Adinkra Foundation of Component Decomposition and the Scan for Superconformal Multiplets in 11D, N = 1 Superspace

For the first time in the physics literature, the Lorentz representations of all 2,147,483,648 bosonic degrees of freedom and 2,147,483,648 fermionic degrees of freedom in an unconstrained eleven dimensional scalar superfield are presented. Comparisons of the conceptual bases for this advance in terms of component field, superfield, and adinkra arguments, respectively, are made. These highlight the computational efficiency of the adinkra-based approach over the others. It is noted at level sixteen in the 11D, N = 1 scalar superfield, the {65} representation of SO(1,10), the conformal graviton, is present. Thus, Adinkra-based arguments suggest the surprising possibility that the 11D, N = 1 scalar superfield alone might describe a Poincare supergravity prepotential in analogy to one of the off-shell versions of 4D, N = 1 superfield supergravity.


Introduction
The standard "workhorse" of Salam-Strathdee superspace [1] is the concept of the "superfield." Previously, we have argued the superfield concept can be augmented by the newer network-centric concept of "adinkras" 4 [2]. From the time of their introduction in one dimensional extended superspaces of the Salam-Strathdee type, GR(d, N ) algebras 5 [3,4] -the adjacency matrices for the adinkra graphs -gave an explicit solution to describing minimal irreducible supermultiplets in one dimensional theories for all values of degree of extension N . On previous occasions (e. g. [5]), we have pointed out that GR(d, N ) matrices are the (G)eneral (R)eal extensions of two-component van der Waerden matrices used in physics. This work also contained a description of how the GR(d, N ) matrices are embedded in Clifford algebras.
As we have noted before, since GR(d, N ) matrices are the adjacency matrices for adinkras, a direct question adinkras answer is, "Given N supercharges in a one dimensional system, what is the minimum number d min of bosons and equal number of fermions required to realize the N supercharges in a linear manner?" In the works of [3,4]  (where we excluded the case of N = 0, i.e. no supersymmetry) was proposed as the answer. Until recently, no derivation of this result that is not related to adinkra-based arguments was known to us.
However, it has been communicated to us [6] that another alternative narrative argument should lead to this same result. W. Siegel has observed that as the form of the one dimensional N -extended supersymmetry algebra implies for N supersymmetry generators q, { q , q } ∝ E , (1.2) this can be regarded as a Clifford algebra. Thus, the quantities q must be spinor representations of SO(N ). Investigating the minimal such irreducible irreps, he argues, must lead to the formula above. To this alternative narrative, we respond the work in [5] (contained in its equations (13)- (16)) precisely provides a derivation aimed at this.
The question raised in the last paragraph can be extended to the more complicated domain of higher dimensional "off-shell" theories by asking "Given N supercharges in a D dimensional system, what is the minimum number d min of bosons and equal number of fermions required to realize the N supercharges in a linear manner without the use of any dynamical assumptions?" Thus, one is led to suspect the existence of a function d min (N , D) in any dimension that gives the answer to the question in general but with the property d min (N , D = 1) = d min (N F(D)) , (1.3) where the function on r.h.s. of the equation is defined in (1.1) and the function F(D) is shown in a few examples in Table 1. The explicit form of d min (N , D) has remained unknown throughout the history of the subject of supersymmetry 6 , but Equation (1.3) gives its boundary condition, i.e. the value when we reduce the dimension to one. The curious reader may question what is the source of the caveat regarding dynamical assumptions? The answer is this is necessary to find prepotential formulations in the Salam-Strathdee superspaces for the theories under study.
When N = 1 and D = 11, one is looking at the low-energy limit of M-Theory [9], the eleven dimensional supergravity theory [17,18].
For decades there has been little understanding created beyond these descriptions of the onshell theories 7 in ordinary Salam-Strathdee superspace. It is thus accurate to describe these as "orphaned" problems currently existing in an "abandoned" state. Using conceptual and computational advances it can be argued there is reason to expect new progress. Analytical progress with regards to the M-Theory corrections to the on-shell theory has been shown in the works of [10,11,12,13,14] and in [13,14] group representation theory was also included in the discussions. By contrast the off-shell situation is as murky as ever.
A special note of attention should be directed to the works of [13,14] as these provide studies that are in a sense "orthogonal" to the direction of our works. These works of Howe et. al. provide a thorough investigation of this class of problems...based on the study of superspace Bianchi identities (referred to as "spinorial cohomology") associated both the geometrical sector in the 11D superspace. These are in accord with the previous analyses of [16,15] with regards to the form of deformations of the superspace torsion tensor. However, the works in [13,14] include analysis of the dual 6-form (and 3-form) sector. The reason these works may be considered "orthogonal" to our efforts is they focus on the Bianchi identities and the objects that appear in them while our efforts are directed toward candidate prepotential superfields.
