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Very roughly speaking, "wall-crossing" refers to a situation where you construct a would-be "invariant" $\Omega(t)$, that would naively be independent of parameters $t$ but actually depends on them in a piecewise-constant way: so starting from any $t_0$, $\Omega(t)$ is invariant under small enough deformations, but jumps at certain real-codimension-1 loci in the parameter space (the walls). You might initially think of this as a kind of quality-control problem in your invariant factory, to be eliminated by some more clever construction of an improved $\Omega(t)$; but at the moment it seems that this is the wrong point of view: there are interesting quantities that really do have wall-crossing behavior.

To name one example of such a quantity: suppose you have a compact Kahler manifold $M$ with an anticanonical divisor $D$, and you want to construct the mirror of a Kahler manifold $M$ using M \setminus D$following the prescription ideas of Strominger-Yau-Zaslow. One As it turns out, one of the crucial essential ingredients in this construction you will need is a study count of holomorphic discs in$M$, with boundary on a special Lagrangian torus$T(t)$in$M$(lying in a family parameterized by$t$). The number of such discs in a given homology class exhibits wall-crossing as$t$varies, and this wall-crossing turns out to be crucial in making the construction work. This story has been developed by Auroux. In physics, the wall-crossing phenomena that have been studied a lot recently arose in the context of "BPS state counting". If you have a supersymmetric quantum field theory of the right sort, depending on parameters$t$, you can define a collection of numbers$\Omega(\gamma, t) \in {\mathbb Z}$: they are superdimensions of certain graded Hilbert spaces attached to the theory (spaces of "1-particle BPS states with charge$\gamma$"). These quantities exhibit wall-crossing as a function of$t$. Moreover,$\Omega(\gamma, t)$are among the relatively few quantities in field theory that we are sometimes actually able to calculate exactly, so naturally they have attracted a lot of interest. In particular, they are the subject of the Ooguri-Strominger-Vafa conjecture of 2004, which in some cases relates their asymptotics to Gromov-Witten invariants; the investigation of this conjecture (mostly by Denef-Moore) is what triggered the current resurgence of interest in wall-crossing from the physics side. A particular case is the$4$-dimensional quantum field theory (or supergravity) associated to a Calabi-Yau threefold$X$(obtained by dimensional reduction of the$10$-dimensional string theory on the$6$-dimensional$X$to leave$10-6=4$dimensional space.) In that case the physically-defined$\Omega(\gamma,t)$are to be identified with the "generalized Donaldson-Thomas invariants" of$X$, studied by Joyce-Song and Kontsevich-Soibelman among others. The mathematical interpretation of$t$in that case is as a point on the space of Bridgeland stability conditions of$X$. (If$X$is compact, the last I heard, this space is not known to be nonempty, but the majority view seems to be that this gap will be filled...) One focal point for the excitement of the last few years is that a pretty remarkable wall-crossing formula has been discovered, the "Kontsevich-Soibelman wall-crossing formula", which completely answers the question of how$\Omega(t)$depends on$t$, and seems to apply (in some form) to all of the situations I described above. The formula was rather surprising to physicists; the process of trying to understand why it is true in the physical setting led to some interesting physical and geometric spin-offs, some of which seem likely to be re-importable into pure mathematics. 2 added 76 characters in body Very roughly speaking, "wall-crossing" refers to a situation where you have some quantity construct a would-be "invariant"$\Omega(t)$\Omega(t)$, that depends on would naively be independent of parameters $t$ but actually depends on them in a piecewise-constant way: so starting from any $t_0$, $\Omega(t)$ is invariant under small enough deformations, but jumps at certain real-codimension-1 loci in the parameter space (the walls). You might initially think of it this as a kind of quality-control problem in your invariant factory, to be eliminated by some more clever construction of an improved $\Omega(t)$; but at the moment it seems that this is the wrong point of view: there are interesting quantities that really do have this wall-crossing behavior.

To name one example of such a quantity: suppose you want to construct the mirror of a Kahler manifold $M$ using the prescription of Strominger-Yau-Zaslow. One of the crucial ingredients in this construction is a study of holomorphic discs in $M$, with boundary on a special Lagrangian torus $T(t)$ in $M$ (lying in a family parameterized by $t$). The number of such discs in a given homology class exhibits wall-crossing as $t$ varies, and this wall-crossing turns out to be crucial in making the construction work. This story has been developed by Auroux.

