7.2 Hypermultiplets revisited

Let us revisit the structure of the full and half hypermultiplets introduced in Sec. 2.1.2 from the viewpoints here. First, let us recall the types of irreducible representations of compact groups:

In a non-supersymmetric theory with a number of real scalars ${\varphi }^i$, $i=1,\dots ,n$, they can have an action of the flavor symmetry $F$ or the gauge symmetry group $G$ if there are real $n\times n$ matrices $T^a$, $a=1,\dots ,dimG$ representing the Lie algebra of $G$:

This representation clearly has an invariant symmetric tensor ${\delta }_{ij}$ as it acts on $n$ real scalars with a kinetic term ${\delta }_{ij}{\partial }_{\mu }{\varphi }^i{\partial }_{\mu }{\varphi }^j$. The representation is therefore strictly real.

In an $\mathcal{𝒩}=1$ supersymmetric theory with a number of real scalars ${\varphi }^i$, $i=1,\dots ,n$ together with $n∕2$ Weyl fermions, the supersymmetry requires existence of a matrix $I$ with $I^2=-1$. The actions of the flavor symmetry $F$ and the gauge symmetry $G$ need to commute with this matrix $I$:

We can declare that a complex number $a+bi$ acts on the real scalars by the matrix $a+bI$. Then the space of scalars becomes a complex vector space, and the symmetries act on them preserving the complex structure. So there are $m=n∕2$ chiral multiplets ${\Phi }^s$, $s=1,\dots ,m$, and both $F$ and $G$ are represented on them in terms of $m\times m$ complex matrices representing their Lie algebras. We can summarize the situation in the following way:

In an $\mathcal{𝒩}=2$ supersymmetric theory with a number of real scalars ${\varphi }^i$, $i=1,\dots ,n$ together with $n∕2$ Weyl fermions, the supersymmetry requires existence of matrices $I$, $J$, $K$ with the commutation relations (7.1.9). The actions of the flavor symmetry $F$ and the gauge symmetry $G$ need to commute with $I$, $J$, $K$:

We can declare that a quaternion $a+bi+cj+dk$ acts on the real scalars by the matrix $a+bI+cJ+dK$. Then the space of scalars becomes a quaternionic vector space, and the symmetries act on them preserving the quaternion structure. This requires $n$ to be automatically a multiple of four, $n=4\ell$ Both $F$ and $G$ are represented on them in terms of $\ell \times \ell$ quaternion matrices representing their Lie algebras. Summarizing, we have

where $ℍ$ is the skew-field of quaternions.

As quaternions are not quite common among physicists, we usually just use $a+bI$ to think of the real scalars as complex scalars. Then we have a complex vector space of dimension $2\ell$, and we have $2\ell$ complex scalars ${\Phi }^s$, $s=1,\dots ,2\ell$, acted on by the flavor symmetry $F$ and the gauge symmetry $G$ in a complex representation $\tilde{R}$. The matrix $J+iK$ then determines an $2\ell \times 2\ell$ antisymmetric matrix ${𝜖}_{st}$, which is invariant under the action of $F$ and $G$. This means that $\tilde{R}$ is a pseudoreal representation:

This is the half-hypermultiplet in representation $\tilde{R}$, introduced briefly at the end of Sec. 2.1.2.

From this point of view, a half-hypermultiplet is more elementary than a full hypermultiplet, which is given as follows. Take an arbitrary complex representation $R$ of $F\times G$ of dimension $m$. Let $i=1,\dots ,m$ be its index. We have an invariant tensor ${\delta }_{i\bar{j}}$. Let $\tilde{R}=R\oplus \bar{R}$. It has an index $s=1,\dots ,n,\bar{1}\dots ,\bar{n}$, and automatically has an antisymmetric invariant tensor

Then the half-hypermultiplet based on this representation $\tilde{R}$ is the full hypermultiplet in the representation $R$.

Concretely, consider four real scalars. This system has a natural symmetry $SO\left(4\right)\simeq SU{\left(2\right)}_l\times SU{\left(2\right)}_r$. Add two Weyl fermions, with a natural symmetry $SU{\left(2\right)}_l$. Then the total system has an $\mathcal{𝒩}=2$ supersymmetry where the $SU{\left(2\right)}_R$ symmetry of the $\mathcal{𝒩}=2$ algebra is the $SU{\left(2\right)}_r$ acting on the scalars. The symmetry $SU{\left(2\right)}_l$ can be used as either a flavor or a gauge symmetry. This whole system consists of just one full hypermultiplet, or one half-hypermultiplet in the $SU{\left(2\right)}_l$ doublet.

Next, let $i=1,\dots ,n$ and $a=1,\dots ,m$ the indices for $U\left(n\right)$ and $U\left(m\right)$ symmetries, respectively. Then, $\mathcal{𝒩}=1$ chiral multiplets $Q_{iā}$, ${\tilde{Q}}_{īa}$ form an $\mathcal{𝒩}=2$ hypermultiplet, in the bifundamental representation of $U\left(n\right)\times U\left(m\right)$. When $U\left(n\right)$ is regarded as a gauge symmetry, $U\left(m\right)$ becomes the flavor symmetry.

Another typical construction is to take $i=1,\dots ,2n$ to be an index for $Sp\left(n\right)$ symmetry and $a=1,\dots ,m$ to be that for $SO\left(m\right)$ symmetry. Consider $\mathcal{𝒩}=1$ chiral multiplets $Q_{ia}$. Regard the pair of indices $ia$ as a single index $s=\left(ia\right)$, running from $1$ to $2nm$. This system has an antisymmetric invariant tensor ${𝜖}_{st}={𝜖}_{\left(ia\right)\left(jb\right)}=J_{ij}{\delta }_{ab}$, thus they make up a hypermultiplet with the symmetry $Sp\left(n\right)\times SO\left(m\right)$, commuting with the superalgebra. When $Sp\left(n\right)$ is made into a gauge symmetry, $SO\left(m\right)$ becomes the flavor symmetry, and vice versa. This explains the fact that when there are $n$ hypermultiplets in the vector representation of gauge $SO\left(m\right)$, we have $Sp\left(n\right)$ flavor symmetry, and when there are $m$ half-hypermultiplets in the fundamental representation of gauge $Sp\left(n\right)$, we have $SO\left(m\right)$ flavor symmetry.