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 quasi-primary 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.
In this paper we consider non-relativistic-conformal group, then we calculate two point function for the fields that are Galilean conformal-invariant, then we show that the correlation function for Galilean conformal-invariant fields in $d=2$ are different from $d\neq2$. Our result in $d\neq2$ is different with previous results in \cite{3}.
Maxwell's Electrodynamics admits two distinct Galilean limits called the Electric and Magnetic limits. We show that the equations of motion in both these limits are invariant under the Galilean Conformal Algebra in D=4, thereby exhibiting non-relativistic conformal symmetries. Remarkably, the symmetries are infinite dimensional and thus Galilean Electrodynamics give us the first example of an infinitely extended Galilean Conformal Field Theory in D>2. We examine details of the theory by looking at purely non-relativistic conformal methods and also use input from the limit of the relativistic t…
In the present work we considered Galilean conformal algebras (GCA), which arises as a contraction relativistic conformal algebras ($x_i\rightarrow εx_i$, $t\rightarrow t$, $ε\rightarrow 0$). We can use the Galilean conformal symmetry to constrain two-point and three-point functions. Correlation functions in space-time without boundary condition were found in \cite{1}. In real situations there are boundary conditions in space-time, so we have calculated correlation functions for Galilean confrormal invariant fields in semi-infinite space with boundary condition in $r=0$. We have calculated two…
The six-dimensional exotic Galilean algebra in (2+1) dimensions with two central charges $m$ and $θ$, is extended when $m=0$, to a ten-dimensional Galilean conformal algebra with dilatation, expansion, two acceleration generators and the central charge $θ$. A realisation of such a symmetry is provided by a model with higher derivatives recently discussed in \cite{peterwojtek}. We consider also a realisation of the Galilean conformal symmetry for the motion with a Coulomb potential and a magnetic vortex interaction. Finally, we study the restriction, as well as the modification, of the Galilean…
We perform a detailed analysis of Galilean field theories, starting with free theories and then interacting theories. We consider non-relativistic versions of massless scalar and Dirac field theories before we go on to review our previous construction of Galilean Electrodynamics and Galilean Yang-Mills theory. We show that in all these cases, the field theories exhibit non-relativistic conformal structure (in appropriate dimensions). The surprising aspect of the analysis is that the non-relativistic conformal structure exhibited by these theories, unlike relativistic conformal invariance, beco…
Conformal Galilei algebra contains so(1,2) subalgebra which is the conformal algebra in one dimension. In this note we generalize methods previously developed for one-dimensional many-body systems and construct a unitary map relating a quantum mechanics invariant under the conformal Galilean transformations to a set of decoupled particles for which the same symmetry is realized in a nonlocal way. Possible applications of the map are discussed.
In this note, we construct a Wess-Zumino-Witten model based on the Galilean conformal algebra in 2-spacetime dimensions, which is a nonrelativistic analogue of the relativistic conformal algebra. We obtain exact background corresponding to σ-models in six dimensions (the dimension of the group manifold) and a central charge c=6. We carry out a Sugawara type construction to verify the conformal invariance of the model. Further, we discuss the feasibility of the background obtained as a physical spacetime metric.
We obtain the entanglement negativity for various bipartite zero and finite temperature pure and mixed state configurations in a class of $(1+1)$-dimensional Galilean conformal field theories. In this context we establish a construction for computing the entanglement negativity for such bipartite states involving a suitable replica technique. Our construction exactly reproduces certain universal features observed for entanglement negativity of corresponding states in relativistic $(1+1)$-dimensional conformal field theories.
