# Self duality in unconventional conformal supersymmetry

Pedro D. Alvarez<EMAIL_ADDRESS>Departamento de Física, Universidad
de Antofagasta, Aptdo. 02800, Chile  Cristóbal Corral<EMAIL_ADDRESS>Instituto de Ciencias Exactas y Naturales, Universidad Arturo Prat, Playa
Brava 3256, 1111346, Iquique, Chile  Facultad de Ciencias, Universidad Arturo
Prat, Avenida Arturo Prat Chacón 2120, 1110939, Iquique, Chile  Jorge Zanelli
<EMAIL_ADDRESS>Centro de Estudios Científicos (CECs), Arturo Prat 514,
Valdivia, Chile,  Universidad San Sebastián, General Lagos 1163, Valdivia,
Chile

###### Abstract

In this work, we study (anti-)self duality conditions in unconventional
conformal supersymmetry. We focus on a theory constructed in a Townsend-
MacDowell-Mansouri form for an $SU(2,2|N)$ gauge connection with matter fields
in the adjoint representation. We found bosonic solutions that correspond to
analytic gravitational instantons with nontrivial torsion. These
configurations can be regarded as the torsional generalization of the Taub-
NUT/Bolt-AdS and Eguchi-Hanson metric and they are (anti-)self-dual with
respect to a generalized dual operator. We explore their global properties and
show that they saturate a BPS bound.

##  I Introduction

In Euclidean Yang-Mills (YM) theory, special field configurations satisfying
an (anti-)self duality condition, i.e. $F=\pm*F$, correspond to absolute
minima of the action.111Here,
$F=\frac{1}{2}F_{\mu\nu}\text{d}x^{\mu}\wedge\text{d}x^{\nu}$ is the YM field
strength $2$-form,
$*F=\frac{1}{2}\varepsilon_{\mu\nu\lambda\rho}F^{\lambda\rho}\text{d}x^{\mu}\wedge\text{d}x^{\nu}$
its dual with and $\varepsilon_{\mu\nu\lambda\rho}$ the alternating Levi-
Civita tensor in four-dimensions, and $\wedge$ denotes the wedge product of
differential forms. These configurations solve the field equations
automatically by virtue of Bianchi identities, saturate a
Bogomol’nyi–Prasad–Sommerfield (BPS) bound, and they describe the tunneling
between topologically distinct vacua of the theory Belavin  _et al._ (1975);
’t Hooft (1976a). They are usually dubbed _instantons_ and they play an
important role in understanding non-perturbative aspects of QCD, such as the
breaking of chiral symmetry ’t Hooft (1976b); Diakonov and Petrov (1986).

Similarly, gravitational instantons are solutions to the Einstein equations of
Euclidean general relativity satisfying (anti-)self-duality conditions, i.e.
$R_{ab}=\pm\tfrac{1}{2}\epsilon_{abcd}R^{cd}\equiv\pm\tilde{R}_{ab}$, where
$R^{ab}=\tfrac{1}{2}R^{ab}_{\ \mu\nu}\text{d}x^{\mu}\wedge\text{d}x^{\nu}$ is
the Riemann curvature $2$-form and $\epsilon_{abcd}$ is the Levi-Civita symbol
Eguchi  _et al._ (1980). Examples of gravitational instantons include the
Euclidean Schwarzschild solution which describes a four-dimensional black
hole, the Taub-NUT configuration Taub (1951); Newman  _et al._ (1963) and the
Eguchi-Hanson instanton Eguchi and Hanson (1978, 1979) that describes an
asymptotically locally Euclidean four-dimensional space whose radial slices
are topologically $\mathbb{S}^{3}/\mathbb{Z}_{2}=\mathbb{RP}^{3}$. The
interest in gravitational instantons was consolidated in the late seventies
with the discovery of multi-center solutions Gibbons and Hawking (1978) and
the realization that those configurations are (hyper) Kähler geometries
Hitchin (1979); Calabi (1979).

The above examples highlight the importance of duality in the non-perturbative
analysis of the topological sectors of the vacuum for the Euclidean
continuation of a gauge theory. In each case, the notion of duality depends on
the presence of a Hodge-like operator in the theory. For the Maxwell or Yang-
Mills action, the relevant dual operation defines the kinetic term, that is,

$\frac{1}{2}\operatorname{Tr}[F\wedge*F]=-\frac{1}{4}\operatorname{Tr}[F_{\mu\nu}F^{\mu\nu}]\,\sqrt{|g|}\,\text{d}^{4}x\,.$
(1)

In the gravitational case, on the other hand, the corresponding kinetic term
reads

$\frac{1}{2}\epsilon_{abcd}\,\mathscr{F}^{ab}\wedge\mathscr{F}^{cd}=\mathscr{F}^{ab}\wedge\tilde{\mathscr{F}}_{ab}\,,$
(2)

where the Yang-Mills field strength has been replaced by the (anti-)de Sitter
curvature two-form, i.e. $\mathscr{F}^{ab}=R^{ab}\pm\lambda\,e^{a}\wedge
e^{b}$, with $\lambda$ being a constant related to the (anti-)de Sitter radius
and the Levi-Civita symbol in curved spacetime $\epsilon_{\mu\nu\lambda\rho}$
has been replaced by that in the tangent space, $\epsilon_{abcd}$ Townsend
(1977); MacDowell and Mansouri (1977). Additionally,
$e^{a}=e^{a}_{\mu}\text{d}x^{\mu}$ denotes the vierbein field representing the
metric structure of spacetime through the relation
$g_{\mu\nu}=\eta_{ab}e^{a}_{\mu}e^{b}_{\nu}$ where
$\eta_{ab}=\operatorname{diag}(-,+,+,+)$ is the Minkowski metric.222Here, we
will work mostly in the Euclidean signature. Therefore, the relation between
the metric and the vierbein reads
$g_{\mu\nu}=\delta_{ab}e^{a}_{\mu}e^{b}_{\nu}$, where
$\delta_{ab}=\operatorname{diag}(+,+,+,+)$ is the Euclidean metric.

In recent years, we have considered a family of supersymmetric theories in
which spin-1/2 fermions –possibly representing quarks and leptons– are
combined with the bosonic non-Abelian gauge fields as parts of a super
connection for a graded Lie algebra Alvarez  _et al._ (2012, 2014a, 2022). In
other words, bosons and fermions belong to the adjoint representation of a
superalgebra rather than to the fundamental representation, as in the
conventional form of supersymmetry. In this way, the resulting theories do not
exhibit matching of fermionic and bosonic degrees of freedom (no
superpartners), the quantum numbers and couplings of all fields are those of
standard QFT, and gravity is naturally built in the theory. In these theories,
the notion of duality for the bosonic fields is essentially the same as (1)
for internal symmetry gauge fields and (2) for the gravitational sector. In
the case of fermions, what plays the role of the Levi-Civita symbol is the
Clifford chirality operator $\gamma_{5}$ defined below.

The resulting _unconventional_ supersymmetric models can be seen as
generalizations of Yang-Mills theory, where the dual operation (denoted by
$\circledast$) allows writing a Lagrangian invariant under local Lorentz and
internal gauge transformations, which are the expected unbroken symmetries of
a standard physics. Such an operator can be found naturally in the
superconformal algebra (see for instance Alvarez  _et al._ (2022)). These
algebras have been used to construct related theories of matter Alvarez  _et
al._ (2020), extended MacDowell-Mansouri supergravities Alvarez  _et al._
(2021) and a grand unified model Alvarez  _et al._ (2023a). Then, a question
that comes to mind is whether the notion of duality represented by
$\circledast$ can also give rise to instanton-like configurations. In this
paper, we explore the possibility afforded by this generalized duality
operator to construct geometric instanton solutions in the matter-free sector.

The paper is organized as follows. In Sec. II, we present the model we are
interested in alongside their field equations and the consistent truncation to
the bosonic sector. Section III is devoted to the (anti-)self duality
condition in an unconventional conformal supersymmetric Townsend-MacDowell-
Mansouri (TMM) theory for the $SU(2,2|N)$ group and to the analysis of its
consequences for the action. In Sec. IV, we obtain (anti-)self-dual
gravitational instantons that represent the torsional generalization of the
Taub-NUT/Bolt-AdS and Eguchi-Hanson solutions in general relativity. Then,
their global properties are analyzed in Sec. V and our findings are summarized
in Sec. VI. Finally, Appendix A provides further details about the fundamental
representation of $SU(2,2|N)$.

##  II Unconventional conformal supersymmetry

Let us illustrate the construction of a supersymmetric model in four
dimensions with fermionic matter fields in the adjoint representation.
Following Refs. Alvarez  _et al._ (2014a, 2020, 2021, 2022), we start with a
gauge connection for $SU(2,2|N)$. Then, one is immediately faced with the fact
that no dynamical gauge-invariant action for such a group exists in four
dimensions. As noticed early on by the authors of Ref. Townsend (1977);
MacDowell and Mansouri (1977), the largest gauge symmetry of the action is a
bosonic subgroup of $SU(2,2|N)$ which contains the Lorentz transformations and
the internal $SU(N)$ gauge group. Therefore, one can expect the generalized
dual operator $\circledast$ to be allowed to explicitly break some of the
symmetries in the action. The $\mathfrak{su}(2,2|N)$ graded Lie algebra333For
more details about the fundamental representation and algebra see Appendix A.
includes the generators of the conformal group,
$\mathds{J}_{a},\mathds{K}_{a},\mathds{J}_{ab}$ and $\mathds{D}$, together
with internal generators $\mathds{T}_{I}$ of $SU(N)$ and $\mathds{Z}$ for
$U(1)$, plus $N$ complex spinorial supercharges, $\mathds{Q}^{\alpha}_{i}$ and
$\overline{\mathds{Q}}^{i}_{\alpha}$. The $SU(2,2|N)$ connection can be
written as

$\mathds{A}=\Omega+\overline{\mathds{Q}}_{\alpha}^{i}(\not{e}\psi_{i})^{\alpha}+(\overline{\psi}^{i}\not{e})_{\alpha}\mathds{Q}_{i}^{\alpha}\
\,,$ (3)

where we have denoted
$(\not{e}\psi_{i})^{\alpha}\equiv(\gamma_{a})^{\alpha}{}_{\beta}e^{a}\psi^{\beta}_{i}$,
and the bosonic part of the connection is

$\Omega=\frac{1}{2}\omega^{ab}\mathds{J}_{ab}+f^{a}\mathds{J}_{a}+g^{a}\mathds{K}_{a}+h\mathds{D}+A^{I}\mathds{T}_{I}+A\mathds{Z}\,.$
(4)

As usual, the field strength curvatures associated to this gauge connection
are given by

$\displaystyle\mathds{F}=$
$\displaystyle\text{d}\mathds{A}+\mathds{A}\wedge\mathds{A}=\frac{1}{2}\mathcal{F}^{ab}\mathds{J}_{ab}+\mathcal{F}^{a}\mathds{J}_{a}+\mathcal{G}^{a}\mathds{K}_{a}+\mathcal{H}\mathds{D}+\mathcal{F}^{I}\mathds{T}_{I}+\mathcal{F}\mathds{Z}+\overline{\mathds{Q}}\mathcal{X}+\overline{\mathcal{X}}\mathds{Q}\,,$
(5)

whose components along the $SU(2,2|N)$ generators are444Contracted spinor
indexes will be omitted whenever unambiguous.

