{\nu}\tilde{F}$. The two Finsler $\tilde{F}$
and $F$ are said to be _projectively equivalent_.

Coming now to the case of an infinitesimal deformation, (4.1) gives

$\delta
G^{\mu}=\frac{1}{2}g^{\mu\rho}(y^{\nu}\partial_{y^{\rho}}\delta_{\nu}-\delta_{\rho})\delta
F^{2}=\frac{1}{2}g^{\mu\rho}(\mathrm{d}\partial_{y^{\rho}}-\delta_{\rho})\delta
F^{2}\,.$ (4.2)

In the second step we have used (3.21), (3.22). ($\delta F^{2}$ is to be
understood as $\delta(F^{2})$.)

Recall from (3.29) that $\rho=\rho^{\mu}{}_{\mu}$ is the analogue of the Ricci
tensor. So to obtain the analogue of (2.2) we need
$\delta\rho=\delta\rho^{\mu}{}_{\mu}$. We can use either (3.28a) or (3.28b) to
obtain

$\delta\rho=D_{\mu}\delta G^{\mu}+L^{\mu}_{\mu\nu}\delta
G^{\nu}-\frac{1}{2}\mathrm{d}\partial_{y^{\mu}}\delta G^{\mu}\,.$ (4.3)

Combining with (4.2):

$\begin{split}\delta\rho=&\frac{1}{2}g^{\mu\nu}\left(D_{\mu}\mathrm{d}\partial_{y^{\nu}}-D_{\mu}\delta_{\nu}-\frac{1}{2}\mathrm{d}^{2}\partial_{y^{\mu}}\partial_{y^{\nu}}\right)\delta
F^{2}\\\ &-\frac{1}{2}\mathrm{d}(L^{\rho}\partial_{y^{\rho}}\delta
F^{2})+\frac{3}{2}\mathrm{d}I_{\nu}\delta
G^{\nu}+\frac{1}{2}I_{\mu}\mathrm{d}\delta G^{\mu}\,.\end{split}$ (4.4)

(4.3) and (4.4) are similar to the first and second step in (2.2).

These results hold for a general deformation of a Finsler structure $F^{2}$.
If the latter is in fact a (pseudo-)Riemannian geometry, (4.2) becomes

$\delta G^{\mu}=\frac{1}{2}(\mathrm{d}\partial_{y_{\mu}}-\nabla^{\mu})\delta
F^{2}\,.$ (4.5)

As for (4.4), since neither $g$ nor $\Gamma^{\rho}_{\mu\nu}$ depend on $y$, we
have $L^{\rho}_{\mu\nu}=0=I_{\mu}$, so the second line in (4.4) vanishes. The
last term on the first line contains
$g^{\mu\nu}\partial_{y^{\mu}}\partial_{y^{\nu}}$; we saw back in (2.10) that
this gives the trace operator $t$ on monomials such as (2.7), and now we can
consider it as extending it on arbitrary (even non-polynomial) functions of
$y$. This point will be crucial later. With this understanding, the first line
of (4.4) is nothing but the Fronsdal operator (2.21):

$\delta\rho=-\frac{1}{2}{\mathcal{F}}(\delta F^{2})\,,$ (4.6)

as anticipated in the introduction.

In the next subsection we will motivate the parameterization (1.1), and apply
it to (4.6).

###  4.2 Parametrization

We will now justify the expansion (1.1). The idea is to start from a
(pseudo-)Riemannian geometry, and to deform it into the more general Finsler
type.

(1.1) is not quite a series expansion, since $F^{2}$ should be homogeneous. In
Euclidean signature, one way to deal with this constraint is to think of it as
a function on a sphere. For fixed $x$, $F^{2}(x,y)$ is a function on
$S^{D-1}\cong\mathbb{R}^{D}/\mathbb{R}_{+}$, where the quotient acts by
rescaling. A basis of functions on the round sphere is given by the spherical
harmonics. These can be viewed as polynomials of the type $\phi_{s}$ in (2.7)
restricted to $\\{2\delta_{2}\equiv\delta_{\mu\nu}y^{\mu}y^{\nu}=1\\}$, or
alternatively as functions on $\mathbb{R}^{D}$ made homogeneous of degree
_zero_ by dividing them by an appropriate power of $\delta_{2}$:

$\frac{1}{s!}\frac{1}{\delta_{2}^{s/2}}\phi_{\mu_{1}\ldots\mu_{s}}\,$ (4.7)

where $s$ is any non-negative integer, and the tensors
$\phi_{\mu_{1}\ldots\mu_{s}}$ are irreducible: $\phi^{\prime}_{s}=0$.

This suggests that a good parameterization for $F^{2}$ can be obtained
generalizing (4.7) by replacing $\delta_{2}$ with
$g_{2}=\frac{1}{2}g^{0}_{\mu\nu}y^{\mu}y^{\nu}$. More precisely, $F^{2}$ is
homogeneous of degree two, not zero; this can be repaired by multiplying by an
overall $g_{2}$. This leads us to

$F^{2}=2g_{2}+\frac{\phi_{3}}{\sqrt{g_{2}}}+\frac{\phi_{4}}{g_{2}}+\ldots=2g_{2}+\sum_{s>2}g_{2}^{1-s/2}\phi_{s}\,.$
(4.8)

This extends (1.1) by including odd $s$. Actually their presence creates a
possible issue in the pseudo-Lorentzian case because of the roots, as we
discussed at the end of Sec. 3.1. It becomes more sensible then to make
$F^{2}$ defined above the light cone by changing
$(g_{2})^{s/2}\to(-g_{2})^{s/2}$. Our discussion in the following is not
influenced by this issue.

As mentioned earlier, the $\phi_{s}$ should be irreducible for an expansion
that is complete and non-redundant. But no harm is done if the $\phi_{s}$ are
not traceless and (4.8) is redundant. In that case, $F^{2}$ is trivially
invariant (even at a finite level) under

$\delta_{\kappa}\phi_{s_{0}}=\kappa_{s_{0}}\,,\qquad\delta_{\kappa}\phi_{s_{0}+2}=-g_{2}\kappa_{s_{0}}\,.$
(4.9)

This redundancy will provide useful cross-checks of our results. It can be
fixed to make the $\phi_{s}$ traceless again, or fixed only partially by
enforcing a weaker constraint such as $\phi^{\prime\prime}_{s}=0$. (This was
imposed in (2.32), but only in order to obtain a gauge-invariant action for
(2.21); here we will not be using that action.) Alternatively one can proceed
by expanding each $\phi_{s}$ in (4.8) in terms of its traceless components as
in (2.14), collecting powers of $g_{2}$. This gives an expansion like (4.8)
whose degree $s$ coefficient is now the traceless

$\sum_{j=0}^{\infty}\phi_{s+2j,j}\,.$ (4.10)

Similar to (4.9), there is an invariance where we deform

$\delta g_{2}=\kappa_{2}\,,\qquad\delta\phi_{s}=(2-s/2)\kappa_{2}g_{s-2}\,.$
(4.11)

One could even use this to connect higher spins around a curved space to
higher spins around flat space, at the price of making the $\phi_{s}$
maximally non-irreducible.

Strictly speaking, a complete parameterization would have included lower spins
in (4.8); we have not done so because of our focus on higher spins. A spin-one
term $\phi_{1}\sqrt{g_{2}}$ term can be generated by changing
$\phi_{3}\to\phi_{3}+g_{2}\phi_{1}$ in (4.8), or in other words (4.9) for
$s_{0}=1$. A spin-zero term $\phi_{0}g_{2}$ is equivalent to rescaling the
first term in (4.8), or (after renaming $g_{2}\to g_{2}(1+\phi_{0})^{-1}$) to
rescaling $\phi_{s}\to(1+\phi_{0})^{s/2-1}$. In the future it might be
interesting to investigate the effect of including this scalar; I have not
done so in what follows. It would be especially intriguing to explore the
limit in which $\phi_{0}\to-1$, and the first term in (4.8) would disappear.

