Galilean Conformal Algebra Research Articles

Galilean fluids from non-relativistic gravity

The 1/c-expansion of general relativity appropriately sourced by matter can be used to derive an action principle for Newtonian gravity. The gravitational part of this action is known as non-relativistic gravity (NRG). It is possible to source NRG differently and in such a way that one can construct solutions that are not described by Newtonian gravity (as they do not admit a notion of absolute time). It is possible to include a negative cosmological constant such that NRG admits a non-relativistic AdS solution. This non-relativistic AdS vacuum has Killing vectors that form the Galilean conformal algebra and a boundary that admits a conformal class of Newton-Cartan geometries. This begs the question of whether there exists an analogue of the fluid/gravity correspondence for NRG. In this paper we derive a non-relativistic AdS brane solution of NRG and confirm that it corresponds to the 1/c2-expansion of the AdS black brane geometry. We perform a Galilean boost of the non-relativistic AdS brane and derive the associated boundary energy-momentum tensor. We then show that this is the energy-momentum tensor of a massless Galilean fluid and explain how this is linked to the conformal isometries of the boundary. Along the way, we also present several new results for the theory of non-relativistic gravity itself. In particular we present a rewriting that greatly shortens and simplifies the equations of motion of the NRG action.

Left-symmetric superalgebra structures on the super Galilean conformal algebra

The compatible left-symmetric superalgebra structures on the super Galilean conformal algebra with some natural grading conditions are completely determined. The results of earlier work on left-symmetric superalgebra structures on the infinite-dimensional Lie superalgebra g studied by Chen and Hassan play an essential role in determining these compatible structures.

Characterization of simple smooth modules

In this paper, we characterize simple smooth modules over some infinite-dimensional Z-graded Lie algebras. More precisely, we prove that if one specific positive root element of a Z-graded Lie algebra g locally finitely acts on a simple g-module V, then V is a smooth g-module. These infinite-dimensional Z-graded Lie algebras include the Virasoro algebra, affine-Virasoro algebras, the (twisted, mirror) Heisenberg-Virasoro algebras, the planar Galilean conformal algebra, and many others. This result for untwisted affine Kac-Moody algebras holds unless we change the condition from “locally finitely” to “locally nilpotently”. We also show that these are not the case for the Heisenberg algebra.

Simple restricted modules for the universal central extension of the planar Galilean conformal algebra

In this paper, we construct and give a characterization of simple restricted modules over the universal central extension G˜ of the planar Galilean conformal algebra G. As a result, we not only recover many known simple G˜-modules (resp. G-modules), but also construct a lot of new ones. In particular, we recover the main results in [13] and [19].

On higher-dimensional Carrollian and Galilean conformal field theories

In this paper, we study the Carrollian and Galilean conformal field theories (CCFT and GCFT) in d>2d>2 dimensions. We construct the highest weight representations (HWR) of Carrollian and Galilean conformal algebra (CCA and GCA). Even though the two algebras have different structures, their HWRs share similar structure, because their rotation subalgebras are isomorphic. In both cases, we find that the finite dimensional representations are generally reducible but indecomposable, and can be organized into the multiplets. Moreover, it turns out that the multiplet representations in d>2d>2 CCA and GCA carry not only the simple chain structure appeared in logCFT or 2d2d GCFT, but also more generally the net structures. We manage to classify all the allowed chain representations. Furthermore we discuss the two-point and three-point correlators by using the Ward identities. We mainly focus on the two-point correlators of the operators in chain representations. Even in this relative simple case, we find some novel features: multiple-level structure, shortage of the selection rule on the representations, undetermined 2-pt coefficients, etc.. We find that the non-trivial correlators could only appear for the representations of certain structure, and the correlators are generally polynomials of time coordinates for CCFT (spacial coordinates for GCFT), whose orders depend on the levels of the correlators.

2-Local derivations on the planar Galilean conformal algebra

This paper is devoted to studying 2-local derivations on the planar Galilean conformal algebra. We prove that every 2-local derivation on the planar Galilean conformal algebra is a derivation.

Lie super-bialgebra structures on the two dimensional supersymmetric Galilean conformal algebra

Let be the two dimensional supersymmetric Galilean conformal algebra. In this paper, we obtain that Lie superalgebra admits only triangular coboundary Lie super-bialgebra structures and the proof is mainly based on the computation of the first cohomology group of with coefficients in the tensor product of its adjoint module.

Maximally symmetric nonlinear extension of electrodynamics with Galilean conformal symmetries

A maximally symmetric nonlinear extension of Maxwell's theory in four dimensions called ModMax has been recently introduced in the literature. This theory preserves both electromagnetic duality and conformal invariance of the linear theory. In this short paper, we introduce a Galilean cousin of the ModMax theory, written in a covariant formalism, that is explicitly shown to be invariant under Galilean conformal symmetries. We discuss the construction of such a theory involving Galilean electromagnetic invariants and show how the classical structure of the theory is invariant under the action of Galilean conformal algebra.

Whittaker modules for the planar Galilean conformal algebra and its central extension

Let be the planar Galilean conformal algebra and be its universal central extension. Then (resp. ) admits a triangular decomposition: (resp. ). In this paper, we study universal and generic Whittaker -modules (resp. -modules) of type where is a Lie algebra homomorphism. We classify the isomorphism classes of universal and generic Whittaker modules. Moreover, we show that a generic Whittaker module of type is simple if and only if is nonsingular. For the nonsingular case, we completely determine the Whittaker vectors in universal and generic Whittaker modules. For the singular case, we concretely construct some proper submodules of generic Whittaker modules.

