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## Noncommutative instantons and twistor transform

Venue: | Commun. Math. Phys |

Citations: | 54 - 4 self |

### Citations

2981 | Noncommutative Geometry,
- Connes
- 1994
(Show Context)
Citation Context ...er-Moyal product. Leaving this question aside for a moment, 1 one can define the exterior differential calculus over A . Differential geometry of noncommutative spaces has been developed by A. Connes =-=[12]-=-. In our situation Connes’ general theory is greatly simplified. For example, the sheaf of 1-forms Ω 1 (A) is simply a bimodule A⊕4 (the relation of this definition with the general theory is explaine... |

794 | String Theory and Noncommutative Geometry.
- Seiberg, Witten
(Show Context)
Citation Context ...g theory, or equivalently4 ANTON KAPUSTIN, ALEXANDER KUZNETSOV, AND DMITRI ORLOV for large string tension. Recently, another kind of low-energy limit of string theory was discussed in the literature =-=[31]-=-. Consider a trivial U(r) -bundle on X = R4 with a connection A whose curvature FA is of the form 1 ⊗ f, , where 1 is the unit section of End(E) , and f is a constant nondegenerate 2-form. For small f... |

344 | Noncommutative Noetherian Rings, - McConnell, Robson - 1987 |

319 |
Lectures on Hilbert schemes of points on surfaces,
- Nakajima
- 1999
(Show Context)
Citation Context ...ion free sheaves on P2 is provided by the twistor variety P 3 � . This gives a geometrical interpretation of Nakajima’s results (the description of the moduli space M(r,0,k) by the deformed ADHM data =-=[27, 26]-=-). We will construct a hyperkähler manifold M parametrizing certain complexes on P 3 � which is isomorphic to M(r,0,k)34 ANTON KAPUSTIN, ALEXANDER KUZNETSOV, AND DMITRI ORLOV (which is also a hyperkä... |

306 |
Noncommutative geometry and matrix theory
- Connes, Douglas, et al.
- 1998
(Show Context)
Citation Context ...limit in which both the B -field and the string tension become infinite. The idea that D-branes in a nonzero B-field are described Yang-Mills theory on a noncommutative space was first put forward in =-=[13]-=- for the case of D-branes wrapped on tori. 1.5. Instanton equations on a noncommutative R 4 . The deformed R 4 that one obtains in the Seiberg-Witten limit is completely characterized by its algebra o... |

296 |
Vector bundles on complex projective spaces. Birkhauser,
- Okonek, Schneider, et al.
- 1980
(Show Context)
Citation Context ...−→ 0 for which the map n is an epimorphism and m is a monomorphism. (Note that there is another more restrictive definition of a monad, according to which the dual map (m) ∗ to be an epimorphism, see =-=[29]-=-). The coherent sheaf E = Ker(n)/Im(m) is called the cohomology of a monad. A morphism between two monads is a morphism of complexes. The following lemma is proved in [29](Lemma 4.1.3) in the commutat... |

293 |
Hyperkahler metrics and supersymmetry,
- Hitchin, Karlhede, et al.
- 1987
(Show Context)
Citation Context ... by (B1,B2,I,J) ↦→ (−B2 † ,B1 † , −J † ,I † ), and, moreover, it is a flat hyperkähler manifold (see [27]). The map µ = (µr,µc) is a hyperkähler moment map for the action of U(k) defined in (26) (see =-=[19]-=-). Since the action of U(k) on µ −1 c (0) ∩ µ −1 r (−2� · 1) is free, the quotient M = µ −1 c (0) ∩ µ −1 r (−2� · 1)/U(k) is a smooth hyperkähler manifold. This manifold parametrizes complexes (22) wi... |

