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Joyce's invariant and Virasoro Constraints for Quot schemes on curves

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims that Quot schemes on curves satisfy Virasoro constraints, giving universal linear relations among all their tautological intersection numbers and reducing every f-class intersection to a- and b-class intersections.

desk verdict A genuinely new extension of Virasoro constraints to Quot schemes, with a strong rank N-1 computation, but the proof of the main theorem has an internal e=1 compatibility gap that needs fixing. read the letter →

arxiv 2608.04795 v1 pith:7YYVBLAS submitted 2026-08-05 math.AG

Parvez Rasul This is my paper · ORCID
classification math.AG MSC 14N3514D2014C1717B6914H60
keywords QuotschemesJoyceinvariantwall-crossingVirasoroconstraintsvertexalgebravirtualfundamentalclasstautologicalintersectionnumbersvectorbundlesoncurves
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 claims that the Quot scheme Quot_{r,d}(E) of quotients of a vector bundle E on a smooth projective curve satisfies Virasoro constraints, i.e., universal linear relations among all tautological intersection numbers. To get there, the paper introduces Joyce's enumerative invariant for the Quot scheme, shows that it equals the pushforward of the virtual fundamental class, and proves a wall-crossing formula expressing this invariant through moduli of sheaves. The invariant is computed explicitly in the rank-one-kernel case, yielding intersection numbers of a- and b-classes. If the central theorem is correct, every f-class intersection on any such Quot scheme can be eliminated in favor of a- and b-class intersections by an explicit recursive formula.

What carries the argument

The load-bearing object is Joyce's invariant [Quot_{r,d}(E)]^{inv}, a class in the shifted homology of the stack of E^∨-pairs; it coincides with the pushforward f_*[Quot_{r,d}(E)]^{vir} and satisfies a wall-crossing formula as an iterated Lie bracket of a point class and sheaf invariants M_α. The Virasoro proof uses the pair vertex algebra $V^{{pa}}$ on the homology of pairs of complexes, whose conformal element produces operators $L_m^{{pa}}$ dual to the descendent Virasoro operators; the Quot class is shown to be a primary state, meaning all positive Virasoro operators kill it. The rank-(N-1) computation uses an explicit description of the homology algebra and the residue formula res_{z=0} $z^{{-(ν+N d_)}}$ ρ(z) σ(N/z - s_{1,2,2}).

What would settle it

Specialize Theorem 6.7 to E = O_C^N and a degree d with dim Q ≥ 0: the theorem predicts ∫_{[Quot_{N-1,d}(O^N)]^{vir}} $a_1^{{dim Q}}$ = N^g. A direct virtual-localization computation of this single integral, using the torus action on O^N, that produces a value different from N^g would disprove the claimed Virasoro constraints and the associated f-class elimination.

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Extended reading notes

Core claim

The central result is Theorem 5.9: for any m ≥ 0 and any descendent D, the integral of ξ_{K^∨}(L_m(D)) over the virtual fundamental class [Quot_{r,d}(E)]^{vir} vanishes. This means the Quot scheme satisfies the Virasoro constraints, so all descendent integrals obey universal linear relations. The paper further proves that any top-degree polynomial in the tautological classes a, b, and f can be replaced by a polynomial in a and b alone, with a recursive formula for the replacement. For the case where the kernel has rank one, it computes the invariant explicitly and obtains the intersection formula ∫ ∏_{i=1}^s b_{j_i}^1 b_{j_i+g}^1 $a_1^{{dim Q - s}}$ = $N^{{g-s}}$, where N = rank(E).

Load-bearing premise

The proof assumes that the sheaf-counting invariants appearing in the wall-crossing sum are primary states killed by all Virasoro operators, and that pushing the wall-crossing formula into the pair vertex algebra respects the Lie bracket; if either fails, the Virasoro vanishing for Quot schemes does not follow.

