This question is motivated by the current interest of Mathematics and Physics community in Wall Crossing. My questions are :

  1. What is wall crossing in Physics, what are the reasons for current interest in it.

  2. What is wall crossing in terms of mathematics, what is the reason for interest, is it just physics or some mathematical motivation.


  • $\begingroup$ It's a wall (not soft!) question. And string theory rather than mathematical physics. Have you followed the links on en.wikipedia.org/wiki/Wall-crossing ? Is this a question for a question? $\endgroup$ – Wadim Zudilin Jan 3 '11 at 8:12
  • $\begingroup$ I too don't think that the question is soft. I've retagged. I didn't add the string theory tag, though. $\endgroup$ – José Figueroa-O'Farrill Jan 3 '11 at 20:52

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 $M \setminus D$ following the ideas of Strominger-Yau-Zaslow. As it turns out, one of the essential ingredients you will need is a 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 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.

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    $\begingroup$ I'd give this two up arrows if I could. Excellent answer. $\endgroup$ – Jeff Harvey Jan 3 '11 at 23:46
  • $\begingroup$ Great answer, Andy. You humbly did not mention your own contributions... $\endgroup$ – Spiro Karigiannis Jan 4 '11 at 3:14
  • $\begingroup$ In general $T(t)$ is only Lagrangian instead of special Lagrangian. $\endgroup$ – YHBKJ Sep 1 '14 at 3:50

In extremely vague, rough, oversimplified and imprecise terms, wall-crossing phenomena can be described as follows.

Given some linear category $\mathcal{C}$ (vector bundles, coherent sheaves, etc.), from a physicist's point of view it is quite natural to consider the space $\mathcal{M}_{\mathcal{C}}$ of all objects of $\mathcal{C}$, and to take averages of naturally defined quantities on $\mathcal{M}_{\mathcal{C}}$, obtaining characteristic numbers $I_\mathcal{C}$ (I've said I'm really oversimplifying things).

To a mathematician, the space $\mathcal{M}_{\mathcal{C}}$ makes no sense in the realm of manifolds or schemes: it is a kind of derived object known as (higer) stack. However, things are quite under control if $\mathcal{M}$ is given a stability condition, i.e., a notion of stable objects. Now, the idea of Tom Bridgeland (derived from Michael Douglas investigations on D-branes) is that there is not a distinguished stability condition, but a whole space $\Sigma$ of stability conditions (which turns out to be a topological manifold). For any $P$ in $\Sigma$ one can then consider the moduli space $\mathcal{M}_P$ of $P$-stable objects, and this leads to a formalization of the physicist's space $\mathcal{M}_\mathcal{C}$ as a bundle of moduli spaces over the space $\Sigma$ of stability conditions: the fiber over $P$ is $\mathcal{M}_P$.

Now, the geometry of $\mathcal{M}_P$ can abruptly change as $P$ moves in $\Sigma$. This is easily explained: if $P$ and $Q$ are two distinct points in $\Sigma$, there could be some object $E$ in $\mathcal{C}$ which is $P$-stable but not $Q$ stable. So in the moduli space $\mathcal{M}_P$ there will be a point representing $E$, but in $\mathcal{M}_Q$ there will be not such a point. Conversely there will be some object $F$ which is $Q$-stable but not $P$-stable. So, if we join $P$ and $Q$ by a path and follow $E$ along this path, then we see that at some point along the path $E$ must become unstable and so disappear from the moduli space of stable objects and be substituted by $F$. And this happens for each path between $P$ and $Q$: there's a wall separating the region where $E$ is stable and $F$ is unstable from the region where $E$ is unstable and $F$ is stable.

As we cross the wall the geometry of the moduli space abruptly changes. However this change in geometry is expected to be quite regular: more precisely it is expected to be a Mukai flop. This is too, quite easy to imagine from abstract nonsense (proving it formally is another kind of affair...). Namely, for a fixed wall, one expects that there is a whole linear space of objects which become unstable on the wall. And since we are considering objects up to isomorphisms, and we are considering linear categories, nonzero scalars will act as isomorphisms and so will have to be quotiented out. This leaves us with a projective space of objects that gets unstabilized by the wall, and so disappears as we cross the wall. But for what we just said above, there will be also a projective space of objects appearing as we cross the wall. So crossing the wall, there is a $\mathbb{P}^n$ replaced by another $\mathbb{P}^n$, and this is, very roughtly and basically, the idea of a Mukai flop. The idea of wall crossing transition of moduli spaces by flops goes back at least to Thaddeus' influential work on the Verlinde formula, studying the dependence of geometric invariant theory quotients on the choice of stability (linearization of the action), and similar wall crossings were important e.g. in Donaldson theory.

