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Mirror functor for deformed preprojective algebras

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Pith's one-line read Bulk deformations of symplectic plumbings turn the localized mirror of the core Lagrangian into a deformed preprojective algebra, with bulk weight $\delta_i$ playing the role of $\lambda_i$.

desk verdict Strong algebraic framework, but the bulk-deformation interpretation of lambda rests on an unproven admissibility assumption for noncompact cotangent fibers. read the letter →

arxiv 2608.05764 v1 pith:YDRSKQ3V submitted 2026-08-06 math.SG math.AG

Hansol Hong , Siu-Cheong Lau , Ju Tan This is my paper · ORCID
classification math.SGmath.AG MSC 53D3716G2014A22
keywords homologicalmirrorsymmetrydeformedpreprojectivealgebrasbulkdeformationsLagrangianimmersionsFukayacategoriesKoszuldualityplumbingsNakajimaquivervarieties
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes a symplectic origin for the deformation parameter in deformed preprojective algebras, the noncommutative quiver algebras attached to Kleinian singularities and Nakajima quiver varieties. For a plumbing of two-spheres, the authors prove that deforming the ambient symplectic manifold by bulk cycles passing through each sphere component turns the bulk-deformed Maurer-Cartan algebra of the Lagrangian core into the deformed preprojective algebra of the plumbing quiver, with the bulk weight $\delta_i$ playing the role of $\lambda_i$. Applying the same construction to framed Lagrangian branes produces Nakajima quiver varieties at nonzero complex moment-map level. The categorical engine is a Koszul-duality quasi-equivalence between the Fukaya subcategory generated by the Lagrangian and finite-dimensional modules over its extended localized mirror, so these identifications are consequences of the mirror functor rather than formal analogies.

What carries the argument

The load-bearing object is the extended localized mirror $\tilde{\mathcal{A}}_L$, the completed reduced cobar construction of the Floer $A_\infty$-algebra $CF(L,L)$ --- equivalently its Koszul dual --- whose degree-zero part is the ordinary Maurer-Cartan algebra $A_L$. For plumbings of $T^*S^2$, the computation that carries the paper is the bulk-deformed Maurer-Cartan equation: after the coordinate change (7.1), $m^{b,b}_0 = \sum_v \left(\delta_v + \sum_{t(a)=v} \epsilon(a) x_{\bar a} x_a\right) P_v$, where $P_v$ is the minimum of the Morse function on the $v$-th sphere. The term $\delta_v P_v$ is produced by a constant disc with an interior marked point constrained to the bulk fiber followed by a Morse trajectory; the terms $x_{\bar a}x_a$ are the constant-polygon contributions of the immersed sectors. The Koszul-dual free resolution $V^! \otimes V^\sharp \otimes V^!$ is used to identify Hochschild invariants, exhibit a Calabi-Yau structure, and prove the categorical quasi-equivalence behind the mirror functor.

What would settle it

Compute the full filtered bulk-deformed $A_\infty$-structure of the core Lagrangian in the standard plumbing of two 2-spheres, keeping all Novikov terms. If the stabilized constant disc at the cotangent fiber contributes terms beyond $\delta_i P_i$ that cannot be removed by an invertible coordinate change, or if the bulk parameter enters the vertex relation nonlinearly, the Maurer-Cartan algebra is not the deformed preprojective algebra and Theorem 7.1 fails.

