Let $ V$ be a braided tensor category and $ C$ a tensor category equipped with a braided tensor functor $G:Vto Z(C)$. For any exact indecomposable $C$-module category $M$, we explicitly construct a right adjoint of the action functor $rho:Z^V(C)to C^*_{M}$ afforded by $M$. Here $Z^V(C)$ is the M"uger's centralizer of the subcategory $G(V)$ inside the center $Z^V(C)$, also known as the relative center. The construction is parallel to the one presented by K. Shimizu, but using instead the relative coend end. This adjunction turns out to be monadic, thus inducing Hopf monads $T_{V}: Cto C$, such that there is a monoidal equivalence of categories $ C_{T_{V}}simeq Z^V(C).$ If $bar{rho}: C^*_{ M}to Z^V(C)$ is the right adjoint of $rho,$ then $bar{rho}(Id_{M})$ is the braided commutative algebra constructed in [R. Laugwitz and C. Walton. Braided commutative algebras over quantized enveloping algebras, Transform. Groups 26(3) (2021), 957--993]. As a consequence of our construction of these algebras, in terms of the right adjoint to $rho$, we can provide a recipe to compute them when $C=Rep(H# T)$ is the category of finite-dimensional representations of a finite-dimensional Hopf algebra $H# T$ obtained by bosonization, and choosing an arbitrary $Rep(H# T)$-module category $M$. We show an explicit example in the case of Taft algebras.
{"title":"Central Hopf Monads and Braided Commutative Algebras","authors":"Noelia Bortolussi, Adriana Mejía Castaño, Martín Mombelli","doi":"arxiv-2409.01918","DOIUrl":"https://doi.org/arxiv-2409.01918","url":null,"abstract":"Let $ V$ be a braided tensor category and $ C$ a tensor category equipped\u0000with a braided tensor functor $G:Vto Z(C)$. For any exact indecomposable\u0000$C$-module category $M$, we explicitly construct a right adjoint of the action\u0000functor $rho:Z^V(C)to C^*_{M}$ afforded by $M$. Here $Z^V(C)$ is the\u0000M\"uger's centralizer of the subcategory $G(V)$ inside the center $Z^V(C)$,\u0000also known as the relative center. The construction is parallel to the one\u0000presented by K. Shimizu, but using instead the relative coend end. This\u0000adjunction turns out to be monadic, thus inducing Hopf monads $T_{V}: Cto C$,\u0000such that there is a monoidal equivalence of categories $ C_{T_{V}}simeq\u0000Z^V(C).$ If $bar{rho}: C^*_{ M}to Z^V(C)$ is the right adjoint of $rho,$\u0000then $bar{rho}(Id_{M})$ is the braided commutative algebra constructed in [R.\u0000Laugwitz and C. Walton. Braided commutative algebras over quantized enveloping\u0000algebras, Transform. Groups 26(3) (2021), 957--993]. As a consequence of our\u0000construction of these algebras, in terms of the right adjoint to $rho$, we can\u0000provide a recipe to compute them when $C=Rep(H# T)$ is the category of\u0000finite-dimensional representations of a finite-dimensional Hopf algebra $H# T$\u0000obtained by bosonization, and choosing an arbitrary $Rep(H# T)$-module\u0000category $M$. We show an explicit example in the case of Taft algebras.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225241","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}
Schur functions have been shown to satisfy certain stability properties and recurrence relations. In this paper, we prove analogs of these properties with Schur's $Q$-functions using vertex operator methods.
{"title":"Plethysm Stability of Schur's $Q$-functions","authors":"John Graf, Naihuan Jing","doi":"arxiv-2409.01479","DOIUrl":"https://doi.org/arxiv-2409.01479","url":null,"abstract":"Schur functions have been shown to satisfy certain stability properties and\u0000recurrence relations. In this paper, we prove analogs of these properties with\u0000Schur's $Q$-functions using vertex operator methods.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225242","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}
Any choice of a spherical fusion category defines an invariant of oriented closed 3-manifolds, which is computed by choosing a triangulation of the manifold and considering a state sum model that assigns a 6j symbol to every tetrahedron in this triangulation. This approach has been generalized to oriented closed 3-manifolds with defect data by Meusburger. In a recent paper, she constructed a family of invariants for such manifolds parametrised by the choice of certain spherical fusion categories, bimodule categories, finite bimodule functors and module natural transformations. Meusburger defined generalised 6j symbols for these objects, and introduces a state sum model that assigns a generalised 6j symbol to every tetrahedron in the triangulation of a manifold with defect data, where the type of 6j symbol used depends on what defect data occur within the tetrahedron. The present work provides non-trivial examples of suitable bimodule categories, bimodule functors and module natural transformation, all over categories of $G$-graded vector spaces. Our main result is the description of module functors in terms of matrices, which allows us to classify these functors when $G$ is a finite cyclic group. Furthermore, we calculate the generalised 6j symbols for categories of $G$-graded vector spaces, (bi-)module categories over such categories and (bi-)module functors.