Representations of the Planar Galilean Conformal Algebra

Representations of the Planar Galilean Conformal Algebra

Structures of generalized loop planar Galilean conformal algebras

In this article, we mainly study the structure theory of generalized loop planar Galilean conformal algebra We determine the derivations, second cohomology groups, and automorphism groups of

Galilean electrodynamics: covariant formulation and Lagrangian

In this paper, we construct a single Lagrangian for both limits of Galilean electrodynamics. The framework relies on a covariant formalism used in describing Galilean geometry. We write down the Galilean conformal algebra and its representation in this formalism. We also show that the Lagrangian is invariant under the Galilean conformal algebra in d = 4 and calculate the energy-momentum tensor.

On Galilean conformal bootstrap

In this work, we develop conformal bootstrap for Galilean conformal field theory (GCFT). In a GCFT, the Hilbert space could be decomposed into quasiprimary states and its global descendants. Different from the usual conformal field theory, the quasiprimary states in a GCFT constitute multiplets, which are block-diagonized under the Galilean boost operator. More importantly the multiplets include the states of negative norms, indicating the theory is not unitary. We compute global blocks of the multiplets, and discuss the expansion of four-point functions in terms of the global blocks of the multiplets. Furthermore we do the harmonic analysis for the Galilean conformal symmetry and obtain an inversion formula. As the first step to apply the Galilean conformal bootstrap, we construct generalized Galilean free theory (GGFT) explicitly. We read the data of GGFT by using Taylor series expansion of four-point function and the inversion formula independently, and find exact agreement. We discuss some novel features in the Galilean conformal bootstrap, due to the non-semisimpleness of the Galilean conformal algebra and the non-unitarity of the GCFTs.

Uniqueness of Galilean conformal electrodynamics and its dynamical structure

We investigate the existence of action for both the electric and magnetic sectors of Galilean Electrodynamics using Helmholtz conditions. We prove the existence of unique action in magnetic limit with the addition of a scalar field in the system. The check also implies the non existence of action in the electric sector of Galilean electrodynamics. Dirac constraint analysis of the theory reveals that there are no local degrees of freedom in the system. Further, the theory enjoys a reduced but an infinite dimensional subalgebra of Galilean conformal symmetry algebra as global symmetries. The full Galilean conformal algebra however is realized as canonical symmetries on the phase space. The corresponding algebra of Hamilton functions acquire a state dependent central charge.

Left-Symmetric Algebra Structures on the Planar Galilean Conformal Algebra

In this paper, we mainly determine the compatible left-symmetric algebra structures on the planar Galilean conformal algebra with some natural grading conditions. The results of earlier work on left-symmetric algebra structures on the twisted Heisenberg-Virasoro algebra play an important role in determining these compatible structures. As a corollary, any such left-symmetric algebra contains an infinite-dimensional non-trivial subalgebra that is also a submodule of the regular module.

Strings with non-relativistic conformal symmetry and limits of the AdS/CFT correspondence

We find a Polyakov-type action for strings moving in a torsional Newton-Cartan geometry. This is obtained by starting with the relativistic Polyakov action and fixing the momentum of the string along a non-compact null isometry. For a flat target space, we show that the world-sheet theory becomes the Gomis-Ooguri action. From a target space perspective these strings are non-relativistic but their world-sheet theories are still relativistic. We show that one can take a scaling limit in which also the world-sheet theory becomes non-relativistic with an infinite-dimensional symmetry algebra given by the Galilean conformal algebra. This scaling limit can be taken in the context of the AdS/CFT correspondence and we show that it is realized by the ‘Spin Matrix Theory’ limits of strings on AdS5 × S5. Spin Matrix theory arises as non-relativistic limits of the AdS/CFT correspondence close to BPS bounds. The duality between non-relativistic strings and Spin Matrix theory provides a holographic duality of its own and points towards a framework for more tractable holographic dualities whereby non-relativistic strings are dual to near BPS limits of the dual field theory.

Supersymmetric Galilean conformal blocks

We set up the bootstrap procedure for supersymmetric Galilean Conformal (SGC) field theories in two dimensions by constructing the SGC blocks in the mathcal{N}=1 and two possible mathcal{N}=2 extensions of the Galilean conformal algebra. In all analyzed cases, we present the bootstrap equations by crossing symmetry of the four point function. In addition, we compute the global SGC blocks analytically by solving the differential equations obtained by acting with the Casimirs of the global subalgebras inside the four point function. These global blocks agree with the general SGC blocks in the limit of large central charge. We comment on possible applications to supersymmetric BMS3 invariant field theories and flat holography.

Biderivations of the Galilean Conformal Algebra and their applications

In this paper, the biderivations of the Galiean conformal algebra are determined. As an application, the forms of the commutative post-Lie algebra structures on the Galiean conformal algebra are obtained.

Flatspace chiral supergravity

We propose a holographic duality between a 2 dimensional (2d) chiral superconformal field theory and a certain theory of supergravity in 3d with flatspace boundary conditions that is obtained as a double scaling limit of a parity breaking theory of supergravity. We show how the asymptotic symmetries of the bulk theory reduce from the "despotic" Super Bondi-Metzner-Sachs algebra (or equivalently the Inhomogeneous Super Galilean Conformal Algebra) to a single copy of the Super-Virasoro algebra in this limit and also reproduce the same reduction from a study of null vectors in the putative 2d dual field theory.

Lie Conformal Algebras of Planar Galilean Type

Motivated by the Lie structure of the planar Galilean conformal algebra, we construct a class of infinite rank Lie conformal algebras C P G ( a , b ) , where a, b are complex numbers. All their conformal derivations are shown to be inner. The rank-one conformal modules and ℤ-graded free intermediate series modules over C P G ( a , b ) are completely classified. The parallel results of the finite Lie conformal subalgebra C P G ( a , b ) of C P G ( a , b ) are also presented.