arXiv 1404.21922014-04hep-thmath-phmath.MPmath.NTmath.RT

SL（2，Z）不变性和D-瞬子对$D^6 R^4$相互作用的贡献

SL(2,Z)-invariance and D-instanton contributions to the $D^6 R^4$ interaction

M. B. Green, S. D. Miller, P. Vanhove

备注：: Latex. 31 pages. v3 : a few clarifications. Version to be published

摘要：

在~IIB型弦理论的低能展开中，$D^6R^4$相互作用的模不变系数被证明是一个非齐次Laplace本征方程的解，它是由11维超引力双圈Feynman图的环向紧化得到的。本文确定了满足适当的适度增长条件的微分方程的精确$SL(2,\mathbb Z)$不变解$f(x+iy)$为$y\to \infty$（弱耦合极限）.该解被表示为具有模式$\widehat{f}_n(y) e^{2πi n x}$的傅里叶级数，其中模式系数$\widehat{f}_n(y)$在$K$ -贝塞尔函数中是双线性的。$SL(2,\mathbb Z)$下的不变性要求这些模式满足小$y$的非平凡边界条件$ \widehat{f}_n(y) =O(y^{-2})$，这唯一地确定了解决方案。$f(x+iy)$的大- $y$展开式包含已知的微扰（幂行为）项，以及精确确定的指数递减贡献，这些贡献具有D-瞬子、反D-瞬子和D-瞬子/反D-瞬子对的预期形式。

英文摘要：

The modular invariant coefficient of the $D^6R^4$ interaction in the low energy expansion of type~IIB string theory has been conjectured to be a solution of an inhomogeneous Laplace eigenvalue equation, obtained by considering the toroidal compactification of two-loop Feynman diagrams of eleven-dimensional supergravity. In this paper we determine the exact $SL(2,\mathbb Z)$-invariant solution $f(x+iy)$ to this differential equation satisfying an appropriate moderate growth condition as $y\to \infty$ (the weak coupling limit). The solution is presented as a Fourier series with modes $\widehat{f}_n(y) e^{2πi n x}$, where the mode coefficients, $\widehat{f}_n(y)$ are bilinear in $K$-Bessel functions. Invariance under $SL(2,\mathbb Z)$ requires these modes to satisfy the nontrivial boundary condition $ \widehat{f}_n(y) =O(y^{-2})$ for small $y$, which uniquely determines the solution. The large-$y$ expansion of $f(x+iy)$ contains the known perturbative (power-behaved) terms, together with precisely-determined exponentially decreasing contributions that have the form expected of D-instantons, anti-D-instantons and D-instanton/anti-D-instanton pairs.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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