{"title":"Generalised 6j symbols over the category of $G$-graded vector spaces","authors":"Fabio Lischka","doi":"arxiv-2409.09055","DOIUrl":"https://doi.org/arxiv-2409.09055","url":null,"abstract":"Any choice of a spherical fusion category defines an invariant of oriented\u0000closed 3-manifolds, which is computed by choosing a triangulation of the\u0000manifold and considering a state sum model that assigns a 6j symbol to every\u0000tetrahedron in this triangulation. This approach has been generalized to\u0000oriented closed 3-manifolds with defect data by Meusburger. In a recent paper,\u0000she constructed a family of invariants for such manifolds parametrised by the\u0000choice of certain spherical fusion categories, bimodule categories, finite\u0000bimodule functors and module natural transformations. Meusburger defined\u0000generalised 6j symbols for these objects, and introduces a state sum model that\u0000assigns a generalised 6j symbol to every tetrahedron in the triangulation of a\u0000manifold with defect data, where the type of 6j symbol used depends on what\u0000defect data occur within the tetrahedron. The present work provides non-trivial\u0000examples of suitable bimodule categories, bimodule functors and module natural\u0000transformation, all over categories of $G$-graded vector spaces. Our main\u0000result is the description of module functors in terms of matrices, which allows\u0000us to classify these functors when $G$ is a finite cyclic group. Furthermore,\u0000we calculate the generalised 6j symbols for categories of $G$-graded vector\u0000spaces, (bi-)module categories over such categories and (bi-)module functors.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253468","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 Frobenius map for the $SL_3$ skein module of any oriented 3-manifold specialized at a root of unity, and describe the map by way of threading certain polynomials along links. The homomorphism is a higher rank version of the Chebyshev-Frobenius homomorphism of Bonahon-Wong. The strategy builds on a previous construction of the Frobenius map for $SL_3$ skein algebras of punctured surfaces, using the Frobenius map of Parshall-Wang for the quantum group $mathcal{O}_q(SL_3).$
{"title":"Miraculous cancellations and the quantum Frobenius for $SL_3$ skein modules","authors":"Vijay Higgins","doi":"arxiv-2409.00351","DOIUrl":"https://doi.org/arxiv-2409.00351","url":null,"abstract":"We construct a quantum Frobenius map for the $SL_3$ skein module of any\u0000oriented 3-manifold specialized at a root of unity, and describe the map by way\u0000of threading certain polynomials along links. The homomorphism is a higher rank\u0000version of the Chebyshev-Frobenius homomorphism of Bonahon-Wong. The strategy\u0000builds on a previous construction of the Frobenius map for $SL_3$ skein\u0000algebras of punctured surfaces, using the Frobenius map of Parshall-Wang for\u0000the quantum group $mathcal{O}_q(SL_3).$","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198766","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}
This is the second part of a project aimed at formalizing Rozansky-Witten models in the functorial field theory framework. In the first part we constructed a symmetric monoidal $(infty, 3)$-category $mathscr{CRW}$ of commutative Rozansky-Witten models with the goal of approximating the $3$-category of Kapustin and Rozansky. In this paper we extend work of Brunner, Carqueville, Fragkos, and Roggenkamp on the affine Rozansky-Witten models: we exhibit a functor connecting their $2$-category of matrix factorizations with the homotopy $2$-category of $mathscr{CRW}$, and calculate the associated TFTs.
{"title":"Higher categories of push-pull spans, II: Matrix factorizations","authors":"Lorenzo Riva","doi":"arxiv-2409.00219","DOIUrl":"https://doi.org/arxiv-2409.00219","url":null,"abstract":"This is the second part of a project aimed at formalizing Rozansky-Witten\u0000models in the functorial field theory framework. In the first part we\u0000constructed a symmetric monoidal $(infty, 3)$-category $mathscr{CRW}$ of\u0000commutative Rozansky-Witten models with the goal of approximating the\u0000$3$-category of Kapustin and Rozansky. In this paper we extend work of Brunner,\u0000Carqueville, Fragkos, and Roggenkamp on the affine Rozansky-Witten models: we\u0000exhibit a functor connecting their $2$-category of matrix factorizations with\u0000the homotopy $2$-category of $mathscr{CRW}$, and calculate the associated\u0000TFTs.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225256","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 explicitly construct a (unitary) $mathbb{Z}/2mathbb{Z}$ permutation gauging of a (unitary) modular category $mathcal{C}$. In particular, the formula for the modular data of the gauged theory is provided in terms of modular data of $mathcal{C}$, which provides positive evidence of the reconstruction program. Moreover as a direct consequence, the formula for the fusion rules is derived, generalizing the results of Edie-Michell-Jones-Plavnik. Our construction explicitly shows the genus-$0$ data of the gauged theory contains higher genus data of the original theory. As applications, we obtain an identity for the modular data that does not come from modular group relations, and we prove that representations of the symmetric mapping class group (associated to closed surfaces) coming from weakly group theoretical modular categories have finite images.
