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Diffeomorphism Invariant Integrable Field Theories and Hypersurface Motions in Riemannian Manifolds
High Energy Physics - Theory · Submitted 1 Dec 1995
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In the Hamiltonian formulation we discover infinitely many hierarchies of integrable, multidimensional, $N$-component theories possessing infinitely many diffeomorphism invariant, Poisson commuting, conserved charges.
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The abstract, sentence by sentence
We discuss hypersurface motions in Riemannian manifolds whose normal velocity is a function of the induced hypersurface volume element and derive a second order partial differential equation for the corresponding time function $\tau(x)$ at which the hypersurface passes the point $x$. Equivalently, these motions may be described in a Hamiltonian formulation as the singlet sector of certain diffeomorphism invariant field theories. At least in some (infinite class of) cases, which could be viewed as a large-volume limit of Euclidean $M$-branesmoving in an arbitrary $M+1$-dimensional Riemannian manifold, the models are integrable: In the time-function formulation the equation becomes linear (with $\tau(x)$ a harmonic function on the embedding Riemannian manifold). We explicitly compute solutions to the large volume limit of Euclidean membrane dynamics in $\Real^3$ by methods used in electrostatics and point out an additional gradient flow structure in $\Real^n$. In the Hamiltonian formulation we discover infinitely many hierarchies of integrable, multidimensional, $N$-component theories possessing infinitely many diffeomorphism invariant, Poisson commuting, conserved charges.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We discuss hypersurface motions in Riemannian manifolds whose normal velocity is a function of the induced hypersurface volume element and derive a second order partial differential equation for the corresponding time function $\tau(x)$ at which the hypersurface passes the point $x$. | contribution 0.87 | 0.93 | 0.16 | asserted 1.00 |
| s1 | Equivalently, these motions may be described in a Hamiltonian formulation as the singlet sector of certain diffeomorphism invariant field theories. | interpretation 0.62 | 0.61 | 0.13 | hedged possibility 0.96 |
| s2 | At least in some (infinite class of) cases, which could be viewed as a large-volume limit of Euclidean $M$-branesmoving in an arbitrary $M+1$-dimensional Riemannian manifold, the models are integrable: In the time-function formulation the equation becomes linear (with $\tau(x)$ a harmonic function on the embedding Riemannian manifold). | result 0.89 | 0.69 | 0.34 | hedged possibility 0.81 |
| s3 | We explicitly compute solutions to the large volume limit of Euclidean membrane dynamics in $\Real^3$ by methods used in electrostatics and point out an additional gradient flow structure in $\Real^n$. | result 0.54 | 0.95 | 0.10 | asserted 1.00 |
| s4 | In the Hamiltonian formulation we discover infinitely many hierarchies of integrable, multidimensional, $N$-component theories possessing infinitely many diffeomorphism invariant, Poisson commuting, conserved charges. | result 0.95 | 0.94 | 0.43 | asserted 1.00 |
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Original abstract
We discuss hypersurface motions in Riemannian manifolds whose normal velocity is a function of the induced hypersurface volume element and derive a second order partial differential equation for the corresponding time function $\tau(x)$ at which the hypersurface passes the point $x$. Equivalently, these motions may be described in a Hamiltonian formulation as the singlet sector of certain diffeomorphism invariant field theories. At least in some (infinite class of) cases, which could be viewed as a large-volume limit of Euclidean $M$-branesmoving in an arbitrary $M+1$-dimensional Riemannian manifold, the models are integrable: In the time-function formulation the equation becomes linear (with $\tau(x)$ a harmonic function on the embedding Riemannian manifold). We explicitly compute solutions to the large volume limit of Euclidean membrane dynamics in $\Real^3$ by methods used in electrostatics and point out an additional gradient flow structure in $\Real^n$. In the Hamiltonian formulation we discover infinitely many hierarchies of integrable, multidimensional, $N$-component theories possessing infinitely many diffeomorphism invariant, Poisson commuting, conserved charges.
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