Higher-Spin Flat Space Cosmologies with Soft Hair

# Higher-Spin Flat Space Cosmologies with Soft Hair

Martin Ammon, Theoretisch-Physikalisches Institut, Friedrich-Schiller University of Jena, Max-Wien-Platz 1, D-07743 Jena, Germany, EuropeInstitute for Theoretical Physics, TU Wien, Wiedner Hauptstrasse 8–10/136, A-1040 Vienna, Austria, EuropeCMCC-Universidade Federal do ABC, Santo André, S.P. BrazilUniversité libre de Bruxelles, Boulevard du Triomphe (Campus de la Plaine), 1050 Bruxelles, Belgium, Europe    Daniel Grumiller, Theoretisch-Physikalisches Institut, Friedrich-Schiller University of Jena, Max-Wien-Platz 1, D-07743 Jena, Germany, EuropeInstitute for Theoretical Physics, TU Wien, Wiedner Hauptstrasse 8–10/136, A-1040 Vienna, Austria, EuropeCMCC-Universidade Federal do ABC, Santo André, S.P. BrazilUniversité libre de Bruxelles, Boulevard du Triomphe (Campus de la Plaine), 1050 Bruxelles, Belgium, Europe    Stefan Prohazka, Theoretisch-Physikalisches Institut, Friedrich-Schiller University of Jena, Max-Wien-Platz 1, D-07743 Jena, Germany, EuropeInstitute for Theoretical Physics, TU Wien, Wiedner Hauptstrasse 8–10/136, A-1040 Vienna, Austria, EuropeCMCC-Universidade Federal do ABC, Santo André, S.P. BrazilUniversité libre de Bruxelles, Boulevard du Triomphe (Campus de la Plaine), 1050 Bruxelles, Belgium, Europe    Max Riegler, Theoretisch-Physikalisches Institut, Friedrich-Schiller University of Jena, Max-Wien-Platz 1, D-07743 Jena, Germany, EuropeInstitute for Theoretical Physics, TU Wien, Wiedner Hauptstrasse 8–10/136, A-1040 Vienna, Austria, EuropeCMCC-Universidade Federal do ABC, Santo André, S.P. BrazilUniversité libre de Bruxelles, Boulevard du Triomphe (Campus de la Plaine), 1050 Bruxelles, Belgium, Europe    and Raphaela Wutte Theoretisch-Physikalisches Institut, Friedrich-Schiller University of Jena, Max-Wien-Platz 1, D-07743 Jena, Germany, EuropeInstitute for Theoretical Physics, TU Wien, Wiedner Hauptstrasse 8–10/136, A-1040 Vienna, Austria, EuropeCMCC-Universidade Federal do ABC, Santo André, S.P. BrazilUniversité libre de Bruxelles, Boulevard du Triomphe (Campus de la Plaine), 1050 Bruxelles, Belgium, Europe
###### Abstract

We present and discuss near horizon boundary conditions for flat space higher-spin gravity in three dimensions. As in related work our boundary conditions ensure regularity of the solutions independently of the charges. The asymptotic symmetry algebra is given by a set of current algebras. The associated charges generate higher-spin soft hair. We derive the entropy for solutions that are continuously connected to flat space cosmologies and find the same result as in the spin-2 case: the entropy is linear in the spin-2 zero-mode charges and independent from the spin-3 charges. Using twisted Sugawara-like constructions of higher-spin currents we show that our simple result for entropy of higher-spin flat space cosmologies coincides precisely with the complicated earlier results expressed in terms of higher-spin zero mode charges.

\preprint

TUW–17–01

## 1 Introduction

Higher-spin theories provide useful insights into aspects of the holographic principle Klebanov:2002ja (); Sezgin:2002rt (); Giombi:2012ms (). Particularly three-dimensional higher-spin theories are useful in this context, since they can be formulated as Chern–Simons theories Blencowe:1988gj () with specific boundary conditions Henneaux:2010xg (); Campoleoni:2010zq (); Gaberdiel:2011wb (); Campoleoni:2011hg (). Developments in three-dimensional higher-spin theories include the discovery of minimal model holography Gaberdiel:2010pz (); Gaberdiel:2012uj (), higher-spin black holes Gutperle:2011kf (); Castro:2011fm (); Ammon:2012wc (); Bunster:2014mua (), non-AdS holography Gary:2012ms (); Gutperle:2013oxa (); Bergshoeff:2016soe (), higher-spin holographic entanglement entropy Ammon:2013hba (); deBoer:2013vca () and particularly flat space higher-spin theories Afshar:2013vka (); Gonzalez:2013oaa (), the main topic of the present work.

