Near horizon dynamics of three dimensional black holes

Daniel Grumiller1*, Wout Merbis2

1 Institute for Theoretical Physics, TU Wien, Wiedner Hauptstr. 8-10/136,
A-1040 Vienna, Austria

2 Université Libre de Bruxelles & International Solvay Institutes, Physique
Théorique et Mathématique, Campus Plaine - CP 231, B-1050 Bruxelles, Belgium

*<EMAIL_ADDRESS>
## Abstract

We perform the Hamiltonian reduction of three dimensional Einstein gravity
with negative cosmological constant under constraints imposed by near horizon
boundary conditions. The theory reduces to a Floreanini–Jackiw type scalar
field theory on the horizon, where the scalar zero modes capture the global
black hole charges. The near horizon Hamiltonian is a total derivative term,
which explains the softness of all oscillator modes of the scalar field. We
find also a (Korteweg–de Vries) hierarchy of modified boundary conditions that
we use to lift the degeneracy of the soft hair excitations on the horizon.

###### Contents

  1. 1 Introduction
  2. 2 Near horizon boundary conditions
    1. 2.1 AdS3 gravity in the bulk
    2. 2.2 Boundary conditions
    3. 2.3 Near horizon boundary conditions
  3. 3 Reduction of the Chern–Simons action
    1. 3.1 General aspects of the Hamiltonian reduction
    2. 3.2 Near horizon action
  4. 4 Boundary Hamiltonians
    1. 4.1 Near horizon Hamiltonian
    2. 4.2 Chiral bosons, Liouville theory and Schwarzian action
    3. 4.3 KdV action and symmetries
  5. 5 KdV scaling limit for the near horizon Hamiltonian
  6. 6 Discussion
  7. A Brown–Henneaux boundary conditions and holonomies
  8. B Gelfand–Dikii differential polynomials and Hamiltonians

##  1 Introduction

Gravity in the presence of asymptotic boundaries has led to numerous surprises
and insights, starting from Bondi, van der Burgh, Metzner and Sachs’ seminal
work in the 1960ies which proved that the asymptotically flat limit of general
relativity does not reduce to special relativity but has an infinite symmetry
enhancement due to supertranslations [1, 2]. Brown and Henneaux’s discovery in
the 1980ies showed that the canonical realization of the asymptotic symmetries
in AdS3 Einstein gravity leads to a classical central extension [3], thereby
providing a precursor of the AdS3/CFT2 correspondence. Since then, their
asymptotic analysis has been generalized e.g. to higher spins [4, 5], lower
spins [6] and within Einstein gravity [7, 8, 9, 10].

A key outcome of all these considerations is the asymptotic symmetry algebra
generated by the boundary charges, since the physical phase space falls into
representations of this algebra and different physical states are labelled by
the values of these charges, see e.g. [11, 12] and references therein.
Depending on the precise form of the asymptotic symmetry algebra, it is
possible to generate descendants of a state, so-called “edge-state
excitations”, by acting on it with raising operators. (The presence of edge
states is also familiar from Quantum Hall physics [13]; in a gravity context
they are often referred to as “boundary gravitons” since the raising operators
have a gravitational interpretation as diffeomorphisms that are not pure gauge
at the boundary.)

Boundary charges also exist in the presence of finite boundaries, which can
arise in two ways: either there is an actual boundary present in the physical
system or one introduces a boundary by cutting out some part of spacetime, see
for instance [14, 15]. The prototypical example of the latter is to cut out
the black hole interior and to replace the black hole by some membrane [16,
17], corresponding to suitable boundary conditions imposed on a (stretched)
horizon [18]. Carlip pioneered the suggestion that such an approach might
account for the black hole entropy [19]. The idea to use the black hole
horizon as boundary thus has a long pre-history, but concrete proposals for
precise boundary conditions and symmetry analyses are relatively recent [20,
21, 22, 23, 24, 25].

In the present work we focus on consequences of the near horizon boundary
conditions proposed in [22] for the boundary theory. More specifically, we
perform, discuss, extend and apply the Hamiltonian reduction of the action
under constraints imposed by these near horizon boundary conditions,
analogously to the asymptotic analysis of Coussaert, Henneaux and van Driel
[26].

Here is a summary of our main results. The near horizon boundary action for
each chiral sector,

$S_{\textrm{\tiny
NH}}[\Phi_{n},\,\Pi_{n}]=\int\mathrm{d}t\,\bigg{(}\sum_{n\geq
0}\Pi_{n}\dot{\Phi}_{n}-H_{\textrm{\tiny NH}}\bigg{)}$ (1)

depends on a scalar field
$\Phi(t,\,\sigma)=\sum_{n\in\mathbb{Z}}\Phi_{n}(t)e^{in\sigma}+\sigma
J_{0}(t)$, the momentum of which essentially is given by its spatial
derivative, $\Pi_{n}(t)=-\frac{k}{2}\,J_{-n}(t)$ with $J_{n}(t)=in\Phi_{n}(t)$
for $n\neq 0$, like in the Floreanini–Jackiw action for self-dual scalars
[27]. Remarkably, the near horizon Hamiltonian depends only on the momentum
zero mode $J_{0}$

$H_{\textrm{\tiny NH}}=\frac{k}{2}\,\zeta J_{0}\,.$ (2)

The equations of motion are solved by

$\Phi(t,\,\sigma)=\Phi_{0}(t)+J_{0}\,\sigma+\sum_{n\neq
0}\frac{J_{n}}{in}\,e^{in\sigma}$ (3)

where $J_{0}$ is related to the BTZ horizon radii and $\Phi_{0}(t)=-\zeta
t+\phi$ with constant $\zeta$ and $\phi$. The mode decomposition (3) is
reminiscent of the ultrarelativistic/tensionless limit of string theory [28].

The near horizon Hamiltonian density being a total derivative term implies the
softness of all near horizon oscillator modes $J_{n}$, which provides yet
another way to see that $J_{n}$ generate soft hair excitations of black holes
[22] in the sense of Hawking, Perry and Strominger [29].

If one desires to attribute black hole entropy to soft hair degeneracy [30,
25] then one of the problems is the infinite degeneracy of soft hair
excitations. The naive physical intuition behind a possible resolution is that
infinite blueshift factors at the horizon multiplying zero energy excitations
could yield a finite result. Producing such a cutoff in a controlled way is
essential for applications to black hole entropy, like in the fluff proposal
[31, 32]. In the present work we find a novel way to generate such a cutoff,
by considering our near horizon boundary conditions as limiting case of an
analytically continued 1-parameter family of boundary conditions. The zeroth,
first and second member of this family yields, respectively, near horizon,
Brown–Henneaux and Korteweg–de-Vries (KdV) boundary conditions. The associated
boundary Hamiltonian densities for any positive integer $N$ generalize the
result (2)

${\cal H}_{N}=\frac{k}{4\pi}\,\zeta_{N}{\cal
J}^{N+1}+\sum_{i=1}^{N-1}h_{i,\,N}{\cal
J}^{N-i-1}\big{(}\partial_{\sigma}^{i}{\cal J}\big{)}^{2}+{\cal
H}_{N}^{\textrm{\tiny nl}}\qquad\qquad{\cal J}=\Phi^{\prime}$ (4)

which no longer is a total derivative term ($h_{i,\,N}$ are field-independent
coefficients and ${\cal H}_{N}^{\textrm{\tiny nl}}$ vanishes for $N\leq 4$).
Setting $N=0$ in (4) recovers (2). If instead we take the limit $N=\epsilon\to
0^{+}$ we get a Hamiltonian with log contribution

${\cal H}_{\textrm{\tiny
log}}=\frac{k}{4\pi}\,\zeta_{\epsilon}\Phi^{\prime}\ln\big{(}\Phi^{\prime}\big{)}$
(5)

that provides a cutoff on the soft hair spectrum.

This work is organized as follows. In Section 2 we review our near horizon
boundary conditions and their relation to Brown–Henneaux boundary conditions.
In section 3 we perform the reduction of the Chern–Simons action to a boundary
action, up to one boundary term. In section 4 we fix this boundary term
explicitly in different ways, corresponding to near horizon, Brown–Henneaux
and KdV boundary conditions as well as generalizations thereof. In section 5
we consider an analytic continuation of this KdV hierarchy and focus on the
limit when continuously approaching near horizon boundary conditions, in order
to make the soft hair excitations slightly non-soft. In section 6 we compare
our results with those of the fluff proposal and tensionless strings.

#### Note added:

After finishing our work we received a preview [33] that also considers a
hierarchy of integrable deformations of the near horizon boundary conditions
[22]. The Gardner hierarchy employed in that work contains the KdV hierarchy
studied in our section 4.3 as special case (for $b=0$ in their notation). As
their work does not address the boundary actions discussed in the present
paper our respective works are complementary.

##  2 Near horizon boundary conditions

In this section we review the near horizon boundary conditions for AdS3
Einstein gravity [22]. We also list our conventions here.

###  2.1 AdS3 gravity in the bulk

Three dimensional Einstein gravity is a topological gauge theory which can be
described as a Chern–Simons theory for the appropriate gauge group [34,
35]111The relation between the Chern–Simons and metric formulations of
Einstein gravity is not fully understood at the quantum level.. The gauge
group reflects the isometries of the maximally symmetric vacuum of the theory,
which in turn depends on the sign of the cosmological constant $\Lambda$. In
the present work we focus on negative cosmological constant,
$\Lambda=-1/\ell^{2}$, mainly due to the fact that the presence of BTZ black
hole solutions [36, 37] requires negative $\Lambda$.

AdS3 gravity is described by SO$(2,2)\cong$ SL$(2,\hbox{\mybb R})\times$
SL$(2,\hbox{\mybb R})$ Chern–Simons theory. The Einstein–Hilbert action
(neglecting boundary terms) is given as

$S_{\textsc{eh}}[A,\bar{A}]=S_{\textsc{cs}}[A]-S_{\textsc{cs}}[\bar{A}]$ (6)

where

$S_{\textsc{cs}}[A]=\frac{k}{4\pi}\int_{{\cal
M}}\mathrm{tr}\big{(}A\wedge\mathrm{d}A+\frac{2}{3}A\wedge A\wedge A\big{)}$
(7)

and the Chern–Simons level is given by $k=\frac{\ell}{4G}$ ($G$ is Newton’s
constant). The gauge connections $A,\bar{A}$ take values in the algebra
$\mathfrak{sl}(2,\hbox{\mybb R})$ with generators $L_{-1},L_{0},L_{+1}$ and
commutators

$[L_{m},L_{n}]=(m-n)L_{m+n}\qquad\qquad m,n=-1,0,+1\,.$ (8)

The trace in the action (7) is a non-degenerate symmetric bilinear form on the
Lie algebra, chosen as

$\mathrm{tr}(L_{1}L_{-1})=-1\qquad\qquad\mathrm{tr}(L_{0}^{2})=\frac{1}{2}\qquad\qquad\mathrm{tr}(L_{\pm
1}L_{0})=0=\mathrm{tr}(L_{\pm 1}^{2})\,.$ (9)

Sometimes an explicit representation in term of $(2\times 2)$ matrices is
needed. We make a standard choice compatible with the commutation and trace
relations above.

$L_{-1}=\left(\begin{array}[]{cc}0&-1\\\ 0&0\end{array}\right)\qquad\qquad
L_{0}=\frac{1}{2}\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\qquad\qquad L_{+1}=\left(\begin{array}[]{cc}0&0\\\
1&0\end{array}\right)$ (10)

The Cartan variables, dreibein and (dualized) spin-connection, are given as
linear combinations of the Chern–Simons connections $A$, $\bar{A}$.

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

The metric is bilinear in the dreibein and thus also bilinear in the
Chern–Simons connections.

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

All solutions of three dimensional gravity are locally gauge equivalent to
each other, but differ up to boundary terms or global identifications. Hence
the specification of boundary conditions is a crucial part of the definition
of the theory under consideration. The boundary conditions will determine
which gauge transformations are proper gauge transformations, in the sense
that they keep the boundary data invariant, and which are improper gauge
transformations that turn into symmetry transformations of the boundary
theory.

###  2.2 Boundary conditions

Suppose that our manifold ${\cal M}$ is topologically an annulus times time,
with the outer boundary being a hypersurface of constant large radius
approaching the asymptotically AdS-boundary and the inner boundary of the
annulus corresponding to a stretched horizon, i.e., a hypersurface of constant
radius $r$ close to the locus of a black hole horizon. We equip the manifold
with a coordinate system $(t,\sigma,r)$, where $t$ is the time coordinate,
$\sigma\sim\sigma+2\pi$ the coordinate around the annulus and $r$ some radial
coordinate so that the asymptotic boundary is at $r\to\infty$ and the inner
boundary $\partial{\cal M}$ at $r\to 0$.

In section 2.3 we are going to impose boundary conditions inspired by near
horizon considerations at the inner boundary. While in principle one could
independently impose a different set of boundary conditions at the asymptotic
boundary, we choose the same boundary conditions there, which guarantees that
the respective boundary conditions are (trivially) consistent with each other.
To make sure of this, it is convenient to express the Chern–Simons connection
in radial gauge

$A=b(r)^{-1}\,\big{(}\mathrm{d}+a(t,\sigma)\big{)}\,b(r)$ (13)

such that ${b(r)\in}$ SL$(2,\mathbb{R})$ depends only on the radial coordinate
and the boundary connection
$a(t,\sigma)=a_{t}\mathrm{d}t+a_{\sigma}\mathrm{d}\sigma$ only has legs in the
$(t,\sigma)$ plane. Moreover, the group element $b(r)$ is not allowed to vary,
$\delta b=0$. The choice (13) makes it manifest that the limits $r\to\infty$
and $r\to 0$ yield the same sets of boundary conditions. This is so because
all the state-dependent information is contained in the boundary connection
$a(t,\sigma)$, which is independent from the radial coordinate.

On an equal time slice, the asymptotic charges on the boundary are obtained by
functionally integrating

$\delta
Q[\varepsilon]=-\frac{k}{2\pi}\oint\mathrm{d}\sigma\;\mathrm{tr}\big{(}\varepsilon\,\delta
a_{\sigma}\big{)}$ (14)

for asymptotic symmetry transformations $\varepsilon$ satisfying $\delta
a_{\sigma}=\partial_{\sigma}\varepsilon+[a_{\sigma},\varepsilon]$ (see e.g.
[38]).

Hence $a_{\sigma}$ contains all information about the asymptotic charges, and
specifying boundary conditions means specifying the form of $a_{\sigma}$ and
its allowed fluctuations. By contrast, the time component $a_{t}$ contains the
information about the sources of the boundary theory. It can always be taken
to be proportional to an asymptotic symmetry transformation with arbitrary
parameter, which would then play the rôle of a chemical potential for the
corresponding boundary charge [39].

The interplay between boundary conditions, holonomies of the gauge connection
and SL$(2)$ conjugacy classes is reviewed in appendix A.

