Conformal Twist Defects in 4D Maxwell Theory

• The topic defines conformal twist defects as codimension-two surfaces that enforce a non-invertible duality swapping electric and magnetic fields.
• They arise via twisted boundary conditions that yield a generalized free-field sector alongside decoupled chiral edge modes with topological anomalies.
• Analysis leverages symTFT frameworks and operator spectrum techniques, deepening insights into defect fusion and modern symmetry operations in gauge theories.

A conformal twist defect in four-dimensional Maxwell theory is a codimension-two (surface) defect that implements a non-invertible electromagnetic duality transformation, exchanging electric and magnetic sectors as fields encircle the defect support. Such defects represent a precise interface between local and non-local symmetry operations, combining duality, higher-form symmetry gauging, and topological field-theoretic manipulations. Their mathematical, physical, and anomaly structure reflects both the simplicity and subtlety of free Maxwell theory and the depth of modern approaches to symmetry and defect CFT.

1. Definition and Duality Action

A conformal twist defect in 4d Maxwell theory is defined via a codimension-2 insertion—typically a surface Σ—whose presence implements a nontrivial duality transformation of the gauge field across Σ. This transformation exchanges the electric and magnetic field strengths, corresponding to the non-trivial element of the SO(2) electromagnetic duality group:

$$\begin{pmatrix} F_L \ i * F_L \end{pmatrix}\Big|_{\Sigma} = \begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix} \begin{pmatrix} F_R \ i * F_R \end{pmatrix}\Big|_{\Sigma}$$

where $F$ is the field strength, $*$ denotes the Hodge dual, and the subscripts $L, R$ denote the values on the left and right of the defect. As one traverses a loop encircling the defect, the Maxwell fields transform by

$F(\theta + 2\pi) = - * F(\theta)$

This twist boundary condition specifies the conformal twist defect as a 'swap' of electric and magnetic fluxes, which can be realized as a monodromy in the gauge field or as a boundary condition for the path-integral (Shao et al., 25 Sep 2025, Niro et al., 2022). Operationally, these defects arise as endpoints of codimension-one topological symmetry defects implementing non-invertible global symmetries.

2. Operator Spectrum and Boundary Conditions

The operator content localized on the twist defect is determined by solving the classical Maxwell equations subject to the above twisted boundary conditions on Σ. In an adapted coordinate system (e.g., mapping $\mathbb{R}^4$ to $\mathbb{R} \times S^3$ so that the defect sits at $S^1 \subset S^3$), field configurations must satisfy a duality-exchanging boundary condition at the defect locus.

The resulting quantization of fluctuations yields two families of primary defect operators labeled by conformal weights $(h, \bar{h})$:

$$\mathcal{V}^s_z: (h, \bar{h}) = \left(1 + \frac{|s|}{2},\, \frac{|s|}{2}\right)$$

$$\mathcal{V}^s_{\bar{z}}: (h, \bar{h}) = \left(\frac{|s|}{2},\, 1 + \frac{|s|}{2}\right)$$

where $s$ is a transverse spin label, taking values in $\mathbb{Z} + \frac{1}{4}$ or $\mathbb{Z} + \frac{3}{4}$ depending on the branch, and arises from the mode expansion around the defect (Shao et al., 25 Sep 2025). The scaling dimensions $\Delta$ are given by the dispersion relation

$$\omega = \begin{cases} 2 + |s| + |n| + 2m, & |s| \in \mathbb{N} + \frac{1}{4} \ |s| + |n| + 2m, & |s| \in \mathbb{N} + \frac{3}{4} \end{cases}$$

where $n$ is an integer and $m \geq 0$.

The defect OPE of the Maxwell field thus closes on a "generalized free field" sector, meaning that higher-point correlators are determined by Wick contractions of the two-point functions, with no interacting (nonfree) defect degrees of freedom (Herzog et al., 2022). For example, the two-point function of a defect vector operator is

$$\langle \widehat{\psi}_+^{(s)}(x) \widehat{\psi}_-^{(-s)}(0) \rangle \sim \frac{1}{(w)^{2+2|s|}}$$

All higher-point correlators factorize accordingly, consistent with the conformal and reflection-positive structure of the defect CFT.

3. Factorization Structure and Edge Sectors

A central result is the factorization of defect correlation functions into two decoupled sectors (Shao et al., 25 Sep 2025):

• Universal generalized free-field sector: This sector consists of defect operators arising from the quantization of the Maxwell field subject to the duality-twisted boundary. All correlation functions in this sector are determined by conformal symmetry and are fixed by Wick's theorem, i.e., they have the structure of a generalized free field theory.
• Chiral current (edge) sector: Owing to the topological nature of the three-dimensional duality defect, edge modes localized at the boundary (i.e., the endpoint at which the 2d twist defect resides) appear. These are described by a chiral boson $\varphi$ with action

$$S_{\text{chiral}}[\varphi] = -\frac{N}{2\pi} \int d^2x\, \partial_{\psi}\varphi\, \partial_t\varphi$$

and holomorphic current

$$J_{\bar{z}} = -\frac{i N}{\pi} \partial_{\bar{z}} \varphi$$

with $N$ a parameter encoding the defect level. This sector is completely decoupled from the generalized free-field sector: correlation functions of bulk or defect generalized free fields and edge currents factorize.

This structure is strongly reminiscent of the factorization in Chern-Simons theory, where bulk topological fields and gapless chiral edge currents co-exist but decouple in correlation functions (Shao et al., 25 Sep 2025).

