We make explicit computations in the formal symplectic geometry of Kontsevich and determine the Euler characteristics of the three cases, namely commutative, Lie and associative ones, up to certain weights.From these, we obtain some non-triviality results in each case. In particular, we determine the integral Euler characteristics of the outer automorphism groups Out F_n of free groups for all n <= 10 and prove the existence of plenty of rational cohomology classes of odd degrees. We also clarify the relationship of the commutative graph homology with finite type invariants of homology 3-spheres as well as the leaf cohomology classes for transversely symplectic foliations. Furthermore we prove the existence of several new non-trivalent graph homology classes of odd degrees. Based on these computations, we propose a few conjectures and problems on the graph homology and the characteristic classes of the moduli spaces of graphs as well as curves.
{"title":"Computations in formal symplectic geometry and characteristic classes of moduli spaces","authors":"S. Morita, Takuya Sakasai, Masaaki Suzuki","doi":"10.4171/QT/61","DOIUrl":"https://doi.org/10.4171/QT/61","url":null,"abstract":"We make explicit computations in the formal symplectic geometry of Kontsevich and determine the Euler characteristics of the three cases, namely commutative, Lie and associative ones, up to certain weights.From these, we obtain some non-triviality results in each case. In particular, we determine the integral Euler characteristics of the outer automorphism groups Out F_n of free groups for all n <= 10 and prove the existence of plenty of rational cohomology classes of odd degrees. We also clarify the relationship of the commutative graph homology with finite type invariants of homology 3-spheres as well as the leaf cohomology classes for transversely symplectic foliations. Furthermore we prove the existence of several new non-trivalent graph homology classes of odd degrees. Based on these computations, we propose a few conjectures and problems on the graph homology and the characteristic classes of the moduli spaces of graphs as well as curves.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"1 1","pages":"139-182"},"PeriodicalIF":1.1,"publicationDate":"2012-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86645444","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Andersen, N. Gammelgaard, Magnus Roed Lauridsen
We give a differential geometric construction of a connection, which we call the Hitchin connection, in the bundle of quantum Hilbert spaces arising from metaplectically corrected geometric quantization of a prequantizable, symplectic manifold, endowed with a rigid family of Kähler structures, all of which give vanishing first Dolbeault cohomology groups. This generalizes work of both Hitchin, Scheinost and Schottenloher, and Andersen, since our construction does not need that the first Chern class is proportional to the class of the symplectic form, nor do we need compactness of the symplectic manifold in question. Furthermore, when we are in a setting similar to the moduli space, we give an explicit formula and show that this connection agrees with previous constructions. Mathematics Subject Classification (2010). 53D50, 32Q55.
{"title":"Hitchin’s connection in metaplectic quantization","authors":"J. Andersen, N. Gammelgaard, Magnus Roed Lauridsen","doi":"10.4171/QT/31","DOIUrl":"https://doi.org/10.4171/QT/31","url":null,"abstract":"We give a differential geometric construction of a connection, which we call the Hitchin connection, in the bundle of quantum Hilbert spaces arising from metaplectically corrected geometric quantization of a prequantizable, symplectic manifold, endowed with a rigid family of Kähler structures, all of which give vanishing first Dolbeault cohomology groups. This generalizes work of both Hitchin, Scheinost and Schottenloher, and Andersen, since our construction does not need that the first Chern class is proportional to the class of the symplectic form, nor do we need compactness of the symplectic manifold in question. Furthermore, when we are in a setting similar to the moduli space, we give an explicit formula and show that this connection agrees with previous constructions. Mathematics Subject Classification (2010). 53D50, 32Q55.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"20 1","pages":"327-357"},"PeriodicalIF":1.1,"publicationDate":"2012-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87020019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
C. Armond, S. Garoufalidis and T.Le have shown that a unicolored Jones polynomial of a B-adequate link has a stable tail at large colors. We categorify this tail by showing that Khovanov homology of a unicolored link also has a stable tail, whose graded Euler characteristic coincides with the tail of the Jones polynomial.
C. Armond, S. Garoufalidis和T.Le已经证明了b -充足链路的单色Jones多项式在大颜色时具有稳定的尾。我们通过证明单色链路的Khovanov同调也有一个稳定的尾巴来对这个尾巴进行分类，这个稳定的尾巴的梯度欧拉特征与琼斯多项式的尾巴一致。
{"title":"Khovanov homology of a unicolored b-adequate link has a tail","authors":"L. Rozansky","doi":"10.4171/QT/58","DOIUrl":"https://doi.org/10.4171/QT/58","url":null,"abstract":"C. Armond, S. Garoufalidis and T.Le have shown that a unicolored Jones polynomial of a B-adequate link has a stable tail at large colors. We categorify this tail by showing that Khovanov homology of a unicolored link also has a stable tail, whose graded Euler characteristic coincides with the tail of the Jones polynomial.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"57 1","pages":"541-579"},"PeriodicalIF":1.1,"publicationDate":"2012-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80221505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the representation theory of the smallest quantum group and its categori- fication. The first part of the paper contains an easy visualization of the3j -symbols in terms of weighted signed line arrangements in a fixed triangle and new binomial expressions for the 3j -symbols. All these formulas are realized as graded Euler characteristics. The3j -symbols appear as new generalizations of Kazhdan-Lusztig polynomials. A crucial result of the paper is that complete intersection rings can be employed to obtain rational Euler characteristics, hence to categorify rational quantum numbers. This is the main tool for our categorification of the Jones-Wenzl projector, ‚-networks and tetrahedron net- works. Networks and their evaluations play an important role in the Turaev-Viro construction of 3-manifold invariants. We categorify these evaluations by Ext-algebras of certain simple Harish-Chandra bimodules. The relevance of this construction to categorified colored Jones invariants and invariants of3-manifolds will be studied in detail in subsequent papers.
