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arxiv: 2506.14359 · v2 · submitted 2025-06-17 · 🧮 math.AG · hep-th

The refined local Donaldson-Thomas theory of curves

Sergej Monavari This is my paper

Pith reviewed 2026-05-19 09:39 UTC · model grok-4.3

classification 🧮 math.AG hep-th
keywords refined Donaldson-Thomas theorylocal curvesK-theoretic invariantsYoung diagramsequivariant vertexHilbert schemesDT/PT correspondencetopological strings
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The pith

The K-theoretically refined Donaldson-Thomas theory of local curves reduces to three universal series computed from the 1-leg K-theoretic equivariant vertex.

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

This paper solves the K-theoretically refined Donaldson-Thomas theory of local curves by avoiding degeneration and instead using direct localisation on the moduli space of sheaves. The refined partition function reduces to the equivariant intersection theory of skew nested Hilbert schemes on smooth projective curves. For any Young diagram, this intersection theory depends only on three universal series that the paper computes explicitly in terms of the 1-leg K-theoretic equivariant vertex. The results also yield a formula for the refined topological string partition function of local curves in the refined limit, and they establish the K-theoretic DT/PT correspondence for local curves in arbitrary genus.

Core claim

We solve the K-theoretically refined Donaldson-Thomas theory of local curves. Our results avoid degeneration techniques, but rather exploit direct localisation methods to reduce the refined Donaldson-Thomas partition function to the equivariant intersection theory of skew nested Hilbert schemes on smooth projective curves. We show that the latter is determined, for every Young diagram, by three universal series, which we compute in terms of the 1-leg K-theoretic equivariant vertex. In the refined limit, our results establish a formula for the refined topological string partition function of local curves proposed by Aganagic-Schaeffer. Analogous structural results hold for the refined Pandhar

What carries the argument

The reduction via direct localisation to the equivariant intersection theory of skew nested Hilbert schemes on smooth projective curves, which factors through three universal series independent of the curve and Young diagram.

If this is right

  • An explicit formula for the refined topological string partition function of local curves is obtained in the refined limit.
  • The K-theoretic DT/PT correspondence holds for local curves in arbitrary genus.
  • Analogous structural results, including determination by three universal series, apply to the refined Pandharipande-Thomas theory of local curves.
  • The explicit results on local curves are expected to serve as input for the refined GW/PT correspondence on all smooth Calabi-Yau threefolds.

Where Pith is reading between the lines

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

  • If the three-series reduction holds, similar factorisations might appear in refined theories for non-local Calabi-Yau threefolds once localisation techniques are extended.
  • The independence from the base curve suggests that the same universal series could organise refined invariants for families of curves or for higher-genus targets.
  • Verification on a few low-degree Young diagrams and low-genus curves would give an immediate numerical test of the claimed universality.

Load-bearing premise

Direct localisation on the moduli space of sheaves reduces the refined DT partition function exactly to the equivariant intersection theory of skew nested Hilbert schemes, and this theory factors completely through three universal series that do not depend on the choice of curve or Young diagram.

What would settle it

Compute the refined DT partition function directly for the total space of the canonical bundle over an elliptic curve with a specific Young diagram and check whether the result agrees with the expression built from the three universal series extracted from the 1-leg K-theoretic vertex.

Figures

Figures reproduced from arXiv: 2506.14359 by Sergej Monavari.

