Wall-crossing formulas via spectral networks

Johannes Horn; Martin M\"oller

REVIEW 2 major objections 6 minor 15 references

Wall-crossing formulas via spectral networks

T0 review · 2 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read Spectral networks alone prove the wall-crossing formula

desk verdict A serious and significant proof of wall-crossing via spectral networks, but its load-bearing local detour enumeration is verified by pictures rather than algebra and should be independently checked. read the letter →

arxiv 2508.07727 v1 pith:S5D574QZ submitted 2025-08-11 math.AG math.GT

Johannes Horn , Martin Möller This is my paper · ORCID

classification math.AGmath.GT MSC 30F3032G1514N3537F34

keywords quadraticdifferentialsspectralnetworkswall-crossingformulaBPSinvariantspathliftingA0-laminationshat-homologysaddleconnections

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 proves the Kontsevich–Soibelman wall-crossing formula entirely through quadratic differentials and their spectral networks, with no input from Donaldson–Thomas theory. It shows that the product of BPS automorphisms over the active directions in a sector is covariantly constant as the quadratic differential moves through moduli space, as long as the sector's boundary rays never hit a saddle connection. The proof handles infinite-area surfaces, including spiral domains where trajectories accumulate, by working in a profinite completion truncated by central charge. An intermediate result extends the path-lifting rule to A0-laminations and shows their lifts generate the hat-homology lattice, which is what makes the BPS automorphism act on the right completed module.

What carries the argument

The completed signed path groupoid $\widehat{\Pi}_\Delta$ and its homological shadow $\widehat{H}_\Delta$: formal linear combinations of path (or homology) classes truncated by the absolute value of the central charge $Z$, which makes infinite detour sums converge even when trajectories are dense. The path-lifting rule $F(\wp,\theta)$ adds to the trivial lift of $\wp$ all elementary detours that follow rays of the spectral network $W_\theta$ to a zero, loop around it, and return on the other sheet; the extension to A0-laminations with a preferred-sheet rule yields enough elements to approximate every hat-homology class. The wall-crossing automorphism $K_\theta$ acts on $\widehat{H}_\Delta$ b

What would settle it

Work out a concrete rank-one configuration (e.g. a saddle connection between two simple zeros on a genus-two surface) and explicitly enumerate all ±-admissible rays and their intersection numbers with the BPS cycle $L(\hat\gamma_0)$, either by hand or by computer; if any admissible detour family is missing from Figures 11–14 or any intersection number in Lemmas 6.5–6.7 is wrong, then the identity $F^+(\wp,\theta_0)=K_{\theta_0}F^-(\wp,\theta_0)$ fails for a single-crossing path. Alternatively, compute $S_\Delta$ around a closed loop in the space of framed quadratic differentials crossing a ran

Watch

Extended reading notes

Core claim

Theorem 1.1: for an infinite-area half-translation surface $(X,q)$ whose singularity signature has no simple poles, the BPS automorphism $S_\Delta$ is covariantly constant along any path of homology-framed quadratic differentials whose boundary rays stay non-active. The proof constructs a path-lifting rule $F(\wp,\theta)$ that assigns to each path (or A0-lamination) a formal sum of lifts of all possible detours along the spectral network, valued in a completed signed path groupoid. At a rank-one active direction $\theta_0$ the one-sided limits satisfy $F^+(\wp,\theta_0)=K_{\theta_0}F^-(\wp,\theta_0)$, where $K_{\theta_0}$ is the BPS automorphism built from saddle connections and ring-domain

Load-bearing premise

The local case analysis of all possible ±-admissible detour paths near a rank-one saddle connection is complete and sign-correct; it is verified by inspecting finitely many figures rather than by a systematic algebraic enumeration, and a missing configuration or sign error would break the wall-crossing identity $F^+=K_\theta F^-$.

