A simple parametrization for G2

We give a simple parametrization of the $G_2$ group, which is consistent with the structure of $G_2$ as a SU(3) fibration. We also explicitly compute the (bi)invariant measure, which turns out to have a simple expression.Comment: 9 page

Uniformization, Unipotent Flows and the Riemann Hypothesis

We prove equidistribution of certain multidimensional unipotent flows in the moduli space of genus $g$ principally polarized abelian varieties (ppav). This is done by studying asymptotics of $\pmb{\Gamma}_{g} \sim Sp(2g,\mathbb{Z})$-automorphic forms averaged along unipotent flows, toward the codimension-one component of the boundary of the ppav moduli space. We prove a link between the error estimate and the Riemann hypothesis. Further, we prove $\pmb{\Gamma}_{g - r}$ modularity of the function obtained by iterating the unipotent average process $r$ times. This shows uniformization of modular integrals of automorphic functions via unipotent flows

Duality invariance in Fayet-Iliopoulos gauged supergravity

We propose a geometric method to study the residual symmetries in $N=2$, $d=4$ $\text{U}(1)$ Fayet-Iliopoulos (FI) gauged supergravity. It essentially involves the stabilization of the symplectic vector of gauge couplings (FI parameters) under the action of the U-duality symmetry of the ungauged theory. In particular we are interested in those transformations that act non-trivially on the solutions and produce scalar hair and dyonic black holes from a given seed. We illustrate the procedure for finding this group in general and then show how it works in some specific models. For the prepotential $F=-iX^0X^1$, we use our method to add one more parameter to the rotating Chow-Comp\`ere solution, representing scalar hair.Comment: 31 pages, uses jheppub.sty. Final version to appear on JHE

Squaring the Magic

We construct and classify all possible Magic Squares (MS's) related to Euclidean or Lorentzian rank-3 simple Jordan algebras, both on normed division algebras and split composition algebras. Besides the known Freudenthal-Rozenfeld-Tits MS, the single-split G\"unaydin-Sierra-Townsend MS, and the double-split Barton-Sudbery MS, we obtain other 7 Euclidean and 10 Lorentzian novel MS's. We elucidate the role and the meaning of the various non-compact real forms of Lie algebras, entering the MS's as symmetries of theories of Einstein-Maxwell gravity coupled to non-linear sigma models of scalar fields, possibly endowed with local supersymmetry, in D = 3, 4 and 5 space-time dimensions. In particular, such symmetries can be recognized as the U-dualities or the stabilizers of scalar manifolds within space-time with standard Lorentzian signature or with other, more exotic signatures, also relevant to suitable compactifications of the so-called M*- and M'- theories. Symmetries pertaining to some attractor U-orbits of magic supergravities in Lorentzian space-time also arise in this framework.Comment: 21 pages, 1 figure, 20 tables; reference adde

E7 groups from octonionic magic square

In this paper we continue our program, started in [2], of building up explicit generalized Euler angle parameterizations for all exceptional compact Lie groups. Here we solve the problem for E7, by first providing explicit matrix realizations of the Tits construction of a Magic Square product between the exceptional octonionic algebra J and the quaternionic algebra H, both in the adjoint and the 56 dimensional representations. Then, we provide the Euler parametrization of E7 starting from its maximal subgroup U=(E6 x U(1))/Z3. Next, we give the constructions for all the other maximal compact subgroups.Comment: 23 pages, added sections with new construction