nLab
supersphere

Contents

Context

Geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

In the context of supergeometry, the supersphere S n|mS^{n \vert m} is the supersubmanifold of the superspace n+1|m\mathbb{R}^{n+1 \vert m} of points at a fixed distance from the origin.

Coset representations

  • The supersphere S 2|2S^{2|2} is the super coset space UOSp(1|2)/U(1)UOSp(1|2)/U(1).

  • The supersphere S r1|2sS^{r-1|2s} is the super coset space OSp(r|2s)/OSp(r1|2s)OSp(r|2s)/OSp(r-1|2s) of orthosymplectic groups (GJS 18).

References

  • Etienne Granet, Jesper Lykke Jacobsen, Hubert Saleur, Spontaneous symmetry breaking in 2D supersphere sigma models and applications to intersecting loop soups, (arXiv:1810.07807)

Last revised on April 24, 2020 at 06:50:17. See the history of this page for a list of all contributions to it.