Paper
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Modular Properties of 3D Higher Spin Theory
Authors
Wei Li,Feng-Li Lin
Chih-Wei Wang,Elizabeth Boulding,JIANXIN LU,Marco Cattaneo,Petr Levchenko,Ralf Kneuper,George Leontaris,Qingcui Bu,Alexander Sevrin,Grazyna Niewiadomska,William Bell,Nathália Giuliatti,Volker Adler,Kirk Winemiller,Ignas Grigelionis,Colin Bernet,John Lord,Dalimil Mazáč,Martin Goodchild,Harvey Burd,Keisuke Yano,Kunal Vyas,Daniel Wiegand,Peter Wolfsteiner,Kristian Jensen,Garrett Poe,Raffaele Savelli,Sian Kathryn Jones,Heather Lynn Lindon,Pedro Aragão,Alain Jaccard,Carolyn Dzierba,Branko Vucijak,Simone Giacomelli,Andrew Degnan,Luis Garcés-Erice,Joseph Fotsing,Pavel Fileviez Perez,Digamber Porob,Srdjan Korac,David Van Wie,Nikolay Bobev,Seok Ki Choi,Luciano Panek,Vishnu Ram OV,José Tormos,Theodore Castro-Santos,Marie Connett,Gijs van Dijck,Shengqiang Shu,Thorsten Alexander Kern,Martin Köhne,Niels-Henrik Møller Hansen,Tomoko Ariga,Gerasimos Spanakis,Bo Feng,Mukund Rangamani,Josep D. Asis,Dan Andersen,Elena Mozhaikina,Vitor Oguri,Akitaka Ariga,Kimmo Tuominen,Hugh Beckie,Nasuf SONMEZ,Uwe Spillmann,Noélia Correia,Rogerio Rocha,Vania Schneider,Ramon Gomez Jimenez,Khairuzzaman Mamun,Jefferson Prado,Brian Nattress,Takahisa Miyatake,José Kullberg,Begoña López Bueno,Torsten Dahms,Ciro Pistillo,Dariusz Banaś,Torben Ferber,Martín Aluja,Dmitry Golubkov,Valentin Geisler-Knünz,Andreas R. Ziegler,Rajamani Narayanan,Aysel Kayis Topaksu,Jana Rubesova,Emanuele ZAMPERINI,Hüseyin Bahtiyar,ILKNUR HOS,Catarina Reis de Oliveira,Andy Buckley,Rachel Bartek,Juan Maria Vazquez,Patricia Sanchez-Baracaldo,Guilherme Da Silva Pereira,Gustavo Alberto Burdman,Marek Jeżabek,Lars Adiels,Marko Stevovic,Mark Jaksa,Neal Weiner,Paulo Assis,Gaurav Gupta,Roberto Auzzi,Jessica Huber,Olivier Marchal,Aude Gehrmann-De Ridder,Koenraad Schalm,Sakura Schafer-Nameki,Nilanjan Brahma,Roberto Emparan,František Sedláček,Dan Melconian,Engel Roza,Ryan Pfeiffer,Dmitrii Nesterenko,Livia Ferro,Pamela Hadley,Justyna Zwolak,Joao Penedones,EMILIO A MARTINEZ,Владимир Елин,Aleksi Vuorinen,Chengkang Zhang,Patipan Uttayarat,Anna ,Leopoldo Pando Zayas,Chaitali Sengupta,Nico Callewaert,Iosif Bena,Nicole Schmitt,Sai Putcha,Daniele Musso,Peter Panfilov,Jonathan Cordeiro,Stephane Coulombe,Patrizia Calefato,David Russell,Antonio Marti,Kapil Dandekar,Matthias Gaberdiel,Krassimira Moutafova,Thomas Gehrmann,Vadim Kostyukhin,Zoltan Elekes,Hooi Jin Ong,Martin Smith,Rafel Escribano,Roberto Mignani,Lian Tao,Francesca Di Lodovico,Neus Lopez March,Arantza Oyanguren,Timothy Hollowood,Matthew Buckley,Phillip Litchfield,Jeffrey Duda,Alfons Weber,Georgios Christodoulou,Joanna Zalipska,Maurizio Martinelli,Ruth Durrer,Leonardo de Lima,Adrian del Rio Vega,David Mitchell,Teresa Gibert,Benjamin Grinstein,Marcin Badziak,Ben Hoare,Francis Johnson,Ronnie Machielsen,Daniel Thompson,Constantinos Papageorgakis,Irina Likhanova,Mario Merola,Rosa Maria VITRANO,Cora Gheorghe-Bulmau,Aleksandra Rasic Markovic,Gianguido Dall'Agata,Arbab Arbab,Mark Pesaresi,Salvatore My,Leif Lönnblad,Xavier Janssen,Junghwan Goh,Brian Walsh,Jeffrey Hutchings,Nuno Anjos,James Ferrando,Michele Cascella,Thomas Panagopoulos,Oktay Doğangün,Wuming Luo,Erhan Gülmez,Josep Flix,Bozydar Wrona,Brian Walker,Yasutaka KUBO,Wei Song,Tomasz Lukowski,Roberto Soria,Valeriu Tudose,Alexander Liu,Lara Lloret Iglesias,Siobhan Howard,Lee Bulla,Luísa Ribas,S. Prem Kumar,Benjamin R. Safdi,Nigel Glover,Anton Martsev,N/A ,Francisco Díaz-Fleischer,Gonzalo Merino,Yakov Yadgarov,Edward Szczerbicki,stefano bianco,gokcen orhan,Zhen-Yi Cai,El-sayed El-Dahshan,Diego Rodriguez-Gomez,IONEL LAZANU,Mikko Laine,Jesús Galech,Máximo Pló Casasús,Anton Poluektov,Sainulabdeen Mohamed Junaideen,Bartlomiej Czech,Humberto Gomez,Tobias Huber,Pedro Arce,Mir Nahidul Ambia,Leo Wiggers,Ryotaku Suzuki,Stefano Ragazzi,José R. 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arXiv:1308.2959v3 [hep-th] 18 Dec 2013
Modular Properties of 3D Higher Spin
Theory
Wei Li,a Feng-Li Lin,b and Chih-Wei Wangb
aMax-Planck-Institut f¨ur Gravitationsphysik, Albert-Einstein-Institut,
Am M¨uhlenberg 1, 14476 Golm, Germany
wei.li@aei.mpg.de
bDepartment of Physics, National Taiwan Normal University,
Taipei, 116, Taiwan
linfengli@phy.ntnu.edu.tw, freeform1111@gmail.com
Abstract
In the three-dimensional sl(N ) Chern-Simons higher-spin theory, we prove that the
conical surplus and the black hole solution are related by the S-transformation of the
modulus of the boundary torus. Then applying the modular group on a given conical
surplus solution, we generate a ‘SL(2, Z)’ family of smooth constant solutions. We then
show how these solutions are mapped into one another by coordinate transformations
that act non-trivially on the homology of the boundary torus.
After deriving a thermodynamics that applies to all the solutions in the ‘SL(2, Z)’
family, we compute their entropies and free energies, and determine how the latter
transform under the modular transformations. Summing over all the modular images
of the conical surplus, we write down a (tree-level) modular invariant partition function.
Modular Properties of 3D Higher Spin
Theory
Wei Li,a Feng-Li Lin,b and Chih-Wei Wangb
aMax-Planck-Institut f¨ur Gravitationsphysik, Albert-Einstein-Institut,
Am M¨uhlenberg 1, 14476 Golm, Germany
wei.li@aei.mpg.de
bDepartment of Physics, National Taiwan Normal University,
Taipei, 116, Taiwan
linfengli@phy.ntnu.edu.tw, freeform1111@gmail.com
Abstract
In the three-dimensional sl(N ) Chern-Simons higher-spin theory, we prove that the
conical surplus and the black hole solution are related by the S-transformation of the
modulus of the boundary torus. Then applying the modular group on a given conical
surplus solution, we generate a ‘SL(2, Z)’ family of smooth constant solutions. We then
show how these solutions are mapped into one another by coordinate transformations
that act non-trivially on the homology of the boundary torus.
