Enlarged super-$\mathfrak{bms}_{3}$ algebra and its flat limit

Javier Matulich∗, Evelyn Rodríguez†,⋆

∗Instituto de Física Teórica UAM/CSIC,
C/ Nicolás Cabrera 13-15, Universidad Autónoma de Madrid, Cantoblanco, Madrid
28049,Spain.
⋆Grupo de Investigación en Física Teórica, GIFT,
Concepción, Chile.
†Departamento de Matemática y Física Aplicadas,
Universidad Católica de la Santísima Concepción,
Alonso de Ribera 2850, Concepción, Chile.
<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

In this paper we analyze the asymptotic symmetries of the three-dimensional
Chern-Simons supergravity for a supersymmetric extension of the semi-simple
enlargement of the Poincaré algebra, also known as AdS-Lorentz superalgebra,
which is characterized by two fermionic generators. We propose a consistent
set of asymptotic boundary conditions for the aforementioned supergravity
theory, and we show that the corresponding charge algebra defines a
supersymmetric extension of the semi-simple enlargement of the
$\mathfrak{bms}_{3}$ algebra, with three independent central charges. This
asymptotic symmetry algebra can alternatively be written as the direct sum of
three copies of the Virasoro algebra, two of which are augmented by
supersymmetry. Interestingly, we show that the flat limit of the obtained
asymptotic algebra corresponds to a deformed super-$\mathfrak{bms}_{3}$
algebra, being the charge algebra of the minimal Maxwell supergravity theory
in three dimensions.

###### Contents

  1. 1 Introduction
  2. 2 Minimal AdS-Lorentz Supergravity theory and flat limit
  3. 3 Asymptotic symmetries
    1. 3.1 Boundary conditions
    2. 3.2 Charge algebra
  4. 4 Minimal deformed super-$\mathfrak{bms}_{3}$ algebra from a flat limit
  5. 5 Discussion

##  1 Introduction

Three dimensional gravity theories formulated through the Chern-Simons (CS)
formalism are considered to be interesting toy models since they had allowed
us to better understand different aspects of the gravitational interaction and
underlying laws of quantum gravity. Furthermore, they share many properties
with higher-dimensional gravity models which, in general, are more difficult
to approach. In the context of the AdS/CFT correspondence [1], three
dimensional models play an important role because there is a well and detailed
understanding both of the bulk theories as well as their two-dimensional dual
[2, 3, 4, 5, 6]. In this scenario, the study of three-dimensional Anti-de
Sitter (AdS) gravity has received a great interest due to its rich boundary
dynamics characterized by the centrally extended conformal algebra in two
dimensions [2]. In the asymptotically flat case, an infinite dimensional
asymptotic symmetry algebra can be found at null infinity, given by the
$\mathfrak{bms}_{3}$ algebra [7]. As it is shown in [8, 9], both aforesaid
asymptotic symmetries are connected through a well-defined flat limit. These
results have been extended to incorporate supersymmetry. For AdS supergravity,
it was shown that the boundary dynamics is governed by the superconformal
algebra [10, 11, 12], while in the flat case, the asymptotic symmetry algebra
is given by a supersymmetric extension of the $\mathfrak{bms}_{3}$ algebra
[13, 14].

On the other hand, the approach of CS (super)gravity theories based on
extensions and deformations of the Poincaré and AdS symmetries has received
great attention and has led to diverse results [15, 16, 17, 18]. These
theories include, in general, new gauge fields besides the usual vielbein and
spin connection, and thus the field content is richer compared to General
Relativity (GR). For instance, the so-called Maxwell algebra, which contains
additional generators and modifies the Poincaré algebra, allows to introduce a
new gauge field denoted as the gravitational Maxwell gauge field. The Maxwell
algebra was first introduced in four spacetime dimensions to describe a
constant Minskowski spacetime in the presence of an electromagnetic background
[19, 20, 21, 22]. In three space-time dimensions, a gravity theory based on
the Maxwell symmetry can be constructed via the CS formalism. This theory
turns out to be given by the Einstein- Hilbert term without the cosmological
constant plus a non back-reacting matter field coupled to the geometry [18].
The Maxwell (super)algebra and its generalizations have been studied in
different contexts leading to interesting results [15, 16, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45].

The presence of an additional gauge field in the Maxwell case leads to new
effects compared to GR. In particular, it was shown that the Maxwell gauge
field modifies not only the vacuum energy and momentum of the stationary
configuration but also the asymptotic structure of the three-dimensional
Maxwell CS gravity [29]. Indeed, it was shown that the asymptotic symmetry of
the theory is given by a deformation and extension of the $\mathfrak{bms}_{3}$
algebra. On the other hand, Maxwell CS gravity can be obtained as a vanishing
cosmological constant limit from the so-called AdS-Lorentz gravity in three-
dimensions [46, 47], whose underlying symmetry corresponds to a semi-simple
extension of the Poincaré algebra. This symmetry allows to introduce a non-
vanishing cosmological constant in the CS action, and the stationary solution
turns out to be a BTZ-like solution [48]. Furthermore, considering the
analysis of asymptotic symmetries, it was also shown in [48] that the deformed
$\mathfrak{bms}_{3}$ algebra, being the asymptotic symmetry of the Maxwell CS
gravity, can be derived as a flat limit of the asymptotic symmetry of the AdS-
Lorentz CS gravity which is given by a semi-simple enlargement of the
$\mathfrak{bms}_{3}$ symmetry.

In this work, we extend the asymptotic analysis done in [29, 48] to the AdS-
Lorentz supergravity theory in three dimensions, whose underlying symmetry
corresponds to a supersymmetric extension of the AdS-Lorentz algebra. As it is
expected, the asymptotic symmetry algebra is given by a supersymmetric
extension of the semisimple enlargement of the $\mathfrak{bms}_{3}$ algebra,
introduced in [49]. The motivation for considering this study is twofold.
First, it is worth it to analyze how the asymptotic structure is modified by
the presence of supersymmetry in the AdS-Lorentz CS gravity theory. Second,
although the enlarged super-$\mathfrak{bms}_{3}$ algebra was first introduced
in [49] as an expansion of the Virasoro algebra, here we will show, by
imposing appropriate boundary conditions, that it indeed corresponds to the
asymptotic symmetry of the minimal AdS-Lorentz CS supergravity theory in three
space-time dimensions. Furthermore, we show that the the deformed
super-$\mathfrak{bms}_{3}$ algebra appears as a vanishing cosmological
constant limit of the enlarged super-$\mathfrak{bms}_{3}$ algebra.

The paper is organized as follows: In section 2, we briefly review the three-
dimensional CS supergravity based on the minimal AdS-Lorentz superalgebra and
discuss about the flat limit at the level of the CS action, invariant tensors,
curvatures and field equations. In section 3, we compute the asymptotic
symmetry algebra for the minimal AdS-Lorentz CS supergravit, which results to
be an enlargement of the super-$\mathfrak{bms}_{3}$ algebra. We provide the
suitable fall-off conditions for the gauge fields at infinity and the gauge
transformations preserving the boundary conditions. We also recall that the
obtained infinite-dimensional symmetry algebra can be written as the direct
sum of three copies of the Virasoro algebra, two of which are augmented by
supersymmetry. In section 4, we analyze the flat limit, which can be performed
throughout all the steps followed in obtaining the asymptotic symmetry algebra
for the AdS-Lorentz supergravity. We show that the vanishing cosmological
constant limit $\ell\rightarrow\infty$ of the enlarged
super-$\mathfrak{bms}_{3}$ algebra leads to an infinite-dimensional lift of
the minimal Maxwell superalgebra. Section 5 is devoted to discussion and
possible future developments.

##  2 Minimal AdS-Lorentz Supergravity theory and flat limit

In this section, we briefly review the so-called minimal AdS-Lorentz
supergravity theory first introduced in [30], and whose vanishing cosmological
constant limit leads to the minimal supersymmetric extension of the Maxwell
gravity [50, 30]. First, we extend the AdS-Lorentz algebra generated by
$\\{J_{a},P_{a},Z_{a}\\}$ with the fermionic generators
$\\{Q_{\alpha},\Sigma_{\alpha}\\}$, obtaining the following superalgebra 111
While there is a supersymmetric extension of the AdS-Lorentz with a single
fermionic generator [51], in the flat limit, the four-momentum generators
$P_{a}$ are no longer expressed as bilinear expressions of the fermionic
generators $Q$. As a consequence, the action constructed from the contracted
superalgebra is an exotic CS supersymmetric action.

$\displaystyle\left[J_{a},J_{b}\right]$ $\displaystyle=$
$\displaystyle\epsilon_{abc}J^{c}\,,\qquad\ \ \
\left[J_{a},P_{b}\right]=\epsilon_{abc}P^{c}\,,$
$\displaystyle\left[P_{a},P_{b}\right]$ $\displaystyle=$
$\displaystyle\epsilon_{abc}Z^{c}\,,\qquad\ \ \
\left[J_{a},Z_{b}\right]=\epsilon_{abc}Z^{c}\,,$
$\displaystyle\left[Z_{a},Z_{b}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{\ell^{2}}\epsilon_{abc}Z^{c}\,,\qquad\left[P_{a},Z_{b}\right]=\frac{1}{\ell^{2}}\epsilon_{abc}P^{c}\,,$
(2.1) $\displaystyle\left[J_{a},Q_{\alpha}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\left(\Gamma_{a}\right)_{\text{
}\alpha}^{\beta}Q_{\beta}\,,\qquad\
\,\left[J_{a},\Sigma_{\alpha}\right]=\frac{1}{2}\,\left(\Gamma_{a}\right)_{\text{
}\alpha}^{\beta}\Sigma_{\beta}\,,$
$\displaystyle\left[P_{a},Q_{\alpha}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\,\left(\Gamma_{a}\right)_{\text{
}\alpha}^{\beta}\Sigma_{\beta}\,,\qquad\
\,\left[P_{a},\Sigma_{\alpha}\right]=\frac{1}{2\ell^{2}}\,\left(\Gamma_{a}\right)_{\text{
}\alpha}^{\beta}Q_{\beta}\,,$ $\displaystyle\left[Z_{a},Q_{\alpha}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2\ell^{2}}\,\left(\Gamma_{a}\right)_{\text{
}\alpha}^{\beta}Q_{\beta}\,,\qquad\left[Z_{a},\Sigma_{\alpha}\right]=\frac{1}{2\ell^{2}}\,\left(\Gamma_{a}\right)_{\text{
}\alpha}^{\beta}\Sigma_{\beta}\,,$
$\displaystyle\left\\{Q_{\alpha},Q_{\beta}\right\\}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\,\left(C\Gamma^{a}\right)_{\alpha\beta}P_{a}\,,\text{
\ \
}\left\\{Q_{\alpha},\Sigma_{\beta}\right\\}=-\frac{1}{2}\,\left(C\Gamma^{a}\right)_{\alpha\beta}Z_{a}\,,\,$
$\displaystyle\left\\{\Sigma_{\alpha},\Sigma_{\beta}\right\\}$
$\displaystyle=$
$\displaystyle-\frac{1}{2\ell^{2}}\,\,\left(C\Gamma^{a}\right)_{\alpha\beta}P_{a}\,,$
(2.2)

where $a,b,\dots=0,1,2$ are Lorentz indices raised and lowered with the (off-
diagonal) Minkowski metric $\eta_{ab}$ and $\epsilon_{abc}$ is the Levi-Civita
tensor. The gamma matrices in three dimensions are denoted by $\Gamma_{a}$ and
$C$ is the charge conjugation matrix, satisfying $C^{T}=-C$ and
$C\Gamma^{a}=(C\Gamma^{a})^{T}$. This supersymmetric extension of the AdS-
Lorentz symmetry, as well as its flat limit $\ell\rightarrow\infty$, is
characterized by the introduction of an additional Majorana spinor charge
$\Sigma$. Let us note that the presence of a second abelian spinorial
generator has already been studied in the context of $D=11$ supergravity [52]
and superstring theory [53].

