arXiv - MATH - Quantum Algebra最新文献_第2页

Khovanov homology and refined bounds for Gordian distances 科瓦诺夫同源性和戈尔迪距离的精炼边界

arXiv - MATH - Quantum Algebra Pub Date : 2024-09-09 DOI: arxiv-2409.05743

Lukas Lewark, Laura Marino, Claudius Zibrowius

引用次数: 0

Skeins on tori 蝶形花

arXiv - MATH - Quantum Algebra Pub Date : 2024-09-09 DOI: arxiv-2409.05613

Sam Gunningham, David Jordan, Monica Vazirani

{"title":"Skeins on tori","authors":"Sam Gunningham, David Jordan, Monica Vazirani","doi":"arxiv-2409.05613","DOIUrl":"https://doi.org/arxiv-2409.05613","url":null,"abstract":"We analyze the $G$-skein theory invariants of the 3-torus $T^3$ and the\u0000two-torus $T^2$, for the groups $G = GL_N, SL_N$ and for generic quantum\u0000parameter. We obtain formulas for the dimension of the skein module of $T^3$,\u0000and we describe the algebraic structure of the skein category of $T^2$ --\u0000namely of the $n$-point relative skein algebras. The case $n=N$ (the Schur-Weyl case) is special in our analysis. We construct\u0000an isomorphism between the $N$-point relative skein algebra and the double\u0000affine Hecke algebra at specialized parameters. As a consequence, we prove that\u0000all tangles in the relative $N$-point skein algebra are in fact equivalent to\u0000linear combinations of braids, modulo skein relations. More generally for $n$\u0000an integer multiple of $N$, we construct a surjective homomorphism from an\u0000appropriate DAHA to the $n$-point relative skein algebra. In the case $G=SL_2$ corresponding to the Kauffman bracket we give proofs\u0000directly using skein relations. Our analysis of skein categories in higher rank\u0000hinges instead on the combinatorics of multisegment representations when\u0000restricting from DAHA to AHA and nonvanishing properties of parabolic sign\u0000idempotents upon them.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198760","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}

引用次数: 0

Complex structure on quantum-braided planes 量子编织平面上的复杂结构

arXiv - MATH - Quantum Algebra Pub Date : 2024-09-09 DOI: arxiv-2409.05253

Edwin Beggs, Shahn Majid

{"title":"Complex structure on quantum-braided planes","authors":"Edwin Beggs, Shahn Majid","doi":"arxiv-2409.05253","DOIUrl":"https://doi.org/arxiv-2409.05253","url":null,"abstract":"We construct a quantum Dolbeault double complex $oplus_{p,q}Omega^{p,q}$ on\u0000the quantum plane $Bbb C_q^2$. This solves the long-standing problem that the\u0000standard differential calculus on the quantum plane is not a $*$-calculus, by\u0000embedding it as the holomorphic part of a $*$-calculus. We show in general that\u0000any Nichols-Woronowicz algebra or braided plane $B_+(V)$, where $V$ is an\u0000object in an abelian $Bbb C$-linear braided bar category of real type is a\u0000quantum complex space in this sense with a factorisable Dolbeault double\u0000complex. We combine the Chern construction on $Omega^{1,0}$ in such a\u0000Dolbeault complex for an algebra $A$ with its conjugate to construct a\u0000canonical metric compatible connection on $Omega^1$ associated to a class of\u0000quantum metrics, and apply this to the quantum plane. We also apply this to\u0000finite groups $G$ with Cayley graph generators split into two halves related by\u0000inversion, constructing such a Dolbeault complex $Omega(G)$ in this case,\u0000recovering the quantum Levi-Civita connection for any edge-symmetric metric on\u0000the integer lattice with $Omega(Bbb Z)$ now viewed as a quantum complex\u0000structure. We also show how to build natural quantum metrics on $Omega^{1,0}$\u0000and $Omega^{0,1}$ separately where the inner product in the case of the\u0000quantum plane, in order to descend to $otimes_A$, is taken with values in an\u0000$A$-bimodule.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198763","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}

