Quantum Gravity Microstates from Fredholm Determinants

A large class of two dimensional quantum gravity theories of Jackiw-Teitelboim form have a description in terms of random matrix models. Such models, treated fully non-perturbatively, can give an explicit and tractable description of the underlying ``microstate'' degrees of freedom. They play a prominent role in regimes where the smooth geometrical picture of the physics is inadequate. This is shown using a natural tool for extracting the detailed microstate physics, a Fredholm determinant ${\rm det}(\mathbf{1}{-}\mathbf{ K})$. Its associated kernel $K(E,E^\prime)$ 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 non-perturbative 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.

A large class of two dimensional quantum gravity theories of Jackiw-Teitelboim form have a description in terms of random matrix models. Such models, treated fully non-perturbatively, can give an explicit and tractable description of the underlying "microstate" degrees of freedom. They play a prominent role in regimes where the smooth geometrical picture of the physics is inadequate. This is shown using a natural tool for extracting the detailed microstate physics, a Fredholm determinant det(1−K). Its associated kernel K(E, E ) 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 non-perturbative definition of JT gravity are constructed explicitly using numerical methods, and the full quenched free energy FQ(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.
Introduction.-Jackiw-Teitelboim (JT) gravity [1] is a two dimensional model of gravity coupled to a scalar φ, with Euclidean action: where R is the Ricci scalar and in the boundary terms, K is the trace of the extrinsic curvature for induced metric h ij and φ b is the boundary value of φ. The constant S 0 multiplies the Einstein-Hilbert action, which yields the Euler characteristic χ(M)=2−2g−b of the spacetime manifold M (with boundary ∂M), with g handles and b boundaries. The partition function of the full quantum gravity theory Z(β) at some inverse temperature β=1/T is given by the path integral over M with a boundary of length β. It has a topological expansion Z(β)= ∞ g=0 Z g (β), where Z g (β) has a factor e −χ(M)S0 . JT gravity is of interest not just as a solvable toy model of gravity, but also because it is a universal sector of the low T near-horizon quantum dynamics of a wide class of higher dimensional black holes [2], including ones in four dimensional asymptotically flat spacetimes. S 0 is the extremal (T =0) Bekenstein-Hawking [3] entropy and φ parameterizes the leading finite T deviation of the geometry from extremality. Insights about this model therefore directly pertain to fundamental questions about quantum gravity and black holes in more "realistic" settings.
Solving the model (1) at leading (disc) order by integrating out φ (locally enforcing R=−2 on M) results in all of the non-trivial dynamics being at the (length β) boundary ∂M. It has a Schwarzian action, and [4]: where the spectral density ρ 0 (E) comes from the Laplace transform: This class of models can be explicitly solved to any order to yield Z g (β) (e.g., yielding corrections to ρ 0 (E)), and correlations thereof. This was shown in Refs. [5,6], along with a striking equivalence (to be recalled below) to models of random N ×N matrices at large N , where the topological expansion parameter e −S0 ∼1/N . This Letter's results go well beyond the topological perturbative expansion to uncover non-perturbative physics, computing the details of individual underlying energy levels of the spectrum. This reveals the physics of the microscopic degrees of freedom of the quantum gravity theory, or, from the higher dimensional perspective, the underlying black hole microstates. The entropy counts microstates via S 0 ∼ log N . Expanding in 1/N around large N will not resolve their individual character. For this, a non-perturbative formulation is needed that allows O(e −N ) physics to be extracted, a program begun in earnest (in this context) in Ref. [7]. The key new tool, introduced in this Letter, will be a Fredholm determinant of an operator that naturally arises from the underlying matrix model description. It yields exquisite details of individual energy levels.
The results for the first six levels for JT gravity are shown in Fig. 1. For illustration, the plots (and all others herein) are for e −S0 =1. For a fixed reference value of E, smaller e −S0 (larger extremal entropy S 0 ) results in an increase in the number of levels found to the left, which is nicely consistent with their microstate interpretation. A key feature is that the knowledge of the spectrum is fundamentally statistical, increasingly so at lower energies. [8] At higher E, the energy levels become more sharply defined (their variance decreases), and also form a continuum. In this regime the spacetime language (and the perturbation theory described above) is a good ap- proximation. Low E is increasingly non-perturbative, a spacetime interpretation falls short, and the statistical description cannot be neglected. This suggests that geometry and statistics are dual phenomena here.