We can see that the aforesaid limit $\ell\rightarrow\infty$ leads to the
minimal Maxwell superalgebra, denoted in [30] as $s\mathcal{M}$, and whose
underlying symmetry corresponds to a supersymmetric extension of the Maxwell
algebra [19, 20, 22], which in turn corresponds to a deformation and extension
of the Poincaré symmetry. As we will see later, the flat limit is also well-
defined at the level of the asymptotic symmetries of the supergravity theories
based on the Maxwell and AdS-Lorentz superalgebras.

There exist an alternative basis in which the minimal AdS-Lorentz superalgebra
can be written as the direct sum of three copies of the Lorentz algebra, two
of which are augmented by supersymmetry [49] . This structure is revealed
after the following redefinitions,

$\displaystyle Z_{a}$
$\displaystyle=\frac{1}{\ell^{2}}\left(J_{a}^{+}+J_{a}^{-}\right)\,,$
$\displaystyle P_{a}$
$\displaystyle=\frac{1}{\ell}\left(J_{a}^{+}-J_{a}^{-}\right)\,,$
$\displaystyle J_{a}$ $\displaystyle=\hat{J}_{a}+J_{a}^{+}+J_{a}^{-}\,,$ (2.3)

along with

$\displaystyle Q_{\alpha}$
$\displaystyle=\frac{1}{\ell^{1/2}}\left(\tilde{Q}_{\alpha}^{+}-i\tilde{Q}_{\alpha}^{-}\right)\,,$
$\displaystyle\Sigma_{\alpha}$
$\displaystyle=\frac{1}{\ell^{3/2}}\left(\tilde{Q}_{\alpha}^{+}+i\tilde{Q}_{\alpha}^{-}\right)\,.$
(2.4)

Then, the minimal AdS-Lorentz superalgebra can be written as the direct sum
$\mathfrak{osp}(2,1)\oplus\mathfrak{osp}(2,1)\oplus\mathfrak{sp}(2)$, where
the $\mathfrak{osp}(2,1)$ algebras are spanned by
$(J_{a}^{+},\tilde{Q}_{\alpha}^{+})$, $(J_{a}^{-},\tilde{Q}_{\alpha}^{-})$ and
the Lorentz algebra is spanned by $\hat{J}_{a}$.

As it was shown in [30], the flat limit can also be performed at the level of
the CS action, supersymmetry transformations and field equations. A CS action
for the minimal AdS-Lorentz superalgebra can be written by considering the
connection one-form

$A=\omega^{a}J_{a}+e^{a}P_{a}+\sigma^{a}Z_{a}+\bar{\psi}Q+\bar{\xi}\Sigma\,,$
(2.5)

where $\omega^{a}$ is the spin connection one-form, $e^{a}$ corresponds to the
dreibein one-form, $\sigma^{a}$ is the gauge field one-form along $Z_{a}$,
while $\psi$ and $\xi$ are fermionic fields. The corresponding curvature two-
form is given by

$F=R^{a}J_{a}+\hat{\mathcal{T}}^{a}P_{a}+\hat{\mathcal{F}}^{a}Z_{a}+\nabla\bar{\psi}Q+\nabla\bar{\xi}\Sigma\,,$
(2.6)

where

$\displaystyle R^{a}$ $\displaystyle=$ $\displaystyle
d\omega^{a}+\frac{1}{2}\epsilon^{abc}\omega_{b}\omega_{c}\,,$
$\displaystyle\hat{\mathcal{T}}^{a}$ $\displaystyle=$ $\displaystyle
de^{a}+\epsilon^{abc}\omega_{b}e_{c}+\frac{1}{\ell^{2}}\epsilon^{abc}\sigma_{b}e_{c}+\frac{1}{4}i\bar{\psi}\Gamma^{a}\psi+\frac{1}{4\ell^{2}}i\bar{\xi}\Gamma^{a}\xi\,,$
$\displaystyle\hat{\mathcal{F}}^{a}$ $\displaystyle=$ $\displaystyle
d\sigma^{a}+\epsilon^{abc}\omega_{b}\sigma_{c}+\frac{1}{2\ell^{2}}\epsilon^{abc}\sigma_{b}\sigma_{c}+\frac{1}{2}\epsilon^{abc}e_{b}e_{c}+\frac{1}{2}i\bar{\psi}\Gamma^{a}\xi\,,$
(2.7)

and the fermionic components are given by

$\displaystyle\nabla\psi$ $\displaystyle=$ $\displaystyle
d\psi+\frac{1}{2}\,\omega^{a}\Gamma_{a}\psi+\frac{1}{2\ell^{2}}\,\sigma^{a}\Gamma_{a}\psi+\frac{1}{2\ell^{2}}\,e^{a}\Gamma_{a}\xi\,,$
$\displaystyle\nabla\xi$ $\displaystyle=$ $\displaystyle
d\xi+\frac{1}{2}\,\omega^{a}\Gamma_{a}\xi\,+\frac{1}{2\ell^{2}}\,\sigma^{a}\Gamma_{a}\xi+\frac{1}{2}\,e^{a}\Gamma_{a}\psi\,.$
(2.8)

Naturally, the curvatures reduce to the minimal Maxwell ones [30] when the
limit $\ell\rightarrow\infty$ is performed.

The non-vanishing components of the invariant tensor for the minimal AdS-
Lorentz superalgebra are given by those for the Maxwell superalgebra, namely

$\displaystyle\left\langle J_{a}J_{b}\right\rangle$
$\displaystyle=\alpha_{0}\eta_{ab}\,,$ $\displaystyle\left\langle
J_{a}P_{b}\right\rangle$ $\displaystyle=\alpha_{1}\eta_{ab}\,,$
$\displaystyle\left\langle P_{a}P_{b}\right\rangle$
$\displaystyle=\alpha_{2}\eta_{ab}\,,$ $\displaystyle\left\langle
J_{a}Z_{b}\right\rangle$ $\displaystyle=\alpha_{2}\eta_{ab}\,,$
$\displaystyle\left\langle Q_{\alpha}Q_{\beta}\right\rangle$
$\displaystyle=\alpha_{1}C_{\alpha\beta}\,,$ $\displaystyle\left\langle
Q_{\alpha}\Sigma_{\beta}\right\rangle$
$\displaystyle=\alpha_{2}C_{\alpha\beta}\,,$ (2.9)

along with

$\displaystyle\left\langle P_{a}Z_{b}\right\rangle$
$\displaystyle=\frac{\alpha_{1}}{\ell^{2}}\eta_{ab}\,,$
$\displaystyle\left\langle Z_{a}Z_{b}\right\rangle$
$\displaystyle=\frac{\alpha_{2}}{\ell^{2}}\eta_{ab}\,,$ (2.10)
$\displaystyle\left\langle\Sigma_{\alpha}\Sigma_{\beta}\right\rangle$
$\displaystyle=\frac{\alpha_{1}}{\ell^{2}}C_{\alpha\beta}\,.$

We can see the flat limit $\ell\rightarrow\infty$ applied to the non-vanishing
components of the invariant tensor for the AdS-Lorentz superalgebra leads us
to the non-degenerate invariant tensor of the minimal Maxwell superalgebra.
Then, using the connection one-form (2.5) and the invariant tensors (2.9) and
(2.10) in the CS action

$I[A]=\frac{k}{4\pi}\int_{\mathcal{M}}\left\langle
AdA+\frac{2}{3}A^{3}\right\rangle\,,$ (2.11)

defined on a three-dimensional manifold $\mathcal{M}$, and where
$k=\frac{1}{4G}$ is the level of the theory related to the gravitational
constant $G$, we get [30]

$\displaystyle I_{sAdS-\mathcal{L}}\,=\,$
$\displaystyle\frac{k}{4\pi}\int\alpha_{0}\left(\,\omega^{a}d\omega_{a}+\frac{1}{3}\,\epsilon_{abc}\omega^{a}\omega^{b}\omega^{c}\right)$
$\displaystyle+\alpha_{1}\left(2e^{a}R_{a}+\frac{2}{\ell^{2}}\,e^{a}F_{a}+\frac{1}{3\ell^{2}}\,\epsilon_{abc}e^{a}e^{b}e^{c}-i\bar{\psi}\nabla\psi-\frac{1}{\ell^{2}}\,i\bar{\xi}\nabla\xi\right)$
$\displaystyle+\alpha_{2}\left(2R^{a}\sigma_{a}+\frac{2}{\ell^{2}}\,F^{a}\sigma_{a}+e^{a}T_{a}+\frac{1}{\ell^{2}}\,\epsilon_{abc}e^{a}\sigma^{b}e^{c}-i\bar{\psi}\nabla\xi-i\bar{\xi}\nabla\psi\right)\,,$
(2.12)

where $T^{a}=de^{a}+\epsilon^{abc}\omega_{b}e_{c}$ is the usual torsion two-
form and

$F^{a}=d\sigma^{a}+\epsilon^{abc}\omega_{b}\sigma_{c}+\frac{1}{2\ell^{2}}\epsilon^{abc}\sigma_{b}\sigma_{c}\,.$
(2.13)

From the previous action we can see that the term proportional to $\alpha_{0}$
contains the gravitational CS Lagrangian [54, 55]. The term along $\alpha_{1}$
contains the Einstein-Hilbert term with negative cosmological constant and the
Rarita-Schwinger term, while $\alpha_{2}$ yields a term involving the field
$\sigma^{a}$, a torsional term and supersymmetric terms involving the second
spinorial gauge field. Naturally, in absence of supersymmetry, the CS action
corresponds to the AdS-Lorentz CS gravity. As we have mentioned before, the
minimal AdS-Lorentz supergravity action (2.12) leads to the minimal Maxwell
supergravity in the vanishing cosmological constant limit
$\ell\rightarrow\infty$. Although the field content in both theories is the
same, the CS action constructed from the AdS-Lorentz superalgebra is richer in
the sense that it allows to introduce a negative cosmological constant. The
field equations reduce to the vanishing of the curvature two-forms (2.7) and
(2.8) provided $\alpha_{2}\neq 0$. This condition on the $\alpha_{2}$ constant
is required, as in the bosonic case, in order to have a non-degenerate
invariant tensor. The CS action (2.12) is invariant by construction under the
gauge transformation $\delta A=d\Lambda+\left[A,\Lambda\right]$. In
particular, the supersymmetry transformation laws read

$\displaystyle\delta\omega^{a}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle\delta e^{a}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,i\bar{\epsilon}\Gamma^{a}\psi\,+\frac{1}{2\ell^{2}}i\,\bar{\varrho}\Gamma^{a}\xi,$
$\displaystyle\delta\sigma^{a}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,i\bar{\epsilon}\Gamma^{a}\xi+\frac{1}{2}\,i\bar{\varrho}\Gamma^{a}\psi\,,$
(2.14) $\displaystyle\delta\psi$ $\displaystyle=$ $\displaystyle
d\epsilon+\frac{1}{2}\,\omega^{a}\Gamma_{a}\epsilon+\frac{1}{2\ell^{2}}\sigma^{a}\Gamma_{a}\epsilon+\frac{1}{2\ell^{2}}e^{a}\Gamma_{a}\varrho\,,$
$\displaystyle\delta\xi$ $\displaystyle=$ $\displaystyle
d\varrho+\frac{1}{2}\,\omega^{a}\Gamma_{a}\varrho\,+\frac{1}{2}e^{a}\Gamma_{a}\epsilon+\frac{1}{2\ell^{2}}\sigma^{a}\Gamma_{a}\varrho\,,$

with gauge parameter
$\Lambda=\rho^{a}J_{a}+\varepsilon^{a}P_{a}+\gamma^{a}Z_{a}+\bar{\epsilon}Q+\bar{\varrho}\Sigma$.
Note that the supersymmetry transformations (2.14) reduce to the Maxwell ones
in the limit $\ell\rightarrow\infty$.

In what follows, we extend the results obtained in [29, 48] to the minimal
AdS-Lorentz supergravity theory and its flat limit, leading to the minimal
Maxwell supergravity. As we will see, the corresponding asymptotic symmetry
algebras of the supergravity theories will correspond to a supersymmetric
extension of the semi-simple enlargement of the $\mathfrak{bms}_{3}$ algebra
in the former case, and to a deformed super-$\mathfrak{bms}_{3}$ in the latter
case. Both charge algebras have three central charges defined in terms of the
coupling constants appearing in the CS action. We will see that both
asymptotic symmetry algebras are related through a well-defined flat limit.

##  3 Asymptotic symmetries

In this section, we compute the asymptotic symmetry algebra for the previously
introduced minimal AdS-Lorentz CS supergravity. To start with, we provide the
suitable fall-off conditions for the gauge fields at infinity and the gauge
transformations preserving our boundary conditions. Then, the charge algebra
is found using the Regge-Teitelboim method [56].

###  3.1 Boundary conditions

We propose the following behaviour of the gauge fields at the boundary

$A=h^{-1}dh+h^{-1}ah\,,$ (3.1)

where the radial dependence is entirely captured by the group element $h=$
e${}^{-rP_{0}}$. The auxiliary gauge field has the form
$a=a_{\phi}d{\phi}+a_{u}du$, with the angular component given by

$\displaystyle a_{\phi}$ $\displaystyle=$ $\displaystyle
J_{1}+\frac{1}{2}{\cal N}P_{0}+\frac{1}{2}\left({\cal M}-\frac{{\cal
F}}{\ell^{2}}\right)\,J_{0}+\frac{1}{2}\mathcal{F}\,Z_{0}+\frac{\psi}{2^{1/4}}\,Q_{+}+\frac{1}{2^{1/4}}\xi\,\Sigma_{+}\,,$
(3.2)

where the functions ${\cal M}$, ${\cal N}$ and $\cal{F}$, and the Grassmann-
valued spinor components $\psi$ and $\xi$ are assume to depend on all boundary
coordinates $x^{i}=(u,\phi)$.

The asymptotic symmetries correspond to the set of gauge transformations
$\delta A=d\lambda+[A,\lambda]$ that preserve the asymptotic conditions (3.1)
invariant. Thus, the angular component $a_{\phi}$ is left invariant for the
Lie-algebra-valued parameter of the form

$\displaystyle\lambda[Y,f,h,\mathcal{E},\mathcal{S}]=$
$\displaystyle\,\left[\frac{1}{2}\left(\mathcal{M}Y+\mathcal{N}Y+\mathcal{N}\frac{h}{\ell^{2}}\right)+\frac{{i}}{2}\mathcal{E}\psi+\frac{{i}}{2\ell}\mathcal{S}\xi-f^{\prime\prime}\right]P_{0}+fP_{1}-f^{\prime}P_{2}$
$\displaystyle+\left[\frac{1}{2}\left(\mathcal{M}-\frac{\mathcal{F}}{\ell^{2}}\right)\,Y-Y^{\prime\prime}\right]\,J_{0}+Y\,J_{1}-Y^{\prime}\,J_{2}$
$\displaystyle+\left[\frac{1}{2}\left(\mathcal{M}h+\mathcal{N}f+\mathcal{F}Y\right)+\frac{{i}}{2}\mathcal{S}\psi+\frac{{i}}{2}\mathcal{E}\xi-h^{\prime\prime}\right]Z_{0}+hZ_{1}-h^{\prime}Z_{2}$
$\displaystyle+2^{-1/4}\left[\left(Y+\frac{h}{\ell^{2}}\right)\psi+\frac{f}{\ell^{2}}\xi-2\mathcal{E}^{\prime}\right]Q_{+}+2^{1/4}\mathcal{E}\,Q_{-}$
$\displaystyle+2^{-1/4}\left[\left(Y+\frac{h}{\ell^{2}}\right)\xi+f\psi-2\mathcal{S}^{\prime}\right]\Sigma_{+}+2^{1/4}\mathcal{S}\Sigma_{-}\,,$
(3.3)

provided the dynamical fields transform in the following way

$\displaystyle\delta{\cal M}$ $\displaystyle=$ $\displaystyle{\cal
M}^{\prime}Y+2{\cal
M}Y^{\prime}-2Y{}^{\prime\prime\prime}+\frac{1}{\ell^{2}}\left({\cal
M}^{\prime}h+2{\cal M}h^{\prime}+{\cal N}^{\prime}f+2{\cal
N}f^{\prime}+i\mathcal{E}\Xi^{\prime}+3i\mathcal{E}^{\prime}\Xi+{i}\mathcal{S}\Psi^{\prime}+3i\mathcal{S}^{\prime}\Psi-2h{}^{\prime\prime\prime}\right)\,,$
$\displaystyle\delta{\cal N}$ $\displaystyle=$ $\displaystyle{\cal
N}^{\prime}Y+2{\cal N}Y^{\prime}+{\cal M}^{\prime}f+2{\cal
M}f^{\prime}+{i}\mathcal{E}\Psi^{\prime}+3{i}\mathcal{E}^{\prime}\Psi-2f{}^{\prime\prime\prime}+\frac{1}{\ell^{2}}\left({\cal
N}^{\prime}h+2{\cal
N}h^{\prime}+i\mathcal{S}\Xi^{\prime}+3i\mathcal{S}^{\prime}\Xi\right)\,,$
$\displaystyle\delta{\cal F}$ $\displaystyle=$ $\displaystyle{\cal
F}^{\prime}Y+2{\cal F}Y^{\prime}+{\cal N}^{\prime}f+2{\cal N}f^{\prime}+{\cal
M}^{\prime}h+2{\cal
M}h^{\prime}+{i}\mathcal{E}\Xi^{\prime}+3{i}\mathcal{E}^{\prime}\Xi+{i}\mathcal{S}\Psi^{\prime}+3{i}\mathcal{S}^{\prime}\Psi-2h{}^{\prime\prime\prime}\,,$
$\displaystyle\delta{\Psi}$ $\displaystyle=$
$\displaystyle\Psi^{\prime}Y+\frac{3}{2}\Psi
Y^{\prime}+\frac{1}{2}\mathcal{M}\mathcal{E}-2\mathcal{E}^{\prime\prime}+\frac{1}{\ell^{2}}\left(\Psi^{\prime}h+\frac{3}{2}\Psi
h^{\prime}+\Xi^{\prime}f+\frac{3}{2}\Xi
f^{\prime}+\frac{1}{2}\mathcal{N}\mathcal{S}\right)\,,$
$\displaystyle\delta{\Xi}$ $\displaystyle=$
$\displaystyle\Xi^{\prime}Y+\frac{3}{2}\Xi
Y^{\prime}+\Psi^{\prime}f+\frac{3}{2}\Psi
f^{\prime}+\frac{1}{2}\mathcal{M}\mathcal{S}+\frac{1}{2}\mathcal{N}\mathcal{E}-2\mathcal{S}^{\prime\prime}+\frac{1}{\ell^{2}}\left(\Xi^{\prime}h+\frac{3}{2}\Xi
h^{\prime}\right)\,.$ (3.4)

In order to accommodate the relevant bosonic solutions of the theory, it is
important to incorporate the Lagrange multipliers for every dynamical field in
the asymptotic form of the gauge field. Therefore, the most general form of
the temporal component is given by

$a_{u}=\lambda[\mu_{\mathcal{N}},\mu_{\mathcal{M}},\mu_{\mathcal{F}},\mu_{\Psi},\mu_{\Xi}]\,,$
(3.5)

where the lagrange multipliers $\mu_{\mathcal{N}}$, $\mu_{\mathcal{M}}$,
$\mu_{\mathcal{F}}$, $\mu_{\Psi}$, $\mu_{\Xi}$ are arbitrary functions of
$(u,\phi)$ and also assumed to be fixed at the boundary [57, 58]. The
preservation of the temporal component (3.5) requires that the field equations
hold asymptotically as well as they provide conditions for the temporal
dependence of the gauge parameter. The explicit form of these conditions are
given by

$\displaystyle\dot{\cal M}$ $\displaystyle=$ $\displaystyle{\cal
M}^{\prime}\mu_{\mathcal{N}}+2{\cal
M}\mu_{\mathcal{N}}^{\prime}-2\mu_{\mathcal{N}}^{\prime\prime\prime}+\frac{1}{\ell^{2}}\left({\cal
M}^{\prime}\mu_{\mathcal{F}}+2{\cal M}\mu_{\mathcal{F}}^{\prime}+{\cal
N}^{\prime}\mu_{\mathcal{M}}+2{\cal
N}\mu_{\mathcal{M}}^{\prime}+i\mu_{\Psi}\Xi^{\prime}+3i\mu_{\Psi}^{\prime}\Xi\right.\,$
$\displaystyle\left.+{i}\mu_{\Xi}\Psi^{\prime}+3i\mu_{\Xi}^{\prime}\Psi-2\mu_{\mathcal{F}}{}^{\prime\prime\prime}\right)\,,$
$\displaystyle\dot{\cal N}$ $\displaystyle=$ $\displaystyle{\cal
N}^{\prime}\mu_{\mathcal{N}}+2{\cal N}\mu_{\mathcal{N}}^{\prime}+{\cal
M}^{\prime}\mu_{\mathcal{M}}+2{\cal
M}\mu_{\mathcal{M}}^{\prime}+{i}\mu_{\Psi}\Psi^{\prime}+3{i}\mu_{\Psi}^{\prime}\Psi-2\mu_{\mathcal{M}}{}^{\prime\prime\prime}+\frac{1}{\ell^{2}}\left({\cal
N}^{\prime}\mu_{\mathcal{F}}+2{\cal{N}}\mu_{\mathcal{F}}^{\prime}\right.$
$\displaystyle\left.+i\mu_{\Xi}\Xi^{\prime}+3i\mu_{\Xi}^{\prime}\Xi\right)\,,$
$\displaystyle\dot{\cal F}$ $\displaystyle=$ $\displaystyle{\cal
F}^{\prime}\mu_{\mathcal{N}}+2{\cal F}\mu_{\mathcal{N}}^{\prime}+{\cal
N}^{\prime}\mu_{\mathcal{M}}+2{\cal N}\mu_{\mathcal{M}}^{\prime}+{\cal
M}^{\prime}\mu_{\mathcal{F}}+2{\cal
M}\mu_{\mathcal{F}}^{\prime}+{i}\mu_{\psi}\Xi^{\prime}+3{i}\mu_{\psi}^{\prime}\Xi+{i}\mu_{\Xi}\Psi^{\prime}+3{i}\mu_{\Xi}^{\prime}\Psi-2\mu_{\mathcal{F}}^{\prime\prime\prime}\,,$
$\displaystyle\dot{\Psi}$ $\displaystyle=$
$\displaystyle\Psi^{\prime}\mu_{{\cal N}}+\frac{3}{2}\Psi\mu_{{\cal
N}}^{\prime}+\frac{1}{2}\mathcal{M}\mu_{\Psi}-2\mu_{\Psi}^{\prime\prime}+\frac{1}{\ell^{2}}\left(\Psi^{\prime}\mu_{\mathcal{F}}+\frac{3}{2}\Psi\mu_{\mathcal{F}}^{\prime}+\Xi^{\prime}\mu_{\mathcal{M}}+\frac{3}{2}\Xi\mu_{\mathcal{M}}^{\prime}+\frac{1}{2}\mathcal{N}\mu_{\Xi}\right)\,,$
$\displaystyle\dot{\Xi}$ $\displaystyle=$
$\displaystyle\Xi^{\prime}\mu_{\mathcal{N}}+\frac{3}{2}\Xi\mu_{\mathcal{N}}^{\prime}+\Psi^{\prime}\mu_{\mathcal{M}}+\frac{3}{2}\Psi\mu_{\mathcal{M}}^{\prime}+\frac{1}{2}\mathcal{M}\mu_{\Xi}+\frac{1}{2}\mathcal{N}\mu_{\psi}-2\mu_{\Xi}^{\prime\prime}+\frac{1}{\ell^{2}}\left(\Xi^{\prime}\mu_{\mathcal{F}}+\frac{3}{2}\Xi\mu_{\mathcal{F}}^{\prime}\right)\,.$
(3.6)

