The quantum connection and its mod p reduction

Abstract: Recent progress on the structure of the quantum connection for monotone symplectic manifolds has used two approaches, which share the common feature of reducing to mod $p$ coefficients. We refine and compare those approaches. In particular, we establish a relation with quantum Steenrod operations which is stronger than that in Chen's work, leading to more precise information about the singularity at $\infty$ of the quantum connection. For the version of the connection relative to a smooth anticanonical divisor, we draw attention to the implications of the categorical mod $p$ Fontaine-Laffaille structure established by Petrov-Vaintrob-Vologodsky.

• The paper proves that the nilpotent parts of formal monodromy at q=∞ lie in the Zariski closures of the corresponding nilpotent quantum cup-product orbits, implying diagonalizable formal factors when quantum multiplication by c₁(M) is diagonalizable.
• The paper establishes an exact identity between the p-th covariant derivative and a quantum Steenrod operation, yielding congruences such as f(∇∂qᵖ) ≡ 0 mod p under the absence of p-torsion in integral cohomology.
• The paper extends mod p Fontaine–Laffaille methods to relative symplectic cohomology and wrapped Fukaya categories, producing canonical comparison maps for odd primes p ≥ n+3 while identifying open problems involving relative Steenrod operations and integral p-adic lifts.

This paper by Dan Pomerleano and Paul Seidel studies the structure of the small quantum connection on a closed monotone symplectic manifold $M^{2n}$ satisfying $[\omega_M] = c_1(M)$, and its analogue relative to a smooth anticanonical divisor. Its unifying theme is reduction to characteristic $p$: two previously distinct approaches to the irregular singularity at $q=\infty$ — one via quantum Steenrod operations (following Chen) and one via categorical methods and the mod $p$ Fontaine–Laffaille structure (following Pomerleano–Seidel) — are refined, compared, and shown to yield complementary quantitative information.

The absolute quantum connection

The central object is the connection $$\nabla_{\partial_q} x = \partial_q x + q^{-1}c_1(M)\ast_q x$$ on $H^*(M)[q^{\pm 1}]$, where $\ast_q$ is the small quantum product weighted by $q^{\int_A c_1}$. It has a regular singularity at $q=0$ with unipotent monodromy; the singularity at $[\omega_M] = c_1(M)$0 is irregular, and its formal normal form was established by Chen: over $[\omega_M] = c_1(M)$1 the connection splits as a direct sum of pieces $[\omega_M] = c_1(M)$2, indexed by eigenvalues $[\omega_M] = c_1(M)$3 of the map $[\omega_M] = c_1(M)$4, with $[\omega_M] = c_1(M)$5 quasi-unipotent. Under the additional assumption that $[\omega_M] = c_1(M)$6 contains a smooth anticanonical divisor (a symplectic hypersurface Poincaré dual to $[\omega_M] = c_1(M)$7 whose complement is Weinstein), Pomerleano–Seidel had earlier bounded the Jordan block sizes of $[\omega_M] = c_1(M)$8 by $[\omega_M] = c_1(M)$9 for integer eigenvalues and $p$0 otherwise.

The paper's first new result strengthens this in a different direction. Let $p$1 be the minimal polynomial of $p$2, so that $p$3. Theorem (Jordan bound refinement): the nilpotent part of each $p$4 lies in the Zariski closure of the conjugacy class of the nilpotent part of $p$5 acting on the generalized $p$6-eigenspace. In particular, if that action is diagonalizable then all $p$7 are diagonalizable, equivalently $p$8 has finite order. This parallels Varchenko's Hodge-theoretic result for Gauss–Manin connections of isolated hypersurface singularities, where the nilpotent monodromy lies in the closure of the conjugacy class of multiplication by $p$9 on the Jacobian ring. The authors note an asymmetry: by Briançon–Skoda one has $q=\infty$0, but no dimension-dependent bound on the degree of $q=\infty$1 (equivalently, on the nilpotency index of $q=\infty$2) is known for quantum cohomology. They also stress that these statements do not contradict the fact that the genuine analytic monodromy always has a Jordan block of size $q=\infty$3, because the formal series realizing the splitting near $q=\infty$4 diverge (Stokes phenomenon).

