9.2 Identification of parameters

9.2.1 Coupling constant

Figure 9.5: In SU(2) Nf = 4, q is related to the UV coupling

Let us first see concretely how q and τUV are related. This can be done by computing a and aD assuming |q| 1. As always, we put the A-cycle at |z| = c, where |q| c 1. Then we easily have

a = 1 2πi Axdz z u. (9.2.1)

The branch points of λ are near 1 and q anyway, and therefore

aD = 1 2πi Bxdz z 1 2πi21qadz z = 2a 2πi log q. (9.2.2)


τU(1) = aD a 2 2πi log q, (9.2.3)

or equivalently

q e2πiτUV (9.2.4)

in the limit of weak coupling; note our convention that τU(1) 2τUV . This relation is often written as

qC = e2πiτUV,C (9.2.5)

with an equality. This should be regarded as a nonperturbative definition of the renormalization and regularization scheme of τUV . Here we added a subscript C to both q and τUV , in order to emphasize that the coupling qC is given by the data on the ultraviolet curve C.

Another common nonperturbative definition of the UV coupling constant is to use the low-energy U(1) coupling τU(1) in the limit when the Coulomb vev is very large |u||μ̃i|:

τUV,Σ := 1 2 lim |u|τU(1) (9.2.6)

This should isolate the SU(2) coupling whose running is stopped at a very large scale given by the Coulomb vev, and can be read off from the complex structure of the Seiberg-Witten curve Σ. That is why we used the subscript Σ here. Let us also define

qΣ = e2πiτUV,Σ. (9.2.7)

This is also a perfectly good scheme, related to the one in (9.2.5) via a finite renormalization.

To explicitly determine the finite renormalization, we note that the Seiberg-Witten curve Σ in the |u| limit is just the torus which is a double-cover of C branched at z = 0, 1,qC,. Then τU(1) in (9.2.6) is given by the complex structure of this Σ. From a basic result in the theory of elliptic functions, we find

qC = λ(τU(1)) = 𝜃2(qΣ2)4 𝜃3(qΣ2)4 = 16qΣ 128qΣ2 + 704qΣ3 3072qΣ4 + . (9.2.8)

This means that τUV,C and τUV,Σ are related by a constant shift of its imaginary part plus instanton corrections.

For more extensive discussions on the non-perturbative finite renormalization, see e.g. Sec. 3.4 and Sec. 3.5 of  [18].

9.2.2 Mass parameters

Next, let us study mass parameters. Recall that λ has mass dimension 1, as its integral give the mass of BPS particles. This means that the five coefficients of the quartic polynomial P(z) are of mass dimension two. We can identify these five coefficients with some combinations of five parameters μi=1,2,3,4 and u. The physical mass parameters are the residues at the poles of λ. Fixing μi fixes four linear combinations of the coefficients of P(z). The sole linear combination which does not change the coefficients of the double poles at z = 0,q, 1, can be identified with the parameter u. Explicitly, we can write

ϕ2(z) = P0(z) (z q)2(z 1)2 dz2 z2 + u (z 1)(z q) dz2 z (9.2.9)

where P0(z) is independent of u.

Let us now go back to the original curve and study the poles of λ. We can compute them from (9.1.5) rather easily when the system is weakly coupled, |q| 1. The residues are μ̃1,2 at z = 0 and μ̃3,4 at z = . When we went from x̃ to x, we subtracted 2 from x. We see that the residues are given by

±μ1 μ2 2 atz = 0, ±μ3 μ4 2 atz = , ±μ1 + μ2 2 atz = q, ±μ3 + μ4 2 atz = 1 (9.2.10)


μi = μ̃i + O(q). (9.2.11)

The variables μ̃i enter rather naturally the Seiberg-Witten curve we guessed in Sec. 6.4.4, whereas the variables μi enter the BPS mass formula. We see that they are related by a finite renormalization.

To understand the combinations in (9.2.10) better, it is helpful to consider the 𝒩=1 superpotential. With four doublet hypermultiplets, we have

W = i(QiΦQ̃i + μiQiQ̃i). (9.2.12)

We combine (Qi,Q̃i) for i = 1, 2, 3, 4 to qI with I = 1,, 8. Then the same term becomes

W qIaqJbΦabδIJ + qIaqJb𝜖abμIJ (9.2.13)

where μIJ is a constant matrix with SO(8) antisymmetric index:

μIJ = μ1 μ1 μ2 μ2 μ3 μ3 μ4 μ4 (9.2.14)

Under the decomposition

SO(8) SO(4) ×SO(4) SU(2)A ×SU(2)B ×SU(2)C ×SU(2)D, (9.2.15)

the entries of SO(8) antisymmetric matrix (9.2.14) decomposes to

SU(2)A SU(2)B SU(2)C SU(3)D diag(±μ1μ2 2 )diag(±μ1+μ2 2 )diag(±μ3+μ4 2 )diag(±μ3μ4 2 ) (9.2.16)

which are exactly the residues we found in (9.2.10) at z = 0,q, 1 and = , respectively. We can regard then that the singularity at z = 0 carries the SU(2)A symmetry, and that the residue of λ there is the mass parameter associated to this SU(2)A symmetry; similarly for SU(2)B at z = q, SU(2)C at z = 1, and SU(2)D at z = .

We call these structures the punctures. From the 6d point of view, we consider a puncture at z = 0 as a four-dimensional object extending along the Minkowski space 3, 1, which somehow carries an SU(2) flavor symmetry on it. We will see various other types of punctures below. To distinguish this one from them, we will call this a regular SU(2) puncture.