Consider the decorated seed:
$$\mathcal S_{\mathrm{dec}} = (\mathcal I, \Gamma, \Pi, \chi,\rho,\mathcal L)$$
which is a candidate motivic object, with $L$-functions arising through trace constructions on $\mathcal S$. To see what that means, consider:
$$ \mathcal S = (\mathcal I, \Gamma, \Pi) $$
which is a $\Pi$-equivariant $4$-sheeted branched cover
$$ p: \mathcal I \longrightarrow \mathbf{ \widehat{C}} $$
equipped with a distinguished embedded graph
$$ \Gamma \subset \mathcal I $$
on the covering surface. $\Gamma$ is combinatorially an octahedral graph. $\Gamma$ contains three distinguished $4$-cycles $\Gamma_x,\Gamma_y, \Gamma_z$ whose union is all of $\Gamma$. The decoration $\chi = \lbrace x,y,z\rbrace$ gives an edge-coloring by $\lbrace x,y,z \rbrace$. This is notably different from Grothendieck's dessins d'enfants, which are downstairs on $\widehat{\mathbf C}$.
Write the holonomy representation
$$ \rho : \pi_1(\Gamma) \longrightarrow U(1) $$
The piecewise mappings $\pi \in \Pi$ (piecewise isometries, polytope exchange transformations in the real case) are given by
$$\pi_i : \mathcal I \to \mathcal I, \quad i \in \{x, y, z\}$$
The action of $\Pi$ on $\mathcal I$ induces automorphisms on $\Gamma$
$$ \pi_{*}: \Gamma \longrightarrow \Gamma $$
which act on the cycle space, in particular on the first homology group
$$ \pi_{*} : H_1(\Gamma, \Bbb Z) \longrightarrow H_1(\Gamma, \Bbb Z).$$
The twisted Ihara zeta function, $\zeta_{\Gamma}(u, \rho)$, sees the cycle space at a coarser level, picking out primitive backtrackless cycles
$$\zeta_\Gamma(u,\rho) = \prod_{[C]} \Bigl(1-\rho(C)\,u^{\ell(C)}\Bigr)^{-1}$$
Since $\Pi$ acts on $\Gamma$, it also acts on $\pi_1(\Gamma)$.
The holonomy representation updates discretely as $\rho \mapsto \rho\circ \pi_*^{-1}$ and we can study orbits such as
$$\mathcal O_{\zeta}=\left\{ \zeta_\Gamma\!\left(u,\rho\circ \pi_*^{-1}\right) : \pi \in \Pi \right\}.$$
The decoration, $\mathcal L$, is a line bundle over $\Gamma$. It also responds to the action of $\Pi$ and gets twisted.
We can think of $\chi,\rho,\mathcal L$ as datum that are responsible for the twisting, as seen with the holonomy twisted Ihara zeta function. We re-organize the datum as:
$$ \mathcal S_{\mathrm{dec}} = (\mathcal S, \mathscr T) $$
where twisting data is now $\mathscr T = (\chi, \rho, \mathcal L)$.
If we suppress twisting data, we recover zeta functions such as the classical Ihara zeta function as traces over the primitive $\mathcal S = (\mathcal I,\Gamma, \Pi)$. I used the Ihara zeta as the guiding example, but $\Gamma$ is not merely combinatorial. It harbors transport, holonomy, line bundles, around which the associated global $L$-function is organized.
We may study $\mathcal S$ as a kind of motivic object. I'm interested in finding a natural cohomology theory for $\mathcal S$.
Consider a singular analytic $L^2$ cohomology, which represents the notion that we have a singular space, $\mathcal I$, and we desire a cohomology theory built from analytic objects that are square integrable near the singularities. I'll suggest an ansatz for the singular set of $\mathcal I$, namely that $\Gamma$ and a set of eight cone points comprise $\mathcal I_{\mathrm{sing}}$.