Editorial extensions

If this is right

The Kontsevich–Soibelman formula is a theorem about quadratic differentials alone, not a corollary of motivic Donaldson–Thomas theory (in the infinite-area case).
The path-lifting functions $F(\wp,\theta)$ are well-defined and homotopy-invariant even when the spectral network has spiral domains, resolving convergence questions left open in earlier treatments.
The extension to A0-laminations shows that lifts of laminations are topologically dense in the hat-homology lattice, with supports controlled inside a fixed cone translate.
The one-sided limits $F^\pm(\wp,\theta_0)$ are genuine limits in the Z-adic topology, and the wall-crossing relation holds for every rank-one direction, for both paths and laminations.
In finite area the theorem is conditional on an approximation statement; under unique ergodicity the non-active-boundary condition forces the path to be a Teichmüller geodesic ray, making the statement nearly void.

Reading between the lines

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

The same completed-groupoid machinery could produce an abelianization of $SL(2,\mathbb{C})$-local systems along spectral networks with dense trajectories, as the authors plan; if successful, this would supply a missing link between non-abelianization and Joyce structures.
A formal, computer-checked enumeration of the admissible detours in Lemmas 6.5–6.7 would remove the main residual doubt in the proof and could be adapted to higher-rank spectral networks, where junctions create trajectories born at interior points and new infinite families.
The paper's exclusion of simple poles looks technical rather than structural; if the approximation statement of Proposition 5.11 can be extended to simple poles, the same argument would give the formula in finite area, where the non-active-boundary condition is very restrictive.
The proof of Proposition 6.10 by induction on Cantor–Bendixon rank suggests that wall-crossing could be organized purely by the accumulation structure of active directions, independent of the specific dynamics.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear Colleague,

You should know this about arXiv:2508.07727: if correct, it is the first fully self-contained proof of the Kontsevich–Soibelman wall-crossing formula inside quadratic differential theory, with the convergence issues from [GMN13b, Question 7] settled for infinite-area surfaces. The interesting new pieces are the profinite convergence framework for path liftings, the extension of the lifting rule to A0-laminations (which generates the hat-homology lattice), and the proof that one-sided limits at a rank-one wall are related by the BPS automorphism. The paper is honest about its boundaries: Theorem 1.1 excludes simple poles, and the finite-area statement (Prop. 1.2) is explicitly conditional. The abstract, however, promises more than the theorem delivers, and should be narrowed.

The main theorem is proven by reducing the global wall-crossing to a local statement, Proposition 6.8, about detours at a single intersection with the spectral network. That reduction is elegant. The soft spot is exactly where the stress-test says it is: the local statement rests on the case-by-case enumeration of ±-admissible detours in Lemmas 6.5–6.7, and those lemmas are verified by inspection of Figures 11–14. The subsequent algebra is sign-sensitive—for instance in Case (4b) the prefactor (1−[eγ0])−1(1−[eγ0]^2)=1+[eγ0] depends on both intersection numbers being right. A missing detour family or a sign error there would break F+=KF−, and the lists are not otherwise derived. There is no reason to suspect such an error, but this is a load-bearing verification-by-picture, and independent checking is needed.

The paper is not circular: the BPS invariants are computed from saddle connections and ring domains via the geometry, matching known values, and there are no fitted parameters. The citation pattern is fine. My overall take is close to the reader's: the architecture is sound and the result is significant, but the proof is not yet fully auditable at the crucial local step.

Who is this for? Anyone working on spectral networks, quadratic differentials and DT theory. It deserves a serious referee, but the referee should be asked to check the detour enumeration with care, and the authors should be asked to either replace figure-inspection by a systematic enumeration or at least record the verification in a way that does not depend on the reader seeing what the authors see. I would engage with it, but not rely on it for downstream results until the local case analysis is tightened.

Best.

Referee Report

2 major / 6 minor

Summary. The paper gives a self-contained proof of the Kontsevich–Soibelman wall-crossing formula for holomorphic quadratic differentials, working directly with spectral networks and path-lifting rules rather than motivic Donaldson–Thomas theory. The main theorem (Theorem 1.1) states that for infinite-area quadratic differentials without simple poles, the BPS automorphism S_Δ is covariantly constant along paths of homology-framed differentials whose boundary rays stay saddle-free, even if intermediate surfaces are not rank one. The proof constructs a completed signed path groupoid, defines path-lifting functions F(℘,θ) with convergence results, extends them to A0-laminations in order to generate the hat-homology lattice, and then compares the one-sided limits of the lifting at rank-one walls via a local detour computation. A finite-area version is stated only under an additional approximation hypothesis (Proposition 1.2), and the authors explicitly note that the finite-area wall-crossing statement is nearly void under non-active boundary rays.