$\displaystyle\mathcal{F}^{ab}$
$\displaystyle=\mathcal{R}^{ab}-\overline{\psi}^{i}\not{e}\wedge\Sigma^{ab}\not{e}\psi_{i},$
$\displaystyle\mathcal{F}^{a}$ $\displaystyle=\text{D}f^{a}+g^{a}\wedge
h+\frac{1}{2}\overline{\psi}^{i}\not{e}\wedge\gamma^{a}\not{e}\psi_{i}\,,$ (6)
$\displaystyle\mathcal{G}^{a}$ $\displaystyle=\text{D}g^{a}+f^{a}\wedge
h-\frac{1}{2}\overline{\psi}^{i}\not{e}\wedge\tilde{\gamma}^{a}\not{e}\psi_{i}\,,$
$\displaystyle\mathcal{H}$ $\displaystyle=\text{d}h+f^{a}\wedge
g_{a}+\frac{1}{2}\overline{\psi}^{i}\not{e}\wedge\gamma_{5}\not{e}\psi_{i}\,,$
(7) $\displaystyle\mathcal{F}^{I}$
$\displaystyle=\text{d}A^{I}+\frac{1}{2}f_{JK}^{I}A^{J}\wedge
A^{K}-i\overline{\psi}^{i}\not{e}\wedge(\lambda^{I})_{i}^{\
j}\not{e}\psi_{j}\,,$ $\displaystyle\mathcal{F}$
$\displaystyle=\text{d}A-\frac{i}{4}\overline{\psi}^{i}\not{e}\wedge\not{e}\psi_{i}\,,$
(8)

and the spinorial components of the field strength are

$\displaystyle\mathcal{X}_{i}^{\alpha}$
$\displaystyle=\text{D}(\not{e}\psi_{i})^{\alpha}+\frac{1}{2}f^{a}\wedge(\gamma_{a}\not{e}\psi_{i})^{\alpha}+\frac{1}{2}g^{a}\wedge(\tilde{\gamma}_{a}\not{e}\psi_{i})^{\alpha}+\frac{1}{2}h\wedge(\gamma_{5}\not{e}\psi_{i})^{\alpha}\,,$
(9a) $\displaystyle\overline{\mathcal{X}}^{i}_{\alpha}$
$\displaystyle=-(\overline{\psi}^{i}\not{e})_{\alpha}\overleftarrow{\text{D}}+\frac{1}{2}(\overline{\psi}^{i}\not{e}\gamma_{a})_{\alpha}\wedge
f^{a}+\frac{1}{2}(\overline{\psi}^{i}\not{e}\tilde{\gamma}_{a})_{\alpha}\wedge
g^{a}+\frac{1}{2}(\overline{\psi}^{i}\not{e}\gamma_{5})_{\alpha}\wedge h\,.$
(9b)

Here, D is the exterior covariant derivative with respect to the Lorentz
connection $\omega^{a}_{\ b}$ and d is the exterior derivative. Moreover,
$\gamma_{a}$ denotes the Dirac matrices satisfying the Clifford algebra
$\left\\{\gamma_{a},\gamma_{b}\right\\}=2\eta_{ab}\mathbb{I}$, the chiral
$\gamma$-matrix is defined as
$\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$, and
$\tilde{\gamma}^{a}=-\gamma_{5}\gamma^{a}$. The generators of the Lorentz
group in the spinorial representation are
$\Sigma_{ab}=\frac{1}{4}\left[\gamma_{a},\gamma_{b}\right]$ while the internal
generators of the $SU(N)$ algebra, $\mathds{T}_{I}$, are anti-hermitian, i.e.
$\mathds{T}_{I}^{\dagger}=-\mathds{T}_{I}$ (also
$\mathds{Z}^{\dagger}=-\mathds{Z}$) and they satisfy the Lie algebra
$[\mathds{T}_{I},\mathds{T}_{J}]=f_{IJ}{}^{K}\mathds{T}_{K}$. The
$\lambda$-matrices satisfy the commutation relation
$[\lambda_{I},\lambda_{J}]=f_{IJK}\lambda^{K}$, where indexes are
raised/lowered with an Euclidean metric $\delta_{IJ}$. Additionally, we have
defined $\mathcal{R}^{ab}=R^{ab}+f^{a}\wedge f^{b}-g^{a}\wedge g^{b}$ with
$R^{ab}=\text{d}\omega^{ab}+\omega^{a}_{\ c}\wedge\omega^{cb}$ being the
Lorentz curvature $2$-form.

The generalized TMM action Townsend (1977); MacDowell and Mansouri (1977) for
the $SU(2,2|N)$ group is,555For a similar construction see Refs. Andrianopoli
and D’Auria (2014); Andrianopoli  _et al._ (2021)

$I=-\int_{\mathcal{M}}\langle\mathds{F}\wedge\circledast\mathds{F}\rangle\,,$
(10)

where $\langle\ \cdot\ \rangle$ stands for an invariant supertrace. It is
sufficient for the dual operator $\circledast$ to act linearly in the
complexified algebra, that is parity odd and to satisfy
$\circledast^{2}=+1(-1)$ in Euclidean (Lorentzian) signature, respectively. In
fact, the conformal algebra contains a grading operator among the bosonic
spacetime generators that allows for a natural definition of the dual operator
(for more details see Alvarez  _et al._ (2022)). A grading among the bosonic
generators of $\mathfrak{su}(2,2|N)$ can be defined by an operator $S$ such
that666The grading $S$ is independent of the grading that distinguishes
fermionic and bosonic operators. A dual operator with properties mentioned
above can be implemented in the spin 1/2 representation of conformal
superalgebra $\mathfrak{su}(2,2|N)$ (see Appendix A).

$S\ G-\varepsilon G\
S=0\,,\qquad\mathrm{where}\quad\begin{cases}\varepsilon=+1\,,\mathrm{if}\
G\in\\{\mathds{J}_{ab},\mathds{D},\mathds{Z},\mathds{T}_{I}\\}\,,\\\
\varepsilon=-1\,,\mathrm{if}\
G\in\\{\mathds{J}_{a},\mathds{K}_{a}\\}\,.\end{cases}\,,$ (11)

define the $S$-even ($\varepsilon=1$) and $S$-odd ($\varepsilon=-1$) subspaces
of the bosonic sector. The spinor representation and the $S$-odd generators
are essential to include matter fields in a geometric way Alvarez  _et al._
(2022). This structure can be used to define actions with a gauge symmetry
$G^{+}\subset SU(2,2)\times SU(N)\times U(1)$, where $G^{+}$ is the group
generated by the $S$-even generators. With the appropriate definition of the
dual operator $\circledast$, the following properties of super-traces
involving products of $S$-even and $S$-odd operators are found

$\displaystyle\langle\mathds{E}\circledast\mathds{O}\rangle=0=\langle\mathds{O}\circledast\mathds{E}\rangle\,,$
$\displaystyle\langle\mathds{E}_{1}\circledast\mathds{E}_{2}\rangle=\langle\mathds{E}_{2}\circledast\mathds{E}_{1}\rangle\,,$
$\displaystyle\langle\mathds{O}_{1}\circledast\mathds{O}_{2}\rangle=-\langle\mathds{O}_{2}\circledast\mathds{O}_{1}\rangle\,,$
(12)

which can be understood as a $\circledast$-grading of the invariant super-
traces of the superalgebra, where $\mathds{E}_{i}$ and $\mathds{O}_{i}$
represent two-form generators that are even and odd in this grading. The set
of $\circledast$-even generators contains the $S$-even bosonic generators and
the supercharges. In particular, Eq. (12) implies that

$\langle\mathds{F}^{-}\wedge\circledast\mathds{F}^{-}\rangle=0\,,$ (13)

where $\mathds{F}^{-}$ are components of the curvature along the $S$-odd
generators, namely,

$\mathds{F}^{-}=\mathcal{F}^{a}\mathds{J}_{a}+\mathcal{G}^{a}\mathds{K}_{a}\,.$
(14)

In this way, the compensating fields $f^{a}$ and $g^{a}$ do not acquire
independent kinetic terms in the action principle (10). Thus, variations with
respect to $f^{a}$ and $g^{a}$ yield algebraic equations for them, which means
that they are actually auxiliary fields.

The field equations of the theory are obtained by demanding arbitrary
variations of the action (10) with respect to the gauge connection, giving

$D_{\mathds{A}}\circledast(\mathds{F}-\mathds{F}^{-})=0\,,$ (15)

where $D_{\mathds{A}}$ stands for the covariant derivative with respect to the
gauge connection $\mathds{A}$, provided the supertrace is nondegenerate [cf.
Eqs. (87)-(90)]. In a purely bosonic background, the field equations take the
simpler form $D_{\Omega}\circledast\mathds{F}^{+}=0$, and in the sector
without internal gauge fields, $A^{I}=0=A$, the field equations, expanded
along the $SU(2,2|N)$ generators, take the form

$\displaystyle\epsilon_{abcd}\text{D}\mathcal{R}^{cd}$ $\displaystyle=0\,,$
(16a) $\displaystyle\text{d}\ast(H+f^{b}\wedge g_{b})$ $\displaystyle=0\,,$
(16b) $\displaystyle g_{a}\wedge\ast(H+f^{b}\wedge
g_{b})-\frac{1}{2}\epsilon_{abcd}f^{b}\wedge\mathcal{R}^{cd}$
$\displaystyle=0\,,$ (16c) $\displaystyle f_{a}\wedge\ast(H+f^{b}\wedge
g_{b})-\frac{1}{2}\epsilon_{abcd}g^{b}\wedge\mathcal{R}^{cd}$
$\displaystyle=0\,,$ (16d)

where $H:=dh$. Now, the same field equations can be obtained varying with
respect to $\omega^{ab}$, $h$, $f^{a}$, and $g^{a}$ the truncated action

$\mathscr{L}_{\rm
bos}=\frac{1}{4}\epsilon_{abcd}\mathcal{R}^{ab}\wedge\mathcal{R}^{cd}-(H+f^{a}\wedge
g_{a})\wedge\ast(H+f^{b}\wedge g_{b})\,.$ (17)

The fields $f^{a}$ and $g^{a}$, associated with translations and special
conformal transformations, are auxiliary (they satisfy algebraic equations).
Since the gauge invariance corresponding to their associated transformations
is broken, their equations can be algebraically solved in a satisfactory
manner, provided they do not break the remaining gauge symmetries. There is a
particular choice for these fields that respects local Lorentz invariance,
that is, $f^{a}=\rho(x)e^{a}$, $g^{a}=\sigma(x)e^{a}$, where $\rho(x)$ and
$\sigma(x)$ are arbitrary scalar functions. Consistent with this assumption
$H$ must vanish, and therefore one can take $h=0$. Then, we get
$\mathcal{R}^{ab}=R^{ab}-\Lambda(x)e^{a}\wedge e^{b}$, where
$\Lambda(x):=\sigma^{2}(x)-\rho^{2}(x)$. With these definitions, the field
equations of the system are

$\displaystyle\epsilon_{abcd}\,\text{D}\mathcal{R}^{cd}$ $\displaystyle=0\,,$
(18a) $\displaystyle\epsilon_{abcd}\,\mathcal{R}^{bc}\wedge e^{d}$
$\displaystyle=0\,.$ (18b)

Equation (18a) resembles the Bianchi identity but its meaning is different
because $\mathcal{R}$ is not the curvature for the Lorentz
connection.777Although the (anti-)self-duality condition requires Euclidean
signature, we continue to refer to the spacetime rotations as Lorentz
transformations or $SO(4)$ rotations indistinctly. Indeed, this equation
reduces to

$\displaystyle\text{d}\ln\Lambda(x)e_{a}\wedge e_{b}=-T_{[a}\wedge e_{b]}\,,$
(19)

which is an algebraic equation for the torsion that can be solved in terms of
derivatives of $\Lambda(x)$. Equation (18b), on the other hand, is basically
Einstein’s equations with variable $\Lambda$. In what follows we will consider
solutions of these equations that satisfy (anti-)self duality conditions.

