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arxiv: 2605.23404 · v1 · pith:T5JKUGR3new · submitted 2026-05-22 · 🧮 math.SG · math.AG· math.DG

Multi-valued Morse homotopy for the SYZ mirror of the complex projective plane

Hayato Nakanishi , Yat-Hin Suen This is my paper

Pith reviewed 2026-05-25 02:52 UTC · model grok-4.3

classification 🧮 math.SG math.AGmath.DG
keywords multi-valued Morse homotopySYZ mirror symmetryA-infinity equivalenceholomorphic vector bundlescomplex projective planeLagrangian multi-sectionstangent bundle sectionsMorse category
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The pith

A category of multi-valued Morse homotopy on the SYZ mirror of CP^2 is A_infinity-equivalent to holomorphic vector bundles on CP^2.

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

The paper defines a category Mo^mult(P) whose objects are multi-valued functions coming from Lagrangian multi-sections on the mirror space P of the complex projective plane. It proves that a full subcategory of this category is A_infinity-equivalent to a full subcategory of the derived category of holomorphic vector bundles on CP^2. The equivalence is then used to give an explicit mirror description of the global sections of the holomorphic tangent bundle on CP^2. A sympathetic reader cares because the construction supplies a concrete Morse-theoretic model for algebraic data that is otherwise difficult to access directly on the complex side. The central step is showing that the proposed A_infinity operations on the multi-valued functions match the composition and higher products in the bundle category.

Core claim

We propose a definition of the category Mo^mult(P) of multi-valued Morse homotopy on P consisting of multi-valued functions associated to Lagrangian multi-sections. We then show that a full subcategory Mo^mult_E(P) of Mo^mult(P) is A_infinity-equivalent to a full subcategory DG^vect_E(CP^2) of the category DG^vect(CP^2) consisting of holomorphic vector bundles over CP^2. As an application, we study the mirror description for global sections of the holomorphic tangent bundle over CP^2.

What carries the argument

The category Mo^mult(P) of multi-valued Morse homotopy, whose objects are multi-valued functions associated to Lagrangian multi-sections and which carries an A_infinity structure that produces the stated equivalence.

If this is right

  • The equivalence maps certain Lagrangian multi-sections on the mirror to holomorphic vector bundles on CP^2.
  • Global sections of the holomorphic tangent bundle on CP^2 receive an explicit description in terms of multi-valued Morse data.
  • The A_infinity operations defined on the multi-valued functions correspond to the composition laws in the category of holomorphic vector bundles.
  • Homological information about vector bundles on CP^2 can be read off from counts of holomorphic disks or gradient trajectories on the mirror side.

Where Pith is reading between the lines

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

  • The same multi-valued construction might apply to other toric Fano manifolds whose SYZ fibrations are known explicitly.
  • The framework could be tested by computing both sides for rank-one bundles and checking agreement on low-degree cohomology groups.
  • If the equivalence holds, it supplies a route to algebraic invariants of CP^2 via combinatorial data on the symplectic mirror without passing through coherent sheaves directly.

Load-bearing premise

The proposed definition of the category of multi-valued functions associated to Lagrangian multi-sections admits a well-defined A_infinity structure that makes the equivalence to holomorphic vector bundles possible.

What would settle it

An explicit computation of the A_infinity products for one low-rank Lagrangian multi-section whose image under the equivalence should be a specific vector bundle on CP^2, showing that the products fail to match the corresponding Ext groups or morphisms.

Figures

Figures reproduced from arXiv: 2605.23404 by Hayato Nakanishi, Yat-Hin Suen.

