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\section*{enhanced gauge symmetry}

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

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

[[!include physicscontents]]

\hypertarget{chernweil_theory}{}\paragraph*{{Chern-Weil theory}}\label{chernweil_theory}

[[!include infinity-Chern-Weil theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{on_coincident_dbranes}{On coincident D-branes}\dotfill \pageref*{on_coincident_dbranes} \linebreak
\noindent\hyperlink{from_massless_bps_states}{From massless BPS states}\dotfill \pageref*{from_massless_bps_states} \linebreak
\noindent\hyperlink{for_mk6_branes_at_adesingularities}{For MK6 branes at ADE-singularities}\dotfill \pageref*{for_mk6_branes_at_adesingularities} \linebreak
\noindent\hyperlink{on_m9branes}{On M9-Branes}\dotfill \pageref*{on_m9branes} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{ReferencesOnCoincidentDBranes}{On coincident D-branes}\dotfill \pageref*{ReferencesOnCoincidentDBranes} \linebreak
\noindent\hyperlink{on_mbranes_at_adesingularities}{On M-branes at ADE-singularities}\dotfill \pageref*{on_mbranes_at_adesingularities} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak


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

Generally, an \emph{enhanced gauge symmetry} refers to the appearance of [[gauge theory]] with a larger [[gauge group]] than may be superficially apparent.

\hypertarget{on_coincident_dbranes}{}\subsubsection*{{On coincident D-branes}}\label{on_coincident_dbranes}

A famous case is the enhanced gauge symmetry in the [[Chan-Paton gauge fields]] of coincident [[D-branes]]. A priori each D-brane carries a [[complex line bundle]]/[[circle principal bundle]], but as $n$ of them coincide the [[gauge group]] is expected to enhance from $(U(1))^n$ to the [[unitary group]] $U(n)$ (or the [[special unitary group]] $SU(n)$).

Mathematically this gauge enhancement on D-branes is modeled by the [[Baum-Douglas geometric cycle]] model for [[K-homology]] (\hyperlink{ReisSzabo05}{Reis-Szabo 05, def. 1.5 and p. 16}, \hyperlink{Szabo08}{Szabo 08, p. 4}): One such cycle is given by a submanifold $W \overset{\phi}{\hookrightarrow} X$ of [[spacetime]] $X$ equipped with [[spin{\tt \symbol{94}}c structure]] and with a [[complex vector bundle]] $E \to X$, and given two such cycles $(W,\phi,E_1)$ and $(W,\phi,E_2)$ with the same underlying manifold (``coincident D-brane [[worldvolume]]'') then their formal [[direct sum]] is identified with the single cycle carrying the [[direct sum of vector bundles]]

\begin{displaymath}
(W,\phi, E_1) + (W, \phi, E_2)
  \;\sim\;
  (W, \phi, E_1 \oplus E_2)
  \,.
\end{displaymath}
If here $E_i$ has [[rank]] $n_i$ and hence [[structure group]] $U(n_i)$, then the [[direct sum of vector bundles]] $E_1 \oplus E_2$ has [[structure group]] the [[product group]] $U(n_1) \times U(n_2)$.

In particular the sum of $n$ coincident cycles each carrying a line bundle yields a rank-$n$ vector bundle with structure group $(U(1))^n$. In this sense then a cycle carrying a general rank $n$-vector bundle, hence with structure group $U(n)$, may be thought of as a ``gauge enhancement'' of making $n$ D-branes coincide.

\hypertarget{from_massless_bps_states}{}\subsubsection*{{From massless BPS states}}\label{from_massless_bps_states}

In the context of the realization of [[super Yang-Mills theories]] as [[effective field theories]] of [[superstring theory]], [[BPS branes]] in the string background becomes BPS states of the SYM theory. These become massless as the [[cycles]] on which the BPS charge is supported shrink away (at boundary points of the given [[moduli space]]). In this case these additional massless states may appear as additional gauge bosons in the compactified gauge theory which thereby develops a larger gauge group. Typically a previously abelian gauge group becomes non-abelian this way.

