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

Source: arXiv:2605.06620 · Published 2026-05-07 · By Kenneth Blakey, Noah Porcelli

TL;DR

This paper addresses the problem of lifting symplectic cohomology—a Floer-theoretic invariant of Liouville manifolds—to complex cobordism through stable homotopical refinements. The main advance is an explicit computation of the complexified lift of symplectic cohomology to the complex cobordism ring spectrum MU as a bulk-deformed version of classical symplectic cohomology via the Chern character. The authors develop a homotopy-coherent version of the Grothendieck-Riemann-Roch (GRR) theorem tailored to complex-oriented flow categories, capturing coherence of Chern character representatives compatible with Floer theory’s gluing operations. They use this to give an explicit chain model and prove an isomorphism of MU-module spectra in homology. Applying this, they establish a cohomological obstruction criterion involving the pair-of-pants product and BV operator that detects when the MU lift cannot be obtained by base change from the sphere spectrum S. They provide examples (notably, cotangent bundles of complex projective spaces of odd dimension) exhibiting this non-base-change phenomenon. Finally, leveraging this obstruction and spectral sequences, they detect nontrivial higher-dimensional complex cobordism classes associated to relative Gromov-Witten type moduli spaces in projective varieties relative to ample divisors.

Their construction uses modern ∞-categorical machinery encapsulating flow categories with complex orientations and stable isomorphisms of tangent bundles to kernels of Cauchy-Riemann type operators, enabling the homotopy-coherent GRR theorem. The bulk-deformations correspond to universal variables indexed by the Chern character summands, which yield a deformation parameter space that interpolates between classical and complex cobordism symplectic cohomology. The results quantitatively characterize the MU-lifted invariant, demonstrating it as a richer and genuinely equivalence-distinct object from classical lifts over S in many geometric examples. This bridges Floer theory, stable homotopy theory, and enumerative geometry by capturing subtle cobordism-level information invisible to ordinary cohomology.

Key findings

• Theorem 1.1 establishes an explicit isomorphism of C[b]-modules: SH_*(M; MU) ⊗Z C ≅ SH_{U_ch}(M; L) ⊗C C[b_n, b, ...], relating the complexified Floer complex bordism to symplectic cohomology bulk-deformed by the Chern character (Equation 1.5).
• Corollary 1.3 gives a computable criterion: if there exists α ∈ SH_*(M; C) with nonzero Lie bracket [α, ch_{ρ_0}(M)] ≠ 0 for some ρ_0, then the Floer complex bordism lift FU cannot be realized as a base change from the sphere spectrum S (Equation 1.8).
• Example 1.5 shows the criterion holds for M = T^* CP^n with n odd ≥ 3, proving FU_{T^*CP^n} ≉ F_{fr} ⊗_S MU for any stable R-polarization, indicating genuinely nontrivial MU lifts distinct from known S-module lifts.
• The obstruction GWU_{T^*CP^n} in complex cobordism associated to relative Gromov-Witten moduli spaces is nonzero for n odd ≥ 3 but vanishes when tensored down to integral bordism, revealing new higher-dimensional complex cobordism phenomena invisible to ordinary homology (Subsection 1.3.2).
• The homotopy coherent Grothendieck-Riemann-Roch theorem (Theorem 1.12) establishes an isomorphism of DG C[b]-modules relating integration of Chern characters along families of Cauchy-Riemann operators to bulk-deformed cohomology, enforcing the coherence needed to lift classical index computations to the Floer categorical setting.
• They provide a concrete chain-level model CM_*(X, ch; C[b]) over the polynomial algebra C[b], encoding chosen cochain representatives of Chern characters compatible with the flow category composition structure and stable isomorphisms of virtual tangent bundles.
• The theory explicitly accounts for Koszul sign conventions and orientations of stratified manifolds with corners, essential for homotopy coherent gluing and compatibility of the Floer moduli spaces in the construction of complex-oriented flow categories (Section 2).
• The spectral sequence associated to the base change C[b] → C degenerates when the bracket condition vanishes but exhibits nontrivial differentials otherwise, obstructing lifts to sphere spectra, elucidating the richer structure captured by MU-modules.