293 |
Gauge field theory and complex geometry,
- Manin
- 1997
(Show Context)
Citation Context ...striction of its twistor transform to the sphere S4 coincides with the instanton bundle corresponding to these ADHM data. The instanton connection can also be reconstructed from the bundle on P3 (see =-=[4, 23]-=- for details). In this section we show that one can consider the noncommutative quadric introduced in section 3 as a noncommutative Grassmannian G(2;4) . We also construct a noncommutative flag variet... |

178 |
Noncommutative projective schemes,
- Artin, Zhang
- 1994
(Show Context)
Citation Context ...nerated graded right modules over this algebra14 ANTON KAPUSTIN, ALEXANDER KUZNETSOV, AND DMITRI ORLOV modulo torsion modules is identified with the category of coherent sheaves on this variety (see =-=[3]-=-, [24], [34]). A different approach to noncommutative geometry has been pursued by A. Connes [12]. 3.3. Noncommutative deformations of commutative varieties. Many important noncommutative varieties ar... |

176 |
Graded algebras of global dimension 3”,
- Artin, Schelter
- 1987
(Show Context)
Citation Context ...inity”. This is in complete analogy with the commutative case. 3.7. Noncommutative P 2 � and P3 � ij by adding a cone “at . Noncommutative deformations of the projective plane have been classified in =-=[1]-=-, [2], [9]. We will need one of them, namely the one whose homogeneous coordinate algebra is a graded algebra PP� = ⊕ PP�i over C generated by i≥0 the elements w1,w2,w3 of degree 1 with the relations:... |

174 |
Faisceaux algebriques coherents,
- Serre
- 1955
(Show Context)
Citation Context ...for any sheaf F we can define a graded A –module Γ(F) = ⊕ H i≥0 0 (X, F(i)). It can be checked that Γ is a functor from the category of coherent sheaves on X to gr(Γ(X)) . coh(X) In a brilliant paper =-=[32]-=-, J-P. Serre described the category of coherent sheaves on a projective scheme X in terms of graded modules over the graded algebra Γ(X) . He proved that the category coh(X) is equivalent to the quoti... |

133 |
Some algebras associated to automorphisms of elliptic curves”
- Artin, Tate, et al.
- 1990
(Show Context)
Citation Context ...”. This is in complete analogy with the commutative case. 3.7. Noncommutative P 2 � and P3 � ij by adding a cone “at . Noncommutative deformations of the projective plane have been classified in [1], =-=[2]-=-, [9]. We will need one of them, namely the one whose homogeneous coordinate algebra is a graded algebra PP� = ⊕ PP�i over C generated by i≥0 the elements w1,w2,w3 of degree 1 with the relations: (8) ... |

122 |
Vector Bundles and Projective Modules.
- Swan
- 1962
(Show Context)
Citation Context ... a compact Hausdorff topological space, then the category of vector bundles over X is equivalent to the category of finitely generated projective modules over the algebra of continuous functions on X =-=[33, 35]-=-. The equivalence is given by the functor which maps a vector bundle to the module of its global sections. It is well known that if A is a commutative noetherian algebra, the category of coherent shea... |

119 | Heisenberg algebra and Hilbert schemes of points on projective surfaces,
- Nakajima
- 1997
(Show Context)
Citation Context ...etric. Furthermore, we reinterpret the deformed ADHM construction of Nekrasov and Schwarz in terms of a noncommutative deformation of the twistor transform. It is interesting to note that H. Nakajima =-=[26]-=- studied the same linear algebra data as Nekrasov and Schwarz and showed that their moduli space coincides with the moduli space of torsion free sheaves on a commutative P2 with a trivialization on a ... |

98 |
Instantons on noncommutative R 4 and (2, 0) superconformal six dimensional theory
- Nekrasov, Schwarz
- 1998
(Show Context)
Citation Context ...mutative R4 is metrically complete. 2. Review of the ADHM construction and summary All instantons on the commutative R4 arise from the so-called ADHM construction. Recently N. Nekrasov and A. Schwarz =-=[28]-=- introduced a modification of this construction whichNONCOMMUTATIVE INSTANTONS AND TWISTOR TRANSFORM 7 produces instantons on the noncommutative R 4 . 2 In the commutative case the completeness of th... |

