arXiv 2102.095372021-02hep-th

$\mathcal{N}=4$ $SU(N)$ SYM中积分关联函数的精确性质

Exact properties of an integrated correlator in $\mathcal{N}=4$ $SU(N)$ SYM

Daniele Dorigoni, Michael B. Green, Congkao Wen

备注：: 54 pages, 5 figures; v2: typos corrected, matches published version in JHEP
DOI：: 10.1007/JHEP05(2021)089

摘要：

我们提出了一个新的表达式的四个超共形原色在$SU(N)$ $\mathcal{N}=4$ SYM的集成相关函数。这种集成的关联函数，这是基于超对称本地化，一直是最近的几个发展的主题。该关联函数被重新表示为对所有$N$和复Yang-Mills耦合$τ$的所有值有效的二维晶格上的和。在这种形式中，它在$SL(2,\mathbb{Z})$ Montonen-Olive二元性下显然是不变的。此外，它还满足一个显着的拉普拉斯差分方程，该方程将$SU(N)$与$SU(N+1)$和$SU(N-1)$分解器联系起来。对于任何固定值的$N$的关联函数是一个无穷级数的非全纯爱森斯坦级数，$E(s;τ,\barτ)$与$s\in \mathbb{Z}$，和合理的系数。积分关联函数的微扰展开是渐近的，$n$环路系数是$ζ(2n+1)$的有理倍数. $n=1$和$n=2$项与直接通过在单圈和双圈微扰SYM中积分表达式确定的结果精确一致。同样，电荷- $k$瞬子的贡献有一个渐近的，但波莱尔可和的微扰修正序列。具有固定$τ$的关联函数的大$N$展开是$N^{1/2-\ell}$（$\ell\in \mathbb{Z}$）的幂级数，其系数是$E_s$与$s\in \mathbb{Z}+1/2$的有理和.这就给出了最近得到的展开式的所有阶导数。我们进一步考虑了t Hooft大- $N$ Yang-Mills理论。各阶系数在$λ$中可展开为收敛级数。对于较大的$λ$，这就变成了一个渐近级数，其系数也是奇zeta值的有理倍数。大- $λ$级数不是Borel可和的，它的再生非微扰完备化是$O(\exp(-2\sqrtλ))$。

英文摘要：

We present a novel expression for an integrated correlation function of four superconformal primaries in $SU(N)$ $\mathcal{N}=4$ SYM. This integrated correlator, which is based on supersymmetric localisation, has been the subject of several recent developments. The correlator is re-expressed as a sum over a two dimensional lattice that is valid for all $N$ and all values of the complex Yang-Mills coupling $τ$. In this form it is manifestly invariant under $SL(2,\mathbb{Z})$ Montonen-Olive duality. Furthermore, it satisfies a remarkable Laplace-difference equation that relates the $SU(N)$ to the $SU(N+1)$ and $SU(N-1)$ correlators. For any fixed value of $N$ the correlator is an infinite series of non-holomorphic Eisenstein series, $E(s;τ,\barτ)$ with $s\in \mathbb{Z}$, and rational coefficients. The perturbative expansion of the integrated correlator is asymptotic and the $n$-loop coefficient is a rational multiple of $ζ(2n+1)$. The $n=1$ and $n=2$ terms agree precisely with results determined directly by integrating the expressions in one- and two-loop perturbative SYM. Likewise, the charge-$k$ instanton contributions have an asymptotic, but Borel summable, series of perturbative corrections. The large-$N$ expansion of the correlator with fixed $τ$ is a series in powers of $N^{1/2-\ell}$ ($\ell\in \mathbb{Z}$) with coefficients that are rational sums of $E_s$ with $s\in \mathbb{Z}+1/2$. This gives an all orders derivation of the form of the recently conjectured expansion. We further consider 't Hooft large-$N$ Yang-Mills theory. The coefficient of each order can be expanded as a convergent series in $λ$. For large $λ$ this becomes an asymptotic series with coefficients that are again rational multiples of odd zeta values. The large-$λ$ series is not Borel summable, and its resurgent non-perturbative completion is $O(\exp(-2\sqrtλ))$.

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