nLab D=6 N=(1,0) SCFT

Idea

superconformal field theory in dimension 6 with number of supersymmetries equal to 1.

Arises from

• M5-branes at D-, and E-type ADE-singularities

• M-theory on S1/G_HW times H/G_ADE

(…)

Related concepts

$d$	$N$	superconformal super Lie algebra	R-symmetry	black brane worldvolume superconformal field theory via AdS-CFT
$\phantom{A}3\phantom{A}$	$\phantom{A}2k+1\phantom{A}$	$\phantom{A}B(k,2) \simeq$ osp$(2k+1 \vert 4)\phantom{A}$	$\phantom{A}SO(2k+1)\phantom{A}$
$\phantom{A}3\phantom{A}$	$\phantom{A}2k\phantom{A}$	$\phantom{A}D(k,2)\simeq$ osp$(2k \vert 4)\phantom{A}$	$\phantom{A}SO(2k)\phantom{A}$	M2-brane D=3 SYM BLG model ABJM model
$\phantom{A}4\phantom{A}$	$\phantom{A}k+1\phantom{A}$	$\phantom{A}A(3,k)\simeq \mathfrak{sl}(4 \vert k+1)\phantom{A}$	$\phantom{A}U(k+1)\phantom{A}$	D3-brane D=4 N=4 SYM D=4 N=2 SYM D=4 N=1 SYM
$\phantom{A}5\phantom{A}$	$\phantom{A}1\phantom{A}$	$\phantom{A}F(4)\phantom{A}$	$\phantom{A}SO(3)\phantom{A}$	D4-brane D=5 SYM
$\phantom{A}6\phantom{A}$	$\phantom{A}k\phantom{A}$	$\phantom{A}D(4,k) \simeq$ osp$(8 \vert 2k)\phantom{A}$	$\phantom{A}Sp(k)\phantom{A}$	M5-brane D=6 N=(2,0) SCFT D=6 N=(1,0) SCFT

(Shnider 88, also Nahm 78, see Minwalla 98, section 4.2)

• D=6 N=(2,0) SCFT

• D=6 supergravity

• D=5 super Yang-Mills theory

• M5-brane elliptic genus

References

General

• Kantaro Ohmori, Six-Dimensional Superconformal Field Theories and Their Torus Compactifications, Springer Theses 2018 (springer:book/9789811330919)

Comprehensive review of expected top-down constructions of $D=6$ $\mathcal{N}=(2,0)$ SCFTs and $D=6$ $\mathcal{N}=(1,0)$ SCFTs

• Jonathan J. Heckman, Tom Rudelius, Top Down Approach to 6D SCFTs, J. Phys. A: Math. Theor. 52 9 (2019) 093001 [arXiv:1805.06467, doi:10.1088/1751-8121/aafc81]

Via the duality between F-theory and heterotic string theory:

• Paul-Konstantin Oehlmann, Fabian Ruehle, Benjamin Sung, The Frozen Phase of Heterotic F-theory Duality [arXiv:2404.02191]

Anomaly cancellation

Discussion of anomaly cancellation and Green-Schwarz mechanism:

• Michael Green, John Schwarz, Pete West, Anomaly-free chiral theories in six dimensions, Nuclear Physics B Volume 254, 1985, Pages 327-348 (doi:10.1016/0550-3213(85)90222-6)

• Augusto Sagnotti, A Note on the Green - Schwarz Mechanism in Open - String Theories, Phys. Lett. B294:196-203, 1992 (arXiv:hep-th/9210127)

• Kantaro Ohmori, Hiroyuki Shimizu, Yuji Tachikawa, Anomaly polynomial of E-string theories, J. High Energ. Phys. 2014, 2 (2014) (arXiv:1404.3887)

  (Green-Schwarz anomaly in 1.2)

• Kantaro Ohmori, Hiroyuki Shimizu, Yuji Tachikawa, Kazuya Yonekura, Anomaly polynomial of general 6d SCFTs, Progress of Theoretical and Experimental Physics, Volume 2014, Issue 10, October 2014, 103B07 (arXiv:1408.5572)

  (Green-Schwarz anomaly on tensor branch of M5-branes on MO9 in 2.18)

