11institutetext: IRAP, Université de Toulouse, CNRS, CNES, UPS, 31400
Toulouse, France

# Strong magnetic fields detected in the cores of 11 red giant stars using
gravity-mode period spacings

S. Deheuvels 11 G. Li 11 J. Ballot 11 F. Lignières 11
<EMAIL_ADDRESS>
Despite their importance in stellar evolution, little is known about magnetic
fields in the interior of stars. The recent seismic detection of magnetic
fields in the core of several red giant stars has given measurements of their
strength and information on their topology. We revisit the puzzling case of
hydrogen-shell burning giants that show deviations from the expected regular
period spacing of gravity modes. These stars also tend to have a too low
measured period spacing compared to their counterparts. We here show that
these two features are well accounted for by strong magnetic fields in the
cores of these stars. For 11 Kepler red giants showing these anomalies, we
place lower limits on the core field strengths ranging from 40 to 610 kG. For
one star, the measured field exceeds the critical field above which gravity
waves no longer propagate in the core. We find that this star shows mixed mode
suppression at low frequency, which further suggests that this phenomenon
might be related to strong core magnetic fields.

###### Key Words.:

Asteroseismology – Stars: magnetic fields

††offprints: S. Deheuvels

##  1 Introduction

Magnetic fields affect stars at all evolutionary stages from star-forming
molecular clouds to white dwarfs and magnetars (McKee & Ostriker 2007, Kaspi &
Beloborodov 2017, Ferrario et al. 2020). In particular, they are expected to
play a central role in the redistribution of angular momentum inside stars
(Maeder & Meynet 2005, Cantiello et al. 2014, Rüdiger et al. 2015), and thus
in the transport of chemical elements. While surface magnetic fields have been
detected and characterized in stars across the HR diagram (Landstreet 1992,
Donati & Landstreet 2009), internal magnetic fields have long remained
inaccessible to direct observations. In red giant stars, the detection of
mixed modes – that is, oscillation modes that behave as gravity (g) modes in
the core and as pressure modes in the envelope – has given strong evidence
that the cores of red giant stars are rotating slowly (e.g., Deheuvels et al.
2012, Mosser et al. 2012, Gehan et al. 2018). This yielded evidence that
angular momentum is redistributed much more efficiently than if only purely
hydrodynamical processes were at work (e.g., Marques et al. 2013). Magnetic
fields could produce the additional transport that is needed (Rüdiger et al.
2015, Jouve et al. 2015, Fuller et al. 2019, Petitdemange et al. 2022).
Observational constraints on the properties of internal magnetic fields are
crucially needed to assess the nature and the efficiency of the magnetic
transport of angular momentum inside stars.

The propagation of magneto-gravity waves is expected to be suppressed when the
magnetic field exceeds a critical strength $B_{\rm c}$ above which Alfvén wave
frequencies become comparable to those of gravity waves. This phenomenon was
invoked by Fuller et al. (2015) to account for the unexpectedly low amplitudes
of dipole mixed modes in a fraction of red giants. For core fields above
$B_{\rm c}$, the authors suggested that the mode energy reaching the
magnetized core would be entirely dissipated and lost, giving rise to purely
p-like dipole modes. This interpretation was questioned by Mosser et al.
(2017), who found that partially suppressed dipole modes still retain a g-like
character. Loi (2020b) later showed that even with strong fields, a fraction
of the incoming waves could remain g-like, which would allow for partial
energy return from the core. The interpretation of suppressed dipole modes
remains debated.

Magnetic fields also produce shifts in the oscillation mode frequencies (Gough
1990). Several studies have recently investigated the impact of internal
fields on the frequencies of mixed modes in red giants (Gomes & Lopes 2020,
Bugnet et al. 2021, Loi 2021). Very recently, Li et al. (2022) detected clear
asymmetries in the rotational multiplets of dipole mixed modes in three Kepler
red giants. They showed that these features can only be accounted for by
internal magnetic fields with intensities ranging from 30 to 130 kG in the
vicinity of the hydrogen burning shell. These findings opened the exciting
opportunity to characterize magnetic fields in the cores of red giants.

We here investigate the irregularity of g-mode period spacings in a group of
red giant branch (RGB) stars, which remains so far unexplained. High-radial
order g modes are expected to be approximately equally spaced in period by
$\Delta\Pi_{l}$, where $l$ is the mode degree. In red giants, dipole mixed
modes can be used to measure $\Delta\Pi_{1}$ using asymptotic expressions of
the mode frequencies (Mosser et al. 2015). While $\Delta\Pi_{1}$ is nearly
constant over the frequency range of observed modes for the vast majority of
RGB stars, some red giants show significant variations of $\Delta\Pi_{1}$
(Mosser et al. 2018, Deheuvels et al. 2022). We here show that this feature is
the signature of strong magnetic fields in the cores of these stars. This
constitutes a new way of detecting and characterizing magnetic field in the
cores of red giants.

In Sect. 2, we present red giants that exhibit deviations from the regular
period spacing pattern of g modes, and we find additional such targets in
Kepler data. We then show in Sect. 3 that strong core magnetic fields can
account for this phenomenon. In Sect. 4, we determine the field strengths that
are required to match the seismic observations. We discuss these measurements
in Sect. 5, before concluding in Sect. 6.

##  2 Red giants with non-constant $\Delta\Pi_{1}$

###  2.1 Previous detections of non-constant $\Delta\Pi_{1}$ in RGB stars

To first order, high-radial-order gravity modes are expected to be equally
spaced in period. Among the 160 RGB stars studied by Mosser et al. (2018),
only one shows clear deviations from a regular period spacing of g modes
(KIC3216736). The authors attributed this irregularity to a buoyancy glitch
(that is, a sharp variation in the Brunt-Väisälä frequency $N$), which induces
periodic variations in the asymptotic period spacing $\Delta\Pi_{1}$.

More recently, Deheuvels et al. (2022) identified additional RGB stars with
non-constant $\Delta\Pi_{1}$, in a different context. These stars appeared
among a peculiar class of RGB stars that are located below the so-called
“degeneracy sequence” in the $(\Delta\nu,\Delta\Pi_{1})$ plane, where RGB
stars regroup when electron degeneracy becomes strong in their core. Most of
the stars in this class are intermediate-mass stars and are thought to result
from mass transfer (Deheuvels et al. 2022). The only four lower-mass stars
with too-low $\Delta\Pi_{1}$ must have a different origin. Contrary to
intermediate-mass stars, they all show clear departures from a constant period
spacing of g modes. Interestingly, the star identified by Mosser et al. (2018)
(KIC3216736) is among these targets. This suggests that there might be a link
between the non-constancy of $\Delta\Pi_{1}$ and the fact that its measured
value is abnormally low. These four stars show only one detected mode per
rotational multiplet.

###  2.2 Additional targets

We searched for other targets showing non-constant $\Delta\Pi_{1}$ among RGB
stars with detected oscillations using the catalog of Yu et al. (2018). To
estimate the period spacings of g modes using dipole mixed modes, we computed
the so-called “stretched” periods $\tau$, defined by the differential equation
$\hbox{d}\tau=\hbox{d}P/\zeta$ (Mosser et al. 2015), where $\zeta$ corresponds
to the fraction of the mode kinetic energy that is enclosed in the g-mode
cavity ($\zeta$ tends to 1 for pure g modes, and 0 for pure p modes). When
building échelle diagrams of these stretched periods, mixed modes are expected
to align in a vertical ridge if $\Delta\Pi_{1}$ is constant and deviations
from a regular period spacing induce curvature in this ridge.

We searched for stars with only one curved ridge detected in order to avoid
the additional complication coming from rotational effects (these effects will
be addressed in a subsequent work). This can mean that these stars are seen
pole-on, so that only the $m=0$ modes can be detected. This could also arise
if the core rotation is too weak to produce detectable rotational splitting in
Kepler data. We thus found seven additional targets, bringing the total of the
sample to 11 stars (see Table 1). Their stretched period échelle diagrams are
shown in Fig. 1 and 6. They were folded using an average value of the
asymptotic period spacing over the frequency range of the observations, which
is further referred to as $\Delta\Pi_{1}^{\rm(meas)}$.

The location of these 11 targets in the $(\Delta\nu,\Delta\Pi_{1})$ plane is
shown in Fig. 2 (black star symbols), where we have used the values
$\Delta\Pi_{1}^{\rm(meas)}$ as the measured asymptotic period spacing. Three
of the seven additional targets lie well below the degeneracy sequence of RGB
stars, which confirms the link between non-constant $\Delta\Pi_{1}$ and low
measured values for these quantities.

Figure 1: Stretched period échelle diagrams of two red giants showing
distortion from the regular g-mode pattern. Blue circles show detected dipole
modes. Red crosses correspond to the best-fit asymptotic mixed mode
frequencies obtained by including a magnetic perturbation.  Figure 2: Location
of RGB stars with non-constant $\Delta\Pi_{1}$ in the
$(\Delta\nu,\Delta\Pi_{1})$ plane. Black star symbols correspond to the values
of $\Delta\Pi_{1}$ that produce the best vertical alignment of modes in the
stretched period échelle diagram (Sect. 2). Colored star symbols correspond to
the corrected values of $\Delta\Pi_{1}$ obtained by taking into account the
magnetic perturbation to the mode frequencies (Sect. 4). Other RGB stars from
Vrard et al. (2016) are show as grey circles (for clarity, stars flagged by
the authors as potential aliases were omitted).

