JT Topological Expansion and Weil–Petersson Volumes

Euclidean Jackiw–Teitelboim (JT) gravity turns a difficult sum over metrics into a sequence of finite-dimensional geometric integrals. At a fixed connected topology, each asymptotic boundary is separated from a compact hyperbolic core by a trumpet. The trumpet carries the Schwarzian boundary dynamics; a Weil–Petersson volume integrates the shapes of the core. This factorization computes every coefficient of the topology expansion exactly in the stated JT convention.

The word coefficient is essential. The Euler-characteristic factor is attached once, disk and cylinder topologies are exceptional, and the all-genus series diverges factorially. Fixed-topology exactness therefore does not by itself define a unique nonperturbative spectrum.

Required background. JT Gravity and the Schwarzian Boundary Mode supplies the Euclidean action and the boundary coupling $C$. Multi-Saddle Sums and Dilute Ensembles supplies the status of a formal saddle sum.

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.

Scope. We consider connected, orientable, pure Euclidean JT surfaces with labeled asymptotic boundaries, AdS radius $L_2=1$, Schwarzian coupling $C>0$, and no matter insertions. Thus $[C]=[\beta]$, while each geodesic length $b$ is dimensionless. Our Weil–Petersson orbifold normalization includes mapping-class-group and automorphism factors; in this normalization, integrating the relative twist at each trumpet–core seam supplies the gluing measure $b\,db$. Other conventions must be translated before formulas are combined.

Evidence cutoff: 25 July 2026.

One Euler weight for each connected topology

The topological term in the JT action is

$$\begin{aligned} I_{\mathrm{top}}&=-S_0\chi(M),\\ \chi(M)&=2-2g-n. \end{aligned}$$

for a connected orientable surface of genus $g$ with $n$ boundaries. Its path-integral weight is therefore

$$e^{-I_{\mathrm{top}}}=e^{S_0(2-2g-n)}.$$

Adding one handle lowers $\chi$ by two and costs $e^{-2S_0}$; adding one boundary at fixed genus lowers $\chi$ by one and costs $e^{-S_0}$. This is a topological coupling, not a thermal Boltzmann factor.

To prevent double counting, define the reduced fixed-topology coefficient $\widehat Z_{g,n}$ with the Euler factor removed. The connected perturbative observable is then

$$\left\langle\prod_{i=1}^{n}Z(\beta_i)\right\rangle_{\!c}^{\mathrm{pert}} \sim \sum_{g=0}^{\infty} e^{S_0(2-2g-n)} \widehat Z_{g,n}(\boldsymbol\beta).$$

The subscript $c$ matters: the left side is a connected cumulant. The full moment also contains products of amplitudes from disconnected surfaces. The separation of the reduced coefficient from its single Euler weight follows Saad, Shenker, and Stanford 2019, § 3, especially Eqs. (63) and (130). A hyperbolic core is called stable when $2g-2+n>0$; the disk and cylinder fail this condition and require separate formulas.

Elementary connected JT topologies and their Euler weights

Surface	g	n	χ	Euler weight	Ordinary stable core?
Disk	0	1	1	eS0	No; exceptional Schwarzian orbit
Double trumpet	0	2	0	1	No; exceptional cylinder
Pair of pants	0	3	−1	e−S0	Yes; zero-dimensional moduli space
Once-holed torus	1	1	−1	e−S0	Yes; two-dimensional moduli space

Trumpets connect thermal boundaries to geodesic seams

An asymptotic JT boundary has regulated proper length $\beta/\epsilon$ and fixed renormalized dilaton. It is not a finite geodesic boundary of a compact hyperbolic surface. To connect the two descriptions, cut the geometry along the unique closed geodesic at the neck of the asymptotic region. The exterior piece is a trumpet: one end is the asymptotic Schwarzian boundary labeled by $\beta$, and the other is a geodesic seam of length $b$.

In the convention inherited from the Schwarzian page, the trumpet path integral is

$$Z_{\mathrm T}(\beta,b) =\sqrt{\frac{C}{2\pi\beta}} \exp\!\left(-\frac{Cb^2}{2\beta}\right).$$

This expression is dimensionless because $C/\beta$ and $b$ are dimensionless. It is the hyperbolic Schwarzian orbit $\operatorname{Diff}(S^1)/U(1)$, not the disk orbit $\operatorname{Diff}(S^1)/\operatorname{PSL}(2,\mathbb R)$.

