On Linearized Nordstr\"om Supergravity in Eleven and Ten Dimensional Superspaces
S. James Gates, Jr., Yangrui Hu, Hanzhi Jiang, and S.-N. Hazel Mak
TL;DR
This paper introduces a superfield formalism to explore the potential construction of scalar versions of supergravity theories in eleven and ten dimensions, addressing the challenge of the unknown full off-shell formulations.
Contribution
It proposes a novel superfield framework as a foundational and experimental approach to develop scalar supergravity theories in higher dimensions.
Findings
Superfield formalism established for higher-dimensional supergravity
Potential pathways for scalar supergravity theories identified
Framework serves as a basis for future theoretical development
Abstract
As the full off-shell theories of supergravity in the important dimensions of eleven and ten dimensions are currently unknown, we introduce a superfield formalism as a foundation and experimental laboratory to explore the possibility that the scalar versions of the higher dimensional supergravitation theory can be constructed.
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Taxonomy
TopicsBlack Holes and Theoretical Physics · Nonlinear Waves and Solitons · Cosmology and Gravitation Theories
HET-1779
On Linearized Nordström Supergravity in Eleven and Ten Dimensional Superspaces
S. James Gates, Jr.,[email protected]a Yangrui [email protected]a, Hanzhi [email protected]a,b, and S.-N. Hazel [email protected]a
aDepartment of Physics, Brown University,
Box 1843, 182 Hope Street, Barus & Holley 545, Providence, RI 02912, USA
bDepartment of Physics & Astronomy,
Rutgers University, Piscataway, NJ 08855-0849, USA *
ABSTRACT
As the full off-shell theories of supergravity in the important dimensions of eleven and ten dimensions are currently unknown, we introduce a superfield formalism as a foundation and experimental laboratory to explore the possibility that the scalar versions of the higher dimensional supergravitation theory can be constructed.
PACS: 11.30.Pb, 12.60.Jv
Keywords: supersymmetry, scalar supergravity, off-shell
1 Introduction
In 1907, Einstein described his “happiest thought” [?] which marked the commencement of the race to create the Theory of General Relativity. Unrealized, he was already decidedly at a disadvantage. As early 1900 the astronomer Karl Schwarzschild (1873-1916) [?] had written about Riemann’s geometrical concepts to describe curved space - but not curved space-time. The latter would not emerge until Hermann Minkowski introduced the concept of “four-geometry” into physics [?,?].
By 1914, there were a number of competitors. At a minimum, these included Max Abraham (1875-1922), Gustav Mie (1868-1957), and Gunnar Nordström (1881-1923). Even the accomplished mathematician David Hilbert (1862-1943) became involved but at the conclusion made the comment in his own work, “The differential equations of gravitation that result are, as it seems to me, in agreement with the magnificent theory of general relativity established by Einstein [?].”
Along the pathway to the end of the race, the idea of scalar gravitational theories was explored. It is of note that Nordström created two such sets of equations [?,?] and even Einstein was taken with idea before discarding it. In the scalar approach, the usual metric of the space-time manifold is replaced by a single scalar field. A way to do this is to begin with the Minkowski metric and simply multiply it by a scalar field. This implies that, geometrically, scalar theories of gravitation are all members of the same conformal class as the usual flat Minkowski metric. Mathematically, scalar gravitation theories are perfectly consistent… they simply do not describe the physical laws observed in our universe.
In the nineteen eighties, superspace geometrical descriptions of supergravity in eleven and ten dimensions were presented in the physics literature. To the best of our knowledge a list of these inaugural publications looks as:
(a.) 11D, = 1 supergravity [?,?],
(b.) 10D, = 2A supergravity, [?],
(c.) 10D, = 2B supergravity [?,?], and
(d.) 10D, = 1 supergravity [?].
Of course these theories had been obtained in other descriptions even earlier. Interested parties are directed to these works for such references.
If we think of the references in [?,?,?,?,?,?] as the eleven and ten dimensional analogs of Einstein’s “happiest thought,” then by analogy all that have occurred since these works are analogs of the race to find the Theory of General Relativity. This also reveals a glaring disappointment. Since the “bell was rung” in this new race, all the competition is still at the starting line.
