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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*{Yang-Mills instanton} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{chernweil_theory}{}\paragraph*{{Chern-Weil theory}}\label{chernweil_theory} [[!include infinity-Chern-Weil theory - contents]] \hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory} [[!include AQFT and operator algebra contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{FromTheMathsToThePhysicsStory}{$SU(2)$-instantons from the correct maths to the traditional physics story}\dotfill \pageref*{FromTheMathsToThePhysicsStory} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{as_gradient_flows_between_flat_connections}{As gradient flows between flat connections.}\dotfill \pageref*{as_gradient_flows_between_flat_connections} \linebreak \noindent\hyperlink{AsDpDpPlus4BraneBoundStates}{As Dp-D(p+4)-brane bound states}\dotfill \pageref*{AsDpDpPlus4BraneBoundStates} \linebreak \noindent\hyperlink{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{as_dpdp4brane_bound_states_2}{As Dp-D(p+4)-brane bound states}\dotfill \pageref*{as_dpdp4brane_bound_states_2} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} In $SU(n)$-[[Yang-Mills theory]] an \emph{[[instanton]]} is a field configuration with non-vanishing second [[Chern class]] that minimizes the Yang-Mills energy. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} Let $(X,g)$ be a [[compact space|compact]] [[Riemannian manifold]] of [[dimension]] 4. Let $G$ be a compact [[Lie group]]. A [[configuration space|field configuration]] of $G$-[[Yang-Mills theory]] on $(X,g)$ is a $G$-[[principal bundle]] $P \to X$ with [[connection on a bundle|connection]] $\nabla$. For $G = SU(n)$ the [[special unitary group]], there is canonically an [[associated bundle|associated]] complex [[vector bundle]] $E = P \times_G \mathbb{C}^n$. Write $F_\nabla \in \Omega^2(X,End(E))$ for the [[curvature]] [[differential form|2-form]] of $\nabla$. One says that $\nabla$ is an \textbf{instanton configuration} if $F_\nabla$ is [[Hodge star operator|Hodge]]-self dual \begin{displaymath} \star F_\nabla = - F_\nabla \,, \end{displaymath} where $\star : \Omega^k(X) \to \Omega^{4-k}(X)$ is the [[Hodge star operator]] induced by the [[Riemannian metric]] $g$. The second [[Chern class]] of $P$, which by the [[Chern-Weil homomorphism]] is given by \begin{displaymath} c_2(E) = \int_X Tr(F_\nabla \wedge F_\nabla) = k \in H^4(X, \mathbb{Z}) \end{displaymath} is called the \textbf{instanton number} or the \textbf{instanton sector} of $\nabla$. Notice that therefore any connection, even if not self-dual, is in some instanton sector, as its underlying bundle has some second Chern class, meaning that it can be obtained from shifting a self-dual connection. The self-dual connections are a convenient choice of ``base point'' in each instanton sector. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{FromTheMathsToThePhysicsStory}{}\subsubsection*{{$SU(2)$-instantons from the correct maths to the traditional physics story}}\label{FromTheMathsToThePhysicsStory} [[!include SU2-instantons from the correct maths to the traditional physics story]] \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{as_gradient_flows_between_flat_connections}{}\subsubsection*{{As gradient flows between flat connections.