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We study the integrable crosscap states of the integrable quantum spin chainsand we classify them for the $\mathfrak{gl}(N)$ symmetric models. We also givea derivation for the exact overlaps between the integrable crosscap states andthe Bethe states. The first part of the derivation is to calculate sum formulafor the off-shell overlap. Using this formula we prove that the normalizedoverlaps of the multi-particle states are ratios of the Gaudin-likedeterminants. Furthermore we collect the integrable crosscap states which canbe relevant in the AdS/CFT correspondence.

Integrable crosscap states in $\mathfrak{gl}(N)$ spin chains