##
**Algebraic deformations of toric varieties. II: Noncommutative instantons.**
*(English)*
Zbl 1262.14002

The paper is dedicated to the construction and description of framed sheaves on noncommutative complex projective plane and moduli spaces of such sheaves. They are matched with the instantons on the noncommutative four-sphere.

In a previous paper [“Algebraic deformations of toric varieties I. General constructions”, arxiv:1001.1242], the authors developed a technique of noncommutative deformation of toric varieties. Their construction uses the combinatorial data of the corresponding fan, but deforms the torus to its noncommutative counterpart. In the present paper these constructions are used for noncommutative projective spaces \(\mathbb{CP}^{n}_{\theta}\). The authors prove that the categories of coherent sheaves on usual projective line \(\mathbb{CP}^1\) and on its noncommutative analogue \(\mathbb{CP}^1_{\theta}\) are naturally equivalent, so one can use standard tools of algebraic geometry (e.g. Birkhoff-Grothendieck theorem) to study sheaves on \(\mathbb{CP}^1_{\theta}\). For higher \(n\) this equivalence does not hold: for \(\theta\in \mathbb{C}\setminus \pi\mathbb{Z}\), categories of coherent sheaves on \(\mathbb{CP}^2\) and on \(\mathbb{CP}^2_{\theta}\) are not equivalent.

The moduli space \(\mathcal{M}_{\theta}(r,k)\) of rank \(r\) framed sheaves is defined as a moduli space of coherent torsion free sheaves \(E\) on \(\mathrm{Open}(\mathbb{CP}^2_{\theta})\) such that there exist isomorphisms \[ H^1(\mathbb{CP}^2_{\theta},E(-1))\simeq \mathbb{C}^k,\;i^{\bullet}(E)\simeq \mathbb{C}^r\otimes \mathcal{O}_{\mathbb{CP}^1_{\theta}}. \] Here \(i^{\bullet}\) denotes the restriction functor at infinity from \(\mathrm{coh}(\mathbb{CP}^2_{\theta})\) to \(\mathrm{coh}(\mathbb{CP}^1_{\theta})\). The moduli space \(\mathcal{M}_{\theta}(r,k)\) degenerates to the classical moduli space \(\mathcal{M}(r,k)\) of framed sheaves at \(\theta=0\).

In Theorem 7.4 the authors show that the corresponding moduli space \(\mathcal{M}_{\theta}(r,k)\) exists, and it is a smooth quasi-projective variety of dimension \(2rk\). The proof is based on the results of [T. A. Nevins and J. T. Stafford, Adv. Math. 210, No. 2, 405–478 (2007; Zbl 1116.14003)], where a more general notion of a noncommutative projective plane was used – a noncommutative deformation is parametrized by an elliptic curve \(\mathcal{E}\subset \mathbb{CP}^2\) and its automorphism. Since present paper uses the torus action, the corresponding degenerate projective curve \(\mathcal{E}\) is a union of three projective lines.

Another result of this paper is a noncommutative deformation of the so-called ADHM construction. In the celebrated paper [M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld and Yu. I. Manin, Phys. Lett. A 65, No. 3, 185–187 (1978; Zbl 0424.14004)], the moduli space \(\mathcal{M}(r,k)\) was realized as a quotient by \(\mathrm{GL}(k)\) of the space of quadruples \[ (B_1,B_2\in \mathrm{End}(\mathbb{C}^k), I\in \mathrm{Hom}(\mathbb{C}^r,\mathbb{C}^k), J\in \mathrm{Hom}(\mathbb{C}^k,\mathbb{C}^r)) \] satifying the equation \[ B_1B_2-B_2B_1+I\circ J=0 \] and a certain stability condition. In the noncommutative situation the defining equation is deformed to \[ B_1B_2-q^2B_2B_1+I\circ J=0, \] where \(q=\exp(\mathbf{i}\theta/2).\) The authors prove that the set of quadruples \((B_1,B_2,I,J)\) satisfying the deformed equation and a stability condition enjoys a free and proper action of \(\mathrm{GL}(k)\), and the corresponding quotient is isomorphic to \(\mathcal{M}_{\theta}(r,k)\). The authors also develop noncommutative version of the twistor transform and use it to relate noncommutative ADHM data with the solutions of the noncommutative anti-selfduality equations on \(S^4_{\theta}\).

It is worth to note that there are other approaches to noncommutative deformations of ADHM data, in particular, the one defined in [N. Nekrasov and A. Schwarz, Commun. Math. Phys. 198, No. 3, 689–703 (1998; Zbl 0923.58062)], where the deformed equation reads as \[ B_1B_2-B_2B_1+I\circ J=\zeta \] It might me interesting to compare and relate these two approaches.

