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Gauge Freedom We say that a pair of monads are equivalent if there is a commuting diagram H ⊗ A(C4 ) ⏐ U⊗id H ⊗ A(C 4 ) σz −−−−→ K ⊗ A(C4 ) ⏐ W ⊗id ˜σz −−−−→ K ⊗ A(C 4 ) for invertible linear maps U : H → H and W : K → K. The effect on the resulting projection is to map P ↦→ W P W ∗ . S. Brain (RU) NCG of Self-Dual Gauge Fields Nijmegen, 12th October 2010 9 / 25
Gauge Freedom We say that a pair of monads are equivalent if there is a commuting diagram H ⊗ A(C4 ) ⏐ U⊗id H ⊗ A(C 4 ) σz −−−−→ K ⊗ A(C4 ) ⏐ W ⊗id ˜σz −−−−→ K ⊗ A(C 4 ) for invertible linear maps U : H → H and W : K → K. The effect on the resulting projection is to map P ↦→ W P W ∗ . Theorem (ADHM) There is a bijective correspondence Mk ∼ = {Monads with index k}/ ∼ . So these equivalences of monads account for all of the gauge freedom in the construction of instantons. S. Brain (RU) NCG of Self-Dual Gauge Fields Nijmegen, 12th October 2010 9 / 25
Page 1 and 2: Noncommutative Geometry of Self-Dua
Page 3 and 4: Motivation What is the difference b
Page 5 and 6: Instantons in Classical Geometry Le
Page 7 and 8: The Moduli Space of Instantons The
Page 9 and 10: Instantons on the Euclidean Four-Sp
Page 11 and 12: The SU(2) Hopf Fibration Define A(C
Page 13 and 14: Quaternionic Structure Note that th
Page 15 and 16: Quaternionic Structure Note that th
Page 17 and 18: Construction of Instantons Instanto
Page 19 and 20: Construction of Instantons Instanto
Page 21: Construction of Instantons Instanto
Page 25 and 26: The Space of Monads Note that, sinc
Page 27 and 28: The Space of Monads Note that, sinc
Page 29 and 30: Functorial Cocycle Deformation Now
Page 31 and 32: Functorial Cocycle Deformation Now
Page 33 and 34: The Deformation Functor More intere
Page 35 and 36: Example: Twisting by a Torus Action
Page 37 and 38: Example: The Connes-Landi Sphere Si
Page 39 and 40: Differential and Hodge Structure on
Page 41 and 42: Differential and Hodge Structure on
Page 43 and 44: A Noncommutative Family of Monads o
Page 45 and 46: A Noncommutative Family of Monads o
Page 47 and 48: The Noncommutative ADHM Constructio
Page 49 and 50: A Noncommutative Family of Instanto
Page 51 and 52: Gauge Freedom in the Noncommutative
Page 53 and 54: Gauge Freedom in the Noncommutative
Page 55 and 56: Understanding the Noncommutative Pa
Page 57 and 58: Understanding the Noncommutative Pa
Page 59 and 60: Summary In obtaining an ADHM constr
Page 61: The Moyal-Groenewold Plane R 4 If

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