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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{NS5-brane} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \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{as_a_black_brane}{As a black brane}\dotfill \pageref*{as_a_black_brane} \linebreak \noindent\hyperlink{relation_to_little_string_theory}{Relation to little string theory}\dotfill \pageref*{relation_to_little_string_theory} \linebreak \noindent\hyperlink{DBranesEndingOnNS5Branes}{D-branes ending on NS5-branes}\dotfill \pageref*{DBranesEndingOnNS5Branes} \linebreak \noindent\hyperlink{open_dbrane_theories_on_the_ns5}{Open D$p$-brane theories on the NS5}\dotfill \pageref*{open_dbrane_theories_on_the_ns5} \linebreak \noindent\hyperlink{d4branes_ending_on_ns5branes}{D4-branes ending on NS5-branes}\dotfill \pageref*{d4branes_ending_on_ns5branes} \linebreak \noindent\hyperlink{D6BranesEndingOnNS5Branes}{D6-branes ending on NS5-branes}\dotfill \pageref*{D6BranesEndingOnNS5Branes} \linebreak \noindent\hyperlink{d6_branes_intersecting_d8branes_over_ns5branes}{D6 branes intersecting D8-branes over NS5-branes}\dotfill \pageref*{d6_branes_intersecting_d8branes_over_ns5branes} \linebreak \noindent\hyperlink{NSHalfBranes}{NS5 half-branes}\dotfill \pageref*{NSHalfBranes} \linebreak \noindent\hyperlink{webs}{Webs}\dotfill \pageref*{webs} \linebreak \noindent\hyperlink{NS5D4D2BoundStates}{NS5/D4/D2 bound states}\dotfill \pageref*{NS5D4D2BoundStates} \linebreak \noindent\hyperlink{relation_to_khovanov_homology}{Relation to Khovanov homology}\dotfill \pageref*{relation_to_khovanov_homology} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{as_a_black_brane_2}{As a black brane}\dotfill \pageref*{as_a_black_brane_2} \linebreak \noindent\hyperlink{as_a_greenschwarz_sigmamodel}{As a Green-Schwarz sigma-model}\dotfill \pageref*{as_a_greenschwarz_sigmamodel} \linebreak \noindent\hyperlink{under_dualities}{Under dualities}\dotfill \pageref*{under_dualities} \linebreak \noindent\hyperlink{relation_to_the_m5brane}{Relation to the M5-brane}\dotfill \pageref*{relation_to_the_m5brane} \linebreak \noindent\hyperlink{intersection_with_o8d8}{Intersection with O8/D8}\dotfill \pageref*{intersection_with_o8d8} \linebreak \noindent\hyperlink{ReferencesNS5D4D2BoundStates}{NS5/D4/D2 bound states}\dotfill \pageref*{ReferencesNS5D4D2BoundStates} \linebreak \noindent\hyperlink{open_dbrane_theories_on_the_ns5_2}{Open D$p$-brane theories on the NS5}\dotfill \pageref*{open_dbrane_theories_on_the_ns5_2} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} In the context of [[string theory]] the \emph{NS5-brane} is a certain extended physical objects -- a [[brane]] -- that appears in/is predicted by the theory. There are different incarnations of this object: For instance the [[effective QFT|effective background QFT]] of the [[type II string theory|type II string]] -- [[type II supergravity]] -- admits solutions to its generalized [[Einstein equations]] which describe higher dimensional analogs of charged [[black holes]] in ordinary gravity. Among them is a 5+1-dimensional ``black brane'' which is [[magnetic charge|magnetically charged]] under the [[Kalb-Ramond field]]. Since the KR field and the field of [[gravity]] constituting this solution of type II supergravity have as quanta the [[worldsheet]] excitations of the [[spinning string]] [[sigma-model]] that sit in what is called the [[Neveu-Schwarz sector]] one calls this the \textbf{NS5-brane}. This is to distinguish it from the [[D-brane|D5-brane]] which is instead charged under the [[RR-field]] whose quanta come from the [[Ramond-Ramond sector]] of the [[superstring]]. There are other incarnations of the NS 5-brane: by the general logic of [[Kalb-Ramond field|higher electromagnetism]] the (1+1)-dimensional string has under [[electric-magnetic duality]] a \emph{magnetic dual} . By dimension counting this is a 5-brane. If we think of the string this way as the structure that supports the [[sigma-model]] that defines perturbative [[string theory]], we also call it the \emph{F1-brane} (the \emph{fundamental} 1-brane). In this sense the the corresponding magnetic dual is the \emph{F5-brane} -- the \emph{fundamental} fivebrane. One can understand the NS5-``black brane'' solution to [[type II supergravity]] as being the solitonic incarnation of the fundamental 5-brane in much the same way as an ordinary [[black hole]] in ordinary [[gravity]] is a solitonic incarnation of the [[fundamental particle]]: as the particle, the black hole it is characterized just by [[mass]], [[charge]] and [[angular momentum]]. Similarly, the ``black'' NS5-brane is characterizes by [[mass]], [[B-field]] charge and [[angular momentum]]. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{as_a_black_brane}{}\subsubsection*{{As a black brane}}\label{as_a_black_brane} [[!include black branes in supergravity -- table]] \hypertarget{relation_to_little_string_theory}{}\subsubsection*{{Relation to little string theory}}\label{relation_to_little_string_theory} By the [[brane scan]], on the [[worldvolume]] of an NS5-brane propagates a [[superstring]]. This is called the \emph{[[little string]]}, see there for more. \hypertarget{DBranesEndingOnNS5Branes}{}\subsubsection*{{D-branes ending on NS5-branes}}\label{DBranesEndingOnNS5Branes} The [[black brane|black]] [[D-branes]] may end on [[black brane|black]] NS5-branes (\hyperlink{CallanHarveyStrominger91}{Callan-Harvey-Strominger 91, sections IV.C and V.B}, \hyperlink{Tseytlin96}{Tseytlin 96}, \hyperlink{ArgurioEnglertHouart97}{Argurio-Englert-Houart 97}, \hyperlink{BrodieHanany97}{Brodie-Hanany 97}, \hyperlink{EGKRS00}{EGKRS 00}). This ought to be this way if [[S-duality]] and [[T-duality]] work as expected, since: \begin{displaymath} \itexarray{ \text{F1 on D5} &\overset{\text{S}}{\leftrightarrow}& \text{D1 on NS5} \\ && \updownarrow\mathrlap{T} \\ && \text{D2 on NS5} \\ && \updownarrow\mathrlap{T} \\ && \text{D3 on NS5} &\overset{\text{S}}{\leftrightarrow}& \text{D3 on D5} \\ && \updownarrow\mathrlap{T} \\ && \vdots } \end{displaymath} This leads to what is called [[geometric engineering of quantum field theory]] on the [[worldvolume]] of these branes (following \hyperlink{HananyWitten97}{Hanany-Witten 97}, review includes \hyperlink{Fazzi17}{Fazzi 17}). \hypertarget{open_dbrane_theories_on_the_ns5}{}\paragraph*{{Open D$p$-brane theories on the NS5}}\label{open_dbrane_theories_on_the_ns5} Combining the above two items, the corresponding [[worldvolume]] theories of the NS5-branes (see also at [[little string theory]]) in the presence of light D$p$-branes ending on/inside them are also referred to as ``open D$p$-brane theories'' or ``OD$p$ theories'' (\hyperlink{GopakumarMinwallaSeibergStrominger00}{Gopakumar-Minwalla-Seiberg-Strominger 00, section 6}, \hyperlink{Harmark00}{Harmark 00, section 4.2}, \hyperlink{Lu06}{Lu 06, section 