Asymptotic dynamics of three dimensional supergravity and higher spin gravity
revisited

Wout Merbisa, Turmoli Neogib and Arash Ranjbarc

a Dutch Institute for Emergent Phenomena & Institute for Theoretical Physics,
University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands.
b Centro de Estudios Científicos (CECs), Av. Arturo Prat 514, Valdivia, Chile.
c Division of Theoretical Physics, Rudjer Bošković Institute
Bijenička 54, 10000 Zagreb, Croatia  email<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>

ABSTRACT

We reconsider the Hamiltonian reduction of the action for three dimensional
AdS supergravity and $W_{3}$ higher spin AdS gravity in the Chern-Simons
formulation under asymptotically anti-de Sitter boundary conditions. We show
that the reduction gives two copies of chiral bosons on the boundary. In
particular, we take into account the holonomy of the Chern-Simons connection
which manifests itself as zero mode of the momentum of the boundary chiral
boson. We provide an equivalent formulation of the boundary action which we
claim to be the geometric action on symplectic leaves of a (super-)Virasoro or
a higher spin $W_{N}$ Poisson manifold in the case of supergravity or higher
spin gravity respectively, where the intersection of leaves (given in terms of
leaves representatives) can be identified as the bulk holonomy. This concludes
the extension to non-linear algebras where the notion of coadjoint
representation is not well-defined. The boundary Hamiltonian depends on a
choice of boundary conditions and is equivalent to the Schwarzian action for
corresponding Brown-Henneaux boundary conditions. We make this connection
explicit in the extended supersymmetric case. Moreover, we discuss the
geometric action in the case of $W_{3}$ AdS3 gravity in both
$\mathfrak{sl}(3)$ highest weight representations based on principal and
diagonal $\mathfrak{sl}(2)$ embeddings.

###### Contents

  1. 1 Introduction
  2. 2 Three dimensional AdS gravity
    1. 2.1 AdS3 supergravity
      1. 2.1.1 Gauge fixing
      2. 2.1.2 Supersymmetric Brown-Henneaux
    2. 2.2 $W_{3}$ higher spin gravity
      1. 2.2.1 Gauge fixing
  3. 3 Hamiltonian reduction of the super Chern-Simons action
    1. 3.1 The symplectic term
    2. 3.2 The Hamiltonian
    3. 3.3 On the holonomy
    4. 3.4 ${\cal N}=1$ super Chern-Simons
      1. 3.4.1 Hyperbolic Holonomy
      2. 3.4.2 Boundary conditions
    5. 3.5 ${\cal N}=2$ super Chern-Simons
      1. 3.5.1 Hyperbolic holonomy
      2. 3.5.2 Boundary condition
    6. 3.6 Comments on General ${\cal N}$
  4. 4 Hamiltonian reduction of $W_{3}$ higher Chern-Simons action
    1. 4.1 Chern-Simons action in the presence of zero modes
      1. 4.1.1 Hyperbolic holonomy
    2. 4.2 Different $SL(3)$ representations and highest weight gauge condition
      1. 4.2.1 Boundary Condition and Hamiltonian Reduction
      2. 4.2.2 Principal Embedding
      3. 4.2.3 Diagonal Embedding
  5. 5 Diffeomorphisms of the supercircle
    1. 5.1 Reparameterizations of the (super)circle
      1. 5.1.1 Geometric action for superconformal groups
    2. 5.2 Bosonic case
    3. 5.3 ${\cal N}=1$ supersymmetry
    4. 5.4 ${\cal N}=2$ supersymmetry
  6. 6 Conclusions
  7. A Conventions
    1. A.1 Matrix Representation for $Osp(2|2)$
  8. B Action on the symplectic leaves
  9. C Geometric quantization for (super)conformal groups
    1. C.1 Geometric actions on the coadjoint orbit
    2. C.2 Hamiltonians

##  1 Introduction

The theory of gravity in three dimensions has the remarkable feature that it
does not carry any dynamical degrees of freedom in the bulk. It is by now well
known that general relativity in three dimensions is a topological gauge
theory which can be described as a Chern-Simons theory for the appropriate
gauge (super)group [1, 2]. The gauge group reflects the isometries of the
maximally symmetric vacuum of the theory, which depends on the presence and
sign of the cosmological constant $\Lambda$. In the present work, we will
focus on negative cosmological constant $\Lambda=-1/\ell^{2}$, mainly due to
the fact that the presence of BTZ black hole solutions of [3] require a non-
vanishing negative $\Lambda$. In three dimensional gravity, all dynamical
degrees of freedom live on the asymptotic boundaries. This interesting
property of the theory has spiked the interest in the theory as a candidate
for a consistent theory of quantum gravity in three dimensions [2, 4].
However, a direct computation of the partition function of AdS3 gravity
remained cumbersome or inconclusive [5, 6].

Recently in [7], it was shown that the complete Poisson structure at the
boundary is not given solely in terms of Kac-Moody currents, but also includes
the zero modes of bulk holonomies, which can be understood as Wilson lines
stretched between the boundaries. Indeed, these Wilson lines are classically
traversable wormholes [8]. In the quantum theory and in the presence of two
boundaries, this is understood as a thermofield double description of two
entangled conformal field theory [9, 10]. This means that the quantum
description of the theory is based on a complete set of observables which are
formed by the currents and the Wilson lines/loops. The Hilbert space does not
factorize into two copies of boundary states, but the boundary theories are
coupled through global zero modes. In [7], a complete analysis was provided of
such Wilson lines in pure Chern-Simons gravity on asymptotically AdS
geometries with two boundaries, as the simplest example. It was shown that, in
the presence of non-trivial holonomies in the bulk, the action obtained from a
Hamiltonian reduction of the 3D Einstein-Hilbert action is exactly the
geometric action on the Virasoro coadjoint orbit, with the holonomy being the
orbit representative.

These results can be promoted to three dimensional supergravity models,
because these are also Chern-Simons theories with boundary conditions of the
Drinfeld-Sokolov type, implementing a Hamiltonian reduction at the boundary
[11]. The resulting asymptotic symmetry algebras are the ${\cal N}$-extended
superconformal algebras of [12, 13, 14, 15], which are linear for ${\cal
N}\leq 2$. A similar discussion in two dimensions has been considered for the
BF formulation of dilaton supergravity [16].

In the case of an annulus topology, we include the holonomies along the lines
of [7, 11] by treating separately the two chiralities. For each chirality, it
leads to a supersymmetric chiral action at each boundary coupled by radial
Wilson lines. One also finds that the system is physically described by two
sets of generators of the superconformal algebras, one at each boundary. These
generators are constrained by the holonomy matching condition and provide,
together with the global modes, a complete description of the system. The
dynamics reduces to the dynamics of these generators and of the global modes,
and can therefore be expressed in terms of geometric actions.

In the geometric construction of the action based on Kirillov-Kostant
prescription [17, 18, 19], the existence of a coadjoint representation is an
utmost necessity. However, this notion is not sufficient and will be
challenged when the boundary algebras are non-linear. This is the case, for
example, when the asymptotic charges of the theory form a non-linear algebra
as in supergravity with $\mathcal{N}>2$ or higher spin gauge theories with
non-linear ${\cal W}$-algebras [20, 21]. Then, the geometric actions can not
be formulated in terms of orbits of the coadjoint representation, since the
phase space does not provide a linear representation, but rather in terms of
the more general concept of symplectic leaves [22, 23]. When the asymptotic
symmetry algebra is linear, the boundary action can be cast in the form of the
Schwarzian action [24, 25] which has been shown to be the geometric action on
the coadjoint orbits of the asymptotic symmetry group. From the point of view
of Hamiltonian reduction, the chiral boundary theory can be also shown to be a
Schwarzian action where the holonomies appear as the constant orbit
representatives.

The generators of the asymptotic symmetry algebra form a Poisson manifold,
with a Poisson bracket that is degenerate if one focuses only on a single
boundary algebra without including the global radial Wilson lines. The
symplectic leaves of this Poisson manifold have a well-defined symplectic
structure, which is the one that enters in the action. Moreover, it happens
that the Hamiltonian reduction in the presence of non-trivial holonomy still
indubiously provides the action on the boundary. Therefore, one is able to
perform a Hamiltonian reduction in supergravity or higher spin gravity in the
presence of bulk holonomies in order to obtain the boundary theory. These
boundary actions should be considered as a candidate for a geometric action on
the symplectic leaves of the Poisson manifold. The holonomies are related to
the leave representatives; in fact, the intersections of symplectic leaves are
determined by the zero modes of asymptotic charges which in turn are given by
the holonomies.

A great advantage of our approach is that holonomies appear in the action as
dynamical fields. They are considered as time-dependent variables in the
action. Their equations of motion set them on-shell to constant variables
determined by the zero modes of the asymptotic charges. These constant
holonomies match with the constant orbit representatives. This happens to be
an important feature once one wants to quantize the theory, since in a genuine
quantum theory of gravity one should consider summing over all possible
solutions of the theory. The presence of holonomies in the action provides a
control over the solution space at the level of action.

Recently, there has been a surge of interest in the computation of the path
integral of two dimensional Jackiw-Teitelboim (JT) gravity. In two dimensions,
the partition function of gravity turns out to have a beautiful mathematical
description. It is the volume of the moduli space of hyperbolic manifolds with
constant curvature and the boundary theory is an ensemble of one dimensional
quantum mechanical models, which can be captured by random matrix theory. An
important piece of information is that in two dimensions, the partition
function is written as the exponential of a Schwarzian action functional [26].
This Schwarzian action is the boundary theory of 2D JT gravity and can be
obtained as the IR limit of the SYK model [27, 28, 29, 30]. In three
dimensions, this piece of information happened to be crucial, as the
Hamiltonian on the boundary of 3D Chern-Simons gravity is given by the sum of
two Schwarzian actions with opposing chirality [31, 25, 7]. There has been a
plethora of works in 3D along this line of thoughts [32, 33, 34, 35, 36, 37].

In this work, we provide a systematic recipe to write down the Schwarzian
action for 3D supergravity and higher spin gravity in the presence of non-
trivial holonomy. While we pay special attention to the role of bulk
holonomies, we focus on the case where the holonomy is in the hyperbolic
conjugacy class. This is because we are mostly interested to exhibit the
result for BTZ black hole solutions and these solutions are in the hyperbolic
holonomy conjugacy class. Upon Hamiltonian reduction, the boundary Hamiltonian
depends on a choice of boundary conditions and is equivalent to the Schwarzian
action for Brown-Henneaux boundary conditions. This Schwarzian action may be
considered as the starting point for computing the partition function of the
corresponding theory. There is a benefit in using this action for computing
the partition function. It already contains information on the Wilson lines,
and therefore it may be more suited in addressing the factorization problem in
3D gravity. In this paper, we only focus on the construction of the boundary
actions and details regarding computation of the partition function will be
discussed elsewhere.

Our paper is organized as follows. In Section 2 we provide a short
introduction to AdS3 supergravity and $W_{3}$ higher spin theories in three
dimensions. In each case, we discuss the gauge fixing and the corresponding
Brown-Henneaux boundary conditions. In Section 3, we perform the Hamiltonian
reduction on the super Chern-Simons theory, or equivalently, AdS3
supergravity. We first show this explicitly for three dimensional AdS
supergravity with ${\cal N}=1,2$ where one can use the Kirilov-Kostant
construction to compare. The boundary action of Chern-Simons AdS3 supergravity
for an arbitrary number of supersymmetry ${\cal N}$ is addressed in Section
3.5. The honolomy part of the action remains undetermined for a general case
and requires a case by case study. We will comment on that in Section 3.6.
Section 4 is aimed at the Hamiltonian reduction of $W_{3}$ AdS3 higher spin
theory. In this case, there are two distinct boundary conditions depending on
the $\mathfrak{sl}(2)$ embedding of the $\mathfrak{sl}(3)$ algebra. We discuss
both the principal and diagonal embedding and provide the boundary action in
each case.

In Section 5, we discuss the geometric action on the coadjoint orbit of the
group of reparameterizations of the supercircle. We show that the result of
Section 3 for ${\cal N}=1,2$ is precisely the one of the geometric action.
Appendix A provides some conventions regarding $Osp(2|2)$ representation,
appendix B comprises a brief view of action on the symplectic leaves and
appendix C is a review of geometric quantization for (super)conformal groups.

##  2 Three dimensional AdS gravity

###  2.1 AdS3 supergravity

Three dimensional AdS supergravity is described by a Chern-Simons theory for a
Lie super-group $G$, where the even part of the group must contain
$SO(2,2)\cong SL(2,\hbox{\mybb R})\times SL(2,\hbox{\mybb R})$. The Einstein
Hilbert action can then be written as the difference of two Chern-Simons
actions

$S_{\textsc{gr}}[{\cal A},\bar{{\cal A}}]=S_{\textsc{cs}}[{\cal
A}]-S_{\textsc{cs}}[\bar{{\cal A}}]\,,$ (1)

where

$S_{\textsc{cs}}[{\cal A}]=\frac{k}{4\pi}\int_{{\cal
M}}\mathrm{Str}\left({\cal A}\wedge d{\cal A}+\frac{2}{3}{\cal A}\wedge{\cal
A}\wedge{\cal A}\right)\,,$ (2)

and $k=\frac{\ell}{4G_{N}}$. The gauge connections ${\cal A},\bar{{\cal A}}$
take values in the superalgebra $\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$ where
$\mathfrak{g}_{0}=\mathfrak{sl}(2,\hbox{\mybb
R})\oplus\tilde{\mathfrak{g}}_{0}$. Here $\tilde{\mathfrak{g}}_{0}$ is the Lie
algebra of the corresponding $R$-symmetry group. The
$\mathfrak{sl}(2,\hbox{\mybb R})$ generators are denoted by
$L_{-1},L_{0},L_{+1}$ and satisfy the commutation relations

$[L_{0},L_{\pm}]=\pm L_{\pm},\qquad[L_{+},L_{-}]=2L_{0}.$ (3)

When $G=OSp({\cal N}|2)\times OSp({\cal N}|2)$, the Lie algebra
$\tilde{\mathfrak{g}}_{0}$ of $R$-symmetry is $\mathfrak{so}({\cal N})$. Let’s
consider $T^{a}$ $(a=1,...,D)$ to be generators of $\mathfrak{so}({\cal N})$
where $D={\cal N}({\cal N}-1)/2$ is the dimension of the algebra
$\mathfrak{so}({\cal N})$. They satisfy the following commutation relations

$\displaystyle[T^{a},T^{b}]$ $\displaystyle=f^{ab}{}_{c}T^{c},$ (4)
$\displaystyle[T^{a},L_{\pm,0}]$ $\displaystyle=0.$ (5)

Fermionic generators transform in the fundamental representation of $SO({\cal
N})$. Therefore the odd part $\mathfrak{g}_{1}$ consists of $2{\cal N}$
generators $Q_{\pm}^{\alpha}$ where $\alpha=1,...,d$. Note that $d={\cal N}$
is the dimension of fundamental representation of $SO({\cal N})$ in which
fermionic generators transform. The algebra $\mathfrak{osp}({\cal
N}|2,\mathbb{R})$ is then defined as

$\displaystyle[L_{0},Q_{\pm}^{\alpha}]$
$\displaystyle=\pm\frac{1}{2}Q_{\pm}^{\alpha},$ (6a)
$\displaystyle[L_{\pm},Q_{\pm}^{\alpha}]$ $\displaystyle=0,$ (6b)
$\displaystyle[L_{\pm},Q_{\mp}^{\alpha}]$ $\displaystyle=+Q_{\pm}^{\alpha},$
(6c) $\displaystyle[T^{a},Q_{\pm}^{\alpha}]$
$\displaystyle=-(\lambda^{a}){}^{\alpha}{}_{\beta}\,Q_{\pm}^{\beta},$ (6d)
$\displaystyle\\{Q_{\pm}^{\alpha},Q_{\pm}^{\beta}\\}$
$\displaystyle=\pm\eta^{\alpha\beta}L_{\pm},$ (6e)
$\displaystyle\\{Q_{\pm}^{\alpha},Q_{\mp}^{\beta}\\}$
$\displaystyle=-\eta^{\alpha\beta}L_{0}\pm\frac{d-1}{2C_{\rho}}(\lambda^{a})^{\alpha\beta}T^{a}.$
(6f)

Here $\lambda^{a}$ are the basis of the representation $\rho$ of $SO({\cal
N})$ in which fermions transform, $\eta^{\alpha\beta}$ is an $SO({\cal N})$
invariant metric, $C_{\rho}$ is the second Casimir in the $\rho$
representation, i.e. one has

$\lambda^{a}\lambda^{a}=-C_{\rho}I,\qquad
tr(\lambda^{a}\lambda^{b})=-\frac{d}{D}C_{\rho}\delta^{ab}.$ (7)

The graded super Jacobi identity for any $X,Y,Z\in\mathfrak{g}$ is written as

$\displaystyle(-1)^{\epsilon(X)\epsilon(Z)}[X,[Y,Z]]+(-1)^{\epsilon(X)\epsilon(Y)}[Y,[Z,X]]+(-1)^{\epsilon(Z)\epsilon(Y)}[Z,[X,Y]]=0,$
(8)

where $\epsilon(X)$ is the parity of $X$ and its value is determined depending
on whether $X$ is Grassmann even or odd. It imposes a condition on
$\lambda^{a}$s,

$(\lambda^{a})^{\beta\gamma}(\lambda^{a})^{\alpha}{}_{\delta}+(\lambda^{a})^{\alpha\gamma}(\lambda^{a})^{\beta}{}_{\delta}=\frac{C_{\rho}}{d-1}(2\eta^{\alpha\beta}\delta^{\gamma}_{\delta}-\eta^{\alpha\gamma}\delta^{\beta}_{\delta}-\eta^{\gamma\beta}\delta^{\alpha}_{\delta}).$
(9)

The matrices $\eta^{\alpha\beta}$ are symmetric, i.e.
$\eta^{\alpha\beta}=\eta^{\beta\alpha}$ while the matrices
$\lambda^{\alpha\beta}$ are antisymmetric, i.e.
$\lambda^{\alpha\beta}=-\lambda^{\beta\alpha}$.

A generic $\mathfrak{osp}({\cal N}|2)$ connection ${\cal A}$ will be
parameterized as

${\cal
A}=A^{n}L_{n}+B^{a}T^{a}+\psi^{\pm}_{\alpha}Q^{\alpha}_{\pm}=A+B+\Psi\,,$ (10)

and likewise for the barred sector.

The relation between the Chern-Simons formulation and the geometric
formulation in terms of the dreibein $e$ and the (dualized) spin connection
$\omega$ is obtained by identifying $sl(2)$ part of the super-connection as
the dreibein and the spin connection, i.e. taking the
$\mathfrak{sl}(2|\hbox{\mybb R})$ connections $A,\bar{A}$ to be

$e=\frac{\ell}{2}\left(A-\bar{A}\right)\,,\qquad\omega=\frac{1}{2}\left(A+\bar{A}\right)\,.$
(11)

The metric is constructed as

$g_{\mu\nu}=2\mathrm{tr}(e_{\mu}e_{\nu})=\frac{\ell^{2}}{2}\mathrm{tr}\left((A-\bar{A})_{\mu}(A-\bar{A})_{\nu}\right)\,.$
(12)

All solutions of three dimensional gravity are locally gauge equivalent to
each other, but differ up to boundary terms or global identifications.
Therefore boundary conditions are crucial in three dimensional gravity, which
we will now digress on at some length.

####  2.1.1 Gauge fixing

The importance of boundary conditions for three dimensional gravity stems from
the fact that locally gauge equivalent solutions can differ by boundary terms.
Imposing suitable boundary conditions in three dimensional gravity then
specifies which bulk solutions are gauge inequivalent and lead to different
boundary charges. Suitable in this context means: leading to integrable
boundary charges. In addition, the boundary conditions should not be too
strict, so that the boundary charges are non-trivial, but not too loose
either, such that the boundary charges are finite.

In the Chern-Simons formulation of three dimensional gravity the boundary
conditions on the gauge connection are most easily represented in a radial
gauge. Let us suppose our manifold ${\cal M}$ has the topology of a filled
cylinder and can be equipped with a coordinate system $(t,\varphi,r)$. There
is a spatial boundary $\partial{\cal M}$ at $r\to\infty$. The radial gauge
fixing is achieved by taking111For simplicity, here we work in one chiral
sector of AdS3 Chern-Simons gravity, unless otherwise specified.

$\partial_{\varphi}{\cal A}_{r}=0\,.$ (13)

This implies that we can solve the Chern-Simons constraint $F_{r\varphi}=0$ by
writing

${\cal A}_{r}=b(r)^{-1}\partial_{r}b(r)\,,\qquad{\cal
A}_{\varphi}=b(r)^{-1}a_{\varphi}(t,\varphi)b(r)\,.$ (14)

Here $b(r)$ is a group element depending only on $r$.

