Dym Boundary Conditions in AdS3 Gravity & PDEs

• Dym boundary conditions are integrable boundary prescriptions in AdS3 gravity and time-dependent PDEs that extend standard Brown–Henneaux conditions with higher-order deformations.
• They yield a hierarchy of evolution equations, such as the Harry Dym equation, generating a spectrum of non-axisymmetric, stationary black hole solutions with well-defined thermodynamic properties.
• In elliptic PDEs, these dynamic boundary conditions ensure well-posedness and asymptotic convergence on non-cylindrical domains by leveraging time-dependent Dirichlet-to-Neumann operators and semigroup methods.

The Dym family of boundary conditions refers to a collection of asymptotic or dynamical boundary prescriptions arising in two major settings: (i) asymptotically anti-de Sitter (AdS$_3$) gravity, where they generalize the Brown–Henneaux boundary conditions through a hierarchy governed by integrable Dym-type equations, and (ii) dynamical (time-dependent) boundary conditions for elliptic PDEs such as the Laplace equation on time-dependent (non-cylindrical) domains. In both contexts, the Dym boundary conditions are intimately tied to integrability, possessing infinitely many conserved charges and, in the gravitational case, yielding a wider spectrum of black hole solutions.

1. Dym Boundary Conditions in AdS$_3$ Gravity

Within three-dimensional AdS gravity, the Dym family of boundary conditions are constructed as $1/c$ deformations (with $c$ the Brown–Henneaux central charge) of the standard Brown–Henneaux (BH) prescription. The gravitational bulk is formulated as a difference of two $\mathfrak{sl}(2,\mathbb{R})$ Chern–Simons actions with connections $A^\pm$. The highest-weight (Drinfeld–Sokolov) gauge is adopted: $$A^\pm = b_\pm^{-1} a^\pm b_\pm + b_\pm^{-1} d b_\pm,\quad b_\pm(\rho) = e^{\pm L_0 \ln(\rho/\ell)},$$ where $a^\pm = a_t^\pm dt + a_\phi^\pm d\phi$ encodes all state dependence on $(t,\phi)$. The BH boundary conditions fix the chemical potentials $\mu^\pm$ to unity and yield Virasoro symmetry.

The Dym boundary conditions are obtained by choosing

$$\mu^\pm = 1 + \sum_{n=1}^N (24\pi / c)^{N-n+1} \frac{\delta H_n^\pm}{\delta \mathcal{L}^\pm},$$

with $H_n^\pm$ the conserved quantities of the Dym integrable hierarchy, and $N$ labels the Dym member (hierarchy order). For $N = 1$, $\mu^\pm = 1 + (12\pi/c)/\sqrt{\mathcal{L}^\pm}$, generating boundary dynamics governed by the Harry Dym equation (Pino et al., 9 Nov 2025, Lara et al., 22 Jan 2024).

2. Dym Integrable Hierarchy and Evolution Equations

Each member of the hierarchy yields distinct evolution equations for the boundary data $\mathcal{L}^\pm$: $$\pm \ell\, \partial_t \mathcal{L}^\pm = 2\mathcal{L}^\pm \partial_\phi \mu^\pm + \mu^\pm \partial_\phi \mathcal{L}^\pm - (c/24\pi)\, \partial_\phi^3 \mu^\pm.$$ For $N=1$ and $\mu = 1 + (12\pi/c)/\sqrt{\mathcal{L}}$, introducing $u = 1/\sqrt{\mathcal{L}}$ recasts the equation as the dynamical (Harry) Dym equation: $u_t = u^3 u_{xxx}.$ Stationary solutions ($\partial_t \mathcal{L} = 0$) generate an ordinary differential equation for $u(\phi)$, whose general solution is a periodic function parametrized by its turning points $u_\pm$: $(u')^2 = -\frac{4A}{u}(u-u_-)(u-u_+).$ The existence of real periodic solutions requires $A>0$, $B>\sqrt{A}$ (with $u_\pm$ dependent on $A$, $B$), and the period quantization $T = 2\pi/m$ for integer $m > 0$. This yields a family of $2\pi$-periodic profiles oscillating between $u_-$ and $u_+$, corresponding to non-axisymmetric boundary stress-tensor configurations (Pino et al., 9 Nov 2025).

3. Black Hole Solutions and Spectral Labeling

The Dym boundary conditions yield a discrete infinity of stationary, generically non-axisymmetric AdS$_3$ black hole solutions. Each sector ($\pm$) is labeled by two continuous parameters ($A^\pm$) and one positive integer ($m^\pm$), along with an arbitrary phase $\varphi_0^\pm$ specifying the oscillation. The parameters have direct geometric interpretation:

• $A^\pm$: controls the mean value of $\mathcal{L}^\pm$ (mass-like charge)
• $m^\pm$: sets the number of "lobes" in the non-axisymmetric profile
• $\Delta\varphi = \varphi_0^+ - \varphi_0^-$: relative orientation of the sectors.

Holonomy constraints are imposed for physical regularity on the Euclidean solid torus: the angular holonomy must be nontrivial, while the Euclidean time holonomy must be trivial up to $-I$. These fix the total boundary charge $H_0^\pm = \int \mathcal{L}^\pm d\phi$ and determine the physical temperature (Pino et al., 9 Nov 2025).