Examples include [[KK-compactification]] of [[M-theory]] on [[K3]] fibers and on [[G2-manifolds]] with [[ADE singularities]], corresponding to [[F-theory]] on [[elliptic fibrations]] making [[K3]]-fibrations with singular fibers, corresponding to [[heterotic string theory]] on singular elliptic fibrations. See at

\begin{itemize}%
\item [[M-theory lift of gauge enhancement on D6-branes]]


\item \emph{[[M-theory on G2-manifolds]]} the section \emph{\href{M-theory+on+G2-manifolds#EnhancedGaugeGroups}{Nonabelian gauge groups and chiral fermions at orbifold singularities}};


\item \emph{[[F-theory]]} the section \emph{\href{F-theory#Fbranescan}{F-brane scan}}


\item \emph{[[duality between F-theory and heterotic string theory]]}.



\end{itemize}
\hypertarget{for_mk6_branes_at_adesingularities}{}\subsubsection*{{For MK6 branes at ADE-singularities}}\label{for_mk6_branes_at_adesingularities}

The [[blow-up]] of an [[ADE-singularity]] is given by a [[union]] of [[Riemann spheres]] that touch each other such as to form the shape of the [[Dynkin diagram]] whose A-D-E label corresponds to that of the given [[finite subgroup of SU(2)]].

This statement is originally due to (\hyperlink{duVal1934I}{duVal 1934 I, p. 1-3 (453-455)}). A description in terms of [[hyper-Kähler geometry]] is due to \hyperlink{Kronheimer89a}{Kronheimer 89a}.

Quick survey of this fact is in \hyperlink{Reid87}{Reid 87}, a textbook account is \hyperlink{Slodowy80}{Slodowy 80}.

 In [[string theory]] this fact is interpreted in terms of [[gauge enhancement]] of the [[M-theory]]-lift of coincident [[black brane|black]] [[D6-branes]] to an [[MK6]] at an ADE-singularity (\href{enhanced+gauge+symmetry#Sen97}{Sen 97}):



\begin{quote}%
graphics grabbed from \href{http://ncatlab.org/schreiber/show/Equivariant+homotopy+and+super+M-branes}{HSS18}


\end{quote}
See at \emph{[[M-theory lift of gauge enhancement on D6-branes]]} for more.

\hypertarget{on_m9branes}{}\subsubsection*{{On M9-Branes}}\label{on_m9branes}

In [[Horava-Witten theory]] there is supposed to be gauge enhancement over the [[M9-branes]] such as to identify their [[worldvolume]] theory with [[E8]]-[[heterotic string theory]].

Under the [[duality in string theory|duality]] between [[M-theory]] and [[type IIA string theory]] the M9-brane may be identified with the [[O8-plane]]:



\begin{quote}%
from \hyperlink{GKSTY02}{GKSTY 02, section 3}


\end{quote}
This may be used to understand the [[gauge enhancement]] to [[E8]]-[[gauge groups]] at the [[heterotic string theory|heterotic]] boundary of [[Horava-Witten theory]]:



\begin{quote}%
from \hyperlink{GKSTY02}{GKSTY 02, section 3}


\end{quote}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[spontaneously broken symmetry]]

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

\hypertarget{ReferencesOnCoincidentDBranes}{}\subsubsection*{{On coincident D-branes}}\label{ReferencesOnCoincidentDBranes}

The idea that on $N$ coincident D-branes there is gauge enhancement to $U(N)$-gauge field theory is due to

\begin{itemize}%
\item [[Edward Witten]], section 3 of \emph{Bound States Of Strings And $p$-Branes}, Nucl.Phys.B460:335-350, 1996 (\href{https://arxiv.org/abs/hep-th/9510135}{arXiv:hep-th/9510135})

\end{itemize}
There, this is called an ``obvious guess'' (first line on p. 8). Subsequently, most authors cite this obvious guess as a fact; for instance the review

\begin{itemize}%
\item [[Robert Myers]], section 3 of \emph{Nonabelian Phenomena on D-branes}, Class.Quant.Grav. 20 (2003) (\href{https://arxiv.org/abs/hep-th/0303072}{arXiv:hep-th/0303072})