###  2.3 Near horizon boundary conditions

The boundary conditions of [22] are formulated as the set of diffeomorphisms
preserving the near horizon expansion of the non-extremal BTZ black hole.
Their purpose was to be able to ask conditional questions given the presence
of a black hole in the bulk, so we restrict ourselves to the BTZ black hole
subsector of solutions. From the analysis in appendix A we see that in radial
gauge this is equivalent to imposing the connections to have hyperbolic
holonomies. So we may write them as ($\mathcal{J}^{\pm}$ are real functions)

$a_{\sigma}=\mathcal{J}^{+}(t,\sigma)L_{0}\qquad\qquad\bar{a}_{\sigma}=-\mathcal{J}^{-}(t,\sigma)L_{0}\,.$
(15)

Under boundary condition preserving gauge transformations they transform as

$\delta_{\varepsilon}{\cal J}^{\pm}L_{0}=\pm\partial_{\sigma}\eta^{\pm}L_{0}$
(16)

where $\varepsilon=\eta^{+}\,L_{0}+\eta^{+}_{\pm}\,L_{\pm 1}$ and
$\bar{\varepsilon}=\eta^{-}\,L_{0}+\eta^{-}_{\pm}\,L_{\pm 1}$. The gauge
parameters $\eta^{\pm}_{\pm}$ do not appear in the transformation laws or in
the charges below, and hence correspond to proper gauge transformations. The
variation of the charges (14) is easily integrated, assuming that $\eta^{\pm}$
are state-independent

$Q[\eta^{\pm}]=-\frac{k}{4\pi}\int\mathrm{d}\sigma\,\eta^{\pm}\mathcal{J}^{\pm}\,.$
(17)

A Fourier decomposition of the charges with respect to the angular coordinate
$\sigma$ yields an asymptotic symmetry algebra consisting of two affine
$\hat{\mathfrak{u}}(1)$ current algebras222We write $i$ times Poisson brackets
so that the right hand sides do not change when passing to commutators.

$\displaystyle
i\\{J^{\pm}_{n},\,J^{\pm}_{m}\\}=\pm\frac{2}{k}\,n\,\delta_{n+m,\,0}\qquad\qquad\\{J^{\pm}_{n},\,J^{\mp}_{m}\\}=0\,.$
(18)

Here
$J^{\pm}_{n}=\tfrac{1}{2\pi}\,\oint\mathrm{d}\sigma\mathcal{J}^{\pm}e^{in\sigma}$
are the Fourier components of $\mathcal{J}^{\pm}$. (Note the conventional
factor $\tfrac{2}{k}$ relative to [22].) This algebra can be rewritten as a
Heisenberg algebra with infinitely many canonical generators $X_{n},P_{n}$ and
two Casimirs ($X_{0}$ and $P_{0}$) at its center. Hence these near horizon
boundary conditions are also referred to as “Heisenberg boundary conditions”.

To see that (15) indeed corresponds to black hole solutions we should
construct the metrics associated with these boundary conditions. First we
write $a_{t}$ and $\bar{a}_{t}$ proportional to a gauge transformation with
arbitrary, state-independent parameters $\zeta^{\pm}$.

$a_{t}=-\zeta^{+}L_{0}\qquad\qquad\bar{a}_{t}=-\zeta^{-}L_{0}$ (19)

Then we should find a suitable group element $b$ to construct the metric (12)
from (13). The choice of [40] is
$b=\exp\big{(}\frac{r}{2\ell}(L_{+}-L_{-})\big{)}$ and $\bar{b}=b^{-1}$. Other
choices for $b(r)$ are suitable as long as they lead to a non-degenerate
metric which contains as much information as the gauge connections
$a,\bar{a}$. This particular choice leads to a metric which, expanded near
$r=0$ (and assuming a co-rotating frame, $\zeta^{+}=\zeta^{-}$) gives Rindler
spacetime,

$\mathrm{d}s^{2}=-\kappa^{2}r^{2}\,\mathrm{d}t^{2}+\mathrm{d}r^{2}+\tfrac{\ell^{2}}{4}\,({\cal
J}^{+}+{\cal J}^{-})^{2}\,\mathrm{d}\varphi^{2}+a\,\big{(}{\cal J}^{+}-{\cal
J}^{-}\big{)}\,r^{2}\,\mathrm{d}t\mathrm{d}\varphi+\dots$ (20)

with Rindler acceleration $\kappa=\zeta^{+}=\zeta^{-}$. For simplicity
henceforth we assume $\zeta^{\pm}$ to be constant, implying on-shell time-
independence of ${\cal J}^{\pm}$ (dot means $\partial_{t}$),

$\dot{\mathcal{J}}^{\pm}=0\,.$ (21)

Finally, writing the functions ${\cal J}^{\pm}(\sigma)$ as

$\mathcal{J}^{\pm}(\sigma)=\frac{\gamma(\sigma)}{\ell}\pm\omega(\sigma)$ (22)

the full metric constructed from this configuration [using (12)] for constant
$\mathcal{J}^{\pm}$ becomes the BTZ metric with inner and outer horizons
$\gamma$ and $\ell|\omega|$,

$r_{+}=\gamma\qquad\qquad r_{-}=\ell|\omega|\,.$ (23)

There are a number of crucial differences as compared to Brown–Henneaux (or
other) boundary conditions:

  * •

Soft Heisenberg hair. The zero modes $J_{0}^{\pm}$ commute with all generators
$J_{n}^{\pm}$. Thus, non-trivial descendants of some state generated by acting
on it with products of $J_{n}^{\pm}$ (with negative $n$) have the same
$J_{0}^{\pm}$ eigenvalues as the original state. Since the near horizon
Hamiltonian is given by the sum of these zero modes [22] this means that all
such descendants are soft in the sense that they do not change the energy
eigenvalue, concurrent with the proposal of Hawking, Perry and Strominger. By
contrast, for Brown–Henneaux boundary conditions Virasoro descendants
generated by acting with $L_{n}^{\pm}$ (with negative $n$) on a given state
will raise the $L_{0}^{\pm}$ eigenvalues of such descendants as compared to
the original state, and hence descendants have a higher energy (as measured by
the Hamiltonian $L_{0}^{+}+L_{0}^{-}$) than the state from which they
originate.

  * •

Fixed temperature. Since Rindler acceleration $a=2\pi T$ and horizon
temperature $T$ are fixed we are automatically in the canonical ensemble.
Their state-independence implies that all states in our theory have the same
temperature. By contrast, for Brown–Henneaux boundary conditions different BTZ
black holes generally have different temperatures.

  * •

Regularity of excitations. All soft hair excitations (with real ${\cal
J}^{\pm}$) of black holes are compatible with regularity conditions, in
particular with the absence of conical defects. Such excited black holes are
sometimes referred to as “black flowers”. By contrast, for Brown–Henneaux
boundary conditions only extremal black holes can carry (Virasoro) excitations
without generating singularities [41], and generally only one BTZ black hole
is free from conical defects for given temperature and angular rotation.

  * •

Reducibility parameter. Gauge transformations that vanish on-shell are called
reducibility parameters (see e.g. [42]). In a gravity context typically they
do not exist outside of mini-superspace models, since this would amount to
vector fields that are Killing for all geometries compatible with a given set
of boundary conditions. For our boundary conditions, however, $\partial_{t}$
is a Killing vector for all geometries, including softly excited ones. Thus,
we do have a non-trivial reducibility parameter. By contrast, for
Brown–Henneaux boundary conditions there is no vector field that is Killing
for all Bañados geometries (118) and hence no reducibility parameter.

  * •

Abelianization. Up to the central extension, the near horizon symmetries (18)
are abelian, which at a technical level is a direct consequence of the
connection (15) residing exclusively in the Cartan subalgebra. By contrast,
Brown–Henneaux boundary conditions lead to non-abelian asymptotic symmetries
(regardless of central extensions).

Before deriving the boundary action we comment on our terminology of ‘near
horizon boundary conditions’. Due to our separation (13) of radial and
boundary coordinates the charges (14) are independent from the radius and thus
can be envisoned to be localized at any $r={\rm const.}$ hypersurface,
including one near the horizon or, alternatively, one near the asymptotic AdS
boundary. Thus, one could also refer to our boundary conditions as ‘asymptotic
boundary conditions’, which is more in line with tradition. However, it is
fair to say that from an asymptotic observer’s perspective our boundary
conditions are somewhat bizarre; as shown in [22, 40] the usual
Fefferman–Graham expansion reveals that the sources depend in a very specific,
but complicated, way on the charges, with no clear asymptotic interpretation,
other than that this leads to a rather simple set of asymptotic symmetries,
given by (18). However, as the list of properties above shows from a near
horizon observer’s perspective the physical meaning of our boundary conditions
is very clear and natural: these boundary conditions guarantee that surface
gravity is state-independent and that all state-dependent excitations in our
theory are compatible with regularity at the horizon. This, together with the
soft hair property, seems like a good reason to associate our boundary
conditions with a near horizon observer rather than an asymptotic one.

##  3 Reduction of the Chern–Simons action

In this section we reduce the Chern–Simons action to a two dimensional field
theory by first reducing AdS3 Einstein gravity to a sum of two chiral
Wess–Zumino–Witten (WZW) models and then imposing the boundary conditions as
constraints on the WZW currents. We shall also discuss the presence of non-
trivial holonomies in the bulk. We consider here the reduction of Chern–Simons
theory to a single boundary of the annulus.

###  3.1 General aspects of the Hamiltonian reduction

Following [43] we rewrite the two Chern–Simons actions in (6) as two WZW
actions for SL$(2,\hbox{\mybb R})$. At this stage, the reduction is very
similar to the one with asymptotically AdS boundary conditions first observed
in [26] (see also [44] for a review). It is most easily obtained after a
Hamiltonian decomposition of the action (6). We focus in the rest of this work
on one chiral sector ($S_{\textsc{cs}}[A]$) and drop $\pm$-superscripts, as
the barred sector is analogous.

The Hamiltonian form of the action (7), supplemented by a boundary term
$I_{\rm bdy}$,

$S_{\textsc{cs}}[A]=\frac{k}{4\pi}\int_{{\cal
M}}\mathrm{d}t\mathrm{d}r\mathrm{d}\sigma\,\mathrm{tr}\big{(}A_{r}\dot{A}_{\sigma}-A_{\sigma}\dot{A}_{r}+2A_{t}F_{\sigma
r}\big{)}+I_{\rm bdy}$ (24)

is our starting point for the reduction. The boundary term $I_{\rm bdy}$ will
be fixed such that the variational principle is well-defined, i.e., the first
variation of the action

$\delta S_{\textsc{cs}}[A]\big{|}_{\textrm{\tiny EOM}}=\delta I_{\rm
bdy}-\frac{k}{2\pi}\int_{\partial{\cal
M}}\mathrm{d}t\mathrm{d}\sigma\,\mathrm{tr}\big{(}A_{t}\delta
A_{\sigma}\big{)}=\delta I_{\rm bdy}-\frac{k}{2\pi}\int_{\partial{\cal
M}}\mathrm{d}t\mathrm{d}\sigma\,\mathrm{tr}\big{(}a_{t}\delta
a_{\sigma}\big{)}$ (25)

vanishes on-shell. In the second equality we used the decomposition (13),
assuming again state-independence of the group element $b$, i.e., $\delta
b=0$. We are going to fix $I_{\rm bdy}$ in the next section and focus on the
symplectic terms in (24) in the remainder of this section.

The constraint $F_{\sigma r}=0$ is solved locally by

$A_{i}=G^{-1}\partial_{i}G\qquad\qquad i=\sigma,r\qquad\qquad
G\in\textrm{SL}(2,\hbox{\mybb R})\,.$ (26)

Globally, there may (and will) be holonomies in the gauge connection. There
are two ways to treat them. One may write the gauge connection as sum of a
periodic group element $g$ plus a term representing the holonomy.
Alternatively, the holonomies can be encoded in the periodicity properties of
the group element $g$. We follow the latter approach and write

$G(t,r,\,\sigma+2\pi)=h(t)G(t,r,\,\sigma)$ (27)

where $h\in$ SL$(2,\mathbb{R})$, with $\mathrm{tr}(h)=H_{\sigma}(a_{\sigma})$
at the boundary, using the holonomy definition (111). We assume in this work
that $h$ depends only on time.

With the assumptions above the action (24) decomposes into two boundary
actions; one at the $r$-boundary and a contribution at the $\sigma$-boundary

$\displaystyle S_{\textsc{cs}}[A]=$ $\displaystyle\;\frac{k}{4\pi}\int_{{\cal
M}}d^{3}x\;\mathrm{tr}\left(\partial_{\sigma}(G^{-1}\partial_{r}GG^{-1}\partial_{t}G)-\partial_{r}(G^{-1}\partial_{\sigma}GG^{-1}\partial_{t}G)\right)$
(28) $\displaystyle-I_{\rm WZ}[G]+I_{\rm bdy}\,,$

with the Wess–Zumino (WZ) term

$I_{\rm WZ}[G]=\frac{k}{12\pi}\int_{{\cal
M}}\mathrm{tr}\big{(}G^{-1}\mathrm{d}G\big{)}^{3}\,.$ (29)

The Wess-Zumino term evaluates to a total derivative as well and hence the
action (28) solely consists out of boundary contributions. In the remainder of
this section we choose $G(t,r,\sigma)|_{\partial_{\cal M}}=g(t,\sigma)$. This
implies $b=1$ at the boundary, which holds particularly for the choice of $b$
two lines below (19).

###  3.2 Near horizon action

To implement the boundary conditions described in section 2.3 it is convenient
to perform a Gauss decomposition of the SL$(2,\hbox{\mybb R})$ group element
in the action (28)

$G=e^{XL_{+}}e^{\Phi L_{0}}e^{YL_{-}}\,,$ (30)

where $\Phi,X,Y$ are fields depending on all spacetime coordinates, while
their pullback to the boundary is $r$-independent since $G|_{\partial{\cal
M}}=g$. In terms of these fields, the action (28) splits into two parts

$S[A]=S[\Phi,\,X,\,Y]=I_{r-{\rm bdy}}[\Phi,\,X,\,Y]+I_{\sigma-{\rm
bdy}}[\Phi,\,X,\,Y]\,,$ (31)

with

$I_{r-{\rm bdy}}[\Phi,\,X,\,Y]=\frac{k}{4\pi}\int_{\partial{\cal
M}}\mathrm{d}t\mathrm{d}\sigma\,\Big{(}\frac{1}{2}\dot{\Phi}\Phi^{\prime}-2e^{\Phi}X^{\prime}\dot{Y}\Big{)}+I_{\rm
bdy}\,.$ (32)

The $\sigma$-boundary terms receive contributions from the first term in (28)
and from the Wess-Zumino term (29). It reads

$I_{\sigma-{\rm
bdy}}=\frac{k}{4\pi}\int\mathrm{d}^{3}x\;\partial_{\sigma}\left[\frac{1}{2}\partial_{r}\Phi\dot{\Phi}-2e^{\Phi}\partial_{r}X\dot{Y}\right]$
(33)

We can simplify the $\sigma$ boundary term by using the periodicity conditions
imposed by the holonomy. For our near horizon boundary conditions, the
holonomy can be encoded by taking

$h(t)=\exp\left(2\pi J_{0}(t)\,L_{0}\right)$ (34)

where $J_{0}$ is defined as the (suitably normalized) zero mode of ${\cal J}$.