The time component of the Chern-Simons connection ${\cal A}_{t}$ is a Lagrange
multiplier for the constraint $F_{r\varphi}=0$ and asking this constraint to
be preserved under time evolution implies that we may write

${\cal A}_{t}=b(r)^{-1}a_{t}(t,\varphi)b(r)\,.$ (15)

In this gauge, the boundary conditions are completely specified by fixing
$a_{\varphi}(t,\varphi)$ and $a_{t}(t,\varphi)$. The functional variation of
the asymptotic charges corresponding to an asymptotic symmetry transformation
$\delta_{\varepsilon}a_{i}=\partial_{i}\varepsilon+[a_{i},\varepsilon\\}$
222Here $[,\\}$ is shorthand notation for the graded commutator. are given by
[38]

$\delta
Q=-\frac{k}{2\pi}\oint\mathrm{d}\varphi\;\mathrm{Str}\left(\varepsilon\delta
a_{\varphi}\right)\,.$ (16)

Hence, to specify suitable boundary conditions means in this context to find a
form of $a_{\varphi}$ such that the above charges are finite and integrable
for all $\varepsilon$ which satisfy
$\partial_{i}\varepsilon+[a_{i},\varepsilon\\}=\delta_{\varepsilon}a_{\varphi}$.
An important aspect of this is to include a specification of state dependence
in $\varepsilon$ and in $a_{\varphi}$ in order to perform the functional
integration of the charges.

It is now clear that $a_{\varphi}$ contains information on the asymptotic
charges. On the other hand $a_{t}$ plays the role of chemical potential, or
the sources. We may always write $a_{t}$ proportional to $\varepsilon$, as the
field equations
$\partial_{t}a_{\varphi}-\partial_{\varphi}a_{t}+[a_{t},a_{\varphi}\\}=0$ will
reduce to the statement that the time derivative of the charges $a_{\varphi}$
is determined by a symmetry transformation and hence weakly vanishes. The on-
shell Chern-Simons action is proportional to $\int_{\partial{\cal
M}}\mathrm{tr}(a_{t}a_{\varphi})$ and so by writing $a_{t}$ proportional to
the gauge parameter $\varepsilon$ one immediately sees that it becomes a
chemical potential for the charges in $a_{\varphi}$.

Now we should specify the boundary conditions by specifying the form of
$a_{\varphi}$. The gauge invariant observables in Chern-Simons theory are
Wilson loops, or the holonomy of the connection around the $\varphi$-cycle.

$H_{\varphi}=\mathrm{tr}\left({\cal P}\exp\left(\oint
a_{\varphi}\mathrm{d}\varphi\right)\right)$ (17)

The holonomies, and therefore the distinct solutions, can be characterized by
the conjugacy classes of the super-group $G$. In Section 5.2, we discuss how
this can be useful in distinguishing different bulk solutions and classifying
them as certain orbits of the asymptotic symmetry group in the bosonic example
of Brown-Henneaux boundary conditions [39] for asymptotically local AdS3
spacetimes.

####  2.1.2 Supersymmetric Brown-Henneaux

The supersymmetric extension of the Brown-Henneaux boundary conditions were
obtained in [11]. In our conventions they are most easily represented by
taking

$a_{\varphi}=L_{-}+{\cal
L}(t,\varphi)L_{+}+\Psi_{\alpha}(t,\varphi)Q^{\alpha}_{+}+{\cal
B}^{a}(t,\varphi)T^{a}$ (18)

An asymptotic symmetry transformation $\delta
a_{\varphi}=\partial_{\varphi}\Lambda+[a_{\varphi},\Lambda\\}$ is
parameterized by

$\Lambda=\chi^{n}L_{n}+\varepsilon^{\pm}_{\alpha}Q^{\alpha}_{\pm}+\omega^{a}T^{a}\,.$
(19)

Three components of $\Lambda$ can be solved for as

$\displaystyle\chi^{0}$
$\displaystyle=-\partial_{\varphi}\chi^{-}\,,\qquad\chi^{+}=-\frac{1}{2}\partial^{2}\chi^{-}+{\cal
L}\chi^{-}+\frac{1}{2}\Psi\cdot\varepsilon^{-}\,,$ (20)
$\displaystyle\varepsilon^{+}_{\alpha}$
$\displaystyle=-\partial_{\varphi}\varepsilon^{-}_{\alpha}+\chi^{-}\Psi_{\alpha}+{\cal
B}^{a}\varepsilon^{-}_{\alpha}\lambda^{a}\,,$ (21)

where $\cdot$ means contraction by $\eta^{\alpha\beta}$. The variation of the
functions ${\cal B}^{a},\Psi_{\alpha}$ and ${\cal L}$ are given by:

$\displaystyle\delta{\cal B}^{a}$
$\displaystyle=\partial_{\varphi}\omega^{a}+f^{abc}B^{b}\omega^{c}-\frac{d-1}{2C_{\rho}}(\Psi\lambda^{a}\varepsilon^{-})\,,$
(22) $\displaystyle\delta\Psi_{\alpha}$
$\displaystyle=-\partial^{2}_{\varphi}\varepsilon^{-}_{\alpha}+{\cal
L}\,\varepsilon^{-}_{\alpha}+\frac{3}{2}\partial_{\varphi}\chi^{-}\Psi_{\alpha}+\chi^{-}\partial_{\varphi}\Psi_{\alpha}+\left(\partial_{\varphi}{\cal
B}^{a}\varepsilon^{-}_{\alpha}+2{\cal
B}^{a}\partial_{\varphi}\varepsilon^{-}_{\alpha}-{\cal B}^{a}{\cal
B}^{b}\varepsilon^{-}_{\alpha}\lambda^{b}\right)\lambda^{a}$
$\displaystyle\qquad+\omega^{a}\Psi_{\alpha}\lambda^{a}-{\cal
B}^{a}\chi^{-}\Psi_{\alpha}\lambda^{a}\,.$ (23) $\displaystyle\delta{\cal L}$
$\displaystyle=\chi^{-}\partial_{\varphi}{\cal
L}+2\partial_{\varphi}\chi^{-}{\cal
L}-\frac{1}{2}\partial^{3}_{\varphi}\chi^{-}+\frac{1}{2}\partial_{\varphi}\Psi\cdot\varepsilon^{-}+\frac{3}{2}\Psi\cdot\partial_{\varphi}\varepsilon^{-}+{\cal
B}^{a}(\Psi\lambda^{a}\varepsilon^{-})\,.$ (24)

The transformation rule for $\Psi$ contains a non-linear term $\sim{\cal
B}^{a}\Psi$ in the last line of (23). This term can be removed by taking the
$R$-symmetry gauge parameter $\omega^{a}$ to be

$\omega^{a}={\cal B}^{a}\chi^{-}+\Omega^{a}\,.$ (25)

This changes $\omega$ into $\Omega$ and removes the last term in (23), while
it also changes (22) to

$\delta{\cal B}^{a}=\partial_{\varphi}({\cal
B}^{a}\chi^{-})+\partial_{\varphi}\Omega^{a}+f^{abc}B^{b}\Omega^{c}-\frac{d-1}{2C_{\rho}}(\Psi\lambda^{a}\varepsilon^{-})\,,$
(26)

It also has an effect on the asymptotic charges where it implements a Sugawara
shift in the $\chi^{-}$ charge. The charges are

$\displaystyle Q[\chi^{-}]$
$\displaystyle=-\frac{k}{2\pi}\int\mathrm{d}\varphi\;\chi^{-}\left({\cal
L}+\frac{C_{\rho}}{d-1}{\cal B}^{a}{\cal
B}^{a}\right)\equiv-\frac{k}{2\pi}\int\mathrm{d}\varphi\;\chi^{-}\hat{{\cal
L}}\,,$ (27) $\displaystyle Q[\varepsilon^{-}]$
$\displaystyle=\frac{k}{2\pi}\int\mathrm{d}\varphi\;\varepsilon^{-}\cdot\Psi\,,$
(28) $\displaystyle Q[\Omega]$
$\displaystyle=-\frac{k}{2\pi}\frac{2C_{\rho}}{d-1}\int\mathrm{d}\varphi\;\Omega^{a}{\cal
B}^{a}\,.$ (29)

The algebra of Fourier modes of the charges, defined as
$L_{n}=Q[\chi^{-}=e^{in\varphi}],G_{n}^{\alpha}=Q[\varepsilon^{-}_{\alpha}=e^{in\varphi}]$
and $T_{n}^{a}=Q[\Omega^{a}=e^{in\varphi}]$, gives the generic ${\cal N}$
superconformal algebra

$\displaystyle[L_{m},L_{n}]$
$\displaystyle=(m-n)L_{m+n}+\frac{k}{2}m^{3}\delta_{m+n,0}\,,$ (30a)
$\displaystyle[L_{m},G_{n}^{\alpha}]$
$\displaystyle=\left(\frac{m}{2}-n\right)G_{m+n}^{\alpha}\,,$ (30b)
$\displaystyle[L_{m},T_{n}^{a}]$ $\displaystyle=-nT_{m+n}^{a}\,,$ (30c)
$\displaystyle\\{G_{m}^{\alpha},G_{n}^{\beta}\\}$
$\displaystyle=\eta^{\alpha\beta}(L_{m+n}+km^{2}\delta_{m+n,0})+i\frac{d-1}{2C_{\rho}}(m-n)(\lambda^{a})^{\alpha\beta}T^{a}_{m+n}$
(30d)
$\displaystyle\qquad-\frac{k}{2}\left(\frac{d-1}{2kC_{\rho}}\right)^{2}\left(\\{\lambda^{a},\lambda^{b}\\}^{\alpha\beta}+\frac{2C_{\rho}}{d-1}\eta^{\alpha\beta}\delta^{ab}\right)(T^{a}T^{b})_{m+n}\,,$
(30e) $\displaystyle[G_{m}^{\alpha},T_{n}^{a}]$
$\displaystyle=i(\lambda^{a})^{\alpha}{}_{\beta}G^{\beta}_{m+n}\,,$ (30f)
$\displaystyle[T_{m}^{a},T_{n}^{b}]$
$\displaystyle=if^{abc}T_{m+n}^{c}+\frac{2C_{\rho}k}{d-1}m\delta^{ab}\delta_{m+n,0}\,.$
(30g)

where $(T^{a}T^{b})_{m+n}$ are the Fourier modes of
$(\frac{k}{2\pi}\frac{2C_{\rho}}{d-1})^{2}B^{a}(\varphi)B^{b}(\varphi)$. This
algebras agrees with [11] and corresponds to the non-linear superconformal
algebras found in [12, 13, 14, 15]. The non-linear terms in the supercharge
anti-commutators are only present for ${\cal N}>2$. For ${\cal N}\leq 2$ the
algebras are superconformal Lie algebras whose group is the centrally extended
group of diffeomorphisms of the supercircle.

###  2.2 $W_{3}$ higher spin gravity

$W_{3}$ gravity in three dimensions can be considered as a higher spin theory
of gravity for a spin-3 field and described by the Chern-Simons theory with
the gauge group $SL(3,\mathbb{R})$. The Einstein Hilbert action is again
written as the difference of two Chern-Simons actions as in (1) and (2), where
now $k$ is the level of the Kac-Moody $\mathfrak{sl}(3)$ algebra, the gauge
connections ${\cal A},\bar{{\cal A}}$ are
$\mathfrak{g}=\mathfrak{sl}(3,\mathbb{R)}$-valued 1-form connections and the
trace is over the $\mathfrak{sl}(3)$ Lie algebra. To define $\mathfrak{g}$ we
use the root space gradation $\mathfrak{g}=\oplus\mathfrak{g}_{(i)}$ with

$\displaystyle\mathfrak{g}_{(0)}$ $\displaystyle=$
$\displaystyle\mathfrak{g}_{0}=\textrm{span}\\{\theta^{a}|a=1,2,...,\textrm{rank}\,\mathfrak{g}\\},$
$\displaystyle\mathfrak{g}_{(j)}$ $\displaystyle=$
$\displaystyle\textrm{span}\\{E^{\alpha}|[\theta^{a},E^{\alpha}]=\alpha^{a}E^{\alpha},a=1,2,...,\textrm{rank}\,\mathfrak{g}\\},$
(31)

where $\mathfrak{g}_{0}$ is the Cartan subalgebra of $\mathfrak{sl}(3)$
spanned by two elements $\theta^{a}$, $a=1,2$ and $E^{\alpha}$ are associated
to the root $\alpha$ of the root space of $\mathfrak{sl}(3)$ Lie algebra. The
Chevalley basis of $\mathfrak{sl}(3)$ are spanned by a set of 8 generators
given by $\left\\{E_{\pm}^{\alpha},\theta^{a}\right\\}$, where
$E_{+}^{\alpha}$, $E_{-}^{\alpha}$ are respectively raising operators
corresponding to positive roots and lowering operators corresponding to
negative roots, and $\theta^{a}$ span the Cartan subalgebra. We recall that
the eigenvalues of the Cartan subalgebra are weights $\omega^{a}$ and the
vector with components $\omega^{a}$ is a weight vector $\vec{\Omega}$, i.e.
$\theta^{a}|\omega\rangle=\omega^{a}|\omega\rangle$ where $|\omega\rangle$ are
eigenvectors of $\theta^{a}$.

Let’s consider the following matrix representation of $sl(3)$

$\displaystyle E_{+}^{1}$ $\displaystyle=\begin{pmatrix}0&1&0\\\ 0&0&0\\\
0&0&0\end{pmatrix},\qquad~{}E_{+}^{2}=\begin{pmatrix}0&0&0\\\ 0&0&1\\\
0&0&0\end{pmatrix},\qquad E_{+}^{3}=\begin{pmatrix}0&0&1\\\ 0&0&0\\\
0&0&0\end{pmatrix},$ $\displaystyle E_{-}^{1}$
$\displaystyle=\begin{pmatrix}0&0&0\\\ 1&0&0\\\ 0&0&0\end{pmatrix},\qquad
E_{-}^{2}=\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 0&1&0\end{pmatrix},\qquad
E_{-}^{3}=\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 1&0&0\end{pmatrix},$ $\displaystyle
H_{1}$ $\displaystyle=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\
0&0&0\end{pmatrix},\qquad H_{2}=\begin{pmatrix}0&0&0\\\ 0&1&0\\\
0&0&-1\end{pmatrix},$ (32)

and

$\displaystyle\theta^{1}=\frac{H_{1}+H_{2}}{\sqrt{2}},\qquad\qquad\theta^{2}=\frac{H_{1}-H_{2}}{\sqrt{6}}.$
(33)

Using the matrix representation of $\mathfrak{sl}(3)$, we find three weight
vectors of $\mathfrak{sl}(3)$

$\vec{\Omega}_{1}=\frac{1}{\sqrt{2}}\left(1,\frac{1}{\sqrt{3}}\right),\qquad\vec{\Omega}_{2}=\frac{1}{\sqrt{2}}\left(0,-\frac{2}{\sqrt{3}}\right),\qquad\vec{\Omega}_{3}=\frac{1}{\sqrt{2}}\left(-1,\frac{1}{\sqrt{3}}\right).$
(34)

Roots are weights of the adjoint representation and root vectors
$\vec{\alpha}_{i}$ can be defined as vectors connecting the weight vectors in
the root lattice, then we can easily write the positive roots

$\displaystyle\vec{\alpha}_{1}$ $\displaystyle=$
$\displaystyle\vec{\Omega}_{1}-\vec{\Omega}_{2}=\frac{1}{\sqrt{2}}\left(1,\sqrt{3}\right),$
$\displaystyle\vec{\alpha}_{2}$ $\displaystyle=$
$\displaystyle\vec{\Omega}_{2}-\vec{\Omega}_{3}=\frac{1}{\sqrt{2}}\left(1,-\sqrt{3}\right),$
$\displaystyle\vec{\alpha}_{3}$ $\displaystyle=$
$\displaystyle\vec{\Omega}_{1}-\vec{\Omega}_{3}=\left(\sqrt{2},0\right).$ (35)

It is obvious that $\vec{\alpha}_{1}$ and $\vec{\alpha}_{2}$ are the simple
roots of $\mathfrak{sl}(3)$, and
$\vec{\alpha}_{3}=\vec{\alpha}_{1}+\vec{\alpha}_{2}$. This is simply
understood from the commutator of the Cartan generators with raising and
lowering operators as

$\displaystyle[\vec{\theta},E^{i}_{\pm}]$
$\displaystyle=\pm\vec{\alpha}_{i}E^{i}_{\pm},$ (36)
$\displaystyle[E^{i}_{\pm},E^{i}_{\mp}]$
$\displaystyle=\vec{\alpha}_{i}.\vec{\theta}$ (37)
$\displaystyle[E^{i}_{\pm},E^{j}_{\pm}]$ $\displaystyle=\pm E^{i+j}_{\pm},$
(38)

where $\vec{\theta}=(\theta_{1},\theta_{2})$. The rest of generators can be
obtained by Serre relation or by the use of Jacobi Identity. Note that here
“.” refers to the inner product on the two dimensional root vector space and
moreover $E^{i}_{\pm}=0$ for $i>3$.

####  2.2.1 Gauge fixing

A general $\mathfrak{sl}(3)$ connection can be written as

${\cal A}=A_{\pm}^{i}E_{\pm}^{i}+\Theta^{i}H^{i},$ (39)

and likewise for $\bar{\mathcal{A}}$. Once again, the gauge inequivalent bulk
solutions are specified by suitable boundary conditions. Following the
discussion of section 2.1.1, we can solve the Chern-Simons constraint in
radial gauge for the connections

$\mathcal{A}_{r}=b(r)^{-1}\partial_{r}b(r),\qquad\qquad\mathcal{A}_{\varphi}=b(r)^{-1}a_{\varphi}(t,\varphi)b(r),$
(40)

where $b(r)$ is a group element which only depends on $r$. Moreover, requiring
that the Chern-Simons constraint remains unchanged under the time evolution
leads to

$\mathcal{A}_{t}=b(r)^{-1}a_{t}(t,\varphi)b(r).$ (41)

This means that the boundary conditions are completely determined by
$a_{\varphi}(t,\varphi)$ and $a_{t}(t,\varphi)$, see [21, 40, 20] for more
details.

The Brown-Henneaux boundary conditions for $W_{3}$ gravity in 3 dimensions
depends on the way that the $SL(2)$ subgroup is embedded in $SL(3)$. We will
explore this in more detail later in section 4.2 where we discuss the
Hamiltonian reduction of $SL(3,\mathbb{R)}$ WZW action.

##  3 Hamiltonian reduction of the super Chern-Simons action

In this section we will discuss the reduction of three dimensional super-
Chern-Simons action to a two dimensional field theory. We first reduce three
dimensional AdS supergravity to a sum of two chiral super Wess-Zumino-Witten
models and then impose the boundary conditions as constraints on the WZW
super-currents. Furthermore, we will discuss the effect of different boundary
conditions and boundary terms on the two dimensional boundary action, paying
special attention to non-trivial holonomies in the bulk. The analysis is kept
general for extended supergravity with any number of supersymmetries. For
$\mathcal{N}\leq 2$, we will show how the boundary action is related to the
geometric action on the coadjoint orbit of the super-Virasoro group. For
larger supersymmetries ($\mathcal{N}>2$), we compute the Schwarzian action
through the reduction, and suggest a form for the geometric action on the
symplectic leaves of corresponding super-Virasoro group manifold.

The first steps of the reduction follows along the lines of [41, 11] (see also
[42] for a recent review). What is new in our approach is that we will not try
to combine the two chiral sectors of the theory, but instead keep them
disconnected. We also discuss the zero modes of the fields and allow for non-
trivial holonomies of the Chern-Simons connection.

We start with the Hamiltonian decomposition of the action (1). Let us focus on
one chiral sector ($S_{\textsc{cs}}[A]$) for now, as the barred sector follows
similarly. The Hamiltonian form of the action (2) is defined on a manifold
${\cal M}$ which we will take to be the disk $D$ times a time direction. The
boundary $S^{1}$ of the disk is at $r\to\infty$ and we use an orientation
$dt\wedge d\varphi\wedge dr$. The Hamiltonian action is

$S[{\cal A}]=\frac{k}{4\pi}\int_{{\cal M}}dtd\varphi
dr\,\mathrm{Str}\left({\cal A}_{r}\dot{{\cal A}}_{\varphi}-{\cal
A}_{\varphi}\dot{{\cal A}}_{r}+2{\cal A}_{t}\mathcal{F}_{\varphi
r}\right)+I_{\Sigma}\,,$ (42)

with

$\mathcal{F}_{r\varphi}=\partial_{r}{\cal A}_{\varphi}-\partial_{\varphi}{\cal
A}_{r}+[{\cal A}_{r},{\cal A}_{\varphi}]\,,$ (43)

and $I_{\Sigma}$ is a boundary term adapted to the boundary conditions under
consideration. This boundary term is added to the action to ensure a well-
defined variational principle. The variation of the bulk action gives

$\delta S[{\cal A}]=\ldots-\frac{k}{2\pi}\int_{\partial{\cal
M}}dtd\varphi\,\mathrm{Str}{\cal A}_{t}\delta{\cal
A}_{\varphi}=\ldots-\frac{k}{2\pi}\int_{\partial{\cal
M}}dtd\varphi\,\mathrm{Str}\;a_{t}\delta a_{\varphi}\,.$ (44)

Here the ellipses stand for terms proportional to the bulk equations of
motion. The boundary term $I_{\Sigma}$ should be such that its variation
cancels the term in (44). We should therefore impose boundary conditions such
that this expression is integrable and finite. We discuss the boundary
conditions in details later in Section 3.2.

