9.5 Generalization

9.5.1 Trivalent diagrams

Figure 9.12: Trivalent diagrams and corresponding ultraviolet curves

The trifundamental hypermultiplet consisting of 𝒩=1 chiral multiplets qaαu for a,α,u = 1, 2 played the central role in the analysis so far. Let us introduce a shorthand notation for it, by representing it by a trivalent vertex with labels A, B, C as in Fig. 9.12, signifying the symmetries SU(2)A, SU(2)B, SU(2)C, acting on the indices a, α, u respectively.

The ultraviolet curve for this system is given by a sphere with three punctures A, B, C, and the Seiberg-Witten curve is given by

Σ : λ2 ϕ(z) = 0 (9.5.1)

where ϕ(z) is given by the condition that the coefficients of the double poles are given by μA2, μB2, μC2 at each of the punctures A,B,C, as in (9.3.7).

Now, the SU(2) theory with four flavors can be obtained by taking two copies of trifundamentals, and coupling an SU(2) gauge multiplet to them. We denote it by taking two trivalent vertices, and connecting them by a line, as shown in Fig. 9.12. We put the exponentiated coupling q on the connecting line.

Starting with this trivalent diagram, we can easily write down the Lagrangian of the theory. The free parameters in the theory are the mass parameters μA,B,C,D and the UV coupling q. The ultraviolet curve of this system is given by a sphere with four punctures A, B, C, D, and the Seiberg-Witten curve is given by

Σ : λ2 ϕ(z) = 0 (9.5.2)

where ϕ(z) is given by the condition that its residues are given by μX2 at each of the punctures X = A,B,C,D as in (9.2.10). The triality of the SU(2) theory with four flavors, shown already in Fig. 9.11, can be depicted in terms of the trivalent diagrams as in Fig. 9.13.

Figure 9.13: Triality, using trivalent diagrams.

This way, we can regard the trivalent diagram as a shorthand to represent the UV Lagrangian. The Seiberg-Witten solution to this given UV Lagrangian theory is given just by replacing each trivalent vertex with a three punctured sphere, and a connecting line with a connecting tube. This is a surprisingly concise method to obtain the Seiberg-Witten solutions to 𝒩=2 gauge theories.

9.5.2 Example: torus with one puncture

Figure 9.14: SU(2) with adjoint plus one free hyper

Let us see a few examples. First, take one trivalent vertex, and connect two out of the three lines by an edge, see Fig. 9.14. We start from a trifundamental described by 𝒩=1 chiral multiplets qaαu, but we couple the same SU(2) gauge multiplet to the index a and α. Then the combination (a,α) is in the tensor product of two spin-12 representations. Therefore we can split it into a triplet and a singlet, with additional index u = 1, 2:

qaαu qiu,qu, (9.5.3)

where i = 1, 2, 3 is the index for the triplet of SU(2).

In total, we just have one full hypermultiplet in the triplet, and another full hypermultiplet in the singlet which is completely decoupled. Therefore this is essentially the 𝒩=2 SU(2) theory, or equivalently the 𝒩=4 SU(2) theory with mass deformation to the hypermultiplet in the adjoint representation. The adjoint mass μ is associated to the remaining one SU(2) flavor symmetry.

Its Seiberg-Witten solution is given by connecting two punctures of a three-punctured sphere by a tube. As shown in Fig. 9.14, the ultraviolet curve is a torus with one puncture. The Seiberg-Witten curve is then

λ2 ϕ(z) = 0 (9.5.4)

where z is now a coordinate of the torus, which we take to be the complex plane with the identification z z + 1 z + τ. As the origin of the coordinate is arbitrary, we put the puncture at the origin. The ϕ(z) is given by the condition that it has a double pole with a given strength at z = 0. This uniquely fixes the form of ϕ(z) to be

ϕ(z) = (μ2(z; τ) + u)dz2 (9.5.5)

where is the Weierstraß function, and u is the Coulomb branch vev u = trΦ22.

