Seminar “Modern Mathematical Physics”

Date and Time: Tuesday, 20 October 2020, at 1:00 PM

Venue: Online conference in Zoom, Bogoliubov Laboratory of Theoretical Physics

Seminar topic: «Classical solutions in flag manifold nonlinear sigma-models»

Speaker: Amari Yuki

Abstract:

Nonlinear sigma (NLσ-) models appear as an effective model of various fundamental theories, e.g. the Yang-Mills theory and Heisenberg model, and can be viewed as a good toy model of the pure Yang-Mills theories. Classical solutions like solitons, instantons and sphalerons in an NLσ-model have been enthusiastically studied as key objects to understand non-perturbative nature of the NLσ-model as well as of fundamental theories from which the model can be derived as an effective model.

In this seminar, we discuss classical solutions in NLσ-models on flag manifolds of SU(N), which are coset spaces of SU(N) including the complex projective space CP^𝑁−1 = SU(N) U(N−1) and full flag space F_N−1 = SU(N)/U(1)N−1. Here, we call the NLσ-model on a manifold M the ℳ-NLσ-model. We first describe how the models can be derived from the pure Yang-Mills theories and spin systems as concrete examples of fundamental theories. Then we review how to construct classical solutions and some properties of them. In two space dimension, each flag manifold NLσ-model possesses 2D instanton or 2D Skyrmion solutions. For the case of CP^𝑁−1, the instantons have been studied since the ’70s.

In contrast, those in the F_N−1 NLσ-model have rarely studied until recently despite their plenty of potential application to physics. This may well be due to complicated structures of the manifold F_N−1. It is well known that instantons in the CP^𝑁−1 model can be obtained by solving first-order differential equations so-called BPS equations. On the other hand, we show that in order to derive instantons in the F_N−1 model we need to solve the BPS equation and an additional first-order equation simultaneously.

In addition, we discuss a three-dimensional topological soliton with knot topology called Hopfion in the F_N−1 NLσ-model with a stabilizing term.