84 |
Existence theorems for dualizing complexes over non-commutative graded and filtered rings,
- Bergh
- 1997
(Show Context)
Citation Context ...a functor from the category gr(A)R to the category gr(A)L . Moreover, its derived functor RHom · A (−,A) gives an anti-equivalence between the derived categories of gr(A)R and gr(A)L (see [38], [39], =-=[37]-=-). If we assume that the composition of the functor HomA(−,A) with the projection gr(A)L −→ qgr(A)L factors through the projection gr(A)R −→ qgr(A)R, then we obtain a functor from qgr(A)R to qgr(A)L w... |

80 |
Polishchuk, Homological properties of associative algebras: the method of helices
- Bondal, E
- 1994
(Show Context)
Citation Context ...is is in complete analogy with the commutative case. 3.7. Noncommutative P 2 � and P3 � ij by adding a cone “at . Noncommutative deformations of the projective plane have been classified in [1], [2], =-=[9]-=-. We will need one of them, namely the one whose homogeneous coordinate algebra is a graded algebra PP� = ⊕ PP�i over C generated by i≥0 the elements w1,w2,w3 of degree 1 with the relations: (8) [w3,w... |

74 |
Generalized Functions vol I
- Gel’fand, Shilov
- 1964
(Show Context)
Citation Context ...x k D m φ(x), k = 0,1,2,... , are finite. Here m = (m1,... ,mn) is an arbitrary polyindex. Convergence on S is defined using the family of norms (39). Then S becomes a complete countably normed space =-=[17]-=-. Proposition 10.4. A function f ∈ S ′ is a multiplier if and only if it is a C ∞ function on R n Proof. Obvious. all of whose derivatives are polynomially bounded. The following theorem proved in [36... |

69 |
Instantons and Geometric Invariant Theory
- Donaldson
- 1984
(Show Context)
Citation Context ...DHM constructions and make a summary of our results. 2.1. Review of the ADHM construction of instantons. First let us outline the ADHM construction of U(r) instantons on the commutative R 4 following =-=[15]-=-. We assume that the constant metric G on R 4 has been brought to the standard form G = diag(1,1,1,1) by a linear change of basis. To construct a U(r) instanton with c2 = k one starts with two Hermite... |

69 |
Dualizing complexes over noncommutative graded algebras
- Yekutieli
- 1992
(Show Context)
Citation Context ...omA(−,A) is a functor from the category gr(A)R to the category gr(A)L . Moreover, its derived functor RHom · A (−,A) gives an anti-equivalence between the derived categories of gr(A)R and gr(A)L (see =-=[38]-=-, [39], [37]). If we assume that the composition of the functor HomA(−,A) with the projection gr(A)L −→ qgr(A)L factors through the projection gr(A)R −→ qgr(A)R, then we obtain a functor from qgr(A)R ... |

62 |
Koszul resolutions,
- Priddy
- 1970
(Show Context)
Citation Context ... It is a well–known fact that d 2 = 0 (see, for example, [24]). Let kA be the trivial right A -module. The Koszul complex K·(A) possesses a natural ε augmentation K· −→ kA −→ 0 . Definition 4.4. (see =-=[30]-=-) A quadratic algebra A = ⊕ Ai i≥0 augmented Koszul complex K·(A) −→ kA −→ 0 is exact. ε is called a Koszul algebra if the In the same manner one can define the left Koszul complex of a quadratic alge... |

53 |
The length of vectors in representation spaces
- Kempf, Ness
- 1978
(Show Context)
Citation Context ...ion [b1,b2] + i · j + 2� · 1 = 0 and the vanishing of the moment map for the action of the group U(k) on the space of quadruples (b1,b2,i,j) . Now it follows from the theorem of Kempf and Ness ([27], =-=[20]-=-) that the map M → M�(r,0,k) is a diffeomorphism. It becomes a complex isomorphism if we replace the natural complex structure of the space M with another one within the P1 of complex structures on M ... |

