Higher-order gaugino condensates on a twisted $$ {\mathbbm{T}}^4 $$
Bibliographic record
Abstract
A bstract We compute the gaugino condensates, $$ \left\langle {\Pi}_{i=1}^k\textrm{tr}\left(\uplambda \uplambda \right)\left({x}_i\right)\right\rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mrow> <mml:msubsup> <mml:mi>Π</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>k</mml:mi> </mml:msubsup> <mml:mi>tr</mml:mi> <mml:mfenced> <mml:mi>λλ</mml:mi> </mml:mfenced> <mml:mfenced> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mfenced> </mml:mrow> </mml:mfenced> </mml:math> for 1 ≤ k ≤ N − 1, in SU( N ) super Yang-Mills theory on a small four-dimensional torus $$ {\mathbbm{T}}^4 $$ <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> , 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 $$ {\mathbbm{T}}^4 $$ <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> . 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, $$ \left\langle {\Pi}_{i=1}^k\textrm{tr}\left(\uplambda \uplambda \right)\left({x}_i\right)\right\rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mrow> <mml:msubsup> <mml:mi>Π</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>k</mml:mi> </mml:msubsup> <mml:mi>tr</mml:mi> <mml:mfenced> <mml:mi>λλ</mml:mi> </mml:mfenced> <mml:mfenced> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mfenced> </mml:mrow> </mml:mfenced> </mml:math> = $$ {\mathcal{N}}^{-1} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> N 2 (16 π 2 Λ 3 ) k , where Λ is the strong-coupling scale and $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> is a normalization constant. We determine the normalization constant, using path integral, as $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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 $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> 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 $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = N 2 correctly reproduces the condensate result, this discrepancy between the Hamiltonian and path-integral formulations calls for reconciliation. We attempt to provide a potential solution we outline in our discussion.
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How this classification was reachedexpand
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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".