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Proceedings TH2002 Supplement (2003) 229 – 234
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Proceedings TH2002
Noncommutative Geometry and Superconnections
for Spontaneous Gauge Symmetry Breakdown
Yuval Ne’eman
Abstract. Superalgebras of differential forms have been used to guarantee off-massshell unitarity in gauge theories. Another type connects the one-forms of the gauge
connections with the zero-forms of the Higgs field in a spontaneously-broken gauge
symmetry and predicts the mass of the Higgs particle.
1 Superalgebras and p-forms in Particle Physics
Z(2)-graded Lie brackets [1] were inspired [2] by integration over manifolds, where
surfaces and volumes imposed the anticommutativity of differentials, then lead to
the construction of Grassmann manifolds in the abstract, to the algebra of p-forms
and to Cartan’s Exterior Calculus . Right from the beginning, this involved both
Z-grading and Z(2)-grading,
α ∧ β == (−1)g(α)g(β) β ∧ α
(1)
where g = 0, 1 for even and odd gradings respectively.
Putting everything on the left then produces the super-abelian Lie superbracket (here a mixed bracket [ , }),
[α, β} = αβ − (−1)g(α) g (β} βα
(2)
The p-forms themselves are antisymmetric tensors contracted with differentials,
1
2
p
αp := αµ1 µ2 ...µp dz µ ∧ dz µ ∧ .. ∧ dz µ
(3)
In Quantum Mechanics and in Relativistic Quantum Field Theory (i.e. in
the relativistic extension of QM), the systematics of Quantum Statistics provide
a relevant Z(2) which we denote as Z(2)QS which is constrained to be correlated
with the spins J, by Z(2)QS = Z(2)2J and correlates with the bracket grading g.
Non-super-abelian superalgebras entered this part of physics in the first String
Theory era, when the superstring, developed by A. Neveu and J.H. Schwarz and
independently by P. Ramond, involved constraints guaranteed by an infinite local
superalgebra.
Yu. Golfand an E. Likhtman [3] applied the idea to the Poincare group,
followed by Volkov and Akulov [4]. A. Salam and J. Strathdee [5] developed the
idea of superspace by taking the quotient of the super-Poincare Group Manifold
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by that of the (homogeneous) Lorentz group, inverting the semi-direct product of
the supergroup’s construction sP 3,1 = SL(2, C) × [R(1, 3)(+)S(1, 3)]. The rules
of the game were clarified by L. Corwin, Y. Ne’eman and S. Sternberg [6], who
also explored the field and collected examples – and by P.G.O. Freund and I.
Kaplansky [7]. Finally, V. Kac classified the (finite) Lie superalgebras [8], while
S. Deser and B. Zumino [9] simultaneously with D.Z. Freedman, S, Ferrara and
P. v. Nieuwenhuizen [10] constructed Supergravity, gauging the super-Poincare
group, and thus involving the super- Lie derivative, renamed anholonomic general
coordinates transformations or AGCT in ref. [11], which also reviewed the dual
relationship between p-forms and superalgebras.
p-forms revealed their presence and various roles very gradually, after the
1974 start (Yang and Wu, etc ) of the second era of geometrization (the first lasted
from Minkowski’s 1908 speech to the 1925 realization of the inconclusiveness of
the Kaluza-Klein model and the birth of Quantum Mechanics).
Two difficulties of RQFT were involved: the apparent loss of unitarity offmass-shell, as proclaimed by G. Chew throughout 1958-1971, and the presence of
anomalies in broken strong interactions global symmetries. The first problem was
solved by Feynman’s introduction of ghost fields (1962) when, after developments
by B.S. De Witt, Slavnov, Taylor, Faddeev and Popov, they were reset in the
algebraic format of the BRST equations.
These then set the stage for the interpretation of the algebraic formulation in
a geometrical context – the BRST equations become the Cartan-Maurer structural
equations of the fibre-bundle, guaranteeing horizontality of the curvature 2-form,
as shown by J. Thierry-Mieg [12-14]. The structural equation is a discrete transformation, not that of a Lie group. I soon provided the complement, which will
open the discussion about the subject of this review. Meanwhile, we note that
the second difficulty in RQFT, namely the anomalies, were also resolved by the
introduction of p-forms [15-16].