Significance. If correct, this is a substantial contribution: it provides a proof of the Kontsevich–Soibelman wall-crossing formula entirely within the geometry of quadratic differentials, resolves convergence questions for spectral networks with dense trajectories and spiral domains, and confirms the richness of the GMN path-lifting rules. The intermediate result that lifts of A0-laminations generate the hat-homology lattice is of independent interest. The paper is carefully structured and does not rely on fitted parameters or external DT input. The main risk to rigor is that the central wall-crossing comparison is reduced to a finite enumeration of detour configurations that is verified by inspection of figures rather than by a systematic proof; this is the load-bearing gap discussed below.

major comments (2)

[§6.2, Lemmas 6.5–6.7 and Proposition 6.8] The completeness and sign correctness of the local detour enumeration is load-bearing for Theorem 6.3, and hence for Theorem 1.1. The proofs of Lemmas 6.5–6.7 are justified 'by inspection' of Figures 11–14, and Proposition 6.8 applies them case-by-case. A missing admissible ray family or a sign error in an intersection number would change the BPS automorphism K and break F^+=KF^-; for instance, Eq. (53) uses both ⟨L(γ0),D⟩ and ⟨L(2γ0),D⟩, and a single sign flip would replace the prefactor (1+[eγ0]) by (1−[eγ0]). I request that the inspection argument be replaced by a systematic enumeration: a complete list of ±-admissible ray types for each configuration, their turn sequences, and all intersection numbers, or a machine-checked verification. As written, the central proof is not fully verifiable without trusting the completeness of the pictures.
[§6.1, Lemma 6.2 and Proposition 6.1] The proof of the one-sided limits relies on Lemma 6.2, which asserts a bijection between detours in nearby directions and ±-admissible detours in the limit. While the rectangle argument for a single ray is plausible, the passage from rays to full detours involves collisions of starting points and the American/British driving rule, which is only discussed heuristically via Figure 10. Since Proposition 6.1 supplies the limits in Theorem 1.3(iv) and is used in Proposition 6.10, please expand this step into a precise statement of the bijection at the level of detour sequences, including multiplicities and the handling of colliding starting points.

minor comments (6)

[Introduction vs. §6.3] The order of the product in the definition of S_Δ is inconsistent: the introduction says the product is taken in clockwise order, while Eq. (54) says counterclockwise order. Since the factors K_θ need not commute in general, please harmonize the convention.
[Proposition 1.2] Typo: 'suppose the there are sufficiently many based path lifting functions' should read 'suppose there are sufficiently many based path lifting functions'.
[§6.2, proof of Case (4b)] The text refers to an 'irrational torus end', but the classification in Lemma 6.4 only speaks of a spiral domain whose interior is a torus; the word 'irrational' is unexplained and potentially misleading.
[Proposition 5.3] In the statement, 'defines an element F(℘,θ)' should be 'F(L,θ)' for the lamination lift.
[Lemma 5.10(iv)] There is a typographical error in the formula: 'c([γ_e]^{-1}])' has an extra closing bracket. Please also clarify the value of c in terms of the base point.
[Lemma 5.6] The product notation with upper and lower indices (e.g. 'Π_{i=1}^k ...') is hard to read; please clarify the summation indices and the ranges of i,j in the displayed formulas.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the wall-crossing theorem is derived from independent geometric definitions and explicit detour computations.

