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arxiv: 2208.03173 · v2 · pith:QN6LOL2Nnew · submitted 2022-08-05 · 🧮 math.RT

Partial compactification of stability manifolds via massless semistable objects

Nathan Broomhead , David Pauksztello , David Ploog , Jon Woolf This is my paper

Pith reviewed 2026-05-25 09:06 UTC · model grok-4.3

classification 🧮 math.RT
keywords Bridgeland stability conditionslax stability conditionsmassless objectsVerdier quotientstriangulated categoriesdeformation equivalencewall-and-chamber structureGrothendieck group
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The pith

Identifying deformation-equivalent lax stability conditions with fixed charge produces a space stratified by quotient stability spaces.

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

The paper extends the space of Bridgeland stability conditions on a triangulated category by allowing semistable objects to have mass zero while retaining a phase. These massless objects form a thick subcategory, and the construction induces a Bridgeland stability condition on the quotient category. It then identifies lax stability conditions that are deformation-equivalent with fixed charge. The resulting space is stratified by the stability spaces of Verdier quotients of the original category by thick subcategories of massless objects. The approach is illustrated explicitly when the Grothendieck group has rank two, where the extended spaces relate to the wall-and-chamber structure.

Core claim

Lax stability conditions permit semistable objects of mass zero but with a defined phase. The subcategory of massless objects is thick and induces a Bridgeland stability condition on the quotient. The space obtained by identifying lax stability conditions that are deformation-equivalent with fixed charge is stratified by the stability spaces of the Verdier quotients of the triangulated category by thick subcategories of massless objects.

What carries the argument

The deformation-equivalence identification of lax stability conditions with fixed charge, which produces a stratification by stability spaces of Verdier quotients by thick subcategories of massless objects.

If this is right

  • When the Grothendieck group has rank two the extended spaces admit explicit descriptions.
  • The construction relates the extended spaces to the wall-and-chamber structure of the original stability space.
  • Deformations of lax stability conditions produce well-defined identifications that yield the stratified space.

Where Pith is reading between the lines

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

  • The stratification may supply a systematic way to approach boundary components where objects become massless.
  • The method could be used to glue stability spaces along different quotient categories arising from different massless subcategories.
  • Explicit low-rank examples suggest the construction may simplify computation of invariants attached to the stability space.

Load-bearing premise

The subcategory of massless objects is thick and induces a Bridgeland stability condition on the quotient category.

What would settle it

A triangulated category together with a lax stability condition in which the massless objects do not form a thick subcategory or in which the induced data on the quotient fails to define a Bridgeland stability condition.

Figures

Figures reproduced from arXiv: 2208.03173 by David Pauksztello, David Ploog, Jon Woolf, Nathan Broomhead.

Figure 1
Figure 1. Figure 1: The definition of support propagation from σ = (P, Z) ∈ StabL (C, N) im￾poses two conditions on the charge of τ = (Q, W). These are illustrated schematically above. We require two conditions because ||·||σ is only a semi-norm. When Z −W is in the subspace Hom(Λ/ΛN, C) on which it restricts to a norm the conditions coincide, and agree with the condition ||W − Z||µN(σ) < sin(πε) for the quotient stability co… view at source ↗
Figure 2
Figure 2. Figure 2: Illustrations of StabQ(C) ∗/C. In each case this is a contractible non-compact Riemann surface. We depict it as a disk, which is holomorphically accurate in the hyperbolic case (Ginzburg algebra Γ2A2, shown orange). In the parabolic cases (purple) it is only topologically accurate, but has the advantage that we can more easily visualise the partial compactification. This is obtained by adding logarithmic s… view at source ↗
Figure 3
Figure 3. Figure 3: The image of Stab(A2)/C → ∆2 : σ 7→ (λmσ(s), λmσ(e), λmσ(t)) where λ = mσ(s) + mσ(e) + mσ(t) is shaded red. The chamber of the stability space in which s, e and t are stable is mapped homeomorphically to the interior. The chamber in which only s and t are stable, together with its bounding wall, are projected down onto the edge x0 − x1 + x2 = 0, and similarly for the other two chambers. The three boundary … view at source ↗
read the original abstract

We introduce two extensions of the space of Bridgeland stability conditions of a triangulated category. First we consider lax stability conditions where semistable objects are allowed to have mass zero but still have a phase. The subcategory of massless objects is thick and there is an induced Bridgeland stability on the quotient category. We study deformations of lax stability conditions. Second we consider the space arising by identifying lax stability conditions which are deformation-equivalent with fixed charge. This second space is stratified by stability spaces of Verdier quotients of the triangulated category by thick subcategories of massless objects. We illustrate our results through examples in which the Grothendieck group has rank $2$. For these, our extended stability spaces can be explicitly described and related to the wall-and-chamber structure of the stability space.