An interesting and potentially confusing aspect of higher-spin theories is that the metric and associated notions like curvature singularities or horizons are not gauge-invariant entities. Nevertheless, there are field configurations that most naturally are interpreted as (higher-spin) black holes or (higher-spin) cosmologies, i.e., as solutions with some characteristic temperature and entropy. Many of the physical questions inspired by black holes and cosmologies addressed in spin-2 gravity can also be addressed in a higher-spin context, sometimes in a straightforward way, but quite often with surprising generalizations and qualitatively new features emerging from the massless higher-spin interactions. In the present work we focus on one particular issue, namely on “soft hair” in flat space higher-spin theories in three dimensions.

The notion of “soft hair” was introduced in a spin-2 context in Hawking:2016msc () and refers to zero energy excitations on black hole horizons. To explicitly construct soft hair excitations, but more generally to address any question that requires the existence of a black hole as part of the question, it is then useful to have boundary conditions that ensure regular horizons for all configurations. While these boundary conditions can be re-interpreted as asymptotic fall-off conditions of Brown–Henneaux type Brown:1986nw (), they take their most natural form if expanded around the horizon. Thus, we shall refer to them as “near horizon boundary conditions”.

In AdS different near horizon boundary conditions were proposed independently in Donnay:2015abr (), Afshar:2015wjm () and Afshar:2016wfy (). In this work we focus on the latter approach, since it leads to the simplest symmetry algebras and due to the Chern–Simons formulation used in Afshar:2016wfy () it is most suitable for generalizations to higher-spins in AdS Grumiller:2016kcp () or flat space Afshar:2016kjj (). The main goal of the present work is to further generalize these results to higher-spins in flat space.

Our main results are new boundary conditions suitable for constructing soft higher-spin hair on flat space cosmologies and a remarkably simple expression for their entropy,

 S=2π(J+0+J−0) (1)

where are the spin-2 zero mode charges. Precisely the same result was found in AdS Einstein gravity Afshar:2016wfy (), in higher derivative gravity Setare:2016vhy (), in AdS higher-spin gravity Grumiller:2016kcp () and in flat space Einstein gravity Afshar:2016kjj (), where the soft hair on black hole horizons is replaced by soft hair on flat space cosmological horizons. The simplicity and universality of the result for the entropy (1) is intriguing.

This paper is organized as follows. In section 2 we present our near horizon boundary conditions and the associated symmetries. In section 3 we provide a map from diagonal to highest weight gauge. In section 5 we calculate the entropy of higher-spin flat space cosmologies and exploit the map from the previous section to match our simple result for entropy with the complicated results appearing in the literature. In section 6 we translate from Chern–Simons into second order formulation and give explicit results for metric and spin-3 field. Before we conclude we discuss in section 7 the generalization to fields with spin greater than . The appendices provide details on and algebras.

## 2 Near horizon boundary conditions and symmetries

Asymptotically flat higher-spin gravity in three spacetime dimensions is conveniently formulated in terms of a Chern–Simons theory. Restricting for simplicity to spin-3 gravity, the action reads

 I[A]=k4π∫⟨CS(A)⟩,withCS(A)=A∧dA+23A∧A∧A, (2)

with Chern-Simons coupling and gauge field valued in . The generators of are denoted by with and . While and generate Lorentz-Transformations and translations, respectively, and generate associated spin-3 transformations. We refer the reader to appendix A for the commutation relations satisfied by the generators as well as for the definition of the non-degenerate invariant symmetric bilinear form . Moreover, we use coordinates , where denotes the radial coordinate, the advanced time and the angular coordinate.

