Universal Structures in C-Linear Enumerative Invariant Theories

Abstract

An enumerative invariant theory in algebraic geometry, differential geometry, or representation theory, is the study of invariants which 'count' τ -(semi)stable objects E with fixed topological invariants E = α in some geometric problem, by means of a virtual class [M ss α (τ )] virt in some homology theory for the moduli spaces M st α (τ ) ⊆ M ss α (τ ) of τ -(semi)stable objects. Examples include Mochizuki's invariants counting coherent sheaves on surfaces, Donaldson-Thomas type invariants counting coherent sheaves on Calabi-Yau 3and 4-folds and Fano 3-folds, and Donaldson invariants of 4-manifolds. We make conjectures on new universal structures common to many enumerative invariant theories. Any such theory has two moduli spaces M, M pl , where the second author (see https://people. maths.ox.ac.uk/ ~joyce/hall.pdf) gives H * (M) the structure of a graded vertex algebra, and H * M pl a graded Lie algebra, closely related to H * (M). The virtual classes [M ss α (τ )] virt take values in H * M pl . In most such theories, defining [M ss α (τ )] virt when M st α (τ ) ̸ = M ss α (τ ) (in gauge theory, when the moduli space contains reducibles) is a difficult problem. We conjecture that there is a natural way to define invariants [M ss α (τ )] inv in homology over Q, with [M ss α (τ )] inv = [M ss α (τ )] virt when M st α (τ ) = M ss α (τ ), and that these invariants satisfy a universal wall-crossing formula under change of stability condition τ , written using the Lie bracket on H * M pl . We prove our conjectures for moduli spaces of representations of quivers without oriented cycles. Versions of our conjectures in algebraic geometry using Behrend-Fantechi virtual classes are proved in the sequel [arXiv:2111.04694].

Virasoro constraints for moduli of sheaves and vertex algebras

“…3 Vertex algebras are representation theoretic objects introduced by Borcherds [10] and they give an axiomatization of conformal field theories in two dimensions [5]. The Lie bracket operation induced from the sheaf-theoretic vertex algebras was used to describe wall-crossing of virtual fundamental classes counting semistable objects, as conjectured in [21] and proven in many cases by Joyce [31]. For surfaces, these wall-crossing formulae are related to the work of Mochizuki [46] where the formulae are presented without vertex algebras.…”

“…D. Joyce recently introduced a vertex algebra and a closely related Lie algebra associated to the derived category D b (X) [21,26,31]. Joyce proposes to use his Lie algebra to study wall-crossing formulae for moduli of sheaves (or, more generally, moduli of semistable objects in a C-linear abelian or triangulated category).…”

“…Remark 2. 21 The part of S k corresponding to the Künneth component 1 ⊗ pt is equal to −rS pt k where r = ch 0 (α) = rk(α). When h 0,q (X) = 0 for q > 0 the appearance of the operator S k is already explained in Remark 2.17 as being related to the ptnormalized universal sheaf.…”

“…Assumption 5.8 Let À be the abelian category of pairs U ⊗ O X → F , where F is a sheaf with dim(F ) ≤ 1 and U a vector space. 21 Then the assumptions in [31, Assumption 5.1-5.3] hold for this category and a surface S such that h 0,2 (S) = 0 with C pe ( À) = {(e, F ) : e = 0, 1 and dim(F An important point in the rank reduction induction that we will use is that there are no contributions of rank 0 objects in the wall-crossing formula above. Lemma 5.10 Let α be such that r(α) > 0.…”

“…Our work relies in a fundamental way on the vertex algebra that D. Joyce recently introduced [21,26,31] to study the wall-crossing of moduli spaces of sheaves. We explain how the Virasoro constraints can be naturally formulated in this language.…”