Collective field formulation of the multi-species Calogero model and its duality symmetries

Abstract

We study the collective field formulation of a restricted form of the multispecies Calogero model, in which the three-body interactions are set to zero. We show that the resulting collective field theory is invariant under certain duality transformations, which interchange, among other things, particles and antiparticles, and thus generalize the well-known strong-weak coupling duality symmetry of the ordinary Calogero model. We identify all these dualities, which form an Abelian group, and study their consequences. We also study the ground state and small fluctuations around it in detail, starting with the two-species model, and then generalizing to an arbitrary number of species.

Solutions of coupled BPS equations for two-family Calogero and matrix models

“…III can be applied to a wide class of coupled BPS equations. In the general case for 1 2 ¼ 2 12 Þ 1, it can be shown that the coupled BPS equations (8) and (9) lead to the following relation between 1 and 2 :…”

Fluctuations around periodic BPS-density waves in the Calogero model

“…Thus, the duality transformations (1.7) exchange particles and antiparticles. With this interpretation we always have 8) and thus the first two terms in (1.1) are manifestly positive. (For more details see e.g.…”

“…A similar observation was also made concerning the more general coupled non-BPS variational equations. All this was the consequence of the invariance of the collective-field Hamiltonian of the twofamily Calogero model under an Abelian group of strong-weak-coupling dualities [8], analogous to (1.7). In [24] and [25] it was shown that a large class of solitons in the two-family Calogero model can be obtained by reducing it to the effectively one-family Calogero model.…”

Waves and solitons in the two-family Calogero model

“…The multispecies Calogero model has been extensively studied in the quantum mechanical [1][2][3][4] and in the collective-field theory context [5][6][7][8][9]. It represents the generalization of the ordinary Calogero model with identical particles to the Calogero model with nonidentical particles.…”

“…(9) and (10), we end up with the coupled BPS equations (6) and (7) for 1 and 2 : In this way, we have reduced the two-family Calogero model to two one-family Calogero decoupled systems. We will prove this decoupling in Sec.…”