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In a nutshell my question is "Are there any easy, educational wall-crossing formulae?".

Recently I often hear the word "(Kontsevich-Soibelman's etc) wall-crossing formula" in algebraic geometry talks. I wonder what they are like but am not really ready to read those relevant papers with high technology. I understand that this kind of formula expresses how your invariants of moduli spaces varies when you change the stability condition to construct the moduli spaces. Could someone provide me with a toy example? I am looking for a "wall-crossing formula" which can be understood by those with basic AG and maybe some GIT.

Thank you very much.

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If "wall-corssing" is really "wall-crossing", then you might have a look at… – Gerry Myerson Mar 2 '13 at 11:10
Thank you for pointing out the typo. I am aware of the other post but I am interested in explicit computations rather than philosophical ideas. Thank you anyways. – Kim Mar 2 '13 at 11:24
up vote 20 down vote accepted

I can tell you the gist of a "wall crossing formula". Typically you have a space of parameters $\newcommand{\eS}{\mathscr{S}}$ $\Lambda$, a configuration space $\newcommand{\eC}{\mathscr{C}}$ $\eC$, and a parametrized moduli space $\newcommand{\eM}{\mathscr{M}}$ $\eM$ which is a subvariety $\eM\subset \eC\times \Lambda$.

We have a natural projection

$$\pi:\eM\to \Lambda. $$

The fiber of this map over $\lambda\in\Lambda$, denoted by $\eM_\lambda$, is called the moduli space corresponding to the parameter $\lambda$.

One could associate to $\eM_\lambda$ various invariants. For example, there might exists a sheaf $\eS\to\eM$ which restricts to a sheaf $\eS_\lambda\to\eM_\lambda$. We denote by $e(\lambda)$ the Euler characteristic of $H^\bullet(\eM_\lambda,\eS_\lambda)$.

It could happend that $e(\lambda)$ depends on $\lambda$, but it could depend in a rather specal way. Namely, there could exists real codimension one subvarieties $W_i\subset \Lambda$, $i\in I$, called walls, so that, if

$$ W:=\bigcup_{i\in I} W_i, $$

then $\lambda\to e(\lambda)$ is continuous on $\Lambda^\ast:=\Lambda\setminus W$. In particular, the function $\lambda\to e(\lambda)$ is constant on the connected components of $\Lambda^*$, which are called chambers.

If two chambers $C_0, C_1$ are adjacent, i.e., they sit on opposite sides of a wall $W_i$ , then $e(\lambda)$ has constant values $e(C_0)$ and $e(C_1)$ in these two chambers and a wall crossing formula will tell you what the difference $e(C_1)-e(C_0)$ is.

Here is a trivial example. Let $\Lambda=\mathbb{R}^2$. We let $(b,c)$ denote the coordinates of a point in $\Lambda$. The configuration space $\eC$ is $\mathbb{R}$ and the parametrized moduli space is

$$\eM= \bigl\lbrace (t,b, c)\in\eC \times \Lambda;\;\;t^2+bt+c=0\,\bigr\rbrace. $$

The moduli space $\eM_{b,c}$ can be identified with the set of real roots of the quadratic polynomial $t^2+bt+c$. We denote by $e(b,c)$ the number of such roots. In other words, $e(b,c)$ is the (topological) Euler characteristic of the space $\eM_{b,c}$.

In this case we have a single wall

$$ W=\bigl\lbrace (b,c)\in \Lambda;\;\; b^2- 4c=0\bigr\rbrace, $$

with two chambers,

$$ C^\pm=\bigl\lbrace \;\pm(b^2-4c)>0\;\bigr\rbrace. $$

In this case the Wall crossing formula is

$$ e(C^+)-e(C^-)= 2. $$

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A simple but very nice example!! Thank you, Liviu!! – Kim Mar 2 '13 at 23:25
Liviu, I beg you pardon, it seems to me that the example you proposed tells us more or less how to solve quadratic equations with real coefficients. You can call this wall-crossing, but this sounds a little bit like giving a new name to a relatively old thing. For me this does not quite answer the question asked by Kim... I suspect that Kontsevich and Soibelman made some more substantial progress? (though I might be wrong...) – aglearner Mar 3 '13 at 22:54
The point of the concrete example is obviously not the final formula. It's the explicit description of the set-up. It is based on three pilars that appear in all situations: the configuration space, the parameter space, and the parametrized moduli space. In the applications I am familiar with (gauge theory) all of them are infinite dimensional. However the fibers of $\pi$ are finite dimensional. The 20th century terminology for the wall crossing formula was bifurcation theory. – Liviu Nicolaescu Mar 4 '13 at 10:16

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