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Reflection states are introduced in the vertical and horizontal modules ofthe Ding-Iohara-Miki (DIM) algebra (quantum toroidal $\mathfrak{gl}_1$). Websof DIM representations are in correspondence with $(p,q)$-web diagrams of typeIIB string theory, under the identification of the algebraic intertwiner ofAwata, Feigin and Shiraishi with the refined topological vertex. Extending thecorrespondence to the vertical reflection states, it is possible to engineerthe $\mathcal{N}=1$ quiver gauge theory of D-type (with unitary gauge groups).In this way, the Nekrasov instanton partition function is reproduced from theevaluation of expectation values of intertwiners. This computation leads to theidentification of the vertical reflection state with the orientifold plane ofstring theory. We also provide a translation of this construction in theIqbal-Kozcaz-Vafa refined topological vertex formalism.

Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver