𝒩 = 4 superconformal Calogero models

Abstract

We continue the research initiated in hep-th/0607215 and apply our method of conformal automorphisms to generate various N =4 superconformal quantum many-body systems on the real line from a set of decoupled particles extended by fermionic degrees of freedom. The su(1, 1|2) invariant models are governed by two scalar potentials obeying a system of nonlinear partial differential equations which generalizes the Witten-Dijkgraaf-Verlinde-Verlinde equations. As an application, the N =4 superconformal extension of the three-particle (A-type) Calogero model generates a unique G 2 -type Hamiltonian featuring three-body interactions. We fully analyze the N =4 superconformal three-and four-particle models based on the root systems of A 1 ⊕ G 2 and F 4 , respectively. Beyond Wyllard's solutions we find a list of new models, whose translational non-invariance of the center-of-mass motion fails to decouple and extends even to the relative particle motion.

Remark on quantum mechanics with conformal Galilean symmetry

“…It also provided an efficient means for constructing various N = 2 and N = 4 superconformal many-body models in one dimension [19,22,23,24]. An elegant geometric interpretation of the similarity transformation as the inversion of the Klein model of the Lobachevsky plane was proposed in [25].…”

“…Firstly, with appropriate modifications the transformation can be used for constructing exactly solvable quantum mechanics models and building the complete set of eigenstates as, for example, in [18,26]. Secondly, as a set of decoupled particle is straightforward to supersymmetrize, the map provides an efficient means for constructing various supersymmetric many-body models in the spirit of [19,22,23,24,27]. Thirdly, with some modification the mapping can be applied in the context of trapped Fermi gases at unitarity (see [28] and references therein).…”

Superconformal Yang-Mills quantum mechanics and Calogero model with $$ {\text{OSp}}\left( {\mathcal{N}|{2},\mathbb{R}} \right) $$ symmetry

“…the AdS 2 /CFT 1 correspondence, is least understood [2]. Although there has been much work on conformal and superconformal mechanics [3][4][5][6][7][8][9][10], the connection to string theory and supergravity is less clear: we do not have the now familiar picture of the gravity side as the near-horizon geometry of coincident branes with the worldvolume gauge theory living on the boundary of the AdS near-horizon region.…”

N=4 superconformal n-particle mechanics via superspace

“…Although conformal multi-particle quantum mechanics (in one space dimension) is a subject with a long and rich history, its N =4 superconformal extension has been achieved only recently [1,2,3,4,5,6,7,8]. Enlarging the conformal algebra su(1, 1) to su(1, 1|2) (with central charge) imposes severe constraints on the particle interactions, which are not easily solved.…”

“…Keeping in mind the transformation properties of the covariant spinor derivatives D a and D a , 5) one may check that the constraints (2.1) are invariant under the N =4 superconformal group if the superfields u A and φ A transform like 6) respectively. Thus, the superfields φ A are superconformal scalars while the u A are vectors.…”