Cap Amplitude in JT Gravity and Matrix Models

• Cap amplitude is a polysemous term in mathematical physics that describes boundary capping in JT gravity, spectral coefficients in matrix models, and localized amplitude estimates in harmonic analysis.
• In JT gravity, it represents the boundary-state overlap that smooths the trumpet geometry into a disk, computed as k sinh(2πk) and enforcing a compact spectral structure.
• In matrix models, it appears as spectral-curve coefficients driving topological recursion and moduli-space volumes, while in harmonic analysis it quantifies amplitude scaling in localized frequency caps.

“Cap amplitude” is a polysemous technical term whose most precise current uses occur in mathematical physics. In finite-cutoff Jackiw–Teitelboim (JT) gravity, it denotes the boundary-state overlap that implements smooth capping of the trumpet geometry and reproduces the finite-cutoff disk amplitude in an open-channel operator formulation (Zhou, 13 Apr 2026). In large-$N$ one-matrix models, the cap amplitude $\psi(b)$ is the expansion coefficient of the spectral-curve 1-form $y\,dx$ and acts as the weight for capping a boundary in the discrete-volume formulation of moduli spaces (Okuyama, 4 Sep 2025). In harmonic analysis, by contrast, “cap amplitude” is used more loosely for the scale parameter or amplitude-dependent structure associated with small caps in decoupling estimates, rather than for a boundary-closing amplitude in the topological sense (Demeter et al., 2019, Johnsrude, 2023).

1. Terminological scope

The term appears in several technically distinct literatures.

Context	Object called “cap amplitude”	Defining relation
Finite-cutoff JT gravity	Cap overlap / cap insertion functional	$$\langle \mathrm{cap}\mid k,N\rangle = k\sinh(2\pi k)$$
Large-$N$ one-matrix models	Coefficients of $y\,dx$ on the spectral curve	$$y\,dx = -\frac{dz}{2z}\sum_{b=0}^{\infty}\psi(b)(z^b+z^{-b})$$
Small-cap decoupling	Cap scale or amplitude-dependent cap analysis	caps of diameter $\sim R^{-\alpha}$ or amplitude-dependent wave envelope estimates

In the JT-gravity and matrix-model settings, the word “cap” refers to an operation that closes a boundary component. In the decoupling literature, a “cap” is instead a localized frequency patch on a curved manifold such as the parabola or cone, and the associated “amplitude” refers to size parameters or amplitude-dependent estimates (Demeter et al., 2019, Johnsrude, 2023). This distinction is essential, because the same phrase does not identify a single cross-disciplinary invariant.

A separate source of confusion is the acronym CAP in optical and wireless communications, where it stands for carrier-less amplitude and phase rather than “cap amplitude.” Non-orthogonal multi-band CAP and staggered CAP are modulation schemes for visible-light communication and are unrelated to the boundary-gluing notion of cap amplitude (Haigh et al., 2018, Haigh et al., 2019).

2. Finite-cutoff JT gravity: cap amplitude as an open-channel boundary overlap

In finite-cutoff JT gravity, the disk amplitude is written as a boundary-state matrix element

$$Z_\varepsilon(\beta)=\langle \mathrm{cap}\mid F_\beta(A_N)\mid \Omega_{\rm geom}\rangle,$$

where $A_N$ is the open-channel Neumann auxiliary Hamiltonian, $\psi(b)$0 is a branch-resolved spectral functional, and $\psi(b)$1 is the trumpet or geometric state (Zhou, 13 Apr 2026).

The defining cap amplitude is the overlap with normalized momentum eigenstates $\psi(b)$2,

$\psi(b)$3

The cited work identifies this as the “target cap overlap.” Its significance is twofold. First, it reproduces the known finite-cutoff disk amplitude when inserted into the open-channel formalism. Second, it is interpreted as geometric input imported from disk–trumpet gluing rather than as something generated by the local auxiliary Hamiltonian itself (Zhou, 13 Apr 2026).

The same work emphasizes that finite cutoff induces a compact-support, double-branch spectral structure. The momentum band is restricted to $\psi(b)$4, and the two energy branches satisfy

$\psi(b)$5

The branch-resolved functional acts as

$\psi(b)$6

so the cap amplitude enters the full disk amplitude with an explicit branch difference (Zhou, 13 Apr 2026).

3. Geometric realization in JT gravity: gluing, analytic jet functionals, and no-go statements

The operator formulation is tied to a rigid length–momentum kernel,

$\psi(b)$7

which connects the open-channel length basis to the momentum basis. Disk–trumpet gluing is implemented by a cap state acting on the trumpet amplitude in the $\psi(b)$8-basis. Smooth capping corresponds to shrinking the geodesic boundary to the disk center, encoded formally by a cap insertion at the imaginary length $\psi(b)$9 (Zhou, 13 Apr 2026).

Within the analytic jet class, the cap insertion is realized as a functional

$y\,dx$0

and the cited work states that the only non-vanishing possibility yielding the target overlap is

$y\,dx$1

Acting on the cosine kernel, this reproduces $y\,dx$2 exactly (Zhou, 13 Apr 2026). In this formulation, the cap amplitude is therefore the operational implementation of regularity at the disk center.

The same paper gives a contour or “third-kind” representation for the branch-resolved operator,

$y\,dx$3

with the two poles at $y\,dx$4 generating the two energy branches and the minus sign in the amplitude arising from the residue difference between the two Riemann sheets (Zhou, 13 Apr 2026).

