non-commutative geometry and supersymmetry

Volume 260, number 3,4 PHYSICS LETTERS B 16 May 1991

Non-commutative geometry and supersymmetry

F a h e e m Hussa i n International Centre for Theoretical Physics, 1-34100 Trieste, Italy

and

George T h o m p s o n lnstitut fdr Physik, Johannes Gutenberg UniversiRit, Staudinger Weg 7, Pf 3980, W-6500 Mainz, FRG

Received 23 January 1991

A direct correspondence between supersymmetric models and a recently proposed field theory based on non-commutative geometry is established.

I. Introduction

In this short note we are concerned with bringing together two strands of thinking about general structures that could underly the theories of fundamental forces. On the one hand supersymmetry has had a large impact on the possibilities o f model building and paves the way for the existence of many particles that have to date not been observed. This is supersymmetry when thought o f as a space-t ime symmetry. It also exists as an internal symmetry, and models have been developed which reduce to the standard model with particular relationships between the coupling constants at tree level [ 1,2 ]. More recently such supersymmetries, with appended Grass- man directions, have been put forward by Delbourgo and collaborators as possible theories that fit well within a Kaluza-Klein framework [ 3 ]. They also, quite respectably, fit the BRST ghosts into a supersymmetric framework which works well for conventional gauge fixing [ 4 ] and also for topological field theories [ 5 ].

On the other hand, Connes and Lott have introduced the notion o f non-commutat ive geometry [6] into physics. Recently, Coquereaux, Esposito-Farese and Vaillant [7] have given a concrete realization of the non- commutat ive programme. It is this version that we consider here. In their construction there appears a Z2 grading which points clearly to a supersymmetric underpinning. In fact there are two pointers: one algebraic (just mentioned above) and one geometric. The idea behind the Coquereaux, Esposito-Farese and Vaillant (CEV) model is to append two points ~l to space-t ime, which are called 0 and 1. Hence possible coordinates are (x ~, 0) and ( x u, 1 ), Now, one-forms on this space are defined by ,~(~x, O) =Audx~, .~¢(8x, 1 ) =Budx~ and d ( x , 1 ) - d ( x , 0) =¢~.

Our view is that the geometric notion, o f the differential of two points, is best treated as the differential in a grassmannian direction. So one doctors the above a little to take points of the form (x u, 0). Then ~¢ (x, 80) = ~ ×d0. But what o f the 772 grading? In CEV the grading is such that with Z2= { 1, q}, q (x ~', O) = ( x ~', 1 ) and at the

Supported by Bundesministerium ftir Forschung und Technologie, Bonn, FRG. ~ This is the simplest version and the one addressed in their paper.

0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland ) 359


level of forms one expects t/AudxU=BudxU. This suggests, as advocated in ref. [7 ], that one has a matrix of forms (A,~, ~ ) with ~/= (o ~), and all of the entries are themselves square matrices. The models of interest B~tdx, u

that are considered are of the form (~- ~) where A and B are (m X m) hermitian matrices and ~is the hermitian conjugate of 0, also rn × m. But such matrices define the supergroup U (m/rn) .

We identify therefore the general models of ref. [ 7 ] with those that arise from taking as the gauge group, U (re~m), for connections defined on a superspace which is conventional space-time plus two (or less - see footnote 2) Grassmann (space-t ime scalar) coordinates.

We begin with the simplest mode, U( 1 / l ). This contains all the relevant features, so it is presented in some detail. U ( m / n ) is then briefly dealt with and a discussion follows.

2. The abelian model and U(I/1)

The U ( 1 ) model of Coquereaux, Esposito-Farese and Vaillant, is based on a matrix connection

) \ 0 B,,dx~ " (1)

The general rule for multiplication is taken to be

( ; C) G ( ; ', C : ) = ( A ^ A ' + ( - 1 ) ° c C ^ D ' CAB,'+(--1)°AAAC'~ \DAA"+'(--1)OBBAD ' BAB +(--1)ODDAC'J ' (2)

where the entries are themselves n × n matrices and the grading 0 is 0 for forms of even degree and 1 otherwise. In ( 1 ), 0 and q~are scalars, so of even grading, while A and B are odd (one-forms). Hence

\ OA-B~ B^B-O0] (3a)

(3b)

(3a) is the general form that we will make use of in subsequent sections. The other ingredient is a generalised exterior derivative,

\ - d D + A - B dB+C+D J" (4)

This differs, by phases, from the one employed by CEV, but this difference is inconsequential. Applying this to ( 1 ) and defining a field strength, as ~-Nc__ dA + s¢O d we find, with ~ = ( ~,' .~2), ~ 2

~NC =FA + (0+0) +00, ~ C = -dO- (A-B)+ (OB-A~) ,

f f~c = -dqT+ (A-B) + (OA-B~), ~NC = F B + ( ~ + ¢ 7 ) + ~ . (5)

The superscript NC indicates non-commutative. Here F A = dA + A ^ A. One may form an action, but we will not need this for the moment.

