63 research outputs found
Interactive 4-D Visualization of Stereographic Images From the Double Orthogonal Projection
The double orthogonal projection of the 4-space onto two mutually
perpendicular 3-spaces is a method of visualization of four-dimensional objects
in a three-dimensional space. We present an interactive animation of the
stereographic projection of a hyperspherical hexahedron on a 3-sphere embedded
in the 4-space. Described are synthetic constructions of stereographic images
of a point, hyperspherical tetrahedron, and 2-sphere on a 3-sphere from their
double orthogonal projections. Consequently, the double-orthogonal projection
of a freehand curve on a 3-sphere is created inversely from its stereographic
image. Furthermore, we show an application to a synthetic construction of a
spherical inversion and visualizations of double orthogonal projections and
stereographic images of Hopf tori on a 3-sphere generated from Clelia curves on
a 2-sphere.Comment: ICGG 2020 submissio
Visualism and technificationâthe patient behind the screen
At stake in this study is the patient's credibility. The Cartesian philosophical standpoint, which holds sway in western thinking, questions with scepticism whether the reported symptoms are âreal.â Do they reside in the body, or are they mentally concocted. However, from the caring perspective any symptom must be both listened and attended to in its own right, not just scrutinized as evidence for an accurate diagnosis
Wall-Crossing in Coupled 2d-4d Systems
We introduce a new wall-crossing formula which combines and generalizes the
Cecotti-Vafa and Kontsevich-Soibelman formulas for supersymmetric 2d and 4d
systems respectively. This 2d-4d wall-crossing formula governs the
wall-crossing of BPS states in an N=2 supersymmetric 4d gauge theory coupled to
a supersymmetric surface defect. When the theory and defect are compactified on
a circle, we get a 3d theory with a supersymmetric line operator, corresponding
to a hyperholomorphic connection on a vector bundle over a hyperkahler space.
The 2d-4d wall-crossing formula can be interpreted as a smoothness condition
for this hyperholomorphic connection. We explain how the 2d-4d BPS spectrum can
be determined for 4d theories of class S, that is, for those theories obtained
by compactifying the six-dimensional (0,2) theory with a partial topological
twist on a punctured Riemann surface C. For such theories there are canonical
surface defects. We illustrate with several examples in the case of A_1
theories of class S. Finally, we indicate how our results can be used to
produce solutions to the A_1 Hitchin equations on the Riemann surface C.Comment: 170 pages, 45 figure
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