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Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups$\\text{Sym}(n)$(in the sofic case) or the finite-dimensional unitary groups$\\text{U}(n)$(in the hyperlinear case)? In the case of$\\text{U}(n)$, the question can be asked with respect to different metrics and norms. This paper answers, for the first time, one of these versions, showing that there exist finitely presented groups which are not approximated by$\\text{U}(n)$with respect to the Frobenius norm$\\Vert T\\Vert _{\\text{Frob}}=\\sqrt{\\sum _{i,j=1}^{n}|T_{ij}|^{2}},T=[T_{ij}]_{i,j=1}^{n}\\in \\text{M}_{n}(\\mathbb{C})$. Our strategy is to show that some higher dimensional cohomology vanishing phenomena implies stability , that is, every Frobenius-approximate homomorphism into finite-dimensional unitary groups is close to an actual homomorphism. This is combined with existence results of certain nonresidually finite central extensions of lattices in some simple$p$-adic Lie groups. These groups act on high-rank Bruhat–Tits buildings and satisfy the needed vanishing cohomology phenomenon and are thus stable and not Frobenius-approximated.

We study Vertex Operator Algebras (VOAs) obtained from the H-twist of 3d N=4 linear quiver gauge theories. We find that H-twisted VOAs can be regarded as the “chiralization” of the extended Higgs branch: many of the ingredients of the Higgs branch are naturally “uplifted” into the VOAs, while conversely the Higgs branch can be recovered as the associated variety of the VOA. We also discuss the connection of our VOA with affine W-algebras. For example, we construct an explicit homomorphism from an affine W-algebra W-n+1(gln,fmin) into the H-twisted VOA for T[1n][2,1n-2][SU(n)] theories. Motivated by the relation with affine W-algebras, we introduce a reduction procedure for the quiver diagram, and use this to give an algorithm to systematically construct novel free-field realizations for VOAs associated with general linear quivers.

We prove that the R-matrix and Drinfeld presentations of the Yangian double in type A are isomorphic. The central elements of the completed Yangian double in type A at the critical level are constructed. The images of these elements under a Harish-Chandra-type homomorphism are calculated by applying a version of the Poincaré-Birkhoff-Witt theorem for the R-matrix presentation. These images coincide with the eigenvalues of the central elements in the Wakimoto modules.

In this paper, we apply some special functions to introduce a new class of control functions that help us define the concept of multi-stability. Further, we investigate the multi-stability of homomorphisms in C∗-algebras and Lie C∗-algebras, multi-stability of derivations in C∗-algebras, and Lie C∗-algebras for the following random operator equation via fixed point methods: μf(ð,x+y2)+μf(ð,x−y2)=f(ð,μx). In particular, for μ=1, the above equation turns out to be Jensen’s random operator equation.