Quiver W-algebras

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Abstract

For a quiver with weighted arrows, we define gauge-theory K-theoretic W-algebra generalizing the definition of Shiraishi et al. and Frenkel and Reshetikhin. In particular, we show that the qq-character construction of gauge theory presented by Nekrasov is isomorphic to the definition of the W-algebra in the operator formalism as a commutant of screening charges in the free field representation. Besides, we allow arbitrary quiver and expect interesting applications to representation theory of generalized Borcherds–Kac–Moody Lie algebras, their quantum affinizations and associated W-algebras.

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Notes

  1. 1.

    The parameters \((q_1^{\frac{1}{2}},q_2^{-\frac{1}{2}})\) are called by (qt) in [1, 5, 9]. However, the parameter t in Nakajima’s (qt)-characters [21], which grades the cohomological degree, has different meaning from the present \(q_2\).

  2. 2.

    Lefschetz—Grothendieck–Hirzebruch–Riemann–Roch—Atiyah–Singer formula.

  3. 3.

    We adopted the normalizations and the zero modes to the conventions of the present paper in which the \(T\)-observables have the simplest canonical form.

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Acknowledgements

We thank for discussions and comments Alexey Sevastyanov, Edward Frenkel, Nikita Nekrasov and Samson Shatashvili. TK is grateful to Institut des Hautes Études Scientifiques for hospitality where a part of this work has been done. The work of TK was supported in part by Keio Gijuku Academic Development Funds, JSPS Grant-in-Aid for Scientific Research (No. JP17K18090), the MEXT-Supported Program for the Strategic Research Foundation at Private Universities “Topological Science” (No. S1511006), JSPS Grant-in-Aid for Scientific Research on Innovative Areas “Topological Materials Science” (No. JP15H05855), and “Discrete Geometric Analysis for Materials Design” (No. JP17H06462). VP acknowledges Grant RFBR 15-01-04217 and RFBR 16-02-01021. The research of VP on this project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (QUASIFT Grant Agreement 677368).

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Correspondence to Vasily Pestun.

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Kimura, T., Pestun, V. Quiver W-algebras. Lett Math Phys 108, 1351–1381 (2018). https://doi.org/10.1007/s11005-018-1072-1

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Keywords

  • Supersymmetric gauge theories
  • Conformal field theories
  • W-algebras
  • Quantum groups
  • Quiver
  • instanton

Mathematics Subject Classification

  • 81T60
  • 81R10
  • 14D21
  • 81R50