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Extensions of Lie superalgebras and supersymmetric abelian gauge fields
Volume 193, number 1
PHYSICS LETTERSB
9 July 1987
EXTENSIONS OF LIE SUPERALGEBRAS AND SUPERSYMMETRIC ABELIAN GAUGE FIELDS G. LANDI Scuola lnternazionale Superiore Studi Avanzati, Strada Costiera I 1.1-34014, Trieste, Italy and lNFN, Sezione di Trieste, 1-34100 Trieste, Italy
and G, MARMO Dipartimento di Fisica, Universit,~degli Studi di Napoli, Mostra d'Oltremare Pad. 19, 1-80125 Naples, Italy and INF, Sezione di Napoli, 1-80125 Naples, Italy
Received 18 March 1987
We constructan extensionof Lie supcralgebraswhich allowsan algebraicdescriptionof the supersymmetricHopf fibration of the supersymmetrictwo-sphere.Moreover,we constructa supcrsymmetricgeneralizationof electromagnetism.
In a previous paper [ 1 ], aiming at an algebraic formulation of gauge theories, much in the same spirit of the "geometry without points" of von Neumann [ 2 ] and of the "non-commutative" geometry used by Witten to deal with string theories [ 3 ], we have used extensions of Lie algebras to give an algebraic description of abelian monopoles and of electromagnetism. One of the main advantages of an algebraic approach is the possibility of generalizing to the supersymmetric case all results which have been obtained in the ordinary case. One has only to replace Lie algebras with Lie superalgebras (or graded Lie algebras). Beside being very important in physics in the context of supersymmetry [4], Lie superalgebras are also important in mathematics where they have been the object of extensive research [ 5 ]. In this letter we consider extensions of Lie superalgebras. The first thing we show is how an exact sequence of Lie superalgebras carries a notion of a covariant derivative in a way which is the analogue of the ordinary (hereafter synonymous of not graded) situation considered in ref. [ 1 ]. We next introduce a "monopole-like" sequence of Lie superalgebras and analyse some of its properties. Finally we describe a supersymmetric generalization of electromagnetism. We refer to a subsequent note [6] for the description of the (graded) Weyl homomorphism and the construction of (graded) Chern-Simons terms. Let us start by briefly describing extensions of Lie superalgebras and intoducing the relevant objects. We refer to ref. [ 5 ] for the definition and the basic properties of Lie superalgebras. In what follows all Lie superalgebras will be Z2-graded. We denote the graded Lie product by [ , ] G and the parity or grading of any homogeneous element by p(" ) • Z2. Extensions of the Lie superalgebras have also been considered in ref. [ 7]. An extension of the Lie superalgebra B by the Lie superalgebra A is an exact sequence of Lie superalgebras I'
/g
0---~ A--~ E--~ B---* 0
(1)
The maps i and it are respectively an injective and a surjective Lie superalgebra homomorphism with 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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im i = k e r n; they are of zero parity, i.e., they preserve the grading; A is an ideal in E. In what follows we shall consider also cases in which A is not graded, i.e., cases in which A has a trivial odd part. With the extension (1) there is associated an exact sequence of Lie superalgebras of derivations by means of the following diagram: i
0
,
A
--~
rt
E
--~
B
--~0
~
P{
Y~
(2)
InG(A)----.
GD(A)---.
OutGD(A)---,0
Here G D ( A ) is the Lie superalgebra of all derivations of A and In G D ( A ) the Lie superalgebra of inner derivations of A, i.e., the adjoint action of A on itself, D a ( a ' ) = [a,a']o for some a eA; this also defines the homomorphism a as a ~ a ( a ) = - D ( a ) . Finally, Out G D ( A ) is the Lie superalgebra of equivalence classes of derivations of A modulo inner ones. The map fl is defined as e-,fl(e): f l ( e ) a = [e,a]c ,
(3)
which is an element of A because A is an invariant subalgebra. Furthermore, 7 is defined to make the last diagram in (2) commute. We shall call 7 the character of extension (1). The maps a, fl and 7 are all of zero parity. , ~ r ~' Two extensions A----E-----. B and A-----.E-----B with the same character 7 are said to be equivalent if there exists an isomorphism fi E---. E' such that i' = f . i and n = f i rt'. We call a connection on the sequence (1) any graded vector space homorphism p: B--,E such that n ° p = i d s .
(4)
As a consequence p is of zero parity. The extent to which p fails to be a Lie superalgebra homomorphism is measured by its curvature F. F is the A-valued skew map of zero parity on B × B defined by F(X., X2)=p([XI,X2IG)-
[p(Xl),p(X:)]o,
V X., X2~B.
(5)
A connection on (1) may be equivalently described by means of an A-valued connection one-form co on E co: E--,A,
Y--*co(Y) = Y - p ( n ( Y ) ) .
