Non-Unique JT Matrix-Integral Completion: Fixed-Theory, Disorder-Average, and Ensemble Distinctions

The JT genus expansion is reproduced by a double-scaled matrix integral, but perturbative topological data do not select one nonperturbative spectrum. A fixed Hamiltonian, a disorder average, a matrix ensemble, and a gravitational sum over connected topologies are distinct claims. Matching their asymptotic expansions does not erase that distinction.

Required background. JT Topological Expansion and Weil–Petersson Volumes supplies the perturbative amplitudes; Random Matrices, Spectral Statistics, and Ensemble Questions supplies ensemble observables.

Helpful background. Fixed-Theory, Ensemble, and Superselection Claims supplies the claim grammar; JT/SYK Spectral Form Factors and Universality Windows shows where averaging enters an observable.

Evidence cutoff: 25 July 2026.

Four objects that must remain distinct

For one Hamiltonian $H_\alpha$,

$$Z_\alpha(\beta)=\operatorname{Tr}e^{-\beta H_\alpha}$$

is a definite function. A disorder ensemble introduces probabilities $p_\alpha$ and moments

$$\overline{Z(\beta_1)\cdots Z(\beta_n)} =\sum_\alpha p_\alpha \prod_{k=1}^n Z_\alpha(\beta_k).$$

A matrix integral is a particular probability measure on spectra. A gravity path integral can produce connected multi-boundary amplitudes,

$$\left\langle Z(\beta_1)\cdots Z(\beta_n)\right\rangle_{\!c} \longleftrightarrow \sum_{\substack{\text{connected}\\\text{JT surfaces}}} e^{S_0\chi}\,\mathcal Z_{g,n},$$

order by order in $e^{-S_0}$. This equality identifies the perturbative JT amplitudes with cumulants of a matrix ensemble Saad, Shenker, and Stanford 2019. It does not imply that every individual matrix is a boundary theory of the same bulk, or that a single fixed theory has nonfactorizing ensemble moments.

First application: two completions with one asymptotic series

The genus expansion determines a formal spectral density

$$\rho(E)\sim\sum_{g=0}^{\infty} e^{(1-2g)S_0}\rho_g(E).$$

Suppose two exact densities differ by

$$\delta\rho(E)=O\!\left(e^{-c e^{S_0}}\right)$$

through the perturbative regime. Every coefficient in the $e^{-S_0}$ expansion agrees, yet the exact low-energy spectrum, positivity properties, and plateau data can differ. Equivalently, different integration contours or Stokes data solve the same string equation asymptotically. This is the elementary mechanism behind nonuniqueness.

Concrete nonperturbative matrix-model proposals impose additional conditions beyond the genus series. Johnson constructed nonperturbative JT candidates designed to avoid instabilities of a direct Hermitian completion Johnson 2020, and later analyzed consistency conditions and alternative completions Johnson 2022. These are extra definitions to test, not consequences of the perturbative coefficients alone.

Factorization and disorder

For a fixed theory with two decoupled boundaries, ordinary factorization gives

$$Z_\alpha(\beta_1)Z_\alpha(\beta_2).$$

An ensemble covariance,

$$\overline{Z(\beta_1)Z(\beta_2)} -\overline{Z(\beta_1)}\,\overline{Z(\beta_2)},$$

can instead be nonzero. A connected Euclidean wormhole naturally computes the latter structure, but interpreting it requires specifying whether gravity defines an ensemble, whether baby-universe superselection selects an $\alpha$-state, or whether some fixed-theory mechanism reproduces the same observable. The bulk topology alone does not decide among these possibilities.

Adversarial control: demand a unique Hamiltonian

Give only all coefficients $\rho_g(E)$ and ask for the exact ordered levels of one finite-dimensional Hamiltonian. The inverse problem has many solutions differing beyond all perturbative orders; consequently exact recurrences and sample-specific plateau fluctuations are undetermined. Adding positivity, a contour, a string equation, or boundary data narrows the possibilities but is additional input.

The strongest established statement is therefore precise: JT perturbative topology has a matrix-integral realization, and its coarse spectral correlations fall in identifiable universality classes. It does not, without further microscopic data, define a unique fixed Hamiltonian or exact finite-$N$ completion. This boundary remains active in the primary literature through the stated cutoff.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Johnson, Clifford V. “Nonperturbative Jackiw–Teitelboim Gravity.” Physical Review D 101, 106023 (2020). DOI. Open PDF.
• Johnson, Clifford V. “Consistency Conditions for Non-Perturbative Completions of JT Gravity.” arXiv:2112.00766 [hep-th] (2022). arXiv.
• Saad, Phil, Stephen H. Shenker, and Douglas Stanford. “JT Gravity as a Matrix Integral.” arXiv:1903.11115 [hep-th] (2019). arXiv.