Symmetry 2012, 4(3), 517-536; doi:10.3390/sym4030517

Review
Supersymmetric Extensions of Non-Relativistic Scaling Algebras
Makoto Sakaguchi 1 and Kentaroh Yoshida 2,*
1
Department of Physics, Ibaraki University, Mito 310-8512, Japan; Email: msakaguc@mx.ibaraki.ac.jp
2
Department of Physics, Kyoto University, Kyoto 606-8502, Japan
*
Author to whom correspondence should be addressed; Email: kyoshida@gauge.scphys.kyoto-u.ac.jp; Tel.: +81-75-753-3879; Fax: +81-75-753-3886.
Received: 3 July 2012; in revised form: 7 August 2012 / Accepted: 8 August 2012 /
Published: 24 August 2012

Abstract

: 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 Symmetry 04 00517 i134 plays an important role in constructing the gravity duals for non-relativistic field theories. Supersymmetric extensions of symmetry algebras including the anisotropic scaling are very helpful to consider holographic relations accurately. We give a short summary on the classification of superalgebras with the anisotropic scaling as subalgebras of the following Lie superalgebras, psu(2,2|4), osp(8|4) and osp (8*|4), which appear in the study of AdS/CFT in type IIB string and M theories. It contains supersymmetric extensions of Schrödinger algebra and Lifshitz algebra.
Keywords:
supersymmetry; non-relativistic limit; scale invariance; AdS/CFT

1. Introduction

One of the most well-studied subjects in string theory is the AdS/CFT correspondence [1,2,3]. Although it has not been rigorously proven yet, it is supported by an enormous amount of evidence and there is no doubt for the correspondence. Assuming that it surely holds, there are two important assets. The one is that AdS/CFT provides us a powerful tool to study the unknown quantum gravity from the well-known quantum field theory without gravity. The other is that the AdS/CFT correspondence is a strong/weak duality and it enables us to study strongly-coupled quantum field theories non-perturbatively by using a description of classical gravitational theories.

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 [4,5]). In this direction there is a motive to figure out the holographic description of non-relativistic field theories because most condensed matter systems are non-relativistic. In particular, having some applications of AdS/CFT to condensed matter systems in our mind, we are interested in non-relativistic fixed points. For example, such fixed points appear in real experiments using ultra-cold atoms. The fixed points exhibit an anisotropic scaling invariance defined by a dynamical critical exponent Symmetry 04 00517 i134 like

Symmetry 04 00517 i001

where λ is a scaling constant and d is the number of spatial directions. The exponent Symmetry 04 00517 i134 measures the anisotropy in the time direction t. When Symmetry 04 00517 i134 = 1, this is the usual scaling symmetry in relativistic field theories. The case with Symmetry 04 00517 i134 ≠ 1 does not respect the Lorentz symmetry any more and the system has to be realized in a non-relativistic manner. The invariance under the anisotropic scaling (1) is a key ingredient to construct the spacetime metrics of the gravity duals [6,7,8]. Then the spacetimes described by these metrics are homogeneous and are represented by cosets [9]. Thus it is of importance to consider symmetry algebras with an isotropic scaling invariance like conformal symmetries in conformal field theories. (For example, the Schrödinger symmetry [10,11] fixes the behavior of two-point functions [12,13].) The scaling symmetry provides us a first clue in looking for the holographic description as in the usual study of AdS/CFT.

There are two typical examples of algebras including a non-relativistic scale invariance with Symmetry 04 00517 i134 ≠ 1. The former is the Schrödinger algebra [10,11] and the latter is the Lifshitz algebra (For the explicit algebra, e.g., see [6,7,9,13]). The Schrödinger algebra comprises the centrally extended Galilean (Bargmann) algebra and the dilatation with Symmetry 04 00517 i134 ≠ 1 . (Rigorously speaking, the Symmetry 04 00517 i134 = 2 case is called the Schrödinger algebra and then the generator of special conformal transformation is contained. However, for convenience, we will call the algebra with Symmetry 04 00517 i134 ≠ 1 the Schrödinger algebra with Symmetry 04 00517 i134 loosely as in most of the recent works.) The Lifshitz algebra consists of time and spatial translations, spatial rotations and the dilatation with Symmetry 04 00517 i134 ≠ 1 (in particular, no Galilean boost). It is well known that the two algebras can be realized as subalgebras of relativistic conformal algebras ( Symmetry 04 00517 i134 = 1) and it would be helpful to see a schematic sequence of the algebras like

