Non-compact Hopf maps and fuzzy ultra-hyperboloids

Abstract

Fuzzy hyperboloids naturally emerge in the geometries of branes, twistor theory, and higher spin theories. In this work, we perform a systematic study of higher dimensional fuzzy hyperboloids (ultra-hyperboloids) based on non-compact Hopf maps. Two types of non-compact Hopf maps; split-type and hybrid-type, are introduced from the cousins of division algebras.We construct arbitrary even-dimensional fuzzy ultra-hyperboloids by applying the Schwinger operator formalism and indefinite Clifford algebras. It is shown that fuzzy hyperboloids, H 2p,2q F , are represented by the coset, H 2p,2q F ≃ SO(2p, 2q + 1)/U (p, q), and exhibit two types of generalized dimensional hierarchy; hyperbolic-type (for q = 0) and hybrid-type (for q = 0). Fuzzy hyperboloids can be expressed as fibre-bundle of fuzzy fibre over hyperbolic basemanifold. Such bundle structure of fuzzy hyperboloid gives rise to non-compact monopole gauge field. Physical realization of fuzzy hyperboloids is argued in the context of lowest Landau level physics.

Cosmological space-times with resolved Big Bang in Yang-Mills matrix models

“…It is natural to wonder about de Sitter branes. There are indeed candidates for fuzzy de Sitter space based on the principal series representations of SO(4, 1) [12,32,[39][40][41][42], some of which should be solutions of the matrix model for m 2 0 = m 2 ; see also [43] for related work. However then the effective metric would coincide with the induced one, without BB.…”

“…We focus on the fuzzy hyperboloid H 4 n as discussed in [32,33]. Analogous to S 4 N , this arises from certain irreducible representations of the noncompact cousin SO(1, 4) of SO (5), and again there are magic representations where this structure group is enhanced to SO(2, 4).…”

“…with R 2 = r 2 (n 2 − 4) [32]. Since X 0 = rM 05 > 0 has positive spectrum, this describes a one-sided hyperboloid in R 1,4 , denoted as H 4 n .…”

“…Since X 0 = rM 05 > 0 has positive spectrum, this describes a one-sided hyperboloid in R 1,4 , denoted as H 4 n . Analogous to fuzzy S 4 N , the semi-classical geometry underlying H 4 n is CP 1,2 [32], which is an S 2 − bundle over H 4 carrying a canonical symplectic structure. In the fuzzy case, this fiber is a fuzzy 2-sphere S 2 n .…”

2D fuzzy anti-de Sitter space from matrix models

“…Fuzzy (A)dS 2 has been studied previously in [11][12][13]; see also [14,15] for related work on fuzzy noncompact coadjoint orbits, and [16] for a different approach to fuzzyness on dS 4 . In the first part of this paper, we elaborate the space of functions on these fuzzy hyperboloids, and provide explicit formulae for the squareintegrable wavefunctions corresponding to unitary irreducible representations of SO (2,1).…”