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We discuss non-Lorentzian Lagrangian field theories in 2

Lorentz symmetry plays a crucial role in many applications of Quantum Field Theory but it is not necessary. Indeed the condensed matter community more often than not looks at theories without it. This opens the door to additional spacetime symmetries such as the Bargmann, Carroll and Schrödinger groups. In particular non-Lorentzian conformal field theories have now received considerable attention and reveal many interesting features, for example see [

It is well-known that one way to construct non-Lorentzian theories with Schrödinger symmetry is to reduce a Lorentzian theory of one higher dimension on a null direction. From the higher dimensional perspective such null reductions are somewhat unphysical but that need not concern us if we are only interested in the features of the reduced theory. Indeed the (null) Kaluza-Klein momentum is often associated with particle number and, in contrast to traditional Kaluza-Klein theories, one need not truncate the action to the zero-modes but rather any given Fourier mode. The resulting theories are interesting in themselves and have applications in Condensed Matter Systems and DLCQ constructions (where one does have to try to make sense of a null reduction).

Here we will explore theories with novel spacetime

We also comment that our interest in these models has arisen through an explicit class of supersymmetric non-Abelian gauge theories in five-dimensions with _{+} is played by the instanton number leading to an additional

In this paper we wish to illustrate some of general aspects of such theories. In

We start with 2^{1}
_{
ij
} is a constant anti-symmetric matrix that satisfies^{
i
} directions so as to bring Ω_{
ij
} to a canonical form; in particular, one can always find orthogonal matrix ^{+} ∈ (−^{+} direction of 2^{+} ∈ [−

As we have seen, the metric _{2n+1}. Indeed, a particular slicing of AdS_{2n+1} that makes this form for the conformal boundary manifest has long been known in the literature [

Let ^{
a
}, _{
ab
} = diag (−1, 1, ^{
a
} provide coordinates on Lorentzian AdS_{2n+1}, with metric given by

Next, we can parameterise solutions to the constraint (^{+}, ^{−}, ^{
i
}). We have^{2}

These coordinates provide a description for AdS_{2n+1} as a one-dimensional fibration over a non-compact form of ^{+} is the coordinate along the fibre. The metric (^{+}, we land precisely on

To go to the conformal boundary, we now restrict to a surface of constant

Finally, let us discuss symmetries. Each isometry in the bulk, described by some Killing vector field, corresponds to a conformal symmetry on the boundary. The full set of such symmetries form the algebra ^{+} direction. We see that this subalgebra can be identified with the subalgebra of bulk isometries that commute with translations along the fibre. It is hence given by

Each continuous spacetime symmetry of a conformal field theory on Minkowski space is generated by an operator

Each operator _{
∂
} of the metric _{
μν
}. Each of these vector fields _{
∂
} then satisfies _{
∂
}, and the Weyl factors

Their non-vanishing commutators are

It is now straightforward to see that translations along ^{+} are an isometry^{3}
^{+} interval. At the level of the symmetry algebra, this amounts to choosing a basis for the space of local operators which diagonalises _{+}. The resulting operators are Fourier modes on the ^{+} interval. They fall into representations of the centraliser of _{+} within

A basis for the subalgebra

where the ^{
α
} are absent for ^{2}−2

Then, these vector fields are indeed conformal Killing vector fields of

Let us identify the subalgebra of pure rotations within ^{
α
}. First, the case of _{+} and hence survive the reduction.

So let us take _{
ij
} = −_{
ji
}, forming _{+} that this rotation commutes with _{+} and thus lies within _{
ij
} = _{
ik
}Ω_{
kj
}−Ω_{
ik
}
_{
kj
} = 0. It then follows from the relation (^{
α
}, ^{2}−2

Thus, for ^{
α
}}. For example, for the first non-trivial case _{
ij
} fall into two classes: those that are anti-self-dual, and those that are self-dual. These two cases correspond to det(_{
ij
}, one can choose for their

Finally, let us state the commutation relations for the algebra ^{
α
}} both amongst themselves and with the other generators can be summarised as follows. As we have seen, {^{
α
}} form a basis for ^{
α
}, so that_{
i
}, _{
j
}] holds down to

Following the discussion in ^{+} interval. In detail, we define_{+} generates the

So let us now consider a (2^{−}, ^{
i
}) = (0, 0), we say it has scaling dimension Δ if it satisfies [

We find that {_{
i
}, _{
i
}}, which raise and lower scalig dimension by one unit, respectively.

Going further, we can generalise results from the ^{−}, ^{
i
}) = (0, 0) by its transformation under the stabiliser of the origin within _{+}, ^{
α
}, _{
i
}, ^{
α
}, so that _{+} is the charge of _{+}. It is clear that in any (2

The key property of such a primary is that it is annihilated by the lowering operators {_{
i
}}, and thus sits at the bottom of a tower of states generated by the raising operators {_{
i
}}, known as usual as

Given any operator Φ(0) at the origin, an operator at some point (^{−}, ^{
i
}) is defined by^{−}
_{−}Φ(^{
i
}
_{
i
}Φ(_{
i
} on Φ(_{
i
}} is non-Abelian.

