JT Topological Expansion and Weil–Petersson Volumes

Euclidean JT amplitudes separate into hyperbolic geometry and a dilaton weight. For fixed genus and asymptotic boundaries, trumpets glue the Schwarzian boundaries to a compact core whose moduli integral is a Weil–Petersson volume. This gives an exact coefficient at each topology, but the genus series is asymptotic and does not choose its own nonperturbative completion.

Required background. JT Gravity and the Schwarzian Boundary Mode supplies the disk boundary theory. Multi-Saddle Sums and Dilute Ensembles supplies the status of topology sums.

Helpful background. Chiral Blocks, Sewing, and Modular Invariance supplies geometric sewing. Evidence and Limits for Resurgence in QFT supplies the distinction between asymptotic data and a completion.

Evidence cutoff: 25 July 2026.

Hyperbolic surfaces and trumpet gluing

For a connected orientable surface of genus $g$ with $n$ asymptotic boundaries,

$$\chi=2-2g-n, \qquad Z_{g,n}\propto e^{S_0\chi}.$$

Cut each asymptotic region along a geodesic of length $b_i$. In the convention

$$Z_{\mathrm{tr}}(\beta,b) =\frac{1}{\sqrt{2\pi\beta}} \exp\!\left(-\frac{b^2}{2\beta}\right),$$

the amplitude is

$$Z_{g,n}(\beta_1,\ldots,\beta_n) =e^{S_0\chi} \int_0^\infty\prod_{i=1}^n \left[b_i\,db_i\,Z_{\mathrm{tr}}(\beta_i,b_i)\right] V_{g,n}(b_1,\ldots,b_n).$$

$V_{g,n}$ is the Weil–Petersson volume of the moduli space of bordered hyperbolic surfaces. The formula records both the topology order and the boundary normalization. Mirzakhani’s recursion determines these volumes from lower-complexity surfaces Mirzakhani 2007.

A three-boundary application

For a pair of pants, $g=0$, $n=3$, there are no continuous internal moduli and

$$V_{0,3}(b_1,b_2,b_3)=1.$$

Each trumpet integral is elementary:

$$\int_0^\infty b\,db\, \frac{e^{-b^2/(2\beta)}}{\sqrt{2\pi\beta}} =\sqrt{\frac{\beta}{2\pi}}.$$

Since $\chi=-1$,

$$Z_{0,3}(\beta_1,\beta_2,\beta_3) =e^{-S_0} \frac{\sqrt{\beta_1\beta_2\beta_3}}{(2\pi)^{3/2}}$$

in this convention. The $e^{-S_0}$ suppression relative to disconnected disks is the expected topology weight. Higher volumes are polynomials in $b_i^2$ and $\pi^2$ determined recursively.

Matrix integrals and perturbative equality

The full formal series is

$$Z_n\sim\sum_{g=0}^{\infty} e^{S_0(2-2g-n)}Z_{g,n}.$$

Its coefficients agree with the topological expansion of a double-scaled matrix integral Saad, Shenker, and Stanford 2019. This is a powerful all-genus identification of perturbative data. It also explains connected multi-boundary amplitudes as connected spectral correlators of an ensemble.

Large-genus adversary

Weil–Petersson volumes grow factorially with genus, so the expansion in $e^{-2S_0}$ has zero radius of convergence. Demand a numerical answer by summing it without a prescription: different nonperturbative completions can share every coefficient and differ by terms invisible at all orders, typically of order $e^{-\mathrm{const}\,e^{S_0}}$ or another model-dependent exponential.

Consequently, fixed-topology amplitudes and their recursion are exact within JT, while the infinite topology sum is formal until contour, spectral support, and nonperturbative definition are supplied. No low-genus wormhole amplitude alone establishes a unique ensemble or fixed boundary Hamiltonian.

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

• Mirzakhani, Maryam. “Simple Geodesics and Weil–Petersson Volumes of Moduli Spaces of Bordered Riemann Surfaces.” Inventiones Mathematicae 167, 179–222 (2007). DOI.
• Saad, Phil, Stephen H. Shenker, and Douglas Stanford. “JT Gravity as a Matrix Integral.” arXiv:1903.11115 [hep-th] (2019). arXiv.