I don't understand how the fermionic zero modes which become monopole operators are being quantized in Witten and Seiberg's paper on gapped boundary phases of TI's via weak coupling: https://arxiv.org/abs/1602.04251.

I'm having trouble with some of the statements made in Appendix B (and used throughout the paper). In Eq. B.16, they list the possible (classical) spins of the zero modes as a function of vorticity: $j'=(v-1)/2, (v-3)/2,\dots,-(v-1)/2$ where $v$ is an integer. I think I understand how this expression is derived -- what happens upon quantizing is the confusing point.

They go on to say that for $v=1$, there is a single zero mode with spin $j'=0$.

For $v=2$, the zero modes have spins $\pm1/2$. At the bottom of page 70, they simply assert that upon quantization this means that upon quantizing, this becomes pair of states with spin $\pm1/4$.

$v=3$ gives three zero modes with spins 1,0,-1. Quantizing the $\pm1$ modes, gives two states with spins $\pm1/2$. The third mode keeps its zero spin.

I've tried comparing with the discussion in Borokhov et al. (arXiv:hep-th/0206054) as well as Sec. 7.1 of Dyer et al. (arXiv:1309.1160), but I don't see how their arguments can be generalized

In particular, Witten and Seiberg give the usual definition of a monopole by performing the path integral in the presence of a Dirac singularity. With fermions present in the action, the semi-classical equations of motion return fermionic zero modes, whose number depends both on the topological charge of the monopole and the charge of the fermions.

Each zero mode is then subsequently treated as an operator which can act of the Fock vacuum $\sim \chi_{i_1}^\dagger\chi_{i_2}^\dagger\cdots\chi_{i_n}^\dagger\left|0\right>$. Enforcing the Gauss constraint places restrictions on the linear combinations which are allowed and these become the monopole operators. My understanding from the Borokhov and Dyers papers is both charge neutrality and that the ground states contain only spin singlets. What am I missing?


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