In a previous paper [“Algebraic deformations of toric varieties I. General constructions”, arxiv:1001.1242], the authors developed a technique of noncommutative deformation of toric varieties. Their construction uses the combinatorial data of the corresponding fan, but deforms the torus to its noncommutative counterpart. In the present paper these constructions are used for noncommutative projective spaces \(\mathbb{CP}^{n}_{\theta}\). The authors prove that the categories of coherent sheaves on usual projective line \(\mathbb{CP}^1\) and on its noncommutative analogue \(\mathbb{CP}^1_{\theta}\) are naturally equivalent, so one can use standard tools of algebraic geometry (e.g. Birkhoff-Grothendieck theorem) to study sheaves on \(\mathbb{CP}^1_{\theta}\). For higher \(n\) this equivalence does not hold: for \(\theta\in \mathbb{C}\setminus \pi\mathbb{Z}\), categories of coherent sheaves on \(\mathbb{CP}^2\) and on \(\mathbb{CP}^2_{\theta}\) are not equivalent.

The moduli space \(\mathcal{M}_{\theta}(r,k)\) of rank \(r\) framed sheaves is defined as a moduli space of coherent torsion free sheaves \(E\) on \(\mathrm{Open}(\mathbb{CP}^2_{\theta})\) such that there exist isomorphisms \[ H^1(\mathbb{CP}^2_{\theta},E(-1))\simeq \mathbb{C}^k,\;i^{\bullet}(E)\simeq \mathbb{C}^r\otimes \mathcal{O}_{\mathbb{CP}^1_{\theta}}. \] Here \(i^{\bullet}\) denotes the restriction functor at infinity from \(\mathrm{coh}(\mathbb{CP}^2_{\theta})\) to \(\mathrm{coh}(\mathbb{CP}^1_{\theta})\). The moduli space \(\mathcal{M}_{\theta}(r,k)\) degenerates to the classical moduli space \(\mathcal{M}(r,k)\) of framed sheaves at \(\theta=0\).

In Theorem 7.4 the authors show that the corresponding moduli space \(\mathcal{M}_{\theta}(r,k)\) exists, and it is a smooth quasi-projective variety of dimension \(2rk\). The proof is based on the results of [T. A. Nevins and J. T. Stafford, Adv. Math. 210, No. 2, 405–478 (2007; Zbl 1116.14003)], where a more general notion of a noncommutative projective plane was used – a noncommutative deformation is parametrized by an elliptic curve \(\mathcal{E}\subset \mathbb{CP}^2\) and its automorphism. Since present paper uses the torus action, the corresponding degenerate projective curve \(\mathcal{E}\) is a union of three projective lines.

Another result of this paper is a noncommutative deformation of the so-called ADHM construction. In the celebrated paper [M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld and Yu. I. Manin, Phys. Lett. A 65, No. 3, 185–187 (1978; Zbl 0424.14004)], the moduli space \(\mathcal{M}(r,k)\) was realized as a quotient by \(\mathrm{GL}(k)\) of the space of quadruples \[ (B_1,B_2\in \mathrm{End}(\mathbb{C}^k), I\in \mathrm{Hom}(\mathbb{C}^r,\mathbb{C}^k), J\in \mathrm{Hom}(\mathbb{C}^k,\mathbb{C}^r)) \] satifying the equation \[ B_1B_2-B_2B_1+I\circ J=0 \] and a certain stability condition. In the noncommutative situation the defining equation is deformed to \[ B_1B_2-q^2B_2B_1+I\circ J=0, \] where \(q=\exp(\mathbf{i}\theta/2).\) The authors prove that the set of quadruples \((B_1,B_2,I,J)\) satisfying the deformed equation and a stability condition enjoys a free and proper action of \(\mathrm{GL}(k)\), and the corresponding quotient is isomorphic to \(\mathcal{M}_{\theta}(r,k)\). The authors also develop noncommutative version of the twistor transform and use it to relate noncommutative ADHM data with the solutions of the noncommutative anti-selfduality equations on \(S^4_{\theta}\).

It is worth to note that there are other approaches to noncommutative deformations of ADHM data, in particular, the one defined in [N. Nekrasov and A. Schwarz, Commun. Math. Phys. 198, No. 3, 689–703 (1998; Zbl 0923.58062)], where the deformed equation reads as \[ B_1B_2-B_2B_1+I\circ J=\zeta \] It might me interesting to compare and relate these two approaches.

Reviewer: Eugene Gorsky (Stony Brook)

### MSC:

14A22 | Noncommutative algebraic geometry |

14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |

16S38 | Rings arising from noncommutative algebraic geometry |