Semiclassical analysis

11.1 Semiclassical analysis

Let us consider $S U (N)$ gauge theory with $N_{f}$ hypermultiplets in the fundamental $N$ -dimensional representation. The $𝒩 = 2$ vector multiplet consists of the $𝒩 = 1$ adjoint chiral multiplet $Φ$ and the $𝒩 = 1$ vector multiplet $W_{α}$ , both $N \times N$ matrices. The hypermultiplets, in terms of $𝒩 = 1$ chiral multiplets, can be written as

Q_{i}^{a}, {\tilde{Q}}_{a}^{i}, a = 1, \dots, N; i = 1, \dots, N_{f} .

(11.1.1)

One branch of the supersymmetric vacuum is given by the condition

[Φ, Φ^{†}] = 0 .

(11.1.2)

This means that $Φ$ can be diagonalized. We denote it by

Φ = d i a g (a_{1}, \dots, a_{N}), \sum a_{i} = 0 .

(11.1.3)

Let us consider a generic situation when $a_{i} \neq a_{j}$ for all $i \neq j$ . Then the gauge group is broken from $S U (N)$ to $U {(1)}^{N - 1}$ . The W-boson mass is given by

M_{W} = | a_{i} - a_{j} |

(11.1.4)

for the W-boson coming from the entry $(i, j)$ of the $N \times N$ matrix. As for the monopole, it is known that the ’t Hooft-Polyakov monopole solution for the breaking from $S U (2)$ to $U (1)$ can be directly regarded as a solution for the breaking from $S U (N)$ to $U (1)$ , by choosing $2 \times 2$ submatrices of $N \times N$ matrices, given by picking the entries at the positions $(i, i)$ , $(i, j)$ , $(j, i)$ and $(j, j)$ for $i \neq j$ . The masses are then

M_{monopole} = | τ_{U V} (a_{i} - a_{j}) | .

(11.1.5)

The one-loop running is

E \frac{d}{d E} τ = \frac{1}{2 π} (2 N - N_{f}) .

(11.1.6)

Then the theory is asymptotically free when $0 \leq N_{f} < 2 N$ . The dynamical scale is then

Λ^{2 N - N_{f}} : = Λ_{U V}^{2 N - N_{f}} e^{2 π i τ_{U V}} .

(11.1.7)

When $N_{f} = 2 N$ , the theory is asymptotically conformal, and $τ_{U V}$ is a dimensionless parameter in the quantum theory.

When there are ﬂavors, the $𝒩 = 1$ superpotential in this vacua is

\begin{matrix} \sum_{i} (Q_{i} Φ {\tilde{Q}}^{i} - μ_{i} Q_{i} {\tilde{Q}}^{i}) = \\ \sum_{i} (Q_{i}^{1}, Q_{i}^{2}, \dots, Q_{i}^{N}) (\begin{matrix} a_{1} - μ_{i} \\ a_{2} - μ_{i} \\ ⋱ \\ a_{N} - μ_{i} \end{matrix}) (\begin{matrix} {\tilde{Q}}_{1}^{i} \\ {\tilde{Q}}_{2}^{i} \\ ⋮ \\ {\tilde{Q}}_{N}^{i} \end{matrix}) . & (11.1.8) \end{matrix}

Then we have one massless charged hypermultiplet component whenever we have $a_{i} - μ_{s} = 0$ for some $i$ and $s$ .

In the strongly-coupled quantum theory, the deﬁnition of $a_{i}$ as the diagonal entries of the gauge-dependent quantity $Φ$ does not make much sense. Instead, as we did in the case of $S U (2)$ gauge theory, we deﬁne $a_{i}$ as the complex numbers entering in the BPS mass formula:

M \geq | n^{i} a_{i} + m_{i} a_{D}^{i} + \sum_{s} f_{s} μ_{s} |

(11.1.9)

where $(n^{i}, m_{i})$ are the electric and the magnetic charges under $U {(1)}^{N - 1}$ infrared gauge group, and $f_{s}$ are the ﬂavor charges. We can also consider gauge-invariant combinations of $Φ$ deﬁned as

x^{N} + u_{2} x^{N - 2} + \dots + u_{N - 1} x + u_{N} : = ⟨ det (x - Φ) ⟩

(11.1.10)

where $x$ is a dummy variable. For $N = 2$ , we had $Φ \sim d i a g (a, - a)$ and therefore $u_{2} = - a^{2}$ up to quantum corrections. Similarly, for general $N$ , $u_{k}$ is the degree $k$ elementary symmetric polynomials of the variables $a_{1}$ , …, $a_{N}$ up to quantum corrections. Our task then is to determine the mapping between $(u_{2}, \dots, u_{N})$ and $(a_{1}, \dots, a_{N})$ including the quantum corrections.

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