Editorial extensions

If this is right

  • Every descendent integral on Quot_{r,d}(E) obeys the Virasoro relations, so tautological intersection numbers satisfy universal linear constraints rather than being independent.
  • Any top-degree polynomial in the a, b, and f classes can be rewritten as a polynomial in a and b alone; the paper gives a recursive algorithm for the rewriting.
  • For rank-one kernels, all a/b intersections are explicit: the product formula ∫ ∏_{i=1}^s b_{j_i}^1 b_{j_i+g}^1 a_1^{dim Q - s} = N^{g-s} holds, and in dimension zero the virtual count of maximal rank-one subbundles is N^g.
  • The same wall-crossing framework proves Virasoro constraints for the related moduli space P(E,α) of stable pairs with sections.
  • With the invariant known for rank-one kernels and the Virasoro constraints in hand, complete tautological intersection numbers on those Quot schemes can be evaluated without computing f-class integrals directly.

Reading between the lines

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

  • The recursive elimination of f-classes suggests that, once any computation of a- and b-intersections is available (for example by virtual localization), the full set of tautological intersection numbers on Quot schemes becomes accessible; the paper itself does not pursue this combination.
  • The explicit residue computation for rank-one kernels indicates that analogous closed-form intersection numbers for higher-rank kernels would follow if a similar invariant computation could be carried out, which the author leaves as future work.
  • The wall-crossing expression may be invertible, potentially allowing the invariant for arbitrary r to be built from lower-rank Quot invariants and sheaf invariants, a route that would bypass Virasoro constraints for explicit numerical evaluations.
  • The N^g count for maximal rank-one subbundles is stable under the choices in the wall-crossing formula; a natural testable extension is whether this count remains unchanged under deformations of E for higher-rank maximal subbundles.
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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

3 major / 6 minor

Summary. The paper applies Joyce's enumerative wall-crossing framework to Quot schemes of a fixed vector bundle on a smooth projective curve. It defines a Joyce invariant for Quot_{r,d}(E), proves a wall-crossing formula expressing it in terms of the point class e_{(0,0),E} and Joyce invariants of sheaf moduli, computes the invariant explicitly in the rank-(N-1) case, and uses it to recover intersection numbers of a- and b-classes. It then formulates Virasoro constraints for Quot schemes following BLM24, proves them in Theorem 5.9, and derives from them a recursive elimination of f-classes in tautological intersection numbers. The paper also contains explicit low-dimensional examples and a comparison with the known count N^{g-s} of maximal subbundles.

Significance. If the main theorem is correct, the paper provides a substantial extension of the sheaf-theoretic Virasoro constraints of BLM24 to Quot schemes, giving universal linear relations among descendent integrals and a method to eliminate f-classes. The explicit wall-crossing formula for Joyce's invariant of Quot schemes and the closed-form computation for rank-one kernels are useful contributions. The paper deserves credit for checking its machinery against an independent baseline: Theorem 6.7 reproduces the known number N^{g-s} of maximal subbundles, and Section 6 derives concrete f-class relations. The main caveat is that the proof of Theorem 5.9 has a gap in the compatibility of the wall-crossing bracket with the pushforward to the pair vertex algebra, and several foundational statements are proved only by reference to Joyce or BLM24.