2 Internal Supersymmetry
My 1979 model [17] based on the supergroup SU (2/1) suggests that electro-weak
unification occurs via the imposition of that supergroup upon (g denotes the Grassmann parity) the octet made of
(a) the gauge bosons (4-potentials) g-odd one-forms for the SU (2) × U (1)
h-even subgroup of SU (2/1) (four states) and
− √23 Z 0
W+
Φ+
νL0
W−
−A0+ √13 Z 0
Φ0
e−
L
Φ̄−
Φ̄0
−A0− √13 Z 0
e−
R
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Noncommutative Geometry for Spontaneous Gauge Symmetry Breakdown 231
(b) the four states of the isospinor Higgs field Φ(x) and its conjugate Φ† (x),
occupying the positions of the K-mesons in the 1961 SU (3) hadron flavor octet,
drawn as 3x3 matrices [18]. In Grassmann systematics, the Higgs field, being a
Lorentz scalar, is a zero-form, thus g-even.
The SU (2/1) defining supermatrix is divided into the “h-even” box-diagonal
{2 × 2}(+){1 × 1} and the two rectangular h-odd {1 × 2}(+){2 × 1} off-diagonal
pieces housing the Higgs field in its primordial insertion; we have thus defined
Z(2)h , yet another Z(2).
The 1979 model thus presents a Lorentz-scalar semi-physical octet made of
the ghosts of the W +/− , Z, A mediator potentials (or of the four J = 1 particles)
and the isospinor Higgs field and its conjugate. I used ”semi-physical” because
seven out of the eight are not entirely ”physical” in the positivist sense, not just the
ghosts. Indeed, we should remember that three of the Higgs field’s four components
end up as contributions to the longitudinal components of the W +/− and Z, who
have thereby acquired their masses.
The same model was independently reached by D.B. Fairlie, though by a
different route (dimensional reduction) [19].
3 The Quillen superconnection
Thus, if we define Z(2)total = Z(2)g ⊗ Z(2)h , the entire supermatrix has odd
total-parity and thus fits in the role of a superconnection – a connection-like role
for a locally-gauged supergroup. This was the kernel of a mathematical construct
devised by D. Quillen [20]. With S. Sternberg [21], we dressed up the 1979 model
in Quillen’s formulation, which was mathematically well-defined and avoided one
conceptual difficulty with the 1979 definitions, namely the need to invent new ghost
states in order to make mixed particle-ghost supermutiplets in all representations,
not just for the adjoint. To accommodate n particles, one needed 2n states, half
of which would be new (harmless ?) ghost states.
As a matter of fact, the division of the defining supermatrix into odd and even
h-gradings in the 3 × 3 case fits a division of the carrier supervector into 2(+)1,
−
which fits beautifully with the chiralities, e.g. (νL0 , e−
L /eR ), with a 4-dimensional
representation for the quarks with a similar good fit to the chiralities [22].
Dynamically, the superalgebra contains h − odd/h − odd anticommutators
so that the curvature h − even component contains quadratic terms of the form
daij Φi (x)Φj (x).
Squaring to make the Hamiltonian thus yields a quartic term for the Higgs
potential, with a coupling scaled by the algebra in terms of the SU (2) × U (1)
couplings. If one adopts one normalization for the h-even part, the system also
scales the two couplings, namely the value of the Weinberg angle, yielding here
sin2 θ = 0.25. Only the negative term, quadratic in Φ, which triggers the non-zero
vacuum expectation value for one Higgs field neutral component, only this term
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has still to be put in “by hand”. The Higgs particle’s mass is predicted [23]1 to
be M (Φ) = 2M (W ); however, an approximate evaluation of the renormalization
group corrections [24] lowers the value to M (Φ) = 130GeV ±10GeV .