The action (42) consists out of two parts. There is a symplectic term and the
boundary term. The latter is responsible for the Hamiltonian of the boundary
theory, while the symplectic part reduces to the geometric action on the
super-Virasoro coadjoint orbit. In the non-supersymmetric theory, it will
reduce to the geometric action on the Virasoro coadjoint orbits of [24]. Let
us discuss the reduction for both these terms in succession.

###  3.1 The symplectic term

Focusing first on the bulk part of one chiral sector of the action, we see
that ${\cal A}_{t}$ is a Lagrange multiplier for the constraint
$\mathcal{F}_{\varphi r}=0$. In general, the spatial sections can have a non-
trivial topology, which implies a non-trivial holonomy along non-contractible
cycles in the manifold. One can take this into account in the following way:

One demands the group elements $G$ to be periodic; as a result of which the
holonomy appears explicitly in the local solution of the constraint
$\mathcal{F}_{\varphi r}=0$,

${\cal A}_{\varphi}=G^{-1}(\partial_{\varphi}+K(t))G\,,\qquad
G(\varphi+2\pi)=G(\varphi).$ (45)

Here, $K(t)$ is a Lie algebra valued function of time that parametrizes the
holonomy333This choice is indeed a restriction we impose at this point, as it
is only possible to eliminate the $\varphi$-dependence in $K(t)$ for simply
connected groups.. Note that for periodic group elements $G$, the integral of
$G^{-1}\partial_{\varphi}G$ over the loop vanishes.

The action with explicit holonomy can be obtained by substituting (45) into
(42). The result is (formula (A.7) of [11])

$\displaystyle S_{\rm SCS}[G,K(t)]=$ $\displaystyle+\frac{k}{4\pi}\int_{{\cal
M}}d^{3}x\;\mathrm{Str}\left(\partial_{r}(G^{-1}\partial_{\varphi}GG^{-1}\partial_{t}G)\right)+\frac{k}{12\pi}\int_{{\cal
M}}\;\mathrm{Str}(G^{-1}dG)^{3}$ (46) $\displaystyle+\frac{k}{4\pi}\int_{{\cal
M}}d^{3}x\;\mathrm{Str}\left(2\partial_{r}(G^{-1}K\partial_{t}G)-\partial_{t}(G^{-1}K\partial_{r}G)\right)+I_{\Sigma}\,,$

up to a total $\varphi$-derivative which is dropped due to the periodicity of
$G$ in $\varphi$. In addition, we have also dropped boundary contributions at
the time boundaries, and we will continue to do so in the following, up to the
point where we discuss them systematically.

In Section 3.3, we continue by representing any group element with the use of
the Gauss decomposition of the group. It enables us to define Maurer-Cartan
one-forms, $G^{-1}dG$, and to further implement constraints from the boundary
conditions, i.e. $a_{\varphi}=G^{-1}\partial_{\varphi}G$, in terms of the
fields parameterizing the group. But first, we will discuss the Hamiltonian
coming from the boundary term in (42).

###  3.2 The Hamiltonian

Imposing different boundary conditions can lead to different boundary
Hamiltonians. Following (44), the boundary term should be taken such that its
variation is

$\delta I_{\Sigma}=\frac{k}{2\pi}\int dtd\varphi\;\mathrm{Str}\left({\cal
A}_{t}\delta{\cal A}_{\varphi}\right)\,.$ (47)

The precise form of boundary action depends on the topology of spatial
surfaces. In the following we focus on a case where the spatial hypersurface
has annulus topology $S^{1}\times[0,1]$.

In that case, we impose the boundary condition ${\cal A}_{-}=0$ of [41] at the
outer boundary. We also choose the Hamiltonians on the respective boundaries
to have the same (positive) sign for definiteness, which one could interpret
as having time evolution on both sides of the black hole run in the same
direction. This is achieved by taking ${\cal A}_{+}=0$ at the inner boundary.
These choices lead to the boundary terms (see section 2.3 of [7]):

$I_{\Sigma_{i,o}}=-\frac{k}{4\pi}\int_{\Sigma_{i,o}}dtd\varphi\,\mathrm{Str}{\cal
A}_{\varphi}^{2}\,.$ (48)

The boundary term in the action is thus seen as the time integral of the
Noether charge of an asymptotic symmetry transformation and hence the
resulting action will remain invariant under the asymptotic symmetries.

$I_{\Sigma}=I_{\Sigma_{i}}+I_{\Sigma_{o}}=-\frac{k}{4\pi}\int dtH_{\rm
bdy}=-\int dt\,Q[\Lambda={\cal A}_{t}],$ (49)

where $\Lambda$ is a Lie algebra valued function of the gauge parameters. If
we choose $\chi^{-}=1$ and $\varepsilon^{-}=0=\Omega^{a}$, then the asymptotic
symmetry corresponds to time translations. The Noether charge for this would
make a suitable Hamiltonian and it’s given by

$I_{\Sigma}=\frac{k}{2\pi}\int\mathrm{d}t\mathrm{d}\varphi\;\hat{{\cal L}}$
(50)

with $\hat{{\cal L}}$ defined in (27).

For the annulus topology, the action includes a contribution from each
boundary, coupled through the holonomy parameterized by $K$.

$\displaystyle S_{\rm CS}[h,l,K(t)]=$
$\displaystyle+\frac{k}{4\pi}\int_{\Sigma_{o}}dtd\varphi\;\mathrm{Str}\left(h^{-1}\partial_{\varphi}hh^{-1}\partial_{-}h+2h^{-1}K\partial_{-}h-K^{2}\right)$
(51)
$\displaystyle-\frac{k}{4\pi}\int_{\Sigma_{i}}dtd\varphi\;\mathrm{Str}\left(l^{-1}\partial_{\varphi}ll^{-1}\partial_{+}l+2l^{-1}K\partial_{+}l+K^{2}\right)+I_{WZ}[G]\,,$

where:

$h=G(t,r=r^{\textrm{outer}},\varphi),\qquad
l=G(t,r=r^{\textrm{inner}},\varphi)\,,$ (52)

and hereon we use the notation
$\partial_{\mp}=\partial_{t}\mp\partial_{\varphi}$.

The Wess-Zumino term

$I_{WZ}[G]=\frac{k}{12\pi}\int_{{\cal M}}\;\mathrm{Str}(G^{-1}dG)^{3}\,,$ (53)

can be written as a total derivative and hence it also only depends on the
boundary values of the group element $G$.

The action (51) is invariant under the gauge symmetry $G\rightarrow\omega(t)G$
and $K\rightarrow\omega(t)K\omega^{-1}(t)$, which implies in terms of the
boundary fields,

$h\rightarrow\omega(t)h,\qquad l\rightarrow\omega(t)l,\qquad
K\rightarrow\omega(t)K\omega^{-1}(t)\,.$ (54)

This gauge invariance results from the redundancy of the parametrization of
the group element $G$ [43] and can straightforwardly be verified in the above
action.

###  3.3 On the holonomy

In this section, we continue the discussion of Section 3.1 by explicitly
computing the reduction for a super-Chern-Simons action with $\mathcal{N}$
extended supersymmetry and the gauge group $OSp({\cal N}|2)$. However before
doing so, we need to clarify few important aspects of the presence of
holonomy. As described in [7], the inequivalent physical contribution of
holonomy $K$ is classified by its conjugacy classes, and moreover, conjugacy
classes are constant in time.

Due to the theorem 3.9 of [44] (see also [45]), the holonomy group of
$OSp({\cal N}|2)$ can be always put in a block diagonalized form. This means
that one can always make the holonomy to be in the even subgroup of the
supergroup. Another way to see this is by looking at the classification of
coadjoint orbits of super Virasoro group [46]. One notices that the coadjoint
orbits that one obtains by considering the holonomy in the odd subspace of the
superspace are not new ones and one could equivalently obtain them by setting
the holonomy to be in the even subgroups.

In the case of $\mathcal{N}=1$ the choice of holonomy is insensitive to the
supersymmetry. This is because for $\mathcal{N}=1$ there is no $R$-symmetry
and as a result the holonomy is classified by the conjugacy classes of
$SL(2)$, similar to the non-supersymmetric case. However, for $\mathcal{N}\geq
2$ there is an $SO({\cal N})$ $R$-symmetry and the holonomy must be classified
by the conjugacy classes of the even subgroup of $OSp({\cal N}|2)$.

Another remark on the choice of holonomy conjugacy classes is that we only
focus on the hyperbolic holonomy class for the rest of this paper. This class
of holonomies includes BTZ solutions and therefore is important in the study
of two-sided eternal BTZ black holes which is the primary (but not the only)
concern of this paper. For the practical reasons and for the sake of
simplicity we only focus on the hyperbolic class. The analysis of the other
classes is similar to the hyperbolic case. For such an analysis in the bosonic
case see [7].

###  3.4 ${\cal N}=1$ super Chern-Simons

We will now continue to discuss the reduction to the boundary action for
${\cal N}=1$ AdS3 supergravity, focusing on the case where the holonomy is in
the hyperbolic conjugacy class of $SL(2)$.

####  3.4.1 Hyperbolic Holonomy

Using the Gauss decomposition, we can parameterize any element of $OSp(1|2)$
as

$\displaystyle h=G[r=r_{o},t,\varphi]$
$\displaystyle=e^{YL_{-}+\psi^{-}Q_{-}}e^{\Phi
L_{0}}e^{XL_{+}+\psi^{+}Q_{+}}\equiv E^{-}E^{0}E^{+}\,$ (55) $\displaystyle
l=G[r=r_{i},t,\varphi]$
$\displaystyle=e^{UL_{+}+\chi^{+}Q_{+}}e^{\tilde{\Phi}L_{0}}e^{VL_{-}+\chi^{-}Q_{-}}\equiv\tilde{E}^{+}\tilde{E}^{0}\tilde{E}^{-}\,$
(56)

where $\Phi,X,Y,\psi^{\pm}$ are fields at outer boundary and
$\tilde{\Phi},U,V,\chi^{\pm}$ are fields defined at inner boundary and all are
depending on spacetime coordinates $(t,\varphi)$444We emphasize that the
parametrization (56) is only at the inner boundary. One can see it as a field
redefinition of the fields which appear in a similar parametrization as (55)..
They are the pullback of the $r$-dependent $G$ to the corresponding
boundaries.

Inserting (55) and (56) inside the action (51), upon using the algebra
$\mathfrak{osp}(1|2,\mathbb{R})$ given in (6) and

$\mathrm{Str}(L_{+}L_{-})=1,\qquad\mathrm{Str}(L_{0}L_{0})=\frac{1}{2},\qquad\mathrm{Str}(Q_{-}Q_{+})=-\mathrm{Str}(Q_{+}Q_{-})=2,$
(57)

the action now reads off as

$S=S_{o}-S_{i}+S_{hol},$ (58)

where555Here and in the rest of the paper we use prime as a shorthand notation
for $\partial_{\varphi}$.

$\displaystyle S_{o}$
$\displaystyle=\frac{k}{4\pi}\int_{\Sigma_{o}}\mathrm{d}t\mathrm{d}\varphi\;\;\Big{[}\frac{1}{2}\Phi^{\prime}\partial_{-}\Phi+2e^{\Phi}(\partial_{-}X-\psi^{+}\partial_{-}\psi^{+})(Y^{\prime}-\psi^{-}\psi^{-}{}^{\prime})-4e^{\Phi/2}\psi^{-}{}^{\prime}\partial_{-}\psi^{+}\Big{]},$
$\displaystyle S_{i}$
$\displaystyle=\frac{k}{4\pi}\int_{\Sigma_{i}}\mathrm{d}t\mathrm{d}\varphi\;\;\Big{[}\frac{1}{2}\tilde{\Phi}^{\prime}\partial_{+}\tilde{\Phi}+2e^{-\tilde{\Phi}}(\partial_{+}V+\chi^{-}\partial_{+}\chi^{-})(U^{\prime}+\chi^{+}\chi^{+}{}^{\prime})-4e^{-\tilde{\Phi}/2}\partial_{+}\chi^{-}\chi^{+}{}^{\prime}\Big{]},$
(59)

and

$\displaystyle
S_{hol}=\frac{k}{4\pi}\int\mathrm{d}t\mathrm{d}\varphi\;\;\Big{[}$
$\displaystyle
k_{0}\bigg{(}\partial_{-}\Phi-\partial_{+}\tilde{\Phi}+2e^{\Phi/2}\psi^{-}\partial_{-}\psi^{+}-2e^{-\tilde{\Phi}/2}\chi^{+}\partial_{+}\chi^{-}$
$\displaystyle-2e^{\Phi}Y(\partial_{-}X-\psi^{+}\partial_{-}\psi^{+})-2e^{-\tilde{\Phi}}U(\partial_{+}V+\chi^{-}\partial_{+}\chi^{-})\bigg{)}-k_{0}^{2}\Big{]}.$
(60)

The Lagrangian is invariant (up to total derivatives) under the residual gauge
symmetry,

$\displaystyle\Phi\to\hat{\Phi}$
$\displaystyle=\Phi+\lambda^{0}\qquad\qquad\tilde{\Phi}\to\hat{\tilde{\Phi}}=\tilde{\Phi}+\lambda^{0},$
(61a) $\displaystyle Y\to\hat{Y}$
$\displaystyle=Ye^{-\lambda^{0}}\qquad\qquad\,U\to\hat{U}=Ue^{\lambda^{0}},$
(61b) $\displaystyle X\to\hat{X}$ $\displaystyle=X\qquad\qquad\qquad
V\to\hat{V}=V,$ (61c) $\displaystyle\psi^{-}\to\hat{\psi}^{-}$
$\displaystyle=\psi^{-}e^{-\lambda^{0}/2}\quad\quad\chi^{-}\to\hat{\chi}^{-}=\chi^{-},$
(61d) $\displaystyle\psi^{+}\to\hat{\psi}^{+}$
$\displaystyle=\psi^{+}\qquad\qquad\,~{}\chi^{+}\to\hat{\chi}^{+}=\chi^{+}e^{-\lambda^{0}/2},$
(61e) $\displaystyle k_{0}\to\hat{k}_{0}$ $\displaystyle=k_{0}\,.$ (61f)

A convenient rewriting of the action is

$S_{\rm CS}[k_{0},Y,\Phi,X,\psi^{\pm},V,\tilde{\Phi},U,\chi^{\pm}]=S_{\rm
bdy}^{\Sigma_{o}}[k_{0},Y,\Phi,X,\psi^{\pm}]-S_{\rm
bdy}^{\Sigma_{i}}[k_{0},V,\tilde{\Phi},U,\chi^{\pm}]\,,$ (62)

with

$\displaystyle S_{\rm bdy}^{\Sigma_{o}}$ $\displaystyle=\frac{k}{4\pi}\int
dtd\varphi\;\Big{[}\frac{1}{2}\partial_{-}\Phi(\Phi^{\prime}+2k_{0})+2e^{\Phi}(\partial_{-}X-\psi^{+}\partial_{-}\psi^{+})(Y^{\prime}-\psi^{-}{\psi^{-}}^{\prime}-k_{0}Y)$
$\displaystyle+4e^{\Phi/2}\partial_{-}\psi^{+}({\psi^{-}}^{\prime}-\frac{k_{0}}{2}\psi^{-})-\frac{1}{2}k_{0}^{2}\Big{]}\,,$
$\displaystyle S_{\rm bdy}^{\Sigma_{i}}$ $\displaystyle=\frac{k}{4\pi}\int
dtd\varphi\;\Big{[}\frac{1}{2}\partial_{+}\tilde{\Phi}({\tilde{\Phi}}^{\prime}+2k_{0})+2e^{-\tilde{\Phi}}(\partial_{+}V+\chi^{-}\partial_{+}\chi^{-})(U^{\prime}+\chi^{+}{\chi^{+}}^{\prime}+k_{0}U)$
$\displaystyle-4e^{-\tilde{\Phi}/2}\partial_{+}\chi^{-}({\chi^{+}}^{\prime}+\frac{k_{0}}{2}\chi^{+})+\frac{1}{2}k_{0}^{2}\Big{]}\,,$

####  3.4.2 Boundary conditions

We are now at the stage where we can impose the reduction conditions on the
super Chern-Simons connection that expresses the asymptotic behaviour of
extended (super) $AdS_{3}$ [11]. We consider explicitly one asymptotic
boundary only (the outer boundary). Similar considerations apply to the inner
boundary. The only difference in their treatment is the choice of
$OSp(N|1,\mathbb{R})$ representation at each boundary. While the boundary
conditions on the fields at outer boundary are in accordance with highest-
weight representation, those on the fields at inner boundary are in accordance
with the lowest-weight representation. As shown in [41] and discussed in [11,
7] the boundary conditions on the fields at $r=r_{2}\equiv r_{o}$ are

${\cal A}_{r}=0,\qquad{\cal A}_{\varphi}=L_{-}+{\cal
L}(t,\varphi)L_{+}+\Psi^{+}(t,\varphi)Q_{+}.$ (64)

Similarly, the boundary conditions on the fields at $r=r_{1}\equiv r_{i}$ are

${\cal A}_{r}=0,\qquad{\cal A}_{\varphi}=L_{+}+{\cal
M}(t,\varphi)L_{-}+\Psi^{-}(t,\varphi)Q_{-}.$ (65)

In terms of the field appearing in the Gauss decomposition, this gives the
conditions

$\displaystyle\begin{array}[]{c||c}\textrm{On Outer Boundary}&\textrm{On Inner
Boundary}\\\ \hline\cr\hline\cr{}\hfil&{}\hfil\\\
e^{\Phi}(Y^{\prime}-\psi^{-}{\psi^{-}}^{\prime}-k_{0}Y)=1&e^{-\tilde{\Phi}}(U^{\prime}+\chi^{+}{\chi^{+}}^{\prime}+k_{0}U)=1\\\
{}\hfil&{}\hfil\\\ \Phi^{\prime}+k_{0}=2X&\tilde{\Phi}^{\prime}+k_{0}=-2V\\\
{}\hfil&{}\hfil\\\
\psi^{+}+e^{\Phi/2}({\psi^{-}}^{\prime}-\frac{k_{0}}{2}\psi^{-})=0&\chi^{-}+e^{-\tilde{\Phi}/2}({\chi^{+}}^{\prime}+\frac{k_{0}}{2}\psi^{+})=0\end{array}$
(73)

Inserting these conditions in the action (LABEL:Sbdy), we find

$\displaystyle S_{\rm bdy}^{\Sigma_{o}}$ $\displaystyle=\frac{k}{4\pi}\int
dtd\varphi\;\Big{[}\frac{1}{2}\partial_{-}\Phi(\Phi^{\prime}+2k_{0})+2\psi^{+}\partial_{-}\psi^{+}-\frac{1}{2}k_{0}^{2}\Big{]}\,,$
$\displaystyle S_{\rm bdy}^{\Sigma_{i}}$ $\displaystyle=\frac{k}{4\pi}\int
dtd\varphi\;\Big{[}\frac{1}{2}\partial_{+}\tilde{\Phi}({\tilde{\Phi}}^{\prime}+2k_{0})-2\chi^{-}\partial_{+}\chi^{-}+\frac{1}{2}k_{0}^{2}\Big{]}\,.$
(74)

Putting these together, the final form of action is written

$\displaystyle S=\frac{k}{4\pi}\int
dtd\varphi\;\Big{[}\frac{1}{2}\partial_{-}\Phi\Phi^{\prime}-\frac{1}{2}\partial_{+}\tilde{\Phi}{\tilde{\Phi}}^{\prime}+k_{0}(\partial_{-}\Phi-\partial_{+}\tilde{\Phi})+2(\psi^{+}\partial_{-}\psi^{+}+\chi^{-}\partial_{+}\chi^{-})-k_{0}^{2}\Big{]}\,.$

This action describes the dynamic of two chiral bosons (one at each boundary)
linked through the holonomy and two free chiral fermions (one at each
boundary). This action is exactly the one described in [7] with the exception
of the free chiral fermions present at each boundary and similarly enjoys
invariance under a redundant gauge symmetry

$\displaystyle\Phi\rightarrow\Phi+\epsilon(t),\qquad\tilde{\Phi}\rightarrow\tilde{\Phi}+\epsilon(t),\qquad
k_{0}\rightarrow k_{0}\,,$
$\displaystyle\psi^{+}\rightarrow\psi^{+},\qquad\chi^{-}\rightarrow\chi^{-}\,.$
(76)

It is of no surprise that fermions remained uncoupled as the holonomy belongs
only to the diagonal subgroup which only allows bosonic degrees of freedom to
couple. In the next section, we should see an extension of this to include
$R$-symmetry.