An alternative to (4.8) would be obtained by applying the same logic directly
to $F$, leading to $F=\sqrt{2g_{2}}+f_{3}/g_{2}+f_{4}(g_{2})^{3/2}+\ldots$.
Adding the spin-one term to this parameterization leads to a generalization of
the Randers choice mentioned in Sec. 3.1.

We will use an expansion similar to (4.8) also for other $y$-dependent
quantities, such as the connection and curvature:

$G^{\mu}=\gamma^{\mu}_{0\,\alpha\beta}y^{\alpha}y^{\beta}+\sum_{s>2}g_{2}^{1-s/2}G^{\mu}_{s}\,,\qquad\rho^{\mu}{}_{\nu}=R^{\mu}_{0\,\alpha\nu\beta}y^{\alpha}y^{\beta}+\sum_{s>2}g_{2}^{1-s/2}\rho^{\mu}_{s\,\nu}\,,$
(4.12)

where as usual a $0$ denotes quantities associated to the (pseudo-)Riemannian
metric $g^{0}_{\mu\nu}$. Notice that the coefficients in this expansion are
typically not traceless even if the $\phi_{s}$ are.

###  4.3 Sum over spins

Let us take the $\phi_{s}$ to be small, and work at linear order in them. In
this case we can use the formalism developed in Sec. 4.1. Namely, we take
$F^{2}=2g_{2}=g_{\mu\nu}y^{\mu}y^{\nu}$ to be of (pseudo-)Riemannian type, and
$\tilde{F}^{2}$ to be (4.8).

Recall that both $D_{\mu}$ and $\delta_{\mu}$ reduce to the usual covariant
derivative $\nabla_{\mu}$ in the (pseudo-)Riemannian case. In particular,
these operators have trivial action on the powers of $g_{2}$ in (4.12). On the
other hand, the $y$ derivatives do act nontrivially.

For $G^{\mu}$, (4.5) gives

$G^{\mu}_{\mathrm{L}\,s}=\frac{1}{2}(\mathrm{d}\phi^{\mu}_{s}-\nabla^{\mu}\phi_{s})+\frac{4-s}{4}y^{\mu}\mathrm{d}\phi_{s-2}\,.$
(4.13)

Here and later, the label L stands for “at linear order in the $\phi_{s}$”.
Recall that $\phi_{s}=0$ for $s<2$, so the last term in (4.13) is actually
absent for $G^{\mu}_{3}$.

$G^{\mu}$ is related to the analogue of a connection, while (4.13) is a
tensor.888One of the advantages of working with (4.5) is that it makes this
property more or less manifest. It is of course possible, but more laborious,
to get the same results from (3.9). The first term in its expansion (4.12)
does contain a connection; the following terms are to be thought of as
deformations, and it is a familiar result that the difference of two
connections is a tensor.

A similar application of (4.8) to (4.6) gives

$\rho^{\mathrm{L}}_{s}\equiv(\rho^{\mathrm{L}}_{s})^{\mu}{}_{\mu}=-\frac{1}{2}{\mathcal{F}}(\phi_{s})+\alpha_{s}\mathrm{d}^{2}\phi_{s-2}\,,\qquad\alpha_{s}\equiv\frac{1}{8}(s-4)(D+s-4)\,,$
(4.14)

again with L denoting linear order in $\phi_{s}$. (4.14) are tensors, as
manifest from (4.4) but not in (3.28). This result was anticipated in the
introduction as (1.2).

It is instructive to check invariance under the trivial transformation (4.9).
Since it does not change $F^{2}$, it should not change the geometric
quantities above; let us focus on (4.14). From (2.27) (with
$\partial_{\mu}\to\nabla_{\mu}$) we obtain the identity

${\mathcal{F}}(-g_{2}\phi_{s-2})=-g_{2}{\mathcal{F}}(\phi_{s-2})+\left(3-\frac{D}{2}-s\right)\mathrm{d}^{2}\phi_{s-2}\,.$
(4.15)

Fixing a particular $s_{0}>4$, there are two affected $\phi_{s}$, which appear
in a total of four terms:

$\displaystyle\delta_{\kappa}\rho^{\mathrm{L}}$
$\displaystyle=g_{2}^{-s_{0}/2}\mathrm{d}^{2}\delta_{\kappa}\phi_{s_{0}}+g_{2}^{1-s_{0}/2}\left(-\frac{1}{2}{\mathcal{F}}(\delta_{\kappa}\phi_{s_{0}})+\alpha_{s_{0}}\mathrm{d}^{2}\delta_{\kappa}\phi_{s_{0}-2}\right)-\frac{1}{2}g_{2}^{2-s_{0}/2}{\mathcal{F}}(\delta_{\kappa}\phi_{s_{0}-2})$
$\displaystyle=g_{2}^{1-s_{0}/2}\left(-\alpha_{s_{0}+2}+\alpha_{s_{0}}+\frac{1}{2}(s_{0}-3)+\frac{D}{4}\right)\mathrm{d}^{2}\kappa_{s_{0}-2}\,.$
(4.16)

In the second step we have used (4.15). Imposing
$\delta_{\kappa}\rho^{\mathrm{L}}=0$ for all $s_{0}$ gives a recurrence
relation for the $\alpha_{s}$, which is indeed satisfied by the value in
(4.14). This also shows that the $\mathrm{d}^{2}\phi_{s-2}$ term in (4.14) is
unavoidable.999A curious possibility is to use (4.15) to express
$\mathrm{d}\phi_{s-2}$ in terms of Fronsdal operators, and to substitute in
(4.14). This results in
$\rho^{\mathrm{L}}_{s}=-\frac{1}{2}{\mathcal{F}}(\psi_{s})$, where
$\psi_{s}=\frac{\alpha_{s+4}}{\alpha_{s+4}-\alpha_{s+2}}\phi_{s}-\frac{\alpha_{s}}{\alpha_{s+2}-\alpha_{s}}g_{2}\phi_{s-2}$,
thus getting rid of the second term in (4.14). Inverting this relation one
would naively seem to arrive at $F^{2}=\sum g_{2}^{1-s/2}\psi_{s}$. However,
the last step only works by summing a series that does not really converge.

A more complicated computation also gives

$\displaystyle(\rho^{\mathrm{L}}_{s})^{\mu}{}_{\rho}=-\frac{1}{2}{\mathcal{F}}^{\mu}{}_{\rho}(\phi_{s})$
$\displaystyle+\frac{4-s}{4}\Big{(}y^{\mu}\nabla_{\rho}\mathrm{d}\phi_{s-2}+g_{\nu\rho}y^{[\nu}\mathrm{d}\nabla^{\mu]}\phi_{s-2}-\frac{1}{2}\delta^{\mu}_{\rho}\mathrm{d}^{2}\phi_{s-2}$
$\displaystyle-
g_{\nu\rho}y^{(\mu}\mathrm{d}^{2}\phi^{\nu)}_{s-2}\Big{)}-\frac{1}{16}(4-s)(6-s)y^{\mu}y_{\rho}\mathrm{d}^{2}\phi_{s-4}\,,$
(4.17)

where we introduced a Fronsdal “precursor”

${\mathcal{F}}^{\mu}{}_{\rho}(\phi_{s})=\nabla_{\rho}\nabla^{\mu}\phi_{s}-\nabla_{\rho}\mathrm{d}\phi^{\mu}_{s}-g^{\mu\nu}\mathrm{d}\nabla_{[\nu}\phi^{s}_{\rho]}+\frac{1}{2}\mathrm{d}^{2}\phi^{\mu}_{s\,\rho}\,$
(4.18)

whose trace is ${\mathcal{F}}(\phi_{s})$. Recall from (3.28) that
$\rho^{\mu}{}_{\nu}$ contains the Finsler analogue of the whole Riemann
tensor. We will actually not need it in this paper, but it might be useful in
further investigations.