In this work, we continue our work on two dimensional Galilean conformal field theory (GCFT$_2$). Our previous work (arXiv:2011.11092) focused on the $ξ\neq 0$ sector, here we investigate the more subtle $ξ=0$ sector to complete the discussion. The case $ξ=0$ is degenerate since there emerge interesting null states in a general $ξ=0$ boost multiplet. We specify these null states and work out the resulting selection rules. Then, we compute the $ξ=0$ global GCA blocks and find that they can be written as a linear combination of several building blocks, each of which can be obtained from a $sl(2,…
We apply the nonlinear realizations method for constructing new Galilean conformal mechanics models. Our starting point is the Galilean conformal algebra which is a non-relativistic contraction of its relativistic counterpart. We calculate Maurer-Cartan one-forms, examine various choices of the relevant coset spaces and consider the geometric inverse Higgs-type constraints which reduce the number of the independent coset parameters and, in some cases, provide dynamical equations. New Galilean conformally invariant actions are derived in arbitrary space-time dimension D=d+1 (no central charges)…
Galilean Conformal Algebras (GCA) have been recently proposed as a different non-relativistic limit of the AdS/CFT conjecture. In this note, we look at the representations of the GCA. We also construct explicitly the two and three point correlators in this non-relativistic limit of CFT and comment on the differences with the relativistic case and also the more studied Schrodinger group.
The Galilean conformal algebra has recently been realised in the study of the non-relativistic limit of the AdS/CFT conjecture. This was obtained by a systematic parametric group contraction of the parent relativistic conformal field theory. In this paper, we extend the analysis to include supersymmetry. We work at the level of the co-ordinates in superspace to construct the N=1 Super Galilean conformal algebra. One of the interesting outcomes of the analysis is that one is able to naturally extend the finite algebra to an infinite one. This looks structurally similar to the N=1 superconformal…
Using the notion of Galilean conformal algebra (GCA) in arbitrary space dimension d, we introduce for d=3 quantized nonrelativistic counterpart of twistors as the spinorial representation of O(2,1){\oplus}SO(3) which is the maximal semisimple subalgebra of three-dimensional GCA. The GC-covariant quantization of such nonrelativistic spinors, which shall be called also Galilean twistors, is presented. We consider for d=3 the general spinorial matrix realizations of GCA, which are further promoted to quantum-mechanical operator representations, expressed as bilinears in quantized Galilean twistor…
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…
In this paper we will present in more detail a construction using Wilson lines and the corresponding dual Galilean conformal field theory calculations for analytically determining holographic entanglement entropy for flat space in $2+1$ dimensions first presented in hep-th/1410.4089. In addition we will show how the construction using Wilson lines can be expanded to flat space higher-spin theories and determine the thermal entropy of (spin-3 charged) flat space cosmologies using this approach.
We obtain the reflected entropy for bipartite states in a class of $(1+1)$-dimensional Galilean conformal field theories ($GCFT_{1+1}$) through a replica technique. Furthermore we compare our results with the entanglement wedge cross section (EWCS) obtained for the dual (2+1) dimensional asymptotically flat geometries in the context of flat holography. We find that our results are consistent with the duality between the reflected entropy and the bulk EWCS for flat holographic scenarios.
In this paper, we construct a single Lagrangian for both limits of Galilean electrodynamics. The framework relies on a covariant formalism used in describing Newton-Cartan 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.