Before proceeding any further lets make a comment on the other conjugacy
classes of $SL(2)$; elliptic and parabolic conjugacy classes. As we just
pointed out since for ${\cal N}=1$, holonomy only couples to the bosonic
fields, the treatment of elliptic and parabolic cases are similar to [7] in
each case. The only difference is the presence of free chiral fermions in the
action which will appear as in the action (3.4.2).

###  3.5 ${\cal N}=2$ super Chern-Simons

The case of ${\cal N}=2$ seems fairly easy to deal with. In fact, one can
write down the $S_{o}$ and $S_{i}$ for a general ${\cal N}>2$. The difficulty
lies in what is the contribution of $S_{hol}$ which depends on the choice of
holonomy. Our guideline motivated by the theorem on supergroup diagonalization
[44, 45] is that the holonomy can be always put in the form where it belongs
to the even subgroup of the supergroup. For ${\cal N}=2$ this simplifies
drastically, since there is only one $R$-symmetry generator and the holonomy
will be given by a direct sum of the ${\cal N}=1$ holonomy (i.e. it belongs to
one of $SL(2,\mathbb{R})$ conjugacy classes) and the $\mathfrak{so}(2)$ Lie
algebra of the ${\cal N}=2$ $R$-symmetry subgroup666This choice of holonomy
can be also shown to be the case for a constant orbit representative, when one
considers the effect of the holonomy implicitly through non-periodicity of the
fields.. This boils down to three distinct classes:

  * •

Hyperbolic holonomy described by hyperbolic conjugacy class of
$\mathfrak{sl}(2)\oplus\mathfrak{so}(2)$. Explicitly, the holonomy group
element is $H=\exp{K}$ with $K(t)=k_{0}(t)L_{0}+k_{r}(t)T$. All elements
conjugate to $K$ belong to this class.

  * •

Elliptic holonomy described by the elliptic conjugacy class of
$\mathfrak{sl}(2)$, or explicitly: all elements conjugate to
$K(t)=\frac{1}{2}k_{e}(t)(L_{-}-L_{+})$.

  * •

Parabolic holonomy, which is described by the parabolic conjugacy class of
$\mathfrak{sl}(2)$, or explicitly: all elements conjugate to
$K(t)=k_{p}(t)L_{+}$.

Before discussing the details for ${\cal N}=2$, let us first write down the
outer and inner contributions to the action $S_{o}$ and $S_{i}$, for general
${\cal N}$.

Once again we are only discussing the hyperbolic holonomy conjugacy classes
which contains BTZ back holes. The Gauss decomposition is a natural
parametrization for the hyperbolic holonomy. It occurs that the $R$-symmetry
generator for ${\cal N}=2$ is also diagonal, which makes the choice of Gauss
decomposition much more suitable for this case.

Using the Gauss decomposition, we can parameterize any element of $OSp({\cal
N}|2)$ as

$\displaystyle h=G[r=r_{o},t,\varphi]$
$\displaystyle=e^{YL_{-}+\psi^{-}_{\alpha}Q^{\alpha}_{-}}e^{\Phi
L_{0}+{\rm{i}}\,C^{a}T^{a}}e^{XL_{+}+\psi^{+}_{\alpha}Q^{\alpha}_{+}}\equiv
E^{-}E^{0}E^{+}\,$ (77) $\displaystyle l=G[r=r_{i},t,\varphi]$
$\displaystyle=e^{UL_{+}+\chi^{+}_{\alpha}Q^{\alpha}_{+}}e^{\tilde{\Phi}L_{0}+{\rm{i}}\,\tilde{C}^{a}T^{a}}e^{VL_{-}+\chi^{-}_{\alpha}Q^{\alpha}_{-}}\equiv\tilde{E}^{+}\tilde{E}^{0}\tilde{E}^{-}\,$
(78)

where $\Phi,X,Y,C^{a},\psi^{\pm}_{\alpha}$ are fields at outer boundary and
$\tilde{\Phi},U,V,\tilde{C}^{a},\chi^{\pm}_{\alpha}$ are fields defined at
inner boundary and all are depending on spacetime coordinates $(t,\varphi)$.
They are the pullback of the $r$-dependent $G$ to the corresponding
boundaries.

Inserting (77) and (78) inside the action (51) and using the
$\mathfrak{osp}(N|2,\mathbb{R})$ algebra given in (6) and

$\displaystyle\mathrm{Str}(L_{+}L_{-})$
$\displaystyle=1,\qquad\qquad\quad~{}~{}\mathrm{Str}(L_{0}L_{0})=\frac{1}{2},$
$\displaystyle\mathrm{Str}(T^{a}T^{b})$
$\displaystyle=\frac{2C_{\rho}}{d-1}\delta^{ab},\qquad\mathrm{Str}(Q^{\alpha}_{-}Q^{\beta}_{+})=-\mathrm{Str}(Q^{\alpha}_{+}Q^{\beta}_{-})=\eta^{\alpha\beta},$
(79)

the action now reads:

$S=S_{o}-S_{i}+S_{hol},$ (80)

where

$\displaystyle
S_{o}=\frac{k}{4\pi}\int_{\Sigma_{o}}\mathrm{d}t\mathrm{d}\varphi\;\;\Big{[}$
$\displaystyle\frac{1}{2}\Phi^{\prime}\partial_{-}\Phi+2e^{\Phi}(\partial_{-}X-\psi^{+}_{\alpha}\frac{\eta^{\alpha\beta}}{2}\partial_{-}\psi^{+}_{\beta})(Y^{\prime}-\psi^{-}_{\alpha}\frac{\eta^{\alpha\beta}}{2}\psi_{\beta}^{-}{}^{\prime})$
$\displaystyle-2e^{\Phi/2}\psi_{\beta}^{-}{}^{\prime}\mathfrak{m}^{\beta\alpha}\partial_{-}\psi^{+}_{\alpha}\Big{]}+S_{o}[u],$
$\displaystyle
S_{i}=\frac{k}{4\pi}\int_{\Sigma_{i}}\mathrm{d}t\mathrm{d}\varphi\;\;\Big{[}$
$\displaystyle\frac{1}{2}\tilde{\Phi}^{\prime}\partial_{+}\tilde{\Phi}+2e^{-\tilde{\Phi}}(\partial_{+}V+\chi^{-}_{\alpha}\frac{\eta^{\alpha\beta}}{2}\partial_{+}\chi^{-}_{\beta})(U^{\prime}+\chi^{+}_{\alpha}\frac{\eta^{\alpha\beta}}{2}\chi_{\beta}^{+}{}^{\prime})$
$\displaystyle+2e^{-\tilde{\Phi}/2}\chi_{\beta}^{+}{}^{\prime}\tilde{\mathfrak{m}}^{\beta\alpha}\partial_{+}\chi^{-}_{\alpha}\Big{]}+S_{i}[w].$
(81)

Here
$\mathfrak{m}\indices{{}^{\alpha}_{\beta}}=e^{\textrm{i}\,C^{a}(\lambda^{a})\indices{{}^{\alpha}_{\beta}}}$,
$\tilde{\mathfrak{m}}\indices{{}^{\alpha}_{\beta}}=e^{\textrm{i}\,\tilde{C}^{a}(\lambda^{a})\indices{{}^{\alpha}_{\beta}}}$,
$\mathfrak{m}^{\alpha\beta}=\mathfrak{m}\indices{{}^{\alpha}_{\gamma}}\eta^{\gamma\beta}$,
and
$\tilde{\mathfrak{m}}^{\alpha\beta}=\tilde{\mathfrak{m}}\indices{{}^{\alpha}_{\gamma}}\eta^{\gamma\beta}$,
with $\lambda^{a}$ the basis of the $\mathfrak{so}({\cal N})$ algebra under
which fermions transform, and

$\displaystyle S_{o}[u]$
$\displaystyle=\frac{k}{4\pi}\int_{\Sigma_{o}}dtd\varphi\;\mathrm{Str}\left(u^{-1}\partial_{\varphi}uu^{-1}\partial_{-}u\right)+I_{WZ}[U]|_{\Sigma_{o}}$
(82) $\displaystyle S_{i}[u]$
$\displaystyle=\frac{k}{4\pi}\int_{\Sigma_{i}}dtd\varphi\;\mathrm{Str}\left(w^{-1}\partial_{\varphi}ww^{-1}\partial_{+}w\right)+I_{WZ}[W]|_{\Sigma_{i}}\,.$

Here, once again, $u(t,\varphi)=e^{\textrm{i}\,C^{a}(t,\varphi)T^{a}}$, and
$w(t,\varphi)=e^{\textrm{i}\,\tilde{C}^{a}(t,\varphi)T^{a}}$ are the
$r$-independent pullback of
$U(t,\varphi,r)=e^{\textrm{i}\,C^{a}(t,\varphi,r)T^{a}}$ and
$W(t,\varphi,r)=e^{\textrm{i}\,\tilde{C}^{a}(t,\varphi,r)T^{a}}$,
respectively.

####  3.5.1 Hyperbolic holonomy

As explained in the previous section, we consider that
$K(t)=k_{0}L_{0}+k_{r}T$. For such a holonomy, the contribution of holonomy in
the action is given by

$S_{hol}=S_{hol}^{k_{0}}+S_{hol}^{k_{r}}$ (83)

with

$\displaystyle
S_{hol}^{k_{0}}=\frac{k}{4\pi}\int\mathrm{d}t\mathrm{d}\varphi\;\;\Big{[}$
$\displaystyle
k_{0}\bigg{(}\partial_{-}\Phi-\partial_{+}\tilde{\Phi}+e^{\Phi/2}(\psi^{-}\cdot\mathfrak{m}\cdot\partial_{-}\psi^{+})-e^{-\tilde{\Phi}/2}(\chi^{+}\cdot\tilde{\mathfrak{m}}\cdot\partial_{+}\chi^{-})$
$\displaystyle-2e^{\Phi}Y(\partial_{-}X-\frac{1}{2}\psi^{+}\cdot\partial_{-}\psi^{+})-2e^{-\tilde{\Phi}}U(\partial_{+}V+\frac{1}{2}\chi^{-}\cdot\partial_{+}\chi^{-})\bigg{)}-k_{0}^{2}\Big{]},$
(84)

and

$\displaystyle
S_{hol}^{k_{r}}=\frac{k}{4\pi}\int\mathrm{d}t\mathrm{d}\varphi\;\;\Big{[}$
$\displaystyle
k_{r}\left(-i\left(\partial_{-}C-\partial_{+}\tilde{C}\right)-e^{\Phi}(\psi^{-}\cdot\lambda\cdot\psi^{-})(\partial_{-}X-\frac{1}{2}\psi^{+}\cdot\partial_{-}\psi^{+})\right.$
$\displaystyle+\left.e^{-\tilde{\Phi}}(\chi^{+}\cdot\lambda\cdot\chi^{+})(\partial_{+}V+\frac{1}{2}\chi^{-}\cdot\partial_{+}\chi^{-})+2e^{\phi/2}(\psi^{-}\cdot\lambda\cdot\mathfrak{m}\cdot\partial_{-}\psi^{+})\right.$
$\displaystyle-\left.2e^{-\tilde{\phi}/2}(\chi^{+}\cdot\lambda\cdot\tilde{\mathfrak{m}}\cdot\partial_{+}\chi^{-})\right)+k_{r}^{2}\Big{]}.$
(85)

Here we have used a shorthand notation “$\cdot$” to avoid writing all
spinorial indices explicitly. As an example, it reads as
$\psi^{+}\cdot\partial_{-}\psi^{+}=\psi^{+}\cdot\eta\cdot\partial_{-}\psi^{+}=\psi^{+}_{\alpha}\eta^{\alpha\beta}\partial_{-}\psi^{+}_{\beta}$
and
$\psi^{-}\cdot\mathfrak{m}\cdot\partial_{-}\psi^{+}=\psi^{-}_{\alpha}\mathfrak{m}^{\alpha\beta}\partial_{-}\psi^{+}_{\beta}$.
Notice that here $\lambda$ (and consequently $\mathfrak{m}$) is given by the
$\mathfrak{osp}(N|2)$ representation of appendix A and we have set
$\frac{2C_{\rho}}{d-1}=-\frac{1}{2}$ accordingly.

The full ${\cal N}=2$ super Chern-Simons action can then be written as

$S_{\rm
CS}[k_{0},K_{r},Y,\Phi,X,\psi^{\pm}_{\alpha},V,\tilde{\Phi},U,\chi^{\pm}_{\alpha}]=S_{\rm
bdy}^{\Sigma_{o}}[k_{0},k_{r},Y,\Phi,X,\psi^{\pm}_{\alpha}]-S_{\rm
bdy}^{\Sigma_{i}}[k_{0},k_{r},V,\tilde{\Phi},U,\chi^{\pm}_{\alpha}]\,,$ (86)

with

$\displaystyle S_{\rm bdy}^{\Sigma_{o}}=\frac{k}{4\pi}\int
dtd\varphi\;\Big{[}$
$\displaystyle\frac{1}{2}\partial_{-}\Phi(\Phi^{\prime}+2k_{0})+\frac{1}{2}\partial_{-}C(C^{\prime}-2ik_{r})$
$\displaystyle+2e^{\Phi}(\partial_{-}X-\frac{1}{2}\psi^{+}\cdot\partial_{-}\psi^{+})(Y^{\prime}-\frac{1}{2}\psi^{-}\cdot{\psi^{-}}^{\prime}-k_{0}Y-\frac{1}{2}k_{r}(\psi^{-}\cdot\lambda\cdot\psi^{-}))$
$\displaystyle-2e^{\Phi/2}({\psi^{-}}^{\prime}-\frac{k_{0}}{2}\psi^{-}-k_{r}\psi^{-}\cdot\lambda)\cdot\mathfrak{m}\cdot\partial_{-}\psi^{+}-\frac{1}{2}(k_{0}^{2}-k_{r}^{2})\Big{]}$
(87)

and

$\displaystyle S_{\rm bdy}^{\Sigma_{i}}=\frac{k}{4\pi}\int
dtd\varphi\;\Big{[}$
$\displaystyle\frac{1}{2}\partial_{+}\tilde{\Phi}(\tilde{\Phi}^{\prime}+2k_{0})+\frac{1}{2}\partial_{+}\tilde{C}(\tilde{C}^{\prime}-2ik_{r})$
$\displaystyle+2e^{-\tilde{\Phi}}(\partial_{+}V+\frac{1}{2}\chi^{-}\cdot\partial_{+}\chi^{-})(U^{\prime}+\frac{1}{2}\chi^{+}\cdot{\chi^{+}}^{\prime}+k_{0}U-\frac{1}{2}k_{r}(\chi^{+}\cdot\lambda\cdot\chi^{+}))$
$\displaystyle+2e^{-\tilde{\Phi}/2}({\chi^{+}}^{\prime}+\frac{k_{0}}{2}\chi^{+}+k_{r}\chi^{+}\cdot\lambda)\cdot\tilde{\mathfrak{m}}\cdot\partial_{+}\chi^{-}+\frac{1}{2}(k_{0}^{2}-k_{r}^{2})\Big{]}.$
(88)

####  3.5.2 Boundary condition

Next, we impose the boundary conditions on the ${\cal N}=2$ super Chern-Simons
connection, corresponding to asymptotically AdS3 spacetimes [11]. The boundary
conditions on the fields at outer boundary are in accordance with highest-
weight representation, those on the fields at inner boundary are in accordance
with the lowest-weight representation. As shown in [41] and discussed in [11,
7], the boundary conditions on the fields at $r=r_{2}\equiv r_{o}$ are

${\cal A}_{r}=0,\qquad{\cal A}_{\varphi}=L_{-}+{\cal
L}(t,\varphi)L_{+}+\Psi^{+}_{\alpha}(t,\varphi)Q_{+}^{\alpha}+{\cal
B}(t,\varphi)T.$ (89)

Similarly, the boundary conditions on the fields at $r=r_{1}\equiv r_{i}$ are

${\cal A}_{r}=0,\qquad{\cal A}_{\varphi}=L_{+}+{\cal
M}(t,\varphi)L_{-}+\Psi^{-}_{\alpha}(t,\varphi)Q_{-}^{\alpha}+{\cal
R}(t,\varphi)T.$ (90)

In terms of the field appearing in the Gauss decomposition, this gives the
conditions

$\displaystyle{\footnotesize{\begin{array}[]{c||c}\textrm{On Outer
Boundary}&\textrm{On Inner Boundary}\\\ \hline\cr\hline\cr{}\hfil&{}\hfil\\\
e^{\Phi}[Y^{\prime}-\frac{1}{2}\psi^{-}\cdot{\psi^{-}}^{\prime}-k_{0}Y-\frac{1}{2}k_{r}(\psi^{-}\cdot\lambda\cdot\psi^{-})]=1&e^{-\tilde{\Phi}}[U^{\prime}+\frac{1}{2}\chi^{+}\cdot{\chi^{+}}^{\prime}+k_{0}U-\frac{1}{2}k_{r}(\chi^{+}\cdot\lambda\cdot\chi^{+})]=1\\\
{}\hfil&{}\hfil\\\ \Phi^{\prime}+k_{0}=2X&\tilde{\Phi}^{\prime}+k_{0}=-2V\\\
{}\hfil&{}\hfil\\\
\psi^{+}_{\alpha}+e^{\Phi/2}[{\psi^{-}}^{\prime}-\frac{k_{0}}{2}\psi^{-}-k_{r}(\psi^{-}\cdot\lambda)]_{\beta}\mathfrak{m}\indices{{}^{\beta}_{\alpha}}=0&\chi^{-}_{\alpha}+e^{-\tilde{\Phi}/2}[{\chi^{+}}^{\prime}+\frac{k_{0}}{2}\psi^{+}+k_{r}(\chi^{+}\cdot\lambda)]_{\beta}\tilde{\mathfrak{m}}\indices{{}^{\beta}_{\alpha}}=0\end{array}}}$
(98)

Inserting these conditions in the action (86), we find

$\displaystyle S_{\rm bdy}^{\Sigma_{o}}=\frac{k}{4\pi}\int
dtd\varphi\;\Big{[}$
$\displaystyle\frac{1}{2}\partial_{-}\Phi(\Phi^{\prime}+2k_{0})+\frac{1}{2}\partial_{-}C(C^{\prime}-2ik_{r})+2\psi^{+}\cdot(\mathfrak{m}-\frac{1}{2}\eta)\cdot\partial_{-}\psi^{+}$
$\displaystyle-\frac{1}{2}(k_{0}^{2}-k_{r}^{2})\Big{]}\,,$ $\displaystyle
S_{\rm bdy}^{\Sigma_{i}}=\frac{k}{4\pi}\int dtd\varphi\;\Big{[}$
$\displaystyle\frac{1}{2}\partial_{+}\tilde{\Phi}(\tilde{\Phi}^{\prime}+2k_{0})+\frac{1}{2}\partial_{+}\tilde{C}(\tilde{C}^{\prime}-2ik_{r})-2\chi^{-}\cdot(\tilde{\mathfrak{m}}-\frac{1}{2}\eta)\cdot\partial_{+}\chi^{-}$
$\displaystyle+\frac{1}{2}(k_{0}^{2}-k_{r}^{2})\Big{]}\,.$ (100)

Putting these together, the final form of action is written with pairs
$(\Phi,\tilde{\Phi})$ and $(C,\tilde{C})$ of chiral bosons linked through the
holonomy and two pairs of chiral fermions with a dilaton coupling to the $C$
field

$\displaystyle S=\frac{k}{4\pi}\int dtd\varphi\;\Big{[}$
$\displaystyle\frac{1}{2}\partial_{-}\Phi\Phi^{\prime}-\frac{1}{2}\partial_{+}\tilde{\Phi}\tilde{\Phi}^{\prime}+k_{0}(\partial_{-}\Phi-\partial_{+}\tilde{\Phi})$
$\displaystyle+\frac{1}{2}\partial_{-}CC^{\prime}-\frac{1}{2}\partial_{+}\tilde{C}\tilde{C}^{\prime}-ik_{r}(\partial_{-}C-\partial_{+}\tilde{C})$
$\displaystyle+2\psi^{+}\cdot(\mathfrak{m}-\frac{1}{2}\eta)\cdot\partial_{-}\psi^{+}+2\chi^{-}\cdot(\tilde{\mathfrak{m}}-\frac{1}{2}\eta)\cdot\partial_{+}\chi^{-}$
$\displaystyle-(k_{0}^{2}-k_{r}^{2})\Big{]}\,.$ (101)

For ${\cal N}=2$, see appendix (A), we have

$\mathfrak{m}-\frac{1}{2}\eta=\begin{pmatrix}0&2e^{-\frac{iC}{2}}-1\\\
2e^{\frac{iC}{2}}-1&0\end{pmatrix}.$ (102)

This action enjoys invariance under a redundant gauge symmetry

$\displaystyle\Phi$
$\displaystyle\rightarrow\Phi+\epsilon_{1}(t),\qquad\tilde{\Phi}\rightarrow\tilde{\Phi}+\epsilon_{1}(t)\,,$
$\displaystyle C$ $\displaystyle\rightarrow
C+\epsilon_{2}(t),\qquad\tilde{C}\rightarrow\tilde{C}+\epsilon_{2}(t)\,,$
$\displaystyle k_{0}$ $\displaystyle\rightarrow
k_{0},\qquad\qquad~{}~{}~{}k_{r}\rightarrow k_{r}\,,$
$\displaystyle\psi^{+}_{\alpha}$
$\displaystyle\rightarrow\psi^{+}_{\alpha},\qquad\quad~{}~{}~{}~{}\chi^{-}_{\alpha}\rightarrow\chi^{-}_{\alpha}\,.$
(103)

The action (101) is one of the main result of this paper. Even though this
action is written for ${\cal N}=2$ supergravity yet it contains general
feature that holds for any ${\cal N}$. The holonomy independent part of the
action always will be written in this form considering a generalization of
$R$-symmetry term. For larger ${\cal N}$ the holonomy dependent part is very
much related to the chosen conjugacy class of holonomy.

