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Noncommutative instantons and twistor transform. (English) Zbl 0989.81127

One-to-one correspondences between framed bundles on a noncommutative \(\mathbb{P}^2\), certain complexes of sheaves on a noncommutative \(\mathbb{P}^3\), and the modified ADHM data of N. Nekrasov and A. Schwarz [Commun. Math. Phys. 198, 689-703 (1998; Zbl 0923.58062)] are given. The method is based on noncommutative algebraic geometry [cf. Yu. I. Manin, “Quantum groups and Noncommutative geometry”, Montreal: Universite de Montréal, centre de Recherches, Mathematiques (CRM) (1988; Zbl 0724.17006)], and use noncommutative twistor transform. It is also shown that the moduli space of framed bundles on the noncommutative \(\mathbb{P}^2\) has a hyperkähler structure and as a complex manifold, is obtained from the corresponding commutative moduli space by a rotation of complex structure.
The outline of the paper is as follows: In section 1, physical motivation of the problem is explained, and noncommutative \(\mathbb{R}^4\) is defined by the Wigner-Moyal product \[ (f*g)(x)= \lim_{y\to x} \exp\left(\frac 12 \hbar \theta_{ij} {\partial^2\over \partial x_i\partial x_j}\right) f(x)g(y), \] where \(\theta\) is a purely imaginary matrix later assumed to have the form \[ \theta= \sqrt{-1}\begin{pmatrix} 0 & a & 0 & 0\\ -a & 0 & 0 & 0\\ 0 & 0 & 0 & b\\ 0 & 0 & -b & 0\end{pmatrix},\quad a+b=1, \] This space appeared in the study of low-energy limit of string theory [N. Seiberg and E. Witten, J. High Energy Phys. 1999, No. 9, Paper No. 32, 93 p., electronic only (1999; Zbl 0957.81085)]. The ADHM construction is reviewed in section 2. Modified ADHM data of Nekrasov and Schwarz are modify ADHM data as follows: \[ [B_1,B_2]+ IJ=0,\;[B_1,B_1^†] +[B_2,B_2^†]+ II^†-J^ † J=-2\hbar 1_{k\times k}. \] Then main result of the paper is summarized as follows:
1. Algebraic bundles on a noncommutative deformation of \(\mathbb{P}^2\) with \(c_2=k\) and a fixed trivialization on the line at infinity.
2. Deformed ADHM data of Nekrasov and Schwarz modulo natural \(U(k)\) action.
3. Certain complexes of sheaves on a noncommutative deformation of \(\mathbb{P}^3\) satisfying reality conditions.
In section 3, geometry of noncommutative varieties is reviewed from the categorical viewpoint. Then noncommutative \(\mathbb{C}^4_\hbar\), noncommutative 4-dimensional quadric \(\mathbb{Q}^4_\hbar\), and noncommutative \(\mathbb{P}^2_\hbar\) and \(\mathbb{P}^3_\hbar\) are introduced. Cohomological properties of sheaves on \(\mathbb{P}^2_\hbar\) and \(\mathbb{P}^3_\hbar\) are studied in section 4 and 5. Bundles on noncommutative projective spaces are defined to be locally coherent sheaves. They are characterized by cohomological properties. Bundles on \(\mathbb{P}^2_\hbar\) are studied in section 6 and the moduli space of noncommutative instanton is shown to have the same description as the Barth’s description of the commutative case [W. Barth, Invent. Math. 42, 63-91 (1997; Zbl 0386.14005)]. The noncommutative \(\mathbb{R}^4_\hbar\) and sphere \(\mathbb{S}^4_\hbar\) are defined by introducing real structures on \(\mathbb{C}^4_\hbar\) and \(\mathbb{Q}^4_\hbar\). The noncommutative real projective space \(\mathbb{P}^3_\hbar (\mathbb{R})\) is similarly defined. Then the noncommutative Penrose map \(\Pi:\mathbb{P}^3_\hbar (\mathbb{R})\to \mathbb{S}^4_\hbar\) is explicitly defined in section 7. The above mentioned relation between commutative and noncommutative moduli spaces of instantons is also proved in this section (Theorem 7.1). The authors say, that this gives a geometrical interpretation of H. Nakajima’s results [H. Nakajima, Ann. Math. (2) 145, No. 2, 379-388 (1997; Zbl 0915.14001); “Lectures on Hilbert schemes of points on surfaces”. University Lecture Series 18. Providence, RI: American Mathematical Society (1999; Zbl 0949.14001)]. Noncommutative Grassmann variety \(G_R(k;V)\) is defined by using a Yang-Baxter operator \(R\) on \(V^*\). Then noncommutative twistor transform is treated as the derived functor of the ordinary twistor transform, and the correspondence between deformed ADHM data and noncommutative instanton bundle is derived (Section 8). Noncommutative differential forms are also defined in section 8 and conjectures similar to the commutative case, the modified ADHM construction of the instanton connection can be interpreted in terms of noncommutative twistor transform.
The noncommutative deformation of \(\mathbb{R}^4\) treated in this paper is related to the Wigner-Weyl product which is the only deformation known to arise in string theory. In section 9 the possibility to extend the results of this paper to another deformation is discussed. The Wigner-Weyl product \(f*g\) may not be defined for arbitrary smooth functions \(f,g\). In the last section 10, applying distribution theory, the *-product is defined on the space of smooth functions with polynomially bounded derivatives.

MSC:

81T75 Noncommutative geometry methods in quantum field theory
81R25 Spinor and twistor methods applied to problems in quantum theory
58B34 Noncommutative geometry (à la Connes)
32L99 Holomorphic fiber spaces
58D27 Moduli problems for differential geometric structures
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81S10 Geometry and quantization, symplectic methods
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
46L55 Noncommutative dynamical systems
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