TUW–19–02

♡♡institutetext: Institute for Theoretical Physics, TU Wien, Wiedner
Hauptstrasse 8–10/136, A-1040 Vienna, Austria♢♢institutetext: School of
physics, Institute for research in fundamental sciences (IPM), P.O.Box
19395-5531, Tehran, Iran♠♠institutetext: The Abdus Salam ICTP, Strada Costiera
11, 34151, Trieste Italy♣♣institutetext: Haute-Ecole Robert Schuman, Rue
Fontaine aux Mûres, 13b, B-6800 Libramont, Belgium

# Interpolating Between Asymptotic and Near Horizon Symmetries

Daniel Grumiller ♢♠ M.M. Sheikh-Jabbari ♣ Cedric Troessaert ♡ and Raphaela
Wutte<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We develop basic tools and matching conditions to interpolate between
asymptotic and near horizon symmetries. We focus on black holes in three
dimensions. In particular, we match Brown–Henneaux boundary conditions at
infinity, which yields two Virasoro algebras, to Heisenberg boundary
conditions at the horizon yielding two $\hat{u}(1)$ current algebras. Our
construction allows to equip BTZ black holes with soft hair excitations at the
horizon invisible to the asymptotic observer.

††preprint: IPM/P-2019/040

##  1 Introduction

It has been advocated for a long time that generic (non-extremal) black hole
horizons are special loci in spacetime and that it could make sense to impose
boundary conditions at these loci Strominger:1997eq ; Carlip:1998wz ;
Carlip:1999cy . In the past couple of years this idea has been implemented in
various ways, leading to different near horizon symmetries Donnay:2015abr ;
Afshar:2015wjm ; Afshar:2016wfy ; Carlip:2017xne ; Haco:2018ske ;
Chandrasekhar:1985kt ; Grumiller:2019fmp ; Aggarwal:2019iay ; Carlip:2019dbu ;
toappear:2019 . In nearly all the cases the near horizon symmetries differ
from the usual asymptotic symmetries. It is thus fair to ask how the former
match with the latter and if we can interpolate between them. This question
provides the main motivation for the present work. To address possible
interpolations between asymptotic and near horizon symmetries we focus on
three-dimensional black holes, since this is the cleanest situation where this
issue arises.

From an asymptotic viewpoint the most natural set of asymptotic symmetries,
the two-dimensional conformal algebra, was derived by Brown and Henneaux
Brown:1986nw . From a near horizon viewpoint a natural set of near horizon
symmetries consists of two copies of $\hat{u}(1)$ current algebras (or,
equivalently, of Heisenberg algebras) Afshar:2016wfy . If one naively
extrapolates either of these boundary conditions, respectively, to the other
boundary one does not recover the appropriate set of symmetries. For instance,
extrapolating the Heisenberg boundary conditions all the way from the horizon
to infinity establishes asymptotic boundary conditions that differ from the
ones of Brown–Henneaux Afshar:2016uax . Similarly, naively moving the cutoff
surface where Brown–Henneaux boundary conditions are imposed all the way from
infinity to the horizon maintains the conformal symmetries and does not lead
to Heisenberg boundary conditions at the horizon.111In the two cases discussed
here (Heisenberg and Brown–Henneaux), the respective symmetries are symplectic
and not just asymptotic Compere:2015knw . This means that one can construct a
family of locally AdS3 solutions which form a phase space, as has been done
respectively in Afshar:2015wjm ; Afshar:2016wfy  and Banados:1998gg . Given
this phase space one may then define the symmetries away from where the
initial boundary conditions have been imposed.

In this work we clarify how a smooth matching works that connects arbitrary
(consistent) sets of boundary conditions at two different boundaries. Phrased
differently, our paper provides a manual how to grow soft hair (in the sense
of Hawking, Perry and Strominger Hawking:2016msc ) on black holes that is
invisible to asymptotic observers. While for concrete applications we imagine
one of these boundaries to be the asymptotic one and the other the (stretched)
horizon, our results are general and apply to any sets of two disconnected
boundary components.

A related gravitational motivation is something one could call “causal patch
holography”, where the boundary considered is a causal patch in some Penrose
diagram (e.g. the diamond-shaped region between two horizons or the region
between the event horizon and the asymptotic boundary). Introducing a finite
cutoff at both boundaries renders the cutoff surfaces timelike in most
applications, so that we are in a situation where we have two disconnected
boundary components, e.g. one at a radial distance close to the black hole
horizon and the other one close to the asymptotic boundary. Since we are going
to employ the Chern–Simons formulation of three-dimensional gravity
Achucarro:1987vz ; Witten:1988hc  our results also have potential applications
beyond gravity.

This paper is organized as follows. In section 2 we review salient features of
Chern–Simons theories. In section 3 we discuss Chern–Simons theories with two
boundaries and address general aspects of the matching conditions. In section
4 we analyze the prime example, connecting asymptotic Brown–Henneaux with near
horizon Heisenberg boundary conditions. In section 5 we treat another example,
connecting asymptotic Compère–Song–Strominger with near horizon Heisenberg
boundary conditions. In section 6 we conclude with a survey of further
applications and generalizations.

##  2 Chern–Simons

We recall now salient features of Chern–Simons theories, without yet
emphasizing their specific gravitational rôle. Three dimensional Chern–Simons
gauge theories with the Lie-algebra valued gauge connection $A$ are governed
by the action

$I_{\textrm{\tiny
CS}}[A]=\frac{k}{4\pi}\int\braket{A\wedge\operatorname{d}\\!A+\frac{2}{3}A\wedge
A\wedge A}\,.$ (1)

The only coupling constant is the dimensionless Chern–Simons level $k$, and
the brackets $\braket{\,,\,}$ denote an invariant, non-degenerate bilinear
form on the gauge algebra. In a derivative expansion the three-dimensional,
parity-odd, Chern–Simons term (1) is the lowest derivative term leading to
gauge-covariant equations of motion

$F=\operatorname{d}\\!A+A\wedge A=0\,.$ (2)

Chern–Simons theories arise universally in three spacetime dimensions.
Prominent examples are quantum Hall systems Zhang:1988wy ; Susskind:2001fb ;
Polychronakos:2001mi  and three-dimensional gravity Deser:1982vy ;
Deser:1982wh ; Achucarro:1986vz ; Witten:1988hc ; Carlip:1995zj ;
Carlip:1998uc . Locally the equations of motion (2) imply gauge flatness of
the connection,

$A=g^{-1}\operatorname{d}\\!g$ (3)

where $g$ is some group element of the underlying gauge group. This means
there are no local physical degrees of freedom in the Chern–Simons theory; it
is a topological quantum field theory of Schwartz type Birmingham:1991ty .

Despite of local triviality, systems described by Chern–Simons theories on
manifolds with boundaries can have physical degrees of freedom, often referred
to as “edge-states” in quantum Hall contexts (see e.g. Balachandran:1991dw )
and “boundary gravitons” in gravitational contexts (see Brown:1986nw  for the
seminal construction). Boundary conditions on the gauge connection $A$ are a
crucial physical input in the theory. Different choices are possible and can
lead to different physical phase spaces and symmetry algebras acting on them,
see Grumiller:2016pqb  for a summary of possibilities in three-dimensional
Einstein gravity with negative cosmological constant.

In all the constructions and applications mentioned above there is a single
(actual or asymptotic) simply connected boundary (“on which the edge
excitations live”). In the next sections we consider Chern–Simons theories on
manifolds with two boundaries.

##  3 Chern–Simons with two boundaries

The goal of this section is to determine whether it is possible to
consistently impose two different sets of falloff conditions on the
Chern–Simons connection at the two boundaries.

For concreteness we focus on the example of three-dimensional Euclidean
Einstein gravity with negative cosmological constant in the Chern–Simons
formulation. We consider a manifold with two boundaries with the topology of a
torus, $S^{1}\times S^{1}$. We use the coordinate system $\tau,r,\varphi$,
where $r$ is the radial coordinate, $\varphi\sim\varphi+2\pi$ is the angular
coordinate and $\tau\sim\tau+\beta$ corresponds to the Euclidean time. One
boundary is located at $r=0$, while the other one is located at $r\to\infty$.
The cycles of the torus are identified through
$(\tau,\varphi)\sim(\tau,\varphi+2\pi)\sim(\tau+\beta,\varphi)$, where $\beta$
is the inverse temperature.222We could have considered the angular velocity
$\Omega$ in the identification
$(\tau,\,\varphi)\sim(\tau+\beta,\,\varphi+\beta\Omega)$. However, the
information about rotation can also be packaged into the values of the
chemical potentials. So, we always have the same torus regardless of the value
of angular velocity. See e.g. Compere:2013nba ; Henneaux:2013dra . The
$\tau$-cycle is contractible, while the $\varphi$-cycle is non-contractible.

In section 3.1 we quickly summarize the main aspects of Euclidean AdS3
Einstein gravity as Chern–Simons theory. In section 3.2 we show how to
interpolate the gauge connection between two boundaries. In section 3.3 we
derive matching conditions that relate some of the charges at the two
boundaries.

###  3.1 AdS3 Einstein gravity as Chern–Simons theory

In the Chern–Simons formulation of AdS3 Einstein gravity Achucarro:1987vz ;
Witten:1988hc  the Chern–Simons level $k$ in the action (1) is (one quarter
of) the ratio between AdS-radius $\ell$ and Newton’s constant $G_{N}$.

