Modular Invariance and Structure of the Exact Wilsonian Action of N=2 Sym

We construct modular invariants on MSU(2), the moduli space of quantum vacua of N = 2 SYM with gauge group SU(2). We also introduce the nonchiral function K(A, Ā) = 2απeSW, where eSW is the Seiberg–Witten metric, e is the Poincaré metric on MSU(2) and α is a regularization scheme–dependent constant. It turns out that K(A, Ā) has all the expected properties of the next to leading term in the Wilsonian effective action S[A, Ā] whose modular properties are considered in the framework of the dimensional regularization. Work supported by the European Commission TMR programme ERBFMRX–CT96–0045. The exact results aboutN = 2 SUSY Yang–Mills obtained by Seiberg and Witten [1] concern the low–energy Wilsonian effective action with at most two derivatives and four fermions. In the SU(2) case, the u–moduli space of quantum vacua is MSU(2), the Riemann sphere with punctures at u = ∞, u = ±Λ2. In [2] results in [1] have been derived from first principles [3, 4]. In particular, the T 2 symmetry u(τ + 2) = u(τ), which rigorously follows from the asymptotic analysis together with the relation [3] u = πi(F − a∂aF/2), (1) and the fact that u(τ) = u(−τ̄), u(τ + 1) = −u(τ), (2) uniquely fix the monodromy group Γ to be Γ(2). The basic observation is that, for real values of u we have the symmetry u(τ) = u(−τ̄ ) which essentially fixes Γ. The reason is that by (1) and Im τ > 0 (except for the singularities where Im τ = 0), u can be seen as uniformizing coordinate. Therefore, MSU(2) ∼= H/Γ where H is the upper half plane (the τ–moduli space; see [5]). This is equivalent to u(γ · τ) = u(τ) with γ ∈ Γ. It follows that there are curves C in the fundamental domains in H such that for τ ∈ C one has γ · τ = −τ̄ . This reasoning together with a proper use of u(τ +1) = −u(τ) essentially implies the results in [2]. In [1] it has been emphasized that the metric ds = Im ( ∂ aF ) |da| = |∂ua|Im ( ∂ aF ) |du|, (3) is at heart of the physics. The natural framework to investigate its properties is uniformization theory [3, 5, 2]. In this Letter we use basic geometrical structures of MSU(2) to derive a modular invariant quantity which fulfills all the expected properties of the next to leading term in the Abelian Wilsonian effective action [6, 7, 8, 9]. Let us now recall the metric introduced in [5]. Let H = {w|Imw > 0} be the upper half plane endowed with the Poincaré metric dsP = (Imw) −2|dw|2. Since τ = ∂ aF is the inverse of the map uniformizing MSU(2), it follows that the positive definite metric dsP = |∂3 aF| 2 (Im τ) |da| = |∂uτ | 2 (Im τ) |du| = e|du|, (4) is the Poincaré metric on MSU(2). This implies that φ satisfies the Liouville equation φuū = e/2.