A & B model approaches to surface operators and Toda theories

Abstract

It has recently been argued by Alday et al that the inclusion of surface operators in 4d N=2 SU(2) quiver gauge theories should correspond to insertions of certain degenerate operators in the dual Liouville theory. So far only the insertion of a single surface operator has been treated (in a semi-classical limit). In this paper we study and generalise this proposal. Our approach relies on the use of topological string theory techniques. On the B-model side we show that the effects of multiple surface operator insertions in 4d N=2 gauge theories can be calculated using the B-model topological recursion method, valid beyond the semi-classical limit. On the mirror A-model side we find by explicit computations that the 5d lift of the SU(N) gauge theory partition function in the presence of (one or many) surface operators is equal to an A-model topological string partition function with the insertion of (one or many) toric branes. This is in agreement with an earlier proposal by Gukov. Our A-model results were motivated by and agree with what one obtains by combining the AGT conjecture with the dual interpretation in terms of degenerate operators. The topological string theory approach also opens up new possibilities in the study of 2d Toda field theories.

43 pages. v2: Added references, including a reference to unpublished work by S.Gukov; minor changes and clarifications.

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• The Refined Topological Vertex
• Loop and surface operators in N=2 gauge theory and Liouville modular geometry
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• Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra
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• Remodeling the B-model
• Gauge Theory Loop Operators and Liouville Theory
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• The gravity duals of N=2 superconformal field theories
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• Topological strings and 5d T_N partition functions
• A direct proof of AGT conjecture at beta = 1
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• Seiberg-Witten theory, matrix model and AGT relation
• 3d & 5d gauge theory partition functions as q-deformed CFT correlators
• Vertices, Vortices & Interacting Surface Operators
• 5D partition functions, q-Virasoro systems and integrable spin-chains
• Surface Operator, Bubbling Calabi-Yau and AGT Relation
• Affine sl(N) conformal blocks from N=2 SU(N) gauge theories
• Quantization of Integrable Systems and a 2d/4d Duality
• Tao Probing the End of the World
• The matrix model version of AGT conjecture and CIV-DV prepotential
• The uses of the refined matrix model recursion
• Flipping the head of T[SU(N)]: mirror symmetry, spectral duality and monopoles
• Towards a proof of AGT conjecture by methods of matrix models
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• From 3d dualities to 2d free field correlators and back
• Central Charges of 2d Superconformal Defects
• Massive Scaling Limit of beta-Deformed Matrix Model of Selberg Type
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• From SO/Sp instantons to W-algebra blocks
• Instanton partition functions in N=2 SU(N) gauge theories with a general surface operator, and their W-algebra duals
• A slow review of the AGT correspondence
• Holomorphic blocks and the 5d AGT correspondence
• Towards a 4d/2d correspondence for Sicilian quivers
• Analysis of correlation functions in Toda theory and AGT-W relation for SU(3) quiver
• S-duality as a beta-deformed Fourier transform
• Non-Perturbative Topological Strings And Conformal Blocks
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• N=2 Generalized Superconformal Quiver Gauge Theory
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• Wilson Surface Central Charge from Holographic Entanglement Entropy
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• Generalized matrix models and AGT correspondence at all genera
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• Seiberg-Witten curve via generalized matrix model
• 2D CFT blocks for the 4D class ${\mathcal{S}}_k$ theories
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• Stable maps to Looijenga pairs
• W-algebras and surface operators in N=2 gauge theories
• Quantum curves and conformal field theory
• Super-quantum curves from super-eigenvalue models
• Modular and duality properties of surface operators in N=2* gauge theories
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• Localization of Vortex Partition Functions in $\mathcal{N}=(2,2)$ Super Yang-Mills theory
• Refined Hopf Link Revisited
• Supersymmetric Partition Functions and a String Theory in 4 Dimensions
• Macdonald topological vertices and brane condensates
• (de)Tails of Toda CFT
• Symmetry enhancements via 5d instantons, qW-algebrae and (1,0) superconformal index
• From CFT to Ramond super-quantum curves
• Orbifolds, Defects and Sphere Partition Function
• On Integrable Structure and Geometric Transition in Supersymmetric Gauge Theories
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• On the Integrability of Four Dimensional N=2 Gauge Theories in the Omega Background
• Topological vertex for Higgsed 5d $T_N$ theories
• Trinion Conformal Blocks from Topological strings
• N=2 superconformal blocks and instanton partition functions
• 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
• Quiver ${\text{W}}_{{\epsilon }_1,{\epsilon }_2}$ algebras of 4d $\mathcal{N}=2$ gauge theories
• Condensates and instanton - torus knot duality. Hidden Physics at UV scale
• Refined toric branes, surface operators and factorization of generalized Macdonald polynomials
• The Condensate from Torus Knots
• Conformal blocks of Chiral fields in N=2 SUSY CFT and Affine Laumon Spaces
• Beta Deformation and Superpolynomials of (n,m) Torus Knots
• Mirror of the refined topological vertex from a matrix model
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• Dessins d'Enfants, Seiberg-Witten Curves and Conformal Blocks
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• On geometric bases for A-polynomials II: ${\mathfrak{su}}_3$ and Kuberberg bracket
• Painlevé Kernels and Surface Defects at Strong Coupling
• Enumerative geometry of surfaces and topological strings