The Coulomb Branch of 3d $${\mathcal{N}= 4}$$ N = 4 Theories

Abstract

Abstract:We propose a construction for the quantum-corrected Coulomb branch of a general 3d gauge theory with N = 4 supersymmetry, in terms of local coordinates associated with an abelianized theory. In a fixed complex structure, the holomorphic functions on the Coulomb branch are given by expectation values of chiral monopole operators. We construct the chiral ring of such operators, using equivariant integration over BPS moduli spaces. We also quantize the chiral ring, which corresponds to placing the 3d theory in a 2d Omega background. Then, by unifying all complex structures in a twistor space, we encode the full hyperkähler metric on the Coulomb branch. We verify our proposals in a multitude of examples, including SQCD and linear quiver gauge theories, whose Coulomb branches have alternative descriptions as solutions to Bogomolnyi and/or Nahm equations.

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

“…Perturbative and non-perturbative quantum corrections modify the geometry and topology of the Coulomb branch, in a way that was precisely described in [5] (see also [77,78]), and which we summarize later in section 2.5. The chiral ring C[M C ] of holomorphic functions on the Coulomb branch is generated by BPS monopole operators, dressed by polynomials in the ϕ vectormultiplet scalars.…”

“…The theory has R-symmetry SU(2) C × SU(2) H , with φ transforming as a triplet of SU(2) C and the hypermultiplet scalars transforming as complex doublets of SU(2) H . 5 We will typically choose a splitting of the vectormultiplet scalars into real and complex parts (σ, ϕ) ∈ g ⊕ g C , together with a splitting of the hypermultiplet scalars into pairs of complex fields (X, Y ) = (X i , Y i ) N i=1 ∈ R ⊕ R * . The SU(2) C × SU(2) H R-symmetry…”

“…where ϕ is the complex adjoint scalar superpartner of the gauge field, which arises from the dimensional reduction of A 4 + iA 5 . These boundary conditions preserve a 2d N = (2, 2) supersymmetry.…”

“…The Ω-deformation is known to localize a non-linear sigma model with hyperkähler target space M to a supersymmetric quantum mechanics whose operator algebraĈ[M] quantizes the Poisson algebra C[M] of holomorphic functions on M [4]. We similarly expect the Ω-deformation to localize a gauge theory to a gauged supersymmetric quantum mechanics, in which a quantization of the chiral ringĈ[M H ] appears as the gauge-invariant part of the operator algebra associated to a quantization of the matter fields [5]. Concretely, our starting point is N copies of the Heisenberg algebra 12) which quantizes the ring C[T * C N ] of hypermultiplet scalars.…”

“…The holomorphic functions on the Higgs and There are two interesting deformations of 3d N = 4 theories that turn the chiral rings into non-commutative algebras: standard and twisted Omega-backgrounds. In the 3d N = 4 context, this was studied in [4,5] (see below for other connections). The Omega backgrounds mix supersymmetry transformations with rotations of some R 2 ⊂ R 3 , and effectively reduce the 3d theory to one-dimensional supersymmetric quantum mechanics supported on the fixed axis of rotations.…”