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.

Comments

  1. We’re surfing a one-parameter family of Hermitian metrics
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    F={L
    t


    =(C
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    , stitched together by the spectral threads of a modified Bessel function:

    🎯 Core Function:
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    )
    🧘 Metric Form:
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    πŸŒͺ Asymptotics:

    As
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    s→0: The metric becomes conical:
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    1

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    g
    t


    (s)∼
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    1


    .

    As
    𝑠


    s→∞: The surface collapses exponentially:
    𝑔
    𝑑
    (
    𝑠
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    𝑒

    4
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    g
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    (s)∼e
    −4
    t
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    .

    These form a warped cylinder with:

    an infinite cusp (collapsing end) at
    𝑠


    s→∞

    a conical explosion at
    𝑠

    0
    s→0

    but crucially not a surface of revolution—expansion is anisotropic! Like a MΓΆbius MΓΆbius MΓΆbius loop with spicy asymmetry.

    🧩 Now the Modular Magic:

    Define a trace-like summation:

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    (
    𝑠
    )
    =

    𝑑
    =
    1

    𝐹
    𝑑
    (
    𝑠
    )
    M(s)=
    t=1




    F
    t


    (s)

    Let:

    Ξ¦
    (
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    =
    1
    +
    2
    𝑀
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    Ξ¦(s)=1+2M(s)

    Then:

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    1
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    =
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    𝛼
    Ξ¦
    (
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    (modular covariance!)
    Ξ¦(1/s)=s
    Ξ±
    Ξ¦(s)(modular covariance!)

    Thus, the global metric:

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    =

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    (s):=∣Ξ¦(s)∣
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    transforms conformally:

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    =

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    (
    𝛼

    2
    )

    𝑔

    (
    𝑠
    )
    g



    (1/s)=∣s∣
    2(Ξ±−2)
    ⋅g



    (s)

    Modular behavior with a twist of spectral Bessel juice! πŸ’«

    πŸ’Ž The Takeaway:

    You're looking at a conformally Hermitian metric on
    𝐢
    ×
    C
    ×
    , with modular-type automorphy under
    𝑠

    1
    /
    𝑠
    s↦1/s, glued together via holomorphic Bessel harmonics. The geometry is controlled by an effective weight
    𝛼
    Ξ±, tied to the Poisson summation duality of
    𝐹
    𝑑
    (
    𝑠
    )
    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.

    ReplyDelete

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