Another observation regarding the action (101) is that it is equivalent to the
sum of two chiral actions of [11] when the holonomies are set to zero. In
another words, this is a generalization of [11] by including the zero modes
and holonomies. In [11] it was shown that for the bosonic part and ignoring
zero modes and holonomies, this is equivalent to the Liouville action. A
detail analysis of such a relation in the case of super Liouville action and
in the presence of zero modes and holonomies will be discussed elsewhere [47].

Last comment regarding the action (101) is that this is indeed the action on
the super Virasoro coadjoint orbit. We will come back to this point in Section
5.4 where we explicitly show that the action (101) is equivalent to the
geometric action on the coadjoint orbit of the group of reparameterizations of
${\cal N}=2$ supercircle.

###  3.6 Comments on General ${\cal N}$

For $\mathcal{N}>2$, the $R$-symmetry group is larger, and there are many more
conjugacy classes. The number of spinors will also be much more, leading to
higher than quadratic orders of interactions. Although the basic idea of
holonomy is the same, the detailed analysis will be much more complicated.

However, for the case when the holonomy is given by $K(t)=k_{0}L_{0}+k_{r}T$,
where $T$ is one of the Cartan elements of the $R$-symmetry algebra, the
action (100) is indeed the reduced Hamiltonian action.

Moreover, for a constant orbit representative, we are able to write down the
complete action for any ${\cal N}$, by considering the holonomy as non-
periodicity in the fields. However, this comes at a price. The drawback of
treating the holonomies as non-periodicity in the fields is that the holonomy
is no longer a dynamical field. In the quantized theory of gravity one expects
to have all the possible solutions such that the dynamics of the theory allows
changing between the solutions. In the geometric picture all these solutions
are determined by different orbits and therefore a sensible classical action
requires to have the orbit representatives (holonomies) as dynamical fields,
such that in the partition function one can sum over all the possible
solutions of them.

All this makes us believe that writing the action in the presence of a
dynamical holonomy requires a case-by-case study that we do not intend to
pursue here.

##  4 Hamiltonian reduction of $W_{3}$ higher Chern-Simons action

As we have repeatedly mentioned, the goal of this paper is to provide a
boundary action through Hamiltonian reduction of various extension of Chern-
Simons theory where the algebra of asymptotic charges is non-linear while
emphasizing the role of zero modes and holonomies. Until now, we discussed the
supersymmetric extension of Chern-Simons theory of gravity where the non-
linear terms will appear in ${\cal N}>2$ supergravity models. Another place
where a non-linear algebra of charges will appear is in higher spin extension
of Chern-Simons gravity [21, 20]. For a spin-$N$ field the asymptotic charges
form a $W_{N}$ algebra. This algebra is non-linear for $N>2$, however we have
much less fields compared to supergravity and a simpler classification of
holonomy conjugacy classes. In this section, we will focus on a Chern-Simons
theory for a spin-3 fields with $W_{3}$ asymptotic symmetry algebra and we
write down the boundary action via Hamiltonian reduction. We will show this
for different representations of $SL(3)$ and we claim that the boundary action
in each case is the geometric action on the symplectic leaves of $W_{3}$
manifold.

###  4.1 Chern-Simons action in the presence of zero modes

In this section, we follow closely the discussion of section 3, however now
the gauge fields are $\mathfrak{sl}(3,\mathbb{R})$ valued. As explained, the
Chern-Simons action (2) can be written as

$S[{\cal A}]=\frac{k}{4\pi}\int_{{\cal M}}dtd\varphi
dr\,\mathrm{tr}\left({\cal A}_{\varphi}\dot{{\cal A}}_{r}-{\cal
A}_{r}\dot{{\cal A}}_{\varphi}+2{\cal
A}_{t}\mathcal{F}_{r\varphi}\right)+I_{\Sigma_{i}}+I_{\Sigma_{o}}\,,$ (104)

with

$\mathcal{F}_{r\varphi}=\partial_{r}{\cal A}_{\varphi}-\partial_{\varphi}{\cal
A}_{r}+[{\cal A}_{r},{\cal A}_{\varphi}]\,.$ (105)

and $I_{\Sigma_{i,o}}$ are boundary terms adapted to the boundary conditions
under consideration.

Imposing the boundary condition ${\cal A}_{-}=0$ at outer boundary and ${\cal
A}_{+}=0$ at inner boundary, we have for the boundary Hamiltonian actions
$I_{\Sigma_{i,o}}$

$I_{\Sigma_{i,o}}=-\frac{k}{4\pi}\int_{\Sigma_{i,o}}dtd\varphi\,\mathrm{tr}{\cal
A}_{\varphi}^{2}\,.$ (106)

The field ${\cal A}_{t}$ is a Lagrange multiplier for the constraint
$\mathcal{F}_{r\varphi}=0$ which can be solved for if

$\displaystyle{\cal
A}_{\varphi}=G^{-1}\left(\partial_{\varphi}+K(t)\right)G,\qquad\qquad{\cal
A}_{r}=G^{-1}\partial_{r}G.$ (107)

Here $G$ is an $SL(3,\mathbb{R})$ element and $K(t)$ is the Lie-algebra valued
holonomy. For our purpose of dealing with hyperbolic holonomy, it is suitable
to choose a representative in the Gauss decomposition of $SL(3,\mathbb{R})$,
i.e.

$\displaystyle G=e^{E_{-}^{\alpha}}e^{\theta^{a}}e^{E_{+}^{\alpha}},$ (108)

where $\\{E_{\pm}^{\alpha},\theta^{a}\\}$, $\alpha=1,2,3$, $a=1,2$ are
generators of $\mathfrak{sl}(3,\mathbb{R})$, which form the Chevalley basis of
$\mathfrak{sl}(3,\mathbb{R})$. A matrix representation in terms of this basis
is given by (2.2) and (33).

As we have discussed in the $\mathfrak{sl}(2)$ case, the zero modes are
related to the non-trivial holonomies in the bulk, which are characterized by
both the conjugacy classes of $\mathfrak{sl}(3)$, and the classification of
symplectic leaves [48, 49]. In the case of $SL(3)$, there are seven different
conjugacy classes; two hyperbolic, one elliptic, three parabolic and the
exceptional conjugacy classes. A complete list of conjugacy classes of the
$SL(3)$ group based on its possible isotropy subgroups can be found in [49].

Once again we focus in our discussion only on holonomies in the non-degenerate
hyperbolic conjugacy class since BTZ black holes have hyperbolic holonomies.
This holonomy class can be represented by diagonalizable matrices with three
distinct real eigenvalues which are determined with two parameters777The
dimension of isotropy subgroups of $SL(3)$ is $2,4$ or $8$ which means the
holonomies can be parametrized with these number of parameters [49].. These
two parameters can be completely fixed by asymptotic symmetry transformations
which set them to the asymptotic charges ${\cal L}$ and $\mathcal{W}$. This is
similar to the Chern-Simons theory with Virasoro symmetry of charges. In that
case the holonomies are determined by isotropy groups of $SL(2)$ group, which
leads to three conjugacy classes. These classes are determined only with one
parameter which upon fixing by an appropriate Virasoro transformation is set
to the Virasoro charge ${\cal L}$. Indeed for the hyperbolic holonomy with a
fixed orbit representative $b_{0}$, the zero modes of ${\cal L}$ is determined
by $b_{0}$ which is related to the only parameter in the holonomy matrix
$k_{0}$. For the details see section 5.2. In the case of $SL(3)$ group, zero
modes of ${\cal L}$ and $\mathcal{W}$ are determined by the intersections of
two symplectic leaves with the representatives $b_{0}$ and $b_{1}$. These
representatives are related to two holonomy parameters $k_{0}$ and $k_{1}$.

####  4.1.1 Hyperbolic holonomy

When dealing with hyperbolic holonomies, a suitable parametrization of group
elements $G$ is given by the Gauss decomposition, where we have:

$\displaystyle
G=e^{\gamma_{3}E_{-}^{3}}e^{\gamma_{2}E_{-}^{2}+\gamma_{1}E_{-}^{1}}e^{\frac{1}{\sqrt{2}}\phi_{1}\theta_{1}+\sqrt{\frac{3}{2}}\phi_{2}\theta_{2}}e^{\xi_{1}E_{+}^{1}+\xi_{2}E_{+}^{2}}e^{\xi_{3}E_{+}^{3}}.$
(109)

Here $\left\\{E_{\pm}^{\alpha},\theta^{a}\right\\}$ are Chevalley basis of
$\mathfrak{sl}(3)$ and are defined in (2.2) and (33).

In terms of these group elements, the action can be written as

$\displaystyle S_{\rm CS}[G,K(t)]=$ $\displaystyle+\frac{k}{4\pi}\int_{{\cal
M}}d^{3}x\;\mathrm{tr}\left(\partial_{r}(G^{-1}\partial_{\varphi}GG^{-1}\partial_{t}G)\right)+\frac{k}{12\pi}\int_{{\cal
M}}\;\mathrm{tr}(G^{-1}dG)^{3}$ (110) $\displaystyle+\frac{k}{4\pi}\int_{{\cal
M}}d^{3}x\;\mathrm{tr}\left(2\partial_{r}(G^{-1}K\partial_{t}G)-\partial_{t}(G^{-1}K\partial_{r}G)\right)+I_{\Sigma_{i}}+I_{\Sigma_{o}}\,.$

Let’s consider an annulus geometry with two boundaries
$r^{o}=r^{\textrm{outer}}$ and $r^{i}=r^{\textrm{inner}}$ such that

$h=G(t,r=r^{\textrm{outer}},\varphi),\qquad
l=G(t,r=r^{\textrm{inner}},\varphi)\,.$ (111)

Similarly to our previous discussions, we would like to impose lowest weight
gauge condition on the inner boundary and it is more practical to use a field
redefinition on the inner boundary in order to parametrize the group elements
$l$ as

$\displaystyle
l=e^{\tilde{\xi}_{3}E_{+}^{3}}e^{\tilde{\xi}_{2}E_{+}^{2}+\tilde{\xi}_{1}E_{+}^{1}}e^{\frac{1}{\sqrt{2}}\tilde{\phi}_{1}\theta_{1}+\sqrt{\frac{3}{2}}\tilde{\phi}_{2}\theta_{2}}e^{\tilde{\gamma}_{1}E_{-}^{1}+\tilde{\gamma}_{2}E_{-}^{2}}e^{\tilde{\gamma}_{3}E_{-}^{3}}.$
(112)

Then, the action on the $r$-boundaries reduces to

$\displaystyle S_{\rm CS}[h,l,K(t)]=$
$\displaystyle+\frac{k}{4\pi}\int_{\Sigma_{o}}dtd\varphi\;\mathrm{tr}\left(h^{-1}\partial_{\varphi}hh^{-1}\partial_{-}h+2h^{-1}K\partial_{-}h-K^{2}\right)$
(113)
$\displaystyle-\frac{k}{4\pi}\int_{\Sigma_{i}}dtd\varphi\;\mathrm{tr}\left(l^{-1}\partial_{\varphi}ll^{-1}\partial_{+}l+2l^{-1}K\partial_{+}l+K^{2}\right)+I_{WZ}[G]\,,$

As usual, the Wess-Zumino term

$I_{WZ}[G]=\frac{k}{12\pi}\int_{{\cal M}}\;\mathrm{tr}(G^{-1}dG)^{3}\,,$ (114)

can be written as a total derivative and hence it also only depends on the
boundary values of the group element $G$.

For the non-degenerate hyperbolic holonomy, we have

$K(t)=k_{0}(t)(H_{1}+H_{2})+k_{1}(t)H_{2}.$ (115)

This is the hyperbolic class with three distinct real eigenvalues
parameterized by two arbitrary functions of $t$.

Now, given the reparametrization (109), the action (110) takes the form

$\displaystyle S_{o}=\frac{k}{4\pi}\int_{\Sigma_{o}}dt\,$ $\displaystyle
d\varphi\bigg{(}2e^{\frac{\phi_{1}+3\phi_{2}}{2}}\left[\gamma_{1}^{\prime}-(k_{0}-k_{1})\gamma_{1}\right]\,\partial_{-}\xi_{1}$
$\displaystyle+2e^{\frac{\phi_{1}-3\phi_{2}}{2}}\left[\gamma_{2}^{\prime}-(k_{0}+2k_{1})\gamma_{2}\right]\,\partial_{-}\xi_{2}$
$\displaystyle+\frac{1}{2}e^{\phi_{1}}F(\gamma_{1},\gamma_{2},\gamma_{3};k_{0},k_{1})\,\left(\xi_{2}\partial_{-}\xi_{1}-\xi_{1}\partial_{-}\xi_{2}-2\partial_{-}\xi_{3}\right)$
$\displaystyle+\frac{1}{2}\left[\left(\phi_{1}^{\prime}+4k_{0}+2k_{1}\right)\partial_{-}\phi_{1}+3\left(\phi_{2}^{\prime}-2k_{1}\right)\partial_{-}\phi_{2}\right]$
$\displaystyle-2\left(k_{0}^{2}+k_{0}k_{1}+k_{1}^{2}\right)\bigg{)},$ (116)

where

$F(\gamma_{1},\gamma_{2},\gamma_{3};k_{0},k_{1})=\gamma_{2}\gamma_{1}^{\prime}-\gamma_{1}\gamma_{2}^{\prime}-2\gamma_{3}^{\prime}+k_{1}(2\gamma_{3}+3\gamma_{1}\gamma_{2})+4k_{0}\gamma_{3}.$
(117)

The action on the inner boundary $S_{i}$ is a completely similar expression to
the action on the outer boundary except written in terms of the fields on
inner boundary $\tilde{\gamma},\tilde{\phi},\tilde{\xi}$ instead of
$\gamma,\phi,\xi$ respectively.

###  4.2 Different $SL(3)$ representations and highest weight gauge condition

The finite dimensional representations of $SL(3)$ can be obtained from those
of $SL(2)$ by considering different non-isomorphic $SL(2)$ subgroups embedded
in $SL(3)$. This can be understood for example from the decomposition of
finite dimensional representation of $SL(3)$ group into the eigenspaces of the
vector space formed from the Cartan elements of $\mathfrak{sl}(3)$ algebra
[50]. These representations come with a unique highest weight. For $SL(3)$
there are two distinct representations: principal and diagonal $SL(2)$
embeddings. In each case, the root space decomposition differ which in turn
leads to a different (but unique in each representation) highest weight. In
the following we discuss each representation and the corresponding boundary
actions. The boundary actions will be different given that the highest weight
gauge condition for the reduction is different in each representation.

####  4.2.1 Boundary Condition and Hamiltonian Reduction

In this section we follow closely the reduction procedure discussed in
sections 3.4 and 3.5. Therefore, we impose an appropriate boundary condition
at each boundary. As before, we consider that the angular component of
$\mathfrak{sl}(3)$ Chern-Simons connection to be fixed by the highest weight
or the lowest weight representations on the outer or inner boundaries
respectively. This is consistent with the asymptotic behaviour of AdS3.

####  4.2.2 Principal Embedding

With the principal embedding, one can choose the basis such that the
$\mathfrak{sl}(3)$ algebra is written as

$\displaystyle[L_{m},L_{n}]$ $\displaystyle=(m-n)L_{m+n},$ (118)
$\displaystyle[L_{m},W_{i}]$ $\displaystyle=(2m-i)W_{m+i},$ (119)
$\displaystyle[W_{i},W_{j}]$
$\displaystyle=-\frac{1}{3}(i-j)(2i^{2}+2j^{2}-ij-8)L_{i+j},$ (120)

where $L_{m}$’s with $m=\pm 1,0$ are the $\mathfrak{sl}(2)$ generators and
$W_{i}$’s with $i=\pm 2,\pm 1,0$ are the rest of $\mathfrak{sl}(3)$
generators. One can obtain these generators directly from the Chevalley basis
[51] as

$\displaystyle L_{0}$
$\displaystyle\equiv\frac{1}{2}\sum_{i=1}^{2}(c_{i}H^{i}),$ (121)
$\displaystyle L_{\pm 1}$
$\displaystyle\equiv\pm\sum_{i=1}^{2}2^{\mp\frac{1}{2}}\sqrt{c_{i}}E_{\mp}^{i},$
(122) $\displaystyle W_{i}$
$\displaystyle\equiv(-1)^{2-i}\frac{(2+i)!}{12}\left(\textrm{adj}_{L_{-1}}\right)^{2-i}\left(L_{+1}\right)^{2},$
(123)

with $c_{i}=2\sum_{j}(C^{-1})_{ij}$ where $C$ is the Cartan matrix of
$\mathfrak{sl}(3)$ algebra

$C=\begin{pmatrix}2&-1\\\ -1&2\end{pmatrix}.$ (124)

An explicit basis for this algebra is then given by [21]

$\displaystyle L_{-1}$ $\displaystyle=\begin{pmatrix}0&-2&0\\\ 0&0&-2\\\
0&0&0\end{pmatrix},\qquad~{}L_{0}=\begin{pmatrix}1&0&0\\\ 0&0&0\\\
0&0&-1\end{pmatrix},\qquad L_{+1}=\begin{pmatrix}0&0&0\\\ 1&0&0\\\
0&1&0\end{pmatrix},$ $\displaystyle W_{-1}$
$\displaystyle=\begin{pmatrix}0&-2&0\\\ 0&0&2\\\ 0&0&0\end{pmatrix},\qquad
W_{0}=\frac{2}{3}\begin{pmatrix}1&0&0\\\ 0&-2&0\\\ 0&0&1\end{pmatrix},\qquad
W_{+1}=\begin{pmatrix}0&0&0\\\ 1&0&0\\\ 0&-1&0\end{pmatrix},$ $\displaystyle
W_{-2}$ $\displaystyle=\begin{pmatrix}0&0&8\\\ 0&0&0\\\
0&0&0\end{pmatrix},\qquad W_{+2}=\begin{pmatrix}0&0&0\\\ 0&0&0\\\
2&0&0\end{pmatrix}.$

The highest and lowest weight gauge conditions in terms of the
$\mathfrak{sl}(2)$-principal embedding basis takes the following form [21, 20]

$\displaystyle{\cal A}_{r}$ $\displaystyle=0,\qquad{\cal
A}_{\varphi}=L_{+1}+{\cal
L}(t,\varphi)L_{-1}+\mathcal{W}(t,\varphi)W_{-2}\qquad\textrm{at outer
boundary}$ (125) $\displaystyle{\cal A}_{r}$ $\displaystyle=0,\qquad{\cal
A}_{\varphi}=L_{-1}+{\cal
L}(t,\varphi)L_{+1}+\mathcal{W}(t,\varphi)W_{+2}\qquad\textrm{at inner
boundary}.$ (126)

