Single particle operators and their correlators in free $$ \mathcal{N} $$ = 4 SYM

We consider a set of half-BPS operators in $$ \mathcal{N} $$ N = 4 super Yang-Mills theory which are appropriate for describing single-particle states of superstring theory on AdS5× S5. These single-particle operators are defined to have vanishing two-point functions with all multi-trace operators and therefore correspond to admixtures of single- and multi-traces. We find explicit formulae for all single-particle operators and for their two-point function normalisation. We show that single-particle U(N) operators belong to the SU(N) subspace, thus for length greater than one they are simply the SU(N) single-particle operators. Then, we point out that at large N, as the length of the operator increases, the single-particle operator naturally interpolates between the single-trace and the S3 giant graviton. At finite N, the multi-particle basis, obtained by taking products of the single-particle operators, gives a new basis for all half-BPS states, and this new basis naturally cuts off when the length of any of the single-particle operators exceeds the number of colours. From the two-point function orthogonality we prove a multipoint orthogonality theorem which implies vanishing of all near-extremal correlators. We then compute all maximally and next-to-maximally extremal free correlators, and we discuss features of the correlators when the extremality is lowered. Finally, we describe a half-BPS projection of the operator product expansion on the multi-particle basis which provides an alternative construction of four- and higher-point functions in the free theory.

An Example on Computing the Irreducible Representation of Finite Metacyclic Groups by Using Great Orthogonality Theorem Method

Representation theory is a study of real realizations of the axiomatic systems of abstract algebra. For any group, the number of possible representative sets of matrices is infinite, but they can all be reduced to a single fundamental set, called the irreducible representations of the group. This paper focuses on an example of finite metacyclic groups of class two of order 16. The irreducible representation of that group is found by using Great Orthogonality Theorem Method.

Mobility and kinematic simulations of cyclically symmetric deployable truss structures

Based on the mechanism of four-fold rigid origami, this study proposes a type of deployable truss structures that consist of repetitive basic parts and retain full cyclic symmetry in the folding/deployment process. On the basis of the irreducible representations and the great orthogonality theorem, symmetry-adapted analysis using group theory is described to identify the symmetry of mobility and kinematic behavior. Equivalent three-dimensional pin-jointed frameworks are employed for the symmetric structures. To verify that the structures can be foldable while retaining their full symmetries, numerical simulations on a series of structures with different symmetries and geometries are carried out. An artificial damping is introduced to stabilize the nonlinear folding behavior with singularity. Symmetry-adapted mobility analysis reveals that the structures of this type can be continuously folded with one degree-of-freedoms. Numerical simulations using the nonlinear iterative method accurately predict the folding behavior, as the results agree very well with the theoretic value.