In physics, the wall-crossing phenomena that have been studied a lot recently arose in the context of "BPS state counting". If you have a supersymmetric quantum field theory of the right sort, depending on parameters $t$, you can define a collection of numbers $\Omega(\gamma, t) \in {\mathbb Z}$: they are superdimensions of certain graded Hilbert spaces attached to the theory (spaces of "1-particle BPS states with charge $\gamma$"). These $\Omega(\gamma, t)$ are among the relatively few quantities in field theory that we are sometimes actually able to calculate exactly, so naturally they have attracted a lot of interest. In particular, they are the subject of the Ooguri-Strominger-Vafa conjecture of 2004, which in some cases relates their asymptotics to Gromov-Witten invariants; the investigation of this conjecture (mostly by Denef-Moore) is what triggered the current resurgence of interest in wall-crossing from the physics side.

A particular case is the $4$-dimensional quantum field theory (or supergravity) associated to a Calabi-Yau threefold $X$ (obtained by dimensional reduction of the $10$-dimensional string theory on the $6$-dimensional $X$ to leave $10-6=4$ dimensional space.) In that case the physically-defined $\Omega(\gamma,t)$ are to be identified with the "generalized Donaldson-Thomas invariants" of $X$, studied by Joyce-Song and Kontsevich-Soibelman among others. The mathematical interpretation of $t$ in that case is as a point on the space of Bridgeland stability conditions of $X$. (If $X$ is compact, the last I heard, this space is not known to be nonempty, but the majority view seems to be that this gap will be filled...)

One focal point for the excitement of the last few years is that a pretty remarkable wall-crossing formula has been discovered, the "Kontsevich-Soibelman wall-crossing formula", which completely answers the question of how $\Omega(t)$ depends on $t$, and seems to apply (in some form) to all of the situations I described above. The formula was rather surprising to physicists; the process of trying to understand why it is true in the physical setting led to some interesting physical and geometric spin-offs, some of which seem likely to be re-importable into pure mathematics.

1

Very roughly speaking, "wall-crossing" refers to a situation where you have some quantity $\Omega(t)$ that depends on parameters $t$ in a piecewise-constant way: so starting from any $t_0$, $\Omega(t)$ is invariant under small enough deformations, but jumps at certain real-codimension-1 loci in the parameter space (the walls). You might initially think of it as a kind of quality-control problem in your invariant factory, to be eliminated by some more clever construction of an improved $\Omega(t)$; but at the moment it seems that this is the wrong point of view: there are interesting quantities that really do have this behavior.

To name one example of such a quantity: suppose you want to construct the mirror of a Kahler manifold $M$ using the prescription of Strominger-Yau-Zaslow. One of the crucial ingredients in this construction is a study of holomorphic discs in $M$, with boundary on a special Lagrangian torus $T(t)$ in $M$ (lying in a family parameterized by $t$). The number of such discs in a given homology class exhibits wall-crossing as $t$ varies, and this wall-crossing turns out to be crucial in making the construction work. This story has been developed by Auroux.

In physics, the wall-crossing phenomena that have been studied a lot recently arose in the context of "BPS state counting". If you have a supersymmetric quantum field theory of the right sort, depending on parameters $t$, you can define a collection of numbers $\Omega(\gamma, t) \in {\mathbb Z}$: they are superdimensions of certain graded Hilbert spaces attached to the theory (spaces of "1-particle BPS states with charge $\gamma$"). These $\Omega(\gamma, t)$ are among the relatively few quantities in field theory that we are sometimes actually able to calculate exactly, so naturally they have attracted a lot of interest. In particular, they are the subject of the Ooguri-Strominger-Vafa conjecture of 2004, which in some cases relates their asymptotics to Gromov-Witten invariants; the investigation of this conjecture (mostly by Denef-Moore) is what triggered the current resurgence of interest in wall-crossing from the physics side.

A particular case is the $4$-dimensional quantum field theory (or supergravity) associated to a Calabi-Yau threefold $X$ (obtained by dimensional reduction of the $10$-dimensional string theory on the $6$-dimensional $X$ to leave $10-6=4$ dimensional space.) In that case the physically-defined $\Omega(\gamma,t)$ are to be identified with the "generalized Donaldson-Thomas invariants" of $X$, studied by Joyce-Song and Kontsevich-Soibelman among others. The mathematical interpretation of $t$ in that case is as a point on the space of Bridgeland stability conditions of $X$. (If $X$ is compact, the last I heard, this space is not known to be nonempty, but the majority view seems to be that this gap will be filled...)

One focal point for the excitement of the last few years is that a pretty remarkable wall-crossing formula has been discovered, the "Kontsevich-Soibelman wall-crossing formula", which completely answers the question of how $\Omega(t)$ depends on $t$, and seems to apply (in some form) to all of the situations I described above. The formula was rather surprising to physicists; the process of trying to understand why it is true in the physical setting led to some interesting physical and geometric spin-offs, some of which seem likely to be re-importable into pure mathematics.