Threat model

The paper is not a security-focused work; the adversary metaphorically corresponds to obstructions within homotopy theory and Floer homotopy classes preventing lifts of symplectic cohomology to certain ring spectra. The adversary is represented by cohomology classes causing spectral sequence differentials obstructing base change from the sphere spectrum, and by failures of coherence in constructing homotopy-theoretic lifts compatible with flow category gluing. There is no external attacker or threat actor model in the classical cybersecurity sense.

Methodology — deep read

1. Threat Model & Assumptions: The authors consider a Floer-theoretic invariant of Liouville manifolds called symplectic cohomology SC_(M), traditionally defined over classical rings like Z or C. They investigate lifts of SC_(M) to more exotic ring spectra, specifically to complex cobordism MU-modules. The adversary, in a metaphorical sense, is the obstruction to lifting flows coherently to MU without degenerating to sphere spectrum base changes. They assume M is graded (2c_1(M) = 0) and that Floer trajectories form flow categories with stable complex orientations.

2. Data: The data arises from moduli spaces X_{xy} of Floer trajectories on a Liouville manifold M, which form stratified smooth manifolds with corners. These moduli spaces carry virtual tangent bundles and come equipped with families of Cauchy-Riemann operators on complex bundles E_{xy} parametrized over R × S^1. The cohomological data includes Chern characters ch(M) with individual summands ch_ρ(M). For examples, they analyze cotangent bundles of CP^n for odd n ≥ 3, and smooth projective varieties with ample divisors.

3. Architecture / Algorithm: They introduce complex-oriented flow categories, i.e., flow categories X with chosen stable isomorphisms of their tangent bundles to kernels of family Cauchy-Riemann operators. This gives rise to MU-module spectra FU associated to X. To capture coherence, they define Grothendieck-Riemann-Roch (GRR) flow categories equipped with virtual bundles and complex orientations allowing construction of chain complexes CM_*(X, ch; C[b]) linear over polynomials C[b] in bulk deformation variables b_ρ. The chains integrate chosen cochain representatives of the Chern characters of the virtual bundles over moduli spaces, coherently compatible with boundary gluings. They prove a homotopy coherent Grothendieck-Riemann-Roch theorem (Thm 1.12) showing an isomorphism of DG C[b]-modules between this chain-level model and the pushforward of Chern characters, accounting for compatibility with ∞-categorical structures and derived equivalences.

4. Training Regime: Not applicable as this is a mathematical construction.

5. Evaluation Protocol: They analyze the resulting modules using spectral sequences associated to base change C[b] → C, tracking the failure of degeneration as an obstruction to base change lifts. They compute explicit pair-of-pants products, BV operators, and Lie brackets on symplectic cohomology (e.g., for T^* CP^n) to test the obstruction condition of Corollary 1.3. Examples are worked out to demonstrate sharpness.

6. Reproducibility: They provide detailed constructions of flow categories, virtual bundles, and their orientations in a homotopy-theoretic setting referencing known frameworks [AB24, HO26]. While code or computational artifacts are not applicable, the paper fully sketches the ∞-categorical and chain-level models to reconstruct their results. The relevant moduli spaces and Floer data are classical objects in symplectic geometry, well documented in the literature.

End-to-End Example: Consider M = T^* CP^n, n odd ≥ 3. They use stable R-polarizations inducing sphere spectrum lifts F_{fr}. By analyzing the Chern character summands ch_ρ(M), they bulk deform SC_(M; C) to SH_^{U_{ch}}(M; L) over C[b], constructing the chain complex CM_(X, ch; C[b]) with coherent representatives of Chern classes on flow simplices. Evaluating the Lie bracket [α, ch_ρ0(M)] for some α ∈ SH_(M; C) yields a nonzero class obstructing base change to S, proving FU_{T^*CP^n} is a genuinely new MU-module and detecting nontrivial complex cobordism classes in relative Gromov-Witten moduli spaces in Z = CP^n × CP^n with divisor D.

The homotopy coherent GRR theorem guarantees compatibility of this construction at the derived level, ensuring the module structure and spectral sequence computations are valid and canonical up to homotopy.

Technical innovations

• A homotopy coherent version of the Grothendieck-Riemann-Roch theorem for complex-oriented flow categories, incorporating ∞-categorical gluing coherence of Chern character representatives, which extends classical index formulas to the Floer homotopy setting.
• An explicit chain-level model CM_*(X, ch; C[b]) linear over the polynomial algebra C[b], encoding bulk-deformations of symplectic cohomology by integrating Chern characters over moduli spaces with coherent choices of cochain representatives.
• A computable obstruction criterion using the Lie bracket on symplectic cohomology involving the pair-of-pants product and BV operator to detect non-base-change of Floer complex bordism lifts from the sphere spectrum to MU.
• Use of complex-oriented flow simplices and decorated oriented flow categories combined with precise Koszul sign conventions and virtual bundle orientation data to ensure stable homotopical coherence in Floer-theoretic invariants.

