Contents

supersymmetry

# Contents

## Idea

The special case of super Yang-Mills theory over a spacetime of dimension 3 and with $\mathcal{N}=4$ 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 $N = 2$, $D = 4$

The $N = 4$, $D = 3$ 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 $N = 4$, $D = 3$ SYM moduli space of vacua and exchanges the Coulomb branch with the Higgs branch. (Intriligator-Seiberg 96)

See also the discussion at symplectic duality.

## 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 $\mathcal{N} = 4$, $d = 3$ Quiver Gauge Theories and Mirror Symmetry, (tesi.cab.unipd.it/46485/)
• Antonio Amariti, Gianmarco Formigoni, A note on $4d$ $\mathcal{N} = 3$ from little string theory (arXiv:2003.05983)

See also:

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

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

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 $\mathcal{N}=4$ 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:

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

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