Pure SU(N) theory

11.2 Pure $S U (N)$ theory

11.2.1 The curve

Without further ado, let us introduce the Seiberg-Witten curves. First, the Seiberg-Witten curve for the pure $S U (N)$ theory is given by

Σ : \frac{Λ^{N}}{z} + Λ^{N} z = x^{N} + u_{2} x^{N - 2} + \dots + u_{N}

(11.2.1)

with the diﬀerential $λ = x d z ∕ z$ as always. The ultraviolet curve $C$ is just a sphere with the complex coordinate $z$ . At each point on the ultraviolet curve $z$ , we have $N$ solutions to the equation above. Therefore, $Σ$ is an $N$ -sheeted cover of $C$ .

Figure 11.1: W-boson of the

S U (N)

theory

Let us check that this curve reproduces the semiclassical behavior. We introduce variables ${\underset{̲}{a}}_{i}$ via

x^{N} + u_{2} x^{N - 2} + \dots + u_{N} = \prod_{i} (x - {\underset{̲}{a}}_{i}) .

(11.2.2)

We declare the A-cycle on the ultraviolet curve to be the unit circle $| z | = 1$ . As the Seiberg-Witten curve is an $N$ -sheeted cover, we can lift this curve to each sheet, which we call the cycle $A_{i}$ . Assume we are in the regime $| {\underset{̲}{a}}_{i} | \sim E$ independent of $i$ , and $E ≫ Λ$ . Then, we can approximately solve (11.2.1) by

x_{i} = {\underset{̲}{a}}_{i} + O (Λ) .

(11.2.3)

It is more convenient to regard $λ = x d z ∕ z$ itself to be the coordinate of the sheets. Then we have

λ_{i} = {\underset{̲}{a}}_{i} \frac{d z}{z} + O (Λ) .

(11.2.4)

The situation is shown in Fig. 11.1. The integral of $λ$ on the cycle $A_{i}$ is easy to evaluate:

a_{i} : = \frac{1}{2 π i} \oint_{A_{i}} λ = {\underset{̲}{a}}_{i} + O (Λ) .

(11.2.5)

Now we can suspend a ring-shaped membrane suspended between the $i$ -th sheet and the $j$ -th sheet. The mass of this object is

| \frac{1}{2 π i} \oint_{A_{i}} λ - \frac{1}{2 π i} \oint_{A_{j}} λ | = | \frac{1}{2 π i} \oint_{A} (λ_{i} - λ_{j}) | = | a_{i} - a_{j} | .

(11.2.6)

This reproduces the mass of the W-boson.

To see the monopoles, we need to understand the structure of the branching of the $N$ -sheeted cover $Σ \to C$ . It is convenient to regard the combination $y = Λ^{N} (z + 1 ∕ z)$ as one coordinate. Then, the equation (11.2.1) can be thought of determining the intersections of the graph of the polynomial

y = P (x) = x^{N} + u_{2} x^{N - 2} + \dots + u_{N}

(11.2.7)

and a horizontal line

y = Λ^{N} (z + \frac{1}{z})

(11.2.8)

as shown in Fig. 11.2. Of course the ﬁgure needs to be complexiﬁed, but the reader should be able to get the idea.

Figure 11.2: There are

(N - 1)

pairs of branch points in

S U (N)

pure theory

As is apparent, two out of $N$ sheets meet at $(N - 1)$ values of $y = Λ (z + 1 ∕ z)$ , each of which becomes a pair $z_{i}^{\pm}$ of branch points on the $z$ -sphere with $z_{i}^{+} z_{i}^{-} = 1$ . Note that the $i$ -th sheet and the $(i + 1)$ -st sheet meet at this pair of branch points. Then we can suspend a disk-shaped membrane between this pair of branch points, as shown in Fig. 11.3.

Figure 11.3: Monopoles of

S U (N)

pure theory

In the semiclassical regime when $| {\underset{̲}{a}}_{i} | \sim | E | ≫ | Λ |$ , we have

| z_{i}^{+} | \sim \frac{E^{N}}{Λ^{N}}, | z_{i}^{-} | \sim \frac{Λ^{N}}{E^{N}} .

(11.2.9)

We call the path connecting $z_{i}^{+}$ and $z_{i}^{-}$ as $B_{i}$ . Then

\begin{align} M_{monopole} & = | \frac{1}{2 π i} \int_{B_{i}} (λ_{i} - λ_{i + 1}) | & (11.2.10) \\ \sim | (a_{i} - a_{i + 1}) \frac{1}{2 π i} \int_{Λ^{N} ∕ E^{N}}^{E^{N} ∕ Λ^{N}} \frac{d z}{z} | & (11.2.11) \\ \sim | (a_{i} - a_{i + 1}) \frac{2 N}{2 π i} log \frac{E}{Λ} | . & (11.2.12) \end{align}

This reproduces the mass of the monopole, by identifying

τ (E) = \frac{2 N}{2 π i} log \frac{E}{Λ} .

