11.3 SU(N) theory with fundamental flavors

11.3.1 Nf = 1

Next, consider the SU(N) theory with one flavor (Q,Q̃) of bare mass μ. The curve is given by

Σ : ΛN 1(x μ) z + ΛNz = xN + u2xN 2 + + uN. (11.3.1)

Recall that in the semiclassical analysis we saw that a light charged hypermultiplet arises when ai μ. Let us check that the curve written above reproduces this behavior.

First, we introduce a̲i as before, and consider the semiclassical regime when all |a̲i| is far larger than |Λ|. The A-cycle on the ultraviolet curve was |z| = 1 as before. Then we find ai ai̲ + O(Λ) just as was in the case of the pure theory.

To see additional singularities in the weakly-coupled region, define z̃ = zΛN 1. The curve is then

x μ z̃ + Λ2N 1z̃ = xN + u2xN 2 + + uN, (11.3.2)

which can be approximated by

x μ z̃ = xN + u2xN 2 + + uN = (x a̲i) (11.3.3)

in the extremely weakly coupled limit. The equation factorizes and the curve separates into two when a̲i = μ; otherwise the curve is a smooth degree-N covering of the z sphere. This shows that when a̲i = 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 zi+ in the large z region is unchanged, as the structure of the Nf = 1 curve in the large z region itself is unchanged from the pure curve. Then

zi+ (EΛ)N. (11.3.4)

In the small z region, the branch points are around where ΛN 1xz and P(x) are of the same order. Assuming |x||a̲i||E|, we see

zi (ΛE)N 1. (11.3.5)

Then the monopole has the mass

Mmonopole = | 1 2πiBiλ| (11.3.6) |(ai ai+1) 1 2πiΛN1EN1ENΛNdz z | (11.3.7) |(ai ai+1)2N 1 2πi log E Λ|. (11.3.8)

This gives

τ(E) = 2N 1 2πi log 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

Σ : ΛN NL i=1NL(x μi) z + zΛN NR i=1NR(x μi) = xN + u2xN 2 + + uN1x + uN(11.3.10)

where NL,NR N. When NL = NR = N, we need to introduce complex numbers f, f as in the curve of SU(2) with four flavors, (9.1.5):

Σ : f i=1N(x μi) z + f z i=1N(x μi) = xN + u2xN 2 + + uN1x + uN.(11.3.11)

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 SU(2) with four flavors. In the following we mainly discuss the case with less than 2N 1 flavors.

Consider the case when μi and μi are all small. Further, consider the regime where |a̲i| Λ. As always we find ai = a̲i + O(Λ). The branch points are at

|zi+|EN NR ΛN NR ,|zi| ΛN NL EN NL. (11.3.12)

Then we find

Mmonopole |(ai ai+1) 1 2πiΛNNLENNLEN NRΛN NRdz z | (11.3.13) |(ai ai+1)2N (NL + NR) 2πi log E Λ|, (11.3.14)

and therefore the one-loop running is

τ(E) = 2N (NL + NR) 2πi log E Λ. (11.3.15)

In the other regime when |μi|,|a̲i| Λ, we can use the redefining trick to find singularities on the Coulomb branch. For example, defining z̃ = zΛN NL, the curve is

i=1NL(x μi) z̃ +z̃Λ2N NR NL i=1NR(xμi) = xN+u2xN 2++uN1x+uN. (11.3.16)

Then the limit Λ 0 can be taken, which gives

i=1NL(x μi) z̃ = i=1N(x a̲i). (11.3.17)

This means that whenever a̲i = μs for some i and s = 1,,NL, the curve splits into two, because the equation can be factorized. The same can be done for the variable w = 1z. Then we also find singularities when a̲i = μs for some i and s = 1,,NR. In total, these reproduce the semiclassical, weakly-coupled physics of SU(N) theory with Nf = NR + NL hypermultiplets in the fundamental representation. The situation is summarized in Fig. 11.5.

Figure 11.5: SU(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 SU(N). Roughly speaking, it arises from N coincident M5-branes.

Consider Argz as the sixth direction x6, and log |z| as the fifth direction x5. Reducing along the x6 direction, we have a 5d theory on a segment. The 5d theory is the maximally supersymmetric Yang-Mills theory with gauge group SU(N). The term

ΛN NL i=1NL(x μi) 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 NL hypermultiplets in the SU(N) fundamental representation there. Similarly, the term

zΛN NR i=1NR(x μi) (11.3.19)

is regarded as the boundary condition such that NR fundamental hypermultiplets there. By further reducing the theory along the fifth direction, we have SU(N) gauge theory with Nf = NL + NR fundamental hypermultiplets in total. We saw that the effect of the boundary conditions becomes noticeable around when

log |zR| (N NR) log E Λ < 0, log |zL| (N NL) log Λ E > 0. (11.3.20)

In the five dimensional Yang-Mills, we have monopole strings, which have ends around |zR| and |zL|. From the four-dimensional point of view, log |zL||zR| 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 Nf into NR and NL is rather arbitrary. In fact, by redefining z, we can easily come to the form of the curve given by

Λ2N Nf iNf(x μi) z + z = xN + u2xN 2 + uN (11.3.21)

where we defined μNL+i := μi. In this form the symmetry exchanging all Nf 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 NL,R, and therefore 2N Nf. This is the condition that the theory is asymptotically free or asymptotically conformal.