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arXiv 2008.02713hep-th
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$\mathcal{N}=4$ Super-Yang-Mills理论中新的模不变量

New Modular Invariants in $\mathcal{N}=4$ Super-Yang-Mills Theory

Shai M. Chester, Michael B. Green, Silviu S. Pufu, Yifan Wang, Congkao Wen

备注：
33 pages plus appendices, v2 accepted to JHEP
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摘要：

我们研究了在${\cal N} = 4$ $SU(N)$ super-Yang-Mills理论的应力张量多重态算子的四点函数中产生的模不变量，在极限中$N$被取为大的，而复杂化的Yang-Mills耦合$τ$被保持固定.我们考虑的特定四点函数是积分型函数，它是通过取${\cal N} = 2^*$理论的压缩球配分函数的四个导数的各种组合，相对于压缩参数$b$和质量参数$m$，在对应于圆球上的${\cal N} = 4$理论的值$b=1$和$m=0$处计算得到的。在$1/N$展开式的每一阶，这些四阶导数都是$(τ, \bar τ)$的模不变函数。证明了在$1/N$中的半整数阶，这些模不变量是非全纯Eisenstein级数的线性组合，而在$1/N$中的整数阶，它们是满足双曲平面上非齐次Laplace本征方程的“广义Eisenstein级数”.这些结果再现了10维平坦空间中IIB型超弦理论中四引力子振幅的低能展开的已知特征，并对$AdS_5\times S^5$中类似展开的结构具有有趣的意义。

英文摘要：

We study modular invariants arising in the four-point functions of the stress tensor multiplet operators of the ${\cal N} = 4$ $SU(N)$ super-Yang-Mills theory, in the limit where $N$ is taken to be large while the complexified Yang-Mills coupling $τ$ is held fixed. The specific four-point functions we consider are integrated correlators obtained by taking various combinations of four derivatives of the squashed sphere partition function of the ${\cal N} = 2^*$ theory with respect to the squashing parameter $b$ and mass parameter $m$, evaluated at the values $b=1$ and $m=0$ that correspond to the ${\cal N} = 4$ theory on a round sphere. At each order in the $1/N$ expansion, these fourth derivatives are modular invariant functions of $(τ, \bar τ)$. We present evidence that at half-integer orders in $1/N$, these modular invariants are linear combinations of non-holomorphic Eisenstein series, while at integer orders in $1/N$, they are certain "generalized Eisenstein series" which satisfy inhomogeneous Laplace eigenvalue equations on the hyperbolic plane. These results reproduce known features of the low-energy expansion of the four-graviton amplitude in type IIB superstring theory in ten-dimensional flat space and have interesting implications for the structure of the analogous expansion in $AdS_5\times S^5$.

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