45 |
Moduli of vector bundles on the projective plane.
- Barth
- 1977
(Show Context)
Citation Context ... was proved above. The uniqueness follows from Lemma 6.3. The equality of dimensions of H and L follows immediately from the exact sequence (17). 6.5. Barth description of monads. Now following Barth =-=[8]-=-, we give a description of the moduli space of vector bundles on P 2 � also [15]). trivial on the line l in terms of linear algebra (see Denote by M�(r,0,k) the moduli space of bundles on the noncommu... |

40 |
Modules projectifs et espaces fibrés à fibre vectorielle. 1958 Séminaire P
- Serre
(Show Context)
Citation Context ... a compact Hausdorff topological space, then the category of vector bundles over X is equivalent to the category of finitely generated projective modules over the algebra of continuous functions on X =-=[33, 35]-=-. The equivalence is given by the functor which maps a vector bundle to the module of its global sections. It is well known that if A is a commutative noetherian algebra, the category of coherent shea... |

36 |
On the subalgebra generated by the one-dimensional elements in the Yoneda Ext-algebra. Algebra, algebraic topology and their interactions
- Löfwall
- 1983
(Show Context)
Citation Context ...can define the left Koszul complex of a quadratic algebra. It is well known that the exactness of the right Koszul complex is equivalent to the exactness of the left Koszul complex (see, for example, =-=[21]-=-). Proposition 4.5. The algebras PS� and PP� are Koszul algebras. Proof. For � = 0 this is a well-known fact about the symmetric algebra S ·(U) . Since the augmented Koszul complex is exact for � = 0,... |

30 | Nekrasov, Space-time foam from non-commutative instantons
- Braden, A
(Show Context)
Citation Context ... formula as Dξ, but is now regarded as an element of HomA((V ⊕ V ⊕ W) ⊗C A,(V ⊕ V ) ⊗C A).NONCOMMUTATIVE INSTANTONS AND TWISTOR TRANSFORM 9 The module Ker D is a projective module over A . Following =-=[10]-=-, we assume that it is isomorphic to a free module of rank r , and v is the corresponding isomorphism v : A⊕r → Ker D. We further assume [10] that the morphism ∆ = DD † ∈ EndA((V ⊕ V ) ⊗ A) is an isom... |

30 | Serre duality for non-commutative projective schemes
- Yekutieli, Zhang
- 1997
(Show Context)
Citation Context ...sts) such that for any two modules M an N we have Ext n+1 A (M,N) = 0 . In the paper [1] the notion of a regular algebra has been introduced. Regular algebras have many good properties (see [3], [2], =-=[39]-=-, etc.). Definition 4.10. A graded algebra A is called regular of dimension d if it satisfies the following conditions: (1) A has global dimension d, (2) A has polynomial growth, i.e. dim An ≤ cn δ fo... |

26 |
Geometry of Yang-Mills Fields
- Atiyah
- 1979
(Show Context)
Citation Context ...striction of its twistor transform to the sphere S4 coincides with the instanton bundle corresponding to these ADHM data. The instanton connection can also be reconstructed from the bundle on P3 (see =-=[4, 23]-=- for details). In this section we show that one can consider the noncommutative quadric introduced in section 3 as a noncommutative Grassmannian G(2;4) . We also construct a noncommutative flag variet... |

16 |
Einstein-Hermitian metrics on noncompact Kähler manifolds, Einstein metrics and Yang-Mills connections
- Bando
- 1990
(Show Context)
Citation Context ... imply that all instantons on R4 arise from the ADHM construction. The correspondence between A and B has been proved by Donaldson [15]. One can also prove the correspondence between A and D directly =-=[7, 11, 18]-=-. The goal of this paper is to extend some of these results to the noncommutative case. We show that there is a natural one-to-one correspondence between the isomorphism classes of the following objec... |