The resulting black hole spacetime is reconstructed as

$$g_{\mu\nu} = \frac{\ell^2}{2} \langle A^+_\mu - A^-_\mu,\, A^+_\nu - A^-_\nu \rangle,$$

yielding metric components $g_{tt}$, $g_{t\phi}$, $g_{\phi\phi}$ etc., depending nontrivially on $\phi$. The horizon, defined by $g(\partial_t,\partial_t) = 0$, generically forms a smooth, single-component, noncircular cross-section curve $\rho_K(\phi)$.

Thermodynamic properties include

• Mass: $$M = (1/\ell)[H_0^+ + H_0^- + (24\pi/c)(H_1^+ + H_1^-)]$$
• Angular momentum: $J = H_0^+ - H_0^-$
• Temperature $T$ derived from the holonomy
• Entropy consistent with the area law and the Chern–Simons free energy or Cardy formula.

4. Asymptotic Symmetry Algebra and Conserved Charges

The asymptotic symmetries of Dym boundary conditions are generated by large gauge transformations preserving $a_\phi^\pm$, parametrized by the basis of Dym conserved charges: $$\eta^\pm(t,\phi) = \epsilon_0^\pm + \sum_{m=1}^M \frac{1}{c^{M-m+1}} \frac{\delta H_m^\pm}{\delta L^\pm},$$ where $H_n^\pm$ are in involution under a bi-Hamiltonian structure: $$\{F,G\}_D = \int d\phi\, \frac{\delta F}{\delta L^\pm} D^\pm \frac{\delta G}{\delta L^\pm},\quad \{F,G\}_E = \int d\phi\, \frac{\delta F}{\delta L^\pm} E \frac{\delta G}{\delta L^\pm},$$ with $D^\pm = 2L^\pm \partial_\phi + \partial_\phi L^\pm$ and $E = \partial_\phi^3$. All $H_n^\pm$ commute with respect to both brackets, yielding an infinite-dimensional abelian algebra (Lara et al., 22 Jan 2024).

This structure generalizes the double Virasoro algebra (BH case, $N=0$) to a Dym hierarchy, underpinning "integrable holography" and suggesting new classes of asymptotic dual field theories.

5. Dym Dynamical Boundary Conditions for Elliptic PDEs

The Dym family also arises in elliptic boundary value problems as dynamic (or dynamical) boundary conditions, addressing well-posedness and evolution for equations such as Laplace’s equation in time-dependent (non-cylindrical) domains.

Given

$$\begin{cases} \Delta_x\,u(t,x)=0,\qquad & (t,x)\in D,\ t>t_0 \ \partial_t u(t,x) = -\partial_n u(t,x) + f(t,x), & (t,x)\in S,\ t>t_0 \ u(t_0,x) = u_0(x), & x\in \Omega_{t_0} \end{cases}$$

where $D = \{(t,x): t>t_0, x\in\Omega_t\}$ is a non-cylindrical space-time domain, $\partial_n u$ is the normal derivative on the moving boundary $S$, and $f$ is a source. The boundary dynamics can be equivalently recast via a time-dependent Dirichlet-to-Neumann operator $A(t)$ mapping $H^{1/2}(\partial\Omega)$ to $H^{-1/2}(\partial\Omega)$: $$\partial_t g(t) + A(t)g(t) = f(t),\quad g(t_0) = g_0,$$ with suitable regularity and sectoriality properties of $A(t)$ (uniform sectorial bounds, Hölder continuity). Existence, uniqueness, and asymptotic convergence to a stationary state are demonstrated via semigroup techniques (Tanabe–Sobolevskii in $H^{-1/2}$, Yagi in $L^2$), with full recovery of the interior solution through variational reconstruction once the boundary data $g(t)$ is known (Lopes et al., 2017).

6. General Properties, Physical Implications, and Future Directions

The Dym family of boundary conditions introduces an integrable deformation perspective to both AdS$_3$ gravity and elliptic PDE boundary problems. In gravity, the family interpolates between standard Virasoro/Brown–Henneaux conditions and higher Dym members, each labeled by the expansion order $N$ in $1/c$. At each step, the boundary dynamics becomes governed by a higher member of the Dym integrable hierarchy, generating ever more intricate asymptotic symmetry algebras and, correspondingly, wider black hole spectra.

Key features include:

• An infinite set of abelian conserved charges $Q_n$ derived from the Dym hierarchy, in contrast to the non-abelian Virasoro algebra.
• Existence of regular, non-axisymmetric, stationary black holes characterized by multiple discrete and continuous parameters, with modified mass-angular momentum relations but standard area law for entropy.
• Bi-Hamiltonian structure of the Dym-generated boundary evolution equations, underpinning integrability.

Potential directions suggested are the holographic implications for AdS$_3$/CFT$_2$ duality, especially the realization of CFTs with abelian symmetry realized via integrable hierarchies, and further exploration of the role of dynamical boundary conditions in parabolic and hyperbolic PDEs on evolving domains. The Dym framework also allows systematic investigation of $1/c$ corrections and their role in nonperturbative gravitational dynamics in lower-dimensional AdS holography.