Computational power as well as algorithmic architecture design have advanced tremendously since the 1980's. Along the lines of new computational paradigms, there now exist breakthroughs in artificial intelligence, neural networks, and deep learning that emerged in the intervening period. By using conceptual and computational advances it can be argued, now is a propitious time to make new progress in many areas.
In the work of [7] a combination of new conceptual and computational tools was deployed to create progress in problems surrounding superspace geometries that describe supergravity in ten dimensions. Results in the work confirmed the analysis of the spectrum [19] of the scalar superfield given by Bergshoeff and de Roo for the component fields in the SG limits of Type-I closed and heterotic string theories. Moreover, this work [7] also gave new similar results in the domains of the Type-IIA and Type-IIB SG limits. Though these problems involve a factor of 65,536 more degrees of freedom than occur in Type-I closed and heterotic string theories, the modern techniques proved to be up to the tasks of complete analysis of these systems. We are now able to abstract the component field content of superfields without using traditional θ-expansions for arbitrarily high dimensions. As the Type-IIA SG system can be obtained from a dimensional reduction of the eleven dimensional system, this was a signal that the supergravity limit of M-Theory should be directly amenable to the same sort of complete analysis to provide complete transparency about the SO(1,10) Lorentz representations. It is the purpose of the present work to fulfill this anticipated result.
The layout of this paper is described below.
Chapter two provides a self-contained description of the "off-shell auxiliary field problem" by beginning at the component level and discussing an ab initio recipe for deriving supersymmetric representations for arbitrary spacetimes. The distinction between "off-shell" and "on-shell" formulations is noted. Next a similar high level discussion of how off-shell supermultiplets, that are equivalent to superfields in the context of Salam-Strathdee superspace, is presented. The final portion of this chapter introduces the concept of the adinkra of a superfield or a supermultiplet, using the example of the 10D, N = 1 scalar superfield, as a network that encodes the Lorentz representations of the field content as well as the orbits of those field representations under the action of supercharges.
Chapter three follows the route of the traditional θ-expansion as applied to the 11D, N = 1 scalar superfield. A discussion involving a recursion formula used to move in a level-by-level manner up the θ-expansion of the superfield is presented and the role of Duffin-Kemmer-Petiau fields is noted. The recursive procedure is applied up to cubic order to show the calculational complications that occur in starting with a superfield and then abstracting the component field content from such a systematic starting point.
The fourth chapter is dedicated to showing that calculational efficiency can occur in the process of extracting the component field content from branching rules of su(32) ⊃ so (11) and the concept of Phethysms instead of the superfield's θ-expansion. In order to implement the use of branching rules, explicit projection matrices needed are presented.
These all powerfully combine so as to make the need for any explicit calculation based on γ-matrices to become totally banished from these considerations. It is the independence of these methods from γ-matrices that allows for substantial computational efficiencies that can be exploited by modern computer based algorithms. All these together become the "secret sauce" that allows unprecedented access to the component level structures that are hidden within superfields.
We apply these tools to extract the SO(1,10) representations from all 1,494 bosonic fields and 1,186 fermionic fields contained in a 11D, N = 1 scalar superfield. To our knowledge, these observations about the numbers of fields (both bosonic and fermionic) as opposed to the number of degrees of freedom have not appeared previously in the literature. Having obtained this information, we follow the path established by Breitenlohner, to look for what superfield can minimally contain the conformal 11D graviton and gravitino. A surprising answer is found.
The fifth chapter is devoted to describing the adinkra of the 11D, N = 1 superfield giving a level-by-level description of the number of fields contained at each level. This relates back to the 1,494 bosonic fields and 1,186 fermionic fields found in the previous chapter. An image of the adinkra up to Level-5 is given.
We include our conclusions where we discuss possible implications for the superfield limit of M-Theory and Type-IIA superstring theory. This is followed by six appendices. Appendix A contains a dictionary between Dynkin Labels and the corresponding representation dimensionalities. Appendices B -D contain technical details of manipulations with 11D γ-matrices. These are included for any researcher who wishes to verify independently the assertions we make about the properties of the γ-matrices that we are able to bypass. Appendix E contains an extended discussion of the role that two distinct types of Young Tableaux play in clarifying the manner in which γmatrices are avoided in this approach. Understanding these plays a role in the final suggestion of the conclusion which is that calculational efficiencies are likely possible if the traditional concept of the Salam-Strathdee superfield is replaced by a newer concept of an "adinkra-field" where the fermionic Young Tableaux play the role of the θ-coordinates and the Dynkin Labels play the role of the fields.
The final appendix presents the decomposition results of the 11D, N = 1 scalar superfield by giving Dynkin labels.