• Kenneth Intriligator, $6d$, $\mathcal{N}=(1,0)$ Coulomb Branch Anomaly Matching, J. High Energ. Phys. 2014, 162 (2014) (arXiv:1408.6745, doi:10.1007/JHEP10(2014)162)

  (Green-Schwarz anomaly on Coulomb branch in 4.1)

• Hiroyuki Shimizu, Aspects of anomalies in 6d superconformal field theories, Tokyo 2018 (spire:1802462, doi:10.15083/00077898)

  (Green-Schwarz anomaly on ADE-singularity in 7.2.8)

• Clay Cordova, Thomas Dumitrescu, Kenneth Intriligator, from p. 18 in: 2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories (arXiv:2009.00138)

Lagrangian description

A class of Lagrangian densities for D=6 N=(1,0) SCFTs with non-abelian gauge group has been proposed in

• Henning Samtleben, Ergin Sezgin, Robert Wimmer, $(1,0)$ superconformal models in six dimensions, J. High Energ. Phys. 2011, 62 (2011) (arXiv:1108.4060, doi:10.1007/JHEP12(2011)062)

reviewed in

• Henning Samtleben, Ergin Sezgin, Robert Wimmer, Linus Wulff, New superconformal models in six dimensions: Gauge group and representation structure, PoS CORFU2011 (2011) 071 (arXiv:1204.0542, spire:1102894)

An attempt to understand the SSW 11-models as higher gauge theories for gauge-L-infinity algebras which are variants of the string Lie 2-algebra is due to:

• Christian Saemann, Lennart Schmidt, The Non-Abelian Self-Dual String and the $(2,0)$-Theory (arXiv:1705.02353)

• Christian Saemann, Lennart Schmidt, Towards an M5-Brane Model I: A 6d Superconformal Field Theory, J. Math. Phys. 59 (2018) 043502 (arxiv:1712.06623)

A description induced form a non-Lorentz invariant 5d Lagrangian:

• Neil Lambert, Tristan Orchard, Non-Lorentzian Avatars of $(1,0)$ Theories, J. High Energ. Phys. 2021 205 (2021) [arXiv:2011.06968, doi:10.1007/JHEP02(2021)205]

• Neil Lambert, Joseph Smith, RG Flows and Symmetry Enhancement in Five-Dimensional Lifshitz Gauge Theories [arXiv:2212.07717]

reviewed in:

• Neil Lambert, A non-Lorentzian View of the M5-brane, talk at M-Theory and Mathematics 2023, NYU Abu Dhabi (2023) [web]

Geometric engineering on 5-branes

On D=6 N=(1,0) SCFTs via geometric engineering on M5-branes/NS5-branes at D-, E-type ADE-singularities, notably from M-theory on S1/G_HW times H/G_ADE, hence from orbifolds of type I' string theory (see at half NS5-brane):

• Ori Ganor, David Morrison, Nathan Seiberg, Branes, Calabi-Yau Spaces, and Toroidal Compactification of the $N=1$ Six-Dimensional $E_8$ Theory, Nucl. Phys. B487:93-127, 1997 (arXiv:hep-th/9610251)

  (emphasis on the KK-compactifications)

• Michele Del Zotto, Jonathan Heckman, Alessandro Tomasiello, Cumrun Vafa, 6d Conformal Matter, JHEP02(2015)054 (arXiv:1407.6359)

• Davide Gaiotto, Alessandro Tomasiello, Holography for $(1,0)$ theories in six dimensions, JHEP12(2014)003 (arXiv:1404.0711)

• Kantaro Ohmori, Hiroyuki Shimizu, $S^1/T^2$ Compactifications of 6d $\mathcal{N} = (1,0)$ Theories and Brane Webs, J. High Energ. Phys. (2016) 2016: 24 (arXiv:1509.03195)

• Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Futoshi Yagi, 6d SCFTs, 5d Dualities and Tao Web Diagrams, JHEP05 (2019)203 (arXiv:1509.03300)

• Ibrahima Bah, Achilleas Passias, Alessandro Tomasiello, $AdS_5$ compactifications with punctures in massive IIA supergravity, JHEP11 (2017)050 (arXiv:1704.07389)

KK-Compactification to $D=4$ $\mathcal{N} = 1$ SYM

The KK-compactification specifically of the D=6 N=(1,0) SCFT to D=4 N=1 super Yang-Mills:

