nLab geometric engineering of quantum field theory

Idea

By embedding quantum field theories into string theory – typically as the worldvolume theories of various branes, e.g. super Yang-Mills theory on D-branes, 6d (2,0)-superconformal QFT on M5-branes, or else at O-planes – the various dualities of string theory will relate different QFTs in way that are typically far from obvious from just looking at these QFTs themselves.

The investigation specifically of N=2 D=4 super Yang-Mills theory and N=1 D=4 super Yang-Mills theory in this fashion has come to be known as geometric engineering of quantum field theory (Katz-Klemm-Vafa 97, Katz-Klemm 96 ).

Specifically, the geometrically engineered QFTs are those on the worldvolume of black D-branes that end on (are suspended between) black NS5-branes (due to Hanany-Witten 97, review includes Fazzi 17). See at D-branes ending on NS5-branes.

graphics grabbed from Fazzi 17, p. 25

graphics grabbed from Fazzi 17, p. 32

For more relations between QFTs found via string theory see at string theory results applied elsewhere.

Related concepts

• quiver gauge theory

• (p,q)5-brane webs

• AdS-QCD correspondence

• M-theory on G₂-manifolds

References

General

The original articles are

• Sheldon Katz, Albrecht Klemm, Cumrun Vafa, Geometric Engineering of Quantum Field Theories, Nucl.Phys. B497 (1997) 173-195 (arXiv:hep-th/9609239)

• Sheldon Katz, Cumrun Vafa, Geometric Engineering of $N=1$ Quantum Field Theories, Nucl.Phys. B497 (1997) 196-204 (arXiv:hep-th/9611090)

• 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:

• Amit Giveon, David Kutasov: Brane Dynamics and Gauge Theory, Rev. Mod. Phys. 71 (1999) 983-1084 [arXiv:hep-th/9802067 doi:10.1103/RevModPhys.71.983]

• Andreas Karch, Field Theory Dynamics from Branes in String Theory, PhD thesis (1998) (doi:10.18452/14371)

• David Morrison, Limitations of Geometric Engineering: Implications for Model Building, talk (2008) [slides:pdf]

• Marco Fazzi, Higher-dimensional field theories from type II supergravity (arXiv:1712.04447)

• Michele Del Zotto: Uncharted Territories in Geometric Engineering, talk at Strings and Geometry 2023 (2023) [slides:pdf, video:YT]

Further developments are in

• Balázs Szendrői, Nekrasov’s Partition Function and Refined Donaldson-Thomas Theory: the Rank One Case, SIGMA, 2012, Volume 8, 088 (doi web)

Examples

Geometric engineering of $D = 6$ $\mathcal{N} = (2,0)$ SCFT

For geometric engineering of the D=6 N=(2,0) SCFT, see at duality between M-theory on Z2-orbifolds and type IIB string theory on K3-fibrations – Geometric engineering of 6d (2,0)-SCFT.

Geometric engineering of $D = 6$ $\mathcal{N} = (1,0)$ SCFT

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):

• 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)