scholarly journals KMS States on the Operator Algebras of Reducible Higher-Rank Graphs

2017 ◽  
Vol 88 (1) ◽  
pp. 91-126 ◽  
Author(s):  
Astrid an Huef ◽  
Sooran Kang ◽  
Iain Raeburn
Keyword(s):  
Operator Algebras ◽  
Kms States ◽  
Higher Rank

scholarly journals Higher rank FZZ-dualities

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Thomas Creutzig ◽  
Yasuaki Hikida
Keyword(s):  
Field Theory ◽  
Reduction Method ◽  
Operator Algebras ◽  
The Self ◽  
Liouville Theory ◽  
Conformal Field ◽  
Self Duality ◽  
Higher Rank ◽  
Corner Vertex ◽  
Liouville Field Theory

Abstract We examine strong/weak dualities in two dimensional conformal field theories by generalizing the Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality between Witten’s cigar model described by the $$ \mathfrak{sl}(2)/\mathfrak{u}(1) $$ sl 2 / u 1 coset and sine-Liouville theory. In a previous work, a proof of the FZZ-duality was provided by applying the reduction method from $$ \mathfrak{sl}(2) $$ sl 2 Wess-Zumino-Novikov-Witten model to Liouville field theory and the self-duality of Liouville field theory. In this paper, we work with the coset model of the type $$ \mathfrak{sl}\left(N+1\right)/\left(\mathfrak{sl}(N)\times \mathfrak{u}(1)\right) $$ sl N + 1 / sl N × u 1 and investigate the equivalence to a theory with an $$ \mathfrak{sl}\left(N+\left.1\right|N\right) $$ sl N + 1 N structure. We derive the duality explicitly for N = 2, 3 by applying recent works on the reduction method extended for $$ \mathfrak{sl}(N) $$ sl N and the self-duality of Toda field theory. Our results can be regarded as a conformal field theoretic derivation of the duality of the Gaiotto-Rapčák corner vertex operator algebras Y0,N,N+1[ψ] and YN,0,N+1[ψ−1].

scholarly journals Spatial realisations of KMS states on the C⁎-algebras of higher-rank graphs

2015 ◽  
Vol 427 (2) ◽  
pp. 977-1003 ◽  
Author(s):  
Astrid an Huef ◽  
Sooran Kang ◽  
Iain Raeburn
Keyword(s):  
Kms States ◽  
C Algebras ◽  
Higher Rank

scholarly journals Uprolling unrolled quantum groups

2021 ◽  
pp. 2150023
Author(s):  
Thomas Creutzig ◽  
Matthew Rupert
Keyword(s):  
Quantum Group ◽  
Lie Algebra ◽  
Quantum Groups ◽  
Vertex Operator ◽  
Operator Algebras ◽  
Vertex Algebra ◽  
Vertex Operator Algebras ◽  
Roots Of Unity ◽  
Simple Lie Algebra ◽  
Higher Rank

We construct families of commutative (super) algebra objects in the category of weight modules for the unrolled restricted quantum group [Formula: see text] of a simple Lie algebra [Formula: see text] at roots of unity, and study their categories of local modules. We determine their simple modules and derive conditions for these categories being finite, non-degenerate, and ribbon. Motivated by numerous examples in the [Formula: see text] case, we expect some of these categories to compare nicely to categories of modules for vertex operator algebras. We focus in particular on examples expected to correspond to the higher rank triplet vertex algebra [Formula: see text] of Feigin and Tipunin and the [Formula: see text] algebras.

scholarly journals Cocycles on groupoids arising from -actions

2021 ◽  
pp. 1-32
Author(s):  
CARLA FARSI ◽  
LEONARD HUANG ◽  
ALEX KUMJIAN ◽  
JUDITH PACKER
Keyword(s):  
Continuous Functions ◽  
Locally Compact Abelian Group ◽  
Compact Metric Space ◽  
Kms States ◽  
Existence And Uniqueness Results ◽  
Commuting Operators ◽  
Uniqueness Results ◽  
Unit Space ◽  
Higher Rank