At the outer boundary, this results in the following conditions

$\displaystyle
e^{\frac{\phi_{1}+3\phi_{2}}{2}}\left[\gamma_{1}^{\prime}-(k_{0}-k_{1})\gamma_{1}\right]$
$\displaystyle=1,$ (127) $\displaystyle
e^{\frac{\phi_{1}-3\phi_{2}}{2}}\left[\gamma_{2}^{\prime}-(k_{0}+2k_{1})\gamma_{2}\right]$
$\displaystyle=1,$ (128) $\displaystyle
F(\gamma_{1},\gamma_{2},\gamma_{3};k_{0},k_{1})$ $\displaystyle=0,$ (129)
$\displaystyle\xi_{2}-(k_{0}+k_{1})-\frac{1}{2}\left(\phi_{1}^{\prime}-\phi_{2}^{\prime}\right)$
$\displaystyle=0,$ (130)
$\displaystyle\xi_{1}-k_{0}-\frac{1}{2}\left(\phi_{1}^{\prime}+\phi_{2}^{\prime}\right)$
$\displaystyle=0,$ (131)

together with the expressions for ${\cal L}$ and $\mathcal{W}$

$\displaystyle{\cal L}$
$\displaystyle=-\frac{1}{4}\left(\xi_{1}^{2}+3\xi_{2}^{2}-2\xi_{1}\xi_{2}+2\xi_{3}+\xi_{1}^{\prime}+\xi_{2}^{\prime}\right),$
(132) $\displaystyle\mathcal{W}$
$\displaystyle=\frac{1}{8}\left(\xi_{3}^{\prime}+\frac{1}{2}(\xi_{2}\xi_{1}^{\prime}-\xi_{2}^{\prime}\xi_{1})+\xi_{2}(\xi_{1}^{2}-2\xi_{1}\xi_{2}+2\xi_{3})\right).$
(133)

Inserting these condition in the action, we obtain the reduced boundary action
on the outer boundary

$\displaystyle S_{\rm
bdy}^{\Sigma_{o}}[k_{0},k_{1},\phi_{1},\phi_{2}]=\frac{k}{4\pi}\int
dtd\varphi\;\bigg{(}$
$\displaystyle\frac{1}{2}\partial_{-}\phi_{1}(\phi^{\prime}_{1}+4k_{0}+2k_{1})+\frac{3}{2}\partial_{-}\phi_{2}(\phi^{\prime}_{2}-2k_{1})$
$\displaystyle-2(k_{0}^{2}+k_{0}k_{1}+k_{1}^{2})\bigg{)}.$ (134)

Following the same steps at the inner boundary, the complete boundary action
can be written as

$\displaystyle S_{\rm
bdy}[k_{0},k_{1},\phi_{1},\phi_{2},\tilde{\phi}_{1},\tilde{\phi}_{2}]=\frac{k}{4\pi}\int
dtd\varphi\,$
$\displaystyle\bigg{(}\frac{1}{2}\partial_{-}\phi_{1}\,\phi^{\prime}_{1}-\frac{1}{2}\partial_{+}\tilde{\phi}_{1}\,\tilde{\phi}^{\prime}_{1}+\frac{3}{2}\partial_{-}\phi_{2}\,\phi^{\prime}_{2}-\frac{3}{2}\partial_{+}\tilde{\phi}_{2}\,\tilde{\phi}^{\prime}_{2}$
$\displaystyle+(2k_{0}+k_{1})(\partial_{-}\phi_{1}-\partial_{+}\tilde{\phi}_{1})-3k_{1}(\partial_{-}\phi_{2}-\partial_{+}\tilde{\phi}_{2})$
$\displaystyle-4(k_{0}^{2}+k_{0}k_{1}+k_{1}^{2})\bigg{)},$ (135)

where $\psi_{1}$ and $\psi_{2}$ are the fields on the inner boundary.

This action has a redundant gauge symmetry

$\displaystyle\phi_{1}$
$\displaystyle\rightarrow\phi_{1}+\epsilon_{1}(t),\qquad\tilde{\phi}_{1}\rightarrow\tilde{\phi}_{1}+\epsilon_{1}(t)\,,$
$\displaystyle\phi_{2}$
$\displaystyle\rightarrow\phi_{2}+\epsilon_{2}(t),\qquad\tilde{\phi}_{2}\rightarrow\tilde{\phi}_{2}+\epsilon_{2}(t)\,,$
$\displaystyle k_{0}$ $\displaystyle\rightarrow
k_{0},\qquad\qquad~{}~{}~{}~{}k_{1}\rightarrow k_{1}\,.$ (136)

In order to write the action (4.2.2), we have eliminated $\gamma_{1}$ and
$\gamma_{2}$ while keeping $\phi_{1}$ and $\phi_{2}$. One can instead keep
$\gamma_{1}$, $\gamma_{2}$ in which case one can write the action as the
$\mathfrak{sl}(3)$ generalization of the Schwarzian action upon the following
field redefinitions:

$\displaystyle\gamma_{1}$
$\displaystyle=e^{-(k_{0}-k_{1})(f(t,\varphi)-\varphi)},$ (137)
$\displaystyle\gamma_{2}$
$\displaystyle=e^{-(k_{0}+2k_{1})(g(t,\varphi)-\varphi)}.$ (138)

At the outer boundary, the action is now

$\displaystyle S$ $\displaystyle=\frac{k}{4\pi}\int
dtd\varphi\,\left[-\frac{4}{3}\left(\left[\frac{\partial_{-}f^{\prime\prime}}{f^{\prime}}-\left(\frac{3}{2}\frac{f^{\prime\prime}}{f^{\prime}}+\frac{1}{4}\frac{g^{\prime\prime}}{g^{\prime}}\right)\frac{\partial_{-}f^{\prime}}{f^{\prime}}\right]\right.\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\left.\left.\left[\frac{\partial_{-}g^{\prime\prime}}{g^{\prime}}-\left(\frac{3}{2}\frac{g^{\prime\prime}}{g^{\prime}}+\frac{1}{4}\frac{f^{\prime\prime}}{f^{\prime}}\right)\frac{\partial_{-}g^{\prime}}{g^{\prime}}\right]\right)\right]$
$\displaystyle+\frac{k}{4\pi}\int
dtd\varphi\,\left[\frac{2}{3}\bigg{(}(k_{0}-k_{1})^{2}f^{\prime}\partial_{-}f+(k_{0}+2k_{1})^{2}g^{\prime}\partial_{-}g\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\left.\frac{(k_{0}-k_{1})(k_{0}+2k_{1})}{2}\left[f^{\prime}\partial_{-}g+g^{\prime}\partial_{-}f\right]\bigg{)}\right]$
$\displaystyle+\frac{k}{4\pi}\int
dtd\varphi\,\left[-\frac{2}{3}\left[(k_{0}-k_{1})f^{\prime}\frac{\partial_{-}g^{\prime}}{g^{\prime}}+(k_{0}+2k_{1})g^{\prime}\frac{\partial_{-}f^{\prime}}{f^{\prime}}\right]\right]\,.$
(139)

This is the action on the outer boundary of an annulus (trumpet) geometry in
the presence of dynamical holonomies (zero modes). We claim that this action
is the action on the symplectic leaves of the $W_{3}$ Poisson manifold for the
constant representatives $k_{0}$ and $k_{1}$. It generalizes the result of
[52] in the presence of non-trivial holonomies. Indeed when there is no non-
trivial holonomy, i.e. $k_{0}=k_{1}=0$, the Hamiltonian reduces to the two-
dimensional generalization of the $W_{3}$ Schwarzian action [52].

At this point lets point out a few interesting observations about these
boundary actions. The first remark is regarding the field redefinitions (137)
and (138). Since $\gamma_{1}$ and $\gamma_{2}$ are periodic, one can
immediately notice that $f(t,\varphi)$ and $g(t,\varphi)$ get a $2\pi$ shift
when going around a circle, i.e.

$\displaystyle f(t,\varphi+2\pi)$ $\displaystyle=f(t,\varphi)+2\pi,$
$\displaystyle g(t,\varphi+2\pi)$ $\displaystyle=g(t,\varphi)+2\pi.$ (140)

This is indeed very similar to $SL(2)$ Chern-Simons theory where such a field
appears in the Schwarzian action. In that case $f(t,\varphi)$ was an element
of reparametrization of the circle or equivalently an element of ${\rm
Diff}(S^{1})$. However, for $SL(3)$ theory the set $(f,g)$ is an element of
diffeomorphisms of $\rm{RP}^{2}$ real projective plane [53]. In fact, for
$SL(n)$ theory the Schwarzian action is written in terms of fields
parametrizing symplectic leaves of ${\rm Diff}(\rm{RP}^{n-1})$. Notice that
$\rm{RP}^{1}\cong S^{1}$.

Another remark is regarding the invariance of the $SL(3)$ Schwarzian action.
In the case of $SL(2)$, there was a $U(1)$ gauge redundancy which is the
isotropy group of $SL(2)$ for the hyperbolic conjugacy class. This resulted in
the invariance of the $SL(2)$ Schwarzian action under ${\rm
Diff}(S^{1})/S^{1}$. For more details, see [7] and section 5 for the same
discussion from the perspective of the geometric action. The isotropy group of
$SL(3)$ for the non-degenerate hyperbolic holonomy is
$\rm{R}^{*}\times\rm{R}^{*}$ where $\rm{R}^{*}$ is the projectively extended
real line (one point compactification of $\rm{R}$). It is topologically
isomorphic to $S^{1}$. Another way to realize this isotropy group is the gauge
redundancy (4.2.2) which is $U(1)\times U(1)$. Therefore, the Schwarzian
action (4.2.2) is invariant under ${\rm
Diff}(\rm{RP}^{2})/\rm{R}^{*}\times\rm{R}^{*}\cong{\rm
Diff}(\rm{RP}^{2})/S^{1}\times S^{1}$ .

####  4.2.3 Diagonal Embedding

We can also choose a basis for the $\mathfrak{sl}(3)$ representation known as
the diagonal embedding, in which the $\mathfrak{sl}(3)$ generators satisfy the
algebra [54]

$\displaystyle[U_{m},U_{n}]$ $\displaystyle=(m-n)U_{m+n},$ (141)
$\displaystyle[U_{m},J_{0}]$ $\displaystyle=0,$ (142)
$\displaystyle[U_{m},G_{n}^{\pm}]$
$\displaystyle=(\frac{m}{2}-n)G_{m+n}^{\pm},$ (143)
$\displaystyle[J_{0},G_{m}^{\pm}]$ $\displaystyle={\pm}\,G_{m}^{\pm},$ (144)
$\displaystyle[G_{m}^{+},G_{n}^{-}]$
$\displaystyle=U_{m+n}-\frac{3}{2}(m-n)J_{0}.$ (145)

A matrix representation for these generators is

$\displaystyle U_{-1}$ $\displaystyle=\begin{pmatrix}0&0&-1\\\ 0&0&0\\\
0&0&0\end{pmatrix},\qquad~{}U_{0}=\frac{1}{2}\begin{pmatrix}1&0&0\\\ 0&0&0\\\
0&0&-1\end{pmatrix},\qquad U_{+1}=\begin{pmatrix}0&0&0\\\ 0&0&0\\\
1&0&0\end{pmatrix},$ $\displaystyle J_{0}$
$\displaystyle=\begin{pmatrix}\frac{1}{3}&0&0\\\ 0&-\frac{2}{3}&0\\\
0&0&\frac{1}{3}\end{pmatrix},\qquad G_{+1/2}^{+}=\begin{pmatrix}0&0&0\\\
0&0&0\\\ 0&1&0\end{pmatrix},\qquad G_{+1/2}^{-}=\begin{pmatrix}0&0&0\\\
1&0&0\\\ 0&0&0\end{pmatrix},$ $\displaystyle G_{-1/2}^{+}$
$\displaystyle=\begin{pmatrix}0&1&0\\\ 0&0&0\\\ 0&0&0\end{pmatrix},\qquad
G_{-1/2}^{-}=\begin{pmatrix}0&0&0\\\ 0&0&-1\\\ 0&0&0\end{pmatrix}.$

The highest and lowest weight gauge conditions in terms of the
$\mathfrak{sl}(2)$-diagonal embedding basis take the following form

$\displaystyle{\cal A}_{r}$ $\displaystyle=0,\,{\cal
A}_{\varphi}=U_{+1}-\frac{8\pi}{k}\left[\left({\cal
L}(t,\varphi)-\frac{6\pi}{k}\mathcal{U}(t,\varphi)\right)U_{-1}+\frac{3}{2}\,\mathcal{U}(t,\varphi)J_{0}+\Psi_{\pm}G^{\pm}_{-1/2}\right]\,\textrm{on}\,\Sigma_{o}$
$\displaystyle{\cal A}_{r}$ $\displaystyle=0,\,{\cal
A}_{\varphi}=U_{-1}-\frac{8\pi}{k}\left[\left({\cal
L}(t,\varphi)-\frac{6\pi}{k}\mathcal{U}(t,\varphi)\right)U_{+1}+\frac{3}{2}\,\mathcal{U}(t,\varphi)J_{0}+\Psi_{\pm}G^{\pm}_{+1/2}\right]\,\textrm{on}\,\Sigma_{i}.$

At the outer boundary, this results in the following conditions

$\displaystyle-\frac{1}{2}e^{\phi_{1}}F(\gamma_{1},\gamma_{2},\gamma_{3};k_{0},k_{1})$
$\displaystyle=1,$ (146) $\displaystyle
e^{\frac{\phi_{1}+3\phi_{2}}{2}}\left[\gamma_{1}^{\prime}-(k_{0}-k_{1})\gamma_{1}\right]-\xi_{2}$
$\displaystyle=0,$ (147) $\displaystyle
e^{\frac{\phi_{1}-3\phi_{2}}{2}}\left[\gamma_{2}^{\prime}-(k_{0}+2k_{1})\gamma_{2}\right]+\xi_{1}$
$\displaystyle=0.$ (148)

Inserting these conditions into the action (4.1.1), we obtain the reduced
action on the outer boundary

$\displaystyle S_{\rm
bdy}^{\Sigma_{o}}[k_{0},k_{1},\phi_{1},\phi_{2}]=\frac{k}{4\pi}\int
dtd\varphi\;$
$\displaystyle\bigg{(}\frac{1}{2}\partial_{-}\phi_{1}(\phi^{\prime}_{1}+4k_{0}+2k_{1})+\frac{3}{2}\partial_{-}\phi_{2}(\phi^{\prime}_{2}-2k_{1})$
$\displaystyle+\left(\xi_{2}\partial_{-}\xi_{1}-\xi_{1}\partial_{-}\xi_{2}\right)-2(k_{0}^{2}+k_{0}k_{1}+k_{1}^{2})\bigg{)}.$
(149)

Following the same steps at the inner boundary, the complete boundary action
can be written as

$\displaystyle S_{\rm bdy}=\frac{k}{4\pi}\int dtd\varphi\,$
$\displaystyle\bigg{(}\frac{1}{2}\partial_{-}\phi_{1}\,\phi^{\prime}_{1}-\frac{1}{2}\partial_{+}\tilde{\phi}_{1}\,\tilde{\phi}^{\prime}_{1}+\frac{3}{2}\partial_{-}\phi_{2}\,\phi^{\prime}_{2}-\frac{3}{2}\partial_{+}\tilde{\phi}_{2}\,\tilde{\phi}^{\prime}_{2}$
$\displaystyle+\left(\xi_{2}\partial_{-}\xi_{1}-\xi_{1}\partial_{-}\xi_{2}\right)-\left(\tilde{\xi}_{2}\partial_{+}\tilde{\xi}_{1}-\tilde{\xi}_{1}\partial_{+}\tilde{\xi}_{2}\right)$
$\displaystyle+(2k_{0}+k_{1})(\partial_{-}\phi_{1}-\partial_{+}\tilde{\phi}_{1})-3k_{1}(\partial_{-}\phi_{2}-\partial_{+}\tilde{\phi}_{2})$
$\displaystyle-4(k_{0}^{2}+k_{0}k_{1}+k_{1}^{2})\bigg{)},$ (150)

where $\tilde{\phi}_{1},\tilde{\phi}_{2},\tilde{\xi}_{1},\tilde{\xi}_{2}$ are
the fields on the inner boundary.

It is interesting to notice that the second line is the free field theory of
spinors with bosonic statistics. It can be understood by comparing this action
to the one of $\mathcal{N}=2$ supergravity (101) and setting the $R$-symmetry
to zero. The sign difference between these two descriptions is because the
fields $\xi_{1},\xi_{2},\tilde{\xi}_{1},\tilde{\xi}_{2}$ are not Grassmann
variables, unlike the corresponding fields in supergravity.

The action (4.2.3) has a gauge redundancy given by

$\displaystyle\phi_{1}$
$\displaystyle\rightarrow\phi_{1}+\epsilon_{1}(t),\qquad\tilde{\phi}_{1}\rightarrow\tilde{\phi}_{1}+\epsilon_{1}(t)\,,$
$\displaystyle\phi_{2}$
$\displaystyle\rightarrow\phi_{2}+\epsilon_{2}(t),\qquad\tilde{\phi}_{2}\rightarrow\tilde{\phi}_{2}+\epsilon_{2}(t)\,,$
$\displaystyle\xi_{1,2}$
$\displaystyle\rightarrow\xi_{1,2},\qquad\qquad~{}\tilde{\xi}_{1,2}\rightarrow\tilde{\xi}_{1,2}\,,$
$\displaystyle k_{0}$ $\displaystyle\rightarrow
k_{0},\qquad\qquad~{}~{}~{}~{}k_{1}\rightarrow k_{1}\,.$ (151)

This is once again related to the fact that the isotropy group of $SL(3)$
group for the non-degenerate holonomy conjugacy class is given by
$\rm{R}^{*}\times\rm{R}^{*}\cong U(1)\times U(1)$. Therefore the action
(4.2.3) is invariant under $\rm{Diff}(\rm{RP}^{2})/S^{1}\times S^{1}$. The
Schwarzian can be obtained by an appropriate change of variables. However, we
will not present it here and we end this discussion by pointing out that given
the similarities between the action (4.2.3) and the action for ${\cal N}=2$
Neveu-Schwarz supergarvity (101) one can find the change of variables by
following the discussion of ${\cal N}=2$ supergravity in section 5.4.

It is important to make a comment about the boundary action (4.2.3). This
action which is obtained through the Drinfeld reduction should be thought as
the candidate for the action on the symplectic leaves of $W_{3}$ manifold.
Even though it is straightforward to write it down from the Hamiltonian
reduction, it is not obvious how to obtain it in the geometric approach. And
moreover we consider the presence of non-trivial holonomies which not only
allows for solutions like BTZ but also provides a geometric sense. This is
because the holonomies are related to the representatives of symplectic leaves
and the invariant charges of the theory, ${\cal
L}(t,\varphi),\mathcal{U}(t,\varphi),\Psi_{\pm}(t,\varphi)$, are determined by
the intersections of the symplectic leaves.

In the next section, we show explicitly the construction of Schwarzian
boundary actions for $\mathcal{N}=1,2$ supergravity with the Kirillov-Kostant
construction of geometric actions. Given that for these theories the
symplectic leaves are the coadjoint orbits, the procedure is tedious but
straightforward. We will show that these geometric actions match the ones
obtained through Hamiltonian reduction where non-trivial holonomies appear as
the orbit representatives.

##  5 Diffeomorphisms of the supercircle

In this section we study explicitly the case of ${\cal N}=0,1$ and $2$ from
the general results of sections 3.4 and 3.5 and we will show that the actions
(3.4.2) and (101) can be understood as the geometric action on the coadjoint
orbit of the group of reparameterizations of the supercircle. The bosonic case
proceeds similarly to the case discussed in [55, 56], although here we clarify
the role of the bulk holonomy as the orbit representative. The ${\cal N}=1$
supersymmetric case was also discussed in [56]. Here we generalize the results
to arbitrary bulk holonomy and extend it to ${\cal N}=2$ supersymmetry.

The case of ${\cal N}>2$ is interesting because the asymptotic symmetry
algebra (30) becomes non-linear. Due to this non-linearity there is no clear
interpretation in terms of coadjoint orbit of some supergroup. Instead, one
can think about these cases by suitably generalizing the concept of symplectic
leaves of a Poisson manifold, which in the case of linear algebras corresponds
to the coadjoint orbits. We will briefly comment on this in Appendix B. In
this section, we start by reviewing the Kirillov-Kostant construction of
geometric actions on the coadjoint orbits in the case of the centrally
extended groups of reparameterizations of the supercircle.

