Pith. sign in

REVIEW 3 major objections 5 minor 41 references

K3 atoms of the cubic fourfold and the BPS structure of the Painlev\'e I determinant line

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper argues that the K3 atom of a cubic fourfold is terminal: no deformation of the variety and no ray of the quantum connection can split, resolve, or bind it, and it proves this through exactly solvable Fano models and a…

desk verdict A useful model computation plus an honest conjecture; the fourfold protection theorem is not proved and the abstract oversells the numerical checks. read the letter →

arxiv 2608.12191 v1 pith:7IIGZITO submitted 2026-08-12 math.AG hep-th

Mark Raugas This is my paper · ORCID
classification math.AGhep-th MSC 14F0814N3534M4034M5534M6058J52
keywords K3atomcubicfourfoldBridgelandstabilitysemiorthogonaldecompositionKuznetsovcomponentPainlevéIdeterminantlineBPSstructure
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 tries to establish that the K3 atom of a cubic fourfold—the rank-24 zero-eigenvalue block of quantum multiplication by the first Chern class—is terminal: it cannot be split, deformed, or coupled to by any ray of the quantum connection. The paper analyzes the exactly solvable models $P^1$ and $P^1 \times P^1$ at resonance, shows the quantum path enters the selection region of the semiorthogonal decomposition at finite time and never leaves, and uses the perturbed resonance to separate chamber walls from scale crossovers. For the cubic fourfold, it combines spectral constancy over moduli with the fact that the Kuznetsov component admits no semiorthogonal decomposition to conclude that the coarse region picture is final. In the coupled case, it develops a determinant-line dictionary for the $A_2$ quiver, identifying the Painlevé I tau function with zeta-regularized spectral determinants and computing quantum periods and curvature currents. If correct, the dynamical protection of the K3 atom becomes a proven fact rather than a conjecture.

What carries the argument

The load-bearing machinery is the rank-24 zero-eigenvalue block of quantum multiplication by $c_1(X)$, called the K3 atom, together with its categorical shadow $A_X$ in the semiorthogonal decomposition $D^b(X)=\langle A_X, O_X, O_X(1), O_X(2)\rangle$. The argument runs through the Sectoriality Lemma: once phase windows of a glued stability condition are separated, every stable object lies in a single sector and no binding BPS state can connect sectors; and through a Serre-duality lemma saying a connected Calabi–Yau category with Serre functor $[2]$ and $HH^0=C$ admits no nontrivial semiorthogonal decomposition. In the models, explicit quantum central charge paths supply the phase gaps; in the cubic fourfold, the centroid obstruction forces $A_X$ into an interior position for every ray. The analytic half is carried by the $A_2$ quiver, whose Riemann–Hilbert problem is solved by the monodromy of the deformed cubic oscillator, with Painlevé I as tau function and the determinant line identified with zeta-regularized spectral data.

What would settle it

Compute, for a smooth cubic fourfold containing no plane, the lifted quantum central charge path along a ray avoiding the degenerate set of Conjecture 6.10, and search the region $t>t_0$ for a stable object whose class lies in the mixed lattice $v+w$ with $v \in \tilde{H}(A_X,\mathbb{Z})$, $v^2 \geq -2$, and $w$ in the span of the $[O(i)]$. Finding such an object, or finding a deformation of the variety that changes the rank-24 spectrum, would falsify Proposition 6.2 and Conjecture 6.10(3).

Watch

Extended reading notes

Core claim

The central claim is Proposition 6.2: for every smooth cubic fourfold and every ray of the quantum connection, the rank-24 zero-eigenvalue block has constant spectrum, so no deformation direction can split it, and any refinement of the limiting semiorthogonal decomposition would induce a nontrivial semiorthogonal decomposition of the connected Calabi–Yau category $A_X$, which cannot exist. Hence the K3 atom is neither resolvable nor destabilizable: the coarse region, with $A_X$ as one unrefined block, is terminal. The supporting discovery is that in the minimal models $P^1$ and the resonant $P^1 \times P^1$, the quantum central charge path crosses a single finite-time wall and then lies forever in the selection region, so the stable spectrum is sectorial and no binding objects form; perturbing the resonance shows the crossover time is not a wall, with all $\varepsilon$-dependence confined to the region layer. In the coupled $A_2$ model, the paper identifies the tau function of the deformed cubic oscillator and Painlevé I with a zeta-regularized determinant, computes closed-form quantum periods through $Z_4$, proves perturbative flatness to all orders, and identifies the curvature current with the tau-divisor current, leaving the nonperturbative jumps as a conjecture.

