Abstract
We study half-BPS surface operators in 5d mathcal{N} = 1 gauge theories compactified on a circle. Using localization methods and the twisted chiral ring relations of coupled 3d/5d quiver gauge theories, we calculate the twisted chiral superpotential that governs the infrared properties of these surface operators. We make a detailed analysis of the localization integrand, and by comparing with the results from the twisted chiral ring equations, we obtain constraints on the 3d and 5d Chern-Simons levels so that the instanton partition function does not depend on the choice of integration contour. For these values of the Chern-Simons couplings, we comment on how the distinct quiver theories that realize the same surface operator are related to each other by Aharony-Seiberg dualities.
Highlights
We make a detailed analysis of the localization integrand, and by comparing with the results from the twisted chiral ring equations, we obtain constraints on the 3d and 5d Chern-Simons levels so that the instanton partition function does not depend on the choice of integration contour
With combined 2d/4d systems, whose low-energy effective action is encoded in a pair of holomorphic functions: the prepotential, which governs the dynamics in four dimensions, and the twisted chiral superpotential, which describes the massive vacua of the quiver theories in two dimensions
We have discussed in detail how two different realizations of a surface defect encoded in the two quiver diagrams of figure 7 and figure 9 correspond, respectively, to the two different JK prescriptions used in the localization approach
Summary
The contour selects those poles in the fundamental domain that are in the upper half plane In this picture we have explicitly shown the 1-instanton case for the SU(3) gauge theory at k = 1. They are related to the partition function of an “ SU(1) ” theory at level ±1 [40, 41],4 and can be interpreted as the contribution of a continuum in the Coulomb branch which has to be suitably taken into account and decoupled in order to properly define the SU(N ) theory at k5d = ±N [40,41,42,43,44] In this way we recover via the contour analysis that the five dimensional Chern-Simons coupling satisfies the constraint obtained by [50]. In the following we will restrict ourselves to k5d as in (2.16)
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