Geometry of special Galileons

A theory known as special Galileon has recently attracted considerable interest due to its peculiar properties. It has been shown that it represents an extremal member of the set of effective field theories with an enhanced soft limit. This property makes its tree-level $S$-matrix fully on-shell reconstructible and representable by means of the Cachazo-He-Yuan representation. The enhanced soft limit is a consequence of new hidden symmetry of the special Galileon action; however, until now, the origin of this peculiar symmetry has remained unclear. In this paper we interpret this symmetry as a special transformation of the coset space $\mathrm{GAL}(D,1)/SO(1,D-1)$ and show that there exists a three-parametric family of invariant Galileon actions. The latter family is closed under the duality which appears as a natural generalization of the above mentioned symmetry. We also present a geometric construction of the special Galileon action using a $D$-dimensional brane propagating in $2D$-dimensional flat pseudo-Riemannian space. Within such a framework, the special Galileon symmetry emerges as a $U(1,D-1)$ symmetry of the target space, which can be treated as a $D$-dimensional Kähler manifold. Such a treatment allows for classification of the higher order invariant Lagrangians needed as counterterms on the quantum level. We also briefly comment on the relation between such higher order Lagrangians and the Lagrangians that are invariant with respect to the polynomial shift symmetry.

Figure 1

The rules for graphical description of the invariant Lagrangians. The basic graphical building blocks correspond to (a), while simple examples are depicted as (b) and (c).

Figure 2

The graphical representation of the inverse effective metric $g^{\mu \nu }$.

Figure 3

The graphical representation of the Riemann tensor $R_{\mu \nu \alpha \beta }$. The blob represents the inverse metric $g^{\rho \sigma }$ (cf. Fig. 2).

Figure 4

The graphical representation of the extrinsic curvature ${\mathcal{K}}_{\alpha \mu \nu }$ and the Christoffel symbol ${\mathrm{\Gamma }}_{\alpha \mu \nu }$.

Figure 5

The measure ${\mathrm{d}}^{D}Z$ in terms of the Galileon field.

Figure 6

The measure $\sqrt{{\mathrm{d}}^{D}Z{\mathrm{d}}^D\overline{Z}}$ in terms of the Galileon field.

Figure 7

The three independent invariants for construction of the general $\delta =4$ Lagrangian. The blob corresponds to the inverse metric tensor (see Fig. 2).

Figure 8

The Lagrangian ${\mathcal{L}}_4^{{\varphi }^4}$ (in the case when the higher order kinetic term vanishes) after integration by parts.