Topologically massive higher- spin gravity Bindusar Sahoo Nikhef - - PowerPoint PPT Presentation

Topologically massive higher- spin gravity

Bindusar Sahoo Nikhef theory group Amsterdam IOP Bhubaneswar 1st March 2013

1

Topologically massive higher- spin gravity

Bindusar Sahoo Nikhef theory group Amsterdam IOP Bhubaneswar 1st March 2013

★Arjun Bagchi, Shailesh Lal, Arunabha Saha and BS, “T

• pologically Massive Higher Spin Gravity,” [ arXiv:1107.0915[hep-

th]], JHEP 1110 (2011) 150

★Arjun Bagchi, Shailesh Lal, Arunabha Saha and BS, “One loop partition function for T

• pologically Massive Higher Spin

Gravity,” [ arXiv:1107.2063[hep-th]], JHEP 1112 (2011) 068

1

Plan of talk

2

Plan of talk

• General introduction and motivation

2

Plan of talk

• General introduction and motivation
• Introduction and overview of higher-spin theories

2

Plan of talk

• General introduction and motivation
• Introduction and overview of higher-spin theories
• Introduction and overview of topologically massive gravity

2

Plan of talk

• General introduction and motivation
• Introduction and overview of higher-spin theories
• Introduction and overview of topologically massive gravity
• Classical aspects of topologically massive higher-spin gravity

2

Plan of talk

• General introduction and motivation
• Introduction and overview of higher-spin theories
• Introduction and overview of topologically massive gravity
• Classical aspects of topologically massive higher-spin gravity
• One loop partition function

2

Plan of talk

• General introduction and motivation
• Introduction and overview of higher-spin theories
• Introduction and overview of topologically massive gravity
• Classical aspects of topologically massive higher-spin gravity
• One loop partition function
• Future directions

2

General Introduction and Motivation

3

General Introduction and Motivation

Understanding the quantum nature of gravity is very crucial

3

General Introduction and Motivation

Understanding the quantum nature of gravity is very crucial

The appropriate degrees of freedom required to formulate quantum gravity resides in one less spatial dimension

• Holographic Principle
• G. ’t Hooft, gr-qc/9310026.
• L. Susskind, J. Math. Phys. 36, 6377 (1995) [hep-th/9409089]

3

General Introduction and Motivation

• In the context of string theory holographic principle is realized via the

celebrated AdS-CFT Conjecture, J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200]

4

General Introduction and Motivation

• In the context of string theory holographic principle is realized via the

celebrated AdS-CFT Conjecture, J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200]

• It relates string theory/(super)gravity theory on negatively curved anti-de

Sitter (AdS) spacetime to a (super) conformal field theory (CFT) on the boundary of the AdS space-time.

4

General Introduction and Motivation

• In the context of string theory holographic principle is realized via the

celebrated AdS-CFT Conjecture, J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200]

• It relates string theory/(super)gravity theory on negatively curved anti-de

Sitter (AdS) spacetime to a (super) conformal field theory (CFT) on the boundary of the AdS space-time.

4

General Introduction and Motivation

• In the context of string theory holographic principle is realized via the

celebrated AdS-CFT Conjecture, J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200]

• It relates string theory/(super)gravity theory on negatively curved anti-de

Sitter (AdS) spacetime to a (super) conformal field theory (CFT) on the boundary of the AdS space-time.

⌧ exp Z φ0O

• CF T

= Zs (φ0)

Edward Witten, Adv.Theor.Math.Phys. 2 (1998) 253-291 [hep-th/9802150]

4

General Introduction and Motivation

5

General Introduction and Motivation

• Applications of AdS-CFT is limited to understanding strongly coupled gauge theories from

gravity

5

General Introduction and Motivation

• Applications of AdS-CFT is limited to understanding strongly coupled gauge theories from

gravity

• Hydrodynamics

5

General Introduction and Motivation

• Applications of AdS-CFT is limited to understanding strongly coupled gauge theories from

gravity

• Hydrodynamics
• QCD

5

General Introduction and Motivation

• Applications of AdS-CFT is limited to understanding strongly coupled gauge theories from

gravity

• Hydrodynamics
• QCD
• Condensed matter physics

5

General Introduction and Motivation

• Applications of AdS-CFT is limited to understanding strongly coupled gauge theories from

gravity

• Hydrodynamics
• QCD
• Condensed matter physics
• Very little progress in understanding quantum gravity from CFT