Symmetry 04 00517 i002

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 subalgebras of the following Lie superalgebras (for other Lie superalgebras, see earlier works [14,15]), psu(2,2|4), osp(8|4) and osp (8*|4), which are concerned with AdS/CFT in type IIB string and M theories. It contains supersymmetric extensions of Schrödinger algebra and Lifshitz algebra. This classification is basically based on the previous works [16,17] but it contains a generalization of the results [16,17] to the arbitrary Symmetry 04 00517 i134 case and a new result on supersymmetric Lifshitz algebras.

This review article is organized as follows. In Section 2 we give general prescriptions to pick up a subalgebra including an anisotropic scaling. In Section 3 possible superalgebras including the anisotropic scaling invariance are classified as subalgebras of psu(2,2|4) . In Section 4 and Section 5 we classify superalgebras of osp(8|4) and osp(8*|4) in the same way. Section 6 is devoted to summary. In Appendices we summarize the notation and convention of psu(2,2|4), osp(8|4) and osp(8*|4) utilized in this article.

2. General Prescriptions

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) ,

Symmetry 04 00517 i003

Here Pµ describes a time translation and spatial translations, Lµυ contains spatial rotations and Lorentz boosts, D is a relativistic dilatation ( Symmetry 04 00517 i134 = 1) and Kµ describes special conformal transformations. For a generator T, the dimension d(T) is measured as

Symmetry 04 00517 i004

It is easy to read the dimensions of the generators

Symmetry 04 00517 i005

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

Symmetry 04 00517 i006

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 D never appears in the right-hand side of commutators of the generators in (3) . Therefore D can be removed and the reduced set of the generators,

Symmetry 04 00517 i007

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 Symmetry 04 00517 i134 . Indeed, the Schrödinger and Lifshitz algebras are obtained from a relativistic conformal algebra by shifting the relativistic dilatation D( Symmetry 04 00517 i134 = 1) with a certain U(1) generator V like

Symmetry 04 00517 i008

The anisotropic dilatation generator Symmetry 04 00517 i009 plays a central role in our discussions.

Furthermore, it would be convenient to introduce two charges dz and υ of Dz and V defined as, respectively,

Symmetry 04 00517 i010
Symmetry 04 00517 i011

where T is a generator. The υ charge enables us to figure out subalgebras pictorially, as we will see later.

3. Non-Relativistic Superalgebras from psu(2,2|4)

We first classify superalgebras with the anisotropic scaling (1) as subalgebras of psu(2,2|4) , following the prescriptions described in Section 2. The notation and convention of psu(2,2|4) are summarized in Appendix A.

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

Symmetry 04 00517 i012

Thus there are two u(1) generators L11 and Symmetry 04 00517 i013, which are the Cartan generators of the two su(2)s. By taking a linear combination of the two generators, a couple of new u(1) generators are defined as

Symmetry 04 00517 i014

The V is used to shift D while U generates the two-dimensional space rotation.

The resulting υ charges of the generators in psu(2,2|4) are summarized as follows:

Symmetry 04 00517 i015

The list of d and υ charges enables us to represent the generators of psu(2,2|4) on the d-υ plane as depicted in Figure 1 (For d charges, see Appendix A). This d-υ plane picture is very helpful to find out subalgebras. For simplicity, it is convenient to introduce the following notation:

Symmetry 04 00517 i016

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

(1) Schrödinger algebra with an arbitrary Symmetry 04 00517 i134 and 24 supercharges (d-υ ≥ 0)

To deduce the algebra, we have to use the new dilatation generator Dz defined in (4) , instead of the relativistic dilatation D. Since the generator V appears in the right-hand side of commutators, V must be included when Symmetry 04 00517 i134 ≠ 2. As we explain in detail as the next example, the Symmetry 04 00517 i134 = 2case is a bit special.