How would one know the race has been successfully ended? As indicated by the title of the work [?] (“Eleven-Dimensional Supergravity on the Mass-Shell in Superspace”), these descriptions possess sets of Bianchi identities that are consistent only when the equations of motion for the component fields in the theory satisfy their mass-shell conditions. This holds true for all of the works in [?,?,?,?,?,?]. So we may take as the sign of the successful completion of the race, if a set of superspace geometries were explicitly found such that their Bianchi identities do not require a mass shell condition. By way of comparison, for 4D, = 1 superspace supergravity, the analogs of the “happiest thought” and the conclusion of the race occurred within one year as is seen via works completed by Wess and Zumino [?,?].
As noted by Misner and Watt [?], though scalar gravitational theories are not realistic, they have value as computational tools in numerical relativity. This raises a very intriguing question. The quote, “History doesn’t repeat itself, but it often rhymes,555Though often attributed to Mark Twain, there is little evidence of this being accurate.” has been stated about many situations. Since scalar gravitation models have value as computational tools for General Relativity, might extending them to eleven and ten dimensional supergeometry offer new ways to replicate Einstein’s path from the “happiest thought” to the higher level of understanding as indicated by his lectures at Göttingen?
It is the purpose of this work to lay a new foundation for such an exploration. We take as a guiding principle the procedure used to provide the first example of a four dimensional supergravity supermultiplet where the closure of the local simple supersymmetry algebra was not predicated on the use of field equations. This was accomplished by Breitenlohner [?] who took as his starting point an off-shell supermultiplet, the so-called “non-abelian vector supermultiplet.” From the vantage of the current time, this is strongly reminiscent of an invocation of the concept of gravity as the double of gauge theory. The final form of Breitenlohner initial presentation realized a reducible representation of supersymmetry. Finally, this first work also did not consider any issues surrounding the construction of actions for the supermultiplet.
With the Breitenlohner approach as a guiding principle to the study of a class of curved supermanifolds containing eleven and ten dimensions, it is thus to be expected the extensions will manifest the same structure of being off-shell, but reducible and not address the issue of the construction of actions. No off-shell gauge vector supermultiplet is known beyond six dimensions, thus one is forced to deviate from completely following the Breitenlohner approach. Since a scalar superfield in any dimension is guaranteed to be off-shell, but reducible, one is naturally led to the study of Nordström supergravity theories in eleven and ten dimensions in this approach.
We organize this current paper in the manner described below.
Chapter two provides a self-consistent introduction to the field-theory and gauge-theory based formulation of gravitation described solely by a metric in dimensions. We use a frame field/spin-connection formulation from the beginning point of our discussion. This eases the transition to the case of superspace for these latter theories as it is an impossibility [?] to introduce a metric/Christoffel Riemannian formulation in the context of a superspace geometry appropriate to supersymmetry. The restriction of the full frame field to retain only the degree of freedom associated with its determinant is presented along with:
(a.) the well-known vanishing of the Weyl tensor, and
(b.) the residual form of the Einstein-Hilbert action under this
restriction.
Chapter three is a transitional one where we review 4D, = 1 supergravity as a paradigm setting arena. We show how the structure of this superspace of this well studied theory suggests pathways that can be pursued for how to carry out construction of scalar supergravitation in all higher dimensions including ten and eleven dimensional theories.
In chapters four through seven, we deploy the lessons found in the third chapter to work in making respective proposals for linearized theories of scalar supergravitation in the 11D, = 1, 10D, = 2A, 10D, = 2B, and 10D, = 1 superspaces.
We follow this work with our conclusions, two appendices, and the bibliography.
2 Gauge Theory Perspective On Ordinary Gravity
The traditional geometrical approach to describing gravity can be regarded as having driven an apparent wedge between general relativity and theory of elementary particles. Instead, a gauge theory and field theory based point of view provides a logical foundation for gravity which permits an alternative to geometry.
