# Noncommutative Instantons and Twistor Transform

@article{Kapustin2001NoncommutativeIA, title={Noncommutative Instantons and Twistor Transform}, author={Anton Kapustin and Alexander Kuznetsov and Dmitri O. Orlov}, journal={Communications in Mathematical Physics}, year={2001}, volume={221}, pages={385-432} }

Abstract: Recently N. Nekrasov and A. Schwarz proposed a modification of the ADHM construction of instantons which produces instantons on a noncommutative deformation of ℝ4. In this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces. We exhibit one-to-one correspondences between three classes of objects: framed bundles on a noncommutative ℙ2, certain complexes of sheaves on a noncommutative ℙ3, and the modified ADHM data. The…

## 120 Citations

Algebraic deformations of toric varieties II. Noncommutative instantons

- Mathematics
- 2011

We continue our study of the noncommutative algebraic and differential geometry of a particular class of deformations of toric varieties, focusing on aspects pertinent to the construction and…

Instanton Expansion of Noncommutative Gauge Theory in Two Dimensions

- Mathematics, Physics

We show that noncommutative gauge theory in two dimensions is an exactly solv-able model. A cohomological formulation of gauge theory defined on the noncom-mutative torus is used to show that its…

Instanton Expansion of Noncommutative Gauge Theory in Two Dimensions

- Mathematics
- 2002

AbstractWe show that noncommutative gauge theory in two dimensions is an exactly solvable model. A cohomological formulation of gauge theory defined on the noncommutative torus is used to show that…

Instantons and vortices on noncommutative toric varieties

- Mathematics
- 2012

We elaborate on the quantization of toric varieties by combining techniques from toric geometry, isospectral deformations and noncommutative geometry in braided monoidal categories, and the…

ON SOLITONS IN NONCOMMUTATIVE GAUGE THEORIES

- Mathematics
- 2000

Recently there has been a revival of interest in noncommutative gauge theories. They are interesting examples of nonlocal field theories which in the certain limit (of large noncommutativity) become…

THE ADHM CONSTRUCTION OF INSTANTONS ON NONCOMMUTATIVE SPACES

- Mathematics
- 2011

We present an account of the ADHM construction of instantons on Euclidean space-time ℝ4 from the point of view of noncommutative geometry. We recall the main ingredients of the classical construction…

A noncommutative deformation of topological field theory

- Mathematics
- 2005

Cohomological Yang–Mills theory is formulated on a noncommutative differentiable four manifold through the θ-deformation of its corresponding BRST algebra. The resulting noncommutative field theory…

Noncommutative localization in noncommutative geometry

- Mathematics
- 2004

The aim of these notes is to collect and motivate the basic localization toolbox for the geometric study of ``spaces'', locally described by noncommutative rings and their categories of one-sided…

## References

SHOWING 1-10 OF 47 REFERENCES

String theory and noncommutative geometry

- Mathematics, Physics
- 1999

We extend earlier ideas about the appearance of noncommutative geometry in string theory with a nonzero B-field. We identify a limit in which the entire string dynamics is described by a minimally…

Noncommutative curves and noncommutative surfaces

- Mathematics
- 1999

In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories.
Roughly speaking and by analogy with the…

Heisenberg algebra and Hilbert schemes of points on projective surfaces

- Mathematics
- 1995

The purpose of this paper is to throw a bridge between two seemingly unrelated subjects. One is the Hilbert scheme of points on projective surfaces, which has been intensively studied by various…

Instantons on Noncommutative ℝ4, and (2,0) Superconformal Six Dimensional Theory

- Physics, Mathematics
- 1998

Abstract:We show that the resolution of moduli space of ideal instantons parameterizes the instantons on noncommutative ℝ4. This moduli space appears to be the Higgs branch of the theory of…

Noncommutative Geometry and Matrix Theory: Compactification on Tori

- Mathematics
- 1997

We study toroidal compactification of Matrix theory, using ideas and results of noncommutative geometry. We generalize this to compactification on the noncommutative torus, explain the classification…

Noncommutative Projective Schemes

- Mathematics
- 1994

An analogue of the concept of projective scheme is defined for noncommutative N-graded algebras using the quotient category C of graded right A-modules modulo its full subcategory of torsion modules.…

Hyperkähler metrics and supersymmetry

- Mathematics
- 1987

We describe two constructions of hyperkähler manifolds, one based on a Legendre transform, and one on a sympletic quotient. These constructions arose in the context of supersymmetric nonlinear…

Noncommutative Noetherian Rings

- Mathematics
- 2001

Articles on the history of mathematics can be written from many dierent perspectives. Some aim to survey a more or less wide landscape, and require the observer to watch from afar as theories develop…

Vector bundles and projective modules

- Mathematics
- 1962

Serre [9, ?50] has shown that there is a one-to-one correspondence between algebraic vector bundles over an affine variety and finitely generated projective modules over its coordinate ring. For some…