The charges dz and υ of the generators are summarized in the following list:

Symmetry 04 00517 i017
Symmetry 04 00517 g001 200
Figure 1. The psu(2,2|4) generators on the d-υ plane.

Click here to enlarge figure

Figure 1. The psu(2,2|4) generators on the d-υ plane.
Symmetry 04 00517 g001 1024

Since dz of time translation generator H is Symmetry 04 00517 i134 , the dynamical critical exponent also becomes Symmetry 04 00517 i134 .

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

Symmetry 04 00517 i018

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 d-υ plane (not dz but d!) as depicted in Figure 1. This region is specified by the Schrödinger condition d-υ ≥ 0. Thus the d-υ plane picture is very useful to understand a subalgebra pictorially.

The commutation relations of the bosonic part are nothing but those of Schrödinger algebra with an arbitrary Symmetry 04 00517 i134 ,

Symmetry 04 00517 i019
Symmetry 04 00517 i020
Symmetry 04 00517 i021

The (anti-)commutation relations including fermionic generators are

Symmetry 04 00517 i022
Symmetry 04 00517 i023
Symmetry 04 00517 i024
Symmetry 04 00517 i025

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 Appendix A.

(2) Schrödinger algebra with Symmetry 04 00517 i134 = 2 and 24 supercharges (d - υ ≥ 0)

As briefly mentioned before, the case with Symmetry 04 00517 i134 = 2 is a bit special because V does not appear in the right-hand side of the (anti-)commutation relations. It implies that the algebra may be closed without V.

Anyway, V can be eliminated in this case and thus the reduced algebra is generated by the following set of the generators,

Symmetry 04 00517 i026

The charge d2 is assigned as in the following list:

Symmetry 04 00517 i027

This algebra contains 24 supercharges and this is nothing but the super Schrödinger algebra found in [16,17]. This super Schrödinger algebra is realized in the AdS5×S5 background with a periodic boundary condition for x--direction corresponding to the generator M [18]. The periodic boundary condition breaks 8 superconformal symmetries.

According to the prescription explained in Section 2, it is possible to find out a smaller subalgebra without the dilatation D2 . This is nothing but a supersymmetric (centrally extended) Galilean algebra.

(3) Schrödinger algebra with an arbitrary Symmetry 04 00517 i134 and 16 supercharges (d - υ ≥ 0 and d ≥ 0)

This algebra is generated by the set of the generators,

Symmetry 04 00517 i028

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 V has been omitted because it does not appear in the right-hand side of the commutators. This is usually referred as the Schrödinger algebra with an arbitrary Symmetry 04 00517 i134 in the literature. This subalgebra is specified by imposing an additional condition d ≥ 0as well as d - υ ≥ 0. The condition comes from the prescription to eliminate negative dimension generators as explained in Section 2.

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 [19,20,21,22,23,24,25,26]). Basically, the Schrödinger symmetry is realized by adding a deformation term of pp-wave type to the AdS spacetime. The deformation term breaks the relativistic superconformal algebra to a smaller one like a super Schrödinger algebra discussed here. It is an open problem to construct gravity solutions preserving the super Schrödingier symmetry including K and V, because the isometry seems to be enhanced to the full relativistic superconformal algebra by adding K and V. Less supersymmetric Schrödinger symmetry also appears in this context. All of the symmetry can be found out by considering smaller subalgebras.

(4) Lifshitz algebra with an arbitrary Symmetry 04 00517 i134 and 16 supercharges (d - υ 0 and d + υ 0)

It is a turn to consider a Lifshitz subalgebra. We consider Dz defined in (4) . The resulting algebra is generated by the set of the generators,

Symmetry 04 00517 i029

Note that V has been omitted because it does not appear in the right-hand side of the commutators. This set contains the usual Lifshitz algebra but there is an additional generator M as the bosonic part, because M is not a center of the algebra any more when Symmetry 04 00517 i134 ≠ 2. The case with Symmetry 04 00517 i134 = 2is a bit special as we will explain in the next example. It also contains 16 supercharges and thus the resulting algebra should be referred to as the super Lifshitz algebra. This subalgebra can be pictorially understood by imposing an additional Lifshitz condition d + υ ≥ 0 as well as the Schrödinger condition d - υ ≥ 0 .