We show that a class of nonrelativistic algebras including non centrally-extended Schrodinger algebra and Galilean Conformal Algebra (GCA) has an affine extension in 2+1 hitherto unknown. This extension arises out of the conformal symmetries of the two dimensional complex plain. We suggest that this affine form may be the symmetry that explains the relaxation of some classical phenomena towards their critical point. This affine algebra admits a central extension and maybe realized in the bulk. The bulk realization suggests that this algebra may be derived by looking at the asymptotic symmetry …
We find an intriguing link between the symmetries of the tensionless limit of closed string theory and the 2-dimensional Galilean Conformal Algebra (2d GCA). 2d GCA has been discussed in the context of the non-relativistic limit of AdS/CFT and more recently in flat-space holography as the proposed symmetry algebra of the field theory dual to 3d Minkowski spacetimes. It is best understood as a contraction of two copies of the Virasoro algebra. In this note, we link this to the tensionless limit of bosonic closed string theory. We show how it emerges naturally as a contraction of the residual ga…
The odd entanglement entropy (OEE) for bipartite states in a class of $(1+1)$-dimensional Galilean conformal field theories ($GCFT_{1+1}$) is obtained through an appropriate replica technique. In this context our results are compared with the entanglement wedge cross section (EWCS) for $(2+1)$-dimensional asymptotically flat geometries dual to the $GCFT_{1+1}$ in the framework of flat holography. We find that our results are consistent with the duality of the difference between the odd entanglement entropy and the entanglement entropy of bipartite states, with the bulk EWCS for flat holographi…
The Galilean Conformal Algebra (GCA) arises from the relativistic conformal algebra in the non-relativistic limit. In two dimensions, one can view it as a limit of linear combinations of the two copies Virasoro algebra. Recently, it has been argued that Topologically Massive Gravity (TMG) realizes the quantum 2d GCA in a particular scaling limit of the gravitational Chern-Simons term. To add strength to this claim, we demonstrate a matching of correlation functions on both sides of this correspondence. A priori looking for spatially dependent correlators seems to force us to deal with high spi…
Non-relativistic versions of the AdS/CFT conjecture have recently been investigated in some detail. These have primarily been in the context of the Schrodinger symmetry group. Here we initiate a study based on a {\it different} non-relativistic conformal symmetry: one obtained by a parametric contraction of the relativistic conformal group. The resulting Galilean conformal symmetry has the same number of generators as the relativistic symmetry group and thus is different from the Schrodinger group (which has fewer). One of the interesting features of the Galilean Conformal Algebra is that it a…
Galilean conformal algebra (GCA) is an Inonu-Wigner (IW) contraction of a conformal algebra, while Newton-Hooke string algebra is an IW contraction of an AdS algebra which is the isometry of an AdS space. It is shown that the GCA is a boundary realization of the Newton-Hooke string algebra in the bulk AdS. The string lies along the direction transverse to the boundary, and the worldsheet is AdS_2. The one-dimensional conformal symmetry so(2,1) and rotational symmetry so(d) contained in the GCA are realized as the symmetry on the AdS_2 string worldsheet and rotational symmetry in the space tran…
The scattering equation formalism for scattering amplitudes, and its stringy incarnation, the ambitwistor string, remains a mysterious construction. In this paper, we pursue the study a gauged-unfixed version of the ambitwistor string known as the null string. We explore the following three aspects in detail; its complexification, gauge fixing, and amplitudes. We first study the complexification of the string; the associated symmetries and moduli, and connection to the ambitwistor string. We then look in more details at the leftover symmetry algebra of the string, called Galilean conformal alg…
Galilean conformal algebra (GCA) in two dimensions arises as contraction of two copies of the centrally extended Virasoro algebra ($t\rightarrow t, x\rightarrowεx$ with $ε\rightarrow 0$). The central charges of GCA can be expressed in term of Virasoro central charges. For finite and non-zero GCA central charges, the Virasoro central charges must behave as asymmetric form $O(1)\pm O(\frac{1}ε)$. We propose that, the bulk description for 2d GCA with asymmetric central charges is given by general massive gravity (GMG) in three dimensions. It can be seen that, if the gravitational Chern-Simons cou…
It is shown that the N-Galilean conformal algebra, with N-odd, is the maximal symmetry algebra of the free Lagrangian involving (N+1)/2-th order time derivative.
