Random Unitary Representations of Surface Groups I: Asymptotic Expansions

In this paper, we study random representations of fundamental groups of surfaces into special unitary groups. The random model we use is based on a symplectic form on moduli space due to Atiyah, Bott and Goldman. Let $$\Sigma _{g}$$ Σ g denote a topological surface of genus $$g\ge 2$$ g ≥ 2 . We establish the existence of a large n asymptotic expansion, to any fixed order, for the expected value of the trace of any fixed element of $$\pi _{1}(\Sigma _{g})$$ π 1 ( Σ g ) under a random representation of $$\pi _{1}(\Sigma _{g})$$ π 1 ( Σ g ) into $$\mathsf {SU}(n)$$ SU ( n ) . Each such expected value involves a contribution from all irreducible representations of $$\mathsf {SU}(n)$$ SU ( n ) . The main technical contribution of the paper is effective analytic control of the entire contribution from irreducible representations outside finite sets of carefully chosen rational families of representations.

Proton-neutron symplectic model description of $^{20}$Ne

A microscopic description of the low-lying positive-parity rotational bands in $^{20}$Ne is given within the framework of the symplectic-based proton-neutron shell-model approach provided by the proton-neutron symplectic model (PNSM). For this purpose a model Hamiltonian is used which includes an algebraic interaction, lying in the enveloping algebra of the $Sp(12,R)$ dynamical group of the PNSM, that introduces both horizontal and vertical mixings of different $SU(3)$ irreducible representations within the $Sp(12,R)$ irreducible collective space of $^{20}$Ne. A good overall description is obtained for the excitation energies of the ground and first two excited $\beta$ bands, as well as for the ground state intraband $B(E2)$ quadrupole collectivity and the known interband $B(E2)$ transition probabilities between the low-lying collective states without the use of an effective charge.

The coordinate algebra of a quantum symplectic sphere does not embed into any C*-algebra

In this note, we generalize a result of Mikkelsen–Szymański and show that, for every [Formula: see text], any bounded ∗-representation of the quantum symplectic sphere [Formula: see text] annihilates the first [Formula: see text] generators. We then classify irreducible representations of its coordinate algebra [Formula: see text].

Characterizing the universal rigidity of generic tensegrities

A tensegrity is a structure made from cables, struts, and stiff bars. A d-dimensional tensegrity is universally rigid if it is rigid in any dimension $$d'$$ d ′ with $$d'\ge d$$ d ′ ≥ d . The celebrated super stability condition due to Connelly gives a sufficient condition for a tensegrity to be universally rigid. Gortler and Thurston showed that super stability characterizes universal rigidity when the point configuration is generic and every member is a stiff bar. We extend this result in two directions. We first show that a generic universally rigid tensegrity is super stable. We then extend it to tensegrities with point group symmetry, and show that this characterization still holds as long as a tensegrity is generic modulo symmetry. Our strategy is based on the block-diagonalization technique for symmetric semidefinite programming problems, and our proof relies on the theory of real irreducible representations of finite groups.

Efficient rules for all conformal blocks

We formulate a set of general rules for computing d-dimensional four-point global conformal blocks of operators in arbitrary Lorentz representations in the context of the embedding space operator product expansion formalism [1]. With these rules, the procedure for determining any conformal block of interest is reduced to (1) identifying the relevant projection operators and tensor structures and (2) applying the conformal rules to obtain the blocks. To facilitate the bookkeeping of contributing terms, we introduce a convenient diagrammatic notation. We present several concrete examples to illustrate the general procedure as well as to demonstrate and test the explicit application of the rules. In particular, we consider four-point functions involving scalars S and some specific irreducible representations R, namely 〈SSSS〉, 〈SSSR〉, 〈SRSR〉 and 〈SSRR〉 (where, when allowed, R is a vector or a fermion), and determine the corresponding blocks for all possible exchanged representations.

Hexagonalization of Fishnet integrals. Part I. Mirror excitations

In this paper we consider a conformal invariant chain of L sites in the unitary irreducible representations of the group SO(1, 5). The k-th site of the chain is defined by a scaling dimension ∆k and spin numbers $$ \frac{\ell_k}{2},\frac{\ell_k}{2} $$ ℓ k 2 , ℓ k 2 The model with open and fixed boundaries is shown to be integrable at the quantum level and its spectrum and eigenfunctions are obtained by separation of variables. The transfer matrices of the chain are graph-builder operators for the spinning and inhomogeneous generalization of squared-lattice “fishnet” integrals on the disk. As such, their eigenfunctions are used to diagonalize the mirror channel of the Feynman diagrams of Fishnet conformal field theories. The separated variables are interpreted as momentum and bound-state index of the mirror excitations of the lattice: particles with SO(4) internal symmetry that scatter according to an integrable factorized $$ \mathcal{S} $$ S -matrix in (1 + 1) dimensions

Splitting fields of real irreducible representations of finite groups

We show that any irreducible representation ρ \rho of a finite group G G of exponent n n , realisable over R \mathbb {R} , is realisable over the field E ≔ Q ( ζ n ) ∩ R E≔\mathbb {Q}(\zeta _n)\cap \mathbb {R} of real cyclotomic numbers of order n n , and describe an algorithmic procedure transforming a realisation of ρ \rho over Q ( ζ n ) \mathbb {Q}(\zeta _n) to one over E E .