BPS/CFT correspondence V: BPZ and KZ equations from qq-characters

Abstract

We illustrate the use of the theory of $qq$-characters by deriving the BPZ and KZ-type equations for the partition functions of certain surface defects in quiver $\mathcal{N}=2$ theories. We generate a surface defect in the linear quiver theory by embedding it into a theory with additional node, with specific masses of the fundamental hypermultiplets. We prove that the supersymmetric partition function of this theory with $SU(2{)}^{r-3}$ gauge group verifies the celebrated Belavin-Polyakov-Zamolodchikov equation of two dimensional Liouville theory. We also study the $SU(N)$ theory with $2N$ fundamental hypermultiplets and the theory with adjoint hypermultiplet. We show that the regular orbifold defect in this theory solves the KZ-like equation of the WZW theory on a four punctured sphere and one-punctured torus, respectively. In the companion paper these equations will be mapped to the Knizhnik-Zamolodchikov equations

arXiv comment39 pages, paper 5 of 5; some typos fixed, refs added

References in corpus (18)

• Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory
• Liouville Correlation Functions from Four-dimensional Gauge Theories
• N=2 dualities
• Loop and surface operators in N=2 gauge theory and Liouville modular geometry
• Supersymmetric vacua and Bethe ansatz
• Quantum integrability and supersymmetric vacua
• Quantum Groups and Quantum Cohomology
• The q-characters of representations of quantum affine algebras and deformations of W-algebras
• Elliptic Calogero-Moser system from two dimensional current algebra
• Deforming SW curve
• BPS/CFT Correspondence III: Gauge Origami partition function and qq-characters
• BPS/CFT correspondence IV: sigma models and defects in gauge theory
• Affine sl(N) conformal blocks from N=2 SU(N) gauge theories
• Spiked Instantons from Intersecting D-branes
• Splitting of surface defect partition functions and integrable systems
• Perverse coherent sheaves on blow-up. I. a quiver description
• VEV of Baxter's Q-operator in N=2 gauge theory and the BPZ differential equation
• BPZ equations for higher degenerate fields and nonperturbative Dyson-Schwinger equations

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• Cohomological Hall algebras, vertex algebras and instantons
• Intersecting defects in gauge theory, quantum spin chains, and Knizhnik-Zamolodchikov equations
• Splitting of surface defect partition functions and integrable systems
• Superspin chains and supersymmetric gauge theories
• On $W$-representations of $\beta$- and $q,t$-deformed matrix models
• Quantum Elliptic Calogero-Moser Systems from Gauge Origami
• Super instanton counting and localization
• Quantum integrability from non-simply laced quiver gauge theory
• Quiver ${\text{W}}_{{\epsilon }_1,{\epsilon }_2}$ algebras of 4d $\mathcal{N}=2$ gauge theories
• Double Inozemtsev Limits of the Quantum DELL System
• Twisted reduction of quiver W-algebras
• Super Topological Recursion and Gaiotto Vectors For Superconformal Blocks
• A note on chiral trace relations from qq-characters
• VEV of $Q$-operator in $U(1)$ linear quiver 5d gauge theories