Novel integrable spin-particle models from gauge theories on a cylinder

Abstract

We find and solve a large class of integrable dynamical systems which includes Calogero-Sutherland models and various novel generalizations thereof. In general they describe N interacting particles moving on a circle and coupled to an arbitrary number, m, of su(N ) spin degrees of freedom with interactions which depend on arbitrary real parameters x j , j = 1, 2, . . . , m. We derive these models from SU(N ) Yang-Mills gauge theory coupled to non-dynamic matter and on spacetime which is a cylinder. This relation to gauge theories is used to prove integrability, to construct conservation laws, and solve these models.Integrable models have always played a central role in classical and quantum mechanics. Most prominent examples, like the Kepler problem, are systems with few (≤ 3) degrees of freedom. An important exception is a class of integrable N-particle models associated with the names Calogero, Moser and Sutherland [1, 2] (for review see [3]). These are models for identical particles moving on one dimensional space and interacting via certain repulsive twobody potentials v(r). A well-known example is v(r) ∝ g 2 / sin 2 (gr) (which includes v(r) ∝ 1/r 2 in the limit g → 0), and we refer to the corresponding model as CS model. It is known that these models allow for interesting generalizations which also have dynamic spin degrees of freedom [4,5]. The CS model and its generalizations have recently received quite some attention in different contexts. Here we only mention their relation to gauged matrix models [6] and gauge theories on a cylinder [7] which will be relevant for us. a a the latter relation is implicit already in earlier work; see e.g. [8]

Finding and solving Calogero–Moser type systems using Yang–Mills gauge theories

“…We will also refer to the former case as Calogero model and the latter as Sutherland model. We then derive and extend the results for the novel systems found in [1] (Section 4.4). A novel class of models which can not be solved in such an explicit manner but still should be integrable is discussed in Section 4.5.…”

“…We now solve the equations of motion for the system given by the Hamiltonian (84) and the Poisson brackets (88)-(89). More general than in [1], we do not assume that q α (t) etc. all are real.…”

“…In a previous paper [1] we presented a novel class of integrable spin-particle systems which contains known integrable systems of Calogero-Moser (CM) type [2] and certain known spin generalization [3,4] thereof as special cases, and many other generalizations which (to our knowledge) were not known before. Our method not only allowed us to find and prove integrability of these models but also to solve them explicitly.…”

Calogero-Moser Models with Noncommutative Spin Interactions

“…Most other models can be obtained as appropriate reductions of this model, taking advantage of its discrete or continuous symmetries [22]. In particular, generalizations involving U(n) noninvariant interactions can be obtained this way, recovering the trigonometric models derived in [23,24] and extending them to the elliptic case [22].…”

Small data global regularity for half-wave maps

“…Here we define the operator p´△q a specification necessary on account of the fact that u does not vanish at infinity, but instead approaches some p P S 2 , while ∇u does vanish at infinity. In fact, the expression p´△q 1 2 u under our current definition is then well-defined since ∇ t,x upt,¨q P L p pR n q for some p P p1, 8q, for all t.…”