Duality for dormant opers of classical types B and C

Abstract: A $\mathfrak{g}$-oper for a simple Lie algebra $\mathfrak{g}$ is a specific type of flat principal bundle on an algebraic curve. When the base field is of prime characteristic $p$, those with vanishing $p$-curvature are called dormant $\mathfrak{g}$-opers, and they form finite and geometrically meaningful moduli spaces. In earlier work, a canonical duality was established between dormant $\mathfrak{sl}<em>n$-opers and dormant $\mathfrak{sl}</em>{p-n}$-opers. This duality has provided effective tools for the study of higher-rank cases, as well as for the computation and structural understanding of the associated enumerative invariants. The main result of this paper extends this duality phenomenon to classical Lie algebras of type B and C. More precisely, under the numerical condition $p-1 = 2 (\ell +m)$, we construct a canonical isomorphism between the moduli spaces of dormant $\mathfrak{so}<em>{2\ell +1}$-opers and dormant and $\mathfrak{sp}</em>{2m}$-opers with prescribed symmetric radii.

• The paper establishes a canonical, base-change-compatible isomorphism between dormant \(\mathfrak{so}_{2\ell+1}\)-opers and \(\mathfrak{sp}_{2m}\)-opers when \(p-1=2(\ell+m)\), with radii related by complement.
• It identifies orthogonal and symplectic opers with self-dual \(\mathfrak{sl}_n\)-opers and proves that the type-A duality preserves self-duality, enabling the classical-type correspondence.
• The duality extends generic étaleness and Verlinde-type formulas for projection degrees across admissible B- and C-type ranks, while explicit genus-zero three-point data remains an open problem.

This paper by Yasuhiro Wakabayashi extends the rank-level duality for dormant opers from type $A$ to the classical Lie algebras of types $B$ and $C$. The central result is a canonical isomorphism between the moduli stacks of dormant $\mathfrak{so}_{2\ell+1}$-opers and dormant $\mathfrak{sp}_{2m}$-opers with prescribed radii, valid whenever the characteristic satisfies $p - 1 = 2(\ell + m)$. This duality then feeds into an enumerative framework, yielding Verlinde-type formulas for the generic degrees of the relevant projection morphisms in ranks that were previously out of reach.

Background and motivation

A $\mathfrak{g}$-oper on an algebraic curve is a flat principal $G$-bundle with a $B$-reduction satisfying a Griffiths–transversality condition together with a nondegeneracy requirement along the simple negative root spaces. In characteristic $p$, the fundamental invariant of a flat bundle is its $B$0-curvature; a flat bundle with vanishing $B$1-curvature is called dormant. Dormant opers are finite in number over a fixed pointed stable curve, and their moduli stacks are proper Deligne–Mumford stacks with finite projection to $B$2 (2605.17981). These objects arise in $B$3-adic Teichmüller theory (via Mochizuki's indigenous bundles), in characteristic-$B$4 non-abelian Hodge theory through the Cartier transform, and in connection with the Grothendieck–Katz conjecture on $B$5-curvatures.

The previously known duality, established in earlier work of the author, gives for each $B$6 an involution

$B$7

over $B$8, where $B$9 is a bijection between the admissible radius sets induced by complementation of subsets of $C$0. This duality permits reduction of high-rank enumerative questions to low-rank ones. The present paper asks whether an analogous statement holds when $C$1 is replaced by $C$2 or $C$3, and answers affirmatively under the numerical constraint $C$4.

Vector-bundle descriptions and self-duality

The paper works throughout in the logarithmic setting: opers are defined on log curves $C$5 arising from $C$6-pointed stable curves, using logarithmic crystalline differential operators $C$7. A key technical device is the description of orthogonal and symplectic opers via vector bundles. Fixing a dormant $C$8-theta characteristic $C$9, one forms the vector bundle $\mathfrak{so}_{2\ell+1}$0 with its natural filtration; a $\mathfrak{so}_{2\ell+1}$1-oper is a connection on this bundle compatible with the determinant data. A $\mathfrak{so}_{2\ell+1}$2-oper (resp. $\mathfrak{so}_{2\ell+1}$3-oper) is then such an oper equipped with a nondegenerate symmetric (resp. skew-symmetric) bilinear form valued in the flat line bundle $\mathfrak{so}_{2\ell+1}$4, compatible with the flag via orthogonality $\mathfrak{so}_{2\ell+1}$5.

The bridge to types B and C is the notion of self-duality. A $\mathfrak{so}_{2\ell+1}$6-oper is self-dual if it is isomorphic to its dual twisted by $\mathfrak{so}_{2\ell+1}$7. The paper proves that, for $\mathfrak{so}_{2\ell+1}$8, orthogonal and symplectic opers correspond bijectively to self-dual $\mathfrak{so}_{2\ell+1}$9-opers, refining an earlier result. The proof of the crucial sign computation—showing that the scalar $\mathfrak{sp}_{2m}$0 relating the two bilinear forms extracted from a self-duality isomorphism equals $\mathfrak{sp}_{2m}$1 in the orthogonal case and $\mathfrak{sp}_{2m}$2 in the symplectic case—is carried out locally on a dense open where $\mathfrak{sp}_{2m}$3 and $\mathfrak{sp}_{2m}$4 are trivial, evaluating the form on sections obtained by applying the connection repeatedly to a generator of $\mathfrak{sp}_{2m}$5. This reduction to self-dual $\mathfrak{sp}_{2m}$6-opers is what allows the type-$\mathfrak{sp}_{2m}$7 duality machinery to be applied verbatim.