for the asymptotic field, and for the parameters we have

$\displaystyle\dot{Y}$
$\displaystyle=Y^{\prime}\mu_{\mathcal{J}}-\mu_{\mathcal{J}}^{\prime}Y$
$\displaystyle\dot{f}$
$\displaystyle=Y^{\prime}\mu_{\mathcal{M}}-\mu_{\mathcal{M}}^{\prime}Y+f^{\prime}\mu_{\mathcal{J}}-\mu_{\mathcal{J}}^{\prime}f-i\mu_{\Psi}\mathcal{E}+\frac{1}{\ell^{2}}\left(f^{\prime}\mu_{Z}-\mu_{Z}^{\prime}f+h^{\prime}\mu_{M}-\mu_{M}^{\prime}h-\mu_{\Xi}\mathcal{S}\right)$
$\displaystyle\dot{h}$
$\displaystyle=Y^{\prime}\mu_{\mathcal{Z}}-\mu_{\mathcal{Z}}^{\prime}Y+f^{\prime}\mu_{\mathcal{M}}-\mu_{\mathcal{M}}^{\prime}f+h^{\prime}\mu_{\mathcal{J}}-\mu_{\mathcal{J}}^{\prime}h-\mu_{\Xi}\mathcal{E}-\mu_{\psi}\mathcal{S}+\frac{1}{\ell^{2}}\left(h^{\prime}\mu_{Z}-\mu_{Z}^{\prime}h\right)$
$\displaystyle\dot{\mathcal{E}}$
$\displaystyle=\mu_{\mathcal{J}}\mathcal{E}^{\prime}-\frac{1}{2}\mathcal{E}\mu_{\mathcal{J}}^{\prime}-\mu_{\Psi}^{\prime}Y+\frac{1}{2}\mu_{\Psi}Y^{\prime}+\frac{1}{\ell^{2}}\left(\mu_{M}\mathcal{S}^{\prime}-\frac{1}{2}\mathcal{S}\mu_{M}^{\prime}+\mu_{Z}\mathcal{E}^{\prime}-\frac{1}{2}\mathcal{E}\mu_{Z}^{\prime}-f\mu_{\Sigma}^{\prime}+\frac{1}{2}\mu_{\Sigma}f^{\prime}-h\mu_{\psi}^{\prime}+\frac{1}{2}\mu_{\psi}h^{\prime}\right)$
$\displaystyle\dot{\mathcal{S}}$
$\displaystyle=\mu_{\mathcal{J}}\mathcal{S}^{\prime}-\frac{1}{2}\mathcal{S}\mu_{\mathcal{J}}^{\prime}+\mu_{\mathcal{M}}\mathcal{E}^{\prime}-\frac{1}{2}\mathcal{E}\mu_{\mathcal{M}}^{\prime}-\mu_{\Xi}^{\prime}Y+\frac{1}{2}\mu_{\Xi}Y^{\prime}-\mu_{\psi}^{\prime}T+\frac{1}{2}\mu_{\psi}T^{\prime}+\frac{1}{\ell^{2}}\left(\mu_{Z}\mathcal{S}^{\prime}-\frac{1}{2}\mathcal{S}\mu_{Z}^{\prime}+\mu_{\Xi}h^{\prime}-\frac{1}{2}h\mu_{\Xi}^{\prime}\right)\,.$
(3.7)

The above structure contains the information of the asymptotic structure of
the minimal AdS-Lorentz CS supergravity and their corresponding algebra.
Indeed, the charge algebra of the theory can be computed following the Regge-
Teitelboim approach [56]. In what follows we will consider this construction.

###  3.2 Charge algebra

Let us now compute the charge algebra of the supergravity theory previously
introduced. As discussed in [59], the algebra is spanned by the conserved
charges $Q[\Lambda]$. Furthermore, the charge algebra in representation of
Poisson brackets can be obtained using the Regge-Teitelboim method directly
from the transformation law

$\delta_{\Lambda_{2}}Q[\Lambda_{1}]=\left\\{Q[\Lambda_{1}],Q[\Lambda_{2}]\right\\}.$
(3.8)

On the other hand, the variation of the charge in CS theory is given by

$\delta
Q[\Lambda]=\frac{k}{2\pi}\int\limits_{\partial\Sigma}\left\langle\Lambda\delta
A\right\rangle\,.$ (3.9)

After applying the gauge transformation (3.1) which introduces the asymptotic
field (3.2) we get

$\delta Q[\lambda]=\frac{k}{2\pi}\int d\phi\left\langle\lambda\delta
a_{\phi}\right\rangle\,.$ (3.10)

Considering the invariant tensors (2.9) and (2.10), and the gauge field $a$
defined in (3.2) in the previous expression, we get

$\displaystyle\delta Q[Y,f,h,\mathcal{E},\mathcal{S}]=\frac{k}{4\pi}\int$
$\displaystyle d\phi\left[Y\left(\alpha_{0}{\delta\cal
M}+\left(\alpha_{2}-\frac{\alpha_{0}}{\ell^{2}}\right)\delta{\cal
Z}+\alpha_{1}\delta{\cal N}\right)+f\left(\alpha_{2}{\delta\cal
N}+\alpha_{1}\delta{\cal M}\right)\right.$
$\displaystyle\left.+h\left(\alpha_{2}{\delta\cal
M}+\frac{\alpha_{1}\delta\mathcal{N}}{\ell^{2}}\right)-2{i}\mathcal{E}\left(\alpha_{1}\delta\Psi+\alpha_{2}\delta\Xi\right)-2{i}\mathcal{S}\left(\alpha_{2}\delta\Psi+\frac{\alpha_{1}}{\ell^{2}}\delta\Xi\right)\right]\,.$
(3.11)

We assume that the functions $Y$, $f$, $h$, $\mathcal{E}$ and $\mathcal{S}$ do
not depend on the fields, in which case it is trivial to integrate the
variation out, finding

$\displaystyle Q[Y,f,h,\mathcal{E},\mathcal{S}]=\frac{k}{4\pi}\int$
$\displaystyle d\phi\left[Y\left(\alpha_{0}{\cal
M}+\left(\alpha_{2}-\frac{\alpha_{0}}{\ell^{2}}\right){\cal Z}+\alpha_{1}{\cal
N}\right)+f\left(\alpha_{2}{\cal N}+\alpha_{1}{\cal M}\right)\right.$
$\displaystyle\left.+h\left(\alpha_{2}{\cal
M}+\frac{\alpha_{1}\mathcal{N}}{\ell^{2}}\right)-2{i}\mathcal{E}\left(\alpha_{1}\Psi+\alpha_{2}\Xi\right)-2{i}\mathcal{S}\left(\alpha_{2}\Psi+\frac{\alpha_{1}}{\ell^{2}}\Xi\right)\right]\,.$
(3.12)

Now we define the asymptotic charges which correspond to the independent terms
in (3.12),

$\displaystyle j[Y]$ $\displaystyle=$ $\displaystyle\frac{k}{4\pi}\int
d\phi\,Y\left(\alpha_{0}{\cal
M}+\left(\alpha_{2}-\frac{\alpha_{0}}{\ell^{2}}\right){\cal Z}+\alpha_{1}{\cal
N}\right)\,,$ $\displaystyle p[f]$ $\displaystyle=$
$\displaystyle\frac{k}{4\pi}\int d\phi\,f\left(\alpha_{2}{\cal
N}+\alpha_{1}{\cal M}\right)\,,$ $\displaystyle z[h]$ $\displaystyle=$
$\displaystyle\frac{k}{4\pi}\int d\phi\,h\left(\alpha_{2}{\cal
M}+\frac{\alpha_{1}\mathcal{N}}{\ell^{2}}\right)\,,$ $\displaystyle
g[\mathcal{E}]$ $\displaystyle=$ $\displaystyle\frac{k}{4\pi}\int
d\phi{i}\,\mathcal{E}\left(\alpha_{1}\Psi+\alpha_{2}\Xi\right)\,,$
$\displaystyle h[S]$ $\displaystyle=$ $\displaystyle\frac{k}{4\pi}\int
d\phi{i}\,\mathcal{S}\left(\alpha_{2}\Psi+\frac{\alpha_{1}}{\ell^{2}}\Xi\right)\,.$
(3.13)

Then, the Poisson brackets of these independent charges can be evaluated using
(3.8) and (3.4). Expanding in Fourier modes,

${\cal J}_{m}=j[e^{im\text{$\phi$}}]\text{\,,\qquad}{\cal
P}_{m}=p[e^{im\text{$\phi$}}]\text{\,,\qquad}{\cal
Z}_{m}=z[e^{im\text{$\phi$}}]\text{\,,\qquad}{\cal
G}_{m}=g[e^{im\text{$\phi$}}]\text{\,,\qquad}{\cal
H}_{m}=h[e^{im\text{$\phi$}}]\,,$ (3.14)

the Poisson brackets read

$\displaystyle i\left\\{\mathcal{J}_{m},\mathcal{J}_{n}\right\\}$
$\displaystyle=\left(m-n\right)\mathcal{J}_{m+n}+\dfrac{c_{1}}{12}\,m^{3}\delta_{m+n,0}\,,$
$\displaystyle i\left\\{\mathcal{J}_{m},\mathcal{P}_{n}\right\\}$
$\displaystyle=\left(m-n\right)\mathcal{P}_{m+n}+\dfrac{c_{2}}{12}\,m^{3}\delta_{m+n,0}\,,$
$\displaystyle i\left\\{\mathcal{P}_{m},\mathcal{P}_{n}\right\\}$
$\displaystyle=\left(m-n\right)\mathcal{Z}_{m+n}+\dfrac{c_{3}}{12}\,m^{3}\delta_{m+n,0}\,,$
$\displaystyle i\left\\{\mathcal{J}_{m},\mathcal{Z}_{n}\right\\}$
$\displaystyle=\left(m-n\right)\mathcal{Z}_{m+n}+\dfrac{c_{3}}{12}\,m^{3}\delta_{m+n,0}\,,$
$\displaystyle i\left\\{\mathcal{P}_{m},\mathcal{Z}_{n}\right\\}$
$\displaystyle=\frac{1}{\ell^{2}}\left(m-n\right)\mathcal{P}_{m+n}+\dfrac{c_{2}}{12\ell^{2}}\,m^{3}\delta_{m+n,0}\,,$
$\displaystyle i\left\\{\mathcal{Z}_{m},\mathcal{Z}_{n}\right\\}$
$\displaystyle=\frac{1}{\ell^{2}}\left(m-n\right)\mathcal{Z}_{m+n}+\dfrac{c_{3}}{12\ell^{2}}\,m^{3}\delta_{m+n,0}\,,$
(3.15)

along with

$\displaystyle i\left\\{\mathcal{J}_{m},\mathcal{G}_{n}\right\\}$
$\displaystyle=\left(\frac{m}{2}-n\right)\mathcal{G}_{m+n}\,,$ $\displaystyle
i\left\\{\mathcal{P}_{m},\mathcal{G}_{n}\right\\}$
$\displaystyle=\left(\frac{m}{2}-n\right)\mathcal{H}_{m+n}\,,$ $\displaystyle
i\left\\{\mathcal{J}_{m},\mathcal{H}_{n}\right\\}$
$\displaystyle=\left(\frac{m}{2}-n\right)\mathcal{H}_{m+n}\,,$ $\displaystyle
i\left\\{\mathcal{P}_{m},\mathcal{H}_{n}\right\\}$
$\displaystyle=\dfrac{1}{\ell^{2}}\left(\frac{m}{2}-n\right)\mathcal{G}_{m+n}\,,$
$\displaystyle i\left\\{\mathcal{Z}_{m},\mathcal{G}_{n}\right\\}$
$\displaystyle=\dfrac{1}{\ell^{2}}\left(\frac{m}{2}-n\right)\mathcal{G}_{m+n}\,,$
$\displaystyle i\left\\{\mathcal{Z}_{m},\mathcal{H}_{n}\right\\}$
$\displaystyle=\dfrac{1}{\ell^{2}}\left(\frac{m}{2}-n\right)\mathcal{H}_{m+n}\,,$
$\displaystyle i\left\\{\mathcal{G}_{m},\mathcal{G}_{n}\right\\}$
$\displaystyle=\mathcal{P}_{m+n}+\dfrac{c_{2}}{6}\,m^{2}\delta_{m+n,0}\,,$
$\displaystyle i\left\\{\mathcal{G}_{m},\mathcal{H}_{n}\right\\}$
$\displaystyle=\mathcal{Z}_{m+n}+\dfrac{c_{3}}{6}\,m^{2}\delta_{m+n,0}\,,$
$\displaystyle i\left\\{\mathcal{H}_{m},\mathcal{H}_{n}\right\\}$
$\displaystyle=\frac{1}{\ell^{2}}\mathcal{P}_{m+n}+\dfrac{c_{2}}{6\ell^{2}}\,m^{2}\delta_{m+n,0}\,,$
(3.16)

where we have used the integral representation of the Kronecker delta
$\delta_{m,n}=\frac{1}{2\pi}\int d\phi\,e^{i(m-n)\text{$\phi$}}$. The previous
algebra defines a supersymmetric extension of the semi-simple enlargement of
the $\mathfrak{bms}_{3}$ algebra [48], and it was first introduced in [49] as
an expansion of the super Virasoro algebra. As it is expected, the asymptotic
symmetry algebra corresponds to an infinite-dimensional lift of the minimal
AdS-Lorentz superalgebra, with three central charges which are related to the
CS level $k$ and to the three coupling constants appearing in the supergravity
action through