After deriving a thermodynamics that applies to all the solutions in the ‘SL(2, Z)’
family, we compute their entropies and free energies, and determine how the latter
transform under the modular transformations. Summing over all the modular images
of the conical surplus, we write down a (tree-level) modular invariant partition function.
Contents
1 Introduction and Summary 1
2 Basics of 3D higher-spin theory 3
2.1 Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Asymptotic symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Smooth solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Thermodynamics 11
3.1 Variational principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 On-shell action, free energy, and entropy . . . . . . . . . . . . . . . . . . . . 17
4 Conical surplus and black hole are S-dual 21
4.1 S-transformation of holonomy condition and on-shell charges . . . . . . . . . 22
4.2 Coordinate transformation between conical surplus and black hole . . . . . . 25
4.3 S-transformation of free energies . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.4 Example-1: N = 3 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5 SL(2, Z) family of smooth solutions 32
5.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.2 Reasoning from dual CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3 Coordinate transformations between members of ‘SL(2, Z)’ family . . . . . . 36
5.4 Mapping of the free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.5 Modular invariant full partition function . . . . . . . . . . . . . . . . . . . . 40
6 Discussion 40
6.1 Modular invariance of integrability condition . . . . . . . . . . . . . . . . . . 41
6.2 Canonical vs. holomorphic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
A Solving σs in terms of µs 44
B SL(2, Z) family in spin-2 case 45
C Example-2: N = 4 46
1 Introduction and Summary
In the theory of the three-dimensional pure Einstein gravity with a negative cosmological
constant, as there is no propagating degree of freedom in the bulk, all asymptotically AdS so-
lutions are locally diffeomorphic and differ only in their global structures [1–4]. In Euclidean
signature, starting with a thermal AdS3, whose conformal boundary is a torus with modulus
τ , we can obtain an ‘SL(2, Z)’ family of solutions via modular transformations (τ 7 → aτ +b
cτ +d )
on the modulus of the boundary torus [5]. In particular, the S-transformation τ 7 → − 1
τ maps
1
1 Introduction and Summary 1
2 Basics of 3D higher-spin theory 3
2.1 Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Asymptotic symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Smooth solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Thermodynamics 11
3.1 Variational principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 On-shell action, free energy, and entropy . . . . . . . . . . . . . . . . . . . . 17
4 Conical surplus and black hole are S-dual 21
4.1 S-transformation of holonomy condition and on-shell charges . . . . . . . . . 22
4.2 Coordinate transformation between conical surplus and black hole . . . . . . 25
4.3 S-transformation of free energies . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.4 Example-1: N = 3 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5 SL(2, Z) family of smooth solutions 32
5.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.2 Reasoning from dual CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3 Coordinate transformations between members of ‘SL(2, Z)’ family . . . . . . 36
5.4 Mapping of the free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.5 Modular invariant full partition function . . . . . . . . . . . . . . . . . . . . 40
6 Discussion 40
6.1 Modular invariance of integrability condition . . . . . . . . . . . . . . . . . . 41
6.2 Canonical vs. holomorphic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
A Solving σs in terms of µs 44
B SL(2, Z) family in spin-2 case 45
C Example-2: N = 4 46
1 Introduction and Summary
In the theory of the three-dimensional pure Einstein gravity with a negative cosmological
constant, as there is no propagating degree of freedom in the bulk, all asymptotically AdS so-
lutions are locally diffeomorphic and differ only in their global structures [1–4]. In Euclidean
signature, starting with a thermal AdS3, whose conformal boundary is a torus with modulus
τ , we can obtain an ‘SL(2, Z)’ family of solutions via modular transformations (τ 7 → aτ +b
cτ +d )
on the modulus of the boundary torus [5]. In particular, the S-transformation τ 7 → − 1
τ maps
1
the AdS3 into a BTZ black hole. The full modular invariant partition function consists of
a sum of all the modular images of the AdS3 partition function: Z[τ ] = ∑ ZAdS3 [aτ +b
cτ +d ] [6].
One can then study the phase structure using Z[τ ], and for example show how Hawking-
Page transition [7] occurs when τ moves across the boundary between different fundamental
domains [5, 6, 8].
Vasiliev’s higher-spin theory is a generalization of the gravity theory; besides the gravi-
ton, it contains massless spin-s fields with s ≥ 3; and it lives in spaces with non-zero constant
curvatures, i.e. AdS or dS spaces [9–12]. In three dimensions, it can be consistently trun-
cated to a Chern-Simons subsector after the scalar in the theory is decoupled [13, 14].1 We
only consider AdS space in this paper. In AdS3, the gauge algebra of this Chern-Simons
theory is an infinite-dimensional Lie algebra hs[λ] [15, 16], which at λ = N reduces to the
finite-dimensional sl(N) [9, 17]. The 3D Chern-Simons high-spin theory is a straightforward
generalization of the sl(2) Chern-Simons theory (the alternative formulation of the 3D pure
gravity with a negative cosmological constant [18,19]) and share its essential features: in par-
ticular, it does not have any propagating degree of freedom in the bulk of the three-manifold
M; the topology of M and the boundary data on ∂M determine the dynamics.
In the Chern-Simons higher-spin theory, two types of smooth solutions have been found
and studied: the conical surplus constructed by [20] and the black hole by [21]. They can be
viewed as the higher-spin-charge-carrying generalizations of the AdS3 and BTZ black hole,
respectively. Since the notion of the boundary torus still exists in the higher-spin version of
the Chern-Simons theory, we can ask whether these two solutions, the conical surplus and
the black hole, are related via an S-transformation of the modulus of the boundary torus.
In some sense, this has to happen since these two solutions reduce to AdS3 and BTZ
when all the higher-spin charges are set to zero. The non-trivial part of the story is this:
the boundary modulus τ can be considered as the thermodynamical conjugate of the spin-2
charge, and in the higher-spin theory, the spin-2 field is coupled with all the higher-spin
fields, therefore a transformation of τ inevitably induces the corresponding transformations
on the chemical potentials of all the higher-spin charges. The crux in generalizing the
modular properties of the spin-2 theory to a higher-spin theory is in determining this induced
transformations on the higher-spin chemical potentials.
For this purpose, one first needs a consistent description of the thermodynamics of the
given solution. Up till now this is absent for the conical surplus; however for the black hole
an extensive literature on its thermodynamics has already emerged: e.g. a ‘holomorphic’
approach represented by [21–26] and a ‘canonical’ one represented by [27–29].
In this paper we choose the ‘canonical’ formalism because, as we will show later, in this
formalism important quantities and equations are manifestly modular invariant or covariant.
1In 3D, the scalar does not sit in the higher-spin multiplet therefore can be consistently decoupled.
2
a sum of all the modular images of the AdS3 partition function: Z[τ ] = ∑ ZAdS3 [aτ +b
cτ +d ] [6].
One can then study the phase structure using Z[τ ], and for example show how Hawking-
Page transition [7] occurs when τ moves across the boundary between different fundamental
domains [5, 6, 8].
Vasiliev’s higher-spin theory is a generalization of the gravity theory; besides the gravi-
ton, it contains massless spin-s fields with s ≥ 3; and it lives in spaces with non-zero constant
curvatures, i.e. AdS or dS spaces [9–12]. In three dimensions, it can be consistently trun-
cated to a Chern-Simons subsector after the scalar in the theory is decoupled [13, 14].1 We
only consider AdS space in this paper. In AdS3, the gauge algebra of this Chern-Simons
theory is an infinite-dimensional Lie algebra hs[λ] [15, 16], which at λ = N reduces to the
finite-dimensional sl(N) [9, 17]. The 3D Chern-Simons high-spin theory is a straightforward
generalization of the sl(2) Chern-Simons theory (the alternative formulation of the 3D pure
gravity with a negative cosmological constant [18,19]) and share its essential features: in par-
ticular, it does not have any propagating degree of freedom in the bulk of the three-manifold
M; the topology of M and the boundary data on ∂M determine the dynamics.
