Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1507.00941

On 2-Dimensional Dijkgraaf-Witten Theory with Defects

Aria L. Doughterty·Hwajin Park·David N. Yetter
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we provide a construction of a state-sum model for finite gauge-group Dijkgraaf-Witten theory on surfaces with codimension 1 defects. The construction requires not only that the triangulation be subordinate to the filtration, but flag-like: each simplex of the triangulation is either disjoint from the defect curve, or intersects it in a closed face. The construction allows internal degrees of freedom in the defect curves by introducing a second gauge-group from which edges of the curve are labeled in the state-sum construction. Edges incident with the defect, but not lying in it, have states lying in a set with commuting actions of the two gauge-groups. We determine the appropriate generalizations of the 2-cocycles specifying twistings of defect-free 2D Dijkgraaf-Witten theory. Examples arising by restriction of group 2-cocycles, and constructed from characters of the 2-dimensional guage group are presented. This research was carried out at Summer Undergraduate Mathematics Research (SUMaR) math REU at Kansas State University, funded by NSF under DMS award #1262877.

Explore related subjects

math.QAmath.GT

Keep this discovery

BibTeXRIS

Aria L. Doughterty, Hwajin Park, David N. Yetter. 2015-07-03. On 2-Dimensional Dijkgraaf-Witten Theory with Defects. https://arxiv.org/abs/1507.00941

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On Diagrammatic Categorification of Verma Modules I: Braiding

In this paper, we study the extensions of KLRW algebras to tensor products of Verma module representations of $\mathfrak{sl}_2$. Our motivation is to construct a theory of Khovanov homology for knot complements in $S^3$ (and which also categorifies the Gukov-Manolescu two-variable series for knot complements), which will be done in the second part of this work. We construct the categorification of R-matrices for Verma modules as functors given by derived tensor products with diagrammatic bimodules and explicitly compute their projective resolutions. We also prove these braiding functors induce an action of the braid group on the relevant categories. Then, we describe how to incorporate strands in finite-dimensional representations of $\mathfrak{sl}_2$, thereby establishing functors that serve as the Khovanov homology on a braid complement. In the case of the unknot, this gives knot homologies in $S^1\times D^2$, which we compare to Annular Khovanov Homology through several examples and show they are very closely related, conjecturing they are of the same dimension. We conclude with a proposal for the categorification of the cups and caps of Verma module colored strands, which we build upon in the next paper.

math.QA↗
arXiv2026

Some finite dimensional representations of shifted quantum affine algebras of type A

In this paper, we study finite dimensional representations of shifted quantum affine algebras of type A. We give an explicit description of the tensor product of simple evaluation modules of the quantum loop algebra and a one-dimensional representation of the shifted quantum affine algebra under the separation condition. As a consequence, we give the q-characters of some finite dimensional simple modules of the shifted quantum affine algebra.

math.QA↗