$c_{i}=12k\alpha_{i-1}\,,\qquad i=1,2,3$ (3.17)

Indeed, the minimal AdS-Lorentz superalgebra is a finite subalgebra of
(3.15)-(3.16) with generators $\left\\{{\cal J_{{\rm-1}}{\rm,}J_{{\rm
0}}{\rm,{\cal J}}_{{\rm 1}}{\rm,{\cal P}_{-1},{\cal P}_{0}}{\rm,{\cal
P}_{1}}{\rm,{\cal Z}_{-1}}{\rm,}{\cal Z}_{{\rm 0}}{\rm,}{\cal
Z}}_{1}{\rm,}\cal Q_{{\rm 1/2}}{\rm,}\cal Q_{{\rm-1/2}}{\rm,}\cal G_{{\rm
1/2}}{\rm,}\cal G_{{\rm-1/2}}\right\\}$. This can be explicitly seen by
identifying the modes in (3.15)-(3.16) with the generators in (2.1)-(2.2) as
follows

$\displaystyle\mathcal{J}_{-1}$ $\displaystyle=-\sqrt{2}J_{0}\,,$
$\displaystyle\mathcal{J}_{1}$ $\displaystyle=\sqrt{2}J_{1}\,,$
$\displaystyle\mathcal{J}_{0}$ $\displaystyle=J_{2}\,,$
$\displaystyle\mathcal{P}_{-1}$ $\displaystyle=-\sqrt{2}P_{0}\,,$
$\displaystyle\mathcal{P}_{1}$ $\displaystyle=\sqrt{2}P_{1}\,,$
$\displaystyle\mathcal{P}_{0}$ $\displaystyle=P_{2}\,,$
$\displaystyle\mathcal{Z}_{-1}$ $\displaystyle=-\sqrt{2}Z_{0}\,,$
$\displaystyle\mathcal{Z}_{1}$ $\displaystyle=\sqrt{2}Z_{1}\,,$
$\displaystyle\mathcal{Z}_{0}$ $\displaystyle=Z_{2}\,,$
$\displaystyle\mathcal{G}_{-1/2}$ $\displaystyle=\sqrt{2}Q_{+}\,,$
$\displaystyle\mathcal{G}_{1/2}$ $\displaystyle=\sqrt{2}Q_{-}\,,$
$\displaystyle\mathcal{H}_{-1/2}$ $\displaystyle=\sqrt{2}\Sigma_{+}\,,$
$\displaystyle\mathcal{H}_{1/2}$ $\displaystyle=\sqrt{2}\Sigma_{-}\,.$ (3.18)

As it was mentioned in [49], the algebra (3.15)-(3.16) is isomorphic to the
direct product of three copies of the Virasoro algebra, two of which are
augmented by supersymmetry. Indeed, after the following redefinitions:

$\displaystyle\mathcal{L}_{m}^{+}$
$\displaystyle=\frac{1}{2}\left(\ell^{2}\mathcal{Z}_{m}+\ell\mathcal{P}_{m}\right)\,,$
$\displaystyle\mathcal{L}_{m}^{-}$
$\displaystyle=\frac{1}{2}\left(\ell^{2}\mathcal{Z}_{-m}-\ell\mathcal{P}_{-m}\right)\,,$
$\displaystyle\hat{\mathcal{L}}_{m}=\mathcal{J}_{-m}-\ell^{2}\mathcal{Z}_{-m}\,,$
$\displaystyle Q_{r}^{+}$
$\displaystyle=\frac{1}{2}\left(\ell^{1/2}\mathcal{G}_{r}+\ell^{3/2}\mathcal{H}_{r}\right)\,,$
$\displaystyle{Q}_{r}^{-}$
$\displaystyle=\frac{i}{2}\left(\ell^{1/2}\mathcal{G}_{r}-\ell^{3/2}\mathcal{H}_{r}\right)\,,$
$\displaystyle c^{\pm}$ $\displaystyle=\frac{1}{2}\left(\ell^{2}c_{3}\pm\ell
c_{2}\right)\,,$ $\displaystyle\hat{c}$
$\displaystyle=\left(c_{1}-\ell^{2}c_{3}\right)\,.$ (3.19)

the direct product $\mathfrak{svir}\oplus\mathfrak{svir}\oplus\mathfrak{vir}$
is revealed,

$\begin{array}[]{lcl}i\left\\{\mathcal{L}_{m}^{\pm},\mathcal{L}_{n}^{\pm}\right\\}&=&\left(m-n\right)\mathcal{L}_{m+n}^{\pm}+\dfrac{c^{\pm}}{12}\,m^{3}\delta_{m+n,0}\,,\\\\[5.0pt]
i\left\\{\hat{\mathcal{L}}_{m},\hat{\mathcal{L}}_{n}\right\\}&=&\left(m-n\right)\mathcal{\hat{L}}_{m+n}+\dfrac{\hat{c}}{12}\,m^{3}\delta_{m+n,0}\,,\\\\[5.0pt]
i\left\\{\mathcal{L}_{m}^{+},\mathcal{Q}_{r}^{+}\right\\}&=&\left(\dfrac{m}{2}-r\right)\mathcal{Q}_{m+r}^{+}\,\,,\\\\[5.0pt]
i\left\\{\mathcal{L}_{m}^{-},\mathcal{Q}_{r}^{-}\right\\}&=&\left(\dfrac{m}{2}-r\right)\mathcal{Q}_{m+r}^{-}\,\,,\\\\[5.0pt]
i\left\\{\mathcal{Q}_{r}^{+},\mathcal{Q}_{s}^{+}\right\\}&=&\mathcal{L}_{r+s}^{+}+\dfrac{c^{+}}{6}\,r^{2}\delta_{r+s,0}\,,\\\\[5.0pt]
i\left\\{\mathcal{Q}_{r}^{-},\mathcal{Q}_{s}^{-}\right\\}&=&\mathcal{L}^{-}_{r+s}+\dfrac{c^{-}}{6}\,r^{2}\delta_{r+s,0}\,.\end{array}$
(3.20)

##  4 Minimal deformed super-$\mathfrak{bms}_{3}$ algebra from a flat limit

As we have previously mentioned, Maxwell CS gravity can be obtained as a
vanishing cosmological constant limit from the so-called AdS-Lorentz gravity
in three-dimensions. Furthermore, considering the analysis of its
corresponding asymptotic symmetries, it was shown in [48] that the deformed
$\mathfrak{bms}_{3}$ algebra, being the asymptotic symmetry of the Maxwell CS
gravity, can be derived as a flat limit of a semi-simple enlargement of the
$\mathfrak{bms}_{3}$ symmetry, which in turn corresponds to the asymptotic
symmetry of the AdS-Lorentz CS gravity.

As we have discussed in the previous sections, in presence of supersymmetry,
it was also shown that the minimal Maxwell supergravity appears as flat limit
of a supersymmetric extension of the AdS-Lorentz algebra [30]. In the same
way, it is possible to show that a minimal deformed super-$\mathfrak{bms}_{3}$
algebra can be obtained from the enlarged super-$\mathfrak{bms}_{3}$ symmetry
obtained above. Indeed, the vanishing cosmological constant limit
$\ell\rightarrow\infty$ can be explicitly taken at every step in section 3,
thus leading to the following asymptotic symmetry algebra

$\displaystyle i\left\\{\mathcal{J}_{m},\mathcal{J}_{n}\right\\}$
$\displaystyle=\left(m-n\right)\mathcal{J}_{m+n}+\dfrac{c_{1}}{12}\,m^{3}\delta_{m+n,0}\,,$
$\displaystyle i\left\\{\mathcal{J}_{m},\mathcal{P}_{n}\right\\}$
$\displaystyle=\left(m-n\right)\mathcal{P}_{m+n}+\dfrac{c_{2}}{12}\,m^{3}\delta_{m+n,0}\,,$
$\displaystyle i\left\\{\mathcal{P}_{m},\mathcal{P}_{n}\right\\}$
$\displaystyle=\left(m-n\right)\mathcal{Z}_{m+n}+\dfrac{c_{3}}{12}\,m^{3}\delta_{m+n,0}\,,$
$\displaystyle i\left\\{\mathcal{J}_{m},\mathcal{Z}_{n}\right\\}$
$\displaystyle=\left(m-n\right)\mathcal{Z}_{m+n}+\dfrac{c_{3}}{12}\,m^{3}\delta_{m+n,0}\,,$
$\displaystyle i\left\\{\mathcal{J}_{m},\mathcal{G}_{n}\right\\}$
$\displaystyle=\left(\frac{m}{2}-n\right)\mathcal{G}_{m+n}\,,$ $\displaystyle
i\left\\{\mathcal{P}_{m},\mathcal{G}_{n}\right\\}$
$\displaystyle=\left(\frac{m}{2}-n\right)\mathcal{H}_{m+n}\,,$ $\displaystyle
i\left\\{\mathcal{J}_{m},\mathcal{H}_{n}\right\\}$
$\displaystyle=\left(\frac{m}{2}-n\right)\mathcal{H}_{m+n}\,,$ $\displaystyle
i\left\\{\mathcal{G}_{m},\mathcal{G}_{n}\right\\}$
$\displaystyle=\mathcal{P}_{m+n}+\dfrac{c_{2}}{6}\,m^{2}\delta_{m+n,0}\,,$
$\displaystyle i\left\\{\mathcal{G}_{m},\mathcal{H}_{n}\right\\}$
$\displaystyle=\mathcal{Z}_{m+n}+\dfrac{c_{3}}{6}\,m^{2}\delta_{m+n,0}\,,$
(4.1)

corresponding to a supersymmetric extension of the deformed
$\mathfrak{bms}_{3}$ algebra [49, 29]. It is worth highlighting that (4.1)
corresponds to the asymptotic symmetry algebra of the minimal Maxwell
supergravity theory in three dimensions.

##  5 Discussion

In this paper, we have studied the asymptotic symmetries of the three-
dimensional Chern-Simons supergravity theories based on the supersymmetric
extension of two different algebras connected by a flat limit, i.e. the AdS-
Lorentz and the Maxwell algebras. When imposing appropriate boundary
conditions, we found that our results describe new asymptotic structures of
these supergravity theories. Similarly to the AdS-Lorentz and Maxwell
superalgebras, the asymptotic generators contain additional generators
$\mathcal{Z}_{m}$ along with fermionic charges which modify the
super-$\mathfrak{bms\mathrm{{}_{3}}}$ symmetry of the simplest $\mathcal{N}=1$
flat supergravity. In the AdS-Lorentz case, the asymptotic symmetry algebra is
given by an enlarged super- $\mathfrak{bms}\mathrm{{}_{3}}$ symmetry. Although
this algebra was already presented in [49], here we have shown that it is
indeed the asymptotic symmetry algebra of the AdS-Lorentz supergravity theory
in three space-time dimensions. The same statement is valid when the flat
limit is considered. Indeed, the flat limit can be performed throughout all
the steps followed in obtaining the asymptotic symmetry algebra for the AdS-
Lorentz supergravity. In that case, the obtained charge algebra would have
been the deformed super- $\mathfrak{bms}\mathrm{{}_{3}}$. Then, our findings
extend the results of [48], where it was explored the flat Maxwell limit at
the level of the asymptotic boundary conditions.