For a stable core,

$$2g-2+n>0,$$

let $\mathcal M_{g,n}(\mathbf b)$ be the moduli space of genus-$g$ hyperbolic surfaces with labeled geodesic boundaries of fixed lengths $\mathbf b=(b_1,\ldots,b_n)$. Put

$$d=3g-3+n, \qquad \dim_{\mathbb R}\mathcal M_{g,n}(\mathbf b)=2d,$$

and define its Weil–Petersson volume by

$$V_{g,n}(\mathbf b) =\int_{\mathcal M_{g,n}(\mathbf b)} \frac{\omega_{\mathrm{WP}}^{\,d}}{d!}.$$

This “volume” measures the space of inequivalent hyperbolic shapes at fixed boundary lengths; it is not the area of one spacetime. Choose $d$ internal curves in a pants decomposition, with Fenchel–Nielsen length–twist coordinates $(\ell_a,\tau_a)$. Then

$$\omega_{\mathrm{WP}} =\sum_{a=1}^{d}d\ell_a\wedge d\tau_a.$$

The external boundary lengths $b_i$ are fixed parameters of $V_{g,n}(\mathbf b)$, not integration coordinates in this moduli-space volume. Gluing a trumpet introduces a separate relative twist $\theta_i$ around its external seam, with $\theta_i\sim\theta_i+b_i$. Integrating one twist period supplies the factor $b_i$; the JT gluing integral then varies the seam length itself:

$$\int_0^{b_i}d\theta_i\,db_i=b_i\,db_i.$$

It is useful to package all exterior factors as

$$d\mu_{\mathrm{glue}} \equiv \prod_{i=1}^{n} b_i\,db_i\,Z_{\mathrm T}(\beta_i,b_i).$$

The reduced coefficient for every stable topology is then

$$\widehat Z_{g,n}(\boldsymbol\beta) =\int_{(0,\infty)^n} d\mu_{\mathrm{glue}}\, V_{g,n}(\mathbf b).$$

The full connected contribution of that topology is $e^{S_0(2-2g-n)}\widehat Z_{g,n}$. Except for the disk, the integrated Euclidean configurations need not be complete classical JT saddles; the fixed-topology path integral is nevertheless defined by the stated measure. The boundary orbit, gluing measure, stable formula, and this saddle qualification are derived in Saad, Shenker, and Stanford 2019, §§ 3.3.5–3.4.2, Eqs. (123)–(138) and reviewed in Mertens and Turiaci 2023, § 4.2, Eqs. (4.16)–(4.21).

What is fixed and what is integrated in the gluing formula

Object	Held fixed when defined	Integrated or summed	Meaning
βi	Asymptotic boundary condition	No	Inverse temperature or renormalized asymptotic length
bi	While defining the core volume	Yes, in JT gluing	Finite geodesic seam length; it is not βi
(ℓa, τa)	Only at one point of core moduli space	Yes, inside Vg,n	Internal Fenchel–Nielsen length and twist; these are not the external bi
θi	Relative seam orientation before gluing	Yes, over one period	External gluing twist; its period supplies the factor bi
Vg,n(b)	g, n, and all seam lengths	Core shapes and twists	Weil–Petersson moduli-space volume, not spacetime area
Ẑg,n	Topology and asymptotic boundary data	Seams and core moduli	Reduced coefficient with no Euler weight
Genus series	n and all βi	Formal sum over g	Asymptotic connected topology expansion

The disk and cylinder are exceptional

The condition $2g-2+n>0$ excludes the disk $(0,1)$ and cylinder $(0,2)$. There are no ordinary moduli spaces $\mathcal M_{0,1}$ or $\mathcal M_{0,2}$ whose Weil–Petersson volumes can simply be inserted into the stable formula. They must be computed from their boundary orbits.

