Abstract
Various two dimensional quantum gravity theories of Jackiw-Teitelboim (JT) form have descriptions as random matrix models. Such models, treated nonperturbatively, can give an explicit and tractable description of the underlying "microstate" degrees of freedom, which play a prominent role in regimes where the smooth geometrical picture of the physics is inadequate. This is shown using a natural tool, a Fredholm determinant det(1-K), which can be defined explicitly for a wide variety of JT gravity theories. To illustrate the methods, the statistics of the first several energy levels of a nonperturbative definition of JT gravity are constructed explicitly using numerical methods, and the full quenched free energy F_{Q}(T) of the system is computed for the first time. These results are also of relevance to quantum properties of black holes in higher dimensions.
Highlights
Introduction.—Jackiw-Teitelboim (JT) gravity [1] is a two dimensional model of gravity coupled to a scalar φ, with Euclidean action on a spacetime manifold M: I
The partition function of the full quantum gravity theory ZðβÞ at some inverse temperature β 1⁄4 1=T is given by the path integral over M expansion ZðβÞ
Reference [5] showed that the JT gravity partition function ZðβÞ is equivalent to a Hermitian matrix model loop expectation: at leading order, ρ0ðEÞ was set as Eq (2), with the parameter identification e−S0 ∼ 1=N, and they proved that order by order the matrix model correlation functions that follow precisely match those obtained from the gravity path integral on manifolds M, a highly nontrivial and remarkable result
Summary
Introduction.—Jackiw-Teitelboim (JT) gravity [1] is a two dimensional model of gravity coupled to a scalar φ, with Euclidean action on a spacetime manifold M: I. In perturbation theory, everything at higher genus is determined by the leading (disc) order results by “topological recursion” properties [21] equivalent to certain “loop equations.” Reference [5] showed that the JT gravity partition function ZðβÞ is equivalent to a Hermitian matrix model loop expectation (of length β): at leading order, ρ0ðEÞ was set as Eq (2), with the parameter identification e−S0 ∼ 1=N, and they proved that order by order the matrix model correlation functions that follow precisely match those obtained from the gravity path integral on manifolds M, a highly nontrivial and remarkable result. After double scaling to an end point, scaled parts of all the key quantities survive to define the physics: X, RðXÞ, and PðX; λÞ have scaled pieces x; uðxÞ, and ψðx; EÞ (where x ∈ R), and Ri’s recursion relation becomes a differential equation for uðxÞ that contains all the information about the model.
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