##  III (anti-)Self-dual configurations

The existence of the generalized dual operator $\circledast$ provides us with
a concrete notion of (anti-)self-duality, that is,

$\circledast(\mathds{F}-\mathds{F}^{-})=\pm(\mathds{F}-\mathds{F}^{-})\,.$
(20)

This condition is motivated by the definition of the operator $\circledast$
and the form of the field equations (15). As with the standard notion of
duality mentioned above, the $\circledast$-graded traces (12) imply that a
(anti-)self-dual configurations (20) solve the field equations (15). These
configurations saturate the BPS bound as well, that is,

$\int_{\mathcal{M}}\langle\mathds{F}\wedge\circledast\mathds{F}\rangle=\pm\int_{\mathcal{M}}\langle\mathds{F}\wedge\mathds{F}\rangle=\mp
8\pi^{2}C\,,$ (21)

where $C$ denotes the Chern-Pontryagin index for $SU(2,2|N)$. Indeed, one can
supplement the action (10) with a topological invariant of the Pontryagin
class without modifying the field equations, in order to set all the
(anti-)self-dual configurations as ground states of the theory (see for
instance Refs. Miskovic and Olea (2009); Araneda  _et al._ (2016)), namely,

$\displaystyle
I\mp\int_{\mathcal{M}}\langle\mathds{F}\wedge\mathds{F}\rangle=\mp\frac{1}{2}\int_{\mathcal{M}}\langle\left(\mathds{F}\pm\circledast\mathds{F}\right)\wedge\circledast\left(\mathds{F}\pm\circledast\mathds{F}\right)\rangle\,.$
(22)

One can check directly that this action vanishes for configurations satisfying
(20) by virtue of (13), extending the moduli space of vacuum configurations to
all (anti-)self-dual states.

We consider the matter-free (anti-)self-dual configurations, that is, with no
fermions and vanishing internal gauge fields: $\psi=0$, ${\overline{\psi}}=0$,
$A^{I}=0$, $A=0$. We take the fields $f^{a}=\rho(x)e^{a}$,
$g^{a}=\sigma(x)e^{a}$, and $h=0$, then condition (20) alongside the relevant
components of the curvature gives

$\displaystyle\mathcal{R}_{ab}$
$\displaystyle=\pm\frac{1}{2}\epsilon_{abcd}\mathcal{R}^{cd}\equiv\pm\tilde{\mathcal{R}}_{ab}\,.$
(23)

Combining this condition with (18b) gives $R^{a}{}_{b}e^{b}=0$, which implies
covariantly constant Torsion, $\text{D}T^{a}=0$. This, in turn, forces
$\tilde{\mathcal{R}}^{ab}=\ast\mathcal{R}^{ab}$, which relates two different
notions of (anti-)self duality on-shell.

Therefore, if Eqs. (23) and (19) are satisfied, the field equations are solved
automatically due to (anti-)self duality. Moreover, one can show that the
Euclidean on-shell action is proportional to the topological Chern-Pontryagin
index when evaluated on (anti-)self-dual configurations satisfying Eq. (23),
since the Lagrangian (17) becomes

$\displaystyle\mathscr{L}_{\rm
bos}=\frac{1}{4}\epsilon_{abcd}\mathcal{R}^{ab}\wedge\mathcal{R}^{cd}=\pm\frac{1}{2}\mathcal{R}_{ab}\wedge\mathcal{R}^{ab}=\pm\frac{1}{2}R_{ab}\wedge
R^{ab}\,,$ (24)

where the last equality is a consequence of the Bianchi identity
$\text{D}T^{a}=R^{a}_{\ b}\wedge e^{b}$ and the condition $\text{D}T^{a}=0$.
Hence, the Euclidean on-shell action is

$\displaystyle I=\int_{\mathcal{M}}\mathscr{L}_{\rm
bos}=\pm\frac{1}{2}\int_{\mathcal{M}}R_{ab}\wedge R^{ab}=\pm
4\pi^{2}P_{1}[\mathcal{M}]$ (25)

where $P_{1}[\mathcal{M}]$ is the $SO(4)$ Chern-Pontryagin index that measures
the differences between the number of harmonic self-dual and anti-self-dual
forms on $\mathcal{M}$.

In what follows, we present a family of gravitational instantons with
nontrivial torsion in unconventional conformal supersymmetric TMM theory; some
of them being (anti-)self-dual in the sense of Eq. (23). They represent
concrete solutions that exhibit the conditions discussed here.

##  IV Gravitational instantons

In order to find (anti-)self-dual solutions to the field equations (18a) and
(18b), we construct an ansatz for the vierbein, the “Lorentz” $SO(4)$
connection and $\Lambda(x)$, based on the $U(1)$-fibration over Kähler
manifolds. To this end, we consider the left-invariant Maurer-Cartan forms of
$SU(2)$, $\sigma_{i}$ with $i=1,2,3$; they are

$\displaystyle\sigma_{1}$
$\displaystyle=\cos\tau\,\text{d}\vartheta+\sin\tau\sin\vartheta\,\text{d}\varphi\,,$
(26a) $\displaystyle\sigma_{2}$
$\displaystyle=-\sin\tau\,\text{d}\vartheta+\cos\tau\sin\vartheta\,\text{d}\varphi\,,$
(26b) $\displaystyle\sigma_{3}$
$\displaystyle=\text{d}\tau+\cos\vartheta\,\text{d}\varphi\,.$ (26c)

These $\mathfrak{su}(2)$-valued $1$-forms satisfy the condition
$\text{d}\sigma_{i}=-\tfrac{1}{2}\epsilon_{ijk}\sigma^{j}\wedge\sigma^{k}$.
Using them as building blocks, we consider a metric ansatz whose radial slices
are conformally related to Berger’s sphere, that is,

$\displaystyle\text{d}s^{2}$
$\displaystyle=\frac{\text{d}r^{2}}{f(r)}+f(r)J^{2}(r)\sigma_{3}^{2}+h(r)\left(\sigma_{1}^{2}+\sigma_{2}^{2}\right)\,,$
(27)

where
$\sigma_{1}^{2}+\sigma_{2}^{2}=\text{d}\vartheta^{2}+\sin^{2}\vartheta\text{d}\varphi^{2}$
is the line element of the unit $2$-sphere. The metric (27) is locally
invariant under the action of the isometry group $SU(2)\times U(1)$ and
possesses the relevant properties for analyzing different (anti-)self-dual
configurations. Additionally, using the relation
$g_{\mu\nu}=\delta_{ab}e^{a}_{\mu}e^{b}_{\nu}$ in Euclidean signature, we
choose a vierbein basis as

$\displaystyle e^{1}$ $\displaystyle=\sqrt{h(r)}\,\sigma_{1}\,,$
$\displaystyle e^{2}$ $\displaystyle=\sqrt{h(r)}\,\sigma_{2}\,,$
$\displaystyle e^{3}$ $\displaystyle=\sqrt{f(r)}\,J(r)\,\sigma_{3}\,,$
$\displaystyle e^{4}$ $\displaystyle=\frac{\text{d}r}{\sqrt{f(r)}}\,.$ (28)

Analogously, demanding the Lorentz connection to remain invariant under the
action of the isometry group of the metric, the connection satisfies

$\displaystyle\omega^{12}$
$\displaystyle=\omega_{1}(r)\,\sigma_{3}+\omega_{2}(r)\,e^{4}+\mathring{\omega}^{12}\,,$
$\displaystyle\omega^{13}$
$\displaystyle=\omega_{3}(r)\,\sigma_{1}-\omega_{4}(r)\,\sigma_{2}\,,$ (29a)
$\displaystyle\omega^{14}$
$\displaystyle=\omega_{5}(r)\,\sigma_{1}-\omega_{6}(r)\,\sigma_{2}\,,$
$\displaystyle\omega^{23}$
$\displaystyle=\omega_{4}(r)\,\sigma_{1}+\omega_{3}(r)\,\sigma_{2}\,,$ (29b)
$\displaystyle\omega^{24}$
$\displaystyle=\omega_{6}(r)\,\sigma_{1}+\omega_{5}(r)\,\sigma_{2}\,,$
$\displaystyle\omega^{34}$
$\displaystyle=\omega_{7}(r)\,\sigma_{3}+\omega_{8}(r)\,e^{4}\,,$ (29c)

where $\omega_{i}(r)$, with $i=1,...,8$, are arbitrary functions of the radial
coordinate only, to be determined from the field equations, and
$\mathring{\omega}^{ab}$ is the torsion-free Levi-Civita connection defined by
$\text{d}e^{a}+\mathring{\omega}^{a}_{\ b}\wedge e^{b}=0$. Equivalently, the
most general field $\Lambda(x)$ compatible with the action of $SU(2)\times
U(1)$ isometry group implies that $\Lambda=\Lambda(r)$. Then, the field
equations for the connection (18a) are solved by

$\displaystyle\omega_{1}$
$\displaystyle=\omega_{2}=\omega_{3}=\omega_{6}=\omega_{8}=0\,,$
$\displaystyle\omega_{4}$ $\displaystyle=-\frac{J}{2}\sqrt{\frac{f}{h}}\,,$
(30a) $\displaystyle\omega_{5}$
$\displaystyle=\frac{\left(h\Lambda\right)^{\prime}}{2\Lambda}\sqrt{\frac{f}{h}}\,,$
$\displaystyle\omega_{7}$ $\displaystyle=\frac{\left(f\Lambda
J^{2}\right)^{\prime}}{2\Lambda J}\,,$ (30b)

where $f=f(r)$, $h=h(r)$, $J=J(r)$, $\Lambda=\Lambda(r)$, and prime denotes
differentiation with respect to the radial coordinate $r$.

The form of these ansätze, together with the solution of the equation for the
connection, allow the construction of two interesting families of
gravitational instantons of the Taub-NUT and Eguchi-Hanson types possessing
nontrivial torsion.

###  IV.1 Taub-NUT/Bolt-AdS-like solutions

The Taub-NUT metric, first found as a one-parameter extension of the
Schwarzschild solution in vacuum general relativity Taub (1951); Newman  _et
al._ (1963), is usually interpreted as a gravitational dyon in the sense that
the new parameter –the NUT charge– introduces a gravitomagnetic interaction
between the background geometry and test particles following geodesics Lynden-
Bell and Nouri-Zonoz (1998). The solution presents a topological defect –the
Misner string Misner (1963)–, which is the gravitational analog of the Dirac
string in the magnetic monopole. The string can be rendered unobservable by
performing an improper coordinate transformation that, in the Lorentzian
signature, introduces closed time-like curves Misner (1963). Thus, the Taub-
NUT metric is usually studied in the Euclidean continuation, which bears a
close resemblance to instantons in Yang-Mills theory Hawking (1977); Eguchi
_et al._ (1978); Gibbons and Hawking (1979); Eguchi  _et al._ (1980). However,
recent developments have led to consider the Lorentzian Taub-NUT metric again
since it was shown that the Misner string is transparent to geodesic observers
Clément  _et al._ (2015), allowing an interpretation of the NUT charge as a
genuine thermodynamic variable Hennigar  _et al._ (2019); Bordo  _et al._
(2019a, b); Ballon Bordo  _et al._ (2019); Durka (2022); Wu and Wu (2019);
Chen and Jiang (2019); Ballon Bordo  _et al._ (2020).

The Taub-NUT solution with nontrivial torsion in the unconventional conformal
supersymmetric model studied here is obtained by replacing the solutions (30)
into (18b) alongside the radial-coordinate choice $h(r)=r^{2}-n^{2}$, with $n$
being the NUT charge. Thus, $h(r)$ represents the area coordinate with the
origin at $r=n$. The hypersurfaces of constant $r$ are conformally the
Berger’s sphere. In general, $n\in\mathbb{R}$. However, in this work, we shall
focus on the case $n\in\mathbb{R}^{+}$.