Now let us rederive these objects using the superalgebra U ( 1 / 1 ) and an expanded space-time. The generators of U ( 1 / 1 ) satisfy

[ R , , R j ] = 0 , [Ri, Q ] = - ( - 1 ) i Q , [R, ,Q]=(-I ) iQ, i = 1 , 2 ,

{ Q , Q } = R , + R 2 , Q2=(~2=0 . (6)

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The fundamental two-dimensional representation is

R l = ( l 0 ~ ) , R2=(00 01) , Q=(00 10), (~=(01 00). (7)

We consider a space-time with two extra Grassmann coordinates (x u, 0, if) ~2. A gauge field in the present content becomes an algebra valued graded one-form with an expansion

q~(x, 0, O)=dxUV,,(x, 0, O) +dO S(x, O, O) +dO~(x, O, O). (8)

[To distinguish with the CEV, non-commutative, model we use q0 for the gauge field in the supersymmetric case ]. The connection with the U (1) model of CEV is established by taking gauge potentials (superfields) of the form

V,,(x,O)=Au(x)R,+Bu(x)R2, S(x,O,O)=O,(x)Q+O2(x)O., Z(x,O,g)=Ckz(x)Q+~l(X)O. (9)

If we put the connection q~(x, 0, O) into its matrix form, we find that

( _ A u % ~ - ~ld0nt-~2d0-~ (A ~ ) ~(x'O'g)=kO2dO+O~dO Budx*' }= ~ " (10)

This is clearly in one to one correspondence with ( 1 ) if we further choose ~2 = 0. From now on we will make this choice in this section and put ~ =~. The field strength is taken to be the graded two-form

.~s(qb) = d ~ + qb~, (11)

where d is the exterior derivation defined as

dX= (dx ~ O~ + dO 0o + d00o) X, ( 12 )

and the multiplication is the usual matrix multiplication. The differentials dx ~ and dO are one-forms for which a graded wedge product is defined such that

(13) dxU^dx"=-cL , c"^dx u, dxuAdO(dO)=-dO(dO)Adx u, d 0 ^ d O = d O ^ d 0 .

Let us see that the product @qb defined here in fact corresponds to the (3 product of CEV (3a), (3b).

(A^A+09^~ AAfp+~o^B) (/)(iD= \(/SAA+BA(/5 (/SA(fl+BAB

( f ~ d 0 d O d0(---AO+~B)~ = \dO(OA-B~) (~0 dO dO J '

(14a)

(14b)

having used (13). So we see that ordinary matrix multiplication in the superfield theory corresponds to the graded multiplication of CEV. In particular, for the U( 1 ) model (14b) matches (3b). Where CEV have functions ~ we have one-forms ¢ dO.

It remains to explain the differences that arise in using d or a. Let us write a as

Then a direct calculation shows that in general (not just for U ( 1 ) entries).

( (° ~NC(S/)~ .~S A+ dO - ~ d 0 d 0 , (16)

,2 One can get away with using only one Grassmann coordinate, but it makes the construction of actions less natural.

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where = denotes equivalence, in the sense that where in CEV one has functions q~, we have one-forms q~ dO. For the U ( 1 ) theory under consideration (16) means that the components of the supersymmetric field strength ~-s differ from those of ~NC by a shift of unity in the Higgs fields ¢ and q~ and the addition of 1 to the composite ¢~7. Explicitly, using ( 11 ) and (14) we have

( d A + 0 ¢ d 0 dff dO(-AO+¢B-d(b)~ f f s ( ~ ) = \ d 0 ( ¢ A - B ~ - d ¢ ) dB+O¢ dO d0 ]

and

k d0 (q~A-B¢+A- B - de)

(17)

dO( -AO+ OB- A + B-ddp ) ~ dB+ (q~¢+ 0+ ¢+ 1)d0 d 0 ] "

(18)

The differences between the two models lie in the fact that the minimum of the Higgs potential of the supersymmetric model is at ¢ = ¢ = 0 while for the non-commutative model it lies at ¢=~7= 1. To get a non-zero vacuum expectation value for ¢, in the supersymmetric case, one needs to add an explicit mass term as (~2S2 ((~))2 only gives a ( ~ ) 2 coupling. By shifting ¢ and q~ by 1 we get the mass term but also a constant term (an infinity for it multiplies the volume of space-time). When this constant term is subtracted, we get simply the term ( ~ c ) 2 . A shift is not enough, because one could simply undo the shift by setting the vacuum expectation value of ¢, ( ¢ ) , to be equal to it. Then ( ¢ ) would not appear again (at tree level).

U(m/n)

In the general case, in the CEV model, the connection ~4= (A C) has blocks which are m×m, re×n, nXm and n × n matrices. Let us first remark on the case of m = n. The appropriate superalgebra in this case is U (m/ m ) which has (m + m) 2 generators, which match the number of components of _d. One may split the generators into odd and even parts and ansatz once more that the correct superfield should be

(1) = A j, ' R i d Y ~' -4- B/, iRid.xlz -~- (YQi dO+ q)iQidO , ( 19 )

which corresponds to the connection d = (2 ¢e) of CEV. The properties of the standard multiplication and the O multiplication of CEV were seen to match in the U (1, 1 ) model. By virtue of the fact that ~ 's are now differential forms, the same holds here. Consequently, eq. (16) also holds in this case. The U (m/m) models and the non-commutative theories based on even matrices are isomorphic.

