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César Galindo

Publications and source records attributed to César Galindo.

At least 37 records · Page 2Linked to original sources
arXiv2024

Braided Zestings of Verlinde Modular Categories and Their Modular Data

Zesting of braided fusion categories is a procedure that can be used to obtain new modular categories from a modular category with non-trivial invertible objects. In this paper, we classify and construct all possible braided zesting data for modular categories associated with quantum groups at roots of unity. We produce closed formulas, based on the root system of the associated Lie algebra, for the modular data of these new modular categories.

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arXiv2024

Abelian Group Quantum Error Correction in Kitaev's Model

In this paper, we present a detailed mathematical description of the error correction process for Kitaev's model for finite Abelian groups. The number of errors Kitaev's model can correct depends on the lattice and its topology. Although there is a theoretical maximum number of errors that can be corrected, we prove that correcting this number of errors, in general, is an NP-complete problem. Consequently, we introduce a polynomial-time correction algorithm that corrects a number of errors below the theoretical maximum.

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arXiv2023

Spherical Morita contexts and relative Serre functors

The Morita context provided by an exact module category over a finite tensor category gives a two-object bicategory with duals. Right and left duals of objects in the module category are given by internal Homs and coHoms, respectively. We express the double duals in terms of relative Serre functors, which leads to a Radford isomorphism for module categories. There is a bicategorical version of the Radford $S^4$ theorem: on the bicategory of a Morita context, the relative Serre functors assemble into a pseudo-functor, and the Radford isomorphisms furnish a trivialization of the square of this pseudo-functor, i.e. of the fourth power of the duals. We also show that the Morita bicategories coming from pivotal exact module categories are pivotal as bicategories, leading to the notion of pivotal Morita equivalence. This equivalence of tensor categories amounts to the equivalence of their bicategories of pivotal module categories. Furthermore, we introduce the notion of a spherical module category; it ensures that all categories in the Morita context of a spherical module category are spherical. Our results are motivated by and have applications to topological field theory.

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arXiv2023

$G$-crossed braided zesting

For a finite group $G$, a $G$-crossed braided fusion category is $G$-graded fusion category with additional structures, namely a $G$-action and a $G$-braiding. We develop the notion of $G$-crossed braided zesting: an explicit method for constructing new $G$-crossed braided fusion categories from a given one by means of cohomological data associated with the invertible objects in the category and grading group $G$. This is achieved by adapting a similar construction for (braided) fusion categories recently described by the authors. All $G$-crossed braided zestings of a given category $\mathcal{C}$ are $G$-extensions of their trivial component and can be interpreted in terms of the homotopy-based description of Etingof, Nikshych and Ostrik. In particular, we explicitly describe which $G$-extensions correspond to $G$-crossed braided zestings.

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arXiv2022

Equivariant fusion subcategories

We provide a parameterization of all fusion subcategories of the equivariantization by a group action on a fusion category. As applications, we classify the Hopf subalgebras of a family of semisimple Hopf algebras of Kac-Paljutkin type and recover Naidu-Nikshych-Witherspoon classification of the fusion subcategories of the representation category of a twisted quantum double of a finite group.

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arXiv2021

Equivariant Morita theory for graded tensor categories

We extend categorical Morita equivalence to finite tensor categories graded by a finite group $G$. We show that two such categories are graded Morita equivalent if and only if their equivariant Drinfeld centers are equivalent as braided $G$-crossed tensor categories.

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arXiv2020

Trivializing group actions on braided crossed tensor categories and graded braided tensor categories

For an abelian group $ A $, we study a close connection between braided crossed $ A $-categories with a trivialization of the $ A $-action and $ A $-graded braided tensor categories. Additionally, we prove that the obstruction to the existence of a trivialization of a categorical group action $T$ on a monoidal category $\mathcal{C}$ is given by an element $O(T)\in H^2(G,\operatorname{Aut}_\otimes(\operatorname{Id}_{\mathcal{C}}))$. In the case that $O(T)=0$, the set of obstructions form a torsor over $\operatorname{Hom}(G,\operatorname{Aut}_\otimes(\operatorname{Id}_{\mathcal{C}}))$, where $\operatorname{Aut}_\otimes(\operatorname{Id}_{\mathcal{C}})$ is the abelian group of tensor natural automorphisms of the identity. The cohomological interpretation of trivializations, together with the homotopical classification of (faithfully graded) braided $A$-crossed tensor categories developed in arXiv:0909.3140, allows us to provide a method for the construction of faithfully $A$-graded braided tensor categories. We work out two examples. First, we compute the obstruction to the existence of trivializations for the braided crossed category associated with a pointed semisimple tensor category. In the second example, we compute explicit formulas for the braided $\mathbb{Z}/2$-crossed structures over Tambara-Yamagami fusion categories and, consequently, a conceptual interpretation of the results in arXiv:math/0011037 about the classification of braidings over Tambara-Yamagami categories.

