Characteristic cohomology of the global genus-two Koszul complex

Determine the cohomology sheaves of the Koszul complex generated by the four commuting infinite-order operators associated with the odd supersymmetry generators on the entire affine Minkowski space of symmetric complex $2\times2$ matrices, rather than only on the genus-two Siegel upper half-plane.

Background

The paper constructs a Koszul complex from the infinite-order operators obtained by exponentiating the four odd generators of the Lie superalgebra (1∣4)(1|4) and proves that, on the genus-two Siegel upper half-plane, its zeroth cohomology is the constant sheaf generated by the Riemann Thetanullwert and all higher cohomology sheaves vanish.

The same operators are defined on the full affine Minkowski superspace, whose underlying even space is the vector space of symmetric 2×22\times2 complex matrices. The authors indicate that the cited work determines the cohomology on this larger space, but explicitly leave that global determination outside the scope of the paper.

References

In fact, proves a more general statement. The operators $D_y{1/2}$, $y\in g_$ make sense on the entire Minkowski superspace $C{3|2}$, not just on its future tube $SH_2$. Therefore the complex of sheaves $C\bullet$ can be defined on the entire $C3 = _2(C)$, not just on $H_2$. The results of amount to determination of $ H\bullet(C\bullet)$ on all of $C3$. We do not address this problem here.

— Supersymmetry, differential operators of infinite order and theta functions  (2608.12846 - Kapranov, 13 Aug 2026) in Section “Analysis of the Koszul complex” following Theorem \ref{theta:susy-2}, final remark of the section