Expanding on the discussion in
Section 3.6 regarding the smaller CM, TM, and VM multiplets, we define a convention for nodal field definitions that is consistent with the CM, TM, and VM that can be applied to the larger
multiplets. First, dynamical fields appear to the left of auxiliary fields. For auxiliary fermions, those of lower mass dimension appear to the left of those of higher mass dimension. For bosonic fields, they are listed in the nodes left to right in the following order: scalars, pseudoscalars, vectors, pseudovectors, tensors, and pseudotensors. Gauge fields appear to the right of non-gauge fields of the same rank. In the case of multiple pseudoscalars for instance, the pseudoscalar that comes in a pair with a scalar (that form a complex scalar as in the fields
K and
L of the dWvH multiplet for instance) appears before non-paired pseudoscalars. Fields with components are listed left to right in numerical order if there is a single component. Fields with more complicated index structure, such as the graviton, gravitino, and antisymmetric tensors, are listed in the orders shown in the specific examples below.
For the dWvH formulation of the (
,1) supermultiplet, we order the bosons according to
for the OS formulation we order the bosons according to
where the ordering for
is as follows for both the dWvH and OS multiplets:
Note that
Finally, for the non-minimal SG bosons,
Next, for both dWvH and OS formulations fermions, we choose
while, for fermions of the non-minimal SG supermultiplet fermions, we use
In
Appendix B, we display the explicit
and
matrices for the
representation of the dWvH multiplet. For all three multiplets, the
and
matrices satisfy the
algebra, the algebra of general, real matrices of size
that encode
N supersymmetries [
1]:
As
and
for the dWvH, OS, and
multiplets, their
and
matrices satisfy more specifically the
algebra.
Recall, the parameter
is defined through the relationship
The parameters
and
are referred to as the isomer parameters. They encode the number
cis-isomer adinkras and the number
trans-isomer adinkras into which a multiplet can be decomposed. The parameter
. For the dWvH, OS, and
multiplets, we find
7.2. , Eigenvalues, and Gadgets for the dWvH, OS, and Multiplets
The explicit matrix forms of
and
are too large to display and be instructive in this paper. We have published them open-source in the files
dWvH.m,
OS.m, and
nmSG.m at the previously mention GitHub data repository. As an example, in
Appendix C we show the explicit form for the
for the
representation of the dWvH multiplet. Unlike the fundamental
,
, and
representations [
17,
18,
20,
21,
23,
24], the
and
for the dWvH, OS, and
representations are all true
representations composed of six linearly independent elements:
In contrast, the
and
for the
,
, and
each form a single, non-trivial
representation, with only three linearly independent algebra elements [
17,
18,
20,
21,
23,
24]. That is either the
or the
vanish and either the either the
or the
vanish for the
,
, and
. This is not the case for the dWvH, OS, and
representations: the
for these are all nontrivial. We see then for the dWvH, OS, and
representations, the
and
all form true
representations, each which separate into two commuting
representations,
and
, respectively, as shown in the previous section. The eigenvalues for
and
for the dWvH, OS, and
multiplets are all
.
All of the dWvH, OS, and
multiplets have gadgets, Equation (38), that are normalized to
:
The gadgets between the three different representations depend on the diagonal Lagrangian parameters
,
,
as well as the superspace supergravity parameter
n. While presenting the results below, we comment on the interesting cases where gadgets between the different representations are zero or five. As described in
Section 3.6, where the gadget is described as the vector analogy of a dot product, a gadget of zero means the multiplets are gadget-orthogonal, which is analogous to two vectors being orthogonal. A gadget value of
is analogous to two vectors being parallel.
First, we define the self-gadget of a representation as the gadget between the same representation with two different values of its Lagrangian parameter: one unprimed, the other primed. We then have the following three sets of parameters to consider, one set for each of the
representations: (
), (
), and (
). We find the following self-gadget values:
This demonstrates interestingly that five is the
minimum value that the dWvH self-gadget can take. The minimum self-gadget value for the OS multiplet is precisely
and the minimum value for the
self-gadget is
. The OS self-gadget equals five for three separate relationships between
and
. The
self-gadget equals five for the precise value of
, two solutions of
n that depend on
and
, and of course the case
. The self-gadgets are summarized in
Table 2 where to more succinctly write the
results, we define the function
It is worth noting that the minimum case
for
corresponds to its reduction to old-minimal supergravity [
33], as described in
Section 6.
The gadgets between the dWvH, OS, and nmSG multiplets are as follows:
Upon closer inspection of these gadgets, we find some interesting facts as to holographic possibilities. For instance, an obvious solution for which dWvH and OS are parallel, i.e., have a gadget value equal to five, is
The form of the gadget between dWvH and OS on the second line of Equation (111), however, indicates perhaps a more natural choice might be
Solutions exist to make dWvH parallel to
, and OS parallel to
, but these solutions are complication conditional solutions on
n so we have published these calculations in the file
Compare20x20Reps.nb at the previously mentioned GitHub data repository. Two obvious cases to investigate are
and
, for which we find
As to orthogonality (gadget value of zero), inspection of Equation (111) reveals that there are no real solutions for
and
that make the dWvH and OS multiplets orthogonal
We do have, however, that
On the other hand, the OS and
multiplets can be made to be orthogonal for various
ranges on
n. As these solutions for OS-
orthogonality are rather complicated and thus not terribly instructive in their entirety, we have published the results in the file
Compare20x20Reps.nb at the previously mentioned GitHub data repository. An interesting case is the following where both the dWvH and OS multiplets each are simultaneously orthogonal to the
multiplet (but not each other):
This leaves the obvious cases
and
to investigate as to orthogonality. In these cases, there is no real solution for dWvH-
orthogonality and only one real solutions for OS-
orthogonality:
Finally, we summarize the dWvH-
gadgets and OS-
gadgets in the physically interesting cases of
,
, and
. In these cases,
is known to reduce to a representation that is part of a tower of higher spin that extends to
SUSY [
33,
36,
37,
38,
39], old-minimal supergravity [
33], and new-minimal supergravity [
46], respectively. Both gadgets
and
diverge for
and
diverges for
, as shown in
Figure 7.
Finite values of the gadget
exist for both
and
and a finite value for the gadget
exists for
.
As these results along with
Figure 7 indicate, the gadget values between these multiplets can be greater than the normalization of five. This is likely from the non-adinkraic nature of the representations.