ScalingStacks

Proof. [03T5]

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Proof. Obviously, connection ∇¯\bar{\nabla} does not change under the transformation xj↦xj+ε​nj,nj∈𝐙x_{j}\mapsto x_{j}+\varepsilon n_{j},n_{j}\in{\bf Z}. The transformation xj∨↦xj∨+nj∨,nj∨∈𝐙x_{j}^{\vee}\mapsto x_{j}^{\vee}+n_{j}^{\vee},n_{j}^{\vee}\in{\bf Z} together with the gauge transformation of ∇¯\bar{\nabla} by h=exp(2πi/ε∑jnj∨xj)h=exp(2\pi i/\varepsilon\sum_{j}n_{j}^{\vee}x_{j}) also preserves ∇¯\bar{\nabla}. This proves the Lemma. ■\blacksquare

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