Quantum Steenrod operations and $q=\infty$5-curvature

The mechanism behind the new bound is an exact identity, conjectured by Etingof and J. H. Lee: for any prime $q=\infty$6,

$q=\infty$7

where $q=\infty$8 denotes the quantum Steenrod operation associated to $q=\infty$9, introduced by Fukaya and developed by Wilkins and Seidel–Wilkins. Chen had only used the weaker statement that the difference of the two sides is nilpotent. The proof exploits the covariant constancy of $p$0 with respect to $p$1, together with the fact that both sides agree at $p$2 (reducing to classical Steenrod powers) and agree when $p$3 (reducing to the $p$4-th quantum power); the difference is strictly degree-decreasing and hence vanishes once it vanishes on the unit, which follows from the Frobenius-twisted isomorphism $p$5 between completed Laurent series spaces built from operations applied to $p$6.

Two corollaries follow without further reference to Steenrod operations. First, if $p$7 has no $p$8-torsion, then $p$9 on integral cohomology — infinitely many congruence constraints on a single operator. Second, over a finite field $$\nabla_{\partial_q} x = \partial_q x + q^{-1}c_1(M)\ast_q x$$0 on which $$\nabla_{\partial_q} x = \partial_q x + q^{-1}c_1(M)\ast_q x$$1 is diagonalizable, the connection splits as a direct sum of trivial connections $$\nabla_{\partial_q} x = \partial_q x + q^{-1}c_1(M)\ast_q x$$2; the proof passes through the vanishing of the $$\nabla_{\partial_q} x = \partial_q x + q^{-1}c_1(M)\ast_q x$$3-curvature on each summand and Katz's theorem that zero $$\nabla_{\partial_q} x = \partial_q x + q^{-1}c_1(M)\ast_q x$$4-curvature implies triviality. A technical caveat is recorded: the splitting argument does not respect the $$\nabla_{\partial_q} x = \partial_q x + q^{-1}c_1(M)\ast_q x$$5-grading unless mild extra hypotheses hold (no $$\nabla_{\partial_q} x = \partial_q x + q^{-1}c_1(M)\ast_q x$$6-torsion, or even-degree cohomology), since the $$\nabla_{\partial_q} x = \partial_q x + q^{-1}c_1(M)\ast_q x$$7-component of $$\nabla_{\partial_q} x = \partial_q x + q^{-1}c_1(M)\ast_q x$$8 must be controlled via lifts to $$\nabla_{\partial_q} x = \partial_q x + q^{-1}c_1(M)\ast_q x$$9 coefficients.

Katz's method refined

The passage from characteristic $H^*(M)[q^{\pm 1}]$0 back to characteristic $H^*(M)[q^{\pm 1}]$1 uses a refinement of Katz's nilpotent connection theorem. For connections over $H^*(M)[q^{\pm 1}]$2, where $H^*(M)[q^{\pm 1}]$3 is a finitely generated subring of $H^*(M)[q^{\pm 1}]$4, the paper proves: if the mod-$H^*(M)[q^{\pm 1}]$5 reductions have $H^*(M)[q^{\pm 1}]$6-curvature lying in a fixed nilpotent orbit closure $H^*(M)[q^{\pm 1}]$7, then over $H^*(M)[q^{\pm 1}]$8 the connection is equivalent to $H^*(M)[q^{\pm 1}]$9 with rational eigenvalues and nilpotent part in $\ast_q$0. The proof combines a truncation argument producing non-resonant normal forms over finitely generated rings, the elementary identity that $\ast_q$1 is conjugate to $\ast_q$2 for nilpotent $\ast_q$3 in characteristic $\ast_q$4, and the Jacobson property of $\ast_q$5. Combined with the Steenrod comparison — which identifies the $\ast_q$6-curvature on each eigenspace summand with the nilpotent part of $\ast_q$7 — this yields the Jordan closure theorem stated above.

