Domain Walls for Two-Dimensional Renormalization Group Flows

TL;DR

The paper develops an explicit algebraic construction for renormalization group domain walls between two-dimensional CFTs, focusing on consecutive Virasoro minimal models. By embedding the product theory into a larger coset theory TB with a hidden current algebra B and using a Z2 twist, the authors produce a non-rational RG boundary whose disk one-point functions reproduce the leading-order UV to IR operator mixing obtained from conformal perturbation theory. They validate the approach with detailed calculations for minimal models and extend the framework to general coset flows, highlighting potential links to holography and integrability. The work provides a concrete, algebraic handle on RG interfaces in two dimensions and opens avenues for further tests and generalizations across broader classes of RCFTs.

Abstract

Renormalization Group domain walls are natural conformal interfaces between two CFTs related by an RG flow. The RG domain wall gives an exact relation between the operators in the UV and IR CFTs. We propose an explicit algebraic construction of the RG domain wall between consecutive Virasoro minimal models in two dimensions. Our proposal passes a stringent test: it reproduces in detail the leading order mixing of UV operators computed in the conformal perturbation theory literature. The algebraic construction can be applied to a variety of known RG flows in two dimensions.

Figures (7)

• Figure 1: An RG flow between theories ${\cal T}_{UV}$ and ${\cal T}_{IR}$ is initiated by perturbing ${\cal T}_{UV}$ by a relevant operator integrated over the whole space. If the relevant operator is integrated over a half-space only, the RG flow will produce a RG interface between ${\cal T}_{UV}$ and ${\cal T}_{IR}$
• Figure 2: The folding trick: a conformal interface between theories ${\cal T}_{UV}$ and ${\cal T}_{IR}$ can be reinterpreted as a conformal boundary condition in the product theory ${\cal T}_{UV} \times {\cal T}_{IR}$. Holomorphic and anti-holomorphic quantities in ${\cal T}_{IR}$ are exchanged. For example, the gluing condition for the stress tensor for a conformal interface $T_{UV} - \bar{T}_{UV}=T_{IR} - \bar{T}_{IR}$ becomes the gluing condition for a conformal boundary $T_{UV} + T_{IR}= \bar{T}_{UV} + \bar{T}_{IR}$
• Figure 3: We produce a non-rational candidate for the RG boundary condition by colliding a rational topological interface between the product theory ${\cal T}_{IR} \times {\cal T}_{UV}$ and the auxiliary theory ${\cal T}_{\cal B}$ and a rational, twisted boundary condition in ${\cal T}_{\cal B}$.
• Figure 4: A generic topological interface with label $a$ can be swept across a bulk field $(b, \bar{b})$, but will transform it to a sum (with appropriate fusion coefficients) of twist fields attached to a segment of topological defect of label $c$. Notice that $c$ must be present in the fusion of $a$ with itself and in the fusion of $b$ and $\bar{b}$.
• Figure 5: a) The RG interface is expected to intertwine the topological defects ${\cal D}^{UV}_{r,1}$ and ${\cal D}^{IR}_{1,r}$. b) A ${\cal D}^{UV}_{r,1}$ can continue topologically across a RG interface and become ${\cal D}^{IR}_{1,r}$ on the other side. c) Equivalently, the product defect ${\cal D}^{IR}_{1,r}{\cal D}^{UV}_{r,1}$ in the product theory ${\cal T}_{UV} \times {\cal T}_{IR}$ can end topologically on the RG boundary in a unique fashion.