###  2.3 Origin of distortions in g-mode pattern

Deviations from a regular $\Delta\Pi_{1}$ are generally attributed to buoyancy
glitches (e.g., Mosser et al. 2015). To produce the observed distortions in
$\Delta\Pi_{1}$, we showed in Appendix A that a buoyancy glitch needs to have
a large amplitude (the local value of $N$ must be multiplied by a factor of at
least six) and be located either deep inside the inert He core, or well above
the H-burning shell. While this cannot be excluded, no known process is
expected to produce such strong features in these regions of an RGB star.
Secondly, the shape of the modulation in $\Delta\Pi_{1}$ that is produced by a
large-amplitude glitch strongly differs from the observations (see Appendix
A). Finally, the hypothesis of a buoyancy glitch would not explain why the
measured values of $\Delta\Pi_{1}$ are unexpectedly low for most of our stars.
It thus seems unlikely that the observed deviations in $\Delta\Pi_{1}$ arise
from buoyancy glitches. In the following sections, we explore the possibility
that the irregularities in $\Delta\Pi_{1}$ are produced by internal magnetic
fields.

##  3 Effects of magnetic fields on g-mode period spacings

The influence of magnetic fields over oscillation mode frequencies has been
studied over the last decades using a perturbative approach (Unno et al. 1989,
Gough 1990). Recently, their effects on mixed modes in red giants have been
addressed in the special case of dipolar fields with specific radial profiles,
either aligned with the rotation axis (Hasan et al. 2005, Gomes & Lopes 2020,
Mathis et al. 2021, Bugnet et al. 2021) or inclined (Loi 2021).

Li et al. (2022) extended these studies to an arbitrary magnetic field and
obtained a general expression for the magnetic frequency shift that is valid
provided that the azimuthal component is not much larger than the radial one
($B_{\phi}/B_{r}\ll\omega_{\rm max}/N$, where $\omega_{\rm max}$ is the
angular frequency at the maximum power of oscillations and $N$ is the Brunt-
Väisälä frequency111The ratio $\omega_{\rm max}/N$ typically exceeds $10^{2}$
for red giants.). Accordingly, the multiplets of $l=1$ pure g modes (that is,
g modes that are not coupled to p modes) undergo an average shift $\omega_{\rm
B}$, given by

$\omega_{\rm B}=\frac{\mathcal{I}}{\mu_{0}\omega^{3}}\int_{r_{\rm i}}^{r_{\rm
o}}K(r)\overline{B_{r}^{2}}\hbox{d}r,$ (1)

where $K(r)$ is a weight function that probes the g-mode cavity and sharply
peaks in the vicinity of the H-burning shell (HBS), $\mathcal{I}$ is a factor
that depends on the core structure (see Eq. 45 and 46 of Li et al. 2022), and
$\overline{B_{r}^{2}}=(4\pi)^{-1}\iint
B_{r}^{2}\sin\theta\,\hbox{d}\theta\,\hbox{d}\phi$. The angular frequency
shifts of the components of g-mode dipole multiplets are then given by

$\displaystyle\delta\omega_{\rm g}(m=0)$
$\displaystyle=\left(1-a\right)\omega_{\rm B}$ (2)
$\displaystyle\delta\omega_{\rm g}(m=\pm 1)$
$\displaystyle=\left(1+\displaystyle\frac{a}{2}\right)\omega_{\rm B},$ (3)

where $a$ is a dimensionless coefficient that depends on the horizontal
geometry of $B_{r}^{2}$ ($a\;\propto\iint
B_{r}^{2}P_{2}(\cos\theta)\sin\theta\,\hbox{d}\theta\hbox{d}\phi$, where
$P_{2}(\cos\theta)$ is the second order Legendre polynomial).

The dependency of magnetic shifts with $\omega^{-3}$ shows that low-frequency
(that is, high-radial-order) g modes are more affected by magnetic fields. For
this reason, magnetic shifts create a deviation from the regular period
spacing of pure g modes, as was already pointed out by Loi (2020a), Bugnet et
al. (2021), and Li et al. (2022). Very recently Bugnet (2022) proposed a
method to detect the signature of magnetic fields exploiting this property.

For illustration, Fig. 1 shows stretched échelle diagrams of mixed modes with
magnetic perturbations (see Sect. 4). The left panel corresponds to a case
where the unperturbed g modes have an asymptotic period spacing of
$\Delta\Pi_{1}=85$ s, and we have added a magnetic perturbation corresponding
to a frequency shift of $3.9\,\mu$Hz at $\nu_{\rm max}$. The ridge appears
strongly curved, similarly to the observations. To visualize the ridge
properly, the stretched échelle diagram was folded using a period spacing of
$\Delta\Pi_{1}^{\rm(meas)}=73.8$ s, which is much lower than the unperturbed
period spacing $\Delta\Pi_{1}$.

Magnetic perturbations thus account for both characteristics of the stars
identified in Sect. 2. First, they produce curved ridges in the period échelle
diagram. Since magnetic shifts are always positive, the period spacings of g
modes decrease with decreasing mode frequency. Thus, the curvature always has
the same shape, the low-frequency part of the ridge being bent to the left
direction of the period échelle diagram. Interestingly, all the targets
identified in Sect. 2 show ridges curved in this direction (see Fig. 6).
Secondly, magnetic perturbations yield a measured period spacing that is
significantly lower than the asymptotic unperturbed period spacing
$\Delta\Pi_{1}$. This can explain why most of the targets identified in Sect.
2 are located below the degenerate sequence in the $(\Delta\nu,\Delta\Pi_{1})$
plane.

##  4 Measurement of magnetic field strengths

We then estimated the field strengths that are required to account for the
observations. For this purpose, we computed asymptotic expressions of the
mixed mode frequencies including magnetic perturbations. We followed the
method that we proposed in Li et al. (2022), which is briefly recalled here.
The effects magnetic fields are taken into account by adding a magnetic
perturbation to the frequencies of pure p and g modes. These perturbed
frequencies are then plugged into the asymptotic expression of mixed mode
frequencies given by Shibahashi (1979). While the frequencies of p modes are
unaffected (Li et al. 2022), the periods of g modes are expressed as

$P_{\rm g}=P_{\rm g,0}\left(1+\frac{\delta\omega_{\rm g}}{2\pi}P_{\rm
g,0}\right)^{-1}$ (4)

where $P_{\rm g,0}=(n_{\rm g}+1/2+\varepsilon_{\rm g})\Delta\Pi_{1}$ is the
first-order asymptotic expression of $l=1$ g modes without perturbation, and
$\delta\omega_{\rm g}$ is the magnetic perturbation to g-mode frequencies.
Using Eq. 1-3, $\delta\omega_{\rm g}$ can be written as $\delta\omega_{\rm
g}=\delta\omega_{0}\left(\omega_{\rm max}/\omega\right)^{3},$ where
$\delta\omega_{0}$ corresponds to the magnetic shift at $\omega_{\rm max}$.

For the 11 stars of our sample, we optimized the values of $\Delta\Pi_{1}$,
$\delta\omega_{0}$, $\varepsilon_{\rm g}$, and $d_{01}$ (defined below) to
match the observations at best using a Markov chain Monte Carlo approach.
Based on the measurements of $\varepsilon_{\rm g}$ for hundreds of Kepler red
giants by Mosser et al. (2018), we assumed a gaussian prior on
$\varepsilon_{\rm g}$ with a mean of 0.28 and a standard deviation of 0.08,
and we considered uniform priors for the other parameters. The characteristics
of pure p modes were derived from the observed radial modes, with the
exception of $d_{01}$, defined as the average small separation $\nu_{{\rm
p},l=0}-\nu_{{\rm p},l=1}+\Delta\nu/2$, which was considered as a free
parameter of the fit.

The optimal parameters of the fit are given in Table 1. The corresponding
asymptotic frequencies are shown as red crosses in Fig. 1 and 6. The agreement
with the observations is very good, the curvature of the ridge being well
reproduced for all the stars. The fit also provides an estimate of the
unperturbed asymptotic period spacing $\Delta\Pi_{1}$ for these stars, which
is, as expected, larger than the apparent period spacing
$\Delta\Pi_{1}^{\rm(meas)}$. We used the newly determined values of
$\Delta\Pi_{1}$ to update the location of the 11 targets in the
$(\Delta\nu,\Delta\Pi_{1})$ plane in Fig. 2 (colored star symbols). It is
striking to observe that they now lie on the degenerate sequence, as expected
for stars in this mass range and evolutionary state. Thus, there is a body of
evidence that the distortions to the g-mode pattern that are observed in the
11 targets of the sample are indeed produced by internal magnetic fields. This
yields the opportunity to characterize these fields.

