Construct the bulk-coupled level-changing interface

Construct the complete two-dimensional interface theory coupled simultaneously to the doubled Chern–Simons theories at adjacent levels, including the bulk couplings that specify how line operators transmit across the interface and cancel the level-jump anomaly.

Background

The paper studies interfaces between the doubled theories Tk=Gk×G−k\mathsf T_k=G_k\times G_{-k} and Tk+1\mathsf T_{k+1}, with the goal of realizing the large-level limit through a chain of adjacent-level slabs. A complete interface must include not only an intrinsic two-dimensional conformal or topological theory, but also its couplings to both bulk theories and the associated line-transmission data.

For adjacent SU(2)SU(2) levels, the diagonal coset provides a candidate gapless chiral algebra, while genus-one consistency yields an obstruction to the desired uniformly refining transport for gapped interfaces. The unresolved problem is to construct the actual bulk-coupled interface rather than only identify candidate intrinsic interface theories or necessary modular constraints.

References

An intrinsic two-dimensional theory without these bulk couplings is therefore not the complete interface datum, and the detailed construction of the doubled bulk-coupled interfaces \mathfrak J_k remains open.

— Large-Level Limit of Doubled Chern-Simons Theory Part I: Regular Sector, Boundaries and Interfaces  (2610.26735 - Guan et al., 19 Oct 2026) in Section 6.1, “The interface chain and refinement”

We conjecture that, after the continuum averaging and normalization required to form the limiting fields, this sector converges as k\to\infty to the Runkel--Watts theory \mathcal M_\infty of. We further conjecture that the physical bulk-coupled interface contains the fixed-charge junction sectors of~eq:RW-fixed-charge-limit, whose limiting conformal-weight set is S in every fixed regular BF[SU(2)] charge sector. For each fixed n, we conjecture that the K\to\infty limit of the finite chain~eq:interface-chain consists of n+1 BF[SU(2)] regions separated by n such bulk-enriched two-dimensional interfaces.

eq:RW-fixed-charge-limit:

h±⟶(t−δ)24,h+−h−=η(yJ−δ)2(k+3)⟶0.h_\pm\longrightarrow\frac{(t-\delta)^2}{4}, \qquad h_+-h_-=\frac{\eta(y_J-\delta)}{2(k+3)} \longrightarrow0.

eq:interface-chain:

TK ∣ΣK+1/2 TK+1 ∣ΣK+3/2 ⋯ ∣ΣK+n−1/2 TK+n ,K≥1 ,\mathsf T_K\ \big|_{\Sigma_{K+1/2}}\ \mathsf T_{K+1}\ \big|_{\Sigma_{K+3/2}}\ \cdots\ \big|_{\Sigma_{K+n-1/2}}\ \mathsf T_{K+n}\,,\qquad K\geq1\,,

— Large-Level Limit of Doubled Chern-Simons Theory Part I: Regular Sector, Boundaries and Interfaces  (2610.26735 - Guan et al., 19 Oct 2026) in Conjecture 6.3, Section 6.3, “The large-level limit of the unit-jump interface”