引用次数: 0

$N_K=1$ SUSY structure of chiral de Rham complex from the factorization structure 从因式分解结构看手性德拉姆复合体的 $N_K=1$ SUSY 结构

arXiv - MATH - Quantum Algebra Pub Date : 2024-09-06 DOI: arxiv-2409.04220

Takumi Iwane, Shintarou Yanagida

引用次数: 0

The strong Haagerup inequality for q-circular systems q 循环系统的强哈格鲁普不等式

arXiv - MATH - Quantum Algebra Pub Date : 2024-09-05 DOI: arxiv-2409.03177

Todd Kemp, Akihiro Miyagawa

引用次数: 0

Mathematical ideas and notions of quantum field theory 量子场论的数学思想和概念

arXiv - MATH - Quantum Algebra Pub Date : 2024-09-04 DOI: arxiv-2409.03117

Pavel Etingof

引用次数: 0

A Non-Invertible Symmetry-Resolved Affleck-Ludwig-Cardy Formula and Entanglement Entropy from the Boundary Tube Algebra 非不可逆性对称解析阿弗莱克-路德维希-卡迪公式和来自边界管代数的纠缠熵

arXiv - MATH - Quantum Algebra Pub Date : 2024-09-04 DOI: arxiv-2409.02806

Yichul Choi, Brandon C. Rayhaun, Yunqin Zheng

引用次数: 0

On the R-matrix realization of the quantum loop algebra. The case of $U_q(D^{(2)}_n)$ 关于量子环代数的 R 矩阵实现.$U_q(D^{(2)}_n)$ 的情况

arXiv - MATH - Quantum Algebra Pub Date : 2024-09-03 DOI: arxiv-2409.02021

A. Liashyk, S. Pakuliak

引用次数: 0

Quantum graphs, subfactors and tensor categories I 量子图、子因子和张量类别 I

arXiv - MATH - Quantum Algebra Pub Date : 2024-09-03 DOI: arxiv-2409.01951

Michael Brannan, Roberto Hernández Palomares

引用次数: 0

Generalized Tube Algebras, Symmetry-Resolved Partition Functions, and Twisted Boundary States 广义管代数、对称解析分部函数和扭曲边界态

arXiv - MATH - Quantum Algebra Pub Date : 2024-09-03 DOI: arxiv-2409.02159

Yichul Choi, Brandon C. Rayhaun, Yunqin Zheng

{"title":"Generalized Tube Algebras, Symmetry-Resolved Partition Functions, and Twisted Boundary States","authors":"Yichul Choi, Brandon C. Rayhaun, Yunqin Zheng","doi":"arxiv-2409.02159","DOIUrl":"https://doi.org/arxiv-2409.02159","url":null,"abstract":"We introduce a class of generalized tube algebras which describe how finite,\u0000non-invertible global symmetries of bosonic 1+1d QFTs act on operators which\u0000sit at the intersection point of a collection of boundaries and interfaces. We\u0000develop a 2+1d symmetry topological field theory (SymTFT) picture of boundaries\u0000and interfaces which, among other things, allows us to deduce the\u0000representation theory of these algebras. In particular, we initiate the study\u0000of a character theory, echoing that of finite groups, and demonstrate how many\u0000representation-theoretic quantities can be expressed as partition functions of\u0000the SymTFT on various backgrounds, which in turn can be evaluated explicitly in\u0000terms of generalized half-linking numbers. We use this technology to explain\u0000how the torus and annulus partition functions of a 1+1d QFT can be refined with\u0000information about its symmetries. We are led to a vast generalization of\u0000Ishibashi states in CFT: to any multiplet of conformal boundary conditions\u0000which transform into each other under the action of a symmetry, we associate a\u0000collection of generalized Ishibashi states, in terms of which the twisted\u0000sector boundary states of the theory and all of its orbifolds can be obtained\u0000as linear combinations. We derive a generalized Verlinde formula involving the\u0000characters of the boundary tube algebra which ensures that our formulas for the\u0000twisted sector boundary states respect open-closed duality. Our approach does\u0000not rely on rationality or the existence of an extended chiral algebra;\u0000however, in the special case of a diagonal RCFT with chiral algebra $V$ and\u0000modular tensor category $mathscr{C}$, our formalism produces explicit\u0000closed-form expressions - in terms of the $F$-symbols and $R$-matrices of\u0000$mathscr{C}$, and the characters of $V$ - for the twisted Cardy states, and\u0000the torus and annulus partition functions decorated by Verlinde lines.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"184 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225238","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}

引用次数: 0