Rank-n logarithmic conformal field theory (LCFT) in the BTZ black hole Rank-n logarithmic conformal field theory (LCFT) in the BTZ black hole Taeyoon Moon (CQUeST) collaboration with Prof. Yun Soo Myung

 Rank-3 finite temperature LCFT  PRD 86, (2012) [arXiv: [hep-th]]  Rank-n LCFT in the BTZ black hole  arXiv : [hep-th] This presentation summarizes..

 Critical gravity - Quadratic gravity in four dimensions - Critical gravity - Log mode in critical gravity - Poly critical gravity  Logarithmic conformal field theory (LCFT)  Rank-3 LCFT on BTZ black hole  Rank-n LCFT on BTZ black hole Contents are…

 Critical gravity - Quadratic gravity in four dimensions - Critical gravity - Log mode in critical gravity - Poly critical gravity  Logarithmic conformal field theory (LCFT)  Rank-3 LCFT on BTZ black hole  Rank-n LCFT on BTZ black hole Contents are… brief review

 Critical gravity - Quadratic gravity in four dimensions - Critical gravity - Log mode in critical gravity - Poly critical gravity  Logarithmic conformal field theory (LCFT)  Rank-3 LCFT on BTZ black hole  Rank-n LCFT on BTZ black hole Contents are… brief review our work

Quadratic gravity in four dimensions (brief review)

For a long time, people have asked the following question:

Quadratic gravity in four dimensions (brief review) For a long time, people have asked the following question: What is the four dimensional gravity theory, which is renormalizable and unitary?

Quadratic gravity in four dimensions (brief review) K. S. Stelle (1977)

Quadratic gravity in four dimensions (brief review) K. S. Stelle (1977) This gravity theory is renormalisable, and it describes in general a massless spin-2 graviton, a massive spin-2 field and a massive scalar.

Quadratic gravity in four dimensions (brief review) K. S. Stelle (1977) This gravity theory is renormalisable, and it describes in general a massless spin-2 graviton, a massive spin-2 field and a massive scalar. The energies of excitations of the massive spin-2 field are negative, while those of the massless graviton are, as usual, positive. Thus although the theory is renormalisable, it suffers from having ghosts.

Quadratic gravity in four dimensions (brief review) K. S. Stelle (1977) This gravity theory is renormalisable, and it describes in general a massless spin-2 graviton, a massive spin-2 field and a massive scalar. The energies of excitations of the massive spin-2 field are negative, while those of the massless graviton are, as usual, positive. Thus although the theory is renormalisable, it suffers from having ghosts. The massive spin-0 is absent in the special case α = −3 β, while the massive spin-2 is absent if instead α = 0.

Quadratic gravity in four dimensions (brief review) K. S. Stelle (1977) This gravity theory is renormalisable, and it describes in general a massless spin-2 graviton, a massive spin-2 field and a massive scalar. The energies of excitations of the massive spin-2 field are negative, while those of the massless graviton are, as usual, positive. Thus although the theory is renormalisable, it suffers from having ghosts. The massive spin-0 is absent in the special case α = −3 β, while the massive spin-2 is absent if instead α = 0. What if AdS background?

Critical gravity in four dimensions (brief review) What if AdS background? K. S. Stelle (1977)

Critical gravity in four dimensions (brief review) What if AdS background? (Lu, Pope, PRL2011)

Critical gravity in four dimensions (brief review) (Lu, Pope, PRL2011)

Critical gravity in four dimensions (brief review) (Lu, Pope, PRL2011)

Critical gravity in four dimensions (brief review) (Lu, Pope, PRL2011)

Critical gravity in four dimensions (brief review) (Lu, Pope, PRL2011)

Critical gravity in four dimensions (brief review) critical point (Lu, Pope, PRL2011)

Critical gravity in four dimensions (brief review) critical point (Lu, Pope, PRL2011) The massive spin-0 is absent in the special case α = −3 β, while the massive spin-2 is absent if instead α = 0.

