Non-perturbative 3D quantum gravity: quantum boundary states and exact partition function

Abstract

We push forward the investigation of holographic dualities in 3d quantum gravity formulated as a topological quantum field theory, studying the correspondence between boundary and bulk structures. Working with the Ponzano-Regge topological state-sum model defining an exact discretization of 3d quantum gravity, we analyze how the partition function for a solid twisted torus depends on the boundary quantum state. This configuration is relevant to the AdS3/CFT2 correspondence. We introduce boundary spin network states with coherent superposition of spins on a square lattice on the boundary surface. This allows for the first exact analytical calculation of Ponzano-Regge amplitudes with extended 2D boundary (beyond the single tetrahedron). We get a regularized finite truncation of the BMS character formula obtained from the one-loop perturbative quantization of 3d gravity. This hints towards the existence of an underlying symmetry and the integrability of the theory for finite boundary at the quantum level for coherent boundary spin network states. Contents Acknowledgement 38A. Boundary linear forms on the 2-torus 38 B. Stationary spin configurations from Critical couplings 40 C. Ponzano-Regge determinants and bi-variate Chebyshev polynomials 41References 42 * Electronic address: christophe.goeller@ens-lyon.fr † Electronic address: etera.livine@ens-lyon.fr ‡ Electronic address: ariello@perimeterinstitute.ca 5 A spin network mathematically refers to the basis of the space of SU(2)-gauge invariant L 2 wave-functions diagonalizing the SU(2) Casimirs and constructed from SU(2) Wigner matrices and intertwiners [49,50].However, it is often used to indicate more broadly a generic wave-function in that Hilbert space. For a discussion of the gauge invariance of spin networks and their extensions under SU(2) transformations, the interested reader can refer to [51].

Extended actions, dynamics of edge modes, and entanglement entropy

“…Recently, many new results revealing important insights into the role of edge modes have been obtained, both at finite distance and at infinity. At finite distance, there have been successful definitions of quasi-local holography through the path integral for quantum gravity [7][8][9][10][11][12][13], leading to boundary models which can be thought of as capturing the dynamics of the edge modes. There have also been efforts to characterize, for local subsystems, the most general boundary symmetry algebras spanned by the edge modes [14][15][16][17][18][19][20][21], with potentially important consequences for quantum gravity [22,23].…”

“…It would be interesting to study richer Abelian theories, such as U(1) N , which admit topological boundary conditions [84] and gluing along heterointerfaces [43], and eventually the non-Abelian case and the classification of all possible boundary theories. 13 As a subtlety, one can observe in (4.3) that the two boundary equations of motion obtained by varying A, j, when combined, simply imply that Da = ± * Da, meaning that a is a gauged chiral field. This is essentially the same equation of motion as that derived from the effective boundary action (4.14).…”

Geometric actions and flat space holography

“…For negative and real M 0 , one would have to analytically continue θ to have a (positive) imaginary part. Recent evidence from a discretized Ponzano-Regge model of three-dimensional flat space [93] indicates that θ indeed obtains a finite imaginary shift.…”

Kinematical gravitational charge algebra

“…There is growing understanding that edge modes must play a key role in quantum gravity. They are central in the quantization of 2d Yang-Mills and 2d gravity [2][3][4], They play a key role in 3d quantum gravity [5][6][7][8]. They are essential to the understanding of boundary dynamics and to the construction of defects operators in Chern-Simons theory [9][10][11][12].…”