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\section*{wall crossing}

\begin{quote}%
This entry is about discontinuities in parameter dependence of (often asymptotic) solutions of [[differential equations]] and similar phenomena (notably in the context of [[BPS states]] formalized via [[Bridgeland stability conditions]]) with stability parameters (and their stability slopes) in [[algebraic geometry]] which are often interpreted as crossing the walls of marginal stability in [[physics]]. For the different notions of the same name in [[Morse theory]] see at \emph{[[Cerf wall crossing]]} and for the (Weyl chamber wall) crossing functors in representation theory see [[wall crossing functor]].


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

\noindent\hyperlink{overview}{Overview}\dotfill \pageref*{overview} \linebreak
\noindent\hyperlink{in_supersymmetric_field_theory}{In supersymmetric field theory}\dotfill \pageref*{in_supersymmetric_field_theory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{introductions_and_lectures}{Introductions and lectures}\dotfill \pageref*{introductions_and_lectures} \linebreak
\noindent\hyperlink{original_articles}{Original articles}\dotfill \pageref*{original_articles} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{in_supergravity}{In supergravity}\dotfill \pageref*{in_supergravity} \linebreak
\noindent\hyperlink{conferences_and_seminars}{Conferences and seminars}\dotfill \pageref*{conferences_and_seminars} \linebreak
\noindent\hyperlink{categorification}{Categorification}\dotfill \pageref*{categorification} \linebreak


\hypertarget{overview}{}\subsection*{{Overview}}\label{overview}

In the study of [[solitons]], one may try a [[WKB approximation|WKB-style approximation]] to a nonlinear [[wave equation]] (see also \emph{[[eikonal equation]]}, \emph{[[Maslov index]]}, etc.). Stokes has observed that when trying to connect the local solutions, one has discontinuities along certain lines, now called \textbf{Stokes lines}. This is called the [[Stokes phenomenon]]. Similar issues appear in study of [[isomonodromic deformation]]s of nonlinear [[ODEs]] in the [[complex plane]], which is also relevant in [[soliton]] theory, and [[integrable systems]], and special functions like [[Painlevé transcendent]]s. This has especially been studied by the Kyoto school (Jimbo, Miwa, Sato, [[Masaki Kashiwara|Kashiwara]] etc.), including the use of [[D-modules]] and [[microlocal analysis]]. The Kyoto school found a connection of isomonodromic theory to what is called [[holonomic quantum field]]s.

The solutions of meromorphic differential equations can be expressed in terms of [[meromorphic connection]]s. Then the slopes related to the solutions can be viewed as features of particular objects in a category of $D$-[[D-module|modules]]. More generally, slope filtrations are structures which appear in many other additive categories, e.g. in [[Hodge theory]], theory of Dieudonn\'e{} modules and so on. Many of those are related to the stability of the objects, which is important in the construction of [[moduli spaces]].

In [[algebraic geometry]], [[Grothendieck]] has shown how to correctly define and construct some fundamental [[moduli spaces]], like Hilbert schemes and Quot schemes for [[coherent sheaves]]. The work has been continued by [[David Mumford]] who geometrized classical invariant theory into [[geometric invariant theory]]. To keep moduli under control, one needs to impose stability conditions on objects and also look at classes with some fixed data: those involve slopes or equivalently phase factors. This is thus similar to the phases of eikonal in the case of Stokes phenomenon. Cf. also Harder-Narasimhan filtration, [[Castelnuovo-Mumford regularity]] (cf. \href{http://en.wikipedia.org/wiki/Castelnuovo%E2%80%93Mumford_regularity}{wikipedia}) etc.

\hypertarget{in_supersymmetric_field_theory}{}\subsubsection*{{In supersymmetric field theory}}\label{in_supersymmetric_field_theory}

In [[super Yang-Mills theory]] the number of [[BPS states]] is locally constant as a function of the parameters of the theory, but it may jump at certain ``walls'' in the [[moduli spaces]] of parameters. The precise behaviour of the BPS states as one crosses these walls is studied as ``wall crossing phenomena''.

