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The special case of super Yang-Mills theory over a spacetime of dimension 4 and with supersymmetry.
(Shnider 88, also Nahm 78, see Minwalla 98, section 4.2)
A speciality of , SYM is that its moduli space of vacua has two “branches” called the Coulomb branch and the Higgs branch. This is the content of what is now called Seiberg-Witten theory (Seiberg-Witten 94). Review includes (Albertsson 03, section 2.3.4).
While confinement in plain Yang-Mills theory is still waiting for mathematical formalization and proof (see Jaffe-Witten), N=2 D=4 super Yang-Mills theory is has been obsrved in (Seiberg-Witten 94).
By dimensional reduction on families of SYM theories interpolate to N=4 D=3 super Yang-Mills theory. (Seiberg-Witten 96).
super Yang-Mills theory can be realized as the worldvolume theory of M5-branes compactified on a Riemann surface (Klemm-Lerche-Mayr-Vafa-Warner 96, Witten 97, Gaiotto 09), hence as a compactifiction of the 6d (2,0)-superconformal QFT on the M5. This in particular gives a geometric interpretation of Seiberg-Witten duality in 4d in terms of the 6d 5-brane geometry.
Specifically the AGT correspondence expresses this relation in terms of the partition function of the theory and a 2d CFT on the Riemann surface on which the 5-brane is compactified. See at AGT correspondence for more on this.
gauge theory induced via AdS-CFT correspondence
M-theory perspective via AdS7-CFT6 | F-theory perspective |
---|---|
11d supergravity/M-theory | |
Kaluza-Klein compactification on | compactificationon elliptic fibration followed by T-duality |
7-dimensional supergravity | |
topological sector | |
7-dimensional Chern-Simons theory | |
AdS7-CFT6 holographic duality | |
6d (2,0)-superconformal QFT on the M5-brane with conformal invariance | M5-brane worldvolume theory |
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 |
topological twist | |
topologically twisted N=2 D=4 super Yang-Mills theory | |
KK-compactification on Riemann surface | |
A-model on , Donaldson theory |
(Shnider 88, also Nahm 78, see Minwalla 98, section 4.2)
Davide Gaiotto, Recent progress in field theory (2009) (pdf)
Gregory Moore, Four-dimensional Field Theory and Physical Mathematics (arXiv:1211.2331)
Greg Moore, Surface Defects and the BPS Spectrum of Theories (pdf)
Yuji Tachikawa, supersymmetric dynamics for pedestrians, Lecture Notes in Physics, vol. 890, 2014 (arXiv:1312.2684, doi:10.1007/978-3-319-08822-8, web version)
Mohammad Akhond, Guillermo Arias-Tamargo, Alessandro Mininno, Hao-Yu Sun, Zhengdi Sun, Yifan Wang, Fengjun Xu, The Hitchhiker’s Guide to 4d Superconformal Field Theories [arXiv:2112.14764]
The terminology “Coulomb branch” and “Higgs branch” first appears in
See also at Seiberg-Witten theory:
The dimensional reduction to was first considered in
The confinement-phenomenon was observed in
Reviews of this confinement mechanism:
Alexei Yung, What Do We Learn about Confinement from the Seiberg-Witten Theory (arXiv:hep-th/0005088)
Cecilia Albertsson, Superconformal D-branes and moduli spaces (arXiv:hep-th/0305188)
Yang-Hui He, Lectures on D-branes, Gauge Theories and Calabi-Yau Singularities, International Summer School in Mathematical Physics (arXiv:hep-th/0408142)
Emily Nardoni, From Supersymmetry to adjoint QCD, talk at Strings 2022 [indico:4940894, pdf, video]
For references on wall crossing of BPS states see the references given there.
SYM including its Seiberg-Witten theory (Seiberg-Witten 94) may be understood as being the compactification of the 6d (2,0)-superconformal QFT on the worldvolume of M5-branes on a Riemann surface: the Riemann surface is identified with the Seiberg-Witten curve of complexified coupling constants. This observation goes back to
Albrecht Klemm, Wolfgang Lerche, Peter Mayr, Cumrun Vafa, Nicholas Warner, Self-Dual Strings and Supersymmetric Field Theory, Nucl. Phys. B 477 (1996) 746-766 [arXiv:hep-th/9604034, doi:10.1016/0550-3213(96)00353-7]
Edward Witten, Solutions Of Four-Dimensional Field Theories Via M Theory, Nucl. Phys. B 500 (1997) 3-42 [arXiv:hep-th/9703166, doi:10.1016/S0550-3213(97)00416-1]
review in:
Csaba Csaki, Joshua Erlich, John Terning, pp. 4 of; Seiberg-Witten Description of the Deconstructed 6D Theory, Phys. Rev. D 67 025019 (2003) [arXiv:hep-th/0208095, doi:10.1103/PhysRevD.67.025019]
Ling Bao, Elli Pomoni, Masato Taki, Futoshi Yagi, p. 5 of: M5-branes, toric diagrams and gauge theory duality, J. High Energ. Phys. 2012 105 (2012) [arXiv:1112.5228, doi:10.1007/JHEP04(2012)105]
Taro Kimura, §4.5.2 in: Seiberg–Witten Geometry, Chapter 4 in: Instanton Counting, Quantum Geometry and Algebra, Mathematical Physics Studies, Springer (2021) [doi:10.1007/978-3-030-76190-5_4]
The further observation that therefore the sewing of Riemann surfaces on which one compactifies the M5-brane yields a gluing operation on N=2 SYM theories is due to
cf. at AGT correspondence.
The topological twisting of the compactification which is used around (2.27) there was previously introduced in section 3.1.2 of:
and is discussed also for instance in section 5.1 of
(This is possibly also the mechanism behind the AGT correspondence, though the details behind that statement seem to be unclear.)
A brief review of these matters is in (Moore 12, section 7). A formalization of the topological twist in perturbation theory formalized by factorization algebras with values in BV complexes is in section 16 of
For more on this see at topologically twisted D=4 super Yang-Mills theory.
An amplification of the relevance of this to the understanding of S-duality/electric-magnetic duality is in
and the resulting relation to the geometric Langlands correspondence is discussed in
.
The corresponding dual theory under AdS-CFT duality is discussed in
Discussion of construction of just N=1 D=4 super Yang-Mills theory this way is in
Relation to the 3-brane in 6d:
Paul Howe, Neil Lambert, Peter West, Classical M-Fivebrane Dynamics and Quantum Yang-Mills, Phys. Lett. B418 (1998) 85-90 (arXiv:hep-th/9710034)
Neil Lambert, Peter West, Gauge Fields and M-Fivebrane Dynamics, Nucl. Phys. B524 (1998) 141-158 (arXiv:hep-th/9712040)
Neil Lambert, Peter West, Superfields and the M-Fivebrane, Phys. Lett. B424 (1998) 281-287 (arXiv:hep-th/9801104)
Neil Lambert, Peter West, Monopole Dynamics from the M-Fivebrane, Nucl. Phys. B556 (1999) 177-196 (arXiv:hep-th/9811025)
and via F-theory in
Last revised on June 28, 2024 at 13:30:13. See the history of this page for a list of all contributions to it.