Static Patch Worldsheet Formulation

• Static Patch Worldsheet Formulation is a method that models 2D de Sitter static patches using a deformed c=1 conformal field theory derived from S-wave reduction of higher-dimensional gravity.
• It employs an SL(2, ℝ) Hamiltonian deformation and stretched horizon boundary conditions to discretize Virasoro modes and establish emergent worldsheet time aligned with static-patch causality.
• The framework unifies de Sitter thermodynamics with near-horizon AdS₂/Schwarzian dynamics, offering a UV-complete perspective on gravitational entropy corrections.

The static patch worldsheet formulation provides an effective approach to capturing the low-energy physics of two-dimensional (2D) de Sitter static patches via a conformal field theory (CFT) framework on a cylinder, deformed by an $\mathrm{SL}(2, \mathbb{R})$ Hamiltonian. This construction emerges naturally via S-wave reduction from higher-dimensional Einstein gravity and offers both a UV-complete description and a connection to near-horizon AdS$_2$/Schwarzian dynamics. Central elements include the presence of a stretched horizon, emergent UV boundary conditions, and the equivalence between worldsheet and static-patch time. The framework also clarifies the relation between de Sitter thermodynamics, Virasoro mode discretization, and AdS$_2$ near-extremal entropy corrections (Das, 3 Dec 2025).

1. Field-Theoretic Structure and Worldsheet Theory

The worldsheet formulation is grounded in a free scalar field $\Phi(\sigma, \tau)$, interpreted as a $c=1$ CFT on the cylinder, with left/right stress tensors $T(z),\,\bar{T}(\bar{z})$. The target space metric is the 2D de Sitter static patch: $$ds^2 = -\left(1-\frac{\rho^2}{r_N^2}\right) d\tau^2 + \frac{d\rho^2}{1-\frac{\rho^2}{r_N^2}},$$ where $\rho = \pm r_N$ mark the cosmological horizons, providing natural boundaries for $\Phi$. The worldsheet action, in the conformal gauge, is

$$S_{\mathrm{ws}} = \frac{1}{4\pi} \int d^2 \sigma \sqrt{-h}h^{ab}\partial_a\Phi\,\partial_b\Phi,$$

with a mode expansion

$$\Phi(\sigma, \tau) = \phi_0 + i\sum_{n\neq 0}\frac{a_n}{n} e^{-in\tau}e^{in\sigma}, \quad [a_n,a_m]=n\delta_{n+m,0}.$$

The canonical Hamiltonian is deformed by $\mathrm{SL}(2, \mathbb{R})$ generators: $$H_c = i\gamma(L_1-L_{-1}) - i\gamma(\bar{L}_1-\bar{L}_{-1}),$$ with $\gamma=1/r_N$, or, equivalently, in strip coordinates $(\tau_E \in [0,\pi],\ \sigma\in\mathbb{R})$,

$$H_o = \frac{\gamma}{\pi} \int_0^\pi d\tau_E \sin \tau_E\, T_{00}(\tau_E).$$

This deformation governs the emergent worldsheet time, encoding the static patch’s causal properties.

2. Dimensional Reduction and Dilaton Gravity Correspondence

Starting from 4D Einstein gravity (with or without cosmological constant $\Lambda$), the metric ansatz

$$ds_4^2 = g_{\mu\nu}(x)dx^\mu dx^\nu + r_N^2\phi(x)d\Omega_2^2$$

enables an S-wave reduction to 2D. After integrating over $S^2$ and applying a Weyl rescaling $g_{\mu\nu}\to\omega^2(\phi)\bar{g}_{\mu\nu}$, local dynamics are governed by

$$I_{\mathrm{eff}} = I_0 + \frac{1}{16\pi G_2}\int d^2x \sqrt{-\bar{g}}\,\left[\frac12 (\nabla\Phi)^2 + \text{(total derivatives)}\right].$$

Here, $\Phi=\phi-\phi_0$ parameterizes dilaton fluctuations around a constant value $\phi_0=1$. The background $\bar{g}_{\mu\nu}$ is fixed to the 2D static patch metric with Ricci curvature $R_2=2/r_N^2$, and the only propagating degree of freedom becomes the massless scalar $\Phi$ on that background.

3. UV Regulation: Stretched Horizon and Boundary Conditions

The worldsheet description introduces “stretched horizons” at $\rho=\pm(r_N-\epsilon)$, implementing a brick-wall regularization. The coordinate transformation $\rho = r_N\tanh(\theta/r_N)$ yields stretched horizon positions at $\theta = \pm \Lambda_c$ with

$$\Lambda_c \equiv r_N \frac12 \ln\left(\frac{2r_N}{\epsilon}\right).$$

Imposing Dirichlet boundary conditions $\Phi(\theta=\pm \Lambda_c, \tau)=0$ ensures vanishing of total-derivative terms and produces a discrete normal-mode spectrum: $$\omega_n = \frac{2\pi n}{r_N \ln(2r_N/\epsilon)},\quad n\in \mathbb{Z}^+.$$ Mode spacing matches the CFT’s Virasoro spectrum under modular quantization. The stretched horizon implements an emergent observer—cutting off UV divergences and yielding an “outside” entropy

$$S_{\text{out}}^\Phi = \frac{1}{24}\ln\left(\frac{2r_N}{\epsilon}\right),$$

for a total generalized entropy $S_{\text{gen}}=\phi_0/(2G_2)+S_{\text{out}}^\Phi$.

