Superfield Component Decompositions and the Scan for Prepotential Supermultiplets in 10D Superspaces

The first complete and explicit SO(1,9) Lorentz descriptions of all component fields contained in $\mathcal{N} = 1$, $\mathcal{N} = 2$A, and $\mathcal{N} = 2$B unconstrained scalar 10D superfields are presented. These are made possible by the discovery of the relation of the superfield component expansion as a consequence of the branching rules of irreducible representations in one ordinary Lie algebra into one of its Lie subalgebras. Adinkra graphs for ten dimensional superspaces are defined for the first time, whose nodes depict spin bundle representations of SO(1,9). An analog of Breitenlohner's approach is implemented to scan for superfields that contain graviton(s) and gravitino(s), which are the candidates for the prepotential superfields of 10D off-shell supergravity theories and separately Yang-Mills theories are similarly treated.


Introduction
On the basis of superspace geometry, recently a study [1] of the problem of describing scalar gravitation in the context of eleven and ten dimensional superspaces at the linearized level was completed. The results of this effort showed with regards to supergeometical concepts there were no significant differences (nor more importantly obstructions) between this problem in the high dimensional superspaces than in a four dimensional superspace. Though one important distinction was noted... there is a long standing lack of a theory in the area of scalar superfield representations of irreducibility. Related to this is the problem of defining a superconformal multiplet in these regimes. We were thus re-energized to look at these problems.
One of the longest unsolved problems in the study of supersymmetry is the fact that an irreducible off-shell formulation containing a finite number of component fields for the ten dimensional (and eleven dimensional) supergravity multiplet has not been presented. This statement accurately describes the current state-of-the-art of the field well after thirty years since it began. Perhaps even more distressing is that even a detailed component level presentation of a reducible off-shell formulation explicitly showing a finite number of component fields for such theories also remains in the realm of the unknown. It is the purpose of this work to report on progress on the second of these puzzles.
In this work we have developed techniques, both algorithmic and analytical, that allow the first complete and explicit SO (1,9) Lorentz descriptions of all component fields contained in 10D, N = 1, N = 2A, and N = 2B unconstrained scalar superfields. They form the maximal reducible supermultiplets.
Next, we will present the maximal reducible representations of supergravity in 10D related to unconstrained scalar superfields by "tensoring" them with different representations of the SO (1,9) Lorentz group and search for graviton(s) and gravitino(s). To achieve this goal, we borrow Breitenlohner's approach [2], which gave the first off-shell 4D, N = 1 supergravity description by using the known off-shell structure of the 4D, N = 1 vector supermultiplet (v a , λ β , d). Since the structure of off-shell supermultiplets in ten dimensions is poorly understood, we implement the approach of Breitenlohner by use of a 10D scalar superfield in place of the vector supermultiplet. This will ensure an off-shell supersymmetry realization.
For the first of these keys, we have built upon efforts that began in the work of [17] wherein the first two images shown in Figure 1.1 were presented. More recently Faux [18] introduced the third image in Figure 1.1. All the supermultiplets depicted are "off-shell," i.e. the closure of the supersymmetry algebra does not require the use of the equation of motion for any component field.
The graphs from [17] are among the first in the literature where the individual nodes of one dimensional adinkras are "aggregated" together into structures that carry non-trivial representations of the Lorentz group. Furthermore, the links in this class of diagrams are also "aggregations" of the links that appear in the one dimensional adinkras. Both of these attributes pointed to the pos- The important principle established by the explicit demonstration of these graphs is that it is a priori possible to construct adinkras in greater than one dimensional contexts. This is not, however, an "organic" constructive process. Namely, these structures were obtained by first completing a component-level analytical construction of the supermultiplet. An "organic" process would dispense with any component-level lead-in. Instead on the basis of a set of principles and tools, there should be a way to construct such adinkras in higher dimensions. This is the primary goal of this current work.
Also on the analytical side, our efforts will build upon a class of works [19,20,21] wherein the concept of the "Tableaux calculus" was introduced by Howe, Stelle, and Townsend. To our knowledge, these were the first works that explicitly discussed the use of Young Tableaux as applied to problems in the representation theory of spacetime supersymmetry.
However, the supermultiplets illustrated in Figure 1.1 are constrained, i.e. subject to supercovariant derivative differential equations. This is also the case of the works by Howe, Stelle, and Townsend. In particular, these authors proposed to use Young Tableaux that are associated with the superspace covariant derivative operators acting on constrained supermultiplets. The concept of associating the superspace covariant derivative operators with Tableaux images will be important in our work. In our exploration, we also find it useful to utilize the more conventional class of Tableaux that are associated with the Lorentz indices carried by fields. We will call the former of these Young Tableaux "fermionic Young Tableaux" (FYT) while distinguishing the latter by the term "bosonic Young Tableaux" (BYT). An important distinction of our work is the supermultiplets are unconstrained.
Our work is also a beneficiary of a report undertaken by N. Yamatsu [22] who presented results for grand unified model-building based on finite-dimensional Lie algebras. In this work, the reader can find results for projection matrices and branching rules between Lie algebras and their subalgebras up to high ranks, as well as Dynkin labels and Weyl dimensional formulae of irreducible representations, and much more. The results we have explicitly called out in the last sentence have proven to be particularly useful when applied to the SO (1,9) representation theory we consider.
The final enabling tool for our work is based on algorithmic efficiency and here the work by Feger and Kephart [23] played an important role. These authors developed and presented a Mathematica application -Lie Algebras and Representation Theory (LieART) which carries out computations typically occurring in the context of Lie algebras and representation theory. This provides a robust algorithm that enables the study of weight systems. One of its features that enhances its efficiency is the use of Dynkin labels for irreducible representations. It is thus ideal for specializing to SO (1,9) representation theory.
We organize this current paper in the manner described below. In Chapter two, we discuss the numbers of independent bosonic and fermionic components in a scalar superfield. Chapter three is a transitional one where we review the θ−expansion of a real scalar superfield in 4D, N = 1 theory and introduce the higher dimensional adinkra technology. In Chapter four, we present the decomposition result of the scalar superfield in 10D, N = 1 theory. Two different approaches that lead to the same result are presented. One we call the "handicraft approach, 4 " where we introduce Bosonic and Fermionic Young Tableaux. The other is the application of branching rules, the restrictions of representations from a Lie algebra g to one of its subalgebra h. The complete ten dimensional N = 1 adinkra is drawn. In Chapter five, we start from the well-studied off-shell description of 4D, N = 1 supergravity established by Breitenlohner and show how to apply his idea to carry out constructions of candidates for the prepotential and Yang-Mills supermultiplets in 10D, N = 1 theory. In Chapter six, we present the methodology and results of constructing the 10D, N = 2A scalar superfield from the 10D, N = 1 scalar superfield, as well as the discussion of the search of prepotential supermultiplet in Type IIA superspace. The ten dimensional IIA adinkra diagram is shown at low orders. However, its complete structure is given in the form of a list of the component field representations it contains. These same steps are repeated in chapter seven that gives the methodology and explicit decomposition results as well as the discussion of constructing prepotential supermultiplet in Type IIB superspace. The ten dimensional IIB adinkra diagram is also shown at low orders.
We follow the presentation of our work with conclusions, three appendices, and the bibliography. The first appendix gives detailed discussions about chiral and vector supermultiplets obtained from 4D, N = 1 unconstrained scalar superfield. The second appendix contains tables of SO (1,9) representations drawn from the work of Yamatsu. The third appendix presents the results of "tensoring" low order bosonic representations of SO (1,9) with the basic unconstrained 10D, N = 1 scalar superfield. Finally, the fourth appendix presents the results of "tensoring" low order fermionic representations of SO (1,9) with the basic unconstrained 10D, N = 1 scalar superfield.

