superﬁeld component decompositions and the scan for

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]

MotivationAn 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

[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

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

4[2] P. Breitenlohner, “A Geometric Interpretation of Local Supersymmetry,” Phys. Lett. 67B (1977) 49, DOI: 10.1016/0370-2693(77)90802-4.

Scalar Superfield Decomposition in 10D, 𝒩 = 1

10D, 𝒩 = 2A 10D, 𝒩 = 2BBreitenlohner Approach [2]

11D, 𝒩 = 1

prepotential candidates

10D YM supermultiplet

minimal scalar superfield decomp

9D to 5D

Young Tableau Approach Branching Rule Approach

Constructions of Superspaces

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spacetime dimensional superspace: , where and . 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

, where

D (xa , θα) a = 0,1,2,…, D − 1α = 1,2,…, d d

2d nB = nF = 2d−1

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General expansion of a scalar superfield in superspace:

θ− 4D, 𝒩 = 1

Goal: decompose the monomials θ−

Scalar Superfield Decomposition4D, 𝒩 = 1

Convention of the color of the irrep: blue if bosonic, red if spinorial

{1} {4} {6}

{4} {1}

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The basis of the vector space spanned by spinors is summarized in the Table below: gamma matrices are 4 × 4

Irreducible monimials:θ−


Adinkra Definition4D, 𝒩 = 1

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[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.

Chiral & Vector Supermultiplet

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The 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)

4D, 𝒩 = 1

Vector Supermultiplet Hodge-dual variants of the chiral supermultiplet

Traditional Path to Superfield Component Decompositions

Write down the general expansion of a scalar superfield

Level-n is the monomial with n If you start from constructing irreducible monomials, what would you write?

θ−

θ− θsθ−

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quadratic monomials11D, 𝒩 = 1

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cubic monomials11D, 𝒩 = 1

Problems: monomials have multiple expressions*

You wouldn’t know two versions of and are identically zero Even for gamma matrix multiplications, you can get multiple expressions. e.g.

θ−{320} {5,280}

means that a single -trace of the expression is by definition equal to zero. * [ ]IR γ32 × 31 × 30

3!= {4,960} = {32} ⊕ {1,408} ⊕ {3,520}

* Explicit expressions and detailed discussions will be presented in the paper which is coming soon.


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In superspace, the grassmann coordinate has 16 real components Write down the general expansion of a scalar superfield in superspace:

10D, 𝒩 = 1θ− 10D, 𝒩 = 1

Goal: decompose level-n into a direct sum of irreducible representations of Lorentz group . SO(1,9)


Young Tableau Approach

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[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).

A Tableau: a filling-in of the boxes with certain symbols Young Tableau: irrep of and In the study of and , new sets of Young Tableaux were presented

[5]: Column strict tableau

[6]: YT with negative boxes

Sn SU(n)O(n) SO(n)

(3,2, − 1, − 2)

Bosonic / Spinorial Young Tableaux

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In , 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:

𝔰𝔬(1,9)

{126}

Then

{126}

{126} {126}


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thus, level-2 = {120}

level-1: Qudratic level:

Tensor product decomposition:

Then from dimensionality:

{16} = □

[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]].

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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! has to belong to has to belong to

Decompositions of all sYT in cubic level:

{560}{672}

level-3 = {560}


Branching Rules

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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

, by

𝔤 𝔥𝔤

𝔥

𝔤

D = 12 → D = 11 O(10) ⊃ O(9)

[9] T. Curtright, Fundamental Supermultiplet In Twelve-dimensions, Front. in Phys. 6, 137 (2018). doi:10.3389/fphy.2018.00137

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level-n branching rule between the n-th fundamental representation of and its maximal S-subalgebra

Projection Matrix: given that

=𝔰𝔲(16) 𝔰𝔬(1,9)

{16} ={16}

Psu(16)⊃so(1,9) =Psu(16)⊃so(1,9)

Weight System of the defining rep of 𝔰𝔲(16)

Weight System of the spinor rep of 𝔰𝔬(1,9)

Branching Rules

(weight vector)T𝔰o(1,9) = Psu(16)⊃so(1,9)(weight vector)T

𝔰u(16)

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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/.

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How to understand the “bar” representations?

Assign to and to → the irrep of

is Irreps corresponding to the component fields are the conjugate of the irreps corresponding to the monimials

{16} χα {16}χ ·α

χα = χ·βCα ·β

χα {16}

θ−

Adinkra10D, 𝒩 = 1

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22



11D, 𝒩 = 1




9D to 5D


Breitenlohner Approach

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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 supergravity was carried out by Breitenlohner: start with the component fields of the WZ gauge vector supermultiplet with SUSY transformation laws and do a series of replacements of the fields

4D, 𝒩 = 1

4D, 𝒩 = 1

Look at the level-2, there is a irrep. Consider the prepotential {4} Ha

There is no irrep in case. {10} 10D, 𝒩 = 1

Bosonic Superfields

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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 embeddings for gravitons and 10 possible embeddings for gravitinos If you find a in level-m, you will find a in level-(m+1): SUSY transformation law of the graviton in the theory:

𝒱 × {120} = 𝒱abc

{54} {144}10D, 𝒩 = 1

Fermionic Superfields

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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 , satisfying 3 possible embeddings for gravitons, 15 possible embeddings for gravitinos, and auxiliary fields

𝒱 × {560} = 𝒱 γab

(σa)γδ𝒱 γab = 0

Yang-Mills Supermultiplet10D, 𝒩 = 1

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In [12], they investigated the structure of 1-form gauge theory 1-form U(1) superspace covariant derivatives Superspace connection In [12]: all off-shell theories of this type must include a bosonic component field The structure of the spinorial gauge connection superfield and its gauge parameter are given by

10D, 𝒩 = 1∇A = DA + igΓAt

ΓA = (Γα , Γa)

f[5]

[12] S. J. Gates, Jr. and S. Vashakidze, “On D = 10, N = 1 Supersymmetry, Superspace Geometry and Superstring Effects”, Nucl. Phys. B 291, 172 (1987). doi:10.1016/0550-3213(87)90470-6.

Yang-Mills Supermultiplet10D, 𝒩 = 1

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Gauge field f[5]

Gaugino field

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11D, 𝒩 = 1




9D to 5D


Scalar Superfield Decomposition10D, 𝒩 = 2A

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How to construct a scalar superfield based on scalar superfields?

10D, 𝒩 = 2A10D, 𝒩 = 1

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Denote level-n of scalar superfield as Level-17 to 32 are the conjugates of Level-15 to 0 Each level is self-conjugate

10D, 𝒩 = 1 ℓn


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72 possible graviton embeddings, 280 gravitino embeddings, and associated auxiliary fields If you find a in level-m, you will find a and in level-(m+1) SUSY transformation law of the graviton in the theory

{54} {144} {144}10D, 𝒩 = 2A


The only reasons for the incompleteness is convenience of presentation.

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How to construct a scalar superfield based on scalar superfields?

10D, 𝒩 = 2B10D, 𝒩 = 1

Scalar Superfield Decomposition10D, 𝒩 = 2B

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Scalar Superfield Decomposition10D, 𝒩 = 2B72 possible graviton embeddings, 280 gravitino embeddings, and associated auxiliary fields If you find a in level-m, you will find a in level-(m+1) SUSY transformation law of the graviton in the theory

{54} {144}10D, 𝒩 = 2B

The only reasons for the incompleteness is convenience of presentation.


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In superspace, the grassmann coordinates have 32 real components

Young Tableau approach can give unique decomposition up to level-5

Branching Rules for to :

Level-n branching rule between the n-th fundamental reps of and its maximal S-subalgebra

Projection Matrix: based on

theory → 10D TypeIIA theory → decomposition for each level: reproduce scalar

superfield decomposition

11D, 𝒩 = 1

𝔰𝔲(32) 𝔰𝔬(1,10)

= 𝔰𝔲(32)𝔰𝔬(1,10)

{32} = {32}

11D, 𝒩 = 1𝔰𝔬(1,10) 𝔰𝔬(1,9) 10D, 𝒩 = 2A

Psu(32)⊃so(1,10) =

Adinkra11D, 𝒩 = 1

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The only reasons for the incompleteness is convenience of presentation. Detailed discussions will be presented in the paper which is coming soon.


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In superspace, the PM grassmann coordinates have 16 real components Start from scalar superfield decomposition: →

decomposition for each level → decomposition for

fundamental representations and trivial representation: scalar superfield decomposition

9D, 𝒩 = 1

10D, 𝒩 = 1𝔰𝔬(1,9)

𝔰𝔬(1,8)

𝔰𝔲(16) 𝔰𝔬(1,8)

9D, 𝒩 = 1

Detailed discussions will be presented in the paper which is coming soon.

Minimal Scalar Superfield Decomposition8D

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In superspace, the PM grassmann coordinates have 16 real components

8D



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In superspace, the SU(2)-Majorana grassmann coordinates have 16 real components

7D



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In superspace, the SU(2)-Majorana-Weyl grassmann coordinates have 8 real components

6D



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In minimal superspace, the SU(2)-Majorana grassmann coordinates have 8 real components

5D


Conclusions

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The first complete and explicit Lorentz descriptions of all component fields contained in

and minimal unconstrained scalar superfields

These decompositions are constructed without information from an off-shell component formulation for the first time A proposal for identifying the corresponding prepotential supermultiplets by applying the Breitenlohner approach in 10D superspaces A new beginning for the search for irreducible off-shell formulations of the 10D Yang-Mills supermultiplet derived from superfields High dimensional adinkras are defined for the first time

10D, 𝒩 = 1, 𝒩 = 2A, 𝒩 = 2B, 11D, 𝒩 = 1,9D to 5D

Outlook

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Dive far more deeply into the relations between analytical expressions of the irreducible monomials, Young tableaux, and Dynkin labels Construct irreducible representations of spacetime SUSY, especially supergravity multiplets.

θ−

Thank You!

superﬁeld component decompositions and the scan for

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