Now it is clear that the theory at the coupling given by τ and the same theory at the coupling given by τ = 1τ are equivalent after exchanging the monopoles and the adjoint quarks. The space of the coupling can be identified with the moduli space 1 of the tori, i.e genus-1 Riemann surfaces, which is given by

1 = SL(2, ) (9.5.6)

where is the upper half plane where τ takes the value in, and SL(2, ) is the modular group exchanging the edges of the torus. The duality group can be identified with the modular group.

9.5.3 Example: sphere with five punctures

Figure 9.15: An SU(2)2 theory and its curve

As another example, take three trivalent vertices and connect them as shown in Fig. 9.15. The leftmost trivalent vertex counts as Nf = 2 flavors for SU(2)1, and the rightmost one counts as Nf = 2 flavors for SU(2)2. In addition, we have a hypermultiplet coming from the central trivalent vertex, qaiu where a is for SU(2)1 and i is for SU(2)2. The remaining index u = 1, 2 is an index for the flavor symmetry. In older literature it was more customary to denote this hypermultiplet charged under SU(2)1 ×SU(2)2 using 𝒩=1 chiral multiplets (Qai,Q̃ia) which are

Qai = qaju=1𝜖ij,Q̃ia = qbiu=2𝜖ab. (9.5.7)

This is usually called the bifundamental multiplet charged under SU(2)1 ×SU(2)2.

The Seiberg-Witten solution to this theory is easily found, as shown in Fig. 9.15. We start from three spheres, described by complex coordinates z1, z2, and z3. The punctures A,B are at z1 = 0, 1; the puncture C is at z2 = 1; the punctures D,E are at z3 = 1,, respectively. To connect z1 = and z2 = 0, we introduce w1 = 1z1 and require the relation w1z2 = q. This simply means that we have via z1 = qz2. Similarly, by connecting z2 = and z3 = 0, we have z2 = qz3. Now we introduce z = z3 to describe the coordinate on the resulting sphere with five punctures. Then the punctures are at z = 0, qq, q, 1 and , each representing an SU(2) flavor symmetry which we call SU(2)A,B,C,D,E respectively. The gauge couplings of SU(2)1 ×SU(2)2 can be identified with q and q.

Let us denote the mass parameters associated to the flavor symmetries by μA,B,C,D,E. The Seiberg-Witten curve is

λ2 ϕ(z) = 0 (9.5.8)

where ϕ(z) needs to satisfy the asymptotic behavior

ϕ(z) μX2 zX2dzX2 (9.5.9)

where zX for X = A,B,C,D,E is a local coordinate on the ultraviolet curve such that the puncture X is at zX = 0. From the conditions at A, B, C, D, we find that ϕ is given by

ϕ(z) = P(z) z2(z 1)2(z q)2(z qq)2dz2 (9.5.10)

where P(z) is a polynomial. To impose the condition at E, we go to the coordinate w = 1z. For ϕ(z) to behave as dw2w2, P(z) can have terms of up to z6. We see that ϕ(z) has seven coefficients. Five combinations are mass parameters, and two linear combinations that do not shift the coefficients of the double poles are the Coulomb branch parameters u = trΦ22 and u = trΦ22 of two gauge multiplets SU(2)1,2. From the Seiberg-Witten solution, we see that this theory has strong-weak coupling dualities where five flavor symmetry groups SU(2)A,B,C,D,E can be arbitrarily permuted, with an appropriate change of the couplings (q,q) of the two gauge groups. This extended duality was first found in [56].

9.5.4 Example: a genus-two surface

Figure 9.16: Two SU(2)3 theories and their curves

As the third example, let us take two trivalent vertices and connect them with three edges. There are two topologically distinct ways to do this, as shown on the left hand side of Fig. 9.16.

The upper theory is an SU(2)l ×SU(2)m ×SU(2)r gauge theory. There are half-hypermultiplets which are in

3 2 1,1 2 3 (9.5.11)

and one full hypermultiplet charged under SU(2)m. Note that the trivalent-graph construction does not allow us to consider theories with non-zero mass term for this last full hypermultiplet. The lower theory is an SU(2)1 ×SU(2)2 ×SU(2)3 theory with two half-hypermultiplets in the trifundamental representation. Again, the trivalent-graph construction does not allow us to introduce non-zero mass term for this full hypermultiplet in the trifundamental.