###  5.1 Reparameterizations of the (super)circle

The coadjoint action on a generic element of the dual space
$b_{0}\in\mathfrak{g}^{*}$ of any Lie algebra defines a particular coadjoint
orbit. The coadjoint orbits are symplectic manifolds isomorphic to the coset
$G/{\cal H}_{b_{0}}$, where $G$ is the Lie group in question and ${\cal
H}_{b_{0}}$ the stabilizer subgroup of the orbit, defined by all elements
$h\in G$ which leave the orbit representative $b_{0}$ invariant under the
coadjoint action ${\rm Ad}_{h}^{*}$. The symplectic form on the orbit is the
Kirillov-Kostant symplectic form $\Omega$. As it is closed, locally it is also
exact $\Omega=\mathrm{d}\alpha$. Then $\alpha$ can be used to define an action
on the orbit $S[g;b_{0}]=\int_{\gamma}\alpha$, whose symplectic structure by
definition coincides with the Kirillov-Kostant symplectic structure. This
action is known as the geometric action on the coadjoint orbit. The addition
of invariant Hamiltonians was discussed in [55] and the general construction
is reviewed in appendix C.1. From there we see that the Kirillov-Kostant
symplectic form for centrally extended groups consists of two terms; one
proportional to the orbit representative $b_{0}$ and one proportional to the
central charge $c$

$\displaystyle\Omega=\mathrm{d}\alpha$
$\displaystyle=\mathrm{d}\langle\operatorname{Ad}^{*}_{g^{-1}}b_{0},Y\rangle-\frac{c}{2}\langle\mathrm{d}S(g),Y\rangle\,$
(152)
$\displaystyle=\mathrm{d}\langle\operatorname{Ad}^{*}_{g^{-1}}(b_{0},c),(Y,n_{Y})\rangle.$

Here $\operatorname{Ad}^{*}_{g^{-1}}b_{0}$ is the coadjoint action of $g\in G$
on the orbit representative $b_{0}$, $\langle,\rangle$ is the pairing between
the Lie algebra and its dual space and hence defines a map
$\mathfrak{g}^{*}\times\mathfrak{g}\to\mathbb{R}$. $S(g)$ is the Souriau
cocycle defining the central extension of $G$ to $\hat{G}=G\times\mathbb{R}$.

In the above equation, $n_{Y}$ solves equation (236) and $Y$ can be obtained
from the following equality

$\mathrm{d}b=-\operatorname{ad}_{Y}^{*}b.$ (153)

For $b=\operatorname{Ad}_{g^{-1}}^{*}b_{0}$ this gives

$\mathrm{d}(\operatorname{Ad}_{g^{-1}}^{*}b_{0})=-\operatorname{ad}_{Y}^{*}(\operatorname{Ad}_{g^{-1}}^{*}b_{0}).$
(154)

From (153) it follows that $Y$ solves the Maurer-Cartan equation
$\mathrm{d}Y=-\frac{1}{2}\operatorname{ad}_{Y}Y$, whose solution is locally
$Y=g^{-1}dg$. So to find the geometric action, one needs to compute the
Maurer-Cartan form $Y$ and write the Kirillov-Kostant symplectic form (152) as
a total exterior derivative, or find $n_{Y}$ as a solution to (236). In
practice it will also be useful to do the former and to obtain $Y$ from (154).

The Kirillov-Kostant symplectic form will give the symplectic part of the
action. The geometric action can be extended by adding the Noether charge for
a global symmetry as Hamiltonian [55]. The resulting action is by construction
invariant under the same symmetries as the symplectic term. The Noether charge
associated to a symmetry generated by a vector field $(X,n)$ in the centrally
extended Lie algebra $\hat{\mathfrak{g}}=\mathfrak{g}\oplus\mathbb{R}$ is
given by

$Q_{(X,n)}=-\langle(b,c),(X,n)\rangle\,.$ (155)

Hence as Hamiltonian we may simply add

$H_{(X,n)}=-\int_{\gamma}\;Q_{(X,n)},$ (156)

for $\gamma$ a path along the orbit. The total geometric action as a function
of group elements $g$ and for a given $(b_{0},c)$ is then [57]

$S_{\hat{G}}[g;b_{0},c]=\int_{\gamma}\alpha-
Q_{(X,n)}=\int_{\gamma}\langle\operatorname{Ad}_{g^{-1}}^{*}(b_{0},c),(Y,n_{Y})-(X,n)\rangle\,.$
(157)

We will now work out the different ingredients in this expression for the
centrally extended group of diffeomorphisms of the supercircle.

####  5.1.1 Geometric action for superconformal groups

When the Lie group is taken to be the centrally extended group of
diffeomorphisms of the supercircle $\widehat{\rm Diff}(S^{1|{\cal N}})$, the
geometric action constructed from (152) corresponds to the kinetic term of the
reduced AdS3 supergravity action (101) up to the ${\cal N}=2$ case, as we will
now demonstrate. Let us denote the general ${\cal N}$-extended superspace by
superspace coordinate $z=\\{\varphi,\theta^{i}\\}$ where $0<\varphi\leq 2\pi$
is a bosonic coordinate (the angle on the circle) and $\theta^{i}$ is a
collection of ${\cal N}$ Grassmann coordinates. The supercovariant derivative
is

$D^{i}=\partial_{\theta^{i}}+\theta^{i}\partial_{\varphi}\,,$ (158)

such that

$\\{D^{i},D^{j}\\}=2\delta^{ij}\partial_{\varphi}\,.$ (159)

Reparameterizations of the supercircle consists of coordinates transformations

$\varphi\to\tilde{\varphi}(\varphi,\theta^{i})\,,\qquad\theta^{j}\to\tilde{\theta}^{j}(\varphi,\theta^{i})\,,$
(160)

subject to the constraint that the supercovariant derivative transforms
covariantly, or

$D^{i}=(D^{i}\tilde{\theta}^{j})\tilde{D}^{j}\,.$ (161)

This condition implies the following useful identities

$\displaystyle D^{i}\tilde{\varphi}$
$\displaystyle=\tilde{\theta}D^{i}\tilde{\theta},$ (162)
$\displaystyle(D^{i}\tilde{\theta}^{j})(D^{k}\tilde{\theta}^{j})$
$\displaystyle=\delta^{ik}(\partial_{\varphi}\tilde{\varphi}+\tilde{\theta}^{j}\partial_{\varphi}\tilde{\theta}^{j}),$
(163) $\displaystyle\det\left(D\tilde{\theta}\right)$
$\displaystyle=\pm(\partial_{\varphi}\tilde{\varphi}+\tilde{\theta}^{j}\partial_{\varphi}\tilde{\theta}^{j})^{{\cal
N}/2}.$ (164)

Elements of the super-Virasoro algebra consists of the pair $(X(z),n)$ and
this is paired with elements of the dual space $(B(z),c)$ (we will denote the
superspace dual elements by capital $B$, keeping $b$ and $b_{0}$ reserved for
the bosonic orbit representative) as

$\langle(B,c),(X,n)\rangle=\langle
b,X\rangle+cn=\int\mathrm{d}z\;B(z)X(z)+cn\,,$ (165)

where
$\mathrm{d}z=\mathrm{d}\varphi\mathrm{d}\theta^{1}\ldots\mathrm{d}\theta^{\cal
N}$. The adjoint and coadjoint action of the centrally extended superconformal
group are

$\displaystyle\operatorname{Ad}_{\tilde{z}}(X(z),n)$
$\displaystyle=\left(\det(D\tilde{\theta})^{-\frac{2}{{\cal
N}}}X(\tilde{z}),n-\int\mathrm{d}z\,S^{({\cal
N})}(z;\tilde{z}^{-1})X(z)\right)\,,$ (166)
$\displaystyle\operatorname{Ad}_{\tilde{z}}^{*}(B(z),c)$
$\displaystyle=\left(\det(D\tilde{\theta})^{\frac{4-{\cal N}}{{\cal
N}}}B(\tilde{z})-cS^{({\cal N})}(z;\tilde{z}),c\right)\,.$ (167)

Here $S^{({\cal N})}(z;\tilde{z})$ are the super Schwarzian derivatives for
${\cal N}$ extended supersymmetry. They satisfy the cocycle condition

$S^{({\cal N})}(z,\tilde{\tilde{z}})=(\det(D\tilde{\theta}))^{\frac{4-{\cal
N}}{{\cal N}}}S^{({\cal N})}(\tilde{z},\tilde{\tilde{z}})+S^{({\cal
N})}(z,\tilde{z})$ (168)

and $S^{({\cal N})}(z;z)=0$ so that $S^{({\cal
N})}(z;\tilde{z})=-\operatorname{Ad}_{\tilde{z}}^{*}S^{({\cal
N})}(z;\tilde{z}^{-1})$. The condition (168) can be solved only for ${\cal
N}\leq 4$ [58] and the limiting ${\cal N}=4$ case is special as there are two
independent non-trivial (and non-local) solutions. The upper bound ${\cal
N}=4$ is natural as the stress tensor sits in a supermultiplet of conformal
dimension $(2-\frac{{\cal N}}{2})$, which becomes negative for ${\cal N}>4$.
In our case, however, the asymptotic symmetry algebras (30) correspond to the
superconformal Lie algebras of $\widehat{\rm Diff}(S^{1|{\cal N}})$ only for
${\cal N}=0,1,2$. For ${\cal N}>2$ there is a non-linear term in the
superconformal algebras (30), which makes the interpretation in terms of
coadjoint orbits difficult. Hence here we will discuss the case up to ${\cal
N}=2$.

The result for the (super)-Schwarzians is

$\displaystyle{\cal N}=0$ $\displaystyle S^{(0)}(\varphi;\tilde{\varphi})$
$\displaystyle=\frac{1}{24\pi}\left(\frac{\tilde{\varphi}^{\prime\prime\prime}}{\tilde{\varphi}^{\prime}}-\left(\frac{\tilde{\varphi}^{\prime\prime}}{\tilde{\varphi}^{\prime}}\right)^{2}\right)\,,$
(169) $\displaystyle{\cal N}=1$ $\displaystyle S^{(1)}(z;\tilde{z})$
$\displaystyle=\frac{1}{12\pi}\left(\frac{\partial_{\varphi}^{2}\tilde{\theta}}{D\tilde{\theta}}-2\frac{D\partial_{\varphi}\tilde{\theta}\partial_{\varphi}\tilde{\theta}}{(D\tilde{\theta})^{2}}\right)\,,$
(170) $\displaystyle{\cal N}=2$ $\displaystyle S^{(2)}(z;\tilde{z})$
$\displaystyle=\frac{1}{24\pi}\epsilon^{ij}\frac{2(D^{i}\partial_{\varphi}\tilde{\theta}^{k})(D^{j}\tilde{\theta}^{k})-2(D^{i}D^{j}\tilde{\theta}^{k})(\partial_{\varphi}\tilde{\theta}^{k})}{(D^{p}\tilde{\theta}^{q})(D^{p}\tilde{\theta}^{q})}\,.$
(171)

To proceed we need to find the Maurer-Cartan form. We can do so by solving for
the $\partial_{\tilde{\varphi}}B_{0}$ part of (154). For the centrally
extended group of diffeomorphisms of the supercircle, the relevant part of
(154) becomes

$\mathrm{d}\left(\det(D\tilde{\theta})^{\frac{4-{\cal N}}{{\cal
N}}}B_{0}(\tilde{z})-cS^{({\cal
N})}(z;\tilde{z})\right)=-\operatorname{ad}_{Y}^{*}\left(\det(D\tilde{\theta})^{\frac{4-{\cal
N}}{{\cal N}}}B_{0}(\tilde{z})-cS^{({\cal N})}(z;\tilde{z})\right).$ (172)

The $B_{0},\partial_{\tilde{\varphi}}B_{0},\tilde{D}^{i}B_{0}$ and $c$
dependent parts of this equation should vanish individually. Expanding this
equation using the following useful formulas

$\displaystyle\operatorname{ad}_{Y}X$
$\displaystyle=Y\partial_{\varphi}X-X\partial_{\varphi}Y+\frac{1}{2}D^{i}YD^{i}X,$
(173) $\displaystyle\mathrm{d}B_{0}(\tilde{\varphi},\tilde{\theta}^{i})$
$\displaystyle=\partial_{\tilde{\varphi}}B_{0}(\tilde{\varphi},\tilde{\theta}^{i})(\mathrm{d}\tilde{\varphi}+\tilde{\theta}^{j}\mathrm{d}\tilde{\theta}^{j})+\mathrm{d}\tilde{\theta}^{j}\tilde{D}^{j}B_{0}(\tilde{\varphi},\tilde{\theta}^{i}),$
(174)

and

$\partial_{\varphi}=\partial_{\varphi}\tilde{\theta}^{k}\tilde{D}^{k}+(\det
D\tilde{\theta})^{2/{\cal N}}\partial_{\tilde{\varphi}},$ (175)

we find an over-constrained set of equations for $Y$ which is solved by

$Y=-\frac{\mathrm{d}\tilde{\varphi}+\tilde{\theta}^{i}\mathrm{d}\tilde{\theta}^{i}}{(\det
D\tilde{\theta})^{2/{\cal N}}}.$ (176)

We now have all the ingredients to write down the geometric actions on the
coadjoint orbit for the ${\cal N}$-extended superconformal group. The
$c$-independent part of the symplectic term can be written down for generic
${\cal N}$. It is (writing $\mathrm{d}=\mathrm{d}t\,\partial_{t}$):

$-\int\mathrm{d}t\;\mathrm{d}z\;\det(D\tilde{\theta})^{\frac{2-{\cal N}}{{\cal
N}}}B_{0}(\tilde{\varphi},\tilde{\theta}^{i})(\dot{\tilde{\varphi}}+\tilde{\theta}^{j}\dot{\tilde{\theta}}^{j})$
(177)

The $c$-dependent part should be obtained by writing the second term in (152)
as a total exterior derivative for the relevant Schwarzian listed in
(169)-(171). We will study this on a case by case basis in the next sections.

Before doing so we comment on the Hamiltonian. In the case of interest the
action is invariant under a shift symmetry generated by vector fields
$-\partial_{\varphi}$. So we may add to the symplectic part a Hamiltonian
(156) with $(X,n)=(-1,0)$.888The symmetry generator $(X,n)=(0,n)$ generate
constant shifts of $n$ which are trivial as the vectors belong to the extended
little algebra [55]. This, together with (167) gives the Hamiltonians

$H^{({\cal
N})}(\tilde{z};B_{0},c)=-\int\mathrm{d}t\,\mathrm{d}z\left(\det(D\tilde{\theta})^{\frac{4-{\cal
N}}{{\cal N}}}B_{0}(\tilde{z})-cS^{({\cal N})}(z;\tilde{z})\right).$ (178)

Due to the cyclic property of the Schwarzian derivative (168), the orbit
representative term of the Hamiltonian (178) can be absorbed into the super-
Schwarzian by a second reparameterization of the supercircle
$\tilde{\tilde{z}}$ such that

$cS^{({\cal N})}(\tilde{z};\tilde{\tilde{z}})=-B_{0}(\tilde{z})\,.$ (179)

The Hamiltonian then becomes $H^{({\cal
N})}(\tilde{\tilde{z}};0,c)=\int\mathrm{d}t\,\mathrm{d}z\left(cS^{({\cal
N})}(z;\tilde{\tilde{z}})\right)$ and the orbit representative terms will
translate into periodicity conditions of the new variable $\tilde{\tilde{z}}$.
The same transformation will remove the orbit representative term (177) from
the symplectic term [55].

Next we will discuss explicitly the ${\cal N}=0,1,2$ cases to show that the
actions (3.4.2) and (101) are indeed the geometric actions on the coadjoint
orbit of the centrally extended group of diffeomorphisms of the supercircle.
The orbit representative terms are given by the holonomy of the Chern-Simons
connection by comparing the periodicities of the fields to those of
$\tilde{\tilde{z}}$ satisfying (179).

###  5.2 Bosonic case

The geometric action on the coadjoint orbit of the Virasoro group of
diffeomorphisms of the circle $\widehat{\rm Diff}(S^{1})$ was worked out by
Alekseev and Shatashvili [24] and reported to be relevant to three dimensional
AdS3 gravity in [55] and later expanded upon in [56]. In [7] it was shown that
for the bosonic theory, the action obtained from the Hamiltonian reduction
coincides with Alekseev-Shatashvili action where the holonomy plays the role
of orbit representative. In order to set the ground for the supersymmetric
case, we briefly review the discussion here.

From the last two subsections, we reproduce the Alekseev-Shatashvili action
from (157) in the bosonic case. This is

$S_{\widehat{\rm Diff}(S^{1})}[f(\varphi);b_{0},c]=\int
dtd\varphi\;\left[\frac{c}{24\pi}\left(\frac{3}{2}\frac{f^{\prime\prime}\partial_{-}f^{\prime}}{f^{\prime
2}}-\frac{\partial_{-}f^{\prime\prime}}{f^{\prime}}\right)+b_{0}(f)f^{\prime}\partial_{-}f\right],$
(180)

where we have taken $\tilde{\varphi}=f(\varphi)$ subject to the periodicity
condition $f(\varphi+2\pi)=f(\varphi)+2\pi$. From (100) we see that in the
bosonic case (i.e., no fermions, no $R$-symmetry and $k_{r}=0$) the reduction
from Chern-Simons theory on the outer boundary gives

$S=\frac{k}{8\pi}\int\mathrm{d}t\,\mathrm{d}\varphi\;\bigg{[}\Big{(}\partial_{-}\log(Y^{\prime}-k_{0}Y)\Big{)}\Big{(}\partial_{\varphi}\log(Y^{\prime}-k_{0}Y)\Big{)}-k_{0}^{2}\bigg{]}\,,$
(181)

up to total derivative terms. Moreover, in order to make a link with the
classification of (super-)Virasoro coadjoint orbits, in this section we focus
on the case where the holonomy is constant and time-independent, which is the
case for constant orbit representatives. However, the field $Y(\varphi)$ here
is periodic, i.e. $Y(\varphi+2\pi)=Y(\varphi)$ which is not the usual
$Y(\varphi+2\pi)=Y(\varphi)+2\pi$ as it is for elements of $\widehat{\rm
Diff}(S^{1})$. Therefore, in order to relate these two actions we consider a
field transformation

$\tilde{\tilde{\varphi}}\equiv
Y(f(\varphi))=e^{-k_{0}(f(t,\varphi)-\varphi)}.$ (182)

such that $f(\varphi+2\pi)=f(\varphi)+2\pi$ while $Y(\varphi)$ is periodic. In
the new variable $f(\varphi)$ the action takes the form of (180) with the
central charge and the orbit representative given as

$c=6k,\qquad\qquad\qquad b_{0}=\frac{k}{8\pi}k_{0}^{2}.$ (183)

In the case in hand, the holonomy is in the hyperbolic conjugacy class and
therefore we have $k_{0}^{2}=4{\cal L}_{0}$. One then finds that the orbit
representative $b_{0}$ is related to the zero mode of the bulk Chern-Simons
solution as

$b_{0}=\frac{c}{12\pi}{\cal L}_{0}=\frac{k}{2\pi}{\cal L}_{0}\,.$ (184)

We have thus illustrated that the zero mode of the bulk solutions correspond
directly to the constant representatives of the Virasoro coadjoint orbits. In
the case of the BTZ solutions, where ${\cal L}_{0}$ is positive, these
correspond to the generic Virasoro orbit with a $U(1)$ little group generated
by the action of $L_{0}$. The Noether charge for this $L_{0}$ action is
exactly the Hamiltonian (178) that was added to the geometric action.

The exceptional orbits of Virasoro have constant representative terms
$b_{0}=-\frac{cn^{2}}{48\pi}$ for integer $n$. At these points, the little
group on the orbit is enlarged to $SL^{(n)}(2,\mathbb{R})$. From the bulk
perspective these orbits correspond to (the left-moving sector of) Bañados
geometries with ${\cal L}_{0}=-\frac{1}{4}n^{2}$ for integer $n$. For $n=1$
this is the global AdS3 ground state, while for larger $n$ the geometries have
an angular access of a multiple of $2\pi$. For those solutions the Hamiltonian
of (180) becomes unbounded from below [59] and hence this is a good reason to
discard them.

The bulk solutions with conical defects correspond to orbits with negative
$b_{0}\neq-\frac{cn^{2}}{48\pi}$. At this point $k_{0}=\sqrt{4{\cal L}_{0}}$
becomes imaginary and hence the holonomy $h=\exp(4\pi\sqrt{{\cal
L}_{0}}L_{0})$ becomes complex and is not an element of $SL(2,\mathbb{R})$
anymore. This is the Chern-Simons perspective for discarding solutions with
conical singularities. For the exceptional orbits where
$b_{0}=-\frac{cn^{2}}{48\pi}$ the holonomy is still an element of
$SL(2,\mathbb{R})$ since exactly at those values we have $h=\exp(2\pi
niL_{0})$. The complete discussion of the geometric action with holonomies in
different conjugacy classes of $SL(2,\mathbb{R})$ can be found in [7].