Critical gravity in four dimensions (brief review) critical point (Lu, Pope, PRL2011) The massive spin-0 is absent in the special case α = −3 β, while the massive spin-2 is absent if instead α = 0. Is this the very gravity action that we want ?

Log mode in critical gravity

critical point

Log mode in critical gravity critical point

Log mode in critical gravity critical point

Log mode in critical gravity critical point

Log mode in critical gravity

vanishes !

Log mode in critical gravity vanishes ! Furthermore, the mass and entropy of black holes at the critical point both vanish ! (Lu, Pope, PRL2011)

Log mode in critical gravity vanishes ! Furthermore, the mass and entropy of black holes at the critical point both vanish ! (Lu, Pope, PRL2011) Crucially, such a log mode can lead to a negative norm state, which can not be defined in Hilbert space. (Porrati, Roberts, PRD2011)

Log mode in critical gravity “Conformal gravity and extensions of critical gravity” Lu, Pang, Pope, PRD84, (2011) “Unitary truncations and critical gravity: a toy model” Bergshoeff et al., JHEP04, 134 (2012)

Log mode in critical gravity “Conformal gravity and extensions of critical gravity” Lu, Pang, Pope, PRD84, (2011) “Unitary truncations and critical gravity: a toy model” Bergshoeff et al., JHEP04, 134 (2012)

Log mode in critical gravity “Conformal gravity and extensions of critical gravity” Lu, Pang, Pope, PRD84, (2011) “Unitary truncations and critical gravity: a toy model” Bergshoeff et al., JHEP04, 134 (2012) “Polycritical Gravities” Nutma, PRD85, (2012)

Poly critical Gravities (Nutma, PRD85, (2012))

“critical” Poly critical Gravities (Nutma, PRD85, (2012))

“critical” Poly critical Gravities (Nutma, PRD85, (2012))

“critical” “tri-critical” Poly critical Gravities (Nutma, PRD85, (2012))

“critical” “tri-critical” Poly critical Gravities (Nutma, PRD85, (2012))

: tri critical point Poly critical Gravities (Nutma, PRD85, (2012))

: tri critical point Poly critical Gravities (Nutma, PRD85, (2012))

: tri critical point Poly critical Gravities (Nutma, PRD85, (2012))

“On unitary subsectors of polycritical Gravities” Nutma et al, arXiv: Poly critical Gravities (Nutma, PRD85, (2012))

Poly critical Gravities

Logarithmic conformal field theory (LCFT)

 LCFT is CFT where correlation functions and operator product expansions may contain logarithms. Logarithmic conformal field theory (LCFT)

 LCFT is CFT where correlation functions and operator product expansions may contain logarithms.  A defining feature of LCFT is that the Hamiltonian does not diagonalise, but rather contains Jordan cells of rank two or higher. Logarithmic conformal field theory (LCFT)

 LCFT is CFT where correlation functions and operator product expansions may contain logarithms.  A defining feature of LCFT is that the Hamiltonian does not diagonalise, but rather contains Jordan cells of rank two or higher.  Another relevant difference to ordinary CFT is that LCFT is not unitary. Logarithmic conformal field theory (LCFT)

Rank-2 Jordan cell

Logarithmic conformal field theory (LCFT)

Dual theory of critical gravity is LCFT !

Logarithmic conformal field theory (LCFT) JHEP 07 (2008), 134

Logarithmic conformal field theory (LCFT) JHEP 07 (2008), 134

Logarithmic conformal field theory (LCFT) JHEP 07 (2008), 134

Logarithmic conformal field theory (LCFT) JHEP 09 (2012), 114

Logarithmic conformal field theory (LCFT) JHEP 09 (2012), 114 4D critical gravity is dual to LCFT. 3

Motivation

Motivation Let us first consider a toy scalar model, describes rank 3 (or n) LCFT 2 in BTZ black hole background.

Motivation Let us first consider a toy scalar model, describes rank 3 (or n) LCFT 2 in BTZ black hole background.

Motivation Let us first consider a toy scalar model, describes rank 3 (or n) LCFT 2 in BTZ black hole background.