Once the full spectrum and statistics are known, any property of the model can be computed. A key example is the quenched free energy F Q (T )=−β −1 log Z(β) , needed to compute thermodynamic quantities down to low T (the more readily computable annealed free energy, F A (T )=−β −1 log Z(β) gives a negative entropy at low enough T ). Its computation has been discussed frequently [9][10][11] [12], but so far not completed. Later in this Letter, the full spectrum will be used to explicitly compute F Q (T ) for the first time. It is displayed in Fig. 2.
Perturbative Matrix Models.-The full topological expansion of a large class of models of JT gravity has been shown by Saad, Shenker and Stanford [5] and Stanford and Witten [6] to be captured [13] by certain random matrix models in the "double scaling limit" of Refs. [14]. In the simplest matrix models, the probability distribution of the N ×N matrix M is p(M )=e −TrV (M ) with Gaussian V (M )= 1 2 M 2 being the most famous prototype [15]. A more general polynomial V (M )= p g p M p is of interest for studying gravity. (See e.g. Ref. [16] for a review.) Treating the matrix partition function Z= p(M )dM as a toy field theory and working at large N , the Feynman diagrammatic expansion can be viewed (following 't Hooft, and Brezin et. al. [17]) as tessellations of 2D Euclidean spacetimes, with each order in the 1/N expansion corresponding to the topology upon which the diagrams can be drawn. The double scaling limit is a combination of sending N →∞ while also tuning the couplings g p to critical values such that surfaces large compared to the scale of the tessellation dominate. This yields univer-sal continuum physics. In this sense the matrix model can be thought of as an alternative method for doing the 2D gravity path integral. However, as already mentioned, fundamental quantum gravity questions will require non-perturbative physics-indeed going beyond geometry. A path integral definition that is intrinsically based on integrating over smooth manifolds M cannot get access to this physics. Matrix models can unlock this non-perturbative sector, since the geometrical spacetime regime is only an "emergent" subsector of the theory.
In the large N limit the distribution of eigenvalues λ i becomes dense and can be described with a smooth coordinate λ. The non-zero eigenvalues fill a finite segment of the real λ line, and a spectral density ρ(λ) describes their distribution. (The focus here will be on V (M ) that produce a single connected component.) The Laplace transform of ρ(λ) is (roughly) the expectation value of a "loop operator" Tr[e M ] , which makes a loop of length in the tessellations. The double-scaling limit (N →∞ with g p →g c p ) focuses on the scaling region in the neighbourhood of an endpoint [18] (at, say, λ=λ 0 ), blowing up that region and sending the other end to infinity, i.e., λ=λ 0 +Eδ p , as parameter δ→0, for some positive p. Finally, the spectral density is some ρ(E) supported on the real line starting at E=0 stretching to infinity. The details of ρ(E) depend upon the gravity theory in question.
In general, the dictionary between the choice of gravity theory and double-scaled potential is not straightforward, but in genus perturbation theory, everything at higher genus is determined by the leading (disc) order results by "topological recursion" properties studied by Mirzakhani, and by Eynard and Orantin, equivalent to certain "loop equations" for correlation functions [19]. Crucially, Ref. [5] showed that the JT gravity partition function Z(β) is equivalent to a loop expectation (of length β). At leading/disc order, the spectral density is set as Eq. (2), and the topological expansion parameter match: e −S0 ∼1/N (after double-scaling). They then checked that order by order in topology, the correlation functions of the matrix model that follow from the leading result equal the explicit results obtained by directly doing the gravitational path integral on manifolds M. It is highly non-trivial and remarkable that it works. Nevertheless, this is still all perturbative. The next step is to study non-perturbative contributions to ρ(E), beyond the reach of the recursion approach.
