Boundary RG Flows of N=2 Minimal Models

TL;DR

The paper develops a boundary RG-flow framework for $N=2$ minimal models using B-type Landau-Ginzburg branes, uncovering a simple matrix-factorization flow that generates boundary flows and identifying the explicit perturbing operators. It shows that B-brane charges form a torsion lattice $ olinebreak \\Lambda_B \cong \mathbb{Z}_{k+2}$, while mirror symmetry maps these to an A-brane lattice of rank $k+1$ in the ${\f Z}_{k+2}$ LG orbifold, reproducing the known A-brane charges. The RG flows are interpreted as tachyon-condensation-driven brane recombination, with explicit flows $$\mathscr{B}_{L_1} \oplus \mathscr{B}_{L_2} \rightarrow \mathscr{B}_{L_1+L_2+1}$$ generated by a fermionic open-string state of dimension $\Delta<1$, and more general perturbations yielding splitting flows consistent with a decreasing boundary entropy $g$. The work connects the LG picture to RCFT boundary states, matches spectra and $g$-functions, and extends to LG orbifolds where the brane-recombination pattern and torsion charges persist under orbifold projections, highlighting a robust, mirror-consistent description of boundary RG flows in these theories.

Abstract

We study boundary renormalization group flows of N=2 minimal models using Landau-Ginzburg description of B-type. A simple algebraic relation of matrices is relevant. We determine the pattern of the flows and identify the operators that generate them. As an application, we show that the charge lattice of B-branes in the level k minimal model is Z_{k+2}. We also reproduce the fact that the charge lattice for the A-branes is Z^{k+1}, applying the B-brane analysis on the mirror LG orbifold.

Figures (9)

• Figure 1: Flow of A-type boundary conditions in the minimal model
• Figure 2: The A-brane $\Scr{A}_{L,M}$. This is the example $L=2$, $M=5$ for $k=6$.
• Figure 3: Cancellation of out-going and in-coming rays of two A-branes.
• Figure 4: The disc picture of the mirror A-branes. The boundary entropy is given by the length of the segment. It decreases under the RG flow $\Scr{B}_{L_1}\oplus\Scr{B}_{L_2}\rightarrow\Scr{B}_{L_1+L_2+1}$ by the triangle inequality.
• Figure 5: The RG flow $\Scr{B}_{L_1}\oplus\Scr{B}_{L_2} \rightarrow \Scr{B}_{{L_1+L_2\over 2}+j+1}\oplus\Scr{B}_{{L_1+L_2\over 2}-j-1}$ generated by the operator corresponding to $|j\rangle_{L_1L_2}^f$ or $|j\rangle_{L_2L_1}^f$. The mid-point after the flow can only be in the shaded regions. The two components correspond to whether the flow is generated by $|j\rangle_{L_1L_2}^f$ or $|j\rangle_{L_2L_1}^f$. It is evident that the sum of lengths decreases under the flow.