In order to specify our boundary conditions we first use some of the gauge freedom at our disposal to fix the radial dependence of the connection as

 A=b−1(a+d)b, (3)

where the radial dependence is encoded in the group element as Afshar:2016kjj ()

 b=exp(1μPM1)exp(r2M−1). (4)

 a=avdv+aφdφ. (5)

We propose the following new near-horizon boundary conditions111The relation to the notation used in Afshar:2016kjj () is given by , , , .

 aφ =JL0+PM0+J(3)U0+P(3)V0, (6a) av =μPL0+μJM0+μ(3)PU0+μ(3)JV0. (6b)

All the functions appearing in (6) are in principle arbitrary functions of the advanced time and the angular coordinate . The functions are identified as chemical potentials and thus are fixed in such a way that . The equations of motion

 F=dA+[A,A]=0 (7)

put further constraints on the functions as well as that can be interpreted as holographic Ward identities. These constraints force the state dependent functions to obey the following time evolution equations

 ∂vJ=∂φμP,∂vP=∂φμJ,∂vJ(3)=∂φμ(3)P,∂vP(3)=∂φμ(3)J. (8)

In particular, for -independent chemical potentials the holographic Ward identities (8) imply conservation of all the state dependent functions.

### 2.1 Canonical charges and near horizon symmetry algebra

The next step in the asymptotic symmetry analysis is to determine the gauge transformations that preserve the boundary conditions (3)–(6). The gauge parameters that encode such transformations are given by

 ϵ=b−1(ϵPL0+ϵJM0+ϵ(3)PU0+ϵ(3)JV0)b. (9)

As a consequence also the infinitesimal transformation behavior of the state dependent functions takes a particularly simple form

 (10)

Moreover, the conserved charges associated to boundary conditions preserving transformations may be computed via the Regge-Teitelboim approach Regge:1974zd (), where their variation is given by

 (11)

Evaluating this expression for our case yields

 δQ[ϵ]=k2π∫dφ⟨ϵδAφ⟩=k2π∫dφ(ϵJδJ+ϵPδP+43ϵ(3)JδJ(3)+43ϵ(3)PδP(3)). (12)

The global charges may now be obtained by functionally integrating (12),

 Q[ϵ]=k2π∫dφ⟨ϵAφ⟩=k2π∫dφ(ϵJJ+ϵPP+43ϵ(3)JJ(3)+43ϵ(3)PP(3)). (13)

After having determined the canonical boundary charges, their Dirac bracket algebra can be read off from their infinitesimal transformation behavior using

 δYQ[X]={Q[X],Q[Y]}. (14)

This yields

 {J(φ),P(¯φ)}=k2π∂φδ(φ−¯φ),{J(3)(φ),P(3)(¯φ)}=2k3π∂φδ(φ−¯φ), (15)

where all other Dirac brackets vanish. Expanding into Fourier modes

 J(φ) =1k∑n∈ZJne−inφ P(φ) =1k∑n∈ZPne−inφ (16a) J(3)(φ) =34k∑n∈ZJ(3)ne−inφ P(3)(φ) =34k∑n∈ZP(3)ne−inφ (16b)

(with the usual decomposition of the -function, ), and replacing the Dirac brackets by commutators using we obtain the following asymptotic symmetry algebra for the boundary conditions (3)–(6)

 [Jn,Pm]=knδn+m,0,[J(3)n,P(3)m]=4k3nδn+m,0 (17)

with all other commutators vanishing. At this point it should also be noted that the algebra (17) can be brought to the same form as in Grumiller:2016kcp () by making the redefinitions

 J±±n=12(Pn±Jn),J(3)±±n=12(P(3)n±J(3)n). (18)

The generators and then satisfy

 [J+n,J+m] =[J−n,J−m]=k2nδn+m,0, [J+n,J−m]=0, (19a) =[J(3)−n,J(3)−m]=2k3nδn+m,0, [J(3)+n,J(3)−m]=0. (19b)

In particular, we obtain in total four current algebras, two of which have level and the remaining two have level

### 2.2 Soft Hair

In this subsection we show that the states generated by acting with arbitrary combinations of near horizon symmetry generators (17) on some reference state all have the same energy and thus correspond to soft hair excitations of that reference state. In order to show this we first determine the Hamiltonian in terms of near horizon variables, then proceed in building modules using (17), and finally show that all states in these modules have the same energy eigenvalue.

