• 沒有找到結果。

# Modular Properties of 3D Higher Spin Theory

N/A
N/A
Protected

Share "Modular Properties of 3D Higher Spin Theory"

Copied!
30
0
0

(1)

### Modular Properties of 3D Higher Spin Theory

(Based on 1308.2959)

### Feng-Li Lin

(National Taiwan Normal University)

### Wei Li

(Max-Planck-Institut)

(2)

## Outline

(3)

### I. Modular Properties

(4)

Same lattice, if

rotation scaling

Modular transformation:

(Passive Point of View)

### Modular Group and Modular Transformation

Complex plane quotient Torus

1

Modulus:

S-dual:

(5)

### 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

(6)

### Thermal AdS and BTZ Black Hole

Focus on the solutions with Euclidean signature.

BTZ black hole:

In general:

(7)

### Modular Invariant Partition Function

defined on torus, must be modular invariant.

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.

(8)

(9)

[Vasiliev `91]

(10)

### 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:

(11)

### 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

(12)

(13)

### 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

(14)

### Conical Surpluses in SL(N)

[Castro et al. `11]

### Characterized by holonomy condition along -cycle

Constraint the vector of eigenvalues:

### When

Contain conical singularity (conical surplus)

Carry higher spin charges with even spin

(15)

### 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:

(16)

### 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:

(17)

### 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

(18)

### 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.

(19)

### Coordinate Transformation

The coordinate transformation that relate CS to some general modular image :

(20)

(21)

### Thermodynamics

(''canonical'' formalism) [de Boer, Jottar `13]

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

(22)

### Boundary Action

[de Boer, Jottar `13]

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.

(23)

### Evaluation of On-Shell Action (Free Energy)

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:

(24)

### 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

(25)

### 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:

(26)

### 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):

(27)

(28)

### 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?

(29)

(30)

## Thank you!

  The The extended nature extended nature of string theory introduces of string theory introduces additional degrees of freedom?. additional degrees of freedom localized

S15 Expectation value of the total spin-squared operator h ˆ S 2 i for the ground state of cationic n-PP as a function of the chain length, calculated using KS-DFT with various

of the spin polarisation L. Bocher et al. submitted (2011).. Mapping plasmons and EM fields Mapping plasmons and EM fields.. New possibilities for studying the low

• In 1976 Hawking argued that the formation and evaporation of black holes leads to a fundamental loss of information from the universe, a breakdown of predictability, as

Specifically, in Section 3, we present a smoothing function of the generalized FB function, and studied some of its favorable properties, including the Jacobian consistency property;

We try to explore category and association rules of customer questions by applying customer analysis and the combination of data mining and rough set theory.. We use customer

Because there is less information production produced in auctions, the information production theory predicts that auctions in IPOs would have higher volatility and less

This research is based on the TRIZ theory, try to enhance the function of pencils and personalize the design1. TRIZ theory includes contradiction matrix, physical contradictions, and