12.5 Tame punctures and Higgsing

Figure 12.26: Change of the type of the puncture.

In Sec. 12.2, we introduced punctures on the ultraviolet curve labeled by Young diagrams in a rather ad hoc manner. Examples for SU(4) case were shown in Fig. 12.7. In this last subsection of the note, we would like to study the meaning of the Young diagram in slightly more detail. For example, how should we understand the process of changing the full puncture to the simple puncture, i.e. the puncture of type (3, 1), shown in Fig 12.26? We will use this particular example of changing the full puncture (1, 1, 1, 1) to the simple puncture (3, 1) as a concrete example throughout this section. The extension to the general punctures should be left as an exercise to the reader. The content of this section is based on an unpublished work with Francesco Benini, done sometime between 2009 and 2010.

The Seiberg-Witten curves are both given by

λ4 + ϕ2(z)λ2 + ϕ3(z)λ + ϕ4(z) = 0. (12.5.1)

In both cases, ϕk(z) has one full puncture at z = 0 and five simples punctures at z = zi. For the first, the puncture at z = was full and for the second, it is a simple puncture, of type (3, 1, 1).

For the first, the fields ϕk(z) are given by

ϕk(z) = uk(1) + uk(2)z + uk(3)z2 + uk(4)z3 i5(z zi) dzk zk 1. (12.5.2)

For the second, they are given by

ϕ2(z) = u2(1) + u2(2)z + u2(3)z2 + u2(4)z3 i5(z zi) dz2 z , ϕ3(z) = u3(1) + u3(2)z + u3(3)z2 i5(z zi) dz3 z2 , ϕ4(z) = u4(1) + u4(2)z i5(z zi) dz4 z3 . (12.5.3)

Here, uk(i) is the dimension-k Coulomb branch operator of the i-th gauge group, and the way to determine them from the pole structure was described around (12.2.17).

It is clear that ϕk(z) in (12.5.3) is obtained by setting u4(3, 4) = u3(4) = 0 in (12.5.2). We will explain below that we can start from the first theory, set the Coulomb branch parameters to this subspace, and then move to the Higgs branch, realizing the second theory.

Figure 12.27: Assignment of new names to the fields.

To facilitate the analysis of the Higgs branch, we introduce new names to the bifundamentals, see Fig 12.27. We name the rightmost SU(N) flavor symmetry SU(N)0, and the gauge groups SU(N)i=1,2,3, from the right to the left. Introduce an auxiliary N-dimensional complex space V i for each of them. For each consecutive pair SU(N)i+1 ×SU(N)i, we have a bifundamental hypermultiplet (Qba,Q̃ab) where a = 1,,N and b = 1,,N are the indices for SU(N)i+1, SU(N)i respectively. We regard Qba as a linear map Ai : V i V i+1 and Q̃ab as a map in the reverse direction Bi : V i+1 V i. Note that each pair (Ai,Bi) comes from one of the several three-punctured spheres comprising the ultraviolet curve, as shown in the figure. Let us say that there are k three-punctured spheres in total.

Let us introduce the notation

Mi := BiAi,Mi := AiBi. (12.5.4)

We will use the trivial identity

trMin = trBiAiBiAi = trAiBiAiBi = trMin (12.5.5)

repeatedly below.

Note that trMi := trMi = trMi is the mass term for the i-th U(1) flavor symmetry, which can be naturally associated to the simple puncture of the i-th three-punctured sphere. We also have two other gauge invariant combinations, namely

M0|traceless := M0 1 NtrM0,Mk|traceless := Mk 1 NtrMk. (12.5.6)

M0|traceless is an adjoint of SU(N) flavor symmetry associated to the full puncture of the rightmost sphere, at z = . Similarly, Mk|traceless is an adjoint of the SU(N) flavor symmetry at the puncture z = 0.

Now, we would like to make a local modification at the puncture z = , by giving a non-zero vev to the adjoint field M0|traceless. Other gauge-invariant combinations trMi for i = 1,,k and Mk|traceless are ‘localized’ at other punctures. So we choose to keep them zero.

The F-term relation from the adjoint scalar in the gauge multiplet for SU(N)i is

Mi+1|traceless = Mi|traceless. (12.5.7)

As we are imposing the condition trMi = 0, we can drop the tracelessness condition and just say

Mi+1 = Mi. (12.5.8)

Then we have the following relations:

trM0n = trM0n = trM1n = trM1n = = trMkn = trMkn = 0 (12.5.9)

for arbitrary n.

This means that the gauge-invariant combination M0, transforming in the adjoint of the SU(N) flavor symmetry, is a nilpotent matrix. They can be put in the Jordan normal form by a complexified SU(N) rotation:

M0 = Jt1 Jt2 , iti = N (12.5.10)

where Jt is the Jordan cell of size t,

Jt = 01 01 01 0 t. (12.5.11)

We again found a partition (ti) of N. We argue below that this partition (ti) is exactly the Young diagram labeling the punctures introduced in Sec. 12.2. To study the effect of the vev (12.5.10), we need to find a choice of hypermultiplet fields (Ai,Bi) solving the F-term and the D-term relations.

Figure 12.28: A graphical notation for matrices.

