12.2 SU(N) quiver theories and tame punctures

12.2.1 Quiver gauge theories

To this aim, we introduce a new diagrammatic notation for 𝒩=2 gauge theories. This notation is related to but distinct from the trivalent one introduced in Sec. 9.5.

A diagram is composed of squares and circles with integers written in them, and edges connecting squares and circles. A square with N stands for a U(N) flavor symmetry, and a circle with N an SU(N) gauge symmetry. An edge connecting two objects with N and M written within them represents a hypermultiplet (Qia,Q̃ai) where i = 1,,N and j = 1,,M. They are in the tensor product of the fundamental representation of SU(N) and SU(M), and is called the bifundamental hypermultiplet. Such a diagram specifies an 𝒩=2 gauge theory. This class of theories is often called quiver gauge theories.

Figure 12.5: SU(N) quiver theory

The simplest cases are when all the squares and circles have the same number N written in them, see Fig. 12.5. The first one in the figure is just a bifundamental hypermultiplet. The second one is the SU(N) theory with 2N flavors. The last one is an SU(N)1 ×SU(N)2 theory, so that

Note that both SU(N)1 and SU(N)2 have zero beta function.

Their Seiberg-Witten solutions can be obtained by combining the knowledge we acquired so far. Namely, each edge corresponds to the bifundamental hypermultiplet of SU(N) ×SU(N), which we know to come from a three punctured sphere of 6d theory of type SU(N), with two full punctures and one simple puncture. All we have to do then is to prepare one such sphere for each edge, and connect pairs of full punctures by tubes. For example, the Seiberg-Witten solution for the third theory in Fig. 12.5 is given by

λN + ϕ2(z)λN 2 + + ϕN(z) = 0 (12.2.1)

where ϕk(z) has five singularities, such that two at z = 0, = are full and the other three at z = 1, q and qq are simple.

For simplicity, let us assume that all the mass parameters are zero. Then, from the condition of the order of the poles of the singularities given in (12.1.11), the fields ϕk(z) are uniquely fixed to be

ϕk(z) = uk(1)z + uk(2) (z 1)(z q)(z qq) dzk zk 1. (12.2.2)

The reader should check that it has the correct behavior at z = . This theory is superconformal, as both SU(1)1 and SU(2)2 have zero one-loop beta function. This is reflected by the fact that the variables appearing in the Seiberg-Witten curve (12.2.1) can be assigned scaling dimensions in a natural way. The differential λ should have scaling dimension one, since its integral gives the BPS mass formula: [λ] = 1. We then set [z] = 0 and [ϕk] = k. This means that uk(i = 1, 2) should be two Coulomb branch operators with scaling dimension k. Indeed, we are dealing with an SU(N)2 gauge theory which is superconformal, and there are exactly one Coulomb branch operator of scaling dimension k for k = 2,,N.

12.2.2 𝒩=2 theory

Figure 12.6: SU(N) plus adjoint: the 𝒩=2 theory.

A rather degenerate situation arises when we take just one bifundamental hypermultiplet (Qia,Q̃ia) and couple one SU(N) gauge multiplet to both indices, see Fig. 12.6. The N × N hypermultiplet components now behave as an adjoint representation plus a singlet. The singlet part is completely decoupled, and therefore the theory is essentially the SU(N) gauge theory with an adjoint hypermultiplet. When massless this is the 𝒩=4 super Yang-Mills, whereas it is called 𝒩=2 theory when massive. The Seiberg-Witten solution can then be obtained by taking a three-punctured sphere and connecting the two full punctures. We end up having a torus with one simple puncture. This solution was first found in [70], to which the readers should refer for details.

12.2.3 Linear quiver theories

So far we learned how to solve gauge theories shown in Fig. 12.5. They have the gauge group

SU(N) × ×SU(N) × ×SU(N) (12.2.3)

with bifundamentals between adjacent SU(N) groups, and additional N flavors each for the first and the last SU(N) groups. All SU(N) groups have zero beta function.

Let us consider a slight generalization of this class of theories. The gauge group is of the following form

SU(N) × ×SU(N) ×SU(Nk) ×SU(Nk1) ×SU(N2) ×SU(N1). (12.2.4)

We put the bifundamental hypermultiplets between adjacent SU(N) and SU(N). Such gauge theories are often called linear quiver gauge theories, since the gauge factors are arranged in a linear fashion.

Here, we introduce additional flavors for every SU group, so that they all have zero beta functions. Define N0 = 0 and Nk+1 = N. Then the condition we need to impose is

Ni1 + Ni+1 + ni = 2Ni,i = 1,,k (12.2.5)

where ni is the number of additional fundamental hypermultiplet for SU(Ni). Since ni 0, we have si si+1 where si = Ni Ni1. Clearly i=1k + 1si = N.