5

General Introduction and Motivation

• Applications of AdS-CFT is limited to understanding strongly coupled gauge theories from

gravity

• Hydrodynamics
• QCD
• Condensed matter physics
• Very little progress in understanding quantum gravity from CFT
• It involves finding CFT duals with analytic tractability and intrinsic complexity so that one

is able to capture some aspects of quantum gravity with quantitative precision

5

General Introduction and Motivation

• Applications of AdS-CFT is limited to understanding strongly coupled gauge theories from

gravity

• Hydrodynamics
• QCD
• Condensed matter physics
• There are problems connecting AdS-CFT to the real world because it relies on

supersymmetry and cannot be defined on de Sitter spacetime

• Very little progress in understanding quantum gravity from CFT
• It involves finding CFT duals with analytic tractability and intrinsic complexity so that one

is able to capture some aspects of quantum gravity with quantitative precision

5

General Introduction and Motivation

• Applications of AdS-CFT is limited to understanding strongly coupled gauge theories from

gravity

• Hydrodynamics
• QCD
• Condensed matter physics
• There are problems connecting AdS-CFT to the real world because it relies on

supersymmetry and cannot be defined on de Sitter spacetime

Higher Spin holography helps !!

• Very little progress in understanding quantum gravity from CFT
• It involves finding CFT duals with analytic tractability and intrinsic complexity so that one

is able to capture some aspects of quantum gravity with quantitative precision

5

Introduction and overview of higher-spin theories in three dimensions

• Higher-spin theories are gravitational theories in the presence of

additional higher-spin (s>2) gauge field

6

Introduction and overview of higher-spin theories in three dimensions

• Higher-spin theories are gravitational theories in the presence of

additional higher-spin (s>2) gauge field

• Theories of this type were first constructed by Vasiliev for [M.A. Vasiliev, Int. J.
• Mod. Phys. A 6 (1991) 1115] and were later generalized to AdS3

AdS4

6

Introduction and overview of higher-spin theories in three dimensions

• Higher-spin theories are gravitational theories in the presence of

additional higher-spin (s>2) gauge field

• Unlike their higher-dimensional cousins, they admit a truncation to an

arbitrary maximal spin N, rather than involving the customary infinite tower

• f higher-spin fields
• Theories of this type were first constructed by Vasiliev for [M.A. Vasiliev, Int. J.
• Mod. Phys. A 6 (1991) 1115] and were later generalized to AdS3

AdS4

6

Introduction and overview of higher-spin theories in three dimensions

• Higher-spin theories are gravitational theories in the presence of

additional higher-spin (s>2) gauge field

• Unlike their higher-dimensional cousins, they admit a truncation to an

arbitrary maximal spin N, rather than involving the customary infinite tower

• f higher-spin fields
• Essentially they can be written as a SL(N,R)×SL(N,R) gauge theory as:
• Theories of this type were first constructed by Vasiliev for [M.A. Vasiliev, Int. J.
• Mod. Phys. A 6 (1991) 1115] and were later generalized to AdS3

AdS4

6

S = ` 16⇡GSCS[A] − ` 16⇡GSCS[ ˜ A] A =

• j a

µ Ja + t a1···as−1 µ

Ta1···as−1

• dxµ

j a

µ =

⇣ ! + e ` ⌘ a

µ ,

˜ j a

µ =

⇣ ! − e ` ⌘ a

µ

t a1···as−1

µ

= ⇣ ! + e ` ⌘ a1···as−1

µ

, ˜ t a1···as−1

µ

= ⇣ ! − e ` ⌘ a1···as−1

µ

SCS[A] = Z tr ✓ A ∧ dA + 2 3A ∧ A ∧ A ◆

Introduction and overview of higher-spin theories in three dimensions

7

Introduction and overview of higher-spin theories in three dimensions

• The asymptotic symmetry algebra for theories with higher spin in

have been examined in [M. Henneaux, S. J. Rey arXiv:1008.4579] and [A. Campoleoni, S. Fredenhagen, S.

