Abstract
We show that the pure gauge anomalies of 6d mathcal{N} = (1, 0) theories compactified on a circle are captured by field-dependent Chern-Simons terms appearing at one-loop in the 5d effective theories. These terms vanish if and only if anomalies are canceled. In order to obtain this result, it is crucial to integrate out the massive Kaluza-Klein modes in a way that preserves 6d Lorentz invariance; the often-used zeta-function regularization is not sufficient. Since such field-dependent Chern-Simons terms do not arise in the reduction of M-theory on a threefold, six-dimensional F-theory compactifications are automatically anomaly free, whenever the M/F-duality can be used. A perfect match is then found between the 5d mathcal{N} = 1 prepotentials of the classical M-theory reduction and one-loop circle compactification of an anomaly free theory. Finally, from this potential, we read off the quantum corrections to the gauge coupling functions.
Highlights
We show that the pure gauge anomalies of 6d N = (1, 0) theories compactified on a circle are captured by field-dependent Chern-Simons terms appearing at one-loop in the 5d effective theories
A crucial ingredient for obtaining this result is the regularization one uses in order to integrate out the massive modes: we find that it needs preserve 6d Lorentz invariance
The M/F-duality tells us that reduction of M-theory on an elliptically fibered smooth CY3 is to be matched with the circle reduction of a 6d (1,0) theory, pushed on the Coulomb branch and with the massive mode integrated out
Summary
The gauge invariant field strengths for the two-forms Bα are given by. Such a form being required by supersymmetry. With anomalies canceled in this way, one can write a supersymmetric action which is invariant at one-loop, whose bosonic part reads. Where the metric for the tensor multiplets is given by gαβ = 2 ˆα ˆβ − Ωαβ. This is a pseudo-action as the kinetic terms of the two-forms should vanish because of their (anti)-self-duality conditions gαβ Gα = Ωαβ Gα. The easiest way out, which suffices for our purposes, is to impose these self-duality conditions at the level of the equations of motion
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