{"title":"Categorifying fractional Euler characteristics, Jones–Wenzl projectors and 3j-symbols","authors":"I. Frenkel, C. Stroppel, Joshua Sussan","doi":"10.4171/QT/28","DOIUrl":"https://doi.org/10.4171/QT/28","url":null,"abstract":"We study the representation theory of the smallest quantum group and its categori- fication. The first part of the paper contains an easy visualization of the3j -symbols in terms of weighted signed line arrangements in a fixed triangle and new binomial expressions for the 3j -symbols. All these formulas are realized as graded Euler characteristics. The3j -symbols appear as new generalizations of Kazhdan-Lusztig polynomials. A crucial result of the paper is that complete intersection rings can be employed to obtain rational Euler characteristics, hence to categorify rational quantum numbers. This is the main tool for our categorification of the Jones-Wenzl projector, ‚-networks and tetrahedron net- works. Networks and their evaluations play an important role in the Turaev-Viro construction of 3-manifold invariants. We categorify these evaluations by Ext-algebras of certain simple Harish-Chandra bimodules. The relevance of this construction to categorified colored Jones invariants and invariants of3-manifolds will be studied in detail in subsequent papers.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"149 1","pages":"181-253"},"PeriodicalIF":1.1,"publicationDate":"2012-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73421008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we develop the theory of bimodules over von Neumann algebras, with an emphasis on categorical aspects. We clarify the relationship between dualizability and finite index. We also show that, for von Neumann algebras with finite dimensional centers, the Haagerup L 2 -space and Connes fusion are functorial with respect to homor- phisms of finite index. Along the way, we describe a string diagram notation for maps between bimodules that are not necessarily bilinear.
{"title":"Dualizability and index of subfactors","authors":"A. Bartels, Christopher L. Douglas, A. Henriques","doi":"10.4171/QT/53","DOIUrl":"https://doi.org/10.4171/QT/53","url":null,"abstract":"In this paper, we develop the theory of bimodules over von Neumann algebras, with an emphasis on categorical aspects. We clarify the relationship between dualizability and finite index. We also show that, for von Neumann algebras with finite dimensional centers, the Haagerup L 2 -space and Connes fusion are functorial with respect to homor- phisms of finite index. Along the way, we describe a string diagram notation for maps between bimodules that are not necessarily bilinear.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"68 1","pages":"289-345"},"PeriodicalIF":1.1,"publicationDate":"2011-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75635272","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The free abelian group C carries both a cohomological grading h and a quantum grading q. The differential dKh increases h by 1 and preserves q, so that the Khovanov cohomology is bigraded. We write F C for the decreasing filtration defined by the bigrading, so F C is generated by elements whose cohomological grading is not less than i and whose quantum grading is not less than j . In general, given abelian groups with a decreasing filtration indexed by Z Z, we will say that a group homomorphism has order .s; t/ if .F i;j / F iCs;jCt . So dKh has order .1; 0/. In [5], a new invariant I .K/ was defined using singular instantons, and it was shown that I .K/is related to Kh.K/ through a spectral sequence. The notation K here denotes the mirror image of K. Building on the results of [5], we establish the following theorem in this paper.