One can in particular apply the transformation rules (

At the level of symmetries, the presence of conformal symmetry in the relativistic theory manifests itself as an enhancement of the Poincaré algebra to the conformal algebra. The analogous statement in non-relativistic theories is an enhancement of the inhomogeneous Galilean algebra—or rather, its central extension, the Bargmann algebra—to the Schrödinger algebra. Let us denote by Schr(

Then, Schr(

Recall, we defined the subalgebra _{+}. It is clear that in the limit that _{+} degenerates to become simply a null translation. Indeed, this is also evident from the form of _{+} in terms of the conventional conformal generators, as in

Hence, in the limit _{+} strictly in the _{
ij
} drops out entirely, and thus the breaking of the rotation subalgebra _{
ij
} at finite

Things therefore work smoothly at the level of the algebra. However, given a theory admitting the Ω-deformed non-relativistic conformal symmetry ^{+} ∼ ^{+} + 2_{+} for some _{+}, which by a Lorentz boost is seen to be unphysical.

A convenient way to arrive at this setup—which from the 2^{+} direction for some _{+} eigenvalue in _{+}≔^{4}

Indeed, this precise DLCQ limit of a

A deep and powerful result tool in the study of relativistic conformal field theory is the operator-state map, relating on one hand conformal primary operators, and on the other, eigenstates of the Hamiltonian of the theory on a sphere. An analogous map exists in conventional non-relativistic conformal field theories [

We will now show that construction applies in an almost identical way to the

We approach the construction of our operator-state map from the perspective of automorphisms of the symmetry algebra, a well-established point of view in relativistic CFTs which has also recently been formulated for non-relativistic CFTs governed by the Schrödinger group [

Given some operator Φ(0) at the origin, we may define a state_{
i
}, _{
i
}} to raise and lower the _{
i
} |Φ⟩ = (Δ−1)_{
i
} |Φ⟩.

We can consider

Thus, we have on one hand primary operators and their descendants, all with definite scaling dimension, and on the other hand, eigenstates of the operator

So let us consider transformed states and operators given by

Explicitly, the transformed operators under (

Up to normalisation these operators (

If we assume unitarity in the original Minkowskian theory, then all states will have non-negative norm. Just as is the case of Lorentzian CFTs, we can use this assumption to place constraints on the eigenvalues of certain operators. The original Minkowskian symmetry generators were all Hermitian operators, but since we are interested in states quantised in the analogue of radial quantisation, we should instead consider the barred generators of _{+} is its central charge, taking values in

Note, we see that a scalar primary must have _{+} ≤ 0. It is interesting to note that this condition appears to be manifestly realised in known supersymmetric interacting gauge theory examples of _{+} is identified as instanton charge, and the dynamics are constrained such that only anti-instantons, corresponding to _{+} ≤ 0, are allowed to propagate [

We can use the positivity of Δ and

We have thus far explored the reduction of symmetries of an even-dimensional conformal field theory when dimensionally reduced along a particular conformally-compactified direction. In six or fewer dimensions the conformal algebra admits extensions to various Lie superalgebras and thus it is natural to extend our analysis to determine the fate of supersymmetry under such dimensional reductions. In particular, any surviving supersymmetry constitutes a Lie superalgebra extension of

The dimensional reduction we have constructed is novel only for

In six-dimensions the only choices for relativistic superconformal algebras are

In Minkowski signature we choose conventions where all Bosonic generators are Hermitian, as before. Their commutation relations are the same as in ^{
AB
} and ^{
αβ
} are the five and six-dimensional charge conjugation matrices, with ^{5}
_{*} = Γ_{012345}._{+}, defined in terms of the six-dimensional (hatted) operators as_{+}. Precisely which set of supercharges this is depends on whether Ω_{
ij
} is self-dual or anti-self-dual; without loss of generality, let us choose the latter case. Then, letting a ± subscript denote chirality under Γ_{05}, the commuting supercharges are_{
ij
} is self-dual, is found simply by swapping all Γ_{05} chiralities. Then, their commutation relations with the bosonic generators are_{±} = 1/2 (1 ±Γ_{*}).

Thus there are 50 = 1 + 15 + 10 + 24 Bosonic generators corresponding to the central extension,

The Fermionic generators can also be transformed by

Rather unusually for such algebras, along with a pair of Fermionic generators that raise and lower the eigenvalue of _{−} and _{+}, we also have generators that do not change this eigenvalue; Θ_{−}. We can see that while _{−} is nilpotent, so each super conformal primary belongs to a family of such states, an original bosonic state, plus those that follow from the action of

A more interesting bound is found from the norm

It is interesting to note that, up to a choice of real form for the respective algebras, the reduction of symmetry from the six-dimensional (2, 0) superalgebra down to centraliser of _{+} is identical to the symmetry breaking pattern of the classical ABJM theory, which realises manifestly only a particular subalgebra of the full three-dimensional

In this section we want to discuss examples of field theories in (2

To begin we consider a free real scalar in (1 + 1)-dimensions, ^{(k)}, for ^{(0)}(^{−}) is expanded in terms of right moving oscillators. One might also consider including winding modes but we will not do so here as the spatial direction is not compact.

