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We construct Gaiotto states with fundamental multiplets in $SU(N)$ gauge theories, in terms of the orthonormal basis of spherical degenerate double affine Hecke algebra (SH in short), the representations of which are equivalent to those of $W_n$ algebra with additional $U(1)$ current.
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Hiraku Nakajima and Kota Yoshioka, “Instanton counting on blowup. I. 4-dimensional pure gauge theory”, Invent. Math. 162, 313-355 (2005)
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A. Mironov and A. Morozov, “On AGT relation in the case of U(3),” Nucl. Phys. B 825
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V. A. Fateev and A. V. Litvinov, “Integrable structure, W-symmetry and AGT relation,” JHEP 1201
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2013
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N. Nekrasov and A. Okounkov, “Seiberg-Witten theory and random partitions,” arXiv:hep-th/0306238
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D. Maulik and A. Okounkov, “Quantum Groups and Quantum Cohomology,” arXiv:1211.1287 [math.AG]
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A. Smirnov, “On the Instanton R-matrix,” arXiv:1302.0799 [math.AG]
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