(6)
The curvature two-form g2 is the A-valued skew map on E × E defined by g2(Y,, Y 2 ) = F ( z t Y , , T t Y 2 ) ,
V Y,, Yz~E,
(7)
and in terms of co "Q( Y, , Y2 ) = [ Y' ,"Q( Y2 ) ]G -- ( -- )P( Y' )P( Y2) [ Y2 ,CO( YI ) I G -- CO( [ YI , Y2 I o ) -- [ CO( Yt ) ,CO( Y2 ) ]G .
(8)
By its definition one sees that g2 is horizontal, i.e. g2( Y~, Y2)=0 whenever one of the Y~'s belongs to A. As in the case of the ordinary situation we assume that A, B and E have as coefficients elements of a 7?2graded commutative algebra ~ = ~ o ) @ ~ ) and require that elements of E act as graded derivations in ~ , with [ X , Y ] c "f= X" ( Y ' f ) - ( - )p¢x)p(r) Y'( X ' f ) , [ X, f Y I G = ( - )Ptf)P~x) f [ X,Y] + ( X ' f ) Y, 62
Vf~;
(9) X, YeE.
(10)
Volume 193, number 1
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9 July 1987
Again, A being an ideal in E, condition (10) forces A to act trivially on ~ . We shall call the sequence (1), when endowed with the previous structure, a Lie superbundle with algebra ~. Once we give a connection p on the sequence (1) we have an action of B on A provided by the map fl.p: B--, GD (A), with fl defined in (3). Using this action we define the "covariant derivative" of an element Ye A along an element Xe B by setting
Vx Y=-fl*p( S) ' Y= [p( X),Y]G
(1 1)
and FxYeA, V XeB, YeA. The map defined by the previous equation has the properties of a graded covariant derivative, namely it is linear in both X and Y and
Vx(fY) = (-)P(f)P(x)fVxY+ (X.f) Y, VfxY=fVxY,
V XeB, YeA, fe ~ .
(12)
We describe now the "monopole-like" sequence of superalgebras (for a related construction see ref. [8]). We take E to be a free algebra of finite type generated with coefficients in ~ by three event elements Yt, Y2, Y3 and by two odd elements Vi, V2 whose graded commutation relations are those of the graded extension GSU(2) of the rotation group [9] (the supergroup GSU(2) is often called UOSP(1,2) [10])
[Yj,Yk]_=ejg¢Yt,
[Yj, V,,]_=½i(crj)a~Va, j , k , l = l , 2 , 3 ,
[V,~,Val+=½i(Ctu),~aYj, a, f l = l , 2 .
(13)
Here a~, a2, tr3 are the Pauli matrices and C is the charge conjugation matrix C = - i a 2 . We assume there is an action (representation) of E on ~. To define the monopole sequence we require the algebra A to be an even algebra generated over ~ by an element Z whose commutators with Y~, ..., V2 all vanish. In particular this implies that A acts trivially on ~ . It Z is of the form
Z=f~Y, +f2Y2+f3Y3+g,V, +g2V2, fj, gc, e ~ ,
p(j~j)=0,
p(g,~)=l.
(14)
By requiring [Z, IT,.]o = [Z, V~] o = 0, the coefficients fj and g~ have to obey the flollowing equations:
Yk'f,, --ek,,,~=O,
(15a)
Yk'g~, + ½i(a k)~,pgp = 0 ,
(15b)
Z,~'f,~-½i(Cam),~pgp=O, Z~.gB-½i(am)ad,,=O,
¥ i , k = l , 2 , 3,
(15c)
V a, f l = l , 2.
(15d)
We shall assume the action of E on ~ to be such that a non-trivial solution of the previous equations does exist. Let us now introduce the ~-valued dual forms. We take three even one-forms ff such that if(Y,) =JJkand if(V~,) = 0 and two more odd one-forms ~ such that ~ ' ( Z ~ ) = J ~ and ~ ( ~ ) = 0. If we take the combination
O=(1/P)ff, O' + g a ~ ) ,
P=ff,)2+~)2+~)2#0,
g2=(g,)2 +(g~)2=O ,
(16)
with the f ' s and the ga'S solutions of eqs. (15), then 0 (Z) -- 1 and the A-valued one-form on E
oJ=Z®O
(17)
is a connection form for the sequence (1). Remark. Since A acts trivially on ~ and is generated by an element which commutes with all elements in E, any A-valued form ~ can be identified with an ~-valued form ~ via ~= Z® 63
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For the curvature t2 defined in ( 8 ) we have $2(X, Y) = { X . O ( Y ) - ( - )p(x)p(r) r ' o ( x ) - O ( [ X , Y ] o ) } Z
-{dO(X, Y)}Z, v X, YeE,
(18)
and the Bianchi identity follows d~=0.
