On the geometry of some unitary Riemann surface braid group representations and Laughlin-type wave functions

Generalized theta functions, projectively flat vector bundles and noncommutative tori

“…Note that, in accordance with the Stone-von Neumann theorem, the representation W is not irreducible: indeed, by Riemann-Roch, its multiplicity is precisely θ dim(H) = (q/r) • r = q, also cf. [18,20]. Corollary 3.2.…”

“…It is known that above algebras are actually strongly Morita equivalent, see e.g. the remarks in [20] and [19]. It would be interesting to explicitly compare the two algebra-valued Hermitian structures involved.…”

“…In this subsection we outline Matsushima's theory [11], tailoring the exposition to our purposes and referring to [10], especially Ch.IV-7 and to [20], Section 3.2, for background material. Here we just recall that an irreducible holomorphic vector bundle (i.e.…”

“…In this Section we reinterpret the Matsushima construction of holomorphic HEvector bundles over a two-dimensional torus C/Λ -with lattice Λ = 1, τ and Im τ > 0-via representations of the two-dimensional noncommutative torus, see also [20]. This will be unfolded through the following series of propositions.…”

“…In this paper we address the well-known intriguing and multifaceted relationship between theta functions and representations of Heisenberg groups (both infinite and finite [11,13,15]), from a blended complex differential geometric and noncommutative geometric viewpoint, possibly bringing in novel insights and, in particular, improving the treatment given in [20]. Specifically, in Section 3, via a series of Propositions, we prove the existence of a representation of the noncommutative torus A 1/θ (θ = q/r, q and r being coprime positive integers) on the space of sections Γ(E r,q ) of a projectively flat Hermitian-Einstein holomorphic vector bundle (or HE -vector bundle for short) E r,q of rank r and degree q on a two-dimensional torus.…”