自由阿贝尔群C同时携带上同调阶h和量子阶q。微分dKh使h增加1并保持q，从而使Khovanov上同调被大阶化。我们将F C表示由级配定义的递减过滤，因此F C是由上同级配不小于i且量子级配不小于j的元素产生的。一般来说，给定以zz为指标的滤除量递减的阿贝尔群，我们称群同态为s阶;t/ if .F i;j / F i;所以dKh的阶是。1;0 /。在[5]中，利用奇异实例定义了一个新的不变量I . k /，并证明了I . k /与Kh有关。K/通过谱序列。这里的符号K表示K的镜像。根据[5]的结果，本文建立了以下定理:
{"title":"Filtrations on instanton homology","authors":"P. Kronheimer, T. Mrowka","doi":"10.4171/QT/47","DOIUrl":"https://doi.org/10.4171/QT/47","url":null,"abstract":"The free abelian group C carries both a cohomological grading h and a quantum grading q. The differential dKh increases h by 1 and preserves q, so that the Khovanov cohomology is bigraded. We write F C for the decreasing filtration defined by the bigrading, so F C is generated by elements whose cohomological grading is not less than i and whose quantum grading is not less than j . In general, given abelian groups with a decreasing filtration indexed by Z Z, we will say that a group homomorphism has order .s; t/ if .F i;j / F iCs;jCt . So dKh has order .1; 0/. In [5], a new invariant I .K/ was defined using singular instantons, and it was shown that I .K/is related to Kh.K/ through a spectral sequence. The notation K here denotes the mirror image of K. Building on the results of [5], we establish the following theorem in this paper.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"5 1","pages":"61-97"},"PeriodicalIF":1.1,"publicationDate":"2011-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89079469","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Erratum to: “A categorification of quantum sl(n)”","authors":"M. Khovanov, Aaron D. Lauda","doi":"10.4171/QT/15","DOIUrl":"https://doi.org/10.4171/QT/15","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"1 1","pages":"97-99"},"PeriodicalIF":1.1,"publicationDate":"2011-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90310420","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that closed differential forms on a smooth manifold X can be interpreted astopological(respectivelyEudlidean)supersymmetricfieldtheoriesofdimension0j1overX. As a consequence, concordance classes of such field theories are shown to represent de Rham cohomology. The main contribution of this paper is to make all new mathematical notions regarding supersymmetric field theories precise.
{"title":"Differential forms and 0-dimensional supersymmetric field theories","authors":"Henning Hohnhold, M. Kreck, S. Stolz, P. Teichner","doi":"10.4171/QT/12","DOIUrl":"https://doi.org/10.4171/QT/12","url":null,"abstract":"We show that closed differential forms on a smooth manifold X can be interpreted astopological(respectivelyEudlidean)supersymmetricfieldtheoriesofdimension0j1overX. As a consequence, concordance classes of such field theories are shown to represent de Rham cohomology. The main contribution of this paper is to make all new mathematical notions regarding supersymmetric field theories precise.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"12 1","pages":"1-41"},"PeriodicalIF":1.1,"publicationDate":"2011-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77738122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define a category of planar diagrams whose Grothendieck group contains an integral version of the infinite rank Heisenberg algebra, thus yielding a categorification of this algebra. Our category, which is a q-deformation of one defined by Khovanov, acts naturally on the categories of modules for Hecke algebras of typeA and finite general linear groups. In this way, we obtain a categorification of the bosonic Fock space. We also develop the theory of parabolic induction and restriction functors for finite groups and prove general results on biadjointness and cyclicity in this setting. Mathematics Subject Classification (2010). Primary: 20C08, 17B65; Secondary: 16D90.
{"title":"Erratum to: Hecke algebras, finite general linear groups, and Heisenberg categorification","authors":"Anthony M. Licata, Alistair Savage","doi":"10.4171/QT/37","DOIUrl":"https://doi.org/10.4171/QT/37","url":null,"abstract":"We define a category of planar diagrams whose Grothendieck group contains an integral version of the infinite rank Heisenberg algebra, thus yielding a categorification of this algebra. Our category, which is a q-deformation of one defined by Khovanov, acts naturally on the categories of modules for Hecke algebras of typeA and finite general linear groups. In this way, we obtain a categorification of the bosonic Fock space. We also develop the theory of parabolic induction and restriction functors for finite groups and prove general results on biadjointness and cyclicity in this setting. Mathematics Subject Classification (2010). Primary: 20C08, 17B65; Secondary: 16D90.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"60 1","pages":"183-183"},"PeriodicalIF":1.1,"publicationDate":"2011-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84977859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
R. Askanazi, S. Chmutov, C. Estill, J. Michel, P. Stollenwerk
For a graph embedded into a surface, we relate many combinatorial parameters of the cycle matroid of the graph and the bond matroid of the dual graph with the topological parameters of the embedding. This will give an expression of the polynomial, defined by M. Las Vergnas in a combinatorial way using matroids as a specialization of the Krushkal polynomial, defined using the symplectic structure in the first homology group of the surface.
对于嵌入曲面的图，我们将图的循环矩阵和对偶图的键阵的许多组合参数与嵌入的拓扑参数联系起来。这将给出由M. Las Vergnas以组合方式定义的多项式的表达式，该多项式使用拟阵作为Krushkal多项式的专门化，使用曲面的第一个同调群中的辛结构定义。
{"title":"Polynomial invariants of graphs on surfaces","authors":"R. Askanazi, S. Chmutov, C. Estill, J. Michel, P. Stollenwerk","doi":"10.4171/QT/35","DOIUrl":"https://doi.org/10.4171/QT/35","url":null,"abstract":"For a graph embedded into a surface, we relate many combinatorial parameters of the cycle matroid of the graph and the bond matroid of the dual graph with the topological parameters of the embedding. This will give an expression of the polynomial, defined by M. Las Vergnas in a combinatorial way using matroids as a specialization of the Krushkal polynomial, defined using the symplectic structure in the first homology group of the surface.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"4 1","pages":"77-90"},"PeriodicalIF":1.1,"publicationDate":"2010-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87975368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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