In the Chern-Simons higher-spin theory, two types of smooth solutions have been found
and studied: the conical surplus constructed by [20] and the black hole by [21]. They can be
viewed as the higher-spin-charge-carrying generalizations of the AdS3 and BTZ black hole,
respectively. Since the notion of the boundary torus still exists in the higher-spin version of
the Chern-Simons theory, we can ask whether these two solutions, the conical surplus and
the black hole, are related via an S-transformation of the modulus of the boundary torus.
In some sense, this has to happen since these two solutions reduce to AdS3 and BTZ
when all the higher-spin charges are set to zero. The non-trivial part of the story is this:
the boundary modulus τ can be considered as the thermodynamical conjugate of the spin-2
charge, and in the higher-spin theory, the spin-2 field is coupled with all the higher-spin
fields, therefore a transformation of τ inevitably induces the corresponding transformations
on the chemical potentials of all the higher-spin charges. The crux in generalizing the
modular properties of the spin-2 theory to a higher-spin theory is in determining this induced
transformations on the higher-spin chemical potentials.
For this purpose, one first needs a consistent description of the thermodynamics of the
given solution. Up till now this is absent for the conical surplus; however for the black hole
an extensive literature on its thermodynamics has already emerged: e.g. a ‘holomorphic’
approach represented by [21–26] and a ‘canonical’ one represented by [27–29].
In this paper we choose the ‘canonical’ formalism because, as we will show later, in this
formalism important quantities and equations are manifestly modular invariant or covariant.
1In 3D, the scalar does not sit in the higher-spin multiplet therefore can be consistently decoupled.
2
In the ‘canonical’ formalism we generalize the thermodynamics of the black hole to include
all smooth stationary solutions.
Once the thermodynamics of the conical surplus is established, we determine the required
transformation on the higher-spin chemical potential accompanying the S-transformation on
the boundary modulus τ 7 → − 1
τ , and prove that the conical surplus and the black hole are
S-dual. We then show that the black hole and the conical surplus are related by a coordinate
transformation (that changes the modular parameter of the boundary torus from τ to − 1
τ ).
Then we generalize from the S-transformation to the full modular group: starting with
a higher-spin-charge-carrying conical surplus solution and applying on it the full modular
group, we can generate an ‘SL(2, Z)’ family of smooth stationary solutions. They are all con-
nected by coordinate transformations that act non-trivially on the homology of the boundary
torus. Their free energies hence their on-shell partition functions are related via modular
transformations. The full modular invariant partition function then involves summing over
all the modular images of the conical surplus.
The paper is organized as follows. In Section 2 we briefly review some basics of 3D sl(N)
higher-spin theory, summarize known stationary smooth solutions, and define new smooth
stationary solutions in the ‘SL(2, Z)’ family. In Section 3 we formulate a thermodynamics
that is universal to all members of the ‘SL(2, Z)’ family (including conical surplus and black
hole). Then in Section 4 we prove that a conical surplus can be mapped into a black hole
via an S-transformation of the modulus of the boundary torus. In section 5 we show how to
generate an ‘SL(2, Z)’ family of smooth solutions. We summarize and discuss open problems
in section 6. In Appendix A we present a detailed proof for a statement that is central to
our paper; in Appendix B we review the spin-2 story; finally in Appendix C we discuss sl(4)
theory as a concrete example.
2 Basics of 3D higher-spin theory
In this section we first review some basics of the three-dimensional sl(N) higher-spin theory
(for more details see the earlier works [20,21,30] and the reviews [31,32]). Then we summarize
known smooth solutions in this theory, i.e. the conical surplus and the black hole, and
meanwhile prepare the readers for our later discussion on general smooth solutions. In this
paper we focus on stationary, axially symmetric, solutions.
3
all smooth stationary solutions.
Once the thermodynamics of the conical surplus is established, we determine the required
transformation on the higher-spin chemical potential accompanying the S-transformation on
the boundary modulus τ 7 → − 1
τ , and prove that the conical surplus and the black hole are
S-dual. We then show that the black hole and the conical surplus are related by a coordinate
transformation (that changes the modular parameter of the boundary torus from τ to − 1
τ ).
Then we generalize from the S-transformation to the full modular group: starting with
a higher-spin-charge-carrying conical surplus solution and applying on it the full modular
group, we can generate an ‘SL(2, Z)’ family of smooth stationary solutions. They are all con-
nected by coordinate transformations that act non-trivially on the homology of the boundary
torus. Their free energies hence their on-shell partition functions are related via modular
transformations. The full modular invariant partition function then involves summing over
all the modular images of the conical surplus.
The paper is organized as follows. In Section 2 we briefly review some basics of 3D sl(N)
higher-spin theory, summarize known stationary smooth solutions, and define new smooth
stationary solutions in the ‘SL(2, Z)’ family. In Section 3 we formulate a thermodynamics
that is universal to all members of the ‘SL(2, Z)’ family (including conical surplus and black
hole). Then in Section 4 we prove that a conical surplus can be mapped into a black hole
via an S-transformation of the modulus of the boundary torus. In section 5 we show how to
generate an ‘SL(2, Z)’ family of smooth solutions. We summarize and discuss open problems
in section 6. In Appendix A we present a detailed proof for a statement that is central to
our paper; in Appendix B we review the spin-2 story; finally in Appendix C we discuss sl(4)
theory as a concrete example.
2 Basics of 3D higher-spin theory
In this section we first review some basics of the three-dimensional sl(N) higher-spin theory
(for more details see the earlier works [20,21,30] and the reviews [31,32]). Then we summarize
known smooth solutions in this theory, i.e. the conical surplus and the black hole, and
meanwhile prepare the readers for our later discussion on general smooth solutions. In this
paper we focus on stationary, axially symmetric, solutions.
3
2.1 Action
In three dimensions, the Einstein-Hilbert action with a negative cosmological constant Λ
can be rewritten in terms of an sl(2, R) ⊕ sl(2, R) Chern-Simons theory (up to a boundary
term) [18, 19]:
S = SCS[A] − SCS[ ¯A] with SCS[A] = k
4π
∫
M
Tr[A ∧ dA + 2
3 A ∧ A ∧ A] (2.1)
where M is a locally AdS3 manifold; A and ¯A are sl(2, R) gauge fields; the trace ‘Tr’ is on
the 2-dimensional representation of sl(2) and the level k = lAdS3
4GN . This trick (of rewriting
Einstein-Hilbert action with a negative Λ into a gauge theory) only works in three dimensions.
On the other hand, the three-dimensional Vasiliev higher-spin theory is also much more
tractable than its higher-dimensional siblings. Besides the gauge fields, the theory has only
one additional scalar field, which can be consistently decoupled since it is not part of the
higher-spin multiplet in 3D. The gauge field subsector can then be written as a Chern-Simons
theory (2.1) with A and ¯A ∈ hs[λ], which is an infinite-dimensional Lie algebra and at λ = N
reduces to sl(N) (after quotiented by an infinite ideal). In this paper we will focus on the
3D sl(N) Chern-Simons theory. The trace ‘Tr’ in the Chern-Simons action (2.1) is now on
the N-dimensional representation of sl(N) and the level becomes k = ℓAdS
4GN
1
2 Tr[(L0)2] .2
Now let us parametrize the base manifold M. Since the theory has a negative cosmologi-
cal constant, we choose the boundary condition to be asymptotically AdS3. A constant-time
slice of an asymptotically AdS3 space M is topologically a disc. Specify a radial coordinate
ρ and an angular coordinate φ, the coordinate is then {ρ, t, φ}. The asymptotic boundary
∂M is at ρ → ∞ and the boundary coordinates are {t, φ}.
In this paper, we focus on Euclidean signature. The Wick rotation into Euclidean signa-
ture is via t 7 → itE and the coordinate becomes {ρ, z, ¯z} with
z ≡ φ + itE . (2.2)
Accordingly, the gauge symmetry becomes sl(N, C). The connection A ∈ sl(N, C); and in
the representation we will choose, ¯A is A’s anti-hermitean conjugate:3
¯A = −A† . (2.3)
Therefore, most of the time we only need to write the A’s side of the expression and the one
2This additional normalization factor 1
2 Tr[(L0)2] is necessary for the spin-2 subsector of sl(N ) higher-spin
theory to match the Einstein gravity.