Primers Before 11D
The component formulation of supersymmetrical systems has traditionally followed a pattern that we will review next.

Component Primer Before 11D
At a general level, one begins with a set of bosonic representations we denote by {R (i) } and a set of fermionic representations we denote by {R (j) }. The range of the indices on the two distinct sets need not be the same, i.e. the values taken on by (i) and (j) are generally different. Next one assumes a set of dynamics codified by specifying a Lagrangian, the schematic form of which is realized as (where ∂ is the spacetime derivative but written in an index-free notation), which is followed by introducing a "supercharge" that we (once more schematically) write as D together with the definitions of its realizations on the bosonic reps {R (i) } and fermionic reps {R (j) } according to For the reader interested in seeing a more explicit discussion of this in examples, the work in [20] is recommended.
When one appropriately calculates an expression that is second order in the D operator, a bifurcation occurs with two possible outcomes where the term ∂L stands for a set of equations of motion that are derivable from L. To reconcile the differences between outcomes (a.) and (b.) in (2.4) above, it is most common to demand that the fields in the system should obey their equations of motion.

Superfield Primer Before 11D
The idea of the superfield, or equivalently an "off-shell supermultiplet" is to modify the starting point in three ways: (a.) the range of the index (i) describing the bosonic representations {R (i) } is allowed to increase, (b.) the range of the index (j) describing the fermionic representations {R (j) } is allowed to increase, (c.) a "height" or "Level" number is introduced for all the bosonic reps {R (i) } and fermionic reps {R (j) }, each with their enhanced range of indices.
The Level numbers are non-negative integers. It is convenient to partition the Level numbers into even and odd integers. In the language of superfields, this corresponds to the monomial of Grassmann coordinates associated with the component field representation in the "θ-expansion of the superfield." As the ranges of the indices the (i) and (j) in this subsection are greater than those associated with (2.1), this means new bosonic reps and new fermionic reps are under consideration. The new bosonic reps are called "auxiliary bosonic fields" while the new fermionic reps are called "auxiliary fermionic fields." Over the totality of the component field reps, one must now define the action of D. For the point covered by the following arguments, we will use the words "superfield" and "adinkra" interchangeably.
For example, if we begin with the "i-th" bosonic representation {R (i) } p at level p in the adinkra, then the action of the spinor covariant (in an index-free notation) derivative D must take the form where {R (j) } p+1 and {R (j) } p−1 correspond to the "j-th" fermionic representations at the p + 1 level and p − 1 level respectively in the adinkra.
In a similar manner, if we begin with the "j-th" fermionic representation {R (j) } p at level p in the adinkra, then the action of the spinor covariant derivative D must take the form where {R (i) } p+1 and {R (i) } p−1 correspond to the "i-th" bosonic representations at the p + 1 level and p − 1 level respectively in the adinkra.
The quantities c (2.5) and (2.6) respectively) are sets of constants typically proportional to γ-matrices, Minkowski metric, Levi-Civita tensor, or powers of any of these.
The overarching point is by starting from the definitions in (2.5) and (2.6) and repeating the calculation described by (2.4), the constants in two equations that define the realization of D are fixed by the condition that they only lead to the condition (b.), i.e. describe an "off-shell" realization of supersymmetry.

Adinkra Primer Before 11D
In the work of [7], the complete descriptions of the component field representations required to describe an off-shell theory of scalar gravitation in 10D, N = 1, N = 2A, and N = 2B superspaces were presented. In the following, it is expedient for us to concentrate on the 10D, N = 1 case and focus on the adinkra. This was done in the form of the adinkra shown in (2.1). As we will discuss later in the next section, the conventional superfield approach while adequate for extracting the necessary information about the bosonic reps {R (i) } p and fermionic reps {R (j) } p , becomes more unwieldy. The adinkra approach offers a way around this.
The utility of this adinkra graph is that it provides a "roadmap" to the writing of the explicit form of the action of the supercharge on any particular component field.

Traditional Path to Superfield Component Decompositions
Before applying the same idea to eleven dimensional superspace, as we did in [7], the traditional method and its problems need to be discussed. If we start from constructing the irreducible θ−monomials to understand the eleven dimensional scalar superfield decomposition, two uniqueness problems will show up: (1) θ−monomials have multiple expressions from the cubic level; (2) gamma matrix multiplications have multiple expressions. To illustrate the first problem, the quadratic level and cubic level will be discussed in detail in the following sections. For the second one, all gamma matrix multiplication results are listed in Appendix B. Moreover, constructing irreducible θ−monomials requires a number of Fierz identities as shown shortly. Compared to the group representation approach embodied by adinkras, the traditional method is much less efficient.
Each higher dimensional superspace with D bosonic dimensions, for purposes of counting is equivalent to some value of d, which is the number of real components of θ. This is shown in a few cases below (where d = F(D)).   So for the case of the 11D, N = 1 theory, the real unconstrained scalar superfield Ψ contains 2,147,483,648 bosonic and 2,147,483,648 fermionic degrees of freedom that are representations of supersymmetry, While superfields easily provide a methodology for finding collections of components in principle, actually obtaining those component fields is not as easy as it might first appear. This is especially true in the eleven dimensional case.
In the rest of this chapter we are going to discuss the complications of applying the most straightforward θ-expansions in the eleven dimensional superspace. The discussion is meant to provide an explicit demonstration of the difficulties one encounters in such a program. For the reader not interested in these details, it is recommended to skip to chapter four.
A naive expansion of a real scalar superfield V can be expressed as abcd ef gh (x) + . . . [indices] (x) indicates the θ-order. In writing this expression we have introduced "auxiliary nilpotent coordinates" defined in (3.8) below.
There exists a recursion formula that can be applied at any non-trivial order n in the θ-expansion to derive the form of the terms at order (n + 1) in the θ-expansion. We can begin this by looking at the term linear in θ, and next observe the quadratic terms may generated by a simple replacement in this expression.
The quantity (B) αβ is a Duffin-Kemmer-Petiau [21,22,23] field. Therefore under the action of this replacement, we find with V(quadratic) given by To continue, we take the component fields ϕ (2) abc (x), ϕ abcd (x) and make the simultaneous replacements which yield the cubic order terms that appear on the second line of (3.1).
The general rule is that if one starts with the component fields at Level-n, where n is even, of the scalar superfield, then to obtain the component fields at Level-(n + 1) one simply replaces the starting component fields by a θ-coordinate whose index is contracted against a new fermionic fields in a manner that is consistent with Lorentz symmetry.
Also a general rule is that if one starts with the component fields at Level-n, where n is odd, of the scalar superfield, then to obtain the component fields at Level-(n + 1) one simply replaces the starting component fields by a new DKP field times a θ-coordinate whose index is contracted against one index on new DKP fields in a manner that is consistent with Lorentz symmetry.
Although one can carry out this procedure to define the component fields to all orders in the θ-expansion, it is highly inefficient and redundant. This redundancy occurs due to the equivalence of many terms obtained as well as the vanishing of many terms both by use of Fierz identities. There is also the issue of irreducibility that must be enforced. We next turn to the issue of irreducibility.

