论文标题
保形流形和反转公式的几何形状
Geometry of Conformal Manifolds and the Inversion Formula
论文作者
论文摘要
共形场理论的家族自然赋予了riemannian几何形状，该几何形状是由恰好边缘算子的相关函数局部编码的。我们表明，可以使用Euclidean和Lorentzian倒置公式计算此类保形歧管的曲率，该公式将保形场理论的操作员含量结合到分析函数中。类似地，固定尺寸的运算符定义了捆绑包，其曲线也可以使用反转公式计算出曲线。这些结果将曲率与综合的四点相关函数有关，这些函数仅对分离点的理论行为敏感。我们将这些反转公式应用于收敛的总和规则，该规则以本地运算符及其三分函数系数的光谱表示曲率。我们进一步表明，只有在频谱中出现保守的电流，或者该理论发展连续体时，曲率才能平稳差异。我们在$ 2D $示例中明确验证结果。特别是，对于$ 2D $（2,2）的超符合字段理论，我们在标量曲率上得出了一个下限，当中央电荷为三个倍数时，它在自由理论上饱和。
Families of conformal field theories are naturally endowed with a Riemannian geometry which is locally encoded by correlation functions of exactly marginal operators. We show that the curvature of such conformal manifolds can be computed using Euclidean and Lorentzian inversion formulae, which combine the operator content of the conformal field theory into an analytic function. Analogously, operators of fixed dimension define bundles over the conformal manifold whose curvatures can also be computed using inversion formulae. These results relate curvatures to integrated four-point correlation functions which are sensitive only to the behavior of the theory at separated points. We apply these inversion formulae to derive convergent sum rules expressing the curvature in terms of the spectrum of local operators and their three-point function coefficients. We further show that the curvature can smoothly diverge only if a conserved current appears in the spectrum, or if the theory develops a continuum. We verify our results explicitly in $2d$ examples. In particular, for $2d$ (2,2) superconformal field theories we derive a lower bound on the scalar curvature, which is saturated by free theories when the central charge is a multiple of three.