Non-Perturbative Matrix Models.-The direct nonperturbative completion of Ref. [5]'s Hermitian matrix model definition of JT gravity has instabilities. One way to see this [5,7] is to note that perturbatively it is a combination [20] of the family of (2k−1, 2) minimal models (coupled to gravity), obtained [14] as Hermitian matrix models: It was known long ago that the k even models are non-perturbatively unstable due to eigenvalue tunneling to infinite negative values. However Ref. [7] supplied a non-perturbative completion of the JT gravity matrix model that preserves the perturbative expansion to all orders. It is built by double-scaling complex matrices M for which the potential is built from M M † [21]. It can be thought of as a model of an Hermitian matrix with a manifestly positive eigenvalue spectrum. The model is unafflicted by non-perturbative instabilities. [22] A useful way of seeing the difference between the two types of model is through their orthogonal polynomial formulation. The Hermitian matrix model integral can be written entirely in terms of a system of N polynomials P i (λ), orthogonal with respect to the measure dλe −V (λ) . The polynomials themselves are related according to λP i (λ)=P i+1 (λ)+R i P i−1 (λ) and the R i satisfy a recursion relation determined by V (λ). Knowing the coefficients R i turns out to be equivalent to solving the partition function integral or any insertion into the integral of traces of powers of M i.e., of λ. In the large N limit, the index i/N becomes a continuous coordinate X that runs from 0 to 1, the endpoints of the density. The orthogonal polynomials become functions of X: P i (λ)→P (X, λ), as do the recursion coefficients: R i →R(X). Recall that the double scaled limit focuses on an endpoint, say X=0. In the limit, scaled versions of all the key quantities survive to define the continuum physics. For example, X has a scaled piece x, which runs over the whole real line, via: X=0−xδ px where p x is some positive power and δ→0 in the limit. The topological expansion parameter 1/N also picks up a scaling piece, denoted , via: 1/N = δ pn . For the recursion coefficients R(X)=R(0)+u(x)δ pu , with p u some other positive power. In the limit (the powers p x , p u etc., are determined by requiring finite physics), the recursion relation for R i becomes a differential equation for u(x). It contains both perturbative and non-perturbative information about the model. The orthogonal polynomials P i (λ) become ψ(x, E), wavefunctions of an Hamiltonian that is the scaled part of λ (acting as an operator inside the integral): λ=λ 0 +Hδ p , where: Solving the double scaled matrix model boils down to determining u(x). This in turn defines a quantum mechanics and wavefunctions ψ(E, x) and energies E via: In fact, the loop operator in this language is [23]: a trace over the exponentiated Hamiltonian. Putting dψ|ψ ψ|=1 into Eq. (5) and using Eq. (4) with ψ≡ ψ|x gives Z(β)= dEρ(E)e −βE with: i.e., knowing the complete ψ(E, x) can be used to construct the full spectral density to all orders and beyond. The difference between the two types of model can now be explained, using the simplest (Gaussian) case as an example. In the Hermitian matrix model, the unscaled P i (λ) are Hermite functions, and in the scaling limit at an endpoint become [24] Airy functions , which satisfy Eq. (4) for potential u(x)=−x. Their energies range over the whole real E line. Using them in Eq. (6) gives the full nonperturbative density: Its support includes E<0. Large E gives the classical (disc order) result ρ 0 =E 1 2 /(π ) supported only on E≥0. In contrast, for the complex matrix models, similar steps [21] give a nonlinear equation for u(x): where R=u(x)+x. The solution of interest has u(x)=−x for x→−∞ and u(x)=0 for x→+∞. So for large energies the behaviour of the wavefunctions (in the x<0 regime, which corresponds to perturbation theory) is again like the Airy form of the Hermitian case, and so the physics is perturbatively identical. However at low energies the wavefunctions become less Airy-like (in fact they join on to a Bessel-like behaviour at positive x), and moreover the energies in the problem are bounded below by zero. The resulting fully non-perturbative spectral density for this problem resembles that of the Airy problem, but naturally truncates [7] at E=0. The non-perturbative states at E<0 are not present, while the perturbative physics remains the same. There is an analogous model for any k with this kind of behaviour, and Ref. [7] showed how to build a non-perturbative completion of JT gravity using them. It is a matter of finding the correct equation for u(x). It is constructed as follows: The general double scaled matrix model yields the form (7) but with R= are the "Gel'fand-Dikii" [25] polynomials in u(x) and its derivatives, beginning as R k [u]=u(x) k + · · · . (Their details are not needed here.) The t k are couplings. The specific values t k =π 2k−2 /k!(k−1)! yield an equation that to leading order gives the JT spectral density (2), with =e −S0 .
Solving the equation for u(x) order by order (expanding about large x<0 perturbation theory) yields the identical physics that Ref. [5] constructed. Meanwhile, the full solution for u(x) (constructed numerically to good accuracy in Ref. [26]) determines H. The spectral problem can be solved numerically to yield wavefunctions ψ(E, x), and the integral (6) yields the non-perturbative ρ(E). It is plotted in Fig. 1 as the solid black curve. At large E it asymptotes to the classical Schwarzian result (2) (dashed line), but at small E there are undulations that are invisible in perturbation theory. [27] They are the precursors of the underlying microphysics.