$k=\frac{\ell}{4G_{N}}$ (4)

In the remainder of this work we set $\ell=1$. The gauge algebra for AdS3
Einstein gravity is $sl(2,{\mathbb{R}})\oplus sl(2,{\mathbb{R}})$. This split
can also be performed on the level of the action,

$I_{\textrm{\tiny CS}}[A^{+},A^{-}]=I_{\textrm{\tiny
CS}}[A^{+}]-I_{\textrm{\tiny CS}}[A^{-}]$ (5)

where $A^{+}$ and $A^{-}$ are elements of $sl(2,{\mathbb{R}})$ each. Since
$A^{+}$ and $A^{-}$ decouple, we restrict our discussion mostly to one
$sl(2,{\mathbb{R}})$ sector in the following and to reduce clutter drop the
superscript plus. Hereafter, $A$ without superscript refers to $A^{+}$. We use
the standard $sl(2,{\mathbb{R}})$ generators ${\tt L}_{1},{\tt L}_{0}$ and
${\tt L}_{-1}$ with commutation relations $[{\tt L}_{n},\,{\tt
L}_{m}]=(n-m)\,{\tt L}_{n+m}$. The non-degenerate, invariant bilinear form on
the algebra is given by $\braket{L_{0},L_{0}}=1/2$,
$\braket{L_{1},L_{-1}}=\braket{L_{-1},L_{1}}=-1$ and $\braket{L_{n},L_{m}}=0$
otherwise.

###  3.2 Gauge connection in presence of two boundaries

To find a flat connection interpolating between a certain set of boundary
conditions at $r=0$ and a (possibly different) set of boundary conditions at
$r\to\infty$ we make the following ansatz. We take a solution to the field
equations (2) subject to _one_ set of boundary conditions at $r\to\infty$,
$A_{\infty}$, and a solution subject to _a different_ set of boundary
conditions at the boundary $r=0$, $A_{0}$. Next we rewrite both of them in the
form (3). In general, this is impossible with single-valued group elements
$g$. However, we allow for multivaluedness of the group element $g$ and thus
continue to employ (3) even globally.

Using the Gauss decomposition, we write the group element $g_{\infty}$ for,
say, the solutions $A_{\infty}$ as333In addition to the Gauss decomposition we
have pulled out an extra factor $e^{r{\tt L}_{0}}$, which is common practice
in order to make the functions $\eta_{\infty}$, $\psi_{\infty}$ and
$\lambda_{\infty}$ $r$-independent. While in our matching the corresponding
functions will be $r$-dependent, we still pull out such a factor in order to
asymptote to well-known expressions.

$g_{\infty}=e^{\eta_{\infty}{\tt L}_{1}}\,e^{\psi_{\infty}{\tt
L}_{0}}\,e^{\lambda_{\infty}{\tt L}_{-1}}\,e^{r{\tt L}_{0}}\,.$ (6)

The functions $\eta_{\infty}=\eta_{\infty}(\tau,\varphi)$,
$\psi_{\infty}=\psi_{\infty}(\tau,\varphi)$ and
$\lambda_{\infty}=\lambda_{\infty}(\tau,\varphi)$ are allowed to be
multivalued. The connection (3) associated to the group element (6) can be
rewritten as

$A_{\infty}=b^{-1}\,(\operatorname{d}\\!+a_{\infty})\,b\,,\qquad\qquad
b=e^{r{\tt L}_{0}}$ (7)

with

$a_{\infty}=e^{\psi_{\infty}}\operatorname{d}\\!\eta_{\infty}\,{\tt
L}_{1}+(\operatorname{d}\\!\psi_{\infty}+2\lambda_{\infty}e^{\psi_{\infty}}\operatorname{d}\\!\eta_{\infty})\,{\tt
L}_{0}+(\operatorname{d}\\!\lambda_{\infty}+\lambda_{\infty}\operatorname{d}\\!\psi_{\infty}+\lambda^{2}_{\infty}e^{\psi_{\infty}}\operatorname{d}\\!\eta_{\infty})\,{\tt
L}_{-1}\,.$ (8)

The connection $A_{0}$ is decomposed in the same way, replacing everywhere
subscripts $\infty$ by $0$.

To obtain the full connection $A$ we use a suitable profile function to
smoothly interpolate between the connection $A_{0}$, parametrized by
$\eta_{0},\psi_{0}$ and $\lambda_{0}$ and the connection $A_{\infty}$,
parametrized by $\eta_{\infty},\psi_{\infty}$ and $\lambda_{\infty}$. A group
element $g$ interpolating between the two boundaries can be built by first
Gauss decomposing as above

$g=e^{\eta{\tt L}_{1}}\,e^{\psi{\tt L}_{0}}\,e^{\lambda{\tt L}_{-1}}\,e^{r{\tt
L}_{0}}$ (9)

with

$\eta=\frac{\eta_{0}+\eta_{\infty}f(r)}{1+f(r)}\,,\qquad\qquad\lambda=\frac{\lambda_{0}+\lambda_{\infty}f(r)}{1+f(r)}\,,\qquad\qquad\psi=\frac{\psi_{0}+\psi_{\infty}f(r)}{1+f(r)}\,,$
(10)

such that the connection $A$ that interpolates between the two boundaries is
given by

$A=b^{-1}\,\big{(}\operatorname{d}\\!+a\big{)}\,b\,,\qquad\qquad b=e^{r{\tt
L}_{0}}$ (11)

with

$a=e^{\psi}\operatorname{d}\\!\eta\,{\tt
L}_{1}+\big{(}\operatorname{d}\\!\psi+2\lambda
e^{\psi}\operatorname{d}\\!\eta\big{)}\,{\tt
L}_{0}+\big{(}\operatorname{d}\\!\lambda+\lambda\operatorname{d}\\!\psi+\lambda^{2}e^{\psi}\operatorname{d}\\!\eta\big{)}\,{\tt
L}_{-1}\,.$ (12)

The information of the specific boundary conditions one tries to match will be
encoded in the interpolating function $f(r)$. Even for a given set of boundary
conditions, there is a large (pure gauge) ambiguity in how to choose this
function. The minimal requirements are $\lim_{r\to\infty}f(r)\to\infty$ and
$\lim_{r\to 0}f(r)\to 0$, ensuring that we match with $A_{\infty}$ and
$A_{0}$, respectively. Moreover, we demand that $f(r)$ is monotonically
increasing as we move from $r=0$ to $r=\infty$.

In order to satisfy the desired sets of boundary conditions at either of the
boundaries the interpolating function $f(r)$ additionally should have
appropriate fall-off behavior at both boundaries to make sure that the
modifications to the connection produced by our gluing are subleading, i.e.,
pure gauge from either boundary theory point of view. As we shall see in the
prime example in section 4 it is simple to find such functions $f(r)$.

###  3.3 Matching conditions

The group element $g$ in (9)-(10) corresponds to a connection (3) satisfying
the equations of motion (2) and both sets of asymptotic conditions. However,
we are not done yet. We need to make sure that by interpolating between
$A_{\infty}$ and $A_{0}$ we do not introduce global inconsistencies. One
simple example of an inconsistency is a configuration $A_{\infty}$ that
requires Euclidean time $\tau$ to have a certain periodicity,
$\tau\sim\tau+\beta_{\infty}$, while $A_{0}$ requires also some periodicity,
$\tau\sim\tau+\beta_{0}$. Unless both periodicities coincide,
$\beta_{\infty}=\beta_{0}$, no smooth matching between $A_{\infty}$ and
$A_{0}$ is possible. In physical terms this means we have to make sure that
corresponding temperatures (and possibly other quantities) match when
interpolating.

More generally, we have to compare holonomies calculated at infinity with
holonomies calculated at zero and demand that the results agree with each
other. If they did not agree this would imply that we introduced sources or
singularities somewhere in the middle of the manifold, which needs to be
avoided for a smooth matching. In our specific setup, that amounts to imposing
that the holonomy around the $\tau$-cycle is trivial and that the holonomies
around the $\varphi$-cycle coincide for the connections we want to match. In
what follows we carry this out by introducing a group element $\tilde{h}$
based on the holonomy.

The group element $g$ and the Wilson line444As usual, $\mathcal{P}$ denotes
path ordering.

$h(r,\tau,\tau_{0},\varphi)=\mathcal{P}\exp\int\limits_{\tau_{0}}^{\tau}A_{\tau}(r,\tau^{\prime},\varphi)\operatorname{d}\\!\tau^{\prime}$
(13)

fulfill the same differential equation,

$A_{\tau}(r,\tau,\varphi)=g^{-1}(r,\tau,\varphi)\partial_{\tau}g(r,\tau,\varphi)\,,\quad
A_{\tau}(r,\tau,\varphi)=h^{-1}(r,\tau,\tau_{0},\varphi)\partial_{\tau}h(r,\tau,\tau_{0},\varphi).$
(14)

Using that $h(r,\tau_{0},\tau_{0},\varphi)=\mathbb{1}$ and that the group
element $g$ solves the differential equation (14) on a constant $\varphi$
slice yields

$h(r,\tau,\tau_{0},\varphi)=g^{-1}(r,\tau_{0},\varphi)g(r,\tau,\varphi)\,.$
(15)

From the holonomy $h(r,\tau+\beta,\tau,\varphi)$, we can build a new group
element $\tilde{h}(r,\tau+\beta,\tau,\varphi)$,

$\tilde{h}(r,\tau+\beta,\tau,\varphi)=g(r,\tau,\varphi)h(r,\tau+\beta,\tau,\varphi)g^{-1}(r,\tau,\varphi)=g(r,\tau+\beta,\varphi)g^{-1}(r,\tau,\varphi),$
(16)

which corresponds to a Wilson line of an auxiliary connection
$gA_{\tau}g^{-1}$, as

$\displaystyle\partial_{\tau}\tilde{h}(r,\tau,\tau_{0},\varphi)$
$\displaystyle=(gA_{\tau}g^{-1})(r,\tau,\varphi)\,\tilde{h}(r,\tau,\tau_{0},\varphi)\,,$
(17) $\displaystyle\tilde{h}^{-1}(r,\tau,\tau_{0},\varphi)$
$\displaystyle=\mathcal{P}\exp\int\limits_{\tau_{0}}^{\tau}(-gA_{\tau}g^{-1})(r,\tau^{\prime},\varphi)\operatorname{d}\\!\tau^{\prime}\,.$
(18)

This group element can be used to control the periodicity. In particular, one
can show that the connection is periodic of period $\beta$ along the
$\tau$-cycle iff $\tilde{h}(r,\tau+\beta,\tau,\varphi)$ is constant, i.e. iff
the following three expressions are constant in $r$, $\tau$ and $\varphi$

$\displaystyle
e^{\frac{\psi_{h}}{2}}=e^{\frac{\Delta\psi}{2}}\big{(}1+\Delta\lambda\,\eta
e^{\psi}\big{)},\qquad\lambda_{h}e^{\frac{\psi_{h}}{2}}=\Delta\lambda
e^{\psi+\frac{\Delta\psi}{2}},$
$\displaystyle\eta_{h}e^{\frac{\psi_{h}}{2}}=e^{\frac{\Delta\psi}{2}}\big{[}\Delta\eta+\eta(1-e^{-\Delta\psi})+\eta\Delta\lambda(\eta+\Delta\eta)e^{\psi}\big{]},$
(19)

where $\Delta X\equiv X(\tau+\beta)-X(\tau)$ and $X_{h}$ correspond to the
Gauss decomposition of $\tilde{h}$.

