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1. Rigid automorphisms of linking systems

   https://doi.org/10.1017/fms.2021.17  

   Glauberman, George; Lynd, Justin (January 2021, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract A rigid automorphism of a linking system is an automorphism that restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that each rigid automorphism of a centric linking system is inner. We prove that the group of rigid outer automorphisms of a linking system at the prime $$2$$ is elementary abelian and that it splits over the subgroup of rigid inner automorphisms. In a second result, we show that if an automorphism of a finite group G restricts to the identity on the centric linking system for G , then it is of $p'$ -order modulo the group of inner automorphisms, provided G has no nontrivial normal $p'$ -subgroups. We present two applications of this last result, one to tame fusion systems. 

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2. Deconfined quantum criticality of nodal $d$ -wave superconductivity, Néel order, and charge order on the square lattice at half-filling

   https://doi.org/10.1103/PhysRevResearch.6.033018  

   Christos, Maine; Shackleton, Henry; Sachdev, Subir; Luo, Zhu-Xi (July 2024, Physical Review Research)

   We consider a SU(2) lattice gauge theory on the square lattice, with a single fundamental complex fermion and a single fundamental complex boson on each lattice site. Projective symmetries of the gauge-charged fermions are chosen so that they match with those of the spinons of the $\pi$-flux spin liquid. Global symmetries of all gauge-invariant observables are chosen to match with those of the particle-hole symmetric electronic Hubbard model at half-filling. Consequently, both the fundamental fermion and fundamental boson move in an average background $\pi$-flux, their gauge-invariant composite is the physical electron, and eliminating gauge fields in a strong gauge-coupling expansion yields an effective extended Hubbard model for the electrons. The SU(2) gauge theory displays several confining/Higgs phases: a nodal $d$-wave superconductor, and states with Néel, valence-bond solid, charge, or staggered current orders. There are also a number of quantum phase transitions between these phases that are very likely described by $(2+1)$-dimensional deconfined conformal gauge theories, and we present large flavor expansions for such theories. These include the phenomenologically attractive case of a transition between a conventional insulator with a charge gap and Néel order, and a conventional $d$-wave superconductor with gapless Bogoliubov quasiparticles at four nodal points in the Brillouin zone. We also apply our approach to the honeycomb lattice, where we find a bicritical point at the junction of Néel, valence bond solid (Kekulé), and Dirac semimetal phases. Published by the American Physical Society2024 

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3. Twisted Fibrations in M/F-theory

   https://doi.org/10.1007/JHEP01(2024)017  

   Anderson, Lara B; Gray, James; Oehlmann, Paul-Konstantin (January 2024, Journal of High Energy Physics)

   In this work we investigate 5-dimensional theories obtained from M-theory on genus one fibered threefolds which exhibit twisted algebras in their fibers. We provide a base-independent algebraic description of the threefolds and compute light 5D BPS states charged under finite sub-algebras of the twisted algebras. We further construct the Jacobian fibrations that are associated to 6-dimensional F-theory lifts, where the twisted algebra is absent. These 6/5-dimensional theories are compared via twisted circle reductions of F-theory to M-theory. In the 5-dimensional theories we discuss several geometric transitions that connect twisted with untwisted fibrations. We present detailed discussions of $$e_6^{(2)}$$, $$so(8)^{(3)}$$ and $$su(3)^{(2)}$$ twisted fibers and provide several explicit example threefolds via toric constructions. Finally, limits are considered in which gravity is decoupled, including Little String Theories for which we match 2-group symmetries across twisted T-dual theories. 

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4. 2, 12, 117, 1959, 45171, 1170086, …: a Hilbert series for the QCD chiral Lagrangian

   https://doi.org/10.1007/JHEP01(2021)142  

   Gráf, Lukáš; Henning, Brian; Lu, Xiaochuan; Melia, Tom; Murayama, Hitoshi (January 2021, Journal of High Energy Physics)

   null (Ed.)

   A bstract We apply Hilbert series techniques to the enumeration of operators in the mesonic QCD chiral Lagrangian. Existing Hilbert series technologies for non-linear realizations are extended to incorporate the external fields. The action of charge conjugation is addressed by folding the $$ \mathfrak{su}(n) $$ su n Dynkin diagrams, which we detail in an appendix that can be read separately as it has potential broader applications. New results include the enumeration of anomalous operators appearing in the chiral Lagrangian at order p 8 , as well as enumeration of CP -even, CP -odd, C -odd, and P -odd terms beginning from order p 6 . The method is extendable to very high orders, and we present results up to order p 16 . (The title sequence is the number of independent C -even and P -even operators in the mesonic QCD chiral Lagrangian with three light flavors of quarks, at chiral dimensions p 2 , p 4 , p 6 , …) 

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5. Topological Symmetry Groups of the Petersen Graphs

   https://doi.org/10.3390/sym15061267  

   Elzie, Deion; Fridhi, Samir; Mellor, Blake; Silva, Daniel; Wilson, Robin T. (June 2023, Symmetry)

   The topological symmetry group of an embedding Γ of an abstract graph γ in S3 is the group of automorphisms of γ that can be realized by homeomorphisms of the pair (S3,Γ). These groups are motivated by questions about the symmetries of molecules in space. The Petersen family of graphs is an important family of graphs for many problems in low-dimensional topology, so it is desirable to understand the possible groups of symmetries of their embeddings in space. In this paper, we find all the groups that can be realized as topological symmetry groups for each of the graphs in the Petersen family. Along the way, we also complete the classification of the realizable topological symmetry groups for K3,3. 

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