The relative theory

The second half concerns deformed symplectic cohomology $\ast_q$8 for a smooth anticanonical divisor $\ast_q$9, defined via a telescope complex over Hamiltonians of increasing slope, deformed by counting continuation-map solutions with prescribed intersection divisors with $q^{\int_A c_1}$0. Additively, $q^{\int_A c_1}$1, and after inverting $q^{\int_A c_1}$2 one recovers $q^{\int_A c_1}$3. Two structures beyond the standard TQFT picture are constructed here:

• A quantum module structure $q^{\int_A c_1}$4 making $q^{\int_A c_1}$5 a module over $q^{\int_A c_1}$6, defined by counting cylinders with an adjacency condition to a pseudo-cycle transverse to $q^{\int_A c_1}$7. The compatibility $q^{\int_A c_1}$8 is proved via a two-marked-point parameter space whose boundary analysis implements the gluing deformation.
• An endomorphism $q^{\int_A c_1}$9, defined using cylinders meeting $q=0$0 at a distinguished point $q=0$1, satisfying $q=0$2; this is proved by taking the pseudo-cycle to be a perturbation of the inclusion $q=0$3 and comparing linearized operators.

Consequently $q=0$4 on $q=0$5, and there is an $q=0$6-equivariant version $q=0$7 carrying a connection $q=0$8 whose $q=0$9 part is $[\omega_M] = c_1(M)$00. For primes with no torsion in $[\omega_M] = c_1(M)$01, one obtains $[\omega_M] = c_1(M)$02 for suitable $[\omega_M] = c_1(M)$03 depending on $[\omega_M] = c_1(M)$04; finite generation over the Weyl algebra makes $[\omega_M] = c_1(M)$05 uniform, and the authors sketch (without completing, since it requires control of low-energy thimble solutions) a bound $[\omega_M] = c_1(M)$06.

The mod $[\omega_M] = c_1(M)$07 Fontaine–Laffaille structure

The categorical input is a generalization of the Petrov–Vaintrob–Vologodsky construction to $[\omega_M] = c_1(M)$08-algebras over coefficient rings. For a dga or homotopically projective $[\omega_M] = c_1(M)$09-algebra $[\omega_M] = c_1(M)$10 over $[\omega_M] = c_1(M)$11 with a deformation $[\omega_M] = c_1(M)$12, admitting a lift to $[\omega_M] = c_1(M)$13, with $[\omega_M] = c_1(M)$14 finite rank over $[\omega_M] = c_1(M)$15, homologically smooth, and with the Kaledin class annihilated by $[\omega_M] = c_1(M)$16 up to Hochschild degree shifts, there is an injective map $[\omega_M] = c_1(M)$17 from localized Hochschild homology to localized periodic cyclic homology satisfying

$[\omega_M] = c_1(M)$18

with image spanning the target over $[\omega_M] = c_1(M)$19 in blocks of length $[\omega_M] = c_1(M)$20. The algebraic section handles several points afresh: replacing Yoneda embedding arguments with cofibrant dg replacements valid over rings, transferring deformations along quasi-isomorphisms via acyclicity of a cone complex, and deriving the required Hochschild homology concentration from Shklyarov duality plus the universal coefficient theorem over $[\omega_M] = c_1(M)$21.

Applied to the wrapped Fukaya category of $[\omega_M] = c_1(M)$22 deformed by the Borman–Sheridan element — where the open-closed and cyclic open-closed maps are isomorphisms onto deformed symplectic cohomology, and the Kaledin class corresponds to $[\omega_M] = c_1(M)$23 under the closed-open compatibility — this yields, for odd $[\omega_M] = c_1(M)$24 with no torsion, a canonical injective Fontaine–Laffaille map

$[\omega_M] = c_1(M)$25

A consequence is that $[\omega_M] = c_1(M)$26 on $[\omega_M] = c_1(M)$27 for some $[\omega_M] = c_1(M)$28 (uniform in $[\omega_M] = c_1(M)$29, though no explicit bound on $[\omega_M] = c_1(M)$30 is given). The authors compare this with the Steenrod-based bound $[\omega_M] = c_1(M)$31: qualitatively similar, quantitatively different, and neither implies the other. Both would follow from a single stronger conjecture, namely that $[\omega_M] = c_1(M)$32 is zero mod $[\omega_M] = c_1(M)$33 outright; a potential route runs through a relative version of the Etingof–Lee conjecture involving hypothetical relative quantum Steenrod operations on $[\omega_M] = c_1(M)$34, which the authors note their current strategy does not readily establish.