6}) \hypertarget{d4branes_ending_on_ns5branes}{}\paragraph*{{D4-branes ending on NS5-branes}}\label{d4branes_ending_on_ns5branes} see \hyperlink{Witten97}{Witten 97} \begin{quote}% graphics grabbed from \hyperlink{EGKRS00}{EGKRS 00} \end{quote} \hypertarget{D6BranesEndingOnNS5Branes}{}\paragraph*{{D6-branes ending on NS5-branes}}\label{D6BranesEndingOnNS5Branes} Consider a [[black brane|black]] NS5-brane with [[near horizon geometry]] $\underset{\sim AdS_7}{\underbrace{ \mathbb{R}^{5,1} \times \mathbb{R}_{\phi} }} \times S^3$ (\hyperlink{EGKRS00}{EGKRS 00, p. 8}): The [[3-sphere]] factor $S^3$ is the [[unit sphere]] around the black NS5-brane [[worldvolume]] $\mathbb{R}^{5,1}$, and $\mathbb{R}_{\phi}$ parameterizes the radial distance from it. Placing a [[D6-brane]] at one point of the $S^3$-factor (\hyperlink{EGKRS00}{EGKRS 00, p. 20}) means to take its [[worldvolume]] to be the factor $\mathbb{R}^{5,1} \times \mathbb{R}_\phi$, hence extending to one side of the NS5-brane (\hyperlink{EGKRS00}{EGKRS 00, p. 7}): Placing another [[D6-brane]] at the corresponding antipodal point means to have it extend also to the other side (\hyperlink{EGKRS00}{EGKRS 00, p. 5}): or else to have embedded the black NS5-brane into a single [[D6-brane]]. Special properties [[D6-branes]] ending on NS5-branes were highlighted in (\hyperlink{BrodieHanany97}{Brodie-Hanany 97, section 2.4}): the [[worldvolume]] theory becomes [[chiral fermion|chiral]]. The [[M-theory]]-lift of this situation should be the [[4-sphere|4-spherical]] [[orbifold]] [[near horizon geometry]] of an [[M5-brane]] (\hyperlink{MFF12}{MFF 12, section 8.3}) \begin{displaymath} \underset{\sim AdS_7}{\underbrace{\mathbb{R}^{5,1} \times \mathbb{R}_\phi}} \times S^4/G_{ADE} \end{displaymath} where $G_{ADE} \subset SU(2)$ is a [[finite group|finite]] [[subgroup]] of [[special unitary group|SU(2)]] (hence in the [[ADE classification]]), [[action|acting]] via the identification $S^4 \simeq S(\mathbb{H} \oplus \mathbb{R})$ (see at \emph{[[4-sphere]]} the section \emph{\href{4-sphere#QuaternionAction}{SU(2)-action}}). For non-trivial $G_{ADE}$, this [[action]] has precisely two [[fixed points]] $S^0 \hookrightarrow S^4 \to S^3$. Hence $\mathbb{R}^{5,1} \times \mathbb{R}_\phi \times S^0$ must be the [[worldvolume]] of two [[KK-monopoles]] of [[11d supergravity]], which is the M-theory lift of the two [[D6-branes]]. While the M-theory lift of the NS5-brane is the [[M5-brane]] with [[worldvolume]] $\mathbb{R}^{5,1}$. See also \hyperlink{Fazzi17}{Fazzi 17, p. 38}: \begin{quote}% The NS5-D6 Hanany–Witten setup engineering six-dimensional $(1,0)$ theories is equivalent to M5-branes at an $A_k$ singularity in eleven dimensions. \end{quote} Next, this construction may be repeated, having the [[D6-branes]] end on different NS5-branes, hence ``suspended between NS5-branes'' (graphics from \hyperlink{Fazzi17}{Fazzi 17, p. 33}): \hypertarget{d6_branes_intersecting_d8branes_over_ns5branes}{}\paragraph*{{D6 branes intersecting D8-branes over NS5-branes}}\label{d6_branes_intersecting_d8branes_over_ns5branes} And on the other end the D6-branes may end on [[D8-branes]] over an NS5-brane (\hyperlink{JanssenMeessenOrtin99}{Janssen-Meessen-Ortin 99, Section 3}). \begin{quote}% graphics grabbed from \hyperlink{GaiottoTomasiello14}{Gaiotto-Tomasiello 14} \end{quote} \hypertarget{NSHalfBranes}{}\paragraph*{{NS5 half-branes}}\label{NSHalfBranes} By the discussion \href{NS5-brane#D6BranesEndingOnNS5Branes}{above}\emph{, a [[black brane|black]] [[D6-brane]] may end on a [[black brane|black]] [[NS5-brane]], and in fact a priori each [[black brane|brane]] [[NS5-brane]] has to be the junction of two [[black brane|black]] [[D6-branes]].