The Signed Monodromy Group of an Adinkra

TL;DR

This work identifies the signed monodromy group arising from Adinkras, via the dash/solid edge data, with Salingaros Vee groups G_n. The unsigned monodromy recovers the familiar elementary 2-group M ≅ F_2^{N−k−1}, while the signing data yields a quotient by a central subgroup giving 𝓗 ≅ G_{N−1} or G_{N−2} depending on whether the all-ones codeword h_1 lies in the code C. The kernel Σ of the sign map is an elementary 2-group whose rank depends on the code dimension k and on this central kernel K, and the GR(d,N) algebra relations are encoded by the signed L_I, R_I matrices. The results illuminate a deep connection between Adinkra combinatorics, signed permutation groups, and Clifford- or Salingaros-vee type groups, with concrete instances recovering the quaternion group Q_8 and links to Belyi-dessins via the monodromy framework. These findings provide a structured algebraic-differential bridge between Adinkra representations of supersymmetry and finite 2-groups, with potential implications for the GR(d,N) algebra and its representations.

Abstract

An ordering of colours in an Adinkra leads to an embedding of this Adinkra into a Riemann surface $X$, and a branched covering map $β_X:X\to\mathbb{CP}^1$. This paper shows how the dashing of edges in an Adinkra determines a signed permutation version of the monodromy group, and shows that it is isomorphic to a Salingaros Vee group.

Figures (7)

• Figure 1: An $N=4$ Adinkra with the topology of a Hamming $4$-cube. Bosons are the white nodes, and fermions are black. Note the colouring and dashing of the edges. The height assignment here, shown literally by height on the page, puts all bosons at height $0$ and all fermions at height $1$.
• Figure 2: The quotient of the $4$-cube Adinkra by the doubly even code generated by $\langle 1111\rangle$. This can be obtained from the Adinkra in Figure \ref{['fig:adinkra40']} by identifying each of the four leftmost bosons (and the four leftmost fermions) with the boson (resp. fermion) four nodes to its right.
• Figure 3: The beachball for $N=5$. The white node is the boson at $0$, the black node is the fermion at $\infty$, the coloured edges are the $z_j(t)$, and the $\times$ are at the $\xi_j$, where the covering map $\beta_X$ is ramified.
• Figure 4: On the left, the Riemann surface $X$ for the $N=4$, $k=1$ Adinkra from Figure \ref{['fig:adinkra41']}. There is an order $4$ branched covering map $\beta_X$ that sends it to the corresponding $N=4$ beachball on the right.
• Figure 5: Views of the beachball $\mathbb{CP}^1$ near $0$ and $\infty$. The path $z_1(t)$ goes along the positive real axis. Note that the ordering of colours goes counterclockwise around $0$ and clockwise around $\infty$.

Theorems & Definitions (66)

• Definition 2.2
• Definition 2.5
• Definition 4.1
• Proposition 4.1
• proof
• Definition 5.1
• Proposition 5.1
• Remark 5.2
• Example 5.3
• Example 5.4