The Hamiltonian is associated to the charge that generates time translations. In the metric formulation this would correspond to the Killing vector . Since the gauge transformations (9) are related on-shell to the asymptotic Killing vectors via , the variation of the charge associated to translations in the advanced time coordinate can be determined via

 δH:=δQ[∂v] =k2π∫dφ⟨ξvAvδAφ⟩=k2π∫dφ⟨AvδAφ⟩ =k2π∫dφ(μJδJ+μPδP+43μ(3)JδJ(3)+43μ(3)PδP(3)). (20)

This expression can be trivially functionally integrated to yield the Hamiltonian

 H=k2π∫dφ(μJJ+μPP+43μ(3)JJ(3)+43μ(3)PP(3)). (21)

For constant chemical potentials and the Hamiltonian reduces to

 (22)

After having determined the Hamiltonian the next step in our analysis is to build modules using (17). There are two ways of building modules relevant to our analysis. One is via highest weight representations wheres the other one uses a construction similar to induced representations.
We first start with modules built from highest weight representations of (17). Assume that there is a highest weight (vacuum) state satisfying

 Jn|0⟩=Pn|0⟩=J(3)n|0⟩=P(3)n|0⟩=0,∀n≥0. (23)

New states can then be constructed from such a vacuum state by repeated application of operators with negative Fourier mode number as

 |ψ({p})⟩∼∏ni>0J−ni∏n(3)i>0J(3)−n(3)i∏mi>0P−mi∏m(3)i>0P(3)−m(3)i|0⟩, (24)

where . Since the Hamiltonian is a linear combination of , , and , it is evident that the Hamiltonian commutes with any element appearing in the asymptotic symmetry algebra (17). Thus when acting with on any one obtains the same value for the energy for all possible ’s. This proves our claim that the states are “soft hair” of the vacuum; similar considerations apply when replacing the vacuum with any other state, such as some flat space cosmology, which can then be decorated with soft spin-2 and spin-3 hair.
Now we investigate the same issue for modules built from representations that are similar in spirit to the induced representations found in flat space holography (see e.g. Campoleoni:2015qrh (); Campoleoni:2016vsh ()). In the following we consider all “boosted” states that can be built from a “rest frame” state via

 |ψ({q})⟩∼∏ni(Jni)∏n(3)i(J(3)n(3)i)|Ω⟩, (25)

where . For a given “rest frame” state one can generate [] “boosted” states as written in (25). In addition this “rest frame” state has to satisfy

 Pn|Ω⟩=P(3)n|Ω⟩=0,∀n∈Z. (26)

One way to argue such representations is via taking an ultra-relativistic limit of the highest-weight representations used in Grumiller:2016kcp (). On the level of generators the ASA222We focus only on the generators and . The argument can be repeated in the exact same way for and . in Grumiller:2016kcp () and the one in (17) are related via an ultra-relativistic boost that can be incorporated as

 J±±n=12(Pnϵ±Jn), (27)

in the limit . By looking at highest-weight representations built from one finds that in terms of the generators and one has

 J±n|Ω⟩=12(P±nϵ±J±n)|Ω⟩=0,∀n≥0. (28)

In order to satisfy these relations when one finds that, indeed, acting with on has to be zero for all values of , whereas one can act with on without spoiling (28). One can now again act with the Hamiltonian (22) on all states in the module (25) and using the same line of argument as for the highest weight representations one finds again that all states have the same energy eigenvalue and can thus be interpreted as soft excitations as well.

Thus, the soft hair property does not depend on whether highest weight or representations of the form (25) are used. Moreover, since the Hamiltonian is an element of the Cartan subalgebra of , one can even conclude that the soft hair property is independent of any representation that can be built via acting on some reference state using the near horizon symmetry generators.

## 3 Relating near horizon and asymptotic symmetries

In order to show that the spin-2 and spin-3 charges of higher-spin cosmological solutions in flat space emerge as composite operators constructed from the ones, we have to relate the boundary conditions we presented in this work (6) with the boundary conditions that describe a flat space cosmology with spin-2 and spin-3 hair. Thus it is first necessary to describe both boundary conditions by the same set of variables.

Flat space cosmologies with spin-2 and spin-3 hair including chemical potentials are given by the following connection Gary:2014ppa ()

