arXiv subjects

Maximiliano Sandoval

Publications and source records attributed to Maximiliano Sandoval.

3 recordsLinked to original sources

arXiv2025

Generalized theta functions, projectively flat vector bundles and noncommutative tori

In this paper, the well-known relationship between theta functions and Heisenberg group actions thereon is resumed by combining complex algebraic and noncommutative geometric techniques in that we describe Hermitian-Einstein vector bundles on 2-tori via representations of noncommutative tori, thereby reconstructing Matsushima's setup and elucidating the ensuing Fourier-Mukai-Nahm (FMN) aspects. We prove the existence of noncommutative torus actions on the space of smooth sections of Hermitian-Einstein vector bundles on 2-tori preserving the eigenspaces of a natural Laplace operator. Motivated by the Coherent State Transform approach to theta functions, we extend the latter to vector valued thetas and develop an additional algebraic reinterpretation of Matsushima's theory making FMN-duality manifest again.

math.QA↗

arXiv2021

The noncommutative geometry of the Landau Hamiltonian: Differential aspects

In this work we study the differential aspects of the noncommutative geometry for the magnetic $C^*$-algebra which is a 2-cocycle deformation of the group $C^*$-algebra of $\mathbb{R}^2$. This algebra is intimately related to the study of the Quantum Hall Effect in the continuous, and our results aim to provide a new geometric interpretation of the related Kubo's formula. Taking inspiration from the ideas developed by Bellissard during the 80's, we build an appropriate Fredholm module for the magnetic $C^*$-algebra based on the magnetic Dirac operator which is the square root (à la Dirac) of the quantum harmonic oscillator. Our main result consist of establishing an important piece of Bellissard's theory, the so-called second Connes' formula. In order to do so, we establish the equality of three cyclic 2-cocycles defined on a dense subalgebra of the magnetic $C^*$-algebra. Two of these 2-cocycles are new in the literature and are defined by Connes' quantized differential calculus, with the use of the Dixmier trace and the magnetic Dirac operator.

math-ph↗

arXiv2020

The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects

This work provides a first step towards the construction of a noncommutative geometry for the quantum Hall effect in the continuum. Taking inspiration from the ideas developed by Bellissard during the 80's we build a spectral triple for the $C^*$-algebra of continuous magnetic operators based on a Dirac operator with compact resolvent. The metric aspects of this spectral triple are studied, and an important piece of Bellissard's theory (the so-called first Connes' formula) is proved.

math-ph↗