一个量子指数定理的注记
Notes for a Quantum Index Theorem
摘要：
我们认为DHR超选择部门与有限的统计作为量子场论类似物的椭圆算子KMS泛函发挥的作用，与热核正则化组成的跟踪。我们扩展了我们的局部全纯维数公式，并证明了类似的指数定理在量子场论的上下文中。解析指数是琼斯指数，更准确地说是最小维数，在四维时空中，DHR定理给出了指数的完整性。我们引入了全纯维数的概念，然后将几何维数定义为在电荷共轭下对称的全纯维数的一部分。我们应用AHKT理论的化学势，我们推广到低维的情况下，通过使用共形域论。关于弯曲时空的量子场论，流形的几何进入维数的表达式。如果一个量子黑洞被描述为一个具有分叉的Killing视界的时空，并且扇区在视界上是可局域的，那么全纯维数的对数与由于电荷的增加而增加的自由能成正比，与温度成反比，因此与Hartle-Hawking KMS态的表面引力成正比。对于此分析，我们考虑通过将场限制在视界（“全息”）而获得的共形网络。与我们以前对林德勒时空的工作相比，这个结果的不同之处在于它涉及真正的黑洞时空，如施瓦茨希尔德-克鲁斯卡尔流形，并且与黑洞本身的熵有关，而不是与外部系统的熵有关。展望与超对称和非交换几何的可能关系。
英文摘要：
We view DHR superselection sectors with finite statistics as Quantum Field Theory analogs of elliptic operators where KMS functionals play the role of the trace composed with the heat kernel regularization. We extend our local holomorphic dimension formula and prove an analogue of the index theorem in the Quantum Field Theory context. The analytic index is the Jones index, more precisely the minimal dimension, and, on a 4-dimensional spacetime, the DHR theorem gives the integrality of the index. We introduce the notion of holomorphic dimension; the geometric dimension is then defined as the part of the holomorphic dimension which is symmetric under charge conjugation. We apply the AHKT theory of chemical potential and we extend it to the low dimensional case, by using conformal field theory. Concerning Quantum Field Theory on curved spacetime, the geometry of the manifold enters in the expression for the dimension. If a quantum black hole is described by a spacetime with bifurcate Killing horizon and sectors are localizable on the horizon, the logarithm of the holomorphic dimension is proportional to the incremental free energy, due to the addition of the charge, and to the inverse temperature, hence to the surface gravity in the Hartle-Hawking KMS state. For this analysis we consider a conformal net obtained by restricting the field to the horizon (``holography''). Compared with our previous work on Rindler spacetime, this result differs inasmuch as it concerns true black hole spacetimes, like the Schwarzschild-Kruskal manifold, and pertains to the entropy of the black hole itself, rather than of the outside system. An outlook concerns a possible relation with supersymmetry and noncommutative geometry.
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