$J_{0}(t)=\frac{1}{2\pi}\oint\mathrm{d}\sigma\,{\cal J}(t,\sigma)$ (35)

The periodicity property (27) implies periodicity conditions on the fields
$\Phi,X$ and $Y$ appearing in the Gauss decomposition

$\displaystyle\Phi(t,\sigma+2\pi)$ $\displaystyle=\Phi(t,\sigma)+2\pi
J_{0}(t)\,,$ (36a) $\displaystyle X(t,\sigma+2\pi)$ $\displaystyle=e^{-2\pi
J_{0}(t)}X(t,\sigma)\,,$ (36b) $\displaystyle Y(t,\sigma+2\pi)$
$\displaystyle=Y(t,\sigma)\,.$ (36c)

This implies that the $\sigma$ boundary term evaluates to a total
$r$-derivative

$I_{\sigma-{\rm
bdy}}=\frac{k}{4}\int\mathrm{d}r\mathrm{d}t\;\partial_{r}\Phi|_{\sigma=0}\dot{J}_{0}(t)\,,$
(37)

where $\Phi|_{\sigma=0}=\Phi(t,r,\sigma=0)$. The action thus gives a corner
contribution to the $r$-boundary action (32).

The action (32) depends only on the boundary values of the fields, so from now
on when referring to $\Phi$, $X$ and $Y$ we exclusively mean their boundary
values. The boundary conditions $a_{\sigma}={\cal J}(t,\,\sigma)L_{0}$ impose
the following conditions on these boundary fields

$\Phi^{\prime}={\cal J}\,,\qquad\qquad X^{\prime}=0\,,\qquad\qquad
Y^{\prime}+Y\Phi^{\prime}=0\,.$ (38)

These constraints remove the second term in the boundary action (32).
Including the $\sigma$-boundary term (37), the total action (31) simplifies to

$S_{\textrm{red}}[\Phi]=-\frac{k}{4\pi}\int_{\partial{\cal
M}}\mathrm{d}t\mathrm{d}\sigma\,\frac{1}{2}\dot{\Phi}\Phi^{\prime}+\frac{k}{4}\int\mathrm{d}t\,\Phi|_{\sigma=0}\dot{J}_{0}+I_{\rm
bdy}\,.$ (39)

And the field $\Phi$ satisfies a generalized periodicity condition

$\Phi(t,\,\sigma+2\pi)=\Phi(t,\,\sigma)+2\pi J_{0}(t)$ (40)

which captures the holonomy in the bulk. We thus see that the holonomies of
the Chern–Simons connection, which correspond to the horizon radii of the BTZ
black hole, appear as linear terms in $\sigma$ (or, equivalently, as momenta)
in the mode expansion of $\Phi$.

When expressed in terms of a periodic field $\tilde{\Phi}$ defined through
$\Phi(t,\sigma)=\tilde{\Phi}(t,\sigma)+\sigma J_{0}(t)$, the boundary action
reads

$S_{\rm
red}[J_{0},\tilde{\Phi}]=-\frac{k}{4\pi}\int_{\Sigma_{i}}\mathrm{d}t\mathrm{d}\sigma\;\left(\frac{1}{2}\dot{\tilde{\Phi}}(\tilde{\Phi}^{\prime}+2J_{0})\right)+I_{\rm
bdy}\,.$ (41)

up to total time derivatives. This action is equivalent to the boundary action
obtained from reducing U(1) Chern–Simons theory to the boundary [45].

##  4 Boundary Hamiltonians

In this section we fix the boundary term $I_{\rm bdy}$ in the Hamiltonian form
of the Chern–Simons action (24). The defining property of $I_{\rm bdy}$ is
that its variation cancels the boundary term obtained from the variation of
the bulk Chern–Simons theory,

$\delta I_{\rm bdy}=\frac{k}{2\pi}\int_{\partial{\cal
M}}\mathrm{d}t\mathrm{d}\sigma\,\mathrm{tr}\big{(}a_{t}\,\delta
a_{\sigma}\big{)}\,.$ (42)

For our near horizon boundary conditions in section 2.3 the only term
contributing to the variation of $a_{\sigma}$ is $\delta{\cal J}L_{0}$ and
hence only the $L_{0}$ component of $a_{t}$ will contribute to the boundary
Hamiltonian. If we write $a_{t}=-\zeta(t,\varphi)L_{0}$ the variation of the
boundary term becomes

$\delta I_{\rm bdy}=-\frac{k}{4\pi}\int_{\partial{\cal
M}}\mathrm{d}t\mathrm{d}\sigma\,\zeta\,\delta{\cal J}\,.$ (43)

The remaining part of the specification of the boundary conditions consists of
stating whether $\zeta$ is allowed to be a functional of ${\cal J}$ and if so,
which one. A minimal requirement that we impose is finiteness and
integrability of the boundary term $I_{\rm bdy}$. Finiteness of (43) is
guaranteed by our choice of radial gauge (13). Integrability imposes a
condition on $\zeta$,

$\zeta({\cal J})=\frac{\delta{\cal H}}{\delta{\cal J}}\,,$ (44)

where ${\cal H}$ is the boundary Hamiltonian density that we shall refer to as
“near horizon Hamiltonian density” when placing the boundary at or near the
horizon.

There are infinitely many different choices of $\zeta({\cal J})$ that lead to
an integrable boundary term. We discuss a few of them in this section and give
their gravitational interpretation, starting with the choice that is most
natural from a near horizon perspective.

###  4.1 Near horizon Hamiltonian

The simplest assumption is $\delta\zeta=0$, corresponding to the near horizon
boundary conditions of [22]; additionally we assume $\zeta=$ constant. The
variation of the boundary term is trivially integrated

$I_{\rm bdy}=-\int\mathrm{d}t\,H_{\textrm{\tiny
NH}}=-\int\mathrm{d}t\mathrm{d}\sigma\,{\cal H}_{\textrm{\tiny
NH}}=-\frac{k}{4\pi}\int\mathrm{d}t\mathrm{d}\sigma\,\zeta{\cal
J}=-\frac{k}{2}\int\mathrm{d}t\,\zeta J_{0}\,,$ (45)

to obtain the near horizon Hamiltonian

$H_{\textrm{\tiny NH}}=\frac{k}{2}\zeta J_{0}\,.$ (46)

As $J_{0}$ commutes with all other modes $J_{n}$ the Hamiltonian (46) assigns
equal energy to each of the $J_{n}$ descendants of a state, which is the
softness property mentioned in section 2.3.

In terms of the boundary scalar field theory (39), the near horizon action is
given by

$S_{\textrm{\tiny
NH}}[\Phi]=-\frac{k}{4\pi}\int\mathrm{d}t\mathrm{d}\sigma\,\frac{1}{2}\dot{\Phi}\Phi^{\prime}-\frac{k}{2}\int\mathrm{d}t\left(\zeta
J_{0}-\frac{1}{2}\Phi|_{\sigma=0}\dot{J}_{0}\right)\,.$ (47)

Note that the last terms do not contribute to the near horizon equations of
motion for $\Phi$, but it fixes the time derivative of $\Phi_{0}$, the zero
mode of $\Phi$.

$\dot{\Phi}^{\prime}=0\,,\qquad\dot{\Phi}_{0}=-\zeta\,.$ (48)

The solution to the first field equations (48) is
$\Phi(t,\sigma)=\Phi_{t}(t)+\Phi_{\sigma}(\sigma)$ which, together with the
periodicity condition (40), implies $\dot{J}_{0}=0$ and gives the mode
decomposition announced in (3),

$\Phi(t,\,\sigma)\big{|}_{\textrm{\tiny
EOM}}=\Phi_{0}(t)+J_{0}\,\sigma+\sum_{n\neq 0}\frac{J_{n}}{in}e^{in\sigma}\,.$
(49)

with $\Phi_{0}(t)=-\zeta t+\phi$ where $\phi$ is an arbitrary constant.

Since the near horizon Hamiltonian density (2) is a total derivative term, the
only non-trivial information in our near horizon theory is captured by the on-
shell value of the Hamiltonian and by the symplectic structure. To discuss the
latter we make off-shell a mode decomposition like (49),

$\Phi(t,\,\sigma)=\Phi_{0}(t)+J_{0}(t)\,\sigma+\sum_{n\neq
0}\Phi_{n}(t)e^{in\sigma}\,,\qquad\qquad\Phi_{n}(t):=\frac{J_{n}(t)}{in}\,,$
(50)

where we allow arbitrary time-dependence of $J_{n}$, and plug it into the near
horizon action (47), obtaining

$S_{\textrm{\tiny
NH}}[\Phi_{0},\,J_{n}]=\frac{k}{2}\,\int\mathrm{d}t\,\Big{(}-\dot{\Phi}_{0}J_{0}-\sum_{n>0}\dot{\Phi}_{n}J_{-n}-\zeta
J_{0}\Big{)}\,.$ (51)

The Hamiltonian action corresponding to (51) simplifies to333The sum extends
only over positive integers to avoid having to go through the Dirac analysis
of systems with second class constraints, i.e., the configuration variables
are defined as positive index quantities $J_{n}$ and the momenta, up to a
scale factor, by negative index quantities $J_{-n}$. This is a trivial
implementation of the more general Faddeev–Jackiw method [46].

$S_{\textrm{\tiny
NH}}[\Phi_{n},\,\Pi_{n}]=\int\mathrm{d}t\,\bigg{(}\sum_{n\geq
0}\Pi_{n}\dot{\Phi}_{n}-H_{\textrm{\tiny NH}}\bigg{)}\,,$ (52)

with the near horizon Hamiltonian (46) and the momenta

$\Pi_{n}=-\frac{k}{2}\,J_{-n}\qquad n\geq 0\,,$ (53)

which are the main results of this paper announced in the introduction
(1)-(3). Canonical Poisson brackets $\\{\Phi_{0},\,\Pi_{0}\\}=1$,
$\\{\Phi_{n},\,\Pi_{m}\\}=\delta_{n,\,m}$ then essentially recover (one chiral
half of) the near horizon symmetry algebra (18).

$i\,\\{J_{n},\,J_{m}\\}=\frac{2}{k}\,n\,\delta_{n+m,\,0}\,,\qquad\qquad
i\,\\{J_{0},\,\Phi_{0}\\}=\frac{2i}{k}\,.$ (54)

The only change as compared to the near horizon symmetry algebra (18) is the
presence of a canonically conjugate, $\Phi_{0}$, for the zero mode $J_{0}$.
The relation (53), which basically states that the momentum $\Pi$ of the
scalar field $\Phi$ is given by its spatial derivative, is a key
characteristic of self-dual scalar fields [47], so the action (52) resembles
the Floreanini–Jackiw action [27]. We shall say a bit more about this relation
in the next subsection when discussing Brown–Henneaux type of boundary
conditions from a near horizon perspective.

For later purposes it is convenient to split the scalar field

$\Phi=\Phi_{+}+\Phi_{-}\,,$ (55)

into positive and negative modes (the subscripts $\pm$ should not be confused
with the superscripts $\pm$ referring to the chiral sectors, which we mostly
suppress in this work)

$\Phi_{+}(t,\,\sigma)=\Phi_{0}(t)+\sum_{n>0}\frac{J_{n}(t)}{in}\,e^{in\sigma}\,,\qquad\qquad\Phi_{-}(t,\,\sigma)=\sigma
J_{0}(t)-\sum_{n>0}\frac{J_{-n}(t)}{in}\,e^{-in\sigma}\,,$ (56)

and define the momentum

$\Pi:=-\frac{k}{4\pi}\,\Phi_{-}^{\prime}=-\frac{k}{4\pi}\,\Big{(}J_{0}(t)+\sum_{n>0}J_{-n}(t)e^{-in\sigma}\Big{)}\,,$
(57)

in terms of which the symplectic part of the action (52) is given by
$\int\mathrm{d}t\mathrm{d}\sigma\,\Pi\,\dot{\Phi}_{+}$.

Since the Hamiltonian $H_{\textrm{\tiny NH}}$ in (46) commutes with all the
oscillators, $J_{n}$ descendants cannot raise or lower the energy of any state
in the theory. We thus recover the expected statement that $J_{n}$ descendants
are soft hair on the black hole horizon.

With possible applications to microstates and black hole entropy in mind this
soft hair degeneracy is a stumbling block, since there is no sensible way to
count the number of soft states contributing to a black hole at a given energy
$J_{0}$. Proposals to lift the degeneracy of the soft hairs were made in the
literature [31, 32]. In section 5 below we propose a new way for lifting the
soft hair degeneracy by constructing a natural hierarchy of integrable
boundary terms (44), and then taking the limit to the near horizon Hamiltonian
above. In order to achieve this, we introduce this hierarchy in the remainder
of this section, starting with a reconsideration of Brown–Henneaux boundary
conditions from a near horizon perspective.

###  4.2 Chiral bosons, Liouville theory and Schwarzian action

To set the stage for a generalization of the boundary action, we discuss the
relation of the above reduction to the more familiar reduction of the action
using Brown–Henneaux boundary conditions, leading to Liouville theory [26].

In [22] it was shown how the near horizon boundary conditions are related to
the usual Brown–Henneaux boundary conditions with chemical potentials. Using
slightly different conventions here, we map the near horizon boundary
conditions $a_{\sigma}^{\rm NH}$ given by (15) to the Brown–Henneaux ones
[$a_{\sigma}^{\rm BH}$ given in (112)] by a suitable gauge transformation $g$,

$a_{\sigma}^{\rm BH}=g^{-1}(\partial_{\sigma}+a_{\sigma}^{\rm NH})g\,.$ (58)

This map relates the charges in both formulations of the boundary conditions
as

${\cal L}=\frac{1}{4}{\cal J}^{2}+\frac{1}{2}{\cal J}^{\prime}\,.$ (59)

The chemical potentials are also related to each other, but in a state-
dependent way; i.e. the relation involves the charges ${\cal J}$

$\zeta=\mu^{\prime}-{\cal J}\mu\,.$ (60)

Assuming $\delta\mu=0$ and integrating (44) yields the Brown–Henneaux boundary
Hamiltonian

$I_{\rm bdy}=\frac{k}{4\pi}\int_{\partial{\cal
M}}\mathrm{d}t\mathrm{d}\sigma\;\mu\,\Big{(}\frac{1}{2}{\cal J}^{2}+{\cal
J}^{\prime}\Big{)}=\frac{k}{2\pi}\int_{\partial{\cal
M}}\mathrm{d}t\mathrm{d}\sigma\;\mu{\cal L}\,.$ (61)

Expressed in terms of the scalar field $\Phi$, one chiral half of the reduced
action with Brown–Henneaux boundary conditions and arbitrary chemical
potential is given by a boundary action

$S[\Phi]=\int_{\partial{\cal
M}}\mathrm{d}t\mathrm{d}\sigma\;\Big{(}\Pi\,\dot{\Phi}_{+}+\frac{\mu\,k}{8\pi}\,\big{(}(\Phi^{\prime})^{2}+2\Phi^{\prime\prime}\big{)}\Big{)}\,,$
(62)

analogous to the the near horizon action (47), but with a different
Hamiltonian density,

${\cal H}_{\textrm{\tiny
BH}}=-\frac{k\mu}{8\pi}\,\big{(}(\Phi^{\prime})^{2}+2\Phi^{\prime\prime}\big{)}\,.$
(63)

It is also possible to derive this boundary action directly from imposing the
constraints on $a_{\sigma}$ and $a_{t}$ following from the Brown–Henneaux
boundary conditions (112) and (117). The constraints on the fields $X,Y$ and
$\Phi$ appearing in the Gauss decomposition (30) for the Brown–Henneaux
boundary conditions are

$X^{\prime}=e^{-\Phi}\,,\qquad\qquad\Phi^{\prime}=-2Y\,,$ (64)

and ${\cal L}$ is related to the field $\Phi$ by the Miura transformation (or,
equivalently, a twisted Sugawara construction)

${\cal
L}=\frac{1}{4}\,\big{(}\Phi^{\prime}\big{)}^{2}+\frac{1}{2}\,\Phi^{\prime\prime}\,.$
(65)

Implementing these constraints into the action (28) gives the action (62) up
to total derivatives. The periodicity condition (40) on the field $\Phi$
implies

$X(t,\sigma+2\pi)=e^{-2\pi J_{0}(t)}\,X(t,\sigma)\,.$ (66)

Here the holonomy has been parameterized by (34), and $J_{0}$ is related to
the zero modes of ${\cal L}_{0}$ via equation (59).