(11.2.13)

This correctly reproduces the one-loop running of the pure $S U (N)$ theory.

11.2.2 Infrared gauge coupling matrix

Let us check that our curve satisﬁes the condition that the coupling matrices of the low-energy $U {(1)}^{N - 1}$ theory is positive deﬁnite. For this purpose we need to understand the geometry of the Seiberg-Witten curve $Σ$ better. This is an $N$ -sheeted cover of $C$ with $2 N - 2$ branch points $z_{i}^{\pm}$ of order 2 and 2 branch points $z = 0, \infty$ of order $N$ . The genus $g$ of $Σ$ is then determined by the Riemann-Hurwitz formula¹⁵ :

χ (Σ) = N χ (C) - (2 N - 2) - 2 (N - 1)

(11.2.14)

where $χ (Σ) = 2 - 2 g$ and $χ (C) = 2$ are the Euler number of the respective surfaces. We ﬁnd $g = N - 1$ . The basis of the 1-cycles consists of $(2 N - 2)$ cycles $A_{i}$ and ${\tilde{B}}^{i}$ , $i = 1, \dots, N - 1$ , where the intersections are given by

A_{i} \cdot A_{j} = 0 = {\tilde{B}}^{i} \cdot {\tilde{B}}^{j}, A_{i} \cdot {\tilde{B}}^{j} = δ_{i}^{j} .

(11.2.15)

Here the dot product counts the number of intersections (including signs) of two one-cycles. The resulting set of cycles is shown in Fig. 11.4.

Figure 11.4: Cycles

A_{i}

and

{\tilde{B}}_{i}

on the Seiberg-Witten curve of the pure

S U (N)

theory.

The ﬁgures 11.3 and 11.4 are drawn in a rather diﬀerent manner. The cycles from $A_{1}$ to $A_{N - 1}$ can be directly identiﬁed. We have

A_{N} = - A_{1} - A_{2} \dots - A_{N - 1}

(11.2.16)

as far as the line integral of holomorphic forms are concerned. Correspondingly, the variables $a_{i}$ as deﬁned in (11.2.5) are not linearly independent, and we have

a_{N} = - a_{1} - \dots - a_{N - 1} .

(11.2.17)

The combination $B_{i} - B_{i + 1}$ in Fig. 11.3 intersects with $A_{i}$ positively and with $A_{i + 1}$ negatively. Then, we see

B_{i} - B_{i + 1} = {\tilde{B}}^{i} - {\tilde{B}}^{i + 1} .

(11.2.18)

Equivalently, ${\tilde{B}}^{i}$ is a closed one-cycle completing the open path $B_{i}$ in a way independent of $i$ . Then we deﬁne

a_{D}^{i} : = \frac{1}{2 π i} \oint_{{\tilde{B}}^{i}} λ

(11.2.19)

on the curve. Let us consider

τ^{i j} : = \frac{\partial a_{D}^{i}}{\partial a_{j}} = X_{D}^{i k} {(X^{- 1})}_{k}^{j}

(11.2.20)

where

X_{i}^{k} : = \frac{\partial a_{i}}{\partial u_{k}}, X_{D}^{j k} : = \frac{\partial a_{D}^{j}}{\partial u_{k}} .

(11.2.21)

Deﬁning

ω_{k} = \frac{\partial}{\partial u_{k}} λ |_{constant z},

(11.2.22)

we ﬁnd

τ^{i j} = X_{D}^{i k} {(X^{- 1})}_{k}^{j} where X_{i}^{k} = \oint_{A_{i}} ω_{k}, X_{D}^{j k} = \oint_{{\tilde{B}}_{j}} ω_{k} .

(11.2.23)

It can be checked that $ω_{2, 3, \dots, N}$ form a basis of holomorphic non-singular one-forms on $Σ$ . The matrix $τ^{i j}$ formed this way is known mathematically as the period matrix of $Σ$ , and is known to satisfy

τ^{i j} = τ^{j i}, I m τ^{i j} is positive deﬁnite .

(11.2.24)

From the ﬁrst condition, we see that there is locally a function $F (a_{i})$ such that

a_{D}^{i} = \frac{\partial F}{\partial a_{i}}, τ^{i j} = \frac{\partial^{2} F}{\partial a_{i} \partial a_{j}} .

(11.2.25)

This justiﬁes that we identify $a_{i}$ , $a_{D}^{i}$ deﬁned this way with the $a_{i}$ appearing in the low-energy description of $U {(1)}^{N - 1}$ gauge theory. The inverse gauge coupling matrix is given by $I m τ^{i j}$ , whose positive deﬁniteness is guaranteed by the mathematical relation (11.2.24).

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11.2 Pure SU(N) theory

11.2.1 The curve

11.2.2 Infrared gauge coupling matrix

11.2 Pure $S U (N)$ theory