| 1. |
- Arnlind, Joakim, et al.
(author)
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Fuzzy Riemann surfaces
- 2009
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In: Journal of High Energy Physics (JHEP). - : Springer Science and Business Media LLC. - 1126-6708 .- 1029-8479. ; :6, s. 047-
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Journal article (peer-reviewed)abstract
- We introduce C-Algebras (quantum analogues of compact Riemann surfaces), defined by polynomial relations in non-commutative variables and containing a real parameter that, when taken to zero, provides a classical non-linear, Poisson-bracket, obtainable from a single polynomial C (onstraint) function. For a continuous class of quartic constraints, we explicitly work out finite dimensional representations of the corresponding C-Algebras.
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| 2. |
- Arnlind, Joakim, et al.
(author)
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Goldfish geodesics and hamiltonian reduction of matrix dynamics
- 2008
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In: Letters in Mathematical Physics. - : Springer Science and Business Media LLC. - 0377-9017 .- 1573-0530. ; 84:1, s. 89-98
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Journal article (peer-reviewed)abstract
- We describe the Hamiltonian reduction of a time-dependent real-symmetric NxN matrix system to free vector dynamics, and also provide a geodesic interpretation of Ruijsenaars-Schneider systems. The simplest of the latter, the goldfish equation, is found to represent a flat-space geodesic in curvilinear coordinates.
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| 3. |
- Arnlind, Joakim, 1979-
(author)
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Graph Techniques for Matrix Equations and Eigenvalue Dynamics
- 2008
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Doctoral thesis (other academic/artistic)abstract
- One way to construct noncommutative analogues of a Riemannian manifold Σ is to make use of the Toeplitz quantization procedure. In Paper III and IV, we construct C-algebras for a continuously deformable class of spheres and tori, and by introducing the directed graph of a representation, we can completely characterize the representation theory of these algebras in terms of the corresponding graphs. It turns out that the irreducible representations are indexed by the periodic orbits and N-strings of an iterated map s:(reals) 2→(reals)2 associated to the algebra. As our construction allows for transitions between spheres and tori (passing through a singular surface), one easily sees how the structure of the matrices changes as the topology changes. In Paper II, noncommutative analogues of minimal surface and membrane equations are constructed and new solutions are presented -- some of which correspond to minimal tori embedded in S7. Paper I is concerned with the problem of finding differential equations for the eigenvalues of a symmetric N × N matrix satisfying Xdd=0. Namely, by finding N(N-1)/2 suitable conserved quantities, the time-evolution of X (with arbitrary initial conditions), is reduced to non-linear equations involving only the eigenvalues of Χ.
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| 4. |
- Arnlind, Joakim, et al.
(author)
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Noncommutative Riemann Surfaces by Embeddings in R-3
- 2009
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In: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 0010-3616 .- 1432-0916. ; 288:2, s. 403-429
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Journal article (peer-reviewed)abstract
- We introduce C-Algebras of compact Riemann surfaces ∑ as non-commutative analogues of the Poisson algebra of smooth functions on ∑. Representations of these algebrasgive rise to sequences of matrix-algebras for which matrix-commutators converge to Poisson-brackets as N → ∞. For a particular class of surfaces, interpolating between spheres and tori, we completely characterize (even for the intermediate singular surface) all finite dimensional representations of the corresponding C-algebras.
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