Finally we record the perturbative expression for the mean Cartan (3.25)

$I^{\mathrm{L}\,s}_{\mu}=\frac{1}{4}\phi^{\prime}_{s\,\mu}+\frac{4-s}{8}\left((D+s-2)\phi^{s-2}_{\mu}+y_{\mu}\phi_{s-2}^{\prime}\right)+\frac{(4-s)(6-s)}{16}(D+s-4)y_{\mu}\phi_{s-4}\,$
(4.19)

and for the Landsberg tensor (3.18):

$\displaystyle L_{\mu\nu\rho}^{\mathrm{L\,s}}$
$\displaystyle=\frac{1}{4}\mathrm{d}\phi^{s}_{\mu\nu\rho}+\frac{3}{8}(4-s)\left(g_{(\mu\nu}\mathrm{d}\phi^{s-2}_{\rho)}+y_{(\mu}\mathrm{d}\phi^{s-2}_{\nu\rho)}\right)$
(4.20)
$\displaystyle+\frac{3(4-s)(6-s)}{16}\left(y_{(\mu}y_{\nu}\mathrm{d}\phi^{s-4}_{\rho)}+y_{(\mu}g_{\nu\rho)}\mathrm{d}\phi^{s-4}\right)+\frac{(4-s)(6-s)(8-s)}{32}y_{\mu}y_{\nu}y_{\rho}\mathrm{d}\phi_{s-6}\,.$

Again here
$I^{\mathrm{L}}_{\nu}=\sum_{s>2}g_{2}^{1-s/2}I^{\mathrm{L}\,s}_{\nu}$,
$L^{\mathrm{L}}_{\mu\nu\rho}=\sum_{s>2}g_{2}^{1-s/2}L^{\mathrm{L}\,s}_{\mu\nu\rho}$.

##  5 Nonlinearities

We now go beyond the linear order in the $\phi_{s}$. This quickly becomes
complicated, but we will at least try to make the structure clear in Sec. 5.1.
In Sec. 5.2 we make the general results more concrete for $\rho_{4}$ and
$\rho_{6}$.

###  5.1 General expansion

We still raise and lower indices with the unperturbed (pseudo-)Riemannian
metric, now denoted by $g^{0}_{\mu\nu}$ to avoid confusion with the
fundamental tensor (3.4) of Finsler geometry. We expand the latter and its
inverse as in (4.8), (4.12):

$g_{\mu\nu}=g^{0}_{\mu\nu}+\sum_{s>2}g_{2}^{1-s/2}q^{s}_{\mu\nu}\,,\qquad
g^{\mu\nu}=g_{0}^{\mu\nu}+\sum_{s>2}g_{2}^{1-s/2}p^{s}_{\mu\nu}\,.$ (5.1)

(4.8) gives

$q^{s}_{\mu\nu}=\frac{1}{2}\phi^{s}_{\mu\nu}-\frac{s-4}{4}(\phi_{s-2}g^{0}_{\mu\nu}+2y_{(\mu}\phi^{s-2}_{\nu)})+\frac{1}{8}(s-4)(s-6)y_{\mu}y_{\nu}\phi_{s-4}\,.$
(5.2)

As usual, $\phi_{s<3}$ is understood to be zero; so
$q^{3}_{\mu\nu}=\frac{1}{2}\phi^{3}_{\mu\nu}$,
$q^{4}_{\mu\nu}=\frac{1}{2}\phi^{4}_{\mu\nu}$,
$q^{5}_{\mu\nu}=\frac{1}{2}\phi^{5}_{\mu\nu}-\frac{1}{4}\phi_{3}g^{0}_{\mu\nu}-\frac{1}{2}y_{(\mu}\phi^{3}_{\nu)}$,
in the usual notation (2.8). The inverse $g^{\mu\nu}$ can be found by
inverting (5.1). This is formally similar to several familiar problems in QFT:
the answer can be written a bit implicitly as

$p_{s}^{\mu\nu}=\sum_{\begin{subarray}{c}(s_{1},\ldots,s_{n})\\\
\text{partitions of }s-2\end{subarray}}(-1)^{n}(q_{s_{1}+2}q_{s_{2}+2}\ldots
q_{s_{n}+2})^{\mu\nu}\,.$ (5.3)

The parenthesis is explicitly
$g_{0}^{\mu\mu_{1}}q^{s_{1}+2}_{\mu_{1}\nu_{1}}g_{0}^{\nu_{1}\mu_{2}}q^{s_{2}+2}_{\mu_{2}\nu_{2}}g_{0}^{\nu_{2}\mu_{3}}\ldots
q^{s_{n}+2}_{\mu_{n}\nu_{n}}g_{0}^{\nu_{n}\nu}$; the partitions are to be
considered in all possible orders. The first two (5.3) are
$p_{3}^{\mu\nu}=-q_{3}^{\mu\nu}$,
$p_{4}^{\mu\nu}=-q_{4}^{\mu\nu}+q^{\mu}_{3\,\rho}q_{3}^{\rho\nu}$.

For $G^{\mu}$, we can work with (4.1) instead of (4.5); it is still more
convenient than the original (3.9). The non-linearity in $G^{\mu}_{s}$ comes
from the inverse $g^{\mu\nu}$ in (4.1); in terms of (4.13) we can write

$G_{s}^{\mu}=G_{\mathrm{L}\,s}^{\mu}+\sum_{s^{\prime}>2}p_{s^{\prime}}^{\mu\nu}G^{s-s^{\prime}}_{\mathrm{L}\,\nu}\,,$
(5.4)

The curvature is quite a bit more complicated:

$\rho^{\mu}_{s\,\rho}=\nabla_{\rho}G^{\mu}_{s}-\frac{1}{2}\mathrm{d}\partial_{y^{\rho}}G^{\mu}_{s}+\frac{s-4}{4}y_{\rho}\mathrm{d}G^{\mu}_{s-2}+Q^{\mu}_{s\,\rho}\,.$
(5.5)

The term $Q^{\mu}_{s}$, quadratic in $G^{\mu}_{s}$, is nastier and reads in
general

$\displaystyle Q^{\mu}_{s\,\rho}=\sum_{s^{\prime}=3}^{s-1}$
$\displaystyle\left(-\frac{1}{4}(\partial_{y^{\rho}}G^{\nu})_{s^{\prime}}(\partial_{y^{\nu}}G^{\mu})_{s-s^{\prime}+2}+\frac{1}{2}G^{\nu}_{s^{\prime}}(\partial_{y^{\nu}}\partial_{y^{\rho}}G^{\mu})_{s-s^{\prime}+2}\right)\,;$
$\displaystyle(\partial_{y^{\rho}}G^{\mu})_{s}$
$\displaystyle=\partial_{y^{\rho}}G^{\mu}_{s}-\frac{s-4}{2}y_{\rho}G^{\mu}_{s-2}\,,$
(5.6) $\displaystyle(\partial_{y^{\nu}}\partial_{y^{\rho}}G^{\mu})_{s}$
$\displaystyle=\partial_{y^{\nu}}\partial_{y^{\rho}}G^{\mu}_{s}-\frac{s-4}{2}(g_{\nu\rho}G^{\mu}_{s-2}+2y_{(\nu}\partial_{y^{\rho)}}G^{\mu}_{s-2})+\frac{1}{4}(s-4)(s-6)y_{\nu}y_{\rho}G^{\mu}_{s-4}\,.$

The expression (5.5) is only linear and quadratic in $G^{\mu}_{s}$, but the
$p_{s}$ appearing in (5.4) are themselves very non-linear in the $q_{s}$ and
hence in the $\phi_{s}$. In particular we see that the non-linearity grows
with $s$.