Figure 1
Figure 1. Figure 1: Respectively from the left, a Young diagram of size 8, a reversed plane partition of size 18 and a skew plane partition of size 22. 2.1.1. Socle. Let λ be a Young diagram. Recall from [28, Def. 2.16] that the socle of λ is the set of maximal elements of λ with respect to the partial order of λ, equivalently, Soc(λ) = { (i, j) ∈ λ | (i + 1, j),(i, j + 1) ∈/ λ } ⊂ λ, and the subsocle of λ is Subsoc(λ) = { (i… view at source ↗
Figure 2
Figure 2. Figure 2: We denote by "•" the (co)socle and by "×" the (co)subsocle of the Young diagram in [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The number of "•" on the right (resp. below) of □ is the arm (resp. leg) length of □. 2.2. Skew nested Hilbert schemes. Let X be a quasi-projective scheme, λ a Young diagram and m = (m□)□∈λ a reverse plane partition of shape λ. Recall that we defined in [43, Def. 2.2] the double nested Hilbert functor of points Hilbm(X) : Schop → Sets, T 7→ ( (Z□)□∈λ ⊂ X × T Z□ a T-flat closed subscheme with Z□|t 0-dimensi… view at source ↗
Figure 4
Figure 4. Figure 4: A 1-leg plane partition of normalised size 5, with asymptotic profile a Young diagram of size 1 [PITH_FULL_IMAGE:figures/full_fig_p031_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A 1-leg PT box arrangement of size 3, with asymptotic profile a Young diagram of size 3. In grey, the infinite leg. In red, the 3 boxes with negative x3-coordinate. Under the above correspondence it is immediate to verify that, given a reverse plane partition m, the vertex term vm reproduces precisely13 the normalised vertex term of [56, Sec. 4.6] of the corresponding 1-leg plane partition σ. As already re… view at source ↗
read the original abstract

We solve the $K$-theoretically refined Donaldson-Thomas theory of local curves. Our results avoid degeneration techniques, but rather exploit direct localisation methods to reduce the refined Donaldson-Thomas partition function to the equivariant intersection theory of skew nested Hilbert schemes on smooth projective curves. We show that the latter is determined, for every Young diagram, by three universal series, which we compute in terms of the 1-leg $K$-theoretic equivariant vertex. In the refined limit, our results establish a formula for the refined topological string partition function of local curves proposed by Aganagic-Schaeffer. In the second part, we show that analogous structural results hold for the refined Pandharipande-Thomas theory of local curves. As an application, we deduce the K-theoretic DT/PT correspondence for local curves in arbitrary genus, as conjectured by Nekrasov-Okounkov. Thanks to the recent machinery developed by Pardon, we expect our explicit results on local curves to play a key role towards the proof of the refined GW/PT conjectural correspondence of Brini-Schuler for all smooth Calabi-Yau threefolds.

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 / 2 minor

Summary. The paper claims to solve the K-theoretically refined Donaldson-Thomas theory of local curves by applying direct localisation methods to the moduli space of sheaves, thereby reducing the refined DT partition function exactly to the equivariant intersection theory of skew nested Hilbert schemes on smooth projective curves. It asserts that this intersection theory is determined, for every Young diagram, by three universal series that are computed explicitly in terms of the 1-leg K-theoretic equivariant vertex. Analogous structural results are stated for the refined Pandharipande-Thomas theory, yielding as an application the K-theoretic DT/PT correspondence for local curves in arbitrary genus (as conjectured by Nekrasov-Okounkov). In the refined limit the results are said to confirm a formula for the refined topological string partition function of local curves proposed by Aganagic-Schaeffer.

Significance. If the central reduction and universality statements hold, the work supplies explicit, degeneration-free formulas for refined DT and PT invariants of local curves and thereby advances several open conjectures in the field. The explicit computation of the three universal series from the 1-leg K-theoretic vertex, together with the application to the DT/PT correspondence, constitutes a concrete technical contribution that could serve as input for broader programs such as the refined GW/PT correspondence for Calabi-Yau threefolds via Pardon’s machinery.

major comments (2)
  1. [§3] §3 (Direct localisation step): The central claim that localisation on the moduli space of sheaves yields precisely the equivariant intersection theory of skew nested Hilbert schemes, with no residual curve-dependent or virtual corrections in the K-theoretic setting, is load-bearing for both the universality and the independence from degeneration techniques. The argument relies on the precise form of the virtual structure sheaf and the K-theoretic Euler class of the normal bundle to the fixed loci; any mismatch would introduce additional factors that depend on the curve or the Young diagram and would undermine the reduction to three universal series.
  2. [§4–5] §4–5 (Extraction of universal series): The assertion that the equivariant intersection theory on skew nested Hilbert schemes is completely determined by three universal series independent of the curve and of the Young diagram is extracted from the 1-leg K-theoretic vertex. While the vertex computation itself is standard, the manuscript must supply an explicit verification that no hidden curve- or diagram-dependent terms survive after the localisation; a low-degree example (e.g., the empty diagram or a single-box diagram on a genus-0 or genus-1 curve) would make this independence manifest.
minor comments (2)
  1. [Notation and definitions] The notation for the skew nested Hilbert schemes and the precise definition of the three universal series would benefit from a short diagram or a reference to the corresponding objects in the ordinary (non-K-theoretic) nested Hilbert scheme literature.
  2. [Refined limit discussion] A few typographical inconsistencies appear in the indexing of the Young diagrams when the series are specialised to the refined limit; these do not affect the logic but should be uniformised.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The major comments raise important points about the localization argument in §3 and the verification of the universal series in §4–5. We address each below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [§3] §3 (Direct localisation step): The central claim that localisation on the moduli space of sheaves yields precisely the equivariant intersection theory of skew nested Hilbert schemes, with no residual curve-dependent or virtual corrections in the K-theoretic setting, is load-bearing for both the universality and the independence from degeneration techniques. The argument relies on the precise form of the virtual structure sheaf and the K-theoretic Euler class of the normal bundle to the fixed loci; any mismatch would introduce additional factors that depend on the curve or the Young diagram and would undermine the reduction to three universal series.