3This is also the convention used by [20, 27, 33]
4
In three dimensions, the Einstein-Hilbert action with a negative cosmological constant Λ
can be rewritten in terms of an sl(2, R) ⊕ sl(2, R) Chern-Simons theory (up to a boundary
term) [18, 19]:
S = SCS[A] − SCS[ ¯A] with SCS[A] = k
4π
∫
M
Tr[A ∧ dA + 2
3 A ∧ A ∧ A] (2.1)
where M is a locally AdS3 manifold; A and ¯A are sl(2, R) gauge fields; the trace ‘Tr’ is on
the 2-dimensional representation of sl(2) and the level k = lAdS3
4GN . This trick (of rewriting
Einstein-Hilbert action with a negative Λ into a gauge theory) only works in three dimensions.
On the other hand, the three-dimensional Vasiliev higher-spin theory is also much more
tractable than its higher-dimensional siblings. Besides the gauge fields, the theory has only
one additional scalar field, which can be consistently decoupled since it is not part of the
higher-spin multiplet in 3D. The gauge field subsector can then be written as a Chern-Simons
theory (2.1) with A and ¯A ∈ hs[λ], which is an infinite-dimensional Lie algebra and at λ = N
reduces to sl(N) (after quotiented by an infinite ideal). In this paper we will focus on the
3D sl(N) Chern-Simons theory. The trace ‘Tr’ in the Chern-Simons action (2.1) is now on
the N-dimensional representation of sl(N) and the level becomes k = ℓAdS
4GN
1
2 Tr[(L0)2] .2
Now let us parametrize the base manifold M. Since the theory has a negative cosmologi-
cal constant, we choose the boundary condition to be asymptotically AdS3. A constant-time
slice of an asymptotically AdS3 space M is topologically a disc. Specify a radial coordinate
ρ and an angular coordinate φ, the coordinate is then {ρ, t, φ}. The asymptotic boundary
∂M is at ρ → ∞ and the boundary coordinates are {t, φ}.
In this paper, we focus on Euclidean signature. The Wick rotation into Euclidean signa-
ture is via t 7 → itE and the coordinate becomes {ρ, z, ¯z} with
z ≡ φ + itE . (2.2)
Accordingly, the gauge symmetry becomes sl(N, C). The connection A ∈ sl(N, C); and in
the representation we will choose, ¯A is A’s anti-hermitean conjugate:3
¯A = −A† . (2.3)
Therefore, most of the time we only need to write the A’s side of the expression and the one
2This additional normalization factor 1
2 Tr[(L0)2] is necessary for the spin-2 subsector of sl(N ) higher-spin
theory to match the Einstein gravity.
3This is also the convention used by [20, 27, 33]
4
for ¯A can then be inferred using (2.3).
The Chern-Simons action (2.1) has gauge degrees of freedom A ∼ A + dΛ, which allows
us to fix the gauge as:
A(ρ, z, ¯z) = b−1 a(z, ¯z) b + b−1 db (2.4)
where b = eρL0 is an SL(N)-valued 0-form, and a is an sl(N)-valued 1-form on the boundary
∂M:
a = az dz + a¯z d¯z . (2.5)
2.2 Asymptotic symmetries
The sl(2) subalgebra corresponds to the spin-2 (i.e. gravity) sector, with generators {L0,±1},
whose commutators are:
[Lm, Ln] = (m − n)Lm+n , m, n = −1, 0, 1 . (2.6)
(We will also sometimes write Ln = W (2)
n .) From the sl(N) gauge symmetry, we first need
to make a choice as to which sl(2) subalgebra corresponds to the gravity sector, namely we
need to choose how the gravity sl(2) embeds in the full gauge algebra sl(N). The choice
of this embedding then determines the spectrum of the theory. The principal embedding
is particularly simple because the field of each spin appears once and only once. In this
paper we only discuss the principal embedding and a generalization to other embeddings is
straightforward.
Next, one can choose the boundary condition for A and determine the asymptotic sym-
metry group. This was done in [30, 34–36]. In the absence of sources, the asymptotic AdS
condition implies
A¯z = 0 . (2.7)
However this boundary condition (2.7) is too weak and gives rise to a phase space that is too
large (with an affine sl(N) algebra as its asymptotic symmetry). An additional boundary
condition was proposed by [30] to supplement (2.7) (see also [37] for the spin-2 case):
(A − AAdS )|ρ→∞ = O(1) , (2.8)
which reduces the phase space by imposing a first-class constraint on the sl(N) affine algebra
and results in a WN algebra as the asymptotic symmetry. This is the bulk realization of the
Drinfeld-Sokolov reduction (the reduction of an affine algebra to a W-algebra) [38].
5
The Chern-Simons action (2.1) has gauge degrees of freedom A ∼ A + dΛ, which allows
us to fix the gauge as:
A(ρ, z, ¯z) = b−1 a(z, ¯z) b + b−1 db (2.4)
where b = eρL0 is an SL(N)-valued 0-form, and a is an sl(N)-valued 1-form on the boundary
∂M:
a = az dz + a¯z d¯z . (2.5)
2.2 Asymptotic symmetries
The sl(2) subalgebra corresponds to the spin-2 (i.e. gravity) sector, with generators {L0,±1},
whose commutators are:
[Lm, Ln] = (m − n)Lm+n , m, n = −1, 0, 1 . (2.6)
(We will also sometimes write Ln = W (2)
n .) From the sl(N) gauge symmetry, we first need
to make a choice as to which sl(2) subalgebra corresponds to the gravity sector, namely we
need to choose how the gravity sl(2) embeds in the full gauge algebra sl(N). The choice
of this embedding then determines the spectrum of the theory. The principal embedding
is particularly simple because the field of each spin appears once and only once. In this
paper we only discuss the principal embedding and a generalization to other embeddings is
straightforward.
Next, one can choose the boundary condition for A and determine the asymptotic sym-
metry group. This was done in [30, 34–36]. In the absence of sources, the asymptotic AdS
condition implies
A¯z = 0 . (2.7)
However this boundary condition (2.7) is too weak and gives rise to a phase space that is too
large (with an affine sl(N) algebra as its asymptotic symmetry). An additional boundary
condition was proposed by [30] to supplement (2.7) (see also [37] for the spin-2 case):
(A − AAdS )|ρ→∞ = O(1) , (2.8)
which reduces the phase space by imposing a first-class constraint on the sl(N) affine algebra
and results in a WN algebra as the asymptotic symmetry. This is the bulk realization of the
Drinfeld-Sokolov reduction (the reduction of an affine algebra to a W-algebra) [38].
5
In the process of the Drinfeld-Sokolov reduction, different gauge choices give different
bases for the W-algebra. A particular convenient choice is the highest-weight gauge, which
gives rise to a W-algebra in which all W (s) are primaries with respect to the lowest spins
[30,36]. Since there is no spin-1 field in the sl(N) Chern-Simons theory, all W (s≥3) fields are
Virasoro primaries:
[Lm, W (s)
n ] = [(s − 1)m − n]W (s)
m+n , s = 3, . . . , N , m, n ∈ Z . (2.9)
This is the gauge we will use throughout this paper.4
Recall that in the spin-2 case the bulk isometry sl(2) is given by the ‘wedge’ subalgebra
(generated by L−1,0,1) of Virasoro algebra. Here the (N2 − 1)-dimensional bulk isometry
sl(N) is generated by W (s)
m with s = 2, . . . , N and m = −s + 1, . . . , s − 1. An explicit
representation of (2.9) for m ≤ |N| is [39]:
W (s)
n = (−1)s−n−1 (s + n − 1)!
(2s − 2)! (AdjL−1 )s−n−1(L1)s−1 (2.10)
where the adjoint action AdjAB = [A, B]. Lastly, we will choose a convention in which
(Lm)† = (−1)mL−m (2.11)
which together with (2.10) implies (W (s)
m )† = (−1)mW (s)
−m for s = 2, . . . , N. In this convention
we have (2.3).