###  5.2 Spin-six nonlinear equation

To illustrate the general results above, we now consider the case where only
even $s$ are present, and give the first two $\rho_{s}$.

The first coefficients of the fundamental tensor and its inverse read

$\begin{split}&q^{4}_{\mu\nu}=\frac{1}{2}\phi^{4}_{\mu\nu}\,,\qquad
q^{6}_{\mu\nu}=\frac{1}{2}\phi^{6}_{\mu\nu}-\frac{1}{2}\phi_{4}g^{0}_{\mu\nu}-y_{(\mu}\phi^{4}_{\nu)}\,;\\\
&p_{4}^{\mu\nu}=-\frac{1}{2}\phi_{4}^{\mu\nu}\,,\qquad
p_{6}^{\mu\nu}=q_{4}^{\mu\rho}q_{4\rho}{}^{\nu}-q_{6}^{\mu\nu}=\frac{1}{4}\phi_{4}^{\mu\rho}\phi_{4\rho}{}^{\nu}+\frac{1}{2}g_{0}^{\mu\nu}\phi_{4}+y^{(\mu}\phi_{4}^{\nu)}-\frac{1}{2}\phi_{6}^{\mu\nu}\,.\end{split}$
(5.7)

The first spray coefficients are

$G_{4}^{\mu}=\frac{1}{2}(\mathrm{d}\phi^{\mu}_{4}-\nabla^{\mu}\phi_{4})\,,\qquad
G_{6}^{\mu}=\frac{1}{2}(\mathrm{d}\phi^{\mu}_{6}-\nabla^{\mu}\phi_{6})-\frac{1}{2}y^{\mu}\mathrm{d}\phi_{4}-\frac{1}{2}g^{\mu\nu}_{4}G_{4\,\nu}\,.$
(5.8)

In particular, the $s=4$ result coincides with the linear result (4.13).

As for the curvature, again the $s=4$ result coincides with the linear (4.17),
(4.18), while for $s=6$ a quadratic term appears:

$\displaystyle\rho^{\mu}_{4\,\rho}$
$\displaystyle=\nabla_{\rho}G^{\mu}_{4}-\frac{1}{2}\mathrm{d}\partial_{y^{\rho}}G^{\mu}_{4}=(\rho^{\mathrm{L}}_{4})^{\mu}{}_{\rho}\,,$
(5.9) $\displaystyle\rho^{\mu}_{6\,\rho}$
$\displaystyle=\nabla_{\rho}G^{\mu}_{6}-\frac{1}{2}\mathrm{d}\partial_{y^{\rho}}G^{\mu}_{6}+\frac{1}{2}y_{\rho}\mathrm{d}G^{\mu}_{4}-\frac{1}{4}\partial_{y^{\rho}}G^{\nu}_{4}\partial_{y^{\nu}}G^{\mu}_{4}+\frac{1}{2}G^{\nu}_{4}\partial_{y^{\nu}}\partial_{y^{\rho}}G^{\mu}_{4}=(\rho^{\mathrm{L}}_{6})^{\mu}{}_{\rho}+{\mathcal{Q}}^{\mu}{}_{\rho}(\phi_{4})\,,$

where

${\mathcal{Q}}^{\mu}{}_{\rho}(\phi_{4})=-\frac{1}{2}\nabla_{\rho}(\phi_{4}^{\mu\nu}G^{4}_{\nu})+\frac{1}{4}\mathrm{d}(\phi_{4}^{\mu\nu}{}_{\rho}G^{4}_{\nu}+\phi_{4}^{\mu\nu}\partial_{y^{\rho}}G^{4}_{\nu})-\frac{1}{4}\partial_{y^{\rho}}G^{\nu}_{4}\partial_{y^{\nu}}G^{\mu}_{4}+\frac{1}{2}G^{\nu}_{4}\partial_{y^{\nu}}\partial_{y^{\rho}}G^{\mu}_{4}\,.$
(5.10)

This ${\mathcal{Q}}$ now denotes the part that is quadratic in $\phi_{4}$
rather than in $G^{\mu}_{4}$, so it contains more terms than the $Q$ in (5.1).

Notice that the $\rho^{\mu}_{s\,\rho}$ are not symmetric if we lower the first
index with $g^{0}_{\mu\nu}$ (as we are doing in this section). Indeed the
symmetric property advertised in 3.2 holds if we lower the index with the full
fundamental Finsler tensor, $\rho_{\mu\rho}\equiv
g_{\mu\nu}\rho^{\nu}{}_{\rho}$. In our spin expansion, the $s=4$ term of this
object reads
$g^{0}_{\mu\nu}\rho^{\nu}_{4\,\rho}+\frac{1}{2}\phi^{4}_{\mu\nu}R^{\nu}{}_{\alpha\rho\beta}y^{\alpha}y^{\beta}$;
this can indeed be shown to be symmetric using (2.6).101010For the much more
laborious $s=6$ check, I used the software package xAct.

Finally, taking the trace:

$\rho_{4}=-\frac{1}{2}{\mathcal{F}}(\phi_{4})\,,\qquad\rho_{6}=-\frac{1}{2}{\mathcal{F}}(\phi_{6})+\frac{1}{4}(D+2)\mathrm{d}^{2}\phi_{4}+{\mathcal{Q}}(\phi_{4})\,.$
(5.11)

This signals a cubic interaction among two $s=4$ and one $s=6$ fields. (Such
an interaction is also present in Vasiliev theories [31].)

##  6 Challenges for Finsler dynamics

Given the appearance of the Fronsdal operator in Sec. 4, we now investigate
whether a Finsler action can indeed propagate degrees of freedom with higher
spins. Unfortunately this will give mixed results.

###  6.1 Equations of motion

For pure GR, Einstein’s equations set to zero the Ricci tensor. This suggests
to take for pure Finsler gravity

$\rho=0\,,$ (6.1)

sometimes called Rutz equation [32]. For us this has the advantage that we
know it already to be related to the Fronsdal operator. A natural variant is
$\rho-kF^{2}=0$, for $k$ a constant, which is similar to pure GR with a
cosmological constant.

On the other hand, it was argued in [33, 34] that this cannot be obtained from
an action. Another natural possibility is to write the closest possible
analogue of the Einstein–Hilbert action for pure GR. $\rho$ also contain $y$
variables, and to obtain an analogue of the Ricci scalar we would like to
somehow trace over them. This can be achieved by integrating over them,
leading to

$S=\int\mathrm{d}^{D}x\,\mathrm{d}^{D-1}\hat{y}\,g\,\frac{\rho}{F^{2}}\,.$
(6.2)

(See [20, 11.3.1] and [35].) The measure is the product of the usual
$\sqrt{g}\mathrm{d}^{D}x$ and of $\sqrt{g}\mathrm{d}^{D-1}\hat{y}$ along the
fiber of the sphere tangent bundle $SM$. The $F^{2}$ is included to make the
integrand of degree zero, so that it is defined on the fiber of that bundle.
The variation of (6.2) gives

$\rho=\frac{1}{D+2}F^{2}g^{\mu\nu}\left(\partial_{y^{\mu}}\partial_{y^{\nu}}\rho+2D_{\mu}\mathrm{d}I_{\nu}-2\mathrm{d}I_{\mu}\mathrm{d}I_{\nu}+2\partial_{y^{\mu}}\mathrm{d}^{2}I_{\nu}\right)\,.$
(6.3)

We will see that (6.1) and (6.3) can be analyzed along similar lines.111111A
third possibility was proposed in [36], which is however chosen in such a way
that its classical solutions coincide with those of ordinary GR. Thus we will
not consider it here.