Superfield Diophantine Considerations
Via the use of simple toroidal compactification, one can count the numbers of independent bosonic and fermionic component fields that occur in a scalar superfield. As a fermionic coordinate cannot be squared, this means that in the Grassmann coordinate expansion of a superfield, any one specified fermionic coordinate can only occur to the zeroth power or the first power. As component fields occur as coefficients of monomials in superspace Grassmann coordinates, counting the latter is the same as the former... as long as the superfield is not subject to any spinorial "supercovariant derivative" constraint.
Each higher dimensional superspace with D bosonic dimensions, for purposes of counting is equivalent to some value of d, where d is the number of independent equivalent real one-dimensional fermionic coordinates on which the superfield depends. The total number of independent monomials is given by 2 d . Next, to count the total number of bosonic components n B in the scalar superfield, we simply divide by a factor of two. This same argument applies to the total number of fermionic components n F in the scalar superfield. While superfields easily provide a methodology for finding collections of component fields that are representations of spacetime supersymmetry, one thing that superfields do not yield so easily is a theory of the constraints that provides irreducible representations. For any constrained superfield, it must be the case that the number of component fields does not depend solely on the parameter d. This is most certainly true for the maximally constrained and therefore minimal irreducible representations. This is illustrated by comparing the final two columns of the first three rows in Tables 1 and 2

4D Scalar Superfield Decomposition
The off-shell four dimensional vector supermultiplet is well understood now to be a part of the unconstrained superfield as given below. In 4D, N = 1 real superspace, the spinor index (the Greek index) on θ α runs from 1 to 4, and we have 8 bosons and 8 fermions, as counted in Table 1. The θ−expansion of a scalar superfield is , and N (x a ) are bosonic component fields with 8 d.o.f. 5 in total; ψ α (x a ) and χ δ (x a ) are fermionic component fields with 8 d.o.f. in total. We will discuss this θ−expansion result from different perspectives.

Group Theory Perspective
From the perspective of group theory, we can translate the scalar superfield decomposition problem to the irreducible decomposition problem of representations in so (4). First, we can write the general expression of the θ−expansion of a superfield

(3.2)
Due to the antisymmetric property of the Grassmann coordinates θ α , the quantities θ α , θ α θ β , θ α θ β θ γ , and θ α θ β θ γ θ δ have 4, 6, 4, and 1 degrees of freedom, respectively. They can be interpreted as representations of so(4) with 4, 6, 4, and 1 dimensions. We use level-n to denote the θ−monomial with n θs. The problem is reduced to do the irreducible decompositions of these representations and the results can be listed as Note that level-4 and 3 are conjugate to level-0 and 1, respectively, while level-2 is self-conjugate. In order to distinguish between bosonic irreps and fermionic irreps, we color their dimensions: blue if bosonic and red if fermionic. In the rest of the paper, we will use these conventions.
Recall that in 4D, N = 1 real superspace, we can create the covariant gamma matrices which are 4 × 4 real matrices. The basis of the space of matrices over these spinors is summarized in Table 3.