Under conformal transformations generated by $\xi$ the energy-momentum flux
component ${\cal L}$ transforms with an infinitesimal Schwarzian derivative,
while $X$ transforms like a weight-0 primary, $\Phi$ like a twisted weight-0
primary,444Entanglement entropy transforms in the same way, see [48]. For
vacuum solutions to the Einstein equations $\Phi$ is essentially equivalent to
entanglement entropy and the equality (65) corresponds to saturation of the
quantum null energy condition [49]. $e^{-\Phi}$ like a weight-1 primary and
$Y$ like a twisted weight-1 primary. The corresponding transformation laws are
given by (115) and

$\displaystyle\delta_{\xi}X$ $\displaystyle=\xi\,X^{\prime}\,,$
$\displaystyle\delta_{\xi}\Phi$
$\displaystyle=\xi\,\Phi^{\prime}-\xi^{\prime}\,,$ (67)
$\displaystyle\delta_{\xi}e^{-\Phi}$ $\displaystyle=\big{(}\xi
e^{-\Phi}\big{)}^{\prime}\,,$ $\displaystyle\delta_{\xi}Y$ $\displaystyle=\xi
Y^{\prime}+\xi^{\prime}\,Y+\tfrac{1}{2}\,\xi^{\prime\prime}\,.$ (68)

The formulas above can be derived starting from the near horizon
transformation law (16), using a relation analogous to (60), namely
$\eta=\xi{\cal J}-\xi^{\prime}$, with ${\cal J}=\Phi^{\prime}$.

In the case of constant chemical potentials (and discarding holonomy
terms)555For a recent analysis including the holonomies and both boundaries of
the annulus, see [45] we may compare with known results in the literature. The
last term in the action (62) is a total derivative and the reduced action is
equal to the Floreanini–Jackiw action [27] of a chiral boson [47] with
propagation speed $\mu$. This is a key difference to near horizon boundary
conditions, where the propagation speed tends to zero.

Setting $\mu=1/\ell$ and combining the two chiral sectors reproduces the
results of the reduction under Brown–Henneaux boundary conditions performed in
[50], before the two chiral WZW models are combined into a non-chiral WZW
model [51].

$S[\Phi^{\pm}]=-\frac{k}{8\pi}\int_{\partial{\cal
M}}\mathrm{d}t\mathrm{d}\sigma\,\Big{(}\dot{\Phi}^{+}\Phi^{+\,\prime}-\dot{\Phi}^{-}\Phi^{-\,\prime}-\frac{1}{\ell}\Phi^{+\,\prime}\Phi^{+\,\prime}-\frac{1}{\ell}\Phi^{-\,\prime}\Phi^{-\,\prime}\Big{)}$
(69)

It was shown in [50] that for vanishing holonomies this action is related to
the Hamiltonian form of Liouville theory by a series of field redefinitions
(see (3.7)-(3.11) of [50] for more details). This connects our work to the
result of [26].

Another representation of the action (62) is obtained by changing variables to
$X$ using the constraint $X^{\prime}=e^{-\Phi}$. Then the kinetic term
$\dot{\Phi}\Phi^{\prime}$ is equal to the geometric action of the Virasoro
group on its coadjoint orbit [52] first derived by Alekseev and Shatashvili
[53]

$S[X]=\frac{k}{4\pi}\int\mathrm{d}t\mathrm{d}\sigma\;\bigg{(}\frac{\dot{X}^{\prime\prime}}{X^{\prime}}-\frac{3}{2}\frac{X^{\prime\prime}\dot{X}^{\prime}}{X^{\prime}{}^{2}}-\mu\\{X,\sigma\\}\bigg{)}\,,$
(70)

where the Hamiltonian term is given by the Schwarzian derivative

$\\{X,\sigma\\}=\frac{X^{\prime\prime\prime}}{X^{\prime}}-\frac{3}{2}\bigg{(}\frac{X^{\prime\prime}}{X^{\prime}}\bigg{)}^{2}\,.$
(71)

To obtain the formulation of the Alekseev–Shatashvili action with non-zero
representative on the coadjoint orbit one should make a field redefinition
$X=\exp(-J_{0}f(t,\sigma))$ such that $f(t,\,\sigma+2\pi)=f(t,\,\sigma)+2\pi$
reproduces the periodicity conditions (66) for $X$. This gives the total
action (for $\mu=1$)

$S[f;J_{0}]=\frac{k}{4\pi}\int\mathrm{d}t\mathrm{d}\sigma\;\bigg{(}\frac{\dot{f}^{\prime\prime}}{f^{\prime}}-\frac{3}{2}\frac{f^{\prime\prime}\dot{f}^{\prime}}{f^{\prime}{}^{2}}-\\{f,\sigma\\}-\frac{1}{2}J_{0}^{2}f^{\prime}(\dot{f}-f^{\prime})\bigg{)}\,.$
(72)

The orbit representative, denoted as $b_{0}$ in [53] is related to the zero
mode charges of the bulk solution as

$b_{0}=\frac{k}{8\pi}J_{0}^{2}=\frac{c}{12\pi}{\cal L}_{0}\,,$ (73)

where ${\cal L}_{0}$ is the zero mode of ${\cal L}$. From this formula it is
clear that the exceptional PSL$(2,\mathbb{R})$ invariant orbit at
$b_{0}=-\frac{c}{48\pi}$ corresponds to the global AdS3 solution with ${\cal
L}_{0}=-\frac{1}{4}$, while the BTZ black holes correspond to orbits with
$b_{0}>0$. Our near horizon boundary conditions do not include the global AdS3
ground state unless we analytically continue $J_{0}$ to imaginary values such
that $J_{0}^{2}=-1$.

The relation between this action and the boundary theory of pure AdS3 gravity
with Brown–Henneaux boundary conditions was reported in [54] and expanded upon
recently in [55]. It is interesting to note that besides the formulation of
the boundary action as the geometric action on the coadjoint orbit of the
Virasoro group, it can also be obtained as the geometric action for an affine
$\hat{u}(1)$ Kac–Moody group. For affine Lie groups the Kirillov–Kostant orbit
method gives the WZW model of the corresponding group [56]. In the case of
$\hat{u}(1)$, the symplectic term of the geometric action is the near-horizon
action (39).666To be more precise, the near-horizon action corresponds to the
term proportional to the central charge of the $\hat{u}(1)$ geometric action.
The orbit representative term can be obtained by a field redefinition to a
periodic field $\varphi$ as $\Phi(t,\sigma)=\varphi(t,\sigma)+J_{0}\,\sigma$.
As discussed in [54], suitable Hamiltonians for geometric actions are
(invariant tensor products of) Noether charges for global symmetries of the
symplectic term. In this context the near-horizon Hamiltonian (46) can be
understood as the Noether charge for the shift symmetry

$\Phi(t,\sigma)\to\Phi(t,\sigma)+\phi\,,$ (74)

where $\phi$ is an arbitrary constant shift. The Brown–Henneaux Hamiltonian
(63) is the square of this Noether charge (up to a total derivative).

###  4.3 KdV action and symmetries

We generalize now the key relation (60) between near horizon and
Brown–Henneaux chemical potentials in a specific non-linear way, while
maintaining finiteness and integrability of the boundary charges as well as
the shift symmetry (74). This will lead to novel boundary actions. The
Hamiltonian of the boundary theory is modified by choosing boundary conditions
where the chemical potentials $\zeta$ depend on the charges ${\cal J}$. The
choice (60) in the previous subsection was a special case of the general
possibility (44). In [57] (see also [58, 59]) similar arguments were used to
derive the KdV hierarchy from boundary conditions on AdS3 gravity, where the
Brown–Henneaux charges ${\cal L}$ solve the KdV equation. In this subsection
we show how the boundary action (39) reduces to the action principle leading
to the KdV equation for the near horizon charges ${\cal J}$ instead of for
${\cal L}$.

The idea is to choose the chemical potentials $\zeta$ such that the boundary
term (25) integrates to a differential polynomial of rank $N$ representing the
integral of motion of the KdV equation. A useful basis of chemical potentials
$\zeta$ are the Gelfand–Dikii differential polynomials $R_{N}({\cal J})$ [60],
as they automatically satisfy the integrability condition (44).

Starting from $R_{0}=1$, these differential polynomials are defined
recursively by

$R_{N+1}^{\prime}=\frac{N+1}{2N+1}\,{\cal D}R_{N}\,,$ (75)

for ${\cal D}=\partial_{\sigma}{\cal J}+2{\cal
J}\partial_{\sigma}+\frac{1}{2}\partial_{\sigma}^{3}$. Taking

$\zeta_{N}=R_{N}({\cal J})\,,$ (76)

and integrating (44) leads to a hierarchy of boundary Hamiltonians

$\displaystyle H_{0}$ $\displaystyle=\int\mathrm{d}\sigma\,{\cal J}\,,$ (77)
$\displaystyle H_{1}$ $\displaystyle=\int\mathrm{d}\sigma\,\frac{1}{2}\,{\cal
J}^{2}\,,$ (78) $\displaystyle H_{2}$
$\displaystyle=\int\mathrm{d}\sigma\,\Big{(}\frac{1}{3}\,{\cal
J}^{3}-\frac{1}{6}\,{\cal J}^{\prime 2}\Big{)}\,,$ (79) $\displaystyle H_{N}$
$\displaystyle=\frac{1}{N+1}\,\int\mathrm{d}\sigma\,R_{N+1}({\cal J})\,.$ (80)

The Hamiltonian $H_{0}$ is identical to the near horizon Hamiltonian (46) for
constant $\zeta$. The Hamiltonian $H_{1}$ (when multiplied by constant $\mu$)
leads to the chiral boson action (62) following from the Brown–Henneaux
boundary conditions, which are now understood as deformed boundary conditions
on the horizon. The Hamiltonian $H_{2}$ [after rescaling by $-k/(4\pi)$ and
using the definitions (55)-(57)] leads to the boundary action

$S_{\rm
KdV}[\Phi]=\int\mathrm{d}t\mathrm{d}\sigma\Bigg{(}\Pi\,\dot{\Phi}_{+}+\frac{k}{4\pi}\Big{(}\frac{1}{3}\,\big{(}\Phi^{\prime}\big{)}^{3}-\frac{1}{6}\,\big{(}\Phi^{\prime\prime}\big{)}^{2}\Big{)}\Bigg{)}\,.$
(81)

The field equations for $\Phi$ following from the KdV action (81) can be
written in terms of ${\cal J}=\Phi^{\prime}$ as

$\dot{{\cal J}}=2{\cal J}{\cal J}^{\prime}+\frac{1}{3}\,{\cal
J}^{\prime\prime\prime}\,.$ (82)

This is the KdV equation for the current ${\cal J}$. In this case the chemical
potential is $\zeta_{2}={\cal J}^{2}+\tfrac{1}{3}\,{\cal J}^{\prime\prime}$
which means the bulk field equations $F_{t\varphi}=0$ also reproduce the KdV
equation (82). Finally, the identity (80) was derived in [60], see their Eq.
(16’) [taking into account the respective normalizations of $R_{N}$] and their
appendix 1. Appendix B contains a discussion of Gelfand–Dikii differential
polynomials and associated Hamiltonian densities for general $N$.

Thus, we have constructed a (KdV) hierarchy of different boundary conditions
labelled by a non-negative integer $N$, with the case $N=0$ corresponding to
near horizon boundary conditions, $N=1$ to Brown–Henneaux boundary conditions,
$N=2$ to KdV boundary conditions and $N>2$ leading to boundary Hamiltonian
densities of the form

${\cal H}_{N}[\Phi]\sim\frac{1}{N+1}{\cal
J}^{N+1}+\sum_{i=1}^{N-1}h_{i,\,N}{\cal
J}^{N-i-1}\big{(}\partial_{\sigma}^{i}{\cal J}\big{)}^{2}+{\cal
H}_{N}^{\textrm{\tiny nl}}\,,\qquad\qquad{\cal J}=\Phi^{\prime}\,,$ (83)

with some rational coefficients $h_{i,\,N}$ and additional terms of similar
form in ${\cal H}_{N}^{\textrm{\tiny nl}}$ when $N\geq 5$, see appendix B for
details. The equations of motion

$\dot{\cal J}=N\,{\cal J}^{N-1}{\cal
J}^{\prime}+\sum_{i=1}^{N-2}\hat{h}_{i,\,N}{\cal
J}^{N-i-1}\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\sigma}^{2i+1}{\cal
J}+\frac{2(N!)^{2}}{(2N)!}\,\partial_{\sigma}^{2N-1}{\cal J}\,,$ (84)

descending from the action

$S_{N}[\Phi]=\,\int\mathrm{d}t\mathrm{d}\sigma\,\Big{(}\Pi\,\dot{\Phi}_{+}+\frac{k}{4\pi}{\cal
H}_{N}[\Phi]\Big{)}\,,$ (85)

generalize the KdV equation (82), where $\hat{h}_{i,\,N}$ are rational multi-
coefficients and $\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}$ means
that some derivatives may act to the left. The expression on the right hand
side of the equations of motion (84) is given by $\partial_{\sigma}R_{N}$, see
appendix B for explicit results up to $N=6$.

The action (85) (up to a total derivative term) again has the shift symmetry
(74). The field equations (84) for $N>1$ have an anisotropic scale invariance
with odd anisotropy coefficient

$N>1:\qquad\qquad
t\to\lambda^{2N-1}\,t\,,\qquad\qquad\sigma\to\lambda\sigma\,,\qquad\qquad\Phi\to\lambda^{-1}\Phi\,.$
(86)

The Lifshitz type scaling behavior (86) resembles the one found in [57], but
differs from it since our basic entity is the spin-1 current ${\cal
J}=\Phi^{\prime}$, whereas in [57] the basic entity was the spin-2 current
${\cal L}$. Note that (86) is an invariance of the equations of motion, but
not of the action (85), which gets multiplied by a factor $1/\lambda^{2}$ (see
[61] for a discussion of such invariances). For $N=1$ the scale invariance
becomes isotropic, and the transformation weight of ${\cal J}$ could be
arbitrary. For $N=0$ additionally the transformation weight of time becomes
arbitrary. To fix this arbitrariness, for $N=0$ and $N=1$ we demand that not
only the equations of motion are invariant, but also the action (85),
obtaining

$N=0\;\textrm{or}\;1:\qquad\qquad
t\to\lambda^{N}\,t\,,\qquad\qquad\sigma\to\lambda\sigma\,,\qquad\qquad\Phi\to\Phi\,.$
(87)

We consider finally the near horizon symmetries induced by deformed boundary
conditions within the KdV hierarchy, starting with the two known cases. For
$N=0$ the near horizon boundary conditions remain undeformed and the near
horizon symmetries are given by spin-1 currents (18). For $N=1$ the near
horizon boundary conditions are deformed to Brown–Henneaux and the near
horizon symmetries are given by spin-2 currents

$L_{n}=\frac{k}{4}\,\sum_{p}J_{n-p}J_{p}+\dots$ (88)

where the ellipsis denotes a possible twist term proportional to $nJ_{n}$. The
Sugawara relation (88) is compatible with the Miura-transformation (65) and
with the boundary Hamiltonian $H_{1}$ (78), and leads to the Poisson brackets

$\displaystyle i\\{L_{n},\,L_{m}\\}$ $\displaystyle=(n-m)\,L_{n+m}+\dots$ (89)
$\displaystyle i\\{L_{n},\,J_{m}\\}$ $\displaystyle=-m\,J_{n+m}+\dots$ (90)

where the ellipses denote possible central extensions depending on the twist
term. Since (89) is nothing but the Virasoro algebra, $L_{n}$ are spin-2
currents, with their usual brackets with the spin-1 current (90).

