Non-Invertible Symmetries, Brane Dynamics, and Tachyon Condensation

TL;DR

This work builds a holographic realization of the SymTFT for a 4d $\mathcal N=1$ $SU(M)$ gauge theory in the Klebanov–Tseytin/KS background, deriving the bulk topological action and matching bulk symmetry generators to D-brane configurations. It shows that defect fusion is governed by brane center-of-mass and relative motions, with non-invertible (condensation) defects arising from tachyon condensation in brane–antibrane systems, and that condensation defects can be obtained by summing over solitonic branes. The authors compute the full fusion algebra for the symmetry defects from brane dynamics, including dielectric polarization and D3/D1 bound states, and interpret these results in terms of both bulk versus boundary data and boundary conditions, revealing how invertible vs non-invertible sectors emerge on the boundary depending on the chosen boundary data. The analysis provides holographic insight into non-invertible and higher-form symmetries, with implications for higher fusion categories and their centers, and suggests further avenues to classify symmetry data via brane dynamics in holographic setups.

Abstract

We study the Symmetry Topological Field Theory in holography associated with 4d $\mathcal{N}=1$ Super Yang-Mills theory with gauge algebra $\mathfrak{su}(M)$. From this, all the bulk symmetry operators are computed and matched to various D-brane configurations. The fusion algebra of the operators emerges from brane dynamics. In particular, we show that the symmetry operators are purely determined from the center-of-mass modes of the branes. We identify the TQFT fusion coefficients with the relative motion of the branes. We also establish the origin of condensation defects, arising from fusion of non-invertible operators, as the consequence of tachyon condensation in brane-anti-brane pairs.

Figures (3)

• Figure 1: Fusion of three stacks of D5-branes. The left figure depicts the fusion $\mathcal{N}_p\otimes(\mathcal{N}_{p'}\otimes\mathcal{N}_{p"})$, while the right depicts the fusion $(\mathcal{N}_p\otimes\mathcal{N}_{p'})\otimes\mathcal{N}_{p"}$. In the former case, there is a center-of-mass mode $\alpha$ for the overall $p+p'+p"$ stack, and two other relative modes $\alpha'$ and $\alpha"$ respectively for the $p$ -- $(p'+p")$ and $p'$ -- $p"$ stacks. The same applies to the figure on the right.
• Figure 2: Condensation defect arising from tachyon condensation on a $\text{D$p$\,-$\overline{\text{D}p}$}$ system wrapping some internal manifold. We schematically decomposed the worldvolume of the branes as a direct product of internal and external manifolds. Chan-Paton modes exist both for open strings ending on the same brane or on different branes. The latter includes tachyonic modes, which, upon integrating out, lead to a condensate of lower branes within the (both internal and external) worldvolume of the original $\text{D$p$\,-$\overline{\text{D}p}$}$ system.
• Figure 3: A schematic picture for the induced charges on the $\mathbb{Z}_M^{(1)}$ generator $\mathcal{O}_q(W^2)$. In addition to the D3- and D1-brane charges from the constituent branes, there are additional induced charges describing F1-strings and D1-branes. The F1-charge is spread throughout the manifold $W^2$, while the D1-charge is localized to codimension-1 submanifolds.