Quadratic Level
We denote the 11 dimensional 32-component Majorana Grassmann coordinate by θ α . Since C αβ , (γ [3] ) αβ and (γ [4] ) αβ are the antisymmetric elements in the covering Clifford algebra over 11D, we can define all possible quadratic θ-monomials as follows. {1} Now if we look at the quadratic θ-terms, we see the total number of bosonic component fields at this level can be found by simply counting the number of independent quadratic θ-monomials so all is well as this equation gives the complete decomposition of the product of two Grassmann coordinates into irreducible representations of the 11D Lorentz group.

Cubic Level
We can construct cubic θ-monomials from all the possible quadratic θ-monomials as listed in Equation (3.8). Since C αβ , (γ [3] ) αβ and (γ [4] ) αβ are the antisymmetric Clifford algebra elements in 11D, we can write all the possible cubic monomials starting with no free Lorentz vector index and going up to four free Lorentz vector indices. All of the possible irreducible cubic θ-monomials can be written as (3.10) where the notation IR simply means that a single γ-trace of the expression is by definition equal to zero. We will discuss each representation (except {5, 280}) one by one in the following subsections. We will explain each dimension from the corresponding irreducibility condition, and prove that different versions in each representation are actually equivalent. We will show that the cubic monomials of {320} vanish. We argue that {5, 280} cubic monomials also vanish in a similar way. Another strong reason for {5, 280} to vanish is 5, 280 > 4, 960 (see equation below). We can then decompose all the cubic θ-monomials by where the left hand side simply counts the number of independent cubic θ-monomials one can write, and the rightmost part contains {32}, {1, 408}, and {3, 520} which are irreducible representations of the 11D Lorentz group as shown in Appendix A.

{32} Cubic Monomials
We have three versions of expressions of cubic θ-monomials with no free vector index and one free spinor index as indicated in Equation (3.10), which are 14) The degrees of freedom of these monomials are thus 32, and so they are in the spinorial representation {32}. Here we define these three objects To find whether V 1, V 2, and V 3 are related, we examine the objects A 1 , A 2 and A 3 . Since C αβ , (γ [3] ) αβ and (γ [4] ) αβ form the complete basis for the antisymmetric elements in 11D Clifford algebra with two spinor indices, we can expand the A objects into this basis, i.e. find the Fierz identities. The Fierz identities in Appendix D tell us that the objects A 1 , A 2 and A 3 are related by a system of linear equations (we suppress spinor indices here for simplicity, as all of them have the same structure), This makes sense as A 1 , A 2 and A 3 are the only three objects of this spinor index structure with no free vector indices and at least one Clifford element being antisymmetric. By solving these linear equations, we get A 2 = 66A 1 , which means all three of the expressions of the cubic θ-monomials are equivalent up to a multiplica- We can therefore take any of the cubic θ-monomial constructed from linear combinations of V 1, V 2 and V 3 as the fermionic irreducible representation {32} of so(11).

{320} Cubic Monomials
For one free vector index, we have two expressions of cubic θ-monomials as suggested in Equation (3.10). We can write all the terms from the index structures as The irreducibility conditions of setting the single γ-traces to zero are Thus, these cubic monomials with one free vector index have 32 × 11 − 32 = 320 degrees of freedom and are in the {320} representation. From the irreducibility conditions, we can fix the relative coefficients to k 1 = − 1 7 and l 1 = − 1 8 . Without loss of generality, we omit the overall coefficientsk 0 andl 0 . Therefore, we can write where we define five objects When we consider all the objects with one free vector index constructed by two Clifford algebra basis elements with one of them being antisymmetric, we find the additional object B as defined above. From our past experience, we know that it has to occur in our Fierz expansions. The relevant Fierz identities are listed in Appendix D. They can be rewritten into a system of linear equations of objects B, C 1 , C 2 , D 1 , and D 2 (vector and spinor indices suppressed) as The solutions are Therefore, from Equations (3.23) and (3.24), it is very clear that This suggests that there exists no cubic θ-monomial in the {320} irreducible representation of so (11).
Another way of seeing that {320} cubic θ-monomials do not exist is the above constructed monomials fail to satisfy the irreducibility condition. From Equation (3.27), we see that V 1 and V 2 are clearly proportional to B, for example. The irreducibility conditions in (3.21) and (3.22) thus read as A 1 = 0 numerically (otherwise, the {32} cubic monomials would also vanish). Doing this with C's or D's will give us other linear combinations of A's, which would not vanish also as all A's are proportional to A 1 .

{1, 408} Cubic Monomials
There are two versions of expressions of cubic θ-monomials with two antisymmetric vector indices as listed in Equation (3.10). They can be expanded in the following basis where the relative coefficients are fixed by the irreducibility conditions From the conditions we know that these cubic θ-monomials have 32× 11×10 2 −32×11 = 1, 408 degrees of freedom, and thus they live in the {1, 408} representation. By omitting the overall coefficients g 0 andh 0 , we can write the two versions as where we define seven objects [5] (γ [4] ) [δ (γ [5] ) β]α .
An additional object E is defined to span the entire basis, which plays a similar role to B in the {320} representation. The Fierz expansions of all these objects as listed in Appendix D give us the system of linear equations (3.37) By solving these linear equations, we obtain which means It is of note that here we also have the freedom to choose any other two objects as our basis. For example, we can choose F 1 and F 2 instead. Then we will find