}}\label{as_gradient_flows_between_flat_connections} We discuss how Yang-Mills instantons may be understood as trajectories of the [[gradient flow]] of the [[Chern-Simons theory]] [[action functional]]. Let $(\Sigma,g_\Sigma)$ be a [[compact space|compact]] 3-[[dimensional]] [[Riemannian manifold]] . Let the [[cartesian product]] \begin{displaymath} X = \Sigma \times \mathbb{R} \end{displaymath} of $\Sigma$ with the [[real line]] be equipped with the product metric of $g$ with the canonical metric on $\mathbb{R}$. Consider field configurations $\nabla$ of [[Yang-Mills theory]] over $\Sigma \times \mathbb{R}$ with finite Yang-Mills action \begin{displaymath} S_{YM}(\nabla) = \int_{\Sigma \times \mathbb{R}} F_\nabla \wedge \star F_\nabla \,\,\lt \infty \,. \end{displaymath} These must be such that there is $t_1 \lt t_2 \in \mathbb{R}$ such that $F_\nabla(t \lt t_1) = 0$ and $F_\nabla(T \gt t_2) = 0$, hence these must be solutions interpolating between two [[curvature|flat]] connections $\nabla_{t_1}$ and $\nabla_{t_2}$. For $A \in \Omega^1(U\times \mathbb{R}, \mathfrak{g})$ the [[Lie algebra valued 1-form]] corresponding to $\nabla$, we can always find a [[gauge transformation]] such that $A_{\partial_t} = 0$ (``[[temporal gauge]]''). In this gauge we may hence equivalently think of $A$ as a 1-parameter family \begin{displaymath} t \mapsto A(t) \in \Omega^1(\Sigma, \mathfrak{g}) \end{displaymath} of connections on $\Sigma$. Then the self-duality condition on a Yang-Mills instanton \begin{displaymath} F_\nabla = - \star F_\nabla \end{displaymath} reads equivalently \begin{displaymath} \frac{d}{d t} A = -\star_{g} F_A \,\,\, \in \Omega^1(\Sigma, \mathfrak{g}) \,. \end{displaymath} \begin{defn} \label{HodgeInnerProduct}\hypertarget{HodgeInnerProduct}{} On the linear [[configuration space]] $\Omega^1(\Sigma, \mathfrak{g})$ of [[Lie algebra valued forms]] on $\Sigma$ define the [[Hodge inner product]] [[metric]] \begin{displaymath} G(\alpha, \beta) := \int_{\Sigma} \langle \alpha \wedge \star_g \beta \rangle \,, \end{displaymath} where $\langle-,-\rangle : \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R}$ is the [[Killing form]] [[invariant polynomial]] on the [[Lie algebra]] $\mathfrak{g}$. \end{defn} \begin{prop} \label{}\hypertarget{}{} The instanton equation \begin{displaymath} \frac{d}{d t} A = -\star_{g} F_A \end{displaymath} is the equation characterizing trajectories of the [[gradient flow]] of the [[Chern-Simons action functional]] \begin{displaymath} S_{CS} : \Omega^1(\Sigma, \mathfrak{g}) \to \mathbb{R} \end{displaymath} \begin{displaymath} A \mapsto \int_\Sigma CS(A) \end{displaymath} with respect to the \hyperlink{HodgeInnerProduct}{Hodge inner product metric} on $\Omega^1(\Sigma,\mathfrak{g})$. \end{prop} \begin{proof} The [[variational calculus|variation]] of the Chern-Simons action is \begin{displaymath} \delta S_{CS}(A) = \int_\Sigma \langle \delta A \wedge F_A\rangle \end{displaymath} (see [[Chern-Simons theory]] for details). In other words, we have the 1-form on $\Omega^1(\Sigma,\mathfrak{g})$: \begin{displaymath} \delta S_{CS}(-)_A = \int_\Sigma \langle - \wedge F_A \rangle \,. \end{displaymath} The corresponding [[gradient vector field]] \begin{displaymath} \nabla S_{CS} := G^{-1} \delta S_{CS} \end{displaymath} is uniquely defined by the equation \begin{displaymath} \begin{aligned} \delta S_{CS}(-) & = G(-,\nabla