The measurement of $\delta\omega_{0}$ can be used to derive an estimate of
$\langle B_{r}^{2}\rangle=\int_{r_{\rm i}}^{r_{\rm
o}}K(r)\overline{B_{r}^{2}}\hbox{d}r$ using Eq. 1-3. The obtained expression
depends on the asymmetry parameter $a$, which can unfortunately not be
measured with only one component detected per multiplet. However, we have
shown that $-1/2\leqslant a\leqslant 1$ (Li et al. 2022), so that we can place
a lower limit on the value of $\langle B_{r}^{2}\rangle$. We obtain

$\langle B_{r}^{2}\rangle_{\rm
min}=\frac{2}{3}\frac{\delta\omega_{0}\omega_{\rm
max}^{3}\mu_{0}}{\mathcal{I}},$ (5)

where $\mu_{0}$ is the magnetic permeability. This expression is valid
regardless of whether the observed modes have an azimuthal number of $m=0$ or
$m=\pm 1$ (indeed, the factors $1-a$ and $1+a/2$ appearing in Eq. 2 and 3,
respectively, are both always inferior to $3/2$). Only in the very specific
case of a field that is entirely concentrated on the poles ($a\rightarrow 1$)
would the measured field be much larger than $\langle B_{r}^{2}\rangle_{\rm
min}$. For instance, if $B_{r}^{2}$ has an axisymmetric dipolar configuration
($a=2/5$), we have $\langle B_{r}^{2}\rangle=5/2\langle B_{r}^{2}\rangle_{\rm
min}$.

To calculate $\langle B_{r}^{2}\rangle_{\rm min}$, the term $\mathcal{I}$ must
be known, for which a model of the stellar internal structure is needed. For
this purpose, we used a pre-computed grid of stellar models of red giants with
various masses, metallicities, and evolutionary stages, built with the
evolution code MESA (Paxton et al. 2011). For each target, we selected models
from the grid that simultaneously reproduce the asymptotic large separation of
p modes $\Delta\nu$ and the asymptotic period spacing of dipole g modes
$\Delta\Pi_{1}$. The models that satisfy this condition all give similar
estimates of $\mathcal{I}$. We thus obtained measurements of $\langle
B_{r}^{2}\rangle_{\rm min}^{0.5}$ ranging from about 40 kG to about 610 kG
(see Table 1).

Figure 3: Minimal field strength $\langle B_{r}^{2}\rangle_{\rm min}^{0.5}$
required to account for the observed distortions in the g-mode period spacing
for the 11 stars of our sample (black circles). They are plotted as function
of the mixed mode density $\mathcal{N}=\Delta\nu/(\Delta\Pi_{1}\nu_{\rm
max}^{2})$, which is a proxy for evolution along the RGB (Gehan et al. 2018).
The red dashed line indicates the critical field $B_{\rm c}$ and the grey
long-dashed line shows the minimal field strength $B_{\rm th}$ required to
detect the distortions in the g-mode pattern. The blue star symbols show the
stars from Li et al. (2022).

##  5 Discussion

###  5.1 Magnetic field strength vs evolution

In Fig. 3, we plot the measured field strengths as a function of the density
of mixed modes $\mathcal{N}=\Delta\nu/(\Delta\Pi_{1}\nu_{\rm max}^{2})$, which
is a good proxy for the evolution along the red giant branch (Gehan et al.
2018). We observe a clear decrease of the measured field intensities along the
evolution. At first sight, this trend is surprising. Indeed, assuming
conservation of the magnetic flux, the contraction of the core as red giants
evolve should increase the field intensity, so that one would have expected
the opposite trend. Before interpreting this trend, we addressed the question
of potential observational biases. In Appendix C, we calculated the threshold
field strength $B_{\rm th}$ that is required to produce detectable variations
in the g-mode period spacing over the observed frequency range. As shown in
Fig. 3, $B_{\rm th}$ decreases along the evolution on the RGB. This explains
why we do not detect lower-intensity fields in unevolved red giants. However,
the lack of higher-intensity fields in more evolved stars cannot be explained
by this observational bias, and thus the decrease in the field strength with
evolution seems real.

###  5.2 Comparison with the critical field

We compared the measured minimal field intensities with the critical field
$B_{\rm c}$. We stress that for fields over $B_{\rm c}$, a local analysis
shows that gravity waves can no longer propagate (Fuller et al. 2015). While
the details of how global modes are affected remain uncertain, it is clear
that they will be impacted. We used the stellar models selected from our grid
in Sect. 4 to estimate $B_{\rm c}$ for each star of the sample. We evaluated
$B_{\rm c}$ in the HBS, where it reaches a sharp minimum (Fuller et al. 2015),
and where our field measurements have the highest sensitivity. Fig. 3 shows
that the value of $B_{\rm c}$ in the HBS decreases with evolution, as was
already pointed out by Fuller et al. (2015). We observe that our minimal field
strength measurements closely follow the trend of $B_{\rm c}$ with evolution.
One possible explanation for the trend observed in Fig. 3 is that the core
field increases with evolution, owing to magnetic flux conservation, and
eventually reaches the critical field $B_{\rm c}$. Above this field, mixed
modes would no longer form, making the seismic detection of core magnetic
fields impossible.

Figure 4: Power spectrum of KIC 6975038 obtained from Kepler data. Color-
shaded areas indicate the location of $l=0$ (blue), $l=1$ (red) and $l=2$
(green) modes.

###  5.3 Link with stars with suppressed dipole mixed modes

The ratio between the minimal measured field strength and the critical field
$B_{\rm c}$ is maximal for KIC 6975038, where it reaches a factor of about
1.7. Interestingly, this star shows clear signs of dipole mixed mode
suppression. Fig. 4 shows the power spectrum of KIC 6975038 built with Kepler
data. The regions of the spectrum where dipole mixed modes are expected are
highlighted in red. While the dipole mixed mode pattern clearly appears at
high frequency, it is nearly absent for frequencies around $\nu_{\rm max}$ and
below.

This type of behavior is expected, assuming that field intensities above the
critical field $B_{\rm c}$ can suppress mixed modes. Indeed, $B_{\rm c}$
varies as $\omega^{2}$, so that for a given field strength, there exists a
transition frequency $\omega_{\rm c}$ below which mixed modes should be
strongly suppressed and above which they should be unaffected (Fuller et al.
2015, Loi 2020b). This can be used to estimate the field strength for stars
where the transition between suppressed and normal modes can be detected. For
KIC6975038, the observed transition frequency $\omega_{\rm c}$ yields a radial
field intensity of about 180 kG in the HBS. This estimate has the same order
of magnitude as the minimal field strength $\langle B_{r}^{2}\rangle_{\rm
min}^{0.5}=301$ kG that was inferred in an independent way using the
perturbations to the g-mode period spacing (Sect. 4). This star thus combines
two different features that have been interpreted as potential indications of
the presence of core magnetic fields and they both lead to comparable
estimates of magnetic field strength. While more stars of this type would be
required to draw conclusions, this is further indication that there might be a
link between mixed mode suppression and strong core fields.

###  5.4 Origin of the detected fields

One possibility is that the detected fields were produced by a dynamo in the
convective core during the main sequence. The stars of our sample have masses
ranging from 1.11 to 1.56 $M_{\odot}$ (see Table 1). Contrary to the three
stars studied in Li et al. (2022), the lowest-mass stars likely had a
radiative core during most the main sequence. However, even these stars
possessed a small initial convective core at the beginning of the main
sequence, owing to the burning of 3He and 12C outside of equilibrium
(Deheuvels et al. 2010). The ohmic diffusion timescale being longer than the
evolution timescale (Cantiello et al. 2016), these fields can have survived
until the red giant phase and relaxed into stable configurations (Braithwaite
& Spruit 2004). By using the stellar models introduced in Sect. 4 and assuming
a conservation of the magnetic flux, we estimated the main-sequence field
strengths that would be required to produce the detected fields (Appendix D).
We found minimal field intensities ranging from 1 to 26 kG inside the main-
sequence convective cores. This is in general lower than the radial magnetic
field strengths found by the numerical simulation of a convective core (Brun
et al. 2005) or order-of-magnitude estimates assuming equipartition with the
convective motion kinetic energy (Cantiello et al. 2016). A dedicated study
will be necessary to determine whether the measured core fields can be
accounted for by this possible origin of the fields, taking into account the
diversity of the dynamo-generated fields and the dissipation provoked by their
relaxation (Becerra et al. 2022) and potential instabilities (Gouhier et al.
2022) in the post-main-sequence phase.

##  6 Conclusion

We here revisited the puzzling case of H-shell burning red giants that exhibit
strong deviations from the regular period spacing that gravity modes should
reach in the high-radial order limit (Deheuvels et al. 2022). We showed that
this peculiarity is unlikely to be produced by buoyancy glitches, and on the
contrary very well accounted for by strong magnetic fields in the core of
these stars. We thus placed lower limits on the strength of the radial field
in the vicinity of the H-burning shell, ranging from 40 to 610 kG for the 11
stars of our sample. We also showed that for one star, the measured field
exceeds the critical field $B_{\rm c}$ above which gravity waves can no longer
propagate in the core (Fuller et al. 2015). Interestingly, this star shows
mixed mode suppression at low frequency, which further suggests that this
phenomenon might be related to strong core magnetic fields, although it should
be noted that the mechanisms leading to mode suppression remain uncertain.
This study focused on red giants with one single component detected per
multiplet, to avoid the additional complication arising from rotational
effects. We plan to search more generally for similar behavior in Kepler data
in the near future.

###### Acknowledgements.

S.D., J.B. and F.L. acknowledge support from from the project BEAMING
ANR-18-CE31-0001 of the French National Research Agency (ANR) and from the
Centre National d’Etudes Spatiales (CNES).

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##  Appendix A Can the deviations in g-mode period spacings be related to
buoyancy glitches?