For $N=2$ the situation is qualitatively different from $N=0$ and $N=1$, since
one obtains infinitely many mutually commuting charges, namely all the $H_{i}$
[60]. Note that, for instance, $H_{0}$ is the zero mode of the tower of $N=0$
charges $J_{n}$, $H_{1}$ is the zero mode of the tower of $N=1$ charges
$L_{n}$, and $H_{i}$ (83) for $i\geq 2$ are zero mode charges given by the
whole hierarchy of boundary Hamiltonians. For $N>2$ the situation is identical
to $N=2$. Similar results were derived for a KdV hierarchy based on the
Brown–Henneaux boundary charges by Perez, Tempo and Troncoso in [57], where
our $\hat{\mathfrak{u}}(1)$ charges ${\cal J}$ are replaced by the Virasoro
charges ${\cal L}$. Most of their conclusions carry over to the present case,
including the statements made in this paragraph.

##  5 KdV scaling limit for the near horizon Hamiltonian

In this section we consider analytic continuation of the family of boundary
Hamiltonian densities (83) to real $N\in[0,\,1]$ with the intention of taking
the limit $N=\epsilon\to 0^{+}$, assuming large ${\cal J}$. We start by
omitting all derivative terms in ${\cal J}$. There are three reasons for this.

  1. 1.

For the boundary points of the interval of interest, $N=0,\,1$, the derivative
terms are absent, so it does make sense to assume also the analytic
continuation between these two points maintains this property.

  2. 2.

In the limit of large black holes (which is necessary for a good semi-
classical description) the quantity ${\cal J}$ parametrically is large, which
means that the first term in (83) dominates over all the remaining terms.

  3. 3.

When continuing analytically a good guiding principle is to maintain as many
symmetries as possible. The scaling symmetry (87) of the boundary action
persists if there are no derivative terms present.

In the continuous family of boundary Hamiltonians

$H_{\epsilon}=\frac{k}{4\pi}\,\frac{\zeta_{\epsilon}}{\epsilon(1+\epsilon)}\,\int\mathrm{d}\sigma\,{\cal
J}^{1+\epsilon}\,,\qquad\qquad 0\leq\epsilon\leq
1\,,\quad\zeta_{\epsilon}\in\mathbb{R}\,,$ (91)

a convenient normalization factor in front of the integral is introduced in
order to have an interesting limit $\epsilon\to 0^{+}$,

$\lim_{\epsilon\to
0^{+}}H_{\epsilon}=\frac{k}{4\pi}\,\zeta_{\epsilon}\,\int\mathrm{d}\sigma\,{\cal
J}\ln{\cal J}=:H_{\textrm{\tiny log}}\,,$ (92)

where we dropped a boundary term before taking the limit. The associated
action at finite $\epsilon$ [using again the definitions (55)-(57)]

$S_{\epsilon}[\Phi]=\,\int\mathrm{d}t\mathrm{d}\sigma\,\Big{(}\Pi\dot{\Phi}_{+}-\frac{k}{4\pi}\frac{\zeta_{\epsilon}}{\epsilon(1+\epsilon)}\,\big{(}\Phi^{\prime}\big{)}^{1+\epsilon}\Big{)}\,,$
(93)

yields the limiting action

$\lim_{\epsilon\to
0^{+}}S_{\epsilon}[\Phi]=\,\int\mathrm{d}t\mathrm{d}\sigma\,\Big{(}\Pi\dot{\Phi}_{+}-\frac{k}{4\pi}\zeta_{\epsilon}\Phi^{\prime}\ln\big{(}\Phi^{\prime}\big{)}\Big{)}=:S_{\textrm{\tiny
log}}[\Phi]\,.$ (94)

The action (93) is invariant under anisotropic scalings

$t\to\lambda^{\epsilon}\,t\,,\qquad\qquad\sigma\to\lambda\,\sigma\,,\qquad\qquad\Phi\to\Phi\,.$
(95)

For $\epsilon=1,0$ the result (87) is recovered. Also the limiting action (94)
has this invariance for constant $\zeta_{\epsilon}$, since the inhomogeneous
term coming from the logarithm is a total derivative term. Gratifyingly, in
the limit of vanishing $\epsilon$ the scale invariance above is compatible
with the one of near horizon boundary conditions [57, 40]

The field equations derived from the limiting action (94),

$\dot{\Phi}^{\prime}=-\zeta_{\epsilon}\frac{\Phi^{\prime\prime}}{\Phi^{\prime}}\,,$
(96)

can easily be integrated once,

$\dot{\Phi}=-\zeta_{\epsilon}(1+\ln\Phi^{\prime})\,,$ (97)

where the integration constant is fixed by the $J_{0}$ field equation.

The off-shell Fourier-like expansion (50) for $\Phi$ permits to decompose the
integrated field equations (97) in the limit of large $J_{0}$,

$\displaystyle\dot{\Phi}_{0}$ $\displaystyle=-\zeta_{\epsilon}\,(1+\ln
J_{0})\,,$ (98a) $\displaystyle\dot{J}_{0}$ $\displaystyle=0\,,$ (98b)
$\displaystyle\dot{J}_{n}$
$\displaystyle=-in\,\zeta_{\epsilon}\frac{J_{n}}{J_{0}}+\dots\,,$ (98c)

with the solution

$\Phi(t,\,\sigma)=\Phi_{0}(t)+J_{0}(t)\,\sigma+\sum_{n\neq
0}\frac{J_{n}(t)}{in}e^{in\sigma}\stackrel{{\scriptstyle\textrm{\tiny
EOM}}}{{=}}\phi(t)+J_{0}\,\sigma+\sum_{n\neq
0}\frac{J_{n}^{(0)}}{in}e^{in\big{(}\sigma-\zeta_{\epsilon}t/J_{0}\big{)}}+\dots$
(99)

where $\phi(t)=-\zeta_{\epsilon}(1+\ln J_{0})t+\phi_{0}$, the ellipses denote
subleading terms in $1/J_{0}$, and $J_{n}^{(0)}$ are integration constants of
individual soft hair Fourier modes. Note that $\dot{J}_{0}$ vanishes as
consequence of the field equations, so it is not an assumption but rather a
result that the quantity determining the holonomy at $\epsilon=0$ is time
independent.

The main physical consequence of the limiting action (94) as compared to the
near horizon action (52) is that the soft hair excitations acquire a positive
energy, which we calculate now.

Plugging the mode expansion (50) into the Hamiltonian (92) yields

$H_{\textrm{\tiny
log}}=\frac{k\zeta_{\epsilon}}{4\pi}\,\int\mathrm{d}\sigma\,\bigg{(}J_{0}(t)+\sum_{n\neq
0}J_{n}(t)e^{in\sigma}\bigg{)}\,\bigg{(}\sum_{n\neq
0}\frac{J_{n}(t)e^{in\sigma}}{J_{0}(t)}-\frac{1}{2}\bigg{(}\sum_{n\neq
0}\frac{J_{n}(t)e^{in\sigma}}{J_{0}(t)}\bigg{)}^{2}+\dots\bigg{)}$ (100)

where the ellipsis refers to higher order terms suppressed at least by
$1/J_{0}^{3}$ and to terms that exclusively depend on the zero mode $J_{0}$
and thus do not contribute to the dynamics of soft hair excitations.
Neglecting these terms, the Hamiltonian (100) integrates to

$H_{\textrm{\tiny
log}}=\frac{k\zeta_{\epsilon}}{2J_{0}(t)}\,\sum_{n>0}J_{n}(t)J_{-n}(t)\stackrel{{\scriptstyle\textrm{\tiny
EOM}}}{{=}}\frac{k\zeta_{\epsilon}}{2J_{0}}\,\sum_{n>0}J_{n}^{(0)}J_{-n}^{(0)}\,.$
(101)

Up to an overall factor, the term displayed in (101) is essentially the
Sugawara stress tensor of a spin-1 current. If expressed in terms of momenta
(53) the limiting Hamiltonian

$H_{\textrm{\tiny
log}}[J_{n},\,\Pi_{n}]=\frac{ik\zeta_{\epsilon}}{4\Pi_{0}}\,\sum_{n>0}nJ_{n}\Pi_{n}$
(102)

is of ‘$xp$-form’. The limiting Hamiltonian action

$S_{\textrm{\tiny
log}}[\Phi_{n},\,\Pi_{n}]=\int\mathrm{d}t\,\Big{(}\sum_{n\geq
0}\dot{\Pi}_{n}J_{n}-H_{\textrm{\tiny log}}\Big{)}\,,$ (103)

recovers the on-shell condition $J_{0}=\rm const.$ and produces equations of
motion for the soft hair excitations

$\dot{J}_{n}=-in\zeta_{\epsilon}\,\frac{J_{n}}{J_{0}}\,,\qquad\qquad\dot{\Pi}_{n}=in\zeta_{\epsilon}\,\frac{\Pi_{n}}{J_{0}}\,,$
(104)

that coincide with (98), up to terms suppressed by $1/J_{0}^{2}$.

Thus, soft hair excitations of the vacuum, $J_{-n}|0\rangle$, are not soft
with respect to $H_{\textrm{\tiny log}}$, but rather are finite energy
eigenstates,777In evaluating the commutator (105) we made the canonical
replacement $i\\{,\\}\to[,]$ and used the left Poisson bracket (54), since we
still have the same symplectic structure as in the undeformed theory.

$H_{\textrm{\tiny log}}J_{-n}|0\rangle=[H_{\textrm{\tiny
log}},\,J_{-n}]|0\rangle=\frac{\zeta_{\epsilon}}{J_{0}}\,n\,J_{-n}|0\rangle\,,$
(105)

with eigenvalues proportional to $n$.

##  6 Discussion

We conclude with a comparison and intriguing relations to previous results and
proposals, in particular the fluff proposal [31, 32], starting with the
latter.

The fluff proposal envisions the BTZ microstates as $\hat{\mathfrak{u}}(1)$
descendents of the vacuum

$\big{|}\textrm{BTZ\;micro}\,(\\{n_{i}^{\pm}\\})\big{>}=\prod
J^{+}_{-n_{i}^{+}}J^{-}_{-n_{i}^{-}}\big{|}0\big{>}\,,\qquad\qquad
J^{\pm}_{n}\big{|}0\big{>}=0\,,\quad\forall n\geq 0\,,$ (106)

labelled by two sets of positive integers $\\{n_{i}^{\pm}\\}$ subject to the
spectral constraints

$\sum n_{i}^{\pm}=c\,\Delta^{\pm}\\\
,,\qquad\qquad\Delta^{\pm}=\frac{1}{2}\,\big{(}\ell M_{\textrm{\tiny{BTZ}}}\pm
J_{\textrm{\tiny{BTZ}}}\big{)}=\frac{c}{24}\,\big{(}J_{0}^{\pm}\big{)}^{2}\,,$
(107)

that provide a cutoff on these descendants. (It is assumed that the products
of Brown–Henneaux central charge $c$ and weights $\Delta^{\pm}$ are large
integers; $M_{\textrm{\tiny{BTZ}}}$ and $J_{\textrm{\tiny{BTZ}}}$ are
Brown–Henneaux mass and angular momentum.) The spectral constraints (107) give
soft hair excitations an effective energy linear in the mode number $n$. This
property leads to a Hardy–Ramanujan counting [62] of the degeneracy of
$\hat{\mathfrak{u}}(1)$ descendants (dubbed “fluff”) leading to the
Bekenstein–Hawking entropy of BTZ black holes. While in [31] the constraints
(107) were imposed essentially by an argument going back to Bañados [63], in
[32] they were derived from independent working assumptions. One of them
required the existence of a weight-1 CFT primary ${\cal W}=\exp{(-\Phi)}$ with
the twisted periodicity property ${\cal W}(\tau,\,\sigma+2\pi)=\exp{(-2\pi
J_{0})}\,{\cal W}(\tau,\,\sigma)$ and non-vanishing commutation relation
between the zero mode operator $\hat{\Phi}_{0}$ of $\Phi$ and the zero mode
operator $\hat{J}_{0}$ of the $\hat{\mathfrak{u}}(1)$ current.

Key aspects of the fluff proposal reproduced by our limiting action (103) with
(102) are

  1. 1.

the fact that soft hair excitations fall into $\hat{\mathfrak{u}}(1)$ current
algebra representations (54)

  2. 2.

the existence of a canonically conjugate to the near horizon zero mode charge
$J_{0}$, namely $\Phi_{0}$, with Poisson bracket (54)

  3. 3.

the existence of a weight-1 CFT primary operator $X^{\prime}={\cal
W}=\exp{(-\Phi)}$ in (64) with generalized periodicity property (40)

  4. 4.

the lift of soft hair degeneracy to energies linear in the mode number $n$,
see (105).

Additionally, the anisotropic scale invariance of the original near horizon
action (47) is shared by the limiting action (94), which played a crucial rôle
in a Cardy-type of counting of BTZ microstates from a near horizon perspective
[64]. We have thus confirmed crucial aspects of the fluff proposal [31, 32].

However, we have not succeeded in deriving all important aspects of the fluff
proposal. In particular, the free parameter $\zeta_{\epsilon}$ appearing as
prefactor in the Hamiltonian (102) is undetermined. In order to obtain the
spectral constraints (107) we need to fix it to the value

$\textrm{desired:}\quad\zeta_{\epsilon}^{\pm}=\frac{J_{0}^{\pm}}{c}$ (108)

in each chiral sector, assuming $H_{\textrm{\tiny
log}}^{\pm}\big{|}\textrm{BTZ\;micro}\,(\\{n_{i}^{\pm}\\})\big{>}=\Delta^{\pm}\big{|}\textrm{BTZ\;micro}\,(\\{n_{i}^{\pm}\\})\big{>}$
for BTZ microstates (106). There are two reasons why the choice (108) is not
obvious. First of all, if we keep $J_{0}$ as state-dependent parameter then it
is impossible to demand (108) without deforming the boundary conditions, since
$\zeta_{\epsilon}$ is a chemical potential. Second, even if we allow for
possible dependence of $J_{0}$ the choice (108) is not the most natural one;
instead, the commutation relation (105) and the solution (99) both may suggest
$\zeta_{\epsilon}=J_{0}$ as ‘natural’.