Figure 1
Figure 1. Figure 1: The moment polytope P ⊂ MR and the fan over NR. is given by Y := T ∗NR/M and pNR : T ∗NR/M → NR. We denote by Y the covering space of Y and the same symbol pNR : Y → NR. The Fubini-Study form ωF S is expressed as ωF S = 2 e 2ξ1 (1 + e 2ξ2 )dξ1 ∧ dy1 − e 2ξ1 e 2ξ2 dξ1 ∧ dy2 − e 2ξ1 e 2ξ2 dξ2 ∧ dy1 + e 2ξ2 (1 + e 2ξ1 )dξ2 ∧ dy2 (1 + e 2ξ1 + e 2ξ2 ) 2 on U. By this expression, the induced metric {g ij} on NR … view at source ↗
Figure 2
Figure 2. Figure 2: For a Lagrangian section L1 = graph(dfL1 ) and a Lagrangian multi-section L2 = graph(df(1) L2 ) ∪ graph(df(2) L2 ), the black lines are the spec￾tral network WL2 and the orange line is the branch cut of L2. The red lines are jagged gradient lines from p to q1, q4, q5. The blue lines from p to q2, q3 are ordinary gradient lines, which does not interact with the (12)-wall of WL2 . Remark 4.4. Note that the d… view at source ↗
Figure 3
Figure 3. Figure 3: An ordinary trivalent gradient tree. The elements of Mjag(v12, v23; v13) are obtained by replacing the edges with jagged gradi￾ent lines. m2 : Momult(P)(L1, L2) ⊗ Momult(P)(L2, L3) → Momult(P)(L1, L3) by m2(v12, v23) := X v13∈Momult(P)(L1,L3) |v13|=|v12|+|v23| X γ∈Mjag(v12,v23;v13) ±e −A(γ) v13, where vij are the bases of Momult(P)(Li , Lj ) and A(γ), A(ℓ) ∈ [0,∞] are the symplectic areas of the piecewise … view at source ↗
Figure 4
Figure 4. Figure 4: The tropical Lagrangian multi-section (L trop TCP 2 , φ trop TCP 2 ) associated to the holomorphic tangent bundle TCP2 . FLT CP 2 = {f (1) LT CP 2 , f(2) LT CP 2 } be the potential function of LTCP 2 . We denote by L (1) TCP 2 (resp. L (2) TCP 2 ) the first (resp. second) sheet of LTCP 2 . For a maximal cone σ of Σ, the fiber coordinates of LTCP 2 at the vertex ˇσ of the moment polytope are given by d(φ tr… view at source ↗
Figure 5
Figure 5. Figure 5: The fiber coordinate of LTCP 2 at each vertex of the moment polytope. The orange dashed lines are the branch cuts. The left side is the first sheet L (1) TCP 2 and the right side is the second sheet L (2) TCP 2 of LTCP 2 = L (1) TCP 2 ∪ L (2) TCP 2 . 0 0 1 1 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The realization of LTCP 2 on the edge ˇρ2. The black lines are the Lagrangian sections L(1) and L(2). The red line is the first sheet L (1) TCP 2 and the blue line is the second sheet L (2) TCP 2 of LTCP 2 = L (1) TCP 2 ∪ L (2) TCP 2 . We denote by E the collection (L(1), LTCP 2 , L(2)) of the Lagrangians which is the same notation as the exceptional collection given in subsection 3. 4.3. The space of morp… view at source ↗
Figure 7
Figure 7. Figure 7: The projection of the intersection L(1) ∩ LTCP 2 to the moment polytope and gradient trajectories. • Momult E (P)(LTCP 2 , L(2)): The projection of the intersection π ′ (LTCP 2∩L(2)) and gradient lines are depicted in [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The projection of the intersection LTCP 2 ∩ L(2) to the moment polytope and gradient lines. polytope, each V(j1,j2) forms a generator whose degree turns out to be |V(j1,j2) | = 0. On the other hand, other intersections do not form generators since each v k (j1,j2) does not satisfy Condition (M). Hence we have H 0 (Momult E (P))(LTCP 2 , L(2)) ∼= Momult E (P)(LTCP 2 , L(2)) ∼= C · V(1,1) ⊕ C · V(1,0) ⊕ C · … view at source ↗
Figure 9