To write down such a choice, it is useful to introduce a further diagrammatic notation, see Fig 12.28. An N-dimensional vector space V has N basis vectors. Let us denote them by a column of N dots. A matrix whose entries are 0 or 1, from V to V can be represented by a set of arrows connecting the a-th dot for V to the b-th dot for V if and only if the (a,b)-th entry of the matrix is 1. In the center of Fig 12.28 we denoted a Jordan block J4 of size 4. The rightmost diagram of the same figure is for a projector to the last two basis vectors.

Figure 12.29: A particular point on the Higgs branch.

For concreteness, let N = 4, and give a nilpotent vev to M0 of type (3, 1), namely it is given by J3 J1. A solution to the F-term relations are given in Fig. 12.29. There, we see that the unbroken gauge group is now SU(4) ×SU(4) ×SU(3) ×SU(2).

In general, a solution to the F-term relations can be constructed as follows. Let us say we would like to set M0 = X, where X is in a Jordan normal form. We identify the vector spaces V 0 = V 1 = V 2 = . Let us introduce the notation Wi = ImXi and denote the projector to Wi by PWi. We then set

A0 = X,A1 = XPW1,A2 = XPW2, (12.5.12)

and take

B0 = PW1,B1 = PW2,B2 = PW3,. (12.5.13)

Clearly, the remaining gauge group is of the form

×SU(N3) ×SU(N2) ×SU(N1) (12.5.14)


Ni = N dim Wi = N rankXi. (12.5.15)

Define si = Ni Ni1. A short combinatorial computation shows that when X has the type described by a Young diagram whose i-th column from the left has height ti, the sequence (s1,s2,) is such that si is the width of the i-th row from the bottom. This is exactly the rule we already introduced in Sec. 12.2 for the gauge group. Now let us determine the massless matter content of the resulting theory.

An indirect but fast way to determine the matter content is as follows. We started from a superconformal theory without any parameters. After the Higgsing, the only parameter with mass dimensions is the vev of the hypermultiplet fields. By the general decoupling of the hypermultiplet and the vector multiplet side of the Lagrangian, which we discussed in Sec. 7.1, we see that there cannot be any mass terms or dynamical scales in the low-energy theory after the Higgsing. Therefore, the resulting theory is also superconformal. We already determined Ni, and we can only have bifundamental fields or fundamental fields. This shows that SU(Ni) should have exactly

ni = 2Ni Ni+1 Ni1 (12.5.16)

fundamental hypermultiplets in addition.

Figure 12.30: Mass terms generated for scalar fields.

Of course this result can also be obtained by a direct computation of the mass terms of the various fields in the system. Note that originally, there is an 𝒩=1 superpotential trAiΦiBi and trBiΦi+1Ai where Φi is the adjoint scalar of the SU(N)i vector multiplet. As we gave vevs to some components to Ai and Bi, we see that certain components of hypermultiplets scalars and vector multiplet scalars pair up, due to the three-point couplings. One example is shown in Fig 12.30. There, the vev of A1 represented by a red down-left arrow gives a mass term of a component of the vector multiplet scalar of the gauge group for V 2 and a component of B1.

Figure 12.31: Remaining fields after the Higgsing.

We see that always a bifundamental in SU(Ni+1) ×SU(Ni) remains massless. But from a careful analysis of the mass terms, we see that sometimes more charged hypermultiplets remain massless. For example, as shown in Fig 12.31, the whole bifundamental between V 3 and V 2 remains massless. At V 2, SU(4) is broken to SU(3). Therefore, from the point of view of the unbroken SU(4) at V 3, we see there are an SU(4) ×SU(3) bifundamental together with a fundamental of SU(4). This can be generalized to see that the number of additional fundamental hypermultiplets of SU(Ni) is given by (12.5.16).

Figure 12.32: Flavor symmetry assignment.

In Sec. 12.2, we said that the puncture at z = carries all the flavor symmetry associated to the additional ni fundamental hypermultiplets attached to SU(Ni). This sounded somewhat counter-intuitive, since the flavor symmetry SU(ni) looks more associated to the i-th node. Now we understand the physical mechanism operating here. Let us take the puncture of type (3, 1) again for concreteness, see Fig 12.32. The vev X = M0, which is from our rule is given by X = J3 J1, is invariant under the U(1) rotation acting on the three basis vectors, as denoted by black dots in the figure. This symmetry, if unaccompanied by the gauge rotation, does not fix the Higgs vevs Ai and Bi. To make the symmetry compatible with the Higgs vev, we need to rotate at the same time all the other basis vectors connected from the original black dots by the arrows representing Ai and Bi.

We see that the Higgs vevs identify the U(1) flavor symmetry rotating three basis vectors of V 0 and the U(1) flavor symmetry rotating the last basis vector of V 3. After the Higgsing, this latter U(1) symmetry is exactly the flavor symmetry carried by the additional one fundamental hypermultiplet of SU(4) at V 3, denoted by green in the figure. This analysis can be generalized to arbitrary types of punctures.

Summarizing, we found a new interpretation of the punctures introduced in Sec. 12.2. Such a puncture can always be obtained from the full puncture, by first choosing the Coulomb branch vevs to the right subspace, and then giving a nilpotent vev to the hypermultiplet combination M0 which transforms in the adjoint of the flavor SU(N) associated to the full puncture. The vev given to M0 causes some of the other hypermultiplet fields Ai, Bi for i > 0 to have non-zero vevs, breaking the original gauge group ×SU(N) ×SU(N) ×SU(N) to ×SU(N3) ×SU(N2) ×SU(N1).