Let us now move on to the construction of the Lagrangian with $\mathcal{\mathcal{N}}=2$ supersymmetry. An $\mathcal{\mathcal{N}}=2$ supersymmetric theory is in particular an $\mathcal{\mathcal{N}}=1$ supersymmetric theory. Therefore it is convenient to use $\mathcal{\mathcal{N}}=1$ superﬁelds to describe $\mathcal{\mathcal{N}}=2$ systems. For this purpose let us quickly recall the $\mathcal{\mathcal{N}}=1$ formalism. In this section only, we distinguish the imaginary unit by writing it as $i$.

An $\mathcal{\mathcal{N}}=1$ vector multiplet consists of a Weyl fermion ${\lambda}_{\alpha}$ and a vector ﬁeld ${A}_{\mu}$, both in the adjoint representation of the gauge group $G$. We combine them into the superﬁeld ${W}_{\alpha}$ with the expansion

$${W}_{\alpha}={\lambda}_{\alpha}+{F}_{\left(\alpha \beta \right)}{\mathit{\theta}}^{\beta}+D{\mathit{\theta}}_{\alpha}+\cdots $$ | (2.1.1) |

where $D$ is an auxiliary ﬁeld, again in the adjoint of the gauge group. ${F}_{\alpha \beta}=\frac{i}{2}{\sigma}^{\mu}{\phantom{\rule{0.0pt}{0ex}}}_{\stackrel{\u0307}{\gamma}}^{\beta}{\stackrel{\u0304}{\sigma}}^{\nu}{\phantom{\rule{0.0pt}{0ex}}}_{\alpha}^{\stackrel{\u0307}{\gamma}}{F}_{\mu \nu}$ is the anti-self-dual part of the ﬁeld strength ${F}_{\mu \nu}$.

The kinetic term for a vector multiplet is given by

$$\int {d}^{2}\mathit{\theta}\frac{-i}{8\pi}\tau tr\phantom{\rule{0.3em}{0ex}}{W}_{\alpha}{W}^{\alpha}+cc.$$ | (2.1.2) |

where

$$\tau =\frac{4\pi i}{{g}^{2}}+\frac{\mathit{\theta}}{2\pi}$$ | (2.1.3) |

is a complex number combining the inverse of the coupling constant and the theta angle. We call it the complexiﬁed coupling of the gauge multiplet. Expanding in components, we have

We use the convention that $tr\phantom{\rule{0.3em}{0ex}}{T}^{a}{T}^{b}=\frac{1}{2}{\delta}^{ab}$ for the standard generators of gauge algebras, which explain why we have the factors $1\u2215\left(2{g}^{2}\right)$ in front of the gauge kinetic term. The $\mathit{\theta}$ term is a total derivative of a gauge-dependent term. Therefore, it does not aﬀect to perturbative computations. It does aﬀect non-perturbative computations, to which we will come back later.

An $\mathcal{\mathcal{N}}=1$ chiral multiplet $Q$ consists of a complex scalar $Q$ and a Weyl fermion ${\psi}_{\alpha}$, both in the same representation of the gauge group. In terms of a superﬁeld we have

$$Q=Q{\left|\right.}_{\mathit{\theta}=0}+2{\psi}_{\alpha}{\mathit{\theta}}^{\alpha}+F{\mathit{\theta}}_{\alpha}{\mathit{\theta}}^{\alpha}$$ | (2.1.5) |

where $F$ is auxiliary. The coeﬃcient 2 in front of the middle component is unconventional, but this choice allows us to remove various annoying factors of $\sqrt{2}$ appearing in the formulas later. The chiral multiplet ${Q}_{1,\dots}$ can be in an arbitrary complex representation $R$ of the gauge group $G$. The kinetic term is then

where $V$ is the vector superﬁeld, ${\rho}_{a}{\phantom{\rule{0.0pt}{0ex}}}_{j}^{i}$ is the matrix representation of the gauge algebra, and $W\left(Q\right)$ is a gauge invariant holomorphic function of ${Q}_{1,\dots}$.

