- Email: [email protected]
Graded fibre bundles and unified field theories
Physica 114A (1982) 389-392 North-Holland
GRADED AND UNIFIED
Publishing Co.
FIBRE
BUNDLES
FIELD
THEORIES
Richard KERNER Depatiement de Micanique, Uniuersitt Pierre et Marie Curie, 4, Place Jussieu, F-75005Paris,
France
A gauge field theory is constructed in terms of connections over fibre bundles using a supergroup as structure group. Superconnections are imposed to eliminate ghost fields with negative energy but these break the conformal symmetry. The theory can be related to classical theory by use of suitable scaling factors: this leads to specified masses for the Higgs multiplet and the spinor multiplet.
In the notations introduced by Wess, Zumino and othersrm3), the graded Poincare algebra is generated by the following differential operators: (1)
ukm
=
i(YkYrn
-
YmYk)-
(2)
The indices (Y,fl are raised and lowered by means of the invariant tensors Q, ~~8 where (Y,6 = 1,2. Th ese operators act upon the smooth functions defined over the product space M4 x { 0) where M, is the space-time and {0) is the linear space spanned by the anticommuting spinors 13”,is. M4 x (0) is usually referred to as a superspace. By analogy, let us construct the supergroup associated with a compact, semisimple Lie group G. Let a, b = 1,2,. . . , N = dim G; let Cg, be the structure constants of G, and g,b its Cartan-Killing metric. The generators of the Clifford algebra associated with gab satisfy YaYb
+
Yb7/o
=
2&bl,
(3)
1 being the identity matrix in K-dimensional
linear space where K = 2rM2’.The matrices unb = $($‘rb - -yb-/) generate the spinor representation of SO(N); the matrice.s T, = iC:q,, give us the spinor representation of the Lie algebra do, and, by exponentiation, also the representation of G itself. All these representations act on the space of group spinors xA where A = 1,2,. . . , K. Let us impose the anticommutation rule for the group spinors XAXB+ XBXA= 0.
(4)
In view of the above-defined action of G on the linear space {x}, the semidirect product G •i {x} will be called the supergroup associated with G (see e.g. ref. 4). If I,, denotes the generators (left-invariant vector fields on G) satisfying [L,, Lb] = C&L,, then the operators sa, = L, + r;fBxBaA satisfy 0378437 1/82/oooooooO /$02.75 @ 1982 North-Holland
390
R.KERNER
[gd,, ‘3,,] = C&L,. There exists an invariant tensor eAB such that if we define &eAD = T~BD,and then take the operators 9, =
~B+~;BX~~,
(5)
then (6)
The generalized structure constants C& = - Cz,, C& = C& = T&, + T&, and C& = -C& = -& provide us with the adjoint representation of the superalgebra (equivalent to the generalized Jacobi identity). Let us consider now a theory of (super)-gauge fields constructed as connections in principal fibre bundles over M4 x (0) with the structure group replaced by our supergroup G O(x) P(W x (01, G 0 1x1).
(8)
The superconnection A is a left-invariant l-form over P with values in the superalgebra. In local coordinates, A decomposes into A = Ai dx’ + A, de” + Aa dt?. On the other hand, it decomposes
(9) in the superalgebra
A = A”$33,+ AB6?8a.
as (10)
The “vertical” components (AZ, A& A:) are completely determined by the generalized Maurer-Cartan equations; the non-trivial (horizontal) part of the curvature tensor is given by FiL = %Ae - (- 1) w(K)~)[~++)I~~A$ + C$AX,At.
(11)
Here K and L represent the horizontal indices (j, (Y,fi), while 4 and 9 denote the vertical indices (a, B); T(K) means the Grassmann parity of K, i.e. r(j) = 0, ~(a) = 0, n(o) = 1, ~(6) = 1, r(B) = 1. The Lagrangian of the theory is, as in usual Yang-Mills theory,
The components of A do not depend explicitly on xA because of the left-invariance with respect to G Cl {x}, which includes translational invariance with respect to {x}; they are polynomials in B’s with coefficients depending on XEM‘$.
In order to have proper relationships between spin and statistics and to eliminate ghost fields with negative energy we must impose the following
GRADED FIBRE BUNDLES
supergauge
AND UNIFIED FIELD THEORIES
391
conditions
i)
(A$) = n($) + r(K),
(13)
ii)
(AZ)+ = A%,(A;)+ = A;
(hermiticity).