Detailed Microphysics.-A new type of tool will now be introduced that will take the construction much further than before. Fredholm determinants and their associated kernels are a powerful tool that have been used in the statistical physics literature for computing random matrix model properties such as correlation functions [28]. The determinantal structure arises because of the van der Monde determinant i =j (λ i −λ j ) 2 that comes as the Jacobian for going from M variables to eigenvalues λ. The orthogonal polynomial basis inherits all this structure, and can be used as the building blocks, in their guise as the ψ(E, x). For example the core object is the "kernel": In fact, it has already played a role, as its diagonal is the non-perturbative density, Eq. (6). Clearly there is a lot more information to be extracted from its off-diagonal terms. Various combinations of the kernel, Laplace transformed, can be used to write multi-point correlation functions [23] for Z(β), but the raw form will be a much more direct tool for studying the spectrum. It is the context of the Fredholm determinant construction that gives K its kernel moniker. In solving problems of the form [29]: Returning to the physics, the Fredholm determinant can be used to focus on one energy/eigenvalue at a time as follows. For the 1st energy level (labelled henceforth as the 0th, for the ground state, E 0 , of JT gravity system), the probability of finding no eigenvalues on the interval (0, s) (chosen since the lowest possible energy is zero) is E(0; s)=det(I−K| (0,s) ). Crucially (for later) this is a cumulative probability density function (CDF). The probability density function (PDF) for finding an energy is F (0; s)=−dE(0; s)/ds. For orientation, in the case of the Airy model the interval would be (−∞, s), and using the Airy kernel yields the famous Tracy-Widom distribution [30] of the smallest eigenvalue. Here, the JT gravity system will reveal a new kind of smallest eigenvalue distribution.
More generally, for the nth energy level, the probability (CDF) for the interval (0, s) is written as: Correspondingly the PDF is: F (n; s)=−dE(n; s)/ds. The major challenge now is to compute the determinant of the infinite dimensional operator in Eq. (9). This requires much more care, and is prone to severe numerical difficulties even though the problem is effectively discrete (∼ 700 energies were used). Useful at this point is the impressive work of Bornemann [31] that shows how to use quadrature on a relatively small number of points in the energy interval (0, s) to compute Fredholm determinants: Using Clenshaw-Curtis quadrature to break up the interval into m points e i and compute weights w i (i = 1 · · · m), an integral s 0 f (E)dE on the interval would be computed as m i w i f (e i ). So similarly the determinant becomes: If the ψ(E, x) are known accurately (such as for Airy and Bessel kernels) the method gives impressive results with modest values of m such as 8, or 16. In the case in hand, the ψ(E, x) were found as approximate numerical solutions to an eigenstate problem for which the potential u(x) was itself a solution to a difficult (15th order [26]) non-linear equation, so some challenging numerical difficulties due to errors can be expected. However, they can be surmounted well enough to get very good results for the definition of JT gravity discussed here [32]. The result for the zeroth level (the ground state) is the focus of the inset of Fig. 1. A small amount of smoothing has been applied to remove numerical noise. To the left, at zero, the CDF E(0; s) shows that there is a nonzero (but small) chance of finding the ground state there, falling steadily to zero to the far right with the increasing unlikelihood of finding the ground state at very high energies. Its derivative, the PDF F (0; s) is also shown in the inset. It peaks at around 0.75. The mean of the distribution gives the average ground state of the ensemble E 0 0.66. (For illustration purposes all computations were done using =1). Computing the results for higher levels is straightforward, although numerical inaccuracies in finding the first level get successively amplified with each level. The first six levels are shown in the main part of Fig. 1. What has been uncovered here with the Fredholm determinant technique are the explicit probability peaks for individual energies/microstates that, when added together, produce the previously found nonperturbative undulations in the spectral density (black line). In principle any quantity can now be computed using this information, as will be demonstrated next.
