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High Energy Physics - Theory, hep-th,Mathematical Physics, math-ph,Mathematics, Mathematical Physics, math.MP,Nonlinear Sciences, Exactly Solvable and Integrable Systems, nlin.SI
Abstract:
Inspired by the integrable structures appearing in weakly coupled planar N=4
super Yang-Mills theory, we study Q-operators and Yangian invariants of
rational integrable spin chains. We review the quantum inverse scattering
method along with the Yang-Baxter equation which is the key relation in this
systematic approach to study integrable models. Our main interest concerns
rational integrable spin chains and lattice models. We recall the relation
among them and how they can be solved using Bethe ansatz methods incorporating
so-called Q-functions. In order to remind the reader how the Yangian emerges in
this context, an overview of its so-called RTT-realization is provided. The
main part is based on the author's original publications. Firstly, we construct
Q-operators whose eigenvalues yield the Q-functions for rational homogeneous
spin chains. The Q-operators are introduced as traces over certain monodromies
of R-operators. Our construction allows us to derive the hierarchy of commuting
Q-operators and the functional relations among them. We study how the
nearest-neighbor Hamiltonian and in principle also higher local charges can be
extracted from the Q-operators directly. Secondly, we formulate the Yangian
invariance condition, also studied in relation to scattering amplitudes of N=4
super Yang-Mills theory, in the RTT-realization. We find that Yangian
invariants can be interpreted as special eigenvectors of certain inhomogeneous
spin chains. This allows us to apply the algebraic Bethe ansatz and derive the
corresponding Bethe equations that are relevant to construct the invariants. We
examine the connection between the Yangian invariant spin chain eigenstates
whose components can be understood as partition functions of certain 2d lattice
models and tree-level scattering amplitudes of the four-dimensional gauge
theory. Finally, we conclude and discuss some future directions.
