nLab
D=3 N=4 super Yang-Mills theory

Contents

Context

Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Super-Geometry

String theory

Contents

Idea

The special case of super Yang-Mills theory over a spacetime of dimension 3 and with 𝒩=4\mathcal{N}=4 number of supersymmetries.

Properties

Coulomb- and Higgs-branches

Both the Coulomb branches and the Higgs branch of D=3 N=4 super Yang-Mills theory are hyperkähler manifolds. In special cases they are compact hyperkähler manifolds (e.g. dBHOO 96).

Reduction from N=2N = 2, D=4D = 4

The N=4N = 4, D=3D = 3 SYM theory can be obtained by dimensional reduction from N=2 D=4 super Yang-Mills theory (Seiberg-Witten 96)

Mirror symmetry

A version of mirror symmetry acts on the N=4N = 4, D=3D = 3 SYM moduli space of vacua and exchanges the Coulomb branch with the Higgs branch. (Intriligator-Seiberg 96)

See also the discussion at symplectic duality.

Topological twist and Rozansky-Witten theory

A topological twist of D=3 N=4 super Yang-Mills theory is Rozansky-Witten theory.

References

General

The construction of D=3 N=4 super Yang-Mills theory by dimensional reduction from N=2 D=4 super Yang-Mills theory was first considered in

Discussion as the worldvolume-theory of D3-D5 brane intersections:

Review of the moduli space of vacua:

  • Federici Carta, Moduli Spaces of 𝒩=4\mathcal{N} = 4, d=3d = 3 Quiver Gauge Theories and Mirror Symmetry, (tesi.cab.unipd.it/46485/)

Via KK-compactification from little string theory:

  • Antonio Amariti, Gianmarco Formigoni, A note on 4d4d 𝒩=3\mathcal{N} = 3 from little string theory (arXiv:2003.05983)

See also:

  • Mikhail Evtikhiev, 𝒩=3\mathcal{N} = 3 SCFTs in 4 dimensions and non-simply laced groups (arXiv:2004.03919)

Mirror symmetry for D=3D=3 𝒩=4\mathcal{N}=4 SYM

On mirror symmetry for D=3 N=4 super Yang-Mills theory

The mirror symmetry operation was discussed in

Discussion with emphasis of Higgs branches/Coulomb branches as Hilbert schemes of points

Lift to M-theory

Lift to M-theory:

Coulomb branch and monopole moduli

Review of Coulomb branches of D=3 N=4 super Yang-Mills theory:

  • Marcus Sperling, chapter III of: Two aspects of gauge theories : higher-dimensional instantons on cones over Sasaki-Einstein spaces and Coulomb branches for 3-dimensional 𝒩=4\mathcal{N}=4 gauge theories (spire:1495766/, pdf, pdf)

Identification of the Coulomb branch of D=3 N=4 super Yang-Mills theory with the moduli space of monopoles in Yang-Mills theory:

On D=3 N=4 super Yang-Mills theories with compact hyperkähler manifold Coulomb branches obtained by KK-compactification of little string theories:

The Rozansky-Witten invariants of these moduli spaces:

On a mathematical definition of quantum Coulomb branches of D=3 N=4 super Yang-Mills theory:

Hilbert schemes and Higgs/Coulomb branches

Identification of Higgs branches/Coulomb branches in D=3 N=4 super Yang-Mills theory with Hilbert schemes of points of complex curves:

Witten index

Discussion of the Witten index of D=3 N=4 super Yang-Mills theory:

using discussion in

See also on the Witten index for D=3 N=2 super Yang-Mills theory:

Wilson loop operators

On Wilson loop operators in D=3 N=4 super Yang-Mills theory:

Last revised on April 9, 2020 at 03:46:29. See the history of this page for a list of all contributions to it.