Holomorphic field realization of SH${}^c$ and quantum geometry of quiver gauge theories

Abstract

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 $\mathcal{N}=2$ supersymmetric gauge theories. In this paper, we rewrite the SH${}^c$ algebra in terms of three holomorphic fields $D_0(z)$, $D_{\pm 1}(z)$ with which the algebra and its epresentations 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.

32 pages, 5 figures, v3: minor corrections

References in corpus (12)

• BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters
• Instanton counting with a surface operator and the chain-saw quiver
• Deforming SW curve
• Gauge theories on Omega-backgrounds from non commutative Seiberg-Witten curves
• Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity
• Conformal blocks of W_N Minimal Models and AGT correspondence
• SH${}^c$ Realization of Minimal Model CFT: Triality, Poset and Burge Condition
• Construction of Gaiotto states with fundamental multiplets through Degenerate DAHA
• (de)Tails of Toda CFT
• Spherical Hecke algebra in the Nekrasov-Shatashvili limit
• Finite ${\epsilon }_2$-corrections to the $\mathcal{N}=2$ SYM prepotential
• Mayer expansion of the Nekrasov pre potential: the subleading ${\epsilon }_2$-order

Cited by in corpus (17)

• W-symmetry, topological vertex and affine Yangian
• Explicit examples of DIM constraints for network matrix models
• Toric Calabi-Yau threefolds as quantum integrable systems. R-matrix and RTT relations
• Coherent states in quantum ${\mathcal{W}}_{1+\infty }$ algebra and qq-character for 5d Super Yang-Mills
• A slow review of the AGT correspondence
• Factorization of differential expansion for antiparallel double-braid knots
• Fiber-base duality from the algebraic perspective
• Bethe/Gauge correspondence in odd dimension: modular double, non-perturbative corrections and open topological strings
• Rectangular superpolynomials for the figure-eight knot
• Quantum Integrable Systems from Supergroup Gauge Theories
• Topological Vertex/anti-Vertex and Supergroup Gauge Theory
• VEV of Baxter's Q-operator in N=2 gauge theory and the BPZ differential equation
• Shifted quantum groups and matter multiplets in supersymmetric gauge theories
• R-matrices and Miura operators in 5d Chern-Simons theory
• Solution of quantum integrable systems from quiver gauge theories
• Elliptic algebra, Frenkel-Kac construction and root of unity limit
• Quantum $W_{1+\infty }$ subalgebras of BCD type and symmetric polynomials