Search Results (5)

Let $\mathcal{M}(Spin(8,\mathbb{C}\left)\right)$ be the moduli space of $Spin(8,\mathbb{C})$-Higgs bundles over a compact Riemann surface X of genus $g\ge 2$. This admits a system called the Hitchin integrable system, induced by the Hitchin map, the fibers of which are Prym varieties. Moreover, the triality automorphism of $Spin(8,\mathbb{C})$ acts on $\mathcal{M}(Spin(8,\mathbb{C}\left)\right)$, and those Higgs bundles that admit a reduction in the structure group to $G_2$ are fixed points of this action. This defines a map of moduli spaces of Higgs bundles $\mathcal{M}\left(G_2\right)\to \mathcal{M}(Spin(8,\mathbb{C}))$. In this work, the action of triality automorphism is extended to an action on the Hitchin integrable system associated with $\mathcal{M}(Spin(8,\mathbb{C}\left)\right)$. In particular, it is checked that the map $\mathcal{M}\left(G_2\right)\to \mathcal{M}(Spin(8,\mathbb{C}))$ is restricted to a map at the level of the Prym varieties induced by the Hitchin map. Necessary and sufficient conditions are also provided for the Prym varieties associated with the moduli spaces of $G_2$ and $Spin(8,\mathbb{C})$-Higgs bundles to be disconnected. Finally, some consequences are drawn from the above results in relation to the geometry of the Prym varieties involved. Full article

This paper surveys recent progress in our search for an appropriate internal space algebra for the standard model (SM) of particle physics. After a brief review of the existing approaches, we start with the Clifford algebras involving operators of left multiplication by octonions. A central role is played by a distinguished complex structure that implements the splitting of the octonions $\mathbb{O}=\mathbb{C}\oplus {\mathbb{C}}^3$, which reflect the lepton-quark symmetry. Such a complex structure on the 32-dimensional space $\mathcal{S}$ of $C{\ell }_{10}$ Majorana spinors is generated by the $C{\ell }_6(\subset C{\ell }_{10})$ volume form, ${\omega }_6={\gamma }_1\cdots {\gamma }_6$, and is left invariant by the Pati–Salam subgroup of $Spin\left(10\right)$, $G_{\mathrm{PS}}=Spin\left(4\right)\times Spin\left(6\right)/{\mathbb{Z}}_2$. While the $Spin\left(10\right)$ invariant volume form ${\omega }_{10}={\gamma }_1\dots {\gamma }_{10}$ of $C{\ell }_{10}$ is known to split $\mathcal{S}$ on a complex basis into left and right chiral (semi)spinors, $\mathcal{P}=\frac{1}{2}(1-i{\omega }_6)$ is interpreted as the projector on the 16-dimensional particle subspace (which annihilates the antiparticles).The standard model gauge group appears as the subgroup of $G_{\mathrm{PS}}$ that preserves the sterile neutrino (which is identified with the Fock vacuum). The ${\mathbb{Z}}_2$-graded internal space algebra $\mathcal{A}$ is then included in the projected tensor product $\mathcal{A}\subset \mathcal{P}C{\ell }_{10}\mathcal{P}=C{\ell }_4\otimes C{\ell }_6^0$. The Higgs field appears as the scalar term of a superconnection, an element of the odd part $C{\ell }_4^1$ of the first factor. The fact that the projection of $C{\ell }_{10}$ only involves the even part $C{\ell }_6^0$ of the second factor guarantees that the color symmetry remains unbroken. As an application, we express the ratio $\frac{m_H}{m_W}$ of the Higgs to the W boson masses in terms of the cosine of the theoretical Weinberg angle. Full article

In $26+1$ space–time dimensions, we introduce a gravity theory whose massless spectrum can be acted upon by the Monster group when reduced to $25+1$ dimensions. This theory generalizes M-theory in many respects, and we name it Monstrous M-theory, or M${}^2$-theory. Upon Kaluza–Klein reduction to $25+1$ dimensions, the M${}^2$-theory spectrum irreducibly splits as 1 ⊕ 196,883, where 1 is identified with the dilaton, and 196,883 is the dimension of the smallest non-trivial representation of the Monster. This provides a field theory explanation of the lowest instance of the Monstrous Moonshine, and it clarifies the definition of the Monster as the automorphism group of the Griess algebra by showing that such an algebra is not merely a sum of unrelated spaces, but descends from massless states for M${}^2$-theory, which includes Horowitz and Susskind’s bosonic M-theory as a subsector. Further evidence is provided by the decomposition of the coefficients of the partition function of Witten’s extremal Monster SCFT in terms of representations of $S{O}_{24}$, the massless little group in $25+1$; the purely bosonic nature of the involved $S{O}_{24}$-representations may be traced back to the unique feature of 24 dimensions, which allow for a non-trivial generalization of the triality holding in 8 dimensions. Last but not least, a certain subsector of M${}^2$-theory, when coupled to a Rarita–Schwinger massless field in $26+1$, exhibits the same number of bosonic and fermionic degrees of freedom; we cannot help but conjecture the existence of a would-be $\mathcal{N}=1$ supergravity theory in $26+1$ space–time dimensions. Full article

The properties of spinors and vectors in (2 + 2) space of split quaternions are studied. Quaternionic representation of rotations naturally separates two $SO(2,1)$ subgroups of the full group of symmetry of the norms of split quaternions, $SO(2,2)$. One of them represents symmetries of three-dimensional Minkowski space-time. Then, the second $SO(2,1)$ subgroup, generated by the additional time-like coordinate from the basis of split quaternions, can be viewed as the internal symmetry of the model. It is shown that the analyticity condition, applying to the invariant construction of split quaternions, is equivalent to some system of differential equations for quaternionic spinors and vectors. Assuming that the derivatives by extra time-like coordinate generate triality (supersymmetric) rotations, the analyticity equation is reduced to the exact Dirac–Maxwell system in three-dimensional Minkowski space-time. Full article

In order to combine internal symmetries and spacetime that has Poincaré symmetry, it is necesary to introduce supersymmetry, Supersymmetry of Connes is based on involution, and that of Cartan is based on triality. Cartan’s supersymmetry allows violation of Lorentz symmetry and time reversal violation can occur. Full article