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\section*{Seiberg duality}

\begin{quote}%
under construction


\end{quote}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory}

[[!include functorial quantum field theory - contents]]

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{realization_in_string_theory}{Realization in string theory}\dotfill \pageref*{realization_in_string_theory} \linebreak
\noindent\hyperlink{formalization_by_derived_quiver_categories}{Formalization by derived quiver categories}\dotfill \pageref*{formalization_by_derived_quiver_categories} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{toric_duality}{Toric duality}\dotfill \pageref*{toric_duality} \linebreak
\noindent\hyperlink{from_dbranes_on_del_pezzo_singularities}{From D-branes on del Pezzo singularities}\dotfill \pageref*{from_dbranes_on_del_pezzo_singularities} \linebreak
\noindent\hyperlink{ExamplesForExceptionalGaugeGroups}{For exceptional gauge groups}\dotfill \pageref*{ExamplesForExceptionalGaugeGroups} \linebreak
\noindent\hyperlink{chiral_and_nonchiral_duals}{Chiral and non-chiral duals}\dotfill \pageref*{chiral_and_nonchiral_duals} \linebreak
\noindent\hyperlink{duality_cascade}{Duality cascade}\dotfill \pageref*{duality_cascade} \linebreak
\noindent\hyperlink{with_few_supercharges}{With few supercharges}\dotfill \pageref*{with_few_supercharges} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{original_articles}{Original articles}\dotfill \pageref*{original_articles} \linebreak
\noindent\hyperlink{lectures_and_reviews}{Lectures and reviews}\dotfill \pageref*{lectures_and_reviews} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Seiberg duality (named after (\hyperlink{Seiberg}{Seiberg})) is a version of [[electric-magnetic duality]] in [[supersymmetry|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

\begin{displaymath}
N_f \gt N_c +1
\end{displaymath}
with [[soliton]]s in a theory of $N_f$ quark flavors and gauge group $SU(\tilde N_c)$, where

\begin{displaymath}
\tilde N_c = N_f - N_c 
  \,.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{realization_in_string_theory}{}\subsubsection*{{Realization in string theory}}\label{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-brane]]s. The duality operation corresponds to exchanging the two NS5-branes.

\begin{itemize}%
\item [[Amihay Hanany]] and [[Edward Witten]], \emph{Type-IIB superstrings, BPS monopoles and three-dimensional gauge dynamics} , Nucl. Phys. B 492 (1997) 152 (\href{}{hep-th/9611230}).


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



\end{itemize}
See also \emph{[[string theory results applied elsewhere]]}.

\hypertarget{formalization_by_derived_quiver_categories}{}\subsubsection*{{Formalization by derived quiver categories}}\label{formalization_by_derived_quiver_categories}

Seiberg duality is formalized by [[equivalence of categories|equivalences]] of [[derived categories]] of [[quiver]] representations.

\begin{itemize}%
\item David Berenstein, [[Michael Douglas]], \emph{Seiberg Duality for Quiver Gauge Theories} (\href{http://arxiv.org/abs/hep-th/0207027}{arXiv:hep-th/0207027})


\item Subir Mukhopadhyay, Koushik Ray, \emph{Seiberg duality as derived equivalence for some quiver gauge theories} (\href{http://arxiv.org/abs/hep-th/0309191}{arXiv:hep-th/0309191})



\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{toric_duality}{}\subsubsection*{{Toric duality}}\label{toric_duality}

\emph{Toric Duality} is Seiberg duality for $N=1$ theories with toric [[moduli spaces]].

\begin{itemize}%
\item Bo Feng, [[Amihay Hanany]] and Y.-H. He, \emph{D-brane gauge theories from toric singularities and toric duality, Nucl. Phys. B 595 (2001) 165 hep-th/0003085.}


\item C.E. Beasley and [[M. Ronen Plesser]], \emph{Toric duality is Seiberg duality}, J. High Energy Phys. 12 (2001) 001 hep-th/0109053. JHEP02(2004)070


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


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


\item Bo Feng, S. Franco, [[Amihay Hanany]] and Y.-H. He, \emph{Unhiggsing the del Pezzo}, J. High Energy Phys. 08 (2003) 058 hep-th/0209228.


\item S. Franco and [[Amihay Hanany]], \emph{Toric duality, Seiberg duality and Picard-Lefschetz transformations} , Fortschr. Phys. 51 (2003) 738 hep-th/0212299.