Graph Theory Perspective: Adinkra
From the perspective of graph theory, particularly adinkra diagrams, we can define a four dimensional adinkra based on Equation (3.3). First of all, the adinkra diagram carries information about component fields rather than θ−monomials, so we need to translate Equation (3.3) to field variable language. Consider a variable with one upstairs spinor index χ α and assign irrep {4} to this field. What is the irrep corresponding to the variable with one downstairs spinor index χ β ? Since where C αβ is the spinor metric, the irrep of χ β is still {4}. Generally speaking, in 4D, N = 1 real notation, the irreps corresponding to component fields are the same as their θ−monomials.
Then we can use open nodes to denote bosonic component fields and put the dimensions of their corresponding irreps in the centers of the open nodes. For fermionic component fields, we use closed nodes with a similar convention for dimensionality. The level number represents the height assignment and it increases with height. Black edges connect nodes in adjacent levels and represent supersymmetric transformation operations on the component fields. We can also interpret the adinkra using the idea in [27], as illustrated in the following table.
Level Adinkra nodes Component fields Irrep(s) in so(4)  Figure 3.1. The graph shows the "f " bosonic component field at the lowest level, the "ψ" fermionic field at level one, the "g," "h," and "v b " bosonic component fields at level two, the "χ" fermionic field at level three, and finally the "N " bosonic component field at level four. By comparison, we see that starting from the 1D, N = 4 adinkra, if we aggregate a set of bosons or fermions into a single node, then a set of corresponding links will be merged (we use black links to replace them), and thus emerges the 4D, N = 1 adinkra. Of course, to reach this goal we must decide to enforce some rules. Four and only four black nodes are merged at the first and third levels. At the second level, the only merging choices are either one or four nodes as permitted in an "aggregated" node. From this procedure, although we have a large number of different 1D, N = 4 adinkras, they all collapse into the same 4D, N = 1 version. It is useful to also recall that the aggregation of nodes leads to "dimensional enhancement" [24,25,26] that allows the adinkra nodes to carry representations of the four dimensional Lorentz group.  It is useful to know one of the nodes at the lowest level of the bottom right graph in Figure 3.3 (i.e. chiral supermultiplet) must be regarded as the spacetime integral of a spin-0 field. The origin of this field was from the initial starting point as being one of the aggregated part of the "4" at the middle level of the V adinkra. The bottom left graph in Figure 3.3 has nodes associated with the component fields of the vector supermultiplet where the gauge field is restricted to the Coulomb gauge. Refer to Appendix A for details.
In 4D, there is a comprehensive understanding of how to start with a reducible representation such as the real scalar superfield V and "break" it apart into its irreducible components. However, the extension of such a procedure is totally unknown for the cases of 10D superfields.
For a general representation of spacetime supersymmetry, there is currently no understanding of how to carry out the process which has been accomplished in the realm of the 4D, N = 1 supersymmetry. Resolving this class of problems in the realm of supersymmetric representation theory is a primary motivation for the adinkra approach to the study of superfields. If we borrow the language of genetics, adinkras play the role of genomes for superfields. The problem of finding the irreducible off-shell representations of all superfields is equivalent to a genomeediting problem. We are still without the mathematical analog of a Cas9 7 capable of beginning with a reducible graph, that is a viable off-shell representation of spacetime supersymmetry, and ending with off-spring irreducible graphs that also provides viable off-shell representations of spacetime supersymmetry.