Load-bearing premise

The load-bearing premise is that the lifted quantum central charge path from the prior literature exists, is glued at every time, runs from a geometric glued stability condition to a quasi-convergent sectorial tail, and induces the interior-$A_X$ mutation; the paper records that quasi-convergence of that path was asserted without verification in the cited source, and if no such path can be built, the finite-time entry time has no ground.

Editorial extensions

If this is right

  • For the cubic fourfold, the K3 atom is terminal: no deformation of the variety and no ray of the quantum connection produces a sub-block, a wall, or a binding stable object inside the rank-24 block.
  • In the $P^1$ and resonant $P^1 \times P^1$ models, the quantum cohomology path reaches the selection region at finite time $t_0$ and stays there; binding BPS states exist only before $t_0$ and decouple at the wall, realizing dynamical protection as the endpoint of wall-crossing.
  • The perturbed resonance shows that the internal crossover time is not a wall: the stable spectrum is $\varepsilon$-independent, and the $\varepsilon$-dependence lives entirely in the region layer of scale crossovers and augmented boundary levels.
  • The cubic threefold analogue predicts finite-time entry into the selection region of the interior-$Ku$ ordering, with the intermediate Jacobian atom protected, and the centroid obstruction makes the Kuznetsov component interior for every ray.
  • For the $A_2$ quiver, the tau function of Painlevé I and the zeta-regularized determinant of the deformed cubic oscillator are identified to all perturbative orders, with curvature current equal to the tau-divisor current; the nonperturbative jumps are conjectured to be Joyce-structure automorphisms.

Reading between the lines

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

  • If terminality survives scrutiny, the K3 atom becomes a deformation-invariant invariant of smooth cubic fourfolds that no Bridgeland wall-crossing can alter, strengthening the rationality invariant extracted from Stokes data.
  • The two-layer separation between chamber walls and scale crossovers suggests a general diagnostic: for Fano degenerations whose degenerate block has $HH^0=C$, expect terminality; for blocks with larger Hochschild cohomology, expect the resonant $P^1 \times P^1$ behavior with exact phase-locking and no internal clock.
  • The determinant/Painlevé I dictionary implies a concrete testable recipe for higher-degree oscillators: zeta-regularized determinants of deformed polynomial potentials should exhibit the same flatness and divisor-current structure, with the golden-ratio constant replaced by other algebraic numbers.
  • The conjectured identification of nonperturbative determinant jumps with Joyce automorphisms, if proven, would make the determinant line a practical computational tool for Stokes data in coupled BPS structures.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the conjectured dynamical protection of the K3 atom in the derived category of a cubic fourfold by analyzing exactly solvable models. For the projective line and the resonant product P^1 x P^1, it proves that the lifted quantum cohomology path exits the geometric chamber at a finite time, enters the selection region of the associated semiorthogonal decomposition, and never leaves; perturbing the resonance shows the resulting internal crossover is not a wall. For the cubic fourfold it proves a terminality statement (Proposition 6.2) for the rank-24 zero-eigenvalue block, formulates a chamber theorem as Conjecture 6.10, and gives a detailed proof architecture. For the A_2 quiver and the deformed cubic oscillator, it develops a tau/determinant dictionary: closed-form quantum periods through Z_4, all-orders perturbative flatness with exact curvature equal to the tau-divisor current, and numerical verification of the identification of the zeta determinant's zero divisor with the Painlev\'e I pole locus along a trajectory. The paper closes with a conjecture that nonperturbative jumps of the determinant line are automorphisms of the underlying Joyce structure.

Significance. If the main results hold, the paper provides a substantial computational and conceptual advance: it gives the first explicit finite-time chamber-entry analysis in the noncommutative MMP for Fano models, proves that the K3 atom block of a smooth cubic fourfold cannot be split by deformation or categorical decomposition (Proposition 6.2), and develops a rich analytic dictionary that may lead to a rigorous tau-function/determinant identification in the minimal coupled case. The model computations in Sections 3--5 are largely self-contained and rigorous given the cited classifications, Lemma 7.1 is a clean algebraic computation, and the numerical work in Section 8 is extensive and reproducible. The central fourfold chamber theorem, however, is a conjecture whose proof depends on an external quasi-convergence statement whose verification is omitted in the cited reference; the tau-divisor identification is verified numerically on one trajectory, not established analytically. The paper is honest about these gaps, but they are load-bearing for the fourfold protection claim.