Watch

Extended reading notes

Core claim

The central claim is Theorem 7.1: for the plumbing of $T^*S^2$ according to a graph $D$, with core Lagrangian $L$ and bulk cycle $b=\sum_i \delta_i C_i$ where $C_i = T^*_{p_i}L_i$ is the cotangent fiber at a point of the $i$-th sphere component, the bulk-deformed Maurer-Cartan algebra of $L$ is isomorphic to the deformed preprojective algebra $\Pi(Q)_\lambda$ of the double quiver of $D$, with $\lambda_i = \delta_i$. The mechanism is that the fiber $C_i$ stabilizes the constant disc through $p_i$, contributing $\delta_i P_i$ to $m^{b,b}_0$; after an invertible coordinate change the relation at vertex $i$ is $\delta_i + \sum_{t(a)=i} \epsilon(a) x_{\bar a} x_a = 0$, exactly the deformed preprojective relation. The framed version of the same computation identifies the bulk-deformed Maurer-Cartan space with the Nakajima quiver variety at complex moment-map level $\mu_C = \delta$. Together with Theorem 4.3, which promotes the extended localized mirror functor to a quasi-equivalence $\mathcal{D}\mathrm{Fuk}_L(X) \cong \mathcal{D}_{\mathrm{fd}}(\tilde{\mathcal{A}}_L)$ under a finiteness and positivity hypothesis on the Floer complex, this gives a categorical, not merely formal, bridge from bulk classes to noncommutative deformations.

Load-bearing premise

The load-bearing premise is that the cotangent fiber $C_i$ at a point of each sphere component is an admissible bulk-deformation cycle and that its entire effect is to add exactly $\delta_i$ to the vertex relation, with all higher-order disc corrections removed by an invertible coordinate change.

Editorial extensions

If this is right

  • The deformation parameter $\lambda$ of a deformed preprojective algebra is realizable as the weight of a codimension-two bulk cycle; changing the bulk class changes the noncommutative deformation of the mirror.
  • For framed plumbings, nonzero complex moment-map levels of Nakajima quiver varieties arise from the same bulk cycles, so the level records the bulk class rather than being independent data.
  • Under the Koszul-duality quasi-equivalence, representations of the deformed preprojective algebra literally describe the Fukaya subcategory generated by the core Lagrangian, so noncommutative deformations acquire a symplectic-topological meaning.
  • In affine ADE plumbings, bulk deformations recover noncommutative deformations of crepant resolutions and their local charts, and framed branes map to the monadic complexes of framed torsion-free sheaves on noncommutative projective surfaces.
  • The generation analysis shows the core Lagrangian split-generates the compact Fukaya category only in nilpotent cases such as ADE Dynkin type for $n=2$ or trees for $n\ge 3$; with cycles in the plumbing graph, the completed local mirror is a genuine local chart and may fail to see the whole category.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves implicit that the same bulk-cycle recipe should apply to other quiver-like symplectic models: replacing the cotangent fiber by a codimension-two cycle in a different homology class should change the vertex weights in a predictable way, giving a testable normal form for bulk-deformed localized mirrors.
  • The invertible-coordinate-change step suggests a broader structural principle: for Lagrangians whose Floer complex has no nonpositive generators and controlled constant-disc moduli, codimension-two bulk insertions act as vertex-weight deformations of the Koszul dual; the plumbing theorem is the first model case of that principle.
  • One could probe the paper's conjecture that total moment-map level measures noncommutativity by computing Hochschild cohomology of the bulk-deformed localized mirror as a function of $\sum_i \delta_i$; the Calabi-Yau and resolution machinery in the paper provides a direct tool for this.
  • The conifold section indicates a three-dimensional analogue: suitable bulk cycles deform the Ginzburg-type algebra of the resolved conifold into a noncommutative crepant resolution, suggesting that the correspondence between bulk classes and deformation parameters extends beyond surfaces.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper develops a localized homological mirror symmetry formalism for an immersed Lagrangian brane L, possibly equipped with a higher-rank flat bundle, and applies it to plumbings of cotangent bundles of spheres. The extended localized mirror \tilde A_L is presented as the Koszul dual of the Floer A-infinity algebra of L, and Theorem 4.3 asserts a quasi-equivalence between the Fukaya subcategory generated by L and the derived category of nilpotent finite-dimensional modules over \tilde A_L. In the second part, the paper introduces bulk deformations by cycles C_i, defined as cotangent fibers over points of the core spheres, and claims in Theorem 7.1 that the bulk-deformed Maurer-Cartan algebra of the core is isomorphic to the deformed preprojective algebra of the plumbing quiver, with applications to Nakajima quiver varieties, Kawamata's noncommutative deformations of A_n resolutions, and noncommutative ADHM data.

