Volume 151B, number 2 PHYSICS LETTERS 7 February 1985

1 N = ~ S U P E R S Y M M E T R Y IN TWO DIMENSIONS

Makoto SAKAMOTO Department of Physics, Kyushu University, Fukuoka 812, Japan

Received 9 May 1984 Revised manuscript received 23 October 1984

1 We show that an N = ~ supersymmetry algebra exists for 2 (mod 8) dimensions. Also we construct N = ½ supersymme- tric lagrangians in two dimensions by use of a superfield formulation. In contrast to usual supersymmetric theories, the theory has a different number of on-shell degrees of freedom between bosons and fermions in a supermultiplet and positive semi-definiteness of the energy cannot be obtained from the algebra.

Supersymmetry is physically and mathematical ly a subject o f considerable interest * 1. A number o f authors have proposed various types of supersymmet- ric lagrangians. Possible forms of lagrangians depend on the dimension of the space - t ime since the degrees o f freedom o f particles are closely related to the di- mension (e.g. the number o f on-shell degrees o f free- dom o f a Dirac spinor and a real vector are 2 [D/2 ] and D - 2 in D dimensions). Properties o f supersym- metric theories also depend on the space - t ime dimension. It is known that there exist Ma jo rana - Weyl fermions for and only for 2 (mod 8) dimensions [2]. We may expect supersymmetric theories in those dimensions to have peculiar features. In this paper,

• 1 we will first show the exastence o f a n N = ~ supersym- metry algebra which is obtained from a n n = 1 super- symmetry algebra for and only for 2 (mod 8) dimen- sions. Next, we will construct two-dimensional

1 models with the N = ~ supersymmetry by use o f a superfield formulation [3] and will discuss their prop- erties.

We start with the N = 1 supersymmetry algebra,

{Qa, at3} = -Pu(PuC)o,~ , (1)

where Q, Pu and C are Majorana supercharge, energy- momentum operators and charge conjugation ma-

.1 For reviews see ref. [1].

trix ,2 , respectively. The algebra (1) exists only for 2 or 4 (mod 8) dimensions (assume the dimension to be even) because the Majorana condit ion is only possible when the dimension of the space - t ime is 2 or 4 (rood 8) [2]. Introduce chiral projection matrices h e by

h+ = ~( I + PD+I) , (2)

where

FD+ 1 = r/F01- ' lr 2 ... 1 - 'D-1 . (3)

Here, r/ is a constant and fixed by the conditions,

( rD+ l ) 2 = 1, (ro+l)t = r o + 1 . (4)

From (3), we have

C-1I'D+I C= - - f 'T+ l , f o r D = 2 (rood 8 ) ,

= +FT+I , for D = 4 (mod 8) . (5)

Using (1 )and (5), we obtain

{a+_c~, a+_#} = -Pu(h+_ PuchT+_),~O

= -Pu(h+I'uC)ao, f o r D = 2 ( m o d 8 ) ,

= 0 , for D = 4 (mod 8 ) , (6)

,2 The charge conjugation matrix C satisfies C -11"#C = - pu T.

115

Volume 151B, number 2 PHYSICS LETTERS 7 February 1985

where

Q+ = h+a . (7)

Furthermore, we have the following relation:

(Q+)C=cQT=Q_+, f o r D = 2 ( m o d 8 ) ,

= Q ; , f o r D = 4 ( m o d 8 ) , (8)

where we have used the Majorana property of the supercharge, i.e.,

a c = COT = a . (9)

Therefore,we conclude that for 2 (mod 8) dimensions there exists the following supersymmetry algebra con- taining only, say, Q+ (not Q_),

{a+a, a+a} = - P u ( h + puC)at~ . (1 O)

We call theories with this algebra (10)N = ½ super- symmetric ones [4]. The reason why the algebra (10) exists only for 2 (rood 8) dimensions is that Majorana- Weyl spinors can be defined only for those dimensions. Note that the existence of the algebra (10) does not mean the existence of non-trivial theories in all 2 (mod 8) dimensions.

