Weak-coupling limit and trifundamentals

9.3 Weak-coupling limit and trifundamentals

Figure 9.6: Weakly-coupled limit of

S U (2)

N_{f} = 4

Let us now take the limit $q \to 0$ to decouple the gauge $S U (2)$ , see Fig. 9.6. On the left hand side, we have a sphere parameterized by $z$ , with four points at $z = 0, q, 1$ and $\infty$ . On the right hand side, we have two spheres, parameterized by $z^{'}$ and $z^{″}$ . We put the points $A$ , $B$ , $G$ on the ﬁrst sphere, at $z^{'} = 0$ , $1$ and $\infty$ , and then the points $G^{'}$ , $C$ , $D$ on the second sphere, at $z^{″} = 0$ , $1$ and $\infty$ . Then we glue the neighborhoods of $G$ and $G^{'}$ by declaring

z^{'} z^{″} = q .

(9.3.1)

Deﬁning $z = z^{″}$ , we see that four points $A, B, C, D$ are exactly as in the ﬁrst description. In this limit, around the tube connecting $G$ and $G^{'}$ , $λ ≃ \pm a d z ∕ z$ . Then in the sphere containing $A$ , $B$ and $G$ , we have three singularities, with residues of $λ$ given by

\pm \frac{μ_{1} + μ_{2}}{2}, \pm \frac{μ_{1} - μ_{2}}{2}, \pm a,

(9.3.2)

each corresponding to the symmetry $S U {(2)}_{A}$ , $S U {(2)}_{B}$ and $S U {(2)}_{G}$ , respectively. Here $S U {(2)}_{G}$ was originally the gauge symmetry.

Figure 9.7: The ultraviolet curve of the trifundamental, together with the BPS paths representing hypermultiplets

We were talking about the $N_{f} = 4$ theory. Then each of the sphere with three punctures should be associated to the $N_{f} = 2$ hypermultiplet system, see Fig. 9.7; note that this is not coupled to any gauge group. Let us recall the structure of the hypermultiplets again. We start from two hypermultiplets $(Q_{i}^{a}, {\tilde{Q}}_{a}^{i})$ in the doublet of $S U (2)$ , $i = 1, 2$ and $a = 1, 2$ . We combine them to $q_{I}^{a}$ , $a = 1, 2$ and $I = 1, \dots, 4$ , making $S U (2) \times S O (4)$ symmetry manifest. We then decompose the $S O (4)$ index $I$ into the pair $(α, u)$ where $α = 1, 2$ and $u = 1, 2$ : we have the trifundamental $q_{a α u}$ . The mass term for this hypermultiplet is

μ^{a b} q_{a α u} q_{b β v} 𝜖^{α β} 𝜖^{u v} + {\tilde{μ}}^{α β} q_{a α u} q_{b β v} 𝜖^{a b} 𝜖^{u v} + {\hat{μ}}^{u v} q_{a α u} q_{b β v} 𝜖^{a b} 𝜖^{α β},

(9.3.3)

where

μ_{b}^{a} = a d i a g (1, - 1), {\tilde{μ}}_{β}^{α} = \frac{μ_{1} - μ_{2}}{2} d i a g (1, - 1), {\hat{μ}}_{β}^{α} = \frac{μ_{1} + μ_{2}}{2} d i a g (1, - 1) .

(9.3.4)

Then $(a, b)$ are the indices for $S U {(2)}_{G}$ , $(α, β)$ for $S U {(2)}_{A}$ , and $(u, v)$ for $S U {(2)}_{B}$ . The physical masses of these ﬁelds are given by

\pm a \pm \frac{μ_{1} - μ_{2}}{2} \pm \frac{μ_{1} + μ_{2}}{2} = {\pm a \pm μ_{1}, \pm a \pm μ_{2}} .

(9.3.5)

which are the masses for the two doublets of $S U (2)$ with bare masses $μ_{1, 2}$ .

The curve of the system, shown in Fig. 9.7 is given by

λ^{2} - ϕ (z) = 0,

(9.3.6)

where $ϕ (z)$ has the asymptotic behavior

ϕ (z) \sim \frac{{\tilde{μ}}^{2}}{z^{2}} d z^{2}, \sim \frac{{\hat{μ}}^{2}}{{(z - 1)}^{2}} d z^{2}, \sim \frac{μ^{2}}{w^{2}} d w^{2}

(9.3.7)

at $z = 0$ , $z = 1$ , $z = \infty$ respectively. Here $w = 1 ∕ z$ as always, and we set $μ = a$ , $\tilde{μ} = (μ_{1} - μ_{2}) ∕ 2$ and $\hat{μ} = (μ_{1} + μ_{2}) ∕ 2$ . Note that these asymptotic conditions uniquely ﬁx the quadratic diﬀerential $ϕ (z)$ to be

ϕ (z) = \frac{μ^{2} z^{2} + ({\hat{μ}}^{2} - {\tilde{μ}}^{2} - μ^{2}) z + {\tilde{μ}}^{2}}{z^{2} {(z - 1)}^{2}} d z^{2}

(9.3.8)

As was discussed before, the BPS particles of this system can be found by solving the BPS equation (6.1.6)

A r g \frac{λ}{d s} = e^{i 𝜃}

(9.3.9)

for a given $𝜃$ . As $ϕ (z)$ given above has two branch points only, the solution to the BPS equation should start from one and end on the other. A computer simulation shows that there are always four and only four such solutions, corresponding to the hypermultiplets with masses given in (9.3.5).

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