Gauged Linear Sigma Model with F-term for A-type ALE Space

We construct yet another ${\mathcal N}=(4,4)$ gauged linear sigma model for the $A_N$-type ALE space. In our construction the toric data of the ALE space are manifest. Due to the $SU(2)_R$ symmetry, the F-term is automatically determined. The toric data, which govern the Kähler structures of the ALE space, are embedded into $U(1)$ charges of charged hypermultiplets. The F-term is also inevitable to determine the complex structures of the ALE space. In the IR limit, we obtain the Kähler potential of the $A_N$-type ALE space. We also find the origin of the ${\mathbb Z}_{N+1}$ orbifold symmetry in the singular limit of the $A_N$-type ALE space. In a special case, we reproduce an explicit form of the Kähler potential of the $A_1$-type ALE space, i.e., the Eguchi-Hanson space.

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