We consider the possibility of breaking the GUT group to the Standard Model gauge group in $F$-theory compactifications by turning on certain $U(1)$ fluxes. We show that the requirement of massless hypercharge is equivalent to a topological constraint on the UV completion of the local model. The possibility of this mechanism is intrinsic to $F$-theory. We address some of the phenomenological signatures of this scenario. We show that our models predict monopoles as in conventional GUT models. We discuss in detail the leading threshold corrections to the gauge kinetic terms and their effect on unification. They turn out to be related to Ray–Singer torsion. We also discuss the issue of proton decay in $F$-theory models and explain how to engineer models which satisfy current experimental bounds.

In this paper, we continue our program, started in Euler angles for $G(2)$, of building up explicit generalized Euler angle parameterizations for all exceptional compact Lie groups. Here we solve the problem for $E_7$, by first providing explicit matrix realizations of the Tits construction of a Magic Square product between the exceptional octonionic algebra $\mathfrak{J}$ and the quaternionic algebra $\mathbb{H}$, both in the adjoint and the 56-dimensional representations. Then, we provide the Euler parametrization of $E_7$ starting from its maximal subgroup $U = (E_6 \times U(1))/\mathbb{Z}_3$. Next, we give the constructions for all the other maximal compact subgroups.

We derive the leading asymptotic behavior and build a new series representation for the Fredholm determinant of integrable integral operators appearing in the representation of the time and distance-dependent correlation functions of integrable models described by a six-vertex $R$-matrix. This series representation opens a systematic way for the computation of the long-time, long-distance asymptotic expansion for the correlation functions of the aforementioned integrable models away from their free fermion point. Our method builds on a Riemann–Hilbert based analysis.

The aim of this paper is to analyze some geometric properties of the rigid Calabi–Yau three-fold $\mathcal{Z}$ obtained by a quotient of $E_3$, where $E$ is a specific elliptic curve. We describe the cohomology of $\mathcal{Z}$ and give a simple formula for the trilinear form on $\mathrm{Pic}(\mathcal{Z})$. We describe some projective models of $\mathcal{Z}$ and relate these to its generalized mirror. A smoothing of a singular model is a Calabi–Yau three-fold with small Hodge numbers which was not known before.

We consider the Cauchy problem for the wave equation in a general class of spherically symmetric black hole geometries. Under certain mild conditions on the far-field decay and the singularity, we show that there is a unique globally smooth solution to the Cauchy problem for the wave equation with data compactly supported away from the horizon that is compactly supported for all times and decays in $L_{loc}^\infty$ as $t$ tends to infinity. We obtain as a corollary that in the geometry of black hole solutions of the SU(2) Einstein/Yang–Mills equations, solutions to the wave equation with compactly supported initial data decay as $t$ goes to infinity.

We continue our study of the noncommutative algebraic and differential geometry of a particular class of deformations of toric varieties, focusing on aspects pertinent to the construction and enumeration of noncommutative instantons on these varieties. We develop a noncommutative version of twistor theory, which introduces a new example of a noncommutative four-sphere. We develop a braided version of the ADHM construction and show that it parameterizes a certain moduli space of framed torsion free sheaves on a noncommutative projective plane. We use these constructions to explicitly build instanton gauge bundles with canonical connections on the noncommutative four-sphere that satisfy appropriate anti-selfduality equations. We construct projective moduli spaces for the torsion free sheaves and demonstrate that they are smooth. We define equivariant partition functions of these moduli spaces, finding that they coincide with the usual instanton partition functions for supersymmetric gauge theories on $\mathbb{C}2$.

Adinkras are diagrams that describe many useful supermultiplets in $D = 1$ dimensions. We show that the topology of the Adinkra is uniquely determined by a doubly even code. Conversely, every doubly even code produces a possible topology of an Adinkra. A computation of doubly even codes results in an enumeration of these Adinkra topologies up to $N = 28$, and for minimal supermultiplets, up to $N = 32$.