三次特殊几何专题
Topics in Cubic Special Geometry
摘要：
我们重新考虑了N=2立方特殊Kaehler几何的次导量子微扰修正。施加在轴子移位下的不变性，所有这些校正（除了虚常数1）都可以通过适当的、较低的单三角辛变换（称为Peccei-Quinn（PQ）变换）来引入或去除。由于PQ变换不属于d=4的U-对偶群G4，在对称情况下，它们通常对G4的电荷表示R的唯一四次不变多项式I4有非平凡作用。这导致了有趣的现象，在理论的极端黑洞吸引子，即可能性，使不同的电荷轨道之间的过渡R，与相应的变化的超对称性的支持吸引子解决方案。此外，PQ变换的适当动作也可以将I4设置为零，反之亦然，它可以生成非零的I4：这对应于“大”和“小”电荷轨道之间的转换，我们在“特殊坐标”辛框架内对其进行了详细分类。最后，PQ变换的行动后，最近建立的对应凯莱超行列式和椭圆曲线的简要说明，我们推导出一个等价的，替代的表达I4，与相关的应用程序黑洞熵。
英文摘要：
We reconsider the sub-leading quantum perturbative corrections to N=2 cubic special Kaehler geometries. Imposing the invariance under axion-shifts, all such corrections (but the imaginary constant one) can be introduced or removed through suitable, lower unitriangular symplectic transformations, dubbed Peccei-Quinn (PQ) transformations. Since PQ transformations do not belong to the d=4 U-duality group G4, in symmetric cases they generally have a non-trivial action on the unique quartic invariant polynomial I4 of the charge representation R of G4. This leads to interesting phenomena in relation to theory of extremal black hole attractors; namely, the possibility to make transitions between different charge orbits of R, with corresponding change of the supersymmetry properties of the supported attractor solutions. Furthermore, a suitable action of PQ transformations can also set I4 to zero, or vice versa it can generate a non-vanishing I4: this corresponds to transitions between "large" and "small" charge orbits, which we classify in some detail within the "special coordinates" symplectic frame. Finally, after a brief account of the action of PQ transformations on the recently established correspondence between Cayley's hyperdeterminant and elliptic curves, we derive an equivalent, alternative expression of I4, with relevant application to black hole entropy.
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