42)
V 1 and V 2 are always proportional to each other, thus they are equivalent and there's only one {1, 408} irreducible representation sitting in the cubic sector.
Let's check the irreducibility condition. If we choose E and F 1 as the basis, the conditions in (3.32) and (3.33) translate to After some quick calculations, we can simplify this condition as which is exactly satisfied by Equation (3.27). If we choose another basis, like F 1 and F 2 , and simplify the irreducible condition, we will get the relation between objects B and D 2 in Equation We can then comment further on why {320} cubic monomials must vanish. We observe that there is one non-vanishing independent object in {32}, therefore the irreducibility condition in {320} implies that there are more than one independent objects in {320}. Meanwhile, there are two independent objects in {1, 408}, and the irreducibility condition in {1, 408} implies that there is only one independent object in {320}. Thus, we reach a contradiction. Therefore, sandwiching from {32} and {1, 408} would force the {320} objects to vanish.
Following this line of logical arguments, the next representation with three free vector indices, {3, 520}, would have to have three independent objects if not vanishing, and its irreducibility conditions should reduce to the relations between objects in {1, 408} in Equation (3.38), as we will see.
Now rewrite the irreducibility conditions in (3.48) and (3.49) to This condition can be simplified as which exactly satisfies Equation (3.38), as predicted in the last subsection.
The main message of this section of our work is that explicit θ-expansion of the eleven dimensional scalar superfield is considerably more complicated than in lower dimensions. One must contend with three separate problems: (a.) there are multiple equivalent ways to express the required θ-monomials, (b.) some apparently reasonable monimal combinations actually vanish, (c.) the requirement of irreducibility of the θ-monomial expansion requires carefully constructed combinations. The resolution of the first two of these problems relies of the derivation of Fierz identities. With regard to the final problem, the only methodology known to us is brute force establishment of their existences.
All of these point to the fact that superfields and their accompanying θ-expansions become increasing unwieldy as a "technological platform" for the study of supermultiplets in higher dimensions. We will return to this in our conclusions.

(4.3)
Since V is a scalar superfield, the conjecture given in the conclusion of the work of [7] implies the following statements for even values of n and  (11) In [7] while we have applied branching rules to find component decompositions of scalar superfields in ten dimensions, we didn't explain the details of branching rule calculations. In this section, we will present the explicit algorithmic [24] calculations needed for finding branching rules, in particular for the case of su(32) ⊃ so (11).
First, a branching rule is a relation between a representation of Lie algebra g and representations of one of its Lie subalgebras h. For a simple Lie algebra g, its Lie subalgebras can be classified as regular subalgebras and special subalgebras. Regular subalgebras can be obtained by deleting dots from extended Dynkin diagrams, while special subalgebras cannot. Moreover, subalgebras can be classified as maximal subalgebras and non-maximal subalgebras. The definition of a maximal subalgebra h of g is that there is no any subalgebra l satisifies h ⊂ l ⊂ g.
Branching rules between g and h are completely determined by the projection matrix P g⊃h . Suppose the rank of Lie algebra g is n and the rank of Lie algebra h is m, then the projection matrix P g⊃h is a m × n matrix. Given a weight vector w g in g, the projected weight vector where weight vectors are row vectors. Thus, the algorithm for calculating a branching rule of an irrep R of g given the projection matrix can be summarized as follows.
1. Write the weight diagram of R; 2. Calculate the projected weight vector in h by Equation (4.6) for every weight vector in the weight diagram of R and get the projected weight diagram; 3. Find irrep(s) in h corresponding to the projected weight diagram.
As for how to obtain the projection matrix P g⊃h , the recipe depends on which class of subalgebra of g does h belong to. If h is a maximal subalgebra of g, one can calculate the projection matrix by a "reverse" process.
1. Find one branching rule of g ⊃ h and write down the weight diagrams of the reps of g and h; 2. Find an appropriate correspondence between weight vectors; 3. Calculate the matrix elements by Equation (4.6).
Since so (11) is a maximal special subalgebra of su(32), we can find the projection matrix from the weight systems of the irreducible representation {32} in su(32) and so (11), as we know one su(32) ⊃ so(11) branching rule or more clearly in Dynkin labels, The definition of the projection matrix for the branching rules of su(32) ⊃ so (11) is (v so(11) ) T = P su(32)⊃so(11) (w su(32) ) T . (4.9) The matrix P su(32)⊃so (11) is a 5×31 matrix, as the ranks of so (11) and su(32) are 5 and 31 respectively.
The weight system of {32} = (1000000000000000000000000000000) in su(32) has 32 weights given by 31 digits  If we put together all the weights in the weight system of {32} in su(32) and so(11) into two matrices, where the superscript indices i = 1, . . . , 32 of w su(32) i and v so (11) i label the weights in the {32} weight system, then V so(11) T = P su(32)⊃so(11) W su(32) T . (4.13) If a matrix A m×n has rank m (m ≤ n), then it has right inverse B n×m such that AB = I m×m . Now the matrix (W su(32) T ) 31×32 has rank 31. Hence, there exists a right inverse (W su(32) T ) −1 32×31 , so we can invert the formula P su(32)⊃so(11) = V so (11) T W su(32) T −1 , (4.14) and find the explicit projection matrix to be