full rationale

The paper's derivation chain is self-contained in the relevant sense. The path lifting rule F is defined in Section 4 from spectral-network detours, independently of the wall-crossing automorphism K. BPS data are introduced geometrically in Section 6.2: the cycles L(γ) are specified case-by-case in (43), the automorphism K is then defined by (44), and the BPS invariants Ω(γ) are read off from these geometric cycles in (46). The central comparison F+(℘, θ) = K F−(℘, θ) is not assumed: it is proved in Theorem 6.3 via the single-intersection verification in Proposition 6.8, whose computations use intersection numbers of detours with the independently defined cycles L(γ). No fitted parameter is renamed as a prediction, and no equation in the proof reduces to its own input. The classification of rank one directions is imported from external work [BS15, Hai24, KW24], and the path-lifting framework from [GMN13b]; these citations are load-bearing background, not self-citations by the present authors, and they do not presuppose the wall-crossing equality. The main real risk flagged by the paper is that Lemmas 6.5–6.7 are verified by inspection of Figures 11–14 rather than by an algebraic enumeration; a missing detour or sign error there would break the proof. That is a correctness or completeness risk, not circularity, because the lemmas compute rather than define the objects they compare.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The central claim rests on standard quadratic differential theory, the classification and CBR statements cited above, and the paper's own new constructions (path-lifting, BPS cycles), none of which are fitted to data.

assumptions (3)

domain assumption Classification of rank one directions into the finite list in Lemma 6.4
Used in Section 6.2 to define BPS cycles and invariants and to set up the case analysis; imported from [BS15], [Hai24] and [KW24].
domain assumption The set of saddle connection directions SC(X,q) has finite Cantor-Bendixon rank
Used in the inductive proof of Proposition 6.10; cited from [Aul18, Theorem 1.5].
standard math Support property for BPS invariants: only finitely many active classes with bounded central charge
Used in Lemma 6.9 to show S_Δ is computable by finitely many factors at each truncation; standard consequence of discreteness of periods.

invented entities (1)

BPS cycles L(bγ0) associated to rank one directions
purpose: Defined in Section 6.2 (Equation 43) to define the BPS automorphism Kθ by intersection with cycles in ring-domain cases; they refine the homology class [bγ0] by including boundary components needed for non-closed paths.
The cycles are internal to the proof. Their homology classes reproduce the known BPS invariants Ω(γ) in (46), but the cycles themselves are not independently observable.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Wall-crossing formulas via spectral networks." pith.science (2026). https://pith.science/paper/S5D574QZ

@misc{pith2026250807727,
  author       = {Pith},
  title        = {Pith review of: Wall-crossing formulas via spectral networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S5D574QZ}},
  note         = {Machine review of arXiv:2508.07727}
}

abstract

We give a self-contained proof of the Kontsevich-Soibelman wall-crossing formula entirely in the scope of quadratic differentials without relying on input from DT theory. Our approach is based on path-lifting rules for spectral networks introduced by Gaiotto, Moore and Neitzke. We provide a framework to justify the convergence of the path liftings, including the cases with spiral domains. In particular, we define path lifting rules for spectral networks associated to holomorphic quadratic differentials. As an intermediate step in the proof of the wall-crossing formula, we show that upon extending the path lifting rules to $\mathcal{A}_0$-laminations we generate the hat-homology lattice.

Figures

Figures reproduced from arXiv: 2508.07727 by the authors.

**Figure 1.** The spectral network at a saddle connection between two distinct zeros. In red a +-admissible ray that is not minusadmissible. In purple a minus-admissible ray that is not +- admissible. (Branch cut as zig-zag line.) is the number of zeros) where the trajectory emanating from ℘± hits a zero. (The reader may compare with the construction of suspensions over interval exchange transformations e.g. in [Yoc10] for a vis… view at source ↗

**Figure 2.** Examples of special detours appearing in F(℘, θ+): The red detour (of the path ℘) follows a +-admissible ray, which does not turn at the first zero it encounters. The purple detour (of the path q) takes two detours at the same point of the spectral network, following the American driving rule. (Superpose the two sheets for the mnemonic.) In the saddle-free case detour rules prohibited multiple detours at the same p… view at source ↗

**Figure 3.** The lift F(℘, θ) for the two homotopic paths ℘ and q around a zero: On top and left all five lifts of ℘ and on the right all three lifts of q. Note that ℘1 and the detour on top in the middle cancels each other. ℘1 ℘2 ℘ [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: The lift F(℘, θ) for a path ℘ crossing a single trajectory twice. Note that the orange and the purple detour differ up to smooth homotopy only by their winding number. The main technical tool in several proofs is the truncated spectral network Wθ,L depending on a direc…