The triviality of the $\tau$-cycle further requires
$h(r,\tau_{0}+\beta,\tau_{0},\varphi)=\tilde{h}(r,\tau_{0}+\beta,\tau_{0},\varphi)=\pm\mathbb{1}$
leading to $\Delta\lambda=\Delta\eta=0,e^{\Delta\psi}=1$, or equivalently,

$\displaystyle\eta(r,\tau+\beta,\varphi)$
$\displaystyle=\eta(r,\tau,\varphi),$
$\displaystyle\lambda(r,\tau+\beta,\varphi)$
$\displaystyle=\lambda(r,\tau,\varphi),$
$\displaystyle\psi(r,\tau+\beta,\varphi)$
$\displaystyle=\psi(r,\tau,\varphi)+2\pi i\,n,\ \quad n\in\mathbb{Z}$ (20)

In particular, these conditions must hold for $\eta_{\infty}$, $\psi_{\infty}$
and $\lambda_{\infty}$ and $\eta_{0}$, $\psi_{0}$ and $\lambda_{0}$ for each
constant $\varphi$ slice. With this, we have shown that the matching of the
periodicity in $\tau$ of the two asymptotic solutions is necessary for the
regularity in $\tau$ of the new connection. One can easily check that, due to
the form (9) of the interpolating group element $g$, this is also sufficient.

We have dealt with the $\tau$-cycle and focus now on the $\varphi$-cycle. The
group element controlling the periodicity in $\varphi$ takes the form

$\tilde{h}_{\varphi}(r,\tau,\varphi+2\pi,\varphi)=g(r,\tau,\varphi+2\pi)g^{-1}(r,\tau,\varphi).$
(21)

The periodicity in $\varphi$ is equivalent to the expressions in (19) being
constant but, this time, $\Delta X=X(\varphi+2\pi)-X(\varphi)$. In particular,
when evaluating this expression for both asymptotic group elements $g_{0}$ and
$g_{\infty}$, we must obtain the same value.

As a consequence of matching the holonomies $\tilde{h}$, the functions
$\eta_{\infty}$, $\psi_{\infty}$, $\lambda_{\infty}$ are not completely
independent from the functions $\eta_{0}$, $\psi_{0}$, $\lambda_{0}$. We shall
see how the matching conditions work in detail when discussing examples. We
start with the first one in the next section.

##  4 Brown–Henneaux/Heisenberg matching

We discuss now our prime example, the matching between Brown–Henneaux boundary
conditions at infinity and Heisenberg boundary conditions at the horizon.

In section 4.1 we present the connection at both boundaries. In section 4.2 we
study consequences of the matching conditions. In section 4.3 we collect the
results and present the full gauge connection that smoothly interpolates
between Brown Henneaux boundary conditions at infinity and Heisenberg boundary
conditions at $r=0$ as well as the corresponding metric. In section 4.4 we
show how to interpolate non-smoothly using the Heaviside function while
maintaining smoothness of curvature.

###  4.1 Connection in presence of two boundaries

Consider Brown–Henneaux boundary conditions at $r\to\infty$ and Heisenberg
boundary conditions at $r=0$. Our first task is to bring the connection into
the form used in section 3.2.

For a theory with just one boundary at $r\to\infty$, the connection subject to
the Brown–Henneaux boundary conditions with constant chemical potentials $\mu$
in the plus-sector, see (5), in Euclidean time is given by Brown:1986nw ;
Banados:1998gg ; Carlip:1998uc

$A_{\infty}=b^{-1}\,\big{(}\operatorname{d}\\!+a_{\infty}\big{)}\,b\,,\qquad\qquad
b=e^{r{\tt L}_{0}}\,,\qquad\qquad a_{\infty}=\big{(}{\tt
L}_{1}-\frac{1}{2}\mathcal{L}L_{-1}\big{)}\operatorname{d}\\!z\,,$ (22)

where $z=i\mu\tau+\varphi$ and $\bar{z}=-i\mu\tau+\varphi$. On-shell
$\mathcal{L}=\mathcal{L}(z,\bar{z})$ obeys

$\partial_{\bar{z}}\mathcal{L}=0\,.$ (23)

As any holomorphic function on a compact connected complex manifold (such as a
torus) must be constant, $\mathcal{L}$ is constant. In the following, we
assume without loss of generality that $\mathcal{L}$ is positive.

Alternatively, for a theory with just one boundary at $r=0$, imposing
Heisenberg boundary conditions with constant chemical potentials
Afshar:2016wfy , the connection can be written as

$A_{0}=b^{-1}\,\big{(}\operatorname{d}\\!+a_{0}\big{)}\,b\,,\qquad b=e^{r{\tt
L}_{0}}\,,\qquad a_{0}=\frac{1}{2}\,\big{(}{\tt L}_{-1}-{\tt
L}_{1}\big{)}\,\big{(}\mathcal{J}(\varphi)\,\operatorname{d}\\!\varphi+i\zeta\,\operatorname{d}\\!\tau\big{)}\,,$
(24)

where $\zeta=\mathrm{const}$.555The form of the connection originally provided
in Afshar:2016wfy ; Afshar:2016kjj  is recovered by a change of basis:
$\tilde{{\tt L}}_{0}=\frac{1}{2}({\tt L}_{-1}-{\tt L}_{1})$, $\tilde{{\tt
L}}_{1}=\frac{1}{2}({\tt L}_{1}+2{\tt L}_{0}+{\tt L}_{-1})$, $\tilde{{\tt
L}}_{-1}=\frac{1}{2}({\tt L}_{1}-2{\tt L}_{0}+{\tt L}_{-1})$. We construct now
a solution of the field equations subject to the Brown–Henneaux boundary
conditions at $r\to\infty$ and subject to the Heisenberg boundary conditions
at $r=0$ along the lines of section 3, without yet specifying the
interpolating function $f(r)$. The variables $\eta_{0/\infty}$,
$\psi_{0/\infty}$ and $\lambda_{0/\infty}$ representing the group elements
$g_{0/\infty}$ can be chosen as

$\displaystyle\eta_{\infty}$
$\displaystyle=\mp\frac{e^{-\psi_{\infty}}}{\sqrt{2\mathcal{L}}}\,,$
$\displaystyle\psi_{\infty}$ $\displaystyle=\pm\sqrt{2\mathcal{L}}z\,,$
$\displaystyle\lambda_{\infty}$
$\displaystyle=\mp\frac{\sqrt{\mathcal{L}}}{\sqrt{2}}\,,$ (25a)
$\displaystyle\eta_{0}$ $\displaystyle=\pm\frac{1}{2}e^{-\psi_{0}}\,,$
$\displaystyle\psi_{0}$
$\displaystyle=\pm\int\mathcal{J}(\varphi^{\prime})\operatorname{d}\\!\varphi^{\prime}\pm
i\zeta\tau\,,$ $\displaystyle\lambda_{0}$ $\displaystyle=\pm 1\,.$ (25b)

These functions obey the following quasi-periodicity conditions

$\displaystyle\psi_{0}(\tau,\varphi+2\pi)$
$\displaystyle=\psi_{0}(\tau,\varphi)+2\pi K_{0}\,,$
$\displaystyle\psi_{\infty}(\tau,\varphi+2\pi)$
$\displaystyle=\psi_{\infty}(\tau,\varphi)+2\pi K_{\infty}\,,$ (26a)
$\displaystyle\eta_{0}(\tau,\varphi+2\pi)$ $\displaystyle=e^{-2\pi
K_{0}}\eta_{0}(\tau,\varphi)\,,$
$\displaystyle\eta_{\infty}(\tau,\varphi+2\pi)$ $\displaystyle=e^{-2\pi
K_{\infty}}\eta_{\infty}(\tau,\varphi)\,,$ (26b)
$\displaystyle\lambda_{0}(\tau,\varphi+2\pi)$
$\displaystyle=\lambda_{0}(\tau,\varphi)\,,$
$\displaystyle\lambda_{\infty}(\tau,\varphi+2\pi)$
$\displaystyle=\lambda_{\infty}(\tau,\varphi)\,.$ (26c)

Using the Fourier decomposition

$J_{n}=\frac{k}{4\pi}\int\operatorname{d}\\!\varphi\,\mathcal{J}(\varphi)\,e^{in\varphi}\,,\qquad\qquad
L_{0}=\frac{k}{4\pi}\int\operatorname{d}\\!\varphi\,\mathcal{L}=\frac{k}{2}\mathcal{L}$
(27)

we find $K_{\infty}=\pm\sqrt{2\mathcal{L}}=\pm\sqrt{\frac{4}{k}L_{0}}$ and
$K_{0}=\pm\frac{2}{k}J_{0}$. The quasi-periodicity relations along the
$\tau$-cycle are analogous to (26),

