Monodromy Defects for Electric-Magnetic Duality, Hyperbolic Space, and Lines

Abstract: In this note we explore monodromy defects for non-invertible symmetries in Maxwell theory, exploiting the conformal mapping to $AdS_{3} \times S^{1}$. With this approach we recover the spectrum of the defect conformal primaries. We also dedicate some time discussing the behaviour of Wilson/'t Hooft lines in the presence of such a monodromy defect, and highlight the following aspects of their behaviour: i) the lines can terminate on the defect, ii) lines of the unit electric (magnetic) charge may seize to be indecomposable, and can be represented as integer powers of some more elementary lines, and iii) they behave as topological objects when brought close to the defect, and this behaviour is governed by a Chern-Simons theory.

• The paper demonstrates that monodromy defects reconfigure the operator spectrum via non-invertible electric-magnetic duality in abelian Maxwell theory.
• Using a conformal mapping from R⁴ to AdS₃ × S¹, the analysis derives explicit spectra for defect-localized conformal primaries and line operators.
• It reveals that the duality-induced infrared sector is governed by an emergent Chern-Simons TQFT, which dictates fusion rules for Wilson and 't Hooft lines.

Monodromy Defects, Non-invertible Duality Symmetries, and Line Operators in Maxwell Theory

Overview

The paper "Monodromy Defects for Electric-Magnetic Duality, Hyperbolic Space, and Lines" (2604.15117) investigates the interplay of codimension-two monodromy defects associated with non-invertible electric-magnetic duality symmetries in abelian Maxwell theory. By leveraging a conformal mapping from $\mathbb{R}^4$ to $AdS_3 \times S^1$, the analysis yields the spectrum of defect-localized conformal primaries and a comprehensive description of Wilson and 't Hooft line operators in the monodromy defect background. The work further demonstrates that duality and triality monodromy defects induce a nontrivial restructuring of the line operator content, governed by emergent Chern-Simons (CS) topological quantum field theories in the defect’s infrared (IR) sector.

Framework for Non-invertible Duality Defects

Non-invertible duality defects in $U(1)$ Maxwell theory are supported on codimension-two manifolds implementing $\mathcal{S}$ and related $SL(2,\mathbb{Z})$ duality transformations as monodromies. The standard duality interface, relevant at special points in coupling space, is generically constructed as a "topological interface" of the form $e^{\frac{iN}{2\pi}\int_\Sigma A d\tilde{A}}$, coupling electric and magnetic gauge fields across the interface surface $\Sigma$. For rational values of the complexified coupling $\tau = \frac{i N_e}{N_m}$, extended interfaces $\mathcal{D}^{(2)}_{N_e, N_m}$ are realized via concatenation of duality transformations with half-space gauging of finite abelian one-form subgroups. Triality defects, associated to $\mathcal{S}\mathcal{T}$ at special values of $AdS_3 \times S^1$0, are also constructed.

Crucially, these topological interfaces are non-invertible for generic rational values and have boundaries that manifest as monodromy defects. The action of these defects on the gauge fields is encoded in explicit gluing conditions reflecting the duality operation, e.g., $AdS_3 \times S^1$1 for the standard $AdS_3 \times S^1$2-duality defect, and analogous relations for more complicated group elements.

AdS Mapping and Operator Spectrum

Employing the conformal mapping $AdS_3 \times S^1$3, the codimension-two defect is mapped to a boundary in $AdS_3 \times S^1$4. After compactification along $AdS_3 \times S^1$5 with a twist dictated by the duality transformation, the effective $AdS_3 \times S^1$6 theory comprises a tower of massive vector Kaluza-Klein modes and a decoupled topological sector.

Solving the equations of motion in this setup, the defect-localized operator spectrum is read off via the AdS/CFT dictionary. The modes are classified by their $AdS_3 \times S^1$7 momentum, with characteristic shifts arising due to the duality twist. For the $AdS_3 \times S^1$8-duality defect, the vector operators $AdS_3 \times S^1$9 acquire conformal dimensions $U(1)$0:

• $U(1)$1 with $U(1)$2, $U(1)$3,
• $U(1)$4 with $U(1)$5, $U(1)$6.