The remaining functions that solve the field equations are found to be
$J(r)=2n$ and

$\displaystyle f(r)$
$\displaystyle=\frac{a\left[r\,b^{2}+3r^{2}(r+b)+bn^{2}\right]}{b\left(r^{2}-n^{2}\right)}-\frac{r(r+b)^{2}}{b\left(r^{2}-n^{2}\right)}+\frac{(r+b)^{3}\left(r\,b+n^{2}\right)}{b\ell^{2}\left(r^{2}-n^{2}\right)}\,,$
(31a) $\displaystyle\Lambda(r)$ $\displaystyle=\frac{a}{(r+b)^{2}}\,,$ (31b)

where $a$, $b$ and $\ell$ are integration constants. This is a four-parameter
family of solutions with nontrivial torsion sourced by $\Lambda(x)$ [see Eq.
(19)]. Explicitly, the components of torsion associated to the solution in Eq.
(31) can be expressed as

$\displaystyle T^{a}=-\frac{\sqrt{f(r)}}{r+b}\;e^{a}\wedge e^{4}\,,$ (32)

which vanishes at the zeroes of the metric function $f(r)$; the nuts and bolts
cases are discussed below. On the other hand, the components of the Lorentz
curvature $2$-form for this solution are

$\displaystyle R^{12}$ $\displaystyle=\mathscr{A}(r)\,e^{1}\wedge
e^{2}-2\mathscr{B}(r)\,e^{3}\wedge e^{4}\,,$ $\displaystyle R^{13}$
$\displaystyle=\mathscr{C}(r)\,e^{1}\wedge e^{3}-\mathscr{B}(r)\,e^{2}\wedge
e^{4}\,,$ (33a) $\displaystyle R^{14}$
$\displaystyle=\mathscr{C}(r)\,e^{1}\wedge e^{4}+\mathscr{B}(r)e^{2}\wedge
e^{3}\,,$ $\displaystyle R^{23}$ $\displaystyle=\mathscr{C}(r)\,e^{2}\wedge
e^{3}+\mathscr{B}(r)\,e^{1}\wedge e^{4}\,,$ (33b) $\displaystyle R^{24}$
$\displaystyle=\mathscr{C}(r)\,e^{2}\wedge e^{4}-\mathscr{B}(r)\,e^{1}\wedge
e^{3}\,,$ $\displaystyle R^{34}$ $\displaystyle=\mathscr{D}(r)\,e^{3}\wedge
e^{4}-2\mathscr{B}(r)\,e^{1}\wedge e^{2}\,,$ (33c)

where

$\displaystyle\mathscr{A}(r)$
$\displaystyle=\frac{1}{r^{2}-n^{2}}\left[1-\frac{f\left(b^{2}r^{2}+3n^{2}r^{2}+8bn^{2}r+3b^{2}n^{2}+n^{4}\right)}{\left(r^{2}-n^{2}\right)(r+b)^{2}}\right]\,,$
(34a) $\displaystyle\mathscr{B}(r)$
$\displaystyle=\left[\frac{fn}{2\left(r^{2}-n^{2}\right)}\right]^{\prime}\,,$
(34b) $\displaystyle\mathscr{C}(r)$
$\displaystyle=-\left[\frac{f\left(br+n^{2}\right)}{2\left(r^{2}-n^{2}\right)(r+b)}\right]^{\prime}-\frac{f\left(b^{2}-n^{2}\right)}{2\left(r^{2}-n^{2}\right)(r+b)^{2}}\,,$
(34c) $\displaystyle\mathscr{D}(r)$
$\displaystyle=-\frac{f^{\prime\prime}}{2}+\frac{f^{\prime}}{r+b}-\frac{f}{(r+b)^{2}}\,.$
(34d)

Additionally, the asymptotic behavior of the solution as $r\to\infty$ is given
by

$\displaystyle f(r)$
$\displaystyle\simeq\frac{r^{2}}{\ell^{2}}+\frac{r}{b\ell^{2}}\left(3a\ell^{2}-\ell^{2}+n^{2}+3b^{2}\right)\
+\frac{1}{\ell^{2}}\left(3a\ell^{2}-2\ell^{2}+4n^{2}+3b^{2}\right)$
$\displaystyle+\frac{1}{r\,b\ell^{2}}\left[\ell^{2}\left(3an^{2}+(a-1)b^{2}-n^{2}\right)+b^{4}+n^{4}+6b^{2}n^{2}\right]+\mathcal{O}(r^{-2})\,,$
(35a) $\displaystyle\Lambda(r)$
$\displaystyle\simeq\frac{a}{r^{2}}-\frac{2ab}{r^{3}}+\frac{3ab^{2}}{r^{4}}+\mathcal{O}(r^{-5})\,.$
(35b)

The metric function exhibits the typical linear behavior in the radial
coordinate as $r\to\infty$ that appears in solutions to conformal gravity
Riegert (1984); Lu  _et al._ (2012); Liu  _et al._ (2013); Corral  _et al._
(2021), producing a space with weakened local AdS asymptotics. However, as
shown by Maldacena in Ref. Maldacena (2011), one can impose Newmann boundary
conditions on the Fefferman-Graham expansion to select only the Einstein modes
of the theory, reducing four-dimensional conformal gravity to Einstein-AdS
gravity. In the case of the torsional Taub-NUT instanton presented here, the
Riemannian curvature, i.e. the one constructed from the Levi-Civita connection
$\mathring{\omega}^{a}_{\ b}$, possesses an asymptotically locally AdS
behavior with curvature radius $\ell^{-2}$. The Lorentz curvature $2$-form is
asymptotically locally flat rather than asymptotically locally AdS. This is
due to the fact that the leading order in the asymptotic expansion of the
torsion, i.e.,

$\displaystyle
T^{a}=-\left[\frac{1}{\ell}+\frac{3a\ell^{2}+b^{2}-\ell^{2}+n^{2}}{2b\ell\,r}+\mathcal{O}(r^{-2})\right]e^{a}\wedge
e^{4}\,,$ (36)

is asymptotically constant. This constant in the contorsion contribution to
the Lorentz curvature cancels the constant term from the Riemann curvature.
Then, the fall-off of the Lorentz curvature $2$-form in Eq. (33) is encoded in
the asymptotic expansion of its components as $r\to\infty$, that is,

$\displaystyle\mathscr{A}(r)$
$\displaystyle=\frac{\ell^{2}-b^{2}-3n^{2}}{\ell^{2}\,r^{2}}+\mathcal{O}(r^{-3})\,,$
(37a) $\displaystyle\mathscr{B}(r)$
$\displaystyle=\frac{n\left(\ell^{2}-n^{2}-3a\ell^{2}-3b^{2}\right)}{2b\ell^{2}r^{2}}+\mathcal{O}(r^{-3})\,,$
(37b) $\displaystyle\mathscr{C}(r)$
$\displaystyle=\frac{3a\ell^{2}+b^{2}-\ell^{2}+3n^{2}}{2\ell
r^{2}}+\mathcal{O}(r^{-3})\,,$ (37c) $\displaystyle\mathscr{D}(r)$
$\displaystyle=\frac{\ell^{2}-b^{2}-3n^{2}}{\ell^{2}r^{2}}+\mathcal{O}(r^{-3})\,.$
(37d)

These configurations are therefore asymptotically locally Lorentz-flat
solutions. Similar locally Lorentz-flat configurations are solutions of three-
dimensional Einstein-AdS gravity Alvarez  _et al._ (2014b), which can also be
formulated as a Chern-Simons theory for the AdS group (for a review see
Zanelli (2005); Hassaine and Zanelli (2016)). On the other hand,
asymptotically locally Lorentz-flat black holes have been reported in Ref.
Alvarez  _et al._ (2023b) in four-dimensional unconventional conformal
supersymmetric TMM theory.

There are two distinct topologies that can be obtained from the solution (31).
They are characterized by the dimensionality of the set of fixed points of the
Killing vector $\xi=\partial_{\tau}$. In particular, they correspond to nuts
or bolts if either of the following conditions are met

Nut: $\displaystyle f(n)$ $\displaystyle=0\,$ and $\displaystyle\beta_{\tau}$
$\displaystyle=\frac{4\pi}{f^{\prime}(r)}\bigg{|}_{r=n}\,,$ (38) Bolt:
$\displaystyle f(r_{h})$ $\displaystyle=0\,$ and $\displaystyle\beta_{\tau}$
$\displaystyle=\frac{4\pi}{f^{\prime}(r)}\bigg{|}_{r=r_{h}}\,,$ (39)

where $\beta_{\tau}$ is the period of the Euclidean time, i.e.
$\tau\sim\tau+\beta_{\tau}$ and $r_{h}>n$ is the bolt radius. The NUT
condition (38) fixes one integration constant as

$\displaystyle a=\frac{(b+n)[\ell^{2}-(b+n)^{2}]}{\ell^{2}(b+3n)}\equiv a_{\rm
nut}\,.$ (40)

Then, the metric function $f(r)$ and the scalar $\Lambda(r)$ become

$\displaystyle f_{n}(r)$
$\displaystyle=\frac{(r-n)(2r+b+n)}{(b+3n)(r+n)}+\frac{(r-n)^{2}\left[r\,b+3n(r+b)+n^{2}\right]}{\ell^{2}(b+3n)(r+n)}\,,$
(41a) $\displaystyle\Lambda_{n}(r)$
$\displaystyle=\frac{(b+n)\left[\ell^{2}-(b+n)^{2}\right]}{\ell^{2}(b+3n)(r+b)^{2}}\,.$
(41b)

Remarkably, this solution is locally self-dual in the sense of (23) and
possesses nontrivial torsion sourced by $\Lambda(x)$. The absence of conical
singularities at $r=n$ implies that the Euclidean-time coordinate has to be
identified as $0\leq\tau\leq 8\pi n$, rendering the topology near the origin
that of $\mathbb{R}^{4}$.

Depending on the region in the parameter space, the metric function $f_{n}(r)$
could have a single root at $r=n$ and, in other cases, a second one at
$r=r_{+}>n$. This produces a compact self-dual Taub-NUT metric. The different
behavior of the solution depending on the region in the parameter space are
given in Figure 1.

Figure 1: Parameter space in the case of NUT. In the leftmost plot, the gray
region corresponds to solutions with two roots to the equation $f(\rho)=0$,
with $\rho\equiv r/n$. The black line corresponds to solutions with only one
root and the white region corresponds to solutions with $f(r)>0$ for $r>n$.
The orange curve is a generic representative of the gray region
($\ell^{2}=n^{2}$, $b=-(5/2)n$), and the blue curve is a generic
representative of the white region ($\ell^{2}=9n$, $b=-2n$). These two cases
are shown in the second plot. The third plot is a zoom-in of the region around
$f(\rho)=0$.