In the present paper we consider the realization of $2-$dimensional Galilean conformal algebra ($GCA_2$) on the boundary of cosmological new massive gravity. At first we consider the contracted BTZ black hole solution. We obtain entropy formula for the $GCA_2$ in term of contracted scaling dimension $Δ$ and central charge $C_1$. This entropy formula exactly matches with the non-relativistic limit of Bekenstein-Hawking entropy of BTZ black hole. Then we extend our study to the contracted warped $AdS_3$ black hole solution of CNMG. We obtain the entropy of dual $GCA_2$ in terms of central charge…
Galilean conformal algebras can be constructed by contracting a finite number of conformal algebras, and enjoy truncated $\mathbb{Z}$-graded structures. Here, we present a generalisation of the Galilean contraction procedure, giving rise to Galilean conformal algebras with truncated $\mathbb{Z}^{\otimesσ}$-gradings, $σ\in\mathbb{N}$. Detailed examples of these multi-graded Galilean algebras are provided, including extensions of the Galilean Virasoro and affine Kac-Moody algebras. We also derive the associated Sugawara constructions and discuss how these examples relate to multivariable extensi…
A maximally symmetric non-linear 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…
The Galilean Conformal Algebra (GCA) arises in taking the non-relativistic limit of the symmetries of a relativistic Conformal Field Theory in any dimensions. It is known to be infinite-dimensional in all spacetime dimensions. In particular, the 2d GCA emerges out of a scaling limit of linear combinations of two copies of the Virasoro algebra. In this paper, we find metrics in dimensions greater than two which realize the finite 2d GCA (the global part of the infinite algebra) as their isometry by systematically looking at a construction in terms of cosets of this finite algebra. We list all p…
The procedure of null reduction provides a concrete way of constructing field theories with Galilean invariance. We use this to examine Galilean gauge theories, viz. Galilean electrodynamics and Yang-Mills theories in spacetime dimensions 3 and 4. Different non-relativistic conformal symmetries arise in these contexts: Schr{ö}dinger symmetry in $d=3$ and Galilean conformal symmetry in $d=4$. A canonical analysis further reveals that the symmetries enhance to their infinite dimensional versions in phase space and pick up central extensions. In addition, for the Abelian theory, we discuss non-re…
It is shown that centrally extended N-Galilean conformal algebra, with N-odd, is the maximal symmetry algebra of the Schrodinger equation corresponding to the free Lagrangian involving (N+1)/2-th order time derivatives.
Boundary conformal field theory (BCFT) is the study of conformal field theory (CFT) in semi-infinite space-time. In non-relativistic limit ($x\rightarrowεx, t\rightarrow t, ε\rightarrow 0$), boundary conformal algebra changes to boundary Galilean conformal algebra (BGCA). In this work, some aspects of AdS/BCFT in non-relatvistic limit were explored. We constrain correlation functions of Galilean conformal invariant fields with BGCA generators. For a situation with a boundary condition at surface $x=0$ ($z=\bar{z}$), our result is agree with non-relativistic limit of BCFT two-point function. We…
We study a new contraction of a d+1 dimensional relativistic conformal algebra where n+1 directions remain unchanged. For n=0,1 the resultant algebras admit infinite dimensional extension containing one and two copies of Virasoro algebra, respectively. For n> 1 the obtained algebra is finite dimensional containing an so(2,n+1) subalgebra. The gravity dual is provided by taking a Newton-Cartan like limit of the Einstein's equations of AdS space which singles out an AdS_{n+2} spacetime. The infinite dimensional extension of n=0,1 cases may be understood from the fact that the dual gravities cont…
We present higher-spin algebras containing a Poincaré subalgebra and with the same set of generators as the Lie algebras that are relevant to Vasiliev's equations in any space-time dimension $D \geq 3$. Given these properties, they can be considered either as candidate rigid symmetries for higher-spin gauge theories in Minkowski space or as Carrollian conformal higher-spin symmetries in one less dimension. We build these Lie algebras as quotients of the universal enveloping algebra of $iso(1,D-1)$ and we show how to recover them as Inönü-Wigner contractions of the rigid symmetries of higher-sp…
