SU(N) theory with fundamental ﬂavors

11.3 $S U (N)$ theory with fundamental ﬂavors

11.3.1 $N_{f} = 1$

Next, consider the $S U (N)$ theory with one ﬂavor $(Q, \tilde{Q})$ of bare mass $μ$ . The curve is given by

Σ : \frac{Λ^{N - 1} (x - μ)}{z} + Λ^{N} z = x^{N} + u_{2} x^{N - 2} + \dots + u_{N} .

(11.3.1)

Recall that in the semiclassical analysis we saw that a light charged hypermultiplet arises when $a_{i} \sim μ$ . Let us check that the curve written above reproduces this behavior.

First, we introduce ${\underset{̲}{a}}_{i}$ as before, and consider the semiclassical regime when all $| {\underset{̲}{a}}_{i} |$ is far larger than $| Λ |$ . The A-cycle on the ultraviolet curve was $| z | = 1$ as before. Then we ﬁnd $a_{i} \sim \underset{̲}{a_{i}} + O (Λ)$ just as was in the case of the pure theory.

To see additional singularities in the weakly-coupled region, deﬁne $\tilde{z} = z ∕ Λ^{N - 1}$ . The curve is then

\frac{x - μ}{\tilde{z}} + Λ^{2 N - 1} \tilde{z} = x^{N} + u_{2} x^{N - 2} + \dots + u_{N},

(11.3.2)

which can be approximated by

\frac{x - μ}{\tilde{z}} = x^{N} + u_{2} x^{N - 2} + \dots + u_{N} = \prod (x - {\underset{̲}{a}}_{i})

(11.3.3)

in the extremely weakly coupled limit. The equation factorizes and the curve separates into two when ${\underset{̲}{a}}_{i} = μ$ ; otherwise the curve is a smooth degree- $N$ covering of the $z$ sphere. This shows that when ${\underset{̲}{a}}_{i} = μ$ , a one-cycle on the Seiberg-Witten curve shrinks, and the membrane suspended there produces a massless hypermultiplet, see Fig. 5.9.

The one-loop running can also be checked. The branch points $z_{i}^{+}$ in the large $z$ region is unchanged, as the structure of the $N_{f} = 1$ curve in the large $z$ region itself is unchanged from the pure curve. Then

z_{i}^{+} \sim {(E ∕ Λ)}^{N} .

(11.3.4)

In the small $z$ region, the branch points are around where $Λ^{N - 1} x ∕ z$ and $P (x)$ are of the same order. Assuming $| x | \sim | {\underset{̲}{a}}_{i} | \sim | E |$ , we see

z_{i}^{-} \sim {(Λ ∕ E)}^{N - 1} .

(11.3.5)

Then the monopole has the mass

\begin{align} M_{monopole} & = | \frac{1}{2 π i} \int_{B_{i}} λ | & (11.3.6) \\ \sim | (a_{i} - a_{i + 1}) \frac{1}{2 π i} \int_{Λ^{N - 1} ∕ E^{N - 1}}^{E^{N} ∕ Λ^{N}} \frac{d z}{z} | & (11.3.7) \\ \sim | (a_{i} - a_{i + 1}) \frac{2 N - 1}{2 π i} log \frac{E}{Λ} | . & (11.3.8) \end{align}

This gives

τ (E) = \frac{2 N - 1}{2 π i} log \frac{E}{Λ}

(11.3.9)

as it should be.

11.3.2 General number of ﬂavors

More generally, we can consider the curve given by

\begin{matrix} Σ : \frac{Λ^{N - N_{L}} \prod_{i = 1}^{N_{L}} (x - μ_{i})}{z} + z Λ^{N - N_{R}} \prod_{i = 1}^{N_{R}} (x - μ_{i}^{'}) \\ = x^{N} + u_{2} x^{N - 2} + \dots + u_{N - 1} x + u_{N} & (11.3.10) \end{matrix}

where $N_{L}, N_{R} \leq N$ . When $N_{L} = N_{R} = N$ , we need to introduce complex numbers $f$ , $f^{'}$ as in the curve of $S U (2)$ with four ﬂavors, (9.1.5):

\begin{matrix} Σ : f \cdot \frac{\prod_{i = 1}^{N} (x - μ_{i})}{z} + f^{'} \cdot z \prod_{i = 1}^{N} (x - μ_{i}^{'}) \\ = x^{N} + u_{2} x^{N - 2} + \dots + u_{N - 1} x + u_{N} . & (11.3.11) \end{matrix}

We also need to distinguish the mass parameters in the curve and the mass parameters in the BPS mass formula, carefully studied in Sec. 9.2 for $S U (2)$ with four ﬂavors. In the following we mainly discuss the case with less than $2 N - 1$ ﬂavors.

Consider the case when $μ_{i}$ and $μ_{i}^{'}$ are all small. Further, consider the regime where $| {\underset{̲}{a}}_{i} | ≫ Λ$ . As always we ﬁnd $a_{i} = {\underset{̲}{a}}_{i} + O (Λ)$ . The branch points are at

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

(11.3.12)

Then we ﬁnd

\begin{align} M_{monopole} & \sim | (a_{i} - a_{i + 1}) \frac{1}{2 π i} \int_{Λ^{N - N_{L}} ∕ E^{N - N_{L}}}^{E^{N - N_{R}} ∕ Λ^{N - N_{R}}} \frac{d z}{z} | & (11.3.13) \\ \sim | (a_{i} - a_{i + 1}) \frac{2 N - (N_{L} + N_{R})}{2 π i} log \frac{E}{Λ} |, & (11.3.14) \end{align}

and therefore the one-loop running is

τ (E) = \frac{2 N - (N_{L} + N_{R})}{2 π i} log \frac{E}{Λ} .