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

(5) Lifshitz algebra with Symmetry 04 00517 i134 = 2 and 16 supercharges (d - υ 0 and d + υ 0)

The generator V may be omitted because it does not appear in the right-hand side of the commutators. Then the subalgebra is generated by the set of the generators,

Symmetry 04 00517 i030

The dynamical critical exponent is given by Symmetry 04 00517 i134 = 2 because d2(H) = 2 . The (anti-)commutation relations are given by (5) with Symmetry 04 00517 i134 = 2, (6), (9), (12) and (40). This is the Lifshitz algebra with 16 supercharges and center M. In the bosonic case M does not appear in the right-hand side of commutators and hence it can be eliminated to give the usual Lifshitz algebra. However, in the supersymmetric case, the anti-commutator of Symmetry 04 00517 i031 and Symmetry 04 00517 i032 gives rise to M. Thus, by restricting to representations with zero central charge M, or by dropping Symmetry 04 00517 i031 and Symmetry 04 00517 i032, the generator M can also be removed. In the latter case, the resulting 8 super Lifshitz algebra is generated by

Symmetry 04 00517 i033

Interestingly, gravity solutions of Lifshitz spacetime preserving 8 super Lifshitz symmetry are found in the literatures [27,28,29,30,31,32,33]. The strategy to construct the solutions is the same as in the Schrödinger spacetime.

Note that the original construction of Lifshitz spacetime with Symmetry 04 00517 i134 = 2 starts from a Schrödinger spacetime with Symmetry 04 00517 i134 = 0 [27,28]. This construction can be explained algebraically by identifying the dilatation D0 in the Schrödinger spacetime with Symmetry 04 00517 i134 = 0with the generator M in the Lifshitz spacetime with Symmetry 04 00517 i134 = 2. That is, the Lifshitz algebra with Symmetry 04 00517 i134 = 2 can also be obtained as a subalgebra of Schrödinger algebra with Symmetry 04 00517 i134 = 0.

Symmetry 04 00517 g002 200
Figure 2. The osp(8|4) generators on the d-υ plane.

Click here to enlarge figure

Figure 2. The osp(8|4) generators on the d-υ plane.
Symmetry 04 00517 g002 1024

4. Non-Relativistic Superalgebras from osp(8|4)

Next let us consider subalgebras of sp(4) ⊂ osp(8|4) . The detail of our convention and notation for this superalgebra is summarized in Appendix B. The result is basically the same as in the case of psu(2,2|4), up to small differences. Thus we will briefly mention the difference and do not try to repeat the same explanation. For example, the relation to gravity solutions is omitted here.

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

Symmetry 04 00517 i034

The charge υ(T) of a generator T can be measured with (6) . The values of υ(T) of the generators are summarized in the list:

Symmetry 04 00517 i035

For d charges see Appendix B. The generators are expressed on the d - υ plane (See Figure 2). For simplicity, it is convenient to introduce the following notation,

Symmetry 04 00517 i036

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

(1) Schrödinger algebra with an arbitrary Symmetry 04 00517 i134 and 24 supercharges (d - υ ≥ 0)

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

Symmetry 04 00517 i037

All of the generators are confined in the lower triangular region (shaded) on the d-υ plane as depicted in Figure 2. This region is specified by the Schrödinger condition d - υ ≥ 0 again as in the case of psu(2,2|4) . The set (20) contains 16 supertranslations and 8 superconformal symmetries. Hence 24 supercharges are included in total. The charges Symmetry 04 00517 i038 and υ of the generators are summarized in the list,

Symmetry 04 00517 i039

It follows that the dynamical critical exponent is Symmetry 04 00517 i134 because dz(H) = Symmetry 04 00517 i134 . The commutation relations of the bosonic subalgebra are given by

Symmetry 04 00517 i040
Symmetry 04 00517 i041

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

Symmetry 04 00517 i042
Symmetry 04 00517 i043
Symmetry 04 00517 i044

Note that u(1)2 and so(8) act on the generators, following (5), (6), and (47) in Appendix B.