2.3 Smooth solutions
2.3.1 Equations of motion
The equation of motion of the Chern-Simons action is the flatness condition for A: F ≡
dA + A ∧ A = 0. In the gauge (2.4), this translates into the flatness of a:
f ≡ da + a ∧ a = 0 . (2.12)
In this paper we will only consider axially-symmetric, stationary, solutions. For these
solutions, A has only ρ-dependence. In the gauge (2.4) , this means that a is constant, hence
throughout this paper we will refer to them as constant solutions (although they can rotate).
4Other gauges are possible and might be more suitable for other questions, for details see [36].
6
bases for the W-algebra. A particular convenient choice is the highest-weight gauge, which
gives rise to a W-algebra in which all W (s) are primaries with respect to the lowest spins
[30,36]. Since there is no spin-1 field in the sl(N) Chern-Simons theory, all W (s≥3) fields are
Virasoro primaries:
[Lm, W (s)
n ] = [(s − 1)m − n]W (s)
m+n , s = 3, . . . , N , m, n ∈ Z . (2.9)
This is the gauge we will use throughout this paper.4
Recall that in the spin-2 case the bulk isometry sl(2) is given by the ‘wedge’ subalgebra
(generated by L−1,0,1) of Virasoro algebra. Here the (N2 − 1)-dimensional bulk isometry
sl(N) is generated by W (s)
m with s = 2, . . . , N and m = −s + 1, . . . , s − 1. An explicit
representation of (2.9) for m ≤ |N| is [39]:
W (s)
n = (−1)s−n−1 (s + n − 1)!
(2s − 2)! (AdjL−1 )s−n−1(L1)s−1 (2.10)
where the adjoint action AdjAB = [A, B]. Lastly, we will choose a convention in which
(Lm)† = (−1)mL−m (2.11)
which together with (2.10) implies (W (s)
m )† = (−1)mW (s)
−m for s = 2, . . . , N. In this convention
we have (2.3).
2.3 Smooth solutions
2.3.1 Equations of motion
The equation of motion of the Chern-Simons action is the flatness condition for A: F ≡
dA + A ∧ A = 0. In the gauge (2.4), this translates into the flatness of a:
f ≡ da + a ∧ a = 0 . (2.12)
In this paper we will only consider axially-symmetric, stationary, solutions. For these
solutions, A has only ρ-dependence. In the gauge (2.4) , this means that a is constant, hence
throughout this paper we will refer to them as constant solutions (although they can rotate).
4Other gauges are possible and might be more suitable for other questions, for details see [36].
6
Their equations of motion (2.12) reduce to
[az , a¯z ] = 0 . (2.13)
Once az is fixed, a¯z can be determined via the equation of motion (2.13), whose solution
is simply a¯z being an arbitrary traceless function of az . By Cayley-Hamilton theorem, an
arbitrary function of a N × N matrix az (that is generic enough) truncates to a polynomial
of az of degree-(N − 1); therefore a¯z has the expansion [51]:
a¯z =
N∑
s=2
σs
[
(az )s−1 − Tr(az )s−1
N 1
]
. (2.14)
Up to now, {σs} are (N − 1) arbitrary complex parameters. Later we will show how they
are fixed in terms of the chemical potentials of the higher-spin charges.
2.3.2 Holonomy condition
In this subsection, we define the condition that characterizes a generic smooth constant
solution. The known solutions, i.e. the conical surplus and the black hole, are the two
special cases.
Any given az together with the a¯z related by it via (2.14) would solve the equation of
motion (2.13). However, requiring the solution be smooth imposes a much more stringent
constraint. Since in the higher-spin theory, the spin-2 field is coupled with all higher-spin
fields hence the Ricci scalar is no longer a gauge invariant entity, the smoothness condition
need to be prescribed in terms of other, gauge-invariant, observables. In the 3D Chern-
Simons theory, the natural candidate is the holonomy around a one-cycle C in M:
HolC (A) ≡ Pe∮
C A . (2.15)
The smoothness condition is then simply that the holonomy around any contractible cycle
(A-cycle) must be trivial, i.e. HolA(A) ∈ center of the gauge group [20, 21].
First let us describe the cycles in this 3D Euclidean spacetime. The asymptotic boundary
of the Euclidean AdS3 is a torus. First we fix its homology basis (α, β) with α ∩ β = 1. Then
we give this torus a complex structure. This allows us to define a holomorphic 1-form ω; we
can choose its basis such that ∮
α ω = 1, then τ ≡ ∮
β ω defines the modulus of the torus. Once
the modulus of the boundary torus is fixed, different bulk geometries correspond to different
ways of filling the solid torus. We first fix the primitive contractible cycle (A-cycle), then
the primitive non-contractible cycle (B-cycle) that satisfies A∩B= 1 is uniquely determined
up to shifts in the A-cycle. The (A,B) homology basis is related to the original (α, β) basis
7
[az , a¯z ] = 0 . (2.13)
Once az is fixed, a¯z can be determined via the equation of motion (2.13), whose solution
is simply a¯z being an arbitrary traceless function of az . By Cayley-Hamilton theorem, an
arbitrary function of a N × N matrix az (that is generic enough) truncates to a polynomial
of az of degree-(N − 1); therefore a¯z has the expansion [51]:
a¯z =
N∑
s=2
σs
[
(az )s−1 − Tr(az )s−1
N 1
]
. (2.14)
Up to now, {σs} are (N − 1) arbitrary complex parameters. Later we will show how they
are fixed in terms of the chemical potentials of the higher-spin charges.
2.3.2 Holonomy condition
In this subsection, we define the condition that characterizes a generic smooth constant
solution. The known solutions, i.e. the conical surplus and the black hole, are the two
special cases.
Any given az together with the a¯z related by it via (2.14) would solve the equation of
motion (2.13). However, requiring the solution be smooth imposes a much more stringent
constraint. Since in the higher-spin theory, the spin-2 field is coupled with all higher-spin
fields hence the Ricci scalar is no longer a gauge invariant entity, the smoothness condition
need to be prescribed in terms of other, gauge-invariant, observables. In the 3D Chern-
Simons theory, the natural candidate is the holonomy around a one-cycle C in M:
HolC (A) ≡ Pe∮
C A . (2.15)
The smoothness condition is then simply that the holonomy around any contractible cycle
(A-cycle) must be trivial, i.e. HolA(A) ∈ center of the gauge group [20, 21].
First let us describe the cycles in this 3D Euclidean spacetime. The asymptotic boundary
of the Euclidean AdS3 is a torus. First we fix its homology basis (α, β) with α ∩ β = 1. Then
we give this torus a complex structure. This allows us to define a holomorphic 1-form ω; we
can choose its basis such that ∮
α ω = 1, then τ ≡ ∮
β ω defines the modulus of the torus. Once
the modulus of the boundary torus is fixed, different bulk geometries correspond to different
ways of filling the solid torus. We first fix the primitive contractible cycle (A-cycle), then
the primitive non-contractible cycle (B-cycle) that satisfies A∩B= 1 is uniquely determined
up to shifts in the A-cycle. The (A,B) homology basis is related to the original (α, β) basis
7
via a modular transformation:
(
B
A
)
=
(
a b
c d
) (
β
α
)
with
(
a b
c d
)
∈ PSL(2, Z) (2.16)
with
PSL(2, Z) ≡ SL(2, Z)/Z2 , SL(2, Z) ≡ {
(
a b
c d
) ∣
∣ a, b, c, d ∈ Z, ad − bc = 1} . (2.17)
Then the torus with (A,B) as the homology basis but with the same holomorphic 1-form ω
has a modular parameter
modular parameter ≡
∫
B ω
∫
A ω = aτ + b
cτ + d , (2.18)
which is a modular transformation of the original modulus τ . Throughout this paper, we
use γ to denote an element of the modular group PSL(2, Z) and define ˆγτ to be its action
on τ :5
γ ≡
(
a b
c d
)
∈ PSL(2, Z) : τ 7 −→ ˆγτ ≡ aτ + b
cτ + d . (2.19)
Also note that throughout this paper, by modular parameter we mean the ratio of the
(complex) length of the non-contractible(B) cycle and that of the contractible(A) cycle, as
defined in (2.18); and we reserve the term modulus for τ .