Methodology 2: Plethysms
Generally, branching rules of su(m) ⊃ so(n) where so(n) is the maximal special subalgebra of su(m) are equivalent to symmetrized tensor powers of the generating irrep with respect to the partition, which is called Plethysm [25,26,27]. For example, for su(10) ⊃ so (10), the generating irrep is the defining representation of so(10) (10000), and for su(16) ⊃ so (10) Plethysm is basically based on the manipulation of the character polynomial. Consider a Lie group G with rank n and a representation of it R. For any group element g ∈ G, the representation D R (g) is a square matrix and the character χ R (g) is defined as the trace of the matrix D R (g). Suppose the Lie group G has m generators T 1 , T 2 , . . . , T m and the first n generators form the Cartan subalgebra. Then the group element g can be expressed as and diagonalized as Since the character is invariant under the conjugacy class, where D R (T 1 ) to D R (T n ) are diagonal matrices. Moreover their diagnoal entries form the weight vectors w i (i = 1, 2, . . . , dim(R)) of representation R: the k-th entry of w i is the i-th diagonal entry of D R (T k ). Therefore where v g = (a 1 , . . . , a n ). Then actually we can treat e ivg as X and rewrite the character as the character polynomial The character polynomials of symmetrized tensor power with respect to the partition λ of R can be obtained by the character polynomials of R. General algorithm can be found in [25]. Here we only list the algorithm for completely antisymmetric and completely symmetric cases [26,27].
• Completely antisymmetric: the character polynomials of k-th completely antisymmetric tensor power of R is the summation of all products of k distinct monomials; • Completely symmetric: the character polynomials of k-th completely antisymmetric tensor power of R is the summation of all products of k monomials.
For example, the character polynomial of antisymmetric square of R is X w 1 +w 2 + X w 1 +w 3 + · · · + X w 1 +w dim(R) + X w 2 +w 3 + X w 2 +w 4 + · · · + X w 2 +w dim(R) + · · · + X w dim(R)−1 +w dim(R) . (4.21) The character polynomial of symmetric square of R is One can quickly check the dimension. The dimension of k-th completely antisymmetric tensor power of R is dim(R)! k![dim(R)−k]! , which is also the number of monomials occur in its character polynomial. The dimension of k-th completely symmetric tensor power of R is which is also the number of monomials occur in its character polynomial.
Thus, based on the character polynomial approach, one can obtain the whole weight system of k-th completely (anti)symmetric tensor power of R directly from the weight system of R, which is much more efficient than the projection matrix approach.

Component Decomposition Results
By using the projection matrix and the Plethysm function with the Susyno Mathematica application [26], we obtain the explicit Lorentz decomposition results of the 11D, N = 1 scalar superfield as follows. The decomposition results can also be expressed in terms of Dynkin Labels, which are listed in Appendix F.         This is also consistent with the existence of the spinor metric C αβ and C αβ . Consider a field with a upstairs spinor index χ α and assign it with the irrep {32}. We can lower the spinor index by χ β = χ α C αβ , (4.23) and the irrep corresponding to χ β is also {32}. That means in 11D, N = 1 case, the position of the spinor index doesn't matter in the context of representation theory. Since Type IIA theory can be obtained by the projection from 11D, N = 1 theory, we can reproduce the scalar superfield decomposition results in 10D, N = 2A superspace, which was listed in Chapter six of [7] by projecting 11D, N = 1 component decomposition results into 10D. In one specified level, we restrict each irreps of so(11) into so(10) and consequently obtain a direct sum of several irreps of so (10). The projection matrix of so(11) ⊃ so(10) is (4.24)

11D, N = 1 Breitenlohner Approach
In 11D, N = 1 superspace, the graviton has (11 × 12)/2 = 66 degrees of freedom and can be split into the conformal part and non-conformal part Similarly, the gravitino has 11 × 32 = 352 degrees of freedom and split as where the non-conformal "spin-1 2 part" of the gravitino is defined as ψ β ≡ (γ a ) βγψa γ . A final interesting note to make concerns the three-form gauge field b abc which is known to occur in the on-shell eleven dimensional supergravity theory. Since this bosonic gauge field is a form, it is already obvious that it is irreducible and it follows as far as representation goes b abc = {165}. This tells us the scalar superfield gives one possible embedding for the graviton, two possible embeddings for the gravitino, along with a number of auxiliary fields. It's possible that the scalar superfield itself plays the roles both as prepotential and conformal compensator. This is not a new phenomenon. In 4D, N = 1 supergravity among a number of off-shell distinct formulations, there exists one where the vector superfield H a provides the superconformal supermultiplet as well as the compensator supermultiplet. Momentarily in our discussion, let us depart the domain of the 11D theory to discuss this particular 4D, N = 1 supergravity theory.
Among the many forms of irreducible off-shell 4D, N = 1 supergravity, there is the one first described in the work [29]. This form of the off-shell theory possesses one prepotential: the conformal prepotential H a and all the component fields of the theory reside in it. The component fields associated with this form of supergravity include the 4D graviton, gravitino, the axial vector auxiliary field, and two auxiliary 3-forms, Although often overlooked, this is one of the original off-shell formulations (the Stelle-West formulation) known in the literature [30]. In this limit the linearized frame superfields take the form as   (4.32) and this representation can be "tensored" with the {10} and {4} representations of the SO(1,3) algebra. Doing this we find so that the final line of (4.31) yields  (4.31). As in the 4D theory, the graviton occurs at second order in the θ-expansion of H a , two spinorial derivatives occur in (4.31). For the proposed 11D theory, the graviton occurs at sixteenth order in the θ-expansion of V, hence sixteen spinorial derivatives occur in (4.35).
We know that the conformal graviton occurs in the scalar superfield. That led us to wonder how frequent it appears in other superfields. We have performed a computer-based search involving tensoring the scalar superfield up to the irrep dimension 260,338. The appearance of the conformal graviton represenation in the case of the 11D, N = 1 theory is very different than the behavior seen in the case of the 10D, N = 1 theory. In the latter case, the conformal graviton representation only occurs in some particular cases of tensoring between either bosonic or spinorial irreps of SO (1,9) and the scalar superfield. In the former case, the conformal graviton represenation appears in every case where either a bosonic or a spinorial irrep of SO(1,10) is tensored with the 11D, N = 1 scalar superfield up to the case of the {255, 255} irrep. Then the next irrep {260, 338} does not contain conformal graviton at any level.
However, if we demand that the graviton {65} must occur at the middle level  only, the gravitino {320} must appear at the next level , and the 3-form {165} must appear at the same level as the graviton , the number of superfields that satisfy all these conditions drops drastically, from 91 to 4. They are listed in Table 3, where the numbers of graviton(s), gravitino(s) and 3-form(s) that occur at

