从扭弦映射得到SYM和ABJM的动量振幅面
The momentum amplituhedron of SYM and ABJM from twistor-string maps
摘要：
研究了${\cal N}=4$ SYM和${\cal N}=6$ ABJM中树振幅的扭弦公式与$D=4$和$D=3$运动学空间中相应的动量振幅面之间的显著联系。基于到正Grassmannian的Veronese映射，定义了从$G_{+}(2,n)$到SYM动量振幅面所在的四维运动空间的一个$(2n{-}4)$维子空间的扭弦映射.我们提供了强有力的证据证明扭弦映射是从$G_+(2,n)$到动量振幅面体内部的一个同构;后者的标准形可以作为前者的标准形的前推而得到，后者已知给出SYM的树振幅。然后，我们转移到三维：基于Veronese映射到正交正格拉斯曼，我们提出了一个类似的twistor-string映射从模空间${\cal M}_{0,n}^+$到三维运动空间的$(n{-}3)$维子空间。图像给出了一个新的正几何，它在理论上充当ABJM的动量振幅面;它的标准形给出了理论中具有减少的超对称性的树振幅。我们还展示了如何紧化的${\cal M}_{0,n}^+$的边界映射到SYM和ABJM对应的振幅因式分解通道的动量振幅面的边界，特别是对于ABJM的情况下，该地图漂亮地排除了所有不需要的通道。
英文摘要：
We study remarkable connections between twistor-string formulas for tree amplitudes in ${\cal N}=4$ SYM and ${\cal N}=6$ ABJM, and the corresponding momentum amplituhedron in the kinematic space of $D=4$ and $D=3$, respectively. Based on the Veronese map to positive Grassmannians, we define a twistor-string map from $G_{+}(2,n)$ to a $(2n{-}4)$-dimensional subspace of the 4d kinematic space where the momentum amplituhedron of SYM lives. We provide strong evidence that the twistor-string map is a diffeomorphism from $G_+(2,n)$ to the interior of momentum amplituhedron; the canonical form of the latter, which is known to give tree amplitudes of SYM, can be obtained as pushforward of that of former. We then move to three dimensions: based on Veronese map to orthogonal positive Grassmannian, we propose a similar twistor-string map from the moduli space ${\cal M}_{0,n}^+$ to a $(n{-}3)$-dimensional subspace of 3d kinematic space. The image gives a new positive geometry which conjecturally serves as the momentum amplituhedron for ABJM; its canonical form gives the tree amplitude with reduced supersymmetries in the theory. We also show how boundaries of compactified ${\cal M}_{0,n}^+$ map to boundaries of momentum amplituhedra for SYM and ABJM corresponding to factorization channels of amplitudes, and in particular for ABJM case the map beautifully excludes all unwanted channels.
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