CR embeddability of quotients of the Rossi sphere via spectral theory

We look at the action of finite subgroups of SU(2) on [Formula: see text], viewed as a CR manifold, both with the standard CR structure as the unit sphere in [Formula: see text] and with a perturbed CR structure known as the Rossi sphere. We show that quotient manifolds from these actions are indeed CR manifolds, and relate the order of the subgroup of SU(2) to the asymptotic distribution of the Kohn Laplacian’s eigenvalues on the quotient. We show that the order of the subgroup determines whether the quotient of the Rossi sphere by the action of that subgroup is CR embeddable. Finally, in the unperturbed case, we prove that we can determine the size of the subgroup by using the point spectrum.

Seeking SUSY fixed points in the 4 − ϵ expansion

We use the 4 − ϵ expansion to search for fixed points corresponding to 2 + 1 dimensional $$ \mathcal{N} $$ N =1 Wess-Zumino models of NΦ scalar superfields interacting through a cubic superpotential. In the NΦ = 3 case we classify all SUSY fixed points that are perturbatively unitary. In the NΦ = 4 and NΦ = 5 cases, we focus on fixed points where the scalar superfields form a single irreducible representation of the symmetry group (irreducible fixed points). For NΦ = 4 we show that the S5 invariant super Potts model is the only irreducible fixed point where the four scalar superfields are fully interacting. For NΦ = 5, we go through all Lie subgroups of O(5) and use the GAP system for computational discrete algebra to study finite subgroups of O(5) up to order 800. This analysis gives us three fully interacting irreducible fixed points. Of particular interest is a subgroup of O(5) that exhibits O(3)/Z2 symmetry. It turns out this fixed point can be generalized to a new family of models, with NΦ = $$ \frac{\mathrm{N}\left(\mathrm{N}-1\right)}{2} $$ N N − 1 2 − 1 and O(N)/Z2 symmetry, that exists for arbitrary integer N≥3.

Two-ended quasi-transitive graphs

The well-known characterization of two-ended groups says that every two-ended group can be split over finite subgroups which means it is isomorphic to either by a free product with amalgamation [Formula: see text] or an HNN-extension [Formula: see text], where [Formula: see text] is a finite group and [Formula: see text] and [Formula: see text]. In this paper, we show that there is a way in order to spilt two-ended quasi-transitive graphs without dominated ends and two-ended transitive graphs over finite subgraphs in the above sense. As an application of it, we characterize all groups acting with finitely many orbits almost freely on those graphs.

On exotic consistent anomalies in (1+1)$d$: A ghost story

We revisit ’t Hooft anomalies in (1+1)dd non-spin quantum field theory, starting from the consistency and locality conditions, and find that consistent U(1) and gravitational anomalies cannot always be canceled by properly quantized (2+1)dd classical Chern-Simons actions. On the one hand, we prove that certain exotic anomalies can only be realized by non-reflection-positive or non-compact theories; on the other hand, without insisting on reflection-positivity, the exotic anomalies present a caveat to the inflow paradigm. For the mixed U(1) gravitational anomaly, we propose an inflow mechanism involving a mixed U(1)\times×SO(2) classical Chern-Simons action with a boundary condition that matches the SO(2) gauge field with the (1+1)dd spin connection. Furthermore, we show that this mixed anomaly gives rise to an isotopy anomaly of U(1) topological defect lines. The isotopy anomaly can be canceled by an extrinsic curvature improvement term, but at the cost of creating a periodicity anomaly. We survey the holomorphic bcbc ghost system which realizes all the exotic consistent anomalies, and end with comments on a subtlety regarding the anomalies of finite subgroups of U(1).

On Periodic Shunkov’s Groups with Almost Layer-finite Normalizers of Finite Subgroups

Layer-finite groups first appeared in the work by S. N. Chernikov (1945). Almost layer-finite groups are extensions of layer-finite groups by finite groups. The class of almost layer-finite groups is wider than the class of layer-finite groups; it includes all Chernikov groups, while it is easy to give examples of Chernikov groups that are not layer-finite. The author develops the direction of characterizing well-known and well-studied classes of groups in other classes of groups with some additional (rather weak) finiteness conditions. A Shunkov group is a group 𝐺 in which for any of its finite subgroups 𝐾 in the quotient group <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msub><mi>N</mi><mi>G</mi></msub><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mi>K</mi></mfrac></math> any two conjugate elements of prime order generate a finite subgroup. In this paper, we prove the properties of periodic not almost layer-finite Shunkov groups with condition: the normalizer of any finite nontrivial subgroup is almost layer-finite. Earlier, these properties were proved in various articles of the author, as necessary, sometimes under some conditions, then under others (the minimality conditions for not almost layer-finite subgroups, the absence of second-order elements in the group, the presence of subgroups with certain properties in the group). At the same time, it was necessary to make remarks that this property is proved in almost the same way as in the previous work, but under different conditions. This eliminates the shortcomings in the proofs of many articles by the author, in which these properties are used without proof.