Pfenninger, S. Theisen arXiv:1008.4744]. They found that a Brown-Henneaux like analysis

for a theory with maximal spin N in the bulk yields a asymptotic symmetry algebra.

WN

AdS3

8

• This has been tested at the one-loop level in [M. R. Gaberdiel, R. Gopakumar and A. Saha arXiv:

1009.6087], using the Heat Kernel techniques developed in [J. R. David, M. R. Gaberdiel and R. Gopakumar arXiv:0911.5085]

Introduction and overview of higher-spin theories in three dimensions

• The asymptotic symmetry algebra for theories with higher spin in

have been examined in [M. Henneaux, S. J. Rey arXiv:1008.4579] and [A. Campoleoni, S. Fredenhagen, S.

Pfenninger, S. Theisen arXiv:1008.4744]. They found that a Brown-Henneaux like analysis

for a theory with maximal spin N in the bulk yields a asymptotic symmetry algebra.

WN

AdS3

8

• This has been tested at the one-loop level in [M. R. Gaberdiel, R. Gopakumar and A. Saha arXiv:

1009.6087], using the Heat Kernel techniques developed in [J. R. David, M. R. Gaberdiel and R. Gopakumar arXiv:0911.5085]

Introduction and overview of higher-spin theories in three dimensions

• The asymptotic symmetry algebra for theories with higher spin in

have been examined in [M. Henneaux, S. J. Rey arXiv:1008.4579] and [A. Campoleoni, S. Fredenhagen, S.

Pfenninger, S. Theisen arXiv:1008.4744]. They found that a Brown-Henneaux like analysis

for a theory with maximal spin N in the bulk yields a asymptotic symmetry algebra.

WN

AdS3

• Finally, this lead to the proposal of a duality between a family of higher-

spin theories in and minimal models in the large N limit in [M. R.

Gaberdiel, R. Gopakumar arXiv:1011.2986], which has subsequently been checked in [M. R. Gaberdiel, T. Hartman arXiv:1101.2910]

AdS3

WN

8

Introduction and overview of topologically massive gravity

• Pure Einstein gravity in 3 dimensions is locally trivial classically while its

quantum status is still not clear.

9

Introduction and overview of topologically massive gravity

• Pure Einstein gravity in 3 dimensions is locally trivial classically while its

quantum status is still not clear.

There is however an interesting deformation to pure 3 dimensional gravity. It is the inclusion

• f gravitational Chern Simons terms

9

Introduction and overview of topologically massive gravity

• Pure Einstein gravity in 3 dimensions is locally trivial classically while its

quantum status is still not clear.

There is however an interesting deformation to pure 3 dimensional gravity. It is the inclusion

• f gravitational Chern Simons terms

S3 = SEH + SCS where SEH = R d3x√−g(R − 2Λ) and SCS =

1 2µ

R d3x✏µνρ ⇣ Γσ

µλ@νΓλ ρσ + 2 3Γσ µλΓλ νθΓθ ρσ

⌘

9

Introduction and overview of topologically massive gravity

• Pure Einstein gravity in 3 dimensions is locally trivial classically while its

quantum status is still not clear.

There is however an interesting deformation to pure 3 dimensional gravity. It is the inclusion

• f gravitational Chern Simons terms

S3 = SEH + SCS where SEH = R d3x√−g(R − 2Λ) and SCS =

1 2µ

R d3x✏µνρ ⇣ Γσ

µλ@νΓλ ρσ + 2 3Γσ µλΓλ νθΓθ ρσ

⌘

Essentially in the Chern-Simon’s language this amount to the modification

9

Introduction and overview of topologically massive gravity

• Pure Einstein gravity in 3 dimensions is locally trivial classically while its

quantum status is still not clear.