(19)
Now ~, being horizontal, projects onto an ~-valued two-form F which, due to (19), is closed. A specific realization of the previous monopole-like sequence of superalgebras satisfying all required assumptions can be given in terms of vector fields associated with the supersymmetric Hopf fibrabion on the twodimensional supersymmetric sphere in a way analogous to the one considered in ref. [ 1]. The total space of the fibration is the supergroup UOSP(I, 2). This group is the exponential map of the Grassmann enevelope of the superalgebra uosp(1, 2). A matrix base for uosp(l, 2) is given by [10] A,=½i
(i°i) 0
, A2=½i
1
R,=½
-1
0
(i 0!) (!0 !) 0
, A3=½i
-i
,
R~=½
0 0
0
0
-
1
,
-
.
(20)
0
Then an element s e UOS(1, 2) is of the form (
l+41t/xr/
s = - ½ ( Z o Z t l + Z ~ t l *) ½(z,ztl-ZotlZ)
-½t/x ZoX(1-~t/xq) -ZoX(l-~t/xt/)
½t/ / Zl(l-~t/xt/) ,
(21)
Zo(1-~tlXrl)/
where Zo, z~ are even elements with ZoZox +z~ z~X= 1 and q an odd element in a complex Grassmann algebra Z =Zo(~Z ]; x: Z--+Z is a semilinear map which is a generalization of complex conjugation and which provides Z with a quaternionic structure (see refs. [ 10,11 ] for details). With s* the adjoint o f s [10], let us take
7t(S) --sa3 St - xkak + ~,~r,~ ,
(22)
where ak= (2/i)Ak and ra= (2/i)Ra. It turn out that the xk's are real even (Xk =Xk x ) and the ~ ' s odd Grassmann variables respectively with the relations
(Xl)2+(X2)2-F(X3)2w-l,
~2 = - - ( ~ 1 ) X .
(23)
We see from this that the image $2, of the map zt is a two-dimensional real "sphere" in Z0 with a "fermionic plane" of real dimension two attached to each point. In order to determine the fibres o f n let us take the subgroup S~, ={WeZo: wwX= 1}. It is possible to give an action of St, on UOSP(1, 2): ifseUOSP(1, 2) depends on the parameters Zo, z], ~/, write s = s ( z o , zt, ~1); then, given weS', define s.w--s(WZo, wz,, wq). It is easy to verify that this is a right action. The fact that w w x= 1 implies n(s. w) = re(s) for any we S~,. We have then constructed an S j. fibration over the "supersymmetric two-sphere" $2, S ~,-~UOSP(1, 2)--,$2,. With this fibration we associate the sequence of vector superfie~ds 64
(24)
Volume 193, number 1 i
0
PHYSICSLETTERSB
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Trt
~R v----~R"---~ R (S 2)---~ 0 ,
(25)
where R" is the Lie superalgebra of n-projectable vector superfields on UOPSP(1, 2) and 1~v is the invariant subsuperalgebra of vector superfields which project onto zero. Now the Yi's and the V~'s as in (13) may be taken to be the canonical right invariant vector fields on UOSP (1, 2) and Z any one of the canonical left invariant vector fields. As for ~: we take the graded algebra of supersmooth functions [ 12 ] on $2,. A solution for eqs. (15) provided by the pullback fkn*Xk and g~ = n * ~ of the coordinate functions on $2,. This can be shown in a way analogous to the one used in ref. [ 1 ]. As a final example we describe superelectromagnetism. To be more precise we give an algebraic description of a GU ( 1)-bundle [ GU (1) is the graded extension of U (1) ] over "superspace" M 4'4 by constructing the relevant extension of algebras associated to this bundle. We endow this sequence with a connection which satisfies "Maxwell-type" equations. To this aim let us choose B in the sequence (1) to be the Lie superalgebra generated by the infinitesimal generators of motion on M 4,4 [ 13 ] X u =O/Ox u , X,~ = 0 / 0 0 '~ + ½ i(O/OxU)(TuO)~,
(26)
with 7u Dirac matrices and 0" an anticommuting Majorana spinor. The vector fields (26) obey the following (anti) commutation relations I X u, X~] _ = I X u, X,~]_ =0,
[X,, Xp] + =2i(O/OxU)(TuC),~p,
(27)
with C the charge conjugation matrix. Moreover, let us take A to be generated by an even element O/Oz and an odd one O/0t/which (anti) commute with everything else (one may think of O/Oz and of 0/0q as the infinitesimal generators of translations along the fibres). We take the following connection p: B~E: p(Xu) = X u - A u ( x , O)O/Oz,
p(X,~)=X,~-A~,(x, O)O/Oq,
(28)
where Au(x, O) and A,~(x, O) are functions (the components of the connection). As for the curvature F defined in (5) one has Fu~ = F( X u, X~) = ( X u .A, - X~.Au)O/Oz , Fu~ = F ( Xu, X,~) = ( X u.A~) a / a q - (X~.A~) a/Oz, F,~p =F(X,~, X~) = -2iAu(TuC),~pO/Oz- (X,~ "Ap + Xp "A~,)a/O~l.