Take the smooth part $\mathcal I_{\mathrm{reg}}=\mathcal I~\backslash ~\mathcal I_{\mathrm{sing}}$, equip $\mathcal I_{\mathrm{reg}}$ with a metric, and examine differential forms $\omega$ where:
$$ \omega \in L^2, \quad d\omega \in L^2.$$
A first pass model is the $k$-th $L^2$ cohomology of the singular space:
$$ H^k_{(2)}(\mathcal I) = \frac{\lbrace \omega \in L^2\Omega^k(\mathcal I_{\mathrm{reg}}):d\omega = 0\rbrace}{d(L^2\Omega^{k-1}(\mathcal I_{\mathrm{reg}}))} $$
which is defined analytically on the regular locus.
While it's not yet clear to me how to develop the cohomological aspect, we can at least build out the algebro-geometric basis of the structure in question, namely, $\mathcal S$, by defining a surface of revolution in intrinsic coordinates $(u,v) \in I\times S^1$ where we use
$$ g_\phi = \frac{1}{\phi(u)}du^2 + \phi(u)dv^2, \quad v \sim v+2\pi $$
with $\phi(u)>0$. Since $\lvert g_{\phi}\rvert = 1$, the Laplace-Beltrami operator is:
$$ \Delta = \partial_u(\phi(u)\partial_u)+ \frac{1}{\phi(u)}\partial^2_v $$
Keep in mind that $$\mathcal{I} = \bigcup_{j=1}^{4} \mathcal{O}_j$$
with the sheets $\mathcal{O}_1, \dots, \mathcal{O}_4$ being four spindle orbifolds (topologically, Riemann spheres $\widehat{\mathbf{C}}$ each with two cone points). However in this example we are examining only a single member, say $\mathcal O_1$ not the full object, so the group $\Pi$, does not come into play yet.
A football orbifold has two conical tips with total cone angle $2\pi \alpha$ at each tip $(0<\alpha\le 1$; for a cone of order $q$, $\alpha = 1/q$). Locally near a tip $\phi(u)\sim \alpha^2 r^2$ in a geodesic radius $r$.
Let the azimuthal circle carry a flat $U(1)$ line bundle, with holonomy $e^{2\pi i \varphi}(\varphi \in \Bbb R/\Bbb Z)$. Sections satisfy the twisted periodicity $\Psi(u,v+2\pi)= e^{2\pi i \varphi}\Psi(u,v).$
Fourier-Bloch decomposition gives
$$ \Psi(u,v) = \sum_{m \in \Bbb Z} R_m(u)e^{i(m+\varphi)v}. $$
So holonomy appears as a shift $m \mapsto m+ \varphi$.
Plugging $\Psi = R_m(u)e^{i(m+\varphi)v}$ into $\Delta\Psi = \lambda \Psi$ yields the Sturm-Liouville problem:
$$ (\phi R'_m)' - \frac{(m+\varphi)^2}{\phi(u)}R_m + \lambda R_m = 0 $$
with regularity at the cone tips.
Near a cone of angle $2\pi \alpha$ one finds Bessel behavior with order $\nu = \frac{|m +\varphi|}{\alpha}$ so the $L^2$ solution behaves like $R_m \sim r^{\nu}$.
If the isotropy at a tip has order $q$, azimuthal modes lie in a fixed coset $q\Bbb Z+r$ for some residue $r \in \lbrace 0,...,q-1\rbrace$. We can encode this as an effective shift
$$ m \in q\Bbb Z + r \quad \iff \quad m + \varphi = qn + (r+ \varphi) \quad (n\in \Bbb Z) $$
We define the theta kernel with characteristic $(q,r;\varphi)$
$$ \theta_{q,r;\varphi}(t) = \sum_{n \in \Bbb Z} e^{-\pi(qn+r+\varphi)^2 t} $$
Poisson summation gives the modular inversion
$$ \theta_{q,r;\varphi}(1/t) = \frac{1}{q}t^{-1/2} \sum_{k\in \Bbb Z} \exp\bigg(-\pi \frac{k^2}{q^2 t}\bigg) e^{\frac{2\pi i k}{q}(r+\varphi)}$$
From here we take the Mellin transform with $t^{\frac{s}{2} -1}$ to obtain a functional equation, and then continue it meromorphically to $s \in \Bbb C$.