We explain now how these issues could be resolved. The first issue is
reminiscent of the dilaton gravity description [65] of the SYK model [66, 67,
68, 69], the key issue being that the dilaton is allowed to fluctuate, while
its zero mode is kept fixed [70, 71]. To resolve this issue, we can simply
impose the additional restriction that $J_{0}$ is kept fixed (like in the
microcanonical ensemble) while all other Fourier excitations $J_{n}$ are
allowed to vary. The asymptotic symmetry algebra (18) is compatible with this
restriction.

The second issue was already encountered in Carlip’s attempt to account for
the BTZ black hole entropy, see [72] and refs. therein. Without further input,
a $\hat{\mathfrak{u}}(1)$ current naturally leads to a Virasoro algebra with
(quantum) central charge $c=1$ rather than a (classical) central charge with
Brown–Henneaux value $c=3\ell/(2G)$, which would lead to a considerable under-
counting of the degeneracy of microstates.

Therefore, for the fluff proposal to work it is important not only to provide
a controlled cutoff on the soft hair spectrum (we have succeeded in doing so
in the present work), but also to produce the desired result (108). In [32]
this issue was resolved by a set of Bohr-type quantization conditions that led
to a counting of a discrete set of conical defect geometries with certain
rational values for the conical defect as building blocks for the BTZ
microstates. It would be desirable to derive these conditions [or to directly
derive (108)] from first principles.

In our way of providing a cutoff for soft hair excitations we used the KdV
hierarchy and the gravity-approximation. Since the latter corresponds to the
large central charge approximation on the CFT side, it could be rewarding to
compare our results with corresponding large $c$ results, such as [73, 74].

Finally, let us point out two different ways of interpreting our near horizon
boundary conditions as ultrarelativistic limit of some other theory.

The first one starts from the Floreanini–Jackiw action (62), where the
parameter $\mu$ has the physical interpretation as propagation speed of the
chiral boson. The ultrarelativistic (or Carrollian) limit is $\mu\to 0$ and
recovers our near horizon action (47). This concurs, at least in spirit, with
the near horizon analysis by Donnay and Marteau [75] and Penna [76], who found
Carrollian structures at black hole horizons.

The second consideration starts from bosonic string theory, where the string
spectrum is given by

$X_{\pm}^{\mu}(t\pm\sigma)=\frac{x^{\mu}}{2}+\frac{\ell_{s}^{2}}{2}\,p^{\mu}_{\pm}\,(t\pm\sigma)+\frac{\ell_{s}}{\sqrt{2}}\,\sum_{n\neq
0}\frac{\alpha_{-n}^{\pm}}{in}e^{in(t\pm\sigma)}\,.$ (109)

In the naive ultrarelativistic limit $t\to\epsilon t$, $\sigma\to\sigma$,
$\epsilon\to 0$ (109) reduces to

$X_{\pm}^{\mu}(\sigma)=\frac{x^{\mu}}{2}\pm\frac{\ell_{s}^{2}}{2}\,p^{\mu}_{\pm}\,\sigma+\frac{\ell_{s}}{\sqrt{2}}\,\sum_{n\neq
0}\frac{\alpha_{-n}^{\pm}}{in}e^{\pm in\sigma}$ (110)

which is equivalent to the on-shell expansion (49) for constant $\Phi_{0}$,
identifying $x^{\mu}=2\Phi_{0}$, $\ell_{s}^{2}p^{\mu}_{+}=2J_{0}$ and
$\ell_{s}\alpha^{+}_{-n}=\sqrt{2}J_{n}$ (and similarly for the other chiral
sector). A more careful ultrarelativistic limit also features the linear term
in time present in (49) and is relevant for tensionless strings [28], which
suggests [77, 78, 79, 80] that (nearly) tensionless strings could play a key
rôle in the near horizon descriptions of generic black holes and in
understanding their microstates.

## Acknowledgements

We are grateful to Hamid Afshar, Martin Ammon, Stephane Detournay, Hernán
González, Philip Hacker, Zahra Mirzaiyan, Alfredo Perez, Stefan Prohazka, Max
Riegler, Shahin Sheikh-Jabbari, David Tempo, Ricardo Troncoso, Raphaela Wutte,
Hossein Yavartanoo and Céline Zwikel for collaboration on near horizon
boundary conditions. We thank Arjun Bagchi, Glenn Barnich, Laura Donnay, Oscar
Fuentealba, Gaston Giribet, Hernán González, Marc Henneaux, Javier Matulich,
Blaja Oblak, Malcolm Perry and Andy Strominger for discussions. We thank
particularly one of our referees, Kristan Jensen, for insisting that making
the holonomies time-dependent could (and actually did) affect some of our
results.

#### Funding information

This work was supported by the FWF projects P30822 and P28751. WM is supported
by the ERC Advanced Grant High-Spin-Grav and by FNRS-Belgium (convention FRFC
PDR T.1025.14 and convention IISN 4.4503.15). Part of this work was completed
during the Programme and Workshop “Higher spins and holography” in March/April
2019 at the Erwin Schrödinger International Institute for Mathematics and
Physics (ESI), and we thank ESI for hosting us.

##  Appendix A Brown–Henneaux boundary conditions and holonomies

As discusssed in the main text, boundary conditions are provided by specifying
the form of $a_{\sigma}$. Being a flat connection, $a_{\sigma}$ can locally
always be written as $g^{-1}\partial_{\sigma}g$. What distinguishes solutions
are the global charges, which in turn are measured by Wilson loops, or
holonomies of the connection around a closed $\sigma$-cycle

$H_{\sigma}=\mathrm{tr}\Big{(}{\cal P}\exp\oint
a_{\sigma}\,\mathrm{d}\sigma\Big{)}\,.$ (111)

Because $a_{\sigma}$ is an element of the $\mathfrak{sl}(2,\hbox{\mybb R})$
algebra, the holonomy around the $\sigma$-cycle is characterized by the
conjugacy classes of SL$(2,\hbox{\mybb R})$. These are

#### Hyperbolic.

Conjugate to dilatation $g\sim\left(\begin{array}[]{cc}e^{2\pi\lambda}&0\\\
0&e^{-2\pi\lambda}\end{array}\right)$, with holonomy
$H_{\sigma}=2\cosh(2\pi\lambda)$

#### Parabolic.

Conjugate to translation $g\sim\left(\begin{array}[]{cc}1&2\pi a\\\
0&1\end{array}\right)$, with holonomy $H_{\sigma}=2$

#### Elliptic.

Conjugate to rotation $g\sim\left(\begin{array}[]{cc}\cos 2\pi\alpha&\sin
2\pi\alpha\\\ -\sin 2\pi\alpha&\cos 2\pi\alpha\end{array}\right)$, with
holonomy $H_{\sigma}=2\cos(2\pi\alpha)$

####

To obtain the boundary conditions of Brown and Henneaux we take $a_{\sigma}$
to be in the ‘highest weight gauge’ (and similarly for $\bar{a}_{\sigma}$)

$a_{\sigma}=L_{+}-{\cal L}(t,\sigma)L_{-}=\left(\begin{array}[]{cc}0&{\cal
L}\\\ 1&0\end{array}\right)\,.$ (112)

This choice indeed leads to an asymptotic Virasoro symmetry algebra with the
Brown–Henneaux central charge, which can be shown by computing the charges and
the variations of $a_{\sigma}$ under boundary condition preserving gauge
transformations $\varepsilon$.

$\delta_{\varepsilon}a_{\sigma}=\partial_{\sigma}\varepsilon+[a_{\sigma},\varepsilon]=\left(\begin{array}[]{cc}0&\delta_{\varepsilon}{\cal
L}\\\ 0&0\end{array}\right)\,.$ (113)

Expanding $\varepsilon=\varepsilon^{n}L_{n}$ yields
$\varepsilon^{+}=\varepsilon^{+}(\varepsilon^{+})$ and

$\displaystyle\varepsilon^{0}$
$\displaystyle=-\partial_{\sigma}\varepsilon^{+}\,,$
$\displaystyle\varepsilon^{-}$
$\displaystyle=\frac{1}{2}\partial_{\sigma}^{2}\varepsilon^{+}-{\cal
L}\varepsilon^{+}\,,$ (114) $\displaystyle\delta_{\varepsilon}{\cal L}$
$\displaystyle=\partial_{\sigma}{\cal L}\varepsilon^{+}+2{\cal
L}\partial_{\sigma}\varepsilon^{+}-\frac{1}{2}\partial_{\sigma}^{3}\varepsilon^{+}\,.$
(115)

Using this result in the expression for the charges (14) and assuming
$\varepsilon^{+}$ is state-independent, i.e., $\delta\varepsilon^{+}=0$, one
may trivially integrate them and finds

$Q(\varepsilon)=\frac{k}{2\pi}\oint\mathrm{d}\sigma\,\varepsilon^{+}{\cal
L}\,.$ (116)

After expanding the charges in Fourier modes and using the fact that the
Poisson brackets of the charges can be computed by
$\\{Q(\varepsilon_{1}),Q(\varepsilon_{2})\\}=\delta_{\varepsilon_{1}}Q(\varepsilon_{2})$,
one can easily see the appearance of the Virasoro algebra with a central
charge $c=6k=\frac{3\ell}{2G}$ from the transformation law (115),

The time component part of the connection can be taken to have the same
structure as the gauge parameter $\varepsilon$, but with an arbitrary function
$\mu$ instead of $\varepsilon^{+}$. Explicitly,

$a_{t}=\varepsilon(\mu)=\mu L_{+}-\partial_{\sigma}\mu
L_{0}+\Big{(}\frac{1}{2}\partial^{2}\mu-{\cal L}\mu\Big{)}L_{-}\,.$ (117)

The metrics (with $\mu=1$) to which these solutions correspond are the so-
called Bañados-metrics [81]. They are obtained by taking
$b(r)=\exp(r/\ell\,L_{0})$ (and similarly for the barred sector) and plugging
the connection into (12)

$\mathrm{d}s^{2}=\mathrm{d}r^{2}-\ell^{2}\left(e^{r/\ell}\mathrm{d}x^{+}-e^{-r/\ell}\bar{{\cal
L}}(x^{-})\mathrm{d}x^{-}\right)\left(e^{r/\ell}\mathrm{d}x^{-}-e^{-r/\ell}{\cal
L}(x^{+})\mathrm{d}x^{+}\right)\,.$ (118)

Here we used light cone coordinates $x^{\pm}=t/\ell\pm\sigma$.

We can characterize the Bañados solutions by considering the holonomies of the
connections $a_{\sigma}$ and $\bar{a}_{\sigma}$. The holonomy of the
connection (112) (for constant ${\cal L}$) is given by

$H_{\sigma}(a_{\sigma})=\mathrm{tr}\big{(}\exp(2\pi\sqrt{4{\cal
L}})L_{0}\big{)}=e^{2\pi\sqrt{{\cal L}}}+e^{-2\pi\sqrt{{\cal L}}}\,,$ (119)

and likewise for the other sector. This implies that for ${\cal L}>0$ the
holonomy falls into the hyperbolic conjugacy class of SL$(2,\hbox{\mybb R})$.
When ${\cal L}=0$, the holonomy is parabolic and for ${\cal L}<0$ it is
elliptic.

The Bañados solutions for

${\cal
L}=\frac{2G}{\ell}(\ell\mathfrak{m}+\mathfrak{j})\,,\qquad\qquad\bar{\cal
L}=\frac{2G}{\ell}(\ell\mathfrak{m}-\mathfrak{j})\,,$ (120)

correspond to BTZ black holes with mass $\mathfrak{m}$ and angular momentum
$\mathfrak{j}$. Thus, generic BTZ black holes have hyperbolic holonomies,
whereas extremal black holes (where either ${\cal L}$ of $\bar{\cal L}=0$)
have one connection with parabolic holonomy while the other is still
hyperbolic. Whenever ${\cal L}$ or $\bar{\cal L}$ is negative, then the
holonomy is conjugate to a complex SL$(2,\mathbb{R})$ element unless ${\cal
L}=-\frac{1}{4}n^{2}$ and likewise for $\bar{\cal L}$. At these points the
bulk solution has an angular periodicity of $2\pi n$, with $n=1$ corresponding
to the global AdS3 solution. Any other negative value of ${\cal L},\bar{\cal
L}$ will give conical singularities.

##  Appendix B Gelfand–Dikii differential polynomials and Hamiltonians

We list here the first couple of Gelfand–Dikii differential polynomials
generated recursively through the defining relation (75)888The coefficient of
the third-derivative term in $\cal D$ defined below (75) is free. We fixed it
to $\tfrac{1}{2}$, which, together with our normalization choice for $R_{N}$,
explains differences to (and between) results in the literature.

$\displaystyle R_{0}$ $\displaystyle=1\,,\qquad\qquad\qquad R_{1}={\cal
J}\,,\qquad\qquad\qquad R_{2}={\cal
J}^{2}{\color[rgb]{1,0,0}+\tfrac{1}{3}\,{\cal J}^{\prime\prime}}\,,$ (121)
$\displaystyle R_{3}$ $\displaystyle={\cal J}^{3}-\tfrac{1}{2}\,{\cal
J}^{\prime\,2}{\color[rgb]{1,0,0}+\big{(}{\cal J}{\cal
J}^{\prime}\big{)}^{\prime}+\tfrac{1}{10}\,{\cal J}^{(4)}}\,,$ (122)
$\displaystyle R_{4}$ $\displaystyle={\cal J}^{4}-2{\cal J}{\cal
J}^{\prime\,2}+\tfrac{1}{5}\,{\cal
J}^{\prime\prime\,2}{\color[rgb]{1,0,0}+\tfrac{2}{3}\,\big{(}{\cal
J}^{3}\big{)}^{\prime\prime}+\tfrac{2}{5}\,\big{(}{\cal J}{\cal
J}^{\prime\prime}\big{)}^{\prime\prime}+\tfrac{1}{35}\,{\cal J}^{(6)}}\,,$
(123) $\displaystyle R_{5}$ $\displaystyle={\cal J}^{5}-5{\cal J}^{2}{\cal
J}^{\prime}{}^{2}+{\cal J}{\cal J}^{\prime\prime\,2}-\tfrac{1}{14}\,{\cal
J}^{\prime\prime\prime\,2}{\color[rgb]{1,0,0}+\tfrac{5}{6}\,\big{(}{\cal
J}^{4}\big{)}^{\prime\prime}+\big{(}{\cal J}^{2}{\cal
J}^{\prime\prime}\big{)}^{\prime\prime}+\tfrac{5}{9}\,\big{(}{\cal
J}^{\prime\,3}\big{)}^{\prime}}$
$\displaystyle\quad{\color[rgb]{1,0,0}+\tfrac{1}{7}\,\big{(}{\cal J}{\cal
J}^{(4)}\big{)}^{\prime\prime}+\tfrac{1}{7}\,\big{(}{\cal J}^{\prime}{\cal
J}^{(4)}\big{)}^{\prime}+\tfrac{13}{42}\,\big{(}{\cal
J}^{\prime\prime\,2}\big{)}^{\prime\prime}+\tfrac{1}{126}{\cal J}^{(8)}}\,,$
(124) $\displaystyle R_{6}$ $\displaystyle={\cal J}^{6}-10{\cal J}^{3}{\cal
J}^{\prime\,2}+3{\cal J}^{2}{\cal J}^{\prime\prime\,2}-\tfrac{3}{7}\,{\cal
J}{\cal J}^{\prime\prime\prime\,2}+\tfrac{1}{42}\,\big{(}{\cal
J}^{(4)}\big{)}^{2}{\color[rgb]{0,0,1}-\tfrac{5}{6}\,{\cal
J}^{\prime\,4}+\tfrac{30}{63}\,{\cal J}^{\prime\prime\,3}}$
$\displaystyle\quad{\color[rgb]{1,0,0}+\big{(}{\cal
J}^{5}\big{)}^{\prime\prime}+2\big{(}{\cal J}^{3}{\cal
J}^{\prime\prime}\big{)}^{\prime\prime}+\tfrac{10}{3}\,\big{(}{\cal J}{\cal
J}^{\prime\,3}\big{)}^{\prime}+\tfrac{1}{7}\,\big{(}3{\cal J}^{2}{\cal
J}^{(4)}+6{\cal J}{\cal J}^{\prime}{\cal J}^{\prime\prime\prime}+7{\cal
J}^{\prime\,2}{\cal J}^{\prime\prime}\big{)}^{\prime\prime}}$
$\displaystyle\quad{\color[rgb]{1,0,0}+\tfrac{1}{21}\,\big{(}60{\cal J}{\cal
J}^{\prime\prime}{\cal J}^{\prime\prime\prime}+9{\cal J}^{\prime}{\cal
J}^{\prime\prime\,2}+{\cal J}{\cal J}^{(7)}+3{\cal J}^{\prime}{\cal
J}^{(6)}+8{\cal J}^{\prime\prime}{\cal J}^{(5)}+11{\cal
J}^{\prime\prime\prime}{\cal J}^{(4)}\big{)}^{\prime}}$
$\displaystyle\quad{\color[rgb]{1,0,0}+\tfrac{1}{462}\,{\cal J}^{(10)}}\,,$
(125) $\displaystyle\vdots\;$ $\displaystyle=\;\vdots$ $\displaystyle R_{N}$
$\displaystyle={\cal
J}^{N}+N\,\sum_{i=1}^{N-2}(-1)^{i}\frac{((i+1)!)^{2}}{(2i+2)!}\,\binom{N-1}{i+1}\,{\cal
J}^{N-i-2}\,\big{(}\partial_{\sigma}^{i}{\cal
J}\big{)}^{2}{\color[rgb]{0,0,1}+N\,{\cal H}_{N-1}^{\textrm{\tiny
nl}}}{\color[rgb]{1,0,0}+\partial_{\sigma}R_{N}^{\textrm{\tiny td}}}\,,$ (126)