$\displaystyle\psi_{0}(\tau+\beta,\varphi)$
$\displaystyle=\psi_{0}(\tau,\varphi)+i\beta V_{0}\,,$
$\displaystyle\psi_{\infty}(\tau+\beta,\varphi)$
$\displaystyle=\psi_{\infty}(\tau,\varphi)+i\beta V_{\infty}\,,$ (28a)
$\displaystyle\eta_{0}(\tau+\beta,\varphi)$ $\displaystyle=e^{-i\beta
V_{0}}\eta_{0}(\tau,\varphi)\,,$
$\displaystyle\eta_{\infty}(\tau+\beta,\varphi)$ $\displaystyle=e^{-i\beta
V_{\infty}}\eta_{\infty}(\tau,\varphi)\,,$ (28b)
$\displaystyle\lambda_{0}(\tau+\beta,\varphi)$
$\displaystyle=\lambda_{0}(\tau,\varphi)\,,$
$\displaystyle\lambda_{\infty}(\tau+\beta,\varphi)$
$\displaystyle=\lambda_{\infty}(\tau,\varphi)\,,$ (28c)

where $V_{\infty}=\pm\mu\sqrt{2\mathcal{L}}=\pm\mu\sqrt{\frac{4}{k}L_{0}}$ and
$V_{0}=\pm\zeta$. The corresponding behaviour for the group element is

$\displaystyle g_{0/\infty}(\tau,\varphi+2\pi)$ $\displaystyle=e^{2\pi
K_{0/\infty}{\tt L}_{0}}g_{0/\infty}(\tau,\varphi)\,,$ (29) $\displaystyle
g_{0/\infty}(\tau+\beta,\varphi)$ $\displaystyle=e^{i\beta V_{0/\infty}{\tt
L}_{0}}g_{0/\infty}(\tau,\varphi)\,.$ (30)

###  4.2 Matching conditions

We study now the matching conditions discussed in section 3.3. We start with
the non-contractible $\varphi$-cycle and consider the corresponding holonomy
$\tilde{h}_{\varphi}$,

$\displaystyle\tilde{h}_{\varphi}(r,\tau,\varphi+2\pi,\varphi)=g(r,\tau,\varphi+2\pi)g^{-1}(r,\tau,\varphi)\,.$
(31)

Demanding for consistency that the holonomy $\tilde{h}_{\varphi}$ is
independent of the coordinates establishes the matching condition

$K_{0}=K_{\infty}\,.$ (32)

While this matching condition requires strict equality between $K_{0}$ and
$K_{\infty}$, we have a $\mathbb{Z}_{2}$ ambiguity in our connections (25)
which allows us to match solutions with a priori opposite values of $K_{0}$
and $K_{\infty}$. Depending on whether

$J_{0}=+\sqrt{kL_{0}}\qquad\mathrm{or}\qquad J_{0}=-\sqrt{kL_{0}}$ (33)

we choose

$K_{0}=\frac{2}{k}J_{0}\,,\qquad
K_{\infty}=\sqrt{\frac{4L_{0}}{k}}\qquad\mathrm{or}\qquad
K_{0}=\frac{2}{k}J_{0}\,,\qquad K_{\infty}=-\sqrt{\frac{4L_{0}}{k}}\,,$ (34)

such that $K_{0}=K_{\infty}$ always. For sake of specificity we consider the
case

$J_{0}=+\sqrt{kL_{0}}\,.$ (35)

Periodicity in $\tau$ and triviality around the $\tau$-cycle are equivalent to
the requirement that the holonomy $h$ at $r=0$ and $r\to\infty$ should lie in
the center of the gauge group, i.e.

$\mathcal{P}\exp{\int\limits_{0}^{\beta}A_{\tau}\operatorname{d}\\!\tau}|_{r=0/\infty}=\pm\mathbb{1}\qquad\Longleftrightarrow\qquad
e^{i\beta V_{0/\infty}{\tt L}_{0}}=\pm\mathbb{1}\,$ (36)

leading to $\frac{V_{0}\beta}{2\pi}=n\in{\mathbb{Z}}$ and
$\frac{V_{\infty}\beta}{2\pi}=m\in{\mathbb{Z}}$. In the following, we impose

$n\equiv m=1\qquad\Rightarrow\qquad V_{0}=V_{\infty}=\frac{2\pi}{\beta}\,.$
(37)

Equation (37) implies the relation

$\zeta=\mu\sqrt{\frac{4}{k}L_{0}}\,,$ (38)

which using (33) becomes $\zeta=\mu\frac{2}{k}J_{0}$.

What we have shown above is that the matching conditions relate only the zero
modes, $K_{0}=K_{\infty}$ and $V_{0}=V_{\infty}$. This means that excitations
above these zero modes — e.g. $\hat{u}(1)$ excitations at the horizon — can be
switched on arbitrarily and independently. Thus, we see that all soft-hairy
horizon descendants at $r=0$ are compatible with one configuration at
$r\to\infty$ described by $\mathcal{L}=\frac{2}{k}L_{0}$.

###  4.3 Gauge connection, metric and asymptotic symmetries

For convenience of the reader we summarize now our results for the first
example and translate them also into the metric formulation. The Chern–Simons
connection compatible with Brown–Henneaux boundary conditions at $r\to\infty$
and near horizon boundary conditions at $r=0$ is given by

$A^{\pm}=b_{\pm}^{-1}\big{(}\operatorname{d}\\!+a^{\pm}\big{)}\,b_{\pm}\,,\qquad\qquad
b_{\pm}=e^{\pm r{\tt L}_{0}}\,,\qquad\qquad
a^{\pm}=g_{\pm}^{-1}\operatorname{d}\\!g_{\pm}\,,$ (39)

Inserting the group element

$\displaystyle g_{\pm}=e^{\eta^{\pm}L_{\pm
1}}e^{\pm\psi^{\pm}L_{0}}e^{\lambda^{\pm}L_{\mp 1}}e^{\pm rL_{0}}$ (40)

yields

$a^{\pm}=e^{\psi^{\pm}}\operatorname{d}\\!\eta^{\pm}{\tt L}_{\pm
1}\pm\big{(}\operatorname{d}\\!\psi^{\pm}+2\lambda^{\pm}e^{\psi^{\pm}}\operatorname{d}\\!\eta^{\pm}\big{)}{\tt
L}_{0}+\big{(}\operatorname{d}\\!\lambda^{\pm}+\lambda^{\pm}\operatorname{d}\\!\psi^{\pm}+\lambda^{\pm\,2}e^{\psi^{\pm}}\operatorname{d}\\!\eta^{\pm}\big{)}{\tt
L}_{\mp 1}$ (41)

and

$\displaystyle A^{\pm}=$ $\displaystyle
e^{\psi^{\pm}}\operatorname{d}\\!\eta^{\pm}e^{r}{\tt L}_{\pm
1}\pm\big{(}\operatorname{d}\\!r+\operatorname{d}\\!\psi^{\pm}+2\lambda^{\pm}e^{\psi^{\pm}}\operatorname{d}\\!\eta^{\pm}\big{)}{\tt
L}_{0}$
$\displaystyle+\big{(}\operatorname{d}\\!\lambda^{\pm}+\lambda^{\pm}\operatorname{d}\\!\psi^{\pm}+\lambda^{\pm\,2}e^{\psi^{\pm}}\operatorname{d}\\!\eta^{\pm}\big{)}e^{-r}{\tt
L}_{\mp 1}\,.$ (42)

The functions

$\displaystyle\eta^{\pm}=\frac{\eta^{\pm}_{0}+\eta^{\pm}_{\infty}\,f(r)}{1+f(r)}\,,$
$\displaystyle\psi^{\pm}=\frac{\psi^{\pm}_{0}+\psi^{\pm}_{\infty}\,f(r)}{1+f(r)}\,,$
$\displaystyle\lambda^{\pm}=\frac{\lambda^{\pm}_{0}+\lambda^{\pm}_{\infty}\,f(r)}{1+f(r)}$
(43)

depend on both the asymptotic functions $\eta^{\pm}_{\infty}$,
$\psi^{\pm}_{\infty},\ \lambda^{\pm}_{\infty}$, and on the near horizon
functions $\eta^{\pm}_{0}$, $\psi^{\pm}_{0},\ \lambda^{\pm}_{0}$. We choose
the interpolating function

$f(r)=\sinh^{4}r$ (44)

to guarantee appropriate fall-off behavior near $r=0$ and $r\to\infty$.

Close to the horizon, $g$ then leads to a connection of the form

$a=a_{0}+{\cal O}(r^{4})\operatorname{d}\\!t+{\cal
O}(r^{4})\operatorname{d}\\!\varphi+{\cal O}(r^{3})\operatorname{d}\\!r$ (45)

and near infinity we get

$a=a_{\infty}+{\cal O}(e^{-4r})\operatorname{d}\\!t+{\cal
O}(e^{-4r})\operatorname{d}\\!\varphi+{\cal
O}(e^{-4r})\operatorname{d}\\!r\,.$ (46)

For completeness we present also the minus-sector of Brown–Henneaux boundary
conditions

$A^{-}_{\infty}=b\,\big{(}\operatorname{d}\\!+a_{\infty}^{-}\big{)}\,b^{-1}\,,\qquad\qquad
b=e^{r{\tt L}_{0}}\,,\qquad\qquad a_{\infty}^{-}=\big{(}{\tt
L}_{-1}-\frac{1}{2}\mathcal{L}_{-}{\tt
L}_{1}\big{)}\operatorname{d}\\!\bar{z}\,,$ (47)

where we defined $\bar{z}=\varphi-i\mu_{-}\tau$ and $z=\varphi+i\mu_{-}\tau$.
The on-shell condition

$\partial_{z}{\mathcal{L}}_{-}=0$ (48)

implies that $\mathcal{L}_{-}=\mathrm{const}$. In the following we assume that
$\mathcal{L}_{-}$ is positive without loss of generality. The minus-sector of
the near horizon boundary conditions reads

$A^{-}_{\infty}=b\,\big{(}\operatorname{d}\\!+a\big{)}\,b^{-1}\,,\qquad
b=e^{r{\tt L}_{0}}\,,\qquad a_{0}=-\frac{1}{2}\,\big{(}{\tt L}_{-1}-{\tt
L}_{1}\big{)}\,\big{(}\mathcal{J}_{-}(\varphi)\,\operatorname{d}\\!\varphi-i\zeta_{-}\,\operatorname{d}\\!\tau)\,.$
(49)

The minus-sector of the functions (25) is

$\displaystyle\eta_{\infty}^{-}$
$\displaystyle=\mp\frac{e^{-\psi_{\infty}^{-}}}{\sqrt{2\mathcal{L}_{-}}}\,,$
$\displaystyle\psi_{\infty}^{-}$
$\displaystyle=\pm\sqrt{2\mathcal{L}_{-}}\bar{z}\,,$
$\displaystyle\lambda_{\infty}^{-}$
$\displaystyle=\mp\frac{\sqrt{\mathcal{L_{-}}}}{\sqrt{2}}\,,$ (50a)
$\displaystyle\eta_{0}^{-}$ $\displaystyle=\pm\frac{1}{2}e^{-\psi_{0}^{-}}\,,$
$\displaystyle\psi_{0}^{-}$
$\displaystyle=\pm\int_{0}^{\varphi}\mathcal{J}_{-}(\varphi^{\prime})\operatorname{d}\\!\varphi^{\prime}\mp
i\zeta_{-}\tau\,,$ $\displaystyle\lambda_{0}^{-}$ $\displaystyle=\pm 1\,.$
(50b)

These functions are subject to the quasi-periodicity conditions (26) and the
matching conditions666The modes in the minus-sector are defined in complete
analogy to the plus-sector (27).