(2) Schrödinger algebra with Symmetry 04 00517 i134 = 2 and 24 super charges (d -υ ≥ 0)

The Symmetry 04 00517 i134= 2 case is a bit special and the generator V does not appear in the right-hand side of the (anti-)commutation relations. This implies that V may be omitted. After eliminating V, the resulting algebra generated by

Symmetry 04 00517 i045

is the Schrödinger algebra with 24 supercharges originally found in [16,17]. Note that the bosonic part is rigorously the Schrödinger algebra because V is not contained.

According to the prescription in Section 2, the set (25) contains a supersymmetric (centrally extended) Galilean algebra as a subalgebra.

(3) Schrödinger algebra with an arbitrary Symmetry 04 00517 i134 and 16 supercharges (d - υ ≥ 0 and d ≥ 0)

This algebra is generated by the set of the generators,

Symmetry 04 00517 i046

The region in the d-υ plane where all of the generator are confined is specified by an additional condition d ≥ 0 as well as the Schrödinger condition d - υ≥ 0 . This set contains 16 supercharges (only supertranslations). The generator V has been omitted because it does not appear in the right-hand side of the commutators. The (anti-)commutation relations are given by (5), (6), (21), (23), (24) and (47). The generator M is now a center.

(4) Lifshitz algebra with an arbitrary Symmetry 04 00517 i134 and 16 supercharges (d - υ ≥ 0 and d + υ ≥ 0)

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,

Symmetry 04 00517 i047

All of the generators are confined the region specified by the Lifshitz condition d + υ ≥ 0 as well as the Schrödinger condition d - υ ≥ 0 . The generator V has been omitted again because it does not appear in the right-hand side of the commutators. The (anti-)commutation relations are given by (5), (6), (23) and (47) .

In a special case with Symmetry 04 00517 i134 = 2 , the generator M becomes a center but it appears in the anti-commutator of QI2’s. However, by restricting to representations with zero central charge M, or by removing supercharges QI2, the exact Lifshitz algebra is reproduced.

5. Non-Relativistic Superalgebras from osp(8*|4)

Finally we consider the case of osp(8*|4) . The notation and convention for osp(8*|4) are summarized in Appendix C. The result is basically the same as in the case of psu(2,2|4) and osp(8|4) again, up to notational differences. We will briefly mention the differences as in the previous section.

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

Symmetry 04 00517 i048

which is contained in so(1,5) ⊂ so*(8). Then the υ charge is assigned to the generators as in the list:

Symmetry 04 00517 i049

Here we have defined

Symmetry 04 00517 i050
Symmetry 04 00517 g003 200
Figure 3. The osp(8*|4) generators on the d-υ plane.

Click here to enlarge figure

Figure 3. The osp(8*|4) generators on the d-υ plane.
Symmetry 04 00517 g003 1024

The U(I) with I = 1,2generate two su(2)’s , respectively,

Symmetry 04 00517 i051

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

The generators are represented on the d-υ plane as depicted in Figure 3 (For d charges see Appendix C). For simplicity, it is convenient to introduce the following notation,

Symmetry 04 00517 i052

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

(1) Schrödinger algebra with an arbitrary Symmetry 04 00517 i134 and 24 supercharges (d - υ ≥ 0)

We use the dilatation generator DZ defined in (4) . The algebra is generated by the set of the generators,

Symmetry 04 00517 i053

All of the generators are confined in the lower triangular region (shaded) on the d - υ plane as depicted in Figure 3. This region is specified by the Schrödinger condition d - υ ≥ 0 . The set (29) contains 16 supertranslations and 8 superconformal symmetries. Thus the total amount of supercharges is 24. The charges dz and υ are summarized in the list:

Symmetry 04 00517 i054

It follows that the dynamical critical exponent is Symmetry 04 00517 i134 because dz(H) = Symmetry 04 00517 i134 . The commutation relations of bosonic subalgebra are

Symmetry 04 00517 i055
Symmetry 04 00517 i056

Here αij are defined as

Symmetry 04 00517 i057

The (anti-)commutation relations including fermionic generators are

Symmetry 04 00517 i058
Symmetry 04 00517 i059
Symmetry 04 00517 i060

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 Appendix C, respectively.

