Microscopic Lagrangian

2.1 Microscopic Lagrangian

2.1.1 $𝒩 = 1$ superﬁelds

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

An $𝒩 = 1$ vector multiplet consists of a Weyl fermion $λ_{α}$ and a vector ﬁeld $A_{μ}$ , both in the adjoint representation of the gauge group $G$ . We combine them into the superﬁeld $W_{α}$ with the expansion

W_{α} = λ_{α} + F_{(α β)} 𝜃^{β} + D 𝜃_{α} + \dots

(2.1.1)

where $D$ is an auxiliary ﬁeld, again in the adjoint of the gauge group. $F_{α β} = \frac{i}{2} σ^{μ}_{\dot{γ}}^{β} {\bar{σ}}^{ν}_{α}^{\dot{γ}} F_{μ ν}$ is the anti-self-dual part of the ﬁeld strength $F_{μ ν}$ .

The kinetic term for a vector multiplet is given by

\int d^{2} 𝜃 \frac{- i}{8 π} τ t r W_{α} W^{α} + c c .

(2.1.2)

where

τ = \frac{4 π i}{g^{2}} + \frac{𝜃}{2 π}

(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

\frac{1}{2 g^{2}} t r F_{μ ν} F^{μ ν} + \frac{𝜃}{16 π^{2}} t r F_{μ ν} {\tilde{F}}^{μ ν} + \frac{1}{g^{2}} t r D^{2} - \frac{2 i}{g^{2}} t r \bar{λ} γ^{μ} \partial_{μ} λ .

(2.1.4)

We use the convention that $t r T^{a} T^{b} = \frac{1}{2} δ^{a b}$ for the standard generators of gauge algebras, which explain why we have the factors $1 ∕ (2 g^{2})$ in front of the gauge kinetic term. The $𝜃$ 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 $𝒩 = 1$ chiral multiplet $Q$ consists of a complex scalar $Q$ and a Weyl fermion $ψ_{α}$ , both in the same representation of the gauge group. In terms of a superﬁeld we have

Q = Q |_{𝜃 = 0} + 2 ψ_{α} 𝜃^{α} + F 𝜃_{α} 𝜃^{α}

(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

\int d^{4} 𝜃 Q^{†}^{j} e^{V^{a} ρ_{a}_{j}^{i}} Q_{i} + \int d^{2} 𝜃 W (Q) + \dots

(2.1.6)

where $V$ is the vector superﬁeld, $ρ_{a}_{j}^{i}$ is the matrix representation of the gauge algebra, and $W (Q)$ 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

δ λ_{α} = 0, δ ψ_{α} = 0

(2.1.7)

which give

D_{a} = 0, F_{i} = 0 .

(2.1.8)

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

Q_{\bar{j}}^{†} ρ_{a}^{\bar{j} i} Q_{i} = 0, \frac{\partial W}{\partial Q_{i}} = 0 .

(2.1.9)

2.1.2 Vector multiplets and hypermultiplets

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

\begin{matrix} \leftrightarrow & λ_{α} & \leftrightarrow & A_{μ} & 𝒩 = 1 vector multiplet, \\ Φ & \leftrightarrow & {\tilde{λ}}_{α} & \leftrightarrow & 𝒩 = 1 chiral multiplet . \end{matrix}

(2.1.10)

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

One easy way to construct the second supersymmetry action is to demand that the theory is symmetric under the $S U (2)$ rotation acting on $λ_{α}$ and ${\tilde{λ}}_{α}$ . A symmetry which does not commute with the supersymmetry generators is called an R-symmetry in general. Therefore this $S U (2)$ symmetry is often called the $S U {(2)}_{R}$ 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 $𝒩 = 2$ supersymmetric theories without $S U {(2)}_{R}$ symmetry: there can just be two sets of supersymmetry generators without $S U (2)$ symmetry relating them, see e.g. [31, 32]. That said, for simplicity, we only deal with $𝒩 = 2$ supersymmetric systems with $S U {(2)}_{R}$ symmetry in this lecture note.

The Lagrangian is then

\frac{I m τ}{4 π} \int d^{4} 𝜃 t r Φ^{†} e^{[V, \cdot]} Φ + \int d^{2} 𝜃 \frac{- i}{8 π} τ t r W_{α} W^{α} + c c .

