A one-dimensional theory for Higgs branch operators

Abstract

We use supersymmetric localization to calculate correlation functions of half-BPS local operators in 3d N = 4 superconformal field theories whose Lagrangian descriptions consist of vectormultiplets coupled to hypermultiplets. The operators we primarily study are certain twisted linear combinations of Higgs branch operators that can be inserted anywhere along a given line. These operators are constructed from the hypermultiplet scalars. They form a one-dimensional non-commutative operator algebra with topological correlation functions. The 2-and 3-point functions of Higgs branch operators in the full 3d N = 4 theory can be simply inferred from the 1d topological algebra. After conformally mapping the 3d superconformal field theory from flat space to a round three-sphere, we preform supersymmetric localization using a supercharge that does not belong to any 3d N = 2 subalgebra of the N = 4 algebra. The result is a simple model that can be used to calculate correlation functions in the 1d topological algebra mentioned above. This model is a 1d Gaussian theory coupled to a matrix model, and it can be viewed as a gauge-fixed version of a topological gauged quantum mechanics. Our results generalize to non-conformal theories on S 3 that contain real mass and Fayet-Iliopolous parameters. We also provide partial results in the 1d topological algebra associated with the Coulomb branch, where we calculate correlation functions of local operators built from the vectormultiplet scalars.

A new duality between $$ \mathcal{N} $$ = 8 superconformal field theories in three dimensions

“…We also match the values of the S 3 partition functions of the two theories. In addition, we provide a new check using the recently proposed supersymmetric localization of 3d N = 4 theories to a topological 1d sector [15]. Using this method, we calculate the two-and three-point functions of lowlying half and quarter BPS operators, which we use to extract their OPE coefficients, listed in (5.27), (5.28), and (B.1).…”

“…For more details, see [15] as well as [25][26][27]. For the particular case of N = 8 SCFTs, the Higgs and Coulomb topological sectors are isomorphic, so without loss of generality we will study the Higgs one.…”

“…For the particular case of N = 8 SCFTs, the Higgs and Coulomb topological sectors are isomorphic, so without loss of generality we will study the Higgs one. In [15], it was shown that for N = 4 SCFTs described by a Lagrangian with a vector multiplet with gauge algebra g and a hypermultiplet in representation R of g, it is possible to use supersymmetric localization to obtain an explicit description of the 1d sector associated with the Higgs branch. When the 1d topological sector is defined on a great circle within S 3 parameterized by ϕ, as above, its explicit description takes the form of a Gaussian 1d theory coupled to a matrix model:…”

“…Indeed, it is not hard to see that the fields ( X, X T ) transform as a doublet under SU(2) F . It is thus convenient to define 15) where the y a are the same polarization variables introduced earlier in (5.7).…”

“…As explained above, the 1d theory corresponding to the BLG theory requires a generalization of [15]. If we are not interested in correlation functions of operators arising from monopole operators in 3d, we conjecture that we can simply insert (5.4) into (5.2) and compute correlation functions of gauge-invariant operators built from Q and Q.…”