(29)
It is easy to verify that F obey the "homogeneous Maxwell equations" dF=O.
(30)
With Xa= ( X u, X~), eqs. (30) explicitly read ( _ )p(Xa)pcxh){[X~,F(Xb, Xc)]c - F ( [ X a , X b ] G , X c ) } + g . c . p . = O .
(31)
Remark. One may think of d as the cohomology operator of the superalgebra B with coefficient in A (for the cohomology of Lie superalgebras see also ref. [14]). In order to obtain the "inhomogeneous equations" we need a metric g on B. Such a metric is a non-degenerate map from B × B to ~ which is graded symmetric, namely g(X, Y) = ( - )p~x)p~r)g( y, X), V X, Y~B. A possible choice for g is the Nath-Arnowitt metric [ 15,16 ] g = ( d x u - ½i07udO) 2 +dO dO.
(32) 65
Volume 193, number 1
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This metric is fiat in the bosonic sector but is not in the fermionic one and it gives rise to non-vanishing Christoffel symbols [ 15,16 ]. Having the metric we define the "codifferential" $ of F'by $F(X) =--gabFxoF(Xb, X)~A,
V X~B.
(33)
Here gab are the components of g - I and V is defined by V xoF( Xb, Xc) = [Xa,F( Xb, Xc) ] - F( V xoXb, Xc) - ( - )P¢x=)p(xb) F ( Xb, V xaXc) ,
(34)
with lTxaXb = F abCXc .
(35)
The Christoffel symbols Fab~ for the metric (32) are explicitly computed in ref. [ 16]. Then, we require the connection (28) to be such that the "inhomogeneous vacuum Maxwell equations" OF=0
(36)
are fulfilled. Remark. Eqs. (36) require that the components of J F along O/Oz and 0/0~/vanish separately. G. Landi would like to thank B. Binegar, J. Blank, M. Blau, V. Dobrev, M. Havlicek, B. Mitra and C. Reina for useful discussions. References [ 1 ] G. Landi and G. Marmo, Lie algebra extension and abelian monopoles, submitted to Phys. Lett. B. [ 2 ] J. Von Neumann, Collected work, Vol. 4, Continuous geometry and other topics (Pergamon Press, London, 1962). [3] E. Witten, Nycl. Phys. B 268 (1986) 253. [4] See e.g.J. Wess and J. Bagger, Supersymmetry and supergravity (Princeton U.P. Princeton, 1983). [5] L. Corwin, Y. Ne'eman and S. Sternberg, Rev. Mod. Phys. 47 (1975) 573; V.G. Kac, Adv. Math 26 (1977) 8. [6] G. Landi and G. Marmo, in preparation. [7] H. Tilgner, J. Math. Phys. 18 (1977) 1987. [ 8 ] A.P. Balachandran, G. Marmo, S.S. Skagerstam and A. Stern, Gauge symmetries and fibre bundles: applications to panicle dynamics, Lecture Notes in Physics, Vol. 188 (Springer, Berlin, 1983). [9] A. Pais and V. Rittenberg, J. Math. Phys. 16 (1975) 2062. [ 10 ] F.A. Berezin and V.N. Tolstoy, Commun. Math. Phys. 78 (1981 ) 409. [ l 1] V. Rittenberg and M. Scheunen, J. Math. Phys. 19 (1978) 709. [ 12 ] B. Konstant, Graded manifolds, graded Lie theory and prequantization, in: Differential geometric methods in mathematical physics, Lecture Notes in Mathematics, "Col. 570 (Springer, Berlin, 1977 ); J. Dell and L. Smolin, Commun. Math. Phys. 66 (1979) 197; D.A. Leites, Russ. Math. Surv. 35 (1980) 1; A. Rogers, J. Math. Phys. 21 (1980) 1352; A. Jadczyk and K. Pilch, Commun. Math. Phys. 78 (1981) 373. [ 13] A. Salam and J. Strathdee, Nucl. Phys. B 76 (1974) 477; Phys. Rev. D 11 (1975) 1521. [ 14] D.A, Leites, Funct. Anal. Appl. 9 (1975) 75; B. Binegar, Lett. Math. Phys. 12 0986) 301. [ 15 ] P. Nath and R. Arnowitt, Phys. Lett. B 56 (1975 ) 177. [16] G. Woo, Lett. Nuovo Cimento 13 (1975) 546.
