Virasoro constraints for moduli of sheaves and vertex algebras. (English) Zbl 07813242
This important paper proves Virasoro constraints for the moduli spaces of (1) slope semistable bundles on smooth projective curves, (2) slope or Gieseker semistable torsion-free sheaves on smooth projective surfaces \(S\) with \(h^{1,0}(S)=h^{2,0}(S)=0\), and (3) slope semistable one-dimensional sheaves on smooth projective surfaces with \(h^{1,0}(S)=h^{2,0}(S)=0\) under some assumption on wall-crossing formulas (Theorem A). The important point is the method of the proof: The sheaf theoretic Virasoro constraint is rephrased in terms of primary states for a natural conformal vector in Joyce’s vertex algebra (Theorem B). This implies that the Virasoro constraints are preserved under wall-crossing (Theorem C), and the cases (1)–(3) can be reduced to the rank one case.
As reviewed in §1.4 and §4 of this paper, Joyce’s vertex algebra was introduced to study the wall-crossing of (higher) moduli stacks \(\mathcal{M}_X\) of semistable objects in the derived category of sheaves on a smooth projective variety \(X\). Joyce introduced a vertex algebra structure on the homology of \(\mathcal{M}_X\) (with shifted grading) in the Ringel-Hall style: The vertex operators \(Y(-,z)\) is defined in terms of the map \(\Sigma\colon \mathcal{M}_X \times \mathcal{M}_X \to \mathcal{M}_X\) taking direct sums and a perfect complex \(\Theta = \mathrm{Ext}^\vee \oplus \sigma^*\mathrm{Ext}\) on \(\mathcal{M}_X \times \mathcal{M}_X\) (see §4.1 for the detail). As explained in §4.2, Joyce’s vertex algebra considered in this paper is isomorphic to a lattice vertex algebra associated to the \(\mathbb{Z}/2\mathbb{Z}\)-graded vector space \(\Lambda = K^0(X) \oplus K^1 (X)\) and the Euler paring. The standard construction gives a conformal vector of the vertex algebra, and it is shown in §4.4 (proof of Theorem 1.9) that the Virasoro constraint is equivalent to that the virtual fundamental class of the moduli is a primary state in the vertex algebra.
The rank reduction mentioned above is executed in §5 via wall-crossing, In each of the 3 cases, the paper considers the moduli spaces of Bradlow pairs depending on a stability parameter \(t>0\). In the small range of \(t\) the Bradlow moduli is a projective bundle over the original moduli, and in the large range the Bradlow moduli is easier to understand. The point is that the Bradlow moduli also satisfy the pair analogue of the Virasoro constraint (Theorem 1.11, proved in §5 and §6), by which one can reduce the rank.
As reviewed in §1.4 and §4 of this paper, Joyce’s vertex algebra was introduced to study the wall-crossing of (higher) moduli stacks \(\mathcal{M}_X\) of semistable objects in the derived category of sheaves on a smooth projective variety \(X\). Joyce introduced a vertex algebra structure on the homology of \(\mathcal{M}_X\) (with shifted grading) in the Ringel-Hall style: The vertex operators \(Y(-,z)\) is defined in terms of the map \(\Sigma\colon \mathcal{M}_X \times \mathcal{M}_X \to \mathcal{M}_X\) taking direct sums and a perfect complex \(\Theta = \mathrm{Ext}^\vee \oplus \sigma^*\mathrm{Ext}\) on \(\mathcal{M}_X \times \mathcal{M}_X\) (see §4.1 for the detail). As explained in §4.2, Joyce’s vertex algebra considered in this paper is isomorphic to a lattice vertex algebra associated to the \(\mathbb{Z}/2\mathbb{Z}\)-graded vector space \(\Lambda = K^0(X) \oplus K^1 (X)\) and the Euler paring. The standard construction gives a conformal vector of the vertex algebra, and it is shown in §4.4 (proof of Theorem 1.9) that the Virasoro constraint is equivalent to that the virtual fundamental class of the moduli is a primary state in the vertex algebra.
The rank reduction mentioned above is executed in §5 via wall-crossing, In each of the 3 cases, the paper considers the moduli spaces of Bradlow pairs depending on a stability parameter \(t>0\). In the small range of \(t\) the Bradlow moduli is a projective bundle over the original moduli, and in the large range the Bradlow moduli is easier to understand. The point is that the Bradlow moduli also satisfy the pair analogue of the Virasoro constraint (Theorem 1.11, proved in §5 and §6), by which one can reduce the rank.
Reviewer: Shintarou Yanagida (Nagoya)
MSC:
| 14F08 | Derived categories of sheaves, dg categories, and related constructions in algebraic geometry |
| 14H60 | Vector bundles on curves and their moduli |
| 14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |
| 17B68 | Virasoro and related algebras |
| 17B69 | Vertex operators; vertex operator algebras and related structures |
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