The Taub-bolt case, on the other hand, leads to an instanton with a horizon
located at the largest root of the polynomial $f(r_{h})=0$. Indeed, the
condition (39) implies that the following relation between the parameters must
hold

$\displaystyle
a=\frac{\left[r_{h}(\ell^{2}-n^{2}-b^{2})-b\left(r_{h}^{2}+n^{2}\right)\right](r_{h}+b)^{2}}{\ell^{2}\left[n^{2}b+b^{2}r_{h}+3r_{h}^{2}(b+r_{h})\right]}\equiv
a_{\rm bolt}\,.$ (42)

This condition, in turn, implies that metric function $f(r)$ and the scalar
$\Lambda(r)$ are

$\displaystyle f_{b}(r)$
$\displaystyle=-\frac{r(r+b)^{2}}{b\left(r^{2}-n^{2}\right)}+\frac{(r+b)^{3}\left(r\,b+n^{2}\right)}{b\ell^{2}\left(r^{2}-n^{2}\right)}$
$\displaystyle-\frac{(r_{h}+b)^{2}\left[r\,b^{2}+b\left(3r^{2}+n^{2}\right)+3r^{3}\right]\left[r_{h}b^{2}+b\left(r_{h}^{2}+n^{2}\right)+r_{h}\left(n^{2}-l^{2}\right)\right]}{b\ell^{2}\left(r^{2}-n^{2}\right)\left[n^{2}b+b^{2}r_{h}+3r_{h}^{2}(b+r_{h})\right]}\,,$
(43a) $\displaystyle\Lambda_{b}(r)$
$\displaystyle=\frac{\left[r_{h}(\ell^{2}-n^{2}-b^{2})-b\left(r_{h}^{2}+n^{2}\right)\right](r_{h}+b)^{2}}{\ell^{2}(r+b)^{2}\left[n^{2}b+b^{2}r_{h}+3r_{h}^{2}(b+r_{h})\right]}\,,$
(43b)

respectively. One can check that the condition $f_{b}(r_{h})=0$ is satisfied.
The absence of conical singularities at the bolt implies that
$0\leq\tau\leq\beta_{\tau}$, where the period in Euclidean time is

$\displaystyle\beta_{\tau}=\frac{4\pi\ell^{2}\left[n^{2}b+b^{2}r_{h}+3r_{h}^{2}(r_{h}+b)\right]}{(r_{h}+b)\left[3r_{h}^{2}(r_{h}+b)+3r_{h}\left(\ell^{2}-n^{2}\right)+b\left(\ell^{2}-3n^{2}\right)\right]}\,.$
(44)

The topology of the Taub-bolt instanton is different from that of Taub-NUT.
The former connects a two-dimensional set of fixed points of the Killing
vector $\partial_{\tau}$ with a locally Lorentz-flat asymptotic region
$r\to\infty$ rather than a zero-dimensional set as in the case of NUT. The
whole space has the topology of $\mathbb{CP}^{2}$ with a point removed and
sent to infinity Page (1978). Even though this space is not locally self-dual
as in the Taub-NUT case, one can check that it is asymptotically locally self-
dual in the sense of Eq. (23).

###  IV.2 Eguchi-Hanson-like solutions

There is another anti-self-dual gravitational instanton in the space of
solutions of this theory. In the context of Einstein gravity, this solution
was first found by Eguchi and Hanson in Eguchi and Hanson (1979, 1978). For
vanishing cosmological constant, it represents a Ricci-flat asymptotically
locally Euclidean space with $\mathbb{R}^{4}/\mathbb{Z}_{2}$ topology; the
solution with nonvanishing cosmological term was studied in Ref. Pedersen
(1985). The hypersurfaces of constant radial coordinate have the topology of
$\mathbb{S}^{3}/\mathbb{Z}_{2}=\mathbb{RP}^{3}$ and its isometry group is
$U(1)\times SU(2)/\mathbb{Z}_{2}\simeq U(2)$. The Eguchi-Hanson (EH) metric
was shown to be relevant in the resolution of orbifold singularities Gibbons
and Hawking (1978, 1979); Nitta and Uzawa (2021) and it has been used to
construct higher-dimensional solutions with nontrivial topology in Einstein
gravity and beyond Clarkson and Mann (2006a, b); Ishihara  _et al._ (2006);
Chng  _et al._ (2006); Wong and Mann (2012a, b); Hendi  _et al._ (2012);
Durgut  _et al._ (2023).

The generalized EH solution in the unconventional conformal supersymmetric
theory studied here is found by choosing $h(r)=r^{2}/4$ in Eq. (27). Then, we
find that the field equations are solved by $J^{2}(r)=h(r)$ and

$\displaystyle f(r)$
$\displaystyle=1-\frac{a^{4}}{r^{4}}-\frac{q^{2}}{r^{2}}+\frac{4pa^{4}\,r^{2}}{q^{2}}+pr^{4}\,,$
(45a) $\displaystyle\Lambda(r)$
$\displaystyle=-\frac{2\left(16pa^{12}-4a^{4}q^{4}-q^{8}\right)}{q^{2}\left(2a^{4}+q^{2}\,r^{2}\right)^{2}}\,.\,$
(45b)

where $a$, $q$, and $p$ are integration constants. This is a three-parameter
family of solutions that can be regarded as a torsional extension of the
metric found in Ref. Corral  _et al._ (2021), in which the nontrivial
components of the torsion for this solution are

$\displaystyle T^{a}$
$\displaystyle=-\frac{2q^{2}r\,\sqrt{f}}{2a^{4}+q^{2}r^{2}}\,e^{a}\wedge
e^{4}\,.$ (46)

Note that the torsion vanishes as $q\to 0$ with $a\neq 0$. In this sense, $q$
can be regarded as a torsional hair whose backreaction resembles that of the
Maxwell field, as it happens in Poincaré gauge theories Cembranos and
Valcarcel (2017); Cembranos and Gigante Valcarcel (2018); Obukhov (2020). Now
the components of the Lorentz curvature $2$-form are

$\displaystyle R^{12}$ $\displaystyle=\mathscr{W}(r)\,e^{1}\wedge
e^{2}-2\mathscr{X}(r)\,e^{3}\wedge e^{4}\,,$ $\displaystyle R^{13}$
$\displaystyle=\mathscr{Y}(r)\,e^{1}\wedge e^{3}-\mathscr{X}(r)\,e^{2}\wedge
e^{4}\,,$ (47a) $\displaystyle R^{14}$
$\displaystyle=\mathscr{Y}(r)\,e^{1}\wedge e^{4}+\mathscr{X}(r)\,e^{2}\wedge
e^{3}\,,$ $\displaystyle R^{23}$ $\displaystyle=\mathscr{Y}(r)\,e^{2}\wedge
e^{3}+\mathscr{X}(r)\,e^{1}\wedge e^{4}\,,$ (47b) $\displaystyle R^{24}$
$\displaystyle=\mathscr{Y}(r)\,e^{2}\wedge e^{4}-\mathscr{X}(r)\,e^{1}\wedge
e^{3}\,,$ $\displaystyle R^{34}$ $\displaystyle=\mathscr{Z}(r)\,e^{3}\wedge
e^{4}-2\mathscr{X}(r)\,e^{1}\wedge e^{2}\,,$ (47c)

where

$\displaystyle\mathscr{W}(r)$
$\displaystyle=\frac{8fa^{4}q^{2}}{\left(2a^{4}+q^{2}r^{2}\right)^{2}}-\frac{4\left(f-1\right)}{r^{2}}\,,$
(48a) $\displaystyle\mathscr{X}(r)$ $\displaystyle=\frac{f^{\prime}}{2r}\,,$
(48b) $\displaystyle\mathscr{Y}(r)$
$\displaystyle=\frac{16fa^{4}q^{2}r-f^{\prime}\left(4a^{8}-q^{4}r^{4}\right)}{2r\left(2a^{4}+q^{2}r^{2}\right)^{2}}\,,$
(48c) $\displaystyle\mathscr{Z}(r)$
$\displaystyle=-\frac{\left[f^{\prime}r\left(2a^{4}+q^{2}r^{2}\right)^{2}\right]^{\prime}+2f^{\prime}\left(2a^{4}-q^{2}r^{2}\right)^{2}-8f^{\prime}q^{4}r^{4}-16fa^{4}q^{2}r}{2r\left(2a^{4}+q^{2}r^{2}\right)^{2}}\,.$
(48d)

If $p\neq 0$, this solution has a curvature singularity for $r\to\infty$,
while its action and topological invariants diverge. This can be seen from the
asymptotic expansion of the torsion $2$-form for large $r$,

$\displaystyle T^{a}$
$\displaystyle=-\left[2\sqrt{p}\;r+\frac{q^{4}-4a^{8}p}{q^{4}\sqrt{p}\;r^{3}}+\mathcal{O}(r^{-5})\right]e^{a}\wedge
e^{4}\,.$ (49)

On the other hand, the fall-off of the Lorentz curvature $2$-form in this non-
coordinate basis as $r\to\infty$ is characterized by

$\displaystyle\mathscr{W}(r)$
$\displaystyle=-4pr^{2}-\frac{8pa^{4}}{q^{2}}+\frac{4\left(2a^{4}q^{4}+q^{8}-8a^{12}p\right)}{q^{6}r^{4}}+\mathcal{O}(r^{-6})\,,$
(50a) $\displaystyle\mathscr{X}(r)$
$\displaystyle=2pr^{2}+\frac{4pa^{4}}{q^{2}}+\frac{q^{2}}{r^{4}}+\mathcal{O}(r^{-6})\,,$
(50b) $\displaystyle\mathscr{Y}(r)$
$\displaystyle=2pr^{2}+\frac{4pa^{4}}{q^{2}}+\frac{8a^{4}q^{4}+q^{8}-32a^{12}p}{q^{6}r^{4}}+\mathcal{O}(r^{-6})\,,$
(50c) $\displaystyle\mathscr{Z}(r)$
$\displaystyle=-4pr^{2}-\frac{8pa^{4}}{q^{2}}+\frac{4\left(2a^{4}q^{4}+q^{8}-8a^{12}p\right)}{q^{6}r^{4}}+\mathcal{O}(r^{-6})\,.$
(50d)

Thus, if $p\neq 0$, this solution cannot be regarded as a gravitational
instanton, whereas if $p=0$ the solution becomes

$\displaystyle f(r)$
$\displaystyle=1-\frac{a^{4}}{r^{4}}-\frac{q^{2}}{r^{2}}\,,$ (51a)
$\displaystyle\Lambda(r)$
$\displaystyle=\frac{2q^{2}\left(4a^{4}+q^{4}\right)}{\left(2a^{4}+q^{2}\,r^{2}\right)^{2}}\,,\,$
(51b)

and it is anti-self dual in the sense of Eq. (23). This solution has been
studied in the context of Einstein gravity in Refs. Xiao (2004); Chen and
Zhang (2021); Yang and Zhang (2023). Indeed, the asymptotic behavior of the
torsion in this case is given by

$\displaystyle T^{a}$
$\displaystyle=-\left[\frac{2}{r}-\frac{q^{4}+4a^{4}}{q^{2}r^{3}}-\frac{\left(q^{4}+4a^{4}\right)\left(q^{4}-8a^{4}\right)}{4q^{4}r^{5}}+\mathcal{O}(r^{-7})\right]e^{a}\wedge
e^{4}\,,$ (52)

while the asymptotic expansion for the Lorentz curvature $2$-form can be read
explicitly from Eq. (50) by taking $p=0$. The solution of Eq. (51) has a bolt
at $r=r_{h}$ defined by the largest real positive root of the polynomial
$f(r_{h})=0$, that is,

$\displaystyle r_{h}^{2}=\frac{1}{2}\left(q^{2}+\sqrt{q^{4}+4a^{2}}\right)\,.$
(53)

This bolt exists for all $a,q\in\mathbb{R}$ as the discriminant is always
positive definite. The absence of conical singularities at the bolt, in turn,
demands that the Euclidean-time coordinate has to be identified according to
$0\leq\tau\leq\beta_{\tau}$, where

$\displaystyle\beta_{\tau}=\frac{8\pi}{r_{h}\,f^{\prime}|_{r=r_{h}}}=\frac{4\pi
r_{h}^{4}}{2a^{4}+q^{2}r_{h}^{2}}\,.$ (54)

In the vanishing torsion limit, i.e. $q\to 0$, the bolt is located at
$r_{h}=a$ and the period of the Euclidean time is $2\pi$, giving a topology of
$\mathbb{S}^{3}/\mathbb{Z}_{2}$ as in the original Eguchi-Hanson metric Eguchi
and Hanson (1979, 1978).