These quite natural conditions leave the PoincarC and gauge invariance but they will break the conformal symmetry which was present in the from the beginning. Under conditions (13) and (14) the most general sion of A contains only the classical gauge field B;(X), the Higgs multiplet 4” and a reducible multiplet of Dirac spinors I/J*
(14) intact, theory expanscalar
(15)
The universal length scale 1 has to be introduced when we unify the x-space with the O-space (note that the dimension of 13is cm”?; there are also two arbitrary dimensionless parameters A and v. When the Lagrangian is calculated, only the terms proportional to 8’02$$ are important, other terms being irrelevant as their integration over the volume element d40 in B-space will vanish. This relevant part of the Lagrangian density is
where Gi is the Yang-Mills Vj4’ = aj+” + CE,BP+“,
tensor of By, and Vj+t = ajet + C;;~BB~$J~.
(17)
In order to identify our expression with the classical theory we have to perform some scalings, namely to put 2” = 32, introduce ia = 24” and choose 8A2+ 4v = 32. Then we obtain the Lagrangian (in an abbreviated notation)
392
W-9 The higgs multiplet has acquired the mass CL+= l/21. In order to make our theory renormalizable, we have to remove the term Vi~VirC,; this leads to 8 - 3h2 - 2A = 0, i.e. either A = -2 either A = 4/3. In the first case the spinorial part of the Lagrangian is &jVj$
+ (yjVjlJ)$
+ $/.lJI$ + ~Vj$lJ~j$
(19)
describing a massless multiplet of Dirac spinors interacting with the massive Higgs scalar through current-current interaction and including a Fermi-like four-point interaction-exactly as in the theory of the weak interactions (neutrino). In the second case the spinor part of the Lagrangian is
Here the spinor multiplet acquires the mass plL = 40/271; the Yukawa coupling appears, as in the theory of strong interactions (e.g. when G = W(2), as the coupling of spinors describing proton and neutron, 6“ describing pions). In order to give this theory its full meaning, the decomposition of $* into irreducible representations of G has to be performed and different coefficients appearing in the mass terms, four-point coupling, Yukawa and current-current couplings have to be computed, giving a non-trivial mass spectrum and removing the degeneracy of couplings.
References 1) 2) 3) 4) 5)
J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39. A. Salam and J. Stiathdee, Nucl. Phys. B76 (1974) 477. R. Arnowitt, P. Nath and B. Zumino, Phys. Lett. B56 (1975) 81. A. Rogers, J. Math. Phys. 22 (1981) 939. R. Kerner and E.M. Da Silva Maia, J. Math. Phys. to be published.
GRADED AND UNIFIED
Publishing Co.
FIBRE
BUNDLES
FIELD
THEORIES
Richard KERNER Depatiement de Micanique, Uniuersitt Pierre et Marie Curie, 4, Place Jussieu, F-75005Paris,
France
A gauge field theory is constructed in terms of connections over fibre bundles using a supergroup as structure group. Superconnections are imposed to eliminate ghost fields with negative energy but these break the conformal symmetry. The theory can be related to classical theory by use of suitable scaling factors: this leads to specified masses for the Higgs multiplet and the spinor multiplet.
In the notations introduced by Wess, Zumino and othersrm3), the graded Poincare algebra is generated by the following differential operators: (1)
ukm
=
i(YkYrn
-
YmYk)-
(2)
The indices (Y,fl are raised and lowered by means of the invariant tensors Q, ~~8 where (Y,6 = 1,2. Th ese operators act upon the smooth functions defined over the product space M4 x { 0) where M, is the space-time and {0) is the linear space spanned by the anticommuting spinors 13”,is. M4 x (0) is usually referred to as a superspace. By analogy, let us construct the supergroup associated with a compact, semisimple Lie group G. Let a, b = 1,2,. . . , N = dim G; let Cg, be the structure constants of G, and g,b its Cartan-Killing metric. The generators of the Clifford algebra associated with gab satisfy YaYb
+
Yb7/o
=
2&bl,
(3)
1 being the identity matrix in K-dimensional
linear space where K = 2rM2’.The matrices unb = $($‘rb - -yb-/) generate the spinor representation of SO(N); the matrice.s T, = iC:q,, give us the spinor representation of the Lie algebra do, and, by exponentiation, also the representation of G itself. All these representations act on the space of group spinors xA where A = 1,2,. . . , K. Let us impose the anticommutation rule for the group spinors XAXB+ XBXA= 0.
