Modular Properties of 3D Higher Spin Theory
(Based on 1308.2959)
Chih-Wei Wang
Feng-Li Lin
(National Taiwan Normal University)Wei Li
(Max-Planck-Institut)Collaborators:
National Taiwan University 01/10/2014
Outline
●
Modular Properties
–
Modular group and Modular transformation
–
Modular invariant partition function
●
Higher Spin Theory
–
Basics of higher spin theory
–
Known smooth solutions
–
General framework
●
Thermodynamic and Modular Invariant Partition Function
●
Summary and Discussion
I. Modular Properties
Same lattice, if
rotation scaling
Modular transformation:
(Passive Point of View)
Modular Group and Modular Transformation
Complex plane quotient Torus
1
Modulus:
S-dual:
Solid Torus (A/B cycles)
1
● A(B)-cycle: (non-)contractible cycle
● Modular Parameter
A B
A
B
Solid Torus 2D torus
General choice of A/B cycles Modular Parameter
Thermal AdS and BTZ Black Hole
Focus on the solutions with Euclidean signature.
BTZ black hole:
Thermal AdS:
In general:
Modular Invariant Partition Function
saddle point approximation
defined on torus, must be modular invariant.
AdS3/CFT2 correspondence:
● If we include only thermal AdS and BTZ black hole, the result can not be modular invariant. However if we start from AdS and sum over all modular images, the result will be modular invariant.
● This means we sum over the contributions from the “distinct” solutions with general A cycles choices:
The goal of this work is to extend the above story to higher spin theory.
II. Higher Spin Theory
●
An extension of ordinary gravity theory including an infinite tower of massless higher spin fields with spin s ≥3 coupled non-linearly.
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The theory lives in AdS (or dS) space. The no-go theorems are evaded.
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In AdS3, the theory can be realized as a Chern-Simon gauge theory with an infinite-dimensional gauge algebra hs[λ].
●
At λ=N, the algebra reduce to sl(N). The result theory is a nature
generalization of the usual sl(2) Chern-Simon theory. This 3D Chern-Simon theory with sl(N) algebra will be the main topic of this talk.
Vasiliev's Higher-Spin Theory
[Vasiliev `91]Basics of 3D Higher Spin Theory
● In D=2+1(or 3), there is a gauge formulation of Einstein gravity in terms of the Chern-Simon Theory:
The action of the Chern-Simon Theory:
sl(2) algebra:
Equation of motion:
● A convenient gauge choice:
is a gauge field lives on the boundary E.O.M. for a constant connection:
Basics of 3D Higher Spin Theory
● Extend to higher spin theory by sl(2) sl(N)
sl(2)● Precise field content will depend on how one embed the gravity sector sl(2) into sl(N) Principal embedding:
sl(N) generators Embeded sl(2):
Higher spin:
● “Singularity” and “Horizon” are no longer gauge-invariant concepts.
● The only gauge invariant quantity: holonomy
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The diffeomorphism and local Lorentz symmetry are contained in the gauge symmetry .
●
Asymptotic to AdS3 :
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Asymptotic symmetry algebra
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For hs[], the asymptotic symmetry algebra is
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Goberdiel and Gopakumar conjecture:
algebra
Some other features of 3D Higher Spin Theory
Higher spin gravity based on hs[]
Holographic Dual
Known smooth solutions in 3D Higher Spin Theory
Black Holes with higher spin charges in SL(3)
[M. Gutperle and P. Kraus. `11]● Identify zero mode of spin 2 field and the modulus as thermodynamic conjugate pair (holomorphic formalism).
● Add higher spin chemical potential :
● How to fix the relation between charges/chemical potentials?
– In normal gravity, using the smooth condition on horizon.
– Holonomy around -cycles match with normal BTZ black hole
.
Ward identity analysis on CFT
● Use integrability to obtain the partition function and entropy.
– Entropy depends on higher spin charges.
