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

TL;DR

The paper establishes a precise bridge between four-dimensional N=2 SU(2) gauge theories with surface operators and two-dimensional CFTs by computing the ramified instanton partition function through localization on the affine Laumon space and matching it to CFT correlation functions with a degenerate Φ_{1,2} insertion. It extends this correspondence to irregular conformal blocks arising in decoupling limits and then reinforces the connection with geometric engineering via open topological strings, using both B-model topological recursion on Seiberg-Witten curves and A-model topological vertices. The work provides comprehensive cross-checks across multiple dual descriptions (COFT, Nekrasov-like partitions, topological strings) for N_f = 0,1,2,3,4, and demonstrates the emergence of irregular blocks and degeneration patterns consistent with gauge-theory decouplings. These results give strong, nonperturbative evidence for the surface operator–degenerate CFT correspondence and illuminate how open/closed string amplitudes encode ramified instanton data in refined backgrounds.

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.

Figures (8)

• Figure 1: Correspondence among instanton counting, CFT and topological string
• Figure 2: Conformal blocks and their degenerations
• Figure 3: Local Hirzebruch surface $K_{{\bf F}_0}$
• Figure 4: Local del Pezzo surface $K_{{dP}_2}$
• Figure 5: D-brane inserted on an inner leg of local Hirzebruch surface