Authors:Pedro Freitas, Petr Siegl Abstract: Reviews in Mathematical Physics, Ahead of Print. We study the spectral properties of the two-dimensional Dirac operator on bounded domains together with the appropriate boundary conditions which provide a (continuous) model for graphene nanoribbons. These are of two types, namely, the so-called armchair and zigzag boundary conditions, depending on the line along which the material was cut. In the former case, we show that the spectrum behaves in what might be called a classical way; while in the latter, we prove the existence of a sequence of finite multiplicity eigenvalues converging to zero and which correspond to edge states. Citation: Reviews in Mathematical Physics PubDate: Wed, 19 Nov 2014 01:59:58 GMT DOI: 10.1142/S0129055X14500184

Authors:Dafeng Zuo et al Abstract: Reviews in Mathematical Physics, Ahead of Print.
In this paper, we are interested in a series of sub-hierarchies of the KP hierarchy introduced by Date et al. in [4], which we call the ℬð’žr-KP hierarchy for r ∈ ℤ≥0. In a unified way, we construct additional symmetries of the ℬð’žr-KP hierarchy and show that all of them form a -algebra. Then, we introduce the constrained ℬð’žr-KP hierarchy, and construct its additional symmetries and show that all of them form a Witt algebra. PubDate: Wed, 19 Nov 2014 01:59:58 GMT

Authors:Francesco D'Andrea et al Abstract: Reviews in Mathematical Physics, Volume 26, Issue 09, October 2014.
We describe a categorical approach to finite noncommutative geometries. Objects in the category are spectral triples, rather than unitary equivalence classes as in other approaches. This enables us to treat fluctuations of the metric and unitary equivalences on the same footing, as representatives of particular morphisms in this category. We then show how a matrix geometry (Moyal plane) emerges as a fluctuation from one point, and discuss some geometric aspects of this space. PubDate: Wed, 15 Oct 2014 07:14:32 GMT

Authors:João P. Nunes Abstract: Reviews in Mathematical Physics, Ahead of Print.
We review some recent results on the problem of the choice of polarization in geometric quantization. Specifically, we describe the general philosophy, developed by the author together with his collaborators, of treating real polarizations as limits of degenerating families of holomorphic polarizations. We first review briefly the general framework of geometric quantization, with a particular focus on the problem of the dependence of quantization on the choice of polarization. The problem of quantization in real polarizations is emphasized. We then describe the relation between quantization in real and Kähler polarizations in some families of symplectic manifolds, that can be explicitly quantized and that constitute an important class of examples: cotangent bundles of Lie groups, abelian varieties and toric varieties. Applications to theta functions and moduli spaces of vector bundles on curves are also reviewed. PubDate: Wed, 24 Sep 2014 02:41:46 GMT

Authors:Lucio S. Cirio et al Abstract: Reviews in Mathematical Physics, Ahead of Print.
We elaborate on the quantization of toric varieties by combining techniques from toric geometry, isospectral deformations and noncommutative geometry in braided monoidal categories, and the construction of instantons thereon by combining methods from noncommutative algebraic geometry and a quantized twistor theory. We classify the real structures on a toric noncommutative deformation of the Klein quadric and use this to derive a new noncommutative four-sphere which is the unique deformation compatible with the noncommutative twistor correspondence. We extend the computation of equivariant instanton partition functions to noncommutative gauge theories with both adjoint and fundamental matter fields, finding agreement with the classical results in all instances. We construct moduli spaces of noncommutative vortices from the moduli of invariant instantons, and derive corresponding equivariant partition functions which also agree with those of the classical limit. PubDate: Wed, 10 Sep 2014 02:52:09 GMT