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Getting around wall-crossing. What is a "wall" really?

+ 3 like - 0 dislike

What follows comes from this paper of Gottsche and Zwegers

Consider a 4-manifold $X$ with $b_{+}=1$ and a Riemannian metric $g$. Then, $g$ determines a ray in 
H^2(X,\mathbb{R})^{+} = \{ H \in H^2(X,\mathbb{R})\, |\, H^2>0 \}.
That is, $H^2(X,\mathbb{R})^{+}$ is the space of self-dual harmonic forms (and I think a common name is also "the positive cone"). A representative of this ray is called period point of $X$ and is denoted by $\omega(g)$ (and I have the impression that if $X$ is symplectic this is the symplectic form, or if Kahler, this is the Kahler form). Now, the quotient space
has two connected components. Why? Choosing an orientation amounts to choosing one of the two components and we will denote the oriented space as $\mathbb{H}_X$. 

Apparently $\mathbb{H}_X$ is a hyperbolic space $r$-space with $r=b_2(X)-1$. $\mathbb{H}_X$. Why is this so? $b_2(X)$ is just the Betti number of the second cohomology group, i.e. the dimension of the second cohomology group of $X$ (and I think it is sufficient to consider de Rham cohomology here).

Coming to the main point of this question we can define what a wall in $\mathbb{H}_X$ is. By a wall in $\mathbb{H}_X$ we mean the intersection of $\mathbb{H}_X$ with the set

$$ W^{\xi} = \{ L \in H^2(X,\mathbb{R}^+) \, | \, \xi L = 0  \} / \mathbb{R}^+ $$

where $\xi \in H^2(X,\mathbb{Q})$ and $\xi^2<0$. Now, this seems some kind of orthogonality condition. I do not have though any clue why these 2-forms are evaluated in different fields. Why the latter require the rationals?  

@UrsSchreiber Is there a physical pictures of what these walls are? Usually, we consider moduli spaces of gauge theories and these walls, also called "walls of marginal stability" if I am not mistaken are places in the moduli stack of connections where the stability of the vector bundle is non trivial. Still, is there some other picture where these cohomology classes could represent for example some sort of charges? 

asked May 25, 2016 in Mathematics by conformal_gk (3,535 points) [ no revision ]

1 Answer

+ 5 like - 0 dislike

The story is about a compact oriented $4$-manifold $X$. In such case, the degree two cohomology $H^2(X,\mathbb{R})$ has a natural non-degenerate symmetric bilinear pairing, given by Poincaré duality. In terms of differential forms, it is simply

$\alpha.\beta= \int_X \alpha \wedge \beta$

and in terms of homology, it is simply the intersection of 2-cycles inside X (in a 4-manifods, two 2-cycles intersect generally in a bunch of points and the number of these intersection points is the value of the pairing between these two cycles). A real non-degenerate symmetric pairing is completely determined by its signature. In the case of $H^2(X,\mathbb{R})$, this signature is denoted by $(b_+,b_-)$ where $b_+$ is the number of positive signs and $b_-$ is the number of negative signs. Of course, $b_+ + b_- = b_2(X)$, the dimension of the total space $H^2(X,\mathbb{R})$.

So the assumption $b_+=1$ means that the signature is $(+,-,...,-)$ with $r=b_2(X)-1$ minus signs. In other words, the space $H^2(X,\mathbb{R})$ equipped with the pairing exactly look like the Minkowski space of special relativity. In particular, one has a "light cone", where $\alpha.\alpha=0$, the interior of the cone is exactly $H^2(X,\mathbb{R})^+$, with two components corresponding to the "past and future cones". Dividing by $\mathbb{R}^+$ is modding out by "time translation". For each of the components, if one has in mind the image of a cone in dimension three (case $r=2$), this is equivalent to intersecting the interior of the cone with an horizontal plane, and one obtains one disk. In higher dimensions, one obtains a ball of dimension $r$, which is exactly $\mathbb{H}_X$. One standard construction of the hyperbolic space of dimension $r$ is as one component of the paraboloid $\alpha.\alpha=1$ in a Minkowski space of dimension $r+1$, which is exactly our setting. One obtain an identification of the hyperbolic space with $\mathbb{H}_X$ by projection of the hyperboloid on a section of the cone. It is the standard way to go from the paraboloid model to the disk model of the hyperbolic space. In paticular, say for $r=2$, $\mathbb{H}_X$ is really a disk and one shoud think to this disk as the disk model of the hyperbolic plane.

One can think of $\mathbb{H}_X$ as a parameter space for metrics, each Riemannian metric on $X$ determines a point in $\mathbb{H}_X$, via the decomposition of 2-foms in self dual and anti self dual parts with respect to the metric. More precisely, the point in $\mathbb{H}_X$ is the unique self dual class with respect to the metric. For a Kähler metric, this is indeed the Kähler class.

The walls are walls for Donaldson or equivalently Seiberg-Witten invariants. Donaldson theory is about count of instantons on $X$, i.e. count of anti self dual connections on $X$ with respect to some gauge group $G$. The notion of instanton depends on the metric of $X$, but as the "count" is given by some index of some elliptic operator, one expects these numbers to be constant under continuous deformation of the metric. This is always true if $b_+>1$ and almost true of $b_+=1$. The problems comes from the existence of abelian instantons, i.e. configurations of $U(1)$ gauge fields with anti self dual curvature. Such $U(1)$ gauge fields lives on some line bundle $L$. For a metric defining a point $\omega$ in $\mathbb{H}_X$, the self-dual part of the curvature $F$ of the $U(1)$ connection is $\omega.F$. So we have an anti sel dual connection exactly when $\omega.F=0$ i.e. when $\omega.c_1(L)=0$ where $c_1(L)$ is the first Chern class of $L$ (which is the class of $F$ up to an easy constant). So the walls in $\mathbb{H}_X$ i.e. the $\omega$ in $\mathbb{H}_X$ such that for the corresponding metric there exists some abelian instantons, are exactly the $\omega$ such that there exists $L$ line bundle such that $\omega.c_1(L)=0$. Remark that $c_1(L) \in H^2(X,\mathbb{Z})$ ("flux quantization"). In particular, there are at most countably many walls dividing $\mathbb{H}_X$ into chambers. In each of these chambers, the Donaldson invariants are constant but when one cross a wall, a $G$-instanton can absorb or emit an abelian instanton and so the number of $G$-instantons jumps hence non-trivial wall-crossing.

There are many places in mathematics and physics where there are "walls", "wall crossing", "stability" and the precise meaning of the terms depends on the precise story. If the last part of the question is about the relation between the wall crossing of Donaldson invariants and other kinds of wall-crossing, then one should be more precise about the latter.

answered May 26, 2016 by 40227 (4,660 points) [ revision history ]

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