where ${\color[rgb]{0,0,1}{\cal H}^{\textrm{\tiny nl}}_{N-1}}$ is defined
below and ${\color[rgb]{1,0,0}R_{N}^{\textrm{\tiny td}}}$ captures total
derivative terms. Their schematic form for $N>2$ is given by

${\color[rgb]{1,0,0}R_{N}^{\textrm{\tiny
td}}=\sum_{\\{b_{i,N}\\}}r_{\\{b_{i,N}\\}}\,{\cal
J}^{b_{0,N}}\prod_{i=1}^{2N-5}\big{(}\partial_{\sigma}^{i}{\cal
J}\big{)}^{b_{i,N}}+\frac{2^{1-N}N!}{(2N-1)!!}\,\partial_{\sigma}^{2N-3}{\cal
J}}\,,$ (127)

where the sets of non-negative integers ${\color[rgb]{1,0,0}\\{b_{i,N}\\}}$
are subject to the constraints

${\color[rgb]{1,0,0}{\sum_{i=0}^{2N-5}\big{(}i+2\big{)}\,b_{i,N}=2N-1}}\,,$
(128)

and ${\color[rgb]{1,0,0}r_{\\{b_{i,N}\\}}}$ are rational coefficients for each
of these sets. The expression ${\color[rgb]{0,0,1}{\cal H}_{N}^{\textrm{\tiny
nl}}}$ denotes terms at least cubic in derivatives of ${\cal J}$. These terms
vanish for $N<5$ and otherwise read

${\color[rgb]{0,0,1}{\cal H}_{N}^{\textrm{\tiny
nl}}=\sum_{\\{a_{i,N}\\}}h_{\\{a_{i,N}\\}}\,{\cal
J}^{a_{0,N}}\prod_{i=1}^{N-4}\big{(}\partial_{\sigma}^{i}{\cal
J}\big{)}^{a_{i,N}}+h_{\\{a_{2,N}=1,\,a_{N-3,N}=2\\}}\,{\cal
J}^{\prime\prime}\big{(}\partial_{\sigma}^{N-3}{\cal J}\big{)}^{2}}\,,$ (129)

where the sets of non-negative integers ${\color[rgb]{0,0,1}\\{a_{i,N}\\}}$
are subject to the constraints

${\color[rgb]{0,0,1}\sum_{i=0}^{N-4}(i+2)\,a_{i,N}=2N+2}\,,\qquad\qquad{\color[rgb]{0,0,1}\sum_{i=1}^{N-4}a_{i,N}\geq
3}\,,\qquad\qquad{\color[rgb]{0,0,1}a_{i^{\textrm{\tiny max}},N}\geq 2}\,,$
(130)

with ${\color[rgb]{0,0,1}a_{i^{\textrm{\tiny max}},N}}$ denoting the number
for the largest value of ${\color[rgb]{0,0,1}i=i^{\textrm{\tiny max}}}$ that
leads to non-vanishing ${\color[rgb]{0,0,1}a_{i,N}}$ within a given set
${\color[rgb]{0,0,1}\\{a_{i,N}\\}}$, and
${\color[rgb]{0,0,1}h_{\\{a_{i,N}\\}}}$ are rational coefficients for each of
these sets. Explicit results for $N=5,6,7$ are

$\displaystyle{\color[rgb]{0,0,1}{\cal H}_{5}^{\textrm{\tiny nl}}}$
$\displaystyle{\color[rgb]{0,0,1}=-\tfrac{5}{36}\,{\cal
J}^{\prime\,4}+\tfrac{5}{63}\,{\cal J}^{\prime\prime\,3}}\,,$ (131)
$\displaystyle{\color[rgb]{0,0,1}{\cal H}_{6}^{\textrm{\tiny nl}}}$
$\displaystyle{\color[rgb]{0,0,1}=-\tfrac{5}{6}\,{\cal J}{\cal
J}^{\prime\,4}+\tfrac{10}{21}\,{\cal J}{\cal
J}^{\prime\prime\,3}+\tfrac{1}{2}\,{\cal J}^{\prime\,2}{\cal
J}^{\prime\prime\,2}-\tfrac{5}{42}\,{\cal J}^{\prime\prime}{\cal
J}^{\prime\prime\prime\,2}}\,,$ (132) $\displaystyle{\color[rgb]{0,0,1}{\cal
H}_{7}^{\textrm{\tiny nl}}}$
$\displaystyle{\color[rgb]{0,0,1}=-\tfrac{35}{12}\,{\cal J}^{2}{\cal
J}^{\prime\,4}+\tfrac{5}{3}\,{\cal J}^{2}{\cal
J}^{\prime\prime\,3}+\tfrac{7}{2}\,{\cal J}{\cal J}^{\prime\,2}{\cal
J}^{\prime\prime\,2}-\tfrac{5}{6}\,{\cal J}{\cal J}^{\prime\prime}{\cal
J}^{\prime\prime\prime\,2}}$
$\displaystyle\quad{\color[rgb]{0,0,1}+\tfrac{7}{24}\,{\cal
J}^{\prime\prime\,4}-\tfrac{1}{4}\,{\cal J}^{\prime\,2}{\cal
J}^{\prime\prime\prime\,2}+\tfrac{7}{132}\,{\cal J}^{\prime\prime}\big{(}{\cal
J}^{(4)}\big{)}^{2}}\,.$ (133)

The split in (126) into ${\color[rgb]{0,0,1}N\,{\cal H}_{N-1}^{\textrm{\tiny
nl}}}$ and ${\color[rgb]{1,0,0}\partial_{\sigma}R_{N}}$ is ambiguous, since we
can add total derivative terms to the former and subtract them from the
latter. Above we made particular choices for ${\color[rgb]{0,0,1}{\cal
H}_{N}^{\textrm{\tiny nl}}}$ that minimize the number of derivatives acting on
the ${\cal J}$-factor with the highest number of derivatives for each term.
(For instance, the term ${\cal J}^{\prime\,2}{\cal J}^{\prime\prime\prime\,2}$
in (133), up to total derivative terms and a term proportional to ${\cal
J}^{\prime\prime\,4}$, is equivalent to $-{\cal J}^{\prime\,2}{\cal
J}^{\prime\prime}{\cal J}^{(4)}$, which has four derivatives on the last
factor, whereas the original term had at most three derivatives.)

A cross-check on the numerical factors in (121)-(126) is the relation [60]

$\frac{\partial R_{N}}{\partial{\cal J}}=N\,R_{N-1}\,.$ (134)

Note that additionally all black terms in (126) (the ones displayed
explicitly) are mapped to each other via (134); the same is true for the red
and blue terms for our choice of ${\color[rgb]{0,0,1}{\cal
H}_{N}^{\textrm{\tiny nl}}}$.

The associated Hamiltonian densities ${\cal H}_{N}$ obeying

$\delta{\cal H}_{N}\sim R_{N}\,\delta{\cal J}\sim\tfrac{1}{N+1}\,\delta
R_{N+1}\,,$ (135)

(where $\sim$ denotes equality up to total derivative terms) are given by

$\displaystyle{\cal H}_{0}$ $\displaystyle={\cal J}\,,\qquad\qquad\qquad{\cal
H}_{1}=\tfrac{1}{2}\,{\cal J}^{2}\,,\qquad\qquad\qquad{\cal
H}_{2}=\tfrac{1}{3}\,{\cal J}^{3}-\tfrac{1}{6}\,{\cal J}^{\prime\,2}\,,$ (136)
$\displaystyle{\cal H}_{3}$ $\displaystyle=\tfrac{1}{4}\,{\cal
J}^{4}-\tfrac{1}{2}\,{\cal J}{\cal J}^{\prime\,2}+\tfrac{1}{20}\,{\cal
J}^{\prime\prime\,2}\,,$ (137) $\displaystyle{\cal H}_{4}$
$\displaystyle=\tfrac{1}{5}\,{\cal J}^{5}-{\cal J}^{2}{\cal
J}^{\prime\,2}+\tfrac{1}{5}\,{\cal J}{\cal
J}^{\prime\prime\,2}-\tfrac{1}{70}\,{\cal J}^{\prime\prime\prime\,2}\,,$ (138)
$\displaystyle{\cal H}_{5}$ $\displaystyle=\tfrac{1}{6}\,{\cal
J}^{6}-\tfrac{5}{3}\,{\cal J}^{3}{\cal J}^{\prime\,2}+\tfrac{1}{2}\,{\cal
J}^{2}{\cal J}^{\prime\prime\,2}-\tfrac{1}{14}\,{\cal J}{\cal
J}^{\prime\prime\prime\,2}+\tfrac{1}{252}\,\big{(}{\cal
J}^{(4)}\big{)}^{2}{\color[rgb]{0,0,1}+{\cal H}_{5}^{\textrm{\tiny nl}}}\,,$
(139) $\displaystyle{\cal H}_{6}$ $\displaystyle=\tfrac{1}{7}\,{\cal
J}^{7}-\tfrac{5}{2}\,{\cal J}^{4}{\cal J}^{\prime\,2}+{\cal J}^{3}{\cal
J}^{\prime\prime\,2}-\tfrac{3}{14}\,{\cal J}^{2}{\cal
J}^{\prime\prime\prime\,2}+\tfrac{1}{42}\,{\cal J}\big{(}{\cal
J}^{(4)}\big{)}^{2}-\tfrac{1}{924}\,\big{(}{\cal
J}^{(5)}\big{)}^{2}{\color[rgb]{0,0,1}+{\cal H}_{6}^{\textrm{\tiny nl}}}\,,$
(140) $\displaystyle\vdots\;$ $\displaystyle=\;\vdots$ $\displaystyle{\cal
H}_{N}$ $\displaystyle=\tfrac{1}{N+1}\,{\cal
J}^{N+1}+\sum_{i=1}^{N-1}(-1)^{i}\frac{((i+1)!)^{2}}{(2i+2)!}\,\binom{N}{i+1}\,{\cal
J}^{N-i-1}\,\big{(}\partial_{\sigma}^{i}{\cal
J}\big{)}^{2}{\color[rgb]{0,0,1}+{\cal H}_{N}^{\textrm{\tiny nl}}}\,.$ (141)

Inspection of the explicit results above shows compatibility with (80), which
is a simple, yet non-trivial, cross-check on the correctness of the equations
displayed in this appendix.

Starting with the general form of the Hamiltonian density (141), analytically
continuing in $N\leq 1$, assuming $N=\varepsilon\to 0^{+}$, inserting ${\cal
J}=\Phi^{\prime}$ and dropping total derivative terms yields the limiting
Hamiltonian density

${\cal H}_{\varepsilon}\sim\varepsilon\,\Phi^{\prime}\,\ln\Phi^{\prime}+{\cal
O}(\varepsilon^{2})\,,$ (142)

which remains finite and non-trivial if rescaled by $1/\varepsilon$.