With the Euler factor removed, the disk coefficient is

$$\widehat Z_{0,1}(\beta) =\frac{C^{3/2}}{\sqrt{2\pi}\,\beta^{3/2}} \exp\!\left(\frac{2\pi^2C}{\beta}\right).$$

Its full connected contribution is $e^{S_0}\widehat Z_{0,1}$. The factor $C^{3/2}/\beta^{3/2}$ is dimensionless, as required.

The cylinder is obtained by gluing two trumpets directly along the same geodesic:

$$\begin{aligned} \widehat Z_{0,2}(\beta_1,\beta_2) &=\int_0^\infty b\,db\, Z_{\mathrm T}(\beta_1,b) Z_{\mathrm T}(\beta_2,b)\\ &=\frac{\sqrt{\beta_1\beta_2}} {2\pi(\beta_1+\beta_2)}. \end{aligned}$$

Its Euler weight is one because $\chi=0$, and the cancellation of $C$ is a useful normalization check. The first line is sometimes made to resemble the stable formula by introducing a distributional convention for $V_{0,2}$; that bookkeeping device is not an ordinary Weil–Petersson volume. The same cylinder becomes the gravitational ramp after the appropriate real-time continuation and averaging, a question treated on JT/SYK Spectral Form Factors and Universality Windows.

The disk orbit and its final answer are derived in Saad, Shenker, and Stanford 2019, § 3.3.5, Eqs. (125) and (129); the double-trumpet gluing is Saad, Shenker, and Stanford 2019, § 3.4.1, Eqs. (133)–(138). Both exceptional answers and their orbit normalizations are reviewed in Mertens and Turiaci 2023, §§ 4.2.1–4.2.2, Eqs. (4.11) and (4.17).

Worked three-boundary amplitude

The first stable core is a pair of pants: $g=0$, $n=3$, and $\chi=-1$. A hyperbolic pair of pants is fixed uniquely by its three boundary lengths, so its moduli space is zero-dimensional and

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

Each exterior integral is elementary:

$$\begin{aligned} \int_0^\infty b\,db\,Z_{\mathrm T}(\beta,b) &=\sqrt{\frac{C}{2\pi\beta}} \left(\frac{\beta}{C}\right)\\ &=\sqrt{\frac{\beta}{2\pi C}}. \end{aligned}$$

Therefore

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

and the full connected contribution is

$$\mathcal Z_{0,3}^{\mathrm{conn}} =e^{-S_0}\widehat Z_{0,3}.$$

This checks both normalization and dimensions: every ratio $\beta_i/C$ is dimensionless. It also exposes an easy topology-bookkeeping trap. Three disconnected disks carry Euler weight $e^{3S_0}$, whereas the connected pair of pants carries $e^{-S_0}$. Their topological-prefactor ratio is therefore

$$\frac{e^{-S_0}}{e^{3S_0}}=e^{-4S_0},$$

not $e^{-S_0}$. The ratio of the full amplitudes additionally contains their different $\beta_i$-dependent coefficients and should not be inferred from topology alone.

More generally, a connected genus-$g$ surface with $n$ boundaries is suppressed relative to $n$ disconnected disks by the topological factor

$$e^{-2S_0(g+n-1)}.$$

Mirzakhani recursion lowers geometric complexity

For stable $(g,n)$, Mirzakhani’s recursion starts from a distinguished boundary and uses a hyperbolic pair of pants adjacent to it. Removing that pair of pants exposes one or two internal geodesics. Their lengths are integrated against explicit hyperbolic kernels, while the remaining surface has lower complexity $2g-2+n$. The three geometric channels are easier to understand than the full kernel formula.

How removing the adjacent pair of pants lowers the topology

Channel	What the two remaining cuffs do	Lower-complexity data
Boundary pair	One cuff is another external boundary; the other is internal	Vg,n−1 with one new internal length
Nonseparating cut	Both cuffs are internal and remain on one connected component	Vg−1,n+1
Separating cut	The two internal cuffs lie on different connected components	Products Vg1,|I|+1Vg2,|J|+1

Starting from the two base cases $V_{0,3}=1$ and $V_{1,1}=(b^2+4\pi^2)/48$, the recursion determines the remaining stable volume polynomials. Each $V_{g,n}$ is symmetric in the $b_i^2$ and has total degree $3g-3+n$; its coefficients are positive normalization factors times mixed tautological intersection numbers. The precise integration identity, kernels, and polynomial theorem are Mirzakhani 2007, Theorem 1.1 and §§ 5–6, printed pp. 203–210.