It is well known that buoyancy glitches induce deviations in the pattern of
high-radial-order g modes (e.g., Miglio et al. 2008). Such deviations were
already found by exploiting the mixed modes of core-helium burning giants
(Mosser et al. 2015). Cunha et al. (2015) and Cunha et al. (2019) provided the
appropriate formalism to determine the properties of buoyancy glitches
(location and amplitude) from their seismic signature. We here investigate
what types of glitches could produce the strong deviations in the period
spacings of g modes that we observed in our sample of Kepler red giants.

A buoyancy glitch produces a periodic modulation in the period spacings of g
modes. The period of this modulation is directly related to the position of
the glitch. This position is generally expressed in terms of its buoyancy
radius or depth. Following the notations of Cunha et al. (2019), the buoyancy
radius $\omega_{\rm g}^{r}$ and depth $\tilde{\omega}_{\rm g}^{r}$ at a radius
$r$ are defined as

$\omega_{\rm
g}^{r}=\int_{r_{1}}^{r}\frac{LN}{r}\,\hbox{d}r\;\;\;\hbox{;}\;\;\;\tilde{\omega}_{\rm
g}^{r}=\int_{r}^{r_{2}}\frac{LN}{r}\,\hbox{d}r,$ (6)

respectively, where $L=[l(l+1)]^{1/2}$, and $r_{1}$ and $r_{2}$ are the inner
and outer turning points of the g-mode cavity. We also introduce the total
buoyancy radius of the g-mode cavity $\omega_{\rm g}\equiv\omega_{\rm
g}^{r_{2}}$. For a glitch located at a radius $r^{\star}$, one period of the
modulation covers $\Delta n$ radial orders, where

$\Delta n=\omega_{\rm g}/\omega_{\rm g}^{r\star},$ (7)

if the glitch is located in the inner half of the cavity ($\omega_{\rm
g}^{r}\star<0.5$). If it is located in the outer half, then $\omega_{\rm
g}^{r\star}$ needs to be replaced by $\tilde{\omega}_{\rm g}^{r\star}$ in Eq.
7. The amplitude of the modulation depends on the sharpness of the variations
in $N$.

###  A.1 Glitch location

The g-mode period spacings can be obtained from the periods of mixed modes by
applying a stretching (see Sect. 2). The difference $\Delta\tau$ between the
stretched periods of consecutive mixed modes (shown as an illustration for
KIC5180345 in Fig. 5) provides an estimate of the g-mode period spacing. For
all the stars in our sample, the observed deviations do not show the periodic
behavior that is expected for buoyancy glitches. If the observed deviations
arise from glitches, the period of the modulation needs to be larger than the
range defined by the observed modes. For example, for KIC5180345 the glitch
period would have to cover at least 40 radial orders. Using Eq. 7 and the
stellar model of KIC5180345 obtained in Sect. 4, this means that the buoyancy
glitch would need to be located either very deep within the g-mode cavity
(below a fractional radius of $10^{-4}$, that is, deep within the inert He
core) or nearly at the outer edge of the g-mode cavity (that is, well above
the H-burning shell).

###  A.2 Glitch amplitude

We also addressed the question of the glitch amplitude that would be required
to reproduce the observations. As shown in Fig. 5, the observed deviations
have an amplitude that reaches about 30% of the average period spacing. For
comparison, Cunha et al. (2019) show the example of a Gaussian-shaped glitch
in an RGB star, with an amplitude of about twice the local value of $N$ and a
width of about $0.001\,R$ (see their Fig. 1). They find that it yields a
modulation in the g-mode period spacing corresponding to only 1.5% of the
asymptotic period spacing (see their Fig. 5).

To roughly estimate the glitch amplitude that would be needed in our case, we
used the formalism of Cunha et al. (2019). We assumed a Gaussian-shaped
buoyancy glitch and we used a Markov chain Monte Carlo (MCMC) to optimize the
glitch properties (amplitude and width) in order to reproduce at best the
observed g-mode period spacings. Fig. 5 shows the best-fit solution (blue
dashed line). This fitting problem appears to be highly degenerated: similar
profiles may be generated by different sets of parameters. However, some
properties of the glitch can be derived from the MCMC. In particular, we
conclude that its amplitude must be greater than six times the local value of
$N$. Any smaller value fails to reproduce the amplitude of the deviations
observed in g-mode period spacings. However, even with the appropriate glitch
amplitude, Fig. 5 clearly shows that the best-fit solution cannot correctly
reproduce the shape of the modulation. Indeed, it is well known (e.g., Miglio
et al., 2008; Cunha et al., 2019) that large-amplitude glitches yield
modulations in the g-mode period spacings that involve sharp localized
features (as opposed to small glitches, which produce sinusoidal modulations),
which seem incompatible with the smoothly varying period spacings that are
observed. On the contrary, a magnetic perturbation to the oscillation modes
provides a very good agreement with the observations (red and black lines in
Fig. 5).

Figure 5: Variations in the g-mode period spacings as a function of mode
frequency for KIC5180345. The observed period spacings (filled circles) were
computed as the difference $\Delta\tau$ between the stretched periods $\tau$
of consecutive dipolar mixed modes (see Sect. 2.2). The blue dashed line
indicates the g-mode period spacings for the best-fit buoyancy glitch
perturbation. The red long-dashed line corresponds to the best-fit magnetic
perturbation (see Sect. 4). For the magnetic perturbation, we also show the
differences $\Delta\tau$, which are directly comparable to the observations
(black solid line).

##  Appendix B Fit of asymptotic frequencies including magnetic perturbation

Fig. 6 shows the stretched échelle diagrams of the detected dipole mixed modes
for 11 red giants in our sample (blue circles). They were folded using the
apparent (perturbed) period spacing $\Delta\Pi_{1}^{\rm(meas)}$. In Sect. 4,
we fit an asymptotic expression of the mode frequencies including a magnetic
perturbation to the observed modes. The optimal solutions are shown in Fig. 6
as red crosses. The agreement is very good. We give in Table 1 the parameters
of the best-fit solutions, along with general stellar properties, for each
star.

Figure 6: Same as Fig. 1 for the remaining stars of the sample.  Table 1: Red
giant branch stars showing strong variations in $\Delta\Pi_{1}$ among the
catalog of Yu et al. (2018).

KIC Id | $\Delta\nu$ | $\nu_{\rm max}$ | $M^{(a)}$ | $\mathcal{N}$ | $\Delta\Pi_{1}^{\rm(meas)}$ | $\Delta\Pi_{1}$ | $\delta\omega_{0}/(2\pi)$ | $\langle B_{r}^{2}\rangle_{\rm min}$
---|---|---|---|---|---|---|---|---
| ($\mu$Hz) | ($\mu$Hz) | $(M_{\odot})$ |  | (s) | (s) | ($\mu$Hz) | kG
6182668 | $16.48\pm 0.12$ | $213.31\pm 2.69$ | $1.24\pm 0.09$ | 4.18 | 80.4 | $86.72\pm 0.15$ | $4.49\pm 0.09$ | 609
9474201 | $14.99\pm 0.02$ | $197.58\pm 1.41$ | $1.53\pm 0.09$ | 4.54 | 76.8 | $84.65\pm 0.07$ | $3.71\pm 0.02$ | 488
6842204 | $14.14\pm 0.04$ | $179.79\pm 0.63$ | $1.26\pm 0.06$ | 5.14 | 81.9 | $85.01\pm 0.05$ | $0.87\pm 0.02$ | 264
8560280 | $13.43\pm 0.07$ | $165.82\pm 0.75$ | $1.11\pm 0.06$ | 5.73 | 73.8 | $85.23\pm 0.05$ | $3.85\pm 0.01$ | 369
8689270 | $13.21\pm 0.03$ | $164.91\pm 0.55$ | $1.14\pm 0.05$ | 5.77 | 79.3 | $84.21\pm 0.05$ | $1.83\pm 0.01$ | 283
3216736 | $12.44\pm 0.02$ | $150.38\pm 0.71$ | $1.23\pm 0.08$ | 6.61 | 74.5 | $83.29\pm 0.03$ | $3.03\pm 0.01$ | 278
5180345 | $11.81\pm 0.03$ | $140.52\pm 0.50$ | $1.17\pm 0.07$ | 7.11 | 74.5 | $84.11\pm 0.04$ | $3.07\pm 0.01$ | 248
6975038 | $10.52\pm 0.02$ | $127.98\pm 0.80$ | $1.29\pm 0.08$ | 7.81 | 63.5 | $82.22\pm 0.08$ | $7.18\pm 0.03$ | 286
3109742 | $9.09\pm 0.01$ | $101.67\pm 0.55$ | $1.32\pm 0.09$ | 11.55 | 71.5 | $76.14\pm 0.05$ | $1.07\pm 0.01$ | 73
6614684 | $8.14\pm 0.01$ | $92.04\pm 0.45$ | $1.56\pm 0.10$ | 13.31 | 71.5 | $72.19\pm 0.01$ | $0.29\pm 0.00$ | 38
7728945 | $8.27\pm 0.02$ | $91.33\pm 0.44$ | $1.51\pm 0.09$ | 13.64 | 71.4 | $72.71\pm 0.03$ | $0.35\pm 0.01$ | 35

Notes. (a): Masses from Yu et al. (2018).

