Ground states of groupoid C*-algebras, phase transitions and arithmetic subalgebras for Hecke algebras

We consider the Hecke pair consisting of the group $P^+_K$ of affine transformations of a number field $K$ that preserve the orientation in every real embedding and the subgroup $P^+_O$ consisting of transformations with algebraic integer coefficients. The associated Hecke algebra $C^*(P^+_K,P^+_O)$ has a natural time evolution $\sigma$, and we describe the corresponding phase transition for KMS$_\beta$-states and for ground states. From work of Yalkinoglu and Neshveyev it is known that a Bost-Connes type system associated to $K$ has an essentially unique arithmetic subalgebra. When we import this subalgebra through the isomorphism of $C^*(P^+_K,P^+_O)$ to a corner in the Bost-Connes system established by Laca, Neshveyev and Trifkovic, we obtain an arithmetic subalgebra of $C^*(P^+_K,P^+_O)$ on which ground states exhibit the `fabulous' property with respect to an action of the Galois group $Gal(K^{ab}/H_+(K))$, where $H_+(K)$ is the narrow Hilbert class field. In order to characterize the ground states of the $C^*$-dynamical system $(C^*(P^+_K,P^+_O),\sigma)$, we obtain first a characterization of the ground states of a groupoid $C^*$-algebra, refining earlier work of Renault. This is independent from number theoretic considerations, and may be of interest by itself in other situations.Comment: 21 pages; v2: minor changes and correction

Complex Multiplication Symmetry of Black Hole Attractors

We show how Moore's observation, in the context of toroidal compactifications in type IIB string theory, concerning the complex multiplication structure of black hole attractor varieties, can be generalized to Calabi-Yau compactifications with finite fundamental groups. This generalization leads to an alternative general framework in terms of motives associated to a Calabi-Yau variety in which it is possible to address the arithmetic nature of the attractor varieties in a universal way via Deligne's period conjecture.Comment: 28 page

Agents, subsystems, and the conservation of information

Dividing the world into subsystems is an important component of the scientific method. The choice of subsystems, however, is not defined a priori. Typically, it is dictated by experimental capabilities, which may be different for different agents. Here we propose a way to define subsystems in general physical theories, including theories beyond quantum and classical mechanics. Our construction associates every agent A with a subsystem SA, equipped with its set of states and its set of transformations. In quantum theory, this construction accommodates the notion of subsystems as factors of a tensor product Hilbert space, as well as the notion of subsystems associated to a subalgebra of operators. Classical systems can be interpreted as subsystems of quantum systems in different ways, by applying our construction to agents who have access to different sets of operations, including multiphase covariant channels and certain sets of free operations arising in the resource theory of quantum coherence. After illustrating the basic definitions, we restrict our attention to closed systems, that is, systems where all physical transformations act invertibly and where all states can be generated from a fixed initial state. For closed systems, we propose a dynamical definition of pure states, and show that all the states of all subsystems admit a canonical purification. This result extends the purification principle to a broader setting, in which coherent superpositions can be interpreted as purifications of incoherent mixtures.Comment: 31+26 pages, updated version with new results, contribution to Special Issue on Quantum Information and Foundations, Entropy, GM D'Ariano and P Perinotti, ed

Complex Multiplication of Exactly Solvable Calabi-Yau Varieties

We propose a conceptual framework that leads to an abstract characterization for the exact solvability of Calabi-Yau varieties in terms of abelian varieties with complex multiplication. The abelian manifolds are derived from the cohomology of the Calabi-Yau manifold, and the conformal field theoretic quantities of the underlying string emerge from the number theoretic structure induced on the varieties by the complex multiplication symmetry. The geometric structure that provides a conceptual interpretation of the relation between geometry and the conformal field theory is discrete, and turns out to be given by the torsion points on the abelian varieties.Comment: 44 page

Universal Spatiotemporal Sampling Sets for Discrete Spatially Invariant Evolution Systems

Let $(I,+)$ be a finite abelian group and $\mathbf{A}$ be a circular convolution operator on $\ell^2(I)$. The problem under consideration is how to construct minimal $\Omega \subset I$ and $l_i$ such that $Y=\{\mathbf{e}_i, \mathbf{A}\mathbf{e}_i, \cdots, \mathbf{A}^{l_i}\mathbf{e}_i: i\in \Omega\}$ is a frame for $\ell^2(I)$, where $\{\mathbf{e}_i: i\in I\}$ is the canonical basis of $\ell^2(I)$. This problem is motivated by the spatiotemporal sampling problem in discrete spatially invariant evolution systems. We will show that the cardinality of $\Omega$ should be at least equal to the largest geometric multiplicity of eigenvalues of $\mathbf{A}$, and we consider the universal spatiotemporal sampling sets $(\Omega, l_i)$ for convolution operators $\mathbf{A}$ with eigenvalues subject to the same largest geometric multiplicity. We will give an algebraic characterization for such sampling sets and show how this problem is linked with sparse signal processing theory and polynomial interpolation theory