Figure 9. Figure 9: The projection of the intersection L(1) ∩ L(2) to the moment polytope. degree are zero. We also have H 0 (Momult E (P))(L(1), L(2)) ∼= Momult E (P)(L(1), L(2)) ∼= C · W(0,0) ⊕ C · W(1,0) ⊕ C · W(0,1), since the differential is trivial for the degree reason. • Momult E (P)(LTCP 2 , LTCP 2 ): The Lagrangian multi-section LTCP 2 has self-intersections only at each vertex of the moment polytope since LTCP 2 ca… view at source ↗
Figure 10
Figure 10. Figure 10: The jagged gradient line emanating from U(−1,0). The flow of grad(f (1) LT CP 2 − fL(1)) is represented by (1) and the flow grad(f (2) LT CP 2 − fL(1)) is represented by (2). If that line reaches W(0,0), then it firstly starts by grad(f (1) LT CP 2 − fL(1)) and finally proceeds by grad(f (2) LT CP 2 − fL(1)). However the gradient line emanating from V(−1,0) proceeds by grad(fL(2) − f (1) LT CP 2 ). Since … view at source ↗
Figure 11
Figure 11. Figure 11: The parliament of polytopes associated to TCP2 . By this expression, we have dim Γ(CP 2 , TCP2 ) = 8 since (0, −1) ⊗ χ (0,0) + (1, 1) ⊗ χ (0,0) + (−1, 0) ⊗ χ (0,0) = 0. On the other hand, we compute the morphisms in the mirror side. The Lagrangian multi-section LTCP 2 associated to the holomorphic tangent bundle TCP2 is given in subsec￾tion 4.2 [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The projection of the intersection L(0) ∩ LTCP 2 to the moment polytope. ary ∂P of the moment polytope, each sheet of the Lagrangian multi-section LTCP 2 is a Lagrangian section and behaves like L(1) or L(2) asymptotically. Hence the degree of V(i1,i2) is equal to 0. In contrast, we can not define the degree of V b (0,0) since the potential function of LTCP 2 around V b (0,0) is not a Morse function. Howe… view at source ↗
Figure 13
Figure 13. Figure 13: The gradient lines emanating from V(0,0) and ending at V b (0,0). m1(V 0 (0,0)) = e −AV b (0,0), m1(V 1 (0,0)) = e −BV b (0,0), m1(V 2 (0,0)) = e −CV b (0,0), where A, B, C ∈ R are the weights associated to the gradient lines. We also have m1(V b (0,0)) = 0 by the degree reason. To summarize, we obtain the following results. Theorem 5.1. The zero-th cohomology H0 (Momult(P))(L(0), LTCP 2 ) is written by M… view at source ↗
Figure 14
Figure 14. Figure 14: The tropical Lagrangian multi-section L trop ΩCP 2 associated to holomorphic cotangent bundle of CP 2 . duality of the tropical Lagrangian multi-sections associated to the toric vector bundles. By taking the potential function −fL2 of the local model instead of fL2 , we can construct the Lagrangian multi-section LΩCP 2 such as LΩCP 2 = −LTCP 2 . Hence, the intersections L(0) ∩ LΩCP 2 coincide with L(0) ∩ … view at source ↗
Figure 15
Figure 15. Figure 15: The projection of the intersection L(0) ∩ LΩCP 2 to the moment polytope. Appendix A. Local model of the Lagrangian multi-sections In this appendix, we review the local model of a Lagrangian multi-section around the simple branch point. For more details see [Fuk05, Sue21, Sue24]. Let B := NR = R 2 ∼= C and Y := T ∗NR ∼= C 2 . We denote by (ξ1, ξ2) the real coordinates of B and (ξ1, ξ2, y1 , y2 ) the real c… view at source ↗
Figure 16
Figure 16. Figure 16: The gradient vector field of fL2 in B. The one on the left is for −π < θ < π, and the one on the right is for π < θ < 3π [PITH_FULL_IMAGE:figures/full_fig_p028_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The gradient lines of fL2 in the moment polytope P. The one on the left is for −π < θ < π, and the one on the right is for π < θ < 3π. References [AJ10] M. Akaho and D. Joyce. Immersed Lagrangian Floer theory, Journal of Differential Geometry, 86(3), (2010), 381–500 [PITH_FULL_IMAGE:figures/full_fig_p029_17.png] view at source ↗
read the original abstract