} \begin{quote}% from \hyperlink{GKSTY02}{GKSTY 02} \end{quote} If in addition the [[black brane|black]] [[NS5-brane]] sits at an [[O8-plane]], hence at the [[orientifold]] [[fixed point]]-locus, then in the ordinary $\mathbb{Z}/2$-[[quotient]] it appears as a ``[[half-brane]]'' with only one copy of [[D6-branes]] ending on it: \begin{quote}% from \hyperlink{GKSTY02}{GKSTY 02} \end{quote} (In \hyperlink{HananyZaffaroni99}{Hanany-Zaffaroni 99} this is interpreted in terms of the [[`t Hooft-Polyakov monopole]].) The lift to [[M-theory]] of this situation is an [[M5-brane]] [[brane intersection|intersecting]] an [[M9-brane]] (see at \emph{[[MO5-plane]]} and at \emph{[[heterotic M-theory on ADE-orbifolds]]}): \begin{quote}% from \hyperlink{GKSTY02}{GKSTY 02} \end{quote} Alternatively the [[O8-plane]] may [[brane intersection|intersect]] the [[black brane|black]] [[D6-branes]] away from the [[black brane|black]] [[NS5-brane]]: \begin{quote}% from \hyperlink{HKLY15}{HKLY 15} \end{quote} In general, some of the NS5 sit away from the [[O8-plane]], while some sit on top of it: \begin{quote}% from \hyperlink{HananyZaffaroni98}{Hanany-Zaffaroni 98} \end{quote} \begin{quote}% graphics grabbed from \hyperlink{ApruzziFazzi17}{Apruzzi-Fazzi 17, p. 18} \end{quote} The lift to [[M-theory]] under [[duality between M-theory and heterotic string theory]] is the [[half M5-brane]]. See also at \emph{[[intersecting D-brane models]]} the section \emph{\href{intersecting+D-brane+model#IntersectionOfD6WithO8}{Intersection of D6s with O8s}}. \hypertarget{webs}{}\paragraph*{{Webs}}\label{webs} corresponding [[brane webs]]: \begin{quote}% graphics grabbed from (Kimura 16 \href{http://www2.yukawa.kyoto-u.ac.jp/~qft.web/2016/slides/kimura.pdf}{pdf}) \end{quote} \hypertarget{NS5D4D2BoundStates}{}\subsubsection*{{NS5/D4/D2 bound states}}\label{NS5D4D2BoundStates} [[bound states]] of NS5-branes with [[D4-branes]] and [[D2-branes]]: \hyperlink{MitraRoy00}{Mitra-Roy 00, section 4}, \hyperlink{MukhiSuryanarayana00}{Mukhi-Suryanarayana 00}, \hyperlink{JiaLuRoy17}{Jia-Lu-Roy 17, p. 12, Table 1}, also \hyperlink{MitraRoy01}{Mitra-Roy 01}, \hyperlink{AlishahihaOz00}{Alishahiha-Oz 00}. \begin{quote}% graphics grabbed from \hyperlink{MukhiSuryanarayana00}{Mukhi-Suryanarayana 00} \end{quote} Notice that this includes configurations with the D4-branes and D2-branes contained strictly inside the NS5: \hyperlink{MitraRoy00}{Mitra-Roy 00, section 4} Review includes \hyperlink{Camino02}{Camino 02, section 4.5} see also \hyperlink{Petri18}{Petri 18} For the lift to [[M-theory]] see at \emph{\href{M2-brane#M2M5BoundStates}{M2-M5 brane bound state}} \hypertarget{relation_to_khovanov_homology}{}\subsubsection*{{Relation to Khovanov homology}}\label{relation_to_khovanov_homology} [[Khovanov homology]] has long been expected to appear as the [[observables]] in a 4-[[dimension]]al [[TQFT]] in higher analogy of how the [[Jones polynomial]] arises as an observable in 3-dimensional [[Chern-Simons theory]]. For instance for $\Sigma : K \to K'$ a cobordism between two [[knots]] there is a natural [[morphism]] \begin{displaymath} \Phi_\Sigma : \mathcal{K}(K) \to \mathcal{K}(K') \end{displaymath} between the Khovanov homologies associated to the two knots. In (\hyperlink{Witten11}{Witten11}) it is argued, following indications in (\hyperlink{GukovSchwarzVafa}{GukovSchwarzVafa}) that this 4d TQFT is related to the [[worldvolume]] theory of the \emph{image} in [[type IIA string theory|type IIA]] of [[D3-branes]] ending on NS5-branes in [[type IIB string theory|type IIB]] after one [[S-duality]] and one [[T-duality]] operation: \begin{displaymath} (D3 - NS4) \stackrel{S}{\mapsto} (D3 - D5) \stackrel{T}{\mapsto} (D4-D6) \,. \end{displaymath} Earlier indication for this had come from the observation that [[Chern-Simons theory]] is the [[effective QFT|effective background theory]] for the [[A-model]] 2d [[TCFT]] (see \href{http://ncatlab.org/nlab/show/TCFT#ActionFunctionals}{TCFT -- Worldsheet and effective background theories} for details). Notice that after the above [[T-duality]] operation the $(D4-D6)$-system wraps the $S^1$ ([[circle]]) along which the T-duality takes place. Lifting that configuration to [[11-dimensional supergravity]] gives [[M5-branes]] (the erstwhile [[D4-brane]]s) on [[Taub-NUT spacetime|Taub-NUT]] ($\times S^1$). The [[M5-branes]] wrap the circle-fiber of Taub-NUT, which shrinks to zero size at the origin (the location of the erstwhile D6, which is where the D4s ``end''). The low-energy theory, on a stack of M5-branes, is the [[6d (2,0)-susy QFT]]. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[brane]], [[string]] \item [[fivebrane structure]], [[differential fivebrane structure]] \end{itemize} [[!include table of branes]] $\backslash$linebreaK \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{as_a_black_brane_2}{}\subsubsection*{{As a black brane}}\label{as_a_black_brane_2} The 5-brane in [[heterotic string theory]] was found as a [[black brane]] in \begin{itemize}% \item [[Andrew Strominger]], \emph{Heterotic solitons}, Nucl.Phys. B343 (1990) 167-184 Nucl.Phys. B353 (1991) 565 (\href{http://inspirehep.net/record/27900}{spire}) \item [[Curtis Callan]], [[Jeffrey Harvey]], [[Andrew Strominger]], \emph{Worldbrane actions for string solitons}, Nuclear Physics B Volume 367, Issue 1, 16 December 1991, Pages 60-82 () \item Marco Cariglia, [[Kurt Lechner]], \emph{NS5-branes in IIA supergravity and gravitational anomalies} (\href{http://arxiv.org/abs/hep-th/0203238}{arXiv:hep-th/0203238}) \end{itemize} The [[brane intersection|intersection]] laws with [[black brane|black]] [[D-branes]] are discussed in \begin{itemize}% \item [[Arkady Tseytlin]], \emph{No-force condition and BPS combinations of p-branes in 11 and 10 dimensions}, Nucl.Phys.B487:141-154,1997 (\href{https://arxiv.org/abs/hep-th/9609212}{arXiv:hep-th/9609212}) \item [[Amihay Hanany]], [[Edward Witten]], \emph{Type IIB Superstrings, BPS Monopoles, And Three-Dimensional Gauge Dynamics}, Nucl. Phys. B492:152-190, 1997 (\href{https://arxiv.org/abs/hep-th/9611230}{arXiv:hep-th/9611230}) \item R. Argurio, [[François Englert]], L. Houart, \emph{Intersection Rules for $p$-Branes}, Phys. Lett. B398:61-68, 1997 (\href{https://arxiv.org/abs/hep-th/9701042}{arXiv:hep-th/9701042}) \item [[Edward Witten]], \emph{Solutions Of Four-Dimensional Field Theories Via M Theory}, Nucl.Phys.B500:3-42,1997 (\href{https://xxx.lanl.gov/abs/hep-th/9703166}{arXiv:hep-th/9703166}) \item [[John Brodie]], [[Amihay Hanany]], \emph{Type IIA Superstrings, Chiral Symmetry, and N=1 4D Gauge Theory Dualities}, Nucl.Phys. B506 (1997) 157-182 (\href{https://arxiv.org/abs/hep-th/9704043}{arXiv:hep-th/9704043}) \item