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arXiv2018

Five-term exact sequence for Kac cohomology

We use relative group cohomologies to compute the Kac cohomology of matched pairs of finite groups. This cohomology naturally appears in the theory of abelian extensions of finite dimensional Hopf algebras. We prove that Kac cohomology can be computed using relative cohomology and relatively projective resolutions. This allows us to use other resolutions, besides the bar resolution, for computations. We compute, in terms of relative cohomology, the first two pages of a spectral sequence which converges to the Kac cohomology and its associated five-term exact sequence. Through several examples, we show the usefulness of the five-term exact sequence in computing groups of abelian extensions.

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arXiv2018

On invariants of Modular categories beyond modular data

We study novel invariants of modular categories that are beyond the modular data, with an eye towards a simple set of complete invariants for modular categories. Our focus is on the $W$-matrix--the quantum invariant of a colored framed Whitehead link from the associated TQFT of a modular category. We prove that the $W$-matrix and the set of punctured $S$-matrices are strictly beyond the modular data $(S,T)$. Whether or not the triple $(S,T,W)$ constitutes a complete invariant of modular categories remains an open question.

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arXiv2018

Reflection fusion categories

We introduce the notion of a $\textit{reflection fusion category}$, which is a type of a $G$-crossed category generated by objects of Frobenius-Perron dimension $1$ and $\sqrt{p}$, where $p$ is an odd prime. We show that such categories correspond to orthogonal reflection groups over $\mathbb{F}_p$. This allows us to use the known classification of irreducible reflection groups over finite fields to classify irreducible reflection fusion categories.

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arXiv2018

Acyclic anyon models, thermal anyon error corrections, and braiding universality

Acyclic anyon models are non-abelian anyon models for which thermal anyon errors can be corrected. In this note, we characterize acyclic anyon models and raise the question if the restriction to acyclic anyon models is a deficiency of the current protocol or could it be intrinsically related to the computational power of non-abelian anyons. We also obtain general results on acyclic anyon models and find new acyclic anyon models such as $SO(8)_2$ and untwisted Dijkgraaf-Witten theories of nilpotent finite groups.

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arXiv2017

Pointed finite tensor categories over abelian groups

We give a characterization of finite pointed tensor categories obtained as de-equivariantizations of finite-dimensional pointed Hopf algebras over abelian groups only in terms of the (cohomology class of the) associator of the pointed part. As an application we prove that every coradically graded pointed finite braided tensor category is a de-equivariantization of a finite dimensional pointed Hopf algebras over an abelian group.

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arXiv2017

Braid Group Representations from Braiding Gapped Boundaries of Dijkgraaf-Witten Theories

We study representations of the braid groups from braiding gapped boundaries of Dijkgraaf-Witten theories and their twisted generalizations, which are (twisted) quantum doubled topological orders in two spatial dimensions. We show that the resulting braid (pure braid) representations are all monomial with respect to some specific bases, hence all such representation images of the braid groups are finite groups. We give explicit formulas for the monomial matrices and the ground state degeneracy of the Kitaev models that are Hamiltonian realizations of Dijkgraaf-Witten theories. Our results imply that braiding gapped boundaries alone cannot provide universal gate sets for topological quantum computing with gapped boundaries.

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arXiv2017

Classification of super-modular categories by rank

We pursue a classification of low-rank super-modular categories parallel to that of modular categories. We classify all super-modular categories up to rank=$6$, and spin modular categories up to rank=$11$. In particular, we show that, up to fusion rules, there is exactly one non-split super-modular category of rank $2,4$ and $6$, namely $PSU(2)_{4k+2}$ for $k=0,1$ and $2$. This classification is facilitated by adapting and extending well-known constraints from modular categories to super-modular categories, such as Verlinde and Frobenius-Schur indicator formulae.

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arXiv2017

Group actions on 2-categories

We study actions of discrete groups on 2-categories. The motivating examples are actions on the 2-category of representations of finite tensor categories and their relation with the extension theory of tensor categories by groups. Associated to a group action on a 2-category, we construct the 2-category of equivariant objects. We also introduce the G-equivariant notions of pseudofunctor, pseudonatural transformation and modification. Our first main result is a coherence theorem for 2-categories with an action of a group. For a 2-category B with an action of a group G, we construct a braided G-crossed monoidal category Z_G(B) with trivial component the Drinfeld center of B. We prove that, in the case of a G-action on the 2-category of representation of a tensor category C, the 2-category of equivariant objects is biequivalent to the module categories over an associated G-extension of C. Finally, we prove that the center of the equivariant 2-category is monoidally equivalent to the equivariantization of a relative center, generalizing results obtained in [S. Gelaki, D. Naidu and D. Nikshych, Centers of graded fusion categories, Algebra Number Theory 3, No. 8 (2009), 959--990.]

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