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 SU(N) and SO(2N) with Nf flavors. The Seiberg-Witten curve for SU(N) with Nf flavors was

ΛN z + zΛN Nf i=1Nf(x μi) = xN + u2xN 2 + + uN1x + uN (11.6.1)

and the Seiberg-Witten curve for SO(2N) with Nf flavors was

x2 Λ2N 2 z + Λ2N 2 2Nfz i=1Nf(x2 μi2) = x2N+u2x2N 2+u4x2N 4++u2N. (11.6.2)

Here, for simplicity, we dropped the flavor terms multiplying 1z 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 SO(2N + 1) with Nf flavors is

x Λ2N 1 z12 + Λ2N 1 2Nfz12 i=1Nf(x2 μi2) = x2N+u2x2N 2+u4x2N 4++u2N, (11.6.3)

and the Seiberg-Witten curve for Sp(N) with Nf flavors is

Λ2N + 2 z12 + 2c + Λ2N + 2 2Nfz12 i=1Nf(x2 μi2) = x2(x2N + u2x2N 2 + u4x2N 4 + + u2N)(11.6.4)

where c2 = Λ4N + 4 2Nf i=1Nf(μi2). The differential is always just λ = xdzz. We again dropped the flavor terms multiplying 1z on the left hand sides.

The curves so far can be always written as

F(x) z + zF̃(x) = P(x) (11.6.5)

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

y2 = P(x)2 4F(x)F̃(x),λ = x 2d log 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± = 1 2(P(x) ± y) (11.6.7)


ζ+ + ζ = P(x),ζ+ζ = F(x)F̃(x). (11.6.8)

Comparing with (11.6.5), we find

ζ+ = F(x) z ,ζ = zF̃(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].