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Quiver gauge theories: beyond reflexivity. (English) Zbl 1437.81062
Summary: Reflexive polygons have been extensively studied in a variety of contexts in mathematics and physics. We generalize this programme by looking at the 45 different lattice polygons with two interior points up to $$\mathrm{SL} (2, \mathbb{Z})$$ equivalence. Each corresponds to some affine toric 3-fold as a cone over a Sasaki-Einstein 5-fold. We study the quiver gauge theories of D3-branes probing these cones, which coincide with the mesonic moduli space. The minimum of the volume function of the Sasaki-Einstein base manifold plays an important role in computing the R-charges. We analyze these minimized volumes with respect to the topological quantities of the compact surfaces constructed from the polygons. Unlike reflexive polytopes, one can have two fans from the two interior points, and hence give rise to two smooth varieties after complete resolutions, leading to an interesting pair of closely related geometries and gauge theories.
##### MSC:
 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 81T25 Quantum field theory on lattices 53Z05 Applications of differential geometry to physics
##### Software:
Calabi-Yau database; PALP; Quiver
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