Motivation Let us first consider a toy scalar model, describes rank 3 (or n) LCFT 2 in BTZ black hole background.

Motivation Let us first consider a toy scalar model, describes rank 3 (or n) LCFT 2 in BTZ black hole background. In addition, if possible, let us compute the retarded Green’s function or absorption cross section for rank 3 (or n) LCFT2

Rank-3 LCFT on BTZ black hole

Bergshoeff et. al [JHEP 1204, 134], T. Moon, Y. S. Myung [PRD 86, (2012) ]

Rank-3 LCFT on BTZ black hole Bergshoeff et. al [JHEP 1204, 134], T. Moon, Y. S. Myung [PRD 86, (2012) ]

Rank-3 LCFT on BTZ black hole

We now focus on the perturbation for the scalar field on the BTZ background metric, the equations are…

Rank-3 LCFT on BTZ black hole

Using the bulk t0 boundary propagator and boundary fields, the solution for the fields are given by

Rank-3 LCFT on BTZ black hole Using the bulk t0 boundary propagator and boundary fields, the solution for the fields are given by

Rank-3 LCFT on BTZ black hole

E. Keski-Vakkuri, Phys. Rev. D 59, (1999)

Rank-3 LCFT on BTZ black hole E. Keski-Vakkuri, Phys. Rev. D 59, (1999)

Rank-3 LCFT on BTZ black hole

In order to calculate two point function, we consider on-shell boundary action. On-shell boundary action (effective action) come from the bulk action which can be written as a surface integral on-shell by integration by part,

Rank-3 LCFT on BTZ black hole In order to calculate two point function, we consider on-shell boundary action. On-shell boundary action (effective action) come from the bulk action which can be written as a surface integral on-shell by integration by part,

Rank-3 LCFT on BTZ black hole In order to calculate two point function, we consider on-shell boundary action. On-shell boundary action (effective action) come from the bulk action which can be written as a surface integral on-shell by integration by part,

Rank-3 LCFT on BTZ black hole

Following AdS/CFT logic, we couple the boundary values of the fields to the dual operators:

Rank-3 LCFT on BTZ black hole Following AdS/CFT logic, we couple the boundary values of the fields to the dual operators:

Rank-3 LCFT on BTZ black hole

Bergshoeff et. al [JHEP 1204, 134] : AdS 3 case

Rank-3 LCFT on BTZ black hole Bergshoeff et. al [JHEP 1204, 134] : AdS 3 case

Rank-3 LCFT on BTZ black hole Bergshoeff et. al [JHEP 1204, 134] : AdS 3 case

Rank-3 LCFT on BTZ black hole Bergshoeff et. al [JHEP 1204, 134] : AdS 3 case

Rank-3 LCFT on BTZ black hole Bergshoeff et. al [JHEP 1204, 134] : AdS 3 case

Rank-3 LCFT on BTZ black hole Bergshoeff et. al [JHEP 1204, 134] : AdS 3 case

Rank-3 LCFT on BTZ black hole

This defines a unitary CFT and thus, the non unitary issue could be resolved by truncating a rank-3 LCFT.

Rank-3 LCFT on BTZ black hole (summary)

We constructed a rank-3 finite temperature LCFT starting from a higher derivative scalar field model in the BTZ black hole background.

Rank-3 LCFT on BTZ black hole (summary) We constructed a rank-3 finite temperature LCFT starting from a higher derivative scalar field model in the BTZ black hole background. Its zero temperature limit reduces to a rank-3 LCFT in the AdS3 background.

Rank-3 LCFT on BTZ black hole (summary) We constructed a rank-3 finite temperature LCFT starting from a higher derivative scalar field model in the BTZ black hole background. Its zero temperature limit reduces to a rank-3 LCFT in the AdS3 background. What else can we do?

Rank-3 LCFT on BTZ black hole One of the important issues in the black hole background is to compute the quasi normal mode.

Rank-3 LCFT on BTZ black hole One of the important issues in the black hole background is to compute the quasi normal mode. Another one is to consider the absorption cross section.

