2.1 Microscopic Lagrangian

2.1.1 𝒩=1 superfields

Let us now move on to the construction of the Lagrangian with 𝒩=2 supersymmetry. An 𝒩=2 supersymmetric theory is in particular an 𝒩=1 supersymmetric theory. Therefore it is convenient to use 𝒩=1 superfields to describe 𝒩=2 systems. For this purpose let us quickly recall the 𝒩=1 formalism. In this section only, we distinguish the imaginary unit by writing it as i.

An 𝒩=1 vector multiplet consists of a Weyl fermion λα and a vector field Aμ, both in the adjoint representation of the gauge group G. We combine them into the superfield Wα with the expansion

Wα = λα + F(αβ)𝜃β + D𝜃α + (2.1.1)

where D is an auxiliary field, again in the adjoint of the gauge group. Fαβ = i 2σμγ̇βσ̄ναγ̇Fμν is the anti-self-dual part of the field strength Fμν.

The kinetic term for a vector multiplet is given by

d2𝜃 i 8π τtrWαWα + cc. (2.1.2)


τ = 4πi g2 + 𝜃 2π (2.1.3)

is a complex number combining the inverse of the coupling constant and the theta angle. We call it the complexified coupling of the gauge multiplet. Expanding in components, we have

1 2g2trFμνFμν + 𝜃 16π2trFμνF̃μν + 1 g2trD2 2i g2trλ̄γμμλ. (2.1.4)

We use the convention that trTaTb = 1 2δab for the standard generators of gauge algebras, which explain why we have the factors 1(2g2) in front of the gauge kinetic term. The 𝜃 term is a total derivative of a gauge-dependent term. Therefore, it does not affect to perturbative computations. It does affect non-perturbative computations, to which we will come back later.

An 𝒩=1 chiral multiplet Q consists of a complex scalar Q and a Weyl fermion ψα, both in the same representation of the gauge group. In terms of a superfield we have

Q = Q𝜃=0 + 2ψα𝜃α + F𝜃α𝜃α (2.1.5)

where F is auxiliary. The coefficient 2 in front of the middle component is unconventional, but this choice allows us to remove various annoying factors of 2 appearing in the formulas later. The chiral multiplet Q1, can be in an arbitrary complex representation R of the gauge group G. The kinetic term is then

d4𝜃QjeV aρajiQi +d2𝜃W(Q) + (2.1.6)

where V is the vector superfield, ρaji is the matrix representation of the gauge algebra, and W(Q) is a gauge invariant holomorphic function of Q1,.

The supersymmetric vacua is obtained by demanding that the supersymmetry transformation of various fields are zero. The nontrivial conditions come from

δλα = 0,δψα = 0 (2.1.7)

which give

Da = 0,Fi = 0. (2.1.8)

By solving the algebraic equations of motion of the auxiliary fields, we find

Qj̄ρaj̄iQi = 0,W Qi = 0. (2.1.9)

2.1.2 Vector multiplets and hypermultiplets

An 𝒩=2 vector multiplet consists of the following 𝒩=1 multiplets, both in the adjoint of the gauge group G:

λα Aμ𝒩=1 vector multiplet, Φ λ̃α 𝒩=1 chiral multiplet. (2.1.10)

Here, the horizontal arrows signify the 𝒩=1 sub-supersymmetry generator manifest in the 𝒩=1 superfield formalism, and the slanted arrows are for the second 𝒩=1 sub-supersymmetry.

One easy way to construct the second supersymmetry action is to demand that the theory is symmetric under the SU(2) rotation acting on λα and λ̃α. A symmetry which does not commute with the supersymmetry generators is called an R-symmetry in general. Therefore this SU(2) symmetry is often called the SU(2)R symmetry. It is by now a standard technique to combine the supersymmetry manifest in a superfield formalism and an R-symmetry to construct a theory with more supersymmetries, see e.g. [30] for an application. It is also to be kept in mind that there can be and indeed are 𝒩=2 supersymmetric theories without SU(2)R symmetry: there can just be two sets of supersymmetry generators without SU(2) symmetry relating them, see e.g. [3132]. That said, for simplicity, we only deal with 𝒩=2 supersymmetric systems with SU(2)R symmetry in this lecture note.

The Lagrangian is then

Imτ 4π d4𝜃trΦe[V,]Φ +d2𝜃 i 8π τtrWαWα + cc. (2.1.11)

The ratio between the prefactors of the Kähler potential and of the gauge kinetic term is fixed by demanding SU(2)R symmetry.

