# Supersymmetric Extensions of Non-Relativistic Scaling Algebras

^{1}

^{2}

^{*}

## 1. Introduction

## 2. General Prescriptions

_{µ}describes a time translation and spatial translations, L

_{µυ}contains spatial rotations and Lorentz boosts, D is a relativistic dilatation ( = 1) and K

_{µ}describes special conformal transformations. For a generator T, the dimension d(T) is measured as

_{z}and υ of D

_{z}and V defined as, respectively,

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

^{1}

_{1}and , 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

#### (1) Schrödinger algebra with an arbitrary and 24 supercharges (d-υ ≥ 0)

_{z}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 ≠ 2. As we explain in detail as the next example, the = 2case is a bit special.

_{z}and υ of the generators are summarized in the following list:

_{z}of time translation generator H is , the dynamical critical exponent also becomes .

_{z}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.

^{2}and su(4) act on these generators as (5), (6), and (40) in Appendix A.

#### (2) Schrödinger algebra with = 2 and 24 supercharges (d - υ ≥ 0)

_{2}is assigned as in the following list:

_{5}×S

^{5}background with a periodic boundary condition for x

^{-}-direction corresponding to the generator M [18]. The periodic boundary condition breaks 8 superconformal symmetries.

_{2}. This is nothing but a supersymmetric (centrally extended) Galilean algebra.

#### (3) Schrödinger algebra with an arbitrary and 16 supercharges (d - υ ≥ 0 and d ≥ 0)

#### (4) Lifshitz algebra with an arbitrary and 16 supercharges (d - υ ≥ 0 and d + υ ≥ 0)

_{z}defined in (4) . The resulting algebra is generated by the set of the generators,

#### (5) Lifshitz algebra with = 2 and 16 supercharges (d - υ ≥ 0 and d + υ ≥ 0)

_{2}(H) = 2 . The (anti-)commutation relations are given by (5) with = 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 and gives rise to M. Thus, by restricting to representations with zero central charge M, or by dropping and , the generator M can also be removed. In the latter case, the resulting 8 super Lifshitz algebra is generated by

_{0}in the Schrödinger spacetime with = 0with the generator M in the Lifshitz spacetime with = 2. That is, the Lifshitz algebra with = 2 can also be obtained as a subalgebra of Schrödinger algebra with = 0.

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

#### (1) Schrödinger algebra with an arbitrary and 24 supercharges (d - υ ≥ 0)

_{z}(H) = . The commutation relations of the bosonic subalgebra are given by

^{2}and so(8) act on the generators, following (5), (6), and (47) in Appendix B.

#### (2) Schrödinger algebra with = 2 and 24 super charges (d -υ ≥ 0)

#### (3) Schrödinger algebra with an arbitrary and 16 supercharges (d - υ ≥ 0 and d ≥ 0)

#### (4) Lifshitz algebra with an arbitrary and 16 supercharges (d - υ ≥ 0 and d + υ ≥ 0)

_{I2}’s. However, by restricting to representations with zero central charge M, or by removing supercharges Q

_{I2}, the exact Lifshitz algebra is reproduced.

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

^{(I)}with I = 1,2generate two su(2)’s , respectively,

^{(1)}, U

^{(2)}} generates an so(4) symmetry after all.

#### (1) Schrödinger algebra with an arbitrary and 24 supercharges (d - υ ≥ 0)

_{Z}defined in (4) . The algebra is generated by the set of the generators,

_{z}and υ are summarized in the list:

_{z}(H) = . The commutation relations of bosonic subalgebra are

_{ij}are defined as

^{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 = 2 and 24 supercharges (d - υ ≥ 0)

#### (3) Schrödinger algebra with an arbitrary and 16 supercharges (d - υ ≥ 0 and d ≥ 0)

#### (4) Lifshitz algebra with an arbitrary and 16 supercharges (d - υ ≥ 0 and d + υ ≥ 0)

## 6. Summary

## A. psu(2,2|4)

^{α}

_{β}(L

^{1}

_{1}=- L

^{2}

_{2}, L

^{1}

_{2}and L

^{2}

_{1})

^{a}

_{b}(a,b,c = 1,2,3,4)

## B. osp(p|2q)

_{2}-graded linear space g = g

^{0}⊕g

^{1}with multiplication, Lie superbracket [, }: g × g→g characterized by the three properties

^{0}(g

^{1}) are called even (odd) .

#### Example: osp(8|4)

_{IJ}= −R

_{JI}(I, J=1, ··· , 8) generates so(8) ,

_{AB}= L

_{BA}(A =1, ··· , 4) generates sp(4) generates sp(4) , and is the symplectic form.

_{Iα}and superconformal symmetry

_{αβ }(=P

_{βα})and special conformal transformation

## C. osp(8*|4)

_{IJ}is symmetric, C

_{IJ}=C

_{JI}. The indices A,B = 1,…,4 are for usp(4) and is the symplectic form. The anti-symmetric matrix R

_{IJ}generates so(2,6) . The symmetric matrix L

_{AB}generates usp(4).

_{AB}

_{aA}and superconformal symmetry

_{ab}(= −P

_{ba}) and special conformal transformation

## 7. Acknowledgments

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Sakaguchi, M.; Yoshida, K.
Supersymmetric Extensions of Non-Relativistic Scaling Algebras. *Symmetry* **2012**, *4*, 517-536.
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**AMA Style**

Sakaguchi M, Yoshida K.
Supersymmetric Extensions of Non-Relativistic Scaling Algebras. *Symmetry*. 2012; 4(3):517-536.
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**Chicago/Turabian Style**

Sakaguchi, Makoto, and Kentaroh Yoshida.
2012. "Supersymmetric Extensions of Non-Relativistic Scaling Algebras" *Symmetry* 4, no. 3: 517-536.
https://doi.org/10.3390/sym4030517