Nonrelativistic conformal groups

Abstract

In this work a systematic study of finite-dimensional nonrelativistic conformal groups is carried out under two complementary points of view. First, the conformal Killing equation is solved to obtain a whole family of finite-dimensional conformal algebras corresponding to each of the Galilei and Newton–Hooke kinematical groups. Some of their algebraic and geometrical properties are studied in a second step. Among the groups included in these families one can identify, for example, the contraction of the Minkowski conformal group, the analog for a nonrelativistic de Sitter space, or the nonextended Schrödinger group.

Extended Galilean symmetries of non-relativistic strings

“…We will write the generator of the most general point transformation and will impose on it the condition of Noether symmetry, which gives the associated non-relativistic Killing equations that one must solve. For a massive non-relativistic particle, the maximal set of symmetries is larger than the Galilei group, and it is in fact the Schrödinger group [20,21], which is the group corresponding to the z = 2 case of an infinite set of z-Galilean conformal algebras [22][23][24].…”

Logarithmic correlators or responses in non-relativistic analogues of conformal invariance

“…t −→ t , r −→ r/c ; c → ∞ (1.2) one obtains the conformal Galilean algebra cga(d), apparently first identified in [38], but independently rediscovered in different contexts [40,82]. It is usually obtained, by a contraction, as the non-relativistic limit of the (d + 2)-dimensional conformal algebra (itself obtained by a non-relativistic holographic construction) [42,76,2,3,71,72,61].…”

“…Alternatively, one may also start from the so-called l-Galilei algebra [40] and recognise the cga as the l = 1 special case [40,82]. The embedding cga(1) ⊂ conf(3) ∼ = B 2 and the associated parabolic extension is illustrated in figure 2c.…”

Aspects of infinite dimensional ℓ-super Galilean conformal algebra

“…These can be regarded as an infinite dimensional extension of the GCAs of finite dimension which were found in [29,30,31]. The structure of the finite dimensional GCAs is a semidirect sum of sl(2, R) ⊕ so(d) and an abelian ideal.…”