Fluxbranes, Generalized Symmetries, and Verlinde's Metastable Monopole

TL;DR

The paper addresses continuous higher-form global symmetries in string derived QFTs and analyzes how coupling to gravity affects Verlinde's metastable monopole. It proposes fluxbranes wrapped at infinity as the natural symmetry operators for these continuous symmetries and derives their topological actions. By studying Verlinde's monopole in both gravity decoupled and gravity on scenarios, it shows how confinement and deconfinement phases emerge and how a lower dimensional TFT sector can arise from fluxbrane interactions. The work offers a top-down mechanism linking generalized symmetry operators to brane dynamics and monopole phase transitions, with implications for non-invertible symmetries and future explorations of partial confinement and gravitational contexts.

Abstract

The stringy realization of generalized symmetry operators involves wrapping "branes at infinity". We argue that in the case of continuous (as opposed to discrete) symmetries, the appropriate objects are fluxbranes. We use this perspective to revisit the phase structure of Verlinde's monopole, a proposed particle which is BPS when gravity is decoupled, but is non-BPS and metastable when gravity is switched on. Geometrically, this monopole is obtained from branes wrapped on locally stable but globally trivial cycles of a compactification geometry. The fluxbrane picture allows us to characterize electric (resp. magnetic) confinement (resp. screening) in the 4D theory as a result of monopole decay. In the presence of the fluxbrane, this decay also creates lower-dimensional fluxbranes, which in the field theory is interpreted as the creation of an additional topological field theory sector.

Figures (3)

• Figure 1: Depiction of the local geometry used to engineer the metastable monopole. Locally, we have $X$ as given by the canonical bundle over a Kähler surface $S$. In this local surface $S$, we have a locally stable 2-cycle $\alpha$ which is the boundary of a non-compact 3-chain $\Gamma$ which extends along the radial direction of $X$ to the boundary $\partial X$, where the image 2-cycle (under the Gysin sequence) is denoted as $\alpha_{\infty}$. In the full compact geometry $Y$, this 3-chain can unwind, so the resulting defect is only metastable. We have also indicated the 3-cycle $\Lambda_3$ which intersects $\Gamma$ at a point in $\partial X$. The internal cycle $\Lambda_3$, when integrated against a 5-form topological term of the fluxbrane results in a 2D TFT in the spacetime which detects the monopole defect.
• Figure 2: Depiction of the monopole / flux tube configuration before and after switching on gravity, as indicated by the value of Newton's constant $G_N$ in the 4D model. In the limit where gravity is switched off (left), we have a heavy line operator as obtained from a D3-brane wrapping a non-compact 3-chain $\Gamma$, and a stringlike flux tube obtained from wrapping a D3-brane on the compact 2-cycle $\alpha$. There is a corresponding symmetry operator obtained from integrating a 5-form over an internal 3-cycle, resulting in a codimension-two topological 1-form symmetry operator $\mathcal{U}_\eta(\Sigma_2)$ which links with the heavy monopole line operator. When gravity is switched on (middle and right), the infinitesimally small 3-ball defining the monopole begins to expand, which in the internal geometry signals the unwinding of $\alpha$ in the full compact geometry $Y$. At early times $t \ll \tau_{\mathrm{mono}}$ below the lifetime of the monopole, this 3-ball is still surrounded by $\Sigma_2$, but at late times $t \gg \tau_{\mathrm{mono}}$ above the lifetime of the monopole, the ball has expanded, and the symmetry operator no longer surrounds a monopole. In this limit, the $U(1)^{(1)}_{\mathrm{mag}}$ is broken, and the electric degrees of freedom have become confined.
• Figure 3: Left: The monopole is linked by the symmetry operator $\mathcal{U}_{\eta}(\Sigma_2)$. Right: After unwinding along $\Gamma_Y$ the monopole has grown beyond $\Sigma_2$ (dotted blue circle). When the flux 4-brane passes through the D5-brane, the flux 2-brane $F2$ associated with NSNS 3-form flux is created. It localizes on $\Sigma_2\times I$ with boundaries on the D5-brane and the flux 4-brane, here $\Sigma_2\times I$ is the difference of two spatial 3-balls.