In the rest of this section we will analyze the classical solutions of this
theory, perturbatively around Minkowski space. In the similar problem for GR,
the equations of motion can be approximated by taking (2.2) for
$g_{\mu\nu}=\eta_{\mu\nu}$. Crucially, the infinitesimal diffeomorphisms (2.4)
can be used to set $\partial^{\nu}h_{\mu\nu}-\frac{1}{2}\partial_{\mu}h=0$,
after which the equation of motion reads simply

$\partial^{2}h_{\mu\nu}=0\,.$ (6.4)

So there are no massive excitations. Diffeomorphisms can be further used to
restrict the massless ones to $\partial^{\mu}h_{\mu\nu}=u^{\mu}h_{\mu\nu}=0$,
with $u$ a timelike vector. These results avoid the presence of modes with the
wrong sign in the action, which would be problematic for quantization. (While
pure GR is ultimately not sensible quantum mechanically because of its non-
renormalizability, it can at least be used as a good effective quantum field
theory.)

We thus expect the role of gauge transformations to be crucial. The presence
of the Fronsdal operator ${\mathcal{F}}$ in (4.6), (4.14) seems promising in
that respect: ${\mathcal{F}}$ does have an infinite-dimensional kernel, or in
other words it has gauge transformations. In the next subsection we examine
whether these can be considered gauge transformations for our Finsler
equations of motion.

###  6.2 Broken Fronsdal gauge transformations

The appearance of the Fronsdal operator in (4.14) might give hope that gauge
transformations of the form $\delta\phi_{s}=\mathrm{d}\lambda_{s-1}$ can be
introduced in Finsler geometry, at least for a small perturbation around flat
space $F^{2}=2\eta_{2}$ (where (2.28) is valid). It is not clear why the
$F^{2}$ parameterization (4.8) should be invariant under such a
transformation.121212[23, (5.9)] notes that the invariance of proper time
$\int\mathrm{d}\tau F$ can be approximately restored by simultaneously
changing the position $x^{\mu}(\tau)$ in a velocity-dependent fashion. In
other words, the gauge transformations of Finsler geometry are only
diffeomorphisms. A priori it might be possible, however, that $F^{2}$ changes
in a way that does not modify the geometry in a physically relevant way. We
would then declare such transformations to be additional gauge
transformations. This might perhaps be similar to the notion of projective
equivalence mentioned in Sec. 4.1.131313The infinitesimal counterpart of (4.1)
is $\delta G^{\mu}=\delta Py^{\mu}$ with $\delta
P=(2g_{2})^{-1}\mathrm{d}\delta F=(2g_{2}^{3})^{-1/2}\mathrm{d}\delta F^{2}$.
From (4.3) we now find $\delta\rho=(1-D/2)\mathrm{d}\delta P$; so $\rho$ is
not invariant under a projective equivalence.

However, the term $\mathrm{d}^{2}\phi_{s-2}$ in (4.14) should give one pause.
Including the Fronsdal term, the total transformation for a Finsler
deformation of Minkowski space reads

$\delta\rho^{\mathrm{L}}_{s}=\mathrm{d}^{3}\left(-\frac{1}{4}\lambda^{\prime}_{s-1}+\alpha_{s}\lambda_{s-3}\right)\,.$
(6.5)

One might try to cancel the two terms in the parenthesis against each other.
The structure looks similar to that of a Stückelberg mechanism, where a mass
term locks together the gauge transformations of two fields of different spin.
It is also reminiscent of the compensator fields in [26, 37, 38], where an
equation of the type ${\mathcal{F}}(\phi_{s})=\mathrm{d}^{3}{\mathcal{H}}_{s}$
appears, with $\delta{\mathcal{H}}_{s}=\lambda^{\prime}_{s-1}$.141414In the
AdS case an extra term is present because of (2.39); we will deal with this at
the end. For theories of Vasiliev type, a certain boundary condition for the
scalar can lead to a Higgs mechanism whose Goldstone field is a bound state of
higher spin fields [39].

Unfortunately, this combined gauge transformation is nothing but a particular
case of the trivial redundancy (4.9), which is only present when the
$\phi_{s}$ are not traceless. This is easiest to see when we only transform
two spins, $\delta\phi_{s_{0}}=\mathrm{d}\lambda_{s_{0}-1}$,
$\delta\phi_{s_{0}-2}=\mathrm{d}\lambda_{s_{0}-3}$, and the two only contain
the same irreducible spin-$(s_{0}-3)$ representation:
$\lambda^{0}_{s_{0}-1}=0$, $\lambda^{\prime\prime}_{s_{0}-1}=0$, and
$\lambda^{\prime}_{s_{0}-3}=0$. Setting $\delta\rho^{\mathrm{L}}$ to zero
gives $\lambda_{s_{0}-3}=\lambda^{\prime}_{s_{0}-1}/(6-D-2s)$, which upon
taking $\mathrm{d}$ reduces to (4.9).

A faster way to proceed is to assemble all the $\lambda_{s}$ into a single

$\Lambda=\sum_{s>1}g_{2}^{1-s/2}\lambda_{s-1}\,,$ (6.6)

so that now

$\delta F^{2}=\mathrm{d}\Lambda\,.$ (6.7)

Notice that $\Lambda$ is homogeneous in $y$ of degree one, since $F^{2}$ is of
degree two. When we perturb around flat (pseudo-)Riemannian space,
$F^{2}=2\eta_{2}$, and use (4.6), the same steps as in (2.28) give

$\delta\rho=-\frac{1}{4}\mathrm{d}^{3}\Lambda^{\prime}\,,\qquad\Lambda^{\prime}=\eta^{\mu\nu}\partial_{y^{\mu}}\partial_{y^{\nu}}\Lambda\,.$
(6.8)

The strategy above now corresponds to demanding $\Lambda^{\prime}=0$. This is
not simply the sum over the individual $\lambda^{\prime}_{s-1}$, because the
factors $g_{2}^{1-s/2}$ in (6.6) also give a contribution when we take the
derivatives with respect to $y^{\mu}$. (This is the subtlety alluded to above
(4.6).) Rather:

$\Lambda^{\prime}=\sum_{s>1}g_{2}^{1-s/2}\left(\lambda^{\prime}_{s-1}-4\alpha_{s}\lambda_{s-3}\right)\,.$
(6.9)

The coefficients in this expansion are not necessarily traceless, so to set
$\Lambda^{\prime}=0$ we need to use the logic leading to (4.10). This gives a
relation $\lambda^{0}_{s-3}+\lambda^{1}_{s-1}+\ldots=0$, which upon using
(4.10) gives $\Lambda=0$. Alternatively, the discussion in Sec. 4.2 tells us
that the parameterization (6.6) is redundant unless we impose that the
$\lambda_{s-1}$ are irreducible:

$\lambda^{\prime}_{s-1}=0\,.$ (6.10)

In other words, this choice can be made without loss of generality. Notice
that this is the infamous traceless condition on the Fronsdal gauge
transformations, which here appears rather naturally. In view of (6.9),
$\Lambda^{\prime}=0$ implies $\lambda_{s-1}=0$ for all $s$.

