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Constant gauge-coupling matrix from the quadratic $N = 2$ prepotential

Physical scope

A finite-dimensional calculation of the gauge-coupling Hessian in the adopted quadratic ultraviolet N=2 model; it does not establish an exact low-energy effective action.

Example

For the adopted microscopic prepotential $F_{U V} (Φ) = \frac{i}{2 g^{2}} κ_{C} (Φ, Φ)$ with $g > 0$ and $κ (T_{a}, T_{b}) = δ_{a b}$ , the Hessian $F_{a b} = \partial_{a} \partial_{b} F_{U V}$ is the constant matrix $i δ_{a b} / g^{2}$ . Its imaginary part is positive definite and independent of the scalar field $Φ$ . This is a property of the adopted quadratic ultraviolet model; it does not assert a non-polynomial low-energy prepotential.

Example references

Four-dimensional $N = 2$ SU(2) vector and hypermultiplet model

Facts & Assumptions

Given: The quadratic ultraviolet prepotential and positive dimensionless coupling in the $N = 2$ model.

[A1]

Four-dimensional $N = 2$ SU(2) vector and hypermultiplet model: $g > 0$ , $κ (T_{a}, T_{b}) = δ_{a b}$ , and $F_{U V} (Φ) = \frac{i}{2 g^{2}} κ_{C} (Φ, Φ)$ .

Verification

Given: [A1] and coordinates $Φ = \sum_{a = 1}^{3} Φ^{a} T_{a}$ on $s u (2)_{C}$ .

1.1givenA1algebra

Bilinearity and the normalized basis give $F_{U V} = \frac{i}{2 g^{2}} \sum_{a = 1}^{3} (Φ^{a})^{2}$ . Differentiating this polynomial first in $Φ^{a}$ and then in $Φ^{b}$ yields $F_{a b} = i δ_{a b} / g^{2}$ , with no dependence on $Φ$ .

2.1givenA1step 1.1algebra∎

Since $g > 0$ , for every nonzero $z \in C^{3}$ the Hermitian quadratic form of the imaginary part is $\sum_{a} ∣ z^{a} ∣^{2} / g^{2} > 0$ . Thus the coupling matrix is positive definite, including at $Φ = 0$ .

Depends on

Four-dimensional $N = 2$ SU(2) vector and hypermultiplet model

Constant gauge-coupling matrix from the quadratic N=2N=2 prepotential