泛函行列式、指数定理和精确量子黑洞熵
Functional determinants, index theorems, and exact quantum black hole entropy
摘要：
BPS黑洞的精确量子熵可以用超引力局域化来计算。在这个程序中，一个重要的成分，一直缺乏到目前为止，是一个循环的效果所产生的二次波动的确切变形（$Q\mathcal{V}$运营商）。利用Atiyah-Bott不动点指数定理和随后的zeta函数正则化方法，计算了四维$\mathcal{N}=2$超引力系统中具有$AdS_{2} \times S^{2}$边界条件的$Q$不变离壳组态的矢量多重态和超多重态的涨落行列式.我们的结果扩展了文献中的大电荷的壳层熵计算的有限电荷的制度。在此基础上，我们给出了$\mathcal{N}=2$超引力中BPS黑洞量子熵的精确公式。我们解释了在arXiv：1111.1161中观察到的关于$\mathcal{N}=8$超引力中$\frac18$-BPS黑洞的抵消。本文还对低能超引力理论中拓扑弦配分函数中对数项的解释作了评述。
英文摘要：
The exact quantum entropy of BPS black holes can be evaluated using localization in supergravity. An important ingredient in this program, that has been lacking so far, is the one-loop effect arising from the quadratic fluctuations of the exact deformation (the $Q\mathcal{V}$ operator). We compute the fluctuation determinant for vector multiplets and hyper multiplets around $Q$-invariant off-shell configurations in four-dimensional $\mathcal{N}=2$ supergravity with $AdS_{2} \times S^{2}$ boundary conditions, using the Atiyah-Bott fixed-point index theorem and a subsequent zeta function regularization. Our results extend the large-charge on-shell entropy computations in the literature to a regime of finite charges. Based on our results, we present an exact formula for the quantum entropy of BPS black holes in $\mathcal{N}=2$ supergravity. We explain cancellations concerning $\frac18$-BPS black holes in $\mathcal{N}=8$ supergravity that were observed in arXiv:1111.1161. We also make comments about the interpretation of a logarithmic term in the topological string partition function in the low energy supergravity theory.
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