The conical surplus solutions that carry higher-spin charges has a contractible cycle
φ ∼ φ + 2π, just like the AdS3 [20].
CS: γ =
(
1 0
0 1
)
=⇒ A-cycle: z ∼ z + 2π ,
B-cycle: z ∼ z + 2πτ . (2.20)
Accordingly, the holonomy around this φ-cycle needs to lie in the center of the gauge group:
Holφ(A) = b−1e2πωφ b ∈ center of G (2.21)
where ωφ is defined as6
ωφ ≡ az + a¯z . (2.22)
5In PSL(2, Z), γ and −γ are identified, hence we can choose c ≥ 0 without loss of generality.
6Throughout the paper we will call such ω ‘holonomy matrix’.
8
(
B
A
)
=
(
a b
c d
) (
β
α
)
with
(
a b
c d
)
∈ PSL(2, Z) (2.16)
with
PSL(2, Z) ≡ SL(2, Z)/Z2 , SL(2, Z) ≡ {
(
a b
c d
) ∣
∣ a, b, c, d ∈ Z, ad − bc = 1} . (2.17)
Then the torus with (A,B) as the homology basis but with the same holomorphic 1-form ω
has a modular parameter
modular parameter ≡
∫
B ω
∫
A ω = aτ + b
cτ + d , (2.18)
which is a modular transformation of the original modulus τ . Throughout this paper, we
use γ to denote an element of the modular group PSL(2, Z) and define ˆγτ to be its action
on τ :5
γ ≡
(
a b
c d
)
∈ PSL(2, Z) : τ 7 −→ ˆγτ ≡ aτ + b
cτ + d . (2.19)
Also note that throughout this paper, by modular parameter we mean the ratio of the
(complex) length of the non-contractible(B) cycle and that of the contractible(A) cycle, as
defined in (2.18); and we reserve the term modulus for τ .
The conical surplus solutions that carry higher-spin charges has a contractible cycle
φ ∼ φ + 2π, just like the AdS3 [20].
CS: γ =
(
1 0
0 1
)
=⇒ A-cycle: z ∼ z + 2π ,
B-cycle: z ∼ z + 2πτ . (2.20)
Accordingly, the holonomy around this φ-cycle needs to lie in the center of the gauge group:
Holφ(A) = b−1e2πωφ b ∈ center of G (2.21)
where ωφ is defined as6
ωφ ≡ az + a¯z . (2.22)
5In PSL(2, Z), γ and −γ are identified, hence we can choose c ≥ 0 without loss of generality.
6Throughout the paper we will call such ω ‘holonomy matrix’.
8
Let’s denote by ‘Λ (ω)’ the vector of eigenvalues of a matrix ω:
U ω U−1 = DiagonalMatrix[λ1, . . . , λN ] =⇒ Λ (ω) ≡ ~λ = (λ1, . . . , λN ) . (2.23)
The center of SL(N, C) is e−2πi m
N 1 with m ∈ ZN , which implies that the eigenvalues of ωφ
satisfies [20]:
Λ (ωφ) = i ~n , (2.24)
where the vector ~n = (n1, . . . , nN ) with ni ∈ Z − m
N , ni 6 = nj for i 6 = j, and ∑N
i ni = 0, and
most importantly ni must come in pairs for the solution to be a conical surplus, i.e. if we
order {ni} into a monotone sequence then
ni + nN +1−i = 0 . (2.25)
This imposes a very strong constraint on m: m = 0 for N odd, and m = 0 or N
2 for N even.
But recall that the center of SL(N, R) is precisely 1 for N odd and ±1 for N even;7 therefore
the constraint (2.25) forces the holonomy to lie in the center of the Lorentzian gauge group
SL(N, R) rather than that of the Euclidean one SL(N, C). In summary the vector ~n obeys
~n = (n1, . . . , nN ) , ni ∈
Z N odd
Z or Z + 1
2 N even
, ni 6 = nj for i 6 = j , ni + nN +1−i = 0,
(2.26)
and can be considered as a ‘topological charge’ of the solution; and we will term it ‘holonomy
vector’. The global AdS3 space corresponds to ~n = ~ρ (the Weyl vector of sl(N), with
ρi = N +1
2 − i), and generic ~n’s satisfying (2.26) give conical surpluses [20]. For discussions
on the conical surplus in hs[λ] Chern-Simons theory see e.g. [40–45]
On the other hand, the (Euclidean) black hole has a contractible cycle z ∼ z + 2πτ .
BH: γ =
(
0 −1
1 0
)
=⇒ A-cycle: z ∼ z + 2πτ ,
B-cycle: z ∼ z − 2π . (2.27)
Accordingly, the trivial holonomy condition is [21]:
Holt(A) = b−1e2πωt b ∈ center of SL(N, R) , (2.28)
with ωt defined as
ωt ≡ τ az + ¯τ a¯z . (2.29)
7The ±1 for N even arises from the fact that the gauge group is actually (SL(N, R)/Z2) × (SL(N, R)/Z2)
[20].
9
U ω U−1 = DiagonalMatrix[λ1, . . . , λN ] =⇒ Λ (ω) ≡ ~λ = (λ1, . . . , λN ) . (2.23)
The center of SL(N, C) is e−2πi m
N 1 with m ∈ ZN , which implies that the eigenvalues of ωφ
satisfies [20]:
Λ (ωφ) = i ~n , (2.24)
where the vector ~n = (n1, . . . , nN ) with ni ∈ Z − m
N , ni 6 = nj for i 6 = j, and ∑N
i ni = 0, and
most importantly ni must come in pairs for the solution to be a conical surplus, i.e. if we
order {ni} into a monotone sequence then
ni + nN +1−i = 0 . (2.25)
This imposes a very strong constraint on m: m = 0 for N odd, and m = 0 or N
2 for N even.
But recall that the center of SL(N, R) is precisely 1 for N odd and ±1 for N even;7 therefore
the constraint (2.25) forces the holonomy to lie in the center of the Lorentzian gauge group
SL(N, R) rather than that of the Euclidean one SL(N, C). In summary the vector ~n obeys
~n = (n1, . . . , nN ) , ni ∈
Z N odd
Z or Z + 1
2 N even
, ni 6 = nj for i 6 = j , ni + nN +1−i = 0,
(2.26)
and can be considered as a ‘topological charge’ of the solution; and we will term it ‘holonomy
vector’. The global AdS3 space corresponds to ~n = ~ρ (the Weyl vector of sl(N), with
ρi = N +1
2 − i), and generic ~n’s satisfying (2.26) give conical surpluses [20]. For discussions
on the conical surplus in hs[λ] Chern-Simons theory see e.g. [40–45]
On the other hand, the (Euclidean) black hole has a contractible cycle z ∼ z + 2πτ .
BH: γ =
(
0 −1
1 0
)
=⇒ A-cycle: z ∼ z + 2πτ ,
B-cycle: z ∼ z − 2π . (2.27)
Accordingly, the trivial holonomy condition is [21]:
Holt(A) = b−1e2πωt b ∈ center of SL(N, R) , (2.28)
with ωt defined as
ωt ≡ τ az + ¯τ a¯z . (2.29)
7The ±1 for N even arises from the fact that the gauge group is actually (SL(N, R)/Z2) × (SL(N, R)/Z2)
[20].
9
Namely
Λ (ωt) = i ~n , (2.30)
with ~n satisfying the same set of conditions as the conical surplus (2.26). ~n = ~ρ corresponds
to the higher-spin-charge-carrying BTZ black hole first constructed in [21]; and other ~n’s
give more generic higher-spin black holes.
