Contents

supersymmetry

Contents

Idea

According to the classification of superconformal symmetry, there should exist superconformal field theories in 6 dimensions…

$d$$N$superconformal super Lie algebraR-symmetryblack brane worldvolume
superconformal field theory
$\phantom{A}3\phantom{A}$$\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)

…with $(2,0)$-supersymmetry, that contain a self-dual higher gauge theory whose field configurations are connections on a 2-bundle (a circle 2-bundle with connection in the abelian case).

This was derived by Claus, Kallosh & van Proeyen 1997, in the abelian case and to low order, as the small fluctuations of the Green-Schwarz sigma-model of the M5-brane around the embedding in the asymptotic boundary of the AdS-spacetime that is the near-horizon geometry of the black M5-brane.

In accord with this the AdS7-CFT6 correspondence predicts that the nonabelian 6d theory is the corresponding theory for $N$ coincident M5-branes.

In the non-abelian case it is expected (Witten 07) that the compactification of such theories are at the heart of the phenomenon that leads to S-duality in super Yang-Mills theory and further to geometric Langlands duality (Witten 09).

Due to the self-duality a characterization of these theories by an action functional is subtle. Therefore more direct descriptions are still under investigation (for instance SSW11). Review includes (Moore11, Moore 12).

Properties

Holographic dual

Under AdS7/CFT6 the 6d $(2,0)$-superconformal QFT is supposed to be dual to M-theory on anti de Sitter spacetime $AdS_7 \times S^4$.

See AdS/CFT correspondence for more on this.

Solitonic 1-branes

The 5d $(2,0)$-SCFT has tensionless 1-brane configurations. From the point of view of the ambient 11-dimensional supergravity these are the boundaries of M2-branes ending on the M5-branes. (GGT)

Compactification on a Riemann surface and AGT correspondence

(graphics taken from (Workshop 14))

The compactification of the 5-brane on a Riemann surface yields as worldvolume theory N=2 D=4 super Yang-Mills theory. See at N=2 D=4 SYM – Construction by compactification of 5-branes.

The AGT correspondence relates the partition function of $SU(2)^{n+3g-3}$-N=2 D=4 super Yang-Mills theory obtained by compactifying the $6d$ M5-brane theory on a Riemann surface $C_{g,n}$ of genus $g$ with $n$ punctures to 2d Liouville theory on $C_{g,n}$.

More generally, this kind of construction allows to describe the 6d (2,0)-theory as a “2d SCFT with values in 4d SYM”. See at AGT correspondence for more on this.

Twistor space description

Famously the solutions to self-dual Yang-Mills theory in dimension 4 can be obtained as images of degree-2 cohomology classes under the Penrose-Ward twistor transform. Since the 6d QFT on the M5-brane contains a 2-form self-dual higher gauge field it seems natural to expect that it can be described by a higher analogy of the twistor transform. For references exploring this idea see at twistor space – References – Application to the self-dual 2-form field in 6d.

gauge theory induced via AdS-CFT correspondence

11d supergravity/M-theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^4$compactificationon elliptic fibration followed by T-duality
7-dimensional supergravity
$\;\;\;\;\downarrow$ topological sector
7-dimensional Chern-Simons theory
$\;\;\;\;\downarrow$ AdS7-CFT6 holographic duality
6d (2,0)-superconformal QFT on the M5-brane with conformal invarianceM5-brane worldvolume theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surfacedouble dimensional reduction on M-theory/F-theory elliptic fibration
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondenceD3-brane worldvolume theory with type IIB S-duality
$\;\;\;\;\; \downarrow$ topological twist
topologically twisted N=2 D=4 super Yang-Mills theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface
A-model on $Bun_G$, Donaldson theory