Substituting into the action we find

By construction the ^{(k)} at fixed _{+} acts as:

However we see that they can be extended to^{−}). Taking _{−1}, _{0}, _{1} form a finite dimensional subalgebra. However just as in the familiar case of the string worldsheet in the quantum theory, where we must normal order the operators ^{(k)}, we will generate a central charge

Let us now consider a free real scalar obtained from reduction from ^{6}
^{+} ∈ [−

Let us consider the reduction of a Fermion. Starting in 2

We see that

This leads to the reduced action_{−}, _{+}, _{
i
} are simply the

We it is helpful to split ^{(k)}, depending on the eigenvalue of _{
ij
}Ω_{
ij
}. It can also vanish if

Finally we observe that in one-dimension we simply find_{
n
} provided that the ^{(k)} are invariant. Furthermore we will encounter a central charge

Let us start with a free four-dimensional Maxwell gauge field_{
μν
} and Fourier expand^{+} we obtain

Finally we consider a free tensor in six-dimensions:_{
μν
} and Fourier expand^{(k)} with components

In this section we would like to see how, by considering the entire Kaluza-Klein tower, we can reconstruct the correlation functions of the 2

Let us start with a tower of scalar fields in one-dimension that are obtained from a two-dimensional scalar as given in

Let us try to compute a two-point function of the original two-dimensional theory. If we try to compute ^{(k)} are the left moving oscillators. Thus we are left with

It is clear that from this treatment we will never be able to reconstruct the right-moving sector as only ^{−} as “time” the right moving modes are forever stuck in one moment of time. Curiously what we have obtained here can be viewed as an action for a chiral Boson, constructed from an infinite number of fields. Note that in this case there is no Ω-deformation. In higher dimensions this is not the case and, as we will now show, it will allow us to reconstruct the full higher dimensional theory.

Now we want to repeat our analysis of 2-point functions but now in higher dimensions. For simplicity we use translational invariance to put one operator at the origin:_{
n,k
}. For

We can now reconstruct the 2^{+}. In terms of

Note that we encounter a problem if we quantize the theory using the action (5.14) with ^{−} as “time” since we obtain the conjugate momentum^{(k)}(^{−}, ^{
i
}), Π^{(k)}(^{−}, 0)] = −2^{−1} [^{(k)}(^{−}, ^{
i
}), ^{(−k)}(^{−}, 0)] is non-zero for

Let us look at this more closely on a case-by-case basis. For ^{(1/2)} and the bound in ^{(k)} with ^{(0)}|0⟩ = 0 for ^{(0)} has a non-zero Lifshitz scaling dimension, ^{(0)}|0⟩ = 0 is also required for the vacuum to preserve

To obtain the 2^{2}/^{2n−3}). In particular for the two cases at hand this means that must have

We also see from ^{+} ∈ [−_{
n.k
} we find (again assuming an

In this appendix we want to present an argument that the normalisation _{
n,k
} introduced in _{
n,k
} given in ^{−}≥ 0 due to the presence of Θ(^{−})) and_{
n,k
} such that^{
iθ
} and observe that_{
θ
} = _{
θ
}

The integral _{
k,n
}. In fact we find that ^{−} = 0 for any ^{−}) cuts off the integral. A natural choice is

In this paper we have examined non-Lorentzian theories with

We then presented examples of free theories in a variety of dimensions with various field contents. Although we kept the Kaluza-Klein tower of fields this is not necessary for

We note that in theories with ^{−} as time. However at the spatial origin such “wrong-sign” terms vanish. Given translational invariance this suggests that the

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

NL is a co-investigator on the STFC grant ST/T000759/1, RM. was supported by David Tong’s Simons Investigator Grant, and TO was supported by the STFC studentship ST/S505468/1.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

We would like to thank A. Lipstein and P. Richmond for initial collaboration on this work, and David Tong for helpful discussions. This paper is to be submitted to the Frontiers Special Edition on Non-Lorentzian Geometry and its Applications.^{7}

It is curious to note that this transformation is similar to the transformation used in [^{+}-dependent rotation by Ω_{
ij
}.

As our focus is on continuous conformal symmetries on the boundary, it is sufficient for our purposes to consider this a local parameterisation of AdS_{2n+2}, and thus neglect global features of this real coordinate choice.

Translations in ^{+} are a conformal symmetry of the original metric

We choose this sign for

Note that Ω_{
AB
} should not be confused with Ω_{
ij
} which we used in the coordinate transformation. To ameliorate this problem we will always explicitly write the indices.

There is also a coupling to the spacetime Ricci scalar but since we are working on a conformally flat metric, this term vanishes.