How can we construct non-commutative models based on odd matrices? The problem is that the grading operator in the CEV model is not easy to implement in this case. From the supersymmetric point of view there are two ways to proceed.

The first is to begin with a U(m/m) and then to embed odd dimensional matrices into this group. For example, the CEV model of the weak interactions, is naturally related to the SU (2/1 ) supergroup which may be embedded in SU (2/2) [ 1 ]. If one does this consistently one arrives precisely at the CEV model. SU (2/1 ) is an obvious candidate as a unifying group as it has eight generators, four bosonic for the gauge potentials and four fermionic for the Higgs fields. The embedding is done as follows.

The 15 generators for S U (2/2 ) are (tlo)(00 to0)(10 t01)00 Ql= - - _ , Q2 = - - _ , Q3 = - - , Q4 =

0 0 0 0 0 0

0 0 0 0 0 0

(t oo) 0 1

0 0

0 0

Qa=Q~ x, a = 1 , 2 , 3 , 4 , (20)

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) l/0+ t The fundamental three-dimensional representation of SU(2 /1 ) is embedded by taking the following eight

generators :

1 1 1 R ° = l - / ~ 3 = - 5 , Ri= 5 - , i = 1,2,3,

and

Qa, Oa, a = l , 2 , ( 2 1 )

All these eight generators have zeros in the last column and row. The generalised connection (19) then can be written as

10 0 A = - - - - , ( 2 2 )

(P~ (P2 - A o 0 0 0 0 0

which is precisely the SU (2/1 ) model of CEV [ 1 ]. By employing the SU (2) × U ( 1 ) trace [ 1 ], one breaks the SU (2/1 ) symmetry and lands on the standard model without the mass term for the Higgs. On the other hand, by using the form of the field strength, eq. (16), through the embedding one arrives at the form of the Higgs potential that forces the symmetry breakdown to occur.

Alternatively, we may work directly in terms of U(m/n) or SU(m/n). Without knowing what a working definition of the non-commutat ive grading is, we may take the relationship, eq. ( 16 ), between the field strengths as a definition of the non-commutat ive field strength. This amounts in the supersymmetric framework to using the derivative d , = d + [~/, ] where q is a shift matrix that one has to use, such that we take the commutator of~/ with even elements and ant icommutators with odd elements as in eq. (15). The field strength is defined by F = d,~d +.J4~4and this agrees with our previous identification. Let us illustrate this with the SU (2/1 ) example. In this case

r] = (Po QI +~oQ1 , (23)

where ~Oo = ~od0, with q 2 = ~ao ~o (R3 -- Ro ). Go is a constant. As one can see from eq. (21), only the Ro survives the S U ( 2 ) trace. Using this F a s the field strength leads to

the Dondi-Jarv is [ 1 ] action supplemented with a mass term which is in agreement with the results of CEV. The details o f the construction of the action may be found in ref. [ 1 ].

On the other hand, one may turn this argument on its head and argue that the correspondence that we have established has given us a way of generating mass terms using the new derivative d, in the supersymmetric models. This enters as a particular choice o f "background" connection, as the shift itself implies. One must

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recognize, o f course, that i t is in a sense m o r e na tura l in the n o n - c o m m u t a t i v e f r amework , as the shift is part o f

the geomet r i c cons t ruc t i on f r o m the outset .

Acknowledgement

The au thors w o u l d like to thank R. C o q u e r e a u x and the T h e o r y G r o u p at M a i n z for useful discussions.

References

[ 1 ] P.H. Dondi and P.D. Jarvis, Phys. Lett. B 84 (1979) 75. [2] J.G. Taylor, Phys. Lett. B 84 (1979) 79. [3] R. Delbourgo and R.B. Zhang, Phys. Lett. B 202 (1988) 296; Phys. Rev. D 38 (1988) 2490. [ 4 ] R. Delbourgo and P.D. Jarvis, J. Phys. A 15 ( 1982 ) 611;

L. Bonora and M. Tonin, Phys. Lett. B 98 ( 1981 ) 48. [5] J.H. Horne, Nucl. Phys. B 318 (1989) 22;

D. Birmingham, M. Blau and G. Thompson, J. Mod. Phys. A, in press. [ 6 ] A. Connes and J. Lott, Particle models and non-commutative geometry, IHES preprint ( 1990 );

M. Dubois-Violette, J. Madore and R. Kerner, Class. Quant. Gray. 6 (1989) 1709. [ 7 ] R. Coquereaux, G. Esposito-Farese and G. Vaillant, Higgs fields as Yang-Mills fields and discrete symmetries, CNRS preprint CPT-

90/P.2407.

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non-commutative geometry and supersymmetry

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