Rank-3 LCFT on BTZ black hole One of the important issues in the black hole background is to compute the quasi normal mode. Another one is to consider the absorption cross section. More concretely we want to check if there is a relation between the rank and the absorption cross section. To this end, we will calculate the absorption cross section by considering the retarded Green’s function approach.

Rank-3 LCFT on BTZ black hole One of the important issues in the black hole background is to compute the quasi normal mode. Another one is to consider the absorption cross section. More concretely we want to check if there is a relation between the rank and the absorption cross section. To this end, we will calculate the absorption cross section by considering the retarded Green’s function approach. The retarded Green’s function is given by

Rank-3 LCFT on BTZ black hole One of the important issues in the black hole background is to compute the quasi normal mode. Another one is to consider the absorption cross section. More concretely we want to check if there is a relation between the rank and the absorption cross section. To this end, we will calculate the absorption cross section by considering the retarded Green’s function approach. The retarded Green’s function is given by Fourier transformation of this leads to

Rank-3 LCFT on BTZ black hole

The absorption cross section is given by

Rank-3 LCFT on BTZ black hole The absorption cross section is given by

Rank-3 LCFT on BTZ black hole The absorption cross section is given by

Rank-3 LCFT on BTZ black hole The absorption cross section is given by

Rank-3 LCFT on BTZ black hole The absorption cross section is given by D. Birmingham, I. Sachs and S. Sen, PLB 413, 281 (1997)

Rank-3 LCFT on BTZ black hole

Naive question

Can we extend rank-3 LCFT to rank-n LCFT in the BTZ black hole ?

Rank-n LCFT on BTZ black hole

In this action, the equations of motions are given by

Rank-n LCFT on BTZ black hole In this action, the equations of motions are given by

Rank-n LCFT on BTZ black hole In this action, the equations of motions are given by We now focus on the perturbation for the scalar field on the BTZ background metric, the equations are…

Rank-n LCFT on BTZ black hole

Using the bulk t0 boundary propagator and boundary fields, the solution for the fields are given by

Rank-n LCFT on BTZ black hole Using the bulk t0 boundary propagator and boundary fields, the solution for the fields are given by

Rank-n LCFT on BTZ black hole Using the bulk t0 boundary propagator and boundary fields, the solution for the fields are given by

Rank-n LCFT on BTZ black hole

Comparing with rank-3(n=3) case,

Rank-n LCFT on BTZ black hole Comparing with rank-3(n=3) case, For i=1 case, j runs over 2, 3 For k=1 case, k becomes k=3 k=2 case, k becomes k=2 / / / /

Rank-n LCFT on BTZ black hole

Bergshoeff et. al [JHEP 1204, 134] : AdS3 case

Rank-n LCFT on BTZ black hole Bergshoeff et. al [JHEP 1204, 134] : AdS3 case

Rank-n LCFT on BTZ black hole

If one truncates the theory to be unitary, one throws away all modes which generate from the (n+1)/2+1 to the n-th column and row of the above matrix (odd rank) or from the n/2+1 to the n-th column and row of the above matrix (even rank). The only non-zero correlation function is given by a reduced matrix as

Rank-n LCFT on BTZ black hole If one truncates the theory to be unitary, one throws away all modes which generate from the (n+1)/2+1 to the n-th column and row of the above matrix (odd rank) or from the n/2+1 to the n-th column and row of the above matrix (even rank). The only non-zero correlation function is given by a reduced matrix as For odd rank, which implies that we have a unitary CFT,

Rank-n LCFT on BTZ black hole If one truncates the theory to be unitary, one throws away all modes which generate from the (n+1)/2+1 to the n-th column and row of the above matrix (odd rank) or from the n/2+1 to the n-th column and row of the above matrix (even rank). The only non-zero correlation function is given by a reduced matrix as For odd rank, For even rank, which contains null states only. which implies that we have a unitary CFT,

Rank-n LCFT on BTZ black hole If one truncates the theory to be unitary, one throws away all modes which generate from the (n+1)/2+1 to the n-th column and row of the above matrix (odd rank) or from the n/2+1 to the n-th column and row of the above matrix (even rank). The only non-zero correlation function is given by a reduced matrix as For odd rank, For even rank, which contains null states only. which implies that we have a unitary CFT,