(2) Schrödinger algebra with Symmetry 04 00517 i134 = 2 and 24 supercharges (d - υ ≥ 0)

The case with Symmetry 04 00517 i134 = 2 is a bit special again. Then V does not appear in the right-hand side of commutators. It implies that V may be eliminated when Symmetry 04 00517 i134 = 2 . The reduced algebra is generated by the set of the generators,

Symmetry 04 00517 i061

It contains 24 supercharges in total and is the same as the result found in [16,17]. A supersymmetric (centrally extended) Galilean algebra can also be found as a subalgebra.

(3) Schrödinger algebra with an arbitrary Symmetry 04 00517 i134 and 16 supercharges (d - υ ≥ 0 and d ≥ 0)

The algebra is generated by the set of the generators,

Symmetry 04 00517 i062

The generators are confined in the region specified by an additional condition d ≥ 0 as well as the Schrödinger condition d - υ ≥ 0. It contains 16 supercharges. The generator V has been omitted because it does not appear in the right-hand side of the commutators. The (anti-)commutation relations are given by (5), (6), (30), (32), (33), (28) and (49).

(4) Lifshitz algebra with an arbitrary Symmetry 04 00517 i134 and 16 supercharges (d - υ ≥ 0 and d + υ ≥ 0)

This algebra is generated by the set of the generators,

Symmetry 04 00517 i063

The generators are in the region fixed by the Lifshitz condition d + υ ≥ 0 and the Schrödinger condition d - υ ≥ 0 . This set contains 16 supersymmetries. Again we have omitted V because it does not appear in the right-hand side of the commutators. The (anti-)commutation relations are given by (5), (6), (28), (32) and (49).

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 Symmetry 04 00517 i134 = 2 , by restricting to representations with zero central charge M or by removing half of supersymmetries.

6. Summary

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, d and υ, and supersymmetric extensions of Schrödinger algebra and Lifshitz algebra have been obtained. This method provides us a pictorial understanding on non-relativistic scaling superalgebras obtained in the previous works [16,17]. Furthermore, we have extracted Schrödinger superalgebras and Lifshitz superalgebras with arbitrary Symmetry 04 00517 i134 . We hope that our result would be useful in constructing gravity solutions preserving superalgebras including the anisotropic scaling and in discussing the relation to the field-theory side.

A. psu(2,2|4)

Let us introduce the superalgebra, psu(2,2|4), following the notation of [34]. The number in a box represents the number of independent generators.

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

• su(2) Lαβ (L11 =- L22, L12 and L21) Symmetry 04 00517 i064

Symmetry 04 00517 i065

• su(2) Symmetry 04 00517 i135 and Symmetry 04 00517 i136 Symmetry 04 00517 i064

Symmetry 04 00517 i066

• su(4) Rab (a,b,c = 1,2,3,4) Symmetry 04 00517 i067

Symmetry 04 00517 i068

• dilatation D Symmetry 04 00517 i069

Symmetry 04 00517 i070

• hypercharge B

Symmetry 04 00517 i071

• central charge C

• supertranslation Symmetry 04 00517 i072 (its conjugate Symmetry 04 00517 i073)

superconformal symmetry Symmetry 04 00517 i074 (its conjugate Symmetry 04 00517 i075) Symmetry 04 00517 i076

Symmetry 04 00517 i077

• translation Symmetry 04 00517 i078 and special conformal transformation Symmetry 04 00517 i079

Symmetry 04 00517 i080

We are interested in psu(2,2|4) and so we set B = C = 0 here.

B. osp(p|2q)

A superalgebra is a ℤ2-graded linear space g = g0g1 with multiplication, Lie superbracket [, }: g × g→g characterized by the three properties

Symmetry 04 00517 i081

where

Symmetry 04 00517 i082

It follows from (1.2) that

Symmetry 04 00517 i083

The elements in g0(g1) are called even (odd) .