S_{CS}) \\ \int_\Sigma \langle - , \star \nabla S_{CS}\rangle \end{aligned} \,. \end{displaymath} With the formula (see [[Hodge star operator]]) \begin{displaymath} \star \star A = (-1)^{1(3+1)} A = A \end{displaymath} we find therefore \begin{displaymath} \nabla S_{CS} = \star_g F_A \,. \end{displaymath} Hence the [[gradient flow]] equation \begin{displaymath} \frac{d}{d t} A + \nabla S_{CS}_A = 0 \end{displaymath} is indeed \begin{displaymath} \frac{d}{d t} A = - \star_g F_A \,. \end{displaymath} \end{proof} Since [[curvature|flat]] connections are the [[critical loci]] of $S_{CS}$ this says that a finite-action Yang-Mills instanton on $\Sigma \times \mathbb{R}$ is a gradient flow trajectory between two \emph{Chern-Simons theory [[vacuum|vacua]]} . Often this is interpreted as saying that ``a Yang-Mills instanton describes the [[tunneling]] between two [[Chern-Simons theory]] [[vacua]]''. \hypertarget{AsDpDpPlus4BraneBoundStates}{}\subsubsection*{{As Dp-D(p+4)-brane bound states}}\label{AsDpDpPlus4BraneBoundStates} Due to the [[higher WZW term]] $\propto \int_{D_{p+4}} C_{p+1} \wedge \langle F \wedge F \rangle$ in the [[Green-Schwarz sigma model]] for [[D-brane|D(p+4)-branes]], [[Yang-Mills instantons]] in the [[Chan-Paton gauge field]] on $D (p+4)$-branes are equivalently [[Dp-D(p+4)-brane bound states]] (see e.g. \hyperlink{Polchinski96}{Polchinski 96, 5.4}, \hyperlink{Tong05}{Tong 05, 1.4}). The lift to [[M-theory]] as [[M5-MO9 brane bound states]] is due to \hyperlink{Strominger90}{Strominger 90}, \hyperlink{Witten96}{Witten 96}. [[!include Dp-D(p+4)-brane bound states -- contents]] \hypertarget{examples_2}{}\subsection*{{Examples}}\label{examples_2} \begin{itemize}% \item In $SU(2)$-YM theory: see \emph{[[BPTS instanton]]} . \item In $SU(3)$-YM theory, [[QCD]]/[[strong nuclear force]]: see \emph{[[instanton in QCD]]} \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[theta vacuum]], [[instanton sea]] \item [[instanton]] \item [[contact instanton]] \item [[Bogomolny equation]], [[Nahm transform]] \item [[caloron]] \item [[non-perturbative effect]] \item [[instanton Floer homology]] \item [[self-dual higher gauge field]] \item [[magnetic monopole]] [[Dirac monopole]], [[Yang monopole]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general}{}\subsubsection*{{General}}\label{general} Introductions and surveys include \begin{itemize}% \item J. Zinn-Justin, \emph{The principles of instanton calculus}, Les Houches (1984) \item M.A. Shifman et al., \emph{ABC of instantons}, Fortschr.Phys. 32,11 (1984) 585 \item [[David Tong]], \emph{TASI Lectures on Solitons} (\href{https://arxiv.org/abs/hep-th/0509216}{arXiv:hep-th/0509216}), \emph{Lecture 1: Instantons} (\href{http://www.damtp.cam.ac.uk/user/tong/tasi/instanton.pdf}{pdf}) \end{itemize} A survey in view of the [[asymptotic series|asymptotic]] nature of the [[Feynman perturbation series]] is in \begin{itemize}% \item [[Igor Suslov]], section 4.5 of \emph{Divergent perturbation series}, Zh.Eksp.Teor.Fiz. 127 (2005) 1350; J.Exp.Theor.Phys. 100 (2005) 1188 (\href{https://arxiv.org/abs/hep-ph/0510142}{arXiv:hep-ph/0510142}) \end{itemize} For a fairly comprehensive list of literature see the bibliography of \begin{itemize}% \item Marcus Hutter, \emph{Instantons in QCD: Theory and Application of the Instanton Liquid