Let us mention that the solutions of the Maxwell and AdS-Lorentz gravity
theories were partially studied in [29] and [48], respectively. Nonetheless,
in order to consider an exhaustive analysis of the energy bounds and
asymptotic Killing spinors it is necessary to fully understand these solutions
and the thermodynamics. We will leave this discussion for a future work which
is already in progress.

It would be worth it to extend our results to the case of
$\mathcal{N}$-supersymmetries. One could expect that the asymptotic symmetry
of the $\mathcal{N}$-extended Maxwell CS supergravity corresponds to a
$\mathcal{N}$-extended deformed super-$\mathfrak{bms}\mathrm{{}_{3}}$[60, 61,
62]. Another aspect that deserves further investigation is the supersymmetric
extension of the Maxwell CS gravity with non-vanishing torsion (or Maxwellian
Teleparallel gravity) [33], and the approach of the corresponding asymptotic
symmetries, extending in this manner the asymptotic analysis done in [33] to
the supersymmetric case. We expect to find an infinite-dimensional lift of the
Maxwellian teleparallel superalgebra as the asymptotic symmetry algebra (work
in progress).

## Acknowledgment

This work was funded by the National Agency for Research and Development ANID
- SIA grant No. SA77210097 and FONDECYT grant 11220486. The authors would like
to thank to P. Concha for enlightening discussions and comments. ER would also
like to thank the Dirección de Investigación and Vice-rectoría de
Investigación of the Universidad Católica de la Santísima Concepción, Chile,
for their constant support. J.M. has been supported by the MCI, AEI, FEDER
(UE) grants PID2021-125700NB-C21 (“Gravity, Supergravity and
Superstrings”(GRASS)) and IFT Centro de Excelencia Severo Ochoa
CEX2020-001007-S.

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open subsets of Painlevé $I\\!I$ to $V\\!I$ phase spaces, and the types
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correspond to the strata $C_{I\\!I}$ to $C_{V\\!I}^{(i|j|k|l)}$ containing the
deformation parameter $\mathbf{c}$.

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# Electron irradiation reveals robust fully gapped superconductivity in
LaNiGa2

S. Ghimire  Ames National Laboratory, Ames, Iowa 50011, USA  Department of
Physics & Astronomy, Iowa State University, Ames, Iowa 50011, USA  K. R. Joshi
Ames National Laboratory, Ames, Iowa 50011, USA  Department of Physics &
Astronomy, Iowa State University, Ames, Iowa 50011, USA  E. H. Krenkel  Ames
National Laboratory, Ames, Iowa 50011, USA  Department of Physics & Astronomy,
Iowa State University, Ames, Iowa 50011, USA  M. A. Tanatar  Ames National
Laboratory, Ames, Iowa 50011, USA  Department of Physics & Astronomy, Iowa
State University, Ames, Iowa 50011, USA  Yunshu Shi  Department of Physics and
Astronomy, University of California, Davis, California 95616, USA  M.
Kończykowski  Laboratoire des Solides Irradiés, CEA/DRF/lRAMIS, École
Polytechnique, CNRS, Institut Polytechnique de Paris, F-91128 Palaiseau,
France  R. Grasset  Laboratoire des Solides Irradiés, CEA/DRF/lRAMIS, École
Polytechnique, CNRS, Institut Polytechnique de Paris, F-91128 Palaiseau,
France  V. Taufour  Department of Physics and Astronomy, University of
California, Davis, California 95616, USA  P. P. Orth  Ames National
Laboratory, Ames, Iowa 50011, USA  Department of Physics & Astronomy, Iowa
State University, Ames, Iowa 50011, USA  Department of Physics, Saarland
University, 66123 Saarbrücken, Germany  M. S. Scheurer  Institute for
Theoretical Physics III, University of Stuttgart, 70550 Stuttgart, Germany  R.
Prozorov<EMAIL_ADDRESS>Ames National Laboratory, Ames, Iowa 50011, USA
Department of Physics & Astronomy, Iowa State University, Ames, Iowa 50011,
USA

###### Abstract

The effects of 2.5 MeV electron irradiation were studied in the
superconducting phase of single crystals of LaNiGa2, using measurements of
electrical transport and radio-frequency magnetic susceptibility. The London
penetration depth is found to vary exponentially with temperature, suggesting
a fully gapped Fermi surface. The inferred superfluid density is close to that
of a single-gap weak-coupling isotropic $s-$wave superconductor.
Superconductivity is extremely robust against nonmagnetic point-like disorder
induced by electron irradiation. Our results place strong constraints on the
previously proposed triplet pairing state by requiring fine-tuned impurity
scattering amplitudes and are most naturally explained by a sign-preserving,
weak-coupling, and approximately momentum independent singlet superconducting
state in LaNiGa2, which does not break time-reversal symmetry. We discuss how
our findings could be reconciled with previous measurements indicating
magnetic moments in the superconducting phase.

##  I Introduction

The centrosymmetric superconductor LaNiGa2 has attracted significant attention
recently [1, 2, 3, 4, 5, 6], as muon spin resonance ($\mu$SR) measurements [2]
reported a breaking of time-reversal symmetry in the superconducting state
below $T_{c}\approx 2$ K. This was interpreted as non-unitary triplet
superconductivity [2]. Specific heat measurements in the superconducting state
suggested weak coupling $s-$wave superconductivity [1, 6], also discussed
theoretically [3, 7], and London penetration depth measurements in
polycrystalline samples indicated nodeless superconductivity [4]. These
results are incompatible with a single-band time-reversal symmetry broken
(TRSB) triplet state, which led to the proposal of a multiband scenario,
specifically an internally antisymmetric nonunitary triplet pairing (INT)
state [4, 5] (see also the recent reviews [8, 9]). Experimental results from
single crystals of LaNiGa2 provided further support for this scenario by
showing that the previously unknown Fermi surface topology of LaNiGa2
possesses Dirac points at the Fermi level [6]. These Dirac points originate
from the non-symmorphic symmetry of the crystal structure (space group Cmcm),
and provide the necessary band degeneracy for INT pairing [6, 10]. The
calculations of the electronic bandstructure reveal a Fermi surface with five
sheets [11, 7, 10].

With this new information about the topologically nontrivial band structure
and the availability of single crystals, it is imperative to further
investigate the superconducting properties of this material. The study of
single crystals is particularly important due to the significant electronic
anisotropy of the superconducting state [6, 10]. Also, while the possibility
of a topology-enabled INT pairing is compatible with a nodeless
superconducting gap with spin-triplet pairing, the effects of such a
superconducting gap structure on physical properties remain to be
investigated. Indeed, investigations of the multiband nature of the
superconducting gap and the effects of impurity scattering require high-
quality single crystals.

In this paper, we report the measurement of the temperature-dependent London
penetration depth $\Delta\lambda(T)$, which directly probes low-energy
quasiparticle excitations at low temperatures. We observe exponential behavior
of $\Delta\lambda(T)$, which confirms a fully gapped superconductiving state.
Although we use the two-band $\gamma-$model to analyze the data, the
superfluid density is close to the single-gap weak-coupling $s-$wave BCS
behavior. Most importantly, we used electron irradiation to introduce non-
magnetic point defects and found that the superconducting gap is robust
against non-magnetic disorder. Our results point towards robust sign-
preserving singlet superconductivity in LaNiGa2.

##  II Experimental methods

Single crystals of LaNiGa2 were grown with a Ga-deficient self-flux [6].
Electrical resistivity was measured using the four-probe technique in Quantum
Design PPMS. The samples were typically 1 to 2 mm long and had a thickness of
less than 0.1 mm. For generic resistivity characterization, contacts to the
samples were made by spot welding. The electrical resistivity at room
temperature was about 40 $\mu\Omega\cdot\text{cm}$, based on the measurements
of seven crystals. The in-plane resistivity measured along the $a$\- and
$c$-directions did not show any anisotropy within the experimental
uncertainty. The $b-$ direction was too short to perform out-of-plane
measurements. The electrical contacts for the samples used in the electron
irradiation experiments were soldered with indium and mechanically reinforced
with Dupont 4929 conductive silver paste [12]. These contacts were found to be
stable during electron irradiation [13].

London penetration depth was measured using a sensitive frequency domain self-
oscillating tunnel diode resonator (TDR) operating at a frequency of around 14
MHz. The measurements were performed in a 3He cryostat down to $\approx$ 400
mK. The experimental setup, measurements, and calibration principles are
described in detail elsewhere [14, 15, 16, 17, 18, 19]. For TDR measurements,
the samples were cut into cuboids of typical size $0.5\times 0.3\times
0.04\;\textrm{mm}^{3}$ where the thinnest direction corresponded to the
crystallographic $b-$axis. In this work precision calibration was achieved
using a well-defined skin depth estimated from the measured resistivity just
above $T_{c}$. The penetration depth was measured in three crystals yielding
practically identical results.

Non-magnetic point-like disorder was introduced at the SIRIUS facility at the
Laboratoire des Solides Irradiés at École Polytechnique, Palaiseau, France.
Electrons accelerated to 2.5 MeV are capable of knocking out ions from their
position in the crystal lattice, creating vacancy-interstitial Frenkel pairs
[20, 21, 22, 23, 24]. To prevent recombination and clustering of freshly
produced defects, irradiation was performed with a sample immersed in liquid
hydrogen at around 22 K. Upon warming, some defects anneal and recombine, but
due to much faster migration rates of the interstitials, a significant
population of vacansies remains at room temperature. The overall effect of
irradiation is characterized by electrical transport measurements that show
resistivity increasing linearly with the irradiation dose. During the
irradiation, the total accumulated dose of electrons propagating through the
sample is measured behind the sample using a Faraday cup. The total beam
current was maintained at 2.7 $\mu$ A through a circular diaphragm of 5 mm in
diameter, which is equivalent to the electron beam flux of $8.6\times 10^{13}$
electrons/($\mathrm{s}\cdot\mathrm{cm^{2}}$). The total acquired irradiation
dose is conveniently measured in C/cm2, where 1 C/cm2 = 6.24 $\times$ 1018
electrons/cm2.

##  III Experimental results

###  III.1 Electrical resistivity

Figure 1: Temperature-dependent resistivity of the LaNiGa2 single crystal
measured with current along the $c$-axis before (blue curve) and after
electron irradiation with a dose of 1.67 C/cm2 (red curve). The upper inset
zooms in on the superconducting transition. Irradiation has suppressed $T_{c}$
by a small amount of 0.025 K (1.24%) while resulting in a significant increase
of $\rho(T_{c})$ from 3.7 to 5.5 $\mu\Omega\cdot\text{cm}$ (48.65%). The lower
inset shows resistivity in a pristine state subtracted from the resistivity
after the irradiation indicating some violation of the Matthiessen rule.

The temperature-dependent resistivity of the LaNiGa2 crystal in the pristine
state before irradiation is shown in Fig. 1 by a blue curve. Measurements were
performed with electrical current along the $c-$axis. Resistivity decreases
monotonically with temperature, with a slight downward deviation from a
$T-$linear behavior below approximately 100 K. The superconducting transition
starts at $T_{c}\approx 2.0$ K (onset) and is reasonably sharp with the offset
at $T_{c,0}\approx 1.96$ K. Electron irradiation with 1.67 C/cm2 (red curve in
Fig. 1) leads to a nearly parallel upward shift of the resistivity curve,
although with some deviation from the Matthiessen rule (lower right inset).
Similar behavior is found in other materials [25, 23]. Upon irradiation, the
superconducting transition decreased by approximately 0.025 K, which is a
change of only 1.24%, while the resistivity increased by a colossal 48.65%.