10D Scalar Superfield Decomposition
In the case of the 10D, N = 1 theory, the number of independent Grassmann coordinates is 2 10/2−1 = 16 due to the Majorana-Weyl condition. Then the superspace has coordinates (x a , θ α ), where a = 0, 1, . . . , 9 and α = 1, . . . , 16. Hence, the θ-expansion of the ten dimensional scalar superfield begins at Level-0 and continues to  where Level-n corresponds to the order O(θ n ). The unconstrained real scalar superfield V contains 2 16−1 bosonic and 2 16−1 fermionic components, as counted in Table 1. Let's express the 10D, N = 1 scalar superfield as We can decompose θ-monomials θ [α 1 · · · θ αn] into a direct sum of irreducible representations of Lorentz group SO (1,9). With the antisymmetric property of Grassmann coordinates, we have All even levels are bosonic representations, while all odd levels are fermionic representations. Note that in a 16-dimensional Grassmann space, the Hodge-dual of a p-form is a (16−p)-form. Therefore, level-(16 − n) is the dual of level-n for n = 0, . . . , 8, and they have the same dimensions. By simple use of the values of the function "16 choose n," these dimensions are found to be the ones that follow,  The θ−monomials contained within the 10D scalar superfield V can be specified by the irreducible representations of so (1,9) as follows. In Appendix B, the irreps with small dimensions are listed. Note that Level-16 to Level-9 are the conjugates of Level-0 to Level-7 respectively, and Level-8 is self-conjugate. That's the meaning of the duality mentioned above. It is also clear that the dimensions given by 16! n!(16−n)! "align" with the order of the irreducible representations for the first four levels (and the corresponding dual levels) of the 10D, N = 1 scalar superfield V. When the dimension of the wedge product does not align with the dimension of an irreducible representation, it must be the case that the sum of a judiciously chosen subset of irreducible representation is equal to 16! n!(16−n)! at the n-th level. One may wonder why "bar" representations occur in 10D, N = 1 theory with the absence of dotted spinor indices. Let us turn to the {16} and {16} representations. If we define χ α as the {16} representation, then it follows that χα is the {16} representation. They both have upper spinor indices. However, it is important to keep in mind that the spinor metric for 10D takes the form C αβ and this means that it is possible to define another spinor that has a subscript index via the equation where obviously χ α has a subscript undotted spinor index. So if we define χ α with a superscript as the {16} representation, then it follows that χ α with a subscript is the {16} representation! This provides a way to understand the meaning of the "bar" representations shown in the studies of the ten dimensional superfield. This also tells us the irreps corresponding to the component fields are nothing but the conjugate of the irreps corresponding to the θ−monomials.
We can also rewrite the scalar superfield decomposition result in Equation (4.4) in Dynkin labels as follows.  In the following subsections, we will present two different approaches to obtain the result in Equation (4.4). We will also draw the 10D, N = 1 adinkra diagram as we did in the 4D, N = 1 case.
In the discussion above, it was noted that the problem of finding a judicious choice of irreducible representations of the Lorentz group appropriate for level-n of the superfield is at the crux of identifying what component fields actually appear at the level-n. In the following discussions we will show two ways to carry out this process.

Handicraft Approach: Fermionic Young Tableaux
In this section, we will utilize the fermionic Young Tableau (FYT) and its application to obtaining the irreducible Lorentz decompositions of the component fields appearing in the θ-expansion of the ten dimensional scalar superfield 8 . Since in superspace, there are not only spacetime coordinates but also Grassmann coordinates, we introduce the fermionic Young Tableau as an extension of the normal (bosonic) Young Tableau. In order to distinguish the bosonic Young Tableaux from the fermionic Young Tableaux, we use different colored boxes: Young Tableaux with blue boxes are bosonic and the ones with red boxes are fermionic. Namely, when calculating the dimension of a representation associated with a Young Tableau, we put "10" into the first box if it is bosonic and "16" if it is fermionic in 10D.
One can start with the quadratic level. In so(10), we can use Young Tableaux to denote reducible representations. The rules of tensor product of two Young Tableaux are still valid. Thus, we have where entries in are completely symmetric in a corresponding set of spinor indices and entries in are completely antisymmetric in these same spinor indices. Therefore, the dimensions of these two representations are 136 and 120 respectively. Moreover, and are the only Young Tableaux that contain two boxes. By using the Mathematica package LieART [23], one obtains 9 the following results about tensor product decomposition in so (10): entirely from dimensionality. This exercise also teaches the lesson that Note that the sigma matrices with five vector indices satisfy the self-dual / anti-self-dual identities (σ [5] ) αβ = 1 5! [5] [5] (σ [5] ) αβ , (σ [5] ) . α .
[5] (σ [5] ). as an image. The level-2 result that we need for the scalar superfield is {16} ∧ {16}, which is the totally antisymmetric piece . 9 Of course simply ways can be used to find these results also.
Next we go to the cubic level. The tensor product of three {16} is (4.14) Therefore we obtain To solve for , first note that where "−" means complement including duplicates if we treat each direct sum of irreps as a set of irreps. Now note that the dimension of is exactly 560. Therefore, we can solve for all the decompositions in cubic level, When we look at the quartic level, we find (4.21) By the hook dimension formula, the tensors with four spinorial indices above have dimensions 3876, 9180, 5440, 7140 and 1820 respectively. It is more useful to express these in terms of tensors with vector indices. By using the second line of Equation (4.21), from Equations (4.9) and (4.10), we However, the r.h.s. of Equation (4.21) and (4.22) are not irreducible representations! By LieART, we obtain the irreducible decomposition In subsequent work, we will develop graphical techniques to get from Equation (4.21) and (4.22) to Equation (4.23).
Similar to Equations (4.15) and (4.16), we have 4 independent equations with 5 different Fermionic Young Tableaux with 4 boxes. Note that in level-n, the number of Young Tableaux with n-boxes is the number of integer partition of n, p(n). In level-5, 6, 7 and 8, there are 7, 11, 15 and 22 different types of Young Tableaux respectively. The number of independent equations in level-n is p(n) − 1. This method becomes increasingly tedious. We won't go through the details of the subsequent levels. Although the systems of equations always seem underconstrained, in all levels the restrictions from dimensionality are able to nail down all the solutions in the 10D, N = 1 case, and give us back the scalar superfield decomposition in Equation (4.4). Thus we call this as a handicraft approach. At level-n of the 10D, N = 1 scalar superfield, we want to decompose the totally antisymmetric product {16} ∧n to so(10) irreps. We are essentially looking at all the one-column Young Tableaux with 16 filled at the first box,