4. Non-Invertible Symmetries and Defect Construction

The twist defect is a concrete realization of a non-invertible global symmetry operation in Maxwell theory (Paznokas, 24 Jan 2025, Niro et al., 2022, Sela, 10 Jan 2024, Meynet et al., 28 Apr 2025). In modern terms:

• The electromagnetic duality group in Maxwell theory is $SL(2,\mathbb{Z})$, which acts on the coupling $\tau$ and exchanges $F$ and $*F$.
• When combined with the gauging of electric and magnetic one-form symmetries (e.g., via discrete subgroups $\mathbb{Z}_N$), the symmetry group is extended to $SL(2,\mathbb{Q})$ (or more generally, to the continuous group $SO(2)$ upon coupling to flat connection symTFTs) (Paznokas, 24 Jan 2025).
• The twist defect implements an $SO(2)$ self-duality rotation locally, but its action on extended operators (Wilson/'t Hooft lines) is non-invertible: the fusion of such defects does not admit a group inverse but is described by a condensation algebra structure (Paznokas, 24 Jan 2025, Niro et al., 2022).

Construction via the 5d symTFT framework proceeds as follows:

• The 4d theory is embedded as a boundary of a 5d "bulk" BF theory with action $S_5 = \frac{i}{2\pi} \int b \wedge dc$ (with two-form fields $b, c$).
• Boundary conditions and higher gauging implement the SO(2) duality rotation as a topological condensation defect, leading to an explicit non-invertible duality interface implementing $F \to *F$ on the boundary theory.

The operator content, fusion rules, and anomaly properties of these defects are determined by the structure of the symTFT, the particular gauging (with discrete torsion), and the defect support.

5. Anomaly Structure and Edge Physics

The 4d Maxwell twist defect exhibits a mixed 't Hooft anomaly involving a chiral $O(2)$ symmetry, as diagnosed via careful examination of edge modes and symmetry transformations (Shao et al., 25 Sep 2025). Specifically:

• The chiral current sector is invariant under a $U(1)$ rotation and charge conjugation, giving an $O(2)$ symmetry.
• The quantization of $U(1)$ charges at the defect is fractionalized: the Hilbert space of the edge theory transforms projectively, with quantum numbers shifted by $\pm \tfrac{1}{2}$ for odd $N$.
• The anomaly manifests as a two-fold degeneracy in every energy level at the defect, robust under RG flows and deformations.

Dynamically, this anomaly constrains the possible RG flows and the allowed operator content of the defect, ensuring the stability of the chiral sector and providing a fundamental obstruction to trivializing the defect via topological manipulations.

6. Relations to 2D CFT and Higher Dimensional Analogues

A close analogy exists with twist fields in the 2d compact boson CFT, where duality defects exchange momentum and winding numbers and lead to a similar structure of factorized correlation functions and edge sectors (Shao et al., 25 Sep 2025, Niro et al., 2022). In that setting, the defect is a zero-dimensional disorder operator attached to a non-invertible topological line defect.

The higher-dimensional construction in Maxwell theory generalizes this structure: the 2d twist defect is attached to the endpoint of a 3d duality (symmetry) defect, and the associated operator spectrum, factorization, and anomaly structures are richer but governed by analogous physical principles.

7. Gauge Theory and CFT Correspondence

There is a precise match between the insertion of conformal twist defects in 4d Maxwell theory and the insertion of topological (Verlinde/loop) defect operators in 2d CFTs (notably Liouville/Toda) under the AGT correspondence (Drukker et al., 2010). In the Abelian case, the defect partition function is:

$$Z^{\text{defect}}_{\text{Maxwell}} = \int da\; e^{2\pi i a} \; |Z_{\text{Nek}}(a)|^2$$

where $e^{2\pi i a}$ is the defect insertion, and $Z_{\text{Nek}}$ is the instanton partition function. This demonstrates the universality of the conformal twist defect construction and its realization across dimensions via modular and topological data.

8. Triviality and Topological Nature in Free Maxwell Theory

For pure Maxwell theory, conformal codimension-two defects—including the Gukov-Witten surface operator—have trivial Weyl anomaly coefficients ($b = d_1 = d_2 = 0$), and the defect operator algebra consists of generalized free fields (Bianchi et al., 2021, Herzog et al., 2022). This reflects the topological character of the defect: no dynamical degrees of freedom propagate along Σ, and local bulk observables are unaffected aside from global monodromy data.

This is in sharp contrast to scalar or fermion theories, where surface defects can support nontrivial central charges, displacement operators, and induced contributions to entanglement entropy.

9. Summary Table: Key Features

Property	Maxwell Twist Defect	Generalized Free Field
Action on fields	$F \rightarrow *F$ at defect	Wick's theorem, trivial OPE structure
Operator spectrum	$\mathcal{V}_z^s$, $\mathcal{V}_{\bar{z}}^s$	Vectors/scalars with fixed dimensions
Conformal anomaly coefficients	$b = d_1 = d_2 = 0$	N/A
Edge sector	Chiral boson, $O(2)$ anomaly	None in bulk
Fusion rule (defect operation)	Non-invertible, condensation algebra	Group-like for closed, sum for open

10. Implications and Outlook

Conformal twist defects in 4d Maxwell theory serve as canonical examples of non-invertible symmetry operators, bridging the gap between topological field theory, defect conformal field theory, and modern approaches to symmetry and duality. They provide a laboratory for studying edge modes, 't Hooft anomalies, non-invertible fusion algebras, and the interplay between geometry, topology, and gauge theory in both mathematical and physical settings. Their classification, anomaly structure, and correspondence with lower-dimensional CFTs continue to motivate developments in the taxonomy and dynamics of defects, interfaces, and higher symmetries in quantum field theory.