Abstract We consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space and study generalized Ruelle operators and $ C^{\ast } $ -algebras associated to these groupoids. We provide a new characterization of $ 1 $ -cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $ k $ -tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle–Perron–Frobenius theory for dynamical systems of several commuting operators ( $ k $ -Ruelle triples and commuting Ruelle operators). Results on KMS states on $ C^{\ast } $ -algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence and uniqueness results for KMS states associated to the graphs.

scholarly journals Separable representations, KMS states, and wavelets for higher-rank graphs

2016 ◽  
Vol 434 (1) ◽  
pp. 241-270 ◽  
Author(s):  
Carla Farsi ◽  
Elizabeth Gillaspy ◽  
Sooran Kang ◽  
Judith A. Packer
Keyword(s):  
Kms States ◽  
Higher Rank

scholarly journals KMS states onC⁎-algebras associated to higher-rank graphs

2014 ◽  
Vol 266 (1) ◽  
pp. 265-283 ◽  
Author(s):  
Astrid an Huef ◽  
Marcelo Laca ◽  
Iain Raeburn ◽  
Aidan Sims
Keyword(s):  
Kms States ◽  
Higher Rank

scholarly journals Monic representations of finite higher-rank graphs

2018 ◽  
Vol 40 (5) ◽  
pp. 1238-1267 ◽  
Author(s):  
CARLA FARSI ◽  
ELIZABETH GILLASPY ◽  
PALLE JORGENSEN ◽  
SOORAN KANG ◽  
JUDITH PACKER
Keyword(s):  
Continuous Functions ◽  
Path Space ◽  
Cyclic Vector ◽  
Kms States ◽  
Infinite Path ◽  
Representation Model ◽  
Higher Rank ◽  
Universal Representation ◽  
Comprehensive Study

In this paper, we define the notion of monic representation for the$C^{\ast }$-algebras of finite higher-rank graphs with no sources, and we undertake a comprehensive study of them. Monic representations are the representations that, when restricted to the commutative$C^{\ast }$-algebra of the continuous functions on the infinite path space, admit a cyclic vector. We link monic representations to the$\unicode[STIX]{x1D6EC}$-semibranching representations previously studied by Farsi, Gillaspy, Kang and Packer (Separable representations, KMS states, and wavelets for higher-rank graphs.J. Math. Anal. Appl. 434 (2015), 241–270) and also provide a universal representation model for non-negative monic representations.

scholarly journals Correlator correspondences for Gaiotto-Rapčák dualities and first order formulation of coset models

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Thomas Creutzig ◽  
Yasuaki Hikida
Keyword(s):  
Field Theory ◽  
Operator Algebras ◽  
Two Dimensions ◽  
Conformal Field Theories ◽  
Generic Level ◽  
First Order ◽  
Conformal Field ◽  
Higher Rank ◽  
Coset Models ◽  
Corner Vertex

Abstract We derive correspondences of correlation functions among dual conformal field theories in two dimensions by developing a “first order formulation” of coset models. We examine several examples, and the most fundamental one may be a conjectural equivalence between a coset (SL(n)k ⊗SL(n)−1)/SL(n)k−1 and $$ \mathfrak{sl}(n) $$ sl n Toda field theory with generic level k. Among others, we also complete the derivation of higher rank FZZ-duality involving a coset SL(n + 1)k /(SL(n)k ⊗ U(1)), which could be done only for n = 2, 3 in our previous paper. One obstacle in the previous work was our poor understanding of a first order formulation of coset models. In this paper, we establish such a formulation using the BRST formalism. With our better understanding, we successfully derive correlator correspondences of dual models including the examples mentioned above. The dualities may be regarded as conformal field theory realizations of some of the Gaiotto-Rapčák dualities of corner vertex operator algebras.

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