Abstract
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 SL(2,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.
Highlights
The worldvolume theory of a stack of D3-branes probing a toric Calabi-Yau (CY) conetype singularity is a 4d N = 1 supersymmetric gauge theory
We focused on polygons with two interior points, which serve as the toric diagrams of certain toric CY3 cones, as well as those of compact base surfaces
We found the quiver gauge theories associated to D3-branes probing these geometries
Summary
Under n T-dualities, the system of D(7 − n)-branes suspended between an NS5 wrapping a holomorphic n-cycle, where the branes meet in a Tn, corresponds to D(7 − 2n)-branes probing CYn+1 [13] These are related to various topics in different dimensions, such as Chern-Simons theory [14–18], brane brick models [19–22], triality [23], quadrality [24] and so forth. : volume function of Y , with bi being components of Reeb vector : number of interior & perimeter points of the toric diagram : number of gauge nodes in the quiver : number of bifundamentals in the quiver : number of perfect matchings in the brane tiling : structure sheaf of X : (Weil) divisor D on X : sheaf of divisor D on X : (total) Chern class of X , with ci denoting the ith Chern class : the ith Chern number (the top Chern number is the Euler number χ). We begin with a lightning review of the key requisite concepts, from toric CY cones to quiver gauge theories
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