**Figure 5.** Figure 5: Homotoping an arc along the boundary: The two arcs of a lamination ℘ and q contribute homotopic detours satisfying the preferred sheet rule. For each of them the convergence proof as in Proposition 4.1 applies, since an infinite number of detours may occur only in spi…

**Figure 6.** Figure 6: The algorithm of Proposition 5.2 applied to e0 with sign +1 (left) and −1 (right) will satisfy the preferred sheet rule. Hence, there are no extra detours contributing to the lift of q in comparison with ℘. The case of ℘ and q differing by a homotopy along their init…

**Figure 7.** Figure 7: Left: One outer vertex is a hole. Right: Both outer vertices are holes. Choose a directed base point ze on ℘e close enough to and pointing away from an incoming boundary component. Then F•(℘e, θ) = [#”γ e + #”ε ] +X k i=1 [PITH_FULL_IMAGE:figures/full_fig_p031_7.png]

**Figure 8.** Figure 8: Left: One inner vertex is a hole. Right: Both inner vertices are holes. iii) One inner vertex is a hole, negative sign: Consider an edge e such that exactly one inner vertex is a hole as depicted on the left in [PITH_FULL_IMAGE:figures/full_fig_p032_8.png]

**Figure 9.** Figure 9: A zero-free rectangle enclosing +-admissible rays approximating the horizontal ray. The purple and the red path are obviously homotopic. q2 q1 × × [PITH_FULL_IMAGE:figures/full_fig_p034_9.png]

**Figure 10.** Figure 10: Two consecutive detours (in purple) at two points of the spectral network. In the limit as θ → θ0 they start over the same point of X. The limit is the detour with American driving rule from [PITH_FULL_IMAGE:figures/full_fig_p034_10.png]

**Figure 11.** Figure 11: A simple saddle connection with a path ℘ crossing various the spectral network at various rays: +-admissible (drawn in slope ϵ) and minus-admissible detours (in slope −ϵ) detours. Only one of the two sheets is drawn for each path. the definition on generators gives a …

**Figure 12.** Figure 12: A simple cylinder and a path crossing the spectral network in a ray: +-admissible (drawn in slope ϵ) and minusadmissible detours (in slope −ϵ) detours. None of the detours comes with cylinder twists here. a1 a1 e1 f1 e1 f1 ℘1 a2 a2 e2 f2 e2 f2 ℘2 [PITH_FULL_IMAGE:fi…

**Figure 13.** Figure 13: A simple cylinder and a path crossing the spectral network in a saddle connection : +-admissible (drawn in slope ϵ) and minus-admissible detours (in slope −ϵ) detours, two representatives from each family of cylinder twists. (ii) Suppose y ∈ Wθ does not belong to a s…

**Figure 14.** Figure 14: A toral ring domain: +-admissible (drawn in slope ϵ) and minus-admissible detours (in slope −ϵ) detours, one representative from each family of cylinder twists (ii) Suppose y ∈ Wθ does not belong to a saddle connection, but to a critical trajectory (see [PITH_FULL_…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 12 canonical work pages

[1]

[AB20] D. G. L. Allegretti and T. Bridgeland. The monodromy of mero- morphic projective structures . Trans. Am. Math. Soc. 373.9 (2020), pp. 6321–6367. 46 JOHANNES HORN, MARTIN M ¨OLLER [ADGZZ20] A. Aggarwal, V. Delecroix, ´E. Goujard, P. Zograf, and A. Zorich. Conjectural large genus asymptotics of Masur-Veech volumes and of area Siegel-Veech constants o...

work page 2020
[5]

Springer-Verlag, Berlin, 1984, pp

Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer-Verlag, Berlin, 1984, pp. xii+184. [Wen07] R. A. Wentworth. Energy of harmonic maps and Gardiner’s for- mula. In: In the tradition of Ahlfors-Bers, IV. Proceedings of the 3rd Ahlfors-Bers colloquium, Ann Arbor, May 19–22, 2005 . Providence, RI: American Mathematical Society (AMS), 2007, pp. 22...

work page 1984
[10]

Clay Math. Proc. Amer. Math. Soc., Providence, RI, 2010, pp. 1–69

work page 2010
[13]