The supersymmetric vacua is obtained by demanding that the supersymmetry transformation of various ﬁelds are zero. The nontrivial conditions come from

$$\delta {\lambda}_{\alpha}=0,\phantom{\rule{2em}{0ex}}\delta {\psi}_{\alpha}=0$$ | (2.1.7) |

which give

$${D}_{a}=0,\phantom{\rule{1em}{0ex}}{F}_{i}=0.$$ | (2.1.8) |

By solving the algebraic equations of motion of the auxiliary ﬁelds, we ﬁnd

$${Q}_{\stackrel{\u0304}{j}}^{\u2020}{\rho}_{a}^{\stackrel{\u0304}{j}i}{Q}_{i}=0,\phantom{\rule{1em}{0ex}}\frac{\partial W}{\partial {Q}_{i}}=0.$$ | (2.1.9) |

An $\mathcal{\mathcal{N}}=2$ vector multiplet consists of the following $\mathcal{\mathcal{N}}=1$ multiplets, both in the adjoint of the gauge group $G$:

Here, the horizontal arrows signify the $\mathcal{\mathcal{N}}=1$ sub-supersymmetry generator manifest in the $\mathcal{\mathcal{N}}=1$ superﬁeld formalism, and the slanted arrows are for the second $\mathcal{\mathcal{N}}=1$ sub-supersymmetry.

One easy way to construct the second supersymmetry action is to demand that the theory is symmetric under the $SU\left(2\right)$ rotation acting on ${\lambda}_{\alpha}$ and ${\stackrel{\u0303}{\lambda}}_{\alpha}$. A symmetry which does not commute with the supersymmetry generators is called an R-symmetry in general. Therefore this $SU\left(2\right)$ symmetry is often called the $SU{\left(2\right)}_{R}$ symmetry. It is by now a standard technique to combine the supersymmetry manifest in a superﬁeld formalism and an R-symmetry to construct a theory with more supersymmetries, see e.g. [30] for an application. It is also to be kept in mind that there can be and indeed are $\mathcal{\mathcal{N}}=2$ supersymmetric theories without $SU{\left(2\right)}_{R}$ symmetry: there can just be two sets of supersymmetry generators without $SU\left(2\right)$ symmetry relating them, see e.g. [31, 32]. That said, for simplicity, we only deal with $\mathcal{\mathcal{N}}=2$ supersymmetric systems with $SU{\left(2\right)}_{R}$ symmetry in this lecture note.

The Lagrangian is then

The ratio between the prefactors of the Kähler potential and of the gauge kinetic term is ﬁxed by demanding $SU{\left(2\right)}_{R}$ symmetry.

An $\mathcal{\mathcal{N}}=2$
hypermultiplet^{4}
consists of the following ﬁelds:

They are both in the same representation $R$ of the gauge group. Therefore, the $\mathcal{\mathcal{N}}=1$ chiral multiplets $Q$ and $\stackrel{\u0303}{Q}$ are in the conjugate representations of the gauge group. We demand again that the theory is symmetric under the $SU\left(2\right)$ rotation acting on $Q$ and ${\stackrel{\u0303}{Q}}^{\u2020}$, to have $\mathcal{\mathcal{N}}=2$ supersymmetry.

For deﬁniteness, let us consider $G=SU\left(N\right)$ and ${N}_{f}$ hypermultiplets ${Q}_{i}^{a}$, ${\stackrel{\u0303}{Q}}_{a}^{i}$ in the fundamental $N$-dimensional representation, where $a=1,\dots ,N$ and $i=1,\dots ,{N}_{f}$. This set of ﬁelds is often called ${N}_{f}$ ﬂavors of fundamentals of $SU\left(N\right)$. The gauge transformation acts on them as

$${Q}_{i}\to {e}^{\Lambda}{Q}_{i},\phantom{\rule{2em}{0ex}}{\stackrel{\u0303}{Q}}^{i}\to {\stackrel{\u0303}{Q}}^{i}{e}^{-\Lambda}$$ | (2.1.13) |

where $\Lambda $ is a traceless $N\times N$ matrix of chiral superﬁelds; the gauge indices are suppressed.

The Lagrangian for the hypermultiplets is

where the gauge index $a$ is suppressed again. The existence of $SU{\left(2\right)}_{R}$ symmetry ﬁxes the ratio of $c$ and ${c}^{\prime}$: it can be done e.g. by comparing the coeﬃcients of ${Q}^{i}\lambda \psi $ from the ﬁrst term and of ${\stackrel{\u0303}{Q}}^{i}\stackrel{\u0303}{\lambda}\psi $ from the second term. We ﬁnd the choice $c={c}^{\prime}$ does the job. In the following we take $c={c}^{\prime}=1$ unless otherwise mentioned. The $SU{\left(2\right)}_{R}$ symmetry also demands that the mass term ${\mu}_{j}^{i}$ satisﬁes $\left[\mu ,{\mu}^{\u2020}\right]=0$. Then $\mu $ can be diagonalized, and consequently the mass term is often written as

$$\sum _{i}\int {d}^{2}\mathit{\theta}{\mu}_{i}{\stackrel{\u0303}{Q}}^{i}{Q}_{i}+cc.$$ | (2.1.15) |

