Abstract
We study the Higgs branches of five-dimensional mathcal{N} = 1 rank-zero theories obtained from M-theory on two classes non-toric non-compact Calabi-Yau threefolds: Reid’s pagodas, and Laufer’s examples. Our approach consists in reducing to IIA with D6-branes and O6-planes, and computing the open-string spectra giving rise to hypermultiplets. Starting with the seven-dimensional worldvolume theories, we switch on T-brane backgrounds to give rise to bound states with angles. We observe that the resulting partially Higgsed 5d theories have discrete gauge groups, from which we readily deduce the geometry of the Higgs branches as orbifolds of quaternionic varieties.
Highlights
We study the Higgs branches of five-dimensional N = 1 rank-zero theories obtained from M-theory on two classes non-toric non-compact Calabi-Yau threefolds: Reid’s pagodas, and Laufer’s examples
The literature focuses on toric threefolds. Those are nice for several reasons: they can be constructed by drawing a two-dimensional diagram, the effective field theory data can be read off such a diagram, the prepotential can be understood in terms of the topological vertex, and one can directly relate it to 5-brane webs in IIB via a chain of dualities, [16]
In order to really know the structure of the Higgs branch, we have to take into account the discrete gauge group we found in (4.5)
Summary
We will introduce the simplest possible example: the Higgs branch of a free hypermultiplet, seen in terms of the moduli space of M-theory on the conifold. The M-theory uplift is given by a C∗-fibration that collapses over the spectral curve of Φ as follows: uv = det z12 − Φ = z2 In this case, the geometry is that of a local K3 with an A1-singularity times a complex plane generated by w. That this is a localized version of the standard string-theoretic Ext computations one would perform for more general objects of the derived category of coherent sheaves Parametrizing both the fluctuation and gauge parameter as follows:. We see that there is a projection from the full hypermultiplet moduli space onto the complex structure moduli space of the conifold π : (φ+, φ−) −→ μ := φ+φ− This map defines the Higgs branch as a C∗-fibration over the complex structure moduli space, whereby the fibers contain the data about the C3 Wilson lines. This uplifts in M-theory to the two Wilson lines of the supergravity C3 form on the S3, and its dual non-compact 3-cycle
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