For gravitational theories in dimensions, the gauge group can be taken as the Poincaré group, and the Lie algebra generators are momentum and spin angular momentum generator . These are taken to satisfy the following commutation relations,
[TABLE]
and it might appear that the definition together with the second and third equations among (2.1) are in contradiction. The resolution to this conundrum is to note
[TABLE]
and the factor of actually corresponds to the vacuum value of the inverse frame field whose first index transforms under the action of and whose second index is inert under the action of the spin angular momentum generator. To distinguish between these two types of quantities, we use the “early” latin letters, , , etc. to denote indices that transform under the action of . Similarly, we use the “late” latin letters, , , etc. to denote indices that do not transform under the action of .
The covariant derivative with respect to this gauge group is
[TABLE]
where is related to the metric through its inverse via . The commutator of generates field strengths torsions and curvatures
[TABLE]
Scalar gravitation can be defined by restricting the form of the frame field to
[TABLE]
where is a finite scalar field. By definition, this defines a class of geometries that is conformally flat in the context of strictly Riemannian spaces. To see this we begin by setting , which implies
[TABLE]
and allows the full Riemann curvature tensor to be solely expressed in terms of the field as
[TABLE]
similarly for the Ricci curvature we find
[TABLE]
and finally for the curvature scalar by ,
[TABLE]
The Weyl tensor is defined by the equation
[TABLE]
and when the results in (2.8) - (2.10) are used, this is found to vanish.
We define and the Einstein-Hilbert action takes the form
[TABLE]
As the full off-shell description of 10D and 11D supergravities are yet unknown, we work with a toy model - scalar supergravity in the higher dimensions, which we expect gives part of the complete solutions. In the subsequent chapters, we replace by , where is an infinitesimal superfield, and study the corresponding linearized supergravity.
3 Nordström Supergravity in 4D, Supergeometry
As a preparatory step for our eventual goals, it is important that we re-visit four dimensional = 1 linearized supergravity as there are important lessons to be gained from asking questions solely in this domain prior to making the leap to eleven and ten dimensions. The formulation of linearized 4D, = 1 supergravity in term of the usual supergravity pre-potential was identified long ago [?]. It is perhaps of importance to note that supergravity pre-potentials bare some resemblance to other better known concepts important for the mathematical description of theories describing gravitation.
One of the most computationally enabling formulations of the dynamics of ordinary gravitation is based on the Arnowitt-Deser-Meisner (ADM) formulation [?] wherein the quadratic form involving the metric takes the form
[TABLE]
in terms of the “lapse” function , “shift” vector , and induced 3-metric . For the equation above to be valid, we can write
[TABLE]
The introduction of frame fields can be accomplished by observing that the quadratic form in (3.1) may also be written as
[TABLE]
by “factorizing” the metric into the product of two frame fields and multiplied by the constant Minkowski metric, , of flat spacetime. Thus, there exist relations between the ADM variables and the frame fields [?,?].
The point of the above discussion is to note that the inverse frame fields may be regarded as functions of the ADM variables, i. e.
[TABLE]
and that for numerical relativity calculations, the latter are far more useful than the frame fields , or even the metric itself. As we will see later, it is the form of 4D, = 1 supergravity often called the “Breitenlohner auxiliary field set” that is relevant to our work. For this formulation, it was first shown in the work of [?] the super-frame superfields are expressed in terms of a more fundamental set of superfields, i.e. the prepotentials and (with the “conformal compensator” explicitly dependent upon a complex linear superfield). In an “echo” of the utility of the ADM variables, the prepotentials are far more useful when component calculations, or quantum calculations are undertaken, with the latter able to utilize the technology of super Feynman graphs.