## References

  * [1] H. Bondi, M. van der Burg and A. Metzner,  _Gravitational waves in general relativity VII. Waves from axi-symmetric isolated systems_ ,  Proc. Roy. Soc. London A269, 21 (1962).
  * [2] R. Sachs,  _Asymptotic symmetries in gravitational theory_ ,  Phys. Rev. 128, 2851 (1962).
  * [3] J. D. Brown and M. Henneaux,  _Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity_ ,  Commun. Math. Phys. 104, 207 (1986),  10.1007/BF01211590.
  * [4] M. Henneaux and S.-J. Rey,  _Nonlinear $W_{infinity}$ as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity_,  JHEP 1012, 007 (2010),  10.1007/JHEP12(2010)007,  1008.4579.
  * [5] A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen,  _Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields_ ,  JHEP 1011, 007 (2010),  10.1007/JHEP11(2010)007,  1008.4744.
  * [6] D. M. Hofman and B. Rollier,  _Warped Conformal Field Theory as Lower Spin Gravity_ ,  Nucl. Phys. B897, 1 (2015),  10.1016/j.nuclphysb.2015.05.011,  1411.0672.
  * [7] G. Barnich and G. Compère,  _Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions_ ,  Class.Quant.Grav. 24, F15 (2007),  10.1088/0264-9381/24/5/F01, 10.1088/0264-9381/24/11/C01,  gr-qc/0610130.
  * [8] G. Compère, W. Song and A. Strominger,  _New Boundary Conditions for $AdS_{3}$_,  JHEP 1305, 152 (2013),  10.1007/JHEP05(2013)152,  1303.2662.
  * [9] D. Grumiller and M. Riegler,  _Most general AdS 3 boundary conditions_,  JHEP 10, 023 (2016),  10.1007/JHEP10(2016)023,  1608.01308.
  * [10] D. Grumiller, W. Merbis and M. Riegler,  _Most general flat space boundary conditions in three-dimensional Einstein gravity_ ,  Class. Quant. Grav. 34(18), 184001 (2017),  10.1088/1361-6382/aa8004,  1704.07419.
  * [11] G. Compère and A. Fiorucci,  _Advanced Lectures on General Relativity_ ,  Lect. Notes Phys. 952 (2019),  10.1007/978-3-030-04260-8,  1801.07064.
  * [12] D. Harlow and J.-q. Wu,  _Covariant phase space with boundaries_ (2019),  1906.08616.
  * [13] D. Tong,  _Lectures on the Quantum Hall Effect_ ,  Lectures given at TIFR, Mumbai (2016), 1606.06687.
  * [14] J. D. Brown and J. W. York, Jr.,  _Quasilocal energy and conserved charges derived from the gravitational action_ ,  Phys. Rev. D47, 1407 (1993).
  * [15] F. Hopfmüller and L. Freidel,  _Gravity Degrees of Freedom on a Null Surface_ ,  Phys. Rev. D95(10), 104006 (2017),  10.1103/PhysRevD.95.104006,  1611.03096.
  * [16] R. H. Price and K. S. Thorne,  _Membrane Viewpoint on Black Holes: Properties and Evolution of the Stretched Horizon_ ,  Phys. Rev. D33, 915 (1986),  10.1103/PhysRevD.33.915.
  * [17] K. S. Thorne, R. Price and D. Macdonald,  _Black holes: The membrane paradigm_ (1986).
  * [18] D. Grumiller and M. M. Sheikh-Jabbari,  _Membrane Paradigm from Near Horizon Soft Hair_ ,  Int. J. Mod. Phys. D27(14), 1847006 (2018),  10.1142/S0218271818470065,  1805.11099.
  * [19] S. Carlip,  _The statistical mechanics of the (2+1)-dimensional black hole_ ,  Phys. Rev. D51, 632 (1995),  gr-qc/9409052.
  * [20] L. Donnay, G. Giribet, H. A. Gonzalez and M. Pino,  _Supertranslations and Superrotations at the Black Hole Horizon_ ,  Phys. Rev. Lett. 116(9), 091101 (2016),  10.1103/PhysRevLett.116.091101,  1511.08687.
  * [21] H. Afshar, S. Detournay, D. Grumiller and B. Oblak,  _Near-Horizon Geometry and Warped Conformal Symmetry_ ,  JHEP 03, 187 (2016),  10.1007/JHEP03(2016)187,  1512.08233.
  * [22] H. Afshar, S. Detournay, D. Grumiller, W. Merbis, A. Perez, D. Tempo and R. Troncoso,  _Soft Heisenberg hair on black holes in three dimensions_ ,  Phys. Rev. D93(10), 101503 (2016),  10.1103/PhysRevD.93.101503,  1603.04824.
  * [23] L. Donnay, G. Giribet, H. A. González and M. Pino,  _Extended Symmetries at the Black Hole Horizon_ ,  JHEP 09, 100 (2016),  10.1007/JHEP09(2016)100,  1607.05703.
  * [24] S. Carlip,  _Black Hole Entropy from Bondi-Metzner-Sachs Symmetry at the Horizon_ ,  Phys. Rev. Lett. 120(10), 101301 (2018),  10.1103/PhysRevLett.120.101301,  1702.04439.
  * [25] S. Haco, S. W. Hawking, M. J. Perry and A. Strominger,  _Black Hole Entropy and Soft Hair_ ,  JHEP 12, 098 (2018),  10.1007/JHEP12(2018)098,  1810.01847.
  * [26] O. Coussaert, M. Henneaux and P. van Driel,  _The Asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant_ ,  Class.Quant.Grav. 12, 2961 (1995),  10.1088/0264-9381/12/12/012,  gr-qc/9506019.
  * [27] R. Floreanini and R. Jackiw,  _Selfdual Fields as Charge Density Solitons_ ,  Phys. Rev. Lett. 59, 1873 (1987),  10.1103/PhysRevLett.59.1873.
  * [28] A. Bagchi,  _Tensionless Strings and Galilean Conformal Algebra_ ,  JHEP 1305, 141 (2013),  10.1007/JHEP05(2013)141,  1303.0291.
  * [29] S. W. Hawking, M. J. Perry and A. Strominger,  _Soft Hair on Black Holes_ ,  Phys. Rev. Lett. 116(23), 231301 (2016),  10.1103/PhysRevLett.116.231301,  1601.00921.
  * [30] S. W. Hawking,  _The Information Paradox for Black Holes_ ,  DAMTP-2015-49 (2015), 1509.01147.
  * [31] H. Afshar, D. Grumiller and M. M. Sheikh-Jabbari,  _Near horizon soft hair as microstates of three dimensional black holes_ ,  Phys. Rev. D96(8), 084032 (2017),  10.1103/PhysRevD.96.084032,  1607.00009.
  * [32] H. Afshar, D. Grumiller, M. M. Sheikh-Jabbari and H. Yavartanoo,  _Horizon fluff, semi-classical black hole microstates — Log-corrections to BTZ entropy and black hole/particle correspondence_ ,  JHEP 08, 087 (2017),  10.1007/JHEP08(2017)087,  1705.06257.
  * [33] A. Perez and E. Ojeda,  _Boundary conditions for General Relativity in three-dimensional spacetimes, integrable systems and the KdV-mKdV hierarchies_ ,  Preprint CECS-PHY-19/02.
  * [34] A. Achucarro and P. K. Townsend,  _A Chern-Simons action for three-dimensional Anti-de Sitter supergravity theories_ ,  Phys. Lett. B180, 89 (1986),  10.1016/0370-2693(86)90140-1.
  * [35] E. Witten,  _(2+1)-dimensional gravity as an exactly soluble system_ ,  Nucl. Phys. B311, 46 (1988),  10.1016/0550-3213(88)90143-5.
  * [36] M. Bañados, C. Teitelboim and J. Zanelli,  _The black hole in three-dimensional space-time_ ,  Phys. Rev. Lett. 69, 1849 (1992),  hep-th/9204099.
  * [37] M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli,  _Geometry of the (2+1) black hole_ ,  Phys. Rev. D48, 1506 (1993),  gr-qc/9302012.
  * [38] M. Riegler and C. Zwikel,  _Canonical Charges in Flatland_ ,  PoS Modave2017, 004 (2018),  10.22323/1.323.0004,  1709.09871.
  * [39] M. Henneaux, A. Perez, D. Tempo and R. Troncoso,  _Chemical potentials in three-dimensional higher spin anti-de Sitter gravity_ ,  JHEP 1312, 048 (2013),  10.1007/JHEP12(2013)048,  1309.4362.
  * [40] H. Afshar, D. Grumiller, W. Merbis, A. Perez, D. Tempo and R. Troncoso,  _Soft hairy horizons in three spacetime dimensions_ ,  Phys. Rev. D95(10), 106005 (2017),  10.1103/PhysRevD.95.106005,  1611.09783.
  * [41] C. Li and J. Lucietti,  _Three-dimensional black holes and descendants_ ,  Phys. Lett. B738, 48 (2014),  10.1016/j.physletb.2014.09.012,  1312.2626.
  * [42] G. Barnich,  _Boundary charges in gauge theories: Using Stokes theorem in the bulk_ ,  Class. Quant. Grav. 20, 3685 (2003),  10.1088/0264-9381/20/16/310,  hep-th/0301039.
  * [43] S. Elitzur, G. W. Moore, A. Schwimmer and N. Seiberg,  _Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory_ ,  Nucl. Phys. B326, 108 (1989),  10.1016/0550-3213(89)90436-7.
  * [44] L. Donnay,  _Asymptotic dynamics of three-dimensional gravity_ ,  PoS Modave2015, 001 (2016),  10.22323/1.271.0001,  1602.09021.
  * [45] M. Henneaux, W. Merbis and A. Ranjbar,  _Asymptotic dynamics of AdS 3 gravity with two asymptotic regions_ (2019),  1912.09465.
  * [46] L. D. Faddeev and R. Jackiw,  _Hamiltonian reduction of unconstrained and constrained systems_ ,  Phys. Rev. Lett. 60, 1692 (1988).
  * [47] J. Sonnenschein,  _Chiral bosons_ ,  Nucl. Phys. B309, 752 (1988),  10.1016/0550-3213(88)90339-2.
  * [48] A. C. Wall,  _Testing the Generalized Second Law in 1+1 dimensional Conformal Vacua: An Argument for the Causal Horizon_ ,  Phys. Rev. D85, 024015 (2012),  10.1103/PhysRevD.85.024015,  1105.3520.
  * [49] C. Ecker, D. Grumiller, W. van der Schee, M. M. Sheikh-Jabbari and P. Stanzer,  _Quantum Null Energy Condition and its (non)saturation in 2d CFTs_ ,  SciPost Phys. 6(3), 036 (2019),  10.21468/SciPostPhys.6.3.036,  1901.04499.
  * [50] G. Barnich and H. A. Gonzalez,  _Dual dynamics of three dimensional asymptotically flat Einstein gravity at null infinity_ ,  JHEP 1305, 016 (2013),  10.1007/JHEP05(2013)016,  1303.1075.
  * [51] E. Witten,  _Nonabelian bosonization in two dimensions_ ,  Commun. Math. Phys. 92, 455 (1984).
  * [52] E. Witten,  _Coadjoint Orbits of the Virasoro Group_ ,  Commun. Math. Phys. 114, 1 (1988),  10.1007/BF01218287.
  * [53] A. Alekseev and S. L. Shatashvili,  _Path Integral Quantization of the Coadjoint Orbits of the Virasoro Group and 2D Gravity_ ,  Nucl. Phys. B323, 719 (1989),  10.1016/0550-3213(89)90130-2.
  * [54] G. Barnich, H. A. Gonzalez and P. Salgado-Rebolledo,  _Geometric actions for three-dimensional gravity_ ,  Class. Quant. Grav. 35(1), 014003 (2018),  10.1088/1361-6382/aa9806,  1707.08887.
  * [55] J. Cotler and K. Jensen,  _A theory of reparameterizations for AdS 3 gravity_,  JHEP 02, 079 (2019),  10.1007/JHEP02(2019)079,  1808.03263.
  * [56] H. Aratyn, E. Nissimov, S. Pacheva and A. H. Zimerman,  _Symplectic actions on coadjoint orbits_ ,  Phys. Lett. B240, 127 (1990),  10.1016/0370-2693(90)90420-B.
  * [57] A. Pérez, D. Tempo and R. Troncoso,  _Boundary conditions for General Relativity on AdS 3 and the KdV hierarchy_,  JHEP 06, 103 (2016),  10.1007/JHEP06(2016)103,  1605.04490.
  * [58] O. Fuentealba, J. Matulich, A. Pérez, M. Pino, P. Rodríguez, D. Tempo and R. Troncoso,  _Integrable systems with BMS 3 Poisson structure and the dynamics of locally flat spacetimes_,  JHEP 01, 148 (2018),  10.1007/JHEP01(2018)148,  1711.02646.
  * [59] H. A. González, J. Matulich, M. Pino and R. Troncoso,  _Revisiting the asymptotic dynamics of General Relativity on AdS 3_,  JHEP 12, 115 (2018),  10.1007/JHEP12(2018)115,  1809.02749.
  * [60] I. M. Gelfand and L. A. Dikii,  _Asymptotic behavior of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-De Vries equations_ ,  Russ. Math. Surveys 30(5), 77 (1975),  10.1070/RM1975v030n05ABEH001522.
  * [61] P. M. Zhang, M. Elbistan, P. A. Horvathy and P. Kosinski,  _A generalized Noether theorem for scaling symmetry_ (2019),  1903.05070.
  * [62] G. Hardy and S. Ramanujan,  _Asymptotic formulae in combinatory analysis_ ,  Proceedings of the London Mathematical Society 2(XVII), 75 (1918).
  * [63] M. Banados,  _Embeddings of the Virasoro algebra and black hole entropy_ ,  Phys. Rev. Lett. 82, 2030 (1999),  10.1103/PhysRevLett.82.2030,  hep-th/9811162.
  * [64] D. Grumiller, A. Perez, D. Tempo and R. Troncoso,  _Log corrections to entropy of three dimensional black holes with soft hair_ ,  JHEP 08, 107 (2017),  10.1007/JHEP08(2017)107,  1705.10605.
  * [65] J. Maldacena, D. Stanford and Z. Yang,  _Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space_ ,  PTEP 2016(12), 12C104 (2016),  10.1093/ptep/ptw124,  1606.01857.
  * [66] A. Kitaev,  _A simple model of quantum holography_ ,  KITP strings seminars, April/May 2015, http://online.kitp.ucsb.edu/online/entangled15/ and http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.
  * [67] S. Sachdev and J. Ye,  _Gapless spin fluid ground state in a random, quantum Heisenberg magnet_ ,  Phys. Rev. Lett. 70, 3339 (1993),  10.1103/PhysRevLett.70.3339,  cond-mat/9212030.
  * [68] S. Sachdev,  _Holographic metals and the fractionalized Fermi liquid_ ,  Phys. Rev. Lett. 105, 151602 (2010),  10.1103/PhysRevLett.105.151602,  1006.3794.
  * [69] J. Maldacena and D. Stanford,  _Remarks on the Sachdev-Ye-Kitaev model_ ,  Phys. Rev. D94(10), 106002 (2016),  10.1103/PhysRevD.94.106002,  1604.07818.
  * [70] D. Grumiller, J. Salzer and D. Vassilevich,  _AdS 2 holography is (non-)trivial for (non-)constant dilaton_,  JHEP 12, 015 (2015),  10.1007/JHEP12(2015)015,  1509.08486.
  * [71] D. Grumiller, R. McNees, J. Salzer, C. Valcárcel and D. Vassilevich,  _Menagerie of AdS 2 boundary conditions_,  JHEP 10, 203 (2017),  10.1007/JHEP10(2017)203,  1708.08471.
  * [72] S. Carlip,  _What we don’t know about BTZ black hole entropy_ ,  Class. Quant. Grav. 15, 3609 (1998),  hep-th/9806026.
  * [73] A. Maloney, G. S. Ng, S. F. Ross and I. Tsiares,  _Generalized Gibbs Ensemble and the Statistics of KdV Charges in 2D CFT_ ,  JHEP 03, 075 (2019),  10.1007/JHEP03(2019)075,  1810.11054.
  * [74] E. M. Brehm and D. Das,  _On KdV characters in large $c$ CFTs_ (2019),  1901.10354.
  * [75] L. Donnay and C. Marteau,  _Carrollian Physics at the Black Hole Horizon_ (2019),  1903.09654.
  * [76] R. F. Penna,  _Near-horizon Carroll symmetry and black hole Love numbers_ (2018),  1812.05643.
  * [77] D. J. Gross and P. F. Mende,  _String Theory Beyond the Planck Scale_ ,  Nucl. Phys. B303, 407 (1988),  10.1016/0550-3213(88)90390-2.
  * [78] E. Halyo, A. Rajaraman and L. Susskind,  _Braneless black holes_ ,  Phys. Lett. B392, 319 (1997),  10.1016/S0370-2693(96)01544-4,  hep-th/9605112.
  * [79] E. J. Martinec,  _The Cheshire Cap_ ,  JHEP 03, 112 (2015),  10.1007/JHEP03(2015)112,  1409.6017.
  * [80] A. Bagchi, S. Chakrabortty and P. Parekh,  _Tensionless Strings from Worldsheet Symmetries_ ,  JHEP 01, 158 (2016),  10.1007/JHEP01(2016)158,  1507.04361.
  * [81] M. Bañados,  _Three-dimensional quantum geometry and black holes_ (1998),  hep-th/9901148.