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

1 major / 0 minor

Summary. The paper introduces lax stability conditions on a triangulated category, allowing semistable objects of mass zero while retaining a phase. It proves that the subcategory of massless objects is thick and induces a Bridgeland stability condition on the Verdier quotient. Deformations of lax stability conditions are studied, and the space obtained by quotienting lax stability conditions by deformation equivalence at fixed central charge is claimed to be stratified by the ordinary stability spaces of these Verdier quotients by thick massless subcategories. The results are illustrated explicitly for categories with rank-2 Grothendieck group, relating the extended spaces to wall-and-chamber structures.

Significance. If the stratification claim holds, the construction yields a partial compactification of Bridgeland stability manifolds with a natural stratification by quotient stability spaces. This extends existing stability theory in a manner that could clarify global structure and degeneration phenomena, particularly in examples where explicit descriptions are feasible. The rank-2 illustrations provide concrete verification of the wall-and-chamber relation.

major comments (1)
  1. [Abstract / construction of the stratified space] The central stratification claim (that the deformation-equivalence quotient at fixed charge is stratified by stability spaces of fixed Verdier quotients C/<massless>) requires that the thick subcategory of massless objects remains constant along connected components of deformation paths with fixed central charge. The abstract asserts thickness and induction of a Bridgeland condition on each quotient but provides no argument that the massless subcategory cannot change when new objects become massless during deformation; if such crossings occur, distinct quotients would be glued, undermining the asserted stratification by fixed quotients.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this key point about the stratification. We address the concern directly below.

read point-by-point responses

  1. Referee: [Abstract / construction of the stratified space] The central stratification claim (that the deformation-equivalence quotient at fixed charge is stratified by stability spaces of fixed Verdier quotients C/<massless>) requires that the thick subcategory of massless objects remains constant along connected components of deformation paths with fixed central charge. The abstract asserts thickness and induction of a Bridgeland condition on each quotient but provides no argument that the massless subcategory cannot change when new objects become massless during deformation; if such crossings occur, distinct quotients would be glued, undermining the asserted stratification by fixed quotients.

    Authors: We agree that an explicit argument is needed to ensure the massless thick subcategory is constant along connected components of deformation paths with fixed central charge, so that the quotient space is indeed stratified by the stability spaces of the corresponding fixed Verdier quotients. Since the central charge Z is fixed, only objects in ker(Z) can possibly be massless. We will add a lemma establishing that, along any continuous path of lax stability conditions with this fixed Z, the thick subcategory generated by the massless semistable objects remains constant on connected components. The proof relies on the upper semicontinuity of the semistable loci and the fact that any change in the generated thick subcategory would require a discrete wall-crossing event that alters the quotient, which is already accounted for in the stratification. With this addition the claimed stratification holds. We will also revise the abstract to reference the new lemma. revision: yes

Circularity Check

0 steps flagged

No circularity; constructions are self-contained extensions of stability theory

full rationale

The paper introduces lax stability conditions and a quotient space by deformation equivalence at fixed charge, asserting that the massless subcategory is thick with an induced Bridgeland stability condition on the quotient. These are presented as definitions and theorems within the new framework rather than reductions to prior fitted inputs or self-citations. No equations or steps reduce a claimed prediction to its own construction by definition, and the stratification claim follows from the stated properties without invoking load-bearing self-citations or ansatzes. The derivation remains independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no information on free parameters, background axioms, or new postulated entities is supplied.

pith-pipeline@v0.9.0 · 5669 in / 1172 out tokens · 35945 ms · 2026-05-25T09:06:27.666914+00:00 · methodology

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    A weak stability condition is built on the categorical resolution of the Kuznetsov component of a singular quadric threefold, producing a Bridgeland stability condition on its derived category via blow-up geometry and...

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