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Hemisphere Partition Functions and Stability Across GLSM Phases

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
In the GLSM setting, is the hemisphere partition function $Z_{D^2}:K_0(D)\to\mathbb C$ a central-charge function for a Bridgeland stability condition on $D$, and for the grade-restriction subcategory $W_n$ are the phase quotient categories $D_{\zeta\gg0}$ and $D_{\zeta\ll0}$ related to $W_n$ by the conjectured commuting equivalences?

Formulation references: [S1], [S2].

Short English statement

In the GLSM setting, is the hemisphere partition function $Z_{D^2}:K_0(D)\to\mathbb C$ a central-charge function for a Bridgeland stability condition on $D$, and for the grade-restriction subcategory $W_n$ are the phase quotient categories $D_{\zeta\gg0}$ and $D_{\zeta\ll0}$ related to $W_n$ by the conjectured commuting equivalences?

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Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Phys-35-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-35

The theorem produces a stability condition with the quintic Gepner symmetry and normalized eigencharge. This page instead asks that the specific hemisphere partition function define a stability condition and that the grade-restriction categories give the specified phase equivalences. The manuscript supplies neither that partition-function identification nor those commuting equivalences. It is related stability information, rather than a claimed partial solution of these GLSM assertions.

Hemisphere Partition Functions and Stability Across GLSM Phases

OpenAI's claim

Every smooth complex quintic threefold is claimed to carry a numerical Bridgeland stability condition with the normalized Gepner charge and phase shift $2/5$ under the specified autoequivalence. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For every smooth quintic hypersurface $X\subset\mathbb P^4_{\mathbb C}$, define on $D^b(\operatorname{Coh}X)$ the autoequivalence \[ \Phi=T_{\mathcal O_X}\circ(-\otimes\mathcal O_X(1)),\qquad T_{\mathcal O_X}(E)=\operatorname{Cone}\left(R\operatorname{Hom}(\mathcal O_X,E)\otimes\mathcal O_X\to E\right). \] OpenAI claims a numerical Bridgeland stability condition $(Z,\mathcal P)$ with support property on the full real numerical Grothendieck group such that \[ Z(\Phi E)=e^{2\pi i/5}Z(E),\quad \Phi(\mathcal P(\varphi))=\mathcal P(\varphi+2/5),\quad Z(\mathcal O_x)=-1 \] for every object $E$, real phase $\varphi$ and closed point $x$. Support means a norm and constant $C$ bound the numerical class of every semistable object by $C|Z(E)|$.

Outline of the argument

Manuscript 1 (family 055, 2026-09-24): Two independent perturbations of the second slope tilt control rank and first Chern character before semistability is defined. The charge bounds the remaining numerical coordinates, enabling finite refinements and Harder--Narasimhan filtrations even for a nondiscrete charge image; compatibility of the tilted aisles supplies the Gepner rotation.

Formal verification

Family 055 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

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