Baselines vs proposed

• Sphere spectrum lift (Ffr): base change realization expected but fails in certain cases such as T^* CP^n for n odd ≥ 3; FU_{T^*CP^n} ≉ Ffr ⊗_S MU (Equation 1.9)
• Ordinary symplectic cohomology SH_*(M; Z/C): recovered by base change from Floer complex bordism over MU (Equation 1.2), but bulk deformation captures strictly richer data after complexification (Theorem 1.1)
• Spectral sequence associated to base change C[b] → C degenerates when Lie bracket obstruction vanishes; otherwise has nontrivial differentials obstructing base change (Corollary 1.3)
• For T^* CP^1 (stably framed), FU_{T^*CP^1} ≅ Ffr,std_{T^*CP^1} ⊗_S MU (Equation 1.13), contrasting with odd-dimensional cases

Limitations

• The main isomorphisms hold after complexification (tensoring with C), thus torsion information in integral MU modules is not captured.
• They do not prove a quasi-isomorphism of chain complexes underlying their isomorphisms, only an equivalence in the derived category up to an autoequivalence constrained by Toën's theorem (Remark 1.14).
• Focus is on graded Liouville manifolds; ungraded or non-exact symplectic manifolds require more complicated periodic analogues or orbifold bordism techniques not fully addressed here.
• No explicit computational algorithms for the higher Chern character representatives or bulk-deformations are provided beyond existence and abstract formulae; explicit evaluation is difficult except in simple cases.
• The obstruction criterion requires nontrivial pair-of-pants product and BV operator structures, which vanish for many closed symplectic manifolds (e.g. K3 surfaces), limiting applicability.
• While illustrating nontrivial higher cobordism classes, the minimal cobordism ring spectrum detecting these classes beyond MU is left as an open question.

Open questions / follow-ons

• Can one explicitly compute FU_{T^*Q} for a general closed manifold Q in terms of Thom spectra associated to MU local systems on free loop spaces, i.e., the ‘spectral MU Viterbo isomorphism’ remains open.
• What is the minimal cobordism ring spectrum B receiving a map MU → B such that the higher-dimensional moduli space classes GWU_{T^*CP^n} remain nontrivial when tensored over B?
• Is it possible to upgrade the derived category isomorphisms to explicit quasi-isomorphisms of chain complexes, potentially via symmetric monoidal structures on flow categories as conjectured in Subsection 4.7?
• How can the constructions and obstruction results be extended to ungraded or periodic cases, or to non-exact symplectic manifolds requiring orbifold bordism or more complicated tangential structures?

Why it matters for bot defense

For bot-defense and CAPTCHA practitioners, this work offers insight into highly sophisticated mathematical structures within homotopy theory and Floer theory, which are far-removed from conventional machine learning or heuristic bot detection methods. However, the general theme of this research—lifting classical invariants to richer homotopical or algebraic structures and detecting nontrivial obstructions via spectral sequences—can be metaphorically inspirational. Analogous to how subtle invariants in geometry require careful coherence and combinatorial orientation considerations, bot-defense mechanisms might explore invariants capturing subtle user interaction patterns resistant to automated replays or synthesis.

Practically, the paper’s computational obstruction criteria and deformation parameters (bulk-deformations) could conceptually translate to designing multi-parameter challenge-response schemes that detect subtle behavioral correlations over time instead of simple snapshot signals. Although the concrete math is not directly applicable, the rigorous homotopy-coherent framework exemplifies the depth of structural checks potentially needed to differentiate bots from humans in adversarial environments.

Cite

@article{arxiv2605_06620,
  title={ Bulk-deformations, Floer complex bordism, and Grothendieck-Riemann-Roch },
  author={ Kenneth Blakey and Noah Porcelli },
  journal={arXiv preprint arXiv:2605.06620},
  year={ 2026 },
  url={https://arxiv.org/abs/2605.06620}
}

Read the full paper

• arXiv:2605.06620 (abstract)
• arXiv:2605.06620 (PDF)