arXiv 1706.044502017-06-14hep-th

共形${\cal N}=4$ SYM四点函数的若干问题

Some aspects of conformal ${\cal N}=4$ SYM four point function

Christopher Rayson

备注：: 2008 Cambridge PhD thesis

摘要：

在${\cal N}=4$超共形Yang Mills中，手征准素BPS算子的四个点函数被表示成明显满足超共形Ward恒等式的形式.他们随后展开的共形分波。关联函数的两对相同的手征原色，一对具有最低的可能的尺度尺寸，被认为是。交叉对称性决定了它们的自由场值，直到数值常量。从不同的超多重峰的分波展开的贡献进行了分析，并确定在自由场的情况下，并与建立的结果在强耦合和弱耦合。在大的$N$中，强耦合极限，自由场值和超引力结果之间存在非平凡的抵消。在微扰的情况下，通过分析某些超几何和对数函数的共形波展开，得到了最低扭曲算子的反常维数和耦合校正的值。接下来，我们试图计算缩短的${\cal N}=4$ SYM算子，首先从基本域构造属于某些低扭曲$SU(4)_R$表示的最一般的算子。独立的解决方案的数量施加在这样的运营商的条件是通过组合的方法。生成函数的数量与自旋$\ell=0,1,2,\dotsc$的运营商。明确的值得到特定的$R$ -对称表示在低扭曲的理论的各个部门。大扭转的渐近行为也被认为是。最后分析了共形场论的算子乘积展开。解决方案的级数展开的发现，最初在两个维度的标量算子，然后更普遍。

英文摘要：

The four point functions of chiral primary BPS operators in ${\cal N}=4$ superconformal Yang Mills are expressed in a form manifestly satisfying the superconformal Ward identities. They are subsequently expanded in terms of conformal partial waves. Correlation functions of two pairs of identical chiral primaries, one pair having the lowest possible scale dimension, are considered. Crossing symmetries determine their free field value up to numeric constants. The contributions from different supermultiplets to the partial wave expansion is analysed, and determined in the case of the free fields and compared with established results at strong and weak coupling. In the large $N$, strong coupling limit, non-trivial cancellations are found between the free field values and results from supergravity. In the perturbative case values are obtained for the anomalous dimensions of lowest twist operators and the correction to the coupling by analysing the conformal wave expansions of certain hypergeometric and logarithmic functions. Next, we attempt to count shortened ${\cal N}=4$ SYM operators, beginning by constructing from fundamental fields the most general operators belonging to certain $SU(4)_R$ representations at low twists. The number of independent solutions to the conditions imposed on such operators is found via a combinatoric approach. Generating functions for the number of operators with spin $\ell=0,1,2,\dotsc$ are derived. Explicit values are obtained for specific $R$-symmetry representations at low twist in various sectors of the theory. The asymptotic behaviour at large twist is also considered. Finally the conformal field theory operator product expansion is analysed. Solutions in terms of series expansions are found, initially for scalar operators in two dimensions, and then more generally.

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