As in the discussion of section 7.5 in [?], we write (with being the superfield linearization of )
[TABLE]
[TABLE]
for the linearized superframe superfield. Similar to the ADM formulation of ordinary gravity, the superframe is expressed in terms of two independent superfields, , and . The remaining structures needed to specify the supergravity supercovariant derivative are the spin-connections which here take the forms
[TABLE]
The superfield introduced above is a general scalar superfield. This implies that the linearized formulation described above is reducible. There are two widely familiar choices that lead to irreducibility. One choice is implemented by picking to depend on , and a chiral superfield (i. e. ). This is the path that leads to the minimal off-shell formulation of 4D, = 1 supergravity. For this choice, the commutator algebra of the superspace supergravity covariant derivative takes the forms
[TABLE]
The other widely known choice “the Breitenlohner auxiliary field set” is implemented by picking to depend on , and a complex linear superfield (i. e. ). This is the path that leads to the non-minimal off-shell formulation of 4D, = 1 supergravity. For this choice, the commutator algebra of the superspace supergravity covariant derivative takes the forms
[TABLE]
The final commutator is found to be explicitly found from the equation
[TABLE]
Under either choice, one can use the definitions of the superframe superfields in (3.5) - (3.7) together with the set of equations of either (3.8) or (3.9) and (3.10) to find the dependence of , , and (for the minimal theory) on , and , or the dependence of , , , and on , and (for the non-minimal theory). These are the standard and well discussed theories of off-shell 4D, = 1 supergravity, i. e. the consistency of the Bianchi identities associated (3.8) or (3.9) and (3.10) for the algebra of the superspace supergravity covariant derivatives do not require on-shell conditions to be imposed on the components fields contained within the superfields.
The process of imposing the Einstein Field Equations in the non-supersymmetrical case in the absence of matter amounts to the condition
[TABLE]
including the cosmological constant. The equivalent in the case of superspace supergravity is accomplished by setting , and (for the minimal theory) to zero or by setting , and (for the non-minimal theory) to zero. The condition = 0 also forces = 0 in the non-minimal theory. Under these conditions, the algebra of superspace supergravity covariant derivatives take the universal form
[TABLE]
At this point, we can take a largely unexplored path as it is possible to consider the limit of these equations wherein = 0. This is the route to the 4D, = 1 superspace version of scalar supergravitation theory à la Nordström in the eleven and ten dimension that are the targets of our study.
The curious reader may wonder from where does the condition = 0 arise? On page 473, of [?] there appears the following text.
Nonsupersymmetric deSitter covariant derivatives can be obtained from gravitational
covariant derivatives by eliminating all field components except the (density) compensat
-ing field (i.e., the determinant of the metric or vierbein). This follows from the fact that
*in deSitter space the Weyl tensor vanishes, which says that there is no conformal (spin 2) *
part to the metric: It is conformally flat .
From the discussion given in chapter two, we saw that Nordström geometries in all dimensions are necessary such as to describe Weyl tensors that vanish and are hence conformally flat.
Also in the work of [?] it is explained that the conformal part of the metric arises solely from . Since the Nordström limit is a conformally flat bosonic space, it must corresponds to setting = 0. Thus, to our knowledge the passage above from “Superspace” marked the first indication of this. Of course, other authors such as in the work of [?] later reaffirmed this point about the structure of superspace of supergravity.
In the limit of our interest, we find
[TABLE]
In response to this restriction, the forms of the algebras in (3.8), (3.9) and (3.10) also change. In particular, the superfield (and consequently ) is identically zero. The latter condition is consistent with the component level description of scalar gravitation in the previous chapter as the Weyl tensor of (2.11) is the leading component field that occurs in and occurs at first order in the -expansion of . The third result in (3.13) also contains two useful bits of information:
(a.) The final term of the equation informs us that the leading term in the -expansion
of corresponds to the linearization of seen in equation (2.6).
(b.) The second term of the equation informs us that the leading term in the -expansion
of corresponds to the spin-1/2 remnant of the gravitino!
Another point to discuss is the dependence of the field strength superfields , and (for the minimal theory) and , , and (for the non-minimal theory) on the superfield . Direct calculation shows that the reality of in both cases implies that it only depends on the difference . The superfield is found to depend on the first spinor derivative (i. e. ) of . Finally, the superfield is found to depend on the second spinorial derivative of an expression linear in and .
We have argued previously [?], the minimal supergravity theory does not extend from four dimensions to eleven dimensions since there is no concept of chirality in the higher dimension. This implies that only the features seen in the non-minimal theory should be expected to occur in the subsequent chapters of this work. As we shall see, this is indeed the case. The commutator algebra for the superspace supergravity covariant derivative responds to the condition = 0, by the elimination of the all terms proportional to and . Thus, we find 4D, = 1 Nordström supergravity that descends from ten or eleven dimensions and only contains the generators associated with 4D, = 1 simple supegravity is described by
[TABLE]
[TABLE]
To our knowledge, the results in (3.14) mark the first time that a superspace description of 4D, = 1 Nordström supergravity has appeared in the literature.
