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'Internal spacetime' is a modification of general relativity that was recently introduced as an approximate spacetime geometric model of quantum nonlocality. In an internal spacetime, time is stationary along the worldlines of fundamental (dust) particles. Consequently, the dimensions of tangent spaces at different points of spacetime vary, and spin wavefunction collapse is modeled by the projection from one tangent space to another. In this article we develop spinors on an internal spacetime, and construct a new Dirac-like Lagrangian $\mathcal{L} = \bar{\psi}(i \slashed \partial - \hat{\omega}) \psi$ whose equations of motion describe their couplings and interactions. Furthermore, we show that hidden within $\mathcal{L}$ is the entire standard model: $\mathcal{L}$ contains precisely three generations of quarks and leptons, the electroweak gauge bosons, the Higgs boson, and one new massive spin-$2$ boson; gluons are considered in a companion article. Specifically, we are able to derive the correct spin, electric charge, and color charge of each standard model particle, as well as predict the existence of a new boson., Comment: 25 pages. Final version. (This paper is a companion to arXiv:1908.09631 and partly replaces its earlier versions; two tables and one figure here also appear in the 'Preliminaries' section of its final version.)

A ghor algebra is the path algebra of a dimer quiver on a surface, modulo relations that come from the perfect matchings of its quiver. Such algebras arise from abelian quiver gauge theories in physics. We show that a ghor algebra $\Lambda$ on a torus is a dimer algebra (a quiver with potential) if and only if it is noetherian, and otherwise $\Lambda$ is the quotient of a dimer algebra by homotopy relations. Furthermore, we classify the simple $\Lambda$-modules of maximal dimension and give an explicit description of the center of $\Lambda$ using a special subset of perfect matchings. In our proofs we introduce formalized notions of Higgsing and the mesonic chiral ring from quiver gauge theory., Comment: 39 pages. I am reorganizing and expanding arXiv:1412.1750 into four shorter papers; this paper is the first of the four

Let $A$ be a dimer algebra and $Z$ its center. It is well known that if $A$ is cancellative, then $A$ and $Z$ are noetherian and $A$ is a finitely generated $Z$-module. Here we show the converse: if $A$ is non-cancellative (as almost all dimer algebras are), then $A$ and $Z$ are nonnoetherian and $A$ is an infinitely generated $Z$-module. Although $Z$ is nonnoetherian, we show that it nonetheless has Krull dimension 3 and is generically noetherian. Furthermore, we show that the reduced center is the coordinate ring for a Gorenstein algebraic variety with the strange property that it contains precisely one 'smeared-out' point of positive geometric dimension. In our proofs we introduce formalized notions of Higgsing and the mesonic chiral ring from quiver gauge theory., Comment: 75 pages. Two new sections and an appendix (originally in 1301.7059v1) have been added

We introduce a geometric realization of noncommutative singularity resolutions. To do this, we first present a new conjectural method of obtaining conventional resolutions using coordinate rings of matrix-valued functions. We verify this conjecture for all cyclic quotient surface singularities, the Kleinian D_n and E_6 surface singularities, the conifold singularity, and a non-isolated singularity, using appropriate quiver algebras. This conjecture provides a possible new generalization of the classical McKay correspondence. Then, using symplectic reduction within these rings, we obtain new, non-conventional resolutions that are hidden if only commutative functions are considered. Geometrically, these non-conventional resolutions result from shrinking exceptional loci to ramified (non-Azumaya) point-like spheres., Comment: 42 pages. Comments welcome

We introduce a theory of geometry for nonnoetherian commutative algebras with finite Krull dimension. In particular, we establish new notions of normalization and height: depiction (a special noetherian overring) and geometric codimension. The resulting geometries are algebraic varieties with positive dimensional points, and are thus inherently nonlocal. These notions also give rise to new equivalent characterizations of noetherianity that are primarily geometric. We then consider an application to quiver algebras whose simple modules of maximal dimension are one dimensional at each vertex. We show that the vertex corner rings of $A$ are all isomorphic if and only if $A$ is noetherian, if and only if the center $Z$ of $A$ is noetherian, if and only if $A$ is a finitely generated $Z$-module. Furthermore, we show that $Z$ is depicted by a commutative algebra generated by the cycles in its quiver. We conclude with an example of a quiver algebra where projective dimension and geometric codimension, rather than height, coincide., Comment: 25 pages. Final version. To appear in J. Algebra Appl

The AdS/CFT correspondence for N=1 Super conformal field theories suggests that dibaryon operators are dual to D-brane states that are point like in AdS and that wrap various cycles in a Sasaki-Einstein manifold. It also suggests that the volume of the D-brane gives the R-charge of the corresponding operator. We elucidate various aspects of this correspondence, paying particular care to study the case of branes at the tip of three different Calabi Yau cones. We show that the arrows in the quiver diagram describing the conformal field theory can be thought of as global sections of a non-trivial holomorphic vector bundle over the Calabi-Yau geometry. We suggest that the zero locus of these sections gives the geometric map that lets us tie a particular dibaryon to a holomorphic cycle, by intersecting the corresponding cycle with the Sasaki-Einstein locus at fixed distance from the origin. We show that this can be compared with the corresponding volumes of the Sasaki-Einstein space and that one gets exact agreement between the volumes of the cycles identified with this procedure and the R-charges of the operators., 30 pages, 1 fig