Deformed GKM conjecture for r-spin enumerative invariants

Consider the deformed generalized Kontsevich matrix model (GKM) with potential given by equation, and allow a shift of its variables. Deformed GKM conjecture. The shifted deformed GKM describes interesting enumerative-geometry invariants in the $r$ -spin case. This is a broad proposed connection between the deformed GKM and $r$ -spin enumerative geometry; the source gives no precise invariant or resolution status.

Sources & referencesView supporting material

Primary source

Alexander Alexandrov, “KP integrability of triple Hodge integrals. II. Generalized Kontsevich matrix model”, arXiv:2009.10961 (2021).

matrix-models r-spin-theory

1 week ago

Progress summary

Refreshed 2 days ago

Partially solved

A 2026 paper confirms the conjecture in a precise cohomological formulation, while its broader wording remains unspecified.

The deformed GKM conjecture proposes that a shifted deformed matrix model encodes meaningful $r$ -spin enumerative geometry. The available source identifies Alexandrov’s conjecture but gives no earlier precise formulation or classical partial results.

2026 CohFT confirmation

For admissible deformed potentials $V_r^\epsilon(z)$ , the paper constructs an $R$ -matrix and $T$ -vector whose action on Witten’s $r$ -spin theory defines a deformed CohFT, and proves the corresponding partition-function identity. It explicitly states that this confirms Alexandrov’s conjecture at the level of CohFTs, using KP integrability, the string equation, initial conditions, and tau-function uniqueness.

Current status (as of August 2026): The conjecture is confirmed for the stated CohFT formulation and admissible deformed potentials, while the original broad claim about unspecified “interesting” invariants is not separately resolved.

Sources

arxiv.org

No solutions have been posted yet.