###  III.2 London penetration depth

Figure 2 shows the low-temperature variation of the London penetration depth,
$\Delta\lambda(T/T_{c})$, with no observable change before and after electron
irradiation with the dose of 1 C/cm2. The solid green line shows an example of
a power-law fit, $\Delta\lambda(T)$ = $A\,T^{n}$, in the temperature range
from $T_{\text{min}}=0.4$ K to $T_{\text{max}}=1.2$ K with exponent, $n=3.8$.
The fitting was repeated for several values of $T_{\text{max}}$. The upper
left inset shows the exponent $n$ as a function of $T_{\text{max}}/T_{c}$.
Clearly, the fitted exponent remains above $n=4$ up to
$T_{\text{max}}=0.5T_{c}$. This type of power-law analysis has proven to be
very useful when a significant variation in the temperature-dependent
penetration depth was observed in iron-based superconductors [26]. It is easy
to verify numerically that the exponent of $n\geq 4$ is practically
indistinguishable from the exponential behavior. Therefore, we observed robust
exponential attenuation of $\Delta\lambda(T)$ at low temperatures, suggesting
a fully gapped Fermi surface in superconducting LaNiGa2 crystals. This result,
now obtained in single crystals, is consistent with the previous study of
polycrystalline samples [4]. We also checked the transition temperature change
upon irradiation from magnetic measurements. The upper right inset shows the
temperature-dependent susceptibility of the same sample in pristine and
irradiated states, focusing on $T_{c}$. Due to natural smearing, it is
difficult to pinpoint $T_{c}$, but clearly only a very small downshift was
induced by irradiation. This is a far cry from what is expected for a
superconductor with broken time-reversal symmetry; see, for example, [27], as
we analyze quantitatively in more detail below.

Figure 2: Low-temperature variation of the London penetration depth
$\Delta\lambda$ as a function of normalized temperature $T/T_{c}$ in pristine
and irradiated with the dose of 1 C/cm2 sample whose data practically
coincide. The solid light green line shows the best power-law fit below
$0.6~{}T_{c}$, with exponent $n=3.8$. The upper right inset shows normalized
magnetic susceptibility indicating a change from $T_{c}=2.08$ K (pristine) to
$2.05$ K (irradiated). The upper left inset shows the exponent $n$ as a
function of the upper limit of the power-law fitting, $T_{\text{max}}/T_{c}$.

###  III.3 Superfluid density

The normalized superfluid density is the quantity needed for theoretical
analysis. It is defined as, $\rho=\lambda(0)^{2}/\lambda(T)^{2}$. Our
technique yields a precise result of the change of the London penetration
depth as a function of temperature $\Delta\lambda$. However, it does not yield
the absolute value, $\lambda(0)$. Hence, we use the data obtained on the
crystals from the same batch that estimated the Ginzburg-Landau values of
$\lambda^{GL}_{a}=174$ nm, $\lambda^{GL}_{b}=509$ nm and
$\lambda^{GL}_{a}=189$ nm [6]. Of these, we need the penetration depth along
the $a-$ and $c-$axes. The average value in a cuboidal crystal is found from
$\lambda_{ac}=\left(a\lambda_{a}+c\lambda_{c}\right)/\left(a+c\right)$ [28].
The London penetration depth we need is related to the Ginzburg-Landau value
via $\lambda(0)=\sqrt{2}\lambda^{GL}$, see Eq. (25) in Ref. 26. We obtain
$\lambda(0)=253$ nm. This is significantly lower than the $\mu$SR value
obtained in polycrystalline samples, $\lambda(0)_{poly}=350$ nm [2], which is
expected considering the anisotropy mentioned above. In a recent work on
single crystals, in-field $\mu$SR measurements estimated $\lambda(0)=151$ nm
[29]. We will also use it for comparison. We note that these experimental
values are for pristine (unirradiated) samples, but not for the theoretically
clean limit, which we will denote as $\lambda_{00}$ (zero scattering and zero
temperature). There is always some disorder in as-grown samples.

Here we extract the London penetration depth in the clean limit and after
irradiation using a known pristine value as well as the resistivity change due
to introduced disorder. This analysis is based on Tinkham’s formula, which is
thought to be approximate. However, we performed a direct comparison with
microscopic calculations of the London penetration depth with non-magnetic
scattering using the Eilenberger approach [26] and found that it gives
practically exact agreement at least for an isotropic Fermi surface and
$s-$wave pairing. According to Tinkham [30],

$\lambda\left(\Gamma\right)=\lambda_{00}\sqrt{1+\frac{\xi_{0}}{\ell}}$ (1)

where $\ell$ is electronic mean free path and $\xi_{0}=\hbar v/\pi\Delta_{0}$
is the Bardeen-Cooper-Schrieffer (BCS) [31] coherence length, not to be
confused with the coherence length determined by the upper critical field,
$\xi^{2}=\phi_{0}/2\pi H_{c2}$, where $\phi_{0}$ is the magnetic flux quantum,
$v$ is Fermi velocity and $\Delta_{0}$ is the superconducting gap at zero
temperature. The clean-limit $\lambda_{00}$ does not depend on any
superconducting parameters and is given by the London theory,

$\lambda_{00}^{2}=\frac{m}{\mu_{0}ne^{2}}$ (2)

where $m$ and $e$ are electron mass and charge, respectively and $n$ is
electron concentration. At the same simple isotropic level, the Drude
resistivity is:

$\rho=\frac{mv}{ne^{2}\ell}$ (3)

and so, we can write,

$\frac{\rho}{\lambda_{00}^{2}}=\frac{\mu_{0}v}{\ell}=\frac{\mu_{0}}{\tau}$ (4)

where $\tau=\ell/v$ is the scattering time. Therefore, the mean free path is
given by,

$\ell=\frac{\mu_{0}v\lambda_{00}^{2}}{\rho}$ (5)

We can now evaluate $\lambda_{00}$ using Eq. (1) where we substitute Eq. (5)
to trivially obtain:

$\lambda_{00}^{2}=\lambda\left(\Gamma\right)^{2}-\frac{\rho\xi_{0}}{\mu_{0}v}$
(6)

Now we can use the measured resistivity after irradiation to estimate the
change in the London penetration depth. This procedure also gives a way to
estimate the current state (clean or dirty) of any sample, irradiated, or
pristine.

Figure 3: Superfluid density obtained from the London penetration depth shown
in Fig. 2 using experimental ($\mu$SR) $\lambda(0)=253$ nm [6] for the
pristine sample (blue circles) and $\lambda(0)=282$ nm for irradiated sample
(green dashed line), obtained as described in the text. The yellow line shows
an excellent $\gamma-$model [32] fit of the data. Standard $d-$ and $s-$wave
curves are shown by dashed and dash-dotted lines, respectively. Inset shows
two superconducting gaps obtained self-consistently from the fit. For
comparison, the superfluid density obtained using $\lambda(0)=151$ nm [29]
along with a $\gamma-$model fit are also shown by the black squares and an
orange solid curve, respectively.

We use literature data for the Fermi velocity, $v=3\times 10^{5}$ m/s [11],
Debye temperature, $T_{D}=166$ K [6], and the value of London penetration
depth determined by the $\mu$SR measurements, $\lambda_{pr}=253$ nm [6]. With
$T_{c}=2.0$ K, we estimate the zero-temperature superconducting gap using a
standard BCS ratio, $\Delta_{0}=1.7638k_{B}T_{c}\approx 0.3$ meV [31].
Therefore, the BCS coherence length is, $\xi_{0}=\hbar v/\pi\Delta_{0}=207$
nm.

As can be seen from Fig.1, the resistivity changed from 3.7
$\mu\Omega\cdot\text{cm}$ to 5.5 $\mu\Omega\cdot\text{cm}$ upon 2.5 MeV
electron irradiation with the total fluence of 1.67 C/cm${}^{2}=1\times
10^{19}$ e-/cm2. First, we estimate the clean-limit penetration depth from Eq.
(6), $\lambda_{00}=209$ nm for the case when a magnetic field is applied along
the $b-$axis. With this value we can now extract the London penetration depth
after the irradiation. Using Eq. (5), the mean free path before irradiation
is, $\ell_{\text{pristine}}=445$ nm, and after $\ell=300$ nm. Therefore, using
Eq. (1), we obtain $\lambda_{\text{irr}}=282$ nm. Note that the criterion
distinguishing between clean and dirty limit is based on the dimensionless
scattering rate, $\Gamma=\hbar v/2\pi T_{c}\ell\approx 0.882\xi_{0}/\ell$. The
latter is simply obtained from the above equations and was obtained long ago
by Helfand and Wetzmeier [33]. This shows the importance of the BCS coherence
length as the relevant scale for scattering. In the present case, we start at
$\Gamma_{\text{pristine}}\approx 0.41$ and after the irradiation, we obtain,
$\Gamma\approx 0.61$. From the analysis we obtain that scattering rate changes
with the dose of electron irradiation as, $d\Gamma/d(\text{dose})\approx 0.12$
where the dose is measured in C/cm2. One may argue that this background amount
of disorder in pristine sample could already be sufficient to drive the system
from the TRSB state into a state with TRS. However, our results imposes severe
limitation on such scenario, especially considering practically no $T_{c}$
change upon irradiation. While this is not an ultra-clean limit, it is
moderately clean (as long as $\Gamma<1$). This is important because now we can
use the original, clean limit, $\gamma-$model [32] to fit the superfluid
density. It is constructed from the measured change of the penetration depth,
$\Delta\lambda(T)$, shown in Fig. 2,

$\rho_{s}\left(T\right)=\left(\frac{\lambda\left(0\right)}{\lambda\left(T\right)}\right)^{2}=\left(1+\frac{\Delta\lambda\left(T\right)}{\lambda\left(0\right)}\right)^{-2}$
(7)

Figure 3 shows the superfluid density before (blue circles) and after (dashed
green line) electron irradiation using Eq. (7) with the experimental ($\mu$SR)
$\lambda(0)=253$ nm [6] for pristine sample, and $\lambda(0)=282$ nm for
irradiated sample, estimated as described above. The yellow line shows an
excellent $\gamma-$model [32] fit of the data for the pristine sample. The
superfluid density of the irradiated sample is shifted to slightly higher
values, consistent with the increased $\lambda(0)$. Considering the close
proximity of the two curves, there was no reason to fit the irradiated sample.
The standard $d-$ and $s-$ wave curves are shown by dashed and dash-dotted
lines, respectively. The inset shows two superconducting gaps obtained self-
consistently from the fit. For comparison, the superfluid density obtained
using $\lambda(0)=151$ nm [29] along with a $\gamma-$model fit are also shown
by the black squares and an orange solid curve, respectively.

###  III.4 Self-consistent $\gamma-$model fitting

Assuming isotropic superconducting order parameters, the $\gamma$-model takes
as input the average Fermi velocities and the densities of states (DOS) on the
effective bands as well as the interaction matrix, $\lambda_{i,j}$ [32, 26].
Here, we consider all of these values as free fitting parameters. Note that
$\lambda_{i,j}$ should not be confused with the penetration depth, $\lambda$.
The 3${}^{\text{rd}}$ interaction parameter is constrained by the value of
$T_{c}$ calculated from the effective interaction constant,
$\lambda_{\text{eff}}$ as
$1.7638k_{B}T_{c}=2\hbar\omega_{D}\exp\left(-\lambda_{\text{eff}}^{-1}\right)$.
Here, the effective interaction constant $\lambda_{\text{eff}}$ is obtained
from the solution of algebraic equations containing all the coefﬁcients,
$\lambda_{i,j}$; see Sec. II A of Ref.[32], and Debye temperature, $T_{D}=166$
K [6], assuming phonon-mediated superconductivity. For a different “glue”,
there will be a different similar quantity. With $T_{c}=2.0$ K and a fixed
$n_{1}=0.5$, we obtained from the fit of the pristine data,
$\lambda_{11}=0.416$, $\lambda_{22}=0.388$, $\lambda_{12}=0.036$,
$\gamma=0.300$ and the effective $\lambda_{\text{eff}}=0.220$. Therefore, we
have two fairly similar, at-first-sight, barely coupled bands, which is
probably why they do not form a single gap everywhere. The key to the
$\gamma-$model, is $\gamma=n_{1}v_{1}^{2}/(n_{1}v_{1}^{2}+n_{2}v_{2}^{2})$,
where $v_{i}$ are the Fermi velocities on different bands. From the fit we see
that $v_{1}<v_{2}$, which may mean that the carriers on band 1 are heavier. As
such, they will contribute the most to the specific heat that appears as if
there were a single band [6]. The opposite is true for superfluid density
where light carriers contribute the most. We note, however, that the specific
heat calculated with above parameters is already quite close to the experiment
considering that the superfluid density is quite close to the single-band
s-wave BCS curve. Fitting the superfluid density calculated with
$\lambda(0)=151$ nm we obtained, $\lambda_{11}=0.440$, $\lambda_{22}=0.401$,
$\lambda_{12}=0.012$, $\gamma=0.300$ and the effective
$\lambda_{\text{eff}}=0.222$, which does not alter the above conclusions in
any way, only showing even weaker interband coupling.

