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arXiv 1809.06280hep-th
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关于大R电荷$\mathcal N=2$手性关联函数和Toda方程

On the large R-charge $\mathcal N=2$ chiral correlators and the Toda equation

Matteo Beccaria

备注：
27 pages. v2: minor clarifications added
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摘要：

我们考虑$\mathcal N=2$ $SU(N)$ SQCD在四维和一个弱耦合制度与大R-电荷最近讨论arXiv：1803.00580。如果$φ$表示$\mathcal N=2$矢量多重态中的伴随标量，则当$g_\text{YM}\to 0$具有大R-电荷时，如$\sim 1/g^2_\text{YM}$，手征原色$(\text{Tr} φ^2)^n$扇区中的2-点函数具有有限极限.关于$\mathcal N=4$关联函数的修正是固定耦合$λ=n\,g^2_\text{YM}$和规范代数秩$N$的非平凡函数$F(λ; N)$。我们将展示如何利用户田方程以下的$tt^*$方程，以控制R-电荷的依赖性。这允许确定$F(λ; N)$的顺序$O(λ^{10})$的通用$N$，大大扩展了以前的结果，并放置在一个坚实的地面上的$SU(2)$情况下提出的猜想。我们表明，一个类似的户田方程，在过去讨论过，可能确实是用于额外的部门$(\text{Tr}φ^2)^n\,\text{Tr}φ^3$由于特殊的混合性质，这些复合运营商的4球。我们讨论了大的R极限在这第二种情况下，并计算相关的缩放函数$F$在顺序$O(λ^7)$和通用$N$。大$N$因式分解也被示出为计算的检查。

英文摘要：

We consider $\mathcal N=2$ $SU(N)$ SQCD in four dimensions and a weak-coupling regime with large R-charge recently discussed in arXiv:1803.00580. If $φ$ denotes the adjoint scalar in the $\mathcal N=2$ vector multiplet, it has been shown that the 2-point functions in the sector of chiral primaries $(\text{Tr} φ^2)^n$ admit a finite limit when $g_\text{YM}\to 0$ with large R-charge growing like $\sim 1/g^2_\text{YM}$. The correction with respect to $\mathcal N=4$ correlators is a non-trivial function $F(λ; N)$ of the fixed coupling $λ=n\,g^2_\text{YM}$ and the gauge algebra rank $N$. We show how to exploit the Toda equation following from the $tt^*$ equations in order to control the R-charge dependence. This allows to determine $F(λ; N)$ at order $O(λ^{10})$ for generic $N$, greatly extending previous results and placing on a firmer ground a conjecture proposed for the $SU(2)$ case. We show that a similar Toda equation, discussed in the past, may indeed be used for the additional sector $(\text{Tr}φ^2)^n\,\text{Tr}φ^3$ due to the special mixing properties of these composite operators on the 4-sphere. We discuss the large R-limit in this second case and compute the associated scaling function $F$ at order $O(λ^7)$ and generic $N$. Large $N$ factorization is also illustrated as a check of the computation.

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