In the bh gauge, two conditions match
Known smooth solutions in 3D Higher Spin Theory
Conical Surpluses in SL(N)
[Castro et al. `11]●
A generalization of AdS3 in higher spin theory
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Characterized by holonomy condition along -cycle
Constraint the vector of eigenvalues:
●
When
●
When
Global AdS3
● Contain conical singularity (conical surplus)
● Carry higher spin charges with even spin
General Framework in sl(N)
[de Boer, Jottar `13, Castro et al. `11]Q and M are linear in charges and chemical potentials respectively
● Smooth solutions are characterized by the holonomy condition along A-cycle:
For a constant gauge field:
Holonomy matrix:
● Condition constraint the vector of the eigenvalues of holonomy matrix:
● Highest/Lowest weight gauge convention:
Uniquely determined by equation of motion:
Modular Images of the Conical Surpluses
For a conical surplus,
For a general modular image ,
Goal: to figure out some transformations of
Using sl(N) algebra and the lowest/highest weight structure of , one can show:
Modular transformation:
Modular Images of the Conical Surpluses
Passive point of view: coordinate transformation and redefinition.
Active point of view: fix coordinate and (in grand canonical ensemble)
In order to sum the partition functions, we need to put them in a particular coordinate and ensemble.
Different solutions, different solid torus
Coordinate Transformation
● Just like the metrics of AdS3 and BTZ black hole are related by a coordinate transformation, the gauge fields of CS and BTZ are related by the following coordinate transformation up to some constant gauge transformation:
● This transformation is actually exactly the coordinate transformation that take the metric of thermal AdS3 to BTZ in Fefferman-Graham form.
Coordinate Transformation
● The coordinate transformation that relate CS to some general modular image :
III. Thermodynamic and Modular Invariant
Partition Function
Thermodynamics
(''canonical'' formalism) [de Boer, Jottar `13]saddle point approximation
● Modulus
act as the chemical potential of spin-2 charge●
s: chemical potential for higher spin charge with s>2Consistent thermodynamic system should have:
add boundary action to impose appropriate boundary condition
● Varying bulk action produce a boundary term:
● When varying the action, one need to vary
(
shape of the torus) explicitly. To do that, we can change the coordinate to the rigid torus and shift
dependence to the gauge field, a, and then vary it.●
involves the variation of charges and chemical potentials including
.
Boundary Action
[de Boer, Jottar `13]Add the following boundary action:
Varying the whole action yield the desired form ( including the part coming from ):
● T is the energy momentum tensor conjugated to the modulus
t.
● T is not holomorphic and will depend on the higher spin charges if the chemical potential is not zero.
● In short, the highest/lowest weight gauge choice of the charge/chemical potential separation plus this particular boundary action yield a consistent thermodynamic system.
Evaluation of On-Shell Action (Free Energy)
[Banados et al. `12]A
B
● Evaluation of the bulk action depends on the choice of A/B cycles.
● Slice the torus along the A-cycle yield the on-shell bulk action:
● For constant gauge fields:
● Using sl(N) algebra and the lowest/highest weight structure of , one can show that the on-shell boundary action is:
● The free energy is:
Black Holes Conical Surpluses
S-dualA/B cycles of black holes and conical surpluses:
The free energy becomes:
Solve the holonomy condition in sl(3) and expand in
Free Energy of SL(2,Z) family of solutions
Explicitly depend on the choice of A/B cycles
Indirectly depend on the choice of A/B cycles through holonomy condition
● Using the relation between the solutions of a conical surplus and a general modular image and the above expression of free energy, one can show that:
Modular Invariant Full Partition Function
● Partition function of a modular image:
● Sum over modular images:
● Sum over :
● Partition function of CS:
● Simple result (obtained non-trivially):
IV. Summary and Discussion
Summary
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We found out how a smooth solution in higher spin theory change under modular transformations through the holonomy condition.
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Using canonical formalism, we showed how to construct free energy (or partition function) in higher spin theory and verified the black holes and conical surpluses are S-dual.
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Starting from a conical surplus, one can generate all solutions related by modular transformations. By summing over all modular images, the
modular invariant partition function can be formally constructed.
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If the partition function can be explicitly constructed, one can use it to study the phase structure (e.g., Hawking-Page transition) in higher spin theory. However...
● How to solve the holonomy condition in general sl(N)?
● How to sum over the modular images?
Discussion
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The partition function obtained by canonical formalism is different from the one in holomorphic formalism (1103.4304) which is deduced from the integrability condition:
●
This integrability is incompatible to the modular transformation we found.
That is even the integrability is satisfied for a black hole, it is no longer true for a conical surplus.
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However, it has been showed that the partition function in holomporphic formalism match with CFT computation (1203.0015). So, what is going on?
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