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

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.

Topological twist and Rozansky-Witten theory

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

• 3d superconformal gauge field theory

  • D=3 N=4 super Yang-Mills theory

• 3d-3d correspondence

• N=2 D=4 super Yang-Mills theory

• N=4 D=4 super Yang-Mills 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

• Nathan Seiberg, Edward Witten, Gauge Dynamics And Compactification To Three Dimensions, In: J.M. Drouffe, J.B. Zuber (eds.) The mathematical beauty of physics: A memorial volume for Claude Itzykson Proceedings, Conference, Saclay, France, June 5-7, 1996 (arXiv:hep-th/9607163, spire:420925)

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

• Amihay Hanany, Edward Witten, Type IIB Superstrings, BPS Monopoles, And Three-Dimensional Gauge Dynamics, Nucl. Phys. B492:152-190, 1997 (arXiv:hep-th/9611230)

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/)

Via KK-compactification from little string theory:

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

and from heterotic string theory on ADE-singularities:

• Michele Del Zotto, Marco Fazzi, Suvendu Giri, A new vista on the Heterotic Moduli Space from Six and Three Dimensions [arXiv:2307.10356]

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

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

The mirror symmetry operation was discussed in

• Ken Intriligator, Nathan Seiberg, Mirror Symmetry in Three Dimensional Gauge Theories, Phys. Lett.B 387 : 513-519,1996 (arXiv:hep-th/9607207)

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

• Jan de Boer, Kentaro Hori, Hirosi Ooguri, Yaron Oz, Mirror Symmetry in Three-Dimensional Gauge Theories, Quivers and D-branes, Nucl. Phys. B493:101-147, 1997 (arXiv:hep-th/9611063)

• Jan de Boer, Kentaro Hori, Hirosi Ooguri, Yaron Oz, Zheng Yin, Mirror Symmetry in Three-Dimensional Gauge Theories, $SL(2,\mathbb{Z})$ and D-Brane Moduli Spaces, Nucl. Phys. B493:148-176, 1997 (arXiv:hep-th/9612131)

• Mathew Bullimore, Andrea Ferrari, Heeyeon Kim, Supersymmetric Ground States of 3d $\mathcal{N}=4$ Gauge Theories on a Riemann Surface (arXiv:2105.08783)

Lift to M-theory

Lift to M-theory:

• M. Porrati, Alberto Zaffaroni, M-Theory Origin of Mirror Symmetry in Three Dimensional Gauge Theories, Nucl. Phys. B490 (1997) 107-120 (arXiv:hep-th/9611201)

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:

• Seiberg-Witten 96

• N. Dorey, V. V. Khoze, M. P. Mattis, David Tong, S. Vandoren, Instantons, Three-Dimensional Gauge Theory, and the Atiyah-Hitchin Manifold, Nucl. Phys. B502 (1997) 59-93 (arXiv:hep-th/9703228)

• David Tong, Three-Dimensional Gauge Theories and ADE Monopoles, Phys. Lett. B448 (1999) 33-36 (arXiv:hep-th/9803148)

• Mathew Bullimore, Tudor Dimofte, Davide Gaiotto, The Coulomb Branch of 3d $\mathcal{N}=4$ Theories, Commun. Math. Phys. (2017) 354: 671 (arXiv:1503.04817)

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

• Kenneth Intriligator, Compactified Little String Theories and Compact Moduli Spaces of Vacua, Phys. Rev. D61:106005, 2000 (arXiv:hep-th/9909219)

The Rozansky-Witten invariants of these moduli spaces:

• Lev Rozansky, Edward Witten, p. 36 of: Hyper-Kähler geometry and invariants of 3-manifolds, Selecta Math., New Ser. 3 (1997), 401–458 (arXiv:hep-th/9612216, doi:10.1007/s000290050016, MR98m:57041)

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

• Hiraku Nakajima, Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional $\mathcal{N} = 4$ gauge theories (arXiv:1706.05154)

• Hiraku Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional $\mathcal{N} = 4$ gauge theories, I (arXiv:1503.03676)

• Alexander Braverman, Michael Finkelberg, Hiraku Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional $\mathcal{N} = 4$ gauge theories, II, Adv. Theor. Math. Phys. 22 (2018) 1071-1147 (arXiv:1601.03586)

• Alexander Braverman, Michael Finkelberg, Hiraku Nakajima, Line bundles over Coulomb branches (arXiv:1805.11826)

  (relation to Hilbert schemes)

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:

• Jan de Boer, Kentaro Hori, Hirosi Ooguri, Yaron Oz, Mirror Symmetry in Three-Dimensional Gauge Theories, Quivers and D-branes, Nucl. Phys. B493:101-147, 1997 (arXiv:hep-th/9611063)

• Jan de Boer, Kentaro Hori, Hirosi Ooguri, Yaron Oz, Zheng Yin, Mirror Symmetry in Three-Dimensional Gauge Theories, $SL(2,\mathbb{Z})$ and D-Brane Moduli Spaces, Nucl. Phys. B493:148-176, 1997 (arXiv:hep-th/9612131)

• Stefano Cremonesi, Amihay Hanany, Alberto Zaffaroni, around (4.4) of: Monopole operators and Hilbert series of Coulomb branches of 3d $\mathcal{N} = 4$ gauge theories, JHEP 01 (2014) 005 (arXiv:1309.2657)

• Alexander Braverman, Michael Finkelberg, Hiraku Nakajima, Line bundles over Coulomb branches (arXiv:1805.11826)

• Mykola Dedushenko, Yale Fan, Silviu Pufu, Ran Yacoby, Section E.2 of: Coulomb Branch Quantization and Abelianized Monopole Bubbling, JHEP 10 (2019) 179 (arXiv:1812.08788)

Witten index

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

• Mathew Bullimore, Andrea E.V. Ferrari, Heeyeon Kim, Twisted Indices of 3d N=4 Gauge Theories and Enumerative Geometry of Quasi-Maps (arXiv:1812.05567)

• Davide Gaiotto, Tadashi Okazaki, Sphere correlation functions and Verma modules (arXiv:1911.11126)

• Mathew Bullimore, Samuel Crew, Daniel Zhang, Boundaries, Vermas, and Factorisation (arXiv:2010.09741)

using discussion in

• Andrei Okounkov, Section 3.4 of: Lectures on K-theoretic computations in enumerative geometry (arXiv:1512.07363)

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

• Mathew Bullimore, Andrea E. V. Ferrari, Heeyeon Kim, The 3d Twisted Index and Wall-Crossing (arXiv:1912.09591)

Wilson loop operators

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

• Tudor Dimofte, Niklas Garner, Michael Geracie, Justin Hilburn, Mirror symmetry and line operators (arXiv:1908.00013)