### 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.

●

### 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

●

### 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

●

### 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

●

### 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>2

Consistent 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-dual}

A/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.

●

### 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:

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### 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?

●