Significance. If the main results hold, Theorem 7.1 gives a novel and potentially influential symplectic interpretation of the deformation parameter of deformed preprojective algebras as weights of codimension-two bulk cycles. The Koszul-duality proof of Theorem 4.3 is explicit and appears largely self-contained, including the construction of a free bimodule resolution and its consequences for Hochschild cohomology and Calabi-Yau structures; these are genuine strengths. The applications to affine ADE plumbings, framed Nakajima quiver varieties, and the conifold are ambitious and would make the paper valuable for both symplectic topology and noncommutative algebraic geometry. However, the paper's main new mechanism, the bulk deformation by cotangent fibers, is not rigorously grounded in the text, and the proof of Theorem 7.1 contains a load-bearing sketched step.

major comments (4)
  1. [Section 7.1, Theorem 7.1] The bulk deformation b = sum_i \delta_i C_i, with C_i = T^*_{p_i}L_i, is not an admissible bulk cycle in the standard FOOO sense as written. In the completed Liouville plumbing, C_i is a properly embedded noncompact Lagrangian disk whose boundary at infinity is a Legendrian circle; it is not a closed codimension-two cycle. The divisor axiom, which is used to discard all insertions with l >= 2 and to obtain the single term \delta_v P_v, requires a genuine cycle class, and the paper does not supply a relative or compactly supported version of the construction. The remark that one should take \delta_i in the Novikov plus-ideal addresses convergence but not closedness or admissibility. Since the identification with the deformed preprojective algebra is deduced from this term, Theorem 7.1 is not established without an admissibility statement or a reformulation in terms of compactly supported closed forms or closed cycles representing the same class.
  2. [Proof of Theorem 7.1, Eq. (7.1)] The coordinate change \tilde x_a = x_a(1 + sum_j a_j (x_{\bar a} x_a)^j), \tilde x_{\bar a} = x_{\bar a} is asserted to remove all higher-order terms and to yield the deformed preprojective relation. The paper does not prove that this change of coordinates is compatible for all arrows simultaneously, does not prove the invertibility of the displayed factor in the noncommutative path algebra, and does not specify how the coefficients a_j depend on Kuranishi perturbations. This is a load-bearing step: the exact form of the relations, for example the coefficients in Corollary 7.2, depends on this coordinate change. A complete proof or a reference for this noncommutative normal-form statement is required.
  3. [Section 7.1, computation of m^{b,b}_0] The computation of the bulk-deformed obstruction term m^{b,b}_0 counts pearl configurations with constant disc components carrying an interior insertion constrained to C_v. In the exact setting all holomorphic discs with boundary on L are constant, and the claimed nonzero contribution \delta_v P_v from the 'stabilized constant disc' is not justified by any specified virtual perturbation scheme. The paper moves from filtered theory with Novikov parameters to polynomial relations over C without proving that the surviving terms are independent of the perturbation and that positive-energy contributions cannot reappear after the coordinate change. This is a second load-bearing gap in the proof of Theorem 7.1.
  4. [Section 6.2, proof of Lemma 6.5] The split-generation claims in Lemma 6.5 rely on the equivalence F(X) \simeq D_fd(G_n(D)) imported from the unpublished preprint [JKL26b]. While the text cites this work, the dependence makes the split-generation applications conditional on a source that a referee cannot verify. This does not affect the central Theorems 4.3 or 7.1, but the manuscript should either prove the needed compatibility or explicitly mark the split-generation statements as depending on [JKL26b].
minor comments (5)
  1. [Example 1.2] The sentence 'Since C has codimension two, the divisor axiom implies that constant discs with more than one interior insertion do not contribute' is an informal preview that presupposes the admissibility point raised in the first major comment; a cross-reference to a rigorous definition would help.
  2. [Section 3.1.1] The symbol V^!_f appears in the discussion of the compactly supported dual but is not defined; please introduce it or remove it.
  3. [Section 6.2, Example 6.6] The name 'affine A_0 graph' for the graph with one vertex and one loop is nonstandard; please clarify the convention or use a more standard terminology.
  4. [References] Several references are to arXiv preprints or works still 'to appear' (for example [JKL26a], [JKL26b], [LT26], [AFO+26], [LNT23]); if any of these have appeared by the time of publication, the bibliography should be updated.
  5. [Proposition 6.2, n >= 3 case] The grading choices for the immersed sectors in the n >= 3 case are delegated to [JKL26b]; a self-contained statement of the needed grading conventions would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the bulk parameter δ_i is a geometric input, and the isomorphism to the deformed preprojective algebra is derived rather than assumed, with only a noncompact bulk-cycle admissibility gap.

