superalgebra and (synthetic ) supergeometry
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In super-representation theory, what is called adinkras (Faux-Gates 04) is a graphical tool for denoting those representations (super multiplets) of the $\mathcal{N}$-extended supersymmetry algebras in one dimension (supersymmetric quantum mechanics with N supersymmetries) for which the supersymmetry generators act, up to derivatives and prefactors, by permutation of superfield components. These are called adinkraic representations (Zhang 13, p. 16).
While adinkraic representations are special among all representations of the 1-dimensional $\mathcal{N}$-extended supersymmetry algebra, the idea is that the dimensional reduction of representations (supermultiplets) of higher dimensional supersymmetry algebras down to 1d are of this form, at least for dimensional reduction from $d = 4$ and $N = \mathbf{4}$ (GGMPPRW 09, section 3, Gates-Hubsch-Stiffler 14).
The classification of adinkras, and hence of adinkraic representations, turns out to be controlled by linear codes (Doran & Faux & Gates & HubschIgaLandweberMiller 11) and to be related to certain special super Riemann surfaces via dessins d'enfants (Doran & Iga & Landweber & Mendez-Diez 13, Doran & Iga & Kostiuk &Mendes-Diez 16).
For background, see at geometry of physics – supersymmetry.
For $N \in \mathbb{N}$, write $\mathbb{R}^{1 \vert N}$ for the super Lie algebra over the real numbers that is spanned by a single generator $P$ in even (i.e., bosonic) degree and $N$ generators $Q_I$, $I \in \{1, 2, \cdots, N\}$ in odd (i.e., fermionic) degree, whose only non-trivial components of the super-Lie bracket are
This is the 1-dimensional $N$-extended super translation super Lie algebra. We may think of this as the super-translational symmetry of 1-dimensional $N$-extended super Minkowski spacetime.
Consider then super Lie algebra representations of $\mathbb{R}^{1 \vert N}$ on super vector spaces of smooth superfields on $\mathbb{R}^{1 \vert N}$ (regarded as a supermanifold) and such that the bosonic generator $P$ acts as the derivative operator on smooth functions on $\mathbb{R}^1$ in each component. If in addition the representation is such that in the canonical linear basis the odd generators $Q_I$ send even/odd basis elements $\phi_i$ to single odd/even basis elements $\psi_j$ (as opposed to linear combinations of them), hence if the $Q_I$ act apart from degree-shift and possibly differentiation by permutations on the components of the superfields, then this representation of $\mathbb{R}^{1\vert N}$ is called adinkraic. (Zhang 13, p. 16).
The corresponding adinkra is the bipartite graph which expresses these permutations:
table grabbed from Doran & Iga & Landweber & Mendez-Diez 13, p. 7
graphics grabbed from Iga-Zhang 15, p. 3
The topology of an adinkra graph together with its edge coloring in $\{1,2, \cdots, N\}$ is called its chromotopology.
The set of adinkra chromotopologies is equivalent to the set of colored $N$-cubs modulo doubly even length-$N$ linear codes (Doran-Faux-Gates-Hubsch-Iga-Landweber-Miller 11)
(A linear code of length $N$ is a linear subspace of $(\mathbb{F}_2)^N$ for $\mathbb{F}_2$ the prime field with two elements and it is doubly even if every element has weight a multiple of 4. )
graphics grabbed from Iga-Zhang 15, p. 4
The dimensional reduction of the smallest supermultiplets of $d = 4, N = \mathbf{4}$ supersymmetry down to 1d yield adinkraic representations (GGMPPRW 09, section 3, Gates-Hubsch-Stiffler 14).
Corresponding adinkras for the chiral scalar supermultiplet (CM), the vector multiplet (VM) and the tensor multiplet (TM) look as follows:
For an introduction to adinkras, see the talk
The concept of adinkras was introduced into supersymmetry representation theory in
and further developed in the article
and many more (“DFGHIL collaboration”). For instance the relation to Clifford supermodules is discussed in
kinetic terms are discussed in
The classification of adinkras in terms of graphs and linear codes is due to
and discussed in mathematical detail in
Yan X Zhang, Adinkras for Mathematicians (arXiv:1111.6055)
Yan X Zhang, The combinatorics of Adinkras, PhD thesis, MIT (2013) (pdf)
The dimensional reduction of the standard supermultiplets of $D = 4, \mathcal{N} = 1$ supersymmetry to adinkraic representations of $D = 1, \mathcal{N}=4$ is due to
Jim Gates, J. Gonzales, B. MacGregor, J. Parker, R. Polo-Sherk, V.G.J. Rodgers, L. Wassink, $4D$, $N = 1$ Supersymmetry Genomics (I), JHEP 0912:008,2009 (arXiv:0902.3830)
Jim Gates, Tristan Hübsch, Kory Stiffler, Adinkras and SUSY Holography, Int. J. Mod. Phys. A29 no. 7, (2014) 1450041 (arXiv:1208.5999)
See also
Discussion for $D=11$, $\mathcal{N}=1$:
The relation of adinkras to special super Riemann surfaces via dessins d'enfants is due to
Charles Doran, Kevin Iga, Greg Landweber, Stefan Méndez-Diez, Geometrization of $\mathcal{N}$-Extended 1-Dimensional Supersymmetry Algebras (arXiv:1311.3736)
Charles Doran, Kevin Iga, Jordan Kostiuk, Stefan Méndez-Diez, Geometrization of $\mathcal{N}$-Extended 1-Dimensional Supersymmetry Algebras II (arXiv:1610.09983)
Further developments includes
Discussion in the context of spectral triples is in
See also
Wes Caldwell, Alejandro Diaz, Isaac Friend, Jim Gates, Jr., Siddhartha Harmalkar, Tamar Lambert-Brown, Daniel Lay, Karina Martirosova, Victor Meszaros, Mayowa Omokanwaye, Shaina Rudman, Daniel Shin, Anthony Vershov, On the Four Dimensional Holoraumy of the $4D$, $\mathcal{N} = 1$ Complex Linear Supermultiplet (arXiv:1702.05453)
Kevin Iga, Adinkras: Graphs of Clifford Algebra Representations, Supersymmetry, and Codes (arXiv:2110.01665)
Wikipedia, Adinkra symbols (physics)