Let us now come back again to the invariants $I_\mathcal{C}$. From the mathematician's perspective we have not a single invariant, but a quantitu $I_P$ for any stability condition $P$. What happens when we cross a wall? the spaces $\mathcal{M}_P$ and $\mathcal{M}_Q$ are geometrically different, so the quantities $I_P$ and $I_Q$ are a priori completely unrelated. On the other hand, the physicist's ill-defined quantity $I_\mathcal{C}$ was defined before stability conditions, so it does not knows about the wall! this means that the well-defined quantities $I_P$ and $I_Q$ must coincide (or better be canonicaly related): in colloquial terms, the invariant $I$ crosses the wall, and the relation between $I_P$ and$I_Q$ ia a wall-crossing phenomenon.

Actually, this is an huge oversimplification even of the physiscist's intuition. A better description of the physicist's point of view is the following (see the excellent answer by Andy Neitzke below for details). The relevant quantity one is interested in does not originally live on $\mathcal{M}$, but in a larger space of superconformal field theories. And it depends on a paramter $t$. Let us denote this quantity by the symbol $\Omega(t)$. As $t$ varies into the parameter space of SCFTs, the quantity $\Omega(t)$ varies continuosly. On the other hand, as said above, $\Omega(t)$ is the average of some quantity $J(t)$, and this average can be computed by localization formulae as suitable integrals on the space of critical points of $J(t)$ (the spaces of vacua of the theory). It turns out that at least certain values of $t$ can be identified with stability conditions on $\mathcal{M}$; if $t=t_P$, this identifies the space of critical points of $J(t_P)$ with the moduli space $\mathcal{M}_P$ and the localization formulae relate in a precise way $\Omega(t_P)$ with $I(P)$. Now, the geometry of the set of critical points of $J(t)$ may abruptly change as the parameter $t$ varies (this is nothing strange: this happens already for families of smooth functions $f_t:\mathbb{R}\to\mathbb{R}$, as every first year calculus student knows). So the formulae for $I_P$ and $I_Q$ will be quite different if there's a wall between $P$ and $Q$. On the other hand, they are related in a precise way to $\Omega(t_P)$ and $\Omega(t_Q)$, and these quantities change in a well-behaved way as $t$ goes from $t_P$ to $t_Q$. Reading this good behaviour at the level of $I_P$ and $I_Q$ gives the wall-crossing formulae.

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    $\begingroup$ +1. Nice answer. Would it be too much to ask to perhaps split this into paragraphs to improve readability? Also you denote the fibre over $P \in \Sigma$ by $\mathcal{P}$, but don't you mean $\mathcal{M}_P$? $\endgroup$ – José Figueroa-O'Farrill Jan 3 '11 at 20:50
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    $\begingroup$ @José. Thanks! I've now followed your advice and split the answer into paragraphs. I've also fixed the $\mathcal{P}$ vs. $\mathcal{M}_P$ typo. $\endgroup$ – domenico fiorenza Jan 3 '11 at 21:16
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    $\begingroup$ It's not quite true that "the physicist's ill-defined quantity ... was defined before stability conditions." Even in the physical description of these quantities, a parameter certainly enters, which plays the role of the stability condition. What is true is that there is some other physically defined quantity $F$ that depends continuously on the parameter, and whose relation to $I$ is understood; this leads to the desired relation between $I_P$ and $I_Q$. $\endgroup$ – Andy Neitzke Jan 3 '11 at 23:18
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    $\begingroup$ Nice answer. I realize you're being vague, but I don't think it's fair to say the space $M_C$ makes no sense without stability -- it's a perfectly nice stack for $C$ abelian (or higher stack in the derived setting), just not a scheme.. $\endgroup$ – David Ben-Zvi Jan 4 '11 at 2:49
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    $\begingroup$ @David. The earliest usage that I'm aware of for walls and wall-crossing (perhaps the original coinage?) is Donaldson's 1987 paper "Irrationality and the h-cobordism conjecture", which proves a wall-crossing formula for an instanton invariant of 4-manifolds with $b^+=1$. The walls are period points for the abelian instantons which one encounters in generic 1-parameter families of conformal structures. $\endgroup$ – Tim Perutz Jan 4 '11 at 16:37

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