Gauge unification in higher dimensions Page: 4 of 22
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construct completely realistic theories.
2 Unified Gauge Symmetry Transformations
on Orbifold Spacetime
In this section we introduce a class of higher dimensional unified field theories, concentrating
on the nature of the gauge symmetry transformation. We then discuss several features which
are generic to this class. For simplicity we consider a single compact extra dimension, y (= x5),
and assume a fixed radius with size given by the unification scale. We take the unified gauge
interactions in 5 dimensions to have gauge group G. The Higgs doublets also propagate in
5 dimensions as components of hypermultiplets. Using 4 dimensional superfield notation, we
write the vector multiplet as (V, E), where V is a 4d vector multiplet and E a chiral adjoint,
and the hypermultiplet as (H, HC), where H and H' are chiral multiplets with opposite gauge
transformations.
The form of the gauge transformations under G can be restricted by compactifying on the
orbifold S'/Z2 with a parity, P, which acts on the vector representation of G, making some
components positive and some components negative: P = (+, +, ...-, -, ...). The orbifold
symmetry on any tensor field 0 is then defined by 0(-y) = P#(y), where P acts separately on
all vector indices of 0, and an overall sign choice for the parity of the multiplet may also be
included in P. It is understood that there is a relative minus sign between the transformation
of HC(E) and H(V), as required by the P invariance of the 5d gauge interactions. In certain
cases P is a discrete gauge transformation, but it need not be.
A non-supersymmetric theory with G = SU(5) and P = (- , -, ,+) was considered
by Kawamura [10]; here we discuss a supersymmetric version. The Higgs bosons are taken
to lie in a hypermultiplet (H, HC) (x, y), with H and HC chiral multiplets transforming as 5
and 5 representations. The orbifold projection accomplishes doublet-triplet splitting, in the
sense that H has a weak doublet zero mode but not a color triplet zero mode. However, the
projections work oppositely for H' which contains only a color triplet zero mode. Similarly,
while the X gauge bosons have negative P and therefore no zero mode, the exotic color triplet,
weak doublet states in the chiral adjoint, Ex, does have a zero mode. Such exotic light states
are generic to orbifolding with a single Z2 and lead to an incorrect prediction for the weak
mixing angle. These exotic states can be removed by introducing two sets of different orbifold
parities, giving additional structures to the spacetime, which we study in the rest of this paper.3
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Hall, Lawrence & Nomura, Yasunori. Gauge unification in higher dimensions, article, January 14, 2001; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc782554/m1/4/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.