Model} (\href{http://arxiv.org/abs/hep-ph/0107098}{arXiv:hep-ph/0107098}) \end{itemize} Detailed argument for the [[theta-vacuum]] structure from [[chiral symmetry breaking]] is offered in \begin{itemize}% \item [[Curtis Callan]], R.F. Dashen, [[David Gross]], \emph{The Structure of the Gauge Theory Vacuum}, Phys.Lett. 63B (1976) 334-340 (\href{http://inspirehep.net/record/3673?ln=en}{spire}) \item G. Morchio, [[Franco Strocchi]], \emph{Chiral symmetry breaking and theta vacuum structure in QCD}, Annals Phys.324:2236-2254, 2009 (\href{https://arxiv.org/abs/0907.2522}{arXiv:0907.2522}) \end{itemize} The multi-instantons in $SU(2)$-Yang-Mills theory ([[BPTS instantons]]) were discovered in \begin{itemize}% \item A. A. Belavin, A.M. Polyakov, A.S. Schwartz, Yu.S. Tyupkin, \emph{Pseudoparticle solutions of the Yang-Mills equations}, Phys. Lett. B 59 (1), 85-87 (1975) \item A. A. Belavin, V.A. Fateev, A.S. Schwarz, Yu.S. Tyupkin, \emph{Quantum fluctuations of multi-instanton solutions}, Phys. Lett. B 83 (3-4), 317-320 (1979) \end{itemize} See also \begin{itemize}% \item [[Michael Atiyah]], [[Nigel Hitchin]], J. M. Singer, \emph{Deformations of instantons}, Proc. Nat. Acad. Sci. U.S. \textbf{74}, 2662 (1977) \item [[Edward Witten]], \emph{Some comments on the recent twistor space constructions}, Complex manifold techniques in theoretical physics (Proc. Workshop, Lawrence, Kan., 1978), pp. 207--218, Res. Notes in Math., 32, Pitman, Boston, Mass.-London, 1979. \end{itemize} Methods of algebraic geometry were introduced in \begin{itemize}% \item M. F. Atiyah, R. S. Ward, \emph{Instantons and algebraic geometry}, Comm. Math. Phys. \textbf{55}, n. 2 (1977), 117-124, \href{http://www.ams.org/mathscinet-getitem?mr=0494098}{MR0494098}, \href{http://projecteuclid.org/euclid.cmp/1103900980}{euclid} \end{itemize} The more general [[ADHM construction]] in terms of linear algebra of vector bundles on projective varieties is proposed in \begin{itemize}% \item M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld, Yu.I. Manin, \emph{Construction of instantons}, Physics Letters 65 A, 3, 185--187 (1978) \href{http://www.new.ox.ac.uk/system/files/ADHM.pdf}{pdf} \end{itemize} Monographs with the standard material include \begin{itemize}% \item [[Dan Freed]], [[Karen Uhlenbeck]], \emph{Instantons and four-manifolds}, Springer-Verlag, (1991) \item [[Robbert Dijkgraaf]], \emph{Topological gauge theories and group cohomology} (\href{staff.science.uva.nl/~rhd/papers/group.ps}{ps}) \item Nicholas Manton, Paul M. Sutcliffe, \emph{Topological solitons}, Cambridge Monographs on Math. Physics, \href{http://books.google.com/books?id=e2tPhFdSUf8C}{gBooks} \end{itemize} Yang-Mills instantons on spaces other than just spheres are explicitly discussed in \begin{itemize}% \item [[Gabor Kunstatter]], \emph{Yang-mills theory in a multiply connected three space}, Mathematical Problems in Theoretical Physics: Proceedings of the VIth International Conference on Mathematical Physics Berlin (West), August 11-20,1981. Editor: R. Schrader, R. Seiler, D. A. Uhlenbrock, Lecture Notes in Physics, vol. 153, p.118-122 (\href{http://adsabs.harvard.edu/abs/1982LNP...153..118K}{web}) \end{itemize} based on \begin{itemize}% \item [[Chris Isham]], [[Gabor Kunstatter]], Phys. Letts. v.102B, p.417, 1981. (\href{http://dx.doi.org/10.1016/0370-2693%2881%2991244-2}{doi}) \item [[Chris Isham]] [[Gabor