(4.24)
They have dimensions 16 n , n = 0, 1, . . . , 16. One natural way is to interpret these Young Tableaux as su (16) irreps (while they do not correspond to irreps when interpreted in the so(10) context), and consider how they branch into so(10) irreps. We summarize the relevant branching rules in the following table. They give us the decomposition in Equation (4.4).  (16) to so(10) branching rules for level-0 to 8 in 10D, N = 1 scalar superfield To our knowledge, the expansion of component fields of a superfield arising from the branching rules of one ordinary Lie algebra over one of its Lie subalgebra has never been noted in any of the prior literature concerning superfields. This is one of the major discoveries we are reporting in this research paper. It provides a clean, precise, and new way to define the component fields in superfields. It is very satisfying also from the point of view of our use of the "handicraft" techniques discussed in the last section. The two methods yield the same conclusions in all levels. However, the method in this section is both labor-saving and in a sense more mathematically rigorous. In a manner similar to how we drew the 4D, N = 1 adinkra, we are now in position to explicitly demonstrate the 10D, N = 1 adinkra by the same process. The adinkra with irrep dimensionality shown in each node is drawn in Figure 4.1, while the one with the corresponding Dynkin labels appears in Figure    With this result, we have achieved two advances we believe to be significant. Considering the case of 4D, N = 1 supersymmetry, the graphs shown in Figure 4.1 and Figure  4.2 for the 10D, N = 1 scalar superfield should be considered as the equivalence to the graph shown in Figure 3.1 for the 4D, N = 1 scalar superfield. The obvious difference in "heights" was to be expected. However, a surprising feature is the maximum width of the 10D, N = 1 scalar superfield adinkra is exactly the same as that of the 4D, N = 1 scalar superfield, namely only three bosonic nodes. Another surprising feature of the 10D, N = 1 scalar superfield is its economy. It only contains 15 independent bosonic field representations and 12 independent fermionic field representations.
A great and obvious challenge here is to discover the analog of the rules that govern the splitting that was described as to how 4D, N = 1 chiral (irreducible) and 4D, N = 1 vector (irreducible) adinkras can be obtained from a 4D, N = 1 real scalar (reducible) adinkra.

The 10D, N = 1 Adinkra & Nordström SG Components
In our previous work of [1], we found that the construction of 10D, N = 1 supergravitation at the level of superfields and in terms of linearized supergeometry is very direct and without obstructions. It involved the introduction of a 10D, N = 1 scalar superfield Ψ that appears in linearized frame operators in the forms Taking the θ → 0 limit to the second of these equations implies that where is a notation used to indicate the taking of the limit. The left hand side of this equation can be defined as with e a m describing the 10D graviton (expressed as a frame field) andψ a β describing the 10D gravitino. Combining the last two equations yields In Nordström theory, only the non-conformal spin-0 part of graviton described by a scalar component field and the non-conformal spin-1 2 part of the gravitino which is described by a {16} component field show up. Both of these are appropriate for a Nordström-type theory.
Two of the results in the work of [1] take the respective forms (4.31) and Application of a spinorial derivative to From the form of the adinkras, we know that application of more than six fermionic derivatives to G αβ (or eight to Ψ) only leads to the dual representations occurring. So for example seven fermionic derivatives applied to G αβ (or nine to Ψ) must lead to the dual representations of those that occurred with the application of five fermionic derivatives (or seven to Ψ). Thus, all the component fields of 10D, N = 1 Nordström SG are obtained from (4.29) as well as applying all possible spinorial derivatives to the superspace covariant supergravity field strength G αβ followed by taking the θ → 0 limit.
Thus, the knowledge of the 10D, N = 1 scalar superfield adinkras provides an avenue to the complete component field description of the corresponding reducible Nordström supergravity theory derived from superspace geometry [1].

10D Breitenlohner Approach
In this chapter, we will apply the Breitenlohner approach to construct candidates for the 10D Type I superconformal multiplet. The principle of this approach is to attach bosonic and fermionic indices on the 10D scalar superfield, and search for the traceless graviton and the traceless gravitino.
Recall that the first off-shell description of 4D, N = 1 supergravity was actually carried out by Breitenlohner [2], who took an approach equivalent to starting with the component fields of the Wess-Zumino gauge 4D, N = 1 vector supermultiplet (v a , λ β , d) together with their familiar SUSY transformation laws . Because the vector supermultiplet is off-shell (up to WZ gauge transformations), the resulting supergravity theory is off-shell and includes a redundant set of auxiliary component fields, i. e. this is not an irreducible description of supergravity. But as seen from (5.2), the supergravity fields are all present, and together with the remaining component fields a complete superspace geometry can be constructed.
The key of this process is that if you look at Equation (3.3), there is a {4} irrep (which is a vector gauge field) in level-2. When you consider the expansion of the superfield with one vector index H a which is also called the prepotential, you will find where {9} is the traceless graviton, as the degree of freedom is 9 = 4×5 2 − 1. In 10D, N = 1 theory, we want to search for the traceless graviton h ab and the traceless gravitino ψ a β . Let us first figure out which irreducible representations do they correspond to. The graviton has two symmetric vector indices, which corresponds to 10×11 2 = 55, and can be decomposed into the traceless part and the trace,h where both {54} and {1} are SO (1,9) irreps, while {55} is not. In the following, we will denote the traceless graviton irrep to be {54}, where the green color is added just to highlight this irrep.
To determine the irreducible representations corresponding to the gravitinoψ a β , first note that it has one vector index which corresponds to {10} and one upper spinor index which corresponds to {16}. We can then decompose the gravitino into its traceless part and its σ-trace part by the tensor product of these two irreps,ψ where the non-conformal spin-1 2 σ-trace part ψ γ is defined in Equation (4.30). The traceless part ψ a β of the gravitino is conformal, and this is what we should search for in the superconformal prepotential multiplet. Now, if we hope to follow the exact same method in Equation Attaching indices onto the scalar superfield would mean where [indices] could be any combination of bosonic and/or fermionic indices, and it would correspond to a (sum of) bosonic or fermionic irreducible representation(s) in SO (1,9). If we let the level-n θ−monomial decompositions of 10D, N = 1 scalar superfield be n for n = 0, 1, . . . , 16, the level-n component field content would be n . For simplicity, consider only the [indices] that corresponds to one and only one irreducible representation {irrep}. Then the level-n component field content of V [indices] would be n ⊗ {irrep}. We denote the component content of the entire superfield by the expression V ⊗ {irrep}, which we will use throughout the following sections.
In the following two sections, we will study the expansions of some bosonic and fermionic superfields respectively to see whether we can identify possible off-shell supergravity supermultiplet(s).

