7.1 General structures of the Higgs branch Lagrangian

First, recall a general 𝒩=1 theory containing only scalars and fermions. Such a theory can be described by the Lagrangian

d4𝜃K(Φ ̄j̄, Φi) = gij̄μϕiνϕ̄j̄ + (7.1.1)


gij̄ = 2K ϕiϕ̄j̄. (7.1.2)

This defines a Kähler manifold. In particular, the manifold is naturally a complex manifold. This fact is almost implicit in our formalism, since the chiral multiplets are by definition complex valued. It is instructive to recall why this was so: we have the basic supersymmetry transformation

δαϕ = ψα,δα̇ψα = iσαα̇μμϕ (7.1.3)

A convention independent fact is that δαδα̇ acting on a complex scalar involves a multiplication by i. In terms of the real and imaginary parts of ϕ, we can schematically write this fact as

δα̇δα Reϕ Imϕ = σα̇αμμI Reϕ Im ϕ (7.1.4)

where the matrix

I = 0 11 0 (7.1.5)

has the property I2 = 1. This is the crucial matrix defining the complex structure of the scalar manifold of an 𝒩=1 theory.

Now, let us consider an 𝒩=2 theory consisting of scalars and fermions only. Note that this means that there are no 𝒩=2 vector multiplets. This theory has two sets of 𝒩=1 supersymmetries δαi = 1, 2. In addition,

δα(c) := c1δα1 + c2δα2 (7.1.6)

also generates an 𝒩=1 sub-supersymmetry when |c1|2 + |c2|2 = 1. Applying the argument in the last paragraph for this 𝒩=1 subalgebra, we find that there are matrices

I(c) = Iana,na = (c̄1,c̄2)σa c1 c2 (7.1.7)

which always satisfy

(I(c))2 = 1. (7.1.8)

Note that na are real and |n1|2 + |n2|2 + |n3|2 = 1, i.e. they are on S2. Denoting (I,J,K) := (I1,I2,I3) for simplicity and expanding (7.1.8), one finds the relations

I2 = J2 = K2 = 1,IJ = K = JI,JK = I = KJ,KI = J = IK. (7.1.9)

This commutation relation of I, J and K is that of a quaternion. A manifold with an action of quaternion algebra on its tangent space is called a hyperkähler manifold. Therefore we found that the scalar manifold of an 𝒩=2 theory without massless vector multiplets is hyperkähler.

Note that the SU(2)R symmetry acts on the doublet (c1,c2), which is restricted to live on the three-sphere |c1|2 + |c2|2 = 1. The map (7.1.7) from this (c1,c2) to na is the standard Hopf fibration S3 S2, and the index a transforms as the triplet of SU(2)R.

Combining with the analysis in Sec. 2.4, we see that general low-energy 𝒩=2 theory has an action of the form

d2𝜃 i 8π τijWαiWjα + cc. +d4𝜃Kv(āj̄,ai) +d4𝜃Kh(q̄t̄,qs) (7.1.10)

such that Kh(q̄t̄,qs) gives a hyperkähler manifold and that there is a prepotential F(ai) giving τij and Kv via the standard formulas (2.4.7), (2.4.8) and (2.4.9).

Note that the hypermultiplet side and the vector multiplet side are completely decoupled. The dependence on the UV gauge coupling is implicitly there in the vector multiplet side. This means that the hypermultiplet side cannot receive quantum corrections depending on the gauge coupling.