$\frac{2}{k}J_{0}^{-}=\pm\sqrt{2\mathcal{L}_{-}}=\pm\sqrt{\frac{4}{k}L_{0}^{-}}\,,\qquad\qquad\zeta_{-}=\pm\mu_{-}\sqrt{2\mathcal{L}_{-}}=\pm\mu_{-}\sqrt{\frac{4}{k}L_{0}^{-}}=\frac{2\pi}{\beta}\,.$
(51)

The metric is obtained from the Chern–Simons gauge field via Banados:1998gg ;
Carlip:1998uc

$g_{\mu\nu}=\frac{1}{2}\langle(A_{\mu}^{+}-A_{\mu}^{-}),(A_{\nu}^{+}-A_{\nu}^{-})\rangle\,.$
(52)

Inserting (39)-(4.3) yields the metric

$\operatorname{d}\\!s^{2}=\big{[}e^{\psi^{+}}e^{r}\operatorname{d}\\!\eta^{+}-e^{-r}(\operatorname{d}\\!\lambda^{-}+\lambda^{-}\operatorname{d}\\!\psi^{-}+{\lambda^{-}}^{2}e^{\psi^{-}}\operatorname{d}\\!\eta^{-})\big{]}\big{[}+\leftrightarrow-\big{]}\\\
+\big{(}\operatorname{d}\\!r+\tfrac{1}{2}\,\big{(}\operatorname{d}\\!\psi^{+}+\operatorname{d}\\!\psi^{-}\big{)}+\lambda^{+}e^{\psi^{+}}\operatorname{d}\\!\eta^{+}+\lambda^{-}e^{\psi^{-}}\operatorname{d}\\!\eta^{-}\big{)}^{2}\,.$
(53)

The explicit form of the near horizon and asymptotic expansions of the metric
for the interpolating function $f(r)=\sinh^{4}(r)$ is given in appendix B.

Here we only show the leading terms in Lorentzian signature ($\tau\rightarrow-
it$). In the asymptotic limit $r\to\infty$ by construction a standard
Fefferman–Graham expansion is recovered ($x^{\pm}=\mu_{\pm}t\pm\varphi$)

$\operatorname{d}\\!s^{2}\big{|}_{r\to\infty}=\operatorname{d}\\!r^{2}-e^{2r}\,\operatorname{d}\\!x^{+}\operatorname{d}\\!x^{-}+\tfrac{1}{2}\,{\mathcal{L}_{+}}\,\operatorname{d}\\!x^{+\,2}+\tfrac{1}{2}\,{\mathcal{L}_{-}}\,\operatorname{d}\\!x^{-\,2}+{\cal
O}(e^{-2r})\,.$ (54)

The near horizon limit $r\to 0$ is compatible with
$\mathcal{J}_{\pm}=\gamma\pm\omega$ , $\zeta_{\pm}=-a$ Afshar:2016wfy ;
Afshar:2016kjj

$\operatorname{d}\\!s^{2}\big{|}_{r\to
0}=\operatorname{d}\\!r^{2}-a^{2}r^{2}\,\operatorname{d}\\!t^{2}+\gamma^{2}\,\operatorname{d}\\!\varphi^{2}\big{(}1+{\cal
O}(r^{2})\big{)}+2a\omega
r^{2}\,\operatorname{d}\\!t\operatorname{d}\\!\varphi+{\cal O}(r^{3})\,.$ (55)

For our matching to work the zero-modes of the state-dependent functions
$\mathcal{J}_{\pm},\mathcal{L}_{\pm}$ and the chemical potentials
$\zeta_{\pm}$ and $\mu_{\pm}$ must be related as in (35), (38) and (51). The
asymptotic symmetry algebra at $r=0$ consists of two $u(1)$-current algebras
Afshar:2016wfy ,

$\big{[}J_{n}^{\pm},\,J_{m}^{\pm}\big{]}=\frac{k}{2}\,n\,\delta_{n+m,\,0}.$
(56)

###  4.4 Matching with a Heaviside function

The discussions in previous sections make it clear that the interpolation
function $f(r)$ can be chosen almost arbitrarily, subject only to some fall-
off conditions. The specific form (43) with some smooth function $f(r)$ is not
the only way the interpolation can happen. In particular, we may drop the
assumption of smoothness and ‘interpolate’ through a Heaviside function

$\displaystyle\eta^{\pm}=\eta_{0}^{\pm}+\theta(r-r_{0})\Delta\eta^{\pm},\,\,\;\psi^{\pm}=\psi_{0}^{\pm}+\theta(r-r_{0})\Delta\psi^{\pm},\,\,\;\lambda^{\pm}=\lambda_{0}^{\pm}+\theta(r-r_{0})\Delta\lambda^{\pm}$
(57)

with some finite critical radius $r_{0}>0$ and the definitions

$\displaystyle\Delta\eta^{\pm}=\eta_{\infty}^{\pm}-\eta_{0}^{\pm}\,,\qquad\qquad\Delta\psi^{\pm}=\psi_{\infty}^{\pm}-\psi_{0}^{\pm}\,,\qquad\qquad\Delta\lambda^{\pm}=\lambda_{\infty}^{\pm}-\lambda_{0}^{\pm}\,,$
(58)

where $\eta_{0/\infty}^{\pm},\psi_{0/\infty}^{\pm}$ and
$\lambda_{0/\infty}^{\pm}$ are given by (25) and (50) and are subject to the
matching conditions (35), (38) and (51). An advantage of such a choice is that
both the asymptotic and near horizon boundary conditions and associated
symmetries apply all the way to the critical radius $r_{0}$, at which point
one set of symmetries changes discontinuously to the other.

To deal with $\theta$-function valued functions and their derivatives we use
elementary aspects of Colombeau theory, see appendix A for a derivation of the
next formula.

$f^{\prime}\big{(}\theta(r-r_{0})\big{)}\theta^{\prime}(r-r_{0})\sim\big{(}f(1)-f(0)\big{)}\,\delta(r-r_{0})\,.$
(59)

Here $\sim$ stands for equality in the sense of Colombeau. It is then
straightforward to write down the explicit form of the Chern–Simons gauge
fields with generic value of soft-hairy excitation at the horizon ($r=0$) and
a constant Brown–Henneaux excitation at the boundary ($r=\infty$).

By construction
$A_{\tau}^{\pm}=A_{\tau,0}^{\pm}+\theta(r-r_{0})(A_{\tau,\infty}^{\pm}-A_{\tau,0}^{\pm})$
and
$A_{\varphi}^{\pm}=A_{\varphi,0}^{\pm}+\theta(r-r_{0})(A_{\varphi,\infty}^{\pm}-A_{\varphi,0}^{\pm})$,
while $A_{r}^{\pm}=\pm{\tt L}_{0}+\delta(r-r_{0}){\cal
A}^{\pm}_{\mathrm{smooth}}$, for some known functions ${\cal
A}^{\pm}_{\mathrm{smooth}}$. Since the explicit form of the $A_{r}^{\pm}$
components is very cumbersome and not illuminating we do not present it here.
Restricting the near horizon excitation functions to zero-modes only777We
focus for simplicity on the particular matching
$\mathcal{J}_{\pm}=\frac{2}{k}J_{0}^{\pm}=\sqrt{2\mathcal{L}_{\pm}}$,
$\zeta_{\pm}=\mu_{\pm}\sqrt{2\mathcal{L}_{\pm}}$ explicitly realized by
$\eta_{0}^{\pm}=\frac{1}{2}\exp(-\psi_{0}^{\pm}),\,\psi_{0}^{\pm}=\mathcal{J}_{\pm}\varphi\pm
i\zeta_{\pm}\tau,\,\lambda_{0}^{\pm}=1$ ,
$\eta_{\infty}^{+}=-\exp(-\psi_{\infty}^{+})/\sqrt{2\mathcal{L}_{+}},\,\psi_{\infty}^{+}=\sqrt{2\mathcal{L}_{+}}z,\,\lambda_{\infty}^{+}=-\sqrt{\mathcal{L_{+}}/2}$
and
$\eta_{\infty}^{-}=-\exp(-\psi_{\infty}^{-})/\sqrt{2\mathcal{L}_{+}},\,\psi_{\infty}^{-}=\sqrt{2\mathcal{L}_{-}}\bar{z},\lambda_{\infty}^{-}=-\sqrt{\mathcal{L_{-}}/2}$,
where $z=i\mu_{+}\tau+\varphi$ and $\bar{z}=-i\mu_{-}\tau+\varphi$. simplifies
the radial component of the connection,

$\displaystyle A_{r}^{+}=$
$\displaystyle-e^{r}\Big{(}\frac{1}{\mathcal{J}_{+}}+\frac{1}{2}\Big{)}\,\delta(r-r_{0}){\tt
L}_{1}+\Big{(}\frac{1}{4}\mathcal{J}_{+}-\frac{1}{\mathcal{J}_{+}}\Big{)}\,\delta(r-r_{0}){\tt
L}_{0}+{\tt L}_{0}$
$\displaystyle-e^{-r}\Big{(}\frac{1}{24}\mathcal{J}_{+}^{2}+\frac{1}{2}\mathcal{J}_{+}+\frac{1}{3\mathcal{J}_{+}}+1\Big{)}\,\delta(r-r_{0}){\tt
L}_{-1}\,.$ (60)

The connection component $A_{r}^{-}$ is obtained from $A_{r}^{+}$ by
exchanging ${\tt L}_{1}\leftrightarrow{\tt L}_{-1},{\tt
L}_{0}\leftrightarrow-{\tt
L}_{0},\mathcal{J}_{+}\leftrightarrow\mathcal{J}_{-}$.