Bosonic Superfields
We will start by attaching bosonic indices on the scalar superfield, which is equivalent to tensoring the corresponding bosonic irreps to the component decomposition of the scalar superfield.

Dynkin Label
Irrep Is there {54}? Of course, the result for V ⊗ {54} is not surprising and can be removed as a candidate for a conformal 10D, N = 1 supergravity prepotential. All of the other entries in the table do provide candidate prepotentials. It is noticeable that several of these candidates ( V ⊗ {210}, V ⊗ {660}, and V ⊗ {770}) allow conformal graviton embeddings at the eighth level. If Wess-Zumino gauges exist to eliminate lower order bosons, these imply "short" supergravity multiplets. Now, one can look into one of those bosonic superfields with traceless gravitons {54} in details as an example. The irreps shown here are component fields. The reader can find other bosonic superfields with traceless gravitons in Appendix C.
The superfield V ⊗ {120} can be interpreted as V abc , a three form (three totally antisymmetric Lorentz indices). It provides four possibilities for the embedding of traceless gravitons (at level-2, level-6, level-10, and level- 14). Note that if one finds a {54} in level-n, one finds a {144} at level-(n + 1). This holds for all bosonic superfields listed in Table 6. We can see this relation between graviton and gravitino clearly by considering the SUSY transformation law of the graviton in 10D N = 1 theory. We know the undotted D-operator acting on the graviton represented by h ab gives a term proportional to the undotted gravitino in the on-shell case (in the off-shell case, there are several auxiliary fields showing up in the r.h.s. besides the undotted gravitino) (5.9) Note that the undotted spinor index on the gravitino is a superscript. According to our convention, ψ b β corresponds to the irrep {144}. Recall that acting the D-operator once will add one to the level. Thus, Equation (5.9) is exactly the mathematical expression of the statement that if you find a {54} in level-n, you will find a {144} in level-(n + 1).
Based on Equation (5.8), we can see four possible embeddings for the graviton, ten possible embeddings for the gravitino, along with a number of auxiliary fields. The superfield V abc is also the simplest nontrivial bosonic superfield that contains traceless gravitons and gravitinos and can be used as the starting point to construct supergravity. In fact, as long ago as 1982, Howe, Nicolai, and Van Proeyen [28] had suggested this particular superfield might be an appropriate point from which to construct a prepotential for 10D, N = 1 supergravity.