66
PHYSICS LETTERSB
9 July 1987
EXTENSIONS OF LIE SUPERALGEBRAS AND SUPERSYMMETRIC ABELIAN GAUGE FIELDS G. LANDI Scuola lnternazionale Superiore Studi Avanzati, Strada Costiera I 1.1-34014, Trieste, Italy and lNFN, Sezione di Trieste, 1-34100 Trieste, Italy
and G, MARMO Dipartimento di Fisica, Universit,~degli Studi di Napoli, Mostra d'Oltremare Pad. 19, 1-80125 Naples, Italy and INF, Sezione di Napoli, 1-80125 Naples, Italy
Received 18 March 1987
We constructan extensionof Lie supcralgebraswhich allowsan algebraicdescriptionof the supersymmetricHopf fibration of the supersymmetrictwo-sphere.Moreover,we constructa supcrsymmetricgeneralizationof electromagnetism.
In a previous paper [ 1 ], aiming at an algebraic formulation of gauge theories, much in the same spirit of the "geometry without points" of von Neumann [ 2 ] and of the "non-commutative" geometry used by Witten to deal with string theories [ 3 ], we have used extensions of Lie algebras to give an algebraic description of abelian monopoles and of electromagnetism. One of the main advantages of an algebraic approach is the possibility of generalizing to the supersymmetric case all results which have been obtained in the ordinary case. One has only to replace Lie algebras with Lie superalgebras (or graded Lie algebras). Beside being very important in physics in the context of supersymmetry [4], Lie superalgebras are also important in mathematics where they have been the object of extensive research [ 5 ]. In this letter we consider extensions of Lie superalgebras. The first thing we show is how an exact sequence of Lie superalgebras carries a notion of a covariant derivative in a way which is the analogue of the ordinary (hereafter synonymous of not graded) situation considered in ref. [ 1 ]. We next introduce a "monopole-like" sequence of Lie superalgebras and analyse some of its properties. Finally we describe a supersymmetric generalization of electromagnetism. We refer to a subsequent note [6] for the description of the (graded) Weyl homomorphism and the construction of (graded) Chern-Simons terms. Let us start by briefly describing extensions of Lie superalgebras and intoducing the relevant objects. We refer to ref. [ 5 ] for the definition and the basic properties of Lie superalgebras. In what follows all Lie superalgebras will be Z2-graded. We denote the graded Lie product by [ , ] G and the parity or grading of any homogeneous element by p(" ) • Z2. Extensions of the Lie superalgebras have also been considered in ref. [ 7]. An extension of the Lie superalgebra B by the Lie superalgebra A is an exact sequence of Lie superalgebras I'
/g
0---~ A--~ E--~ B---* 0
(1)
The maps i and it are respectively an injective and a surjective Lie superalgebra homomorphism with 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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Volume 193, number 1
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9 July 1987
im i = k e r n; they are of zero parity, i.e., they preserve the grading; A is an ideal in E. In what follows we shall consider also cases in which A is not graded, i.e., cases in which A has a trivial odd part. With the extension (1) there is associated an exact sequence of Lie superalgebras of derivations by means of the following diagram: i
0
,
A
--~
rt
E
--~
B
--~0
~
P{
Y~
(2)
InG(A)----.
GD(A)---.
OutGD(A)---,0
Here G D ( A ) is the Lie superalgebra of all derivations of A and In G D ( A ) the Lie superalgebra of inner derivations of A, i.e., the adjoint action of A on itself, D a ( a ' ) = [a,a']o for some a eA; this also defines the homomorphism a as a ~ a ( a ) = - D ( a ) . Finally, Out G D ( A ) is the Lie superalgebra of equivalence classes of derivations of A modulo inner ones. The map fl is defined as e-,fl(e): f l ( e ) a = [e,a]c ,
(3)
which is an element of A because A is an invariant subalgebra. Furthermore, 7 is defined to make the last diagram in (2) commute. We shall call 7 the character of extension (1). The maps a, fl and 7 are all of zero parity. , ~ r ~' Two extensions A----E-----. B and A-----.E-----B with the same character 7 are said to be equivalent if there exists an isomorphism fi E---. E' such that i' = f . i and n = f i rt'. We call a connection on the sequence (1) any graded vector space homorphism p: B--,E such that n ° p = i d s .
(4)
As a consequence p is of zero parity. The extent to which p fails to be a Lie superalgebra homomorphism is measured by its curvature F. F is the A-valued skew map of zero parity on B × B defined by F(X., X2)=p([XI,X2IG)-
[p(Xl),p(X:)]o,
V X., X2~B.