An even nicer way of seeing the failure of this strategy is to notice that
$\Lambda^{\prime}=\eta^{\mu\nu}\partial_{y^{\mu}}\partial_{y^{\nu}}\Lambda=0$
is demanding $\Lambda$ to be a harmonic function of the $y$, while below (6.6)
we observed it to be homogeneous of degree one. These two demands are only met
by a linear function $\Lambda=\lambda_{1}$, which corresponds to $\delta
g_{2}=\mathrm{d}\lambda_{1}$, the familiar diffeomorphisms. Ignoring this
possibility as we did above, we recover that $\Lambda^{\prime}=0$ implies
$\Lambda=0$.

We have concluded that we cannot set to zero the parenthesis in (6.5), or
$\Lambda^{\prime}$ in (6.8). In other words, we have tried to find gauge
transformations with an algebraic constraint (similar to the traceless
constraint for Fronsdal), and we failed. We could still hope that the full
(6.5) can somehow be made zero, by playing with the differential operators.
However, $\mathrm{d}^{3}$ has a finite-dimensional kernel, consisting of
quadratic polynomials in the $x^{\mu}$. Thus we also cannot find gauge
transformations this way.151515We can apply a similar argument to projective
equivalence to flat space. The variation of (4.5) gives $\delta
G_{\mathrm{L}}^{\mu}=\frac{1}{2}\mathrm{d}^{2}\partial_{y_{\mu}}\Lambda$. From
below (4.1) we find $\delta G^{\mu}=Py^{\mu}$. If we ignore the finite-
dimensional kernel of $\mathrm{d}^{2}$, we are led to postulate
$\partial_{y_{\mu}}\Lambda=y^{\mu}Q$, with $\mathrm{d}^{2}Q=2P$. This implies
a “purely radial” dependence, $\Lambda=\Lambda(\eta_{2})$, and the homogeneity
property lets us conclude $\Lambda=\lambda_{0}\sqrt{\eta_{2}}$, which would
correspond to the gauge transformations of a spin-1 field, had we included it
in (4.8). This is compatible with [18, Prop. 3.4.8]; notice however that many
other Finsler metrics projectively equivalent to flat space are given in that
book, not obtained by gauge transformations of Fronsdal type.

We now also analyze this possibility using the parameterization in terms of
sum over spins. It is lengthier, but instructive, and also a good warm-up for
the next subsection. We earlier concluded that we can always fix
$\lambda^{\prime}_{s-1}=0$. Moreover we recall from Sec. 4.2 that
$\phi_{s}^{\prime}=0$ can similarly be assumed without loss of generality.
Since
$(\mathrm{d}\lambda)^{\prime}=\mathrm{d}\lambda^{\prime}+2\partial_{\mu}\lambda^{\mu}$,
we also conclude

$\partial_{\mu}\lambda^{\mu}_{s-1}=0\,.$ (6.11)

Setting $\delta\rho^{\mathrm{L}}=0$ does not necessarily imply that the
coefficients $\delta\rho^{\mathrm{L}}_{s}$ of its $g_{2}$ expansion vanish.
The latter are given by (6.5) and are not necessarily traceless; thus their
trace parts can mix in the total sum
$\sum_{s>2}g_{2}^{1-s/2}\delta\rho^{\mathrm{L}}_{s}$. As discussed near
(4.10), the series can be equivalently rewritten with coefficients
$\sum_{j=0}^{\infty}\delta\rho^{\mathrm{L}}_{s+2j,j}$ (recalling the notation
(2.15)), which are now automatically traceless and can be set to zero. Given
(6.10), (6.11) and recalling that we are taking $g_{2}=\eta_{2}$, we compute
$(\mathrm{d}^{3}\lambda)^{\prime}=6\mathrm{d}\partial^{2}\lambda$,
$(\mathrm{d}^{3}\lambda)^{\prime\prime}=0$. So gauge invariance of
$\rho^{\mathrm{L}}$ requires

$\left[\alpha_{s}\mathrm{d}^{3}\lambda_{s-3}-\frac{6\alpha_{s+2}}{D+2s}\mathrm{d}\partial^{2}\lambda_{s-1}\right]_{0}=0\,.$
(6.12)

One might hope that the two terms can sum to zero; this would lead to a
recursive law on the $\lambda_{s-1}$. We now show that this cannot happen. The
action of $\mathrm{d}$ can be diagonalized after going to Fourier transform.
Consider first $p^{2}<0$. We can take $p^{\mu}=(p^{0},\,0,\,\ldots,\,0)$, so

$\mathrm{d}=-\mathrm{i}y^{0}p_{0}\,,\qquad\mathrm{d}^{\dagger}=\mathrm{i}p_{0}\partial_{y^{0}}\,.$
(6.13)

(6.11) gives us $\partial_{y^{0}}\lambda_{s-1}=0$. The traceless projection in
(6.12) adds a slight complication, which is reviewed in App. A. The two terms
in (6.12) can be viewed (forgetting the constants) as
$(z\mathrm{d})^{3}\lambda_{s-3}$ and $(z\mathrm{d})\lambda_{s-1}$, with
$z\lambda\equiv\lambda_{0}$ the traceless projector. On our
$y^{0}$-independent $\lambda$’s, the action of $(z\mathrm{d})^{j}$ is
proportional to multiplication by $y_{0}^{j}$. This means that the two terms
in (6.12) have $y^{0}$ dependence proportional to $y_{0}^{3}$ and $y_{0}$;
thus the two cannot be equal, unless they are both zero. In the case
$p^{2}=0$, the second term in (6.12) is directly zero. Moreover,
$z\mathrm{d}^{3}=(z\mathrm{d})^{3}$ is invertible, as also shown in App. A.
Thus there are no solutions to (6.12).

We can also try a similar analysis on a maximally symmetric space such AdS.
Here is what changes. First, covariant derivatives don’t commute, and that
complicates the trace $(\mathrm{d}^{3}\lambda)^{\prime}$, which is now
$(6\mathrm{d}(2\nabla^{2}-\Delta)-2[\mathrm{d},\nabla^{2}])\lambda_{s-1}$.
Second, now $\delta{\mathcal{F}}\neq 0$ even if $\lambda^{\prime}_{s-1}=0$, as
we saw back in (2.39). Third, it makes sense to consider now the variant of
(6.1) discussed below it, $\rho-kF^{2}=0$. This however can be reabsorbed in
(2.39) by redefining $\Lambda$. All in all (6.12) is changed to

$\left[-\frac{1}{2}\Lambda\mu_{s}^{2}\mathrm{d}\lambda_{s-1}+\alpha_{s}\mathrm{d}^{3}\lambda_{s-3}-\frac{\alpha_{s+2}}{D+2s}\mathrm{d}(6\nabla^{2}+\gamma_{s})\lambda_{s-1}\right]_{0}=0\,.$
(6.14)

where $\gamma_{s}=\Lambda(6(s-1)(D+s-3)+D+2s-3))/(D-1)$. Now we cannot perform
a Fourier transform to analyze the resulting condition; we should work in a
suitable eigenbasis for AdS. I have not carried out this analysis, but the
problem looks similar enough to (6.12) that at small distances (where Fourier
transform can be used at least approximately) it should reduce to the one for
flat space.