The authors also raise, without resolving, whether the full integral $[\omega_M] = c_1(M)$35-adic Hodge structure (over $[\omega_M] = c_1(M)$36, with a rescaled connection $[\omega_M] = c_1(M)$37) admits an analogous map $[\omega_M] = c_1(M)$38; they verify consistency in the simplest example below.

The case of $[\omega_M] = c_1(M)$39

For $[\omega_M] = c_1(M)$40 with odd $[\omega_M] = c_1(M)$41, explicit eigenvectors $[\omega_M] = c_1(M)$42 of $[\omega_M] = c_1(M)$43 with eigenvalues $[\omega_M] = c_1(M)$44 are written down, with determinant $[\omega_M] = c_1(M)$45 invertible over $[\omega_M] = c_1(M)$46, verifying the trivial splitting predicted by the diagonalizable case; the congruence $[\omega_M] = c_1(M)$47 for $[\omega_M] = c_1(M)$48 follows. The Etingof–Lee identity itself is verified by combining the highest-order terms of the quantum Steenrod matrix with its commutation with the connection.

For the relative theory, $[\omega_M] = c_1(M)$49 reduces via homological mirror symmetry to the Landau–Ginzburg model $[\omega_M] = c_1(M)$50 over $[\omega_M] = c_1(M)$51. Hochschild–Kostant–Rosenberg type computations give

$[\omega_M] = c_1(M)$52

where $[\omega_M] = c_1(M)$53 removes the critical points $[\omega_M] = c_1(M)$54, with $[\omega_M] = c_1(M)$55 and $[\omega_M] = c_1(M)$56. Iterating the $[\omega_M] = c_1(M)$57-action shows $[\omega_M] = c_1(M)$58 on the equivariant localized group, confirming the Fontaine–Laffaille corollary exactly, and the kernel structure identifies the image of $[\omega_M] = c_1(M)$59 with the kernel of $[\omega_M] = c_1(M)$60, suggesting the formula $[\omega_M] = c_1(M)$61. For the speculative $[\omega_M] = c_1(M)$62 extension, the authors exhibit a coordinate change $[\omega_M] = c_1(M)$63 with $[\omega_M] = c_1(M)$64 via Hensel lifting, and propose $[\omega_M] = c_1(M)$65 as a candidate satisfying both required identities — leaving its actual existence and implications open.

Limitations and open questions

Several assumptions constrain the results. The Jordan closure theorem and the congruence results require absence of $[\omega_M] = c_1(M)$66-torsion in integral cohomology (or, for the splitting statement, allow torsion at the cost of losing the $[\omega_M] = c_1(M)$67-graded refinement). The Fontaine–Laffaille theorem requires odd $[\omega_M] = c_1(M)$68. Whether Proposition-level Jordan bounds hold without the anticanonical divisor hypothesis remains unknown, as does a dimension-dependent bound on $[\omega_M] = c_1(M)$69. The relative Etingof–Lee conjecture, the relative quantum Steenrod operations it presupposes, the pair-of-pants product on deformed symplectic cohomology, the strong conjecture $[\omega_M] = c_1(M)$70, the completed bound on the constant $[\omega_M] = c_1(M)$71, and the integral ($[\omega_M] = c_1(M)$72) Fontaine–Laffaille map are all left open. An equivalent definition of the relative quantum connection via punctured Gromov–Witten invariants, useful for computation, is also anticipated but not developed.

Conclusion

The paper consolidates the mod $[\omega_M] = c_1(M)$73 approach to the irregular singularity of the quantum connection on two fronts. On the absolute side, the exact identification of $[\omega_M] = c_1(M)$74-curvature with a quantum Steenrod operation, fed through a refined Katz orbit-closure theorem, upgrades qualitative quasi-unipotence of $[\omega_M] = c_1(M)$75 to a precise conjugacy-class constraint tied to the quantum product with $[\omega_M] = c_1(M)$76. On the relative side, the categorical Fontaine–Laffaille machinery is extended to ring coefficients and combined with newly constructed module structures on deformed symplectic cohomology, yielding canonical maps whose consequences are comparable to, but independent of, the Steenrod-based ones. The remaining gaps — the relative Etingof–Lee conjecture, the strong common strengthening, and the integral $[\omega_M] = c_1(M)$77-adic refinement — are formulated precisely enough to serve as concrete targets for further work.