Abstract
Recently, a physical derivation of the Alday-Gaiotto-Tachikawa correspondence has been put forward. A crucial role is played by the complex Chern-Simons theory arising in the 3d-3d correspondence, whose boundary modes lead to Toda theory on a Riemann surface. We explore several features of this derivation and subsequently argue that it can be extended to a generalization of the AGT correspondence. The latter involves codimension two defects in six dimensions that wrap the Riemann surface. We use a purely geometrical description of these defects and find that the generalized AGT setup can be modeled in a pole region using generalized conifolds. Furthermore, we argue that the ordinary conifold clarifies several features of the derivation of the original AGT correspondence.
Highlights
Corresponds to crossing symmetry of the Liouville four-point function, which was rigorously proven some time ago [9]
We propose to use the conifold geometry as an approximation to the pole region of a full supergravity background that would be needed to account for a defect in a squashed S4 background
The generalized conifolds K1,m allow us to outline a generalization of the derivation of the original AGT correspondence, which we denote by AGTλ, involving the inclusion of surface operators on the gauge theory side and a generalization of Toda theory to Todaλ on the two-dimensional side
Summary
Since the story is rather intricate and hinges on some important assumptions, we will briefly sketch the main logic and possible pitfalls of our arguments here. A crucial element in the original derivation is that the Nahm pole on the scalars transforms under Weyl rescaling to the relevant Drinfeld-Sokolov boundary condition on the Chern-Simons connection. We expect that carefully examining the Weyl rescaling of the full supergravity background and the corresponding worldvolume supersymmetry equations should allow one to translate the Nahm pole arising in the 4d-2d frame to the Drinfeld-Sokolov boundary condition in the 3d-3d frame. We propose to use the conifold geometry as an approximation to the pole region of a full supergravity background that would be needed to account for a defect in a squashed S4 background.3 This approximation suffices for our purposes, it comes with a particular value of the squashing parameter that leads to a curvature singularity corresponding to the conifold point. It would be interesting to formulate a similar correspondence for complex Todaλ theories
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