(4)
In view of the above-defined action of G on the linear space {x}, the semidirect product G •i {x} will be called the supergroup associated with G (see e.g. ref. 4). If I,, denotes the generators (left-invariant vector fields on G) satisfying [L,, Lb] = C&L,, then the operators sa, = L, + r;fBxBaA satisfy 0378437 1/82/oooooooO /$02.75 @ 1982 North-Holland
390
R.KERNER
[gd,, ‘3,,] = C&L,. There exists an invariant tensor eAB such that if we define &eAD = T~BD,and then take the operators 9, =
~B+~;BX~~,
(5)
then (6)
The generalized structure constants C& = - Cz,, C& = C& = T&, + T&, and C& = -C& = -& provide us with the adjoint representation of the superalgebra (equivalent to the generalized Jacobi identity). Let us consider now a theory of (super)-gauge fields constructed as connections in principal fibre bundles over M4 x (0) with the structure group replaced by our supergroup G O(x) P(W x (01, G 0 1x1).
(8)
The superconnection A is a left-invariant l-form over P with values in the superalgebra. In local coordinates, A decomposes into A = Ai dx’ + A, de” + Aa dt?. On the other hand, it decomposes
(9) in the superalgebra
A = A”$33,+ AB6?8a.
as (10)
The “vertical” components (AZ, A& A:) are completely determined by the generalized Maurer-Cartan equations; the non-trivial (horizontal) part of the curvature tensor is given by FiL = %Ae - (- 1) w(K)~)[~++)I~~A$ + C$AX,At.
(11)
Here K and L represent the horizontal indices (j, (Y,fi), while 4 and 9 denote the vertical indices (a, B); T(K) means the Grassmann parity of K, i.e. r(j) = 0, ~(a) = 0, n(o) = 1, ~(6) = 1, r(B) = 1. The Lagrangian of the theory is, as in usual Yang-Mills theory,
The components of A do not depend explicitly on xA because of the left-invariance with respect to G Cl {x}, which includes translational invariance with respect to {x}; they are polynomials in B’s with coefficients depending on XEM‘$.
In order to have proper relationships between spin and statistics and to eliminate ghost fields with negative energy we must impose the following
GRADED FIBRE BUNDLES
supergauge
AND UNIFIED FIELD THEORIES
391
conditions
i)
(A$) = n($) + r(K),
(13)
ii)
(AZ)+ = A%,(A;)+ = A;
(hermiticity).
These quite natural conditions leave the PoincarC and gauge invariance but they will break the conformal symmetry which was present in the from the beginning. Under conditions (13) and (14) the most general sion of A contains only the classical gauge field B;(X), the Higgs multiplet 4” and a reducible multiplet of Dirac spinors I/J*
(14) intact, theory expanscalar
(15)
The universal length scale 1 has to be introduced when we unify the x-space with the O-space (note that the dimension of 13is cm”?; there are also two arbitrary dimensionless parameters A and v. When the Lagrangian is calculated, only the terms proportional to 8’02$$ are important, other terms being irrelevant as their integration over the volume element d40 in B-space will vanish. This relevant part of the Lagrangian density is
where Gi is the Yang-Mills Vj4’ = aj+” + CE,BP+“,
tensor of By, and Vj+t = ajet + C;;~BB~$J~.
(17)
In order to identify our expression with the classical theory we have to perform some scalings, namely to put 2” = 32, introduce ia = 24” and choose 8A2+ 4v = 32. Then we obtain the Lagrangian (in an abbreviated notation)
392
W-9 The higgs multiplet has acquired the mass CL+= l/21. In order to make our theory renormalizable, we have to remove the term Vi~VirC,; this leads to 8 - 3h2 - 2A = 0, i.e. either A = -2 either A = 4/3. In the first case the spinorial part of the Lagrangian is &jVj$
+ (yjVjlJ)$
+ $/.lJI$ + ~Vj$lJ~j$
(19)
describing a massless multiplet of Dirac spinors interacting with the massive Higgs scalar through current-current interaction and including a Fermi-like four-point interaction-exactly as in the theory of the weak interactions (neutrino). In the second case the spinor part of the Lagrangian is
Here the spinor multiplet acquires the mass plL = 40/271; the Yukawa coupling appears, as in the theory of strong interactions (e.g. when G = W(2), as the coupling of spinors describing proton and neutron, 6“ describing pions). In order to give this theory its full meaning, the decomposition of $* into irreducible representations of G has to be performed and different coefficients appearing in the mass terms, four-point coupling, Yukawa and current-current couplings have to be computed, giving a non-trivial mass spectrum and removing the degeneracy of couplings.
References 1) 2) 3) 4) 5)
J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39. A. Salam and J. Stiathdee, Nucl. Phys. B76 (1974) 477. R. Arnowitt, P. Nath and B. Zumino, Phys. Lett. B56 (1975) 81. A. Rogers, J. Math. Phys. 22 (1981) 939. R. Kerner and E.M. Da Silva Maia, J. Math. Phys. to be published.