We end this subsection with the remark that despite of having $\theta$\- and
$\delta$-functions in the gauge fields $A_{\mu}^{\pm}$, the associated field
strengths (by construction) are zero everywhere. Thus, in the metric
formulation the spacetime curvature is locally AdS3 everywhere, even though
the metric component $g_{rr}$ contains terms quadratic in the
$\delta$-function.

##  5 Compère–Song–Strominger/Heisenberg matching

As our second example we consider the matching of near horizon boundary
conditions at $r=0$ to Compère–Song–Strominger (CSS) boundary conditions
Compere:2013bya  at $r=\infty$. In this section we present both the $A^{+}$
and the $A^{-}$ sector, since the sectors are qualitatively different. The
connection subject to CSS boundary conditions at $r\to\infty$ in Euclidean
signature reads

$\displaystyle A^{\pm}$ $\displaystyle=b^{\mp
1}\,\big{(}\operatorname{d}\\!+a^{\pm}\big{)}\,b^{\pm 1}\,,\qquad\qquad
b=e^{r{\tt L}_{0}}\,,$ (61a) $\displaystyle a^{+}$
$\displaystyle=\big{(}-\partial_{z}P\operatorname{d}\\!z-\operatorname{d}\\!\bar{z}\big{)}\Big{(}{\tt
L}_{1}-\frac{\Delta}{k}\,{\tt L}_{-1}\Big{)}\,,$ (61b) $\displaystyle a^{-}$
$\displaystyle=\Big{(}\frac{L}{2}{\tt L}_{1}-{\tt
L}_{-1}\Big{)}\operatorname{d}\\!z\,,$ (61c)

where $z=i\tau+\varphi$ and $\bar{z}=-i\tau+\varphi$ and
$\partial_{\bar{z}}P=0$. Here, $\Delta$ is a constant while $P(z,\bar{z})$ and
$L(z,\bar{z})$ are subject to the on-shell conditions

$\partial_{\bar{z}}\partial_{z}P=0\,,\qquad\qquad\partial_{\bar{z}}L=0\,.$
(62)

As before, this implies that $L$ and $\partial_{z}P$ are constants. This
yields

$P(z,\bar{z})=P_{0}z+P_{1}\,,\qquad\qquad L(z)=L\,.$ (63)

In the following for simplicity and without loss of generality we assume that
$\Delta$ and $L$ are positive. The functions $\eta^{\pm},\psi^{\pm}$ and
$\lambda^{\pm}$ for the group element $g$ read

$\displaystyle\lambda_{\infty}^{+}$
$\displaystyle=\pm\sqrt{\frac{\Delta}{k}}\,,$ $\displaystyle\psi_{\infty}^{+}$
$\displaystyle=\pm 2\sqrt{\frac{\Delta}{k}}(\bar{z}+P_{0}z+P_{1})\,,$
$\displaystyle\eta_{\infty}^{+}$
$\displaystyle=\pm\sqrt{\frac{k}{\Delta}}\frac{e^{-\psi_{\infty}^{+}}}{2}\,,$
(64) $\displaystyle\lambda_{\infty}^{-}$
$\displaystyle=\pm\sqrt{\frac{L}{2}}\,,$ $\displaystyle\psi_{\infty}^{-}$
$\displaystyle=\pm\sqrt{2L}z\,,$ $\displaystyle\eta_{\infty}^{-}$
$\displaystyle=\pm\frac{e^{-\psi_{\infty}^{-}}}{\sqrt{2L}}\,.$ (65)

The solution satisfies the periodicity conditions (26) and (28) with

$K_{\infty}^{+}=\pm 2\sqrt{\frac{\Delta}{k}}(P_{0}+1)\,,\qquad
V_{\infty}^{+}=\pm 2\sqrt{\frac{\Delta}{k}}(P_{0}-1)\,,\qquad
K_{\infty}^{-}=V_{\infty}^{-}=\pm\sqrt{2L}\,.$ (66)

The matching conditions analogous to section 4.2 read
$V_{\infty}^{\pm}=V_{0}^{\pm}=\frac{2\pi}{\beta}$ and
$K_{\infty}^{\pm}=K_{0}^{\pm}$. Using the results from sections 4.1 and 4.3
yields

$\displaystyle\zeta^{+}$ $\displaystyle=\pm
2\sqrt{\frac{\Delta}{k}}(P_{0}-1)=\frac{2\pi}{\beta}\,,$
$\displaystyle\zeta^{-}$ $\displaystyle=\pm\sqrt{2L}=\frac{2\pi}{\beta}\,,$
(67a) $\displaystyle\frac{2}{k}J_{0}^{+}$ $\displaystyle=\pm
2\sqrt{\frac{\Delta}{k}}(P_{0}+1)\,,$ $\displaystyle\frac{2}{k}J_{0}^{-}$
$\displaystyle=\pm\sqrt{2L}\,.$ (67b)

In summary, we can smoothly flow from the CSS boundary conditions at infinity
with fixed (zero-mode) excitations to the Heisenberg boundary conditions at
the horizon with generic excitations. Thus, also black holes in AdS3 with CSS
boundary conditions can be equipped with arbitrary soft hair excitations.

##  6 Concluding remarks

We showed how to interpolate between two different sets of boundary conditions
in AdS3 Einstein gravity using the Chern–Simons formulation. The key result of
section 3 is that with our procedure any two sets of boundary conditions can
be interpolated as long as the matching conditions hold. In particular, the
holonomies around both cycles should yield the same when evaluated near either
of the boundaries.

In our two examples, Brown–Henneaux/Heisenberg matching in section 4 and
Compère–Song–Strominger/Heisenberg matching in section 5, we interpolated
between the asymptotic region and the near horizon region. As we saw in both
examples, a physical consequence of the matching conditions is that the mass
and angular momentum at infinity must match with corresponding charges near
the horizon. However, the tower of Fourier modes of excitations at the horizon
is totally unconstrained by this matching, so that any amount of soft hair
excitations is allowed for any given values of mass and angular momentum, as
measured by an asymptotic observer.

We conclude now with a list of further possible applications and
generalizations.

  * •

Lorentzian signature. Our Euclidean results generalize to Lorentzian
signature. Reconsider our prime example of section 4, but with Lorentzian
signature. In that case classical solutions can have generic Brown–Henneaux
excitations asymptotically, specified by two arbitrary periodic functions
$\mathcal{L}_{\pm}(x^{\pm})$. Nonetheless, the matching conditions relate only
the zero modes of the Brown–Henneaux and Heisenberg charges, while all other
boundary gravitons can be excited independently at either of the boundaries.
This dovetails with the detailed analysis of Bañados geometries Banados:1998gg
and that the entropy and angular momentum are orbit invariant quantities
Sheikh-Jabbari:2016unm .

  * •

Auto-interpolation. It is of course possible to impose the same set of
boundary conditions at either of the boundaries. For instance, one can impose
Brown–Henneaux boundary conditions at the conformal boundary of AdS3 and at an
arbitrary constant radius-slice, e.g. a stretched horizon. We do not present
the details of the analysis here but mention the final result for the
Lorentzian case. Starting with a Bañados geometry Banados:1998gg  near one
boundary one may flow to another Bañados geometry at the other boundary and
the matching conditions tell us that the zero modes must not change. In more
algebraic wording, recall that the Bañados geometries are in one-to-one
relation with Virasoro coadjoint orbits Sheikh-Jabbari:2016unm . Our matching
procedure then means that one can flow within a given coadjoint orbit from one
representant to another, but cannot move across the orbits.

  * •

Flow of boundary action. While our focus was on the interpolation of solutions
with different boundary conditions, it should be possible to interpolate
between corresponding boundary actions. For instance, the Brown–Henneaux
boundary conditions lead to a Liouville-like boundary theory Coussaert:1995zp
; Barnich:2017jgw ; Cotler:2018zff , which near the horizon should flow to a
Cangemi–Jackiw type of scalar field theory Grumiller:2019tyl . It would be
nice to verify this and also to generalize it to flows between other sets of
boundary actions.

  * •

Fluff proposal. An application of soft hair excitations Hawking:2016msc  is
the fluff proposal Afshar:2016uax ; Afshar:2017okz ; Sheikh-Jabbari:2016npa ,
which relies on several working assumptions. One of them is that there are
non-trivial physical boundary excitations near the horizon that are invisible
to an asymptotic observer, which is precisely what we have shown in the
present work. We hope to address the remaining working assumptions in the
future, in particular a mechanism that provides a cut-off on the soft hair
spectrum.

  * •

Higher spin and asymptotic flat theories. Our results generalize
straightforwardly to other gravity or gravity-like theories, like higher spin
gravity Henneaux:2010xg ; Campoleoni:2010zq  or asymptotically flat gravity
Barnich:2006av  or both Afshar:2013vka ; Gonzalez:2013oaa , as long as there
is a Chern–Simons formulation. In particular, higher spin black holes
Gutperle:2011kf  with Brown–Henneaux-like boundary conditions (or their flat
space cosmology cousins Gary:2014ppa ; Matulich:2014hea ) can again be
interpolated to near horizon Heisenberg boundary conditions Grumiller:2016kcp
; Ammon:2017vwt .

