This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

An exciting subject in string theory is to consider some applications of the AdS/CFT correspondence to realistic systems like condensed matter systems. Since most of such systems are non-relativistic, an anisotropic scaling symmetry with the general value of dynamical critical exponent

One of the most well-studied subjects in string theory is the AdS/CFT correspondence [

Based on the latter aspect, many applications of AdS/CFT to realistic systems in nature like QCD and condensed matter physics have been explored enthusiastically (for comprehensive reviews, for example, see [

where λ is a scaling constant and

There are two typical examples of algebras including a non-relativistic scale invariance with

Thus the non-relativistic algebras are intimately related each other from the point of view of the algebraic structure.

The purpose of this review article is to give a short summary on the classification of superalgebras with the anisotropic scaling (1) as

This review article is organized as follows. In

We explain general prescriptions to pick up a subalgebra with an anisotropic scaling in order to make our discussion clear.

The first is a prescription to pick up subalgebras. As a warm-up, let us consider a relativistic conformal algebra in four dimensions, that is a portion of the bosonic part of psu(2,2|4) ,

Here _{µ}_{µυ}_{µ}

It is easy to read the dimensions of the generators

from the commutation relations of conformal algebra. The dimensions of the generators in the subset

are non-negative and the set (3) forms a subalgebra of (2). Thus we can find out a subalgebra by eliminating negative-dimension generators. This is the case in general and hence should be regarded as a general prescription to pick up a subalgebra, which is known as Borel subalgebra. In fact, this prescription picks up less supersymmetric subalgebra, as we will see later.

Furthermore, a smaller subalgebra of (3) can be found because the dilatation

forms a subalgebra. This is nothing but the Poincaré algebra. This is also the case in general even if the starting algebra contains many U(1) charges.

The last is how to introduce an anisotropic scaling generator with an arbitrary

The anisotropic dilatation generator

Furthermore, it would be convenient to introduce two charges _{z}_{z}

where

We first classify superalgebras with the anisotropic scaling (1) as subalgebras of psu(2,2|4) , following the prescriptions described in

Two su(2) subalgebras are contained in psu(2,2|4) like

Thus there are two u(1) generators ^{1}_{1} and

The

The resulting

The list of

Then we shall give some examples of subalgebras of psu(2,2|4) .

To deduce the algebra, we have to use the new dilatation generator _{z}

The charges _{z}

The psu(2,2|4) generators on the

Since _{z}

The resulting algebra is generated by the set of the charges,

The set (8) contains 16 supertranslations and 8 superconformal generators. Hence the amount of supercharges are 24 in total. Note that the generators in (8) are confined in the lower triangular region (shaded) on the _{z}

The commutation relations of the bosonic part are nothing but those of Schrödinger algebra with an arbitrary

The (anti-)commutation relations including fermionic generators are

Here trivial (anti-)commutation relations have been omitted. In addition, u(1)^{2} and su(4) act on these generators as (5), (6), and (40) in

As briefly mentioned before, the case with

Anyway,

The charge _{2}

This algebra contains 24 supercharges and this is nothing but the super Schrödinger algebra found in [_{5}×S^{5} background with a periodic boundary condition for ^{-}-direction corresponding to the generator

According to the prescription explained in _{2} . This is nothing but a supersymmetric (centrally extended) Galilean algebra.

This algebra is generated by the set of the generators,

This is the Schrödinger algebra including 16 supercharges. The (anti-)commutation relations are given by (5), (6), (9), (10), (12), (13) and (40). Note that

Before moving to Lifshitz examples, we would like to comment on gravity solutions preserving super Schrödinger symmetry. Such solutions are reported in many literatures (For example, see [

It is a turn to consider a Lifshitz subalgebra. We consider _{z}

Note that

The (anti-)commutation relations are given only by (5), (6), (9), (12) and (40).

The generator

The dynamical critical exponent is given by _{2}(

Interestingly, gravity solutions of Lifshitz spacetime preserving 8 super Lifshitz symmetry are found in the literatures [

Note that the original construction of Lifshitz spacetime with _{0} in the Schrödinger spacetime with

The osp(8|4) generators on the

Next let us consider subalgebras of sp(4) ⊂ osp(8|4) . The detail of our convention and notation for this superalgebra is summarized in

First let us note that so(1,2) is contained as a subalgebra of osp(8|4) . Then the diagonal u(1) generator is contained in so(1,2) and it is represented by

The charge

For

As in the case of psu(2,2|4), it is possible to find out some subalgebras as follows.