\end{itemize}
Discussion in terms of [[boundary conformal field theory]] is in

\begin{itemize}%
\item [[Jürgen Fuchs]], P. Kaste, [[Wolfgang Lerche]], C. Lutken, [[Christoph Schweigert]], J. Walcher, \emph{Boundary Fixed Points, Enhanced Gauge Symmetry and Singular Bundles on K3}, Nucl.Phys.B598:57-72, 2001 (\href{https://arxiv.org/abs/hep-th/0007145}{arXiv:hep-th/0007145})

\end{itemize}
By actual computation in [[open string field theory]] ``convincing evidence'' for the nonabelian [[Yang-Mills theory]] on coincident D-branes was found, numerically, in (see bottom of p. 22)

\begin{itemize}%
\item Erasmo Coletti, Ilya Sigalov, [[Washington Taylor]], \emph{Abelian and nonabelian vector field effective actions from string field theory}, JHEP 0309 (2003) 050 (\href{https://arxiv.org/abs/hep-th/0306041}{arXiv:hep-th/0306041})

\end{itemize}
Similar numerical derivation, as well as exact derivation at zero momentum, is in

\begin{itemize}%
\item [[Nathan Berkovits]], [[Martin Schnabl]], \emph{Yang-Mills Action from Open Superstring Field Theory}, JHEP 0309 (2003) 022 (\href{https://arxiv.org/abs/hep-th/0307019}{arXiv:hep-th/0307019})

\end{itemize}
The first full derivation seems to be due to

\begin{itemize}%
\item [[Taejin Lee]], \emph{Covariant open bosonic string field theory on multiple D-branes in the proper-time gauge}, Journal of the Korean Physical Society December 2017, Volume 71, Issue 12, pp 886–903 (\href{https://arxiv.org/abs/1609.01473}{arXiv:1609.01473})

\end{itemize}
which is surveyed in

\begin{itemize}%
\item [[Taejin Lee]], \emph{Deformation of the Cubic Open String Field Theory}, Phys. Lett. B 768 (2017) 248 (\href{https://arxiv.org/abs/1701.06154}{arXiv:1701.06154})

\end{itemize}
That on [[D0-branes]] this reproduces the [[BFSS matrix model]] and on [[D(-1)-branes]] the [[IKKT matrix model]] is shown in

\begin{itemize}%
\item [[Taejin Lee]], \emph{Covariant Open String Field Theory on Multiple D$p$-Branes} (\href{https://arxiv.org/abs/1703.06402}{arXiv:1703.06402})

\end{itemize}
Discussion of gauge enhancement on coincident D-branes in terms of [[Baum-Douglas geometric cycles]] for [[K-homology]] is in

\begin{itemize}%
\item Rui Reis, [[Richard Szabo]], \emph{Geometric K-Homology of Flat D-Branes} ,Commun.Math.Phys. 266 (2006) 71-122, Journal of the Korean Physical Society December 2017, Volume 71, Issue 12, pp 886–903 (\href{https://arxiv.org/abs/hep-th/0507043}{arXiv:hep-th/0507043})

\end{itemize}
reviewed in

\begin{itemize}%
\item [[Richard Szabo]], \emph{D-branes and bivariant K-theory}, Noncommutative Geometry and Physics 3 1 (2013): 131. (\href{http://arxiv.org/abs/0809.3029}{arXiv:0809.3029})

\end{itemize}
Derivation via [[rational parameterized stable homotopy theory]] applied to [[schreiber:The brane bouquet]] is in

\begin{itemize}%
\item [[Vincent Braunack-Mayer]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Gauge enhancement of Super M-Branes]]} (\href{https://arxiv.org/abs/1806.01115}{arXiv:1806.01115})

\end{itemize}
\hypertarget{on_mbranes_at_adesingularities}{}\subsubsection*{{On M-branes at ADE-singularities}}\label{on_mbranes_at_adesingularities}

The picture of [[M-theory lift of gauge enhancement on D6-branes]] is due to

\begin{itemize}%
\item [[Ashoke Sen]], Section 2, \emph{A Note on Enhanced Gauge Symmetries in M- and String Theory}, JHEP 9709:001,1997 (\href{http://arxiv.org/abs/hep-th/9707123}{arXiv:hep-th/9707123})

