From Khovanov homology, we extract a new lower bound for the Gordian distance of knots, which combines and strengthens the previously existing bounds coming from Rasmussen invariants and from torsion invariants. We also improve the bounds for the proper rational Gordian distance.
{"title":"Khovanov homology and refined bounds for Gordian distances","authors":"Lukas Lewark, Laura Marino, Claudius Zibrowius","doi":"arxiv-2409.05743","DOIUrl":"https://doi.org/arxiv-2409.05743","url":null,"abstract":"From Khovanov homology, we extract a new lower bound for the Gordian distance\u0000of knots, which combines and strengthens the previously existing bounds coming\u0000from Rasmussen invariants and from torsion invariants. We also improve the\u0000bounds for the proper rational Gordian distance.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198808","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}
We analyze the $G$-skein theory invariants of the 3-torus $T^3$ and the two-torus $T^2$, for the groups $G = GL_N, SL_N$ and for generic quantum parameter. We obtain formulas for the dimension of the skein module of $T^3$, and we describe the algebraic structure of the skein category of $T^2$ -- namely of the $n$-point relative skein algebras. The case $n=N$ (the Schur-Weyl case) is special in our analysis. We construct an isomorphism between the $N$-point relative skein algebra and the double affine Hecke algebra at specialized parameters. As a consequence, we prove that all tangles in the relative $N$-point skein algebra are in fact equivalent to linear combinations of braids, modulo skein relations. More generally for $n$ an integer multiple of $N$, we construct a surjective homomorphism from an appropriate DAHA to the $n$-point relative skein algebra. In the case $G=SL_2$ corresponding to the Kauffman bracket we give proofs directly using skein relations. Our analysis of skein categories in higher rank hinges instead on the combinatorics of multisegment representations when restricting from DAHA to AHA and nonvanishing properties of parabolic sign idempotents upon them.
我们分析了在 G = GL_N, SL_N$ 群和一般量参数下，3-torus $T^3$ 和 2-torus $T^2$ 的 $G$ 琴键理论不变式。我们得到了 $T^3$ 的矢量模块维数公式，并描述了 $T^2$ 矢量范畴的代数结构--即 $n$ 点相对矢量代数。在我们的分析中，$n=N$ 的情况（舒尔-韦尔情况）比较特殊。我们在专门参数下构建了 $N$ 点相对矢代数与双链赫克代数之间的同构关系。因此，我们证明了相对 $N$ 点辫子代数中的所有缠结实际上等价于辫子的线性组合，模数为辫子关系。更一般地说，对于 $n$ 是 $N$ 的整数倍的情况，我们构建了一个从适当的 DAHA 到 $n$ 点相对辫子代数的推射同态。在与考夫曼括号相对应的$G=SL_2$的情况下，我们直接使用斯琴关系给出了证明。我们对高阶阶乘范畴的分析是基于从 DAHA 限制到 AHA 时多段表示的组合学，以及在它们之上的抛物线符号empotents 的非消失性质。
{"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}
We construct a quantum Dolbeault double complex $oplus_{p,q}Omega^{p,q}$ on the quantum plane $Bbb C_q^2$. This solves the long-standing problem that the standard differential calculus on the quantum plane is not a $*$-calculus, by embedding it as the holomorphic part of a $*$-calculus. We show in general that any Nichols-Woronowicz algebra or braided plane $B_+(V)$, where $V$ is an object in an abelian $Bbb C$-linear braided bar category of real type is a quantum complex space in this sense with a factorisable Dolbeault double complex. We combine the Chern construction on $Omega^{1,0}$ in such a Dolbeault complex for an algebra $A$ with its conjugate to construct a canonical metric compatible connection on $Omega^1$ associated to a class of quantum metrics, and apply this to the quantum plane. We also apply this to finite groups $G$ with Cayley graph generators split into two halves related by inversion, constructing such a Dolbeault complex $Omega(G)$ in this case, recovering the quantum Levi-Civita connection for any edge-symmetric metric on the integer lattice with $Omega(Bbb Z)$ now viewed as a quantum complex structure. We also show how to build natural quantum metrics on $Omega^{1,0}$ and $Omega^{0,1}$ separately where the inner product in the case of the quantum plane, in order to descend to $otimes_A$, is taken with values in an $A$-bimodule.
{"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}
We elucidate the comment in (Kapranov-Vasserot, Adv. Math., 2011, Remark 5.3.4) that the $1|1$-dimensional factorization structure of the formal superloop space of a smooth algebraic variety $X$ induces the $N_K=1$ SUSY vertex algebra structure of the chiral de Rham complex of $X$.
