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1. Higher-group symmetry of (3+1)D fermionic $$\mathbb{Z}_2$$ gauge theory: Logical CCZ, CS, and T gates from higher symmetry

   https://doi.org/10.21468/SciPostPhys.16.5.122  

   Barkeshli, Maissam; Hsin, Po-Shen; Kobayashi, Ryohei (January 2024, SciPost Physics)

   It has recently been understood that the complete global symmetry of finite group topological gauge theories contains the structure of a higher-group. Here we study the higher-group structure in (3+1)D\mathbb{Z}_2 ${\mathbb{Z}}_2$gauge theory with an emergent fermion, and point out that pumping chiralp+ip $p+ip$topological states gives rise to a\mathbb{Z}_{8} ${\mathbb{Z}}_8$0-form symmetry with mixed gravitational anomaly. This ordinary symmetry mixes with the other higher symmetries to form a 3-group structure, which we examine in detail. We then show that in the context of stabilizer quantum codes, one can obtain logical CCZ and CS gates by placing the code on a discretization ofT^3 $T^3$(3-torus) andT^2 \rtimes_{C_2} S^1 $T^2{⋊}_{C_2}{S}^1$(2-torus bundle over the circle) respectively, and pumpingp+ip $p+ip$states. Our considerations also imply the possibility of a logicalT $T$gate by placing the code on\mathbb{RP}^3 ${ℝ\mathbb{P}}^3$and pumping ap+ip $p+ip$topological state. 

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2. Local Well-Posedness of the Skew Mean Curvature Flow for Small Data in $$d\geqq 2$$ Dimensions

   https://doi.org/10.1007/s00205-023-01952-y  

   Huang, Jiaxi; Tataru, Daniel (January 2024, Archive for Rational Mechanics and Analysis)

   Abstract The skew mean curvature flow is an evolution equation forddimensional manifolds embedded in$${\mathbb {R}}^{d+2}$$ $R^{d+2}$(or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In an earlier paper, the authors introduced a harmonic/Coulomb gauge formulation of the problem, and used it to prove small data local well-posedness in dimensions$$d \geqq 4$$ $d\geqq 4$. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension$$d\geqq 2$$ $d\geqq 2$. This is achieved by introducing a new, heat gauge formulation of the equations, which turns out to be more robust in low dimensions. 

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3. On the generalization of the Kruskal–Szekeres coordinates: a global conformal charting of the Reissner–Nordström spacetime

   https://doi.org/10.1088/1361-6382/ad4dff  

   Aly, Fawzi; Stojkovic, Dejan (May 2024, Classical and Quantum Gravity)

   Abstract The Kruskal–Szekeres coordinate construction for the Schwarzschild spacetime could be interpreted simply as a squeezing of thet-line into a single point, at the event horizon $r=2M$. Starting from this perspective, we extend the Kruskal charting to spacetimes with two horizons, in particular the Reissner–Nordström manifold, $M_{\text{RN}}$. We develop a new method to construct Kruskal-like coordinates through casting the metric in new null coordinates, and find two algebraically distinct ways to chart $M_{\text{RN}}$, referred to as classes: type-I and type-II within this work. We pedagogically illustrate our method by crafting two compact, conformal, and global coordinate systems labeled ${\mathrm{GK}}_{\text{I}}$and ${\mathrm{GK}}_{\text{II}}$as an example for each class respectively, and plot the corresponding Penrose diagrams. In both coordinates, the metric differentiability can be promoted to $C^{\mathrm{\infty }}$in a straightforward way. Finally, the conformal metric factor can be written explicitly in terms of thetandrfunctions for both types of charts. We also argued that the chart recently reported in Soltani (2023 arXiv:2307.11026) could be viewed as another example for the type-II classification, similar to ${\mathrm{GK}}_{\text{II}}$. 

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4. Instantons and the large $N=4$ algebra

   https://doi.org/10.1088/1751-8121/ada64d  

   Witten, Edward (January 2025, Journal of Physics A: Mathematical and Theoretical)

   Abstract We investigate the differential geometry of the moduli space of instantons on ${\mathrm{S}}^3\times {\mathrm{S}}^1$. Extending previous results, we show that a sigma-model with this target space can be expected to possess a large $N=4$superconformal symmetry, supporting speculations that this sigma-model may be dual to Type IIB superstring theory on ${\mathrm{AdS}}_3\times {\mathrm{S}}^3\times {\mathrm{S}}^3\times {\mathrm{S}}^1$. The sigma-model is parametrized by three integers—the rank of the gauge group, the instanton number, and a ‘level’ (the integer coefficient of a topologically nontrivialB-field, analogous to a WZW level). These integers are expected to correspond to two five-brane charges and a one-brane charge. The sigma-model is weakly coupled when the level, conjecturally corresponding to one of the five-brane changes, becomes very large, keeping the other parameters fixed. The central charges of the large $N=4$algebra agree, at least semiclassically, with expectations from the duality. 

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5. The linearization of the boundary rigidity problem for MP-systems and generic local boundary rigidity

   https://doi.org/10.1088/1361-6420/ad8d77  

   Muñoz-Thon, Sebastián (November 2024, Inverse Problems)

   Abstract We consider an $\mathrm{MP}$-system, that is, a compact Riemannian manifold with boundary, endowed with a magnetic field and a potential. On simple $\mathrm{MP}$-systems, we study the $\mathrm{MP}$-ray transform in order to obtain new boundary rigidity results for $\mathrm{MP}$-systems. We show that there is an explicit relation between the $\mathrm{MP}$-ray transform and the magnetic one, which allows us to apply results from Dairbekovet al(2007Adv. Math.216535–609) to our case. Regarding rigidity, we show that there exists a generic set $G^m$of simple $\mathrm{MP}$-systems, which is open and dense, such that any two $\mathrm{MP}$-systems close to an element in it and having the same boundary action function, must bek-gauge equivalent. 

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