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The structure of $3/2^{-}$ and $1/2^{+}$ states in $^{11}$B is investigatedwith an $\alpha+\alpha+t$ orthogonality condition model (OCM) based on theGaussian expansion method. Full levels up to the $3/2^{-}_{3}$ and $1/2^{+}_2$states around the $\alpha+\alpha+t$ threshold ($E_x$=11.1 MeV) are reproducedconsistently with the experimental energy levels. It is shown that the$3/2_{3}^{-}$ state located around the $^{7}$Li+$\alpha$ threshold has an$\alpha+\alpha+t$ cluster structure, whereas the $3/2_{1}^{-}$ and$3/2_{2}^{-}$ states have a shell-model-like compact structure. We found thatthe $3/2_{3}^{-}$ state does not possess an $\alpha$-condensate-like naturesimilar to the $0^{+}_{2}$ state of $^{12}$C (Hoyle state) which has a dilute$3\alpha$-condensate structure described by a $(0S_{\alpha})^3$ configurationwith about $70$\% probability, although the monopole transition strength of theformer is as large as that of the latter. We discuss the reasons why the$3/2_{3}^{-}$ state does not have the condensate character. On the other hand,the $1/2^{+}_{1}$ state just below the $^{7}$Li+$\alpha$ threshold has acluster structure which can be interpreted as a parity-doublet partner of the$3/2^{-}_3$ state. We indicate that the $12.56$-MeV state($J^{\pi}=1/2^{+}_{2}$) just above the $\alpha+\alpha+t$ threshold observed inthe $^7$Li($^{7}$Li,$^{11}$B$^*$)$t$ reaction etc. is of thedilute-cluster-gas-like, and is a strong candidate for the Hoyle-analogue statewhich has a configuration of $(0S_{\alpha})^{2}(0S_{t})$ with about $65$\%probability from the analyses of the single-cluster motions in $^{11}$B. Thestructure property of the $1/2^{+}$ resonant state is analyzed with the complexscaling method.

$\alpha+\alpha+t$ cluster structures and Hoyle-analogue states in $^{11}$B