Bootstrapping Pure Quantum Gravity in AdS 3 S UNGJAY L EE Korea Institute for Advanced Study in collaboration with Jin-Beom Bae and Kimyeong Lee Strings and Fields @ YITP August 12 th , 2016
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 AdS 3 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 FIND A DUAL CFT = QUANTUM GRAVITY IN AdS 3 ON THE BOUDARY OF AdS 3 [Maldacena][Witten] To attack this problem, Witten and Maloney attempted to define and compute the thermal partition function of 3d quantum gravity in AdS 3
Extremal Conformal Field Theories Focus on this today ! 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
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 … - Z (0,1) ( t ) is not modular invariant - Other states in 3d pure gravity: BTZ black holes (non-perturbative states) 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 M 3 with the same conformal boundary, i.e., a torus T 2 . This is because the Euclidean time becomes periodic. - Semi-classical approximation (= large central charge limit in 2d CFT) Loop-corrections around saddle points : saddle points solving the E.O.M subject to a given conformal boundary
Partition Function of 3D Pure Gravity Classification of M 3 =M (c,d) - They are solid tori that fill in the boundary two-torus T 2 ( t ) - Characterized by a cycle on T 2 that becomes degenerate in the interior of M 3 M (c,d) contractible cycle
Partition Function of 3D Pure Gravity Examples of M (c,d) M (0,1) M (1,0) BTZ Black Holes Thermal AdS 3 Partition function can then be written as
Partition Function of 3D Pure Gravity Thermal AdS 3 : 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 … The zeta-function regularization turns out to break the modular invariance of the partition function, and loose the Hamiltonian interpretation Holomorphic factorization ? Need to consider complex geometries… 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
Does 3D Pure Quantum Gravity Exist? c=24: Famous monster CFT, Z 2 orbifold of free bosons on the Leech lattice [Frenkel,Lepowski,Meurman] Question 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 : includes a primary operator O and its Virasoro descendants : chiral operator OPE Virasoro coefficients conformal block
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 [Rattazzi,Rychkov,Tonni,Vichi] , contradict to the bootstrap equation
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 Theorem = polynomial : positive semidefinite & real symmetric matrix 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
Semi-Definite Programming Translate the numerical bootstrap problem into SDP - First note that (we will show it later) positive Polynomial function of h O of h O - Enough to find a vector such that for - 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 Unlike the global conformal block, the closed form of Virasoro conformal block is still unknown Zamolodchikov’s recursive relation: where satisfies the recursive relation below
Virasoro Conformal Block and other terms are given by
Virasoro Conformal Block Remarks on the recursive relation [1] Seed of the recursion: Virasoro block in the semi-classical limit [2] Poles? the existence of null states [3] Residue? - Level mn null states are regarded as primaries, which implies that the residue has to be proportional to the Virasoro block for its own. - R mn (c,h) vanishes at c=c mn , the minimal model central charge to avoid the poles
Numerical Results
Numerical Results Parameter setting: To solve the SDP below Find such that we set various parameters as follows [1] Central charge: where k is an arbitrary positive integer [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? - Iteration by M times, approximate the function H as a polynomial of degree 2M - Virasoro block quickly converges as the number of iterations increases k=10 Virasoro Block # of iteration
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
Numerical Results Bootstrap Condition to SDP: Polynomial of x Common Denominators [1] , [2] From the recursive relation, the Virasoro block takes the following form Note that for
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