Topic
Toric variety
About: Toric variety is a research topic. Over the lifetime, 2630 publications have been published within this topic receiving 65604 citations. The topic is also known as: torus embedding.
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TL;DR: In this article, a detailed investigation of the combinatorial and geometric objects associated with on-shell diagrams is performed, mainly focusing on their relation to polytopes and toric geometry, the Grassmannian and its stratification.
Abstract: The fundamental role of on-shell diagrams in quantum field theory has been recently recognized. On-shell diagrams, or equivalently bipartite graphs, provide a natural bridge connecting gauge theory to powerful mathematical structures such as the Grassmannian. We perform a detailed investigation of the combinatorial and geometric objects associated to these graphs. We mainly focus on their relation to polytopes and toric geometry, the Grassmannian and its stratification. Our work extends the current understanding of these connections along several important fronts, most notably eliminating restrictions imposed by planarity, positivity, reducibility and edge removability. We illustrate our ideas with several explicit examples and introduce concrete methods that considerably simplify computations. We consider it highly likely that the structures unveiled in this article will arise in the on-shell study of scattering amplitudes beyond the planar limit. Our results can be conversely regarded as an expansion in the understanding of the Grassmannian in terms of bipartite graphs.
49 citations
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49 citations
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TL;DR: In this article, it was shown that any ring of differential operators on X twisted by an invertible sheaf is a factor ring of the fixed ring D(Y)G by an ideal generated by central elements.
48 citations
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TL;DR: In this article, the moduli kspace of the D-brane world-volume gauge theory was investigated by using toric geometry and gauged linear sigma models, and it was shown that there are five phases, which are topologically distinct and connected by flops to each other.
48 citations
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TL;DR: In this paper, the authors give a characterisation of the amoeba, based on the triangle inequality, which they call testing for lopsidedness, and show that if a point is outside the amoba of an affine algebraic variety V⊂(C*)r, there is an element of the defining ideal which witnesses this fact by being lopsides.
Abstract: The amoeba of an affine algebraic variety V⊂(C*)r is the image of V under the map (z1,…,zr)→(log|z1|,…,log|zr|). We give a characterisation of the amoeba, based on the triangle inequality, which we call testing for lopsidedness. We show that if a point is outside the amoeba of V, there is an element of the defining ideal which witnesses this fact by being lopsided. This condition is necessary and sufficient for amoebas of arbitrary codimension as well as for compactifications of amoebas inside any toric variety. Our approach naturally leads to methods for approximating hypersurface amoebas and their spines by systems of linear inequalities. Finally, we remark that our main result can be seen as a precise analogue of a Nullstellensatz statement for tropical varieties
48 citations