Fusion of conformal interfaces

TL;DR

The paper develops a comprehensive framework for the fusion of conformal interfaces in the c=1 CFT, uncovering a black-hole–like structure with topological interfaces acting as entropy-minimizing, attractor-stabilized, solution-generating objects. By using boundary-state methods and the unfolding trick, it derives explicit fusion rules in which interface charges multiply and depend only on topological data, while the intermediate bulk radius decouples in the squeezed limit. The authors prove a topological reduction of fusion and provide a universal entropy-release formula, linking stability to RG flow and suggesting a rich algebra of interfaces akin to BPS state structures, with potential extensions to quantum (non-geometric) interfaces. These results illuminate large, potentially solvable algebras of interfaces in string theory and hint at connections to condensed-matter systems and number-theoretic structures via the logarithmic charges. Overall, the work establishes a robust, multiplicative fusion algebra for topological interfaces and demonstrates their stable, attractor-like behavior within the c=1 moduli space.

Abstract

We study the fusion of conformal interfaces in the c=1 conformal field theory. We uncover an elegant structure reminiscent of that of black holes in supersymmetric theories. The role of the BPS black holes is played by topological interfaces, which (a) minimize the entropy function, (b) fix through an attractor mechanism one or both of the bulk radii, and (c) are (marginally) stable under splitting. One significant difference is that the conserved charges are logarithms of natural numbers, rather than vectors in a charge lattice, as for BPS states. Besides potential applications to condensed-matter physics and number theory, these results point to the existence of large solution-generating algebras in string theory.

Figures (4)

• Figure 1: Folding trick.
• Figure 2: Fusion of two interfaces between three CFTs with radii $R_1$, $R_2$ and $R_3$. In the limit of vanishing separation, $\varepsilon\to 0$, the result should not depend on the value of the radius in the middle region.
• Figure 3: The action of a $[1, 3]^{(+)}$ operator on the D0-brane of theory 2 (black dot) produces three D0-branes of theory 1 (light-colored dots). The latter are arranged periodically around the circle.
• Figure 4: By stripping off their topological parts, we can relate the singularity in the fusion of any two conformal interfaces to the singularity in the product of two basic radius-changing operators.