Fermionic Superfields
Now we investigate fermionic superfields by attaching spinor indices onto the scalar superfield.
Ever since the work of [29], it has been known that the prepotential superfield description of 4D, N = 2 supergravity is described by a fundamental dynamical entity that is fermionic. As a 4D, N = 2 supersymmetrical theory can be descended from a 6D minimally supersymmetrical one, this clearly points out the possibility that higher dimensional theories can possess fermionic supergravity prepotentials.
There is a second reason why the study of fermionic superfields is suggested. Some time ago [7], efforts were undetaken to investigate the structure of 1-form 10D, N = 1 gauge theories as theories involving superspace superconnections over fiber bundles. This study reached a conclusion that all off-shell theories of this type must include a bosonic component field {126}. This observation was later confirmed by a number of subsequent studies [30,31,32,33]. This leads to a powerful restriction if the scanning process is applied to the 1-form 10D, N = 1 gauge theories. At the same level in the expansion in Grassmann coordinates, both the {10} and the {126} SO (1,9) representations must be present. In addition, there must also occur the {16} at one level higher in the expansion to accommodate an accompanying gaugino. In a subsequent section we will discuss the relevance of the features uncovered to the case of the 1-form 10D, N = 1 gauge theory. Furthermore, in order to reach the maximal level of simplication in our presentation, we will only consider the abelian case for the 1-form 10D, N = 1 gauge theory.
We will begin by only considering the feature revealed by our tensoring that are relevant to the case of 10D, N = 1 supergravity theory. The summary of the expansions for the fermionic cases is listed in Table 7. The third column shows which tensor product contributes {54} and the decomposition results show that each one only contributes one {54}. It provides three possible embeddings for traceless gravitons in total (in level-5, level-9, and level-13). Note that if you find a {54} in level-n, you can find a {144} in level-(n + 1). This holds for all fermionic superfields listed in Table 7, which is consistent with Equation (5.9).
There exists a work [7] in the literature from 1987, where the proposal for the use of a fermionic 10D, N = 1 supergravity prepotential was first given. However, the proposed prepotential was of the form V a γ which is equivalent to V ⊗ {144}.  Based on Equation (5.11), we see three possible embeddings for gravitons, fifteen possible embeddings for gravitinos, and auxiliary fields. The superfield V ab γ is also the simplest nontrivial fermionic superfield that contains traceless gravitons and gravitinos and can be used to construct supergravity.
. . . The work previously described in this chapter on fermionic superfields also opens the doorway for using the techniques developed so far in this paper to the issue of scanning the component level description of a superspace connection Γ A = (Γ α , Γ a ) used to define a 1-form U(1) superspace covariant derivatives where t denotes the U(1) generator and g is a coupling constant. The commutators of two covariant derivatives take the forms [7] [∇ α , ∇ β } = i(σ c ) αβ ∇ c + g 1 5! (σ [5] ) αβ f [5] t , (5.13) The equations below are consistent with the solutions of Bianchi identities [7] i [4] .

(5.18)
The results immediately above are the most important. They show that if the component field variable f [5] is set to zero on the RHS of these equations, then the spinorial photino field W β and the bosonic Maxwell field F bc must both satisfy their equations of motion. Stated another way, for these two component fields to be off-shell, it requires the presence of the condition f [5] = 0.
By comparing this with Equation (5.19), we read that in level-0 φ α corresponds to {16}; level-1 A c , φ [3] and f [5] correspond to {10}, {120} and {126}; level-2 Σ α [3] corresponds to {120} ⊗ {16} = 2 ⊗ {16} which is exactly the decomposition at level-2 of (5.21), where 2 is the level-2 θ−monomial of the scalar superfield. One point to note is that f [5] is {126} but not {126}. This is because it is contracted with (σ [5] ) αβ which a self-dual 5-form by Equation One can introduce two sets of 10D spinor coordinates denoted by θ α and θα so that a 10D, N = (1, 1) superfield can be expressed in the form V(x a , θ α , θα). It is possible to organize the expansion so that it takes the form αβ (x a , θ α ) + . . . , (6.1) and the point is that each of the expansion coefficients V (0) (x a , θ α ), V α (x a , θ α ), V αβ (x a , θ α ), . . . is a 10D, N = 1 superfield. More explicitly, from Equation (4.1) one can write Attaching n totally antisymmetric dotted θ's to an undotted θ−monomial corresponds to tensoring the {16} ∧n representation on it, i.e. tensoring the conjugate of the nth level of the θ−monomial of the 10D, N = 1 scalar superfield on it. Therefore, if we denote level-n θ−monomial of 10D, N = 1 scalar superfield as n , we can obtain the 10D, N = 2A scalar superfield θ−monomial decomposition by . . .  • Level-0: {1} In this section, we present the ten dimensional N = 2A adinkra diagram up to the cubic level. Based on the results listed in Section 6.2, we can draw the complete adinkra diagram in principle: 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. Due to the limited space of the paper, we only draw the adinkra up to the cubic level. The Type IIB theory means N = (2, 0), i.e. we have two sets of spinor coordinates denoted by θ α 1 and θ α 2 . Following the same logic as in Type IIA, one can organize the N = 2B scalar superfield as , θ α 1 ), . . . are 10D, N = 1 superfields. Attaching another copy of n totally antisymmetric undotted θ's to an undotted θ−monomial corresponds to tensoring the {16} ∧n representation on it, i.e. the nth level of the θ−monomial of the 10D, N = 1 superfield itself, without any conjugate. Therefore, the N = 2B scalar superfield θ−monomial decomposition can be summarized as . . . Based on Equations (4.4) and (7.2), we can directly obtain the scalar superfield decomposition in 10D, N = 2B. The results from level-0 to level-16 are listed below. Same as in the Type IIA case, we can find graviton embeddings and gravitino embeddings in the θ−expansion of the scalar superfield. We use green color to highlight the irrep {54} which corresponds to the traceless graviton in 10D as well. We can translate irreps into component fields and see 72 graviton embeddings, 280 gravitino embeddings, and accompanying auxiliary fields. The scalar superfield is the simplest superfield that contains traceless graviton and gravitino embeddings that can be used as a starting point to construct supergravity. The numbers of gravitons and gravitinos are the same as those in Type IIA case.
Note that if one find a {54} in level-n, one can find {144} in level-(n + 1), which can be interpreted by SUSY transformation laws of the graviton h ab in 10D, N = 2B theory. Since in Type IIB theory we have two copies of θ α rather than θ α and θ . α , we have D-operators D α i where i = 1, 2. Acting the D-operators on the graviton gives a term proportional to the gravitino ψ b β i in the on-shell case (in the off-shell case, there are several auxiliary fields showing up in the r.h.s. besides the undotted gravitino) Note that the spinor indices on the gravitinos are superscript. It suggests that the gravitinos correspond to irrep {144} in our convention for both i = 1 and 2. In the context of θ−monomials, the conjugate of {54} is still {54}, and the conjugate of {144} is {144}.
Here we draw the ten dimensional N = 2B adinkra diagram up to the cubic level.

