Musings

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 supertrace¹, $$\Omega(\gamma;u) = -\frac{1}{2} Tr_{\mathcal{H}_{\gamma,\text{BPS}}} {(-1)}^{2 J_3}(2 J_3)^2$$ 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 $T^{2r}$ fibration² 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 $g$.