major comments (3)
  1. [§6.3, Remark 6.11] The claim that “the wall program does not rely on” quasi-convergence of the [27, Thm. 4.7] path is not justified. Clause (1) of Conjecture 6.10 asserts the existence of a finite entry time t0 such that the path lies in the selection region. Without quasi-convergence of the sectorial tail, there is no guarantee that the path approaches the limiting semiorthogonal decomposition at any finite time, and hence no basis for the entry time. The solvable models in §§3–5 prove the analoguous statement for P1 and P1×P1 using specific vanishings (e.g., backward Ext1 groups) that do not hold for the cubic fourfold, as the paper itself acknowledges in Remark 6.11c. The sentence “the wall program does not rely on it” is therefore unsupported and requires either a proof of finite-time entry independent of quasi-convergence or a reformulation of Conjecture 6.10 that does not assert finite-time entry.
  2. [§8.6, Proposition 8.6] The identification of the zeta determinant's zero divisor with the Painlevé I tau divisor is only verified numerically along a single trajectory, not established as a theorem. The proof of Proposition 8.6 is a sketch that assumes Borel summability in the closed Stokes regions adjacent to the ray a>0, stated as following from “ℏ-homogeneity” and results of [10, App. A], but no complete argument is supplied. Consequently the assertion that the matching constant is ℜκ0 = log 2 rests on an unverified analytic input. The numerical evidence is strong, but the central dictionary statement of §7.6—that the determinant line reproduces the tau function—remains at the level of a well-tested conjecture, not a proved theorem.
  3. [§6, Proposition 6.2 and Conjecture 6.10] The paper’s route from the terminality of the coarse region to dynamical protection of the K3 atom requires Conjecture 6.10; Proposition 6.2 alone does not imply that the quantum path enters the selection region. The introduction and abstract describe the compatibility question as “the missing ingredient” and present the models as settling the minimal cases, but for the fourfold the statement that the path enters the selection region at finite time is not proved. The distinction between the chamber layer and the region layer in §5 is useful, but it does not supply the missing entry argument. The paper should either limit its claims to the model cases and the terminality proposition, or provide a proof of the fourfold entry statement under the hypotheses of [27].
minor comments (5)
  1. [§5.3] The displayed paragraph after “two-level point” is malformed: the list reads “(1) The coarse separation (8) at rates (2) The internal separation at rates f” without defining the rates or the symbol f. This should be rewritten as a clear display defining the phase and mass clocks for the two levels.
  2. [§8.4, Table] The header of the numerical table, “n ℜa_n −ℑa_n β|a|^{5/4} + B(1/4,1/2)/16 |a|^{-5/4} = 2π(n+1/2)”, mixes a formula with the columns and is hard to parse. Please split the columns explicitly and move the quantization condition to the caption or the text.
  3. [Abstract and §8.7] The abstract states “identifying it with the zeta determinant’s zero divisor along a Painlevé I trajectory,” but the paper verifies this numerically on one trajectory; it is not an analytic identification. Please qualify the statement (e.g., “numerically verifies”) to avoid overstating the result.
  4. [§8.3] The notation switches between the eigenvalue-ray zeta ζ(s) and the full-divisor zeta ζ_D(s) without a clear relation. Please clarify which divisor each zeta function is associated with and how they are related.
  5. [§6.3, Remark 6.11a] The phrase “theO(1) offsets” has a typo and should read “the O(1) offsets.” Similar spacing issues occur in a few places (e.g., “theO(1/t 0) correction” in §3).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's new derivations are self-contained, and the flagged gaps are external correctness risks rather than circular reductions.

full rationale

The claimed derivations do not reduce to their inputs. Sections 3–5 prove chamber entry for P1 and resonant P1×P1 from Okada's classification and explicit Künneth/gluing inequalities, with the Sectoriality Lemma (Lemma 2.2) giving sectoriality from phase gaps; none of these results assumes the target. Proposition 6.2's terminality is argued from deformation invariance of small quantum cohomology and the Serre-functor/HH^0 argument of Lemma 6.1, not from the author's prior conjecture. Conjecture 6.10 is explicitly a conjecture, and Remark 6.11a honestly records that quasi-convergence of the [27, Thm. 4.7] path is asserted with verification omitted in [27], and that the wall program does not rely on it; this is an external correctness risk, not a circularity. The numerical fits in Section 8.4 compare free fit constants to predicted closed forms (e.g., log 2, B(1/4,1/2)/16 sqrt(2)) rather than setting those constants, and Proposition 8.6 supplies an independent argument under stated Borel-summability assumptions. The only self-citation is [36, Conj. 8.1], invoked as the motivating target and interpretive frame, not as a premise of any new proof. There is no exhibited equation or fitted parameter equivalent by construction to an output, so the paper is not circular.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The free parameter is the numerical fit of the determinant asymptotics; the axioms are standard stability and ODE facts plus external existence results from [27] and [10]. No new physical or categorical entities are postulated. The central cubic claim additionally depends on unproved conjectural paths from [27].

