Ring Relations and Mirror Map from Branes

Abstract

We study the space of vacua of three-dimensional $\mathcal{N}=4$ theories from a novel approach building on the type IIB brane realization of the theory and in which the insertion of local chiral operators in the path integral is obtained from integrating out light modes in appropriate brane setups. Most of our analysis focuses on abelian quiver theories which can be realized as the low-energy theory of D3-D5-NS5 brane arrays. Their space of vacua contains a Higgs branch, parametrized by the vevs of half-BPS meson operators, and a Coulomb branch, parametrized by the vevs of half-BPS monopole operators. We show that the Higgs operators are inserted by adding F1 strings and D3 branes, while the Coulomb operators are inserted by adding D1 strings and D3 branes, with specific orientations, to the initial brane setup of the theory. This approach has two main advantages. First the ring relations describing the Higgs and Coulomb branches can be derived by looking at specific brane setups with multiple interpretations in terms of operator insertions. This provides a new derivation of the Coulomb branch quantum relations. Secondly the map between the Higgs and Coulomb operators of mirror dual theories is obtained in a trivial way from IIB S-duality.

76 pages, including 41 figures. v2: discussion on the Higgs branch of non-abelian theories added, published version

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Cited by in corpus (7)

• Coulomb Branch Operators and Mirror Symmetry in Three Dimensions
• Coulomb Branch Quantization and Abelianized Monopole Bubbling
• N=2 Moduli of AdS4 vacua: A fine-print study
• The Space of Vacua of 3d $\mathcal{N}=3$ Abelian Theories
• Nilpotent orbit Coulomb branches of types AD
• Quantized Coulomb Branches, Monopole Bubbling and Wall-Crossing Phenomena in 3d $\mathcal{N}=4$ Theories
• Holomorphic boundary conditions for topological field theories via branes in twisted supergravity