Monopoles, Scattering, and Generalized Symmetries

Abstract

We reconsider the problem of electrically charged, massless fermions scattering off magnetic monopoles. The interpretation of the outgoing states has long been a puzzle as, in certain circumstances, they necessarily carry fractional quantum numbers. We argue that consistency requires such outgoing particles to be attached to a topological co-dimension 1 surface, which ends on the monopole. This surface cannot participate in a 2-group with the magnetic 1-form symmetry and is often non-invertible. Equivalently, the outgoing radiation lies in a twisted sector and not in the original Fock space. The outgoing radiation therefore not only carries unconventional flavor quantum numbers, but is often trailed by a topological field theory. We exemplify these ideas in the 1+1 dimensional, chiral 3450 model which shares many of the same features. We comment on the effects of gauge field fluctuations on the lowest angular momentum fermion scattering states in the presence of a magnetic monopole. While, to leading order, these zero modes can penetrate into the monopole core, in the full theory some of the zero modes are lifted and develop a small centrifugal barrier. The dynamics of the zero modes is that of a multi-flavor Schwinger model with a space-dependent gauge coupling. Symmetries and anomalies constrain the fate of the pseudo-zero modes.

Figures (4)

• Figure 1: On the left, a spherically symmetric wave of radiation propagates towards the monopole, at the center of the sphere. On the right, the spherically symmetric outgoing radiation is accompanied by a non-trivial $3d$ TQFT in the interior of the spherical shell.
• Figure 2: Radial quantization on the plane is equivalent to a cylinder geometry by a conformal transformation. The topological line implements a twist in the Hilbert space of the fermions living at its endpoint.
• Figure 3: We consider two twist operators $V_1,V_2$ that live at the end of two invertible lines $g_1,g_2$, assumed anomaly-free. First, we move $g_2$ across $V_1$, picking up a factor of $\langle V_1,V_2\rangle\in U(1)$, which denotes the monodromy of $V_2$ around $V_1$ (i.e., the discontinuity across the branch cut). Then, we apply an $F$-move to recombine the lines. The end result is a $g_2$-line wrapping $V_1$, which gives by definition the charge of this operator under the line.
• Figure 4: Spacetime diagram for the scattering process of the lowest-lying partial wave mode with one angular dimension suppressed. The left-handed component $\Psi_L$ is a local in-falling excitation, whereas the outgoing right-handed component $\Psi_R$ is trailed by a surface with a Wilson line excitation.