Bootstrapping Pure Quantum Gravity in AdS 3 S UNGJAY L EE Korea - - PowerPoint PPT Presentation

Bootstrapping Pure Quantum Gravity in AdS3

SUNGJAY LEE

Korea Institute for Advanced Study in collaboration with Jin-Beom Bae and Kimyeong Lee August 12th, 2016 Strings and Fields @ YITP

3D Pure Gravity

Action/Lagrangian Negative Cosmological Constant

• No bulk propagating modes
• Perturbatively finite: essentially no counter-term except the vacuum energy

These facts seems to suggest that the 3d pure quantum gravity with negative cosmological constant is TRIVIAL Might hope to have an exact description of quantum theory of gravity?

3D Pure Gravity

Two surprises in 3d pure gravity [1] Black Holes in this seemingly trivial theory [Banados,Teitelboim,Zanelli]

• Finite-size horizon & Bekenstein-Hawking Entropy

3D Pure Gravity

Two surprises in 3d pure gravity [2] Asymptotic Virasoro Algebras [Brown,Henneaux]

• Quantizing the phase space of classical solutions asymptotic to AdS3 modulo

small diffeomorphism

• Two copies of Virasoro algebra acting on the above Hilbert space
• Part of much richer structure, known as AdS/CFT correspondence

3D pure gravity has something more than trivial!

Can 3D pure gravity exist as a consistent quantum theory both perturbatively and non-perturbatively?

3D Pure Quantum Gravity

SOLVE THE PURE QUANTUM GRAVITY IN AdS3

=

FIND A DUAL CFT ON THE BOUDARY OF AdS3 [Maldacena][Witten] To attack this problem, Witten and Maloney attempted to define and compute the thermal partition function of 3d quantum gravity in AdS3

Extremal Conformal Field Theories

Assumption: [Witten] Dual CFT factorizes into a holomorphic CFT and an anti-holomorphic CFT Holomorphic factorization + Modular invariance then imply

• Central charge : AdS scale is quantized in Planck unit
• Every operator has an integer conformal weight

Then, BTZ black holes have quantized mass in AdS size unit Focus on this today !

Extremal Conformal Field Theories

Vacuum and its Virasoro descendants [1] Vacuum energy [2] Virasoro descendants of vacuum = Boundary gravitons [Brown,Henneaux] Perturbative States In 3D pure gravity This cannot be the full answer …

Extremal Conformal Field Theories

Not the end of story …

• Other states in 3d pure gravity: BTZ black holes (non-perturbative states)
• Z(0,1)(t) is not modular invariant

Need primary operators above the vacuum, identified as the BTZ black holes Classical BTZ black holes Exclude BH with no horizon & no entropy

Extremal Conformal Field Theories

Full partition function of 3D pure quantum gravity BTZ BHs

• LARGE GAP in the spectrum
• Modular invariance then determines the partition function uniquely

poles at q=0 + modularity determines non-polar terms

Extremal Conformal Field Theories

Examples of partition function with k Extremal Conformal Field Theories Holomorphic CFT with c=24k that has a large gap (=k+1) in the spectrum, and its partition function is given by

Extremal Conformal Field Theories

BTZ BH entropy from the partition function

• Consider ECFT with k=1, which implies that
• Bekenstein-Hawking entropy of the lightest BTZ black hole
• From the partition function , one can show

Is there a more direct way to obtain the partition function of ECFT from 3D gravity?

3D Pure Quantum Gravity

Thermal partition function of 3d pure gravity [Maloney,Witten]

• Sum over all geometries M3 with the same conformal boundary, i.e.,

a torus T2. This is because the Euclidean time becomes periodic.

• Semi-classical approximation (= large central charge limit in 2d CFT)

: saddle points solving the E.O.M subject to a given conformal boundary Loop-corrections around saddle points

Partition Function of 3D Pure Gravity

Classification of M3=M(c,d)

• Characterized by a cycle on T2 that becomes degenerate in the interior of M3
• They are solid tori that fill in the boundary two-torus T2(t)

M(c,d)

contractible cycle

Partition Function of 3D Pure Gravity

Examples of M(c,d) M(0,1) M(1,0)

Thermal AdS3 BTZ Black Holes

Partition function can then be written as

Partition Function of 3D Pure Gravity

Thermal AdS3: Z(0,1)(t) has a Hamiltonian interpretation

• Trace is over boundary gravitons, perturbative states in 3D pure gravity
• One-loop partition function [Giombi,Maloney,Yin]

[1] Vacuum energy [2] Boundary gravitons = Virasoro descendants of vacuum [3] One-loop exact: vacuum Virasoro character

Partition Function of 3D Pure Gravity

The partition function of the 3d pure gravity can be described as However the above sum is badly divergent… Holomorphic factorization? Need to consider complex geometries… The zeta-function regularization turns out to break the modular invariance of the partition function, and loose the Hamiltonian interpretation Doesn’t help much…

Partition Function of 3D Pure Gravity

SUSY localization [Iizuka,Tanaka,Terashima]

• Use the SUSY localization method to evaluate the partition function

(Chern-Simons gauge theory description)

• Agreed with the partition function of the ECFT with k=1 (c=24)
• Needs further study to have agreement beyond c=24

Partition function from sum over geometries still remains an open problem

[Frenkel,Lepowski,Meurman] Question

Does 3D Pure Quantum Gravity Exist?

c=24: Famous monster CFT, Z2 orbifold of free bosons on the Leech lattice Can extremal CFTs with c=24k exist for k>1? Investigations into extremal CFTs with k>1 have been inconclusive

Attack this problem using the Numerical Conformal Bootstrap!!

