Decomposition, condensation defects, and fusion

TL;DR

This work develops a coherent framework connecting decomposition with condensation defects and their fusion in theories with higher-form symmetries, including non-invertible cases. It constructs condensation defect projectors $S_R(oldsymbol{ Sigma})$ and explicit projection operators $oldsymbol{ Pi}_R$ that realize universe selection along defects, and analyzes fusion using preserved symmetry data and theta angles. Through detailed 2d and 3d examples (e.g., ${f Z}_2$, ${ m D}_4$, ${ m H}$, and ${f Z}_p$ orbifolds, as well as 2-group orbifolds in 3d), it shows how fusion coefficients can reduce to integer multiplicities when underlying TFTs decompose into invertible sectors, while also revealing situations where nontrivial TFT coefficients persist. The paper also proposes new defects by gauging a form symmetry along a worldvolume, offering a broader toolkit for constructing and manipulating (potentially non-topological) higher-dimensional defects and their fusion. Overall, it clarifies how theta angles and higher-form gauging interplay with decomposition to yield computable, sometimes universal, fusion rules for both condensation and novel defects across dimensions.

Abstract

In this paper we outline the application of decomposition to condensation defects and their fusion rules. Briefly, a condensation defect is obtained by gauging a higher-form symmetry along a submanifold, and so there is a natural interplay with notions of decomposition, the statement that d-dimensional quantum field theories with global (d-1)-form symmetries are equivalent to disjoint unions of other quantum field theories. We will also construct new (sometimes non-invertible) defects, and compute their fusion products, again utilizing decomposition. An important role will be played in all these analyses by theta angles for gauged higher-form symmetries, which can be used to select individual universes in a decomposition.

Figures (1)

• Figure 1: We illustrate the computation of a $T^2$ correlation function in the orbifold theory $[X/\Gamma]$ with one-cycle $L$ wrapped by a condensation defect, as shown in (a). In (b)-(d) the cycle is shaded gray for visualization purposes, but the only insertions along the cycle are the point operators shown. The prescription is to insert a projection point operator $\Pi$ at every vertex point in some sufficiently fine triangulation of $L$, as in (b). Since $L$ is connected, it is sufficient to insert $\Pi$ at a single point, and we can write $\Pi$ as a sum over twist fields $p(z)$, shown in (c). Finally, to compute these correlation functions in terms of the parent theory $[X]$, we lift each diagram and sum over all consistent ways of inserting $\Gamma$ lines. In the lift, the $p(z)$ operator becomes an operator $\sigma(z)$ sitting on the end of a $z$ line. We can choose where that line joins the other lines. One choice (where they all meet at a junction of degree five) is shown in (d) and it is implicit in our lift that the product of lines around the junction should be the identity, giving the requirement $gh=hgz$.