There is however an interesting deformation to pure 3 dimensional gravity. It is the inclusion

• f gravitational Chern Simons terms

S3 = SEH + SCS where SEH = R d3x√−g(R − 2Λ) and SCS =

1 2µ

R d3x✏µνρ ⇣ Γσ

µλ@νΓλ ρσ + 2 3Γσ µλΓλ νθΓθ ρσ

⌘

Essentially in the Chern-Simon’s language this amount to the modification

S3 = aLSCS[A] − aRSCS[ ˜ A] + (aL − aR) R tr ⇣ β ∧ ⇣ F − ˜ F ⌘⌘ aL + aR =

` 8⇡G

aL − aR = 2

µ 9

Introduction and overview of topologically massive gravity

• Pure Einstein gravity in 3 dimensions is locally trivial classically while its

quantum status is still not clear.

There is however an interesting deformation to pure 3 dimensional gravity. It is the inclusion

• f gravitational Chern Simons terms

S3 = SEH + SCS where SEH = R d3x√−g(R − 2Λ) and SCS =

1 2µ

R d3x✏µνρ ⇣ Γσ

µλ@νΓλ ρσ + 2 3Γσ µλΓλ νθΓθ ρσ

⌘

Essentially in the Chern-Simon’s language this amount to the modification

S3 = aLSCS[A] − aRSCS[ ˜ A] + (aL − aR) R tr ⇣ β ∧ ⇣ F − ˜ F ⌘⌘ aL + aR =

` 8⇡G

aL − aR = 2

µ

This theory is called topologically massive gravity

9

Introduction and overview of topologically massive gravity

Wei Li, Wei Song, Andrew Strominger, arXiv 0801.4566

TMG at a generic point typically has negative energy excitations and the theory does not make sense. At the chiral point μl = 1, the ghost like excitation goes away and the theory is conjectured to be dual to a “Chiral CFT”

10

Introduction and overview of topologically massive gravity

Wei Li, Wei Song, Andrew Strominger, arXiv 0801.4566

TMG at a generic point typically has negative energy excitations and the theory does not make sense. At the chiral point μl = 1, the ghost like excitation goes away and the theory is conjectured to be dual to a “Chiral CFT”

Daniel Grumiller, Niklas Johansson, arXiv:0805.2610, Daniel Grumiller, Niklas Johansson, arXiv:0808.2575

Even at the chiral point, there were negative energy

• excitations. This was in contradiction with the earlier chiral

CFT conjecture. But this negative energy excitation was nicer in the sense that it had all the properties for the theory to be dual to well known non unitary CFT’s known as “Logarithmic CFT’s” or LCFT’s in short.

10

Introduction and overview of topologically massive gravity

Wei Li, Wei Song, Andrew Strominger, arXiv 0801.4566

TMG at a generic point typically has negative energy excitations and the theory does not make sense. At the chiral point μl = 1, the ghost like excitation goes away and the theory is conjectured to be dual to a “Chiral CFT”

Daniel Grumiller, Niklas Johansson, arXiv:0805.2610, Daniel Grumiller, Niklas Johansson, arXiv:0808.2575

Even at the chiral point, there were negative energy

• excitations. This was in contradiction with the earlier chiral

CFT conjecture. But this negative energy excitation was nicer in the sense that it had all the properties for the theory to be dual to well known non unitary CFT’s known as “Logarithmic CFT’s” or LCFT’s in short.

What is LCFT?

• An LCFT is a generalization of the concept of

(usually two-dimensional) conformal field theory in which the correlators of the basic fields are allowed to be multiply valued and be functions of the logarithm of the separation of the operators

10

Introduction and overview of topologically massive gravity

Why are LCFT’s interesting?

LCFT’s are very useful to study systems at (or near) a critical point with quenched disorder, like spin glasses/quenched random magnets, dilute self-avoiding polymers or percolation [see

arxiv: 1001.0002, Daniel Grumiller and Niklas Johansson for details]

11

Introduction and overview of topologically massive gravity

Why are LCFT’s interesting?

LCFT’s are very useful to study systems at (or near) a critical point with quenched disorder, like spin glasses/quenched random magnets, dilute self-avoiding polymers or percolation [see

arxiv: 1001.0002, Daniel Grumiller and Niklas Johansson for details]

Kostas Skenderis, Marika Taylor, Balt C. van Rees, arXiv: 0906.4926, Daniel Grumiller, Ivo Sachs arXiv:0910.5241

Correlation function computed using rigorous holographic renormalization procedure is consistent with an LCFT dual. But chiral CFT exists as a consistent subsector of the full LCFT and is dual to TMG with truncated boundary conditions.

