Jackiw–Teitelboim gravity
Model of gravity with dilation
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In theoretical physics, Jackiw–Teitelboim gravity, also known as JT gravity or the R=T model, is a theory of gravity with a dilaton in one spatial and one time dimension (1+1D). It was first formulated by Roman Jackiw and Claudio Teitelboim. [1] The theory is notable for being a toy model of quantum gravity that is exactly solvable, and it has found applications in understanding near-extremal black holes and the AdS/CFT correspondence.[2] [3]
The dynamics of JT gravity in the bulk are relatively simple, leading to a constant negative curvature spacetime (Anti-de Sitter space). [4] However, the theory possesses interesting dynamics on the boundary of this spacetime, which are described by the Schwarzian theory.[5] This boundary theory captures the low-energy behavior of the Sachdev-Ye-Kitaev (SYK) model, a model of quantum chaos.[2]
Action
The action for Jackiw–Teitelboim gravity is defined on a 2-dimensional manifold M with a metric gμν and a dilaton field Φ. It is given by:
where GN is the two-dimensional Newton constant, R is the Ricci scalar of the metric gμν, and the cosmological constant is chosen such that the equations of motion fix the spacetime to be constantly curved.
Boundary Terms
For a manifold M with a boundary ∂M, it is necessary to include boundary terms in the action to have a well-posed variational problem. [6] The appropriate boundary terms for JT gravity are a Gibbons–Hawking–York boundary term (GHY) for the metric and a corresponding term for the dilaton field:
Here, h is the induced metric on the boundary, K is the extrinsic curvature, and Φb is the value of the dilaton at the boundary.[7] The second term is a counterterm that normalizes the action. [8]
The full action for Jackiw–Teitelboim gravity with boundary is therefore:
![{\displaystyle S={\frac {1}{16\pi G_{N}}}\left[\int _{M}d^{2}x{\sqrt {-g}}\Phi (R+2)+2\int _{\partial M}dt{\sqrt {h}}\Phi _{b}(K-1)\right]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/707aa6ebc08cad420262e4b4c29e49ab33a983bc)
Schwarzian Boundary Theory
A key feature of JT gravity is that its dynamics can be reduced to a theory on the one-dimensional boundary of the two-dimensional spacetime. This is achieved by integrating out the bulk fields (the metric gμν and the dilaton Φ), leaving an effective action for the boundary degrees of freedom.[9] The resulting effective action is the Schwarzian action, which describes the reparametrizations of the boundary.
Derivation
The derivation of the Schwarzian boundary theory involves several steps:
- Solving the bulk equations of motion: Variation of the JT action with respect to the dilaton Φ imposes the constraint R = -2, which means the bulk geometry is a slice of Anti-de Sitter space (AdS2). The equation of motion for the metric connects it to the dilaton's stress-energy tensor.[10]
- Gauge fixing and boundary conditions: A convenient gauge choice is to fix the metric to be of the form . The boundary of this space is located at z = ε for some small ε. The boundary curve is then parameterized by a time coordinate t(u), where u is a physical time on the boundary.
- Evaluating the on-shell action: The bulk term of the action vanishes on-shell due to the equation of motion for the dilaton. The entire action is then given by the boundary term, evaluated on the solutions to the equations of motion.
- The Schwarzian derivative: After carefully evaluating the boundary term with the imposed boundary conditions and gauge fixing, the action reduces to: where C is a constant proportional to Φb/GN, β is the periodicity of the boundary time coordinate, and {f,u} denotes the Schwarzian derivative of a function f with respect to u:
This effective action for the boundary reparametrization t(u) is the renowned Schwarzian action. [11] This theory describes the spontaneous and explicit breaking of conformal symmetry on the boundary, a feature also seen in the low-energy limit of the SYK model.[2]