This definition of smooth solution by the holonomy around the contractible cycle can be
easily generalized to include solutions whose modular parameter is a generic ˆγτ (other than
τ or − 1
τ ). For a generic γ =
(a b
c d
)
∈ PSL(2, Z), the A/B cycles are
Contractible(A)-cycle: z ∼ z + 2π(cτ + d) ,
Non-contractible(B)-cycle: z ∼ z + 2π(aτ + b) . (2.31)
Accordingly, for a smooth solution, the holonomy around the A-cycle should be trivial
HolA(A) = b−1e2πωA b ∈ center of SL(N, R) , (2.32)
with the holonomy matrix around the A-cycle given by
ωA ≡ 1
2π
∮
A
a = (cτ + d)az + (c¯τ + d)a¯z , (2.33)
namely
Λ (ωA) = i ~n (2.34)
again with ~n given by (2.26). Here we also write down the holonomy matrix around the
B-cycle for comparison and for later use:
ωB ≡ 1
2π
∮
B
a = (aτ + b)az + (a¯τ + b)a¯z . (2.35)
For given ~n and τ , varying γ ∈ PSL(2, Z) then generates a ‘SL(2, Z)’ family of solutions (a
term coined in [5]). Since the T-transformation τ 7 → τ + 1 does not change the A/B cycle
(2.31), we should consider the subgroup of Γ ≡ PSL(2, Z)
Γ∞ ≡ {
(
1 m
0 1
) ∣
∣ m ∈ Z}/Z2 ⊂ Γ ≡ PSL(2, Z) . (2.36)
to be the stabilizer; hence the ‘SL(2, Z)’ family is actually the quotient Γ∞\Γ. Since a γ
in Γ∞\Γ is uniquely given by the lower row (c, d) (which always satisfies gcd(c, d) = 1), an
10
Λ (ωt) = i ~n , (2.30)
with ~n satisfying the same set of conditions as the conical surplus (2.26). ~n = ~ρ corresponds
to the higher-spin-charge-carrying BTZ black hole first constructed in [21]; and other ~n’s
give more generic higher-spin black holes.
This definition of smooth solution by the holonomy around the contractible cycle can be
easily generalized to include solutions whose modular parameter is a generic ˆγτ (other than
τ or − 1
τ ). For a generic γ =
(a b
c d
)
∈ PSL(2, Z), the A/B cycles are
Contractible(A)-cycle: z ∼ z + 2π(cτ + d) ,
Non-contractible(B)-cycle: z ∼ z + 2π(aτ + b) . (2.31)
Accordingly, for a smooth solution, the holonomy around the A-cycle should be trivial
HolA(A) = b−1e2πωA b ∈ center of SL(N, R) , (2.32)
with the holonomy matrix around the A-cycle given by
ωA ≡ 1
2π
∮
A
a = (cτ + d)az + (c¯τ + d)a¯z , (2.33)
namely
Λ (ωA) = i ~n (2.34)
again with ~n given by (2.26). Here we also write down the holonomy matrix around the
B-cycle for comparison and for later use:
ωB ≡ 1
2π
∮
B
a = (aτ + b)az + (a¯τ + b)a¯z . (2.35)
For given ~n and τ , varying γ ∈ PSL(2, Z) then generates a ‘SL(2, Z)’ family of solutions (a
term coined in [5]). Since the T-transformation τ 7 → τ + 1 does not change the A/B cycle
(2.31), we should consider the subgroup of Γ ≡ PSL(2, Z)
Γ∞ ≡ {
(
1 m
0 1
) ∣
∣ m ∈ Z}/Z2 ⊂ Γ ≡ PSL(2, Z) . (2.36)
to be the stabilizer; hence the ‘SL(2, Z)’ family is actually the quotient Γ∞\Γ. Since a γ
in Γ∞\Γ is uniquely given by the lower row (c, d) (which always satisfies gcd(c, d) = 1), an
10
enumeration of all the members in this family is thus [46]:
∀(c, d) with c, d ∈ Z , c ≥ 0 , gcd(c, d) = 1 . (2.37)
Lastly, the holonomy vector for ¯A is always related to that of A via
~¯n = ~n (2.38)
in the anti-hermitean basis we choose. We summarize the discussion of this subsection in
the following table:
EAdS3 and CS black hole Smooth solution γ
A-cycle z ∼ z + 2π z ∼ z + 2πτ z ∼ z + 2π(cτ + d)
B-cycle z ∼ z + 2πτ z ∼ z − 2π z ∼ z + 2π(aτ + b)
modular parameter τ − 1
τ
aτ +b
cτ +d
A-cycle holonomy ωφ = az + a¯z ωt = τ az + ¯τ a¯z ωA = (cτ + d)az + (c¯τ + d)a¯z
3 Thermodynamics
The conical surplus solution constructed in [20] has higher-spin charges but with no chemical
potential turned on, and hence there is no study on its thermodynamics yet. Meanwhile, the
thermodynamics of the black hole in the sl(N) Chern-Simons theory has been extensively
studied, and depending on the choice of the spin-2 conserved charge (the zero-mode of the
energy-momentum tensor) there are two main approaches. In the ‘holomorphic’ formalism
(initiated in [21] and used in [22–26]), the spin-2 conserved charge (for the left-mover A)
T is holomorphic.8 In the ‘canonical’ formalism, the spin-2 conserved charge T is obtained
either via a canonical approach [28, 29] `a la Regge-Teitelboim [50], or via a direct derivation
from the variational principle [27] (for a precursor see [51]) which gives the same result; in
this formalism T is not holomorphic and receives contribution from the right-mover ¯A. The
different definitions of (T, ¯T ) in turn leads to different results for the integrability condition,
the entropy, and finally the free energy. For more details see the discussion in [27]. (For
other discussions on the black hole thermodynamics see [52, 53].)
Now we would like to generalize the result of the thermodynamics of the black hole to the
conical surplus and to all smooth constant solutions in the ‘SL(2, Z)’ family. Which of the
two formalism is better suited for this purpose? Usually the modularity (w.r.t. PSL(2, Z))
8For the CFT computation in this formalism see [47–49].
11
∀(c, d) with c, d ∈ Z , c ≥ 0 , gcd(c, d) = 1 . (2.37)
Lastly, the holonomy vector for ¯A is always related to that of A via
~¯n = ~n (2.38)
in the anti-hermitean basis we choose. We summarize the discussion of this subsection in
the following table:
EAdS3 and CS black hole Smooth solution γ
A-cycle z ∼ z + 2π z ∼ z + 2πτ z ∼ z + 2π(cτ + d)
B-cycle z ∼ z + 2πτ z ∼ z − 2π z ∼ z + 2π(aτ + b)
modular parameter τ − 1
τ
aτ +b
cτ +d
A-cycle holonomy ωφ = az + a¯z ωt = τ az + ¯τ a¯z ωA = (cτ + d)az + (c¯τ + d)a¯z
3 Thermodynamics
The conical surplus solution constructed in [20] has higher-spin charges but with no chemical
potential turned on, and hence there is no study on its thermodynamics yet. Meanwhile, the
thermodynamics of the black hole in the sl(N) Chern-Simons theory has been extensively
studied, and depending on the choice of the spin-2 conserved charge (the zero-mode of the
energy-momentum tensor) there are two main approaches. In the ‘holomorphic’ formalism
(initiated in [21] and used in [22–26]), the spin-2 conserved charge (for the left-mover A)
T is holomorphic.8 In the ‘canonical’ formalism, the spin-2 conserved charge T is obtained
either via a canonical approach [28, 29] `a la Regge-Teitelboim [50], or via a direct derivation
from the variational principle [27] (for a precursor see [51]) which gives the same result; in
this formalism T is not holomorphic and receives contribution from the right-mover ¯A. The
different definitions of (T, ¯T ) in turn leads to different results for the integrability condition,
the entropy, and finally the free energy. For more details see the discussion in [27]. (For
other discussions on the black hole thermodynamics see [52, 53].)
Now we would like to generalize the result of the thermodynamics of the black hole to the
conical surplus and to all smooth constant solutions in the ‘SL(2, Z)’ family. Which of the
two formalism is better suited for this purpose? Usually the modularity (w.r.t. PSL(2, Z))
8For the CFT computation in this formalism see [47–49].
11
requires the holomorphicity of the theory. Therefore naively one would expect that the
‘holomorphic’ formalism be the choice whereas the modular property be absent or at least
obscured in the ‘canonical’ formalism.
