A Geometric Approach to Boundaries and Surface Defects in Dijkgraaf–Witten Theories

Abstract

Dijkgraaf-Witten theories are extended three-dimensional topological field theories of Turaev-Viro type. They can be constructed geometrically from categories of bundles via linearization. Boundaries and surface defects or interfaces in quantum field theories are of interest in various applications and provide structural insight. We perform a geometric study of boundary conditions and surface defects in Dijkgraaf-Witten theories. A crucial tool is the linearization of categories of relative bundles. We present the categories of generalized Wilson lines produced by such a linearization procedure. We establish that they agree with the Wilson line categories that are predicted by the general formalism for boundary conditions and surface defects in three-dimensional topological field theories that has been developed in [FSV].

Theory of defects in Abelian topological states

“…The subgroup M has been referred to as a "Lagrangian subgroup." 27,28,32 The first condition requires that every two particles in M are mutually bosonic, so that they can be condensed simultaneously. The second condition requires that all other quasiparticles not in M are confined after the condensation of M .…”

“…However, we would like to emphasize that it is not clear whether the topological boundary conditions of TQFTs that are classified in Ref. 27 are equivalent to gapped edges of local Hamiltonians, which are the focus of this paper and of Ref. 28.…”

“…[13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] An extrinsic defect is a point-like or line-like defect either in a topological state, or on the interface between two topologically distinct states, which leads to new topological properties that are absent in the topological state without the defect. A crucial distinction between extrinsic defects and the more familiar quasiparticle excitations is that the former are not deconfined excitations of the system; rather, the energy cost for separating point defects, (a) Genons in bilayer systems 13,14 (b) domain walls between ferromagnetic and superconducting backscattering at the edge of a fractional quantum spin Hall (FQSH) state, [15][16][17] (c) lattice dislocations in solvable models of ZN topological order.…”

Ground-state degeneracy for Abelian anyons in the presence of gapped boundaries

“…Gapped boundary conditions correspond to topological boundary conditions in TQFT; for Abelian ChernSimons theory they have been studied in [5][6][7] while a more general theory based on fusion categories was developed in [8,9]. The problem of computing the ground-state degeneracy was previously addressed in [10], but only in the case when there are no boundary domain walls.…”

Orientation Twisted Homotopy Field Theories and Twisted Unoriented Dijkgraaf–Witten Theory

“…It would be interesting to compare these algebraic structures with the input data of the recently proposed unoriented extension of Turaev-Viro-Barrett-Westbury theory [5], [7]. Finally, our geometric approach admits a natural generalization to allow for defects and boundary conditions, along the lines of the oriented case [18]. Calculations in the case of Dijkgraaf-Witten theory will shed light on the as-of-yet undeveloped theory of defects in unoriented theories.…”