In this subsection we have concluded that in spite of (4.6), (4.14), the
Fronsdal gauge transformations $\delta\phi_{s}=\mathrm{d}\lambda_{s-1}$ don’t
survive in Finsler geometry except as the trivial redundancies (4.9). It is
still possible that there is some other gauge transformation, perhaps obtained
by complementing the Fronsdal ones by additional terms involving
$\mathrm{d}^{\dagger}$ or other operators. In any case, in the next subsection
we will sketch a linearized analysis of the Finsler equations of motion.

###  6.3 Perturbations

Let us first consider (6.1). We use the power series parameterization (4.8),
taking $\phi_{s}^{\prime}=0$. We found the linearized $\rho^{\mathrm{L}}$ in
(4.14). Just like for the earlier discussion of gauge transformations,
$\rho^{\mathrm{L}}=0$ does not necessarily imply that the
$\rho^{\mathrm{L}}_{s}$ vanish. As discussed near (4.10), the series can be
equivalently rewritten with coefficients
$\sum_{j=0}^{\infty}\rho^{\mathrm{L}}_{s+2j,j}$, which are now automatically
traceless and can be set to zero. Recalling again (2.14), we use (2.25) to
compute

$(\mathrm{d}^{2}\phi_{s})^{\prime\prime}=-4({\mathcal{F}}\phi_{s})^{\prime}=8(\mathrm{d}^{\dagger})^{2}\phi_{s}\,,\qquad({\mathcal{F}}\phi_{s})^{\prime\prime}=0\,.$
(6.15)

So $\rho^{\mathrm{L}}_{s}$ only has single and double traces, and
$\rho^{\mathrm{L}}_{s+2j,j}=0$ for $j\geqslant 3$. This leads to the
linearized equation of motion

$\left[-\frac{1}{2}{\mathcal{F}}\phi_{s}+\alpha_{s}\mathrm{d}^{2}\phi_{s-2}+f^{3}_{s}({\mathcal{F}}\phi_{s+2})^{\prime}+f^{4}_{s}(\mathrm{d}^{2}\phi_{s})^{\prime}\right]_{0}=0\,,$
(6.16)

where $f^{3}_{s}=(2(D+2s))^{-1}(1-4\alpha_{s+4}/(D+2s+2))$,
$f^{4}_{s}=-\alpha_{s+2}/(D+2s)$.

We now turn to (6.3). The linearization of the second term is simply
$\rho^{\prime}$. Among the terms involving $I_{\mu}$, we can ignore the
quadratic one; the others give

$\displaystyle[g^{\mu\nu}(D_{\mu}\mathrm{d}I_{\nu}$
$\displaystyle+\partial_{y^{\mu}}\mathrm{d}^{2}I_{\nu})]^{\mathrm{L}}=(\nabla_{\mu}\mathrm{d}\partial_{y_{\mu}}+\partial_{y_{\mu}}\mathrm{d}^{2}\partial_{y_{\mu}})t\mathrm{d}F^{2}=(2\nabla_{\mu}\mathrm{d}\partial_{y_{\mu}}+\mathrm{d}\mathrm{d}^{\dagger}+\mathrm{d}^{2}t)t\delta
F^{2}=$
$\displaystyle=\left({\mathcal{F}}-\frac{1}{2}t\mathrm{d}^{2}\right)t\delta
F^{2}\,.$ (6.17)

We used
$1/2[t,\mathrm{d}^{2}]=\nabla^{2}+\nabla_{\mu}\mathrm{d}\partial_{y_{\mu}}+\mathrm{d}\mathrm{d}^{\dagger}$,
which in turn follows again from (2.25). Overall the linearized version of
(6.3) reads

$\left({\mathcal{F}}-\frac{2}{D+2}g_{2}(t{\mathcal{F}}-4{\mathcal{F}}t+2t\mathrm{d}^{2}t)\right)\delta
F^{2}=0\,.$ (6.18)

A lengthier computation gives an equation again of the form (6.16), with more
complicated coefficients.

Given this qualitative similarity, from now on we will focus on (6.16). We
take the pseudo-Riemannian geometry to be flat, $g_{2}=\eta_{2}$. This
simplifies
$(\mathrm{d}^{2}\phi_{s})^{\prime}=(2\partial^{2}+\mathrm{d}\mathrm{d}^{\dagger})\phi_{s}$.
Schematically the equation is now of the form
$[(\partial^{2}+\mathrm{d}\mathrm{d}^{\dagger})\phi_{s}+\mathrm{d}^{2}\phi_{s-2}+(\mathrm{d}^{\dagger})^{2}\phi_{s+2}]_{0}=0$.
This can be clarified further by going to momentum space as in the previous
subsection. The simplest case is $p^{2}<0$, corresponding to massive modes. In
GR these are gauged away, as reviewed near (6.4), but for Finsler we haven’t
found a gauge transformation that can achieve a similar result. We can use
(6.13) and the explicit decomposition in App. A, which tackles the traceless
projection in (6.16). Each $\phi_{s}$ is decomposed under the spatial
$\mathrm{SO}(D-1)$ as a sum $\sum_{k}Y_{sk}\hat{\phi}^{0}_{s,k}$, where
$\hat{\phi}^{0}_{s,k}$ is traceless, degree $k$ and $y^{0}$-independent, while
$Y_{sk}$ is a polynomial whose highest-order $y_{0}$ term is $y_{0}^{s-k}$.
The operators $z\mathrm{d}$ and $z\mathrm{d}^{\dagger}$ (where $z$ is the
traceless projector, $z\phi_{s}=\phi_{s,0}$) are proportional to the identity
on each $Y_{sk}\hat{\phi}^{0}_{s,k}$.

For concreteness let us look at the lowest degrees in (6.16), keeping only
even $s$. For $s=2$, the equation reads
$[(\mathrm{d}^{\dagger})^{2}\phi_{4}]_{0}=0$. The $\hat{\phi}_{4,4}$ and
$\hat{\phi}_{4,3}$ don’t appear, while $\hat{\phi}_{4,k\leqslant 2}=0$. For
$s=4$, schematically we have
$[(\mathrm{d}^{\dagger})^{2}\phi_{6}+(\partial^{2}+\mathrm{d}\mathrm{d}^{\dagger})\phi_{4}]_{0}=0$.
Now $\hat{\phi}_{6,6}$ and $\hat{\phi}_{6,5}$ don’t appear, while for
$k\leqslant 4$, $\hat{\phi}_{6,k}\propto(\partial^{2}+c_{6,k})\phi_{4,k}$, for
$c_{6,k}$ some constants; in particular, $\hat{\phi}_{6,k\leqslant 2}=0$.
Continuing in this fashion we see that $\hat{\phi}_{s,k\leqslant 2}=0$, while
the $\hat{\phi}_{s,s}$ and $\hat{\phi}_{s,s-1}$ are free, and determine the
$\hat{\phi}_{s,k}$ with $3\leqslant k\leqslant s-2$.

Massive modes would then exist for any mass, and they would depend on many
free variables. In particular there would be many $\phi_{s}$ not transverse to
$p^{\mu}$. This would create issues when trying to quantize this theory. The
analysis for massless modes is more complicated, but gives a similar result.

The result of this perturbative analysis is disappointing, but perhaps also
expected. Without a gauge transformation, the Finsler equations of motion
would be problematic from a perturbative QFT standpoint.

###  6.4 Discussion

The appearance of Fronsdal operators in the Finsler Ricci curvature is
encouraging. But the associated higher spin gauge transformations are
broken.161616Holography indicates that higher spin gauge transformations are
exactly preserved only in the Vasiliev theory for a particular choice of
boundary conditions [15]; even their weak breaking at the quantum level is
quite constrained [40, 16, 41, 39]. Finsler geometry only has diffeomorphisms
as gauge transformations.

