The quotient line bundle associated with a planar exchange character

Use the pointed coordinate identification $C_2({\mathbb{R}}^2)\cong {\mathbb{R}}^2\times (0,\mathrm{\infty })\times \mathbb{R}\mathrm{/}\mathbb{Z},$ which sends $q_0=\{(1,0),(-1,0)\}$ to $(0,1,[0])$, to write $X:=C_2({\mathbb{R}}^2)\cong {\mathbb{R}}^2\times (0,\mathrm{\infty })\times \mathbb{R}\mathrm{/}\mathbb{Z}\mathrm{.}$ Put $\tilde{X}:={\mathbb{R}}^2\times (0,\mathrm{\infty })\times \mathbb{R}$ and define $p(c,\rho ,t):=(c,\rho ,[t])\in X,T_n(c,\rho ,t):=(c,\rho ,t+n)(n\in \mathbb{Z})\mathrm{.}$ Define $U(1):=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$ under complex multiplication. Let $\chi :\mathbb{Z}\to U(1)$ satisfy $\chi (0)=1$ and $\chi (m+n)=\chi (m)\chi (n)$ for all $m,n\in \mathbb{Z}$. Let $\mathbb{Z}$ act on $\tilde{X}\times \mathbb{C}$ by $n\cdot ((c,\rho ,t),z):=((c,\rho ,t+n),\chi (n{)}^{-1}z)\mathrm{.}$ The homomorphism law for $\chi$ makes this a group action, and its orbits define the quotient relation. Define the character quotient and its projection by $L_{\chi }:=(\tilde{X}\times \mathbb{C})\mathrm{/}\mathbb{Z},{\mathrm{\Pi }}_{\chi }([\tilde{x},z]):=p(\tilde{x}),$ with the quotient topology on $L_{\chi }$. Since $p(T_n\tilde{x})=p(\tilde{x})$, it induces a continuous map ${\mathrm{\Pi }}_{\chi }:L_{\chi }\to X$. Once $p$ is known to be a covering map, the monodromy of this quotient around a based loop means the fiber map obtained by lifting the loop and keeping its complex coordinate $z$ constant along the lift.

For this item, a unitary flat complex line bundle over a space $X$ means a continuous surjection $\mathrm{\Pi }:L\to X$ with local trivializations ${\mathrm{\Pi }}^{-1}(U)\cong U\times \mathbb{C}$ over open sets $U$, whose changes of fiber coordinate on overlaps are multiplication by functions $g:U\cap V\to U(1)$ that are locally constant (constant on some neighbourhood of each point). The definition above names the quotient data; the following theorem verifies that $L_{\chi }$ has these line-bundle and flatness properties.

The planar two-point configuration space retracts to the quotient circle

The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection

The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$

Real and imaginary parts, complex conjugation, and modulus

Left group actions, transitive actions, and faithful actions

Based loops and the fundamental group

Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings

The inverse in the deck action is chosen so that parallel transport by keeping the lifted fiber coordinate fixed gives monodromy $\chi (n)$ around the loop whose lift ends at $T_n\tilde{x}$.