Conclusion
This work gives the complete decompositions of scalar superfields to irreducible component field representations of the 10D Lorentz group and a proposal for identifying the corresponding superconformal multiplets by applying the Breitenlohner approach in N = 1, N = 2A, and N = 2B superspaces. The new results here provide a foundation for future extensions. Our efforts also mark a new beginning for the search for irreducible off-shell formulations of the 10D Yang-Mills supermultiplet derived from superfields.
We believe it is important to comment on the graphs shown in Figure 1 There is a certain tension in the path we are pursuing with the use of adinkras and superfields in comparison with the well established results in the literature. Many years ago, Nahm [34] pointed out the absence of a superconformal current above six dimensions. This most certainly suggests an obstruction may exist.
On the other hand, in works going back decades [35,36,37,38,39], there have been increasing explorations of the concepts of conformal symmetry within the context of 10D superspaces. Our scans suggest the possibilities of a number of superfields for embedding the component-level conformal gravitons into 10D superfields. This supports the idea of the eventual success of these efforts. Though we do not understand how this tension will be resolved... if it can... we would point the reader to what may prove to be a similar situation.
In the works of [40,41], the Witt algebra (i.e. the "centerless Virasoro algebra") was investigated. It was found that the form of the Witt algebra undergoes a radical change dependent on the number of supercharges under investigation. When the number of supercharges is four or less, the form of the Witt algebra is simple and uniform. However, when the number of supercharges exceeds four, the form of the Witt algebra changes dramatically with the appearance of new generators that are not present for the cases when the number of supercharges is less than four. It seems likely to us this phenomenon may provide a route by which these embeddings that are obvious from the supergeometrical side could lead to conformal supergravity theories in higher dimensions.
In a future work, we will also dive far more deeply into the relations between analytical expressions of the irreducible θ−monomials, Young Tableaux, and Dynkin labels. Along a different direction of future activities lies the extension of our results to the case of 11D, N = 1 supergravity. The results in this work regarding the case of the 10D, N = 2A supergravity theory already contain a lot of information about the eleven dimensional theory as it is equivalent upon toroidal compactification to the 10D, N = 2A supergravity theory. In principle, it is straightforward to construct the component level contents from our approach that uses branching rules. However, in practice this is considerably more computationally challenging than the equivalent work in the 10D, N = 2A supergravity theory. Currently the work on this is underway.
"Our knowledge can only be finite, while our ignorance must necessarily be infinite."

-Karl Popper
A Chiral and Vector Supermultiplets from the 4D, N = 1 Unconstrained Scalar

Superfield
In this appendix, we present an expanded discussion of the mathematical structures that support the validity of the "splitting" of the adinikras presented in Figure 3.3. The vector supermultiplet shown in the figure is well familiar, but this is not so for the chiral supermultiplet shown. We will begin with the vector supermultiplet. The notational conventions used in the following discussion are those of Superspace [42].
The vector supermultiplet contains a 1-form gauge potential A a that is part of a super 1-form with different sectors given by The vector supermultiplet field strength superfield W α is determined in terms of an unconstrained real scalar superfield V by The definition of W α is invariant under gauge transformations with a chiral parameter Λ that takes the form The components of the prepotential superfield V can be defined by the projection method All the components of V can be gauged away by nonderivative gauge transformations except for α , and D , where the vector and spinor are the physical components and D is auxiliary component. We find the action to be However, a superfield with the same structure as V can also be used to describe an entirely different supermultiplet. To distinguish this second supermultiplet from the first, we will use the symbol V to denote its prepotential. This second supermultiplet contains a 3-form gauge potential A abc that is part of a super 3-form with different sectors given by This super 3-form theory has been discussed previously in the works of [42,43] and more recently in [44] in terms of component fields.
The chiral supermultiplet that is contained in a real unconstrained scalar superfield has a gauge invariant superfield field strength of the form where the prepotential V has gauge transformations The physical component fields of this gauge 3-form multiplet are The quantity f is the field strength of the component gauge 3-form A abc , so that the field strength f is invariant. Interpreted the other way, the chiral supermultiplet is that with component con- α , h, A abc ). This is done by taking one of the auxiliary fields f to its Hodge-dual gauge 3-form A abc as shown in the last line of (A.9).
The action for this supermultiplet is given by

B SO(10) Irreducible Representations
Here we list the SO(10) irreducible representations by Dynkin labels and dimensions [22].

C Bosonic Superfield Decompositions
In this appendix, we list a few bosonic superfields that contain the traceless graviton {54} h ab and the traceless gravitino {144} ψ a γ and {144} ψ aγ . Here every irrep corresponds to a component field.

D Fermionic Superfield Decompositions
In this appendix, we list a few fermionic superfields that contain the traceless graviton {54} h ab and the traceless gravitino {144} ψ a γ and {144} ψ aγ . Here every irrep corresponds to a component field.