The loop gravity string
Laurent Freidel, Alejandro Perez, Daniele Pranzetti
TL;DR
The paper studies canonical gravity in finite regions with a generalized Gibbons-Hawking boundary term that includes the Immirzi parameter, revealing new boundary degrees of freedom localized at punctures where gauge and diffeomorphism symmetry are restored at the boundary.These boundary degrees of freedom are described by a 3-dimensional internal string attached to each puncture, with currents encoding the frame field, angular momentum encoding the area flux, and the boundary metric encoded in the stress tensor, yielding a rich boundary CFT structure.At each puncture, the boundary charges realize a twisted U(1)^3 Kac-Moody algebra, from which a Virasoro algebra with central charge c=3 per puncture is obtained via the Sugawara construction; for integer curvature k this enhances to SU(2) (or remains U(1)) symmetry, linking boundary and bulk gauge structures.The results connect to prior LQG literature on corners and spin networks, propose a normalisable Fock-like vacuum with Virasoro structure, and have potential implications for Hamiltonian dynamics and black hole boundary states, suggesting deep ties between boundary CFT and loop gravity in finite regions.
Abstract
In this work we study canonical gravity in finite regions for which we introduce a generalisation of the Gibbons-Hawking boundary term including the Immirzi parameter. We study the canonical formulation on a spacelike hypersuface with a boundary sphere and show how the presence of this term leads to an unprecedented type of degrees of freedom coming from the restoration of the gauge and diffeomorphism symmetry at the boundary. In the presence of a loop quantum gravity state, these boundary degrees of freedom localize along a set of punctures on the boundary sphere. We demonstrate that these degrees of freedom are effectively described by auxiliary strings with a 3-dimensional internal target space attached to each puncture. We show that the string currents represent the local frame field, that the string angular momenta represent the area flux and that the string stress tensor represents the two dimensional metric on the boundary of the region of interest. Finally, we show that the commutators of these broken diffeomorphisms charges of quantum geometry satisfy at each puncture a Virasoro algebra with central charge $c=3$. This leads to a description of the boundary degrees of freedom in terms of a CFT structure with central charge proportional to the number of loop punctures. The boundary $SU(2)$ gauge symmetry is recovered via the action of the $U(1)^3$ Kac-Moody generators (associated with the string current) in a way that is the exact analog of an infinite dimensional generalization of the Schwinger spin-representation. We finally show that this symmetry is broken by the presence of background curvature.