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25. Hu2019.Pdf Superfield Component Decompositions and the Scan for Prepotential Supermultiplets in High Dimensional Superspaces Yangrui Hu Dec 16, Miami 2019 Conference Based on the work with S.J. Gates, Jr. and S.-N. Hazel Mak [arXiv: 1911.00807], [arXiv: 1912.xxxx], and [arXiv: 1912.xxxx] Motivation An irreducible (even a reducible) off-shell formulation containing a finite number of component fields for the ten and eleven dimensional supergravity multiplet has not been presented. Our purpose: reducible off-shell formulation Scalar superfield: compensator Prepotential candidates [1] S.J. Gates, Jr., Y. Hu,, H. Jiang, and , S.-N. Hazel Mak, A codex on linearized Nordström supergravity in eleven and ten dimensional superspaces, JHEP 1907 (2019) 063, DOI: 10.1007/ JHEP07(2019)063 2 Linearized Nordström SUGRA: In Nordström theory, only non-conformal spin-0 part of graviton and non-conformal spin-1/2 part of gravitino show up All component fields of Nordström SG are obtained from spin-0 graviton, spin-1/2 gravitino, and all possible spinorial derivatives to the field strength Gαβ [1] S.J. Gates, Jr., Y. Hu,, H. Jiang, and , S.-N. Hazel Mak, A codex on linearized Nordström supergravity in eleven and ten dimensional superspaces, JHEP 1907 (2019) 063, DOI: 10.1007/ JHEP07(2019)063 3 Scalar Superfield Decomposition in 10D, � = 1 Breitenlohner 10D, � = 2A 10D, � = 2B Approach [2] 9D to 5D minimal 11D, � = 1 scalar superfield decomp prepotential 10D YM candidates supermultiplet Young Tableau Approach Branching Rule Approach [2] P. Breitenlohner, “A Geometric Interpretation of Local Supersymmetry,” Phys. Lett. 67B (1977) 49, DOI: 10.1016/0370-2693(77)90802-4. 4 Constructions of Superspaces D spacetime dimensional superspace: (xa , θα), where a = 0,1,2,…, D − 1 and α = 1,2,…, d. d is the number of real components of the spinors. Recall: one specified Grassmann coordinate cannot be squared→ can only occur to the zeroth power or the first power Number of independent components in unconstrained scalar superfields is d, where d−1 2 nB = nF = 2 5 4D, � = 1 Scalar Superfield Decomposition General θ−expansion of a scalar superfield in 4D, � = 1 superspace: {1} {4} {6} {4} {1} Goal: decompose the θ−monomials Convention of the color of the irrep: blue if bosonic, red if spinorial 6 4D, � = 1 Scalar Superfield Decomposition The basis of the vector space spanned by spinors is summarized in the Table below: gamma matrices are 4 × 4 Irreducible θ−monimials: 7 4D, � = 1 Adinkra Definition [3] C. F. Doran, M. G. Faux, S. J. Gates, Jr., T. Hubsch, K. M. Iga, G. D. Landweber, “On the Matter of N=2 Matter”, Phys.Lett.659B: 441-446,2008, DOI: 10.1016/j.physletb.2007.11.001. [4] M. Faux and S. J. Gates, Jr., “Adinkras: A Graphical technology for supersymmetric representation theory,” Phys. Rev. D 71, 065002 (2005), doi:10.1103/PhysRevD.71.065002 The use of symbols to connote ideas which defy simple verbalization is perhaps one of the oldest of human traditions. 8 Chiral & Vector Supermultiplet The 4D, � = 1 scalar superfield provides a reducible representation of supersymmetry →irreducible representations: chiral and vector supermultiplets How to carry out the process for a general representation of spacetime supersymmetry is unknown! (Motivation for the adinkra approach to the study of superfields) Vector Supermultiplet Hodge-dual variants of the chiral supermultiplet 9 Traditional Path to Superfield Component Decompositions Write down the general θ−expansion of a scalar superfield Level-n is the θ−monomial with n θs If you start from constructing irreducible θ−monomials, what would you write? 11D, � = 1 quadratic monomials 10 11D, � = 1 cubic monomials means that a single -trace of the expression is by definition equal to zero. * [ ]IR γ 32 × 31 × 30 = {4,960} = {32} ⊕ {1,408} ⊕ {3,520} Problems: 3! θ−monomials have multiple expressions* You wouldn’t know two versions of {320} and {5,280} are identically zero Even for gamma matrix multiplications, you can get multiple expressions. e.g. * Explicit expressions and detailed discussions will be presented in the paper which is coming soon. 11 10D, � = 1 Scalar Superfield Decomposition In 10D, � = 1 superspace, the grassmann coordinate has 16 real components Write down the general θ−expansion of a scalar superfield in 10D, � = 1 superspace: Goal: decompose level-n into a direct sum of irreducible representations of Lorentz group SO(1,9). Young Tableau Approach Branching Rule Approach 12 Young Tableau Approach A Tableau: a filling-in of the boxes with certain symbols Young Tableau: irrep of and Sn SU(n) In the study of O(n) and SO(n), new sets of Young Tableaux were presented [5]: Column strict tableau [6]: YT with negative boxes (3,2, − 1, − 2) [5] R. C. King, Weight multiplicities for the classical groups, in Lecture Notes in Physics, Vol. 50, pp. 490-499, Springer-Verlag, Berlin/New York, 1976 [6] G. Girardi, A. Sciarrino and P. Sorba, GENERALIZED YOUNG TABLEAUX AND KRONECKER PRODUCTS OF SO(n) REPRESENTATIONS, Physica 114A, 365 (1982). 