major comments (3)
  1. [Theorem 5.9, proof] The proof asserts that the point class e_{(0,0),E} can be pushed forward through Ξ and that the iterated Lie brackets in formula (22) are compatible with the pair vertex algebra bracket. The justification is that the restrictions of \check{Ext} and (Ξ×Ξ)^*Ext^pa agree on \check{M}×M, and hence state-field correspondences coincide for u∈\check{V}_* and v∈V_*. However, e_{(0,0),E} has e=1 and lies in \check{M}_{(0,0),1}, which is not contained in V_* (whose classes have e=0). Thus every bracket involving e_{(0,0),E}, in either order, violates the stated hypothesis v∈V_*. Since formula (10) and its rewrite (22) use exactly those brackets, the equality in qV^pa and the subsequent lift via Lemma 5.8(2) are not justified as written. This is an internal gap in the proof of the central theorem, not merely a deferral to a cited result.
  2. [Theorem 3.2] Theorem 3.2 is the foundation of the entire wall-crossing construction, but its proof consists of saying that Joyce's Assumptions 4.4 and 5.1-5.3 "follow similarly" to [Joy21, Sections 8.2.1-8.2.4] after replacing the line bundle with a vector bundle. This is load-bearing because Theorem 3.5 and formula (10) depend on it. Please provide a detailed verification of the finiteness, permissible-class, and stability assumptions for E^∨-pairs for arbitrary vector bundles, or state precisely which results in [Joy21] already cover this case.
  3. [Theorem 6.3, Eq. (27)] The displayed recursive formula does not appear to follow from (24) and (25). Substituting (25) into S_{1,2,l+1} from (24) with m=l gives expressions involving μ_{l+1} and f_i for i≤l+1, but the right-hand side of (27) contains only ∑_{i=1}^l ∂μ_l/∂a_i f_i and a second derivative sum over i,k≤l+1 of μ_l, which is not defined. As written, the formula cannot be used recursively to eliminate f_{l+1}. Please correct the indices and clarify the convention for μ_0 and μ_l, or explain if a different indexing is intended.
minor comments (6)
  1. [Section 3, Theorem 3.1] The graded vertex algebra structure is invoked from [Joy21] before the grading shift has been fully explained for the reader; a short reminder of the parity convention would improve readability.
  2. [Equation (14)] The pairing formula is difficult to parse because the ranges of the products and the meaning of the total order are compressed; please spell out the indexing more explicitly.
  3. [Section 2, stability definition] The stability condition μδ is defined only for rank F > 0; for e=0 pairs and for F=0 the reader must infer the limiting convention. Please state the definition uniformly.
  4. [Example 6.4] The class S_{1,0,0} is used implicitly in the Virasoro relation but is never defined; please state that S_{1,0,0}=rank K^∨ under the geometric realization.
  5. [Theorem 6.7] The residue computation ends with "This can be easily calculated to be N^{g-s}"; since this is the only place the numerical factor N^{g-s} is obtained, please include the intermediate steps.
  6. [References] Reference [Joy19] is cited by a URL; if a stable published or arXiv version exists, please cite it in addition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Quot Virasoro proof is a valid reduction to external Joyce wall-crossing and [BLM24] results, and the computed N^{g-s} intersection is checked against an independent baseline.

full rationale

The paper's derivation chain is not circular. Theorem 3.5 defines Joyce's Quot invariant as the pushforward of the virtual fundamental class and derives its wall-crossing formula by applying Joyce's external wall-crossing machinery to the E_-pair category; no constant is fitted to any later answer. Theorem 4.1 computes the rank-(N-1) invariant using the explicit vertex-algebra formula (16) and the known class for the rank-one sheaf invariant from [Bu23], which is an external computation. Theorem 5.9 proves the Virasoro constraints by reducing them to the external theorem [BLM24, Theorem 5.12] that sheaf invariants are primary states, together with the lifting lemma [BLM24, Lemma 5.8]; this is a legitimate reduction of one moduli problem to another, not a self-citation chain or a definitional equivalence. Theorem 6.3 and the f-class elimination follow algebraically from the Virasoro relations once those constraints are established. Theorem 6.7 reproduces the independently known value N^{g-s}, so the computation is checked against an external baseline rather than being forced by construction. The skeptical concern about whether the pushforward along Xi respects Lie brackets for the e=1 point class identifies a possible internal gap in the proof of Theorem 5.9, but a proof gap is not circularity: the disputed assertion is not an output being identified with an input by definition, and no parameter has been fitted to the predicted integrals. The paper also does not rely on self-citations by its own author; the cited [BLM24] and [Bu23] results are independent published mathematical facts. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central construction leans on three external pillars: Joyce's wall-crossing theory, Grothendieck's description of moduli-stack cohomology, and BLM24's primary-state theorem. No free parameters are fitted; the stability thresholds are chosen for chamber arguments, not tuned to data.

assumptions (4)
  • domain assumption Joyce's enumerative invariant and wall-crossing theory for abelian categories (Joy21, Theorems 3.12, 5.9, 8.24).
    The paper builds the Quot scheme invariant directly on Joyce's general wall-crossing formalism; none of this machinery is proved in the paper.
  • domain assumption The homology of moduli stacks of complexes on a curve is generated by tautological classes (Gro20, Theorem 4.15).
    Used in Section 4 to describe H^*(M_α) and H^*(\bar{M}) as polynomial algebras.
  • domain assumption The sheaf invariants M_α are primary states in the pair vertex algebra (BLM24, Theorem 5.12).
    This is the crucial input in Theorem 5.9; if false, the Virasoro proof fails.
  • ad hoc to paper The moduli stack of E^∨-pairs satisfies Joyce's Assumptions 4.4 and 5.1-5.3.
    Theorem 3.2 asserts this without detailed verification, saying it 'follows similarly' to Joyce's line bundle case.

