Worldsheet Description of Exotic Five-brane with Two Gauged Isometries

We study the string worldsheet description of the background geometry of the exotic $5^2_2$-brane where two isometries are gauged. This is an extension of the gauged linear sigma model (GLSM) for the exotic $5^2_2$-brane with a single gauged isometry. The new GLSM with two gauged isometries has only ${\mathcal N}=(2,2)$ supersymmetry rather than ${\mathcal N}=(4,4)$ supersymmetry of the original GLSM. This is caused by a conflict between two different $SU(2)_R$ associated with the two gauge symmetries. However, if we take a certain limit, we can find the genuine string sigma model of the background geometry of the exotic $5^2_2$-brane with ${\mathcal N}=(4,4)$ supersymmetry. We also investigate the worldsheet instanton corrections to the background geometry of the exotic $5^2_2$-brane. The worldsheet instanton corrections to the string sigma model can be traced in terms of the two gauge fields in the new GLSM. This new GLSM gives rise to a different feature of the quantum corrections from the one in the GLSM with the single gauged isometry.