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1. Spin(7) orientifolds and 2d $$ \mathcal{N} $$ = (0, 1) triality

   https://doi.org/10.1007/JHEP01(2022)058  

   Franco, Sebastián; Mininno, Alessandro; Uranga, Ángel M.; Yu, Xingyang (January 2022, Journal of High Energy Physics)

   A bstract We present a new, geometric perspective on the recently proposed triality of 2d $$ \mathcal{N} $$ N = (0 , 1) gauge theories, based on its engineering in terms of D1-branes probing Spin(7) orientifolds. In this context, triality translates into the fact that multiple gauge theories correspond to the same underlying orientifold. We show how Spin(7) orientifolds based on a particular involution, which we call the universal involution, give rise to precisely the original version of $$ \mathcal{N} $$ N = (0 , 1) triality. Interestingly, our work also shows that the space of possibilities is significantly richer. Indeed, general Spin(7) orientifolds extend triality to theories that can be regarded as consisting of coupled $$ \mathcal{N} $$ N = (0 , 2) and (0 , 1) sectors. The geometric construction of 2d gauge theories in terms of D1-branes at singularities therefore leads to extensions of triality that interpolate between the pure $$ \mathcal{N} $$ N = (0 , 2) and (0 , 1) cases. 

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2. Mirror symmetry for the Landau–Ginzburg $$A$$ -model $$M=\mathbb{C}^{n}$$ , $$W=z_{1}\cdots z_{n}$$

   https://doi.org/10.1215/00127094-2018-0036  

   Nadler, David (January 2019, Duke Mathematical Journal)
3. 2d $$ \mathcal{N} $$ = (0, 1) gauge theories and Spin(7) orientifolds

   https://doi.org/10.1007/JHEP03(2022)150  

   Franco, Sebastián; Mininno, Alessandro; Uranga, Ángel M.; Yu, Xingyang (March 2022, Journal of High Energy Physics)

   A bstract We initiate the geometric engineering of 2d $$ \mathcal{N} $$ N = (0 , 1) gauge theories on D1-branes probing singularities. To do so, we introduce a new class of backgrounds obtained as quotients of Calabi-Yau 4-folds by a combination of an anti-holomorphic involution leading to a Spin(7) cone and worldsheet parity. We refer to such constructions as Spin(7) orientifolds . Spin(7) orientifolds explicitly realize the perspective on 2d $$ \mathcal{N} $$ N = (0 , 1) theories as real slices of $$ \mathcal{N} $$ N = (0 , 2) ones. Remarkably, this projection is geometrically realized as Joyce’s construction of Spin(7) manifolds via quotients of Calabi-Yau 4-folds by anti-holomorphic involutions. We illustrate this construction in numerous examples with both orbifold and non-orbifold parent singularities, discuss the role of the choice of vector structure in the orientifold quotient, and study partial resolutions. 

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4. A constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scales

   https://doi.org/10.1051/0004-6361/202347050  

   Jiménez-Donaire, M. J.; Usero, A.; Bešlić, I.; Tafalla, M.; Chacón-Tanarro, A.; Salomé, Q.; Eibensteiner, C.; García-Rodríguez, A.; Hacar, A.; Barnes, A. T.; et al (August 2023, Astronomy & Astrophysics)

   Nitrogen hydrides such as NH3 and N2H+ are widely used by Galactic observers to trace the cold dense regions of the interstellar medium. In external galaxies, because of limited sensitivity, HCN has become the most common tracer of dense gas over large parts of galaxies. We provide the first systematic measurements of N2H+ (1-0) across different environments of an external spiral galaxy, NGC 6946. We find a strong correlation (r > 0.98, p < 0.01) between the HCN (1-0) and N2H+ (1-0) intensities across the inner ∼8 kpc of the galaxy, at kiloparsec scales. This correlation is equally strong between the ratios N2H+ (1-0)/CO (1-0) and HCN (1-0)/CO (1-0), tracers of dense gas fractions (fdense). We measure an average intensity ratio of N2H+ (1-0)/HCN (1-0) = 0.15 ± 0.02 over our set of five IRAM-30m pointings. These trends are further supported by existing measurements for Galactic and extragalactic sources. This narrow distribution in the average ratio suggests that the observed systematic trends found in kiloparsec-scale extragalactic studies of fdense and the efficiency of dense gas (SFEdense) would not change if we employed N2H+ (1-0) as a more direct tracer of dense gas. At kiloparsec scales our results indicate that the HCN (1-0) emission can be used to predict the expected N2H+ (1-0) over those regions. Our results suggest that, even if HCN (1-0) and N2H+ (1-0) trace different density regimes within molecular clouds, subcloud differences average out at kiloparsec scales, yielding the two tracers proportional to each other. 

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5. $$\mathbb {A}^1$$ A 1 -connected varieties of rank one over nonclosed fields

   https://doi.org/10.1007/s00208-015-1257-1  

   Chen, Qile; Zhu, Yi (April 2016, Mathematische Annalen)