In the following, we consider the case of two di- mensions and construct the superfield formulation for theories with only scalar and spinor fields. Assume that the superspace is parameterized by the two-vector xu and the two-component Majorana spinor 0_ with a negative chirality. The infinitesimal supertransla- tions are defined by

8x u =~_iTu0_, 6 0 _ = e_ , (11)

where e_ is a Majorana spinor with negative chirality. In order to obtain supersymmetric lagrangians, we in- troduce superfields which depend on x u and 0 . We find two types o f superfields, one is a scalar superfield and the other a spinor superfield, i.e.,

¢bi(x, 0 _) = a i ( x ) + O_ ~i+(x) , (12)

xlti_(X , 0 _) = Xi_(X) + 0 _F, (x ) , (13)

where A i and F i are real scalars and qJi+, ×i- are two- component Majorana spinors with positive and nega- tive chirality, respectively. The behavior of compo- nent fields under the infinitesimal supertranslations (11) is given by

6Ai = ~_~bi+ , 6t~i+= -i.eTe_Ai , (14)

and

6Xi_= e_F , , 8F, = - e _ i J ~ x , _ . (15)

Because our space is spanned by x u and 0_, the ac- tion should be expressed in the form,

s= fd2x f (16)

The supersymmetric "lagrangian" La must be a Lorentz spinor with a negative chirality since the ac- tion is a Lorentz scalar and d0_ has the same trans- formation property as 0+. Furthermore, the mass- dimension of L,~ must be 3/2 since that of 0 _ is -1/2. (The mass-dimensions of A i , t~i+ , X i - and F i are de- fined by 0, 1/2, 1/2 and 1, respectively.) If we require the renormalizability of the theory and the absence of derivative couplings, it turns out that the most gen- eral supersymmetric "lagrangian" is given by

Lc~ = --½ ~. ~i ( i f fD+)o~, -- ½ ~ (~,-D+)~,-c~ t i

+ ~xPi_c ,W,(cb ) , (17) i

where the W i are arbitrary functions of ~/and

D+ = a/80_ - i~'0_ . (18)

If we admit derivative couplings, we can add terms, say, V(~)it3(D + U(qb) o r ~, V(¢) ( ~ i - D+) ~ i - , to (17). In terms of component fields, the action (17) is re- written as

S=~(d2x(½ ~i ~i+~ff@i++l ~"

l l

+ ~ F i W i ( A ) - ~ f( i-~]+ (19) i 6 ~Aj I"

From the supersymmetry algebra (10) and the 1 super fields (12) and (13), we can show that the N =

supersymmetric theory has various peculiar properties in contrast to usual supersymmetric theories. In the Majorana representation, the algebra (10) is written as

{ Q + ~ , Q + ~ } = ( H + P 01 (20) 0 ate"

116

Volume 151B, number 2 PHYSICS LETTERS 7 February 1985

Usual supersymmetric theories have the property that _ 1 the energy is positive semi-def'mite. For the N -

supersymmetric theory, we can not, however, con- clude positivity of the energy only from the algebra (20). We instead have the following inequality,

(Q+I)2=H+P>~O (21)

although we can obtain positivity of the energy if we use the knowledge of translational invariance o f the vacuum. Furthermore, we observe that bosonic and fermionic states are paired unless E +P = 0. (For usual supersymmetric theories, the states of non-zero energy (E 4: 0) are paired.) Let lb> be any bosonic state with non-zero eigenvalue o f H +P. If a fermionic state If) is defined by

If> = (E + P ) - I Q+I I b >, (22)

If> has the same eigenvalue o f H + P as Ib). On the other hand, the zero-eigenstates o f H + P are not nec- essarily paired because any bosonic or fermionic state with zero eigenvalue satisfies Q+I Ib> = 0 or Q+I If> = 0 [5]. To make this point more clear, it is convenient to introduce light-cone coordinates, i.e.,

x -+ = 2-1/2(x 0 + x 1) , (23)

and the corresponding gamma matrices

T -+ = 2-1/2(3' 0 -+ 3,1). (24)

Using the relation * 3

3,0 I(1 _+ 3,3) = +3,1 ½(1 + 3,3)' (25)

we can rewrite (11) as

8x +=0, 8 x - = ~_ i3 , -0_ , 60 = e . (26)

Hence, x ÷ does not change under the supertransla- tions. This is the reason why states with E +P = 0 are not paired between bosons and fermions. Free mass- less Weyl fermions with a negative chirality describe left-moving particles because the Dirac equation

j ~ _ = 0 (27)

is equivalent to

a_qJ_ = (a/at - a/ax) ¢ _ = o . (2s)