full rationale

The central new claim, Theorem 1.3 / Theorem 7.1, is that for a plumbing of two-spheres, bulk deformation by b = Σ δ_i C_i turns the Maurer-Cartan algebra of the core Lagrangian L into the Crawley-Boevey–Holland deformed preprojective algebra of the plumbing quiver. This is a derivation, not a renaming: the target algebra is defined externally in [CBH98], while the geometric input is the triple (L, b, Morse data). The paper explicitly computes m^{b,b}_0, exhibiting δ_v as the contribution of a constant disc with one bulk insertion followed by a Morse trajectory to P_v, and the remaining terms are the λ=0 preprojective relations; the higher-order factors are removed by the invertible coordinate change of Equation (7.1). The bulk parameter δ_i is a coefficient of an input cycle, not a parameter fitted to reproduce λ, so the isomorphism is not forced by construction. The self-citations ([CHL21], [HLT24], [HKL23], [LNT23], [LT26]) supply background tools: the localized mirror functor, the λ=0 preprojective Floer computation, and a local involution lemma. These are not the source of the new δ-dependence, and the main equivalence Theorem 4.3 is proved in the paper via Koszul duality with explicit resolutions. External benchmarks, such as Ginzburg algebra formality [Her16], the derived McKay correspondence [KV00, BKR01], and the monadic complexes of [BGK02], support the surrounding applications. The real weakness is correctness-related rather than circular: the cotangent fiber C_i is noncompact in the completed Liouville manifold, and the paper does not justify that it is an admissible closed bulk cycle for the FOOO divisor axiom, nor does it discuss intersection numbers β·C_i for the zero disc class. The paper even notes convergence restrictions on δ_i without resolving the closedness issue. That gap concerns the validity of the geometric premise, not a reduction of the conclusion to the hypothesis. Under the rule that circularity must be exhibited by specific equations or fitted parameters, no circular step is present, so a low score is appropriate.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central derivation rests on standard Floer-theoretic machinery rather than on numbers fitted to data. The bulk parameters delta_i enter as inputs and the perturbation-dependent coefficients a_j are removed by a coordinate change, so they are not counted as free parameters. The cited external results are used as tools, not as assumed versions of the target result.