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Pith. "Pith review of Joyce's invariant and Virasoro Constraints for Quot schemes on curves." pith.science (2026). https://pith.science/paper/7YYVBLAS

@misc{pith2026260804795,
  author       = {Pith},
  title        = {Pith review of: Joyce's invariant and Virasoro Constraints for Quot schemes on curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7YYVBLAS}},
  note         = {Machine review of arXiv:2608.04795}
}
abstract

Let $C$ be a smooth projective curve over $\mathbb C$ and let $E$ be a vector bundle over $C$. Let $\text{Quot}_{r,d}(E)$ denote the Quot scheme which parametrizes quotients of $E$ of rank $r$ and degree $d$. Following Joyce's recipe [Joy21], we introduce Joyce's enumerative invariant for the Quot scheme $\text{Quot}_{r,d}(E)$. The invariant can be viewed as a generalization of the virtual fundamental cycle of the Quot scheme. We evaluate intersection pairings on the Quot scheme $\text{Quot}_{\text{rank}(E)-1,d}(E)$ by computing its invariant explicitly. Following the reformulation of sheaf-theoretic Virasoro constraints in terms of Joyce's vertex algebra framework in [BLM24], we give a proof of the Virasoro constraints for the Quot scheme $\text{Quot}_{r,d}(E)$. With the help of these constraints, we compute the (virtual) intersection numbers of $f$-classes on the Quot schemes.

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Works this paper leans on

31 extracted references · 11 canonical work pages

  1. [8]

    [Bu23] Chenjing Bu

    doi:10.4310/MRL.2021.v28.n4.a2. [Bu23] Chenjing Bu. Counting sheaves on curves. Adv. Math. , 434:Paper No. 109334, 87,

    doi:10.4310/mrl.2021.v28.n4.a2 2021

  2. [10]

    [FMR21] Nadir Fasola, Sergej Monavari, and Andrea T

    doi:10.1016/S0370-2693(97)00401-2. [FMR21] Nadir Fasola, Sergej Monavari, and Andrea T. Ricolfi. Higher rank K-theoretic Donaldson-Thomas theory of points. Forum Math. Sigma , 9:Paper No. e15, 51,

    work page doi:10.1016/s0370-2693(97)00401-2

  3. [11]

    [GJT22] Jacob Gross, Dominic Joyce, and Yuuji Tanaka

    doi:10.1017/fms.2021.4. [GJT22] Jacob Gross, Dominic Joyce, and Yuuji Tanaka. Universal structures in C-linear enumerative in- variant theories. SIGMA Symmetry Integrability Geom. Methods Appl. , 18:Paper No. 068, 61,

    doi:10.1017/fms.2021.4 2021

  4. [13]

    [Gro20] Jacob Gross

    doi:10.1142/S0129167X94000024. [Gro20] Jacob Gross. The homology of moduli stacks of complexes, 2020, 1907.03269. URL https://arxiv. org/abs/1907.03269. [GS24] Chandranandan Gangopadhyay and Ronnie Sebastian. Picard groups of some Quot schemes. Int. Math. Res. Not. IMRN , (11):9194–9217,

    arXiv 2020

  5. [14]

    [Hol04] Yogish I

    doi:10.1093/imrn/rnae028. [Hol04] Yogish I. Holla. Counting maximal subbundles via Gromov-Witten invariants. Math. Ann. , 328(1- 2):121–133,

    doi:10.1093/imrn/rnae028

  6. [22]

    [MO07a] Alina Marian and Dragos Oprea

    doi:10.1007/s00222-006-0013-2. [MO07a] Alina Marian and Dragos Oprea. Counts of maps to Grassmannians and intersections on the moduli space of bundles. J. Differential Geom. , 76(1):155–175,

    doi:10.1007/s00222-006-0013-2

  7. [23]