We show that an extended $3D$ Schrödinger algebra introduced in [1] can be reformulated as a $3D$ Poincaré algebra extended with an SO(2) R-symmetry generator and an $SO(2)$ doublet of bosonic spin-1/2 generators whose commutator closes on $3D$ translations and a central element. As such, a non-relativistic Chern-Simons theory based on the extended Schrödinger algebra studied in [1] can be reinterpreted as a relativistic Chern-Simons theory. The latter can be obtained by a contraction of the $SU(1,2)\times SU(1,2)$ Chern-Simons theory with a non principal embedding of $SL(2,\mathbb R)$ into $S…
In this paper, we elaborate on aspects of the recently introduced BMS bootstrap programme. We consider two-dimensional (2d) field theories with BMS3 symmetry and extensively use highest weight representations to uncover the BMS version of crossing symmetry in 4-point functions that are constrained by symmetry. The BMS bootstrap equation is formulated and then analytic expressions for BMS blocks are constructed by looking at the limit of large central charges. These results are also applicable to 2d Galilean Conformal Field Theories through the isomorphism between the BMS3 and 2d Galilean Confo…
We show that BTZ black hole solution of Cosmological Topological Massive Gravity (CTMG) have a hidden conformal symmetry. In this regard, we consider the wave equation of a massless scalar field propagating in BTZ spacetime and find the wave equation could be written in terms of the $SL(2,R)$ quadratic Casimir. From the conformal coordinates, the temperatures of the dual CFTs could be read directly. Moreover, we compute the microscopic entropy of the dual CFT by Cardy formula and find a perfect match to Bekenstein-Hawking entropy of BTZ black hole. Then we consider Glilean conformal algebras (…
A Galilean contraction is a way to construct Galilean conformal algebras from a pair of infinite-dimensional conformal algebras, or equivalently, a method for contracting tensor products of vertex algebras. Here, we present a generalisation of the Galilean contraction prescription to allow for inputs of any finite number of conformal algebras, resulting in new classes of higher-order Galilean conformal algebras. We provide several detailed examples, including infinite hierarchies of higher-order Galilean Virasoro algebras, affine Kac-Moody algebras and the associated Sugawara constructions, an…
Representation theory of an infinite dimensional Galilean conformal algebra introduced by Martelli and Tachikawa is developed. We focus on the algebra defined in (2+1) dimensional spacetime and consider central extension. It is then shown that the Verma modules are irreducible for non-vanishing highest weights. This is done by explicit computation of Kac determinant. We also present coadjoint representations of the Galilean conformal algebra and its Lie group. As an application of them, a coadjoint orbit of the Galilean conformal group is given in a simple case.
Firstly we discuss briefly three different algebras named as nonrelativistic (NR) conformal: Schroedinger, Galilean conformal and infinite algebra of local NR conformal isometries. Further we shall consider in some detail Galilean conformal algebra (GCA) obtained in the limit c equal to infinity from relativistic conformal algebra O(d+1,2) (d - number of space dimensions). Two different contraction limits providing GCA and some recently considered realizations will be briefly discussed. Finally by considering NR contraction of D=4 superconformal algebra the Galilei conformal superalgebra (GCSA…
We consider Schrodinger equations for a non-relativistic particle obeying N+1-th order higher derivative classical equation of motion. These equations are invariant under N(odd)-extended Galilean conformal (NGC) algebras in general d+1 dimensions. In 2+1 dimensions, the exotic Schrodinger equations are invariant under N(even)-GCA.
The usual Galilean contraction procedure for generating new conformal symmetry algebras takes as input a number of symmetry algebras which are equivalent up to central charge. We demonstrate that the equivalence condition can be relaxed by inhomogeneously contracting the chiral algebras and present general results for the ensuing asymmetric Galilean algebras. Several examples relevant to conformal field theory are discussed in detail, including superconformal algebras and W-algebras. We also discuss how the Sugawara construction is modified in the asymmetric setting.