Additionally, if $p\to 0$ and $q\to 0$ while keeping
$p/q^{2}\equiv-\Lambda_{0}/(24a^{4})$ fixed, the solution presented in Eq.
(45) is continuously connected to the Eguchi-Hanson metric of general
relativity with cosmological constant found in Ref. Pedersen (1985), that is,

$\displaystyle
f(r)=1-\frac{a^{4}}{r^{4}}-\frac{\Lambda_{0}\,r^{2}}{6}\;\;\;\;\;\mbox{and}\;\;\;\;\;\Lambda(r)=\Lambda_{0}\,,$
(55)

where $\Lambda_{0}$ is an integration constant. Indeed, one can check using
Eq. (19) that the torsion vanishes in this case. If $\Lambda_{0}\neq 0$, the
hypersurfaces of constant radius have the topology of
$\mathbb{S}^{3}/\mathbb{Z}_{k}$ with $k$ an integer if $0\leq\tau\leq 4\pi/k$
and

$\displaystyle r_{h}=\sqrt{-\frac{2(k-2)}{\Lambda_{0}}}\,.$ (56)

If $\Lambda_{0}>0$, the existence of a bolt demands that $k<2$. In such a
case, the radial coordinate is bounded since it appears a cosmological
horizon. On the contrary, if $\Lambda_{0}<0$, then $k>2$, and the radial
coordinate is unbounded from above. The case $\Lambda_{0}=0$ needs to be
treated separately. In that case, the metric (55) becomes the standard EH
metric of Ref. Eguchi and Hanson (1979, 1978) with $r_{h}=a$ and
$\mathbb{S}^{3}/\mathbb{Z}_{2}$ topology.

##  V Global properties

In this section, we obtain the global properties of the solutions such as
their Euclidean on-shell actions and topological invariants. The first is an
important quantity that allows us to obtain the partition function of the
system to first order in the saddle-point approximation through the relation
$\ln\mathcal{Z}\approx-I_{E}$. Here, we only consider (anti-)self-dual
instantons in the sense of Eq. (23) for the sake of simplicity, as the value
of their Euclidean on-shell action is proportional to the Chern-Pontryagin
index as we showed in Eq. (25). In the case of (anti-)self-dual Taub-NUT/AdS
solution in Eq. (41), its action is

$\displaystyle I_{\rm on-
shell}=4\pi^{2}P_{1}[\mathcal{M}]=\frac{8\pi^{2}\left(\ell^{2}-4n^{2}\right)\left(b^{2}+6nb^{2}+17n^{2}\right)}{\ell^{4}(b+3n)^{2}}\,.$
(57)

Note that if $\ell=2n$ the Chern-Pontryagin index vanishes for the self-dual
solution in Eq. (41). This is related to the fact that at this point in
parameter space the solution becomes global (Euclidean) AdS with a nontrivial
slicing Johnson (2014a, b), which can be rendered as the ground state of the
system. On the other hand, the Euclidean on-shell action of the torsional
anti-self-dual EH instanton (51) is

$\displaystyle I_{\rm on-
shell}=-4\pi^{2}P_{1}[\mathcal{M}]=\frac{4\pi^{2}\left(q^{4}-4r_{h}^{2}q^{2}+6r_{h}^{4}\right)}{r_{h}^{2}\left(q^{2}-2r_{h}^{2}\right)}\,.$
(58)

In the vanishing torsion limit, i.e. $q\to 0$, the action is equal to
$12\pi^{2}$ giving a Chern-Pontryagin index $P_{1}[\mathcal{M}]=-3$, a well-
known result of the original EH space Eguchi and Hanson (1979, 1978).

The Euler characteristic of these configurations can be computed as Eguchi
_et al._ (1980)

$\displaystyle\chi(\mathcal{M})$
$\displaystyle=-\frac{1}{32\pi^{2}}\left[\int_{\mathcal{M}}\epsilon_{abcd}R^{ab}\wedge
R^{cd}-2\int_{\partial\mathcal{M}}\epsilon_{abcd}\,\theta^{ab}\wedge\left(R^{cd}-\frac{2}{3}\,\theta^{a}_{\
c}\wedge\theta^{cd}\right)\right]\,,$ (59)

where $\theta^{ab}=\omega^{ab}-\omega_{0}^{ab}$ is the second fundamental form
possessing only normal components to the boundary $\partial\mathcal{M}$ and
$\omega_{0}^{ab}$ denotes the Lorentz connection of the product space having
only tangential components at the boundary. To compute the second fundamental
form, we evaluate the torsional components at $r=r_{0}$ with $r_{0}$ constant,
such that the full Lorentz connection $\omega_{0}^{ab}$ has tangential
components at the boundary. In the case of the novel torsional self-dual Taub-
NUT/AdS of Eq. (41), the nontrivial components of $\theta^{ab}$ are given by

$\displaystyle\theta^{14}$
$\displaystyle=\frac{(br_{0}+n^{2})\Phi(r_{0})}{\ell\left(r_{0}^{2}-n^{2}\right)(b+r_{0})}\,e^{1}\,,$
(60a) $\displaystyle\theta^{24}$
$\displaystyle=\frac{(br_{0}+n^{2})\Phi(r_{0})}{\ell\left(r_{0}^{2}-n^{2}\right)(b+r_{0})}\,e^{2}\,,$
(60b) $\displaystyle\theta^{34}$
$\displaystyle=-\left[\frac{(br_{0}+n^{2})\Phi(r_{0})}{\ell\left(r_{0}^{2}-n^{2}\right)(b+r_{0})}-\frac{\left(\ell^{2}-4n^{2}\right)(r_{0}-n)(b+r_{0})}{\ell(r_{0}+n)^{2}(3n+b)\Phi(r_{0})}\right]e^{3}\,,$
(60c)

where we have defined

$\displaystyle\Phi(r_{0})=\sqrt{\frac{(r_{0}-n)\left[b(r_{0}^{2}+\ell^{2})+2\ell^{2}r_{0}+n(3r_{0}^{2}+\ell^{2}+2br_{0})-n^{2}(2r_{0}+3b)-n^{3}\right]}{(r_{0}+n)(3n+b)}}\,.$
(61)

Using this result, direct evaluation of the Euler characteristic for the self-
dual Taub-NUT/AdS instanton yields

$\displaystyle\chi(\mathcal{M})=1\,.$ (62)

A similar procedure carried out for the Taub-bolt instanton gives
$\chi(\mathcal{M})=2$. Then, the Euler characteristics of the Taub-NUT and
Taub-bolt spaces differ by one unit, which reflects their different number of
boundaries. Moreover, these values match those of the Riemannian case, showing
that the affine properties of these spaces do not change the topological
invariants of the Euler class.

In the case of the anti-self-dual EH solution, we follow the same strategy: we
consider a product space of constant radial coordinate, i.e. $r=r_{0}$, such
that the Lorentz connection has only tangential components at the boundary.
Then, the nontrivial components of the second-fundamental form are

$\displaystyle\theta^{14}$
$\displaystyle=\frac{\left(2a^{4}-q^{2}r_{0}^{2}\right)\sqrt{r_{0}^{4}-q^{2}r_{0}^{2}-a^{4}}}{r_{0}^{3}\left(2a^{4}+q^{2}r_{0}^{2}\right)}\,e^{1}\,,$
(63a) $\displaystyle\theta^{24}$
$\displaystyle=\frac{\left(2a^{4}-q^{2}r_{0}^{2}\right)\sqrt{r_{0}^{4}-q^{2}r_{0}^{2}-a^{4}}}{r_{0}^{3}\left(2a^{4}+q^{2}r_{0}^{2}\right)}\,e^{2}\,,$
(63b) $\displaystyle\theta^{34}$
$\displaystyle=\frac{2a^{8}+3a^{4}q^{2}r_{0}^{2}+2a^{4}r_{0}^{4}+2q^{4}r_{0}^{4}-q^{2}r_{0}^{6}}{r_{0}^{3}\left(2a^{4}+q^{2}r_{0}^{2}\right)\sqrt{r_{0}^{4}-q^{2}r_{0}^{2}-a^{4}}}\,e^{3}\,.$
(63c)

Then, definition (59) yields the Euler characterstic of the torsional
extension of the EH geometry as

$\displaystyle\chi(\mathcal{M})=2\,.$ (64)

Computing the index theorem for the Dirac operator is more involved. For non-
compact manifolds, one needs to introduce the codimension-1 boundary integral
of the Chern-Simons form that can be evaluated directly from
$\int_{\partial\mathcal{M}}\theta^{a}_{\ b}\wedge R^{b}_{\ a}$. Additionally,
one needs to include the Atiyah-Patodi-Singer $\eta$-invariant that measures
the difference between the positive and negative eigenvalues of the tangential
components of the Dirac operator Atiyah  _et al._ (1973). In absence of
torsion, one can use the Hitchin formula Hitchin (1974) to compute the
$\eta$-invariant directly. Nevertheless, in the presence of torsion, this
formula does not apply and one needs to evaluate this invariant from its
definition by using the Riemann-zeta function regularization procedure Atiyah
_et al._ (1973). Moreover, the Dirac index receives corrections coming from
the Nieh-Yan invariant Hojman  _et al._ (1980); Nelson (1980); Nieh and Yan
(1982), since it has been shown that this term contributes nontrivially to the
axial anomaly Chandia and Zanelli (1997). This is certainly an interesting
aspect that goes beyond the scope of this work and we plan to revisit it in
the future.

##  VI Summary

In this work we have considered the definition of (anti-)self duality brought
in by unconventional supersymmetry. In the bosonic sector of the theory, this
(anti-)self duality corresponds to conditions on the bosonic curvature
components that saturate a BPS bound and, if the action is augmented by a
topological term of the Pontryagin class, the moduli space of vacuum solutions
of the theory is enlarged to include all the (anti-)self-dual configurations
as ground states.

In the bosonic sector of the theory, we obtained explicit solutions of
conformal unconventional supersymmetry that are (anti-)self dual. In
particular, we focused on gravitational instantons constructed out of the
$U(1)$-fibration of Einstein-Kähler manifolds that generalize the Taub-
NUT/Bolt-AdS and Eguchi-Hanson spaces of general relativity including torsion.
In both cases, the torsion is sourced by the compensating fields associated to
the AdS boosts and special conformal transformations, but these fields are
auxiliary and do not represent propagating deegrees of freedom. These
solutions are shown to be continuously connected to their Einsteinian
counterparts in the vanishing torsion limit.

Additionally, the global properties of these solutions such as the action and
their topological invariants have been computed. In the bosonic sector, the
action turns out to be proportional to the Chern-Pontryagin index of $SO(3,1)$
that is explicitly computed; this is usually the case forr instantons in Yang-
Mills theory. Although the geometry of these solutions is different from that
of their Riemannian counterparts as they include nontrivial torsion, the value
of the Euler characteristic is insensitive to torsion, showing that the
different affine properties of these spaces do not modify this aspect of their
topology.

Interesting questions remain open. For instance, the contribution of these
(anti-)self-dual configurations to the Dirac index is certainly of great
interest. This could provide a connection between torsion and the axial
anomaly as shown in Chandia and Zanelli (1997). Moreover, since the solutions
presented here are non-compact, one would expect a boundary contribution to
the topological invariants of the Pontryagin class. Although bulk
contributions of torsion to the axial anomaly have been explored, the presence
of boundaries implies that an extended analysis is needed. Finally, since
these configurations saturate a BPS bound, it can be expected that they
preserve a fraction of the supersymmetry only. We plan to study the existence
of Killing spinors in these spaces to address this aspect in the future.

###### Acknowledgements.

We thank to Fabrizio Canfora, Oscar Castillo-Felisola, Nelson Merino, Omar
Valdivia, and Mauricio Valenzuela for insightful comments and discussions.
This work has been partially funded by Agencia Nacional de Investigación y
Desarrollo (ANID) through FONDECYT grants 11200025, 1201208, 1210500, 1220862
and 1230112.