free parameters (1)
  • numerical fit coefficients for logD(a) = c_lead=-0.6777706, c_log=6e-5, c1=+0.2317591, c0=+0.693147182, c2=+0.535757, c3 approx -7.69
    Least-squares fits to numerically computed logD(a) on a grid in Section 8.4. They are used as verification against predicted constants, not as inputs to the derivation, but they are numbers fitted to data in the paper.
assumptions (6)
  • domain assumption Existence of lifted quantum central charge paths and a geometric glued endpoint for cubic fourfolds, from [27, Theorems 2.72, 4.6, 4.7].
    Conjecture 6.10 depends on these paths; Remark 6.11 notes that quasi-convergence of Theorem 4.7 is asserted without verification in [27].
  • domain assumption Small quantum cohomology is deformation invariant, and all smooth cubic fourfolds form a single deformation class.
    Used in Proposition 6.2 to conclude that the rank-24 zero-eigenvalue block is constant over moduli; cited to [26], [37], [22].
  • domain assumption The Kuznetsov component AX is a connected Calabi-Yau category of dimension two with Serre functor [2] and HH0(AX) = C.
    Used in Lemma 6.1 and Proposition 6.2 to exclude semiorthogonal decompositions of AX; cited to [31].
  • domain assumption Ku(Y) for the cubic threefold is indecomposable and its stability space is a single orbit.
    Used in Conjecture 6.7 and Remark 6.9; cited to [16] and [31].
  • domain assumption Bridgeland's theorem [10]: framed Stokes data of the deformed cubic oscillator solve the A2 Riemann-Hilbert problem, and variation in (a,b) is equivalent to Painlevé I.
    This is the foundation of Section 7 and of the determinant dictionary; the paper builds on it rather than proving it.
  • domain assumption Borel summability of the WKB solutions in closed Stokes regions adjacent to the ray a > 0.
    Explicitly assumed in Proposition 8.6; stated as standard exact-WKB input with partial support from [10, Appendix A].

how reviews work

0 comments
Cite this review

Pith. "Pith review of K3 atoms of the cubic fourfold and the BPS structure of the Painlev\'e I determinant line." pith.science (2026). https://pith.science/paper/7IIGZITO

@misc{pith2026260812191,
  author       = {Pith},
  title        = {Pith review of: K3 atoms of the cubic fourfold and the BPS structure of the Painlev\'e I determinant line},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7IIGZITO}},
  note         = {Machine review of arXiv:2608.12191}
}
abstract

The compatibility of the semiorthogonal decomposition of a cubic fourfold with Bridgeland stability beyond a generic point is the missing ingredient in the conjectured dynamical protection of the K3 atom $\AX$. We analyze it in the exactly solvable models of the noncommutative minimal model program. In the uncoupled Fano models ($\PP^1$; $\PP^1 \times \PP^1$ at resonance) the quantum cohomology path exits the geometric chamber at an explicit finite time, enters the selection region of the gluing, and never leaves; perturbing the resonance shows the $\varepsilon$-crossover is not a wall. The $A_2$ quiver provides a coupled counterpart. We examine the deformed cubic oscillator, Painlev\'e~I, and formulate a tau-JLO dictionary as a determinant identification and establish its perturbative layer, including computing closed-form quantum periods through $Z_4$, all-orders flatness with exact curvature the tau-divisor current, identifying it with the zeta determinant's zero divisor along a Painlev\'e~I trajectory. We close by conjecturing that the determinant's nonperturbative jumps are automorphisms of the underlying Joyce structure.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

41 extracted references · 30 canonical work pages

  1. [27]

    Karube, A.-A

    T. Karube, A.-A. Robotis, V. Zuliani,Toward the noncommutative minimal model program for Fano varieties, arXiv:2601.20739

    work page arXiv

  2. [1]

    Addington, R

    N. Addington, R. Thomas,Hodge theory and derived categories of cubic fourfolds, Duke Math. J.163(2014), 1885–1927

    work page 2014

  3. [2]