11

Questions??

12

Classical aspects of Topologically massive higher- spin gravity

• Motivated by the features of topologically massive gravity recounted

previously, a natural question to ask is what happens when these higher- spin theories are similarly deformed by the addition of a Gravitational Chern-Simons like term.

13

Classical aspects of Topologically massive higher- spin gravity

• Motivated by the features of topologically massive gravity recounted

previously, a natural question to ask is what happens when these higher- spin theories are similarly deformed by the addition of a Gravitational Chern-Simons like term.

This amounts to the modification

S3 = aLSCS[A] − aRSCS[ ˜ A] + (aL − aR) R tr ⇣ β ∧ ⇣ F − ˜ F ⌘⌘ aL + aR =

` 8⇡G

aL − aR = 2

µ 13

Classical aspects of Topologically massive higher- spin gravity

• Motivated by the features of topologically massive gravity recounted

previously, a natural question to ask is what happens when these higher- spin theories are similarly deformed by the addition of a Gravitational Chern-Simons like term.

This amounts to the modification

S3 = aLSCS[A] − aRSCS[ ˜ A] + (aL − aR) R tr ⇣ β ∧ ⇣ F − ˜ F ⌘⌘ aL + aR =

` 8⇡G

aL − aR = 2

µ

A, ˜ A ∈ SL(N, R)

13

Classical aspects of Topologically massive higher- spin gravity

φM1···Ms = ¯ eA1

(M1 · · · ¯

eAs−1

Ms−1hMs)A1···As−1

• We will study the linearized equation of motion of this theory in terms of

the physical field

14

Classical aspects of Topologically massive higher- spin gravity

φM1···Ms = ¯ eA1

(M1 · · · ¯

eAs−1

Ms−1hMs)A1···As−1

• We will study the linearized equation of motion of this theory in terms of

the physical field

S [φ] = 1

2

R d3x√−gφMNP ⇣ ˆ FMNP − 1

2g(MN ˆ

FP ) ⌘

The action is

14

Classical aspects of Topologically massive higher- spin gravity

φM1···Ms = ¯ eA1

(M1 · · · ¯

eAs−1

Ms−1hMs)A1···As−1

• We will study the linearized equation of motion of this theory in terms of

the physical field

S [φ] = 1

2

R d3x√−gφMNP ⇣ ˆ FMNP − 1

2g(MN ˆ

FP ) ⌘

The action is Where

ˆ FMNP = ˜ FMNP +

1 6µεQR(MrQ ˜

FR NP ), ˜ FMNP = FMNP 2

`2 g(MNφP )A A

FMNP [φ] = r2φMNP r(MrAφNP )A + 1

2r(MrNφP )A A

14

Classical aspects of Topologically massive higher- spin gravity

• The above action has the gauge symmetry

φM1···Ms ! φM1···Ms + r(M1ξM2···Ms)

15

Classical aspects of Topologically massive higher- spin gravity

• The above action has the gauge symmetry

φM1···Ms ! φM1···Ms + r(M1ξM2···Ms)

• Using the field redefinition and gauge condition

φMNP = ˜ φMNP 1 9g(MN ˜ φP ), rQ ˜ φQMN = 1 2r(M ˜ φN).

15

Classical aspects of Topologically massive higher- spin gravity

• The above action has the gauge symmetry

φM1···Ms ! φM1···Ms + r(M1ξM2···Ms)

• Using the field redefinition and gauge condition

φMNP = ˜ φMNP 1 9g(MN ˜ φP ), rQ ˜ φQMN = 1 2r(M ˜ φN).

