Surface Defects and Resolvents

Abstract

We study a large class of BPS surface defects in 4d N=2 gauge theories. They are defined by coupling a 2d N=(2,2) gauged linear sigma model to the 4d bulk degrees of freedom. Our main result is an efficient computation of the effective twisted superpotential for all these models in terms of a basic object closely related to the resolvent of the 4d gauge theory, which encodes the curve describing the 4d low energy dynamics. We reproduce and extend the results of brane constructions and compute the effective twisted superpotential for general monodromy surface defects. We encounter novel, puzzling field theory phenomena in the low energy dynamics of the simplest surface defects and we propose some local models to explain them. We also study in some detail the behavior of surface defects near monopole points of the bulk theory's Coulomb branch. Finally, we explore the effect on the defect of breaking the bulk supersymmetry from N=2 to N=1 and show that certain quantities are independent of this breaking.

Figures (8)

• Figure 1: The basic ingredients of brane engineering. We depict the directions transverse to the four-dimensional space-time only. A system of D4 branes ending on NS5 branes gives rise to the four-dimensional degrees of freedom. Extra D2 branes ending on the NS5 brane add two-dimensional degrees of freedom. The $4-4'$ strings become $4d$ hypers. The $2-4$ and $2-4'$ strings become $2d$ chirals of opposite $2d$ flavor charge. The three types of fields are coupled by a cubic superpotential.
• Figure 2: The basic surface defect in pure $SU(2)$. The two D4 branes engineer the bulk gauge theory, while the D2 brane ending on the leftmost NS5 brane engineers the chiral doublet. a) If we end the D2 brane on another NS5 brane far from the system in the $x^7$ direction (not depicted), we arrive at the full $\mathbb{CP}^1$ GLSM. We indicate the geometric directions corresponding to $m$ (i.e. $\sigma$) and to $t$ (i.e. $\partial_m {\cal W}$). Upon lifting to M-theory, the NS5 branes and D4 branes merge into an M5 brane wrapping the curve in the $(\sigma, t)$ coordinates. The D2 brane becomes an M2 brane ending on the smooth M5 brane and can move from one NS5 to the other. b) The second branch of $T(\sigma)$ corresponds to the D2 brane ending on the rightmost NS5 brane. The brane construction provides an alternative description in terms of a chiral doublet of opposite $2d$ flavor charge.
• Figure 3: The basic surface defect in pure $SU(N)$ (here we depict $N=6$). The two branches of $T(\sigma)$ corresponds to the D2 brane ending on either NS5 brane. The brane construction provides an alternative description of the second branch in terms of a chiral anti-fundamental of opposite $2d$ flavor charge as the original chiral fundamental.
• Figure 4: The basic surface defect in $SU(6)$$N_f=4$ SQCD. The extra semi-infinite D4 branes on the right add the bulk flavors to the gauge theory. a) If the D2 brane ends on the leftmost NS5 brane, we have the usual set of $2d$ fundamental chirals of $2d$ flavor charge $1$. b) The second branch of $T(\sigma)$ corresponds to the D2 brane ending on the rightmost NS5 brane. The brane construction provides an alternative description in terms of a chiral $SU(6)$ anti-fundamental of $2d$ flavor charge $-1$, a chiral fundamental of the $U(4)$ flavor group and a superpotential coupling to the bulk hypermultiplets.
• Figure 5: Another simple surface defect in $SU(6)$$N_f=4$ SQCD. Moving the semi-infinite D4 brane from the right to the left does not change the bulk theory. It adds an extra $2d$ chiral to the description of the basic surface defect a), and removes it from the dual description on the second branch b).