An 𝒩=2 hypermultiplet4 consists of the following fields:

Q ψ 𝒩=1 chiral multiplet ψ̃ Q̃ 𝒩=1 antichiral multiplet (2.1.12)

They are both in the same representation R of the gauge group. Therefore, the 𝒩=1 chiral multiplets Q and Q̃ are in the conjugate representations of the gauge group. We demand again that the theory is symmetric under the SU(2) rotation acting on Q and Q̃, to have 𝒩=2 supersymmetry.

For definiteness, let us consider G = SU(N) and Nf hypermultiplets Qia, Q̃ai in the fundamental N-dimensional representation, where a = 1,,N and i = 1,,Nf. This set of fields is often called Nf flavors of fundamentals of SU(N). The gauge transformation acts on them as

Qi eΛQi,Q̃i Q̃ieΛ (2.1.13)

where Λ is a traceless N × N matrix of chiral superfields; the gauge indices are suppressed.

The Lagrangian for the hypermultiplets is

cd4𝜃(QieV Qi+Q̃ieV Q̃i)+c(d2𝜃Q̃iΦQi+cc.)+(d2𝜃μjiQ̃jQi+cc.) (2.1.14)

where the gauge index a is suppressed again. The existence of SU(2)R symmetry fixes the ratio of c and c: it can be done e.g. by comparing the coefficients of Qiλψ from the first term and of Q̃iλ̃ψ from the second term. We find the choice c = c does the job. In the following we take c = c = 1 unless otherwise mentioned. The SU(2)R symmetry also demands that the mass term μji satisfies [μ,μ] = 0. Then μ can be diagonalized, and consequently the mass term is often written as

id2𝜃μiQ̃iQi + cc. (2.1.15)

As another example, let us consider the case when we have a hypermultiplet (Z,Z̃) in the adjoint representation, i.e. they are both N × N traceless matrices. The following discussion can easily be generalized to arbitrary gauge group too. When the hypermultiplet is massless, the total Lagrangian has the form

d2𝜃 i 8π τtrWαWα + cc. + Imτ 4π d4𝜃trΦe[V,]Φ + Imτ 4π d4𝜃(Ze[V,]Z + Z̃e[V,]Z̃) + Imτ 4π d2𝜃Z̃[Φ,Z] + cc.(2.1.16)

where we made a different choice of c = c in (2.1.14). This Lagrangian clearly has SU(3)F flavor symmetry rotating Φ, Z and Z̃. This commutes with the 𝒩=1 supersymmetry manifest in the superfield formalism. We also know that this theory has an SU(2)R symmetry rotating Z and Z̃. These two symmetries SU(3)F and SU(2)R does not commute: we find that there is an SO(6)R symmetry, acting on

ReΦ,ImΦ,ReZ,ImZ,ReZ̃,ImZ̃. (2.1.17)

Note that SO(6)R can also be regarded as SU(4)R, as SO(6) and SU(4) have the same Lie algebra. Then the SU(4)R symmetry acts on the four Weyl fermions

λ,λ̃,ψ,ψ̃ (2.1.18)

in the system, where λ and λ̃ are in the 𝒩=2 vector multiplet, and ψ, ψ̃ are in the 𝒩=2 hypermultiplet. We conclude that this system has in fact 𝒩=4 supersymmetry, whose four supersymmetry generators are acted on by SU(4)R SO(6)R. The argument here is another application of the combination of the manifest and non-manifest symmetries in the superfield formalism.

We can add the mass term d2𝜃μZZ̃ + cc. to (16). This preserves the 𝒩=2 supersymmetry but it breaks 𝒩=4 supersymmetry. The resulting theory is sometimes called the 𝒩=2 theory.

Before closing this section, we should mention the concept of half-hypermultiplet. Let us start from a full hypermultiplet (Qa,Q̃a) so that Qa and Q̃a are in the representations R, R̄, respectively. When R is pseudo-real, or equivalently when there is an antisymmetric invariant tensor 𝜖ab, we can impose the constraint

Qa = 𝜖abQ̃b (2.1.19)

compatible with 𝒩=2 supersymmetry, which halves the number of degrees of freedom in the multiplet. The resulting multiplet is called a half-hypermultiplet in the representation R. We will come back to this in Sec. 7.2.