• The equation of motion can be brought into the form

D(M)D(L)D(R)φMNP = 0

15

Classical aspects of Topologically massive higher- spin gravity

• Where

D(M)φMNP ⌘ φMNP +

1 6µεQR(MrQ ˜

φR

NP )

D(L,R)φMNP ⌘ φMNP ± `

6εQR(MrQ ˜

φR

NP )

16

Classical aspects of Topologically massive higher- spin gravity

• Where

D(M)φMNP ⌘ φMNP +

1 6µεQR(MrQ ˜

φR

NP )

D(L,R)φMNP ⌘ φMNP ± `

6εQR(MrQ ˜

φR

NP )

• Unlike spin-2 the trace is not zero, rather it is a particular spin-zero

component of the trace that is zero

16

Classical aspects of Topologically massive higher- spin gravity

• Where

D(M)φMNP ⌘ φMNP +

1 6µεQR(MrQ ˜

φR

NP )

D(L,R)φMNP ⌘ φMNP ± `

6εQR(MrQ ˜

φR

NP )

• Unlike spin-2 the trace is not zero, rather it is a particular spin-zero

component of the trace that is zero

rMφM = 0

16

Classical aspects of Topologically massive higher- spin gravity

• Where

D(M)φMNP ⌘ φMNP +

1 6µεQR(MrQ ˜

φR

NP )

D(L,R)φMNP ⌘ φMNP ± `

6εQR(MrQ ˜

φR

NP )

• Unlike spin-2 the trace is not zero, rather it is a particular spin-zero

component of the trace that is zero

rMφM = 0

D(M), D(L) and D(R)

• Since

are mutually commuting operators, the full solution can be split up into massive, Left and Right modes

16

Classical aspects of Topologically massive higher- spin gravity

• Where

D(M)φMNP ⌘ φMNP +

1 6µεQR(MrQ ˜

φR

NP )

D(L,R)φMNP ⌘ φMNP ± `

6εQR(MrQ ˜

φR

NP )

• Unlike spin-2 the trace is not zero, rather it is a particular spin-zero

component of the trace that is zero

rMφM = 0

D(M), D(L) and D(R)

• Since

are mutually commuting operators, the full solution can be split up into massive, Left and Right modes

• We can also split up the solution into “trace” modes and “traceless”

modes

16

Classical aspects of Topologically massive higher- spin gravity

• The background metric has SL(2,R)×SL(2,R) isometry. Using the

relations between Laplacian and the casimir of the isometry group, the solutions could be further classified as “primaries” and their descendants

AdS3

17

Classical aspects of Topologically massive higher- spin gravity

• “Primary” conditions

L0φMNP = hφMNP ¯ L0φMNP = ¯ hφMNP L1φMNP = ¯ L1φMNP = 0

• The background metric has SL(2,R)×SL(2,R) isometry. Using the

relations between Laplacian and the casimir of the isometry group, the solutions could be further classified as “primaries” and their descendants

AdS3

17

Classical aspects of Topologically massive higher- spin gravity

• “Primary” conditions

L0φMNP = hφMNP ¯ L0φMNP = ¯ hφMNP L1φMNP = ¯ L1φMNP = 0

1 + 3µ` 3µ` 2 + µ` −1 + µ`

Modes Left weight Right weight Massive Trace Left Trace 4 3 Right Trace 3 4 Massive Traceless Left Traceless 3 Right Traceless 3

• The background metric has SL(2,R)×SL(2,R) isometry. Using the

relations between Laplacian and the casimir of the isometry group, the solutions could be further classified as “primaries” and their descendants

AdS3

17

Classical aspects of Topologically massive higher- spin gravity

At the chiral point µ` = 1 we see that “massive” and “left” modes coincide and hence the basis of solutions become insufficient to describe the dynamics.

18

Classical aspects of Topologically massive higher- spin gravity

At the chiral point µ` = 1 we see that “massive” and “left” modes coincide and hence the basis of solutions become insufficient to describe the dynamics. However at the Chiral point, a new mode emerges which is annihilated by D(L)2 and not by D(L)

Φ(new) = limµ`→1

Φ(M)(µ`)−Φ(L) µ`−1

= dΦ(M)(✏)

d✏

|✏=0

18

Classical aspects of Topologically massive higher- spin gravity

At the chiral point µ` = 1 we see that “massive” and “left” modes coincide and hence the basis of solutions become insufficient to describe the dynamics. However at the Chiral point, a new mode emerges which is annihilated by D(L)2 and not by D(L)

Φ(new) = limµ`→1

Φ(M)(µ`)−Φ(L) µ`−1

= dΦ(M)(✏)

d✏

|✏=0

This is known as “logarithmic” mode

18

Classical aspects of Topologically massive higher- spin gravity

Using this procedure, the logarithmic traceless and trace modes can be evaluated and they satisfy

L0 ˆ χMNP = 3χ(L)

MNP + 4ˆ

χMNP , ¯ L0 ˆ χMNP = 3χ(L)

MNP + 3ˆ

χMNP , L0 ˆ ΣMNP = Σ(L)

MNP + 3ˆ

ΣMNP ¯ L0 ˆ ΣMNP = Σ(L)

MNP .