    [MO07b] Alina Marian and Dragos Oprea

    URL http://projecteuclid.org/ euclid.jdg/1180135668. [MO07b] Alina Marian and Dragos Oprea. The level-rank duality for non-abelian theta functions. Invent. Math., 168(2):225–247,

    work page arXiv

  8. [24]

    [MO07c] Alina Marian and Dragos Oprea

    doi:10.1007/s00222-006-0032-z. [MO07c] Alina Marian and Dragos Oprea. Virtual intersections on the Quot scheme and Vafa-Intriligator formulas. Duke Math. J. , 136(1):81–113,

    doi:10.1007/s00222-006-0032-z

Show all 31 references
  1. [25]

    [MO10] Alina Marian and Dragos Oprea

    doi:10.1215/S0012-7094-07-13613-5. [MO10] Alina Marian and Dragos Oprea. GL Verlinde numbers and the Grassmann TQFT. Port. Math. , 67(2):181–210,

    doi

  2. [28]

    [PR03] Mihnea Popa and Mike Roth

    doi:10.2140/gt.2021.25.3425. [PR03] Mihnea Popa and Mike Roth. Stable maps and Quot schemes. Invent. Math. , 152(3):625–663,

    2021 doi

  3. [31]

    [Wit91] Edward Witten

    doi:10.1016/j.ansens.2007.05.001. [Wit91] Edward Witten. Two-dimensional gravity and intersection theory on moduli space. In Surveys in differential geometry (Cambridge, MA,

    2007 doi

  4. [1986]

    [BR21] Sjoerd V

    doi:10.1073/pnas.83.10.3068. [BR21] Sjoerd V. Beentjes and Andrea T. Ricolfi. Virtual counts on Quot schemes and the higher rank local DT/PT correspondence. Math. Res. Lett. , 28(4):967–1032,

    doi

  5. [1991]

    [Joy19] Dominic Joyce

    doi:10.1142/S0217732391004097. [Joy19] Dominic Joyce. Ringel–hall style lie algebra structures on the homology of moduli spaces,

    doi

  6. [1994]

    [BF97] K

    doi:10.1142/S0129167X94000401. [BF97] K. Behrend and B. Fantechi. The intrinsic normal cone. Invent. Math. , 128(1):45–88,

    doi

  7. [1996]

    [Ber94] Aaron Bertram

    doi:10.1090/S0894-0347-96-00190-7. [Ber94] Aaron Bertram. Towards a Schubert calculus for maps from a Riemann surface to a Grassmannian. Internat. J. Math. , 5(6):811–825,

    doi

  8. [1997]

    [Bla16] Anthony Blanc

    doi:10.1007/s002220050136. [Bla16] Anthony Blanc. Topological K-theory of complex noncommutative spaces. Compos. Math. , 152(3):489–555,

    doi

  9. [2003]

    [Ric20] Andrea T

    doi:10.1007/s00222-002-0279-y. [Ric20] Andrea T. Ricolfi. Virtual classes and virtual motives of Quot schemes on threefolds. Adv. Math. , 369:107182, 32,

    doi

  10. [2004]

    [Int91] Kenneth Intriligator

    doi:10.1007/s00208-003-0475-0. [Int91] Kenneth Intriligator. Fusion residues. Modern Phys. Lett. A , 6(38):3543–3556,

    doi

  11. [2007]

    [Mar25] Alina Marian

    doi:10.1515/CRELLE.2007.066. [Mar25] Alina Marian. The Segre-Verlinde correspondence for the moduli space of stable bundles on a curve. Comm. Math. Phys. , 406(1):Paper No. 5, 14,

    2007 doi

  12. [2009]

    [OP21] Dragos Oprea and Rahul Pandharipande

    doi:10.1090/pspum/080.1/2483941. [OP21] Dragos Oprea and Rahul Pandharipande. Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics. Geom. Topol., 25(7):3425–3505,

    doi

  13. [2010]

    [Mor25] Miguel Moreira

    doi:10.4171/PM/1864. [Mor25] Miguel Moreira. On the intersection theory of moduli spaces of parabolic bundles, 2025, 2503.08898. URL https://arxiv.org/abs/2503.08898. [OP09] A. Okounkov and R. Pandharipande. Gromov-Witten theory, Hurwitz numbers, and matrix models. In Algebrai...