{"title":"$N_K=1$ SUSY structure of chiral de Rham complex from the factorization structure","authors":"Takumi Iwane, Shintarou Yanagida","doi":"arxiv-2409.04220","DOIUrl":"https://doi.org/arxiv-2409.04220","url":null,"abstract":"We elucidate the comment in (Kapranov-Vasserot, Adv. Math., 2011, Remark\u00005.3.4) that the $1|1$-dimensional factorization structure of the formal\u0000superloop space of a smooth algebraic variety $X$ induces the $N_K=1$ SUSY\u0000vertex algebra structure of the chiral de Rham complex of $X$.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198761","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}
The first author and Speicher proved the inequality for operator norms of holomorphic homogeneous polynomials in freely independent $mathscr{R}$-diagonal elements, which improves the bound obtained by Haagerup. We prove a similar inequality for $q$-circular systems, which are neither freely independent nor $mathscr{R}$-diagonal.
{"title":"The strong Haagerup inequality for q-circular systems","authors":"Todd Kemp, Akihiro Miyagawa","doi":"arxiv-2409.03177","DOIUrl":"https://doi.org/arxiv-2409.03177","url":null,"abstract":"The first author and Speicher proved the inequality for operator norms of\u0000holomorphic homogeneous polynomials in freely independent\u0000$mathscr{R}$-diagonal elements, which improves the bound obtained by Haagerup.\u0000We prove a similar inequality for $q$-circular systems, which are neither\u0000freely independent nor $mathscr{R}$-diagonal.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225237","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}
These are expanded notes of a course on basics of quantum field theory for mathematicians given by the author at MIT.
这是作者在麻省理工学院开设的量子场论格式数学基础课程的扩充笔记。
{"title":"Mathematical ideas and notions of quantum field theory","authors":"Pavel Etingof","doi":"arxiv-2409.03117","DOIUrl":"https://doi.org/arxiv-2409.03117","url":null,"abstract":"These are expanded notes of a course on basics of quantum field theory for\u0000mathematicians given by the author at MIT.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198764","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}
We derive a refined version of the Affleck-Ludwig-Cardy formula for a 1+1d conformal field theory, which controls the asymptotic density of high energy states on an interval transforming under a given representation of a non-invertible global symmetry. We use this to determine the universal leading and sub-leading contributions to the non-invertible symmetry-resolved entanglement entropy of a single interval. As a concrete example, we show that the ground state entanglement Hamiltonian for a single interval in the critical double Ising model enjoys a Kac-Paljutkin $H_8$ Hopf algebra symmetry when the boundary conditions at the entanglement cuts are chosen to preserve the product of two Kramers-Wannier symmetries, and we present the corresponding symmetry-resolved entanglement entropies. Our analysis utilizes recent developments in symmetry topological field theories (SymTFTs).
{"title":"A Non-Invertible Symmetry-Resolved Affleck-Ludwig-Cardy Formula and Entanglement Entropy from the Boundary Tube Algebra","authors":"Yichul Choi, Brandon C. Rayhaun, Yunqin Zheng","doi":"arxiv-2409.02806","DOIUrl":"https://doi.org/arxiv-2409.02806","url":null,"abstract":"We derive a refined version of the Affleck-Ludwig-Cardy formula for a 1+1d\u0000conformal field theory, which controls the asymptotic density of high energy\u0000states on an interval transforming under a given representation of a\u0000non-invertible global symmetry. We use this to determine the universal leading\u0000and sub-leading contributions to the non-invertible symmetry-resolved\u0000entanglement entropy of a single interval. As a concrete example, we show that\u0000the ground state entanglement Hamiltonian for a single interval in the critical\u0000double Ising model enjoys a Kac-Paljutkin $H_8$ Hopf algebra symmetry when the\u0000boundary conditions at the entanglement cuts are chosen to preserve the product\u0000of two Kramers-Wannier symmetries, and we present the corresponding\u0000symmetry-resolved entanglement entropies. Our analysis utilizes recent\u0000developments in symmetry topological field theories (SymTFTs).","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"69 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198762","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}
The connection between the R-matrix realization and Drinfeld's realization of the quantum loop algebra $U_q(D^{(2)}_n)$ is considered using the Gaussian decomposition approach proposed by J. Ding and I. B. Frenkel. Our main result is a description of the embedding $U_q(D^{(2)}_{n-1})hookrightarrow U_q(D^{(2)}_n)$ that underlies this connection. Explicit relations between all Gaussian coordinates of the L-operators and the currents are presented.
