arXiv subjects

Esteban Isasi

Publications and source records attributed to Esteban Isasi.

5 recordsLinked to original sources

arXiv2018

The $SU(3)$ $A_1$ graph and its associated quantum groupoid

An explicit and complete construction of the $SU(3)$ $A_1$ associated quantum groupoid is presented in this work, inspired by the approach taken by Trinchero for the $SU(2)$ $A_l$ graphs. New creation and annihilation operators were defined in order to consider the $3$ different types of back-tracks which appear due to the specific structure of $SU(3)$. The $C^{\star}$ bialgebra and the realization of a Temperley-Lieb algebra is studied thoroughly. Finally, it is shown that the construction of the quantum groupoids associated to the $A_{1}$ $SU(N)$ graphs are easily obtained for any value of $N$ using the results of this work. The generalization for higher levels $A_l$ graphs are still an unsolved challenge, but now we count with enough tools, some insight about how to attack this problem, and the first steps towards solving it.

math-ph↗

arXiv2015

Alternative formulation for the operator algebra over the space of paths in a ADE $SU(3)$ graph

In this work we discuss the elements required for the construction of the operator algebra for the space of paths over a simply laced $SU(3)$ graph. These operators are an important step in the construction of the bialgebra required to find the partition functions of some modular invariant CFTs. We define the cup and cap operators associated with back-and-forth sequences and add them to the creation and annihilation operators in the operator algebra as they are required for the calculation of the full space of essential paths prescribed by the fusion algebra. These operators require collapsed triangular cells that had not been found in previous works; here we provide explicit values for these cells and show their importance in order for the cell system to fulfill the Kuperberg relations for $SU(3)$ tangles. We also find that demanding that our operators satisfy the Temperley-Lieb algebra leads one naturally to consider operators that create and annihilate closed triangular sequences, which in turn provides an alternative the cup and cap operators as they allow one to replace back-and-forth sequences with closed triangular ones. We finally show that the essential paths obtained by using closed triangles are equivalent to those obtained originally using back-and-forth sequences.

math-ph↗

arXiv2015

Entanglement sudden-death time: a geometric quantity

We study the entanglement evolution of the set of Bell diagonal states for a two-qubit system coupled to two independent vacuum noise sources. This set can be represented geometrically as the set of points inside a tetrahedron in a three-dimensional Euclidean space and contains the maximally entangled states for bipartite systems. We show that the set of entangled Bell diagonal states can be divided into two bounded subsets in this representation: states that evolve into separable states in a finite time and states that lose their entanglement asymptotically. Additionally, we find that the finite time in which the Bell diagonal states lose their entanglement depends only on the distances from their position in the three-dimensional representation to the boundaries of both, the set of separable states and the set of states that remains always entangled.

quant-ph↗

arXiv2010

Notes on TQFT Wire Models and Coherence Equations for SU(3) Triangular Cells

After a summary of the TQFT wire model formalism we bridge the gap from Kuperberg equations for SU(3) spiders to Ocneanu coherence equations for systems of triangular cells on fusion graphs that describe modules associated with the fusion category of SU(3) at level k. We show how to solve these equations in a number of examples.

math.QA↗

arXiv2005

On quantum symmetries of the non--ADE graph F4

We describe quantum symmetries associated with the F4 Dynkin diagram. Our study stems from an analysis of the (Ocneanu) modular splitting equation applied to a partition function which is invariant under a particular congruence subgroup of the modular group.

hep-th↗