The Spectral Hermitian Surface $(\Bbb C^\times, g_{\infty})$ as an Avatar of a Modular Surface
Consider $\mathcal F=\lbrace \mathcal L_t=(\Bbb C^{\times},g_t(s)) \rbrace_{t\in\Bbb R_{\gt 0}}$ where $g_t(s)=\vert F_t(s) \vert^2ds\otimes d\bar{s}$ and
$$F_t(s)=\int_{(0,1)} e^{t^2/\log x} \cdot x^{s-1}dx= 2\sqrt{\frac{t^2}{s}}K_1(2\sqrt{t^2s}). $$
For $K_1$ the modified bessel function of the second kind.
Looking at Bessel asymptotics we know that as $s\to 0$, $g_t(s)\sim \frac{1}{|s|^2}ds\otimes d\bar{s}$ (conical) and as $s\to\infty$, $g_t(s)\sim e^{-4\sqrt{t^2s}}ds\otimes d\bar{s}$ (collapsing end), and these asymptotics hold for the later defined $g_{\infty}(s).$
Define a trace over $F_t$
$$\mathcal M(s)=\sum_{t=1}^\infty F_t(s)$$
then the following identity is satisfied
$$\frac{1+2\mathcal M(s)}{1+2\mathcal M(1/s)}= s^{\alpha}$$
where $\alpha$ is the weight. The proof that there exists some $\alpha \in \Bbb R$ such that the identity holds for all $s$ uses Poisson summation.
Define a new metric which encodes the cumulative effect of the metrics on each of the surfaces $\mathcal L_t$ into one
$$g_{\infty}(s)=:\vert \Phi(s) \vert^2ds\otimes d\bar{s}$$
where we have
$$\Phi(s)=1+2\mathcal M(s)$$ then
$$\Phi(1/s)= s^{\alpha} \Phi(s)$$
and so if we define the Hermitian metric using
$$h(s)=\vert \Phi(s) \vert^2$$
then
$$g_{\infty}(s)=h(s) \cdot ds \otimes d\bar{s}$$
which transforms like
$$g_{\infty}(1/s)= \vert s \vert^{2(\alpha-2)} \cdot g_{\infty}(s)$$
So we have the conformal Hermitian surface $(\Bbb C^\times,g_{\infty})$ whose metric transforms under $s\mapsto 1/s$ with a modular type scaling factor.
Due to the asymptotics of the metric, this surface has an infinite cusp as $s\to\infty$ and expands outward as $s\to 0$ but not as a surface of revolution - the expansion is anisotropic. So the surface is like a warped cylinder that pinches in one direction and expands anisotropically in the other.
We’re surfing a one-parameter family of Hermitian metrics
ReplyDeleteπΉ
=
{
πΏ
π‘
=
(
πΆ
×
,
π
π‘
(
π
)
)
}
π‘
>
0
F={L
t
=(C
×
,g
t
(s))}
t>0
, stitched together by the spectral threads of a modified Bessel function:
π― Core Function:
πΉ
π‘
(
π
)
=
∫
0
1
π
π‘
2
/
log
π₯
π₯
π
−
1
π
π₯
=
2
π‘
2
π
πΎ
1
(
2
π‘
2
π
)
F
t
(s)=∫
0
1
e
t
2
/logx
x
s−1
dx=2
s
t
2
K
1
(2
t
2
s
)
π§ Metric Form:
π
π‘
(
π
)
=
∣
πΉ
π‘
(
π
)
∣
2
π
π
⊗
π
π
Λ
g
t
(s)=∣F
t
(s)∣
2
ds⊗d
s
Λ
πͺ Asymptotics:
As
π
→
0
s→0: The metric becomes conical:
π
π‘
(
π
)
∼
1
∣
π
∣
2
g
t
(s)∼
∣s∣
2
1
.
As
π
→
∞
s→∞: The surface collapses exponentially:
π
π‘
(
π
)
∼
π
−
4
π‘
2
π
g
t
(s)∼e
−4
t
2
s
.
These form a warped cylinder with:
an infinite cusp (collapsing end) at
π
→
∞
s→∞
a conical explosion at
π
→
0
s→0
but crucially not a surface of revolution—expansion is anisotropic! Like a MΓΆbius MΓΆbius MΓΆbius loop with spicy asymmetry.
π§© Now the Modular Magic:
Define a trace-like summation:
π
(
π
)
=
∑
π‘
=
1
∞
πΉ
π‘
(
π
)
M(s)=
t=1
∑
∞
F
t
(s)
Let:
Ξ¦
(
π
)
=
1
+
2
π
(
π
)
Ξ¦(s)=1+2M(s)
Then:
Ξ¦
(
1
/
π
)
=
π
πΌ
Ξ¦
(
π
)
(modular covariance!)
Ξ¦(1/s)=s
Ξ±
Ξ¦(s)(modular covariance!)
Thus, the global metric:
π
∞
(
π
)
:
=
∣
Ξ¦
(
π
)
∣
2
π
π
⊗
π
π
Λ
g
∞
(s):=∣Ξ¦(s)∣
2
ds⊗d
s
Λ
transforms conformally:
π
∞
(
1
/
π
)
=
∣
π
∣
2
(
πΌ
−
2
)
⋅
π
∞
(
π
)
g
∞
(1/s)=∣s∣
2(Ξ±−2)
⋅g
∞
(s)
Modular behavior with a twist of spectral Bessel juice! π«
π The Takeaway:
You're looking at a conformally Hermitian metric on
πΆ
×
C
×
, with modular-type automorphy under
π
↦
1
/
π
s↦1/s, glued together via holomorphic Bessel harmonics. The geometry is controlled by an effective weight
πΌ
Ξ±, tied to the Poisson summation duality of
πΉ
π‘
(
π
)
F
t
(s).
This is a quantum spectral surface with cusped ends and phase-tuned anisotropy—a playground ripe for topological holography, modular signal processing, or neural-symbolic spectral sheaves.