 ~A=~b−1(~a+d)~b, (29)

with and

 ~aφ =L1−M4L−1−N2M−1+V2U−2+ZV−2 (30a) ~av =a(0)v+a(μ\tiny M)v+a(μ\tiny L)v+a(μ\tiny V)v+a(μ% \tiny U)v (30b)

where

 a(0)v =M1−M4M−1+V2V−2 (31a) a(μ\tiny M)v =μ\tiny MM1−μ′\tiny MM0+12(μ′′\tiny M−12Mμ\tiny M)M−1+12Vμ\tiny MV−2 (31b) a(μ\tiny L)v =a(μ\tiny M)v∣∣M→L−12Nμ\tiny LM−1+Zμ\tiny LV−2 (31c) a(μ\tiny V)v =μ\tiny VV2−μ′\tiny VV1+12(μ′′\tiny V−Mμ\tiny V)V0+16(−μ′′′\tiny V+M′μ\tiny V% +52Mμ′\tiny V)V−1 +124(μ′′′′\tiny V−4Mμ′′\tiny V−72M′μ′\tiny V+32M2μ%V−M′′μ\tiny V)V−2−4Vμ\tiny VM−1 (31d) a(μ\tiny U)v =a(μ\tiny V)v∣∣M→L−8Zμ\tiny UM−1−Nμ\tiny UV0+(56Nμ′\tiny U+13N′μ\tiny U)V−1 +(−13Nμ′′\tiny U−724N′μ′\tiny U−112N′′μ\tiny U+14MNμ\tiny U)V−2. (31e) All functions appearing in (30) and (31) are free functions of v and φ. As such a prime denotes a derivative with respect to φ and a dot a derivative with respect to v. The subscript M→L denotes that in the corresponding quantity all generators and chemical potentials are replaced as Mn→Ln, Vn→Un, μ\tiny M→μ\tiny L and μ\tiny V→μ\tiny U, i.e. a(μ\tiny M)v∣∣M→L =μ\tiny LL1−μ′\tiny LL0+12(μ′′\tiny L−12Mμ\tiny L)L−1+12Vμ\tiny LU−2 (31f) a(μ\tiny V)v∣∣M→L =μ\tiny UU2−μ′\tiny UU1+12(μ′′\tiny U−Mμ\tiny U)U0+16(−μ′′′\tiny U+M′μ\tiny U% +52Mμ′\tiny U)U−1 +124(μ′′′′\tiny U−4Mμ′′\tiny U−72M′μ′\tiny U+32M2μ%U−M′′μ\tiny U)U−2−4Vμ\tiny UL−1. (31g)

The next step is to find an appropriate gauge transformation that maps the connection in (6) to the connection in (30) via . After a fair amount of algebraic manipulation one can find the following group element that provides the appropriate map as with

 g(1) =exp[lL1+mM1+u1U1+v1V1+u2U2+v2V2] (32a) g(2) =exp[−J2L−1−J(3)3U−1+16(JJ(3)+J(3)′2)U2 (32b) −J2M−1−P(3)3V−1+16(PJ(3)+JP(3)+P(3)′2)V−2]. (32c)

The functions , , and depend on and only and have to satisfy

 l′ =1+lJ+2u1J(3) (33a) m′ =lP+mP+2u1P(3)+2v1J(3) (33b) u′1 =u1J+2lJ(3) (33c) v′1 =u1P+v1J+2lP(3)+2mJ(3) (33d) u′2 =−u12+2u2J (33e) v′2 =−v12+2u2P+2v2J, (33f)

and

 μ\tiny L (34a) μ\tiny M =43μ\tiny UP(3)+43μ\tiny VJ(3)−μPm−μJl−2μ(3)Pv1−2μ(3)Ju1+˙m (34b) μ\tiny U =−2μPu2+μ(3)Pl2+μ(3)Pu21+12u1˙l−12l˙u1+˙u2 (34c) μ\tiny V =−2μPv2−2μJu2+2lmμ(3)P+2u1v1μ(3)P+μ(3)Jl2−μ(3)Ju21 +12v1˙l+12u1˙m−12m˙u1−12l˙v1+˙v2. (34d)

Consistency with the on-shell relations (8) also requires that

 μP =μ\tiny LP+83μ\tiny U% JJ(3)+43μ\tiny UJ′−23μ′\tiny UJ−μ′% \tiny L (35a) μJ =μ\tiny MP+83μ\tiny U% PJ(3)+83μ\tiny UJP(3)+83μ\tiny VJJ(3) +43μ\tiny UP′+43μ\tiny VJ′−23μ′\tiny U% P−23μ′\tiny VJ−μ′\tiny M (35b) μ(3)P (35c) μ(3)J =μ\tiny LP(3)+μ\tiny MJ(3)+2μ\tiny UPJ+μ% \tiny VJ2−83μ\tiny UP(3)J(3)−43μ\tiny V(J(3))2 −μ\tiny VJ′−μ% \tiny UP′−32μ′\tiny VJ−32μ′\tiny UP+12μ′′\tiny V. (35d)