To summarize, the limit of off-shell 4D, = 1 superfield supergravity where we only retain the conformal compensator provides a superspace extension of the Nordström supergravitation theory that is discussed in chapter three. We will make a working assumption that such an approach is universally applicable to all superspaces. In particular, in the subsequent chapters we will apply this assumption to superspaces whose bosonic subspaces possess either eleven or ten dimensions.
4 Linearized Nordström Supergravity in 11D, Supergeometry
We begin our discussion by reviewing the work of [?,?] where it was shown that the entire structure of the torsions, curvatures, and 4-form field strengths could be written in terms of a single superfield denoted by . Using the conventions of [?], we can write
[TABLE]
In addition to the torsion and curvature supertensors, the formulation above includes the 4-form supertensor, . It should be noted that these equations in (4.1) are the eleven dimensional analog of the equations in (3.12). In other words, the supergeometry in (4.1) is an “on-shell” supergeometry. We must find a supergeometry consistent with the Norström theory as the analogs of (3.14).
We now wish to construct the linearized torsion and curvature supertensors with property that when all fermions are set to zero, the theory smoothly maps to the linearization of the non-supersymmetrical theory described in chapter two.
For this purpose we introduce eleven dimensional supergravity covariant derivatives linear in the infinitesimal conformal compensator given by
[TABLE]
where the “bare” superderivative operators satisfy
[TABLE]
and the torsion tensors and Riemann curvature tensors can be obtained via
[TABLE]
The commutation relations of the operators with the 11D Lorentz generators satisfy
[TABLE]
in addition to the relations seen in (2.1) and (2.2).
By imposing the constraints
[TABLE]
we obtain the following parameterization results:
[TABLE]
In turn these lead to a set of results that express the torsion and curvature tensors solely in terms of and its derivatives. We give these in the following two subsections.
For the components of the torsion we find the results seen in (4.10)
- (4.15).
[TABLE]
For the components of the curvature we find the results seen in (4.16)
- (4.18).
[TABLE]
In reaching (4.10) - (4.18), we used the Fierz identities (A.24) - (A.27) listed in Appendix A.
It is the last equation that ensures that we have reached our goal. Namely, the choice of constraints in (4.9) has led to a linearized super Riemann curvature tensor expressed solely in terms of an infinitesimal superfield that has the exact form of the first term in the non-supersymmetrical Riemann curvature tensor given in (2.8). Recall that the supersymmetrical theory here is linearized, so to make a proper comparison to the bosonic theory, that should also be linearized. When this is done, there is a matching of the terms.
We should note the work in [?] also constructs a fully non-linear 11D supergeometry in terms of a finite scalar compensator. However, its linearization is different from the one obtained here. In the next three chapters, we will obtain new and never before presented results of this nature for the 10D, = 2A, 10D, = 2B, and 10D, = 1 supergeometries that possess the purely bosonic linearized results as in the linearization of (2.8). The Fierz identities used for simplifying the torsions and curvatures are listed in Appendix B.
5 Linearized Nordström Supergravity in 10D, = 1 Supergeometry
We begin this discussion by pointing out the on-shell description of 10D, = 1 superspace supergravity. A set of torsion and curvature supertensors can be written in the form
[TABLE]
as was noted in the work of [?,?]. In these expression refers to the supercovariantized field strength of a two-form . The results in (5.1) are the 10D, = 1 analogs of the results in (3.12) for the 4D, = 1 superspace geometry. That is the component fields embedded in this supergeometry must obey a set of mass-shell conditions. To release these conditions, one must find the 10D, = 1 analogs of the equations in (3.9) and (3.10). However, as our goal once more is to find a supergeometry consistent with the Norström theory, we seek the analogs of (3.14).
The covariant derivatives linear in the conformal compensator are given by
[TABLE]
and similar to the case of the eleven dimensional theory, here we have
[TABLE]
The commutation relations of Poincare generators in 10D
[TABLE]
is similar to the eleven dimensional case. Also the equation in (4.7) is valid in all ten dimensional theories. There will be some slight modifications for the dotted and barred spinor indices in type IIA and IIB supergravity, respectively.