_ 1 This fact suggests that the N - ~ supersymmetric

:~3 The me t r i c isgtaV = diag (1, - 1 ) and 3'3 =3"03"1.

theory in two dimensions may mismatch bosons and fermions. The numbers of off-shell degrees o f free- dom o f A i , Fi, ~i+ and Xi- are 1, 1, 1 and 1, respec- tively because Ai, F i are real scalar fields and ~i+, Xi- are two-component Majorana-Weyl spinors. On the other hand, the numbers of on-shell degrees o f free- dom o f A i , F i , ~i+ and Xi- are 1,0, 1/2 and 1/2 be- cause the F i are auxiliary fields and the number o f on-shell degrees of freedom of fermions is half in the sense that only left moving states are allowed. There- fore, scalar superfields (12) and spinor superfields (13) have the same number of off-shell degrees of freedom between bosons and fermions but a different number of on-shell ones. This property is quite pecu- liar in contrast to the usual supersymmetric theories. It should be noted that all characteristic properties o f N = ~ supersymmetry are due to the left-r ight asymmetric interactions - Y" Y~i- t~j +( 3 Wi/aAj) in (19). It turns out that if 3Wi/aA j = 3~. /aA i (in the l e f t - right symmetric case), the theory has N = 1 super- symmetry. Indeed, if so, the action (19) is invariant under the transformations

6Ai = -~- ~i+ + ~+Xi- ,

6 ~i+ = - i ~ e - A i + e+Fi,

8Xi - = - i ~e+A i + e_Fi ,

8F i = -~+i~¢i+ - e_ij~xi_ ,

where e is an anticommuting Majorana spinor. Hence, _ 1 N - ~ supersymmetry includes N = 1 supersymmetry

as a special case. 1 The N = ~ supersymmetric theory in two dimen-

sions is faced with no problem in the classical level but with a problem in the quantum level. Because the superfields (12) and (13) contain (Majorana-) Weyl spinors, the theory has the gravitational anom- aly [6] unless the number of q;i+ is equal to that o f x i _ . Hence, in order to maintain conservation o f the ene rgy-momentum tensor, we must introduce the same number o f scalar and spinor superfields.

In ten dimensions, we have not yet found (non- trivial) N = ~ supersymmetric models with only scalars and spinors. However, if we introduce gauge

1 fields, we can have an N - ~ supersymmetfic model,

' iXaJ~X a (29) L = -kF~uvFuva+~

117

Volume 151B, number 2 PHYSICS LETTERS 7 February 1985

where the X a are Majorana-Weyl spinors in the adjoint representation o f the gauge group and

+ _zabc Ab~c = a . A a - ,

DpXa = ~pxa _ gS-"abcAb"c.ztpr~ . (30)

This is known as the supersymmetric Yang-Mil ls theory in ten dimensions [7]. The theory h a s N = supersymmetry because the supercharge is a Ma jo rana - Weyl spinor. There are, in addit ion, supergravity and superstring theories [8] containing Majorana-Weyl spinors in ten dimensions. The relation between those theories in ten dimensions and the N = ~ supersym- metric theory in two dimensions is not clear at the moment .

We would like to thank M. Kobayashi for useful conversations and for reading through the manuscript . We are also grateful for valuable discussions with K. Fujikawa.

R e f e r e n c e s

[1] P. Fayet and S. Ferrara, Phys. Rep. 32C (1977) 249; A. Salam and J. Strathdee, Fortschr. Phys. 26 (1978) 57, J. Wess and J. Bagger, Supersymmetry and supergravity (Princeton U.P., Princeton, NJ, 1983).

[2] F. Gliozzi, J. Scherk and D. Olive, Nucl. Phys. B122 (1977) 253.

[3] A. Salam and J. Strathdee, Nucl. Phys. B76 (1974) 477; S. Ferrara, J. Wess and B. Zumino, Phys. Lett. 51B (1974) 239.

[4] M. Sakamoto, KEK preprint (1984) KEK-TH79. [5] E. Witten, Nucl. Phys. B202 (1982) 253. [6] L. Alvarez-Gaum6 and E. Witten, Nuct. Phys. B234

(1983) 269. [7] L. Brink, J.H. Schwarz and J. Scherk, Nucl. Phys. B121

(1977) 77. [8] J.H. Schwarz, Phys. Pep. 89 (1982) 223.

118