{"title":"Representation Theory of Solitons","authors":"Clay Cordova, Nicholas Holfester, Kantaro Ohmori","doi":"arxiv-2408.11045","DOIUrl":"https://doi.org/arxiv-2408.11045","url":null,"abstract":"Solitons in two-dimensional quantum field theory exhibit patterns of\u0000degeneracies and associated selection rules on scattering amplitudes. We\u0000develop a representation theory that captures these intriguing features of\u0000solitons. This representation theory is based on an algebra we refer to as the\u0000\"strip algebra\", $textrm{Str}_{mathcal{C}}(mathcal{M})$, which is defined in\u0000terms of the non-invertible symmetry, $mathcal{C},$ a fusion category, and its\u0000action on boundary conditions encoded by a module category, $mathcal{M}$. The\u0000strip algebra is a $C^*$-weak Hopf algebra, a fact which can be elegantly\u0000deduced by quantizing the three-dimensional Drinfeld center TQFT,\u0000$mathcal{Z}(mathcal{C}),$ on a spatial manifold with corners. These\u0000structures imply that the representation category of the strip algebra is also\u0000a unitary fusion category which we identify with a dual category\u0000$mathcal{C}_{mathcal{M}}^{*}.$ We present a straightforward method for\u0000analyzing these representations in terms of quiver diagrams where nodes are\u0000vacua and arrows are solitons and provide examples demonstrating how the\u0000representation theory reproduces known degeneracies and selection rules of\u0000soliton scattering. Our analysis provides the general framework for analyzing\u0000non-invertible symmetry on manifolds with boundary and applies both to the case\u0000of boundaries at infinity, relevant to particle physics, and boundaries at\u0000finite distance, relevant in conformal field theory or condensed matter\u0000systems.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Boundary symmetries of (2+1)D topological orders","authors":"Kylan Schatz","doi":"arxiv-2408.10832","DOIUrl":"https://doi.org/arxiv-2408.10832","url":null,"abstract":"We elaborate an algebraic framework for describing internal topological\u0000symmetries of gapped boundaries of (2+1)D topological orders. We present a\u0000categorical obstruction to the coherence of bulk group symmetry and boundary\u0000symmetries in terms of liftings of categorical actions on the bulk theory to a\u0000certain 2-group of boundary symmetries.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Not too little intervals for quantum mechanics","authors":"Damien Calaque","doi":"arxiv-2408.10033","DOIUrl":"https://doi.org/arxiv-2408.10033","url":null,"abstract":"This short paper illustrates the general framework introduced in the paper\u0000\"Not too little discs\" (arXiv:2407.18192), joint with Victor Carmona, on yet\u0000another one dimensional example. It exhibits a discrete model for the free\u0000scalar field on the real line, adapting the treatment from the book of\u0000Costello--Gwilliam to the discrete setting.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"105 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Ribbon Elements of Drinfeld Double of Radford Hopf Algebra","authors":"Hua Sun, Yuyan Zhang, Libin Li","doi":"arxiv-2408.09737","DOIUrl":"https://doi.org/arxiv-2408.09737","url":null,"abstract":"Let $m$, $n$ be two positive integers, $Bbbk$ be an algebraically closed\u0000field with char($Bbbk)nmid mn$. Radford constructed an $mn^{2}$-dimensional\u0000Hopf algebra $R_{mn}(q)$ such that its Jacobson radical is not a Hopf ideal. We\u0000show that the Drinfeld double $D(R_{mn}(q))$ of Radford Hopf algebra\u0000$R_{mn}(q)$ has ribbon elements if and only if $n$ is odd. Moreover, if $m$ is\u0000even and $n$ is odd, then $D(R_{mn}(q))$ has two ribbon elements, if both $m$\u0000and $n$ are odd, then $D(R_{mn}(q))$ has only one ribbon element. Finally, we\u0000compute explicitly all ribbon elements of $D(R_{mn}(q))$.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On geometric bases for {it quantum} A-polynomials of knots","authors":"Dmitry Galakhov, Alexei Morozov","doi":"arxiv-2408.08181","DOIUrl":"https://doi.org/arxiv-2408.08181","url":null,"abstract":"A simple geometric way is suggested to derive the Ward identities in the\u0000Chern-Simons theory, also known as quantum $A$- and $C$-polynomials for knots.\u0000In quasi-classical limit it is closely related to the well publicized\u0000augmentation theory and contact geometry. Quantization allows to present it in\u0000much simpler terms, what could make these techniques available to a broader\u0000audience. To avoid overloading of the presentation, only the case of the\u0000colored Jones polynomial for the trefoil knot is considered, though various\u0000generalizations are straightforward. Restriction to solely Jones polynomials\u0000(rather than full HOMFLY-PT) is related to a serious simplification, provided\u0000by the use of Kauffman calculus. Going beyond looks realistic, however it\u0000remains a problem, both challenging and promising.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A categorical interpretation of Morita equivalence for dynamical von Neumann algebras","authors":"Joeri De Ro","doi":"arxiv-2408.07701","DOIUrl":"https://doi.org/arxiv-2408.07701","url":null,"abstract":"$DeclareMathOperator{G}{mathbb{G}}DeclareMathOperator{Rep}{Rep}\u0000DeclareMathOperator{Corr}{Corr}$Let $G$ be a locally compact quantum group\u0000and $(M, alpha)$ a $G$-$W^*$-algebra. The object of study of this paper is\u0000the $W^*$-category $Rep^{G}(M)$ of normal, unital $G$-representations of $M$\u0000on Hilbert spaces endowed with a unitary $G$-representation. This category has\u0000a right action of the category $Rep(G)= Rep^{G}(mathbb{C})$ for which it\u0000becomes a right $Rep(G)$-module $W^*$-category. Given another\u0000$G$-$W^*$-algebra $(N, beta)$, we denote the category of normal $*$-functors\u0000$Rep^{G}(N)to Rep^{G}(M)$ compatible with the $Rep(G)$-module structure\u0000by $operatorname{Fun}_{Rep(G)}(Rep^{G}(N), Rep^{G}(M))$ and we denote\u0000the category of $G$-$M$-$N$-correspondences by\u0000$operatorname{Corr}^{G}(M,N)$. We prove that there are canonical functors $P:\u0000Corr^{G}(M,N)to operatorname{Fun}_{Rep(G)}(Rep^{G}(N), Rep^{G}(M))$\u0000and $Q: operatorname{Fun}_{Rep(G)}(Rep^{G}(N), Rep^{G}(M))to\u0000operatorname{Corr}^{G}(M,N)$ such that $Q circ Pcong operatorname{id}.