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1. Fractionalization of coset non-invertible symmetry and exotic Hall conductance

   https://doi.org/10.21468/SciPostPhys.17.3.095  

   Hsin, Po-Shen; Kobayashi, Ryohei; Zhang, Carolyn (January 2024, SciPost Physics)

   We investigate fractionalization of non-invertible symmetry in (2+1)D topological orders. We focus on coset non-invertible symmetries obtained by gauging non-normal subgroups of invertible0 $0$-form symmetries. These symmetries can arise as global symmetries in quantum spin liquids, given by the quotient of the projective symmetry group by a non-normal subgroup as invariant gauge group. We point out that such coset non-invertible symmetries in topological orders can exhibit symmetry fractionalization: each anyon can carry a “fractional charge” under the coset non-invertible symmetry given by a gauge invariant superposition of fractional quantum numbers. We present various examples using field theories and quantum double lattice models, such as fractional quantum Hall systems with charge conjugation symmetry gauged and finite group gauge theory from gauging a non-normal subgroup. They include symmetry enrichedS_3 $S_3$andO(2) $O\left(2\right)$gauge theories. We show that such systems have a fractionalized continuous non-invertible coset symmetry and a well-defined electric Hall conductance. The coset symmetry enforces a gapless edge state if the boundary preserves the continuous non-invertible symmetry. We propose a general approach for constructing coset symmetry defects using a “sandwich” construction: non-invertible symmetry defects can generally be constructed from an invertible defect sandwiched by condensation defects. The anomaly free condition for finite coset symmetry is also identified. 

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2. Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings

   https://doi.org/10.1007/s40993-022-00388-9  

   Adžaga, Nikola; Chidambaram, Shiva; Keller, Timo; Padurariu, Oana (October 2022, Research in Number Theory)

   Abstract We complete the computation of all$$\mathbb {Q}$$ $Q$-rational points on all the 64 maximal Atkin-Lehner quotients$$X_0(N)^*$$ $X_0{\left(N\right)}^{\ast }$such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty–Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method (from a forthcoming preprint of the fourth-named author) combined with the Mordell-Weil sieve. Additionally, for square-free levelsN, we classify all$$\mathbb {Q}$$ $Q$-rational points as cusps, CM points (including their CM field andj-invariants) and exceptional ones. We further indicate how to use this to compute the$$\mathbb {Q}$$ $Q$-rational points on all of their modular coverings. 

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3. Second order $$L_p$$ estimates for subsolutions of fully nonlinear equations

   https://doi.org/10.1007/s00526-025-02929-3  

   Dong, Hongjie; Kitano, Shuhei (February 2025, Calculus of Variations and Partial Differential Equations)

   Abstract We obtain new$$L_p$$ $L_p$estimates for subsolutions to fully nonlinear equations. Based on our$$L_p$$ $L_p$estimates, we further study several topics such as the third and fourth order derivative estimates for concave fully nonlinear equations, critical exponents of$$L_p$$ $L_p$estimates and maximum principles, and the existence and uniqueness of solutions to fully nonlinear equations on the torus with free terms in the$$L_p$$ $L_p$spaces or in the space of Radon measures. 

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4. Analytical sphere–thin rod interaction potential

   https://doi.org/10.1140/epje/s10189-025-00480-9  

   Wang, Junwen; Cheng, Shengfeng (April 2025, The European Physical Journal E)

   AbstractA compact analytical form is derived through an integration approach for the interaction between a sphere and a thin rod of finite and infinite lengths, with each object treated as a continuous medium of material points interacting by the Lennard-Jones 12-6 potential and the total interaction potential as a summation of the pairwise potential between material points on the two objects. Expressions for the resultant force and torque are obtained. Various asymptotic limits of the analytical sphere–rod potential are discussed. The integrated potential is applied to investigate the adhesion between a sphere and a thin rod. When the rod is sufficiently long and the sphere sufficiently large, the equilibrium separation between the two (defined as the distance from the center of the sphere to the axis of the rod) is found to be well approximated as$$a+0.787\sigma $$ $a+0.787\sigma$, whereais the radius of the sphere and$$\sigma $$ $\sigma$is the unit of length of the Lennard–Jones potential. Furthermore, the adhesion between the two is found to scale with$$\sqrt{a}$$ $\sqrt{a}$. Graphic abstract) 

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5. Stochastic quantisation of Yang–Mills–Higgs in 3D

   https://doi.org/10.1007/s00222-024-01264-2  

   Chandra, Ajay; Chevyrev, Ilya; Hairer, Martin; Shen, Hao (August 2024, Inventiones mathematicae)

   Abstract We define a state space and a Markov process associated to the stochastic quantisation equation of Yang–Mills–Higgs (YMH) theories. The state space$$\mathcal{S}$$ $S$is a nonlinear metric space of distributions, elements of which can be used as initial conditions for the (deterministic and stochastic) YMH flow with good continuity properties. Using gauge covariance of the deterministic YMH flow, we extend gauge equivalence ∼ to$$\mathcal{S}$$ $S$and thus define a quotient space of “gauge orbits”$$\mathfrak {O}$$ $O$. We use the theory of regularity structures to prove local in time solutions to the renormalised stochastic YMH flow. Moreover, by leveraging symmetry arguments in the small noise limit, we show that there is a unique choice of renormalisation counterterms such that these solutions are gauge covariant in law. This allows us to define a canonical Markov process on$$\mathfrak {O}$$ $O$(up to a potential finite time blow-up) associated to the stochastic YMH flow. 

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