The equation of eigen-values in a general Euclidean Schwarzschild metric

New self-dual solutions of SU(2) Yang-Mills theory in Euclidean Schwarzschild space

We present a systematic study of spherically symmetric self-dual solutions of SU(2) Yang-Mills theory on Euclidean Schwarzschild space. All the previously known solutions are recovered and a new one-parameter family of instantons is obtained. The newly found solutions have continuous actions and interpolate between the classic Charap and Duff instantons. We examine the physical properties of this family and show that it consists of dyons of unit (magnetic and electric) charge.

欧几里得施瓦西黑洞中的广义第二定律

The generalized second law in Euclidean Schwarzschild black hole

本研究针对施瓦西黑洞的广义熵问题，在欧几里得截面下引入含加性无序场的标量量子有效模型，用分布ζ函数法计算平均自由能，导出广义薛定谔算符，最终得到含物质贡献的广义熵密度并验证广义第二定律。

From vortices to instantons on the Euclidean Schwarzschild manifold

The first irreducible solution of the $\mathrm{SU} (2)$ self-duality equations on the Euclidean Schwarzschild (ES) manifold was found by Charap and Duff in 1977, only 2 years later than the famous BPST instantons on $\mathbb{R}^4$ were discovered. While soon after, in 1978, the ADHM construction gave a complete description of the moduli spaces of instantons on $\mathbb{R}^4$, the case of the Euclidean Schwarzschild manifold has resisted many efforts for the past 40 years. By exploring a correspondence between the planar Abelian vortices and spherically symmetric instantons on ES, we obtain: a …

Spectral Properties of Schwarzschild Instantons

We study spectral properties of the Dirac and scalar Laplace operator on the Euclidean Schwarzschild space, both twisted by a family of abelian connections with anti-self-dual curvature. We show that the zero-modes of the gauged Dirac operator, first studied by Pope, take a particularly simple form in terms of the radius of the Euclidean time orbits, and interpret them in the context of geometric models of matter. For the gauged Laplace operator, we study the spectrum of bound states numerically and observe that it can be approximated with remarkable accuracy by that of the exactly solvable ga…

Geometric Interpretation of Schwarzschild Instantons

In this note we address the problem of finding Abelian instantons of finite energy on the Euclidean Schwarzschild manifold. This amounts to construct self-dual L^2 harmonic 2-forms on the space. Gibbons found a non-topological L^2 harmonic form in the Taub-NUT metric, leading to Abelian instantons with continuous energy. We imitate his construction in the case of the Euclidean Schwarzschild manifold and find a non-topological self-dual L^2 harmonic 2-form on it. We show how this gives rise to Abelian instantons and identify them with SU(2)-instantons of Pontryagin number 2n^2 found by Charap a…

The quantum development of an asymptotically Euclidean Cauchy hypersurface

In our model of quantum gravity the quantum development of a Cauchy hypersurface is governed by a wave equation derived as the result of a canonical quantization process. To find physically interesting solutions of the wave equation we employ the separation of variables by considering a temporal eigenvalue problem which has a complete countable set of eigenfunctions with positive eigenvalues and also a spatial eigenvalue problem which has a complete set of eigendistributions. Assuming that the Cauchy hypersurface is asymtotically Euclidean we prove that the temporal eigenvalues are also spatia…