A pessimistic conclusion would be that the Fronsdal operator appears only for
purely mathematical reasons, driven by its similarity with the spin-two
situation.

More optimistically, perhaps some sort of gauge transformation can still be
found. While the most natural guess for gauge transformations was excluded in
Sec. 6.2, I am not excluding that more exotic possibilities might work.
Actually the presence of many free parameters in the previous subsection seems
to indicate that some sort of gauge transformation should exist. It should be
possible to find operators that have those free paramters as their image. We
would also like, however, for such a gauge transformation to have some sort of
geometrical interpretation. It would be difficult otherwise to imagine that it
can survive beyond the linear order. After all, the possibility of some
natural interacting structure is the most interesting possible advantage of
using Finsler geometry for higher spins, as illustrated in Sec. 5.

While the Finsler Ricci $\rho$ seems important in view of its connection to
the Fronsdal operator, another possibility is that it might have to be
modified by additional terms. Even more radically, one might have to modify
the central tenets of Finsler geometry altogether, as advocated in [24]. It
might be helpful to try and recast in terms of Finsler geometry the existing
higher spin theories such as those by Vasiliev [8, 9, 10, 11, 12], the simpler
alternatives in three dimensions [42, 43], or the more recent self-dual
theories [44, 45]. In any case, after finding a geometrical quantity that is
invariant under a set of infinite transformations that extends diffeomorphism
invariance, we would declare such transformations to be additional gauge
transformations, identifying different Finsler geometries that are equivalent
as far as physics is concerned. Time will tell whether any of this can be
achieved.

## Acknowledgements

It’s a pleasure to thank S. Giombi, C. Hull, E. Minguzzi, C. Pfeifer, A.
Sagnotti, A. Zaffaroni for very useful discussions and correspondence. My
interest in Finsler geometry was rekindled by an ongoing collaboration with B.
De Luca, N. De Ponti, A. Mondino, to whom I am very grateful. I would also
like to thank the participants in the 2024 Cook’s Branch Workshop, to whom a
preliminary version of these findings was presented. I am supported in part by
INFN and by MIUR-PRIN contract 2022YZ5BA2.

##  Appendix A More details on the symmetrized derivative

In this appendix we give a more explicit discussion of some properties of the
operator $\mathrm{d}=y^{\mu}\nabla_{\mu}$.

We introduce

$z\phi=(\phi)_{0}\,,$ (A.1)

where the 0 denotes the traceless part as in (2.15). We now act with
$\mathrm{d}$ on the decomposition (2.14), and then we take the traceless part
of the result. Since $\phi^{\prime}_{s,k}=0$, we have
$(\mathrm{d}\phi_{s,k})^{\prime}=2\mathrm{d}^{\dagger}\phi_{s,k}$,
$(\mathrm{d}\phi_{s,k})^{\prime\prime}=0$. Then

$(\mathrm{d}\phi_{s})_{0}=(\mathrm{d}\phi_{s,0})_{0}=\mathrm{d}\phi_{s,0}+2t_{s+1,1}g_{2}\mathrm{d}^{\dagger}\phi_{s,0}\,.$
(A.2)

We can also rewrite this as

$[z,\mathrm{d}]\phi_{s}=2t_{s+1,1}g_{2}\mathrm{d}^{\dagger}z\phi_{s}\,.$ (A.3)

In particular it follows

$[z,\mathrm{d}]z\phi_{s}=0\,,$ (A.4)

which in turn implies

$(z\mathrm{d})^{k}=z\mathrm{d}^{k}\,,\qquad
z\mathrm{d}z\mathrm{d}^{\dagger}=z\mathrm{d}\mathrm{d}^{\dagger}\,.$ (A.5)

We now specialize to flat space. Working in Fourier transform, there are two
cases to consider: $p^{\mu}$ massive or massless. In the former case, by a
Lorentz transformation we can always take the momentum to be purely along
time, $p^{0}\neq 0$, $p^{i}=0$, $i=1,\,\ldots,\,D-1$. Then
$\mathrm{d}=-\mathrm{i}y^{0}p_{0}$,
$\mathrm{d}^{\dagger}=\mathrm{i}p_{0}\partial_{y^{0}}$. A traceless
$\phi_{s,0}$ can be decomposed in $\mathrm{SO}(d-1)$ representations as

$\phi_{s,0}=\sum_{k}Y_{sk}\hat{\phi}^{0}_{s,k}(\vec{y})\,,\qquad
Y_{sk}=y_{0}^{s-k}+a_{1sk}\delta_{2}y_{0}^{s-k-2}+\ldots$ (A.6)

where $\hat{\phi}^{0}_{s,k}$ is traceless and degree $k$; the $s$ is kept only
as a label. The hat denotes dependence on the $y^{i}$ alone,
$\partial_{y^{0}}\hat{\phi}_{s,k}=0$, and the coefficients are explicitly

$a_{1sk}=\frac{(s-k)(s-k-1)}{D+2k-1}\,,\qquad
a_{jsk}=\frac{(s-k-2j+2)(s-k-2j+1)}{j(D+2j+2k-3)}a_{j-1,sk}\,.$ (A.7)

Here $\delta_{2}=\frac{1}{2}\delta_{ij}y^{i}y^{j}$, and the coefficients are
determined by $(\phi_{s,0})^{\prime}=0$. (An expansion similar to (A.6) can be
found in [2].) One finds

$(\mathrm{d}Y_{sk}\hat{\phi}^{0}_{sk})_{0}=-\mathrm{i}p_{0}\frac{D+s+k-2}{D+2s-2}Y_{s+1,k}\hat{\phi}^{0}_{s,k}\,\,,\qquad(\mathrm{d}^{\dagger}Y_{sk}\hat{\phi}^{0}_{sk})_{0}=\mathrm{i}p_{0}(s-k)Y_{s+1,k}\hat{\phi}^{0}_{s,k}\,,$
(A.8)

reproducing for $D=4$ formulas in [2]. Using (A.5) one can also find similar
expressions for
$(\mathrm{d}^{k}Y_{sk}\hat{\phi}^{0}_{sk})_{0}=(z\mathrm{d})^{k}\hat{\phi}^{0}_{sk}$.

For massless momentum we can take $p^{-}=0=p^{i}$; now
$\mathrm{d}=-\mathrm{i}y^{-}p_{-}$. (A.6) is replaced by a decomposition in
terms of $\mathrm{SO}(d-2)$ representations:

$\phi_{s,0}=\sum_{k}Z_{j_{+}sk}\hat{\phi}^{0}_{s,k}(\vec{y})\,,\qquad
Z_{j_{+}sk}=y_{+}^{j_{+}}y_{-}^{s-k-
j_{+}}+b_{1sk}\delta_{2}y_{+}^{j_{+}-1}y_{-}^{s-k-j_{+}-1}+\ldots\,,$ (A.9)

where now $\delta_{2}$ is the transverse $(D-2)$-dimensional metric, and the
coefficients are given by

$b_{1sk}=\frac{j_{+}(s-k-j_{+})}{D+2k-2}\,,\qquad b_{jsk}=\frac{(j_{+}-j)(s-k-
j_{+}-j)}{j(D+2j-4+2k)}b_{j-1,sk}\,.$ (A.10)

(A.8) is replaced by

$(\mathrm{d}Z_{j_{+}sk}\hat{\phi}^{0}_{sk})_{0}=2\mathrm{i}p_{-}\frac{D+2k+2j_{+}}{D+2s}Z_{j_{+}+1,s+1,k}\hat{\phi}^{0}_{s,k}\,.$
(A.11)

From (A.8), (A.11) it now follows that $\mathrm{d}$ has no kernel.

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