###  5.3 ${\cal N}=1$ supersymmetry

Let us now consider the centrally extended group of diffeomorphisms of the
${\cal N}=1$ supercircle $\widehat{\rm Diff}(S^{1|1})$. Writing the geometric
action (157) using the ${\cal N}=1$ super-Schwarzian (170) we find the
superspace action worked out in [57], up to total derivative terms

$\displaystyle S_{\widehat{\rm Diff}(S^{1|1})}(\tilde{z};B_{0},c)$
$\displaystyle=\int\mathrm{d}t\,\mathrm{d}z\bigg{[}\frac{c}{12\pi}\left(\frac{\partial_{\varphi}D\tilde{\theta}\,\partial_{-}\tilde{\theta}}{(D\tilde{\theta})^{2}}\right)+B_{0}(\tilde{z})D\tilde{\theta}(\partial_{-}\tilde{\varphi}+\tilde{\theta}\partial_{-}\tilde{\theta})\bigg{]}$
(185)

In order to compare this to the action obtained from the Chern-Simons
reduction (100), we need a suitable parameterization of $\tilde{\theta}$ to
perform the superspace integral and write the ${\cal N}=1$ superspace action
in components.

The ${\cal N}=1$ constraint $D\tilde{\varphi}=\tilde{\theta}D\tilde{\theta}$
is solved by parameterizing the diffeomorphism of the supercircle as

$\displaystyle\tilde{\varphi}$ $\displaystyle=f(\varphi+\theta\eta(\varphi))$
(186) $\displaystyle\tilde{\theta}$
$\displaystyle=\sqrt{f^{\prime}(\varphi)}\left(\eta(\varphi)+\theta\left(1+\frac{1}{2}\eta\eta^{\prime}\right)\right)$
(187)

Here $f(\varphi+2\pi)=f(\varphi)+2\pi$. Taking the orbit representative term
to be purely bosonic, i.e. $B_{0}(\tilde{z})=b_{0}(f)\tilde{\theta}$ and using
the above parameterization, the action (185) becomes (up to total derivatives)

$\displaystyle S_{\widehat{\rm Diff}(S^{1|1})}(f,\eta;b_{0},c)=\int
dtd\varphi$
$\displaystyle\;\Bigg{[}-\frac{c}{24\pi}\bigg{(}\frac{\partial_{-}{f}^{\prime\prime}}{f^{\prime}}-\frac{3}{2}\frac{\partial_{-}f^{\prime}f^{\prime\prime}}{f^{\prime}{}^{2}}-S^{(0)}(\varphi,f)\,\eta\partial_{-}{\eta}+2\eta^{\prime}\partial_{-}{\eta}^{\prime}$
$\displaystyle-\frac{1}{2}\eta\eta^{\prime}\eta^{\prime\prime}\partial_{-}\eta\bigg{)}+b_{0}(f)(f^{\prime}\partial_{-}f-f^{\prime}{}^{2}\eta\partial_{-}\eta)\Bigg{]}\,.$
(188)

In order to compare the ${\cal N}=1$ Chern-Simons action (100) with the above
action, we can write (100), up to total derivatives, as

$\displaystyle S^{{\cal N}=1}=\frac{k}{8\pi}\int$
$\displaystyle\mathrm{d}t\,\mathrm{d}\varphi\;\bigg{[}\Big{(}\partial_{-}\log(Y^{\prime}-k_{0}Y-\psi^{-}\psi^{-}{}^{\prime})\Big{)}\Big{(}\partial_{\varphi}\log(Y^{\prime}-k_{0}Y-\psi^{-}\psi^{-}{}^{\prime})\Big{)}-k_{0}^{2}$
$\displaystyle+4\frac{\Big{(}\psi^{-}{}^{\prime}-\frac{k_{0}}{2}\psi^{-}\Big{)}\Big{(}\partial_{-}(\psi^{-}{}^{\prime}-\frac{k_{0}}{2}\psi^{-})-\partial_{-}\log(Y^{\prime}-k_{0}Y-\psi^{-}\psi^{-}{}^{\prime})^{\frac{1}{2}}\Big{)}}{Y^{\prime}-k_{0}Y-\psi^{-}\psi^{-}{}^{\prime}}\bigg{]}\,,$
(189)

where we have considered a single pair of fermionic generators and no
$R$-symmetry. We have also set $\eta^{\alpha\beta}=-2,\mathfrak{m}=1$ and
since there is no $R$-symmetry we have $\lambda^{a}=0$.

Like in the bosonic case, both fields $Y(\varphi)$ and $\psi^{-}(\varphi)$ are
periodic. In order to relate these fields to those parameterizing the super
circle, i.e. fields with the periodicity condition of elements of the
$\widehat{\rm Diff}(S^{1|1})$, we consider the following field transformations

$\displaystyle Y(f(\varphi))$
$\displaystyle=e^{-k_{0}(f(t,\varphi)-\varphi)},$ (190)
$\displaystyle\psi^{-}(f(\varphi),\eta(\varphi))$
$\displaystyle=\sqrt{Y^{\prime}-k_{0}Y}\eta(\varphi)=\sqrt{-k_{0}f^{\prime}}e^{-\frac{k_{0}}{2}(f(t,\varphi)-\varphi)}\eta(\varphi),$
(191)

which can be obtained from

$\displaystyle\tilde{\tilde{\varphi}}=e^{-k_{0}(\tilde{\varphi}-\varphi)},\qquad\qquad\tilde{\tilde{\theta}}=\sqrt{-k_{0}e^{-k_{0}(\tilde{\varphi}-\varphi)}}\tilde{\theta},$
(192)

with $\tilde{\tilde{\varphi}}=Y(\varphi+\theta\chi)$.

For the new fields $f(\varphi)$ and $\eta(\varphi)$ we have

$f(\varphi+2\pi)=f(\varphi)+2\pi,\qquad\qquad\eta(\varphi+2\pi)=\eta(\varphi),$
(193)

while $Y(\varphi)$ and $\psi^{-}(\varphi)$ are periodic.

In the new variables, the action takes the form (5.3) with the central charge
and the orbit representative given as

$c=6k,\qquad\qquad\qquad b_{0}=\frac{k}{8\pi}k_{0}^{2}.$ (194)

And so we find the bulk holonomy is still related to the orbit representative
$b_{0}$ as in (184).

This proves our earlier claim that ${\cal N}=1$ boundary action on the outer
boundary (74) obtained from the Hamiltonian reduction in the presence of non-
trivial holonomy parametrized with $k_{0}$ is the geometric action on the
coadjoint orbit of group of reparametrization of $S^{1|1}$ where the orbit
representative $b_{0}$ is identified with the holonomy as in (194).

###  5.4 ${\cal N}=2$ supersymmetry

Here we will discuss the case of ${\cal N}=2$ supersymmetry. The ${\cal N}=2$
supercircle can be studied in a complex basis where

$\theta=\frac{1}{\sqrt{2}}(\theta^{1}+i\theta^{2})\,,\qquad\bar{\theta}=\frac{1}{\sqrt{2}}(\theta^{1}-i\theta^{2})\,,$
(195)

Defining the supercovariant derivatives as

$D=\partial_{\theta}+\bar{\theta}\partial_{\varphi}\,,\qquad\bar{D}=\partial_{\bar{\theta}}+\theta\partial_{\varphi}\,,$
(196)

we have $D^{2}=0=\bar{D}^{2}$ and $\\{D,\bar{D}\\}=2\partial_{\varphi}$.
Equation (161) now implies that

$D=D\tilde{\theta}\tilde{D}\,,\qquad\bar{D}=\bar{D}\tilde{\bar{\theta}}\tilde{\bar{D}}\,,\quad\text{and}\quad
D\tilde{\bar{\theta}}=0=\bar{D}\tilde{\theta}$ (197)

and $\det{D\tilde{\theta}}=D\tilde{\theta}\bar{D}\tilde{\bar{\theta}}$.

The ${\cal N}=2$ Schwarzian derivative in this notation becomes

$S^{(2)}(z;\tilde{z})=\frac{1}{24\pi}\left(\frac{\partial_{\varphi}\bar{D}\tilde{\bar{\theta}}}{\bar{D}\tilde{\bar{\theta}}}-\frac{\partial_{\varphi}D\tilde{\theta}}{D\tilde{\theta}}-2\frac{\partial_{\varphi}\tilde{\theta}\partial_{\varphi}\tilde{\bar{\theta}}}{D\tilde{\theta}\bar{D}\tilde{\bar{\theta}}}\right)$
(198)

and the Maurer-Cartan form is
$Y=-\frac{\mathrm{d}\tilde{\varphi}+\tilde{\theta}\mathrm{d}\tilde{\bar{\theta}}+\tilde{\bar{\theta}}\mathrm{d}\tilde{\theta}}{D\tilde{\theta}\bar{D}\tilde{\bar{\theta}}}\equiv-\frac{\delta\tilde{l}}{D\tilde{\theta}\bar{D}\tilde{\bar{\theta}}}$.
Useful relations to write (152) as a total exterior derivative are
$\mathrm{d}(\delta\tilde{l})=2\mathrm{d}\tilde{\theta}\mathrm{d}\tilde{\bar{\theta}}$,
$D(\delta\tilde{l})=2D\tilde{\theta}\mathrm{d}\tilde{\bar{\theta}}$ and
$\bar{D}(\delta\tilde{l})=2\bar{D}\tilde{\bar{\theta}}\mathrm{d}\tilde{\theta}$.
The answer for the geometric action of the ${\cal N}=2$ theory is

$\displaystyle S_{\widehat{\rm
Diff}(S^{1|2})}(\tilde{z};B_{0},c)=\int\mathrm{d}t\mathrm{d}z$
$\displaystyle\bigg{(}\frac{c}{24\pi}\left(\frac{\partial_{\varphi}\tilde{\theta}\partial_{-}\tilde{\bar{\theta}}-\partial_{\varphi}\tilde{\bar{\theta}}\partial_{-}\tilde{\theta}}{D\tilde{\theta}\bar{D}\tilde{\bar{\theta}}}\right)$
$\displaystyle+B_{0}(\tilde{z})(\partial_{-}\tilde{\varphi}+\tilde{\theta}\partial_{-}\tilde{\bar{\theta}}+\tilde{\bar{\theta}}\partial_{-}\tilde{\theta})\bigg{)}\,.$
(199)

In order to relate this to the component action coming from the reduction of
the Chern-Simons theory we need a suitable parameterization of
$\tilde{\varphi},\tilde{\theta}$ and $\tilde{\bar{\theta}}$. We can obtain
this by writing the most general diffeomorphism of the ${\cal N}=2$ super-
circle compatible with the constraints (161), which in the complex basis read:

$\displaystyle D\tilde{\bar{\theta}}=0\,,\qquad
D\tilde{\varphi}=\tilde{\bar{\theta}}D\tilde{\theta}\,,$ (200)
$\displaystyle\bar{D}\tilde{\theta}=0\,,\qquad\bar{D}\tilde{\varphi}=\tilde{\theta}\bar{D}\tilde{\bar{\theta}}\,.$
(201)

A general super-reparameterization satisfying these constraints can be written
in terms of two bosonic fields: $f(t,\varphi)$, which parameterizes the
$\widehat{\rm Diff}(S^{1})$ element and satisfies
$f(\varphi+2\pi)=f(\varphi)+2\pi$ and $\sigma(t,\varphi)$ which is a $U(1)$
$R$-symmetry field. There are also two Grassmann valued fields
$\psi_{\alpha}(t,\varphi)$ with $\alpha={1,2}$,

$\displaystyle\tilde{\varphi}$
$\displaystyle=f(t,\varphi)-\theta\psi_{1}(t,\varphi)\bar{g}(t,\varphi)+\bar{\theta}\psi_{2}(t,\varphi)g(t,\varphi)+h(t,\varphi)\theta\bar{\theta}\,,$
(202a) $\displaystyle\tilde{\theta}$
$\displaystyle=\psi_{2}(t,\varphi)+\theta\left(\bar{g}(t,\varphi)+\bar{\theta}\psi_{2}^{\prime}(t,\varphi)\right)\,,$
(202b) $\displaystyle\tilde{\bar{\theta}}$
$\displaystyle=-\psi_{1}(t,\varphi)+\bar{\theta}\left(g(t,\varphi)-\theta\psi_{1}^{\prime}(t,\varphi)\right)\,,$
(202c)

where

$\displaystyle g(t,\varphi)$
$\displaystyle=\sqrt{f^{\prime}(t,\varphi)}e^{i\sigma(t,\varphi)/2}\left(1-\frac{1}{4}\frac{\psi\cdot\psi^{\prime}}{f^{\prime}}-\frac{1}{32}\left(\frac{\psi\cdot\psi^{\prime}}{f^{\prime}}\right)^{2}\right)$
(203) $\displaystyle\bar{g}(t,\varphi)$
$\displaystyle=\sqrt{f^{\prime}(t,\varphi)}e^{-i\sigma(t,\varphi)/2}\left(1-\frac{1}{4}\frac{\psi\cdot\psi^{\prime}}{f^{\prime}}-\frac{1}{32}\left(\frac{\psi\cdot\psi^{\prime}}{f^{\prime}}\right)^{2}\right)$
(204) $\displaystyle h(t,\varphi)$
$\displaystyle=-\frac{1}{2}\left(\psi\lambda\psi\right)^{\prime}$ (205)

where we remind the reader that $\cdot$ is contraction with
$\eta^{\alpha\beta}$ and
$\psi\lambda\psi=\psi_{\alpha}\lambda^{\alpha\beta}\psi_{\beta}$. Our
conventions for $\eta^{\alpha\beta}$ and $\lambda^{\alpha\beta}$ are listed in
the appendix A.

In order to show that the action obtained from the Hamiltonian reduction is
indeed the geometric action, we proceed with a bit different approach.

When the geometric action for ${\cal N}=2$ (5.4) is expanded using the above
parameterization, the $c$-dependent term becomes equal to the action obtained
from Chern-Simons theory (100) for ${\cal N}=2$ on the outer boundary, when
the holonomy contributions set to zero with
$\tilde{Y}(t,\varphi),\tilde{\psi}^{-}_{\alpha}(t,\varphi)$ and
$\tilde{C}^{a}(t,\varphi)$ replaced by $f(t,\varphi),\psi_{\alpha}$ and
$\sigma(t,\varphi)$ respectively. That confirms the equivalence of the
theories. However, the fields $\tilde{Y}(t,\varphi)$,
$\tilde{\psi}^{-}_{\alpha}(t,\varphi)$ and $\tilde{C}^{a}(t,\varphi)$ are no
longer periodic. The amount of non-periodicity is encoded in the phase that
these fields pick up under the action of the holonomy on the group elements.
All there is to do now is to find the transformation which reinstates the
$B_{0}(\tilde{z})$ term and read of the relation between the zero modes and
the orbit representative $B_{0}$. To this end we should solve (179) for ${\cal
N}=2$. The orbit representative has only non-zero bosonic values, which we
will take to be constant.

$B_{0}(\tilde{z})=b_{\sigma}+\tilde{\theta}\tilde{\bar{\theta}}b_{0}\,.$ (206)

A super-reparameterization which solves (179) is

$\tilde{\tilde{\varphi}}=e^{-\mu\tilde{\varphi}}\,,\qquad\tilde{\tilde{\theta}}=e^{\frac{12\pi
b_{\sigma}}{c}\tilde{\varphi}}\tilde{\theta}\sqrt{-\mu
e^{-\mu\tilde{\varphi}}}\,,\qquad\tilde{\tilde{\bar{\theta}}}=e^{-\frac{12\pi
b_{\sigma}}{c}\tilde{\varphi}}\tilde{\bar{\theta}}\sqrt{-\mu
e^{-\mu\tilde{\varphi}}}\,,$ (207)

where now

$\mu=\sqrt{48\pi\left(\frac{b_{0}}{c}+12\pi\frac{b_{\sigma}^{2}}{c^{2}}\right)}$
(208)

This transformation relates the different components of $\tilde{\tilde{z}}$ to
the components of $\tilde{z}$ as given in (202). If one parameterizes
$\tilde{\tilde{z}}$ in the same way, but with fields
$\tilde{Y}(t,\varphi),\tilde{C}(t,\varphi)$ and $\tilde{\psi}^{-}(t,\varphi)$
instead of $f(t,\varphi),\sigma(t,\varphi)$ and $\psi_{\alpha}$, then the
field redefinitions which reinstate the orbit representative term, as deduced
from (207) are

$\displaystyle\tilde{Y}(t,\varphi)$ $\displaystyle=e^{-\mu f(t,\varphi)}\,,$
(209a) $\displaystyle\tilde{C}(t,\varphi)$
$\displaystyle=\sigma(t,\varphi)+\frac{24\pi
ib_{\sigma}}{c}f(t,\varphi)+\mu\psi_{1}(t,\varphi)\psi_{2}(t,\varphi)\,,$
(209b) $\displaystyle\tilde{\psi}^{-}_{1}(t,\varphi)$
$\displaystyle=e^{-\frac{12\pi b_{\sigma}}{c}f(t,\varphi)}\sqrt{-\mu e^{-\mu
f(t,\varphi)}}\psi_{1}(t,\varphi)$ (209c)
$\displaystyle\tilde{\psi}^{-}_{2}(t,\varphi)$ $\displaystyle=e^{\frac{12\pi
b_{\sigma}}{c}f(t,\varphi)}\sqrt{-\mu e^{-\mu
f(t,\varphi)}}\psi_{2}(t,\varphi)$ (209d)

The periodicities of the fields $\tilde{Y}(t,\varphi),\tilde{C}(t,\varphi)$
and $\tilde{\psi}^{-}(t,\varphi)$ as dictated by the holonomy of the Chern-
Simons connection are for ${\cal N}=2$

$\displaystyle\tilde{Y}(\varphi+2\pi)$ $\displaystyle=e^{-2\pi\sqrt{4{\cal
L}_{0}}}\tilde{Y}(\varphi)\,,$ (210a) $\displaystyle\tilde{C}(\varphi+2\pi)$
$\displaystyle=\tilde{C}(\varphi)-2\pi i{\cal B}_{0}$ (210b)
$\displaystyle\tilde{\psi}^{-}_{\alpha}(\varphi+2\pi)$
$\displaystyle=e^{-\pi\sqrt{4{\cal
L}_{0}}}\tilde{\psi}^{-}_{\beta}(\varphi)e^{-2\pi{\cal
B}_{0}\lambda^{\beta}{}_{\alpha}}\,,$ (210c)

This is compatible with (209) whenever

$\hat{{\cal L}_{0}}={\cal L}_{0}-\frac{1}{4}{\cal
B}_{0}^{2}=\frac{12\pi}{c}b_{0}\,,\qquad{\cal
B}_{0}=-\frac{24\pi}{c}b_{\sigma}$ (211)

So we see that the orbit representative $b_{0}$ corresponds to the zero-mode
of the Sugawara-shifted stress tensor $\hat{{\cal L}}$ (27).

One can now go back to the original periodic fields
$Y(t,\varphi),C(t,\varphi)$ and $\psi^{-}(t,\varphi)$ with the following field
redefinitions

$\displaystyle Y(\varphi)$
$\displaystyle=e^{\mu\varphi}\tilde{Y}(t,\varphi)\,,$ (212a) $\displaystyle
C(\varphi)$ $\displaystyle=\tilde{C}(t,\varphi)+i{\cal B}_{0}\varphi,$ (212b)
$\displaystyle\psi^{-}_{\alpha}(\varphi)$
$\displaystyle=\tilde{\psi}^{-}_{\beta}(t,\varphi)e^{\frac{\mu}{2}\varphi}e^{{\cal
B}_{0}\lambda^{\beta}{}_{\alpha}\varphi}.$ (212c)

Here one sets

$\displaystyle\mu=\sqrt{k_{0}^{2}+k_{r}^{2}},$ (213)

where

$b_{0}=\frac{k}{8\pi}k_{0}^{2},\qquad\qquad
b_{\sigma}^{2}=\frac{k^{2}}{16\pi^{2}}k_{r}^{2}.$ (214)

We have now shown explicitly that the Hamiltonian reduction of Chern-Simons
theory for the $Osp(2|2)$ group under the highest-weight boundary conditions
(18) gives a two dimensional field theory equivalent to the geometric action
of the centrally extended group of diffeomorphisms of the ${\cal N}=2$
supercircle on its coadjoint orbit. The orbit representatives are related to
the zero modes of the Chern-Simons charges by (211).

##  6 Conclusions

The main result of this paper is to address in a precise and detailed way the
boundary actions for extended supergravity and higher spin $W_{3}$ gravity in
three dimensions in the presence of non-trivial bulk holonomies. We provide
the complete analysis where the topology of the manifold is $\mathbb{R}\times
S^{1}\times[0,1]$. The spatial surfaces with the annulus topology
$S^{1}\times[0,1]$ are then describing an AdS3 space with two boundaries. Even
though we discuss the details for this case, the extension to the topologies
with more boundaries is straightforward.

Similar to three dimensional Chern-Simons pure gravity, the boundary actions
both for extended supergravity and higher spin $W_{3}$ gravity are written in
terms of two free chiral theories living on each boundary, coupled through the