Another example are the [[moduli spaces]] of [[Higgs bundles]], studied by [[Carlos Simpson]] and others, which have special cases with interpretations both in geometry and in the gauge theory (instantons). It appears that sometimes they can be linked to the geometric picture. [[Riemann-Hilbert correspondence]], spectral transform and similar correspondences again play a major role.

Surely, one often works at the derived level. An adaptation of the notion of stability into the setup of [[triangulated categories]] has been introduced by Bridgeland. Bridgeland stability for the derived categories of (boundary conditions of) D-branes (B-model) are relevant for string theory.

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

\begin{itemize}%
\item [[BPS state]], [[D-module]], [[cluster algebra]], [[quiver]], [[representation theory]], [[Donaldson-Thomas invariant]].

\end{itemize}
[[!include field theory with boundaries and defects - table]]

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

\hypertarget{introductions_and_lectures}{}\subsubsection*{{Introductions and lectures}}\label{introductions_and_lectures}

\begin{itemize}%
\item Sergio Cecotti, \emph{Trieste lectures on wall-crossing invariants} (2010) (\href{http://people.sissa.it/~cecotti/ictptext.pdf}{pdf})


\item [[Greg Moore]], \emph{PiTP Lectures on BPS states and wall-crossing in $d = 4$, $\mathcal{N} = 2$ theories} (\href{http://www.physics.rutgers.edu/~gmoore/PiTP_July26_2010.pdf}{pdf})


\item [[Tudor Dimofte]], \emph{Refined wall crossing} (\href{http://thesis.library.caltech.edu/5808/4/TD_part1.pdf}{pdf}), part I of \emph{Refined BPS invariants, Chern-Simons theory, and the quantum dilogarithm}, 2010 (\href{http://thesis.library.caltech.edu/5808/1/DimofteTDofficial.pdf}{pdf}, \href{http://thesis.library.caltech.edu/5808/}{web})



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

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

\begin{itemize}%
\item [[Maxim Kontsevich]], [[Yan Soibelman]], \emph{Stability structures, motivic Donaldson-Thomas invariants and cluster transformations}, \href{http://arxiv.org/abs/0811.2435}{arXiv:0811.2435}; \emph{Motivic Donaldson-Thomas invariants: summary of results}, \href{http://arxiv.org/abs/0910.4315}{0910.4315}; \emph{Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and Mirror Symmetry}, \href{http://arxiv.org/abs/1303.3253}{arxiv/1303.3253}


\item Arend Bayer, [[Yuri Manin|Yuri I. Manin]], \emph{Stability conditions, wall-crossing and weighted Gromov-Witten invariants}, \href{http://arxiv.org/abs/math.AG/0607580}{math.AG/0607580}, Mosc. Math. J. \textbf{9} (1), 2009.


\item [[Mina Aganagic]], Hirosi Ooguri, [[Cumrun Vafa]], Masahito Yamazaki, \emph{Wall crossing and M-theory}, \href{http://arxiv.org/abs/0908.1194}{arxiv/0908.1194}


\item [[Mina Aganagic]], \emph{Wall crossing, quivers and crystals}, \href{http://arxiv.org/abs/1006.2113}{arxiv/1006.2113}


\item S. Cecotti, [[Cumrun Vafa]], \emph{BPS wall crossing and topological strings}, \href{http://arxiv.org/abs/0910.2615}{arXiv/0910.2615}


\item [[Davide Gaiotto]], [[Greg Moore]], [[Andrew Neitzke]], \emph{Wall-crossing, Hitchin systems, and the WKB approximation, (\href{http://arxiv.org/abs/0907.3987}{arxiv/0907.3987})}


\item E. Diaconescu, [[Greg Moore]], \emph{Crossing the wall: branes vs. bundles}, \href{http://arxiv.org/abs/0706.3193}{arXiv/0706.3193}