16 |
Quantum Groups and Noncommutative Geometry, Les Publ. du Centre de Récherches Math., Universite de Montreal
- Manin
- 1988
(Show Context)
Citation Context ...ded) modules over some (graded) algebra. On the other hand, “as A. Grothendieck taught us, to do geometry you really don’t need a space, all you need is a category of sheaves on this would-be space” (=-=[24]-=-, p.83). For this reason, in the realm of algebraic geometry it is natural to regard a noncommutative noetherian algebra as a coordinate algebra of a noncommutative affine variety; then the category o... |

14 |
Une demonstration algebrique d’un theoreme de
- Demazure
- 1968
(Show Context)
Citation Context ... to the same relations in degrees 2ei (id,−Rk j ,k i ) −−−−−−−−→ Λ ki R V ∗ ⊗ Λ kj R V ∗ −−−→ Σ (ki,kj) R and to V ∗ ) for all i > j . This definition is suggested by the Borel-Weil-Bott theorem (see =-=[14]-=-). In particular, for R = R0 we get the algebra corresponding to the commutative flag variety. We define the R -Carthesian product GR(k1;V ) × ... × GR(kr;V ) and the noncomR R mutative flag variety F... |

13 |
Hopf algebras and vector-symmetries
- Lyubashenko
- 1986
(Show Context)
Citation Context ... able to fill in the proofs. 8.2. Tensor categories. A good way to construct noncommutative varieties with properties similar to those of commutative varieties is to start with a tensor category (see =-=[24, 22]-=-). Let T be an abelian tensor category. Consider a tensor functor Φ : T → Vect to the abelian tensor category of vector spaces compatible with the associativity constraint but not compatible with the ... |

12 |
Instantons on nCP2
- BUCHDAHL
- 1993
(Show Context)
Citation Context ... imply that all instantons on R4 arise from the ADHM construction. The correspondence between A and B has been proved by Donaldson [15]. One can also prove the correspondence between A and D directly =-=[7, 11, 18]-=-. The goal of this paper is to extend some of these results to the noncommutative case. We show that there is a natural one-to-one correspondence between the isomorphism classes of the following objec... |

1 |
On an analytic proof of a result by
- Guo
- 1996
(Show Context)
Citation Context ... imply that all instantons on R4 arise from the ADHM construction. The correspondence between A and B has been proved by Donaldson [15]. One can also prove the correspondence between A and D directly =-=[7, 11, 18]-=-. The goal of this paper is to extend some of these results to the noncommutative case. We show that there is a natural one-to-one correspondence between the isomorphism classes of the following objec... |

1 |
Noncommutative curves and noncommutative surfaces, preprint math.RA/9910082
- Stafford, Bergh
(Show Context)
Citation Context ...ded right modules over this algebra14 ANTON KAPUSTIN, ALEXANDER KUZNETSOV, AND DMITRI ORLOV modulo torsion modules is identified with the category of coherent sheaves on this variety (see [3], [24], =-=[34]-=-). A different approach to noncommutative geometry has been pursued by A. Connes [12]. 3.3. Noncommutative deformations of commutative varieties. Many important noncommutative varieties arise as defor... |

1 |
Convolution in K{Mp} spaces, Rocky Mountain
- Swartz
- 1972
(Show Context)
Citation Context ...17]. Proposition 10.4. A function f ∈ S ′ is a multiplier if and only if it is a C ∞ function on R n Proof. Obvious. all of whose derivatives are polynomially bounded. The following theorem proved in =-=[36]-=- describes the subspace of convolutes of S ′ : Theorem 10.5. A distribution f ∈ S ′ is a convolute if and only if it has the form f = ∑ D α fα(x), |α|<r where r is a positive integer, and fα are C 0 f... |