丁杰和弗伦克尔（I. B. Frenkel）提出的高斯和分解方法，考虑了量子环代数$U_q(D^{(2)}_n)$的R矩阵实现和德林菲尔德实现之间的联系。我们的主要结果是对嵌入 $U_q(D^{(2)}_{n-1})/hookrightarrowU_q(D^{(2)}_n)$ 的描述，它是这种联系的基础。本文提出了 L 运算器的全高斯坐标与电流之间的明确关系。
{"title":"On the R-matrix realization of the quantum loop algebra. The case of $U_q(D^{(2)}_n)$","authors":"A. Liashyk, S. Pakuliak","doi":"arxiv-2409.02021","DOIUrl":"https://doi.org/arxiv-2409.02021","url":null,"abstract":"The connection between the R-matrix realization and Drinfeld's realization of\u0000the quantum loop algebra $U_q(D^{(2)}_n)$ is considered using the Gaussian\u0000decomposition approach proposed by J. Ding and I. B. Frenkel. Our main result\u0000is a description of the embedding $U_q(D^{(2)}_{n-1})hookrightarrow\u0000U_q(D^{(2)}_n)$ that underlies this connection. Explicit relations between all\u0000Gaussian coordinates of the L-operators and the currents are presented.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225239","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}
We develop an equivariant theory of graphs with respect to quantum symmetries and present a detailed exposition of various examples. We portray unitary tensor categories as a unifying framework encompassing all finite classical simple graphs, (quantum) Cayley graphs of finite (quantum) groupoids, and all finite-dimensional quantum graphs. We model a quantum set by a finite-index inclusion of C*-algebras and use the quantum Fourier transform to obtain all possible adjacency operators. In particular, we show every finite-index subfactor can be regarded as a complete quantum graph and describe how to find all its subgraphs. As applications, we prove a version of Frucht's Theorem for finite quantum groupoids, and introduce a version of path spaces for quantum graphs.
{"title":"Quantum graphs, subfactors and tensor categories I","authors":"Michael Brannan, Roberto Hernández Palomares","doi":"arxiv-2409.01951","DOIUrl":"https://doi.org/arxiv-2409.01951","url":null,"abstract":"We develop an equivariant theory of graphs with respect to quantum symmetries\u0000and present a detailed exposition of various examples. We portray unitary\u0000tensor categories as a unifying framework encompassing all finite classical\u0000simple graphs, (quantum) Cayley graphs of finite (quantum) groupoids, and all\u0000finite-dimensional quantum graphs. We model a quantum set by a finite-index\u0000inclusion of C*-algebras and use the quantum Fourier transform to obtain all\u0000possible adjacency operators. In particular, we show every finite-index\u0000subfactor can be regarded as a complete quantum graph and describe how to find\u0000all its subgraphs. As applications, we prove a version of Frucht's Theorem for\u0000finite quantum groupoids, and introduce a version of path spaces for quantum\u0000graphs.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"220 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198765","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}
We introduce a class of generalized tube algebras which describe how finite, non-invertible global symmetries of bosonic 1+1d QFTs act on operators which sit at the intersection point of a collection of boundaries and interfaces. We develop a 2+1d symmetry topological field theory (SymTFT) picture of boundaries and interfaces which, among other things, allows us to deduce the representation theory of these algebras. In particular, we initiate the study of a character theory, echoing that of finite groups, and demonstrate how many representation-theoretic quantities can be expressed as partition functions of the SymTFT on various backgrounds, which in turn can be evaluated explicitly in terms of generalized half-linking numbers. We use this technology to explain how the torus and annulus partition functions of a 1+1d QFT can be refined with information about its symmetries. We are led to a vast generalization of Ishibashi states in CFT: to any multiplet of conformal boundary conditions which transform into each other under the action of a symmetry, we associate a collection of generalized Ishibashi states, in terms of which the twisted sector boundary states of the theory and all of its orbifolds can be obtained as linear combinations. We derive a generalized Verlinde formula involving the characters of the boundary tube algebra which ensures that our formulas for the twisted sector boundary states respect open-closed duality. Our approach does not rely on rationality or the existence of an extended chiral algebra; however, in the special case of a diagonal RCFT with chiral algebra $V$ and modular tensor category $mathscr{C}$, our formalism produces explicit closed-form expressions - in terms of the $F$-symbols and $R$-matrices of $mathscr{C}$, and the characters of $V$ - for the twisted Cardy states, and the torus and annulus partition functions decorated by Verlinde lines.
{"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}
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