\item E. Andriyash, [[Frederik Denef]], [[Daniel Jafferis]], [[Greg Moore|G. W. Moore]], \emph{Wall-crossing from supersymmetric galaxies}, \href{http://arxiv.org/abs/1008.0030}{arxiv/1008.0030}


\item Tom Bridgeland, [[Valerio Toledano-Laredo]], \emph{Stability conditions and Stokes factors}, \href{http://arxiv.org/abs/0801.3974}{arxiv/0801.3974}


\item M. C. N. Cheng, E. P. Verlinde, \emph{Wall crossing, discrete attractor flow and Borcherds algebra}, SIGMA 4 (2008), 068, 33 pages, \href{http://www.emis.de/journals/SIGMA/2008/068/sigma08-068.pdf}{pdf}


\item Masahito Yamazaki, \emph{Crystal melting and wall crossing phenomena}, Ph.D. thesis, \href{http://arxiv.org/abs/1002.1709}{arxiv/1002.1709}


\item Michele Cirafici, Annamaria Sinkovics, [[Richard Szabo]], \emph{Instanton counting and wall-crossing for orbifold quivers}, \href{http://arxiv.org/abs/1108.3922}{arxiv/1108.3922}


\item H.-Y. Chen, N. Dorey, K. Petunin, \emph{Moduli space and wall-crossing formulae in higher-rank gauge theories}, JHEP 11 (2011) 020, ; \emph{Wall crossing and instantons in compactified gauge theory}, JHEP 06 (2010) 024 \href{ttp://arxiv.org/abs/1004.0703}{arXiv:1004.0703}



\end{itemize}
\hypertarget{in_supergravity}{}\subsubsection*{{In supergravity}}\label{in_supergravity}

\begin{itemize}%
\item [[Frederik Denef]], \emph{Supergravity flows and D-brane stability}, JHEP 0008:050,2000 (\href{http://arxiv.org/abs/hep-th/0005049}{arXiv:hep-th/0005049})


\item [[Frederik Denef]], \emph{Quantum Quivers and Hall/Hole Halos}, JHEP 0210:023,2002 (\href{http://arxiv.org/abs/hep-th/0206072}{arXiv:hep-th/0206072})


\item [[Daniel Jafferis]], [[Gregory Moore]], \emph{Wall crossing in local Calabi Yau manifolds} (\href{http://arxiv.org/abs/0810.4909}{arXiv:0810.4909})



\end{itemize}
\hypertarget{conferences_and_seminars}{}\subsubsection*{{Conferences and seminars}}\label{conferences_and_seminars}

\begin{itemize}%
\item (past) [[Kontsevich]] in Aarhus, August 2010, \href{http://qgm.au.dk/events/show/artikel/masterclass-aug-2010/}{master class on wall crossing}; we will keep a [[wall crossing in Aarhus|nlab page]] on it


\item (past) \href{http://member.ipmu.jp/domenico.orlando/FocusInvariants.html}{Focus Week on New Invariants and Wall Crossing}, May 18-22, 2009, Kashiwa Campus of the University of Tokyo


\item (past) \href{http://www.math.uiuc.edu/wallcrossing}{Wall-crossing in Mathematics and Physics}, May 24-28, 2010, Department of Mathematics, University of Illinois at Urbana-Champaign


\item Description of seminar on stability conditions and Stokes factors in Bonn, \href{http://www.math.uni-bonn.de/people/compgeo/Hall.pdf}{pdf}



\end{itemize}
Also (\hyperlink{GaiottoMooreWitten15}{Gaiotto-Moore-Witten 15}).

\hypertarget{categorification}{}\subsubsection*{{Categorification}}\label{categorification}

A [[categorification]] of wall crossing formulas to an [[(infinity,2)-category]] of sorts is discussed in

\begin{itemize}%
\item [[Davide Gaiotto]], [[Gregory Moore]], [[Edward Witten]], \emph{An Introduction To The Web-Based Formalism} (\href{http://arxiv.org/abs/1506.04086}{arXiv.1506.04086})

\end{itemize}
[[!redirects wall crossing]] [[!redirects wall-crossing]]



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