##  Appendix C Minimal field strength required to detect magnetic distortion
in g-mode pattern

We search for the minimal field strength that produces a detectable deviation
in the regular period spacing of pure gravity modes. For this purpose, we
consider typical oscillation properties for red giants. More refined estimates
could be obtained on a star-to-star basis, but we are here interested in
deriving broad estimates of magnetic intensity thresholds in order to
investigate observational biases.

For a given red giant with a large separation $\Delta\nu$ and a frequency of
maximum power of the oscillations $\nu_{\rm max}$, we consider that the modes
can be detected in a frequency interval ranging from $f_{\rm min}=\nu_{\rm
max}-2\Delta\nu$ and $f_{\rm max}=\nu_{\rm max}+2\Delta\nu$. The asymptotic
expression of unperturbed pure gravity modes is given by
$P_{n}=\Delta\Pi_{1}(n+1/2+\varepsilon_{\rm g})$, so that we expect to detect
g modes with radial orders ranging from $n_{\rm min}=1/(\Delta\Pi_{1}f_{\rm
max})-\varepsilon_{\rm g}-1/2$ and $n_{\rm max}=1/(\Delta\Pi_{1}f_{\rm
min})-\varepsilon_{\rm g}-1/2$. We then consider the asymptotic periods
$P^{\prime}_{n}$ of perturbed g modes in the presence of a field that produces
a frequency shift $\delta\nu_{0}$ at $\nu_{\rm max}$. Assuming that the
perturbation remains small compared to the mode periods themselves (this is
well verified at the detection limit for all stars of the sample), we have

$P^{\prime}_{n}\approx P_{n}(1-\delta\nu_{0}P_{n}^{4}\nu_{\rm max}^{3}).$ (8)

When analyzing the seismic data in red giants, high-radial order gravity modes
are assumed to be regularly spaced in period and they are thus fit by an
expression of the type
$\overline{P^{\prime}_{n}}=\Delta\Pi_{1}^{\rm(meas)}(n+1/2+\varepsilon_{\rm
g}^{\rm(meas)})$. Magnetic perturbations to the g-mode periods can be detected
if the deviations compared to a regular spacing in period, expressed as
$\delta P_{n}=\overline{P^{\prime}_{n}}-P^{\prime}_{n}$, are sufficiently
large. Since the unperturbed periods $P_{n}$ vary linearly with $n$, the
deviations $\delta P_{n}$ can be written as

$\delta P_{n}=\left(\alpha n+\beta+P_{n}^{5}\nu_{\rm
max}^{3}\right)\delta\nu_{0}$ (9)

where $\alpha$ and $\beta$ are the parameters of a linear regression of the
term $P_{n}^{5}\nu_{\rm max}^{3}$ as a function of $n$. Eq. 9 shows that the
intensity of the deviation from a regular period spacing is proportional to
$\delta\nu_{0}$. This can also be seen in the measured period spacing
$\Delta\Pi_{1}^{\rm(meas)}$, which here corresponds to
$\Delta\Pi_{1}+\alpha\delta\nu_{0}$.

Figure 7: Departures from a regular period spacing of gravity modes in the
presence of a magnetic field, shown as frequency differences $\delta\nu_{n}$
as a function of the radial order $n$ of gravity modes.

The period differences can be translated into frequency differences as
$\delta\nu=-\delta P_{n}/P_{n}^{2}$. Fig. 7 shows the variations in
$\delta\nu_{n}$ as a function of $n$ for an illustration case with
$\Delta\nu=10.6\,\mu$Hz, $\Delta\Pi_{1}=79.9$ s, $\varepsilon_{\rm g}=0.3$,
and $\delta\nu_{0}=0.4\,\mu$Hz. The maximal values of $|\delta\nu_{n}|$ are
reached at the boundaries of the interval, more particularly for $n=n_{\rm
min}$.

To determine whether these differences are detectable, we need to compare them
with the frequency resolution of the measurements of oscillation mode
frequencies. At the edge of the frequency interval where oscillations are
detected, typical uncertainties reach several tens of nHz. We thus considered
here that a deviation from a regular period spacing can be detected if
$\delta\nu_{n_{\rm min}}$ exceeds a threshold $\delta\nu_{\rm th}=100$ nHz. We
thus obtained the following expression for the minimal detectable magnetic
perturbation

$\delta\nu_{0,\rm min}=\delta\nu_{\rm th}\left(\frac{\alpha n_{\rm
min}+\beta}{P_{n_{\rm min}}^{2}}+P_{n_{\rm min}}^{3}\nu_{\rm
max}^{3}\right)^{-1}$ (10)

We then used Eq. 5 to translate the minimal magnetic frequency shifts into
minimal detectable field intensities $B_{\rm th}$, which are shown in Fig. 3.

##  Appendix D Can detected fields result from dynamo action in previous
convective cores?

The stars in which we detected strong core magnetic fields in this study have
masses ranging from 1.11 to 1.56 $M_{\odot}$ (see Table 1). The lowest-mass
stars of the sample have a radiative core during the bulk of their main-
sequence evolution, but even these stars possessed a small convective core at
the beginning of the main sequence because of the burning of 3He and 12C
outside of equilibrium. We tried to determine to what extent the detected
fields are compatible with dynamo-generated fields in the main-sequence
convective core.

For this purpose, we assumed that a uniform field $B_{r,{\rm MS}}$ was
produced during the main sequence, over a distance corresponding to the
maximal extent of the convective core. After the withdrawal of the convective
core, we assumed a conservation of the magnetic flux in each layer, so that
the field intensity varies as $1/r(m)^{2}$ for a layer at Lagrangian
coordinate $m$. We could then calculate the average field strength $\langle
B_{r}^{2}\rangle^{0.5}=\left(\int_{r_{\rm i}}^{r_{\rm
o}}K(r)\overline{B_{r}^{2}}\,\hbox{d}r\right)^{0.5}$ at each step of the
evolution. For each star of the sample, we calculated the intensity of the
main-sequence radial field $B_{r,{\rm MS}}$ that is required to produce the
minimal average fields $\langle B_{r}^{2}\rangle_{\rm min}^{0.5}$ that we
measured in Sect. 4. We thus found main-sequence field strengths ranging from
about 1 to 26 kG (see Table 2). These values correspond to lower limits on
$B_{r,{\rm MS}}$, the actual value depending on the geometry of the current
radial magnetic field. For instance, if $B_{r}^{2}$ has an axisymmetric
dipolar configuration (asymmetry parameter $a=2/5$), the recovered values of
$B_{r,{\rm MS}}$ range from 1 to 40 kG.

Table 2: Minimal main-sequence field intensities required to account for current measured fields in the cores of our sample of red giants, assuming conservation of the magnetic flux along the evolution.  KIC Id |  $B_{r,{\rm MS}}$ (kG)
---|---
6182668 | 18.1
9474201 | 16.2
6842204 | 18.7
8560280 | 25.8
8689270 | 19.6
3216736 | 18.9
5180345 | 17.8
6975038 | 17.7
3109742 | 5.3
6614684 | 0.8
7728945 | 0.8


The Third Way to Interacting $p$-form Theories

Matteo Broccoli$\,{}^{a}$, Nihat Sadik Deger$\,{}^{b}$ and Stefan
Theisen$\,{}^{a}$

aMax-Planck-Insitut für Gravitationsphysik, Albert-Einstein-Institut
Am Mühlenberg 1, D-14476 Potsdam, Germany
bDepartment of Mathematics, Bogazici University,
Bebek, 34342, Istanbul, Turkey

Abstract

We construct a class of interacting $(d-2)$-form theories in $d$ dimensions
that are ‘third way’ consistent. This refers to the fact that the interaction
terms in the $p$-form field equations of motion neither come from the
variation of an action nor are they off-shell conserved on their own.
Nevertheless the full equation is still on-shell consistent. Various
generalizations, e.g. coupling them to $(d-3)$-forms, where 3-algebras play a
prominent role, are also discussed. The method to construct these models also
easily recovers the modified 3$d$ Yang-Mills theory obtained earlier and
straightforwardly allows for higher derivative extensions.

##  1 Introduction

Study of three-dimensional gravity theories brought out a new class of
theories, labelled as “third way consistent”, whose field equations contain
tensors whose divergences vanish only on-shell [1]. These models do not have
actions which contain only the metric field and the Bianchi identity alone
does not guarantee the consistency of the equations of motion. The first such
example was the Minimal Massive Gravity (MMG) [2] which avoids the clash
between bulk and boundary unitarity which is a feature of generic three-
dimensional higher derivative gravity theories; in its parameter space there
is a region where both left and right central charges of the dual CFT and the
energy of the massive bulk graviton mode are positive [3]. These desirable
features motivated further investigation of this type of theories and more
gravity examples were constructed [4, 5]. But this structure is not unique to
gravity, in [6] a Yang-Mills theory with this property was found.111See also
[7, 8] where special cases of the systems discussed in [6] had been
constructed previously. Since all these are in three dimensions, it is natural
to ask whether such models exist in higher dimensions. In this paper we answer
this question affirmatively for $p$-form theories.