The gauge fields and are then mapped to each other provided the following (twisted) Sugawara-like relations hold between the near horizon state-dependent functions , , , and the asymptotic state-dependent functions , , , :

 M =J2+43(J(3))2+2J′ (36a) N =JP+43J(3)P(3)+P′ (36b) V (36c) Z =136(6J2P(3)−8P(3)(J(3))2+3P(3)J′+3J(3)P′ +9JP(3)′+9PJ(3)′+12PJJ(3)+3P(3)′′). (36d)

In addition one can explicitly check that the equations of motion

 ˙M =−2μ′′′\tiny L+2Mμ′\tiny L+M′μ\tiny L+24Vμ′\tiny U+16V′μ\tiny U (37a) ˙N =12˙M∣∣L→M+2Nμ′%L+N′μ\tiny L+24Zμ′\tiny U+16Z′μ\tiny U (37b) ˙V =112μ′′′′′\tiny U−512Mμ′′′\tiny U−58M′μ′′\tiny U−38M′′μ′\tiny U+13M2μ′\tiny U −112M′′′μ% \tiny U+13MM′μ\tiny U+3Vμ′\tiny L+V′μ\tiny L (37c) ˙Z =12˙V∣∣L→M−512Nμ′′′\tiny U−58N′μ′′\tiny U−38N′′μ′\tiny U+23MNμ′\tiny U% −112N′′′μ% \tiny U+13(MN)′μ\tiny U+3Zμ′\tiny L+Z′μ\tiny L, (37d) with 12˙M∣∣L→M =−μ′′′\tiny M+Mμ′\tiny M+12M′(1+μ% \tiny M)+12Vμ′\tiny V+8V′μ\tiny V (37e) 12˙V∣∣L→M =124μ′′′′′\tiny V−524Mμ′′′\tiny V−516M′μ′′\tiny V−316M′′μ′\tiny V+16M2μ′\tiny V −124M′′′μ% \tiny V+16MM′μ\tiny V+32Vμ′\tiny M+12V′(1+μ\tiny M), (37f)

indeed reduce to the simple ones given by (8). The relations (35) show that the “asymptotic chemical potentials” , , , depend not only on the “near horizon chemical potentials” , , , but also on the state-dependent functions , , , , which is one way to see that our near horizon boundary conditions (3)–(6) are inequivalent to the asymptotic ones of Afshar:2013vka (); Gonzalez:2013oaa (). Moreover, the same relations directly map the corresponding gauge parameters that preserve the respective boundary conditions by replacing , , , as well as , , and . Therefore, also the infinitesimal transformation laws for , , and can be directly read off from (37) by replacing e.g. by as well as all occurrences of chemical potentials and by the corresponding gauge parameters and , respectively.

Thus, one can readily see that the fields , , and transform exactly in such a way that they satisfy the algebra. Note, however, that their associated canonical charges still satisfy the (semidirect sum of four) current algebras as before. This can be seen by looking at the variation of the canonical boundary charge. In particular, after using the infinitesimal gauge transformations encoded in (37), the relations between the chemical potentials (35) and the Miura-like transformations (36) reduces to

 δQ =k2π∫dφ(ϵδN+τ2δM+8\raisebox0.0pt$χ$δZ+4κδV) =k2π∫dφ(ϵJδJ+ϵPδP+43ϵ(3)JδJ(3)+43ϵ(3)PδP(3)). (38)

## 4 FW-Algebras from Heisenberg

In this section we relate the -algebra to the near-horizon Heisenberg (or current) algebras. Using the (twisted) Sugawara-like relations (36) between the state-dependent functions as well as their Fourier mode expansions (16) and

 N(φ) =1k∑n∈ZLne−inφ M(φ) =2k∑n∈ZMne−inφ (39a) Z(φ) =√38k∑n∈ZUne−inφ V(φ) =√34k∑n∈ZVne−inφ (39b)

one finds that the (twisted) Sugawara construction for the algebra is given by

 Ln =1k∑p∈Z(Jn−pPp+34J(3)n−pP(3)p)−inPn (40a) Mn =12k∑p∈Z(Jn−pJp+38J(3)n−pJ(3)p)−inJn (40b) Un =√3k2∑p,q∈Z[(Jn−p−qJp−34J(3)n−p−qJ(3)p)J(3)