arxiv: 2605.06620 · v1 · submitted 2026-05-07 · 🧮 math.SG · math.AG· math.AT

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Bulk-deformations, Floer complex bordism, and Grothendieck-Riemann-Roch

Kenneth Blakey , Noah Porcelli

Authors on Pith no claims yet

Pith reviewed 2026-05-08 03:05 UTC · model grok-4.3

classification 🧮 math.SG math.AGmath.AT

keywords symplectic cohomologycomplex cobordismbulk deformationGrothendieck-Riemann-RochFloer homologyLiouville manifoldGromov-Witten invariantspair-of-pants product

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The pith

For a Liouville manifold the complexification of the MU-lift of symplectic cohomology equals the bulk-deformed symplectic cohomology by the Chern character.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that a Floer-homotopical invariant, the complexified lift of symplectic cohomology to complex cobordism, can be computed directly from the classical invariant given by symplectic cohomology bulk-deformed by the Chern character. It reaches this equality by constructing an explicit model for the complexified homotopy groups of the associated MU-module spectrum and by establishing a homotopy-coherent form of the Grothendieck-Riemann-Roch theorem. A reader would care because the result converts an abstract bordism-valued invariant into something expressed in terms of already-studied operations on symplectic cohomology, and it supplies a concrete criterion for when the lift cannot arise by base change from the sphere spectrum.

Core claim

Given a Liouville manifold, the complexification of the lift of symplectic cohomology to complex cobordism equals symplectic cohomology bulk-deformed by the Chern character. The equality is obtained by providing an explicit model for the complexified homotopy groups of the MU-module spectrum associated to a complex-oriented flow category and by proving a homotopy-coherent version of the Grothendieck-Riemann-Roch theorem for that model.

What carries the argument

Explicit model for the complexified homotopy groups of the MU-module spectrum of a complex-oriented flow category, together with the homotopy-coherent Grothendieck-Riemann-Roch theorem relating it to bulk-deformed symplectic cohomology.

If this is right

A computable criterion in terms of the pair-of-pants product and BV operator detects when the MU lift does not arise by base change from the sphere spectrum.
Explicit examples exist where this non-base-change criterion holds.
The criterion detects non-trivial higher-dimensional complex cobordism classes realized by relative Gromov-Witten moduli spaces of a smooth complex projective variety relative to an ample smooth divisor.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same model and GRR relation could be applied to other complex-oriented invariants such as string topology or Lagrangian Floer theory to produce bordism-valued computations.
The non-base-change criterion supplies a test that might distinguish symplectic cohomology from its classical counterparts in mirror-symmetric settings.
Detection of non-trivial cobordism classes in GW moduli spaces suggests that higher bordism invariants of these spaces are accessible without full virtual fundamental class constructions.

Load-bearing premise

An explicit model for the complexified homotopy groups of the MU-module spectrum attached to a complex-oriented flow category exists and the homotopy-coherent Grothendieck-Riemann-Roch theorem holds for that model.

What would settle it

A concrete Liouville manifold for which the complexified MU-lift of symplectic cohomology differs from the bulk-deformed symplectic cohomology by the Chern character, or a complex-oriented flow category where the homotopy-coherent GRR fails to hold.

read the original abstract

Given a Liouville manifold, we compute a Floer-homotopical invariant -- the complexification of the lift of symplectic cohomology to complex cobordism -- in terms of a classical Floer-theoretic invariant, namely, symplectic cohomology bulk-deformed by the Chern character. We do this by giving an explicit model for the complexified homotopy groups of the MU-module spectrum associated to a complex-oriented flow category and proving a ''homotopy coherent'' version of the classical Grothedieck-Riemann-Roch theorem. Using the aforementioned relation, we establish a computable cohomological criterion, in terms of the pair-of-pants product and the BV operator on symplectic cohomology, for when this MU lift cannot be obtained via base change from the sphere spectrum; moreover, we give examples where this holds. Finally, we use this non-base change criterion to detect examples of non-trivial higher-dimensional complex cobordism classes of relative Gromov-Witten type moduli spaces in the context of a smooth complex projective variety relative to an ample smooth divisor.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper equates the complexified MU-lift of symplectic cohomology to its bulk-deformed version via an explicit model and homotopy-coherent GRR, then uses that to give a computable non-base-change criterion and examples in relative GW moduli spaces.

read the letter

The core advance is a direct computation: for a Liouville manifold the complexification of the symplectic cohomology lift to complex cobordism equals the symplectic cohomology bulk-deformed by the Chern character. They obtain this by building an explicit model for the complexified homotopy groups of the MU-module spectrum coming from a complex-oriented flow category and proving a homotopy-coherent version of GRR for that model. From there they extract a criterion, phrased in terms of the pair-of-pants product and BV operator, that detects when the MU lift is not obtained by base change from the sphere spectrum, and they exhibit examples where this happens. They close by applying the criterion to produce non-trivial higher-dimensional complex cobordism classes in relative Gromov-Witten moduli spaces for a smooth projective variety relative to an ample divisor. That package is new; earlier literature has separate pieces on Floer cohomology, MU-modules, and GRR, but not this explicit identification plus the resulting computable test. The construction is technically coherent on paper and the examples are concrete enough to be useful inside the subfield. The load-bearing step is the explicit model and the homotopy-coherent GRR: if those are correctly set up for Liouville manifolds and flow categories, the rest follows without circularity. The paper does not appear to hide any fitting or unstated assumptions that would undermine the claim. Readers already working with homotopical refinements of Floer theory or with cobordism-valued Gromov-Witten invariants will find the criterion and the examples directly usable. It is worth sending to a serious referee who knows both the symplectic and homotopy sides; the work is specialized but the link it draws is precise enough to merit checking the details of the model and the coherent GRR.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to compute the complexification of the lift of symplectic cohomology to complex cobordism (MU) for Liouville manifolds in terms of the bulk-deformed symplectic cohomology by the Chern character. This is done by providing an explicit model for the complexified homotopy groups of the MU-module spectrum associated to a complex-oriented flow category and proving a homotopy-coherent version of the Grothendieck-Riemann-Roch theorem. The resulting relation is used to establish a computable cohomological criterion, based on the pair-of-pants product and BV operator, for when the MU lift is not obtained via base change from the sphere spectrum, along with examples. Finally, this criterion is applied to detect non-trivial higher-dimensional complex cobordism classes of relative Gromov-Witten type moduli spaces for a smooth complex projective variety relative to an ample smooth divisor.

