2015
DOI: 10.1007/978-3-319-18769-3_6
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Localization for $$\mathcal {N}=2$$ Supersymmetric Gauge Theories in Four Dimensions

Abstract: We review the supersymmetric localization of N = 2 theories on curved backgrounds in four dimensions using N = 2 supergravity and generalized conformal Killing spinors. We review some known backgrounds and give examples of new geometries such as local T 2 -bundle fibrations. We discuss in detail a topological foursphere with generic T 2 -invariant metric. This review is a contribution to the special volume on recent developments in N = 2 supersymmetric gauge theory and the 2d-4d relation.

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Cited by 34 publications
(67 citation statements)
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“…Thus Ω is an equivariant extension of the canonical volume form. Indeed one can argue that the lowest component Ω 0 of the equivariant extension Ω of any volume form on S 4 (e.g., that corresponding to the squashed S 4 considered in [16,17] ) always has a different sign at the two poles. Applying our observations to Pestun's theory on a round S 4 [1], we can see that modulo the θ-term the supersymmetric action can be rewritten as…”
Section: Example: Smentioning
confidence: 99%
“…Thus Ω is an equivariant extension of the canonical volume form. Indeed one can argue that the lowest component Ω 0 of the equivariant extension Ω of any volume form on S 4 (e.g., that corresponding to the squashed S 4 considered in [16,17] ) always has a different sign at the two poles. Applying our observations to Pestun's theory on a round S 4 [1], we can see that modulo the θ-term the supersymmetric action can be rewritten as…”
Section: Example: Smentioning
confidence: 99%
“…The equations describing such configurations are obtained by setting to zero the supergravity variations of the gravitino and other fermionic fields in the supergravity multiplet. 12 For N = 2 supersymmetric theories the relevant equations have been analyzed in [28,29], and in particular applied to the (squashed) four-sphere. 13 On the four-sphere, the result of the above analysis (or the shortcut described in footnote 13) is that the Killing spinors ξ I ,ξ I describing supersymmetry variations should solve the 12 One can similarly couple the theory to a nontrivial background for flavor symmetries.…”
Section: N = 2 Supersymmetry On Smentioning
confidence: 99%
“…Indeed, given the monopole chargem = 1 2 , {Y m jm |j ∈ N + |m|, m ∈ {−j, −j + 1, .. . , +j}} forms an orthonormal basis for the space of sections Γ(S − ) of the nontrivial anti-chiral spinor bundle S − 29. The first order differential operator ℘ can be viewed as ∂ − iaz on Γ(S − ), where a is the U (1) connection on S − , taking the monopole profile a = 1 2 (±1 − cos ρ)dχ on U N/S .…”
mentioning
confidence: 99%
“…On the four-ball B 4 , we choose a supergravity background, such that a theory with 4d N = 2 supersymmetry can be put on it, preserving some supercharges. Such backgrounds are reviewed in [71], the Omega-background being a particular example. 44 It is expected that the supersymmetric observables do not depend on much of the details of the background.…”
Section: Conformal Blocks Corners and Liouvillementioning
confidence: 99%