Let us consider the set of all (p+2q) × (p+2q)complex matrices

Symmetry 04 00517 i084

which forms a superalgebra gl(p|2q; ℂ). By imposing the condition that the supertrace should vanish,

Symmetry 04 00517 i085

the superalgebra reduces to sl((p|2q; ℂ). The superalgebra osp((p|2q; ℂ) is a subalgebra of sl((p|2q; ℂ) under an additional condition

Symmetry 04 00517 i086

Here we have introduced the following quantities,

Symmetry 04 00517 i087

Namely,

Symmetry 04 00517 i088

which imply that A∈so(p) and D∈sp(2q) . Furthermore, one may impose the reality condition

Symmetry 04 00517 i089

This is compatible with (43) and then the superalgebra is osp(p|2q; ℝ) with A* = A, B* = B, C* = C and D* = D.

It is convenient to consider the following matrix,

Symmetry 04 00517 i090

where F ∈ osp(p|2q). It follows that

Symmetry 04 00517 i091

Let us denote the components of M like

Symmetry 04 00517 i092

where I,J = 1,…,p and A = 1,…,2q . Then the (anti-)commutation relations of osp(p|2q) are

Symmetry 04 00517 i093

Example: osp(8|4)

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

Symmetry 04 00517 i094

where

RIJ = −RJI (I, J=1, ··· , 8) generates so(8) Symmetry 04 00517 i095,

LAB = LBA (A =1, ··· , 4) generates sp(4) generates sp(4) Symmetry 04 00517 i096 , and Symmetry 04 00517 i097 is the symplectic form.

Let us decompose A into ( Symmetry 04 00517 i098) with Symmetry 04 00517 i098 = 1, 2 . Then the related quantities are rewritten as

Symmetry 04 00517 i099

where we take q = 2 for osp(8|4) and Symmetry 04 00517 i100 Note that

Symmetry 04 00517 i101

and hence Symmetry 04 00517 i102 and Symmetry 04 00517 i103 are regarded as independent generators. It is straightforward to rewrite the (anti-)commutation relations of osp(8|4) in terms of these generators.

• so(8) Symmetry 04 00517 i104

Symmetry 04 00517 i105

• so(1,2) ( Symmetry 04 00517 i106 and Symmetry 04 00517 i103) Symmetry 04 00517 i064

Symmetry 04 00517 i107

• dilatation Symmetry 04 00517 i069

Symmetry 04 00517 i108

• supertranslation Q and superconformal symmetry Symmetry 04 00517 i109

Symmetry 04 00517 i110

• translation Pαβ (=Pβα)and special conformal transformation Symmetry 04 00517 i111

Symmetry 04 00517 i112

C. osp(8*|4)

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

Symmetry 04 00517 i113

It is convenient to work with matrices with spinor indices for so(2,6) below [35] . The (anti-)commutation relations of osp(8*|4) are given by

Symmetry 04 00517 i114

where I,J =1,…8 are spinor indices for chiral subspace of so(2,6) and thus (1 + 5) -dimensional spinor indices [36]. The charge conjugation matrix CIJ is symmetric, CIJ =CJI . The indices A,B = 1,…,4 are for usp(4) and Symmetry 04 00517 i115 is the symplectic form. The anti-symmetric matrix RIJ generates so(2,6) . The symmetric matrix LAB generates usp(4).

Let us decompose the index I into Symmetry 04 00517 i116 with a = 1,…4 and Symmetry 04 00517 i117. Then the related quantities are rewritten as [37].

Symmetry 04 00517 i118

We use here the spinor basis in which

Symmetry 04 00517 i119

Note that Symmetry 04 00517 i120 and Symmetry 04 00517 i121 .

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

• so(1,5) Symmetry 04 00517 i122

Symmetry 04 00517 i123

• usp(4) ≃so(5) LAB Symmetry 04 00517 i124

Symmetry 04 00517 i125

• dilatation D Symmetry 04 00517 i069

Symmetry 04 00517 i126

• supertranslation QaA and superconformal symmetry Symmetry 04 00517 i127

Symmetry 04 00517 i128

• translation Pab (= −Pba) and special conformal transformation Symmetry 04 00517 i129

Symmetry 04 00517 i130

7. Acknowledgments

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.

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