The duality theorem

The construction proceeds through the canonical dormant $\mathfrak{sp}_{2m}$8-oper associated to any line bundle $\mathfrak{sp}_{2m}$9: the quotient $p - 1 = 2(\ell + m)$0, where $p - 1 = 2(\ell + m)$1 records the failure of $p - 1 = 2(\ell + m)$2-th powers to commute with differentiation. Given a dormant $p - 1 = 2(\ell + m)$3-oper, the kernel of the natural surjection from this canonical oper onto the given oper carries a canonical dormant $p - 1 = 2(\ell + m)$4-oper structure, where $p - 1 = 2(\ell + m)$5 is a canonically determined $p - 1 = 2(\ell + m)$6-theta characteristic. Applying this construction twice returns the original oper twisted by the flat line bundle $p - 1 = 2(\ell + m)$7, so at the level of $p - 1 = 2(\ell + m)$8-opers one obtains an involution.

The essential new observation is that this involution preserves self-duality. The proof reduces to the case of a strict log curve with trivialized tangent bundle, where dormant projective connections can be manipulated as differential operators: if $p - 1 = 2(\ell + m)$9 is the $\mathfrak{g}$0-projective connection defining the oper, dormancy forces the existence of a complementary operator $\mathfrak{g}$1 with $\mathfrak{g}$2, and the paper shows both that $\mathfrak{g}$3 as well and that the dual oper corresponds to $\mathfrak{g}$4. Self-duality of the original oper then transfers through the short exact sequence of flat bundles determined by factorization of $\mathfrak{g}$5.

Combining this preservation result with the self-duality correspondence yields the main theorem: for positive integers $\mathfrak{g}$6 with $\mathfrak{g}$7, there is a bijection between isomorphism classes of dormant $\mathfrak{g}$8-opers and dormant $\mathfrak{g}$9-opers on $G$0, restricting to a bijection between those of radii $G$1 and those of radii $G$2, functorial in base change. At stack level this upgrades to an isomorphism

$G$3

satisfying $G$4. Two degenerate cases are also determined completely: the stack of dormant $G$5-opers is empty unless all radii equal the unique element $G$6, in which case the projection to $G$7 is an isomorphism; the unique such oper is, in the smooth unpointed case, the Raynaud-type oper constructed from the sheaf of locally exact differentials. An analogous statement holds for $G$8.

Consequences for moduli and enumeration

Two consequences follow immediately. First, generic étaleness of the projections $G$9 for $B$0 or $B$1 with $B$2 holds in full generality: the cases $B$3 and $B$4 were known, and the remaining high-rank cases reduce to them via the duality. Consequently, the generic degrees $B$5 are well-defined enumerative invariants across the entire range.

Second, these degrees obey a pseudo-fusion rule. For each admissible $B$6, the paper constructs a pseudo-fusion ring $B$7 on the free abelian group $B$8, with multiplication defined by triple-degeneration degrees. When $B$9, the ring for $p$0 is obtained by transporting the ring for the dual-rank symplectic algebra along the radius bijection, and the equality $p$1 ensures the multiplication takes the same form. The resulting Verlinde-type formula reads

$p$2

where $p$3 is the set of ring homomorphisms $p$4 and $p$5. This extends to types B and C, in all ranks below $p$6, formulas previously available only in low rank or for small $p$7 ratios, and parallels the known relationships with conformal-block Verlinde formulas, Grassmannian Gromov–Witten invariants, and Ehrhart quasi-polynomials.

Limitations and open questions

The duality is conditional on the arithmetic constraint $p$8; no statement is made for pairs $p$9 violating it, and the method—passing through self-dual $B$00-opers and the canonical $B$01-oper—does not obviously extend beyond it. Exceptional types are not treated. Most significantly, the paper notes that explicit determination of the degrees requires detailed analysis of the three-pointed genus-zero stacks $B$02, since their values determine the ring-theoretic structure of $B$03; carrying out this analysis for types B and C is identified as an important open problem. The use of pseudo-fusion rules rather than genuine fusion rules reflects the absence of appropriate references establishing the latter framework in this generality.

Conclusion

The paper establishes that the rank-level duality phenomenon for dormant opers is not specific to type $B$04: under the condition $B$05, dormant orthogonal and symplectic opers of complementary ranks are canonically equivalent as moduli problems, with radii transformed by the complement involution. The proof strategy—identifying both classes with self-dual $B$06-opers and verifying compatibility of the $B$07-duality with self-duality—is economical and functorial. Combined with prior finiteness and generic étaleness results, this yields complete Verlinde-type enumerative formulas for dormant opers of types B and C in all admissible ranks, contingent on future computation of the genus-zero fusion data.