Shmuel Elitzur, Amit Giveon, [[David Kutasov]], Eliezer Rabinovici, Gor Sarkissian, \emph{D-Branes in the Background of NS Fivebranes}, JHEP 0008 (2000) 046 (\href{https://arxiv.org/abs/hep-th/0005052}{arXiv:hep-th/0005052}) \end{itemize} Review includes \begin{itemize}% \item [[Marco Fazzi]], \emph{Higher-dimensional field theories from type II supergravity} (\href{https://arxiv.org/abs/1712.04447}{arXiv:1712.04447}) \end{itemize} The [[M-theory]]-lift of the black NS5-brane embedded into a [[D6-brane]] should be the configuration from section 8.3 of \begin{itemize}% \item Paul de Medeiros, [[José Figueroa-O'Farrill]], \emph{Half-BPS M2-brane orbifolds}, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. (\href{http://arxiv.org/abs/1007.4761}{arXiv:1007.4761}, \href{https://projecteuclid.org/euclid.atmp/1408561553}{Euclid}) \end{itemize} \hypertarget{as_a_greenschwarz_sigmamodel}{}\subsubsection*{{As a Green-Schwarz sigma-model}}\label{as_a_greenschwarz_sigmamodel} The [[Green-Schwarz action functionals]] for the NS5-brane: In [[heterotic string theory]] (see also at \emph{[[dual heterotic string theory]]}): \begin{itemize}% \item [[Kurt Lechner]], [[Mario Tonin]], \emph{Worldvolume and target space anomalies in the D=10 super--fivebrane sigma--model} (\href{http://arxiv.org/abs/hep-th/9603094}{arXiv:hep-th/9603094}) \item J. Mourad, \emph{Anomalies of the $SO(32)$ five-brane and their cancellation}, Nucl.Phys. B512 (1998) 199-208 (\href{https://arxiv.org/abs/hep-th/9709012}{arxiv:hep-th/9709012}) \item [[Kurt Lechner]], \emph{Quantum properties of the heterotic five-brane}, Phys.Lett.B693:323-329, 2010 (\href{https://arxiv.org/abs/1005.5719}{arxiv:1005.5719}) \end{itemize} In [[type IIA string theory]]: \begin{itemize}% \item [[Igor Bandos]], Alexei Nurmagambetov, [[Dmitri Sorokin]], \emph{The type IIA NS5--Brane} (\href{http://arxiv.org/abs/hep-th/0003169}{arXiv:hep-th/0003169}) \item Daniel Persson, \emph{Fivebrane Instantons and Hypermultiplets} (2010) (\href{http://string.lpthe.jussieu.fr/QKPHYS2010/Persson.pdf}{pdf}) \end{itemize} as a [[higher WZW model]]: \begin{itemize}% \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]} \end{itemize} \hypertarget{under_dualities}{}\subsubsection*{{Under dualities}}\label{under_dualities} Discussion of the effect of [[T-duality]] on NS5-branes includes \begin{itemize}% \item Eduardo Eyras, Bert Janssen, Yolanda Lozano, \emph{5-branes, KK-monopoles and T-duality}, Nucl.Phys. B531 (1998) 275-301 (\href{https://arxiv.org/abs/hep-th/9806169}{arXiv:hep-th/9806169}) \item [[David Tong]], \emph{NS5-Branes, T-Duality and Worldsheet Instantons}, JHEP 0207:013,2002 (\href{https://arxiv.org/abs/hep-th/0204186}{arXiv:hep-th/0204186}) \end{itemize} \hypertarget{relation_to_the_m5brane}{}\subsubsection*{{Relation to the M5-brane}}\label{relation_to_the_m5brane} Most of the following references are more on the [[M5-brane]]. The fact that the worldvolume theory of the M5-brane should support fields that are [[self-dual higher gauge theory|self-dual]] [[connections on a 2-bundle]] ($\sim$ a [[gerbe]]) is discussed in \begin{itemize}% \item [[Edward Witten]], \emph{[[Conformal field theory in four and six dimensions]]}, in [[Ulrike Tillmann]], \emph{Topology, Geometry and Quantum Field Theory: Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal}, London Mathematical Society Lecture Note Series (2004) (\href{http://arxiv.org/abs/0712.0157}{arXiv:0712.0157}) \end{itemize} as