\end{itemize}
Review of the standard lore of gauge enhancement in M-theory includes

\begin{itemize}%
\item Adam B. Barrett, section 2.3 (following \hyperlink{AcharyaGukov04}{Acharya-Gukov 04}) of \emph{M-Theory on Manifolds with $G_2$ Holonomy}, 2006 (\href{http://arxiv.org/abs/hep-th/0612096}{arXiv:hep-th/0612096})

\end{itemize}
(\ldots{})

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Original articles include

\begin{itemize}%
\item [[Edward Witten]], section 4.6 of \emph{[[String Theory Dynamics In Various Dimensions]]}, Nucl.Phys.B443:85-126,1995 (\href{http://arxiv.org/abs/hep-th/9503124}{arXiv:hep-th/9503124})


\item [[Chris Hull]], [[Paul Townsend]], \emph{Enhanced Gauge Symmetries in Superstring Theories}, Nucl.Phys. B451 (1995) 525-546 (\href{http://arxiv.org/abs/hep-th/9505073}{arXiv:hep-th/9505073})


\item [[Paul Aspinwall]], \emph{Enhanced Gauge Symmetries and K3 Surfaces}, Phys.Lett. B357 (1995) 329-334 (\href{http://arxiv.org/abs/hep-th/9507012}{arXiv:hep-th/9507012})


\item [[Chris Hull]], \emph{Duality, Enhanced Symmetry and Massless Black Holes}, in Proceedings of Strings'95: Future Perspectives in String Theory (World Scientific, 1996, I. Bars et al. eds.), p. 230 (\href{http://physics.usc.edu/Strings95/Proceedings/pdf/hull.pdf}{pdf})


\item M. Bershadsky, [[Ken Intriligator]], [[Shamit Kachru]], [[David Morrison]], V. Sadov, [[Cumrun Vafa]], \emph{Geometric Singularities and Enhanced Gauge Symmetries}, Nucl.Phys.B481:215-252,1996 (\href{http://arxiv.org/abs/hep-th/9605200}{arXiv:hep-th/9605200})


\item [[Philip Candelas]], Eugene Perevalov, Govindan Rajesh, \emph{Toric Geometry and Enhanced Gauge Symmetry of F-Theory/Heterotic Vacua}, Nucl.Phys. B507 (1997) 445-474 (\href{http://arxiv.org/abs/hep-th/9704097}{arXiv:hep-th/9704097})


\item [[Mirjam Cvetic]], [[Chris Hull]], \emph{Wrapped Branes and Supersymmetry}, Nucl.Phys.B519:141-158,1998 (\href{http://arxiv.org/abs/hep-th/9709033}{arXiv:hep-th/9709033})


\item [[Bobby Acharya]], [[Sergei Gukov]], \emph{M theory and Singularities of Exceptional Holonomy Manifolds}, Phys.Rept.392:121-189,2004 (\href{http://arxiv.org/abs/hep-th/0409191}{arXiv:hep-th/0409191})


\item [[James Halverson]], [[David Morrison]], \emph{On Gauge Enhancement and Singular Limits in $G_2$ Compactifications of M-theory} (\href{http://arxiv.org/abs/1507.05965}{arXiv:1507.05965})


\item [[Neil Lambert]], Miles Owen, \emph{Charged Chiral Fermions from M5-Branes} (\href{https://arxiv.org/abs/1802.07766}{arXiv:1802.07766})



\end{itemize}
Discussion via duality of [[M9-branes]] to [[O-planes]] is in

\begin{itemize}%
\item E. Gorbatov, V.S. Kaplunovsky, J. Sonnenschein, [[Stefan Theisen]], S. Yankielowicz, section 3 of \emph{On Heterotic Orbifolds, M Theory and Type I' Brane Engineering}, JHEP 0205:015, 2002 (\href{https://arxiv.org/abs/hep-th/0108135}{arXiv:hep-th/0108135})

\end{itemize}
[[!redirects enhanced gauge symmetries]]

[[!redirects gauge enhancement]] [[!redirects gauge enhancements]]



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