As another example, let us consider the case when we have a hypermultiplet $\left(Z,\stackrel{\u0303}{Z}\right)$ in the adjoint representation, i.e. they are both $N\times N$ traceless matrices. The following discussion can easily be generalized to arbitrary gauge group too. When the hypermultiplet is massless, the total Lagrangian has the form

$$\begin{array}{cc}& \int {d}^{2}\mathit{\theta}\frac{-i}{8\pi}\tau tr\phantom{\rule{0.3em}{0ex}}{W}_{\alpha}{W}^{\alpha}+cc.+\frac{Im\tau}{4\pi}\int {d}^{4}\mathit{\theta}tr\phantom{\rule{0.3em}{0ex}}{\Phi}^{\u2020}{e}^{\left[V,\cdot \right]}\Phi \\ & +\frac{Im\tau}{4\pi}\int {d}^{4}\mathit{\theta}\left({Z}^{\u2020}{e}^{\left[V,\cdot \right]}Z+\stackrel{\u0303}{Z}{e}^{-\left[V,\cdot \right]}{\stackrel{\u0303}{Z}}^{\u2020}\right)+\frac{Im\tau}{4\pi}\int {d}^{2}\mathit{\theta}\stackrel{\u0303}{Z}\left[\Phi ,Z\right]+cc.& \text{(2.1.16)}\end{array}$$where we made a diﬀerent choice of $c={c}^{\prime}$ in (2.1.14). This Lagrangian clearly has $SU{\left(3\right)}_{F}$ ﬂavor symmetry rotating $\Phi $, $Z$ and $\stackrel{\u0303}{Z}$. This commutes with the $\mathcal{\mathcal{N}}=1$ supersymmetry manifest in the superﬁeld formalism. We also know that this theory has an $SU{\left(2\right)}_{R}$ symmetry rotating $Z$ and ${\stackrel{\u0303}{Z}}^{\u2020}$. These two symmetries $SU{\left(3\right)}_{F}$ and $SU{\left(2\right)}_{R}$ does not commute: we ﬁnd that there is an $SO{\left(6\right)}_{R}$ symmetry, acting on

$$Re\Phi ,Im\Phi ,ReZ,ImZ,Re\stackrel{\u0303}{Z},Im\stackrel{\u0303}{Z}.$$ | (2.1.17) |

Note that $SO{\left(6\right)}_{R}$ can also be regarded as $SU{\left(4\right)}_{R}$, as $SO\left(6\right)$ and $SU\left(4\right)$ have the same Lie algebra. Then the $SU{\left(4\right)}_{R}$ symmetry acts on the four Weyl fermions

$$\lambda ,\stackrel{\u0303}{\lambda},\psi ,\stackrel{\u0303}{\psi}$$ | (2.1.18) |

in the system, where $\lambda $ and $\stackrel{\u0303}{\lambda}$ are in the $\mathcal{\mathcal{N}}=2$ vector multiplet, and $\psi $, $\stackrel{\u0303}{\psi}$ are in the $\mathcal{\mathcal{N}}=2$ hypermultiplet. We conclude that this system has in fact $\mathcal{\mathcal{N}}=4$ supersymmetry, whose four supersymmetry generators are acted on by $SU{\left(4\right)}_{R}\simeq SO{\left(6\right)}_{R}$. The argument here is another application of the combination of the manifest and non-manifest symmetries in the superﬁeld formalism.

We can add the mass term $\int {d}^{2}\mathit{\theta}\mu Z\stackrel{\u0303}{Z}+cc.$ to (16). This preserves the $\mathcal{\mathcal{N}}=2$ supersymmetry but it breaks $\mathcal{\mathcal{N}}=4$ supersymmetry. The resulting theory is sometimes called the $\mathcal{\mathcal{N}}={2}^{\ast}$ theory.

Before closing this section, we should mention the concept of half-hypermultiplet. Let us start from a full hypermultiplet $\left({Q}_{a},{\stackrel{\u0303}{Q}}^{a}\right)$ so that ${Q}_{a}$ and ${\stackrel{\u0303}{Q}}^{a}$ are in the representations $R$, $\stackrel{\u0304}{R}$, respectively. When $R$ is pseudo-real, or equivalently when there is an antisymmetric invariant tensor ${\mathit{\epsilon}}_{ab}$, we can impose the constraint

$${Q}_{a}={\mathit{\epsilon}}_{ab}{\stackrel{\u0303}{Q}}^{b}$$ | (2.1.19) |

compatible with $\mathcal{\mathcal{N}}=2$ supersymmetry, which halves the number of degrees of freedom in the multiplet. The resulting multiplet is called a half-hypermultiplet in the representation $R$. We will come back to this in Sec. 7.2.