However, the fact that in the ‘canonical’ formalism T lacks holomorphicity therefore
modularity does not pose any problem in a discussion of the modular properties in this
formalism. First of all, although the spin-2 conserved charge T , and hence the entropy,
is not modular covariant, this is to be expected since they are not holomorphic to start
with. Moreover they are only intermediate quantities. As we will show presently, all the
other final quantities — the connection, the holonomy condition, and the free energy — are
modular invariant or covariant. We will derive manifestly modular covariant expressions for
them. We will also show later in Section 6 that the crucial consistency condition — the
integrability condition (relating conserved charges of different spins) — is modular invariant
in the ‘canonical’ formalism.
On the other hand, as we will discuss in Section 6, in the ‘holomorphic’ formalism,
it is not clear to us how to write down a simple modular transformation such that the
various important relations — the holonomy condition, the integrability condition, etc —
are modular invariant or covariant.
This leads us to choose the ‘canonical’ formalism developed in [27] for our generalization
of thermodynamics of the black hole to all members of the ‘SL(2, Z)’ family. In this section,
we generalize the procedure in [27, 51]
1. Vary the bulk action and identify the source and charge terms in the connection.
2. Write down the suitable boundary action to ensure the variational principle.
3. Identify the conjugate pair of energy and temperature, compute the free energy and
entropy, and check the first law of thermodynamics.
to generic smooth constant solutions (including the conical surplus). In particular, we com-
pute the on-shell action for generic solutions and write down the modular covariant expres-
sions for the entropy and free energy.
One clarification: the construction of these general smooth constant solutions will only
be shown later, in Section 5. However, since we first need to know the thermodynamics of
the conical surplus solution (in order to consistently turn on its chemical potentials) before
we can discuss its relation with the black hole solution, and since the thermodynamics of all
these smooth solutions can be discussed in an unified way (and with no need to know the
full details of the solutions), we will study all of them at once now, and postpone the explicit
construction of these solutions to Section 5.
12
‘holomorphic’ formalism be the choice whereas the modular property be absent or at least
obscured in the ‘canonical’ formalism.
However, the fact that in the ‘canonical’ formalism T lacks holomorphicity therefore
modularity does not pose any problem in a discussion of the modular properties in this
formalism. First of all, although the spin-2 conserved charge T , and hence the entropy,
is not modular covariant, this is to be expected since they are not holomorphic to start
with. Moreover they are only intermediate quantities. As we will show presently, all the
other final quantities — the connection, the holonomy condition, and the free energy — are
modular invariant or covariant. We will derive manifestly modular covariant expressions for
them. We will also show later in Section 6 that the crucial consistency condition — the
integrability condition (relating conserved charges of different spins) — is modular invariant
in the ‘canonical’ formalism.
On the other hand, as we will discuss in Section 6, in the ‘holomorphic’ formalism,
it is not clear to us how to write down a simple modular transformation such that the
various important relations — the holonomy condition, the integrability condition, etc —
are modular invariant or covariant.
This leads us to choose the ‘canonical’ formalism developed in [27] for our generalization
of thermodynamics of the black hole to all members of the ‘SL(2, Z)’ family. In this section,
we generalize the procedure in [27, 51]
1. Vary the bulk action and identify the source and charge terms in the connection.
2. Write down the suitable boundary action to ensure the variational principle.
3. Identify the conjugate pair of energy and temperature, compute the free energy and
entropy, and check the first law of thermodynamics.
to generic smooth constant solutions (including the conical surplus). In particular, we com-
pute the on-shell action for generic solutions and write down the modular covariant expres-
sions for the entropy and free energy.
One clarification: the construction of these general smooth constant solutions will only
be shown later, in Section 5. However, since we first need to know the thermodynamics of
the conical surplus solution (in order to consistently turn on its chemical potentials) before
we can discuss its relation with the black hole solution, and since the thermodynamics of all
these smooth solutions can be discussed in an unified way (and with no need to know the
full details of the solutions), we will study all of them at once now, and postpone the explicit
construction of these solutions to Section 5.
12
3.1 Variational principle
In the presence of higher-spin conserved charges Qs with s = 3, . . . , N, the partition function
(evaluated as an Euclidean path-integral) is a function of the boundary modulus τ and the
chemical potential µs conjugate to the higher-spin charge Qs:
Z [τ ; µs] ≡
∫
DAD ¯A e−I(E)
(3.1)
The free energy of the system is
− βF [τ ; µs] = ln Z [τ ; µs] . (3.2)
In this paper, we take the saddle point approximation (i.e. only include the classical result):
each classical solution contributes e−I(E)|on-shell . For each classical solution, its free energy F
(in the saddle point approximation) is given by
− βF = −I(E)|on-shell . (3.3)
In this section, we study the on-shell action of individual solutions; we will discuss the
contributions from all saddle points to the partition function later in Section 5.
3.1.1 Variation of bulk action
For the discussion in this section, it is enough to know that a smooth constant solution can
be defined by the condition that its holonomy around the A-cycle is trivial, i.e. equation
(2.31), (2.32), and (2.34); and it is determined by the PSL(2, Z) element γ.
The thermodynamics for the case of γ =
(0 −1
1 0
)
(the higher-spin black hole with modular
parameter − 1
τ ) was already given in [27]. We now generalize its derivation to the generic
smooth solution with modular parameter γ =
(a b
c d
)
. The thermodynamical relation comes
out of a direct variational calculation of the Chern-Simons action, which tells us how to add
the boundary term once the choice of source/field is made. In this variational calculation, the
modulus of the boundary torus should actively vary since it carries the physical information
of the inverse temperature (and the twist along the angular direction) [27, 56]. However, in
the coordinate system (z, ¯z) which we have been using, the modular parameter ˆγτ is hidden
in the identification of the A/B cycle; to make it appear explicitly we need to first switch
to a coordinate system (w, ¯w) that lives on a rigid torus with fixed modulus τ = i. The
coordinate transformation from (z, ¯z) to (w, ¯w) is
z = (cτ + d)(1 − i ˆγτ
2 w + 1 + i ˆγτ
2 ¯w) (3.4)
13
In the presence of higher-spin conserved charges Qs with s = 3, . . . , N, the partition function
(evaluated as an Euclidean path-integral) is a function of the boundary modulus τ and the
chemical potential µs conjugate to the higher-spin charge Qs:
Z [τ ; µs] ≡
∫
DAD ¯A e−I(E)
(3.1)
The free energy of the system is
− βF [τ ; µs] = ln Z [τ ; µs] . (3.2)
In this paper, we take the saddle point approximation (i.e. only include the classical result):
each classical solution contributes e−I(E)|on-shell . For each classical solution, its free energy F
(in the saddle point approximation) is given by
− βF = −I(E)|on-shell . (3.3)
In this section, we study the on-shell action of individual solutions; we will discuss the
contributions from all saddle points to the partition function later in Section 5.
3.1.1 Variation of bulk action
For the discussion in this section, it is enough to know that a smooth constant solution can
be defined by the condition that its holonomy around the A-cycle is trivial, i.e. equation
(2.31), (2.32), and (2.34); and it is determined by the PSL(2, Z) element γ.
The thermodynamics for the case of γ =
(0 −1
1 0
)
(the higher-spin black hole with modular
parameter − 1
τ ) was already given in [27]. We now generalize its derivation to the generic
smooth solution with modular parameter γ =
(a b
c d
)
. The thermodynamical relation comes
out of a direct variational calculation of the Chern-Simons action, which tells us how to add
the boundary term once the choice of source/field is made. In this variational calculation, the
modulus of the boundary torus should actively vary since it carries the physical information
of the inverse temperature (and the twist along the angular direction) [27, 56]. However, in
the coordinate system (z, ¯z) which we have been using, the modular parameter ˆγτ is hidden
in the identification of the A/B cycle; to make it appear explicitly we need to first switch
to a coordinate system (w, ¯w) that lives on a rigid torus with fixed modulus τ = i. The
coordinate transformation from (z, ¯z) to (w, ¯w) is
z = (cτ + d)(1 − i ˆγτ
2 w + 1 + i ˆγτ
2 ¯w) (3.4)
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