Conformal Bootstrap

Main idea: study constraints from Associativity of OPE and Unitarity Virasoro conformal block

OPE coefficients Virasoro conformal block : includes a primary operator O and its Virasoro descendants : chiral operator

Conformal Bootstrap

Crossing symmetry: associativity of OPE implies that

Can a given set be a spectrum of a consistent conformal field theory?

Numerical Conformal Bootstrap

Difficult Problem: find OPE coefficients that solve the bootstrap condition Easier Problem: find a linear functional a[*] satisfying

• If you can, then there is NO consistent CFT whose spectrum is given by

Unitary Parity-Preserving CFTs

, contradict to the bootstrap equation [Rattazzi,Rychkov,Tonni,Vichi]

Numerical Conformal Bootstrap

Convenient choice of a functional a

• Find a vector such that

for all

• Can be translated into a standard semi-definite programming (SDP)

Semi-Definite Program

polynomial : positive semidefinite & real symmetric matrix

=

Theorem To have a sense of the above theorem, recall that such a polynomial f(x) can be described as follows [Hilbert]

Positive Semi-definite Matrix

• Enough to find a vector such that for

Semi-Definite Programming

Translate the numerical bootstrap problem into SDP

• First note that (we will show it later)

positive function of hO Polynomial

• f hO
• The problem can then be transformed into SDP

Find such that

There are a number of SDP solvers in the market I use a SDP solver made by D. Simmons-Duffin Before that, we first need to know how to compute the Virasoro conformal blocks

Virasoro Conformal Block

where satisfies the recursive relation below Zamolodchikov’s recursive relation: Unlike the global conformal block, the closed form of Virasoro conformal block is still unknown

Virasoro Conformal Block

and other terms are given by

Virasoro Conformal Block

Remarks on the recursive relation [2] Poles? the existence of null states [3] Residue? [1] Seed of the recursion: Virasoro block in the semi-classical limit

• Level mn null states are regarded as primaries, which implies that the residue

has to be proportional to the Virasoro block for its own.

• Rmn(c,h) vanishes at c=cmn, the minimal model central charge to avoid the poles

Numerical Results

Numerical Results

Parameter setting: To solve the SDP below [1] Central charge: where k is an arbitrary positive integer Find such that we set various parameters as follows [2] Conformal weight of external operator: (lightest BTZ black hole) [3] Spectrum of extremal CFTs:

Numerical Results

Virasoro conformal block: To set a solvable numerical problem,

• Approximate the function H as a polynomial of finite degree in q
• Truncate the number of iteration to solve the recursion relation

Terms of order up to 2M in the function H become accurate after repeating the iteration by M times ( )

Numerical Results

How many iteration do we need?

• Virasoro block quickly converges as the number of iterations increases
• Iteration by M times, approximate the function H as a polynomial of degree 2M

# of iteration Virasoro Block k=10

Numerical Results

Virasoro conformal block: To set a solvable numerical problem, one has to truncate the number of iterations to solve the Zamolodchikov recursive relation

• Not enough. Need to see when can converge

k=30

[1] , Bootstrap Condition to SDP:

Numerical Results

[2] From the recursive relation, the Virasoro block takes the following form

Polynomial of x Common Denominators

Note that for

Numerical Results

SDP Problem

• Add `n’ discrete primary states of low-lying integer conformal weights

for

• If a solution {a} is found, no consistent extremal CFTs

Numerical Results

An Alternative Problem (Optimal Problem) for

• If an optimal solution {a} with negative maximum value is found, then

no feasible solution to the previous SDP with

Numerical Results

Numerical Results: When n=0, no solution found.

• When n=2000, SDPB found optimal solutions with negative maximum value

for in the shaded region

• Around n=2000, SDPB starts to find solutions to 1st problem for k=20-30,40,..70

~ ~

Numerical Results

Criterion to conclude that a feasible solution a is found:

• D-error is smaller than 10-200 (precision)

Results: Our numerical results suggests

• Extremal CFT do not exist as a consistent CFT for
• Near extremal CFT also seems to be ruled out for sufficiently large c

(which includes small BH as spectrum, and gap = k) [Benjamin,Dyer,Fitzpatrick,Maloney,Perlmutter]

Summary

No extremal conformal field theory for k>19 3D pure gravity may not exist as a consistent quantum theory on weakly curved space

under the assumption of holomorphic factorization modulo numerical errors