A central negative result is that the compact-support branch-difference amplitude is not the ordinary thermal trace of any single lower-bounded self-adjoint $y\,dx$5-independent Hamiltonian. The detailed argument further states that local Friedel densities grow at most as $y\,dx$6, whereas the required cap weight behaves as $y\,dx$7 at large $y\,dx$8, so the needed measure cannot be recovered from local open-channel dynamics alone (Zhou, 13 Apr 2026). This directly rebuts the misconception that the cap amplitude is merely a spectral-density effect.

4. Large-$y\,dx$9 one-matrix models: cap amplitude as a spectral-curve coefficient

In large-$$\langle \mathrm{cap}\mid k,N\rangle = k\sinh(2\pi k)$$0 one-matrix models, the cap amplitude is denoted $$\langle \mathrm{cap}\mid k,N\rangle = k\sinh(2\pi k)$$1 and defined as the expansion coefficient of the 1-form $$\langle \mathrm{cap}\mid k,N\rangle = k\sinh(2\pi k)$$2 on the spectral curve: $$\langle \mathrm{cap}\mid k,N\rangle = k\sinh(2\pi k)$$3 Here $$\langle \mathrm{cap}\mid k,N\rangle = k\sinh(2\pi k)$$4 is the uniformization or Joukowsky parameter, and $$\langle \mathrm{cap}\mid k,N\rangle = k\sinh(2\pi k)$$5 is interpreted as the discrete length of a boundary (Okuyama, 4 Sep 2025).

The normalization is fixed by the residue condition $$\langle \mathrm{cap}\mid k,N\rangle = k\sinh(2\pi k)$$6, which implies

$$\langle \mathrm{cap}\mid k,N\rangle = k\sinh(2\pi k)$$7

The same source also gives the orthogonality relations

$$\langle \mathrm{cap}\mid k,N\rangle = k\sinh(2\pi k)$$8

These formulas place the cap amplitude among the primary spectral data of the model, because the coefficients $$\langle \mathrm{cap}\mid k,N\rangle = k\sinh(2\pi k)$$9 encode the Laurent expansion of the spectral-curve differential itself (Okuyama, 4 Sep 2025).

The paper further states that the moments $N$0, $N$1, and thus the higher discrete volumes $N$2, are determined from $N$3. In particular,

$N$4

with $N$5 universal polynomials specified there (Okuyama, 4 Sep 2025). In this setting, the cap amplitude is not a momentum-space boundary overlap but a coefficient system that seeds topological recursion and enumerative geometry data.

5. Boundary capping, the dilaton equation, and free energies in matrix models

The geometric role of $N$6 is made explicit through the discrete-volume dilaton equation,

$N$7

The left-hand side sums over all ways of gluing a cap of boundary length $N$8 to one boundary of an $N$9-boundary surface, while the right-hand side yields the genus-$y\,dx$0, $y\,dx$1-boundary amplitude multiplied by the Euler characteristic factor $y\,dx$2 (Okuyama, 4 Sep 2025).

For the one-boundary sector, the genus-$y\,dx$3 free energy is obtained by capping the remaining boundary: $y\,dx$4 This identifies the cap amplitude as the weight for the cobordism from $y\,dx$5 to the empty set in the discrete volume formalism (Okuyama, 4 Sep 2025).

The JT-gravity and matrix-model uses are not identified as the same object in the cited works. Nevertheless, both encode capping as a boundary-reducing operation: in JT gravity through a boundary-state matrix element, and in matrix models through coefficients of $y\,dx$6 that implement the dilaton equation. This suggests a shared structural role for cap amplitudes as gluing data, while leaving their precise relation model-dependent.

6. Other uses, adjacent terminology, and common confusions

In harmonic analysis, the relevant object is a small cap in frequency space rather than a topological cap. For the parabola, caps of diameter $y\,dx$7 and thickness $y\,dx$8 enter small-cap decoupling estimates, and the exponent $y\,dx$9 is described in one summary as the “amplitude” of the cap (Demeter et al., 2019). The associated inequalities take forms such as

$$y\,dx = -\frac{dz}{2z}\sum_{b=0}^{\infty}\psi(b)(z^b+z^{-b})$$0

while later logarithmic refinements control the subpolynomial loss by a power of $$y\,dx = -\frac{dz}{2z}\sum_{b=0}^{\infty}\psi(b)(z^b+z^{-b})$$1 through an amplitude-dependent wave envelope estimate (Johnsrude, 2023). In this literature, “cap amplitude” refers to cap size or high-amplitude localization, not to a capping functional.

A separate but frequent confusion comes from optical communications. Carrier-less amplitude and phase (CAP) is a modulation family for visible-light communication, including multi-band CAP, non-orthogonal multi-band CAP, and staggered CAP (Haigh et al., 2018, Haigh et al., 2019). Here CAP is an acronym; it does not denote a cap insertion, a spectral-curve coefficient, or a moduli-space capping amplitude.

The same caution applies to CAPA, the acronym for continuous-aperture arrays in 6G-oriented wireless theory. CAPA papers study continuous current distributions $$y\,dx = -\frac{dz}{2z}\sum_{b=0}^{\infty}\psi(b)(z^b+z^{-b})$$2 over an aperture and their optimization for channel capacity and beamforming (Liu et al., 2024, Wang et al., 13 May 2026). Although these works discuss amplitude on an aperture, they do not define “cap amplitude” in the gravitational or matrix-model sense.

The main misconception, therefore, is lexical rather than conceptual: identical or near-identical strings can denote wholly different objects. In current arXiv usage, the technically sharp sense of cap amplitude belongs primarily to boundary-capping constructions in JT gravity and random matrix models (Zhou, 13 Apr 2026, Okuyama, 4 Sep 2025).