A decreasing sequence of integers s1 s2 sk+1 whose sum is N is called a partition of N. Then we can phrase our finding here by saying that this type of gauge theory can be characterized by a partition of N. A partition can be graphically represented by a Young diagram. Here we draw it by arranging boxes so that the widths of the rows are given by si. Examples are shown for N = 4 on the left hand side of Fig. 12.7. There, additional ni flavors are shown by connecting a box ni to a circle Ni.

Figure 12.7: SU(N) tame punctures

What is the Seiberg-Witten solutions of this class of theories? There are a few independent methods to arrive at the solutions. Originally they are obtained using a configuration of branes in type IIA string theory and lifting it to M-theory [6]. We now also have a field theoretical derivation in terms of instanton computation [71]. In this subsection, we just state the results, and give a few justification. We will come back to this point in more details in Sec. 12.5.

The Seiberg-Witten solution is obtained by the following procedure. First, consider a sphere of 6d theory of type SU(N), realizing the theory where all Ni is equal to N. It was given by the Seiberg-Witten curve of the form

λN + ϕ2(z)λN 2 + + ϕN(z) = 0. (12.2.6)

As explained above, we have two full punctures and a number of simple punctures. We then replace one full puncture at z = with a new type of puncture labeled by the Young diagram, see the right hand side of Fig. 12.7. These new types of punctures, together with the simple and the full punctures introduced already, are called tame SU(N) punctures.

The change of the type of the puncture is the change of the structure of the singularities of the fields ϕk(z). We can also write the curve (12.2.6) as

det(λ φ(z)) = 0 (12.2.7)

where φ(z) is a meromorphic one-form which is a traceless N × N matrix, as we did for the SU(2) case in Sec. 9.5.5. Then ϕk(z) is given by an elementary symmetric degree-k polynomial combination of the eigenvalues of φ(z). Then the structure of the singularities of ϕk(z) can be described also by the structure of the residue of φ(z).

12.2.4 Tame punctures

We already saw a full puncture carries the flavor symmetry SU(N), and a simple puncture U(1). To correctly reproduce the flavor symmetry of the total theory, the singularity at z = labeled by the Young diagram s1 s2 sk+1 needs to be associated to the flavor symmetry

S[U(n1) ×U(n2) ×U(nk)] (12.2.8)

where the S[] means that we remove the diagonal U(1) of the following unitary gauge groups.

Figure 12.8: The Young diagram shown here has (si) = (4, 2, 2), (νk) = (1, 1, 1, 1, 2, 2, 3, 3), (pk)k=18 = (0, 1, 2, 3, 3, 4, 4, 5) and (ti) = (3, 3, 1, 1). The standard convention is to use the column heights (ti) to label punctures.

The description becomes complete once we describe how the fields ϕk(z) behave at this new puncture. When the hypermultiplets are all massless, the rule is given as follows. Given a Young diagram with row widths s1 s2 , define pk = k νk where

(ν1,ν2,,νN) = (1,, 1 s1, 2,, 2 s2,, ) (12.2.9)

Then ϕk(z) should have a pole of order pk at the puncture. For an example, see Fig. 12.8.

In terms of the N × N matrix-valued one-form φ(z) the statement is somewhat simpler. Namely, the residue of φ(z) at the puncture should be given by

Resφ(z) Js1 Js2 Jsk (12.2.10)

where Js is an s × s Jordan block:

Js = 01 01 01 0 s. (12.2.11)

It is a good exercise to check that the pole orders pk of ϕk(z) can be reproduced by plugging in (12.2.10) into (12.2.7) and comparing it with (12.2.6).

When the hypermultiplets are massive, the rule goes instead as follows. Take the same Young diagram, but describe it with column heights t1 t2 tx where x is the number of columns. Then λ should have N residues with following structure:

(μ1,,μ1 t1,μ2,,μ2 t2,, ) (12.2.12)

where we need to impose

tiμi = 0. (12.2.13)

This is equivalent to say that the residue of the matrix-valued one-form φ(z) should be conjugate to a diagonal matrix with entries given by (12.2.12).

We identify these residues with the mass parameters associated to the flavor symmetry (12.2.8). There are ni mass parameters μa(i), a = 1,,ni for each U(ni). We then make the identification

(μ1(1),,μn1(1); μ1(2),,μn2(2); ; μ1(k),,μnk(k)) = (μ1,μ2,,μx). (12.2.14)

Note that ni equals the number of columns x. The individual ni corresponds to the number of columns of a certain given height, say h, then there is an index a such that

ta = ta+1 = = ta+ni1 = h. (12.2.15)

Then the Weyl group of the U(ni) flavor symmetry can be identified with the permutation of the columns of height h.

It is conventional in the 𝒩=2 literature to label the punctures using column heights (ti). The full puncture is then associated to the Young diagram (1, 1,, 1), and the simple puncture has the Young diagram (N 1, 1). We can indeed check that the general formulas (12.2.9) and (12.2.12) reproduce (12.1.6) and (12.1.11). Note also that the puncture of type (N) does not have poles at all. This corresponds to an absence of the puncture.