##  Appendix A Fundamental representation of $SU(2,2|N)$

Throughout this work, we consider the following representation of $SU(2,2|N)$

$\displaystyle(\mathds{J}_{a})^{A}_{\ B}$
$\displaystyle=\frac{s}{2}(\gamma_{a})^{\alpha}_{\ \beta}\delta^{A}_{\
\alpha}\delta^{\beta}_{\ B}=\frac{s}{2}(\gamma_{a})^{A}_{\ B}\,,$
$\displaystyle(\mathds{J}_{ab})^{A}_{\ B}$
$\displaystyle=\frac{1}{4}[\gamma_{a},\gamma_{b}]^{A}_{\
B}=(\Sigma_{ab})^{A}_{\ B}\,,$ (65) $\displaystyle(\mathds{K}_{a})^{A}_{\ B}$
$\displaystyle=\frac{1}{2}(\tilde{\gamma}_{a})^{A}_{\ B}\,,$
$\displaystyle(\mathds{D})^{A}_{\ B}$
$\displaystyle=\frac{1}{2}(\gamma_{5})^{A}_{\ B}\,,$ (66)
$\displaystyle(\mathds{T}_{I})^{A}_{\ B}$
$\displaystyle=\frac{i}{2}(\lambda_{I}^{t})_{\ B}^{A}\,,$
$\displaystyle(\mathds{Q}^{\alpha}_{i})^{A}_{\ B}$
$\displaystyle=\delta^{A}_{i}\delta^{\alpha}_{B}\,,$ (67)
$\displaystyle(\overline{\mathds{Q}}_{\alpha}^{i})^{A}_{\ B}$
$\displaystyle=\delta^{A}_{\alpha}\delta^{i}_{B}\,,$
$\displaystyle\mathds{Z}^{A}_{\ B}$
$\displaystyle=\left(i\delta^{A}_{\alpha}\delta^{\alpha}_{B}+\frac{4i}{N}\delta^{A}_{i}\delta^{i}_{B}\right)\,,$
(68)

where $\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$, satisfying
$(\gamma_{5})^{2}=\mathds{1}$, and we have defined
$\tilde{\gamma}^{a}=-\gamma_{5}\gamma^{a}$. The $\gamma$-matrices are in a
$4\times 4$ spinor-representation ($\alpha,\beta,\cdots$ run from 1 to 4). The
indices of the tangent space are $a,b=0,1,2,3$ while indices in the adjoint
representation of $\mathfrak{su}(N)$ take values $I,J=1,2,\ldots,N^{2}-1$, and
in the fundamental representation they take the values $i,j=1,2,\ldots,N$. The
$\gamma$-matrices are endomorphisms and they act on spinors as

$\psi^{\alpha}\stackrel{{\scriptstyle\gamma_{a}}}{{\longrightarrow}}(\gamma_{a})^{\alpha}_{\
\beta}\psi^{\beta}\,.$ (69)

In a similar way the $\lambda$-matrices are also endomorphisms and they act on
spinors as

$\psi^{\alpha}_{i}\stackrel{{\scriptstyle\lambda_{I}}}{{\longrightarrow}}(\lambda_{I})_{i}^{\
j}\psi^{\alpha}_{j}\,,$ (70)

and they satisfy the algebra $[\lambda_{I},\lambda_{J}]=f_{IJK}\lambda^{K}$,
where indexes are raised/lowered with an Euclidean metric $\delta_{IJ}$.
Indices of the adjoint representation of $SU(2,2|N)$ are denoted by
$A,B=1,\cdots,N+4$, so we have a $N+4\times N+4$ representation. We find
convenient to split them as $A=(\alpha,i)$. Then, all the possible products
that mix spaces like $p^{i}_{\ A}q^{A}_{\ \alpha}$ are trivial. Thus, the
following relations are understood

$\displaystyle(\gamma_{a})^{A}_{\
B}=\delta^{A}_{\alpha}(\gamma_{a})^{\alpha}_{\
\beta}\delta^{\beta}_{B}\;\;\;\;\;\mbox{and}\;\;\;\;\;C_{\alpha
A}=C_{\alpha\beta}\delta^{\beta}_{A}\,.$ (71)

The generators $\mathds{J}_{a}$ and $\mathds{J}_{ab}$ form a AdS4 algebra and,
alongside $\mathds{D}$ and $\mathds{K}_{a}$, they form the conformal algebra,
that is,

$\displaystyle[\mathds{J}_{a},\mathds{J}_{b}]$
$\displaystyle=\mathds{J}_{ab}\,,$
$\displaystyle[\mathds{J}_{a},\mathds{J}_{bc}]$
$\displaystyle=\eta_{ab}\mathds{J}_{c}-\eta_{ac}\mathds{J}_{b}\,,$ (72)
$\displaystyle[\mathds{J}_{ab},\mathds{J}_{cd}]$
$\displaystyle=-(\eta_{ac}\mathds{J}_{bd}-\eta_{ad}\mathds{J}_{bc}-\eta_{bc}\mathds{J}_{ad}+\eta_{bd}\mathds{J}_{ac})\,,$
$\displaystyle[\mathds{K}_{a},\mathds{K}_{b}]$
$\displaystyle=-\mathds{J}_{ab}\,,$ (73)
$\displaystyle[\mathds{J}_{a},\mathds{K}_{b}]$
$\displaystyle=\eta_{ab}\mathds{D}\,,$
$\displaystyle[\mathds{K}_{a},\mathds{J}_{bc}]$
$\displaystyle=\eta_{ab}\mathds{K}_{c}-\eta_{ac}\mathds{K}_{b}\,,$ (74)
$\displaystyle[\mathds{D},\mathds{K}_{a}]$ $\displaystyle=-\mathds{J}_{a}\,,$
$\displaystyle[\mathds{D},\mathds{J}_{a}]$ $\displaystyle=-\mathds{K}_{a}\,.$
(75)

Including $\mathds{Q}^{\alpha}_{i}$ and $\overline{\mathds{Q}}^{i}_{\alpha}$,
the commutators close in a $\mathfrak{su}(2,2|N)$ superalgebra given by

$\displaystyle[\mathds{J}_{a},\overline{\mathds{Q}}_{\alpha}^{i}]$
$\displaystyle=\frac{1}{2}\overline{\mathds{Q}}_{\beta}^{i}(\gamma_{a})^{\beta}_{\
\alpha}\,,$ $\displaystyle[\mathds{J}_{a},\mathds{Q}^{\alpha}_{i}]$
$\displaystyle=-\frac{1}{2}(\gamma_{a})^{\alpha}_{\
\beta}\mathds{Q}^{\beta}_{i}\,,$ (76)
$\displaystyle[\mathds{J}_{ab},\overline{\mathds{Q}}_{\alpha}^{i}]$
$\displaystyle=\overline{\mathds{Q}}_{\beta}^{i}(\Sigma_{ab})^{\beta}_{\
\alpha}\,,$ $\displaystyle[\mathds{J}_{ab},\mathds{Q}^{\alpha}_{i}]$
$\displaystyle=-(\Sigma_{ab})^{\alpha}_{\ \beta}\mathds{Q}^{\beta}_{i}\,,$
(77) $\displaystyle[\mathds{K}_{a},\overline{\mathds{Q}}_{\alpha}^{i}]$
$\displaystyle=\frac{1}{2}\overline{\mathds{Q}}_{\beta}^{i}(\tilde{\gamma}_{a})^{\beta}_{\
\alpha}\,,$ $\displaystyle[\mathds{K}_{a},\mathds{Q}^{\alpha}_{i}]$
$\displaystyle=-\frac{1}{2}(\tilde{\gamma}_{a})^{\alpha}_{\
\beta}\mathds{Q}^{\beta}_{i}\,,$ (78)
$\displaystyle[\mathds{D},\overline{\mathds{Q}}_{\alpha}^{i}]$
$\displaystyle=\frac{1}{2}\overline{\mathds{Q}}_{\beta}^{i}(\gamma_{5})^{\beta}_{\
\alpha}\,,$ $\displaystyle[\mathds{D},\mathds{Q}^{\alpha}_{i}]$
$\displaystyle=-\frac{1}{2}(\gamma_{5})^{\alpha}_{\
\beta}\mathds{Q}^{\beta}_{i}\,,$ (79)
$\displaystyle[\mathds{T}_{I},\overline{\mathds{Q}}_{\alpha}^{i}]$
$\displaystyle=-\frac{i}{2}\overline{\mathds{Q}}_{\alpha}^{j}(\lambda_{I})^{\
i}_{j}\,,$ $\displaystyle[\mathds{T}_{I},\mathds{Q}^{\alpha}_{i}]$
$\displaystyle=\frac{i}{2}(\lambda_{I})^{\ j}_{i}\mathds{Q}^{\alpha}_{j}\,,$
(80) $\displaystyle[\mathds{Z},\overline{\mathds{Q}}_{\alpha}^{i}]$
$\displaystyle=-i(4/N-1)\overline{\mathds{Q}}_{\alpha}^{i}\,,$
$\displaystyle[\mathds{Z},\mathds{Q}^{\alpha}_{i}]$
$\displaystyle=i(4/N-1)\mathds{Q}^{\alpha}_{i}\,,$ (81)

and

$\displaystyle\\{\mathds{Q}^{\alpha}_{i},\overline{\mathds{Q}}_{\beta}^{j}\\}$
$\displaystyle=\left(\frac{1}{2s}(\gamma^{a})^{\alpha}_{\
\beta}\mathds{J}_{a}-\frac{1}{2}(\Sigma^{ab})^{\alpha}_{\
\beta}\mathds{J}_{ab}-\frac{1}{2}(\tilde{\gamma}^{a})^{\alpha}_{\
\beta}\mathds{K}_{a}+\frac{1}{2}(\gamma_{5})^{\alpha}_{\
\beta}\mathds{D}\right)\delta^{j}_{i}$
$\displaystyle\qquad\qquad\qquad+\delta^{\alpha}_{\beta}\left(-i(\lambda_{I})_{i}^{\
j}\mathds{T}_{I}-\frac{i}{4}\delta_{i}^{j}\mathds{Z}\right)\,.$ (82)

The graduation operator is given by

$\mathcal{G}^{A}_{\ B}=\delta^{A}_{\alpha}\delta^{\alpha}_{\
B}-\delta^{A}_{i}\delta^{i}_{\ B}\,,$ (83)

and it classifies generators in bosonic
$B=\\{\mathds{J}_{a},\mathds{J}_{ab},\mathds{K}_{a},\mathds{D},\mathds{T}_{I},\mathds{Z}\\}$
or fermionic $F=\\{Q^{\alpha}_{i},\overline{\mathds{Q}}^{i}_{\alpha}\\}$
representations by $[B,\mathcal{G}]=0=\\{F,\mathcal{G}\\}$, and it squares to
one, i.e., $\mathcal{G}^{2}=1$. With the graduation operator we can define an
invariant supertrace

$\langle G\rangle\equiv\operatorname{Tr}(\mathcal{G}G)=0\,.$ (84)

The supertrace have the following properties

$\langle B_{1}B_{2}\rangle=\langle B_{2}B_{1}\rangle\,,\quad\langle
BF\rangle=\langle FB\rangle\,,\quad\langle F_{1}F_{2}\rangle=-\langle
F_{2}F_{1}\rangle\,.$ (85)

All the generators $G$ in the representation are s-traceless

$\langle
G\rangle=0\,,\;\;\;\;\;\mbox{where}\;\;\;\;\;G=\\{\mathds{J}_{a},\mathds{J}_{ab},\mathds{K}_{a},\mathds{D},\mathds{T}_{I},\mathds{Z},Q^{\alpha}_{i},\overline{\mathds{Q}}^{i}_{\alpha}\\}\,.$
(86)

The quadratic combinations that give nontrivial traces are

$\displaystyle\langle\mathds{J}_{a}\mathds{J}_{b}\rangle$
$\displaystyle=\eta_{ab}\,,$
$\displaystyle\langle\mathds{J}_{ab}\mathds{J}_{cd}\rangle$
$\displaystyle=-(\eta_{ac}\eta_{bd}-\eta_{bc}\eta_{ad})\,,$ (87)
$\displaystyle\langle\mathds{K}_{a}\mathds{K}_{b}\rangle$
$\displaystyle=-\eta_{ab}\,,$ $\displaystyle\langle\mathds{D}^{2}\rangle$
$\displaystyle=+1\,,$ (88)
$\displaystyle\langle\mathds{T}_{I}\mathds{T}_{J}\rangle$
$\displaystyle=\frac{1}{2}\delta_{IJ}\,,$
$\displaystyle\langle\mathds{Z}^{2}\rangle$ $\displaystyle=4(4/N-1)\,,$ (89)
$\displaystyle\langle\mathds{Q}^{\alpha}_{i}\overline{\mathds{Q}}^{j}_{\beta}\rangle$
$\displaystyle=-\delta^{\alpha}_{\beta}\delta^{j}_{i}=-\langle\overline{\mathds{Q}}^{j}_{\beta}\mathds{Q}^{\alpha}_{i}\rangle\,.$
(90)

The $S$ operator is fundamental in getting the usual expressions for the
kinetic terms. In this case, the nontrivial traces are

$\displaystyle\langle S\mathds{D}\rangle$ $\displaystyle=2i\varepsilon_{s}\,,$
$\displaystyle\langle S\mathds{J}_{ab}\mathds{J}_{cd}\rangle$
$\displaystyle=-\varepsilon_{s}\epsilon_{abcd}=\langle\mathds{J}_{ab}S\mathds{J}_{cd}\rangle\,,$
(91) $\displaystyle\langle\mathds{J}_{a}S\mathds{K}_{b}\rangle$
$\displaystyle=-i\varepsilon_{s}\eta_{ab}=-\langle\mathds{K}_{a}S\mathds{J}_{b}\rangle\,,$
$\displaystyle\langle\mathds{Z}S\mathds{D}\rangle$
$\displaystyle=-2\varepsilon_{s}=\langle\mathds{D}S\mathds{Z}\rangle\,,$ (92)

where $\varepsilon_{s}=\pm 1$.