{"title":"On $mathbb{Z}/2mathbb{Z}$ permutation gauging","authors":"Zhengwei Liu, Yuze Ruan","doi":"arxiv-2408.17195","DOIUrl":"https://doi.org/arxiv-2408.17195","url":null,"abstract":"We explicitly construct a (unitary) $mathbb{Z}/2mathbb{Z}$ permutation\u0000gauging of a (unitary) modular category $mathcal{C}$. In particular, the\u0000formula for the modular data of the gauged theory is provided in terms of\u0000modular data of $mathcal{C}$, which provides positive evidence of the\u0000reconstruction program. Moreover as a direct consequence, the formula for the\u0000fusion rules is derived, generalizing the results of\u0000Edie-Michell-Jones-Plavnik. Our construction explicitly shows the genus-$0$\u0000data of the gauged theory contains higher genus data of the original theory. As\u0000applications, we obtain an identity for the modular data that does not come\u0000from modular group relations, and we prove that representations of the\u0000symmetric mapping class group (associated to closed surfaces) coming from\u0000weakly group theoretical modular categories have finite images.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225257","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 solve the problem of how to classify the first-order vertex-algebraic deformations for any grading-restricted vertex algebra $V$ that is freely generated by homogeneous elements of positive weights. We approach by computing the second cohomology $H^2_{1/2}(V, V)$ constructed by Yi-Zhi Huang. We start with the cocycle on two generators and show that its cohomology class is completely determined by its singular part. To extend the cocycle to any pair of elements in $V$, we take a generating function approach, formulate the cocycle equation, and show that all the complementary solutions are coboundaries. Then we use a very general procedure to construct a particular solution. The procedure applies to vertex algebras that are not freely generated. As a by-product, we show that $H^2_{1/2}(V, V) = H^2_infty(V, V)$. Using these results, we explicitly determine the first-order deformations of the universal Virasoro VOA $Vir_c$, universal affine VOA $V^l(mathfrak{g})$, Heisenberg VOA $V^l(mathfrak{h})$, and the universal Zamolodchikov VOA $W_3^c$.
{"title":"First-order deformations of freely generated vertex algebras","authors":"Vladimir Kovalchuk, Fei Qi","doi":"arxiv-2408.16309","DOIUrl":"https://doi.org/arxiv-2408.16309","url":null,"abstract":"We solve the problem of how to classify the first-order vertex-algebraic\u0000deformations for any grading-restricted vertex algebra $V$ that is freely\u0000generated by homogeneous elements of positive weights. We approach by computing\u0000the second cohomology $H^2_{1/2}(V, V)$ constructed by Yi-Zhi Huang. We start\u0000with the cocycle on two generators and show that its cohomology class is\u0000completely determined by its singular part. To extend the cocycle to any pair\u0000of elements in $V$, we take a generating function approach, formulate the\u0000cocycle equation, and show that all the complementary solutions are\u0000coboundaries. Then we use a very general procedure to construct a particular\u0000solution. The procedure applies to vertex algebras that are not freely\u0000generated. As a by-product, we show that $H^2_{1/2}(V, V) = H^2_infty(V, V)$.\u0000Using these results, we explicitly determine the first-order deformations of\u0000the universal Virasoro VOA $Vir_c$, universal affine VOA $V^l(mathfrak{g})$,\u0000Heisenberg VOA $V^l(mathfrak{h})$, and the universal Zamolodchikov VOA\u0000$W_3^c$.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198769","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}
Alonso Perez-Lona, Daniel Robbins, Eric Sharpe, Thomas Vandermeulen, Xingyang Yu
In this paper we discuss gauging noninvertible zero-form symmetries in two dimensions, extending our previous work. Specifically, in this work we discuss more general gauged noninvertible symmetries in which the noninvertible symmetry is not multiplicity free, and discuss the case of Rep$(A_4)$ in detail. We realize Rep$(A_4)$ gaugings for the $c = 1$ CFT at the exceptional point in the moduli space and find new self-duality under gauging a certain non-group algebra object, leading to a larger noninvertible symmetry Rep$(SL(2, Z_3))$. We also discuss more general examples of decomposition in two-dimensional gauge theories with trivially-acting gauged noninvertible symmetries.