Analogous results are derived for triality and more general duality defects. The tower of defect conformal primaries is robust and independent of the integers $U(1)$7, indicating its universal character as the "universal sector" associated with the duality transformation.

One-form Symmetry Breaking and Defect Operator Algebra

The construction demonstrates that monodromy defects break the bulk one-form electric and magnetic $U(1)$8 symmetries at their locus, as seen in the modified Maxwell equations. The defect supports a chiral current (tangential component), and the breaking pattern induces a defect-localized $U(1)$9 symmetry under which certain operators are charged. This allows bulk Wilson and 't Hooft lines, charged under the broken symmetry, to terminate on the monodromy defect, realizing a form of endpoint operator algebra reminiscent of condensed matter edge theories.

Topological Sector and Line Operator Fusion Rules

The IR sector of the effective $\mathcal{S}$0 theory is governed by an emergent CS theory, with the CS level determined by the structure of the monodromy defect:

• For $\mathcal{S}$1, a $\mathcal{S}$2 CS theory controls the defect. Wilson lines of charge $\mathcal{S}$3 flow to CS lines with the same charge mod $\mathcal{S}$4, and lines with charge a multiple of $\mathcal{S}$5 become trivial.
• For more general defects $\mathcal{S}$6, the CS theory is described by a $\mathcal{S}$7-matrix and possesses $\mathcal{S}$8 primary operators.
• Wilson/'t Hooft lines are shown to decompose into integer powers of more elementary defect-localized lines, a marked departure from their bulk indecomposability.

The paper provides explicit fusion rules for moving bulk lines onto the defect, and demonstrates that the spectrum is invariant under duality action, i.e., certain combinations of Wilson lines become equivalent to 't Hooft lines when transported around the defect. These topological lines on the defect form a Verlinde algebra structure, confirming their inherited topological nature from the IR CS description.

Near-defect Topological Dynamics

When line operators are brought close to the monodromy defect (or boundary in $\mathcal{S}$9), their long-distance physics is described entirely by the defect CS theory. The mapping is exhaustive: all bulk lines can be realized as defect lines in the IR, with fusion rules and algebraic relations dictated by the topological sector. For the standard $SL(2,\mathbb{Z})$0-duality defect, all lines can end on the monodromy defect, and their endpoints correspond to chiral primary field insertions in the defect CFT.

Implications and Outlook

The work establishes that monodromy defects for non-invertible symmetries in four-dimensional gauge theory induce a rich structure of defect-localized operator spectra and topological line operator algebra, controlled by CS theory data. Wilson and 't Hooft lines can terminate on the defect, and their decomposability properties are altered in the presence of non-invertible monodromy defects. All bulk lines can be pushed and fused onto the defect, with explicit fusion rules.

Theoretically, this analysis solidifies the correspondence between duality/topological boundaries and operator algebras in higher-dimensional quantum field theory, melding conformal, defect, and TQFT perspectives. Practically, the results provide concrete tools for the calculation of correlation functions and the exploration of duality defects in more general settings, including possible applications in topological phases of matter and the study of generalized global symmetries.

Potential future directions include:

• Generalization to more intricate non-invertible defects at arbitrary rational points of the coupling space.
• Extension to nonabelian gauge theories (notably, $SL(2,\mathbb{Z})$1 SYM), where supersymmetry preservation, the role of $SL(2,\mathbb{Z})$2-symmetry monodromy, and higher BPS sectors become relevant.
• Further exploration of the defect central charge and other DCFT observables using holography.
• Explicit computation of correlators involving end-point operators and defect-localized primaries in the presence of line insertions.

Conclusion

The paper achieves a comprehensive classification of monodromy defects associated with non-invertible electric-magnetic duality in Maxwell theory, elucidates the induced defect spectrum via conformal mapping and AdS/CFT methods, and clarifies how the lines and their algebra are reorganized by the presence of the defect. The reduction to a CS TQFT in the IR and the explicit fusion rules for lines capture a deep structure relating duality, topology, and operator content in abelian gauge theory. This framework has wide applicability for the study of generalized symmetries, conformal defects, and topological phases in both mathematics and high-energy/condensed matter physics.