This is a short note based on the paper "Surface Defects and Resolvents" by Gaiotto, Gukov and Seiberg. I will illustrate how one can compute (twisted) chiral ring relations for a supersymmetric $2d$ theory coupled to a $4d$ theory. Here the focus is on the two-dimensional aspect, and a future post will shed another light on this problem, from the four-dimensional point of view.

#### $\mathcal{N}=(2,2)$ supersymmetry in two dimensions

Let us begin with a very quick review of some basic notions of two-dimensional theories with $\mathcal{N}=(2,2)$ supersymmetry. A good place to learn about $\mathcal{N}=(2,2)$ supersymmetry is the big book on mirror symmetry, available freely here. Let us first present the basic fields, and how we can construct supersymmetric actions.

It is useful to introduce the superspace formalism : the superfields depend on the time coordinate $x^0$, the space coordinate $x^1$ and the four Grassmann coordinates $\theta^{\pm}$ and $\bar{\theta}^{pm}$. The space-time coordinates are combined into $x^{\pm} = x^0 \pm x^1$. Then we can define eight operators $$D_{\pm} = \frac{\partial}{\partial \theta^{\pm}} - i \bar{\theta}^{\pm} \partial_{\pm} \, , \qquad Q_{\pm} = \frac{\partial}{\partial \theta^{\pm}} + i \bar{\theta}^{\pm} \partial_{\pm}$$ and similar definitions for $\bar{D}_{\pm}$ and $\bar{Q}_{\pm}$. We then say that a superfield $\Phi$ is

• Chiral if $\bar{D}_{\pm} \Phi = 0$ ;
• Anti-chiral if $D_{\pm} \Phi = 0$ ;
• Twisted chiral if $\bar{D}_+ \Phi = D_- \Phi = 0$ ;
• Twisted anti-chiral if $\bar{D}_- \Phi = D_+ \Phi = 0$.

Using these superfields, we can construct supersymmetric invariant actions by combining the following terms :

• The D-terms $\int \mathrm{d}^2 x \, \mathrm{d}^4 \theta \, K (\Phi_i)$ for any superfields $\Phi_i$, where $\mathrm{d}^4 \theta = \mathrm{d} \theta^+ \mathrm{d} \theta^- \mathrm{d} \bar{\theta}^+ \mathrm{d} \bar{\theta}^-$.
• The F-terms $\int \mathrm{d}^2 x \, \mathrm{d}^2 \theta \, W (\Phi_i)$ for any chiral superfields $\Phi_i$. Here $\mathrm{d}^2 \theta = \mathrm{d} \theta^+ \mathrm{d} \theta^-$.
• The twisted F-terms $\int \mathrm{d}^2 x \, \mathrm{d}^2 \tilde{\theta} \, \tilde{W} (U_i)$ for any twisted chiral superfields $U_i$. Here $\mathrm{d}^2 \tilde{\theta} = \mathrm{d} \bar{\theta}^- \mathrm{d} \theta^+$.

#### $\mathcal{N}=(2,2)$ gauge theories

If we want to construct gauge theories, we need another ingredient, namely the vector superfield $V$, which is a real superfield that transforms as usual under gauge transformations (see section 15.2.1 in the Mirror symmetry book for the details). We consider here a $U(1)$ gauge field. The important point is that the super-field-strength that we construct from $V$ is a twisted chiral superfield $$\Sigma = \bar{D}_+ D_- V \, .$$ The Lagrangian for the gauge part is then $$\mathcal{L}_{\mathrm{gauge}} = - \frac{1}{2e^2} \int \mathrm{d}^2 x \, \mathrm{d}^4 \theta \, \bar{\Sigma} \Sigma \, .$$ The kinetic terms for the charged chiral superfields $\Phi$ has the form $$\mathcal{L}_{\mathrm{kin}} = \int \mathrm{d}^2 x \, \mathrm{d}^4 \theta \, \bar{\Phi} e^V \Phi \, .$$

Now a point which is specific to two dimensions is that we can add a linear twisted superpotential $$\tilde{W}_{\mathrm{FI} , \theta} = -t \Sigma$$ where $t=r-i \theta$ with $r$ the Fayet-Iliopoulos parameter and $\theta$ the $theta$-angle. Finally, if it is possible to form a polynomial $W(\Phi)$ which is gauge invariant, we can of course add such a (non-twisted) superpotential term in the action.