$ We\u0000use these functors to show that the $G$-dynamical von Neumann algebras $(M,\u0000alpha)$ and $(N, beta)$ are equivariantly Morita equivalent if and only if\u0000$Rep^{G}(N)$ and $Rep^{G}(M)$ are equivalent as\u0000$Rep(G)$-module-$W^*$-categories. Specializing to the case where $G$ is a\u0000compact quantum group, we prove that moreover $Pcirc Q cong\u0000operatorname{id}$, so that the categories $Corr^{G}(M,N)$ and\u0000$operatorname{Fun}_{Rep(G)}(Rep^{G}(N), Rep^{G}(M))$ are equivalent.\u0000This is an equivariant version of the Eilenberg-Watts theorem for actions of\u0000compact quantum groups on von Neumann algebras.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Vertex operator expressions for Lie algebras of physical states","authors":"Thomas Driscoll-Spittler","doi":"arxiv-2408.07597","DOIUrl":"https://doi.org/arxiv-2408.07597","url":null,"abstract":"We study the Lie algebra of physical states associated with certain vertex\u0000operator algebras of central charge 24. By applying the no-ghost theorem from\u0000string theory we express the corresponding Lie brackets in terms of vertex\u0000algebra operations. In the special case of the Moonshine module this result\u0000answers a question of Borcherds, posed in his paper on the Monstrous moonshine\u0000conjecture.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A proof of The Radial Limit Conjecture for Costantino--Geer--Patureau-Mirand Quantum invariants","authors":"William Elbæk Mistegård, Yuya Murakami","doi":"arxiv-2408.07423","DOIUrl":"https://doi.org/arxiv-2408.07423","url":null,"abstract":"For a negative definite plumbed three-manifold, we give an integral\u0000representation of the appropriate average of the GPPV invariants of\u0000Gukov--Pei--Putrov--Vafa, which implies that this average admits a resurgent\u0000asymptotic expansion, the leading term of which is the\u0000Costantino--Geer--Patureau-Mirand invariant of the three-manifold. This proves\u0000a conjecture of Costantino--Gukov--Putrov.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"SU(3) higher roots and their lattices","authors":"Robert Coquereaux","doi":"arxiv-2409.02926","DOIUrl":"https://doi.org/arxiv-2409.02926","url":null,"abstract":"After recalling the notion of higher roots (or hyper-roots) associated with\u0000\"quantum modules\" of type $(G, k)$, for $G$ a semi-simple Lie group and $k$ a\u0000positive integer, following the definition given by A. Ocneanu in 2000, we\u0000study the theta series of their lattices. Here we only consider the higher\u0000roots associated with quantum modules (aka module-categories over the fusion\u0000category defined by the pair $(G,k)$) that are also \"quantum subgroups\". For\u0000$G=SU{2}$ the notion of higher roots coincides with the usual notion of roots\u0000for ADE Dynkin diagrams and the self-fusion restriction (the property of being\u0000a quantum subgroup) selects the diagrams of type $A_{r}$, $D_{r}$ with $r$\u0000even, $E_6$ and $E_8$; their theta series are well known. In this paper we take\u0000$G=SU{3}$, where the same restriction selects the modules ${mathcal A}_k$,\u0000${mathcal D}_k$ with $mod(k,3)=0$, and the three exceptional cases ${mathcal\u0000E}_5$, ${mathcal E}_9$ and ${mathcal E}_{21}$. The theta series for their\u0000associated lattices are expressed in terms of modular forms twisted by\u0000appropriate Dirichlet characters.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pawel Caputa, Souradeep Purkayastha, Abhigyan Saha, Piotr Sułkowski
{"title":"Musings on SVD and pseudo entanglement entropies","authors":"Pawel Caputa, Souradeep Purkayastha, Abhigyan Saha, Piotr Sułkowski","doi":"arxiv-2408.06791","DOIUrl":"https://doi.org/arxiv-2408.06791","url":null,"abstract":"Pseudo-entropy and SVD entropy are generalizations of the entanglement\u0000entropy that involve post-selection. In this work we analyze their properties\u0000as measures on the spaces of quantum states and argue that their excess\u0000provides useful characterization of a difference between two (i.e. pre-selected\u0000and post-selected) states, which shares certain features and in certain cases\u0000can be identified as a metric. In particular, when applied to link complement\u0000states that are associated to topological links via Chern-Simons theory, these\u0000generalized entropies and their excess provide a novel quantification of a\u0000difference between corresponding links. We discuss the dependence of such\u0000entropy measures on the level of Chern-Simons theory and determine their\u0000asymptotic values for certain link states. We find that imaginary part of the\u0000pseudo-entropy is sensitive to, and can diagnose chirality of knots. We also\u0000consider properties of these entropy measures for simpler quantum mechanical\u0000systems, such as generalized SU(2) and SU(1,1) coherent states, and tripartite\u0000GHZ and W states.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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