(11.3.15)

In the other regime when $| μ_{i} |, | {\underset{̲}{a}}_{i} | ≫ Λ$ , we can use the redeﬁning trick to ﬁnd singularities on the Coulomb branch. For example, deﬁning $\tilde{z} = z ∕ Λ^{N - N_{L}}$ , the curve is

\frac{\prod_{i = 1}^{N_{L}} (x - μ_{i})}{\tilde{z}} + \tilde{z} Λ^{2 N - N_{R} - N_{L}} \prod_{i = 1}^{N_{R}} (x - μ_{i}^{'}) = x^{N} + u_{2} x^{N - 2} + \dots + u_{N - 1} x + u_{N} .

(11.3.16)

Then the limit $Λ \to 0$ can be taken, which gives

\frac{\prod_{i = 1}^{N_{L}} (x - μ_{i})}{\tilde{z}} = \prod_{i = 1}^{N} (x - {\underset{̲}{a}}_{i}) .

(11.3.17)

This means that whenever ${\underset{̲}{a}}_{i} = μ_{s}$ for some $i$ and $s = 1, \dots, N_{L}$ , the curve splits into two, because the equation can be factorized. The same can be done for the variable $w = 1 ∕ z$ . Then we also ﬁnd singularities when ${\underset{̲}{a}}_{i} = μ_{s}^{'}$ for some $i$ and $s = 1, \dots, N_{R}$ . In total, these reproduce the semiclassical, weakly-coupled physics of $S U (N)$ theory with $N_{f} = N_{R} + N_{L}$ hypermultiplets in the fundamental representation. The situation is summarized in Fig. 11.5.

Figure 11.5:

S U (N)

theory with ﬂavors

We have a sphere $C$ described by the coordinate $z$ . The curve $Σ$ is an $N$ -sheeted cover of $C$ . We have one M5-brane wrapping $Σ$ . We call the 6d theory living on $C$ the $𝒩 = (2, 0)$ theory of type $S U (N)$ . Roughly speaking, it arises from $N$ coincident M5-branes.

Consider $A r g z$ as the sixth direction $x_{6}$ , and $log | z |$ as the ﬁfth direction $x_{5}$ . Reducing along the $x_{6}$ direction, we have a 5d theory on a segment. The 5d theory is the maximally supersymmetric Yang-Mills theory with gauge group $S U (N)$ . The term

\frac{Λ^{N - N_{L}} \prod_{i = 1}^{N_{L}} (x - μ_{i})}{z}

(11.3.18)

in the curve can be thought of deﬁning a certain boundary condition on the left side of the ﬁfth direction. We regard it as giving $N_{L}$ hypermultiplets in the $S U (N)$ fundamental representation there. Similarly, the term

z Λ^{N - N_{R}} \prod_{i = 1}^{N_{R}} (x - μ_{i}^{'})

(11.3.19)

is regarded as the boundary condition such that $N_{R}$ fundamental hypermultiplets there. By further reducing the theory along the ﬁfth direction, we have $S U (N)$ gauge theory with $N_{f} = N_{L} + N_{R}$ fundamental hypermultiplets in total. We saw that the eﬀect of the boundary conditions becomes noticeable around when

log | z_{R} | \sim (N - N_{R}) log \frac{E}{Λ} < 0, log | z_{L} | \sim (N - N_{L}) log \frac{Λ}{E} > 0 .

(11.3.20)

In the ﬁve dimensional Yang-Mills, we have monopole strings, which have ends around $| z_{R} |$ and $| z_{L} |$ . From the four-dimensional point of view, $log | z_{L} | ∕ | z_{R} |$ then controlled the mass of the monopoles, which then gave the one-loop running of the theory.

Note that from the four-dimensional point of view, the split of $N_{f}$ into $N_{R}$ and $N_{L}$ is rather arbitrary. In fact, by redeﬁning $z$ , we can easily come to the form of the curve given by

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

(11.3.21)

where we deﬁned $μ_{N_{L} + i} : = μ_{i}^{'}$ . In this form the symmetry exchanging all $N_{f}$ mass parameters is manifest.

From the higher-dimensional perspective, it is however sometimes convenient to stick to the situation where the equation of the Seiberg-Witten curve $Σ$ is of degree $N$ regarded as a polynomial in $x$ . This guarantees that $Σ$ is always an $N$ -sheeted cover of the ultraviolet curve $C$ . Numerically, this condition means that the boundary condition such as (11.3.18) and (11.3.19) also has degrees less than or equal to $N$ . This imposes the constraint $N \geq N_{L, R}$ , and therefore $2 N \geq N_{f}$ . This is the condition that the theory is asymptotically free or asymptotically conformal.

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11.3 SU(N) theory with fundamental ﬂavors

11.3.1 Nf = 1

11.3.2 General number of ﬂavors

11.3 $S U (N)$ theory with fundamental ﬂavors

11.3.1 $N_{f} = 1$