Condensation Defects

Definition

A condensation defect is a topological defect built by gauging a (higher-form) symmetry on a positive-codimension submanifold — “higher gauging” (Roumpedakis–Seifnashri–Shao, 2022). Summing the symmetry’s own operators over a network on a closed submanifold $Y$, with a normalization, produces an operator supported on $Y$ that is topological but generically non-invertible: it fuses with its orientation reversal into itself times a decoupled factor, $S\times \bar{S}=Z(Y)S,$ where $Z(Y)$, the partition function on $Y$ of the gauge theory of the condensed group, depends on the normalization chosen for $S$, and no inverse defect exists. In $2+1$d ${\mathbb{Z}}_N$ gauge theory, summing the Wilson lines of a subgroup $H$ on a closed surface gives a sheet with $Z({\Sigma }_g)=|H{|}^{2g-1}$ in the course’s normalization. One dimension lower, gauging ${\mathbb{Z}}_2$ on a closed curve of the Ising model gives $1+\eta$, which is what the Kramers–Wannier duality defect fuses into, $D\times \bar{D}=1+\eta$. Duality defects come from gauging on half of spacetime at a self-dual point, condensation defects from gauging on a closed submanifold, and both are non-invertible.

Gauging a symmetry on all of spacetime produces a new theory; gauging it on a closed submanifold of positive codimension, when the restricted symmetry is free of anomaly, produces a defect inside the original theory. Gauging a $p$-form symmetry on submanifolds of the various dimensions this allows generates many of the non-invertible symmetries now organizing the field.

Role in Research

Condensation defects are the modern target of the generalized-symmetries course and the setting of the group’s manuscript in preparation, Finite-Cost Currents at Higher-Gauging Walls in Four-Dimensional ${\mathbb{Z}}_N$ Gauge Theory. Its object is the wall obtained by gauging a subgroup $H=⟨k⟩\subset {\mathbb{Z}}_N$ of the electric 1-form symmetry on a closed three-dimensional hypersurface $Y$, $D_H(Y)=\frac{1}{{\mathcal{N}}_Y}{\sum }_{S\in Z_2(Y;H)}U[S],$ a sum over closed $H$-labelled sheet networks. A charge-$q$ Wilson line crosses the wall intact exactly when $q\in \mathrm{Ann}(H)\simeq {\mathbb{Z}}_r$, with $r=\gcd (N,k)$; any other line meets the wall at a junction, and a line along the wall carrying the endpoint charge $q$ mod $N/r$ joins it to a partner junction. The manuscript takes this surface sum, its charge filter and its fusion as prior work (Roumpedakis–Seifnashri–Shao; Choi–Córdova–Hsin–Lam–Shao; Bah–Leung–Waddleton, the closest construction). What it adds is a cost per unit length for the lines along the wall, motivated by the activation cost that the Julia–Toulouse picture keeps for proliferating defects. An exact character transform turns these lines into a ${\mathbb{Z}}_{N/r}$ clock model in which a Wilson–wall junction is a local clock operator. The clock phases decide whether junctions are confined along the wall, and at large cost the tension between junctions, per unit cost, tends to $\bar{\alpha }/2$, where $\bar{\alpha }$ is the smallest absolute value of an integer representing the endpoint charge.

The restricted Villain ensemble is a result of the course rather than of the manuscript: in $2+1$ dimensions a charge-$k$ condensate restricted to a surface reproduces, in its London limit, the condensation sheet of higher gauging. Together these results connect the group’s older Julia–Toulouse language with the modern language of higher-form symmetries, higher gauging and condensation defects.

Relations

• julia-toulouse-mechanism — the defect-condensation picture that motivates a finite cost for the lines along the wall
• higher-form-symmetries — higher gauging = gauging a higher-form symmetry on a submanifold
• villain-action — a charge-$k$ Villain condensate restricted to a surface gives the condensation sheet in the London limit
• kramers-wannier-duality — the duality defect fuses into the simplest condensation defect, $1+\eta$
• wilson-loop — the charge filter and the endpoint rule for lines crossing the wall
• dual-superconductor — the physical condensation picture in the pure-gauge case

Papers

Roumpedakis–Seifnashri–Shao, “Higher gauging and non-invertible condensation defects,” Commun. Math. Phys. 401 (2023) 3043 [arXiv:2204.02407]; Choi–Córdova–Hsin–Lam–Shao, Phys. Rev. D 105 (2022) 125016 [arXiv:2111.01139] and Commun. Math. Phys. 402 (2023) 489 [arXiv:2204.09025]; Bah–Leung–Waddleton, JHEP 09 (2026) 272 [arXiv:2506.04346]; Shao (TASI), arXiv:2308.00747. The group’s manuscript (in preparation; version of 2026-09-20) contributes the finite-cost line sector. See the course bibliography.

Notes

• The novelty boundary is explicit in the manuscript: the subgroup surface sum, its filter and its fusion are prior work, and the contribution is the line sector at a fixed charge filter, “the endpoint prescription with a fixed crossing filter, the resulting observable dictionary, and the uniform bound on the stable charge cost.” It does not derive the line weight from a continuum condensate; that derivation, the relation of the wall tension to a complete Wilson observable and the full topological worldvolume amplitude are among its open problems.
• Bulk condensation (gauging on all of spacetime: a new theory, in which the lines outside $\mathrm{Ann}(H)$ are no longer genuine) versus hypersurface condensation (a defect of the original theory, at which those lines end on junctions) is the distinction between “gauging a subgroup” and “inserting a condensation wall”; the exact sequence $0\to \mathrm{Ann}(H)\to {\mathbb{Z}}_N\to \hat{H}\to 0$ organizes both.
• This page is the concept anchor for the companion master-project on condensation defects in ${\mathbb{Z}}_N$ lattice gauge theory.