Meinhardt

[Mei17] S. Meinhardt. An introduction to (motivic) Donaldson-Thomas the- ory. Confluentes Math. 9.2 (2017), pp. 101–158. [MS91] H. Masur and J. Smillie. Hausdorff dimension of sets of nonergodic measured foliations. Ann. of Math. (2) 134.3 (1991), pp. 455–543. [MZ08a] H. Masur and A. Zorich. Multiple saddle connections on flat surfaces and the principal b...

work page 2017
[179]

[FG07] V. V. Fock and A. B. Goncharov. Dual Teichm¨ uller and lamination spaces. In: Handbook of Teichm¨ uller theory. Volume I. Z¨ urich: Euro- pean Mathematical Society (EMS), 2007, pp. 647–684. [GM91] F. P. Gardiner and H. Masur. Extremal length geometry of Teich- m¨ uller space. Complex Variables, Theo. Appl. 16.2-3 (1991), pp. 209–

work page 2007
[237]

Gaiotto, G

[GMN13a] D. Gaiotto, G. W. Moore, and A. Neitzke. Framed BPS states. Adv. Theor. Math. Phys. 17.2 (2013), pp. 241–397. [GMN13b] D. Gaiotto, G. W. Moore, and A. Neitzke. Spectral networks. Ann. Henri Poincar´ e 14.7 (2013), pp. 1643–1731. [GMN13c] D. Gaiotto, G. W. Moore, and A. Neitzke. Wall-crossing, Hitchin sys- tems, and the WKB approximation . Adv. Ma...

work page 2013
[278]

Donaldson-Thomas theory for categories of homological dimension one with potential

[CM20] J. Chaika and H. Masur. The set of non-uniquely ergodic d-IETs has Hausdorff codimension 1/2. Invent. Math. 222.3 (2020), pp. 749–832. [CMZ18] D. Chen, M. M¨ oller, and D. Zagier. Quasimodularity and large genus limits of Siegel-Veech constants . J. Amer. Math. Soc. 31.4 (2018), pp. 1059–1163. [CS23] A. Calderon and N. Salter. Framed mapping class ...

work page Pith review arXiv 2020
[403]

W ALL-CROSSING 47 [Gou15] E. Goujard. Siegel-Veech constants for strata of moduli spaces of qua- dratic differentials. Geom. Funct. Anal. 25.5 (2015), pp. 1440–1492. [Hai24] F. Haiden. 3-D Calabi-Yau categories for Teichm¨ uller theory. Duke Math. J. 173.2 (2024), pp. 277–346. [HN16] L. Hollands and A. Neitzke. Spectral networks and Fenchel-Nielsen coordi...

arXiv 2015

Show all 15 references

[1995]

Kontsevich and Y

[KS08] M. Kontsevich and Y. Soibelman. Stability structures, motivic Don- aldson–Thomas invariants and cluster transformations . Preprint, arXiv:0811.2435

arXiv
[2008]

Kontsevich and Y

[KS24] M. Kontsevich and Y. Soibelman. Holomorphic Floer theory I: ex- ponential integrals in finite and infinite dimensions . Preprint, arXiv: 2402.07343

arXiv
[2015]

Davison and S

[DM20] B. Davison and S. Meinhardt. Cohomological Donaldson-Thomas the- ory of a quiver with potential and quantum enveloping algebras . In- vent. Math. 221.3 (2020), pp. 777–871. [EMZ03] A. Eskin, H. Masur, and A. Zorich. Moduli spaces of Abelian differen- tials: the principa...

2020
[2019]

2021, pp. 1–66. [Bri22] T. Bridgeland. Joyce structures on spaces of quadratic differentials . Preprint, arXiv:2203.17148

2021 arXiv
[2021]

[Joh80] D. Johnson. Spin structures and quadratic forms on surfaces. J. Lond. Math. Soc., II. Ser. 22 (1980), pp. 365–373. [Kea77] M. Keane. Non-ergodic interval exchange transformations . Israel J. Math. 26.2 (1977), pp. 188–196. [Kec95] A. S. Kechris. Classical descriptive s...