We propose a definition of the category $\mathit{Mo}^{\mathrm{mult}}(P)$ of multi-valued Morse homotopy on $P$ consisting of multi-valued functions associated to Lagrangian multi-sections. We then show that a full subcategory $\mathit{Mo}^{\mathrm{mult}}_{\mathcal{E}}(P)$ of $\mathit{Mo}^{\mathrm{mult}}(P)$ is $A_\infty$-equivalent to a full subcategory $\mathit{DG}_{\mathcal{E}}^{\mathrm{vect}}(\mathbb{C}P^2)$ of the category $\mathit{DG}^{\mathrm{vect}}(\mathbb{C}P^2)$ consisting of holomorphic vector bundles over the complex projective plane $\mathbb{C}P^2$. As an application, we study the mirror description for global sections of the holomorphic tangent bundle over $\mathbb{C}P^2$.

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.

Referee Report

2 major / 1 minor

Summary. The paper proposes a definition of the category Mo^mult(P) consisting of multi-valued functions associated to Lagrangian multi-sections, equipped with an A_∞ structure. It claims that the full subcategory Mo^mult_E(P) is A_∞-equivalent to the full subcategory DG^vect_E(CP^2) of holomorphic vector bundles on CP^2, and applies the result to give a mirror description of global sections of the holomorphic tangent bundle on CP^2.

Significance. If the A_∞ structure on multi-valued Morse homotopy is rigorously defined and the equivalence holds, the result would supply a concrete bridge between SYZ mirror symmetry constructions and the derived category of coherent sheaves on CP^2, potentially allowing explicit computations of mirror maps for vector bundles. The application to the tangent bundle is a natural test case.

major comments (2)
  1. [Definition of Mo^mult(P) and its A_∞ structure] The definition of the A_∞ operations μ_n on the objects of Mo^mult(P) (multi-valued functions tied to Lagrangian multi-sections) is not supplied with explicit formulas or sign conventions. Without these, it is impossible to verify closure under the A_∞ relations or compatibility with the multi-valuedness; this is load-bearing for the claimed equivalence Mo^mult_E(P) ≃ DG^vect_E(CP^2).
  2. [Proof of the equivalence] The proof of the A_∞-equivalence is stated in the abstract but the manuscript supplies no chain maps, homotopy equivalences, or verification that the functors preserve the higher operations. A concrete check on generators (e.g., the structure sheaf or tangent bundle) is required to substantiate the claim.
minor comments (1)
  1. [Introduction] Notation for the multi-valued functions and the subscript E in both categories should be defined at first use rather than left implicit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper to incorporate the requested clarifications and expansions.

read point-by-point responses

  1. Referee: [Definition of Mo^mult(P) and its A_∞ structure] The definition of the A_∞ operations μ_n on the objects of Mo^mult(P) (multi-valued functions tied to Lagrangian multi-sections) is not supplied with explicit formulas or sign conventions. Without these, it is impossible to verify closure under the A_∞ relations or compatibility with the multi-valuedness; this is load-bearing for the claimed equivalence Mo^mult_E(P) ≃ DG^vect_E(CP^2).

    Authors: We acknowledge that while the general form of the A_∞ operations is indicated in the manuscript, the explicit formulas for μ_n together with complete sign conventions are not written out in sufficient detail. In the revised version we will add these formulas in a new subsection of Section 3, including the precise sign conventions obtained by adapting the standard Morse-homotopy signs to the multi-valued setting. This will make verification of the A_∞ relations and compatibility with multi-valuedness straightforward. revision: yes

  2. Referee: [Proof of the equivalence] The proof of the A_∞-equivalence is stated in the abstract but the manuscript supplies no chain maps, homotopy equivalences, or verification that the functors preserve the higher operations. A concrete check on generators (e.g., the structure sheaf or tangent bundle) is required to substantiate the claim.

    Authors: We agree that the current proof sketch in Section 4 does not supply the explicit chain maps, homotopy data, or generator-level verification. We will expand the proof to construct the A_∞ functor explicitly, exhibit the required homotopy equivalences, and perform a direct check on the structure sheaf and on the tangent bundle (the latter being the main application). These additions will be placed in a dedicated subsection. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The abstract proposes a definition of the category Mo^mult(P) from multi-valued functions on Lagrangian multi-sections and claims an A_infty-equivalence for a full subcategory to DG^vect_E(CP^2). No equations, fitted parameters, or self-citations are exhibited that reduce the claimed equivalence or the well-definedness of the A_infty operations to a tautology by construction. The verification that the proposed objects carry operations satisfying the A_infty relations is an independent step, not a renaming or self-definition of the input data. This matches the default expectation of a non-circular paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the new definition of multi-valued functions and the assumption that the SYZ mirror P and its Lagrangian multi-sections are suitably defined; no free parameters or invented physical entities appear.

axioms (2)
  • domain assumption The SYZ mirror of CP^2 exists as a space P equipped with Lagrangian multi-sections.
    Invoked to define the objects of Mo^mult(P).
  • ad hoc to paper An A_infty structure can be placed on the category of multi-valued Morse homotopy.
    Part of the proposed definition in the abstract.
invented entities (1)
  • multi-valued functions associated to Lagrangian multi-sections no independent evidence
    purpose: Objects of the new category Mo^mult(P).
    Introduced by the paper to extend ordinary Morse homotopy.

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Reference graph

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