arXiv 1709.048972017-09-14hep-th

6d（DELL）系统的模性质

Modular properties of 6d (DELL) systems

G. Aminov, A. Mironov, A. Morozov

备注：: 23 pages
发表信息：: JHEP, 23 (2017) 2017
DOI：: 10.1007/JHEP11(2017)023

摘要：

如果超杨-米尔斯理论具有精确的共形不变性，那么在复裸电荷$τ= \fracθ{2π}+ \frac{4π\imath}{g^2}\longrightarrow -\frac{1}τ$的变化下，还存在一个附加的模不变性。然而，低能的Seiberg-Witten预势${\cal F}(a)$并不是明确不变的，因为平坦模量也改变了$a \longrightarrow a_D = \partial{\cal F}/\partial a$。结果表明，前势不是模形式，它也依赖于反常的爱森斯坦级数$E_2$。这种依赖性通常用普适MNW模反常方程来描述。我们证明，在具有{ \it two}独立模参数$τ$和$\hat τ$的$6d$ $SU(N)$理论中，模异常方程改变，因为$τ$的模变换伴随有（$N$依赖的！）$\hatτ$位移，反之亦然。这是双椭圆方程组的一个新的性质，值得进一步研究。

英文摘要：

If super-Yang-Mills theory possesses the exact conformal invariance, there is an additional modular invariance under the change of the complex bare charge $τ= \fracθ{2π}+ \frac{4π\imath}{g^2}\longrightarrow -\frac{1}τ$. The low-energy Seiberg-Witten prepotential ${\cal F}(a)$, however, is not explicitly invariant, because the flat moduli also change $a \longrightarrow a_D = \partial{\cal F}/\partial a$. In result, the prepotential is not a modular form and depends also on the anomalous Eisenstein series $E_2$. This dependence is usually described by the universal MNW modular anomaly equation. We demonstrate that, in the $6d$ $SU(N)$ theory with {\it two} independent modular parameters $τ$ and $\hat τ$, the modular anomaly equation changes, because the modular transform of $τ$ is accompanied by an ($N$-dependent!) shift of $\hatτ$ and vice versa. This is a new peculiarity of double-elliptic systems, which deserves further investigation.

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