Figure 12.9: SU(2) tame punctures

Let us apply this general discussion to the particular case N = 2 which we discussed extensively in Sec. 9. There, we introduced a different diagrammatic notation using trivalent vertices, reflecting special properties of SU(2), see Fig. 12.9. In the current approach, we see that both the full puncture and the simple puncture for N = 2 have the Young diagram (1, 1), thus losing the distinction. The only other type of puncture is (2), which corresponds to the absence of puncture in the first place. Therefore the construction in this section does not give anything new for N = 2.

12.2.5 Tame punctures and the number of Coulomb branch operators

Let us check that the prescription described above reproduces the expected number of Coulomb branch operators. Compare, for example, the first and the fourth rows of Fig. 12.7. The Seiberg-Witten solutions are both given by

λ4 + ϕ2(z)λ2 + ϕ3(z)λ + ϕ4(z) = 0. (12.2.16)

In both cases, ϕk(z) has one full puncture at z = 0 and five simple punctures at z = zi. The puncture at z = changes types. For the theory at the first row, the puncture at z = is a full puncture, where ϕk(z) has poles of order k 1. This determines the fields ϕk(z) to be given by

ϕk(z) = uk(1) + uk(2)z + uk(3)z2 + uk(4)z3 i5(z zi) dzk zk 1. (12.2.17)

Note that the degree of the polynomial in the numerator is fixed by the order of the pole at z = . We identify uk(i) as the dimension-k Coulomb branch operator of the i-th SU(4) gauge group.

Now change the type of the puncture at z = . The allowed order of the pole there is reduced by νk as given in (12.2.9). In this particular case, the orders of the poles for ϕ2(z), ϕ3(z), ϕ4(z) are reduced by 0, 1, 2 respectively. This reduces the degree of the polynomials in the numerator of (12.2.17) by 0, 1, 2 respectively, resulting in

ϕ2(z) = u2(1) + u2(2)z + u2(3)z2 + u2(4)z3 i5(z zi) dz2 z (12.2.18) ϕ3(z) = u3(1) + u3(2)z + u3(3)z2 i5(z zi) dz3 z2 (12.2.19) ϕ4(z) = u4(1) + u4(2)z i5(z zi) dz4 z3 . (12.2.20)

We identify uk(i) as a dimension k Coulomb branch operator for the i-th gauge group. We see that the third gauge group now has the Coulomb branch operators of dimension 2 and of dimension 3, and that the fourth gauge group only has the Coulomb branch operator of dimension 2. This agrees with our claim that the gauge group is now SU(4) ×SU(4) ×SU(3) ×SU(2).

This analysis of the number of the Coulomb branch operators can be extended to arbitrary N and to arbitrary Young diagram. By a straightforward but somewhat cumbersome combinatorial computation we see that the pole structure (12.2.9) reproduces the structure of the gauge group as given in (12.2.4).

12.2.6 Tame punctures and the decoupling

Now let us study what happens when we make the coupling of the last gauge group in (12.2.4) very weak. When we completely turn off the coupling, we lose the last gauge group SU(Nk). The new last gauge group is SU(Nk1), which is now coupled to Nk + nk1 hypermultiplets in the fundamental representation. Note that Nk of them originally came from the bifundamental hypermultiplet for SU(Nk1) ×SU(Nk).

Figure 12.10: Decoupling one.

This process for the quiver tail characterized by the Young diagram (3, 1) is shown on the right hand side of Fig. 12.10. In terms of the ultraviolet curve, turning off the coupling of the last gauge group corresponds to splitting off the last two punctures. When we completely decouple the gauge group, we find a new puncture emerging. The type of this new puncture can be determined by the rule explained above, from the resulting gauge theory with one less gauge group. In this case, the newly appearing puncture on the left has the Young diagram (2, 1, 1). The decoupled three-punctured sphere on the right hand side represents one hypermultiplet in the doublet representation of SU(2). We intentionally do not discuss the new puncture arising on this decoupled three-punctured sphere on the right; for more details, see [7273].

Figure 12.11: Decoupling the next.

We can continue the process. Decoupling the next gauge group, the Young diagram becomes (1, 1, 1, 1), i.e. the full puncture. The situation is shown in Fig. 12.11. The decoupled three-punctured sphere on the right hand side represents two hypermultiplets in the triplet representation of SU(3).

Note that SU(3) gauge group before the complete decoupling can be thought of as gauging the SU(3) subgroup of the SU(4) flavor symmetry of the full puncture, as shown in the second row of the figure. This splits four fundamental flavors coupled to SU(4) into a set of three flavors and an additional one flavor. The SU(3) gauge group makes the first three into the bifundamental hypermultiplet of SU(4) ×SU(3), and one flavor remains to couple just to SU(4) on the upper row.

Another example of decoupling process for the puncture of type (2, 2) is shown in Fig. 12.12. The decoupled three-punctured sphere on the right hand side represents an empty theory.

Figure 12.12: Another example of decoupling.