Seiberg-Witten solutions for various other simple gauge groups

11.6 Seiberg-Witten solutions for various other simple gauge groups

Let us close this section by very brieﬂy mentioning the Seiberg-Witten solution of various other gauge theories. First, let us copy the solutions for $S U (N)$ and $S O (2 N)$ with $N_{f}$ ﬂavors. The Seiberg-Witten curve for $S U (N)$ with $N_{f}$ ﬂavors was

\frac{Λ^{N}}{z} + z Λ^{N - N_{f}} \prod_{i = 1}^{N_{f}} (x - μ_{i}) = x^{N} + u_{2} x^{N - 2} + \dots + u_{N - 1} x + u_{N}

(11.6.1)

and the Seiberg-Witten curve for $S O (2 N)$ with $N_{f}$ ﬂavors was

x^{2} [\frac{Λ^{2 N - 2}}{z} + Λ^{2 N - 2 - 2 N_{f}} z \prod_{i = 1}^{N_{f}} (x^{2} - μ_{i}^{2})] = x^{2 N} + u_{2} x^{2 N - 2} + u_{4} x^{2 N - 4} + \dots + u_{2 N} .

(11.6.2)

Here, for simplicity, we dropped the ﬂavor terms multiplying $1 ∕ z$ on the left hand sides. For the full expressions, see (10) and (21), respectively.

Without derivations, we present here the Seiberg-Witten curves for other classical gauge groups. The Seiberg-Witten curve for $S O (2 N + 1)$ with $N_{f}$ ﬂavors is

x [\frac{Λ^{2 N - 1}}{z^{1 ∕ 2}} + Λ^{2 N - 1 - 2 N_{f}} z^{1 ∕ 2} \prod_{i = 1}^{N_{f}} (x^{2} - μ_{i}^{2})] = x^{2 N} + u_{2} x^{2 N - 2} + u_{4} x^{2 N - 4} + \dots + u_{2 N},

(11.6.3)

and the Seiberg-Witten curve for $S p (N)$ with $N_{f}$ ﬂavors is

\begin{matrix} \frac{Λ^{2 N + 2}}{z^{1 ∕ 2}} + 2 c + Λ^{2 N + 2 - 2 N_{f}} z^{1 ∕ 2} \prod_{i = 1}^{N_{f}} (x^{2} - μ_{i}^{2}) \\ = x^{2} (x^{2 N} + u_{2} x^{2 N - 2} + u_{4} x^{2 N - 4} + \dots + u_{2 N}) & (11.6.4) \end{matrix}

where $c^{2} = Λ^{4 N + 4 - 2 N_{f}} \prod_{i = 1}^{N_{f}} (- μ_{i}^{2})$ . The diﬀerential is always just $λ = x d z ∕ z$ . We again dropped the ﬂavor terms multiplying $1 ∕ z$ on the left hand sides.

The curves so far can be always written as

\frac{F (x)}{z} + z \tilde{F} (x) = P (x)

(11.6.5)

for some polynomials $F (x)$ , $\tilde{F} (x)$ and $P (x)$ . In the older literature, it is more common to ﬁnd the curve and the diﬀerential in the form

y^{2} = P {(x)}^{2} - 4 F (x) \tilde{F} (x), λ = \frac{x}{2} d log \frac{P (x) - y}{P (x) + y} .

(11.6.6)

To relate (11.6.5) and (11.6.6), note that the equation (11.6.6) implies that the combination

ζ_{\pm} = \frac{1}{2} (P (x) \pm y)

(11.6.7)

satisﬁes

ζ_{+} + ζ_{-} = P (x), ζ_{+} ζ_{-} = F (x) \tilde{F} (x) .

(11.6.8)

Comparing with (11.6.5), we ﬁnd

ζ_{+} = \frac{F (x)}{z}, ζ_{-} = z \tilde{F} (x) .

(11.6.9)

This also explains the diﬀerential given in (11.6.6).

This older form is mathematically easier to deal with in certain situations, because the branch cut of the function $y (x)$ is at most of order 2. Mathematically, such Riemann surfaces are called hyperelliptic, and have a few special properties compared to more general Riemann surfaces. That said, the forms we use in the rest of the lecture note is much more physical and usually more useful.

A good summary of the curves for classical gauge groups listed above can be found e.g. in [67]. For exceptional gauge groups, the situation is more complicated. Although one can write the Seiberg-Witten curve, it is more natural to study the Seiberg-Witten geometry, which is a complex 3-dimensional space, ﬁbered over the ultravioletcurve $C$ . A very nice presentation for the pure theory can be found in [68]. With matter hypermultiplets, a modern reference is [69].

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