###  III.5 Upper critical field

Finally, we discuss the upper critical field, $H_{c2}$, displayed in Fig.4.
The $H_{c2}(T)$ was estimated from the onset of a diamagnetic signal in the
TDR $\Delta\lambda(T)$ curves shown in the upper right and lower left insets,
respectively. While all other properties show quite conventional behavior,
$H_{c2}(T)$ exhibits an unusual convex shape. However, we note that the TDR
curves not only shift, but also broaden significantly, making it difficult to
extract the precise $H_{c2}$ values. A similar behavior, convex curvature and
broadening, was previously observed in AC susceptibility measurements in
polycrystalline samples [4].

Figure 4: Upper critical field, $H_{c2}$, measured in pristine and irradiated
samples obtained from the onset of a diamagnetic signal in the TDR
$\Delta\lambda(T)$ curves shown in the upper-right and bottom-left insets,
respectively.

Clearly, there is practically no change in $H_{c2}(T)$ after electron
irradiation. However, both this observation and the convex curvature can be
easily understood within multiband superconductivity [34, 35, 36] only
confirming what we already learned from the analysis of the superfluid
density. However, an important observation is that the upper critical field is
small. If triplet pairing was present, one would expect a large upper critical
field, like in UTe2. On the other hand, LaNiGa2 is not a heavy fermion
material, so it would be subject to a smaller orbital limit [6].

##  IV Discussion

We now discuss the relation of our findings to the pairing states of previous
works [2, 4]. The fact that our penetration depth data can be fitted with a
fully gapped order parameter that has two different magnitudes on two
different (subsets of) bands in the system is consistent with previous works:
in the previous paper on polycrystalline samples, the two gap values were
$\Delta_{1}(0)/k_{B}T_{c}=2.04$ and $\Delta_{2}(0)/k_{B}T_{c}=1.29$ [4];
meanwhile we obtained closer ratios of 1.89 and 1.57, respectively. The
difference is most likely due to a significant anisotropy of the penetration
depth. While having several rather than one single gap magnitude is, in fact,
the expected behavior for a complex multiband superconductor, previous $\mu$SR
measurements in LaNiGa2 [2] have indicated broken time-reversal symmetry in
its superconducting state—a far less common phenomenon. In that context, our
measurements of the effects of irradiation provide particularly strong
constraints, which we discuss next.

Figure 5: (a) Comparison of the disorder sensitivity of the transition
temperature in LaNiGa2 with other superconductors (symbols). The solid lines
show the expectations for two isotropic gaps with different gap ratios.
Corresponding parameter $\zeta$ is shown. (b) Dimensionless disorder-
sensitivity $\zeta$, see Eq. (9), shown as a function of
$\hat{w}_{j}=w_{j}/|\bm{w}|$ (inset) and $|w_{0}|/\sqrt{w_{1}^{2}+w_{3}^{2}}$
(main panel, red), and extracted from the measurements of irradiated samples
(dashed, main panel).

As illustrated in Fig. 5(a), showing the suppression of the critical
temperature of LaNiGa2 along with that of unconventional superconductors and
the Abrikosov-Gorkov (AG) curve for magnetic scattering in a BCS state, the
suppression of $T_{c}$ with disorder is very weak. More quantitatively, we
extract a value of the dimensionless sensitivity to disorder scattering
$\zeta$ [13] given by

$\zeta=2\pi^{-2}\left[d(\Delta T_{c}/T_{c0})/d\Gamma\right]\approx 0.0124\,.$
(8)

To demonstrate that this is difficult to reconcile with time-reversal-symmetry
breaking in the superconducting state, we next focus on the most favorable
scenario of an order parameter that does not depend on momentum. Due to the
low point group ($D_{2h}$) symmetries of LaNiGa2, non-unitary triplet pairing
has been argued to be the most natural realization of broken time-reversal
symmetry [2]. Since a triplet superconducting order parameter is symmetric in
spin space, the absence of nodes necessitates involving more than one orbital
degree of freedom [4, 5] to make the entire order parameter antisymmetric; the
simplest limit of such an INT state involves only two orbitals (Pauli matrices
$\sigma_{j}$), reading as $\Delta=\bm{d}\cdot\bm{s}\,i\sigma_{y}$, where
$\bm{s}=(s_{x},s_{y},s_{z})$ are the spin Pauli matrices and $\bm{d}$ is the
triplet vector. Although only $\bm{d}$ with $\bm{d}^{*}\times\bm{d}\neq 0$ is
consistent with the broken time-reversal symmetry, we will not further specify
$\bm{d}$ as the following statements do not depend on the specific form of
$\bm{d}$.

To model disorder for electron-irradiation experiments, we make the common
assumption of non-magnetic and spin-rotation-invariant impurities that are
point-like, i.e., scatter between all momenta within the Brillouin zone with
equal amplitude; the associated scattering matrix (a $4\times 4$ matrix in
spin and orbital space) can then be parametrized as
$W=s_{0}\otimes(w_{0}\sigma_{0}+w_{1}\sigma_{x}+w_{3}\sigma_{z})$, where
$w_{j}\in\mathbbm{R}$. Intuitively, $w_{0}\pm w_{3}$ captures the amplitude of
the scattering within each of the two bands ($\pm$) while $w_{1}$ describes
the interband scattering. Further making the most favorable assumption for
superconductivity that the two bands are approximately degenerate (on the
scale of $\Delta$), which in fact is imposed by the non-symmorphic symmetry
[10], we can readily apply the generalized Anderson theorem of [13] and find

$\zeta=\frac{w_{1}^{2}+w_{3}^{2}}{w_{0}^{2}+w_{1}^{2}+w_{3}^{2}}.$ (9)

We can see that both interband ($w_{1}$) scattering as well as a scattering
imbalance ($w_{3}$) between the two bands are pair breaking. It might hold
$w_{3}\ll w_{0}$, but it seems less plausible that $w_{1}\ll w_{0}$. In fact,
as also illustrated visually in Fig. 5(b), the obtained $\zeta\approx 0.0124$
gives a very strong constraint on the matrix elements—quantitatively, we find
$|w_{0}|/\sqrt{w_{1}^{2}+w_{3}^{2}}=\sqrt{\zeta^{-1}-1}\simeq 8.9$. We note
that generalizing to momentum dependent superconducting order parameters would
make the pairing state further susceptible to $w_{0}$, as one can immediately
conclude from the generalized Anderson theorem [13]. As such, it would even
increase its sensitivity and does not help with reconciling broken time-
reversal symmetry and our irradiation data.

One alternative possibility is that the time-reversal-symmetry-breaking
superconducting state is the leading instability only by a small margin and
there is an almost degenerate sub-leading state in the spin singlet channel
[recall that Eq. (9) holds for $\Delta=\bm{d}\cdot\bm{s}\,i\sigma_{y}$
irrespective of the form of $\bm{d}$]. Then either our “pristine” sample
already exhibits enough disorder to favor that subleading singlet state or
only the amount of disorder resulting from irradiation did. Note that this
subleading singlet state must preserve time-reversal symmetry since the point
group $D_{2h}$ of the system only admits one-dimensional irreducible
representations. Taking the critical temperatures $T_{c}\approx
2.1\,\textrm{K}$ (onset), $1.96\,\textrm{K}$, and $1.94\,\textrm{K}$ from [2],
our pristine sample, and our irradiated sample, respectively, we conclude that
this degeneracy must be within at most $0.16\,\textrm{K}$, i.e., about only
$8\%$ of $T_{c}$. Therefore, also this scenario requires some fine-tuning
assumption, similar to the constraint on the disorder matrix elements,
$|w_{0}|/\sqrt{w_{1}^{2}+w_{3}^{2}}\approx 8.9$, noted above.

Finally, the third possibility is that the perfectly homogeneous system
prefers a time-reversal preserving spin-singlet state (consistent with our
irradiation study) which, due to a subleading triplet or singlet
superconducting instability, exhibits time-reversal-symmetry-breaking local
patterns around impurities [37, 38]. These could be picked up by local probes
like $\mu$SR. To exclude or confirm this scenario further irradiation studies
in combination with $\mu$SR would be required.

##  V Conclusions

To summarize the experimental findings, thermodynamic and transport
measurements in high-quality LaNiGa2 crystals are quantitatively consistent
with a non-sign-changing multigap superconducting state with almost equal but
barely coupled bands, one heavier and another lighter. The most striking
observation is that superconductivity is extremely robust to disorder and the
transition temperature did not change significantly after electron
irradiation; meanwhile, the resistivity changed by about 50%, indicating that
the irradiation indeed produced non-magnetic scattering centers. More
quantitatively, we find a disorder sensitivity parameter $\zeta\approx
0.0124$, much smaller than the value $1/2$ for a (single-band) superconducting
order parameter transforming under a non-trivial irreducible representation
[13]. Being sensitive to the relative phase of the superconducting order
parameter on different parts of the Fermi surface, the remarkably weak
disorder sensitivity yields very strong constraints on superconductivity: we
find that the INT state [4, 5] is only consistent with this observation if
pair-breaking scattering due to interband transitions and a scattering
imbalance between the bands is about nine times weaker than non-pair-breaking
scattering. This implies that intra-band matrix elements of the impurities
must be at least nine times larger than the inter-band contributions—a rather
unwonted scenario. Therefore, we identified and discussed two alternative
possibilities to reconcile these findings with previous $\mu$SR measurements
[2], finding a time-reversal-symmetry-breaking state. The first scenario is
based on the assumption that the non-unitary INT superconductor is indeed the
leading instability but very closely followed by a time-reversal-symmetric
singlet phase that is favored by the disorder we introduce. Our estimates,
however, reveal that this requires a similar amount of fine-tuning than the
aforementioned scenario, involving the impurity matrix elements. In the second
alternative scenario the homogeneous bulk superconducting phase is a time-
reversal-symmetry-preserving singlet state while the moments picked up by
$\mu$SR are related to disorder-induced local fields associated with the
admixture of a competing superconducting state with non-trivial complex phases
[37, 38]. Our findings reveal that the microscopic form of the superconducting
state in LaNiGa2 is still an open question and call for further experimental
and theoretical work to pinpoint which scenario is realized.

Note added.—Just before posting our article, another work appeared [39] in
which nuclear magnetic resonance (NMR) measurements of LaNiGa2 crystals found
no enhancement of paramagnetism or magnetic fluctuations and yielded a
Korringa ratio different from that of other time-reversal-symmetry-breaking
superconductors. These observations are aligned with our conclusions.

##  VI Acknowledgements

This work was supported by the U.S. Department of Energy (DOE), Office of
Science, Basic Energy Sciences, Materials Science and Engineering Division.
Ames Laboratory is operated for the U.S. DOE by Iowa State University under
contract DE-AC02-07CH11358. Electron irradiation was supported by
“Investissements d’Avenir” LabEx PALM (ANR-10-LABX-0039-PALM). The authors
acknowledge support from the EMIR&A French network (FR CNRS 3618) on the
platform SIRIUS at Ecole Polytechnique in Palaiseau, France. Sample synthesis
at UC Davis was supported by the UC Laboratory Fees Research Program
(LFR-20-653926). M.S.S. acknowledges funding by the European Union
(ERC-2021-STG, Project 101040651—SuperCorr). Views and opinions expressed are
however those of the authors only and do not necessarily reflect those of the
European Union or the European Research Council Executive Agency. Neither the
European Union nor the granting authority can be held responsible for them.

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