(5)
A connection on (1) may be equivalently described by means of an A-valued connection one-form co on E co: E--,A,
Y--*co(Y) = Y - p ( n ( Y ) ) .
(6)
The curvature two-form g2 is the A-valued skew map on E × E defined by g2(Y,, Y 2 ) = F ( z t Y , , T t Y 2 ) ,
V Y,, Yz~E,
(7)
and in terms of co "Q( Y, , Y2 ) = [ Y' ,"Q( Y2 ) ]G -- ( -- )P( Y' )P( Y2) [ Y2 ,CO( YI ) I G -- CO( [ YI , Y2 I o ) -- [ CO( Yt ) ,CO( Y2 ) ]G .
(8)
By its definition one sees that g2 is horizontal, i.e. g2( Y~, Y2)=0 whenever one of the Y~'s belongs to A. As in the case of the ordinary situation we assume that A, B and E have as coefficients elements of a 7?2graded commutative algebra ~ = ~ o ) @ ~ ) and require that elements of E act as graded derivations in ~ , with [ X , Y ] c "f= X" ( Y ' f ) - ( - )p¢x)p(r) Y'( X ' f ) , [ X, f Y I G = ( - )Ptf)P~x) f [ X,Y] + ( X ' f ) Y, 62
Vf~;
(9) X, YeE.
(10)
Volume 193, number 1
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Again, A being an ideal in E, condition (10) forces A to act trivially on ~ . We shall call the sequence (1), when endowed with the previous structure, a Lie superbundle with algebra ~. Once we give a connection p on the sequence (1) we have an action of B on A provided by the map fl.p: B--, GD (A), with fl defined in (3). Using this action we define the "covariant derivative" of an element Ye A along an element Xe B by setting
Vx Y=-fl*p( S) ' Y= [p( X),Y]G
(1 1)
and FxYeA, V XeB, YeA. The map defined by the previous equation has the properties of a graded covariant derivative, namely it is linear in both X and Y and
Vx(fY) = (-)P(f)P(x)fVxY+ (X.f) Y, VfxY=fVxY,
V XeB, YeA, fe ~ .
(12)
We describe now the "monopole-like" sequence of superalgebras (for a related construction see ref. [8]). We take E to be a free algebra of finite type generated with coefficients in ~ by three event elements Yt, Y2, Y3 and by two odd elements Vi, V2 whose graded commutation relations are those of the graded extension GSU(2) of the rotation group [9] (the supergroup GSU(2) is often called UOSP(1,2) [10])
[Yj,Yk]_=ejg¢Yt,
[Yj, V,,]_=½i(crj)a~Va, j , k , l = l , 2 , 3 ,
[V,~,Val+=½i(Ctu),~aYj, a, f l = l , 2 .
(13)
Here a~, a2, tr3 are the Pauli matrices and C is the charge conjugation matrix C = - i a 2 . We assume there is an action (representation) of E on ~. To define the monopole sequence we require the algebra A to be an even algebra generated over ~ by an element Z whose commutators with Y~, ..., V2 all vanish. In particular this implies that A acts trivially on ~ . It Z is of the form
Z=f~Y, +f2Y2+f3Y3+g,V, +g2V2, fj, gc, e ~ ,
p(j~j)=0,
p(g,~)=l.
(14)
By requiring [Z, IT,.]o = [Z, V~] o = 0, the coefficients fj and g~ have to obey the flollowing equations:
Yk'f,, --ek,,,~=O,
(15a)
Yk'g~, + ½i(a k)~,pgp = 0 ,
(15b)
Z,~'f,~-½i(Cam),~pgp=O, Z~.gB-½i(am)ad,,=O,
¥ i , k = l , 2 , 3,
(15c)
V a, f l = l , 2.
(15d)
We shall assume the action of E on ~ to be such that a non-trivial solution of the previous equations does exist. Let us now introduce the ~-valued dual forms. We take three even one-forms ff such that if(Y,) =JJkand if(V~,) = 0 and two more odd one-forms ~ such that ~ ' ( Z ~ ) = J ~ and ~ ( ~ ) = 0. If we take the combination
O=(1/P)ff, O' + g a ~ ) ,
P=ff,)2+~)2+~)2#0,
g2=(g,)2 +(g~)2=O ,
(16)
with the f ' s and the ga'S solutions of eqs. (15), then 0 (Z) -- 1 and the A-valued one-form on E
oJ=Z®O
(17)
is a connection form for the sequence (1). Remark. Since A acts trivially on ~ and is generated by an element which commutes with all elements in E, any A-valued form ~ can be identified with an ~-valued form ~ via ~= Z® 63
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For the curvature t2 defined in ( 8 ) we have $2(X, Y) = { X . O ( Y ) - ( - )p(x)p(r) r ' o ( x ) - O ( [ X , Y ] o ) } Z
-{dO(X, Y)}Z, v X, YeE,
(18)
and the Bianchi identity follows d~=0.
