Surface Operator, Bubbling Calabi-Yau and AGT Relation

Abstract

Surface operators in N=2 four-dimensional gauge theories are interesting half-BPS objects. These operators inherit the connection of gauge theory with the Liouville conformal field theory, which was discovered by Alday, Gaiotto and Tachikawa. Moreover it has been proposed that toric branes in the A-model topological strings lead to surface operators via the geometric engineering. We analyze the surface operators by making good use of topological string theory. Starting from this point of view, we propose that the wave-function behavior of the topological open string amplitudes geometrically engineers the surface operator partition functions and the Gaiotto curves of corresponding gauge theories. We then study a peculiar feature that the surface operator corresponds to the insertion of the degenerate fields in the conformal field theory side. We show that this aspect can be realized as the geometric transition in topological string theory, and the insertion of a surface operator leads to the bubbling of the toric Calabi-Yau geometry.

36 pages, 14 figures. v2: minor changes and typos corrected

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• A direct proof of AGT conjecture at beta = 1
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• Affine sl(N) conformal blocks from N=2 SU(N) gauge theories
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• Mass-deformed $T_N$ as a linear quiver
• Towards a proof of AGT conjecture by methods of matrix models
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• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• Central Charges of 2d Superconformal Defects
• The Liouville side of the Vortex
• Exact partition functions of Higgsed 5d $T_N$ theories
• A slow review of the AGT correspondence
• Intersecting Surface Defects and Two-Dimensional CFT
• Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String
• Holomorphic Blocks for 3d Non-abelian Partition Functions
• Topological vertex formalism with O5-plane
• Quantum Hitchin Systems via beta-deformed Matrix Models
• On W-algebras and the symmetries of defects of 6d N=(2,0) theory
• W(1+infinity) algebra as a symmetry behind AGT relation
• Refined geometric transition and $qq$-characters
• Refined Chern-Simons Theory and Knot Homology
• On a hyperholomorphic line bundle over the Coulomb branch
• Generalized Whittaker states for instanton counting with fundamental hypermultiplets
• Modular and duality properties of surface operators in N=2* gauge theories
• Macdonald topological vertices and brane condensates
• (de)Tails of Toda CFT
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• On Integrable Structure and Geometric Transition in Supersymmetric Gauge Theories
• Orbifolds, Defects and Sphere Partition Function
• Surface defects on E-string from 5-brane webs
• N=2 superconformal blocks and instanton partition functions
• Topological Vertex/anti-Vertex and Supergroup Gauge Theory
• Surface operators in 5d gauge theories and duality relations
• The integral representation of solutions of KZ equation and a modification by $\mathcal{K}$ operator insertion
• Surface Operators from M-strings
• The Condensate from Torus Knots
• Conformal blocks of Chiral fields in N=2 SUSY CFT and Affine Laumon Spaces
• Refined topological vertex for a 5D $Sp(N)$ gauge theory with antisymmetric matter
• Refined toric branes, surface operators and factorization of generalized Macdonald polynomials
• Geometric transition in Non-perturbative Topological string
• Quantum spectral curve for (q,t)-matrix model
• The Enhancement of Supersymmetry in M-strings
• Note on refined topological vertex, Jack polynomials and instanton counting
• MacMahon KZ equation for Ding-Iohara-Miki algebra
• Orbifold Vortex and Super Liouville Theory
• Towards a gauge theory interpretation of the real topological string
• Deformed planar topological open string amplitudes on Seiberg-Witten curve
• Refined open topological strings revisited
• Quantum mirror curve of periodic chain geometry
• Non-Stationary Difference Equation and Affine Laumon Space: Quantization of Discrete Painlevé Equation
• $q$-Difference Kac-Schwarz Operators in Topological String Theory
• Chiral rings for surface operators in 4d and 5d SQCD
• Painlevé Kernels and Surface Defects at Strong Coupling