The first core with a genuine moduli integral is the once-holed torus. In the orbifold convention used by the JT gluing formula,

$$V_{1,1}(b)=\frac{b^2+4\pi^2}{48}.$$

Mirzakhani’s original displayed volume is twice this expression and places the compensating automorphism factor in the recursion. Mixing that unhalved input with the orbifold recursion would double the amplitude. The convention translation is spelled out in Saad, Shenker, and Stanford 2019, § 2.5, Eq. (59) and n. 18; § 3.1, Eq. (67); § 3.4, Eq. (135).

Using the two Gaussian moments

$$\begin{aligned} \int_0^\infty b\,db\,Z_{\mathrm T} &=\sqrt{\frac{\beta}{2\pi C}},\\ \int_0^\infty b^3\,db\,Z_{\mathrm T} &=\frac{2\beta^{3/2}} {\sqrt{2\pi}\,C^{3/2}}. \end{aligned}$$

gives

$$\begin{aligned} \widehat Z_{1,1}(\beta) &=\frac{1}{48} \sqrt{\frac{\beta}{2\pi C}} \left(\frac{2\beta}{C}+4\pi^2\right)\\ &=\frac{\beta^{3/2}} {24\sqrt{2\pi}\,C^{3/2}} +\frac{\pi^{3/2}\sqrt\beta} {12\sqrt2\,\sqrt C}. \end{aligned}$$

The full genus-one one-boundary term is $e^{-S_0}\widehat Z_{1,1}$. As an independent normalization check, the matrix loop equation gives exactly the second line Mertens and Turiaci 2023, § 4.3, Eqs. (4.42)–(4.43).

Geometric recursion becomes matrix topological recursion

The disk coefficient has the reduced leading spectral density

$$\rho_0(E) =\frac{C}{2\pi^2} \sinh\!\left(2\pi\sqrt{2CE}\right),$$

whose Laplace transform is $\widehat Z_{0,1}(\beta)$. The physical leading density carries the additional disk factor $e^{S_0}$.

After Laplace transforming the boundary lengths, Mirzakhani’s recursion for $V_{g,n}$ becomes the loop-equation, or topological-recursion, hierarchy of a double-scaled Hermitian matrix ensemble with this leading density. Eynard and Orantin proved the recursion equivalence for Weil–Petersson volumes Eynard and Orantin 2007, § 2, especially Theorem 2.1; the JT boundary factors and density complete the matching in Saad, Shenker, and Stanford 2019, § 3.5.

The licensed statement is precise:

• at every fixed $(g,n)$, the connected JT coefficient equals the corresponding coefficient in the double-scaled matrix loop expansion;
• the gravity connected amplitudes map to ensemble cumulants of $Z(\beta)=\operatorname{Tr}e^{-\beta H}$;
• the equality is order by order in the topological coupling $g_{\mathrm{top}}=e^{-S_0}$.

It does not identify every matrix realization with one boundary theory, turn an ensemble cumulant into a fixed-Hamiltonian product, or select a unique nonperturbative matrix integral. Those distinctions are developed on Random Matrices, Spectral Statistics, and Ensemble Questions and Non-Unique JT Matrix-Integral Completion.

Large genus defeats literal summation

The divergence is not a qualitative guess. For fixed $n$, Mirzakhani and Zograf proved that the punctured volumes have the asymptotic expansion

$$\begin{aligned} V_{g,n}(\mathbf0) &=\frac{C_{\mathrm{MZ}}}{\sqrt g} (2g-3+n)!\\ &\quad\times(4\pi^2)^{2g-3+n}\\ &\quad\times \left[1+\frac{c_n^{(1)}}{g}+O(g^{-2})\right]. \end{aligned}$$

where $C_{\mathrm{MZ}}>0$ is a universal constant in their convention; their theorem does not determine its value Mirzakhani and Zograf 2015, Theorem 1.2. The transfer to JT coefficients is immediate and does not require a resurgent assumption. Positivity of the volume polynomial gives $V_{g,n}(\mathbf b)\geq V_{g,n}(\mathbf0)$, so at fixed positive $\beta_i/C$,

$$\widehat Z_{g,n}(\boldsymbol\beta) \geq V_{g,n}(\mathbf0) \prod_{i=1}^n \sqrt{\frac{\beta_i}{2\pi C}}.$$