Let us consider the dilatation generator defined in (4) . The algebra is generated by the set of the generators,

All of the generators are confined in the lower triangular region (shaded) on the

It follows that the dynamical critical exponent is _{z}

The (anti-)commutation relations including fermionic generators are given by

Note that u(1)^{2} and so(8) act on the generators, following (5), (6), and (47) in

The

is the Schrödinger algebra with 24 supercharges originally found in [

According to the prescription in

This algebra is generated by the set of the generators,

The region in the

In the same way as in the case of psu(2,2|4) , it is possible to find out a super Lifshitz algebra. This algebra is generated by the set of the generators,

All of the generators are confined the region specified by the Lifshitz condition

In a special case with _{I2}’s. However, by restricting to representations with zero central charge _{I2}, the exact Lifshitz algebra is reproduced.

Finally we consider the case of osp(8*|4) . The notation and convention for osp(8*|4) are summarized in

Let us consider the diagonal u(1) generator defined as

which is contained in so(1,5) ⊂ so*(8). Then the

Here we have defined

The osp(8

The ^{(I)}

This implies that {^{(1)}, ^{(2)}} generates an so(4) symmetry after all.

The generators are represented on the

Let see some examples of subalgebras of osp(8*|4) below.

We use the dilatation generator _{Z}

All of the generators are confined in the lower triangular region (shaded) on the _{z}

It follows that the dynamical critical exponent is _{z}

Here _{ij}

The (anti-)commutation relations including fermionic generators are

Note that the bosonic symmetry generators of u(1)^{2}, so(4) and so(5) act on the generators in the obvious way, following (5), (6), (28), and (49) in

The case with

It contains 24 supercharges in total and is the same as the result found in [

The algebra is generated by the set of the generators,

The generators are confined in the region specified by an additional condition

This algebra is generated by the set of the generators,

The generators are in the region fixed by the Lifshitz condition

In the same way as in the case of psu(2,2|4) and osp(8|4) , it is possible to realize the exact Lifshitz algebra when

We have presented a classification of superalgebras with the anisotropic scaling (1) as subalgebras of the superalgebras: psu(2,2|4), osp(8|4) and osp (8*|4), which are concerned with AdS/CFT in type IIB string and M theories. Our method to extract subalgebras is basically to find Borel subalgebras of these superalgebras. It enables us to derive non-relativistic scaling algebras systematically. In particular, we have considered two u(1) charges,

Let us introduce the superalgebra, psu(2,2|4), following the notation of [

We begin with u(2,2|4), in which the (anti-)commutation relations are given by

• su(2) ^{α}_{β}^{1}_{1} =^{2}_{2}, L^{1}_{2}^{2}_{1}

• su(2)

• su(4) ^{a}_{b}

• dilatation

• hypercharge

• central charge

• supertranslation

superconformal symmetry

• translation

We are interested in psu(2,2|4) and so we set

A superalgebra is a ℤ_{2}-graded linear space ^{0} ⊕^{1} with multiplication, Lie superbracket [, }:

where

It follows from (1.2) that

The elements in ^{0}(^{1}) are called even (odd) .

Let us consider the set of all (

which forms a superalgebra gl(

the superalgebra reduces to sl((

Here we have introduced the following quantities,

Namely,

which imply that

This is compatible with (43) and then the superalgebra is osp(

It is convenient to consider the following matrix,

where

Let us denote the components of

where

As a concrete example, let us consider osp(8|4) . Its (anti-)commutation relations are given by

where

• _{IJ}_{JI}

• _{AB}_{BA}

Let us decompose

where we take

and hence

• so(8)

• so(1,2) (

• dilatation

• supertranslation _{Iα}

• translation _{αβ }_{βα}

We introduce here the superalgebra, osp(8*|4) . Its bosonic part is

It is convenient to work with matrices with spinor indices for so(2,6) below [

where _{IJ}_{IJ}_{JI}_{IJ}_{AB}

Let us decompose the index

We use here the spinor basis in which

Note that

It is straightforward to rewrite the (anti-)commutation relations of osp(8*|4) in terms of these generators.

• so(1,5)

• usp(4) ≃so(5) _{AB}

• dilatation

• supertranslation _{aA}

• translation _{ab}_{ba}

This work was supported by the scientific grants from the ministry of education, science, sports, and culture of Japan (No. 22740160), and in part by the Grant-in-Aid for the Global COE Program “The Next Generation of Physics, Spun from Universality and Emergence” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.

The triality of SO(8) and hence of SO*(8) makes this possible. Concretely, we take

A chiral spinor of SO(2,6) is decomposed into