Holomorphic field realization of SH c and quantum geometry of quiver gauge theories

  • Jean-Emile Bourgine
  • Yutaka Matsuo
  • Hong Zhang
Open Access
Regular Article - Theoretical Physics


In the context of 4D/2D dualities, SH c algebra, introduced by Schiffmann and Vasserot, provides a systematic method to analyse the instanton partition functions of \( \mathcal{N}=2 \) supersymmetricgaugetheories. Inthispaper,werewritetheSH c algebrainterms of three holomorphic fields D 0(z), D ±1(z) with which the algebra and its representations are simplified. The instanton partition functions for arbitrary \( \mathcal{N}=2 \) super Yang-Mills theories with A n and A n (1) type quiver diagrams are compactly expressed as a product of four building blocks, Gaiotto state, dilatation, flavor vertex operator and intertwiner which are written in terms of SH c and the orthogonal basis introduced by Alba, Fateev, Litvinov and Tarnopolskiy. These building blocks are characterized by new conditions which generalize the known ones on the Gaiotto state and the Carlsson-Okounkov vertex. Consistency conditions of the inner product give algebraic relations for the chiral ring generating functions defined by Nekrasov, Pestun and Shatashvili. In particular we show the polynomiality of the qq-characters which have been introduced as a deformation of the Yangian characters. These relations define a second quantization of the Seiberg-Witten geometry, and, accordingly, reduce to a Baxter TQ-equation in the Nekrasov-Shatashvili limit of the Omega-background.


Conformal and W Symmetry Differential and Algebraic Geometry Gauge Symmetry String Duality 


Open Access

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Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Jean-Emile Bourgine
    • 1
  • Yutaka Matsuo
    • 2
  • Hong Zhang
    • 3
  1. 1.INFN BolognaUniversità di BolognaBolognaItaly
  2. 2.Department of PhysicsThe University of TokyoTokyoJapan
  3. 3.Department of Physics and Center for Quantum Spacetime (CQUeST)Sogang UniversityMapo-guKorea

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