WedgeHolography@NagoyaHolography in a Wedge
Yukawa Institute for Theoretical Physics Zixia Wei
Codimension two holography for wedges Ibrahim Akal, Yuya Kusuki, Tadashi Takayanagi, Zixia Wei arXiv: 2007.06800, Phys.Rev.D 102 (2020) 12, 126007
@Nagoya U. virtually Dec.10th
• Background
• Consistency Checks
AdS/CFT Correspondence [Maldacena; ’97] Background
• Quantum Gravity in (d+1)-D asymptotically anti de Sitter spacetime (AdS) d-D conformal field theory (CFT)
• Quantum gravity = Quantum Theory without gravity Important in the studies of quantum gravity!
• AdS’s asymptotic boundary = the manifold on which the CFT is defined CFTboundary theory AdSbulk theory
! x
AdS/CFT Correspondence [Maldacena; ’97] Background
• AdS/CFT is a codimension-1 holography in the sense that AdS has 1 more dimension.
• QuestionCan we have, for example, a codimension-2 holohraphy?
• AnswerYes (d+1)-D gravity in a wedge (d-1)-D CFT
! x
z
Wedge Holography as a Codimension-2 Holography
• WThe wedge region sandwiched by two branes Q1 and Q2 in (d+1)-D Poincare AdS
• Σ(d-1)-D spacetime where Q1 and Q2 intersect
• Gravity in W CFT in Σ > codimension-2 Holography!
• W itself is not an asymptotic AdS!
W (shaded region)
AdSBCFT [Takayanagi, ’11] Boundary CFTs and their holographic duals
• Boundary CFT (BCFT)CFT defined on a manifold with boundary
• BCFT has a boundary > AdS should also have a boundary! > Brane Q (sometimes end-of-the-world brane)
• Neumann b.c. on Q
Introduce a boundary
AdSBCFT [Takayanagi, ’11] Boundary CFTs and their holographic duals
• Ex.) BCFT living on a 2D half plane: > Half of the Poincare AdS with a brane Q : attached on it.
>
" = "*
• 1. A CFT living onΣ(d-D flat space) > (d+1)-D AdS
Σ Σ
• 2. Introduce a boundary toΣ > Half of AdS Brane Q1
Q1
Deriving Wedge Holography From AdS/BCFT to Wedge Holography
• 3. Introduce another boundary toΣ > a portion of AdS Brane Q1&Q2
Q1Q2
• 4. Bring the two boundaries of Σ closer and closer!
Q1Q2
Σ Σ
• 5. Σ: (d-1)-D flat space > (d+1)-D Wedge region W sandwiched by Q1&Q2
Q1Q2
Σ
Σ
W
We derived the codimension-2 wedge holography
as a limit of AdS/BCFT
Wedge Holography From another point of view
W
Σ
Q1Q2
• Braneworld holography [Randall, Sundrum; ‘99] Gravity living on (d+1)-D AdS with ETW branes = Another gravitational theory living on the branes
• From this point of view, wedge holography: Gravity in W: (d+1)-D ←Braneworld holography Another gravity in Q1 Q2: d-D ←Q1&Q2 are AdS, AdS/CFT CFT on Σ: (d-1)-D
& • Neumann b.c. on Q1, Q2 Kab ! Khab = ! Thab
Wedge Holography Different point of views
W : wedge of AdSd+1 d+1
Σ
Q1Q2
CFTd!1
Σ
Justifications for Wedge Holography
• Wedge in AdS and CFT both have a SO(1,d) symmetry.
• Wedge in AdS can be regarded as a warped product of
AdS and direction [Emparan, Johnson, Myers; ’99] [Miao; ’20]
CFT on the corner of wedge may be composed from
conventional holographic CFT induced from AdS / CFT
+ a sector corresponding to fluctuations in direction.
d+1 d!1
• Consistency check from physical quantities in wedge AdS and CFT :
• Free Energy Holographic Entanglement Entropy 2-point functions of scalar primaries
• [Miao, ’20] Holographic Renyi Entropy 2-point functions of energy-stress tensor
d+1 d!1
Free Energy 4D wedge / 2D CFT as an example
• The Free Energy of a 4D wedge which corresponds to a 2D CFT
on is S2
GN sinh "*
L log # + . .
• The Free Energy of a 2D CFT can be written as
where is the Weyl anomaly and
.
$(') $(S2) = 2
,
matrix .
.
• Example: Consider a Bell state in a two qubit system
Then the entanglement entropy of the first qubit is
.
( !0) + !0) + !1) + !1))
A SA = log 2
Entanglement Entropy in CFT2
• Let us consider the vacuum state of a 1+1D CFT defined on an infinite line.
• Let us take a single interval with length as the subsystem .
• It is well-known that, for general CFTs,
l A
#
Holographic Entanglement Entropy in AdS/CFT
• (Ryu-Takayanagi Formula) The EE of subsystem in CFT can be evaluated by the area of the minimal codimension-2 surface
ending on in AdS
Holographic Entanglement Entropy in AdS/BCFT
• (Ryu-Takayanagi Formula) The EE of subsystem in CFT can be evaluated by the area of the minimal codimension-2 surface
ending on or ETW brane Q in AdS
A
-A
.A
• Let us start from a time slice of AdS /BCFT .
Then, we take the limit to get a time slice of AdS /CFT .
4 3
4 2
• Accordingly, the entanglement entropy of is evaluated by a double minimization:
A
GN : Newton constant in AdS4
Comparing EE in CFT and HEE from wedge AdS2 4 GN : Newton constant in AdS4
• HEE evaluated from wedge AdS 4
SA = L2
GN sinh "*
• The EE of a single interval with length A l
SA = c 3 log l
#
c = 3L2
GN sinh "*
• We use the wedge structure to construct a codimension-2 holography.
• Some justifications and consistency checks are given.
• codimension-N holography?
• 1D CFT?
The Next Topic
• Background
• Consistency Checks
Reminder: AdSBCFT Boundary CFTs and their holographic duals
• Ex.) BCFT living on a 2D half plane: > Half of the Poincare AdS with a brane Q : attached on it.
>
" = "*
Map to another BCFT
T = & 'L
Q
Q
T = & 'L
Map to another BCFT Radius: &2 ! '2
Note that we can interpret either the shaded region or the unshaded region as
the physical region.
Q
T = & 'L
z
x !
!T ! > 1/L
Radius: &2 ! '2 Radius:
&2 ! '2 = i '2 ! &2
What does imaginary radius mean? Radius: &2 ! '2 0 r Radius:
&2 ! '2 = i '2 ! &2 0 ir1
Now we would like to interpret the
unshaded region as the physical region.
x r2/x
x ))R2
r
r1
r1
• CFT with an imaginary boundary,
• RT formula in AdS .
A = {x !x 3 a}
2
3
#r + Sbdy
r1 A
• We extended the tension in a specific example of AdS/BCFT.
• We gave it a formulation using CFT with imaginary boundary.
• By analytic continuation, this describes CFT spacetime nucleation.
• It corresponds to spacetime nucleation in AdS.
• The ETW brane in AdS is dS!!! > relation to dS holography?
• More justifications?