# Matt Kerr, $K_2$ and quantum curves (Thursday, July 21)

> Holy moly, Matt has a lot of equations and writes super fast! I was not able to record even half of what was written.

## A miraculous coincidence

:::{.remark}
Suppose $f\in \Mero(\CC, \CC)$ with no poles in the horizontal strip $\abs{\Im(z)} < 2\pi$, then $e^{\pm 2\pi i \dd{}{z}}$ is a shift operator $f(z) \mapsto f(z+2\pi i)$.
Example: $e^{2\pi i \dd{}{z}} z^k = \sum_\ell \qty{ (2\pi i)^\ell \over \ell! } \qty{\dd{}{z}}^\ell z^k = (z+ 2\pi i)^k$.
Consider 
\[
\hat\phi \da e^r + e^{-r} + e^{2\pi \dd{}{r}} + e^{-2\pi i \dd{}{r} }\actson L^2(\RR)
.\]
This is some kind of quantum Hamiltonian for a Fermi gas?
Note that this is unbounded, but its inverse is bounded and self-adjoint with spectrum that can be enumerated.
This suggests a curve of the form $x+x\inv+y+y\inv = \lambda$, is there any AG or Hodge theory arising in this spectral problem?
:::

:::{.remark}
Consider elliptic curves, pullbacks of the Legendre family $y^2 = x(x-1)(x-16t^2)$.
This has two periods $A(t), B(t)$ given by integrating over the two torus cycles $\alpha,\beta$, which are complicated but annihilated by the Picard-Fuchs operator $L = D_t^2 - 16t^2 (D_t + 1)^2$.
Solve $L(\wait) = c$ a constant using variation of parameters to get 
\[
C(t) \approx B(t) \int A(t) {\dt \over t} - A(t) \int B(t) {\dt \over t}
.\]
Can choose constant terms such that the monodromy of $C$ lies in $\ZZ(2) \gens{A(t), B(t)}$ where $\ZZ(2) = 2\pi i \ZZ$.
Set $\nu(t) = {1\over \pi^2} \log^2(-t) + {7\over 6} + \bigo(t\log(t))$ and define $t_n$ by $\nu(t_n) = n+2$.
Then $-\log(-t_n) = \pi\sqrt{n + {5\over 6}}$, which yields a sequence matching the exponents in the eigenvalues of $\hat \phi$.
Something is going on!
:::

:::{.remark}
Local mirror symmetry conjecture linking inhomogeneous Picard-Fuchs equations of Hori-Vafa models and enumerative geometry of local CYs: the $B$ model mirror of $K_{\PP^1\times \PP^1}$ is the $\VMHS$ of $H^3$ of $Y_a = V(x+x\inv + y+y\inv + a + uv) \subseteq (\CCstar)^2\times \times \CC^2$ with coordinates $\tv{x, y}$ and $\tv{u, v}$.
See Frobenius duality for differential operators?
The Geneva conjectures connected difference operators and quantum curves to enumerative geometry of CYs.
:::

## Beilinson's Formula

:::{.remark}
Like $X\slice \CC$ smooth projective, and note
\[
\K_2(\CC(X)) = 
{
\CC(X)\units \wedge \CC(X)\units \over 
\gens{F \wedge (1-F)} 
} 
\mapsvia{\text{T}} \bigoplus _{p\in ?} \CCstar
\]
where $T$ is the *tame symbol*.
We want to compute the regulator 
\[
\ker(T) = H_M^2(X, \ZZ(2)) \mapsvia{R_X} H^2_D(X, \ZZ(2)) \cong H^1(X^\an, \CC/\ZZ(2))
.\]
Beilinson shows that if $\ts{f, g}\in \ker(T)$ and $\gamma\in \pi_1(X\sm ?, x_0)$ then 
\[
\inner{R_X\ts{f, g}}{[\gamma]} \equiv \int_\gamma \log(f) {dg\over g} - \log g(x_0) \int_\gamma {df\over f}
.\]
:::

:::{.remark}
Consider $X \mapsvia{\pi} S$ a family of smooth curves and $z\in H_m^2(X, \ZZ(2))$ we can consider their fibers $z_2\in H_M^2(X_s, \ZZ(2))$.
This maps to 
\[
R(S) = H^1(X_s, \CC/\ZZ(2)) \cong \Ext^1_{\MHS}(\ZZ(2), H^1(X_s, \ZZ))
.\]
By Griffiths transversality, $\nabla R = F^1 H^1_{X\slice S} \tensor \omega_S^1$.
If $z$ lifts to $\ts{F, G}$ (?) then $\nabla R = {dF\over F}\wedge {dG\over G}$.
:::

## Families of elliptic curves

:::{.remark}
Let $\Delta \subseteq \RR\times [-1, 1]$ be a reflexive polytope, e.g. the polytope for $\PP^n$, let $\Delta^\circ$ be is polar dual.
Somehow construct an elliptic surface $\mce$ out of this!
The symbol $\ts{-x, -y}\in \K_2(\CC(\mce))$ lifting to $z\in H^2_M(\Sigma\sm \mce_\infty, \ZZ(2))$ and take its regulator in $H^1(E_?, \CC/\ZZ(2))$.

Idea: regulators of $\K_2$ of curves relate to Hori-Vafa models, so we've reduced to relating regulators to quantum curves:

![](figures/2022-07-21_10-53-34.png)

:::

## Quantum Curves

:::{.remark}
Theorems:

- For all $a\in V$, there is a $\psi_a\in L^2(\RR)\smz$ such that $\hat\phi \psi_a = a\psi _a$.
- Asymptotically, 100% of $\sigma(\hat \phi)$ belong to $V$.

The first primarily uses Beilinson's formula, the second uses two-variable Fourier transforms.
These establish a link between the spectrum $\sigma(\hat\phi)$ (quantum curve side) to the $\ZZ\dash$locus of truncated HNFs (regulator side).
:::

:::{.remark}
See Grothendieck period conjecture.
:::