Higher-order gaugino condensates on a twisted $$ {\mathbbm{T}}^4 $$
Bibliographic record
Abstract
A bstract We compute the gaugino condensates, $$ \left\langle {\Pi}_{i=1}^k\textrm{tr}\left(\uplambda \uplambda \right)\left({x}_i\right)\right\rangle $$ Π i = 1 k tr λλ x i for 1 ≤ k ≤ N − 1, in SU( N ) super Yang-Mills theory on a small four-dimensional torus $$ {\mathbbm{T}}^4 $$ T 4 , subject to ’t Hooft twisted boundary conditions. Two recent advances are crucial to performing the calculations and interpreting the result: the understanding of generalized anomalies involving 1-form center symmetry and the construction of multi-fractional instantons on the twisted $$ {\mathbbm{T}}^4 $$ T 4 . These self-dual classical configurations have topological charge k / N and can be described as a sum over k closely packed lumps in an instanton liquid. Using the path integral formalism, we perform the condensate calculations in the semi-classical limit and find, assuming gcd( k , N ) = 1, $$ \left\langle {\Pi}_{i=1}^k\textrm{tr}\left(\uplambda \uplambda \right)\left({x}_i\right)\right\rangle $$ Π i = 1 k tr λλ x i = $$ {\mathcal{N}}^{-1} $$ N − 1 N 2 (16 π 2 Λ 3 ) k , where Λ is the strong-coupling scale and $$ \mathcal{N} $$ N is a normalization constant. We determine the normalization constant, using path integral, as $$ \mathcal{N} $$ N = N 2 , which is N times larger than the normalization used in our earlier publication [1]. This finding resolves the extra-factor-of- N discrepancy encountered there, aligning our results with those obtained through direct supersymmetric methods on ℝ 4 . The normalization constant $$ \mathcal{N} $$ N can be understood within the Euclidean path-integral framework as the Witten index I W . From the Hamiltonian approach, it is well-established that I W = N . While the value $$ \mathcal{N} $$ N = N 2 correctly reproduces the condensate result, this discrepancy between the Hamiltonian and path-integral formulations calls fo
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.051 | 0.004 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".