Surface Operator, Bubbling Calabi-Yau and AGT Relation

TL;DR

This work presents a topological string framework for surface operators in 4D $ ext{N}=2$ gauge theories and links them to the AGT correspondence. It demonstrates that open-string wave-functions computed with toric A-branes reproduce ramified Nekrasov partition functions and encode Gaiotto curves, with degenerate Liouville insertions realized through geometric transitions that bubble the Calabi–Yau geometry. The analysis extends to $SU(N_c)$ theories with $2N_c$ flavors, the $ ext{N}=2^{*}$ theory, and their 5D uplifts, including decoupling limits to pure SU(2) Yang–Mills, thereby providing a unified string-theoretic picture of ramified instantons. The bubbling mechanism offers an efficient approach to compute ramified partition functions and clarifies the open/closed duality underlying the extended AGT correspondence.

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.

Figures (14)

• Figure 1: The toric geometry which engineers the $SU(2)$ gauge theory with four flavors (left) and the corresponding Gaiotto curve (right). This geometry is constructed by gluing two geometries that engineers the four free hypermultiplets theory respectively.
• Figure 2: The open A-model topological strings with a toric A-brane on the leg. The background geometry is the toric Calabi-Yau three-fold which engineers the $\mathcal{T}_{3,0}$ theory. The dotted line here denotes the non-compact Lagrangian branes. The Young diagram $Y$ labels the representations of the Wilson loop for the boundary of the world-sheet.
• Figure 3: The D3 brane wraps the non-compact two-cycle whose circle fibration degenerates on the base Riemann surface $H(x,y)=0$.
• Figure 4: The toric diagram of the Calabi-Yau mirror to $uv+H(x,y)=0$. The toric skeleton is just the thinned Riemann surface $H(x,y)=0$.
• Figure 5: $\mathcal{T}_{2,1}$ v.s. $\mathcal{T}_{1,1}$ : torus with two punctures and torus with a puncture and a degenerate field insertion. The X denotes the insertion of a degenerate field.