Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String

Abstract

Following a recent paper by Alday and Tachikawa, we compute the instanton partition function in the presence of the surface operator by the localization formula on the moduli space. For SU(2) theories we find an exact agreement with CFT correlation functions with a degenerate operator insertion, which enables us to work out the decoupling limit of the superconformal theory with four flavors to asymptotically free theories at the level of differential equations for CFT correlation functions (irregular conformal blocks). We also argue that the K theory (or five dimensional) lift of these computations gives open topological string amplitudes on local Hirzebruch surface and its blow ups, which is regarded as a geometric engineering of the surface operator. By computing the amplitudes in both A and B models we collect convincing evidences of the agreement of the instanton partition function with surface operator and the partition function of open topological string.

73 pages, 8 figures. v2: minor changes in section 6 and Appendix A, typos corrected and references added. v3: sections 1--3 revised, conventions fixed, references added. v4: minor changes. v5: minor changes

References in corpus (9)

• The Refined Topological Vertex
• Vortex Counting and Lagrangian 3-manifolds
• Open string amplitudes and large order behavior in topological string theory
• Asymptotically free N=2 theories and irregular conformal blocks
• Deformed Prepotential, Quantum Integrable System and Liouville Field Theory
• Gauge Theory, Ramification, And The Geometric Langlands Program
• Surface Operator, Bubbling Calabi-Yau and AGT Relation
• Instanton counting via affine Lie algebras I: Equivariant J-functions of (affine) flag manifolds and Whittaker vectors
• Invariants of algebraic curves and topological expansion

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