assumptions (6)
  • domain assumption Floer theory on the relevant exact plumbings admits the Kuranishi or Morse model computations used for the Maurer-Cartan equations.
    Invoked throughout Section 7 and in the use of FOOO09b and HLT24, Theorem 5.2, for the formulas for p_v.
  • domain assumption The core Lagrangian L is exact, so every holomorphic polygon with boundary on L is constant.
    Used in Proposition 6.2 and Theorem 7.1 to reduce all counts to constant polygons plus Morse flow lines.
  • domain assumption The Floer complex V satisfies Assumption 3.1: finite rank, Z-graded with nonnegative degrees, unital with V^0 isomorphic to k, and no nonpositive degree immersed generators.
    Explicit assumption in Section 3.1 that underlies Theorem 4.3 and Lemma 4.4.
  • standard math Wrapped Fukaya categories of plumbings are equivalent to dg module categories over Ginzburg algebras as in EL17, EL19, and KL25.
    Used in Lemma 6.5 and Corollary 6.4 as an external benchmark to identify the compact Fukaya category with finite-dimensional dg modules.
  • domain assumption The Novikov parameter can be specialized to complex numbers after restricting to the objects for which the bulk parameters introduce no convergence issues.
    The paper states this restriction after Theorem 7.1; it is needed to state the bulk-deformed algebras over C.
  • standard math The Morita equivalence between the deformed preprojective algebra and the deformed skew group ring B_tau, and the derived McKay correspondence, hold as in CBH98, BKR01, and KV00.
    Used in the proof of Theorem 7.9 and Proposition 7.15 to identify the mirror algebra with the noncommutative surface.

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Pith. "Pith review of Mirror functor for deformed preprojective algebras." pith.science (2026). https://pith.science/paper/YDRSKQ3V

@misc{pith2026260805764,
  author       = {Pith},
  title        = {Pith review of: Mirror functor for deformed preprojective algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YDRSKQ3V}},
  note         = {Machine review of arXiv:2608.05764}
}
abstract

We study localized homological mirror symmetry associated to an immersed Lagrangian brane $\mathbb{L}$, possibly equipped with a higher rank flat bundle, of a symplectic manifold $X$. Under a certain finiteness assumption on the Floer theory of $\mathbb{L}$, we deduce a quasi-equivalence $\mathcal{D}\mathrm{Fuk}_\mathbb{L}(X) \cong \mathcal{D}_{\mathrm{fd}}(\tilde{\mathcal A}_\mathbb{L})$ using Koszul duality, where $\tilde{\mathcal{A}}_{\mathbb{L}}$ is the dual differential graded quiver algebra called the extended localized mirror. We apply this to obtain some HMS results for plumbings of cotangent bundles of spheres, and to reproduce known results of split-generation of compact objects. In the second part of the paper, we consider bulk deformation cycles of $X$ that have non-trivial intersections with $\mathbb{L}$. This gives rise to noncommutative deformations of the mirror. When applied to (framed) plumbings, we obtain mirror functors to deformed preprojective algebras (or Nakajima quiver varieties at a general complex moment-map level). For the ADHM and affine $ADE$-type immersions, our construction produces mirror functors to the noncommutative spaces studied by Kapustin-Kuznetsov-Orlov, Baranovsky-Ginzburg-Kuznetsov and Kawamata.

Figures

Figures reproduced from arXiv: 2608.05764 by the authors.

Figure 2
Figure 2. The image of L under the conic fibration of X. The degree-one immersed sector from S 2 k+1 to S 2 k is denoted by αk,k+1, while the sector in the opposite direction is denoted by βk+1,k. In addition, the degree-one self-immersed sectors of Sk are denoted by Xk and Yk. By the Weinstein neighborhood theorem, the plumbing space constructed above for the affine An Dynkin graph can be identified with a neighborhood of L … view at source ↗
Figure 3
Figure 3. Double quiver of affine D4 and the underlying quiver of A2 We localize A0 at the matrix of arrows a1 a2  , that is, we insert the reverse arrows α 1 and α 2 so that [PITH_FULL_IMAGE:figures/full_fig_p040_3.png] view at source ↗
Figure 4
Figure 4. The double conic fibration of X, with singular fibers over ±1. We now choose two rays in the base, which start from the origin in the base and go to infinity, and they are chosen so that they meet the two components of L respectively. The first bulk cycle C1 is obtained by choosing, in a regular fiber over one of the rays, a real three-dimensional submanifold whose compact circle direction is ∆(z), and then parallel… view at source ↗

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