By adoption of the constraints
[TABLE]
we obtain the following parameterization results:
[TABLE]
As the consequence of this choice of parameters, we find the torsion supertensors given in (5.8) - (5.13).
[TABLE]
For the components of the curvatures, we find the results seen in (5.14) - (5.16).
[TABLE]
It has long been suggested [?] that a superfield with the structure of should appear in the off-shell structure of 10D, = 1 supergeometry and that it was related by a superdifferential operator to an underlying unconstrained prepotential analogous to that appears in 4D, = 1 supergravity. However, there are reasons to believe [?,?] that must be related to an even more fundamental spinorial prepotential . In the equations of (5.11) and (5.14) the superfield has precisely the structure suggested in the work by Howe, Nicolai, and Van Proeyen.
6 Linearized Nordström Supergravity in 10D, = 2A Supergeometry
We repeat the discussions as seen in the previous two chapters with a beginning of the on-shell description of 10D, = 2A superspace supergravity. A set of torsion and curvature supertensors can be written in the form
[TABLE]
as was noted in the work of [?]. In these expressions refers to the supercovariantized field strengths of a two-form gauge field, and
[TABLE]
with and denoting supercovariantized field strength for a gauge 1-form and a gauge 3-form respectively. The results in (6.1) are the 10D, = 2A analogs of the results in (3.12) for the 4D, = 1 superspace geometry. Those are the component fields embedded in this supergeometry must obey a set of mass-shell conditions. To release these conditions, one must find the 10D, = 2A analogs of the equations in (3.9) and (3.10). Again the goal must be to find a supergeometry consistent with the Norström theory, we seek the analogs of (3.14). In analogy with the 3-form gauge field sector of 11D, = 1 supergravity the gauge fields components are:
[TABLE]
denotes a dilaton superfield, and is its partner dilatino. All of the equations in (6.1) - (6.3) describe the on-shell 10D, = 2A theory, i.e. these are the analogs of (3.12).
The covariant derivatives linear in the conformal compensator are given by
[TABLE]
where the Type IIA supersymmetry algebra
[TABLE]
is satisfied by the bare derivative operators.
By adoption the constraints
[TABLE]
we obtain the following paramaterization values:
[TABLE]
As the consequence of this choice of parameters, we find the torsion supertensors given in (6.10)
- (6.27).
[TABLE]
For the components of the curvatures, we find the results seen in (6.28) - (6.33).
[TABLE]
7 Linearized Nordström Supergravity in 10D, = 2B Supergeometry
Now for a final time we replicate the discussions as seen in the previous three chapters with a beginning of the on-shell description of 10D, = 2B superspace supergravity here. A set of torsion and curvature supertensors can be written in the form
[TABLE]
[TABLE]
[TABLE]
[TABLE]
as was noted in the portion of the work in [?] devoted to type IIB supergravity. We will end our discussion here. As the astute reader can note the expressions are of increasing complication. But the central message of the expressions in (7.1) - (7.4) is that the on-shell description of the 10D, = 2B theory exists in perfect analogy with the on-shell description of 4D, = 1 superspace given by the equations in (3.12).
Now for the covariant derivative operators linear in the conformal compensator and necessary for a Nordström theory may be given by
[TABLE]
with the Type IIB supersymmetry algebra
[TABLE]
By adopting the constraints
[TABLE]
we have the following parameterization results:
[TABLE]
[TABLE]
As the consequence of this choice of parameters, we find the torsion supertensors given in (7.11)
- (7.28).
[TABLE]
For the components of the curvatures, we find the results seen in (7.29) - (7.34).
[TABLE]
8 Conclusion
This work gives a proposal for descriptions of Nordström supergravity in eleven and ten dimensions. This work is but a foundation as in future extensions of this work we plan to continue this exploration. One obvious direction is to extract from this superfield work the implied component level descriptions that follow from the superfield equations we have presented. Our work is based on the assumption that in each of the cases of 11D, = 1, 10D, = 1, 10D, = 2A and 10D, = 2B, a single scalar superfield is required to provide such a description as this was the case for both ordinary gravitation as well as 4D, = 1 supergravity.
