Wall-Crossing in Coupled 2d-4d Systems

Abstract

We introduce a new wall-crossing formula which combines and generalizes the Cecotti-Vafa and Kontsevich-Soibelman formulas for supersymmetric 2d and 4d systems respectively. This 2d-4d wall-crossing formula governs the wall-crossing of BPS states in an N=2 supersymmetric 4d gauge theory coupled to a supersymmetric surface defect. When the theory and defect are compactified on a circle, we get a 3d theory with a supersymmetric line operator, corresponding to a hyperholomorphic connection on a vector bundle over a hyperkahler space. The 2d-4d wall-crossing formula can be interpreted as a smoothness condition for this hyperholomorphic connection. We explain how the 2d-4d BPS spectrum can be determined for 4d theories of class S, that is, for those theories obtained by compactifying the six-dimensional (0,2) theory with a partial topological twist on a punctured Riemann surface C. For such theories there are canonical surface defects. We illustrate with several examples in the case of A_1 theories of class S. Finally, we indicate how our results can be used to produce solutions to the A_1 Hitchin equations on the Riemann surface C.

Figures (42)

• Figure 1: The definition of $A(\sphericalangle)$ as a product of ${\cal S}$-factors associated to the BPS rays $\ell_{ij}$.
• Figure 2: A partial depiction of the groupoid ${\mathbb{V}}$, in case ${\cal V}$ has three elements, here labeled $i$, $j$, $k$. Each morphism space is a torsor for $\Gamma$, of which we have shown only a single element.
• Figure 3: The configuration of BPS rays which participate in the wall-crossing formula \ref{['eq:example-4']}. As we approach the wall, all of the rays shown here become aligned. On the other side of the wall their ordering is reversed.
• Figure 4: The Riemann surface $C$, together with a basis of cycles generating $\Gamma_u$, for $u$ in the weak coupling domain. We choose a square root $x \sim \sqrt{-2u}$ on the complement of the cuts. Paths on this sheet are drawn with solid lines while the paths on the sheet with $x \sim -\sqrt{-2u}$ are drawn with dashed lines. Wiggly orange lines denote branch cuts emanating from the branch points $t_\pm$.
• Figure 5: The $u$-plane for $SU(2)$ Seiberg-Witten theory. The paths ${\cal P }_{\pm \Lambda^2}$ are based at $u$ in the weak-coupling regime. Define ${\cal P }_{\infty}$ as the path $u\to e^{2\pi {\mathrm i}} u$. It is homotopic to the path given by traversing first ${\cal P }_{-\Lambda^2}$ and then ${\cal P }_{\Lambda^2}$.