(19)
Now ~, being horizontal, projects onto an ~-valued two-form F which, due to (19), is closed. A specific realization of the previous monopole-like sequence of superalgebras satisfying all required assumptions can be given in terms of vector fields associated with the supersymmetric Hopf fibrabion on the twodimensional supersymmetric sphere in a way analogous to the one considered in ref. [ 1]. The total space of the fibration is the supergroup UOSP(I, 2). This group is the exponential map of the Grassmann enevelope of the superalgebra uosp(1, 2). A matrix base for uosp(l, 2) is given by [10] A,=½i
(i°i) 0
, A2=½i
1
R,=½
-1
0
(i 0!) (!0 !) 0
, A3=½i
-i
,
R~=½
0 0
0
0
-
1
,
-
.
(20)
0
Then an element s e UOS(1, 2) is of the form (
l+41t/xr/
s = - ½ ( Z o Z t l + Z ~ t l *) ½(z,ztl-ZotlZ)
-½t/x ZoX(1-~t/xq) -ZoX(l-~t/xt/)
½t/ / Zl(l-~t/xt/) ,
(21)
Zo(1-~tlXrl)/
where Zo, z~ are even elements with ZoZox +z~ z~X= 1 and q an odd element in a complex Grassmann algebra Z =Zo(~Z ]; x: Z--+Z is a semilinear map which is a generalization of complex conjugation and which provides Z with a quaternionic structure (see refs. [ 10,11 ] for details). With s* the adjoint o f s [10], let us take
7t(S) --sa3 St - xkak + ~,~r,~ ,
(22)
where ak= (2/i)Ak and ra= (2/i)Ra. It turn out that the xk's are real even (Xk =Xk x ) and the ~ ' s odd Grassmann variables respectively with the relations
(Xl)2+(X2)2-F(X3)2w-l,
~2 = - - ( ~ 1 ) X .
(23)
We see from this that the image $2, of the map zt is a two-dimensional real "sphere" in Z0 with a "fermionic plane" of real dimension two attached to each point. In order to determine the fibres o f n let us take the subgroup S~, ={WeZo: wwX= 1}. It is possible to give an action of St, on UOSP(1, 2): ifseUOSP(1, 2) depends on the parameters Zo, z], ~/, write s = s ( z o , zt, ~1); then, given weS', define s.w--s(WZo, wz,, wq). It is easy to verify that this is a right action. The fact that w w x= 1 implies n(s. w) = re(s) for any we S~,. We have then constructed an S j. fibration over the "supersymmetric two-sphere" $2, S ~,-~UOSP(1, 2)--,$2,. With this fibration we associate the sequence of vector superfie~ds 64
(24)
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Trt
~R v----~R"---~ R (S 2)---~ 0 ,
(25)
where R" is the Lie superalgebra of n-projectable vector superfields on UOPSP(1, 2) and 1~v is the invariant subsuperalgebra of vector superfields which project onto zero. Now the Yi's and the V~'s as in (13) may be taken to be the canonical right invariant vector fields on UOSP (1, 2) and Z any one of the canonical left invariant vector fields. As for ~: we take the graded algebra of supersmooth functions [ 12 ] on $2,. A solution for eqs. (15) provided by the pullback fkn*Xk and g~ = n * ~ of the coordinate functions on $2,. This can be shown in a way analogous to the one used in ref. [ 1 ]. As a final example we describe superelectromagnetism. To be more precise we give an algebraic description of a GU ( 1)-bundle [ GU (1) is the graded extension of U (1) ] over "superspace" M 4'4 by constructing the relevant extension of algebras associated to this bundle. We endow this sequence with a connection which satisfies "Maxwell-type" equations. To this aim let us choose B in the sequence (1) to be the Lie superalgebra generated by the infinitesimal generators of motion on M 4,4 [ 13 ] X u =O/Ox u , X,~ = 0 / 0 0 '~ + ½ i(O/OxU)(TuO)~,
(26)
with 7u Dirac matrices and 0" an anticommuting Majorana spinor. The vector fields (26) obey the following (anti) commutation relations I X u, X~] _ = I X u, X,~]_ =0,
[X,, Xp] + =2i(O/OxU)(TuC),~p,
(27)
with C the charge conjugation matrix. Moreover, let us take A to be generated by an even element O/Oz and an odd one O/0t/which (anti) commute with everything else (one may think of O/Oz and of 0/0q as the infinitesimal generators of translations along the fibres). We take the following connection p: B~E: p(Xu) = X u - A u ( x , O)O/Oz,
p(X,~)=X,~-A~,(x, O)O/Oq,
(28)
where Au(x, O) and A,~(x, O) are functions (the components of the connection). As for the curvature F defined in (5) one has Fu~ = F( X u, X~) = ( X u .A, - X~.Au)O/Oz , Fu~ = F ( Xu, X,~) = ( X u.A~) a / a q - (X~.A~) a/Oz, F,~p =F(X,~, X~) = -2iAu(TuC),~pO/Oz- (X,~ "Ap + Xp "A~,)a/O~l.