We consider a realization of the Galilean conformal algebra (GCA) in two dimensional space-time on the AdS boundary of a particular three dimensional gravity theory, the so-called cosmological topologically massive gravity (CTMG), which includes the gravitational Chern-Simons term and the negative cosmological constant. The infinite dimensional GCA in two dimensions is obtained from the Virasoro algebra for the relativistic CFT by taking a scaling limit $t\to t$, $x\toεx$ with $ε\to 0$. The parent relativistic CFT should have left and right central charges of order $\mathcal{O}(1/ε)$ but oppos…
We demonstrate that the Navier-Stokes equation can be covariantized under the full infinite dimensional Galilean Conformal Algebra (GCA), such that it reduces to the usual Navier-Stokes equation in an inertial frame. The covariantization is possible only for incompressible flows, i.e when the divergence of the velocity field vanishes. Using the continuity equation, we can fix the transformation of pressure and density under GCA uniquely. We also find that when all chemical potentials vanish, $c_{s}$, which denotes the speed of sound in an inertial frame comoving with the flow, must either be a…
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 ful…
We present the analytical calculation of entanglement entropy for a class of two dimensional field theories governed by the symmetries of the Galilean conformal algebra, thus providing a rare example of such an exact computation. These field theories are the putative holographic duals to theories of gravity in three-dimensional asymptotically flat spacetimes. We provide a check of our field theory answers by an analysis of geodesics. We also exploit the Chern-Simons formulation of three-dimensional gravity and adapt recent proposals of calculating entanglement entropy by Wilson lines in this c…
Logarithmic representations of the conformal Galilean algebra (CGA) and the Exotic Conformal Galilean algebra ({\sc ecga}) are constructed. This can be achieved by non-decomposable representations of the scaling dimensions or the rapidity indices, specific to conformal galilean algebras. Logarithmic representations of the non-exotic CGA lead to the expected constraints on scaling dimensions and rapidities and also on the logarithmic contributions in the co-variant two-point functions. On the other hand, the {\sc ecga} admits several distinct situations which are distinguished by different sets…
The pseudo-conformal universe is an alternative to inflation in which the early universe is described by a conformal field theory on approximately flat space-time. The fields develop time-dependent expectation values, spontaneously breaking the conformal symmetries to a de Sitter subalgebra, and fields of conformal weight zero acquire a scale invariant spectrum of perturbations. In this paper, we show that the pseudo-conformal scenario can be naturally realized within theories that would ordinarily be of interest for DBI inflation, such as the world-volume theory of a probe brane in an AdS bul…
We discuss non-relativistic conformal algebras generalizing the Schrödinger algebra. One instance of these algebras is a conformal, acceleration-extended, Galilei algebra, which arises also as a contraction of the relativistic conformal algebra. In two dimensions, this admits an "exotic" central extension, whereby boosts do not commute. We study general properties of non-relativistic conformal field theories with such symmetry. We realize geometrically the symmetry in terms of a metric invariant under the exotic conformal Galilei algebra, although its signature is neither Lorentzian nor Euclid…
It is shown how to obtain conformal blocks from embedding space with the help of the operator product expansion. The minimal conformal block originates from scalar exchange in a four-point correlation functions of four scalars. All remaining conformal blocks are simple derivatives of the minimal conformal block. With the help of the orthogonality properties of the conformal blocks, the analytic conformal bootstrap can be implemented directly in embedding space, leading to a Jacobi-like definition of conformal field theories.
Bootstrap equations for conformal correlators that mimic the early theory of conformal bootstrap are written down in frames of the AdS/CFT approach. The simplified version of these equations, that may be justified if Schwinger-Keldysh formalism is used in AdS/CFT instead of conventional Feynman-Witten diagrams technique, permits to calculate values of conformal dimensions in the $O(N)$ symmetric model with conformal or composite Hubbard-Stratonovich field.
When the interaction potential is suitably reordered, the Moyal field theory admits two types of Galilean symmetries, namely the conventional mass-parameter-centrally-extended one with commuting boosts, but also the two-fold centrally extended ``exotic'' Galilean symmetry, where the commutator of the boosts yields the noncommutative parameter. In the free case, one gets an ``exotic'' two-parameter central extension of the Schroedinger group. The conformal symmetry is, however, broken by the interaction.
The axiomatic Wightman formulation for nonderivative conformal field theory is adopted to derive conformal bootstrap equation for the four point function. The equivalence between PCT theorem and {\it weak local commutativity}, due to Jost, play a very crucial role in axiomatic field theory. The theorem is suitably adopted for conformal field theory to derive the desired equations in CFT. We demonstrate that the two Wightman functions are analytic continuation of each other.
We present a low entry-level introduction to the Conformal Bootstrap. We review and obtain several basic bounds using Linear Programming in machine precision in Mathematica, making the results accessible even to the most uneducated computer user. The Extremal Functional Method and the method of determinants are also explored. Finally, we make an attempt at writing modular invariance statements as crossing symmetry equations for twist operators and perform Modular Bootstrap using 4-pt correlators.