11D, N = 1 Adinkra Diagram
In [7], we have developed ten dimensional adinkra diagrams for the first time. In this chapter, we will apply the same technique to define the 11D, N = 1 adinkra diagram.
Let us first list the number of independent component fields at each level up to Level-16. Count   0  1  1  1  2  3  3  3  4  8  5  9  6  19  7  23  8  49  9  55  10  99  11  106  12  173  13  171  14  247  15  225  16 296 Table 4: Number of Independent Fields at Each Level As usual, beyond the middle level in a superfield (and thus its adinkra), the number of fields at Level-n when 17 ≤ n ≤ 32 is equal to the number Level-(32 − n) since 32 is the top level of the expansion. We thus find 1,198 bosonic fields in the even levels 0-14 together with the even levels 18-32, and 296 at the middle level. So the total number of bosonic fields in the 11D, N = 1 scalar superfield is 1,494 fields. There are 1,186 fermionic fields in the odd levels 1-15 together with the odd levels 17-31. The equality in the number of degrees of freedom is accomplished by, on average, having fermions appear in representations that are larger than that of the bosons. So the total number of fields in the 11D N = 1 scalar superfield is 1,494 bosonic fields and 1,186 fermionic fields. Now we come to the adinkra itself.

Level # Component Field
Based on the component decomposition results shown in Sec. 4.3, we can explicitly demonstrate the 11D, N = 1 adinkra by the same process as we described in [7]: use open nodes to denote bosonic component fields and put their corresponding irreps in the center. For fermionic component fields, use closed nodes. The number of level represents the height assignment. Black edges connect nodes in the adjacent levels, meaning SUSY transformations. While in principle it is possible to draw the adinkra exactly showing all 1,494 bosonic nodes and all 1,186 fermionic nodes, for reason of practicality we will only draw it up to the quintic level.
The Adinkra diagram for 11D, N = 1 up to level-5 can be represented using dimensions in Figure 5.1 or Dynkin labels in Figure 5.2.

Conclusion
As seen from Table 3, the 11D, N = 1 scalar superfield is the simplest bosonic superfield that contains all the on-shell states of eleven dimensional supergravity and has the unique attribute of containing a single candidate for the graviton. This raises delightful possibilitities that we cast into the form of conjectures.

Conjecture # 1:
Let V denote the scalar superfield in a Lorentz superspace of signature SO (1, 10), the facts that at the middle level of its adinkra both the conformal graviton and gauge 3-form (as well the conformal gravitino at one higher level) show up, imply V is a superfield limit of M-Theory, with V being a supergravity prepotential superfield or possibly a semiprepotential superfield.
To our knowledge, there exists no previous suggestions of these possibilities.
However, our calculations, discussions, and explorations also point to something else.
In the section entitled "Traditional Path to Superfield Component Decompositions," we showed explicitly at low orders in the θ-expansion the practical difficulty of using the conventional θexpansion to access the component field content of high dimensional superfields. This suggests the possible value of searching for expansions over quantities other than the Grassmann θ-coordinates of the Salam-Strathdee superspace.
As discussed extensively in Appendix E the approach of introducing two classes of Young Tableaux, one for bosonic representations and one for fermionic representations, is quite useful in both conceptual and calculational efficiency. As Young Tableaux have a well understood definition of multiplication, one can build upon this fact. Our bosonic Young Tableaux correspond to a set of tableaux that obey the usual multiplication rules of such objects. On the other hand, only the totally antisymmetric products of the spinorial Young Tableaux are considered, i.e. single column Tableaux. Furthermore, as shown in this appendix, as single column fermionic Tableaux are also equivalent to a sum of bosonic Young Tableaux, they yield a more efficient method for representing supermultiplets consist of replacing θ-expansions by products of elements taken from the two classes of Young Tableaux.
Thus, we are in position to define "adinkra fields" as an alternative to superfields. These would take the form of conventional superfields, but with the differences that the θ variables raised to all possible powers would be replaced by products of the fermionic Tableaux and the Dynkin Labels play the role of the component fields. The Dynkin Labels implicitly carry the indices on the component field variable coefficients which saturate the indices represented by the boxes of the Tableaux. This is a topic for future study.
"True ignorance is not the absence of knowledge, but the refusal to acquire it."

Added Note In Proof
Recall the decomposition of the inverse frame and gravitino fields in 11D, yields where h (ab) is the conformal graviton, h is the trace, and h [ab] is the two-form; and where ψ a α is the conformal gravitino and ψ β ≡ (γ a ) αβψa α is the γ-trace. Since on-shell 11D supergravity also contains the three-form with d.o.f. {165}, the prepotential superfield must contain irreps must appear at Level-18. Table 5 summarizes the occurrences of these important component fields we care about. Note that Ψ α satisfies the criterion and we conject that V is a supergravity semi-prepotential superfield and Ψ α is a supergravity prepotential superfield.