Having obtained the results for the theories in ten and eleven dimension superspaces, we can compare those results with the ones seen in chapter three. Looking back at (3.8), with a bit of effort, one can show that the condition = 0 cause only modification in the form of the equations. Namely the terms will vanish under this restriction. It is thus pointedly seen that all the basic superfields (i. e. and ) in the algebra of the superspace supergravity covariant derivative are bosonic. This is to be compared to the results shown in (3.14) where a fermionic superfield appears. In all of the higher dimensional theories such superfields appear ubiquitously.
In the future we will also address the very important quest of whether there exists a superspace action for the Nordström supergravity theories in high dimension. It is clear that in order for this to be the case, it is necessary that the scalar superfield should satisfy some superdifferential constraints. The expectation is suggested by the structure of the 4D, = 1 theory. We remind the reader that the irreducible theories require that the superfield is subject to some differential constraints. So it is natural to expect this to extend into the higher dimensional theory.
Our approach also raises an interesting question about Superstring Theories, M-Theory, and F-Theory. Do these theories also possess consistent truncation limits that include Nordström supergravity theories in their low energy limits? If the answer is affirmative, such limits might provide laboratories in which to investigate these more complicated mathematical structures.
*“The most effective way to do it, is to do it.”
* - Amelia Earhart
**Dedication
** SJG wishes to dedicate this work to the memory of Shota Ivan Vashakidze, a valued friend and collaborator in the exploration of ten dimensional superspace geometry.
**Acknowledgements
** The research of S. J. Gates, Jr., Y. Hu, and S.-N. Mak is supported by the endowment of the Ford Foundation Professorship of Physics at Brown University and partially supported by the U.S. National Science Foundation grant PHY-1315155.
Appendix A 11D Clifford Algebra Representation
In this section we briefly summerize the convention that we adopted for 11D gamma matrices. Our 32 32 gamma matrices are defined by the Clifford algebra:
[TABLE]
where denotes the 3232 identity matrix and the inverse metric follows the ”most plus” signature:
[TABLE]
It is known that D-dimensional space-time Dirac spinor has components when D is odd, and components when D is even. Hence in 11D, the spinor indices of the gamma matrices, denoted by , and so forth, run from 1 to 32.
One can raise and lower the spinor indices via the ”spinor metric”, , which satisfies:
[TABLE]
The gamma matrices with multiple vector indices are defined through the equations:
[TABLE]
The symmetric relations of the gamma matrices are given by:
[TABLE]
From the definitions, one can easily work out the following trace identities:
[TABLE]
as well as the following Fierz identities:
[TABLE]
Finally, we list the explicit representations of 11D gamma matrices in terms of tensor products of Pauli matrices:
Spinor metric:
[TABLE]
Gamma matrices:
[TABLE]
Appendix B 10D Clifford Algebra Representation
In this section we briefly summerize the convention that we adopted for 10D sigma matrices. The Clifford algebra is
[TABLE]
where the inverse metric is:
[TABLE]
In 10D, the Dirac spinor has components. We use undotted Greek index to denote 16 component left-handed Majorana spinor, and dotted index to denote right-handed ones,
[TABLE]
where and . We raise and lower the spinor indices by spinor metric as follows:
[TABLE]
The sigma matrices are bispinors. There are three types of them: purely left-handed:
[TABLE]
purely right-handed (related to purely left-handed by the following):
[TABLE]
and mixed bispinors:
[TABLE]
which have relations
[TABLE]
Definition of -matrices with more Lorentz indices:
[TABLE]
and
[TABLE]
The sigma matrices with five vector indices satisfy the self-dual / anti-self-dual identities:
[TABLE]
The symmetric relations of the gamma matrices are given by:
[TABLE]
From the definition, we can easily work out the trace identities:
[TABLE]
and
[TABLE]
From the definition, we can also derive the following 10D sigma matrices identities:
[TABLE]
as well as the following Fierz identities:
[TABLE]
Finally, we list the explicit (real) representations of the sigma matrices in terms of tensor products of Pauli matrices:
[TABLE]
and
[TABLE]
[TABLE]
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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