Quenched Free Energy.-It is important to compute the quantity F Q (T )=−β −1 log Z(β) for JT gravity, but it is difficult. Ref. [9] pointed out that connected diagrams with multiple boundaries ("replica wormholes", as they were implementing a gravitational replica trick) should play a crucial role. Ref. [10] observed that in addition a non-perturbative formulation was needed, such as a matrix model. There have been various useful partial results from toy models [11,12], at low T . Ref. [11]  noted that a better tool was needed, in order to incorporate the properties of individual peaks. The Fredholm determinant has performed handily in that regard, and so computation of F Q (T ) is now rather straightforward.
A numerical approach is natural (since all the explicit data about the levels are numerical), simply directly sampling ensembles. In fact, this was done recently in Ref. [11] for the comparatively simple cases of the Airy model and a variety of Bessel models. There, all that was needed was Gaussian random matrices (readily numerically generated) plus scaling to an endpoint. A matrix model for JT gravity has much more exotic probability distributions however, so direct sampling seems doomed. However, the individual probability distributions for each level can now be generated by reverse-engineering the above results for the E(n; s), which (recall) are CDFs.
A key result from the theory of statistics is that any probability distribution function can be generated from a uniform probability distribution by mapping from the associated CDF. So, ensembles can be generated as follows: For a particular sample, generate the nth energy level E n with the appropriate probability (using uniform PDFs on a computer and converting using the CDF E(n; s) to sample the correct PDF F (n; s)). Then compute log(Z(β))= log( n e −βEn ). This was done for an ensemble of just 5000 samples, and then averaged. For contrast, the annealed quantity F A (T )=−β −1 log Z(β) can be computed too (averaging the partition function over the ensemble and taking the log at the end).
Using just the first six levels yields the content of the inset of Fig. 2. As higher levels are added, the details of the curves settle swiftly. Only successively higher T details are affected by adding successively higher levels, the process asymptoting to the classical result at large T . In the main plot of Fig. 2, a result with 150 levels is given. It is constructed by approximating the E(n; s) for lev-els above about n=10. In this regime the approximation is already very much under control. The point is that even by 6 levels (see Fig. 1), the peaks have narrowed and overlap significantly, as the system returns to the continuum (classical) regime. The n>10 peaks are well approximated by narrow Gaussians, with their location, mean, and standard deviation determined by the classical density curve (2). So CDFs (error functions for Gaussians) were used for high levels. As a test of this, the red dashed line is the annealed free energy computed using the Z(β) obtained by simply Laplace transforming ρ(E) (integrated to the appropriate cutoff energy corresponding to the 150th level). The agreement with the result computed by direct ensemble averaging (red crosses) is remarkably good. This shows that the result for F Q (T ) is accurate. Overall, rather nicely, F Q (T ) is monotonically decreasing (i.e., the entropy S(T ) is manifestly positive), with zero slope at T =0, corresponding to entropy S 0 at extremality. Additionally, this result confirms an earlier suggestion [10] that there is no sign of the replica symmetry breaking transition conjectured in Ref. [9].
Final Remarks.-The studies of this Letter (with several JT supergravity examples to appear) show that matrix models of JT gravity are completely tractable models of a key quantum gravity phenomenon: A cross-over from smooth, classical spacetime geometry (large E (or T ) in Fig. 1 (or 2)) to a regime where spacetime is not enough, and a description in terms of discrete microstates takes over (small E (or T )). The microstates are always there, of course, but the regime where their individuality is inevitable is when the variance or randomness becomes pronounced, while inter-spacing is large. This is a regime beyond smooth spacetimes, even with many handles and wormholes. It is no longer geometrical, but the matrix model computes the physics just as readily. The key tool used here was the Fredholm determinant, whose kernel is built from sewing together two copies of the wavefunctions ψ(E, x). Interestingly, the ψ(E, x) have an interpretation [33] as a type of D-brane probe, in the language of minimal string theory. In a sense then, the Fredholm tool is a D-brane probe that detects an increased spreading out or variance (or itself spreads out) as it moves to lower energies. This is reminiscent of aspects of the D-brane probes involved in the "enhançon mechanism" [34]. This might be relevant when considering what lessons these phenomena might teach about higher dimensional quantum gravity. Finally, as mentioned in the introduction, the microstates uncovered in detail here also model the microstates of the higher dimensional black holes whose near-horizon low T dynamics is controlled by a JT gravity. It will be interesting to see what other features of JT gravity and black hole physics (in various dimensions) might be accessible using matrix model technology.
CVJ thanks the US Department of Energy for support under grant DE-SC 0011687, and, especially during the pandemic, Amelia for her support and patience.