Equivariant Interpolations in Topological Holography

Abstract We revisit equivariant Gromov-Witten theories on P1 and on P1 × C2 . One can introduce three equivariant parameters associated to rotations of the sphere as well as the two planes. A number of points in the parameter space have known holographic duals. These include the symmetric orbifold point dual to the AdS3 × S3 × C2 string theory at string scale radius of curvature, the grand canonical Hurwitz theory and the product of two Kontsevich models. Within this framework, we discuss interpolations in the equivariant parameters. Firstly, we move between the small and large equivariant parameter regimes in Gromov-Witten theory on P1 . At large equivariant parameter, the model is dominated by the pure topological gravity theories at the two fixed points while at small equivariant parameter the theory is equivalent to the grand canonical Hurwitz theory. The deformation is a solvable analogue for the interpolation in the transposition coupling in the moduli space of the AdS3 /CFT2 duality. Moreover, we propose that the full equivariant correspondence between the Gromov-Witten theory on P1 × C2 and the symmetric orbifold of the equivariant plane can be embedded in string theory. On the boundary side of that correspondence, we analyze the scaling limit from the equivariant to the ordinary cohomology ring for the Hilbert scheme of points on the plane in terms of Jack symmetric polynomials. We explicitly compute several structure constants of the equivariant cohomology ring and point out their intriguing positivity and integrality properties.