(2.1.11)

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

An $𝒩 = 2$ hypermultiplet⁴ consists of the following ﬁelds:

\begin{matrix} \leftrightarrow & Q & \leftrightarrow & ψ & 𝒩 = 1 chiral multiplet \\ {\tilde{ψ}}^{†} & \leftrightarrow & {\tilde{Q}}^{†} & \leftrightarrow & 𝒩 = 1 antichiral multiplet \end{matrix}

(2.1.12)

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

For deﬁniteness, let us consider $G = S U (N)$ and $N_{f}$ hypermultiplets $Q_{i}^{a}$ , ${\tilde{Q}}_{a}^{i}$ 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 $S U (N)$ . The gauge transformation acts on them as

Q_{i} \to e^{Λ} Q_{i}, {\tilde{Q}}^{i} \to {\tilde{Q}}^{i} e^{- Λ}

(2.1.13)

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

The Lagrangian for the hypermultiplets is

c \int d^{4} 𝜃 (Q^{†}^{i} e^{V} Q_{i} + {\tilde{Q}}^{i} e^{- V} {\tilde{Q}}^{†}_{i}) + c^{'} (\int d^{2} 𝜃 {\tilde{Q}}^{i} Φ Q_{i} + c c .) + (\int d^{2} 𝜃 μ_{j}^{i} {\tilde{Q}}^{j} Q_{i} + c c .)

(2.1.14)

where the gauge index $a$ is suppressed again. The existence of $S U {(2)}_{R}$ symmetry ﬁxes the ratio of $c$ and $c^{'}$ : it can be done e.g. by comparing the coeﬃcients of $Q^{i} λ ψ$ from the ﬁrst term and of ${\tilde{Q}}^{i} \tilde{λ} ψ$ from the second term. We ﬁnd the choice $c = c^{'}$ does the job. In the following we take $c = c^{'} = 1$ unless otherwise mentioned. The $S U {(2)}_{R}$ symmetry also demands that the mass term $μ_{j}^{i}$ satisﬁes $[μ, μ^{†}] = 0$ . Then $μ$ can be diagonalized, and consequently the mass term is often written as

\sum_{i} \int d^{2} 𝜃 μ_{i} {\tilde{Q}}^{i} Q_{i} + c c .

(2.1.15)

As another example, let us consider the case when we have a hypermultiplet $(Z, \tilde{Z})$ 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{matrix} \int d^{2} 𝜃 \frac{- i}{8 π} τ t r W_{α} W^{α} + c c . + \frac{I m τ}{4 π} \int d^{4} 𝜃 t r Φ^{†} e^{[V, \cdot]} Φ \\ + \frac{I m τ}{4 π} \int d^{4} 𝜃 (Z^{†} e^{[V, \cdot]} Z + \tilde{Z} e^{- [V, \cdot]} {\tilde{Z}}^{†}) + \frac{I m τ}{4 π} \int d^{2} 𝜃 \tilde{Z} [Φ, Z] + c c . & (2.1.16) \end{matrix}

where we made a diﬀerent choice of $c = c^{'}$ in (2.1.14). This Lagrangian clearly has $S U {(3)}_{F}$ ﬂavor symmetry rotating $Φ$ , $Z$ and $\tilde{Z}$ . This commutes with the $𝒩 = 1$ supersymmetry manifest in the superﬁeld formalism. We also know that this theory has an $S U {(2)}_{R}$ symmetry rotating $Z$ and ${\tilde{Z}}^{†}$ . These two symmetries $S U {(3)}_{F}$ and $S U {(2)}_{R}$ does not commute: we ﬁnd that there is an $S O {(6)}_{R}$ symmetry, acting on

R e Φ, I m Φ, R e Z, I m Z, R e \tilde{Z}, I m \tilde{Z} .

(2.1.17)

Note that $S O {(6)}_{R}$ can also be regarded as $S U {(4)}_{R}$ , as $S O (6)$ and $S U (4)$ have the same Lie algebra. Then the $S U {(4)}_{R}$ symmetry acts on the four Weyl fermions

λ, \tilde{λ}, ψ, \tilde{ψ}

(2.1.18)

in the system, where $λ$ and $\tilde{λ}$ are in the $𝒩 = 2$ vector multiplet, and $ψ$ , $\tilde{ψ}$ are in the $𝒩 = 2$ hypermultiplet. We conclude that this system has in fact $𝒩 = 4$ supersymmetry, whose four supersymmetry generators are acted on by $S U {(4)}_{R} ≃ S O {(6)}_{R}$ . 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 $\int d^{2} 𝜃 μ Z \tilde{Z} + c c .$ to (16). This preserves the $𝒩 = 2$ supersymmetry but it breaks $𝒩 = 4$ supersymmetry. The resulting theory is sometimes called the $𝒩 = 2^{*}$ theory.

Before closing this section, we should mention the concept of half-hypermultiplet. Let us start from a full hypermultiplet $(Q_{a}, {\tilde{Q}}^{a})$ so that $Q_{a}$ and ${\tilde{Q}}^{a}$ are in the representations $R$ , $\bar{R}$ , respectively. When $R$ is pseudo-real, or equivalently when there is an antisymmetric invariant tensor $𝜖_{a b}$ , we can impose the constraint

Q_{a} = 𝜖_{a b} {\tilde{Q}}^{b}

(2.1.19)

compatible with $𝒩 = 2$ 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.

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2.1 Microscopic Lagrangian

2.1.1 𝒩=1 superﬁelds

2.1.2 Vector multiplets and hypermultiplets

2.1.1 $𝒩 = 1$ superﬁelds