有理映射的三维散射
Scattering in Three Dimensions from Rational Maps
摘要：
四维${\cal N}=4$超Yang-Mills和${\cal N} = 8$超引力的完备树级S-矩阵在某些有理映射的模空间上具有紧凑的积分形式。在本文中，我们利用当振幅用适当的波函数修饰时，降维变得简单的事实，导出了三维振幅的公式。这一过程导致公式的合理映射的三维最大超对称杨米尔斯和重力理论。新公式的被积函数包含三种基本结构：Parke-Taylor-like因子、Vandermonde行列式和结式。将格拉斯曼方向的某些部分积分出来，就得到了小于最大超对称性的理论公式，这就暴露了第四种结构。结合所有四种基本结构，我们开始搜索三维中的一致S-矩阵。很好，最自然的是那些对应于ABJM和BLG理论。我们还作出了一个连接的权力之间的结果在被积函数，表示的庞加莱群，红外线的行为和保形性的理论。扩展到其他理论在三维和任意尺寸也进行了讨论。
英文摘要：
The complete tree-level S-matrix of four dimensional ${\cal N}=4$ super Yang-Mills and ${\cal N} = 8$ supergravity has compact forms as integrals over the moduli space of certain rational maps. In this note we derive formulas for amplitudes in three dimensions by using the fact that when amplitudes are dressed with proper wave functions dimensional reduction becomes straightforward. This procedure leads to formulas in terms of rational maps for three dimensional maximally supersymmetric Yang-Mills and gravity theories. The integrand of the new formulas contains three basic structures: Parke-Taylor-like factors, Vandermonde determinants and resultants. Integrating out some of the Grassmann directions produces formulas for theories with less than maximal supersymmetry, which exposes yet a fourth kind of structure. Combining all four basic structures we start a search for consistent S-matrices in three dimensions. Very nicely, the most natural ones are those corresponding to ABJM and BLG theories. We also make a connection between the power of a resultant in the integrand, representations of the Poincaré group, infrared behavior and conformality of a theory. Extensions to other theories in three dimensions and to arbitrary dimensions are also discussed.
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