### Wall Crossing

Many of the magical result in string theory and supersymmetric gauge theories have been motivated by consideration of the spectrum of BPS states in these theories. For 16 supercharges ($\mathcal{N}=4$), this spectrum varies continuously as one moves through the moduli space. This fact is at the heart of the S-duality conjectures. For 8 supercharges ($\mathcal{N}=2$ supersymmetry), the spectrum of BPS states jumps discontinuously, as one crosses “walls of marginal stability”. As a consequence, there is usually no manifest S-duality for $\mathcal{N}=2$ theories.

Describing exactly *how* the spectrum of BPS states jumps is, however, a complicated business. and a lot of effort has gone into deriving “wall crossing formulæ.” Recently considerable progress was, apparently (I say “apparently” because the paper has not yet appeared on the arXivs), made by Kontsevich and Soibelman, who proposed a wall-crossing formula for $\mathcal{N}=2$ Seiberg-Witten Theory. Gaiotto, Moore and Neitzke provide a beautiful physical explanation for Kontsevich and Soibelman’s result.

Consider a $d=4$, $\mathcal{N}=2$ supersymmetric gauge theory, with gauge group $G$, of rank $r$. The vector multiplet moduli space, $\mathcal{M}_v$, is a complex manifold of dimension $r$, parametrizing the vacuum expectation values of the scalars in the vector multiplet. At a generic point on $\mathcal{M}_v$, the gauge group is broken to the maximal torus, ${U(1)}^r$ and there is a $2r$-dimensional lattice, $\Gamma$ of allowed electric and magnetic charges. $\langle \cdot, \cdot\rangle: \Gamma\otimes \Gamma \to \mathbb{Z}$ is a nondegenerate skew-bilinear pairing. The lattice is fibered over $\mathcal{M}_v$, degenerating over some divisor, $D$, where some BPS particles become massless. Over ${\mathcal{M}_v\setminus D}$, $\Gamma\otimes \mathbb{C}$ is a flat symplectic vector bundle, with monodromies as one circles $D$. There is a holomorphic section, $Z(u)$ of $\Gamma^*\otimes \mathbb{C}$, such that the central charge of a state of charge $\gamma\in\Gamma$ is $Z_\gamma(u) = Z(u)\cdot \gamma$

$Z(u)$ determines both the structure of the low energy effective Lagragian for the massless sector and the masses of the BPS states. Locally, we can choose a symplectic basis, $(\alpha^i, \beta_i)$, $i=1,\dots,r$, for $\Gamma$. The special coordinates $a^i(u) = Z_{\beta_i}(u)$ are the electric central charges and one can define a holomorphic prepotential, $\mathcal{F}(a^i)$, such that the magnetic central charges $Z_{\alpha^i}(u) = \frac{\partial \mathcal{F}}{\partial a^i}$ The holomorphic gauge couplings $\tau_{i j}(u) = \frac{\partial^2\mathcal{F}}{\partial a^i \partial a^j}$ and the Kähler potential on $\mathcal{M}_v$ is $K = - Im\left(\overline\overline{a}^i \frac{\partial \mathcal{F}}{a^i}\right)$

The BPS bound on the masses of one-particle states of charge $\gamma\in \gamma$,
$M_\gamma \geq |Z_\gamma|$
is saturated by the BPS states. These states form a vector space, $\mathcal{H}_{\gamma,\text{BPS}}$. Understanding how $\mathcal{H}_{\gamma,\text{BPS}}$ varies, as one varies $u\in \mathcal{M}_v$ is the central object of study in this subject. It helps to define the helicity supertrace^{1},
$\Omega(\gamma;u) = -\frac{1}{2} Tr_{\mathcal{H}_{\gamma,\text{BPS}}} {(-1)}^{2 J_3}(2 J_3)^2$
The wall-crossing formula expresses how $\Omega(\gamma;u)$ changes as one crosses a wall of marginal stability. Such a wall arises whenever
$arg(Z_{\gamma_1}(u)) = arg(Z_{\gamma_2}(u))$
for linearly independent $\gamma_1,\gamma_2\in \Gamma$.

Kontsevich and Soibelman define a Lie algebra, with generators $e_\gamma$, $\gamma\in\Gamma$, $[e_{\gamma_1},e_{\gamma_2}] = {(-1)}^{\langle\gamma_1,\gamma_2\rangle}\langle\gamma_1,\gamma_2\rangle e_{\gamma_1+\gamma_2}$ Define the group element $U_\gamma = exp \sum_{n=1}^\infty \tfrac{1}{n^2} e_{n\gamma}$ To each BPS particle, associate a ray in the complex plane, determined by its central charge, $\ell_\gamma = Z_\gamma(u)/\mathbb{R}_-$ Near a wall of marginal stability form the product of all of the $U_\gamma$, corresponding to BPS particles whose central charges become aligned at the wall

where the product is taken with a clockwise ordering of the rays, $\ell_\gamma$.

When we cross a wall of marginal stability, the ordering of the factors in (1) changes, and the $\Omega(\gamma;u)$ jump. The wall-crossing formula (for the jump in $\Omega(\gamma;u)$) of Kontsevich and Soibelman is the statement that $A$ is unchanged.

What Gaiotto *et al* do is provide a physical explanation of (and computational check on ) (1).

Compactify on $\mathbb{R}^3\times S^1$, where the $S^1$ has radius $R$. As shown by Seiberg and Witten, the low energy theory is a $d=3$, $\mathcal{N}=4$ supersymmetric $\sigma$-model, with hyperkähler target space $(\mathcal{M},g)$.
$\mathcal{M}$ is a $T^{2r}$ fibration^{2} over $\mathcal{M}_v$.

The hyperkähler metric, $g$, receives instanton corrections due to BPS particles whose (Euclidean) worldlines wrap the $S^1$. When a BPS particle disappears from the spectrum, so does the corresponding 1-instanton contribution to $g$. It must be replaced by suitable multi-instanton contributions, such that the hyperkähler metric, $g$, is smooth. The wall-crossing formula (1) turns out to be just what is needed to ensure the smoothness of $g$.

^{1} $\Omega(\gamma;u)=+1$ for a hypermultiplet, and $\Omega(\gamma;u)=-2$ for a “short” massive vector multiplet.

^{2} $r$ periodic scalars come from the Wilson lines

around the $S^1$. $r$ additional periodic scalars come from dualizing the $r$ abelian gauge fields on $\mathbb{R}^3$.