    Authors: We appreciate the referee highlighting the load-bearing nature of this step. The localization formula in §3 is obtained by applying the K-theoretic localization theorem to the moduli space equipped with its virtual structure sheaf, which is defined via the K-theoretic Euler class of the obstruction theory. For the torus-fixed loci, identified with skew nested Hilbert schemes, the K-theoretic Euler class of the normal bundle to the fixed locus cancels exactly against the contributions arising from the virtual structure sheaf. This exact cancellation holds because the local curve is the total space of the canonical bundle over a smooth projective curve, and the obstruction theory is equivariant with respect to the torus action; no residual curve-dependent or Young-diagram-dependent factors appear. We will add a clarifying paragraph in §3 that spells out this cancellation step by step. revision: partial

  2. Referee: [§4–5] §4–5 (Extraction of universal series): The assertion that the equivariant intersection theory on skew nested Hilbert schemes is completely determined by three universal series independent of the curve and of the Young diagram is extracted from the 1-leg K-theoretic vertex. While the vertex computation itself is standard, the manuscript must supply an explicit verification that no hidden curve- or diagram-dependent terms survive after the localisation; a low-degree example (e.g., the empty diagram or a single-box diagram on a genus-0 or genus-1 curve) would make this independence manifest.

    Authors: We agree that an explicit low-degree verification would make the independence clearer. While the general argument in §4–5 shows that the equivariant intersection theory on the skew nested Hilbert schemes factors uniformly through the 1-leg K-theoretic vertex (whose computation is independent of the base curve and of the specific Young diagram), we will add a new subsection containing explicit computations for the empty diagram and the single-box diagram. These will be performed on both genus-0 and genus-1 curves, confirming that the resulting series coincide with the three universal series and contain no additional curve- or diagram-dependent terms. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from localisation to independent vertex extraction

full rationale

The paper reduces the K-theoretic refined DT partition function via direct localisation on the moduli space of sheaves to the equivariant intersection theory of skew nested Hilbert schemes on curves. It then determines this intersection theory, for each Young diagram, by three universal series explicitly computed from the 1-leg K-theoretic equivariant vertex. These steps rely on established equivariant vertex techniques and virtual structure sheaf contributions rather than redefining the target quantities in terms of themselves, fitting parameters to the output, or invoking self-citations as the sole justification for the reduction. The central claims remain independent of the final formulas and do not collapse by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard tools of equivariant algebraic geometry and K-theory whose validity is assumed from the literature; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • standard math Equivariant intersection theory on Hilbert schemes and nested Hilbert schemes is well-defined and can be computed via localisation.
    Invoked when reducing the DT partition function to intersection numbers on skew nested Hilbert schemes.
  • domain assumption The 1-leg K-theoretic equivariant vertex generates the universal series that control all higher-leg or higher-genus cases for local curves.
    Used to compute the three universal series for every Young diagram.

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Forward citations

Cited by 1 Pith paper

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  1. On the combinatorics of the refined 1-leg DT/PT correspondence

    math.CO 2026-03 unverdicted novelty 6.0

    New combinatorial proof of Bessenrodt's generating series relation for reversed and skew plane partitions, plus dual to Gansner's theorem, new hook identities, and Fock space interpretation.

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