(29)
It is easy to verify that F obey the "homogeneous Maxwell equations" dF=O.
(30)
With Xa= ( X u, X~), eqs. (30) explicitly read ( _ )p(Xa)pcxh){[X~,F(Xb, Xc)]c - F ( [ X a , X b ] G , X c ) } + g . c . p . = O .
(31)
Remark. One may think of d as the cohomology operator of the superalgebra B with coefficient in A (for the cohomology of Lie superalgebras see also ref. [14]). In order to obtain the "inhomogeneous equations" we need a metric g on B. Such a metric is a non-degenerate map from B × B to ~ which is graded symmetric, namely g(X, Y) = ( - )p~x)p~r)g( y, X), V X, Y~B. A possible choice for g is the Nath-Arnowitt metric [ 15,16 ] g = ( d x u - ½i07udO) 2 +dO dO.
(32) 65
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This metric is fiat in the bosonic sector but is not in the fermionic one and it gives rise to non-vanishing Christoffel symbols [ 15,16 ]. Having the metric we define the "codifferential" $ of F'by $F(X) =--gabFxoF(Xb, X)~A,
V X~B.
(33)
Here gab are the components of g - I and V is defined by V xoF( Xb, Xc) = [Xa,F( Xb, Xc) ] - F( V xoXb, Xc) - ( - )P¢x=)p(xb) F ( Xb, V xaXc) ,
(34)
with lTxaXb = F abCXc .
(35)
The Christoffel symbols Fab~ for the metric (32) are explicitly computed in ref. [ 16]. Then, we require the connection (28) to be such that the "inhomogeneous vacuum Maxwell equations" OF=0
(36)
are fulfilled. Remark. Eqs. (36) require that the components of J F along O/Oz and 0/0~/vanish separately. G. Landi would like to thank B. Binegar, J. Blank, M. Blau, V. Dobrev, M. Havlicek, B. Mitra and C. Reina for useful discussions. References [ 1 ] G. Landi and G. Marmo, Lie algebra extension and abelian monopoles, submitted to Phys. Lett. B. [ 2 ] J. Von Neumann, Collected work, Vol. 4, Continuous geometry and other topics (Pergamon Press, London, 1962). [3] E. Witten, Nycl. Phys. B 268 (1986) 253. [4] See e.g.J. Wess and J. Bagger, Supersymmetry and supergravity (Princeton U.P. Princeton, 1983). [5] L. Corwin, Y. Ne'eman and S. Sternberg, Rev. Mod. Phys. 47 (1975) 573; V.G. Kac, Adv. Math 26 (1977) 8. [6] G. Landi and G. Marmo, in preparation. [7] H. Tilgner, J. Math. Phys. 18 (1977) 1987. [ 8 ] A.P. Balachandran, G. Marmo, S.S. Skagerstam and A. Stern, Gauge symmetries and fibre bundles: applications to panicle dynamics, Lecture Notes in Physics, Vol. 188 (Springer, Berlin, 1983). [9] A. Pais and V. Rittenberg, J. Math. Phys. 16 (1975) 2062. [ 10 ] F.A. Berezin and V.N. Tolstoy, Commun. Math. Phys. 78 (1981 ) 409. [ l 1] V. Rittenberg and M. Scheunen, J. Math. Phys. 19 (1978) 709. [ 12 ] B. Konstant, Graded manifolds, graded Lie theory and prequantization, in: Differential geometric methods in mathematical physics, Lecture Notes in Mathematics, "Col. 570 (Springer, Berlin, 1977 ); J. Dell and L. Smolin, Commun. Math. Phys. 66 (1979) 197; D.A. Leites, Russ. Math. Surv. 35 (1980) 1; A. Rogers, J. Math. Phys. 21 (1980) 1352; A. Jadczyk and K. Pilch, Commun. Math. Phys. 78 (1981) 373. [ 13] A. Salam and J. Strathdee, Nucl. Phys. B 76 (1974) 477; Phys. Rev. D 11 (1975) 1521. [ 14] D.A, Leites, Funct. Anal. Appl. 9 (1975) 75; B. Binegar, Lett. Math. Phys. 12 0986) 301. [ 15 ] P. Nath and R. Arnowitt, Phys. Lett. B 56 (1975 ) 177. [16] G. Woo, Lett. Nuovo Cimento 13 (1975) 546.
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