Thus the glued coefficients inherit at least the same factorial growth, which is already enough to prove zero radius of convergence. This conclusion is not asserted uniformly when $n$, $b_i$, or $\beta_i/C$ scales with $g$.

Recent resurgent analyses refine length-dependent large-genus and transseries data in matrix-model settings Griguolo et al. 2024, § 3.3 and Johnson and Rodrigues 2026, § 4; these refinements add nonperturbative structure rather than removing the need to specify it.

The coefficient ratio makes the failure operational. Suppose, at fixed $n$ and $\boldsymbol\beta$,

$$\widehat Z_{g,n} \sim K A^{-2g}\Gamma(2g+\nu), \qquad A>0.$$

The genus-$g$ term is $T_g\propto e^{-2gS_0}\widehat Z_{g,n}$ up to a $g$-independent prefactor, so

$$\frac{T_{g+1}}{T_g} \sim \frac{(2g+\nu)(2g+\nu+1)} {A^2e^{2S_0}}.$$

For any finite $S_0$, this ratio eventually exceeds one: the series has zero radius of convergence. Its terms are smallest near

$$g_*\simeq\frac{A}{2}e^{S_0},$$

and optimal truncation leaves a remainder of order

$$\exp\!\left(-A e^{S_0}\right)$$

up to powers, phases, and observable-dependent Stokes data. Adding more genera after $g_*$ makes the approximation worse, not better. Saad, Shenker, and Stanford’s extraction of particular Borel singularities from matrix-eigenvalue instantons is explicitly a resurgent completion argument, not a theorem that the genus coefficients choose a unique contour Saad, Shenker, and Stanford 2019, § 5.6, Eqs. (212)–(224).

The elementary ambiguity is already decisive. If $\mathcal S F_{\mathrm{pert}}(g_{\mathrm{top}})$ is one resummation, then

$$F_\lambda(g_{\mathrm{top}}) =\mathcal S F_{\mathrm{pert}}(g_{\mathrm{top}}) +\lambda\exp\!\left(-\frac{A}{g_{\mathrm{top}}}\right)$$

has the same perturbative genus coefficients for every $\lambda$. The coefficients cannot determine $\lambda$, the integration contour, spectral support, or exact level statistics. Explicit matrix constructions make this more than a formal observation: a stable complex-matrix/string-equation completion can retain the perturbative JT sector while changing the low-energy spectrum Johnson 2020, §§ III–IV, and a later family varies the nonperturbative spectral boundary while leaving the perturbative regime unchanged Johnson 2022, §§ III–IV.

Strongest surviving claim. Oriented pure JT determines the connected fixed-topology coefficients and their matrix-loop representation exactly in the displayed measure and boundary convention. The failed hypothesis is that an all-orders asymptotic sequence uniquely specifies an exact function or spectrum. A nonperturbative definition requires additional contour, boundary-condition, positivity, and spectral data; general wormhole and fixed-theory tests belong to Wormholes, Gravitational Path Integrals, and Ensembles.

Common pitfalls

Counting $e^{S_0\chi}$ twice. Either include the Euler weight in the full fixed-topology amplitude or place it outside a reduced coefficient. Never do both.

Setting $C=1$ silently. The trumpet, disk, and genus-one formulas are sensitive to the Schwarzian normalization. Keeping $C$ explicit makes dimensions and neighboring-page conventions checkable.

Confusing $\beta$ with $b$. The former labels an asymptotic thermal boundary; the latter is a finite geodesic seam integrated during gluing.

Calling a Weil–Petersson volume a spacetime volume. It is an integral over inequivalent shapes and twists of hyperbolic surfaces with fixed seam lengths.

