---
title: Cap Amplitude in JT Gravity and Matrix Models
url: https://www.emergentmind.com/topics/cap-amplitude
type: topic
---

# Cap Amplitude in JT Gravity and Matrix Models

“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 [2604.10977]. 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 [2509.03930]. 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 [1908.09166, 2305.00125].

## 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 [1908.09166, 2305.00125]. 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 [1806.08302, 1904.07971].

## 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, $F_\beta(A_N)$ is a branch-resolved spectral functional, and $\mid\Omega_{\rm geom}\rangle$ is the trumpet or geometric state [2604.10977].

The defining cap amplitude is the overlap with normalized momentum eigenstates $\mid k,N\rangle$,
\[
\boxed{\langle \mathrm{cap}\mid k,N\rangle = k\sinh(2\pi k).}
\]
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 [2604.10977].

The same work emphasizes that finite cutoff induces a compact-support, double-branch spectral structure. The momentum band is restricted to $k\leq R=\phi_r/\varepsilon$, and the two energy branches satisfy
\[
k^2 = 2\phi_r E-\varepsilon^2E^2,\qquad
E_\pm(k)=\frac{\phi_r\pm\sqrt{\phi_r^2-\varepsilon^2k^2}}{\varepsilon^2}.
\]
The branch-resolved functional acts as
\[
F_\beta(k)=\chi_{[0,R]}(k)\left(e^{-\beta E_-(k)}-e^{-\beta E_+(k)}\right),
\]
so the cap amplitude enters the full disk amplitude with an explicit branch difference [2604.10977].

## 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,
\[
\langle b\mid k,N\rangle = f_k(b)=\sqrt{\frac{2}{\pi}}\cos(bk),
\]
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 $b$-basis. Smooth capping corresponds to shrinking the geodesic boundary to the disk center, encoded formally by a cap insertion at the imaginary length $b=2\pi i$ [2604.10977].

Within the analytic jet class, the cap insertion is realized as a functional
\[
\langle \mathrm{cap}\mid f\rangle=\sum_{n=0}^\infty c_n f^{(n)}(2\pi i),
\]
and the cited work states that the only non-vanishing possibility yielding the target overlap is
\[
\boxed{\langle \mathrm{cap}\mid f\rangle = i\sqrt{\frac{\pi}{2}}\,f'(2\pi i).}
\]
Acting on the cosine kernel, this reproduces $k\sinh(2\pi k)$ exactly [2604.10977]. 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,
\[
F_\beta(A_N)=\frac{1}{2\pi i}\oint_\Gamma e^{-\beta(\mathcal{E}_0-\zeta)}\,d\log\!\left(\frac{\zeta-B}{\zeta+B}\right),
\]
with the two poles at $\zeta=\pm B$ generating the two energy branches and the minus sign in the amplitude arising from the residue difference between the two Riemann sheets [2604.10977].

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 $\beta$-independent Hamiltonian. The detailed argument further states that local Friedel densities grow at most as $\log k$, whereas the required cap weight behaves as $k\sinh(2\pi k)\sim e^{2\pi k}$ at large $k$, so the needed measure cannot be recovered from local open-channel dynamics alone [2604.10977]. This directly rebuts the misconception that the cap amplitude is merely a spectral-density effect.

## 4. Large-$N$ one-matrix models: cap amplitude as a spectral-curve coefficient

In large-$N$ one-matrix models, the cap amplitude is denoted $\psi(b)$ and defined as the expansion coefficient of the 1-form $y\,dx$ on the spectral curve:
\[
y\,dx = -\frac{dz}{2z}\sum_{b=0}^{\infty}\psi(b)\big(z^b+z^{-b}\big).
\]
Here $z$ is the uniformization or Joukowsky parameter, and $b\in\mathbb{Z}_+$ is interpreted as the discrete length of a boundary [2509.03930].

The normalization is fixed by the residue condition $\operatorname{Res}_{z=0} y\,dx=-1$, which implies
\[
\psi(0)=1.
\]
The same source also gives the orthogonality relations
\[
\sum_{b=0}^\infty \psi(b)=0,\qquad
\sum_{b=0}^\infty \psi(b)(-1)^b=0.
\]
These formulas place the cap amplitude among the primary spectral data of the model, because the coefficients $\psi(b)$ encode the Laurent expansion of the spectral-curve differential itself [2509.03930].

The paper further states that the moments $M_k$, $J_k$, and thus the higher discrete volumes $N_{g,n}$, are determined from $\psi(b)$. In particular,
\[
M_k^{\,k+1}=-\sum_{b=0}^\infty \psi(b)\,c_k(b),
\]
with $c_k(b)$ universal polynomials specified there [2509.03930]. 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 $\psi(b)$ is made explicit through the discrete-volume dilaton equation,
\[
\sum_{b=0}^\infty \psi(b)\,N_{g,n+1}(b,b_1,\dots,b_n)
=
(2-2g-n)\,N_{g,n}(b_1,\dots,b_n).
\]
The left-hand side sums over all ways of gluing a cap of boundary length $b$ to one boundary of an $(n+1)$-boundary surface, while the right-hand side yields the genus-$g$, $n$-boundary amplitude multiplied by the Euler characteristic factor $2-2g-n$ [2509.03930].

For the one-boundary sector, the genus-$g$ free energy is obtained by capping the remaining boundary:
\[
F_g=\frac{1}{2-2g}\sum_{b=0}^\infty \psi(b)\,N_{g,1}(b),\qquad g\geq 2.
\]
This identifies the cap amplitude as the weight for the cobordism from $S^1$ to the empty set in the discrete volume formalism [2509.03930].

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 $y\,dx$ 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 $\sim R^{-\alpha}$ and thickness $\sim R^{-1}$ enter small-cap decoupling estimates, and the exponent $\alpha$ is described in one summary as the “amplitude” of the cap [1908.09166]. The associated inequalities take forms such as
\[
\operatorname{Dec}(\mathcal{T}_\alpha(R^{-1}),p,p)\lesssim_\epsilon R^{\alpha(\frac12-\frac1p)+\epsilon},
\]
while later logarithmic refinements control the subpolynomial loss by a power of $\log R$ through an amplitude-dependent wave envelope estimate [2305.00125]. 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 [1806.08302, 1904.07971]. 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 $J(\mathbf{r})$ over an aperture and their optimization for channel capacity and beamforming [2412.00894, 2605.12910]. 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 [2604.10977, 2509.03930].

Source: https://www.emergentmind.com/topics/cap-amplitude