Let us consider the supersymmetric gauged linear sigma-model (GLSM) Lagrangian $$\mathcal{L}_{\textrm{GLSM}} = \mathcal{L}_{\mathrm{gauge}} + \mathcal{L}_{\mathrm{kin}} + \frac{1}{2} \left( -t \int \mathrm{d}^2 x \, \mathrm{d}^2 \tilde{\theta} \, \Sigma + c.c. \right) \, .$$ We can compute the low-energy twisted effective superpotential as follows :

1. Integrate out the massive chiral matter fields. For a field of total mass $x$, one obtains a contribution $2 \pi i \tilde{W} = - x \log (x/e)$. The total mass can in general be written as $m+\sigma$ where $m$ is the bare mass and $\sigma$ is the scalar part of the vector multiplet.
2. Extremize the total effective $\tilde{W} [\sigma , m]$ with respect to $\sigma$. The equation $\partial_{\sigma} \tilde{W} [\sigma , m] = 0$ are the twisted chiral ring relations. When solved, the solutions plugged back into $\tilde{W} [\sigma , m]$ give the twisted effective superpotential.

As a final comment, note that a GLSM can be used to realize a non-linear sigma-model on certain target Kähler manifolds. For instance,

#### Four-dimensional theory with surface defects

We now show how the procedure of the previous paragraph can be applied to the computation of the twisted effective superpotential for a four-dimensional theory with surface defects. For simplicity we consider the example of pure $\mathcal{N}=2$ $SU(2)$ gauge theory in four dimensions, with a defect that breaks $SU(2) \rightarrow U(1)$. In the UV, this can be defined as a $CP(1)$ sigma-model coupled to the pure $SU(2)$ theory. This means that the $2d$ theory contains an $SU(2)$ doublet of fields  of unit charge under some $U(1)$ gauge field, and after integrating them out, we obtain an effective twisted superpotential $$2 \pi i \tilde{W} [\sigma , \Phi] = t \sigma - \mathrm{Tr} \, \left[ (\sigma + \Phi ) \log \frac{\sigma + \Phi }{e}\right] \, .$$ It is important to note that the $4d$ fields $\Phi$ play the role of $2d$ twisted masses, and we recall that the term $t \sigma$ was introduced in the previous paragraph as the original twisted superpotential.

Now the effects of the $4d$ gauge dynamics are introduced by noting that the second derivative gives $$- 2 \pi i \partial_{\sigma}^2 \tilde{W} [\sigma , \Phi] = \mathrm{Tr} \, \left[ \frac{1}{\sigma + \Phi }\right]$$ and that this quantity, which is usually denoted $T(\sigma)$, can be computed, for instance, using the Dijkgraaf-Vafa method. Naively, if we use the classical relation $\mathrm{Tr} \, \Phi^{2k} = 2u^k$, we would say that $T(x) = 2 \sigma / (\sigma^2 - u)$. But there are instanton corrections, and the result is $$T(x) = \frac{2 \sigma}{\sqrt{(x^2 - u)^2 - 4 \Lambda^4}} \, ,$$ where $\Lambda$ is as usual the scale factor. With this explicit expression at hand, we can integrate once to obtain $\partial_{\sigma} \tilde{W} [\sigma , \Phi]$, and therefore the twisted chiral ring relations. Here, this gives $$\sigma^2 = e^t + u + \Lambda^4 e^{-t} \, .$$