Our construction is based on shifting a flat Yang-Mills gauge connection with
a Lie algebra valued 1-form. The resulting equation from the vanishing field
strength condition is third-way consistent thanks to the Jacobi identity and
cannot be derived from an action. This shifting idea is inspired by the fact
that in the first order formulation of the third way consistent gravity
examples [2, 4, 5] their spin connection is similarly shifted. In the next
section we first show that in $d=3$ choosing the shift appropriately it is
possible to recover the model constructed in [6]. With this approach finding
extensions of [6] becomes straightforward and systematic for which we give one
example. In section 3, we study this issue in general $d$-dimensions. If the
shifting 1-form is a dual field strength of a $(d-2)$-form, then one obtains a
theory of $(d-2)$-forms coupled Yang-Mills gauge fields. However, requiring
the compatibility of the gauge symmetries enforces to set the Yang-Mills field
to zero and one ends up with an interacting $(d-2)$-form theory, which in
$d=3$ corresponds to the ungauged version of [6]. $p$-form theories have been
studied extensively in the past and they are important ingredients of
supergravity and string theory. References to some of the early work include
[9, 10, 11, 12, 13, 14, 16, 15, 17]. The novelty of our model is that
interaction is possible without modifying the standard $p$-form gauge
transformation unlike, for example, the well-known 4-dimensional Freedman-
Townsend (FT) model with 2-forms [9, 10, 11]. Although, the field equation of
this model has the same form as ours, namely a flatness condition, theirs
comes from a non-polynomial action. We also discuss extensions of our model
with additional fields and in particular with $(d-3)$-forms in which
3-algebras enter when one imposes the consistency condition. We conclude in
Section 4 with some future directions.

##  2 Shifting the connection

In this section we review the third way consistent three dimensional gauge
theories which were found in [6]. We present them in a way which can be easily
generalized. Some generalizations will be presented in this section, others,
in particular those to higher dimensions with higher form gauge field, will be
the subject of the next section.

The starting point is a flat connection of an arbitrary gauge group $G$, i.e.
a connection which solves the equation

$F(A)=dA+A^{2}=0\,.$ (2.1)

As it simplifies the calculations, we will mostly use form notation. Here
$A=A_{\mu}^{i}T_{i}dx^{\mu}$ is the gauge field one-form and $i=1,\dots,{\rm
dim}(G)$ is a gauge index. The $T_{i}$ are (anti-hermitian) representation
matrices which satisfy $[T_{i},T_{j}]=f^{k}{}_{ij}T_{k}$. The structure
constants satisfy the Jacobi-identity
$f^{i}{}_{j\underline{k}}f^{j}{}_{\underline{mn}}=0$, where we anti-symmetrize
over the underlined indices. We will be mostly considering compact semi-simple
gauge groups where we can lower the first index with the Cartan-Killing metric
such that $f_{ijk}$ is totally anti-symmetric.

Consider now an arbitrary Lie-algebra valued one-form
$C=C^{i}_{\mu}T_{i}dx^{\mu}$ and the equation

$F(A+C)=F(A)+D_{A}C+C^{2}=0\,,$ (2.2)

where $D_{A}C=dC+AC+CA$ is the covariant derivative of $C$.

For consistency, eq.(2.2) should satisfy $D_{A}F(A+C)=0$. While
$D_{A}F(A)\equiv 0$ (Bianchi identity), we now find the condition

$D_{A}^{2}C+D_{A}C^{2}=[F(A),C]+[D_{A}C,C]=0\,,$ (2.3)

where we have used $D_{A}^{2}C=[F(A),C]$ and $D_{A}C^{2}=[D_{A}C,C]$. Using
the equation of motion (2.2) and $[C^{2},C]=0$, we find that (2.3) is indeed
satisfied on-shell.

In three dimensions a simple generalization of (2.2) exists when $C$ is
conserved. In this case

$F(A+C)+\tau*\\!C=0\qquad\hbox{with}\qquad D_{A}*\\!C=0\,,$ (2.4)

is also on-shell consistent, using $[*C,C]=0$. Here $\tau$ is an arbitrary
constant parameter. Equations (2.2) and (2.4) are gauge covariant if $C$
transforms under gauge transformations as $F$, i.e. $C\to g^{-1}Cg$, $g\in G$.

Note that our starting point, namely (2.1), is the equation of motion
following from the Chern-Simons action. For the choices of $C$ which we will
discuss below, and which depend on the field strength $F(A)$, this is not the
case for (2.2) and (2.4). In other words, they cannot be derived from a local
action unless one introduces auxiliary fields similar to the model discussed
in [6]. If they could be derived from an action, consistency would be
automatic.

We now consider some special choices for $C$. This recovers models constructed
in [6], but also new third way consistent theories.

  1. 1.

The choice $C=\kappa*\\!F(A)\equiv\kappa\,\tilde{F}$, which satisfies
$D_{A}*C=0$; $\kappa$ is an arbitrary parameter of dimension 1/mass. This from
(2.4) leads to the third way consistent equation

$(1-\kappa\,\tau)F+\kappa\,D\tilde{F}+\kappa^{2}\,\tilde{F}^{2}=0\,,$ (2.5)

where here and below $D=D_{A}$. In components this becomes

$\varepsilon^{\mu\nu\rho}\left(D_{\nu}\tilde{F}_{\rho}+\frac{1}{2m}\tilde{F}_{\nu}\times\tilde{F}_{\rho}\right)+\mu\,\tilde{F}^{\mu}=0\,,$
(2.6)

To make contact with the notation used in [6], we have defined
$\tilde{F}_{\mu}\times\tilde{F}_{\nu}=[\tilde{F}_{\mu},\tilde{F}_{\nu}]$,
$\kappa=1/m$ and $\tau=m-\mu$, where $m$ and $\mu$ are mass parameters. For
this choice of $C$, the shift $A\to A+\kappa*\\!F(A)$ was used as a parity
transformation of the gauge field $A$ in [6]. For the special case $m=2\mu$
this system had already appeared in [8].

  2. 2.

If we use instead $C=\kappa\tilde{F}+\alpha\,j$, where $j$ is a conserved
matter current, i.e. $D*j=0$ and, therefore, $D*\\!C=0$, we find from (2.4)
the equation

$F(A+\kappa\tilde{F}+\alpha j)+\tau*\\!(\kappa\tilde{F}+\alpha j)=0\,.$ (2.7)

To make contact with ref.[6] we define $\kappa=1/m,\,\tau=m-\mu$ and normalize
the matter current by choosing $\alpha=\kappa/\tau$. We then find the
following component expression

$\varepsilon^{\mu\nu\rho}\left(D_{\nu}\tilde{F}_{\rho}+\frac{1}{2m}\tilde{F}_{\nu}\times\tilde{F}_{\rho}\right)\,+\mu\tilde{F}^{\mu}=J^{\mu}\,,$
(2.8)

where

$J^{\mu}=j^{\mu}-\frac{\varepsilon^{\mu\nu\rho}}{m-\mu}\left(D_{\nu}j_{\rho}+\frac{\tilde{F}_{\nu}\times
j_{\rho}}{m}+\frac{j_{\nu}\times j_{\rho}}{2m(m-\mu)}\right)\,.$ (2.9)

This agrees222Up to the sign of the last term. with ref.[6], but it has now
been derived without any effort. Note that any choice of $C$ in (2.2) which is
not conserved would result in a third way consistent model that is not of the
form (2.8).

  3. 3.

So far we have shown how to obtain the models constructed in ref.[6] by the
method of ‘shifting the connection’. Since this method is very general, we can
easily construct new third way consistent gauge models in three dimensions.
For instance the choice $C=\kappa*D\tilde{F}$ in (2.4), which satisfies
$D*C=\kappa D^{2}\tilde{F}=\kappa[F,\tilde{F}]=0$, leads to333In odd
dimensions with Minkowski signature, $**=-1$ for any $p$-form.

$F(A+\kappa*D\tilde{F})-\tau\kappa\,D\tilde{F}=0\,.$ (2.10)

In components we find

$\varepsilon^{\mu\nu\rho}\left(-\tau\,D_{\nu}\tilde{F}_{\rho}+\frac{1}{2}\tilde{F}_{\nu}\times\tilde{F}_{\rho}+\frac{1}{2}\kappa\,\varepsilon_{\nu\alpha\beta}\varepsilon_{\rho\sigma\delta}D^{\alpha}\tilde{F}^{\beta}\times
D^{\sigma}\tilde{F}^{\delta}\right)+\frac{1}{\kappa}\tilde{F}^{\mu}=-D^{2}\tilde{F}^{\mu}\,,$
(2.11)

which is a higher derivative extension of (2.6). We remark that the
coefficient of $\tilde{F}^{\mu}$ cannot be set to zero unlike in (2.8).

##  3 The third way with $p$-forms

The discussion so far was in three spacetime dimensions. An immediate question
is whether the construction given there, which uses a shifted connection, also
works dimensions other than three. This is indeed the case. In $d\geq 3$
dimensions we choose $C$ as the dual of a Lie algebra valued $(d-1)$-form $H$,
i.e. $C\equiv\kappa\,\tilde{H}=\kappa*\\!H$ with $H=H^{i}T_{i}$; $\kappa$ is a
coupling constant. From $F(A+\kappa\tilde{H})=0$ we obtain the equation of
motion

$F(A)^{i}+\kappa\,D_{A}\tilde{H}^{i}=-\frac{1}{2}\kappa^{2}f^{i}{}_{jk}\,\tilde{H}^{j}\wedge\tilde{H}^{k}\,,$
(3.1)

which is the $d$-dimensional version of (2.2) that we considered above.444Note
that for $d\neq 3$ there is no analogue of (2.4). Moreover, the choice $C=*F$
is not available either. Hence, in higher dimensions our method does not lead
to a third way extended version of the standard Yang-Mills theory. But,
choosing $C=*D*F$ is possible in all dimensions which will give rise to a
modified flatness condition with higher derivative terms.  As is stands, the
equation is Yang-Mills gauge covariant if $\tilde{H}^{i}$ transforms as
$\delta_{\lambda}\tilde{H}^{i}=f^{i}{}_{jk}\tilde{H}^{j}\lambda^{k}$ under
$\delta_{\lambda}A^{i}=d\lambda^{i}+f^{i}{}_{jk}A^{j}\lambda^{k}$.

