Generalized dynamical spin chain and 4-loop integrability in [formula omitted] superconformal Chern–Simons theory

Abstract

We revisit unitary representation of centrally extended psu ( 2 | 2 ) excitation superalgebra. We find most generally that ‘pseudo-momentum’, not lattice momentum, diagonalizes spin chain Hamiltonian and leads to generalized dynamic spin chain. All known results point to lattice momentum diagonalization for N = 4 super-Yang–Mills theory. Having different interacting structure, we ask if N = 6 superconformal Chern–Simons theory provides an example of pseudo-momentum diagonalization. For SO ( 6 ) sector, we study maximal shuffling and next-to-maximal shuffling terms in the dilatation operator and compare them with results expected from psu ( 2 | 2 ) superalgebra and integrability. At two loops, we rederive maximal shuffling term (3-site) and find perfect agreement with known results. At four loops, we first find absence of next-to-maximal shuffling term (4-site), in agreement with prediction based on integrability. We next extract maximal shuffling term (5-site), the most relevant term for checking the possibility of pseudo-momentum diagonalization. Curiously, we find that result agrees with integrability prediction based on lattice momentum, as in N = 4 super-Yang–Mills theory. Consistency of our results is fully ensured by checks of renormalizability up to six loops.