A new mathematical theorem gives a broad class of quasimap spaces a single global description: the fixed-evaluation quasimap moduli stack for a Nakajima variety is identified with a critical-locus stack, meaning the part of a representation space selected by the critical-point equations of an explicit function. The statement covers any Nakajima variety X, any chosen point p in X and degree data d. In algebraic-geometric terms, the two stacks are isomorphic under the construction.
A critical-point model
The construction begins on the quiver representation space R(v,w;d), where it defines the explicit function W_p. Stable quiver representations are handled through a stack quotient by the non-reductive group GL(U) ⋉ Hom(U,V), and W_p descends to that quotient. This gives the critical-locus description a concrete set of equations and a specified quotient structure.
Building the correspondence
To build the map from quasimaps to this quiver model, the paper defines auxiliary bundles from higher direct images. For each vertex i, it uses R1 p_* (V_i tensor q^* O_P1(-1)) and R1 p_* (V_i tensor q^* O_P1(-2)). These cohomological constructions provide the bridge between the bundles in a quasimap family and the quiver data used on the critical-locus side.
A key geometric step defines V_i as the kernel of ζ_i − zπ_i. The paper states that this kernel is a vector bundle exactly when the pointwise stability condition holds. Stability is therefore the condition that makes the required family of bundles available in the construction.
The paper also gives a rank formula for these cohomology bundles: for each vertex i and any integer k greater than zero, R1 p_* (V_i tensor q^* O_P1(-k)) has rank d_i + (k−1)v_i. The formula makes the dependence of the auxiliary geometry on the degree and quiver data explicit.
The quotient comes with a caveat
The proof checks the correspondence in both directions. It constructs maps between quasimap families and the critical-locus quiver data, and reports that the two composites send families to uniquely isomorphic families. That verification completes the proof of the main isomorphism.
One qualification concerns the quotient. The non-reductive quotient by GL(U) ⋉ Hom(U,V) is the main global presentation. An equivalent quotient by GL(U) alone becomes available only after further choices depending on p and auxiliary data, and the paper explicitly describes it as noncanonical. The simpler quotient is therefore a choice-dependent reformulation, not a canonical global construction over X.
A notable special case
The note also records a narrower identification. When X is T*(GL_n/P) and the evaluation point lies in the zero section, the critical locus is identified with a handsaw quiver variety. The statement is tied to that specialization and does not automatically extend to every Nakajima variety.
The theorem’s boundaries
The introduction links the critical-locus picture to a natural derived enhancement of the quasimap space carrying a (−1)-shifted symplectic structure. Further details of that derived structure are deferred to a companion paper, so the supplied note does not provide the full account of it.
This is a mathematical preprint, not an empirical study. The supplied document is arXiv:2608.25938v1, dated 26 Aug 2026, and it reports no sample, statistical analysis or numerical comparison. Its evidence is a theorem about moduli spaces and quiver data, with no human-population findings. The result is stated for quasimaps from the specified source curve P1, leaving open how the construction behaves for other source curves.
Paper data and sources
Original title: Quasimaps to Nakajima varieties as critical loci
Authors: Spencer Tamagni
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text