1980
[2022]

Bridgeland and I

[BS15] T. Bridgeland and I. Smith. Quadratic differentials as stability con- ditions. Publ. Math., Inst. Hautes ´Etud. Sci. 121 (2015), pp. 155–

2015
[2024]

Kidwai and N

[KW24] O. Kidwai and N. Williams. Donaldson-Thomas invariants for the Bridgeland-Smith correspondence. Preprint, arXiv:2404.00700

arXiv

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

[AB20] D. G. L. Allegretti and T. Bridgeland. The monodromy of mero- morphic projective structures . Trans. Am. Math. Soc. 373.9 (2020), pp. 6321–6367. 46 JOHANNES HORN, MARTIN M ¨OLLER [ADGZZ20] A. Aggarwal, V. Delecroix, ´E. Goujard, P. Zograf, and A. Zorich. Conjectural large genus asymptotics of Masur-Veech volumes and of area Siegel-Veech constants o...

work page 2020

[2] [5]

Springer-Verlag, Berlin, 1984, pp

Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer-Verlag, Berlin, 1984, pp. xii+184. [Wen07] R. A. Wentworth. Energy of harmonic maps and Gardiner’s for- mula. In: In the tradition of Ahlfors-Bers, IV. Proceedings of the 3rd Ahlfors-Bers colloquium, Ann Arbor, May 19–22, 2005 . Providence, RI: American Mathematical Society (AMS), 2007, pp. 22...

work page 1984

[3] [10]

Clay Math. Proc. Amer. Math. Soc., Providence, RI, 2010, pp. 1–69

work page 2010

[4] [13]

Meinhardt

[Mei17] S. Meinhardt. An introduction to (motivic) Donaldson-Thomas the- ory. Confluentes Math. 9.2 (2017), pp. 101–158. [MS91] H. Masur and J. Smillie. Hausdorff dimension of sets of nonergodic measured foliations. Ann. of Math. (2) 134.3 (1991), pp. 455–543. [MZ08a] H. Masur and A. Zorich. Multiple saddle connections on flat surfaces and the principal b...

work page 2017

[5] [179]

[FG07] V. V. Fock and A. B. Goncharov. Dual Teichm¨ uller and lamination spaces. In: Handbook of Teichm¨ uller theory. Volume I. Z¨ urich: Euro- pean Mathematical Society (EMS), 2007, pp. 647–684. [GM91] F. P. Gardiner and H. Masur. Extremal length geometry of Teich- m¨ uller space. Complex Variables, Theo. Appl. 16.2-3 (1991), pp. 209–

work page 2007

[6] [237]

Gaiotto, G

[GMN13a] D. Gaiotto, G. W. Moore, and A. Neitzke. Framed BPS states. Adv. Theor. Math. Phys. 17.2 (2013), pp. 241–397. [GMN13b] D. Gaiotto, G. W. Moore, and A. Neitzke. Spectral networks. Ann. Henri Poincar´ e 14.7 (2013), pp. 1643–1731. [GMN13c] D. Gaiotto, G. W. Moore, and A. Neitzke. Wall-crossing, Hitchin sys- tems, and the WKB approximation . Adv. Ma...

work page 2013

[7] [278]

Donaldson-Thomas theory for categories of homological dimension one with potential

[CM20] J. Chaika and H. Masur. The set of non-uniquely ergodic d-IETs has Hausdorff codimension 1/2. Invent. Math. 222.3 (2020), pp. 749–832. [CMZ18] D. Chen, M. M¨ oller, and D. Zagier. Quasimodularity and large genus limits of Siegel-Veech constants . J. Amer. Math. Soc. 31.4 (2018), pp. 1059–1163. [CS23] A. Calderon and N. Salter. Framed mapping class ...

work page Pith review arXiv 2020

[8] [403]

W ALL-CROSSING 47 [Gou15] E. Goujard. Siegel-Veech constants for strata of moduli spaces of qua- dratic differentials. Geom. Funct. Anal. 25.5 (2015), pp. 1440–1492. [Hai24] F. Haiden. 3-D Calabi-Yau categories for Teichm¨ uller theory. Duke Math. J. 173.2 (2024), pp. 277–346. [HN16] L. Hollands and A. Neitzke. Spectral networks and Fenchel-Nielsen coordi...

arXiv 2015