    2025 arXiv

  14. [2012]

    [Kon92] Maxim Kontsevich

    doi:10.1090/S0065-9266-2011-00630-1. [Kon92] Maxim Kontsevich. Intersection theory on the moduli space of curves and the matrix Airy function. Comm. Math. Phys. , 147(1):1–23,

    2011 doi

  15. [2016]

    [BLM24] Arkadij Bojko, Woonam Lim, and Miguel Moreira

    doi:10.1112/S0010437X15007617. [BLM24] Arkadij Bojko, Woonam Lim, and Miguel Moreira. Virasoro constraints for moduli of sheaves and vertex algebras. Invent. Math. , 236(1):387–476,

    doi

  16. [2018]

    JOYCE’S INV ARIANT AND VIRASORO CONSTRAINTS FOR QUOT SCHEMES 39 [LM24] Woonam Lim and Miguel Moreira

    doi:10.2140/pjm.2018.294.123. JOYCE’S INV ARIANT AND VIRASORO CONSTRAINTS FOR QUOT SCHEMES 39 [LM24] Woonam Lim and Miguel Moreira. Virasoro constraints and representations for quiver moduli spaces, 2024, 2403.13982. URL https://arxiv.org/abs/2403.13982. [Mar07] Alina Marian. ...

    2018 arXiv

  17. [2019]

    [Joy21] Dominic Joyce

    URL https://people.maths.ox.ac.uk/joyce/hall.pdf. [Joy21] Dominic Joyce. Enumerative invariants and wall-crossing formulae in abelian categories, 2021, 2111.04694. URL https://arxiv.org/abs/2111.04694. [JS12] Dominic Joyce and Yinan Song. A theory of generalized Donaldson-Thom...

    2021 arXiv

  18. [2020]

    [TV07] Bertrand Toën and Michel Vaquié

    doi:10.1016/j.aim.2020.107182. [TV07] Bertrand Toën and Michel Vaquié. Moduli of objects in dg-categories. Ann. Sci. École Norm. Sup. (4), 40(3):387–444,

    2020

  19. [2021]

    [BDW96] Aaron Bertram, Georgios Daskalopoulos, and Richard Wentworth

    doi:10.1016/j.geomphys.2021.104154. [BDW96] Aaron Bertram, Georgios Daskalopoulos, and Richard Wentworth. Gromov invariants for holomor- phic maps from Riemann surfaces to Grassmannians. J. Amer. Math. Soc. , 9(2):529–571,

    2021

  20. [2022]

    [GP94] Oscar García-Prada

    doi:10.3842/SIGMA.2022.068. [GP94] Oscar García-Prada. Dimensional reduction of stable bundles, vortices and stable pairs. Internat. J. Math. , 5(1):1–52,

    2022 doi

  21. [2023]

    [EHX97] Tohru Eguchi, Kentaro Hori, and Chuan-Sheng Xiong

    doi:10.1016/j.aim.2023.109334. [EHX97] Tohru Eguchi, Kentaro Hori, and Chuan-Sheng Xiong. Quantum cohomology and Virasoro algebra. Phys. Lett. B , 402(1-2):71–80,

    2023

  22. [2024]

    [Boj24] Arkadij Bojko

    doi:10.1007/s00222-024-01245-5. [Boj24] Arkadij Bojko. Universal virasoro constraints for quivers with relations, 2024, 2310.18311. URL https://arxiv.org/abs/2310.18311. [Boj25] Arkadij Bojko. Wall-crossing for calabi-yau fourfolds: framework, tools, and applications, 2025, 25...

    2024 arXiv

  23. [2025]

    [Mir07] Maryam Mirzakhani

    doi:10.1007/s00220-024-05171-8. [Mir07] Maryam Mirzakhani. Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces. Invent. Math. , 167(1):179–222,

    doi

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