We now want to elevate $H^{i}$ to a dynamical field. To this end we define it
to be the Yang-Mills covariant field strength of a $(d-2)$-form $B^{i}$, i.e.
$H^{i}=(DB)^{i}=dB^{i}+f^{i}{}_{jk}A^{j}B^{k}$. $B^{i}$ transforms covariantly
under Yang-Mills transformations, i.e. like $H^{i}$. Being the curl of a
$p=(d-2)$-form, we would like $H^{i}$ to have the $p$-form gauge symmetry. To
make it compatible with the Yang-Mills symmetry, we need to define the
$p$-form transformation rules for $B^{i}$ and $A^{i}$ as

$\delta_{\xi}B^{i}=D\xi^{i}=d\xi^{i}+f^{i}{}_{jk}A^{j}\xi^{k}\,,\qquad\delta_{\xi}A^{i}=0\,,$
(3.2)

where $\xi^{i}$ are $(d-3)$-forms. But this implies

$\delta_{\xi}H^{i}=f^{i}_{jk}F^{j}\xi^{k}\,.$ (3.3)

Covariance of $H^{i}$ and of the equations of motion require
$F(A)=0$.555Including the term $f^{i}{}_{jk}\tilde{H}^{j}\xi^{k}$ in the
transformation of $B^{i}$ as in the FT-model [11] does not change this
conclusion. Also note that $d=3$ case is exceptional where $B^{i}$ and $A^{i}$
can be identified. The simplest way to achieve this is by setting $A=0$. This
leaves an equation which involves only $H^{i}=dB^{i}$ in (3.1), namely

$d\tilde{H}^{i}=-\tfrac{1}{2}{\kappa}\,\tilde{H}^{j}\wedge\tilde{H}^{k}\,f^{i}{}_{jk}\,.$
(3.4)

This is a system of second order equations for a collection of interacting
$(d-2)$-forms $B^{i}$. If the coupling constant $\kappa=0$, we get a
collection of free $(d-2)$-forms.

Third way consistency of the deformed system follows from the construction and
can be easily checked explicitly: acting with the exterior derivative, the
left hand side vanishes identically due to $d^{2}=0$, but the right hand side
vanishes only on-shell, after using the Jacobi identity. Obviously $H^{i}$ is
invariant under $\delta B^{i}=d\xi^{i}$. The system of equations has a global
symmetry under $G$, under which $B^{i}$ transforms as before, but with
constant parameters $\lambda^{i}$. However, gauging this symmetry fails for
the reasons given above.

It is not difficult to show that the equations of motion cannot be obtained as
the variation of an action which does not contain additional (auxiliary)
fields. The equations are of the form
$\mathscr{L}(B)^{i}_{\underline{\mu}}=0$, and one needs to check the
integrability condition, i.e. whether $\frac{\delta}{\delta
B^{j}_{\underline{\nu}}(y)}\mathscr{L}(B)^{i}_{\underline{\mu}}(x)-(x\leftrightarrow
y,i\leftrightarrow j,\underline{\mu}\leftrightarrow\underline{\nu})$ vanishes.
If it doesn’t, then there is no action from which the equations of motion
follow. In this way one verifies that neither (3.4) nor the other equations in
this section can be derived from an action which does not contain additional,
auxiliary fields.

Note that (3.4) has the form of a flatness condition666 We may also construct
higher derivative extensions of this model by adding further terms in the
shift $C$, e.g. $\kappa*d*(\tilde{H}\wedge\tilde{H})$. The equation of motion
(3.5) will be modified to
$F(\tilde{H}+\kappa*d*(\tilde{H}\wedge\tilde{H}))=0$. As long as the shift is
built from the field strength $H$, $p$-form symmetry is guaranteed to work
off-shell.

$F(\tilde{H})=0\,,$ (3.5)

pretending that $\tilde{H}$ is a gauge field. From this it follows that
$\tilde{H}$ is pure gauge, i.e.

$\tilde{H}=U^{-1}dU$ (3.6)

where $U$ is a group element and where we have set $\kappa=2$ for simplicity.
The Bianchi identity is $dH=0$. This immediately reminds the principal chiral
sigma model (for a review see e.g. [18]) which is defined by the Lagrangian

$L=\textrm{Tr}(\partial_{\mu}\hat{U}^{-1}\partial^{\mu}\hat{U})\,,$ (3.7)

where $\hat{U}$ is an arbitrary Lie group element. One can define a connection
1-form that takes values in the corresponding Lie algebra as
$\hat{A}=\hat{U}^{-1}d\hat{U}$ and then the equations of motion derived from
(3.7) state that $\hat{A}$ is co-closed, i.e. $d*\hat{A}=0$. Moreover,
flatness $F(\hat{A})=d\hat{A}+\hat{A}\wedge\hat{A}=0$ is satisfied
identically, as $\hat{A}$ is pure gauge. We therefore see that that conditions
which $\hat{A}$ and $\tilde{H}$ satisfy are the same, but with the role of the
equations of motion and the Bianchi identity reversed. In this sense the two
models are dual to each other.

This brings us to the comparison of our third way consistent model of
$p=(d-2)$ forms $B^{i}$ with the Lagrangian model of Freedman and Townsend
[11] in $d=4$. They also considered interacting Lie-algebra valued anti-
symmetric tensor fields $B^{i}$, which are coupled to a two-form current
$j=*(*dB^{i}\wedge*dB^{i})$. Minimal coupling, which modifies the equations of
motion, destroys current conservation and requires further non-minimal
couplings, eventually leading to a non-polynomial action and equations of
motion. Current conservation guarantees the invariance of the action under a
deformed, field dependent $p$-form symmetry and the commutator of two
variations on $B$ vanishes on-shell.777The composite field $*dB^{i}+{\cal
O}(B^{2})$ of Freedman and Townsend (called $A^{i}$ there) can be obtained
from the definition $H=DB$ if $D$ is a covariant derivative with ‘gauge field’
$\tilde{H}^{i}$ and solving for $\tilde{H}$. Their $\hat{A}$ is obtained if
one uses a covariant derivative with connection $\tilde{H}+v$ instead, where
$v$ is an independent gauge field (which we called $A$ in (3.1)). The
equations of motion in the FT-model reduce to the ones we have been
considering, namely (3.5), if one formally identifies their composite ‘gauge
field’ $A^{i}$ with our $\tilde{H}^{i}$. This amounts to dropping all higher
order terms in $A^{i}$, which has an infinite power series expansion in the
$p$-form field $B^{i}$. The model is dual to the principal chiral model in the
sense that both can be derived from the same first order action. However, the
equation of motion of the principal chiral model,
$\partial^{\mu}A_{\mu}^{i}=0$, is not satisfied identically in the dual FT
model, but it holds on-shell.

We now discuss various generalizations of our third way consistent system of
equations (3.4), which consist of including additional fields.

Coupling to gravity

To couple the third way consistent system of $(d-2)$-forms to an external
gravitational field is trivial. In that case (3.4) becomes

$\nabla_{\mu}\tilde{H}^{i}_{\nu}-\nabla_{\nu}\tilde{H}^{i}_{\mu}=-\,\kappa\,f^{i}{}_{jk}\tilde{H}^{j}_{\mu}\,\tilde{H}^{k}_{\nu}\,.$
(3.8)

Due to the symmetry of the Christoffel symbols, they drop out and the only
appearance of the metric is through the Hodge star implicit in
$\tilde{H}^{i}$. On-shell consistency can easily be shown for (3.8). We act
with $\nabla_{\rho}$ and anti-symmetrize in all three indices. The l.h.s is
identically zero: we replace the product of the two covariant derivatives by
half of their commutator and use the Bianchi identity
$R_{\underline{\mu\nu\rho}}{}^{\sigma}=0$ to show that it vanishes. The on-
shell vanishing of the r.h.s. is as in the flat case.

If gravity is dynamical, we need to construct an energy-momentum tensor which
is on-shell conserved. A possible candidate is
$T_{\mu\nu}=\tilde{H}^{i}_{\mu}\tilde{H}^{i}_{\nu}-\frac{1}{2}g_{\mu\nu}\tilde{H}^{i\,\rho}\tilde{H}^{i}_{\rho}$
[6]. On-shell conservation is straightforward to demonstrate.