## References

  * Belavin  _et al._ (1975) A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Y. S. Tyupkin, Phys. Lett. B 59, 85 (1975).
  * ’t Hooft (1976a) G. ’t Hooft, Phys. Rev. Lett. 37, 8 (1976a).
  * ’t Hooft (1976b) G. ’t Hooft, Phys. Rev. D 14, 3432 (1976b), [Erratum: Phys. Rev. D 18, 2199 (1978)].
  * Diakonov and Petrov (1986) D. Diakonov and V. Y. Petrov, Nucl. Phys. B 272, 457 (1986).
  * Eguchi  _et al._ (1980) T. Eguchi, P. B. Gilkey, and A. J. Hanson, Phys. Rept. 66, 213 (1980).
  * Taub (1951) A. H. Taub, Annals Math. 53, 472 (1951).
  * Newman  _et al._ (1963) E. Newman, L. Tamburino, and T. Unti, J. Math. Phys. 4, 915 (1963).
  * Eguchi and Hanson (1978) T. Eguchi and A. J. Hanson, Phys. Lett. B 74, 249 (1978).
  * Eguchi and Hanson (1979) T. Eguchi and A. J. Hanson, Annals Phys. 120, 82 (1979).
  * Gibbons and Hawking (1978) G. W. Gibbons and S. W. Hawking, Phys. Lett. B 78, 430 (1978).
  * Hitchin (1979) N. J. Hitchin, Math. Proc. Cambridge Phil. Soc. 85, 465 (1979).
  * Calabi (1979) E. Calabi, Annales scientifiques de l’École Normale Supérieure 4e série, 12, 269 (1979).
  * Townsend (1977) P. K. Townsend, Phys. Rev. D 15, 2795 (1977).
  * MacDowell and Mansouri (1977) S. W. MacDowell and F. Mansouri, Phys. Rev. Lett. 38, 739 (1977), [Erratum: Phys. Rev. Lett. 38, 1376 (1977)].
  * Alvarez  _et al._ (2012) P. D. Alvarez, M. Valenzuela, and J. Zanelli, JHEP 04, 058 (2012).
  * Alvarez  _et al._ (2014a) P. D. Alvarez, P. Pais, and J. Zanelli, Phys. Lett. B 735, 314 (2014a).
  * Alvarez  _et al._ (2022) P. D. Alvarez, R. A. Chavez, and J. Zanelli, JHEP 02, 111 (2022).
  * Alvarez  _et al._ (2020) P. D. Alvarez, M. Valenzuela, and J. Zanelli, JHEP 07, 205 (2020).
  * Alvarez  _et al._ (2021) P. D. Alvarez, L. Delage, M. Valenzuela, and J. Zanelli, JHEP 07, 176 (2021).
  * Alvarez  _et al._ (2023a) P. D. Alvarez, R. A. Chavez, and J. Zanelli, JHEP 02, 050 (2023a).
  * Andrianopoli and D’Auria (2014) L. Andrianopoli and R. D’Auria, JHEP 08, 012 (2014).
  * Andrianopoli  _et al._ (2021) L. Andrianopoli, B. L. Cerchiai, R. Matrecano, O. Miskovic, R. Noris, R. Olea, L. Ravera, and M. Trigiante, JHEP 02, 141 (2021).
  * Miskovic and Olea (2009) O. Miskovic and R. Olea, Phys. Rev. D 79, 124020 (2009).
  * Araneda  _et al._ (2016) R. Araneda, R. Aros, O. Miskovic, and R. Olea, Phys. Rev. D 93, 084022 (2016).
  * Lynden-Bell and Nouri-Zonoz (1998) D. Lynden-Bell and M. Nouri-Zonoz, Rev. Mod. Phys. 70, 427 (1998).
  * Misner (1963) C. W. Misner, J. Math. Phys. 4, 924 (1963).
  * Hawking (1977) S. W. Hawking, Phys. Lett. A 60, 81 (1977).
  * Eguchi  _et al._ (1978) T. Eguchi, P. B. Gilkey, and A. J. Hanson, Phys. Rev. D 17, 423 (1978).
  * Gibbons and Hawking (1979) G. W. Gibbons and S. W. Hawking, Commun. Math. Phys. 66, 291 (1979).
  * Clément  _et al._ (2015) G. Clément, D. Gal’tsov, and M. Guenouche, Phys. Lett. B 750, 591 (2015), 1508.07622 .
  * Hennigar  _et al._ (2019) R. A. Hennigar, D. Kubizňák, and R. B. Mann, Phys. Rev. D 100, 064055 (2019).
  * Bordo  _et al._ (2019a) A. B. Bordo, F. Gray, and D. Kubizňák, JHEP 07, 119 (2019a).
  * Bordo  _et al._ (2019b) A. B. Bordo, F. Gray, R. A. Hennigar, and D. Kubizňák, Class. Quant. Grav. 36, 194001 (2019b).
  * Ballon Bordo  _et al._ (2019) A. Ballon Bordo, F. Gray, R. A. Hennigar, and D. Kubizňák, Phys. Lett. B 798, 134972 (2019).
  * Durka (2022) R. Durka, Int. J. Mod. Phys. D 31, 2250021 (2022).
  * Wu and Wu (2019) S.-Q. Wu and D. Wu, Phys. Rev. D 100, 101501 (2019).
  * Chen and Jiang (2019) Z. Chen and J. Jiang, Phys. Rev. D 100, 104016 (2019).
  * Ballon Bordo  _et al._ (2020) A. Ballon Bordo, F. Gray, and D. Kubizňák, JHEP 05, 084 (2020).
  * Riegert (1984) R. J. Riegert, Phys. Rev. Lett. 53, 315 (1984).
  * Lu  _et al._ (2012) H. Lu, Y. Pang, C. N. Pope, and J. F. Vazquez-Poritz, Phys. Rev. D 86, 044011 (2012).
  * Liu  _et al._ (2013) H.-S. Liu, H. Lü, C. N. Pope, and J. F. Vázquez-Poritz, Class. Quant. Grav. 30, 165015 (2013).
  * Corral  _et al._ (2021) C. Corral, G. Giribet, and R. Olea, Phys. Rev. D 104, 064026 (2021).
  * Maldacena (2011) J. Maldacena, (2011), arXiv:1105.5632 [hep-th] .
  * Alvarez  _et al._ (2014b) P. D. Alvarez, P. Pais, E. Rodríguez, P. Salgado-Rebolledo, and J. Zanelli, Phys. Lett. B 738, 134 (2014b).
  * Zanelli (2005) J. Zanelli, in  _7th Mexican Workshop on Particles and Fields_ (2005) arXiv:hep-th/0502193 .
  * Hassaine and Zanelli (2016) M. Hassaine and J. Zanelli,  _Chern-Simons (super)gravity_, 2016th ed., 100 years of general relativity, Vol. 2 (World Scientific, Hackensack, 2016).
  * Alvarez  _et al._ (2023b) P. D. Alvarez, C. Corral, and J. Zanelli, JHEP 01, 009 (2023b).
  * Page (1978) D. N. Page, Phys. Lett. B 78, 249 (1978).
  * Pedersen (1985) H. Pedersen, Classical and Quantum Gravity 2, 579 (1985).
  * Nitta and Uzawa (2021) M. Nitta and K. Uzawa, Eur. Phys. J. C 81, 513 (2021).
  * Clarkson and Mann (2006a) R. Clarkson and R. B. Mann, Class. Quant. Grav. 23, 1507 (2006a).
  * Clarkson and Mann (2006b) R. Clarkson and R. B. Mann, Phys. Rev. Lett. 96, 051104 (2006b).
  * Ishihara  _et al._ (2006) H. Ishihara, M. Kimura, K. Matsuno, and S. Tomizawa, Phys. Rev. D 74, 047501 (2006).
  * Chng  _et al._ (2006) B. Chng, R. B. Mann, and C. Stelea, Phys. Rev. D 74, 084031 (2006).
  * Wong and Mann (2012a) A. W. C. Wong and R. B. Mann, Phys. Rev. D 85, 046010 (2012a).
  * Wong and Mann (2012b) A. W. C. Wong and R. B. Mann, Phys. Rev. D 86, 124002 (2012b).
  * Hendi  _et al._ (2012) S. H. Hendi, R. B. Mann, N. Riazi, and B. Eslam Panah, Phys. Rev. D 86, 104034 (2012).
  * Durgut  _et al._ (2023) T. Durgut, R. A. Hennigar, H. K. Kunduri, and R. B. Mann, JHEP 03, 114 (2023).
  * Cembranos and Valcarcel (2017) J. A. R. Cembranos and J. G. Valcarcel, JCAP 01, 014 (2017).
  * Cembranos and Gigante Valcarcel (2018) J. A. R. Cembranos and J. Gigante Valcarcel, Phys. Lett. B 779, 143 (2018).
  * Obukhov (2020) Y. N. Obukhov, Phys. Rev. D 102, 104059 (2020).
  * Xiao (2004) Z. Xiao, Communications in Theoretical Physics 42, 235 (2004).
  * Chen and Zhang (2021) J. Chen and X. Zhang, J. Geom. Phys. 161, 104010 (2021).
  * Yang and Zhang (2023) Y. Yang and X. Zhang, Eur. Phys. J. C 83, 574 (2023).
  * Johnson (2014a) C. V. Johnson, Class. Quant. Grav. 31, 235003 (2014a).
  * Johnson (2014b) C. V. Johnson, Class. Quant. Grav. 31, 225005 (2014b).
  * Atiyah  _et al._ (1973) M. F. Atiyah, V. K. Patodi, and I. M. Singer, Bulletin of the London Mathematical Society 5, 229 (1973).
  * Hitchin (1974) N. Hitchin, Advances in Mathematics 14, 1 (1974).
  * Hojman  _et al._ (1980) R. Hojman, C. Mukku, and W. A. Sayed, Phys. Rev. D 22, 1915 (1980).
  * Nelson (1980) P. C. Nelson, Phys. Lett. A 79, 285 (1980).
  * Nieh and Yan (1982) H. T. Nieh and M. L. Yan, J. Math. Phys. 23, 373 (1982).
  * Chandia and Zanelli (1997) O. Chandia and J. Zanelli, Phys. Rev. D 55, 7580 (1997).