well as sections 3 and 4 of \begin{itemize}% \item [[Edward Witten]], \emph{Geometric Langlands From Six Dimensions} (\href{http://arxiv.org/abs/0905.2720}{arXiv:0905.2720}) . \end{itemize} A review of some aspects is in \begin{itemize}% \item [[Robbert Dijkgraaf]], \emph{The mathematics of fivebranes} (\href{http://arxiv.org/PS_cache/hep-th/pdf/9810/9810157v1.pdf}{pdf}) \end{itemize} The relation to [[Khovanov homology]] is discussed in \begin{itemize}% \item [[Edward Witten]], \emph{Fivebranes and knots} (\href{http://arxiv.org/abs/1101.3216}{arXiv:1101.3216}) \item [[Sergei Gukov]], [[Albert Schwarz]], [[Cumrun Vafa]], \emph{Khovanov-Rozansky Homology And Topological Strings} , Lett. Math. Phys. 74 (2005) 53-74, (\href{http://arxiv.org/abs/hep-th/0412243}{arXiv:hep-th/0412243}) \end{itemize} See also \begin{itemize}% \item [[Greg Moore]], \emph{On the role of sixdimensional $(2,0)$-theories in recent developments in Physical Mathematics} , talk at \emph{Strings2011} (\href{http://www-conference.slu.se/strings2011/presentations/3%20Wednesday/930_Moore.pdf}{pdf slides}) \end{itemize} The above discussion makes use of some blog comments (notably by [[Jacques Distler]]) appearing at \begin{itemize}% \item [[Urs Schreiber]], \emph{4d QFT for Khovanov Homology} (\href{http://golem.ph.utexas.edu/category/2011/02/4d_qft_for_khovanov_homology.html}{web}) \end{itemize} \hypertarget{intersection_with_o8d8}{}\subsubsection*{{Intersection with O8/D8}}\label{intersection_with_o8d8} Intersection of [[black brane|black]] NS5-branes with [[O8-planes]]/[[black brane|black]] [[D8-branes]] is discussed in \begin{itemize}% \item [[Amihay Hanany]], [[Alberto Zaffaroni]], \emph{Branes and Six Dimensional Supersymmetric Theories}, Nucl.Phys. B529 (1998) 180-206 (\href{https://arxiv.org/abs/hep-th/9712145}{arXiv:hep-th/9712145}) \item Bert Janssen, Patrick Meessen, Tomas Ortin, \emph{The D8-Brane Tied up: String and Brane Solutions in Massive Type IIA Supergravity}, Phys. Lett. B453 (1999) 229-236 (\href{https://arxiv.org/abs/hep-th/9901078}{arXiv:hep-th/9901078}) \item [[Amihay Hanany]], [[Alberto Zaffaroni]], \emph{Monopoles in String Theory}, JHEP 9912 (1999) 014 (\href{https://arxiv.org/abs/hep-th/9911113}{arXiv:hep-th/9911113}) \item E. Gorbatov, V.S. Kaplunovsky, J. Sonnenschein, [[Stefan Theisen]], S. Yankielowicz, \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}) \item [[Davide Gaiotto]], [[Alessandro Tomasiello]], \emph{Holography for $(1,0)$ theories in six dimensions} (\href{https://arxiv.org/abs/1404.0711}{arXiv:1404.0711}) \item Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Futoshi Yagi, \emph{6d SCFTs, 5d Dualities and Tao Web Diagrams}, JHEP05(2019)203 (\href{https://arxiv.org/abs/1509.03300}{arXiv:1509.03300}) \item Fabio Apruzzi, Marco Fazzi, \emph{$AdS_7/CFT_6$ with orientifolds}, J. High Energ. Phys. (2018) 2018: 124 (\href{https://arxiv.org/abs/1712.03235}{arXiv:1712.03235}) \item [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], Example 2.2.7 of: \emph{[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]}, Communications in Mathematical Physics (2019) (\href{https://arxiv.org/abs/1805.05987}{arXiv:1805.05987}, \href{http://link.springer.com/article/10.1007/s00220-019-03442-3}{doi:10.1007/s00220-019-03442-3}) \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 4 of: \emph{[[schreiber:Super-exceptional embedding construction of the