  * •

Higher derivative theories. It could be of interest to generalize our
discussion to higher derivative theories of gravity, which were studied
vigorously in the past decade, see e.g. Hinterbichler:2011tt . In three
dimensions a large class of these theories allows a formulation as
Chern–Simons-like theories Merbis:2014vja ; Bergshoeff:2014bia . It seems
likely that most of our conclusions carry over to such theories.

  * •

Higher dimensional theories. While our technical implementation of the
interpolation relied on the Chern–Simons formulation, we could have done
everything in the metric formulation. On general grounds, we expect that it
must be possible to interpolate between different sets of boundary conditions
also in higher dimensions, though it is unclear to us what the precise
matching conditions will be. For instance, interpolating between
asymptotically AdSD boundary conditions (see e.g. Skenderis:2002wp ) and near
horizon boundary conditions Grumiller:2019fmp  will be rewarding to better
understand non-extremal black holes and their soft hairy counterparts.

  * •

Non-gravitational applications. Since our technical implementation of the
interpolation relied on the Chern–Simons formulation, our results apply also
to non-gravitational applications of Chern–Simons in presence of two
boundaries. A potential application is a Quantum-Hall system with two
disconnected boundary components and different spectra of edge excitations at
either of the boundaries.

  * •

More boundaries. Finally, it seems interesting to generalize the discussion to
more than two disconnected boundary components. For instance, the recent
holographic model for black hole evaporation Akers:2019nfi  involves
configurations with three and more boundary components, and it could be
worthwhile to impose boundary conditions separately on each of these
components.

We intend to come back to some of these generalizations in the future.

## Acknowledgements

We are grateful to Hamid Afshar, Wout Merbis, Mateusz Piorkowski, and
Friedrich Schöller for useful discussions.

DG was supported by the Austrian Science Fund (FWF), projects P 28751, P 30822
and P 32581. MMShJ would like to thank the hospitality of ICTP HECAP where
this work finished. He also acknowledges the support by INSF grant No 950124
and Saramadan grant No. ISEF/M/98204. RW was supported by a DOC Fellowship of
the Austrian Academy of Sciences and by the Austrian Science Fund FWF under
the Doctoral Program W1252-N27 Particles and Interactions. DG and MMShJ
acknowledge the Iran-Austria IMPULSE project grant, supported and run by
Khawrizmi University. During part of this work, CT was supported by the Max
Planck Institute for Gravitational Physics (Albert Einstein Institute) in
Potsdam.

##  Appendix A Aspects of Colombeau theory

Colombeau theory gives a mathematical framework of multiplying distributions
colombeau1990 . In a general relativistic context the need for Colombeau
theory can arise when considering distributional metrics Clarke:1996pp ;
Steinbauer:1996fv ; Balasin:1996mq ; Steinbauer:2006qi , e.g. for solutions of
Einstein gravity describing gravitational shock-waves (though often Colombeau-
theory can be avoided by choosing suitable coordinates).

In our work we are interested in making sense of first derivatives of smooth
functions of the Heaviside function. The essence of Colombeau theory applied
to this case is to replace the Heaviside function by some smooth family of
functions, $\theta_{\epsilon}$, that in the limit of vanishing $\epsilon$
yields the Heaviside function $\theta$, and to manipulate the expressions in
such a way that the final result does not depend on the precise family.

The result (59) then follows from a chain of equalities

$\lim_{\epsilon\rightarrow
0}\int\limits_{-\infty}^{\infty}\\!f^{\prime}(\theta_{\epsilon}(x))\theta_{\epsilon}^{\prime}(x)\psi(x)\operatorname{d}\\!x=-\lim_{\epsilon\rightarrow
0}\int\limits_{-\infty}^{\infty}\\!f(\theta_{\epsilon}(x))\psi^{\prime}(x)\operatorname{d}\\!x\\\
=-\lim_{\epsilon\rightarrow
0}\int\limits_{-\infty}^{\infty}(f(\theta_{\epsilon}(x))-f(0))\psi^{\prime}(x)\operatorname{d}\\!x=-\int\limits_{0}^{\infty}\left(f(1)-f(0)\right)\psi^{\prime}(x)\operatorname{d}\\!x\\\
=\left(f(1)-f(0)\right)\psi(0)=\left(f(1)-f(0)\right)\int\limits_{-\infty}^{\infty}\delta(x)\psi(x)\operatorname{d}\\!x\,,$
(68)

where $\psi(x)$ is a test-function with the usual properties. The result (68)
is independent of the approximation used for the Heaviside function and allows
to associate $f^{\prime}(\theta(x))\,\theta^{\prime}(x)$ to the distribution
$\delta(x)$ times a prefactor that we just derived

$f^{\prime}(\theta(x))\theta^{\prime}(x)\sim\left(f(1)-f(0)\right)\,\delta(x)\,.$
(69)

The choice $f^{\prime}(\theta(x))=\theta^{n}(x)$ recovers the classic example
by Colombeau colombeau1990

$\theta^{n}(x)\theta^{\prime}(x)\sim\frac{1}{n+1}\,\delta(x)\,.$ (70)

##  Appendix B Explicit form of the interpolating metric

In this appendix we provide the near horizon and the asymptotic expansion of
the metric that interpolates between Brown Henneaux boundary conditions at
infinity and Heisenberg boundary conditions at $r=0$. We display our results
in Lorentzian signature and choose our connection such that
$\frac{2}{k}J_{0}^{\pm}=\sqrt{2\mathcal{L}_{\pm}}$ and
$\zeta_{\pm}=\mu_{\pm}\sqrt{2\mathcal{L}_{\pm}}=-a$, which is explicitly
realized by

$\displaystyle\eta_{\infty}^{+}$
$\displaystyle=-\frac{e^{-\psi_{\infty}^{+}}}{\sqrt{2\mathcal{L}_{+}}}\,,$
$\displaystyle\psi_{\infty}^{+}$
$\displaystyle=\sqrt{2\mathcal{L}_{+}}(\varphi+\mu_{+}t)\,,$
$\displaystyle\lambda_{\infty}^{+}$
$\displaystyle=-\frac{\sqrt{\mathcal{L}_{+}}}{\sqrt{2}}\,,$ (71a)
$\displaystyle\eta_{0}^{+}$ $\displaystyle=\frac{1}{2}e^{-\psi_{0}^{+}}\,,$
$\displaystyle\psi_{0}^{+}$
$\displaystyle=-at+\int_{0}^{\varphi}(\gamma(\varphi^{\prime})+\omega(\varphi^{\prime}))\operatorname{d}\\!\varphi^{\prime}\,,$
$\displaystyle\lambda_{0}^{+}$ $\displaystyle=1\,,$ (71b)
$\displaystyle\eta_{\infty}^{-}$
$\displaystyle=-\frac{e^{-\psi_{\infty}^{-}}}{\sqrt{2\mathcal{L}_{-}}}\,,$
$\displaystyle\psi_{\infty}^{-}$
$\displaystyle=\sqrt{2\mathcal{L}_{-}}(\varphi-\mu_{-}t)\,,$
$\displaystyle\lambda_{\infty}^{-}$
$\displaystyle=-\frac{\sqrt{\mathcal{L_{-}}}}{\sqrt{2}}\,,$ (71c)
$\displaystyle\eta_{0}^{-}$ $\displaystyle=\frac{1}{2}e^{-\psi_{0}^{-}}\,,$
$\displaystyle\psi_{0}^{-}$
$\displaystyle=at+\int_{0}^{\varphi}(\gamma(\varphi^{\prime})-\omega(\varphi^{\prime}))\operatorname{d}\\!\varphi^{\prime}\,,$
$\displaystyle\lambda_{0}^{-}$ $\displaystyle=1\,.$ (71d)

The interpolating function $f(r)=\sinh^{4}(r)$ together with (28) after Taylor
expanding around $r=0$ yields

$\displaystyle g_{tt}$
$\displaystyle=-a^{2}r^{2}-\frac{a^{2}r^{4}}{3}+O(r^{5})$ (72a) $\displaystyle
g_{\varphi\varphi}$
$\displaystyle=\gamma^{2}(\varphi)+r^{2}\left(\gamma^{2}(\varphi)-\omega^{2}(\varphi)\right)+r^{4}\biggl{(}\gamma(\varphi)\left(\sqrt{2\mathcal{L}_{-}}+\sqrt{2\mathcal{L}_{+}}\right)-5\frac{\gamma^{2}(\varphi)}{3}-\frac{\omega^{2}(\varphi)}{3}\biggr{)}$
$\displaystyle\quad+O(r^{5})$ (72b) $\displaystyle g_{rr}$
$\displaystyle=1+r^{3}\left(8\left(-\frac{e^{\tilde{\mathcal{J}}_{-}(\varphi)}}{\sqrt{2\mathcal{L}_{-}}}-\frac{e^{\tilde{\mathcal{J}}_{+}(\varphi)}}{\sqrt{2\mathcal{L}_{+}}}\right)-4\tilde{\mathcal{J}}_{-}(\varphi)-4\tilde{\mathcal{J}}_{+}(\varphi)-8\right)+O\left(r^{5}\right)$
(72c) $\displaystyle g_{\varphi r}$
$\displaystyle=-r^{3}\gamma(\varphi)\left(2\tilde{\mathcal{J}}_{-}(\varphi)+2\tilde{\mathcal{J}}_{+}(\varphi)+\sqrt{2\mathcal{L}_{-}}+\sqrt{2\mathcal{L}_{+}}+4\right)+O\left(r^{4}\right)$
(72d) $\displaystyle g_{tr}$
$\displaystyle=\frac{3}{4}ar^{4}\left(\left(-\frac{4e^{\tilde{\mathcal{J}}_{-}(\varphi)}}{\sqrt{2\mathcal{L}_{-}}}+\frac{4e^{\tilde{\mathcal{J}}_{+}(\varphi)}}{\sqrt{2\mathcal{L}_{+}}}-\sqrt{2\mathcal{L}_{-}}+\sqrt{2\mathcal{L}_{+}}\right)-2\tilde{\mathcal{J}}_{-}(\varphi)+2\tilde{\mathcal{J}}_{+}(\varphi)\right)+O\left(r^{5}\right)$
(72e) $\displaystyle g_{t\varphi}$
$\displaystyle=ar^{2}\omega(\varphi)+\frac{1}{3}ar^{4}\omega(\varphi)+O\left(r^{5}\right)\,,$
(72f)

where
$\tilde{\mathcal{J}}_{\pm}(\varphi)=-\sqrt{2\mathcal{L}_{\pm}}+\int_{0}^{\varphi}\mathcal{J}_{\pm}(\varphi^{\prime})\operatorname{d}\\!\varphi^{\prime}=\int_{0}^{\varphi}\frac{2}{k}\sum_{n\neq
0}J_{n}^{\pm}e^{-in\varphi^{\prime}}\operatorname{d}\\!\varphi^{\prime}$. We
see that $r=0$ is a Killing horizon of the vector $\xi=\partial_{t}$.
Expanding around $r\to\infty$ yields

