skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: On dualizability of braided tensor categories
We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided multi-fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively two-, three- and four-dimensional framed local topological field theories. In particular, we produce a framed three-dimensional local topological field theory attached to the category of representations of a quantum group at any value of $$q$$ .  more » « less
Award ID(s):
2000093
PAR ID:
10253365
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Compositio Mathematica
Volume:
157
Issue:
3
ISSN:
0010-437X
Page Range / eLocation ID:
435 to 483
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; ; (, SciPost Physics)
    Topological domain walls separating 2+1 dimensional topologically ordered phases can be understood in terms of Witt equivalences between the UMTCs describing anyons in the bulk topological orders. However, this picture does not provide a framework for decomposing stacks of multiple domain walls into superselection sectors — i.e., into fundamental domain wall types that cannot be mixed by any local operators. Such a decomposition can be understood using an alternate framework in the case that the topological order is anomaly-free, in the sense that it can be realized by a commuting projector lattice model. By placing these Witt equivalences in the context of a 3-category of potentially anomalous (2+1)D topological orders, we develop a framework for computing the decomposition of parallel topological domain walls into indecomposable superselection sectors, extending the previous understanding to topological orders with non-trivial anomaly. We characterize the superselection sectors in terms of domain wall particle mobility, which we formalize in terms of tunnelling operators. The mathematical model for the 3-category of topological orders is the 3-category of fusion categories enriched over a fixed unitary modular tensor category. 
    more » « less
  2. (, Communications of the American Mathematical Society)
    This paper is the first of a pair that aims to classify a large number of the type I I II quantum subgroups of the categories C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . In this work we classify the braided auto-equivalences of the categories of local modules for all known type I I quantum subgroups of C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . We find that the symmetries are all non-exceptional except for four cases (up to level-rank duality). These exceptional cases are the orbifolds C ( s l 2 , 16 ) Rep ⁡ ( Z 2 ) 0 \mathcal {C}(\mathfrak {sl}_{2}, 16)^0_{\operatorname {Rep}(\mathbb {Z}_{2})} , C ( s l 3 , 9 ) Rep ⁡ ( Z 3 ) 0 \mathcal {C}(\mathfrak {sl}_{3}, 9)^0_{\operatorname {Rep}(\mathbb {Z}_{3})} , C ( s l 4 , 8 ) Rep ⁡ ( Z 4 ) 0 \mathcal {C}(\mathfrak {sl}_{4}, 8)^0_{\operatorname {Rep}(\mathbb {Z}_{4})} , and C ( s l 5 , 5 ) Rep ⁡ ( Z 5 ) 0 \mathcal {C}(\mathfrak {sl}_{5}, 5)^0_{\operatorname {Rep}(\mathbb {Z}_{5})} . We develop several technical tools in this work. We give a skein theoretic description of the orbifold quantum subgroups of C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . Our methods here are general, and the techniques developed will generalise to give skein theory for any orbifold of a braided tensor category. We also give a formulation of orthogonal level-rank duality in the type D D - D D case, which is used to construct one of the exceptionals. We uncover an unexpected connection between quadratic categories and exceptional braided auto-equivalences of the orbifolds. We use this connection to construct two of the four exceptionals. In the sequel to this paper we will use the classified braided auto-equivalences to construct the corresponding type I I II quantum subgroups of the categories C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . This will essentially finish the type I I II classification for s l n \mathfrak {sl}_n modulo type I I classification. When paired with Gannon’s type I I classification for r ≤ 6 r\leq 6 , our results will complete the type I I II classification for these same ranks. This paper includes an appendix by Terry Gannon, which provides useful results on the dimensions of objects in the categories C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . 
    more » « less
  3. ; (, Transformation Groups)
    We prove that commutative algebras in braided tensor categories do not admit faithful Hopf algebra actions unless they come from group actions. We also show that a group action allows us to see the algebra as the regular algebra in the representation category of the acting group. 
    more » « less
  4. ; ; ; ; ; ; (, Communications in Mathematical Physics)
    Abstract We study a commutant-closed collection of von Neumann algebras acting on a common Hilbert space indexed by a poset with an order-reversing involution. We give simple geometric axioms for the poset which allow us to construct a braided tensor category of superselection sectors analogous to the construction of Gabbiani and Fröhlich for conformal nets. For cones in$$\mathbb {R}^2$$ R 2 , we weaken our conditions to a bounded spread version of Haag duality and obtain similar results. We show that intertwined nets of algebras have isomorphic braided tensor categories of superselection sectors. Finally, we show that the categories constructed here are equivalent to those constructed by Naaijkens and Ogata for certain 2D quantum spin systems. 
    more » « less
  5. ; ; ; ; (, Journal of the London Mathematical Society)
    Abstract For a finite group , a ‐crossed braided fusion category is a ‐graded fusion category with additional structures, namely, a ‐action and a ‐braiding. We develop the notion of ‐crossed braided zesting: an explicit method for constructing new ‐crossed braided fusion categories from a given one by means of cohomological data associated with the invertible objects in the category and grading group . This is achieved by adapting a similar construction for (braided) fusion categories recently described by the authors. All ‐crossed braided zestings of a given category are ‐extensions of their trivial component and can be interpreted in terms of the homotopy‐based description of Etingof, Nikshych, and Ostrik. In particular, we explicitly describe which ‐extensions correspond to ‐crossed braided zestings. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email