Applying the stable formula to disk or cylinder. $V_{0,1}$ and $V_{0,2}$ are not ordinary stable volumes. Use the disk orbit and double-trumpet integral separately.

Equating a connected amplitude with a full moment. Connected wormholes compute cumulant-like terms. Products of disconnected disks are separate contributions with different Euler weights.

Inferring convergence from exact recursion. Mirzakhani recursion determines every coefficient; it does not make their all-genus sum convergent or unique.

Promoting an ensemble realization to one fixed theory. Order-by-order matrix-integral equality is a powerful computational statement with a narrower claim domain than exact fixed-Hamiltonian duality.

Exercises

1. Euler bookkeeping

Compute $\chi$ and the Euler weight for a disk, double trumpet, pair of pants, and once-holed torus. Then compare the pair-of-pants weight with three disconnected disks.

Solution

Using $\chi=2-2g-n$ gives

$$\begin{aligned} \chi_{\mathrm{disk}}&=1, &\chi_{\mathrm{cylinder}}&=0,\\ \chi_{\mathrm{pants}}&=-1, &\chi_{\mathrm{torus,1}}&=-1. \end{aligned}$$

Their weights are respectively $e^{S_0}$, $1$, $e^{-S_0}$, and $e^{-S_0}$. Three disconnected disks have total characteristic $3$ and weight $e^{3S_0}$. Hence the connected pair-of-pants topology is suppressed relative to that disconnected topology by $e^{-4S_0}$. This compares only topological prefactors; full amplitudes contain additional boundary-dependent factors.

2. Pair-of-pants gluing

Evaluate $\int_0^\infty b\,db\,Z_{\mathrm T}(\beta,b)$ and use $V_{0,3}=1$ to reproduce $\widehat Z_{0,3}$. Check its dimensions.

Solution

For $a=C/(2\beta)$,

$$\int_0^\infty b e^{-ab^2}db=\frac{1}{2a}=\frac{\beta}{C}.$$

Multiplying by the trumpet prefactor gives

$$\int_0^\infty b\,db\,Z_{\mathrm T}(\beta,b) =\sqrt{\frac{\beta}{2\pi C}}.$$

The three independent seams therefore give

$$\widehat Z_{0,3} =\prod_{i=1}^3\sqrt{\frac{\beta_i}{2\pi C}} =\frac{\sqrt{\beta_1\beta_2\beta_3}} {(2\pi C)^{3/2}}.$$

Since every $\beta_i/C$ is dimensionless, so is the reduced coefficient. The full connected term is $e^{-S_0}\widehat Z_{0,3}$.

3. The exceptional double trumpet

Compute the direct gluing integral for two trumpets and explain why no ordinary $V_{0,2}$ appears.

Solution

Multiplying the two trumpets gives

$$\begin{aligned} \kappa &\equiv\frac C2 \left(\frac1{\beta_1}+\frac1{\beta_2}\right),\\ Z_{\mathrm T}(\beta_1,b)Z_{\mathrm T}(\beta_2,b) &=\frac{C}{2\pi\sqrt{\beta_1\beta_2}}\\ &\quad\times e^{-\kappa b^2}. \end{aligned}$$

Using the same Gaussian integral yields

$$\widehat Z_{0,2} =\frac{\sqrt{\beta_1\beta_2}} {2\pi(\beta_1+\beta_2)}.$$

The cylinder has $2g-2+n=0$, so it lies outside the stable moduli-space domain. Its one length and one twist describe how the two trumpets are identified; there is no separate compact stable core whose ordinary Weil–Petersson volume should be inserted.

4. The once-holed torus

Insert $V_{1,1}(b)=(b^2+4\pi^2)/48$ into the gluing formula and compute $\widehat Z_{1,1}(\beta)$.

Solution

Besides the first Gaussian moment, one needs

$$\int_0^\infty b^3 e^{-Cb^2/(2\beta)}db =\frac{2\beta^2}{C^2}.$$

Including the trumpet prefactor gives

$$\int_0^\infty b^3db\,Z_{\mathrm T} =\frac{2\beta^{3/2}} {\sqrt{2\pi}\,C^{3/2}}.$$

Consequently,

$$\widehat Z_{1,1} =\frac{1}{48} \sqrt{\frac{\beta}{2\pi C}} \left(\frac{2\beta}{C}+4\pi^2\right).$$

The full contribution is $e^{-S_0}\widehat Z_{1,1}$. The result is dimensionless and agrees with the independent matrix-loop calculation.

