functorial quantum field theory
Reshetikhin?Turaev model? / Chern-Simons theory
FQFT and cohomology
superalgebra and (synthetic ) supergeometry
According to the classification of superconformal symmetry, there should exist superconformal field theories in 6 dimensions…
$d$ | $N$ | 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)
…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).
In (Claus-Kallosh-Proeyen 97) such has been derived, 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).
For geometric engineering of the 6d (2,0)-superconformal QFT, see at duality between M-theory on Z2-orbifolds and type IIB string theory on K3-fibrations – Geometric engineering of 6d (2,0)-SCFT.
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.
See at AdS-QCD correspondence.
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)
(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.
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
M-theory perspective via AdS7-CFT6 | F-theory perspective |
---|---|
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 invariance | M5-brane worldvolume theory |
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface | double dimensional reduction on M-theory/F-theory elliptic fibration |
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondence | D3-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 |
$\,$
gauge theory induced via AdS5-CFT4 |
---|
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 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)
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:
Greg Moore, On the role of six‐dimensional $(2,0)$-theories in recent developments in Physical Mathematics, talk at Strings 2011 (pdf slides)
Greg Moore, Applications of the six-dimensional (2,0) theories to Physical Mathematics, Felix Klein lectures Bonn (2012) (pdf, pdf)
Qiaochu Yuan: lecture notes for Mathematical Aspects of Six-Dimensional Quantum Field Theories, Berkeley, December 8th-12th, 2014 at the University of California, Berkeley
Discussion of anomaly cancellation:
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)
Hiroyuki Shimizu, Aspects of anomalies in 6d superconformal field theories, Tokyo 2018 (spire:1802462, pdf)
Review of construction from F-theory KK-compactification:
New approach to construction of candidate Lagrangian densities for D=6 N=(2,0) SCFTs:
See also the references and discussion at M5-brane.
KK-compactification on circle fibers to D=5 super Yang-Mills theory is discussed in (see also at Perry-Schwarz Lagrangian):
Nathan Seiberg, Sec. 7 of Notes on Theories with 16 Supercharges, Nucl. Phys. Proc. Suppl. 67:158-171, 1998 (arXiv:hep-th/9705117)
Michael Douglas, On D=5 super Yang-Mills theory and (2,0) theory, JHEP 1102:011, 2011 (arXiv:1012.2880)
Neil Lambert, Constantinos Papageorgakis, Maximilian Schmidt-Sommerfeld, M5-Branes, D4-Branes and Quantum 5D super-Yang-Mills, JHEP 1101:083, 2011 (arXiv:1012.2882)
Edward Witten, Sections 4 and 5 of Fivebranes and Knots (arXiv:1101.3216)
Shan Hu, 6D $(2,0)$ theory and M5 branes: A KK mode approach, 2013 (hdl:1969.1/151094)
Chris Hull, Neil Lambert, Emergent Time and the M5-Brane, JHEP06(2014)016 (arXiv:1403.4532)
Andreas Gustavsson, Five-dimensional Super-Yang-Mills and its Kaluza-Klein tower. JHEP01(2019)222 (arXiv:1812.01897)
Neil Lambert, Sec. 3.1 and 3.4.3. of Lessons from M2’s and Hopes for M5’s, Proceedings of the LMS-EPSRC Durham Symposium: Higher Structures in M-Theory, August 2018 Fortschritte der Physik Volume 67, Issue 8-9 2019 (arXiv:1903.02825, doi:10.1002/prop.201910011, slides pdf, video recording)
KK-compactification on torus fibers to D=4 super Yang-Mills theory and the related electric-magnetic duality/S-duality in 4-dimensions is discussed in
and the resulting relation to the geometric Langlands correspondence is disucssed in
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 N=1 from 6d D-type N=(1,0) (arXiv:1907.00536)
Discussion of the ADE classification of the 6d theories includes, after (Witten95)
Julie D. Blum, Kenneth Intriligator, New Phases of String Theory and 6d RG Fixed Points via Branes at Orbifold Singularities, Nucl.Phys.B506:199-222,1997 (arXiv:hep-th/9705044)
Jonathan Heckman, David Morrison, Cumrun Vafa, On the Classification of 6D SCFTs and Generalized ADE Orbifolds (arXiv:1312.5746)
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.
Comments along these lines have been made in
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
The basics of the relation of the 6d theory to a 7d theory under AdS-CFT is reviewed holographic duality
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
Edward Witten, Five-Brane Effective Action In M-Theory J. Geom. Phys.22:103-133,1997 (arXiv:hep-th/9610234)
Edward Witten, AdS/CFT Correspondence And Topological Field Theory JHEP 9812:012,1998 (arXiv:hep-th/9812012)
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
See also
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)
Jerome Gauntlett, Joaquim Gomis, Paul Townsend, BPS Bounds for Worldvolume Branes (arXiv:hep-th/9711205)
Paul Howe, Neil Lambert, Peter West, The Threebrane Soliton of the M-Fivebrane (arXiv:hep-th/9710033)
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):
Edward Witten, Section 1 of: Geometric Langlands From Six Dimensions, in Peter Kotiuga (ed.) A Celebration of the Mathematical Legacy of Raoul Bott, CRM Proceedings & Lecture Notes Volume: 50, AMS 2010 (arXiv:0905.2720, ISBN:978-0-8218-4777-0)
Daniel Freed, 4-3-2 8-7-6, talk at ASPECTS of Topology Dec 2012 (pdf, pdf)
Daniel Freed, p. 32 of: The cobordism hypothesis, Bulletin of the American Mathematical Society 50 (2013), pp. 57-92, (arXiv:1210.5100, doi:10.1090/S0273-0979-2012-01393-9)
Daniel Freed, Constantin Teleman, Relative quantum field theory, Commun. Math. Phys. 326, 459–476 (2014) (arXiv:1212.1692, doi:10.1007/s00220-013-1880-1)
David Ben-Zvi, Theory $\mathcal{X}$ and Geometric Representation Theory, talks at Mathematical Aspects of Six-Dimensional Quantum Field Theories IHES 2014, notes by Qiaochu Yuan (pdf I, pdf II, pdf III)
David Ben-Zvi, Algebraic geometry of topological field theories, talk at Reimagining the Foundations of Algebraic Topology April 07, 2014 - April 11, 2014 (web video)
Lukas Müller, Extended Functorial Field Theories and Anomalies in Quantum Field Theories (arXiv:2003.08217)
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