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European Mathematical Society Publishing House
2024-03-19 03:21:20
3
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=QT&vol=7&iss=2&update_since=2024-03-19
Quantum Topology
Quantum Topol.
QT
1663-487X
1664-073X
General
10.4171/QT
http://www.ems-ph.org/doi/10.4171/QT
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
7
2016
2
Orbifold completion of defect bicategories
Nils
Carqueville
Universität Wien, WIEN, AUSTRIA
Ingo
Runkel
Universität Hamburg, HAMBURG, GERMANY
Pivotal bicategories, TQFT, matrix factorisations
Orbifolds of two-dimensional quantum field theories have a natural formulation in terms of defects or domain walls. This perspective allows for a rich generalisation of the orbifolding procedure, which we study in detail for the case of topological field theories. Namely, a TFT with defects gives rise to a pivotal bicategory of "world sheet phases" and defects between them. We develop a general framework which takes such a bicategory $\mathcal B$ as input and returns its "orbifold completion" $\mathcal B_{\mathrm {orb}}$. The completion satisfies the natural properties $\mathcal B \subset \mathcal B_{\mathrm {orb}}$ and $(\mathcal B_{\mathrm {orb}})_{\mathrm{orb}} \cong \mathcal B_{\mathrm {orb}}$, and it gives rise to various new equivalences and nondegeneracy results. When applied to TFTs, the objects in $\mathcal B_{\mathrm {orb}}$ correspond to generalised orbifolds of the theories in $\mathcal B$. In the example of Landau–Ginzburg models we recover and unify conventional equivariant matrix factorisations, prove when and how (generalised) orbifolds again produce open/closed TFTs, and give nontrivial examples of new orbifold equivalences.
Category theory; homological algebra
General
Manifolds and cell complexes
203
279
10.4171/QT/76
http://www.ems-ph.org/doi/10.4171/QT/76
Geometric filtrations of string links and homology cylinders
James
Conant
University of Tennessee, KNOXVILLE, UNITED STATES
Robert
Schneiderman
Lehman College, City University of New York, BRONX, UNITED STATES
Peter
Teichner
Max Planck Institut für Mathematik, BONN, GERMANY
Artin representation, clasper, homology cylinder, Johnson filtration, string link, Whitney tower, Y-filtration
We show that the group of string links modulo order $n$ twisted Whitney tower concordance is an extension of the image of the nilpotent Artin representation by a finite 2-group. Moreover, this 2-group is generated by band sums of iterated Bing-doubles of any string knot with nonzero Arf invariant. We also analyze the Goussarov–Habiro clasper fi ltration of the group of 3-dimensional homology cylinders modulo homology cobordism, importing techniques from our work on Whitney towers to improve on results of J. Levine. In particular, we classify the graded group associated to the Goussarov–Habiro fi ltration in all orders except $4n + 1$. In this last case, it is classi fied up to unknown 2-torsion with a precise upper bound. These calculations confi rm conjectures of Levine in the even cases, and improve on his conjectures in the odd cases. In the last section of this paper we connect the settings of string links and homology cylinders by analyzing a geometric map, originally formulated by N. Habegger.
Manifolds and cell complexes
281
328
10.4171/QT/77
http://www.ems-ph.org/doi/10.4171/QT/77
An odd categorification of $U_q (\mathfrak{sl}_2)$
Alexander
Ellis
University of Oregon, EUGENE, UNITED STATES
Aaron
Lauda
University of Southern California, LOS ANGELES, UNITED STATES
Covering algebras, categorified quantum groups, cyclotomic quotients, odd nil-Hecke algebra, odd Khovanov homology
We define a 2-category that categorifies the covering Kac–Moody algebra for $\mathfrak{sl}_2$ introduced by Clark and Wang. This categorification forms the structure of a super-2-category as formulated by Kang, Kashiwara, and Oh. The super-2-category structure introduces a $\mathbb{Z} \times \mathbb{Z}_{2}$-grading giving its Grothendieck group the structure of a free module over the group algebra of $\mathbb{Z} \times \mathbb Z_2$. By specializing the $\mathbb{Z}_{2}$-action to +1 or to −1, the construction specializes to an “odd” categorification of $\mathfrak{sl}_2$ and to a supercategorification of $\mathfrak{osp}_{1|2}$, respectively.
Group theory and generalizations
Nonassociative rings and algebras
Manifolds and cell complexes
329
433
10.4171/QT/78
http://www.ems-ph.org/doi/10.4171/QT/78