$\displaystyle g_{tt}$
$\displaystyle=-\mu_{-}\mu_{+}e^{2r}+\frac{1}{2}\left(\mu_{-}^{2}\mathcal{L}_{-}+\mu_{+}^{2}\mathcal{L}_{+}\right)+\frac{1}{4}\mu_{-}\mu_{+}e^{-2r}\biggl{(}-64\tilde{\mathcal{J}}_{-}(\varphi)-64\tilde{\mathcal{J}}_{+}(\varphi)-\mathcal{L}_{-}\mathcal{L}_{+}$
$\displaystyle\quad+32\sqrt{2\mathcal{L}_{-}}e^{-\tilde{\mathcal{J}}_{-}(\varphi)}+32\sqrt{2\mathcal{L}_{+}}e^{-\tilde{\mathcal{J}}_{+}(\varphi)}+128\biggr{)}+O\left(e^{-4r}\right)$
(73a) $\displaystyle g_{\varphi\varphi}$
$\displaystyle=e^{2r}+\frac{1}{2}\left(\mathcal{L}_{-}+\mathcal{L}_{+}\right)+\frac{1}{4}e^{-2r}\biggl{(}32e^{-\tilde{\mathcal{J}}_{-}(\varphi)}(\omega(\varphi)-\gamma(\varphi))-32e^{-\tilde{\mathcal{J}}_{+}(\varphi)}(\gamma(\varphi)+\omega(\varphi))$
$\displaystyle\quad+64\tilde{\mathcal{J}}_{-}(\varphi)+64\tilde{\mathcal{J}}_{+}(\varphi)+\mathcal{L}_{-}\mathcal{L}_{+}-128\biggr{)}+O\left(e^{-4r}\right)$
(73b) $\displaystyle g_{rr}$ $\displaystyle=1+O\left(e^{-4r}\right)$ (73c)
$\displaystyle g_{\varphi r}$
$\displaystyle=16e^{-2r}\biggl{(}\sinh(\tilde{\mathcal{J}}_{-}(\varphi))-\cosh(\tilde{\mathcal{J}}_{-}(\varphi))+\sinh(\tilde{\mathcal{J}}_{+}(\varphi))-\cosh(\tilde{\mathcal{J}}_{+}(\varphi))$
$\displaystyle\quad+\sqrt{2}\left(-\frac{1}{\sqrt{\mathcal{L}_{-}}}-\frac{1}{\sqrt{\mathcal{L}_{+}}}\right)\biggr{)}+O\left(e^{-4r}\right)$
(73d) $\displaystyle g_{tr}$
$\displaystyle=16e^{-2r}\left(-\mu_{+}e^{-\tilde{\mathcal{J}}_{-}(\varphi)}+\mu_{-}e^{-\tilde{\mathcal{J}}_{+}(\varphi)}+\sqrt{2}\left(\frac{\mu_{-}}{\sqrt{\mathcal{L}_{+}}}-\frac{\mu_{+}}{\sqrt{\mathcal{L}_{-}}}\right)\right)+O\left(e^{-4r}\right)$
(73e) $\displaystyle g_{t\varphi}$
$\displaystyle=\frac{\mu_{+}-\mu_{-}}{2}e^{2r}+\frac{1}{2}\left(\mu_{+}\mathcal{L}_{+}-\mu_{-}\mathcal{L}_{-}\right)+e^{-2r}\biggl{(}4\mu_{+}e^{-\tilde{\mathcal{J}}_{-}(\varphi)}(\omega(\varphi)-\gamma(\varphi))$
$\displaystyle\quad-8(\tilde{\mathcal{J}}_{+}(\varphi)+\tilde{\mathcal{J}}_{-}(\varphi))(\mu_{-}-\mu_{+})+4\mu_{-}\sqrt{2\mathcal{L}_{-}}e^{-\tilde{\mathcal{J}}_{-}(\varphi)}+4\mu_{-}e^{-\tilde{\mathcal{J}}_{+}(\varphi)}(\gamma(\varphi)+\omega(\varphi))$
$\displaystyle\quad-4\mu_{+}\sqrt{2\mathcal{L}_{+}}e^{-\tilde{\mathcal{J}}_{+}(\varphi)}-16(\mu_{+}-\mu_{-})+\frac{(\mu_{+}-\mu_{-})\mathcal{L}_{-}\mathcal{L}_{+}}{8}\biggr{)}+O\left(e^{-4r}\right)\,.$
(73f)

Restricting the solution to zero modes
$\mathcal{J}_{\pm}=\frac{2}{k}J_{0}^{\pm}=\gamma\pm\omega$
($\tilde{\mathcal{J}}_{\pm}(\varphi)=0$) the near horizon expansion (72)
reduces to

$\displaystyle g_{tt}$
$\displaystyle=-a^{2}r^{2}-\frac{a^{2}r^{4}}{3}+O(r^{5})$ (74a) $\displaystyle
g_{\varphi\varphi}$
$\displaystyle=\gamma^{2}+r^{2}\left(\gamma^{2}-\omega^{2}\right)+\frac{1}{3}r^{4}\left(\gamma^{2}-\omega^{2}\right)+O(r^{5})$
(74b) $\displaystyle g_{rr}$
$\displaystyle=1-r^{3}\left(\frac{8}{\gamma+\omega}-\frac{8}{\omega-\gamma}+8\right)+O\left(r^{5}\right)$
(74c) $\displaystyle g_{\varphi r}$
$\displaystyle=-2(\gamma(\gamma+2))r^{3}+O\left(r^{4}\right)$ (74d)
$\displaystyle g_{tr}$
$\displaystyle=\frac{3ar^{4}\left(\gamma^{2}\omega-\omega^{3}-4\omega\right)}{2(\gamma-\omega)(\gamma+\omega)}+O\left(r^{5}\right)$
(74e) $\displaystyle g_{t\varphi}$
$\displaystyle=ar^{2}\omega+\frac{1}{3}ar^{4}\omega+O\left(r^{5}\right)\,.$
(74f)

and the asymptotic expansion (73) reduces to

$\displaystyle g_{tt}$
$\displaystyle=-\mu_{-}\mu_{+}e^{2r}+\frac{1}{2}\left(\mu_{-}^{2}\mathcal{L}_{-}+\mu_{+}^{2}\mathcal{L}_{+}\right)+\frac{1}{4}\mu_{-}\mu_{+}e^{-2r}\biggl{(}-\mathcal{L}_{-}\mathcal{L}_{+}+32\sqrt{2\mathcal{L}_{-}}$
$\displaystyle\quad+32\left(\sqrt{2\mathcal{L}_{+}}+4\right)\biggr{)}+O\left(e^{-4r}\right)$
(75a) $\displaystyle g_{\varphi\varphi}$
$\displaystyle=e^{2r}+\frac{1}{2}\left(\mathcal{L}_{-}+\mathcal{L}_{+}\right)+\frac{1}{4}e^{-2r}\left(\mathcal{L}_{-}\mathcal{L}_{+}-32\sqrt{2}(\sqrt{\mathcal{L}_{-}}+\sqrt{\mathcal{L}_{+}})-128\right)+O\left(e^{-4r}\right)$
(75b) $\displaystyle g_{rr}$ $\displaystyle=1+O\left(e^{-4r}\right)$ (75c)
$\displaystyle g_{\varphi r}$
$\displaystyle=e^{-2r}\left(-\frac{16\sqrt{2}}{\sqrt{\mathcal{L}_{-}}}-\frac{16\sqrt{2}}{\sqrt{\mathcal{L}_{+}}}-32\right)+O\left(e^{-4r}\right)$
(75d) $\displaystyle g_{tr}$
$\displaystyle=16e^{-2r}\left(\mu_{-}-\mu_{+}\left(\frac{\sqrt{2}}{\sqrt{\mathcal{L}_{-}}}+1\right)+\frac{\sqrt{2}\mu_{-}}{\sqrt{\mathcal{L}_{+}}}\right)+O\left(e^{-4r}\right)$
(75e) $\displaystyle g_{t\varphi}$
$\displaystyle=\frac{\mu_{+}-\mu_{-}}{2}e^{2r}+\frac{1}{2}\left(\mu_{+}\mathcal{L}_{+}-\mu_{-}\mathcal{L}_{-}\right)+\frac{1}{8}e^{-2r}(\mu_{-}-\mu_{+})\biggl{(}-\mathcal{L}_{-}\mathcal{L}_{+}+32\sqrt{2\mathcal{L}_{-}}$
$\displaystyle\quad+32\left(\sqrt{2\mathcal{L}_{+}}+4\right)\biggr{)}+O\left(e^{-4r}\right)\,.$
(75f)

From this last set of equations it is clear that we have a usual
Fefferman–Graham expansion at large radii, namely a fixed metric to order
${\cal O}(e^{2r})$ and state-dependent contributions to ${\cal O}(1)$.
However, to subleading order ${\cal O}(e^{-2r})$ we deviate from the standard
form of asymptotically AdS3 solutions due to the various square-root terms,
which are an echo of the near horizon structure that asymptotically is
perceived as pure gauge.

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