Rank-n LCFT on BTZ black hole If one truncates the theory to be unitary, one throws away all modes which generate from the (n+1)/2+1 to the n-th column and row of the above matrix (odd rank) or from the n/2+1 to the n-th column and row of the above matrix (even rank). The only non-zero correlation function is given by a reduced matrix as For odd rank, For even rank, which contains null states only. which implies that we have a unitary CFT,

Rank-n LCFT on BTZ black hole If one truncates the theory to be unitary, one throws away all modes which generate from the (n+1)/2+1 to the n-th column and row of the above matrix (odd rank) or from the n/2+1 to the n-th column and row of the above matrix (even rank). The only non-zero correlation function is given by a reduced matrix as For odd rank, For even rank, which contains null states only. which implies that we have a unitary CFT,

Rank-n LCFT on BTZ black hole If one truncates the theory to be unitary, one throws away all modes which generate from the (n+1)/2+1 to the n-th column and row of the above matrix (odd rank) or from the n/2+1 to the n-th column and row of the above matrix (even rank). The only non-zero correlation function is given by a reduced matrix as For odd rank, For even rank, which contains null states only. which implies that we have a unitary CFT,

Rank-n LCFT on BTZ black hole If one truncates the theory to be unitary, one throws away all modes which generate from the (n+1)/2+1 to the n-th column and row of the above matrix (odd rank) or from the n/2+1 to the n-th column and row of the above matrix (even rank). The only non-zero correlation function is given by a reduced matrix as For odd rank, For even rank, which contains null states only. which implies that we have a unitary CFT,

Rank-n LCFT on BTZ black hole If one truncates the theory to be unitary, one throws away all modes which generate from the (n+1)/2+1 to the n-th column and row of the above matrix (odd rank) or from the n/2+1 to the n-th column and row of the above matrix (even rank). The only non-zero correlation function is given by a reduced matrix as For odd rank, For even rank, which contains null states only. which implies that we have a unitary CFT,

Rank-n LCFT on BTZ black hole In rank 3 case, the retarded Green’s functions are given by

Rank-n LCFT on BTZ black hole In rank 3 case, the retarded Green’s functions are given by

Rank-n LCFT on BTZ black hole In rank 3 case, the retarded Green’s functions are given by In rank n case, they become

Rank-n LCFT on BTZ black hole In rank 3 case, the retarded Green’s functions are given by In rank n case, they become

Rank-n LCFT on BTZ black hole

In rank 3 case, the absorption cross sections are given by

Rank-n LCFT on BTZ black hole In rank 3 case, the absorption cross sections are given by

Rank-n LCFT on BTZ black hole In rank 3 case, the absorption cross sections are given by In rank n case, they can be written as

Rank-n LCFT on BTZ black hole

Summary and future work

Our main work was to construct rank-n LCFT in the BTZ black hole by considering a higher derivative scalar field model.

Summary and future work Our main work was to construct rank-n LCFT in the BTZ black hole by considering a higher derivative scalar field model. With this, we could compute higher rank Green’s function and absorption cross section by using two point correlation functions.

Summary and future work Our main work was to construct rank-n LCFT in the BTZ black hole by considering a higher derivative scalar field model. With this, we could compute higher rank Green’s function and absorption cross section by using two point correlation functions. In addition, it was shown that through the truncation analysis for odd rank, physical system keeping only unitary subset remains CFT part at the two point function, Green function, and absorption cross section level.

Summary and future work Our main work was to construct rank-n LCFT in the BTZ black hole by considering a higher derivative scalar field model. With this, we could compute higher rank Green’s function and absorption cross section by using two point correlation functions. In addition, it was shown that through the truncation analysis for odd rank, physical system keeping only unitary subset remains CFT part at the two point function, Green function, and absorption cross section level. Future work is to check if we can extract a true physical mode in Rank-3 or Rank-n LCFT through the BRST formalism.

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