M5-brane|Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5]]} (\href{https://arxiv.org/abs/1908.00042}{arXiv:1908.00042}) \end{itemize} On the [[orientifold]] [[T-duality|T-dual]] of half M5-branes: \begin{itemize}% \item Bo Feng, [[Yang-Hui He]], [[Andreas Karch]], [[Angel Uranga]], \emph{Orientifold dual for stuck NS5 branes}, JHEP 0106:065, 2001 (\href{https://arxiv.org/abs/hep-th/0103177}{arXiv:hep-th/0103177}) \end{itemize} For more see at \emph{[[M-theory on S1/G}HW times H/G\_ADE]]\_. \hypertarget{ReferencesNS5D4D2BoundStates}{}\subsubsection*{{NS5/D4/D2 bound states}}\label{ReferencesNS5D4D2BoundStates} [[bound states]] of NS5-branes with [[D4-branes]] and [[D2-branes]] (see also at \emph{\href{M2-brane#ReferencesDyonic}{M2-branes -- Dyonic membranes}}) \begin{itemize}% \item Indranil Mitra, Shibaji Roy, \emph{(NS5,Dp) and (NS5,D(p+2),Dp) bound states of type IIB and type IIA string theories}, JHEP 0102:026, 2001 (\href{https://arxiv.org/abs/hep-th/0011236}{arXiv:hep-th/0011236}) \item Mohsen Alishahiha, Yaron Oz, \emph{Supergravity and ``New'' Six-Dimensional Gauge Theories}, Phys.Lett.B495:418-426, 2000 (\href{https://arxiv.org/abs/hep-th/0008172}{arXiv:hep-th/0008172}) \item Indranil Mitra, Shibaji Roy, \emph{(NS5,D5,D3) bound state, OD3, OD5 limits and SL(2,Z) duality}, Phys.Rev. D65 (2002) 106001 (\href{https://arxiv.org/abs/hep-th/0107127}{arXiv:hep-th/0107127}) \item [[Sunil Mukhi]], Nemani V. Suryanarayana, \emph{Stable Non-BPS States and Their Holographic Duals}, Int. J. Mod. Phys. A16 (2001) 966-975 (\href{https://arxiv.org/abs/hep-th/0011185}{arXiv:hep-th/0011185}) \item J. M. Camino, section 4.5 of \emph{Worldvolume Dynamics of Branes} (\href{https://arxiv.org/abs/hep-th/0210249}{arXiv:hep-th/0210249}) \item Qiang Jia, J. X. Lu, Shibaji Roy, \emph{On 1/4 BPS ((F, D1), (NS5, D5)) bound states of type IIB string theory}, JHEP 08 (2017) 007 (\href{https://arxiv.org/abs/1704.01463}{arXiv:1704.01463}) \item Giuseppe Dibitetto, Nicolò Petrim, \emph{6d surface defects from massive type IIA}, JHEP 01 (2018) 039 (\href{https://arxiv.org/abs/1707.06154}{arXiv:1707.06154}) \item Nicolò Petri, \emph{Surface defects in massive IIA}, talk at \href{http://physics.ipm.ac.ir/conferences/stringtheory3/}{Recent Trends in String Theory and Related Topics} 2018 (\href{http://physics.ipm.ac.ir/conferences/stringtheory3/note/N.Petri.pdf}{pdf}) \end{itemize} \hypertarget{open_dbrane_theories_on_the_ns5_2}{}\subsubsection*{{Open D$p$-brane theories on the NS5}}\label{open_dbrane_theories_on_the_ns5_2} \begin{itemize}% \item [[Rajesh Gopakumar]], [[Shiraz Minwalla]], [[Nathan Seiberg]], [[Andrew Strominger]], \emph{OM Theory in Diverse Dimensions}, JHEP 0008:008, 2000 (\href{https://arxiv.org/abs/hep-th/0006062}{arXiv:hep-th/0006062}) \item Troels Harmark, \emph{Open Branes in Space-Time Non-Commutative Little String Theory}, Nucl.Phys. B593 (2001) 76-98 (\href{https://arxiv.org/abs/hep-th/0007147}{arXiv:hep-th/0007147}) \item J. X. Lu, \emph{$(1 + p)$-Dimensional Open $D(p - 2)$ Brane Theories}, JHEP 0108:049, 2001 (\href{https://arxiv.org/abs/hep-th/0102056}{arXiv:hep-th/0102056}) \end{itemize} [[!redirects NS-5-brane]] [[!redirects F5-brane]] [[!redirects F-5-brane]] [[!redirects 5-brane]] [[!redirects fivebrane]] [[!redirects NS5-branes]] [[!redirects NS-5-branes]] [[!redirects F5-branes]] [[!redirects F-5-branes]] [[!redirects 5-branes]] [[!redirects fivebranes]] [[!redirects half NS5-brane]] [[!redirects half NS5-branes]] \end{document}