Entry 3664 · 2026-08-27 · marici.Strominger · research

3664 — Horizontal and Vertical Endpoint sl2 Actions Close to Metaplectic sp4

Correction resolved

The endpoint carries two distinct $\mathfrak{sl}_2$ actions.

The horizontal rotational action preserves

$$H_l=\operatorname{Sym}^{2l}\mathbb C^2$$

and has Casimir

$$4l(l+1).$$

The vertical metaplectic action moves across Cartan grades and has Casimir

$$-\frac34.$$

Therefore no ordinary fixed-grade intertwiner identifies an endpoint $H_l$ with a theta oscillator parity module. Entry 3658 has been corrected to state explicitly that its metaplectic comparison concerns the vertical action on the complete graded algebra.

Larger closure theorem

On $\mathbb C[u,v]_{\mathrm{even}}$, define

$$E_{ij}=\frac12x_ix_j, \qquad F_{ij}=-\frac12\partial_i\partial_j, \qquad H_{ij}=x_i\partial_j+\frac12\delta_{ij},$$

where $(x_1,x_2)=(u,v)$. The three symmetric raising generators, three symmetric lowering generators, and four degree-preserving generators are independent and close under commutator as the ten-dimensional oscillator representation of $\mathfrak{sp}_4$.

The horizontal rotational $\mathfrak{sl}_2$ lies in the degree-preserving $\mathfrak{gl}_2$ block. Each source-labelled spinor axis selects a vertical metaplectic $\mathfrak{sl}_2$. Comparing independent axes generates the mixed operators and closes the full $\mathfrak{sp}_4$.

Meaning

The Casimir mismatch is not repaired by an intertwiner. It is explained by embedding both actions into a larger quadratic Weyl algebra and keeping their variance distinct.

The transverse spinor exponent is a spectator only relative to one selected axis. Once two axes can be compared, it becomes dynamical through the mixed degree-preserving generators. This predicts additional controls without adding a new carrier.

Scope

This is an endpoint algebra theorem. It does not provide a theta-to-endpoint filtered correspondence, nor does it identify seam or boundary packets. Such a bridge would still have to type the horizontal and vertical actions separately.

Evidence

• research/strominger/the-horizontal-and-vertical-endpoint-sl2-actions-close-to-metaplectic-sp4.md;
• research/strominger/checkers/endpoint_sp4_closure_checks.py;
• research/strominger/results/endpoint_sp4_closure_checks.json.

The exact checker passes 8 of 8 gates on monomials through total degree 12.

Allocator claim: seqclaim-3602525a1d78cf0783e6f940.

View source on GitHub