Defect Fusion and Casimir Energy in Higher Dimensions

TL;DR

This work extends defect fusion to higher-dimensional CFTs by defining a commutative or noncommutative fusion product for parallel p-dimensional defects and isolating a physical Casimir-energy density that governs their short-distance interactions. The authors connect fusion to wedge geometries and hyperbolic-space techniques, enabling efficient computation of Casimir energies for defects in both bosonic O(N) and fermionic Gross-Neveu–Yukawa CFTs, using $d=4-\epsilon$ and $d=2+\epsilon$ expansions as well as large-$N$ numerics. They uncover nontrivial fusion algebras, including nonassociativity and symmetry enhancement, and provide explicit fusion data for line defects and interfaces, along with a classical-mechanical analogy for extraordinary boundary conditions. The results illuminate how defect interactions encode long-distance physics and open avenues for bootstrap constraints and monodromy defect generalizations in higher dimensions. The methods offer a robust toolkit for probing defect dynamics in a broad class of CFTs with potential applications to holography and condensed-matter impurity problems, and they establish concrete Casimir-energy predictions across multiple models and geometries.

Abstract

We study the operator algebra of extended conformal defects in more than two spacetime dimensions. Such algebra structure encodes the combined effect of multiple impurities on physical observables at long distances as well as the interactions among the impurities. These features are formalized by a fusion product which we define for a pair of defects, after isolating divergences that capture the effective potential between the defects, which generalizes the usual Casimir energy. We discuss general properties of the corresponding fusion algebra and contrast with the more familiar cases that involve topological defects. We also describe the relation to a different defect setup in the shape of a wedge. We provide explicit examples to illustrate these properties using line defects and interfaces in the Wilson-Fisher CFT and the Gross-Neveu(-Yukawa) CFT and determine the defect fusion data thereof.

Figures (8)

• Figure 1: Slab and wedge geometry. In the limit of small angle $\theta$, the wedge imitates a parallel slab.
• Figure 2: Weyl transformation of the wedge in flat space to hyperbolic space.
• Figure 3: Feynman diagrams that contribute to the one-point function $\langle {\mathcal{D}}_1 {\mathcal{D}}_2 \phi^1 (x) \rangle$. The filled circle represents defect coupling while the filled box represents the bulk coupling.
• Figure 4: Feynman diagrams that contribute to the defect two-point function $\langle \langle {\mathcal{D}}_1(\hat{n}) {\mathcal{D}}_2(\hat{m}) \rangle \rangle$ in the critical $O(N)$ vector model.
• Figure 5: Feynman diagrams that contribute to the defect two-point function $\langle \langle {\mathcal{D}}_1 {\mathcal{D}}_2 \rangle \rangle$ in the Gross-Neveu-Yukawa model. Here solid lines represent fermion propagators.