The Seiberg-Witten solution is again easily obtained. To obtain the ultraviolet curve, we replace two trivalent vertices with three-punctured spheres, and connect pairs of punctures with tubes. We see that both are given by a smooth genus-2 surface. The Seiberg-Witten curve is a further double cover given by

λ2 ϕ(z) = 0 (9.5.12)

where z is a complex coordinate of the genus-2 surface, and ϕ(z) is a smooth quadratic differential on it. The space of quadratic differentials on a fixed genus-2 surface is complex three dimensional, which we identify with the Coulomb branch vevs ui = trΦi22.

Now it is clear that we can continuously deform the upper theory to the lower theory by tuning the gauge couplings. The non-perturbative space of couplings can be identified with the moduli space 2 of genus-2 Riemann surfaces, which is complex three dimensional. The duality group is identified with the mapping class group 𝒢2 of the genus-2 surface, and 2 = 𝒯2𝒢2 where 𝒯2 is the Teichmüller space of the genus-2 Riemann surface, compare the genus-1 case (9.5.6).

Now a somewhat surprising mathematical fact is that 𝒯2 is equivalent to three copies of the upper half plane 3 in the smooth sense, but not in the holomorphic sense12 :

𝒯2 3in the smooth sense, 𝒯23 in the holomorphic sense. (9.5.13)

Naive perturbative analysis tells us that the space of the couplings of SU(2)3 is just three copies of the upper half plane:

(τ1,τ2,τ3) 3 (9.5.14)

including the complex structure. Therefore, we find that the non-perturbative corrections can make a rather drastic change in the complex structure of the parameter space of supersymmetric theories.

9.5.5 The curve and the Hitchin field

We learned that writing the Seiberg-Witten curve in the form

Σ : λ2 ϕ(z) = 0 (9.5.15)

is very useful for the understanding of the system. This way of presenting the curve is closely related to the so-called Hitchin system on the ultraviolet curve C. Explaining this technique would make another lecture note. A starting point for the reader is e.g. Sec. 3 of [51].

Let us at least present a very crude aspect of it. We consider a meromoprhic one-form φ(z) which is a traceless 2 × 2 matrix. Then, we can form an equation of the form

det(λ φ(z)) = 0 (9.5.16)

for a one-form λ. Identifying (9.5.15) and (9.5.16), we find

1 2trφ(z)2 = ϕ(z). (9.5.17)

This matrix field φ(z) is called the Hitchin field.13 It is possible to understand the existence of such a complex adjoint field on the ultraviolet curve using string duality, but explaining it is outside the aim of this lecture note.

The condition on the field ϕ(z) at a puncture at zX = 0 associated to a mass term μX was given in (9.5.9):

ϕ(z) = μX2dzX2 zX2 + (less singular terms). (9.5.18)

This translates to the condition for the Hitchin field φ(z) given by

φ(z) μX 0 0 μX dzX zX +(less singular terms). (9.5.19)

Here, the symbol means that the terms on the left and the right are conjugate.

Now, let us consider what happens when we turn off μX to zero. In the description (9.5.18), we find that the boundary condition becomes

ϕ(z) = cdzX2 zX + (less singular terms) (9.5.20)

for some constant c. In the description (9.5.19), it is not that the residue just becomes a zero matrix. We note that

μX 0 0 μX μX 1 0 μX (9.5.21)

as long as μX0, and we can take the limit μX 0 on the right hand side. This means that the massless limit results in the boundary condition of the form

φ(z) 01 0 0 dzX zX +(less singular terms). (9.5.22)

Using (9.5.17), one finds that this reproduces the condition (9.5.20).

At this stage, one might not find the advantange of using φ(z) instead of ϕ(z) very much. It turns out, however, that when we discuss a generalization to SU(N) gauge theories or gauge theories with more complicated gauge groups, it turns out to be crucial.