A SO(11) Irreducible Representations
In this appendix, we list some of the SO(11) irreducible representations by Dynkin labels and dimensions and thus give a dictionary between the two methods for describing irreps [24].  Sometimes, multiplication of our general 11D γ matrices can yield multiple equivalent expressions. The following cases of this phenomenon are relevant to know about in the discussion of irreducible monomials.

C Additional Useful Identities for 11D Gamma Matrices
Over and above previous results [8], the list of identities below are useful for any reader who wishes to reproduce the results given in Chapter 3 particularly with regards to the discussion on deriving irreducible θ monomials.
First, we introduce the spinorial Young Tableau as an extension of the normal Young Tableau which is a useful tool in group theory. In order to distinguish the bosonic Young tableaux and spinorial Young tableaux, we apply different colors to the boxes: Young Tableaux with blue boxes are bosonic and the ones with red boxes are spinorial. Namely, when calculating the dimension of a representation associated with any Young Tableau, we put "11" into the box at the uppermost left corner of the tableau if it is bosonic and "32" if it is spinorial in 11D. We also color the irreps: blue if it's bosonic and red if it's spinorial.
At every level of the θ−expansion, the d.o.f. (degree of freedom) of each component field or θ−monomial corresponds to one irreducible representation of so (11). The zeroth level is {1} which is the trivial representation of so (11). The first level is also trivially irreducible, since {32} is already an irreducible representation corresponding to the spinor representation. However, in the higher levels, the story is a nontrivial one. In the following subsections, we will present the step-by-step calculations in quadratic, cubic, and quartic level. We will also show the results of the quintic level. This method will only give us unique solutions up to the quintic level.

E.1 Quadratic Level
Starting with the quadratic level first, We can still use Young Tableaux to denote reducible representations of so (11). The rules of tensor product of two Young tableaux are still valid. Thus, we have where the entries in are completely anti-symmetric spinor indices and the entries in are completely symmetric spinor indices. Therefore the dimensions of these two reducible representations are 496 and 528 respectively. Moreover, and are all Young tableaux that contain two boxes. By using the Mathematica application LieART (Lie Algebras and Representation Theory) provided by [28], the following result for the tensor product decomposition in SO(11) is seen: Now we know what are the decompositions of and : since there is only one way to pick numbers that add up in the r.h.s. of equation (E.2) such that their sum is 496 (or 528). We have a Python code to do this type of searching. The Python code is attached in the end of this appendix E.5 and a brief instruction is also included.
This lowest order example shows us something of interest. If we were to impose as a definition the rule that is equivalent to the Grassmann coordinate θ α , such that only the totally antisymmetric product is meaningful, then we would immediate retain only the single column Tableau and its {1}, {165}, and {330} representations.
Note that in Sec. 3 we discuss the quadratic level from the analytical aspect and all possible quadratic θ-monomials are {1}, {165}, and {330}, which are consistent with equation (E.3).

E.2 Cubic Level
Using similar logic as in the quadratic level, we construct the following tensor product decomposition first: From this process we can obtain some important pieces of information: Since we know 32 × 31 × 30/3! = 4, 960, we can use the program discussed in Appendix E.5 with two assumptions and obtain the unique solution for the decomposition of the completely antisymmetric part: (E.8) The two assumptions are: (a.) assume the cubic level must include the linear level, i.e. {32} must show up in the solution, (b.) assume each irrep only appears once. Note in Sec. 3 we discuss the cubic level from analytical aspect and all possible cubic θ-monomials are {32}, {1, 408}, and {3, 520}, which are consistent with equation (E.8).
Next we can solve the linear equations (E.6), (E.7) and obtain the decompositions of all possible spinorial Young Tableaux (SYT) at the cubic level as following: and of course, this is without applying the rule that only the single column SYT contributes at the end. It is often convenient to not impose this condition until the end of a calculation at a given level. If one checks the dimension, it will be found the dimensions calculated by YT rules in the l.h.s of the equation (E.9) are exactly the sums of numbers on the r.h.s.. Note that, the following equations are the only two independent equations we can find that all SYT in the r.h.s. have three boxes. However, there are three kinds of SYT containing three boxes. So we have two equations with three undetermined variables. If we want to know all of the irreducible decompositions of these three SYT, we have to introduce extra information. This situation is general, as it will be shown that at the quartic level we find the same problem.

E.3 Quartic Level
As before, we construct the following tensor product decomposition first: Then, our task is to find which solution is correct, and we can apply the SYT analysis: Since we understand the quadratic and cubic levels very well, we can derive the following set of linear equations. (E.46) We will close this appendix here. However, there is one matter for future study that is raised by the results presented here. In equations (3.2) -(3.7), there was given a step-by-step recursive argument given for expanding a superfield. It is possible that adapting that θ-coordinate based argument into the language of Young Tableaux might provide extra information in the context of the handicraft approach. In this section, the original script for the program we used to do the searching is attached. The function of this program is explained as following.
First we have a set of numbers called "candidates" in the code. For example, in the code attached below, this set of numbers is {1, 11, 55, 165, 330, 462}, which contains the numbers showing up in the r.h.s. of equation (E.2). Then we have a target sum, and in this case our target sum is 496. This program basically solves the equation