# nLab Seiberg duality

under construction

# Contents

## Idea

Seiberg duality (named after (Seiberg)) is a version of electric-magnetic duality in supersymmetric gauge theory.

For supersymmetric QCD it identifies in the infrared (long distance limit, and only there) the quarks and gluons in a theory with $N_f$ quark flavors and $SU(N_c)$ gauge group for

$N_f \gt N_c +1$

with solitons in a theory of $N_f$ quark flavors and gauge group $SU(\tilde N_c)$, where

$\tilde N_c = N_f - N_c \,.$

## Properties

### Realization in string theory

Seiberg duality follows from phenomena in string theory, where gauge theories arise as the worldvolume theories of D-branes (geometric engineering of quantum field theory). Seiberg duality is obtained for gauge theories of D-branes that stretch between two NS5-branes. The duality operation corresponds to exchanging the two NS5-branes.

• A. Hanany and Edward Witten, Type-IIB superstrings, BPS monopoles and three-dimensional gauge dynamics , Nucl. Phys. B 492 (1997) 152 (hep-th/9611230).

• S. Elitzur, A. Giveon, D. Kutasov, E. Rabinovici and A. Schwimmer, Brane dynamics and $N = 1$ supersymmetric gauge theory, Nucl. Phys. B 505 (1997) 202 (hep-th/9704104)

### Formalization by derived quiver categories

Seiberg duality is formalized by equivalences of derived categories of quiver representations.

## Examples

### Toric duality

Toric Duality is Seiberg duality for $N=1$ theories with toric moduli spaces.

• Bo Feng, Amihay Hanany and Y.-H. He, D-brane gauge theories from toric singularities and toric duality, Nucl. Phys. B 595 (2001) 165 [hep-th/0003085].

• C.E. Beasley and M.R. Plesser, Toric duality is Seiberg duality, J. High Energy Phys. 12 (2001) 001 [hep-th/0109053]. JHEP02(2004)070

• Bo Feng, A. Hanany and Y.-H. He, Phase structure of D-brane gauge theories and toric duality , J. High Energy Phys. 08 (2001) 040 [hep-th/0104259].

• Bo Feng, A. Hanany, Y.-H. He and A.M. Uranga, Toric duality as Seiberg duality and brane diamonds, J. High Energy Phys. 12 (2001) 035 [hep-th/0109063].

• Bo Feng, S. Franco, A. Hanany and Y.-H. He, Unhiggsing the del Pezzo, J. High Energy Phys. 08 (2003) 058 [hep-th/0209228].

• S. Franco and A. Hanany, Toric duality, Seiberg duality and Picard-Lefschetz transformations , Fortschr. Phys. 51 (2003) 738 [hep-th/0212299].

### From D-branes on del Pezzo singularities

• Christopher P. Herzog, Seiberg Duality is an Exceptional Mutation (arXiv:hep-th/0405118)

• Subir Mukhopadhyay, Koushik Ray, Seiberg duality as derived equivalence for some quiver gauge theories Journal of High Energy Physics Volume 2004 JHEP02(2004)

### For exceptional gauge groups

Seiberg duality for gauge groups which are exceptional Lie groups:

But see

### Chiral and non-chiral duals

Due to

• Igor Klebanov, Matthew Strassler, Supergravity and a Confining Gauge Theory: Duality Cascades and $\chi$SB-Resolution of Naked Singularities (arXiv)

A review is in

Discussion in connection with non-conformal variants of AdS/CFT is in

• Eduardo Conde, Jerome Gaillard, Carlos Núñez, Maurizio Piai, Alfonso V. Ramallo, Towards the string dual of tumbling and cascading gauge theories (arXiv:1112.3346)

### With few supercharges

• a 3d Yang-Mills Chern-Simons theory with two supercharges ($N = 1$ SUSY in 3d)

(Adi Armoni, Amit Giveon, Dan Israel, Vasilis Niarchos, 2009

• a non-supersymmetric theory in \$4d§

(Adi Armoni, Dan Israel, Gregory Moraitis, Vasilis Niarchos, 2008).

Discussed in

• Adi Armoni, Two Examples of Seiberg Duality in Gauge Theories With Less Than Four Supercharges (pdf)

duality in physics

## References

### Original articles

The original article is

The “cascade” of Seiberg dualities is due to

### Lectures and reviews

Surveys and reviews include