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Notes on Noncommutative Instantons
High Energy Physics - Theory · Submitted 1 Aug 2001
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In particular, we show for the first time how to construct explicitly regular U(N) instanton solutions on R_{NC}^4 and on R_{NC}^2 x R^2.
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The abstract, sentence by sentence
We study in detail the ADHM construction of U(N) instantons on noncommutative Euclidean space-time R_{NC}^4 and noncommutative space R_{NC}^2 x R^2. We point out that the completeness condition in the ADHM construction could be invalidated in certain circumstances. When this happens, regular instanton configuration may not exist even if the ADHM constraints are satisfied. Some of the existing solutions in the literature indeed violate the completeness condition and hence are not correct. We present alternative solutions for these cases. In particular, we show for the first time how to construct explicitly regular U(N) instanton solutions on R_{NC}^4 and on R_{NC}^2 x R^2. We also give a simple general argument based on the Corrigan's identity that the topological charge of noncommutative regular instantons is always an integer.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study in detail the ADHM construction of U(N) instantons on noncommutative Euclidean space-time R_{NC}^4 and noncommutative space R_{NC}^2 x R^2. | contribution 0.93 | 0.91 | 0.05 | asserted 0.98 |
| s1 | We point out that the completeness condition in the ADHM construction could be invalidated in certain circumstances. | result 0.78 | 0.86 | 0.23 | hedged possibility 1.00 |
| s2 | When this happens, regular instanton configuration may not exist even if the ADHM constraints are satisfied. | result 0.69 | 0.59 | 0.19 | hedged possibility 0.99 |
| s3 | Some of the existing solutions in the literature indeed violate the completeness condition and hence are not correct. | result 0.82 | 0.43 | 0.14 | asserted 0.98 |
| s4 | We present alternative solutions for these cases. | contribution 0.99 | 0.93 | 0.12 | asserted 1.00 |
| s5 | In particular, we show for the first time how to construct explicitly regular U(N) instanton solutions on R_{NC}^4 and on R_{NC}^2 x R^2. | contribution 0.96 | 0.97 | 0.50 | asserted 1.00 |
| s6 | We also give a simple general argument based on the Corrigan's identity that the topological charge of noncommutative regular instantons is always an integer. | result 0.63 | 0.93 | 0.16 | asserted 1.00 |
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The paper
Original abstract
We study in detail the ADHM construction of U(N) instantons on noncommutative Euclidean space-time R_{NC}^4 and noncommutative space R_{NC}^2 x R^2. We point out that the completeness condition in the ADHM construction could be invalidated in certain circumstances. When this happens, regular instanton configuration may not exist even if the ADHM constraints are satisfied. Some of the existing solutions in the literature indeed violate the completeness condition and hence are not correct. We present alternative solutions for these cases. In particular, we show for the first time how to construct explicitly regular U(N) instanton solutions on R_{NC}^4 and on R_{NC}^2 x R^2. We also give a simple general argument based on the Corrigan's identity that the topological charge of noncommutative regular instantons is always an integer.
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4 arXiv papers cite this in 9 sentences
Papers citing it · 3 sentences without a confident label
Compares with it 1 paper
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Usually the single instanton solution in $ℝ_NC^2 ×ℝ_C^2$ with $U(1)$ gauge group is normalized as [ref] [equation] But in our solution this technique cannot be used because in this case $Ψ_0^†Ψ_0=N_1-Λ_-/N_1-1δ(δ-1)$ vanishes when it operates on the state $|0⟩$ and the inverse of $Ψ_0^†Ψ_0$ does not exist.
Noncommutative Vortices and Instantons from Generalized Bose Operators · $ℝ^2_NC×ℝ^2_C$: · 0.86
Cites it as background 4 papers
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Let us first discuss the usual single anti-self-dual $U(1)$ instanton solutions ($k=1, N=1$) in $ℝ^2_NC×ℝ^2_C$ [ref] and in $ℝ^2_NC×ℝ^2_NC$ [ref] .
Noncommutative Vortices and Instantons from Generalized Bose Operators · Noncommutative ADHM Construction · 0.91
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Let us first discuss the usual single anti-self-dual $U(1)$ instanton solutions ($k=1, N=1$) in $R^2_NC×R^2_C$ [ref] and in $R^2_NC×R^2_NC$ [ref] .
Topics in Noncommutative Gauge Theories and Deformed Relativistic Theories · Noncommutative ADHM Construction · 0.89
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In this subsection we briefly review the noncommutative ADHM construction, as explained in e.g. [ref] .
Fayet-Iliopoulos Terms in AdS/CFT with Flavour · Noncommutative U(1) Instantons and modified ADHM Construction · 0.98
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