Abstract
We investigate an alternative approach to the correspondence of four dimensional mathcal{N} = 2 superconformal theories and two-dimensional vertex operator algebras, in the framework of the Ω-deformation of supersymmetric gauge theories. The twodimensional Ω-deformation of the holomorphic-topological theory on the product four manifold is constructed at the level of supersymmetry variations and the action. The supersymmetric localization is performed to achieve a two-dimensional chiral CFT. The desired vertex operator algebra is recovered as the algebra of local operators of the resulting CFT. We also discuss the identification of the Schur index of the mathcal{N} = 2 superconformal theory and the vacuum character of the vertex operator algebra at the level of their path integral representations, using our Ω-deformation point of view on the correspondence.
Highlights
Which were checked at the level of the equivalence of the superconformal indices and the vacuum characters
The Ω-deformation is implemented by modifying the theory as a cohomological field theory with respect to the supersymmetry which squares to an isometry of the underlying manifold
Recalling that the deformed supercharge squares to the isometry on C⊥, we see that the localization with respect to this supercharge would not work if the U(1)r R-symmetry is anomalous
Summary
Let us first review how the Donaldson-Witten twist comes about. For a curved metric on C⊥, we twist the holonomy U(1)C⊥ by taking the diagonal subgroup. We preserve the N = (2, 2) supersymmetry on C whose fermionic generators are. Where w = x1 + ix and w = x1 − ix are the coordinates on C⊥ This supercharge squares to the isometry of C⊥ generated by V = w∂w − w∂w. One can still construct a deformation of the theory which has a supercharge which squares to the isometry on C⊥. We can start from the theory on R4, write the variations of component fields with respect to the naive supercharge (2.7), and seek a way of re-writing them in metric-independent fashion so that deformed supersymmetry variations are consistently defined on arbitrary product manifold C × C⊥. The action of the theory has to be modified correspondingly to ensure the invariance under the deformed supersymmetry
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