Adding lower form fields

A simple generalization, which is inspired by [16, 15], is to add $p$-forms
$b^{a}$ (with $p<d-2$) whose field strengths and dual field strengths we
denote by $h^{a}$ and $\tilde{h}^{a}$ respectively. They come with their own
gauge symmetry. The following system of equations is easily shown to be third
way consistent

$d\tilde{H}^{i}=-\tfrac{1}{2}{\kappa}f^{i}{}_{jk}\tilde{H}^{j}\,\tilde{H}^{k}\,,\qquad
d\tilde{h}^{a}=-\kappa\,t_{i~{}b}^{a}\tilde{H}^{i}\,\tilde{h}^{b}\,,$ (3.9)

provided $[t_{j},t_{k}]=f^{i}{}_{jk}t_{i}$, i.e. $b^{a}$ is a vector in some
representation of $G$.

Other generalizations containing $p$-forms of degree lower than $d-2$ are also
possible. Consider, for instance, in addition to the $B^{i}$, a single
$(d-3)$-form $b$ with field strength $h=db$ and dual field strength
$\tilde{h}$. Postulate its equation of motion as

$d\tilde{h}=\tilde{H}^{i}\wedge\tilde{H}^{j}\wedge\tilde{H}^{k}\,f_{ijk}\,,$
(3.10)

together with (3.4). Consistency now requires
$d\big{(}\tilde{H}^{i}\wedge\tilde{H}^{j}\wedge\tilde{H}^{k}\,f_{ijk}\big{)}=0$.
Indeed, on-shell, we have

$\displaystyle
d\big{(}\tilde{H}^{i}\wedge\tilde{H}^{j}\wedge\tilde{H}^{k}\,f_{ijk}\big{)}=-\frac{3\kappa}{2}\,\tilde{H}^{m}\wedge\tilde{H}^{n}\wedge\tilde{H}^{j}\wedge\tilde{H}^{k}\,f^{i}{}_{mn}f_{ijk}=0\,,$
(3.11)

with the help of the Jacobi identity, where we used (3.4).

A slightly less trivial generalization is to include a collection of
$(d-3)$-forms $b^{i}$ and to impose the equations of motion

$d\tilde{h}^{i}=f^{i}{}_{jkl}\tilde{H}^{j}\wedge\tilde{H}^{k}\wedge\tilde{H}^{l}\,,$
(3.12)

where $f^{i}{}_{jkl}$ is totally antisymmetric in its lower indices.
Consistency now requires

$0=d(f^{i}{}_{jkl}\tilde{H}^{j}\wedge\tilde{H}^{k}\wedge\tilde{H}^{l})=-\frac{3\kappa}{2}\,f^{i}{}_{jkl}\,f^{j}{}_{mn}\tilde{H}^{m}\wedge\tilde{H}^{n}\wedge\tilde{H}^{k}\wedge\tilde{H}^{l}\,,$
(3.13)

which is satisfied if

$f^{i}{}_{j\underline{kl}}\,f^{j}{}_{\underline{mn}}=0\,.$ (3.14)

A simple special solution of this condition is

$f^{i}{}_{jkl}=v^{i}\,f_{jkl}\,,$ (3.15)

where, as before, $f_{jkl}$ are the structure constants of a simple Lie
algebra. Then (3.14) is satisfied by virtue of the Jacobi identity. $v^{i}$ is
an arbitrary vector which transforms in the adjoint representation of the Lie
algebra. If we choose it to have only one non-zero component, say
$v^{i}=\delta^{i1}$, the system reduces to (3.10) plus a collection of free
$(d-3)$-forms.

A more interesting solution to (3.14) can be obtained as follows. The
appearance of $f^{i}{}_{jkl}$ suggests a relation to three-algebras, whose
essential features we briefly recall; for review and references see [19] and
[20]. Denoting its generators by $T_{i}$, one has

$[i,j,k]\equiv[T_{i},T_{j},T_{k}]=f^{l}{}_{ijk}T_{l}\,,$ (3.16)

and the $f^{i}{}_{\underline{jkl}}$ satisfy the fundamental identity

$[i,j,[k,l,m]]=[[i,j,k],l,m]+[k,[i,j,l],m]+[k,l,[i,j,m]]\,.$ (3.17)

Equivalently

$f^{n}{}_{klm}f^{p}{}_{ijn}=f^{n}{}_{ijk}f^{p}{}_{nlm}+f^{n}{}_{ijl}f^{p}{}_{knm}+f^{n}{}_{ijm}f^{p}{}_{kln}\,.$
(3.18)

Following [21], one introduces a bi-invariant metric $h_{ij}$ on the three-
algebra to lower the first index and arrives at $f_{ijkl}$, which is anti-
symmetric in all its indices. Still following [21] we choose it to be
$h_{ij}={\rm diag}(-1,1,\dots,1)$. One then splits the indices into
$i=(0,a,\phi)$ and chooses

$f_{0abc}=f_{\phi abc}=f_{abc}\,,\qquad f_{0\phi ab}=f_{abcd}=0\,,$ (3.19)

and $f_{abc}$ are the structure constants of a compact Lie algebra. This
solves the fundamental identity (3.18). Note that the above choices imply that
the Cartan-Killing metric on the Lie-algebra is $\delta_{ab}$.

We choose furthermore that all components of $f^{i}{}_{jk}$ vanish except for
$f^{a}{}_{bc}$, which are totally anti-symmetric structure constants
$f^{a}{}_{bc}=f_{abc}$. With these choices one verifies that (3.14) is also
satisfied.

The vanishing of some of the components of the structure constants
$f^{i}{}_{jk}$ and $f^{i}{}_{jkl}$ implies that $b^{0}+b^{\phi}$ is a free and
completely decoupled field, while $B^{0}$ and $B^{\phi}$ satisfy source-free
equations, but they appear as sources in the equation for $b^{a}$.

Instead of adding $(d-3)$-form fields we can add a set of $(d-q)$-forms
$c^{i}$ with dual field strengths $\tilde{k}^{i}$ and postulate the equations
of motion

$d\tilde{k}^{i}=\tilde{H}^{j_{1}}\wedge\cdots\wedge\tilde{H}^{j_{q}}\,f^{i}{}_{j_{1}\dots
j_{q}}.$ (3.20)

Consistency now requires the following relation for the ‘structure constants’:

$f^{i}{}_{j\underline{j_{2}\dots j_{q}}}f^{j}{}_{\underline{mn}}=0\,.$ (3.21)

We have not explored this possibility, i.e. the existence of such algebraic
structures.

##  4 Outlook

In this paper, we constructed third way consistent interacting $(d-2)$-form
theories in $d$-dimensions and studied some of their properties. Third way
consistency allows to have interactions without modifying the standard
$p$-form gauge symmetry but the price is the lack of an action. However, it
might be possible to find an action by introducing auxiliary fields as it
happened in [6]. It would be useful to determine such an action which would
allow studying its dynamical couplings with other fields such as gravity [22,
23] which may lead to some interesting applications of these models. Another
reason why an action is desirable is that it might be a viable starting point
for the (path-integral) quantization of these models, a question which is
otherwise not clear how to address. Constructing supersymmetric versions of
our model is another open problem.

Higher order form fields are sources for branes. Adding $(d-3)$-brane sources
to the equations of motion for $(d-2)$-forms and studying the resulting
systems of branes would also be of interest.

Constructing third-way interacting $p$-form theories where all transformations
are compatible with non-trivial charges is another direction worth exploring.
To achieve this, the so-called embedding tensor formalism [24, 25], which is
used in the classification of gauged supergravities and where several
$p$-forms are required, might be useful. In $d=4$ only one- and two-forms are
needed [26] and hence this could be a suitable set-up to couple our model
covariantly to Yang-Mills fields.888We thank the referee for this suggestion.

The 3-dimensional Yang-Mills model of [6] is closely related to non-Abelian
Chern-Simons theories considered in [7] where a novel Higgs mechanism was
found. It would be interesting to see if something similar happens for our
model and try to explain the interaction terms we have as spontaneous breaking
of a local symmetry as it was shown for the MMG model[2] in [27].

Finding third way consistent gravity theories in $d>3$ is another challenge.
In $d=3$, using the first order formulation of gravity was useful [2, 4, 5]
but in higher dimensions the metric description might be more practical. Note
that the Christoffel symbol $\Gamma^{\nu}_{\mu\rho}$ can also be thought of as
the $(\nu,\rho)$ component of a Yang-Mills field:
$\Gamma^{\nu}_{\mu\rho}=(A_{\mu})^{\nu}_{\,\,\,\rho}$ [28] and hence to obtain
such models we expect a shift in the Christoffel connection starting from the
vacuum Einstein equation or a modification of it that comes from an action.
Indeed, we checked that the MMG model [2] can be obtained via the shift:
$\Gamma^{\nu}_{\mu\rho}\rightarrow\Gamma^{\nu}_{\mu\rho}+\kappa\,\varepsilon_{\mu}^{\,\,\,\nu\alpha}S_{\alpha\rho}$
where $S_{\alpha\rho}=R_{\alpha\rho}-\frac{1}{4}Rg_{\alpha\rho}$ is the
Schouten tensor. However, there is an important difference with the gauge
theory case that we studied in this paper, namely not every shift leads to a
third way consistent theory. For example, doing this shift with the Ricci
instead of the Schouten tensor does not work. We hope to come back to these
issues in the near future.

## Acknowledgements

NSD is grateful to the Albert-Einstein-Institute, Potsdam, where this work was
initiated, for generous hospitality and financial support. MB is supported by
the International Max Planck Research School for Mathematical and Physical
Aspects of Gravitation, Cosmology and Quantum Field Theory. We thank P.
Townsend and E. Bergshoeff for feedback on the manuscript.

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