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1. Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phases

   https://doi.org/10.1103/PhysRevB.110.195149  

   Pace, Salvatore D; Zhu, Chenchang; Beaudry, Agnès; Wen, Xiao-Gang (November 2024, Physical Review B)

   We study the nonlinear $$\sigma$$-model in $${(d+1)}$$-dimensional spacetime with connected target space $$K$$ and show that, at energy scales below singular field comfigurations (such as vortices), it has an emergent non-invertible higher symmetry. The symmetry defects of the emergent symmetry are described by the $$d$$-representations of a discrete $$d$$-group $$\mathbb{G}^{(d)}$$ (i.e. the emergent symmetry is the dual of the invertible $$d$$-group $$\mathbb{G}^{(d)}$$ symmetry). The $$d$$-group $$\mathbb{G}^{(d)}$$ is determined such that its classifying space $$B\mathbb{G}^{(d)}$$ is given by the $$d$$-th Postnikov stage of $$K$$. In $(2+1)$D and for finite $$\mathbb{G}^{(2)}$$, this symmetry is always holo-equivalent to an invertible $${0}$$-form---ordinary---symmetry with potential 't Hooft anomaly. The singularity-free disordered phase of the nonlinear $$\sigma$$-model spontaneously breaks this symmetry, and when $$\mathbb{G}^{(d)}$$ is finite, it is described by the deconfined phase of $$\mathbb{G}^{(d)}$$ higher gauge theory. We consider examples of such disordered phases. We focus on a singularity-free $S^2$ nonlinear $$\sigma$$-model in $${(3+1)}$$D and show that it has an emergent non-invertible higher symmetry. As a result, its disordered phase is described by axion electrodynamics and has two gapless modes corresponding to a photon and a massless axion. Notably, this non-perturbative result is different from the results obtained using the $S^N$ and $$\mathbb{C}P^{N-1}$$ nonlinear $$\si$$-models in the large-$$N$$ limit. 

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2. Marcinkiewicz Averages of Smooth Orthogonal Projections on Sphere

   https://doi.org/10.1007/s00041-022-09966-y  

   Bownik, Marcin; Dziedziul, Karol; Kamont, Anna (October 2022, Journal of Fourier Analysis and Applications)

   Abstract We construct a single smooth orthogonal projection with desired localization whose average under a group action yields the decomposition of the identity operator. For any full rank lattice $$\Gamma \subset \mathbb {R}^d$$ Γ ⊂ R d , a smooth projection is localized in a neighborhood of an arbitrary precompact fundamental domain $$\mathbb {R}^d/\Gamma $$ R d / Γ . We also show the existence of a highly localized smooth orthogonal projection, whose Marcinkiewicz average under the action of SO ( d ), is a multiple of the identity on $$L^2(\mathbb {S}^{d-1})$$ L 2 ( S d - 1 ) . As an application we construct highly localized continuous Parseval frames on the sphere. 

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3. On the Convolution Inequality f ≥ f ⋆ f

   https://doi.org/10.1093/imrn/rnaa350  

   Carlen, Eric A; Jauslin, Ian; Lieb, Elliott H; Loss, Michael P (January 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract We consider the inequality $$f \geqslant f\star f$$ for real functions in $$L^1({\mathbb{R}}^d)$$ where $$f\star f$$ denotes the convolution of $$f$$ with itself. We show that all such functions $$f$$ are nonnegative, which is not the case for the same inequality in $L^p$ for any $$1 < p \leqslant 2$$, for which the convolution is defined. We also show that all solutions in $$L^1({\mathbb{R}}^d)$$ satisfy $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x \leqslant \tfrac 12$$. Moreover, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$, then $$f$$ must decay fairly slowly: $$\int _{{\mathbb{R}}^{\textrm{d}}}|x| f(x)\ \textrm{d}x = \infty $$, and this is sharp since for all $r< 1$, there are solutions with $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$ and $$\int _{{\mathbb{R}}^{\textrm{d}}}|x|^r f(x)\ \textrm{d}x <\infty $$. However, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x =: a < \tfrac 12$$, the decay at infinity can be much more rapid: we show that for all $$a<\tfrac 12$$, there are solutions such that for some $$\varepsilon>0$$, $$\int _{{\mathbb{R}}^{\textrm{d}}}e^{\varepsilon |x|}f(x)\ \textrm{d}x < \infty $$. 

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4. Spherical convex hull of random points on a wedge

   https://doi.org/10.1007/s00208-023-02704-9  

   Besau, Florian; Gusakova, Anna; Reitzner, Matthias; Schütt, Carsten; Thäle, Christoph; Werner, Elisabeth_M (August 2023, Mathematische Annalen)

   Abstract Consider two half-spaces$$H_1^+$$ $H_1^{+}$and$$H_2^+$$ $H_2^{+}$in$${\mathbb {R}}^{d+1}$$ $R^{d+1}$whose bounding hyperplanes$$H_1$$ $H_1$and$$H_2$$ $H_2$are orthogonal and pass through the origin. The intersection$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$ $S_{2,+}^d:=S^d\cap H_1^{+}\cap H_2^{+}$is a spherical convex subset of thed-dimensional unit sphere$${\mathbb {S}}^d$$ $S^d$, which contains a great subsphere of dimension$$d-2$$ $d-2$and is called a spherical wedge. Choosenindependent random points uniformly at random on$${\mathbb {S}}_{2,+}^d$$ $S_{2,+}^d$and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of$$\log n$$ $logn$. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on$${\mathbb {S}}_{2,+}^d$$ $S_{2,+}^d$. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere. 

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5. On finite energy monopoles on $$C x \Sigma$$.

   Wang, Donghao (July 2019, ArXiv.org)

   Let $$X=\mathbb{C}\times\Sigma$$ be the product of the complex plane and a compact Riemann surface. We establish a classification theorem of solutions to the Seiberg-Witten equation on $$X$$ with finite analytic energy. The spin bundle $$S^+\to X$$ splits as $$L^+\oplus L^-$$. When $$2-2g\leq c_1(S^+)[\Sigma]<0$$, the moduli space is in bijection with the moduli space of pairs $$((L^+,\bar{\partial}), f)$$ where $$(L^+,\bar{\partial})$$ is a holomorphic structure on $L^+$ and $$f: \mathbb{C}\to H^0(\Sigma, L^+,\bar{\partial})$$ is a polynomial map. Moreover, the solution has analytic energy $$-4\pi^2d\cdot c_1(S^+)[\Sigma]$$ if $$f$$ has degree $$d$$. When $$c_1(S^+)=0$$, all solutions are reducible and the moduli space is the space of flat connections on $$\bigwedge^2 S^+$$. We also estimate the decay rate at infinity for these solutions. 

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