\end{itemize}
\hypertarget{from_dbranes_on_del_pezzo_singularities}{}\subsubsection*{{From D-branes on del Pezzo singularities}}\label{from_dbranes_on_del_pezzo_singularities}

\begin{itemize}%
\item Christopher P. Herzog, \emph{Seiberg Duality is an Exceptional Mutation} (\href{http://arxiv.org/abs/hep-th/0405118}{arXiv:hep-th/0405118})


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



\end{itemize}
\hypertarget{ExamplesForExceptionalGaugeGroups}{}\subsubsection*{{For exceptional gauge groups}}\label{ExamplesForExceptionalGaugeGroups}

Seiberg duality for [[gauge group]]s which are [[exceptional Lie group]]s:

\begin{itemize}%
\item [[Jacques Distler]], [[Andreas Karch]], \emph{$N=1$ Dualities for Exceptional Gauge Groups and Quantum Global Symmetries} (\href{http://arxiv.org/abs/hep-th/9611088}{arXiv:hep-th/9611088})

\end{itemize}
But see

\begin{itemize}%
\item Peter Cho, \emph{Moduli in Exceptional SUSY Gauge Theories} (\href{http://arxiv.org/abs/hep-th/9712116}{arXiv:hep-th/9712116})

\end{itemize}
\hypertarget{chiral_and_nonchiral_duals}{}\subsubsection*{{Chiral and non-chiral duals}}\label{chiral_and_nonchiral_duals}

\begin{itemize}%
\item P. Pouliot, \emph{Chiral Duals of Non-Chiral SUSY Gauge Theories} (\href{http://arxiv.org/abs/hep-th/9507018}{arXiv:hep-th/9507018})

\end{itemize}
\hypertarget{duality_cascade}{}\subsubsection*{{Duality cascade}}\label{duality_cascade}

Due to

\begin{itemize}%
\item [[Igor Klebanov]], [[Matthew Strassler]], \emph{Supergravity and a Confining Gauge Theory: Duality Cascades and $\chi$SB-Resolution of Naked Singularities} (\href{http://arxiv.org/abs/hep-th/0007191}{arXiv})

\end{itemize}
A review is in

\begin{itemize}%
\item [[Matthew Strassler]], \emph{The Duality Cascade} (\href{http://arxiv.org/abs/hep-th/0505153}{arXiv:hep-th/0505153})

\end{itemize}
Discussion in connection with non-conformal variants of [[AdS/CFT]] is in

\begin{itemize}%
\item Eduardo Conde, Jerome Gaillard, [[Carlos Núñez]], Maurizio Piai, Alfonso V. Ramallo, \emph{Towards the string dual of tumbling and cascading gauge theories} (\href{http://arxiv.org/abs/1112.3346}{arXiv:1112.3346})

\end{itemize}
\hypertarget{with_few_supercharges}{}\subsubsection*{{With few supercharges}}\label{with_few_supercharges}

\begin{itemize}%
\item a 3d [[Yang-Mills theory|Yang-Mills]] [[Chern-Simons theory]] with two [[supercharges]] ($N = 1$ SUSY in 3d)

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


\item a non-supersymmetric theory in $4d$;

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



\end{itemize}
Discussed in

\begin{itemize}%
\item Adi Armoni, \emph{Two Examples of Seiberg Duality in Gauge Theories With Less Than Four Supercharges} (\href{http://pyweb.swan.ac.uk/~pyarmoni/seiberg.pdf}{pdf})

\end{itemize}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[quiver gauge theory]]

\end{itemize}
[[duality in physics]]

\begin{itemize}%
\item [[S-duality]]

\begin{itemize}%
\item [[electric-magnetic duality]]

\begin{itemize}%
\item [[Montonen-Olive duality]]

\end{itemize}

\item \textbf{Seiberg duality}


\item [[geometric Langlands duality]]



\end{itemize}


\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{original_articles}{}\subsubsection*{{Original articles}}\label{original_articles}

The original article is

\begin{itemize}%
\item [[Nathan Seiberg]], \emph{Electric-Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories} (\href{http://arxiv.org/abs/hep-th/9411149}{arXiv:hep-th/9411149})

\end{itemize}
The ``cascade'' of Seiberg dualities is due to

\begin{itemize}%
\item [[Igor Klebanov]], [[Matthew Strassler]], \emph{Supergravity and a Confining Gauge Theory: Duality Cascades and $\chi$SB-Resolution of Naked Singularities} (\href{http://arxiv.org/abs/hep-th/0007191}{arXiv:hep-th/0007191})

\end{itemize}
\hypertarget{lectures_and_reviews}{}\subsubsection*{{Lectures and reviews}}\label{lectures_and_reviews}

Surveys and reviews include

\begin{itemize}%
\item [[Matthew Strassler]], \emph{The Duality Cascade} (2005) (\href{http://arxiv.org/abs/hep-th/0505153}{arXiv:hep-th/0505153})


\item M. Chaichian, W.F. Chen, C. Montonen, \emph{New Superconformal Field Theories in Four Dimensions and N=1 Duality} (\href{http://arxiv.org/abs/hep-th/0007240}{arXiv:hep-th/0007240})


\item Flip Tanedo, \emph{Notes on Seibergology} (\href{http://www.lns.cornell.edu/~pt267/files/notes/Seibergology.pdf}{pdf})



\end{itemize}
See also section 22 of

\begin{itemize}%
\item Philip Argyres, \emph{Introduction to supersymmetry} (\href{http://www.physics.uc.edu/~argyres/661/susy1996.pdf}{pdf})

\end{itemize}
[[!redirects Seiberg dualities]]



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