    Bayer, M

    A. Bayer, M. Lahoz, E. Macr` ı, P. Stellari,Stability conditions on Kuznetsov components, Ann. Sci. ´Ec. Norm. Sup´ er. (4)56(2023), 517–570. arXiv:1703.10839

    arXiv 2023

  4. [3]

    Bayer, M

    A. Bayer, M. Lahoz, E. Macr` ı, H. Nuer, A. Perry, P. Stellari,Stability conditions in families, Publ. Math. IH ´ES133(2021), 157–325. arXiv:1902.08184

    arXiv 2021

  5. [4]

    Bayer, E

    A. Bayer, E. Macr` ı,MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations, Invent. Math.198(2014), 505–590

    2014

  6. [5]

    Bonelli, O

    G. Bonelli, O. Lisovyy, K. Maruyoshi, A. Sciarappa, A. Tanzini,On Painlev´ e/gauge theory cor- respondence, Lett. Math. Phys.107(2017), 2359–2413. arXiv:1612.06235

    arXiv 2017

  7. [6]

    Bridgeland,Stability conditions on triangulated categories, Ann

    T. Bridgeland,Stability conditions on triangulated categories, Ann. of Math. (2)166(2007), 317–345

    work page 2007

  8. [7]

    Bridgeland,Riemann–Hilbert problems from Donaldson–Thomas theory, Invent

    T. Bridgeland,Riemann–Hilbert problems from Donaldson–Thomas theory, Invent. Math.216 (2019), 69–124

    work page 2019

Show all 41 references
  1. [8]

    Bridgeland,Riemann–Hilbert problems for the resolved conifold, J

    T. Bridgeland,Riemann–Hilbert problems for the resolved conifold, J. Differential Geom.115 (2020), 395–435

    2020

  2. [9]

    Bridgeland,Tau functions from Joyce structures, SIGMA20(2024), 112

    T. Bridgeland,Tau functions from Joyce structures, SIGMA20(2024), 112. arXiv:2303.07061

    2024 arXiv

  3. [10]

    Bridgeland (appendix by D

    T. Bridgeland (appendix by D. Masoero),On the monodromy of the deformed cubic oscillator, Math. Ann.385(2023), 193–258. arXiv:2006.10648

    2023 arXiv

  4. [11]

    Bridgeland, I

    T. Bridgeland, I. Smith,Quadratic differentials as stability conditions, Publ. Math. IH ´ES121 (2015), 155–278

    2015

  5. [12]

    Cotti, B

    G. Cotti, B. Dubrovin, D. Guzzetti,Helix structures in quantum cohomology of Fano varieties, Lecture Notes in Math. 2356, Springer, 2024

    2024

  6. [13]

    Collins, A

    J. Collins, A. Polishchuk,Gluing stability conditions, Adv. Theor. Math. Phys.14(2010), 563–

    2010

  7. [14]

    Cecotti, C

    S. Cecotti, C. Vafa,Topological–anti-topological fusion, Nuclear Phys. B367(1991), 359–461

    1991

  8. [15]

    Delabaere, H

    E. Delabaere, H. Dillinger, F. Pham,R´ esurgence de Voros et p´ eriodes des courbes hyperelliptiques, Ann. Inst. Fourier43(1993), 163–199. 42 MARK RAUGAS

    1993

  9. [16]

    Feyzbakhsh, L

    S. Feyzbakhsh, L. Pertusi,Serre-invariant stability conditions and Ulrich bundles on cubic three- folds, ´Epij. G´ eom. Alg´ ebrique7(2023), Art. 1. arXiv:2109.13549

    2023

  10. [17]

    Getzler, A

    E. Getzler, A. Szenes,On the Chern character of a theta-summable Fredholm module, J. Funct. Anal.84(1989), 343–357

    1989

  11. [18]

    Happel, I

    D. Happel, I. Reiten, S. Smalø,Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc.120(1996), no. 575

    1996

  12. [19]

    Halpern-Leistner,The noncommutative minimal model program, arXiv:2301.13168

    D. Halpern-Leistner,The noncommutative minimal model program, arXiv:2301.13168

    arXiv

  13. [20]

    Halpern-Leistner, J

    D. Halpern-Leistner, J. Jiang, A.-A. Robotis,Stability conditions and semiorthogonal decompo- sitions I: quasi-convergence, arXiv:2401.00600

    arXiv

  14. [21]

    Halpern-Leistner, A.-A

    D. Halpern-Leistner, A.-A. Robotis,The space of augmented stability conditions, in preparation

  15. [22]

