Automorphic spectra and the conformal bootstrap
Automorphic spectra and the conformal bootstrap
复制标题
自守谱和共形自举
DOI：
10.1090/cams/26
复制
发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Sridip Pal
中科院分区：
文献类型：
--
作者：
P. Kravchuk;D. Mazáč;Sridip Pal
给出了一种约束双曲曲面和2-轨道的拉普拉斯谱的新方法。其主要成分是四种自同构形式积的积分谱分解的一致性。运用表象理论的结合
p
s
l
2 （
r
）
\mathrm PSL＿2{（}\mathbb R{）
}
和半定规划，该方法给出了拉普拉斯谱间隙的严格上界。在几个例子中，边界几乎是尖锐的。例如，所有属2曲面上的界是
λ
1
≤
3.8388976481
\lambda ＿1 \leq 3.8388976481
而Bolza表面则是
λ
1
≈
3.838887258
\lambda ＿1 \approx 3.838887258
． 边界还允许我们确定所有双曲2-轨道所获得的光谱间隙集。我们的方法可以推广到高维双曲流形，并在二维情况下给出更强的界。这些想法受到现代共形引导的密切启发。
We describe a new method for constraining Laplacian spectra of hyperbolic surfaces and 2-orbifolds. The main ingredient is consistency of the spectral decomposition of integrals of products of four automorphic forms. Using a combination of representation theory of
P
S
L
2
(
R
)
\mathrm {PSL}_2(\mathbb {R})
and semidefinite programming, the method yields rigorous upper bounds on the Laplacian spectral gap. In several examples, the bound is nearly sharp. For instance, our bound on all genus-2 surfaces is
λ
1
≤
3.8388976481
\lambda _1\leq 3.8388976481
, while the Bolza surface has
λ
1
≈
3.838887258
\lambda _1\approx 3.838887258
. The bounds also allow us to determine the set of spectral gaps attained by all hyperbolic 2-orbifolds. Our methods can be generalized to higher-dimensional hyperbolic manifolds and to yield stronger bounds in the two-dimensional case. The ideas were closely inspired by modern conformal bootstrap.