19

Classical aspects of Topologically massive higher- spin gravity

Using this procedure, the logarithmic traceless and trace modes can be evaluated and they satisfy

L0 ˆ χMNP = 3χ(L)

MNP + 4ˆ

χMNP , ¯ L0 ˆ χMNP = 3χ(L)

MNP + 3ˆ

χMNP , L0 ˆ ΣMNP = Σ(L)

MNP + 3ˆ

ΣMNP ¯ L0 ˆ ΣMNP = Σ(L)

MNP .

This block diagonal character of is typical of “logarithmic conformal field theories”

L0

19

Classical aspects of Topologically massive higher- spin gravity

Using this procedure, the logarithmic traceless and trace modes can be evaluated and they satisfy

L0 ˆ χMNP = 3χ(L)

MNP + 4ˆ

χMNP , ¯ L0 ˆ χMNP = 3χ(L)

MNP + 3ˆ

χMNP , L0 ˆ ΣMNP = Σ(L)

MNP + 3ˆ

ΣMNP ¯ L0 ˆ ΣMNP = Σ(L)

MNP .

This block diagonal character of is typical of “logarithmic conformal field theories”

L0

We can now compute the energy using the Ostrogradsky method

H = R d2x ⇣ ˙ ˜ φMNP Π(1)MNP + ˙ KMNP Π(2)MNP L ⌘ KMNP = r0 ˜ φMNP , Π(2)MNP ⌘

δS δ ˙ KMNP

19

Classical aspects of Topologically massive higher- spin gravity

20

Classical aspects of Topologically massive higher- spin gravity

µ` < 1

Modes Sign of Energy at Sign of Energy at Sign of Energy at Massive trace +

• Zero

Left trace

• +

Zero Right trace + + + Logarithmic trace + Massive traceless

• +

Zero Left traceless +

• Zero

Right traceless + + + Logarithmic traceless

• µ` > 1

µ` = 1

20

Classical aspects of Topologically massive higher- spin gravity

µ` < 1

Modes Sign of Energy at Sign of Energy at Sign of Energy at Massive trace +

• Zero

Left trace

• +

Zero Right trace + + + Logarithmic trace + Massive traceless

• +

Zero Left traceless +

• Zero

Right traceless + + + Logarithmic traceless

• µ` > 1

µ` = 1

Very suggestive of a higher-spin logarithmic conformal field theory dual

20

One loop partition function

• Our previous analysis suggested that the TMHSG at the chiral point is

dual to a higher spin extension of a logarithmic conformal field theory. Specifically, we found that the space of solutions developed an extra logarithmic branch at the chiral point, which carried negative energy.

21

One loop partition function

• Our previous analysis suggested that the TMHSG at the chiral point is

dual to a higher spin extension of a logarithmic conformal field theory. Specifically, we found that the space of solutions developed an extra logarithmic branch at the chiral point, which carried negative energy.

• A necessary condition for a LCFT dual is a partition function which

cannot be holomorphically factorized.

21

One loop partition function

• Our previous analysis suggested that the TMHSG at the chiral point is

dual to a higher spin extension of a logarithmic conformal field theory. Specifically, we found that the space of solutions developed an extra logarithmic branch at the chiral point, which carried negative energy.

• A necessary condition for a LCFT dual is a partition function which

cannot be holomorphically factorized.

• We shall evaluate the one-loop determinants utilising the machinery

developed in [J. R. David, M. R. Gaberdiel and R. Gopakumar, arXiv:0911.5085 [hep-th]]., according to which the relevant determinant takes the following form

21

One loop partition function

• Our previous analysis suggested that the TMHSG at the chiral point is

dual to a higher spin extension of a logarithmic conformal field theory. Specifically, we found that the space of solutions developed an extra logarithmic branch at the chiral point, which carried negative energy.