Significance. If the central identification holds, the work provides a significant bridge between Floer-homotopical invariants and classical symplectic cohomology, enabling the computation of complex cobordism classes using standard Floer operations. The cohomological criterion offers a practical tool for detecting when higher invariants are non-trivial, with potential applications in both symplectic geometry and algebraic geometry through Gromov-Witten theory. The explicit model and homotopy-coherent GRR, if rigorously proven, represent a technical advancement in handling MU-module spectra in the context of flow categories.

major comments (2)

The construction of the explicit model for the complexified homotopy groups of the MU-module spectrum is central to the identification. However, it is unclear whether this model is independent of prior definitions or if it reduces to them by construction, which could undermine the novelty of the computation. A concrete verification against a known case, such as when the flow category is trivial, would strengthen the claim.
The homotopy-coherent version of GRR is asserted to hold for the model in the setting of Liouville manifolds. Since this is load-bearing for equating the two invariants, the proof should explicitly address the generality required for complex-oriented flow categories arising from symplectic cohomology, including any homotopy coherence issues specific to the pair-of-pants product and BV operator.

minor comments (2)

The spelling 'Grothedieck' in the abstract should be corrected to 'Grothendieck'.
Ensure consistent notation for the bulk-deformation and the Chern character to avoid ambiguity in the criterion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment below and will incorporate clarifications in a revised version to strengthen the exposition.

read point-by-point responses

Referee: The construction of the explicit model for the complexified homotopy groups of the MU-module spectrum is central to the identification. However, it is unclear whether this model is independent of prior definitions or if it reduces to them by construction, which could undermine the novelty of the computation. A concrete verification against a known case, such as when the flow category is trivial, would strengthen the claim.

Authors: The explicit model is constructed directly from the data of a complex-oriented flow category (via the complexified homotopy groups of the associated MU-module spectrum) and is not a mere reduction of prior definitions; its explicitness is what enables the subsequent homotopy-coherent GRR and the computable criterion in the non-trivial symplectic setting. To address the concern, we will add a verification subsection for the trivial flow category case in the revision, confirming that the model recovers the standard complexification of MU-homotopy groups and thereby clarifying both consistency and the novelty of its application to symplectic cohomology. revision: yes
Referee: The homotopy-coherent version of GRR is asserted to hold for the model in the setting of Liouville manifolds. Since this is load-bearing for equating the two invariants, the proof should explicitly address the generality required for complex-oriented flow categories arising from symplectic cohomology, including any homotopy coherence issues specific to the pair-of-pants product and BV operator.

Authors: The homotopy-coherent GRR is proven in the manuscript for general complex-oriented flow categories, with the Liouville manifold case obtained by specializing to the flow category of symplectic cohomology (which is complex-oriented by construction). The pair-of-pants product and BV operator inherit the required homotopy coherence from the standard axioms and coherence data of symplectic cohomology. In the revision we will add an explicit subsection detailing these coherence properties in the symplectic setting, making the generality and applicability fully transparent without changing the core argument. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained via new explicit model and homotopy coherent GRR proof

full rationale

The paper computes the complexification of the MU-lift of symplectic cohomology in terms of bulk-deformed symplectic cohomology by constructing an explicit model for the complexified homotopy groups of the MU-module spectrum for a complex-oriented flow category and proving a homotopy coherent version of the Grothendieck-Riemann-Roch theorem. These constructions are presented as original contributions within the paper, with no reduction to prior self-citations or fitted inputs visible in the abstract or described claims. The subsequent cohomological criterion and examples follow from this relation, but the core identification is grounded in the new model and theorem rather than circularly defined inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard background in Floer theory for Liouville manifolds, properties of complex cobordism spectra, and the existence of a complex-oriented flow category; the new content is the explicit model and homotopy-coherent GRR, which are not further decomposed in the abstract.

axioms (2)

domain assumption Liouville manifold admits a complex-oriented flow category whose associated MU-module spectrum has well-defined complexified homotopy groups.
Invoked in the computation of the Floer-homotopical invariant.
ad hoc to paper A homotopy-coherent version of the Grothendieck-Riemann-Roch theorem holds for the MU-module spectrum in this setting.
This is the key new ingredient used to relate the two invariants.

pith-pipeline@v0.9.0 · 5486 in / 1704 out tokens · 56200 ms · 2026-05-08T03:05:00.992217+00:00 · methodology

discussion (0)

Reference graph

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