$\,$

type II string theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^5$
$\;\;\;\; \downarrow$ topological sector
5-dimensional Chern-Simons theory
$\;\;\;\;\downarrow$ AdS5-CFT4 holographic duality
N=4 D=4 super Yang-Mills theory
$\;\;\;\;\; \downarrow$ topological twist
topologically twisted N=4 D=4 super Yang-Mills theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface
A-model on $Bun_G$ and B-model on $Loc_G$, geometric Langlands correspondence
$d$$N$superconformal super Lie algebraR-symmetryblack brane worldvolume
superconformal field theory
$\phantom{A}3\phantom{A}$$\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)

References

General

The first indication of a 6d theory with a self-dual 2-form field appears in

Derivation of the abelian 6d theory to low order as the small fluctuations of the Green-Schwarz sigma-model of the M5-brane around a solution embedding as the asymptotic boundary of the AdS-spacetime near-horizon geometry of a black 5-brane is due to

General survey:

Discussion of anomaly cancellation:

Review of construction from F-theory KK-compactification:

New approach to construction of candidate Lagrangian densities for D=6 N=(2,0) SCFTs:

• Neil Lambert, $(2,0)$ Lagrangian Structures, Physics Letters B Volume 798, 10 November 2019, 134948 (arXiv:1908.10752)

Compactification to 5d super-Yang-Mills

KK-compactification on circle fibers to D=5 super Yang-Mills theory is discussed in (see also at Perry-Schwarz Lagrangian):

Compactification to 4d super-Yang-Mills

KK-compactification of the $D=6$ $\mathcal{N} = (2,0)$-CFT on torus fibers to D=4 super Yang-Mills theory and the related electric-magnetic duality/S-duality in 4-dimensions:

and the resulting relation to the geometric Langlands correspondence:

For more references on this see at N=2 D=4 super Yang-Mills theory the section References - Constructions from 5-branes.

Relation to BFSS matrix model on tori:

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]

Compactification to 2d CFT

In relation to M5-brane elliptic genus:

and in relation to defects:

Discussion of the ADE classification of the 6d theories includes, after (Witten95)

Models and special properties

Realization of the 6d theory in F-theory is discussed in (Heckmann-Morrison-Vafa 13).

A proposal for related higher nonabelian differential form data is made in

Since by transgression every nonabelian principal 2-bundle/gerbe gives rise to some kind of nonabelian 1-bundle on loop space it is clear that some parts (but not all) of the nonabelian gerbe theory on the 5-brane has an equivalent reformulation in terms of ordinary gauge theory on the loop space of the 5-brane and possibly for gauge group the loop group of the original gauge group.

In fact, via the strict 2-group version of the string 2-group there is a local gauge in which the loop group variables appear already before transgression of the 5-brane gerbe to loop space. This is discussed from a holographic point of view in

On the holographic dual

The basics of the relation of the 6d theory to a 7d theory under AdS-CFT is reviewed holographic duality

• Juan Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2:231, 1998, hep-th/9711200; Wilson loops in Large N field theories, Phys. Rev. Lett. 80 (1998) 4859, hep-th/9803002

The argument that the abelian 7d Chern-Simons theory of a 3-connection yields this way the conformal blocks of the abelian self-dual higher gauge theory of the 6d theory on a single brane is due to

The nonabelian generalization of this 7d action functional that follows from taking the quantum corrections (Green-Schwarz mechanism and flux quantization) of the supergravity C-field into account is discussed in

• Eric D'Hoker, John Estes, Michael Gutperle, Darya Krym,

Exact Half-BPS Flux Solutions in M-theory I Local Solutions (arXiv:0806.0605)

Exact Half-BPS Flux Solutions in M-theory II: Global solutions asymptotic to $AdS_7 \times S^4$ (arXiv:0810.4647)

The $D=6$, $\mathcal{N}=(2,0)$ SCFT as an extended functorial field theory

On the (conjectural) suggestion to view at least some aspects of the D=6 N=(2,0) SCFT (such as its quantum anomaly or its image as a 2d TQFT under the AGT correspondence) as a functorial field theory given by a functor on a suitable cobordism category, or rather as an extended such FQFT, given by an n-functor (at least a 2-functor on a 2-category of cobordisms):

Last revised on February 12, 2024 at 12:37:51. See the history of this page for a list of all contributions to it.