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Register a new userHyperinstantons, the Beltrami Equation, and Triholomorphic Maps
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We consider the Beltrami equation for hydrodynamics and we show that its solutions can be viewed as instanton solutions of a more general system of equations. The latter are the equations of motion for an ${cal N}=2$ sigma model on 4-dimensional worldvolume (which is taken locally HyperKahler) with a 4-dimensional HyperKahler target space. By means of the 4D twisting procedure originally introduced by Witten for gauge theories and later generalized to 4D sigma-models by Anselmi and Fre, we show that the equations of motion describe triholomophic maps between the worldvolume and the target space. Therefore, the classification of the solutions to the 3-dimensional Beltrami equation can be performed by counting the triholomorphic maps. The counting is easily obtained by using several discrete symmetries. Finally, the similarity with holomorphic maps for ${cal N}=2$ sigma on Calabi-Yau space prompts us to reformulate the problem of the enumeration of triholomorphic maps in terms of a topological sigma model.
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An effective algorithm is presented for solving the Beltrami equation fzbar = mu fz in a planar disk. The algorithm involves no evaluation of singular integrals. The strategy, working in concentric rings, is to construct a piecewise linear mu-conformal mapping and then correct the image using a known algorithm for conformal mappings. Numerical examples are provided and the computational complexity is analyzed.
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Liviu Tu{a}taru
, Ion V.Vancea (Department of Theoretical Physics
, n University of Cluj-Napoca
1996
A method for the calculation of the BRST cohomology, recently developed for 2D gravity theory and the bosonic string in the Beltrami parametrization,is generalised to the superstring theories quantized in super-Beltrami parametrization.
We study the 1-form diffeomorphism cohomologies within a local conformal Lagrangian Field Theory model built on a two dimensional Riemann surface with no boundary. We consider the case of scalar matter fields and the complex structure is parametrized by Beltrami differential. The analysis is first performed at the Classical level, and then we improve the quantum extension, introducing the current in the Lagrangian dynamics, coupled to external source fields. We show that the anomalies which spoil the current conservations take origin from the holomorphy region of the external fields, and only the differential spin 1 and 2 currents (as well their c.c) could be anomalous.
In this paper we study some properties of the newly found Arnold-Beltrami flux-brane solutions to the minimal $D=7$ supergravity. To this end we first single out the appropriate Free Differential Algebra containing both a gauge $3$-form $mathbf{B}^{[3]}$ and a gauge $2$-form $mathbf{B}^{[2]}$: then we present the complete rheonomic parametrization of all the generalized curvatures. This allows us to identify two-brane configurations with Arnold-Beltrami fluxes in the transverse space with exact solutions of supergravity and to analyze the Killing spinor equation in their background. We find that there is no preserved supersymmetry if there are no additional translational Killing vectors. Guided by this principle we explicitly construct Arnold-Beltrami flux two-branes that preserve $0$, $1/8$ and $1/4$ of the original supersymmetry. Two-branes without fluxes are instead BPS states and preserve $1/2$ supersymmetry. For each two-brane solution we carefully study its discrete symmetry that is always given by some appropriate crystallographic group $Gamma$. Such symmetry groups $Gamma$ are transmitted to the $D=3$ gauge theories on the brane world--volume that occur in the gauge/gravity correspondence. Furthermore we illustrate the intriguing relation between gauge fluxes in two-brane solutions and hyperinstantons in $D=4$ topological sigma-models.
A relativistic wave equation for spin 1/2 particles in the Melvin space-time, a space-time where the metric is determined by a magnectic field, is obtained. The effects of very intense magnetic fields in the energy levels, as intense as the ones expected to be produced in ultra-relativistic heavy-ion collisions, are investigated.
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