Quantum Gravity in 2+1 Dimensions I

Transcription

1 Quantum Gravity in 2+1 Dimensions I Alex Maloney, McGill University Nordic Network Meeting, A. M. & { S. Giombi, W. Song, A. Strominger, E. Witten, A. Wissanji, X. Yin}

2 Empirical Evidence that Canada is a Nordic country:

3 Overview

4 The Problem: Gravity is notoriously difficult to study as a quantum theory... Classical general relativity: The degrees of freedom are the metric g of space-time, which tells you how space-time curves. The equations of motion are Einstein s equations. Lagrangian formulation: ( ) 1 S(g) = g G R + Λ +... M Classical general relativity doesn t answer all sorts of questions: What happens inside a black hole? How can we think about the big bang singularity?

5 Quantum Gravity Basic conceptual questions: What are the degrees of freedom? What is the Hilbert space H? What is the Hamiltonian H? Are these notions even applicable to gravity? Can a Unitary theory describe singularities, inflation,...? These questions are hard! Lots of solutions of general relativity. Many additional degrees of freedom (the Standard Model,...) Λ appears to be positive. So let s study a simpler theory of gravity.

6 Gravity in 2+1 Dimensions A toy model where we can begin to answer basic conceptual questions about gravity: H, H,... Only two spatial dimensions (fewer solutions). Only metric degrees of freedom. Λ < 0. Einstein gravity is parameterized by The cosmological constant Λ 1/l 2 Newton s constant G which can be combined into a dimensionless coupling constant k = l/16g.

7 Solutions of 2+1 Gravity In fact, the theory is nearly trivial! Einstein s equations completely fix the metric (locally) g has constant negative curvature R = 1/l 2 Locally, the space-time is Anti-de Sitter space: AdS 3 You might think the theory is trivial, but the there are still global degrees of freedom, which lead to a rich structure: Black Holes Cosmologies

8 In 2+1 quantum gravity we can address precisely many issues: Does a purely metric theory of quantum gravity exist? Is quantum gravity a sum over geometries? What is the spectrum of the theory? What is black hole entropy? What are the microstates of a black hole? What is quantum cosmology?

9 The Plan for Today: The Classical Spectrum of AdS 3 Gravity The Sum over Geometries The Quantum Spectrum of AdS 3 gravity Friday: Einstein Gravity vs. Chiral Gravity Saturday: Geometric Interpretation of Black Hole Microstates

10 AdS 3 Gravity

11 T Classical Theory: The prototypical negative curvature metric is AdS 3 ds 2 = dr 2 cosh 2 r dt 2 + sinh 2 r dφ 2 X The boundary cylinder R 1 S 1 lies at infinite distance. We consider only metrics that look like AdS 3 at infinity.

12 Spectrum The two killing vectors H = t generates time translations J = φ generates rotations States are labelled by an energy H and angular momentum J. It is convenient to use the CFT language: L 0 = H + J, L 0 = H J The theory has two types of states: Boundary Gravitons Black Holes

13 Boundary Gravitons Even though there are no local gravitons, there are metric perturbations associated with the boundary. Naively, two metrics describe the same state if they are related by a diffeomorphism. But with AdS boundary conditions, two metrics describe the same state only if they are related by diffeomorphism which vanishes at infinity Diffeo s which act on the boundary give new boundary graviton states These infinitesimal diffeomorphisms generate two copies of the Witt algebra [L m, L n ] = (m n)l m+n This is reminiscent of the Virasoro algebra in a CFT. Brown & Henneaux

14 AdS/CFT Correspondence The isometry group of AdS 3 is SO(2, 2). This is the same as the group of rigid conformal transformations in two dimensions. Indeed, these isometries of AdS 3 act as rigid conformal transformations on the boundary cylinder. The group of asymptotic symmetries is generated by the Virasoro algebra. These asymptotic symmetries act as local conformal transformations on the boundary. The algebra of charges has a central extension [L m, L n ] = (m n)l m+n + c 12 (m3 m)δ m+n with c = 24k. So we expect AdS 3 gravity to be dual to a CFT!

15 BTZ Black Holes There are also black holes, which are locally AdS 3 but differ by global identifications. This is a quotient AdS 3 /Z. ds 2 = dr 2 sinh 2 r dt 2 + cosh 2 r dφ 2 The area L of the black hole horizon is the size of the φ circle

16 The Classical Spectrum In terms of the L 0 eigenvalue = H + J, the spectrum includes A ground state with = k (this is a choice of normalization) A tower of discrete boundary graviton states at integer > k A continuum of black holes with > 0 We want to compute the exact quantum spectrum...

17 Partition Function

18 Partition Function We want to compute the number of states N(E, J) of given energy E and angular momentum J. To do this, we will compute the partition function at finite temperature β 1 and angular potential iθ: ( Z(β, θ) = Tr H e βh+iθj) = N(E, J) e βe+iθj E,J Write this in CFT language by letting τ = θ + iβ Z(τ) = Tr H q L0 q L 0, q = e 2πiτ How can we compute this exactly?

19 Euclidean Path Integral To compute a thermal partition function we take t it E The boundary goes from R 1 S 1 to T 2. The partition function is computed by a Euclidean path integral Z(τ) = Dg e ks(g) over 3-manifolds which are a torus at infinity. The geometry of the boundary T 2 depends on τ = θ + iβ. The two directions of the torus (t E, φ) are identified (t E, φ) (t E, φ + 2π) (t E + β, φ + θ) The parameter τ specifies the conformal structure of the boundaryt 2.

20 Saddle Points In the saddle point approximation Z(τ) = Dg e ks(g) e kvol(m 3) where M 3 is a constant curvature 3-manifold with T 2 boundary. M 3 is a solid torus which fills in the boundary T 2 : T X

21 Saddle Points II There are many such M 3 = M c,d. One for each choice of cycle ct + dx which is contractible in the interior: T X When X is contractible, the M 0,1 is thermal AdS. When T is contractible, the M 1,0 is the Euclidean black hole. In fact, the M c,d are all the solutions of the equations of motion with T 2 boundary.

22 Sum over Geometries The full partition function is a sum over saddle points Z(τ) = c,d Z c,d Z c,d is the contribution from an individual saddle, including perturbative corrections: So we just Z c,d = e kvol(m c,d )+S (1) +k 1 S (2) +... Compute the infinite series of corrections around each saddle. Do the sum over saddles. It can be done!