• Ibrahima Bah, Christopher Beem, Nikolay Bobev, Brian Wecht, Four-Dimensional SCFTs from M5-Branes (arXiv:1203.0303)

• Shlomo S. Razamat, Cumrun Vafa, Gabi Zafrir, $4d$ $\mathcal{N} = 1$ from $6d (1,0)$, J. High Energ. Phys. (2017) 2017: 64 (arXiv:1610.09178)

• Ibrahima Bah, Amihay Hanany, Kazunobu Maruyoshi, Shlomo S. Razamat, Yuji Tachikawa, Gabi Zafrir, $4d$ $\mathcal{N}=1$ from $6d$ $\mathcal{N}=(1,0)$ on a torus with fluxes (arXiv:1702.04740)

• Hee-Cheol Kim, Shlomo S. Razamat, Cumrun Vafa, Gabi Zafrir, E-String Theory on Riemann Surfaces, Fortsch. Phys. (arXiv:1709.02496)

• Hee-Cheol Kim, Shlomo S. Razamat, Cumrun Vafa, Gabi Zafrir, D-type Conformal Matter and SU/USp Quivers, JHEP06(2018)058 (arXiv:1802.00620)

• Hee-Cheol Kim, Shlomo S. Razamat, Cumrun Vafa, Gabi Zafrir, Compactifications of ADE conformal matter on a torus, JHEP09(2018)110 (arXiv:1806.07620)

• Shlomo S. Razamat, Gabi Zafrir, Compactification of 6d minimal SCFTs on Riemann surfaces, Phys. Rev. D 98, 066006 (2018) (arXiv:1806.09196)

• Jin Chen, Babak Haghighat, Shuwei Liu, Marcus Sperling, 4d $\mathcal{N}=1$ from 6d D-type $\mathcal{N}=(1,0)$, J. High Energ. Phys. 2020 152 (2020) [arXiv:1907.00536, doi:10.1007/JHEP01(2020)152]

$SU(2)$-flavor symmetry on heterotic M5-branes

Emergence of SU(2) flavor-symmetry on single M5-branes in heterotic M-theory on ADE-orbifolds (in the D=6 N=(1,0) SCFT on small instantons in heterotic string theory):

• Abhijit Gadde, Babak Haghighat, Joonho Kim, Seok Kim, Guglielmo Lockhart, Cumrun Vafa, Section 4.2 of: 6d String Chains, J. High Energ. Phys. 2018, 143 (2018) (arXiv:1504.04614, doi:10.1007/JHEP02(2018)143)

• Kantaro Ohmori, Section 2.3.1 of: Six-Dimensional Superconformal Field Theories and Their Torus Compactifications, Springer Theses 2018 (springer:book/9789811330919)

Argument for this by translation under duality between M-theory and type IIA string theory to half NS5-brane/D6/D8-brane bound state systems in type I' string theory:

Reviewed in:

• Santiago Cabrera, Amihay Hanany, Marcus Sperling, Section 2.3 of: Magnetic Quivers, Higgs Branches, and 6d $\mathcal{N}=(1,0)$ Theories, JHEP06(2019)071, JHEP07(2019)137 (arXiv:1904.12293)

The emergence of flavor in these half NS5-brane/D6/D8-brane bound state systems, due to the semi-infinite extension of the D6-branes making them act as flavor branes:

• Amihay Hanany, Alberto Zaffaroni, Branes and Six Dimensional Supersymmetric Theories, Nucl.Phys. B529 (1998) 180-206 (arXiv:hep-th/9712145)

• Ilka Brunner, Andreas Karch, Branes at Orbifolds versus Hanany Witten in Six Dimensions, JHEP 9803:003, 1998 (arXiv:hep-th/9712143)

Reviewed in:

• Fabio Apruzzi, Marco Fazzi, Section 2.1 of: $AdS_7/CFT_6$ with orientifolds, J. High Energ. Phys. (2018) 2018: 124 (arXiv:1712.03235)

See also:

• Amihay Hanany, Noppadol Mekareeya, The Small $E_8$ Instanton and the Kraft Procesi Transition, JHEP07 (2018) 098 (arXiv:1801.01129)