/sci/ - Science & Math

>>8770675
It's quite intuitive actually. Fix a Lie group [math]G[/math] and we define the set of maps [math]S^1 \rightarrow G[/math] as the loop group [math]LG[/math] on [math]G[/math]. For a semisimple [math]G[/math], we can find a Lie algebra-valued 2-cocycle [math]\omega \in \Omega^1(LG_\mathbb{C}) \otimes \hat{L\mathfrak{g}} [/math] that induces a Hermitian line bundle [math]L \rightarrow LG_{\mathbb{C}}[/math], where the subscript [math]\mathbb{C}[/math] denotes the complexification and the hat denotes the central extension. Now fix a smooth section [math]s:U_\alpha \rightarrow L[/math] with the connection [math]\nabla = d+i \theta_\alpha[/math], where [math]\bigcup_\alpha U_\alpha = LG_{\mathbb{C}}[/math] is a coordinate patch and [math]\theta_\alpha[/math] is the connection 1-form [math]d\theta_\alpha = \omega[/math]. Define the first Chern class [math]c_1(\nabla)[/math] as the equivalence class of [math]\theta_\alpha[/math]. For the identity [math]e∈LG_\mathbb{C}[/math], set [math]u∈s(e)[/math]. The parallel transport of [math]u[/math] around a loop [math]\gamma⊂L[/math] is given by [eqn]v = u\exp\left(2\pi i\int_D c_1(\nabla)\right)[/eqn], where [math] \partial D = \gamma[/math]. This is the holonomy of [math]\nabla[/math] around [math]\gamma[/math].

>>8770675
It's quite intuitive actually. Fix a Lie group [math]G[/math] and we define the set of maps [math]S^1 \rightarrow G[/math] as the loop group [math]LG[/math] on [math]G [/math]. For a semisimple [math]G[/math], we can find a Lie algebra-valued 2-cocycle [math]\omega \in \Omega^{1}(LG_{\mathbb{C}}) \hat{L\mathfrak{g}}[/math] that induces a Hermitian line bundle [math]L \rightarrow LG_{\mathbb{C}[math], where the subscript [math]\mathbb{C}[/math] denotes the complexification and the hat denotes the central extension. Now fix a smooth section [math]s: U_\alpha \rightarrow L[/math] with the connection [math]\nabla = d + i \theta_\alpha[/math], where [math]\bigcup _{\alpha} U_\alpha = LG_{\mathbb{C}}[/math] and [math]\theta_\alpha[/math] is the connection 1-form [math]d\theta_\alpha = \omega[/math]. Define the first Chern class [math]c_1(\nabla)[/math] as the equivalence class of [math]d\theta_\alpha[/math]. For the identity [math]e \in LG_\mathbb{C}[/math], set [math]u \in s(e)[/math] and the parallel transport of [math]u[/math] around a loop [math]\gamma \subset L[/math] is given by [eqn]v = u\exp(2\pi i\int_D c_1(\nabla)[/math], where [math] \partial D = \gamma[/math]. This is the holonomy of [math]\nabla[/math] around [math]\gamma[/math].