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Paul Seidel

Publications and source records attributed to Paul Seidel.

At least 19 recordsLinked to original sources
arXiv2026

The quantum connection and its mod p reduction

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.

math.SG↗
arXiv2025

P-adic Gamma classes and overconvergent Frobenius structures for quantum connections

Consider the small quantum connection on a monotone symplectic manifold, with p-adic coefficients. We conjecture that this always admits an overconvergent Frobenius structure, whose constant term is given by a characteristic class associated to Morita's p-adic Gamma function. We prove this conjecture for toric Fano varieties and Grassmannians, and also supply additional experimental evidence.

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arXiv2025

P-adic splittings of the quantum connection

We introduce operations with p-adic integer coefficients, associated to idempotents in the quantum cohomology of a monotone symplectic manifold, and apply them to the structure of the quantum connection.

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arXiv2024

Symplectic cohomology relative to a smooth anticanonical divisor

For a monotone symplectic manifold and a smooth anticanonical divisor, there is a formal deformation of the symplectic cohomology of the divisor complement, defined by allowing Floer cylinders to intersect the divisor. We compute this deformed symplectic cohomology, in terms of the ordinary cohomology of the manifold and divisor; and also describe some additional structures that it carries.

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arXiv2023

The quantum connection, Fourier-Laplace transform, and families of A-infinity-categories

Take a closed monotone symplectic manifold containing a smooth anticanonical divisor. The quantum connection on its cohomology has singularities at zero and infinity (in the quantum parameter). At zero it has a regular singular point, by definition. We show that the singularity at infinity is of unramified exponential type. The argument involves: realizing cohomology as a deformation of the symplectic cohomology of the divisor complement; the corresponding deformation of the wrapped Fukaya category; a new categorical interpretation of the Fourier-Laplace transform of D-modules; and the regularity theorem of Petrov-Vaintrob-Vologodsky in noncommutative geometry.

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arXiv2022

Covariant constancy of quantum Steenrod operations

We prove a relationship between quantum Steenrod operations and the quantum connection. In particular there are operations extending the quantum Steenrod power operations that, when viewed as endomorphisms of equivariant quantum cohomology, are covariantly constant. We demonstrate how this property is used in computations of examples.

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arXiv2021

Fukaya A_\infty-structures associated to Lefschetz fibrations. VIII

We use Lefschetz pencil methods to derive structural results about Fukaya categories of Calabi-Yau hypersurfaces; in particular, concerning their dependence on the Novikov parameter. Warning to the reader: the arguments in this paper cite two preprints (papers 6.5 and 7 in the series) which do not at present exist. Until that is remedied, the proofs cannot be considered complete, and this paper should be regarded as a preliminary research announcement.

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arXiv2020

Fukaya A_\infty-structures associated to Lefschetz fibrations. III

Floer cohomology groups are usually defined over a field of formal functions (a Novikov field). Under certain assumptions, one can equip them with connections, which means operations of differentiation with respect to the Novikov variable. This allows one to write differential equations for Floer cohomology classes. Here, we apply that idea to symplectic cohomology groups associated to Lefschetz fibrations, and obtain a relation with enumerative geometry.

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arXiv2019

Formal groups and quantum cohomology

We use chain level genus zero Gromov-Witten theory to associate to any closed monotone symplectic manifold a formal group (loosely interpreted), whose Lie algebra is the odd degree cohomology of the manifold (with vanishing bracket). When taken with coefficients mod p, the p-th power map of the formal group is related to quantum Steenrod operations. The motivation for this construction comes from derived Picard groups of Fukaya categories, and from arithmetic aspects of mirror symmetry.

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arXiv2018

Fukaya A_\infty-structures associated to Lefschetz fibrations. VI

To a symplectic Lefschetz pencil on a monotone symplectic manifold, we associate an algebraic structure, which is a pencil of categories in the sense of noncommutative geometry. One fibre of this "noncommutative pencil" is related to the Fukaya category of the open (meaning, with the base locus removed, and hence exact symplectic) fibre of the original Lefschetz pencil; the other fibres are newly constructed kinds of Fukaya categories.

math.SG↗
arXiv2018

Fukaya A_\infty-structures associated to Lefschetz fibrations. IV

We consider Hamiltonian Floer cohomology groups associated to a Lefschetz fibration, and the structure of operations on them. As an application, we will (under an important additional assumption) equip those groups with connections, which differentiate with respect to the Novikov variable.

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arXiv2017

Fukaya A_\infty structures associated to Lefschetz fibrations. II

We consider the Fukaya category associated to a basis of vanishing cycles in a Lefschetz fibration. We show that each element of the Floer cohomology of the monodromy around infinity gives rise to a natural transformation from the Serre functor to the identity functor. This complements previously known constructions, in a way which fits in well with homological mirror symmetry.

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arXiv2016

Fukaya A_\infty structures associated to Lefschetz fibrations. I

This (partially expository) paper discusses Lagrangian Floer cohomology in the context of Lefschetz fibrations, with emphasis on the algebraic structures encountered there. In addition to the well-known directed A_infinity algebras which appear in this situation, one has additional information encoded in a certain bimodule homomorphism. There are two approaches to constructing this homomorphism: in terms of the (noncompact) Lefschetz thimbles in the total space, or else in terms of vanishing cycle in the fibre. We prove a comparison result, which shows that (up to a certain remaining ambiguity) the two approaches are equivalent.

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arXiv2015

Fukaya A_\infty-structures associated to Lefschetz fibrations. II 1/2

We consider a version of the relative Fukaya category for anticanonical Lefschetz pencils. There are direct connections between the behaviour of this category and enumerative geometry: some of these are results announced here, others remain conjectural. The ultimate aim of this approach is to determine the Fukaya category of the Calabi-Yau hypersurfaces that constitute the pencil.

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arXiv2015

Picard-Lefschetz theory and dilating C^*-actions

We consider C^*-actions on Fukaya categories of exact symplectic manifolds. Such actions can be constructed by dimensional induction, going from the fibre of a Lefschetz fibration to its total space. We explore applications to the topology of Lagrangian submanifolds, with an emphasis on ease of computation.

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