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Bibliographic record
Abstract
A bstract We embed the multi-fractional instantons of SU( N ) gauge theories on $$ {\mathbbm{T}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> with ’t Hooft twisted boundary conditions into U( N ) bundles and use the Nahm transform to study the corresponding configurations on the dual $$ {\hat{\mathbbm{T}}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mn>4</mml:mn> </mml:msup> </mml:math> . We first show that SU( N ) fractional instantons of topological charge $$ Q=\frac{r}{N},r\in \left\{1,2,\dots, N-1\right\} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mi>r</mml:mi> <mml:mi>N</mml:mi> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>∈</mml:mo> <mml:mfenced> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> <mml:mo>…</mml:mo> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfenced> </mml:math> , are mapped to fractional instantons of SU( $$ \hat{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>N</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> ) of charge $$ \hat{Q}=\frac{r}{\hat{N}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>Q</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mi>r</mml:mi> <mml:mover> <mml:mi>N</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:mfrac> </mml:math> , where $$ \hat{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>N</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> = Nq 1 q 3 − rq 3 + q 1 and q 1 , 3 are integer-quantized U(1) fluxes. We then explicitly construct the Nahm transform of constant field strength fractional instantons of SU( N ) and find the SU( $$ \hat{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>N</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> ) configurations they map to. Both the $$ {\mathbbm{T}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> instantons and their $$ {\hat{\mathbbm{T}}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mn>4</mml:mn> </mml:msup> </mml:math> images are self-dual for appropriately tuned torus periods. The Nahm duality can be extended to tori with detuned periods, with detuning parameter ∆, mapping solutions with ∆ > 0 on $$ {\mathbbm{T}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> to ones with $$ \hat{\Delta } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mtext>∆</mml:mtext> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> < 0 on $$ {\hat{\mathbbm{T}}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mn>4</mml:mn> </mml:msup> </mml:math> . We also recall that fractional instantons appear in string theory precisely via the U( N ) embedding, suggesting that studying the end point of tachyon condensation for ∆ ≠ 0 is needed — and is perhaps feasible in a small-∆ expansion, as in field theory studies — in order to understand the appearance and role of fractional instantons in D -brane constructions.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it