20,475 research outputs found
Mean field equations, hyperelliptic curves and modular forms: II
A pre-modular form of weight is
introduced for each , where , such that for , every
non-trivial zero of , namely ,
corresponds to a (scaling family of) solution to the mean field equation
\begin{equation} \tag{MFE} \triangle u + e^u = \rho \, \delta_0 \end{equation}
on the flat torus with singular strength .
In Part I (Cambridge J. Math. 3, 2015), a hyperelliptic curve , the Lam\'e curve, associated to the MFE was
constructed. Our construction of relies on a detailed study
on the correspondence induced
from the hyperelliptic projection and the addition map.
As an application of the explicit form of the weight 10 pre-modular form
, a counting formula for Lam\'e equations of degree
with finite monodromy is given in the appendix (by Y.-C. Chou).Comment: 32 pages. Part of content in previous versions is removed and
published separately. One author is remove
Scalar field fluctuations in Schwarzschild-de Sitter space-time
We calculate quantum fluctuations of a free scalar field in the
Schwarzschild-de Sitter space-time, adopting the planar coordinates that is
pertinent to the presence of a black hole in an inflationary universe. In a
perturbation approach, doing expansion in powers of a small black hole event
horizon compared to the de Sitter cosmological horizon, we obtain time
evolution of the quantum fluctuations and then derive the scalar power
spectrum.Comment: 16 pages and 4 figures, accepted by Classical and Quantum Gravit
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