On the anomalous U(1) in free fermionic superstring models Page: 11 of 25
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3 The origin of the anomalous U(1)
To see how the anomalous U(1) arises in the NAHE-based realistic free fermionic
models, let us recall the E6 model of section 2.1. This model is generated by the
NAHE set basis vectors plus the vector X in Eq. (2.4) or alternatively by the set
(2.6). Let us now consider the model with the vector X in Eq. (2.6) replaced with
the vector 2ry. In this case we obtain a model with SO(10) x U(1)3 x SO(4)3 x SO(16)
gauge group. The sectors b3 still produce the 24 multiplets in the 16 representation of
SO(10) with U(1)3 charges Qj = 1/2. In this model the sectors b, +2ry now produce
24 multiplets in the 16 vectorial representation of the hidden SO(16) gauge group,
which also carry charges Qj = 1/2, Qk = 1/2, i # j # k. Thus, we observe that in
this model U + U2 + U3, which is the U(1) combination (2.9) embedded in E6 in the
E6 model, becomes the anomalous U(1) combination with the total Tr QA = -576.
The anomalous U(1) is therefore seen to arise in this model due to the breaking of
the right-moving N = 2 world-sheet supersymmetry.
Occurrence of a generic anomalous U(1) can depend on both boundary conditions
and free phases. An interesting variation of an anomalous model from the basis set
(2.6) can still occur with the phase choice c(x) = 1. While this generates the
E$ x SO(12) x Eg gauge group and (2,2) world-sheet supersymmetry from the first
four basis vectors, changing the sign of c(b ') and/or that of c(b2) in (2.7) from - to
+ produces an anomalous model when basis vectors bl and b2 are included. In this
case the final gauge group is E6 x U(1)2 x SO(4)3 x SO(16). While U(1)E6 defined in
(2.9) remains non-anomalous (since it is still embedded in E6), the orthogonal U(1)'
and U(1)" in (2.10,2.11) become anomalous.
Changing the sign of c('i) effectively transforms the SO(16) spinor components
of the E8 gauge vectors, coming from the -sector, into a massless matter state (the
128 of SO(16)), thereby breaking E8 to SO(16). Simultaneously, under sign change
of c( 'i), the eight copies of the 27 rep of the observable sector E6 (along with some
states carrying only SO(4)3 and U(1)i charges) originating in (bi, bi+X) are replaced
by eight copies of a 16 rep of SO(16)hid-
If only the sign of c( b1) and not that of c(b1) is changed (or vice versa), then the
sector b3 = 1 + bi + b2 + produces exactly eight copies of the 16 rep of SO(16)hid
that also carry U(1)' and U(1)" charge. If both signs change, then b3 yields only
SO(4)3 x U(1)i reps. In the former case (with c( ) transformed),
Tr (r1, ,2,73) = (-96,192, -96), (3.1)
Here the anomaly can be placed entirely in U(1)A . U(1)1 - 2U(1)2 + U(1)3, also
leaving U(1)' - U(1)1 - U(1)3 as a good symmetry. In the latter case,
Tr (1, h,2, 73) = (-96, 192, -96) + (192, -96, -96) (3.2)
= (96, 96, -192), (3.3)9
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Cleaver, G.B. & Faraggi, A.E. On the anomalous U(1) in free fermionic superstring models, report, November 1, 1997; Gainesville, Florida. (https://digital.library.unt.edu/ark:/67531/metadc699280/m1/11/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.