Kunstatter]], J. Math. Phys. v.23, p.1668, 1982. (\href{http://dx.doi.org/10.1063/1.525552}{doi}) \end{itemize} In \begin{itemize}% \item Henrique N. S\'a{} Earp, \emph{Instantons on $G_2$$-$manifolds} PhD thesis (2009) (\href{http://www.ime.unicamp.br/~hqsaearp/index_files/HENRIQUE_SA_EARP_THESIS_UPDATED.PDF}{pdf}) \end{itemize} is a discussion of Yang-Mills instantons on a 7-dimensional [[manifold with special holonomy]]. \begin{itemize}% \item [[Michael Atiyah]], [[R. Bott]], \emph{The Yang-Mills equations over Riemann surfaces}, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523--615, \href{http://www.ams.org/mathscinet-getitem?mr=702806}{MR85k:14006}, \href{http://dx.doi.org/10.1098/rsta.1983.0017}{doi}. \end{itemize} \hypertarget{as_dpdp4brane_bound_states_2}{}\subsubsection*{{As Dp-D(p+4)-brane bound states}}\label{as_dpdp4brane_bound_states_2} The argument that [[Yang-Mills instantons]] in the [[Chan-Paton gauge field]] on a [[D-brane|D(p+4)-brane]] are equivalent to [[Dp-D(p+4) brane bound states]] goes back to \begin{itemize}% \item [[Edward Witten]], \emph{Small Instantons in String Theory}, Nucl. Phys. B460:541-559, 1996 (\href{https://arxiv.org/abs/hep-th/9511030}{arXiv:hep-th/9511030}) \item [[Michael Douglas]], \emph{Gauge Fields and D-branes}, J. Geom. Phys. 28 (1998) 255-262 (\href{https://arxiv.org/abs/hep-th/9604198}{arXiv:hep-th/9604198}) \end{itemize} following \begin{itemize}% \item [[Andrew Strominger]], \emph{Heterotic solitons}, Nucl. Phys. B343 (1990) 167-184 () Erratum: Nucl. Phys. B353 (1991) 565-565 () (\href{http://inspirehep.net/record/27900}{spire:27900}) \end{itemize} Review is in: \begin{itemize}% \item [[Joseph Polchinski]], Section 5.4 of: \emph{TASI Lectures on D-Branes} (\href{https://arxiv.org/abs/hep-th/9611050}{arXiv:hep-th/9611050}) \item [[David Tong]], Section 1.4 of \emph{TASI Lectures on Solitons} (\href{https://arxiv.org/abs/hep-th/0509216}{hep-th/0509216}) \end{itemize} Discussion specifically of [[D0-D4-brane bound states]]: \begin{itemize}% \item [[Cumrun Vafa]], \emph{Instantons on D-branes}, Nucl. Phys. B463 (1996) 435-442 (\href{https://arxiv.org/abs/hep-th/9512078}{arXiv:hep-th/9512078}) \end{itemize} with emphasis to the resulting [[configuration spaces of points]], as in \begin{itemize}% \item [[Cumrun Vafa]], [[Edward Witten]], Section 4.1 of: \emph{A Strong Coupling Test of S-Duality}, Nucl. Phys. B431:3-77, 1994 (\href{https://arxiv.org/abs/hep-th/9408074}{arXiv:hep-th/9408074}) \end{itemize} Discussion specifically of [[D1-D5-brane bound states]] \begin{itemize}% \item [[Neil Lambert]], \emph{D-brane Bound States and the Generalised ADHM Construction}, Nucl. Phys. B519 (1998) 214-224 (\href{https://arxiv.org/abs/hep-th/9707156}{arXiv:hep-th/9707156}) \end{itemize} Discussion specifically of [[D4-D8-brane bound states]]: In the [[Witten-Sakai-Sugimoto model]] [[geometric engineering of QFT|geometrically engineering]] [[QCD]], where the [[D4-branes]] get interpreted as [[baryons]]: \begin{itemize}% \item [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], Section 5.7 of: \emph{Low energy hadron physics in holographic QCD}, Prog. Theor. Phys.113:843-882, 2005 (\href{https://arxiv.org/abs/hep-th/0412141}{arXiv:hep-th/0412141}) \end{itemize} [[!redirects Yang-Mills instantons]] [[!redirects instanton number]] [[!redirects instanton numbers]] \end{document}