[Paper Review] On 2-Dimensional Dijkgraaf-Witten Theory with Defects

Aria L. Doughterty, Hwajin Park+1 more|arXiv (Cornell University)|Jul 3, 2015

Black Holes and Theoretical Physics5 references3 citations

TL;DR

This paper constructs a state-sum topological quantum field theory for 2-dimensional Dijkgraaf-Witten theory with codimension-1 defects by introducing a flag-like triangulation condition and a second gauge group for defect edges. The key contribution is a topological invariant for surfaces with curves, defined via twisted cocycles on three groups (H, X, G), generalizing the untwisted Dijkgraaf-Witten invariant to include internal degrees of freedom on defects.

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.

Motivation & Objective

• To extend 2D Dijkgraaf-Witten theory to include codimension-1 defects (curves) in surfaces.
• To define a state-sum model that incorporates internal degrees of freedom on the defect curve via a second gauge group.
• To generalize the 2-cocycle twistings of the untwisted Dijkgraaf-Witten invariant to accommodate defect structures.
• To ensure topological invariance under flag-like triangulation moves, establishing a well-defined invariant for surfaces with embedded curves.

Proposed method

• Introduces flag-like triangulations where each 2-simplex intersects the defect curve in a face or not at all, ensuring compatibility with defect structure.
• Assigns labels from three groups: H (for defect edges), X (for incident edges), and G (for bulk edges), with states transforming under commuting actions of H and G.
• Defines a state-sum invariant using three functions α, β, γ derived from group 2-cocycles, satisfying generalized cocycle conditions (1)–(4) for consistency.
• Imposes admissibility conditions on labelings such that the product of local state values over all simplices yields a topological invariant.
• Uses flag-like Alexander moves and extended Pachner moves to prove invariance under triangulation changes, ensuring topological robustness.
• Constructs examples via restriction of global 2-cocycles and via characters of the gauge groups, demonstrating non-trivial initial data.

Experimental results

Research questions

• RQ1 How can 2D Dijkgraaf-Witten theory be extended to include codimension-1 defects with internal degrees of freedom?
• RQ2 What conditions must the twisting cocycles α, β, γ satisfy to ensure invariance under triangulation moves?
• RQ3 Can non-trivial initial data for the state-sum model be constructed from group characters or restricted cocycles?
• RQ4 How does the flag-like triangulation condition ensure compatibility with defect structures and topological invariance?
• RQ5 What is the role of the three groups H, X, G in encoding the defect and bulk gauge symmetries?

Key findings

• The state-sum invariant Z(Σ, C) is independent of the choice of flag-like triangulation, establishing a topological invariant for surfaces with embedded curves.
• The construction generalizes the untwisted Dijkgraaf-Witten invariant by introducing a second gauge group H for defect edges and a set X with commuting H×G actions for incident edges.
• The generalized cocycle conditions (1)–(4) ensure consistency of the state-sum, with (1)–(3) reducing to the 2-cocycle condition and (4) ensuring compatibility under group actions.
• Non-trivial initial data are constructed via restriction of a global 2-cocycle on a larger group Γ to subgroups H, G and a subset X closed under H and G actions.
• Additional examples are built using characters of G and H, where β and γ are defined via orbit-invariant characters on X, with α set to 1.
• The invariant is invariant under flag-like Alexander moves, and vertex ordering does not affect the result due to invariance under extended Pachner moves.