Lattice Realizations of Flat Gauging and T-duality Defects at Any Radius

Abstract: We analyze non-invertible topological interfaces and defects in the two-dimensional compact boson, focusing on the more exotic ones obtained by gauging continuous symmetries with flat connections on a half-space. These include interfaces between mutually irrational radii and T-duality symmetries at arbitrary boson radius. Using the modified Villain discretization on both a Euclidean two-dimensional square lattice and a quantum one-dimensional chain, we show that all these topological interfaces survive discretization and give rise to non-compact edge modes localized at the defect sites. Such non-compact edge modes imply a continuous defect spectrum and an infinite quantum dimension. In the special case of rational radii, we show how the defect action or Hamiltonian can be modified in order to compactify the edge modes and produce more standard defects with finite quantum dimension.

• The paper demonstrates a unified lattice framework that realizes flat gauging and T-duality defects for the 2D compact boson.
• The framework employs a modified Villain prescription to preserve shift and winding symmetries, yielding non-compact edge modes and a continuous spectrum.
• For rational radii, the lattice regularization recovers compact edge modes with finite quantum dimension, aligning with standard duality defects.

Lattice Realizations of Flat Gauging and T-duality Defects at Any Radius

Introduction and Motivation

This paper establishes, through a unified framework, how non-invertible topological defects and interfaces—especially those arising from flat gauging of continuous symmetries—are realized on the lattice for the two-dimensional compact boson. The authors focus particularly on interfaces relevant to T-duality at arbitrary compactification radius, including the generic irrational case $R^2\not\in\mathbb Q$. This approach clarifies the physical consequences of flat gauging, originally understood in the continuum via non-invertible symmetries, when regularized on both Euclidean and Hamiltonian lattices.

Notably, the study demonstrates that, even after discretization, such topological defects generate non-compact edge modes at the defect site, resulting in a continuous local spectrum and infinite quantum dimension for the defect operator. In contrast, for rational radii, the defects can be regularized to support only compact edge modes and thus exhibit finite quantum dimension, reducing to standard lattice duality defects.

Flat Symmetry Gauging and Topological Interfaces

In the continuum, T-duality and more general dualities in the compact boson arise from flat gauging of $U(1)$ momentum and winding symmetries. The authors employ a sequence of dual manipulations: first, they decompactify by gauging the winding symmetry with a flat $U(1)$ connection, then recompactify by gauging a discrete $Z \subset \mathbb R$ subgroup of the shift symmetry. Crucially, these manipulations can be performed not on the entire spacetime but over a half-space, creating a generalized topological interface between distinct compact boson theories at radii $R$ and $R'$. When $R = R'$, the interface closes to a non-invertible symmetry defect; for generic $R/R' \in \mathbb R$ it constitutes a radius-changing interface.

By a precise path integral construction, the authors show that the interface is necessarily endowed with at least one non-compact edge mode, unless $R^2$ is rational—where compactification can further be imposed. These edge degrees of freedom are responsible for the infinite quantum dimension of the defect, as the expectation value of the corresponding operator diverges on homologically trivial loops.

Modified Villain Lattice Prescription

The lattice regularization of the compact boson is performed using the modified Villain prescription, which retains both shift and winding $U(1)$ symmetries at the discrete level. The key technical feature here is the explicit elimination of lattice vortices using a flatness constraint on $U(1)$0-valued link variables, ensuring global winding sectors are preserved non-perturbatively.

Within this framework, the authors construct lattice analogues of the topological interfaces and T-duality defects. This involves introducing a boundary where degrees of freedom change from compact to non-compact, and vice versa, with matching conditions engineered to preserve the topological character of the interface. The analysis rigorously establishes that the topological properties of these interfaces persist under arbitrary lattice deformations, modulo the constraints of gauge invariance.

Non-Invertible T-duality Defects: Spectrum and Quantum Dimension

When implementing the T-duality defect on the lattice at generic radius ratios, the edge supports a non-compact scalar, which directly leads to a defect Hilbert space with a continuous spectrum. This is seen both in the representation of the partition function and in the explicit Hamiltonian formulation.

The paper emphasizes that the infinite quantum dimension—a hallmark of non-invertible defects arising from flat gauging—corresponds physically to the non-normalizability (infinite volume divergence) of the edge mode integration. Furthermore, the non-invertible nature of the T-duality defect at generic radius (irrational or not commensurate rationals) is shown by demonstrating that no nontrivial $U(1)$1 topological operators survive after fusion with the defect.

For rational $U(1)$2, the edge mode can be compactified by modifying the Hamiltonian or defect action. The minimal defect then supports only a finite set of boundary states, and the quantum dimension is finite; these become the canonical topological duality defects familiar from rational CFT.

Hamiltonian Lattice Realizations

The analysis is extended to the quantum Hamiltonian chain, where the compact boson is represented using canonical variables subject to integer-valued gauge constraints. Again, the Villain prescription enables a formulation retaining all the essential symmetries.

Interfaces interpolating between different radii, and the T-duality defect itself, are constructed via local modifications to the Hamiltonian and the set of Gauss laws. The presence or absence of non-compact edge modes is manifest in the spectrum of the defect Hamiltonian: for irrational $U(1)$3 the spectrum is continuous, while for rational $U(1)$4 it becomes discrete.

The explicit mapping between the original and dual variables, as well as the fusion of two T-duality defects (giving the identity), are realized with full control in this framework, matching continuum expectations.

Implications, Outlook, and Future Directions

This work rigorously demonstrates that non-invertible topological defects associated with flat gauging of continuous (including non-compact) symmetries survive discretization on both Euclidean and quantum lattices, provided the modified Villain model is adopted. The appearance of non-compact edge modes leading to infinite quantum dimension is revealed to be a robust and universal feature, not an artifact of continuum field theory. For special radii, the framework recovers the known topological duality defects with finite-dimensional Hilbert spaces.

Practically, this means that lattice regularizations of theories with generalized symmetries must treat the fate of non-invertibility, quantum dimension, and emergent topological modes with care: naive discretizations may obscure such global properties. The results here are immediately relevant for both condensed matter lattice systems (where dualities and edge states control topological order) and high-energy lattice simulations of CFTs with generalized symmetry.

Theoretically, this approach opens the door to systematic classification and construction of non-invertible lattice defects in broader classes of models, including those with non-Abelian or higher-form symmetries. An important avenue for future research is to generalize these constructions to higher dimensions, coupled matter systems, or interacting theories, and to explore the constraint such defects impose on RG flows and phase structure.

Conclusion

This paper provides a comprehensive and mathematically precise treatment of flat gauging, T-duality, and non-invertible defects of the compact boson, both in the continuum and on the lattice. The key result is the universal emergence of non-compact edge modes and infinite quantum dimension for defects arising from flat gauging, with a clear transition to finite quantum dimension at special (rational) points. These findings establish a foundational baseline for understanding topological symmetries and their defects in discrete quantum systems, with implications for both fundamental quantum field theory and quantum matter.