5. Recursion anatomy

Remove the pair of pants adjacent to boundary 1. Identify the lower-complexity volume in the boundary-pair, nonseparating, and separating channels.

Solution

If one remaining cuff is external boundary $j$, the other becomes a boundary of a genus-$g$ surface with $n-1$ total boundaries, so the channel uses $V_{g,n-1}$. If both cuffs are internal and the remainder stays connected, cutting them lowers the genus by one and adds two boundaries; after removing boundary 1 with the pair of pants, the result is $V_{g-1,n+1}$. If the cuffs separate the remainder, the other $n-1$ external boundaries split into sets $I$ and $J$, producing

$$V_{g_1,|I|+1}V_{g_2,|J|+1}, \qquad g_1+g_2=g.$$

In every case the total complexity is lower than the original $2g-2+n$, which makes recursion possible.

6. Optimal truncation and nonuniqueness

Assume $\widehat Z_g\sim K A^{-2g}\Gamma(2g+\nu)$ and $T_g=e^{-2gS_0}\widehat Z_g$. Find the least-term genus and construct a change that is invisible to every perturbative coefficient.

Solution

The ratio is

$$\frac{T_{g+1}}{T_g} \sim \frac{(2g+\nu)(2g+\nu+1)} {A^2e^{2S_0}}.$$

It is approximately one when $2g\simeq Ae^{S_0}$, so

$$g_*\simeq\frac A2e^{S_0}.$$

Stirling’s formula then gives a least term of order $e^{-Ae^{S_0}}$ up to powers. For any constant $\lambda$,

$$F_\lambda =F_0+\lambda e^{-Ae^{S_0}}$$

has the same expansion in every power of $e^{-S_0}$ as $F_0$. The perturbative coefficients therefore cannot determine $\lambda$ or any spectral data that depend on it.

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

• Eynard, Bertrand, and Nicolas Orantin. “Weil–Petersson Volume of Moduli Spaces, Mirzakhani’s Recursion and Matrix Models.” arXiv:0705.3600 [math-ph] (2007). Open PDF.
• Griguolo, Luca, Jacopo Papalini, Lorenzo Russo, and Domenico Seminara. “Asymptotics of Weil–Petersson Volumes and Two-Dimensional Quantum Gravities.” SciPost Physics 17, 156 (2024). DOI and open article. Open PDF.
• Johnson, Clifford V. “Non-Perturbative JT Gravity.” Physical Review D 101, 106023 (2020). DOI and open article. Open PDF.
• Johnson, Clifford V. “Consistency Conditions for Non-Perturbative Completions of JT Gravity.” arXiv:2112.00766 [hep-th], version 2 (2022). Open PDF.
• Johnson, Clifford V., and João Rodrigues. “Non-Perturbative Data for Weil–Petersson Volumes and Intersection Numbers Using Ordinary Differential Equations.” arXiv:2601.03351 [hep-th] (2026). Open PDF.
• Mertens, Thomas G., and Gustavo J. Turiaci. “Solvable Models of Quantum Black Holes: A Review on Jackiw–Teitelboim Gravity.” Living Reviews in Relativity 26, 4 (2023). DOI and open article.
• Mirzakhani, Maryam. “Simple Geodesics and Weil–Petersson Volumes of Moduli Spaces of Bordered Riemann Surfaces.” Inventiones Mathematicae 167, 179–222 (2007). DOI.
• Mirzakhani, Maryam, and Peter Zograf. “Towards Large Genus Asymptotics of Intersection Numbers on Moduli Spaces of Curves.” Geometric and Functional Analysis 25, 1258–1289 (2015). DOI. Open PDF.
• Saad, Phil, Stephen H. Shenker, and Douglas Stanford. “JT Gravity as a Matrix Integral.” arXiv:1903.11115 [hep-th], version 4 (2019). Open PDF.