13 Bosonic / Spinorial Young Tableaux In ��(1,9), we can use Young Tableaux to denote reducible representations. Rules of tensor product are still valid. Color Young Tableaux: blue for bosonic; red for spinorial Consider the self-dual / anti-self-dual identities: {126} {126} Then {126} {126} 14 Young Tableau Approach level-1: {16} = □ Qudratic level: Tensor product decomposition: Then from dimensionality: thus, level-2 = {120} [7] N. Yamatsu, “Finite-dimensional Lie algebras and their representations for unified model building,” arXiv:1511.08771 (2015) (unpublished) [8] R. Feger and T. W. Kephart, “LieART—A Mathematica application for Lie algebras and representation theory,” Comput. Phys. Commun. 192, 166 (2015) doi:10.1016/j.cpc.2014.12.023 [arXiv: 1206.6379 [math-ph]]. 15 Young Tableau Approach Cubic Level: we can write two independent equations that sYT in the l.h.s. have three boxes and sYT in the r.h.s. have one or two boxes: 3 variables, 2 equations! {560} has to belong to {672} has to belong to Decompositions of all sYT in cubic level: level-3 = {560} 16 Branching Rules A Branching Rule is a relation between a representation of a Lie algebra � and representations of its Lie subalgebra � Branching Rules between a Lie algebra � and its Lie subalgebra � are determined by a single projection matrix The projection matrix is fixed by the weight diagrams of a branching rule of them, where weight diagrams can be written down by Cartan matrix of � and Dynkin labels [9]: Branching of massless on-shell states upon reducing D = 12 → D = 11, by O(10) ⊃ O(9) [9] T. Curtright, Fundamental Supermultiplet In Twelve-dimensions, Front. in Phys. 6, 137 (2018). doi:10.3389/fphy.2018.00137 17 Branching Rules level-n = branching rule between the n-th fundamental representation of ��(16) and its maximal S-subalgebra ��(1,9) Projection Matrix: given that {16} ={16} weight vector T weight vector T ( )�o(1,9) = Psu(16)⊃so(1,9)( )�u(16) Weight System of the Weight System of the defining rep of ��(16) spinor rep of ��(1,9) Psu(16)⊃so(1,9) = Psu(16)⊃so(1,9) 18 Branching Rule Results Applying symmetric group can speed up the calculation (plethysm function) [10,11] [10] R. M. Fonseca, “Calculating the Renormalisation Group Equations of a SUSY Model with Susyno,” Comput. Phys. Commun. 183 (2012) 2298–2306, arXiv:1106.5016 [hep-ph] [11] LiE, A Computer Algebra Package for Lie Group Computations, http://www-math.univ-poitiers.fr/ maavl/ LiE/. 19 10D, � = 1 Scalar Superfield Decomposition How to understand the “bar” representations? Assign to α and · {16} χ {16} to χα · β · the irrep of χα = χ Cαβ → is χα {16} Irreps corresponding to the component fields are the conjugate of the irreps corresponding to the θ− monimials 20 10D, � = 1 Adinkra 21 Scalar Superfield Decomposition in 10D, � = 1 Breitenlohner 10D, � = 2A 10D, � = 2B Approach [2] 9D to 5D minimal 11D, � = 1 scalar superfield decomp prepotential 10D YM candidates supermultiplet Young Tableau Approach Branching Rule Approach 22 Breitenlohner Approach The principle of this approach is to attach bosonic and spinor indices on the scalar superfield and look for the traceless graviton and gravitino Recall that the first off-shell description of 4D, � = 1 supergravity was carried out by Breitenlohner: start with the component fields of the WZ gauge 4D, � = 1 vector supermultiplet with SUSY transformation laws and do a series of replacements of the fields Look at the level-2, there is a {4} irrep. Consider the prepotential Ha There is no {10} irrep in 10D, � = 1 case. 23 Bosonic Superfields Goal: study the expansions of some bosonic superfields and search for traceless graviton and traceless gravitino How to get components of the superfields with various bosonic (or fermionic) indices? The first nontrivial one is : 4 possible � × {120} = �abc embeddings for gravitons and 10 possible embeddings for gravitinos If you find a {54} in level-m, you will find a {144}in level-(m+1): SUSY transformation law of the graviton in the 10D, � = 1 theory: 24 Fermionic Superfields Goal: study the expansions of some fermionic superfields and search for traceless graviton and traceless gravitino Attaching some spinor indices on the scalar superfield is like tensoring a spinorial irrep to it The first nontrivial one is γ, � × {560} = �ab satisfying a γ (σ )γδ�ab = 0 3 possible embeddings for gravitons, 15 possible embeddings for gravitinos, and auxiliary fields 25 10D, � = 1 Yang-Mills Supermultiplet In [12], they investigated the structure of 1-form 10D, � = 1 gauge theory 1-form U(1) superspace covariant derivatives ∇A = DA + igΓAt Superspace connection ΓA = (Γα , Γa) In [12]: all off-shell theories of this type must include a bosonic component field f[5] The structure of the spinorial gauge connection superfield and its gauge parameter are given by [12] S.
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