Generalised permutation branes

TL;DR

The paper introduces a new class of non-factorising D-branes in product group manifolds $G\times G$ with unequal fluxes, called generalised permutation branes, which reduce to ordinary permutation branes when the levels $k_1=k_2$. Using a Lagrangian/WZNW framework and Dirac-Born-Infeld analysis, the authors derive the brane geometry, boundary two-forms, and quantisation conditions, showing a discrete set of branes labelled by elements $f$ of the Cartan torus and preserving the diagonal $G$-symmetry. In the concrete case $G=SU(2)$, they analyse $SU(2)_{k_1}\times SU(2)_{k_2}$, compute tensions and open-string spectra, and demonstrate that the allowed charges reproduce the K-theory group $^{\tau}K(SU(2)\times SU(2))=\mathbb{Z}_{k}\oplus\mathbb{Z}_{k}$ with $k=\gcd(k_1,k_2)$, including torsion. They also generalise the construction to higher rank groups and discuss possible extensions to cosets and connections to $N=2$ minimal models, offering a geometrically motivated mechanism to account for all K-theory charges via branes that break the larger product symmetry to a diagonal subgroup. The work provides both semiclassical evidence and a framework for future exact CFT descriptions of these branes, with implications for boundary conditions in product CFTs and related defect constructions.

Abstract

We propose a new class of non-factorising D-branes in the product group GxG where the fluxes and metrics on the two factors do not necessarily coincide. They generalise the maximally symmetric permutation branes which are known to exist when the fluxes agree, but break the symmetry down to the diagonal current algebra in the generic case. Evidence for the existence of these branes comes from a Lagrangian description for the open string world-sheet and from effective Dirac-Born-Infeld theory. We state the geometry, gauge fields and, in the case of SU(2)xSU(2), tensions and partial results on the open string spectrum. In the latter case the generalised permutation branes provide a natural and complete explanation for the charges predicted by K-theory including their torsion.

Figures (6)

• Figure 1: The intersection of $\mathcal{D}_\tau^{\text{red}}(f)$ with the slice $\{e \}\times SU (2)$ for "reduced" levels $k'_{1}=1$ and $k'_{2}=3$: The group manifold of $SU (2)$ which is a three-sphere $S^{3}$ is drawn as a two-sphere, the circles represent spherical conjugacy classes.
• Figure 2: The brane geometry at the critical angle.
• Figure 3: The brane geometry above the critical angle.
• Figure 4: The brane of figure \ref{['fig:genperma']} might decay into this brane of lower angle (which is obtained by translation in the second factor of $SU(2)\times SU(2)$).
• Figure 5: An illustration of the geometry of the three-dimensional brane for different values of $k'_{1}$ and $k'_{2}$ as an $S^{2}$-fibration over the interval $[0,\pi]$. The function $R (\psi)$ measures the radius of the two-sphere sitting over $\psi\in [0,\pi ]$, $d\hat{s}^{2}\propto d\psi^{2}+R (\psi)^{2} (d\theta^{2}+\sin^{2}\theta d\phi^{2})$.