{"title":"Notes on gauging noninvertible symmetries, part 2: higher multiplicity cases","authors":"Alonso Perez-Lona, Daniel Robbins, Eric Sharpe, Thomas Vandermeulen, Xingyang Yu","doi":"arxiv-2408.16811","DOIUrl":"https://doi.org/arxiv-2408.16811","url":null,"abstract":"In this paper we discuss gauging noninvertible zero-form symmetries in two\u0000dimensions, extending our previous work. Specifically, in this work we discuss\u0000more general gauged noninvertible symmetries in which the noninvertible\u0000symmetry is not multiplicity free, and discuss the case of Rep$(A_4)$ in\u0000detail. We realize Rep$(A_4)$ gaugings for the $c = 1$ CFT at the exceptional\u0000point in the moduli space and find new self-duality under gauging a certain\u0000non-group algebra object, leading to a larger noninvertible symmetry Rep$(SL(2,\u0000Z_3))$. We also discuss more general examples of decomposition in\u0000two-dimensional gauge theories with trivially-acting gauged noninvertible\u0000symmetries.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"105 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198767","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}
In this paper, we investigate the differential smoothness of bi-quadratic algebras on three generators with PBW basis.
在本文中，我们研究了具有 PBW 基的三发电机上的双四边形布拉的微分平滑性。
{"title":"Smooth geometry of bi-quadratic algebras on three generators with PBW basis","authors":"Andrés Rubiano, Armando Reyes","doi":"arxiv-2408.16648","DOIUrl":"https://doi.org/arxiv-2408.16648","url":null,"abstract":"In this paper, we investigate the differential smoothness of bi-quadratic\u0000algebras on three generators with PBW basis.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198768","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}
In this work we study the analogues of R-matrices that arise in 5d non-commutative topological-holomorphic Chern-Simons theory, which is known to describe twisted M-theory. We first study the intersections of line and surface operators in 5d Chern-Simons theory, which correspond to M2- and M5-branes, respectively. A Feynman diagram computation of the correlation function of this configuration furnishes an expression reminiscent of an R-matrix derivable from 4d Chern-Simons theory. We explain how this object is related to a Miura operator that is known to realize (matrix-extended) $W_{infty}$-algebras. For 5d Chern-Simons theory with nonabelian gauge group, we then perform a Feynman diagram computation of coproducts for deformed double current algebras and matrix-extended $W_{infty}$-algebras from fusions of M2-branes, M5-branes, and M2-M5 intersections.
在这项工作中，我们研究了在 5d 非交换拓扑-多态 Chern-Simons 理论中出现的 R 矩的类似物，该理论被称为描述扭曲的 M 理论。我们首先研究了 5d Chern-Simons 理论中线和面运算符的交点，它们分别对应于 M2 和 M5-branes。通过费曼图计算这种配置的相关函数，我们得到了一个类似于可从 4d Chern-Simons 理论推导出的 R 矩阵的表达式。我们解释了这个对象是如何与已知可以实现（矩阵扩展的）$W_{infty}$-代数的三浦算子相关联的。对于具有非阿贝尔规规群的5d切尔-西蒙斯理论，我们将从M2-branes、M5-branes和M2-M5交集的融合中，对变形双电流代数和矩阵扩展的$W_{infty}$代数的共乘进行费曼迪图计算。
{"title":"R-matrices from Feynman Diagrams in 5d Chern-Simons Theory and Twisted M-theory","authors":"Meer Ashwinkumar","doi":"arxiv-2408.15732","DOIUrl":"https://doi.org/arxiv-2408.15732","url":null,"abstract":"In this work we study the analogues of R-matrices that arise in 5d\u0000non-commutative topological-holomorphic Chern-Simons theory, which is known to\u0000describe twisted M-theory. We first study the intersections of line and surface\u0000operators in 5d Chern-Simons theory, which correspond to M2- and M5-branes,\u0000respectively. A Feynman diagram computation of the correlation function of this\u0000configuration furnishes an expression reminiscent of an R-matrix derivable from\u00004d Chern-Simons theory. We explain how this object is related to a Miura\u0000operator that is known to realize (matrix-extended) $W_{infty}$-algebras. For\u00005d Chern-Simons theory with nonabelian gauge group, we then perform a Feynman\u0000diagram computation of coproducts for deformed double current algebras and\u0000matrix-extended $W_{infty}$-algebras from fusions of M2-branes, M5-branes, and\u0000M2-M5 intersections.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"2013 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198771","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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