• A necessary condition for a LCFT dual is a partition function which

cannot be holomorphically factorized.

• We shall evaluate the one-loop determinants utilising the machinery

developed in [J. R. David, M. R. Gaberdiel and R. Gopakumar, arXiv:0911.5085 [hep-th]]., according to which the relevant determinant takes the following form

− log det(−∆ + m2

s

`2 )T T (s) =

R ∞

dt t K(s)(τ, ¯

τ; t)e−s(s−3)t

21

One loop partition function

Where is the spin-s heat kernel given by

Ks

K(s)(τ, ¯ τ; t) = P∞

m=1 τ2 √ 4πt|sin mτ

2 |2 cos(smτ1)e− m2τ2 2 4t e−(s+1)t

22

One loop partition function

Where is the spin-s heat kernel given by

Ks

K(s)(τ, ¯ τ; t) = P∞

m=1 τ2 √ 4πt|sin mτ

2 |2 cos(smτ1)e− m2τ2 2 4t e−(s+1)t

The full partition function at the chiral point after including contribution from all the spins starting from 2 till N is

ZT MHSG = QN

s=2

hQ∞

n=s 1 |1−qn|2

Q∞

m=s

Q∞

¯ m=0 1 (1−qm ¯ q ¯

m)

i × hQN

s=3

Qs−1

t=2

Q∞

p=r(s,t)

Q∞

¯ p=k(s,t) 1 (1−qp ¯ q ¯

p)

i

22

One loop partition function

Where is the spin-s heat kernel given by

Ks

K(s)(τ, ¯ τ; t) = P∞

m=1 τ2 √ 4πt|sin mτ

2 |2 cos(smτ1)e− m2τ2 2 4t e−(s+1)t

The full partition function at the chiral point after including contribution from all the spins starting from 2 till N is

ZT MHSG = QN

s=2

hQ∞

n=s 1 |1−qn|2

Q∞

m=s

Q∞

¯ m=0 1 (1−qm ¯ q ¯

m)

i × hQN

s=3

Qs−1

t=2

Q∞

p=r(s,t)

Q∞

¯ p=k(s,t) 1 (1−qp ¯ q ¯

p)

i

Where

k(s, m) = s(s−1)−(s−m+1)(s−m−1)

2(s−m)

and r(s, m) = k(s, m) + s − m

22

Future Directions

• As discussed, from our bulk analysis, we found “clues” that TMHSG at

chiral point is dual to a higher-spin LCFT which has richer structures compared to a normal LCFT. Finding such a dual would put us on a firm footing to compare our bulk results, in particular the one-loop partition function.

23

Future Directions

• As discussed, from our bulk analysis, we found “clues” that TMHSG at

chiral point is dual to a higher-spin LCFT which has richer structures compared to a normal LCFT. Finding such a dual would put us on a firm footing to compare our bulk results, in particular the one-loop partition function.

• Computation of correlation function from the bulk side. In particular, it

would be interesting to see how the trace modes and its logarithmic partner behaves. This could then be compared with field theory result subject to the work discussed in the previous point

23

Future Directions

• As discussed, from our bulk analysis, we found “clues” that TMHSG at

chiral point is dual to a higher-spin LCFT which has richer structures compared to a normal LCFT. Finding such a dual would put us on a firm footing to compare our bulk results, in particular the one-loop partition function.

• Computation of correlation function from the bulk side. In particular, it

would be interesting to see how the trace modes and its logarithmic partner behaves. This could then be compared with field theory result subject to the work discussed in the previous point

• Analogous to [A. Castro, R. Gopakumar, M. Gutperle and J. Raeymaekers, [arXiv:1111.3381 [hep-th]] ], we would like

to identify classical solutions in TMHSG which are locally warped and have conical defects, but nevertheless due to the higher-spin gauge transformations become smooth solutions of the underlying theory. We would also like to identify the corresponding states in the proposed dual LCFT’s (subject to the first point). This will help us to understand our TMHSG-higher spin LCFT conjecture in a better way.

AdS3

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