4. Deformation and Quantization Parameters

The deformed CFT construction is parameterized by:

• Central charge $c$: $c=1$ for a single free boson, generalizable to $c$ matter fields.
• Deformation parameter $\gamma$: $\gamma=1/r_N$ sets the flow generator $H_c$ and hence the emergent worldsheet time scale.
• UV cutoff $\epsilon$: Introduced through the stretched horizon, yielding modular quantization with

$$\Lambda_c = \frac{1}{\gamma}\ln\left(\frac{2}{\epsilon}\right),\quad c_\mathrm{eff} = \frac{c}{2\pi\gamma}\ln\left(\frac{2}{\epsilon}\right),\quad k_n=\frac{2\pi n\gamma}{\ln(2/\epsilon)}.$$

This parameterization controls spectral gaps, entropy, and the accessible Virasoro tower.

Parameter	Physical Role	Static Patch Meaning
$c$	Central charge of CFT	Number of free bosonic species
$\gamma$	SL(2,$\mathbb{R}$) deformation	Inverse dS radius, $\gamma=1/r_N$
$\epsilon$	UV cutoff (stretched horizon)	Proper distance from cosmological horizon

5. Emergent Time and Bulk–Worldsheet Correspondence

The direction of Hamiltonian flow on the worldsheet cylinder is generated by the deformed $H_c$, defining a Euclidean time $s_E$ via

$$i\gamma(l_1 - l_{-1} - \bar{l}_1 + \bar{l}_{-1}) = \partial_{s_E}.$$

The resulting coordinates $(z,\bar{z})= -\coth[\gamma(\sigma \pm i s_E)]$ give $\tau_E\in[0,\pi]$ as the strip time. Analytic continuation to $s_E\to i s$ produces the Milne wedge metric, and further Wick rotation $s_E\to i\theta+\pi/(2r_N)$, $\sigma\to i\tau$ (with $2\gamma r_N=1$) recovers the conformally flat form of the static patch: $$ds^2 = \frac{-d\tau^2 + d\theta^2}{\cosh^2(\theta/r_N)}.$$ This suggests that the bulk static-patch clock emerges as a stroboscopic worldsheet time variable in the deformed CFT, determined by the modular quantization structure imposed by $H_c$.

6. Connection to Schwarzian Theory and AdS$_2$ Near-Horizon Limits

A corresponding Weyl-rescaled reduction in magnetically charged AdS$_4$ backgrounds approaching AdS$_2\times S^2$ yields a free scalar $\Phi$ on rigid Poincaré–AdS$_2$, coupled to a Schwarzian boundary action: $$I_{\text{boundary}} = \frac{\Phi_r}{8\pi G_2}\int ds\,\mathrm{Sch}\{f(s),s\}.$$ There is a semiclassical duality between:

• JT gravity with $c=1$ CFT and large dilaton boundary cutoff $\Phi_r$
• Worldsheet $c=1$ deformed CFT with UV cutoff $\Lambda\propto\Phi_r$ and Hamiltonian $H_c$

Both frameworks reproduce the same near-extremal entropy correction,

$\Delta S = \frac{\pi^2 c}{3}(T \Phi_r)+\dots,$

whether computed from Schwarzian saddles or CFT partition function on an annulus of width $\sim\Lambda$. This unifies the treatment of de Sitter static-patch thermodynamics and AdS$_2$/near-extremal holography within a single modular CFT architecture (Das, 3 Dec 2025).

7. Interpretation and Broader Significance

The static patch worldsheet formulation equates a free-boson CFT on the cylinder—deformed by an $\mathrm{SL}(2, \mathbb{R})$ generator—to the S-wave reduced 2D de Sitter static-patch dilaton gravity at low energies. The stretched horizon with Dirichlet boundary conditions enforces an emergent observer, discretizes the Virasoro spectra, and models the Gibbons–Hawking entropy. The modular quantization induced by the deformed Hamiltonian provides a physically meaningful notion of worldsheet time aligned with bulk causal structure. A plausible implication is that this CFT/worldsheet prescription delivers a unified UV-complete template for gravitational entropy in both cosmological (de Sitter) and near-extremal black hole (AdS$_2$) contexts, aligning semiclassical and CFT accounts of low-energy quantum gravity phenomena (Das, 3 Dec 2025).