    Hinault, T

    T. Hinault, T. Y. Yu, C. Zhang, S. Zhang,Decomposition and framing ofF-bundles and appli- cations to quantum cohomology, arXiv:2411.02266

    arXiv

  16. [23]

    Iritani,An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv

    H. Iritani,An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math.222(2009), 1016–1079

    2009

  17. [24]

    Jaffe, A

    A. Jaffe, A. Lesniewski, K. Osterwalder,QuantumK-theory, I. The Chern character, Comm. Math. Phys.118(1988), 1–14

    1988

  18. [25]

    Karube,The noncommutative MMP for blowup surfaces, arXiv:2410.18446

    T. Karube,The noncommutative MMP for blowup surfaces, arXiv:2410.18446

    arXiv

  19. [26]

    Katzarkov, M

    L. Katzarkov, M. Kontsevich, T. Pantev, T. Y. Yu,Birational invariants from Hodge structures and quantum multiplication, arXiv:2508.05105

  20. [28]

    Keller,Deformed Calabi–Yau completions, J

    B. Keller,Deformed Calabi–Yau completions, J. Reine Angew. Math.654(2011), 125–180. With an appendix by M. Van den Bergh

    2011

  21. [29]

    Kontsevich, Y

    M. Kontsevich, Y. Soibelman,Stability structures, motivic Donaldson–Thomas invariants and cluster transformations, arXiv:0811.2435

    arXiv

  22. [30]

    Kuznetsov,Derived categories of cubic fourfolds, in: Cohomological and geometric approaches to rationality problems, Progr

    A. Kuznetsov,Derived categories of cubic fourfolds, in: Cohomological and geometric approaches to rationality problems, Progr. Math. 282, Birkh¨ auser, 2010, 219–243

    2010

  23. [31]

    Kuznetsov,Calabi–Yau and fractional Calabi–Yau categories, J

    A. Kuznetsov,Calabi–Yau and fractional Calabi–Yau categories, J. Reine Angew. Math.753 (2019), 239–267

    2019

  24. [32]

    Lisovyy, J

    O. Lisovyy, J. Roussillon,On the connection problem for Painlev´ e I, J. Phys. A50(2017), 255202. arXiv:1612.08382

    2017 arXiv

  25. [33]

    Macr` ı,Stability conditions on curves, Math

    E. Macr` ı,Stability conditions on curves, Math. Res. Lett.14(2007), 657–672; see also arXiv:math/0411613

    2007 arXiv

  26. [34]

    Masoero,Poles of int´ egrale tritronqu´ ee and anharmonic oscillators, J

    D. Masoero,Poles of int´ egrale tritronqu´ ee and anharmonic oscillators, J. Phys. A43(2010), 095201

    2010

  27. [35]

    Okada,Stability manifold ofP 1, J

    S. Okada,Stability manifold ofP 1, J. Algebraic Geom.15(2006), 487–505

    2006

  28. [36]

    Raugas,Relating Hodge atoms, spectral triples, and BPS flows, arXiv:2603.16639

    M. Raugas,Relating Hodge atoms, spectral triples, and BPS flows, arXiv:2603.16639

  29. [37]

    Sanda, Y

    F. Sanda, Y. Shamoto,An analogue of Dubrovin’s conjecture, Ann. Inst. Fourier (Grenoble)70 (2020), 621–682. arXiv:1705.05989

    2020 arXiv

  30. [38]

    Sibuya,Global theory of a second order linear ordinary differential equation with a polynomial coefficient, North-Holland, 1975

    Y. Sibuya,Global theory of a second order linear ordinary differential equation with a polynomial coefficient, North-Holland, 1975

    1975

  31. [39]

    Sutherland,The modular curve as the space of stability conditions of a quiver, arXiv:1111.4184

    T. Sutherland,The modular curve as the space of stability conditions of a quiver, arXiv:1111.4184

    arXiv

  32. [40]

    Toda,Moduli stacks and invariants of semistable objects on K3 surfaces, Adv

    Y. Toda,Moduli stacks and invariants of semistable objects on K3 surfaces, Adv. Math.217 (2008), 2736–2781

    2008

  33. [41]

    Voros,The return of the quartic oscillator: the complex WKB method, Ann

    A. Voros,The return of the quartic oscillator: the complex WKB method, Ann. Inst. H. Poincar´ e Sect. A39(1983), 211–338. Pacific Northwest National Laboratory, 1100 Dexter Ave N, Seattle, WA 98109 Email address:raugas@pnnl.gov

    1983

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.