4 Noncommutative Geometry and Connes-Lott Space
Meanwhile, yet another branch of mathematics was born and became relevant to
our issue, namely A. Connes’ noncommutative geometry [25,26]. I shall only discuss
aspects relevant to this model. Connes and Lott applied the NCG methodology
just to rederive the Weinberg-Salam Electroweak theory in a purely geometric set
up [27]. The basic assumption is that the base space of the gauge bundle is a
product of a 2-point “space” Z(2) by Minkowski space. The points are L and R
and project out the relevant piece of the Lorentz group. As a result, the product
reduces to a sum ML ⊕ MR .
It was also shown that our above SU (2/1) model can be derived in this
manner with two advantages beyond the superalgebra’s constraining action on the
couplings:
(a) the quantum statistics issue is entirely resolved, there are no unwanted
ghosts, etc. The carrier space Z(2) can be the chiral Z(2)ch indeed, and be correlated with the Z(2)superbracket
(b) the carrier space is the result of adjoining two fibres sitting over the two
manifolds M L (+)M R . Parallel-transport considerations fix the composition of the
various covariant derivatives. Going within the fibre (say from a point in ν 0 L (x) to
another point on the same fibre eL (y) is done with the same covariant derivative
built on the unbroken gauge symmetry. However, passing from ν 0 L (x) to a point
on e− R (y) requires first a discrete jump between the two spaces, which should
follow the operator symbolizing the mass, which breaks (chiral) barriers.
This is the matrix µ6 , relating eL with eR . In the covariant derivative there
will now be new elements ∆Φ, with ∆ a ”matrix derivative”, here a constant
matrix in the µ6 direction, imaginary because of anti-hermiticity of all derivatives.
In the Hamiltonian R ∧ ∗ R these terms will produce the quadratic term in the
Higgs potential, with the negative sign.
νL0
i
i
e−
L
e−
R
The matrix-derivative
1 This calculation involves an additional assumption without which the mass is closer to
400 GeV, reducing to about 300 GeV by renormalization.
Vol. 4, 2003
Noncommutative Geometry for Spontaneous Gauge Symmetry Breakdown 233
I have derived a solution of this noncommutative geometry type for Riemannian Geometry (Einstein Gravity) to emerge from spontaneous breakdown of a
primordial Metric-Affine Geometry with SL(4, R) as symmetry of its local frames
[28]. The relevant subgroup is P (4, R).
References
[1] Nijenhuis, A., Proc. K. Ned. Akad. Wet. A58 (1955) 3.
[2] See, for example, Y. Choquet-Bruhat, C. De Witt-Morette and M. DillardBleick, Analysis Manifolds and Physics, North Holland pub., Amsterdam
(1977), 543 pp., p. 30-52.
[3] Golfand, Yu., and E. Likhtman, JETP Lett., 13 (1971) 323-326.
[4] Volkov D.V. and V.P. Akulov, Phys. Lett. B46 (1973) 109.
[5] Salam A. and J. Strathdee, Nucl. Phys. B76 (1974) 477.
[6] Corwin, L., Y. Ne’eman and S. Sternberg, Rev.Mod.Phys. 47 (1975) 573-603.
[7] Freund, P.G.O. ad I. Kaplansky, J. Math. Phys., 17 (1976) 228-231.
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3214-3218.
[11] Ne’eman Y., and T. Regge, Rivista d. Nuovo Cim., ser. 3, 1, #5 (1978) 1-43.
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(1978).
[14] Beaulieu, L. and J. Thierry-Mieg, Phys. Lett. B145 (1984) 53.
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[20] Quillen, D., Topology 24(1985) 89.
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[22] Ne’eman, Y. and J. Thierry-Mieg, in Methods in Mathematical Physics (Proc.
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N. Math. 836, p. 318-348.
[23] Ne’eman, Y., Phys. Lett, B427 (1986) 19-25;
[24] Hwang, S., C.Y. Lee and Y. Ne’eman, Int. J. Mod. Phys. A11 (1996) 35093552.
[25] Connes, A., Noncommutative Geometry, Academic Press (1994)
[26] Madore, J., An Introduction to Noncommutative Differential Geometry and
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[28] Ne’eman, Y., Phys. Lett., B427 (1998) 19-25.
Yuval Ne’eman
School of Physics and Astronomy
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel-Aviv University
Tel-Aviv, Israel 69978