Abstract
If super-Yang-Mills theory possesses the exact conformal invariance, there is an additional modular invariance under the change of the complex bare charge . The low-energy Seiberg-Witten prepotential ℱ(a), however, is not explicitly invariant, because the flat moduli also change a − → aD = ∂ℱ/∂a. In result, the prepotential is not a modular form and depends also on the anomalous Eisenstein series E2. This dependence is usually described by the universal MNW modular anomaly equation. We demonstrate that, in the 6d SU(N) theory with two independent modular parameters τ and widehat{tau} , the modular anomaly equation changes, because the modular transform of τ is accompanied by an (N -dependent!) shift of widehat{tau} and vice versa. This is a new peculiarity of double-elliptic systems, which deserves further investigation.
Highlights
If super-Yang-Mills theory possesses the exact conformal invariance, there is an additional modular invariance under the change of the complex bare charge τ
We demonstrate that the first modular anomaly equation along with the convergency condition for the limit σ ( | τ) → ∞ can be used to calculate this prepotential as a series in the mass parameter
We discussed a number of questions concerning the low energy effective action of the 6d SYM theory with two compactified Kaluza-Klein dimensions and the adjoint matter hypermultiplet
Summary
According to [43], there exist non-linear equations for the Seiberg-Witten prepotential, which have exactly the N -particle double-elliptic system as its generic solution. With the help of these equations, the expression for the N = 3 double-elliptic Seiberg-Witten prepotential was derived. In and depend on the both elliptic parameters only through the Eisenstein series. In in (2.1), since otherwise not all the coefficients Ci1,...,in ( , τ, τ) are independent: there are some relations between the functions σ (β αk · a | τ). Since the functions Ci1,...,in,k,(m) (τ ) are quasimodular forms, they can be realized as polynomials in the Eisenstein series E2, E4, and E6:. The constant terms in the expansions of Ci1,...,in,k,(m) (τ ) in powers of q = exp (2πı τ ) correspond to the perturbative part of the prepotential F Dell. Where aij ≡ ai − aj and the functions F (k) = F (k) (a, , β, τ) describing the instanton corrections do not depend on the first elliptic parameter τ. At the righthand sides of (2.7) there are some specific a-independent terms that are essential for the computation of the limit σ ( | τ) → ∞
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