Vacua

2.2 Vacua

The combined system of the vector multiplet and the hypermultiplets has the Lagrangian which is the sum of (2.1.11) and (2.1.14). The supersymmetric vacua are given by the following conditions.

First, the variation of the $D$ auxiliary ﬁelds gives

\frac{1}{g^{2}} [Φ^{†}, Φ] + (Q_{i} Q^{†}^{i} - {\tilde{Q}}^{†}_{i} {\tilde{Q}}^{i}) |_{traceless} = 0,

(2.2.1)

where $X |_{traceless}$ for an $N \times N$ matrix is deﬁned by

X |_{traceless} = X - \frac{1}{N} t r X .

(2.2.2)

We use the convention that a scalar is multiplied by a unit matrix when necessary.

Second, the variation of the $F$ auxiliary ﬁeld of $Φ$ gives

Q_{i} {\tilde{Q}}^{i} |_{traceless} = 0

(2.2.3)

and the $F$ auxiliary ﬁelds of $Q_{i}$ , ${\tilde{Q}}^{i}$ give

Φ Q_{i} + μ_{i}^{j} Q_{j} = 0, {\tilde{Q}}^{i} Φ + μ_{j}^{i} {\tilde{Q}}^{j} = 0

(2.2.4)

for all $i$ . The total scalar potential is a weighted sum of absolute values squared of (2.2.1), (2.2.3) and (2.2.4).

So far we only used the supersymmetry condition with respect to the $𝒩 = 1$ supersymmetry manifest in the superﬁeld notation. By massaging the cross terms between the ﬁrst term and the second term of (2.2.1) and combining them with the squares of (2.2.4), we can re-write the total scalar potential as a weighted sum of the following objects. First, we have one term purely of $Φ$ :

[Φ^{†}, Φ] = 0 .

(2.2.5)

Second, we have terms purely of $Q$ and $\tilde{Q}$ : one is

(Q_{i} Q^{†}^{i} - {\tilde{Q}}^{†}_{i} {\tilde{Q}}^{i}) |_{traceless} = 0

(2.2.6)

and another is (2.2.3). Finally, we have terms mixing $Φ$ and $Q$ , which are (2.2.4) together with

Φ^{†} Q_{i} + μ^{†}_{i}^{j} Q_{j} = 0, {\tilde{Q}}^{i} Φ^{†} + μ^{†}_{j}^{i} {\tilde{Q}}^{j} = 0 .

(2.2.7)

Note that (2.2.5) and (2.2.6) are the $S U {(2)}_{R}$ singlet and triplet parts of the equation (2.2.1), respectively. Furthermore, the equation (2.2.6) together with the real and the imaginary parts of the equation (2.2.3) form the triplet of $S U {(2)}_{R}$ . Finally, the equations (2.2.4) and (2.2.7) transform as a doublet of $S U {(2)}_{R}$ .

Let us summarize. We ﬁrst demanded that one $𝒩 = 1$ sub-supersymmetry is unbroken in (2.2.1), (2.2.3) and (2.2.4). We found the equations satisﬁed are automatically $S U {(2)}_{R}$ invariant, and therefore we see that all the $𝒩 = 2$ supersymmetry is automatically unbroken.

One easy way to have supersymmetry is to demand (2.2.5) and set $Q = \tilde{Q} = 0$ . This subspace of the supersymmetric vacuum moduli is called the Coulomb branch, since there usually remain a number of Abelian gauge ﬁelds in the infrared.

Another extreme is just to demand (2.2.6) and (2.2.3), and set $Φ = 0$ . This is called the Higgs branch. Some people in the ﬁeld reserve the word the Higgs branch for the branch where the gauge group is completely broken, but theoretically the Higgs branch as deﬁned here behaves more uniformly under various operations.

The branches with when both the hypermultiplet scalars $Q$ , $\tilde{Q}$ and the vector multiplet scalars $Φ$ are nonzero are called the mixed branches.

From (2.2.5) we see that $Φ$ can be diagonalized in the supersymmetric vacua. For deﬁniteness let $G = S U (2)$ . Then $Φ = d i a g (a, - a)$ . When $a \neq 0$ this breaks the gauge group to $U (1)$ . As there is a Coulomb ﬁeld remaining in the infrared, these vacua are called the Coulomb branch. Let us compute the mass of the resulting W-bosons. From

\frac{1}{g^{2}} t r | D_{μ} Φ |^{2} = \frac{1}{g^{2}} t r {(\partial_{μ} Φ + [A_{μ}, Φ])}^{2}

(2.2.8)

we have a term

\frac{1}{g^{2}} t r {[A_{μ}, ⟨ Φ ⟩]}^{2}

(2.2.9)

in the Lagrangian, which gives a mass to the vector ﬁeld. Writing

A_{μ} = {(\begin{matrix} A^{0} & W^{+} \\ W^{-} & - A^{0} \end{matrix})}_{μ},

(2.2.10)

we ﬁnd

[(\begin{matrix} 0 & W_{μ}^{+} \\ 0 & 0 \end{matrix}), (\begin{matrix} a & 0 \\ 0 & - a \end{matrix})] = - 2 a (\begin{matrix} 0 & W_{μ}^{+} \\ 0 & 0 \end{matrix}) .

(2.2.11)

The kinetic term in our convention is $t r F_{μ ν} F_{μ ν} ∕ (2 g^{2})$ , and therefore this gives the mass

M_{W} = | 2 a | .

(2.2.12)

The mass terms of the ﬁelds $Q_{i}$ , ${\tilde{Q}}^{i}$ for ﬁxed $i$ are

{\tilde{Q}}^{i} Φ Q_{i} + μ_{i} {\tilde{Q}}^{i} Q_{i} = ({\tilde{Q}}_{1}^{i}, {\tilde{Q}}_{2}^{i}) (\begin{matrix} a + μ_{i} & 0 \\ 0 & - a + μ_{i} \end{matrix}) (\begin{matrix} Q_{1}^{i} \\ Q_{2}^{i} \end{matrix}) .

(2.2.13)

Therefore we have

M_{Q_{i, 1}} = | a + μ |, M_{Q_{i, 2}} = | - a + μ | .

(2.2.14)

We studied the classical mass of the monopole in this model in (1.3.12) when $𝜃 = 0$ . In general, this is given by

M_{monopole} = | 2 τ a | .

(2.2.15)

Classically, there is a general inequality for the mass of a particle

M \geq | n a + m (2 τ a) + \sum_{i} f_{i} μ_{i} |

(2.2.16)

where $n$ , $m$ , $f_{i}$ are the electric, magnetic and ﬂavor charges of the particle. Here the $i$ -th ﬂavor charges are associated to the symmetry

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

(2.2.17)

This inequality, called the Bogomolnyi-Prasad-Sommerﬁeld (BPS) bound, persists in the quantum system, once quantum corrections are taken into account to $a$ and $2 τ a$ . Let us study this point next.

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