Forgetting the twisting by setting $q=1$, $r=0$ and $\varphi=0$, we recover the standard completed Riemann factor $\pi^{-s/2}\Gamma(s/2)\zeta(s)$.
Let
$$\mathcal S_{\mathrm{dec}}=(\mathcal I,\Gamma,\Pi,\chi,\rho,\mathcal L)$$
be a decorated seed, where $\mathcal I$ is the singular geometric object, $\Gamma\subset \mathcal I$ is the distinguished skeletal/degeneration locus, $\Pi$ is the symmetry group, and $(\chi,\rho,\mathcal L)$ is the twisting datum.
Assume that there exists a twisted singular $L^2$-cohomology theory
$$H^\bullet_{\mathrm{seed}}(\mathcal S_{\mathrm{dec}})$$
attached to the pair $\mathscr I = (\mathcal I,\Gamma)$, engaging both the cone/singular geometry of $\mathcal I$ and the holonomy/twisting data $(\rho,\mathcal L)$.
Define for $x\in(0,1)$ and $t\ge 0$, the one-parameter family
$$\varphi_t(x)=e^{t/\ln x}.$$
Then $\{\varphi_t\}_{t\ge 0}$ forms a multiplicative semigroup in the parameter $t$
$$\varphi_{t_1}(x)\varphi_{t_2}(x)=\varphi_{t_1+t_2}(x), \qquad t_1,t_2\ge 0$$
and determines a strongly continuous one parameter semigroup
$$U_t:H^\bullet_{\mathrm{seed}}(\mathcal S_{\mathrm{dec}})\longrightarrow H^\bullet_{\mathrm{seed}}(\mathcal S_{\mathrm{dec}})$$
by multiplication
$$U_t\omega=\varphi_t\,\omega.$$
Its infinitesimal generator is the unbounded operator
$$\mathcal A \omega=\left.\frac{d}{dt}\right|_{t=0}U_t\omega=\frac{1}{\ln x}\,\omega$$
defined on the natural dense domain of classes admitting such differentiation.
Additionally the cone point regime $x\to 0^+$ is asymptotically tame for the generator since
$$\frac{1}{\ln x}\to 0$$
The skeletal regime $x\to 1^-$ corresponding to the degeneration locus $\Gamma$, is the singular regime of the generator since
$$\frac{1}{\ln x}\to -\infty.$$
And, under the logarithmic change of variable
$$u=-\frac{1}{\ln x}>0$$
the seed evolution becomes the ordinary exponential semigroup
$$\varphi_t(x)=e^{-tu}$$
so that the seed dynamics is Laplace type in the $u$-coordinate.
Here, $x$ is encoding distance to degeneration on $\mathscr I$, which is supported by the above asymptotics. It is not a global coordinate on $\mathcal I$. Here $u$ is the spectral, or Laplace coordinate.
The geometry degenerates at $\Gamma$. And the change of variable to the spectral variable, $u$, converts this blowup into large parameter decay.
Let
$$ r: \mathcal I_{\mathrm{reg}} \longrightarrow [0,\infty) $$
be a canonical geometric function vanishing exactly on the degeneration locus $\Gamma$. Define
$$ x(p):= \exp\bigg( -\frac{1}{r(p)} \bigg), \quad r(p)>0. $$
Then $$ -\frac{1}{\ln x(p)}=r(p), $$
and hence
$$ \varphi_t(p) = e^{t/\ln x(p)}=e^{-tr(p)}. $$
The twisted singular $L^2$-cohomology $H^{\bullet}_{\mathrm{seed}}(\mathcal S_{\mathrm{dec}})$ provides the natural Hilbert space of admissible twisted harmonic sectors on the seed, while the semigroup generated by $\varphi_t(x) = e^{t/\ln x}$ furnishes the intrinsic flow on those sectors that spectrally resolves transport between the cone point geometry and the singular skeletal locus $\Gamma$.
Comments
Post a Comment