Five-branes, Seven-branes and Five-dimensional E_n field theories

We generalize the (p,q) 5-brane web construction of five-dimensional field theories by introducing (p,q) 7-branes, and apply this construction to theories with a one-dimensional Coulomb branch. The 7-branes render the exceptional global symmetry of these theories manifest. Additionally, 7-branes allow the construction of all E_n theories up to n=8, previously not possible in 5-brane configurations. The exceptional global symmetry in the field theory is a subalgebra of an affine symmetry on the 7-branes, which is necessary for the existence of the system. We explicitly determine the quantum numbers of the BPS states of all E_n theories using two simple geometrical constraints.Comment: 28 pages, LaTeX, 8 figure

The Gauss map and secants of the Kummer variety

Fay's trisecant formula shows that the Kummer variety of the Jacobian of a smooth projective curve has a four dimensional family of trisecant lines. We study when these lines intersect the theta divisor of the Jacobian, and prove that the Gauss map of the theta divisor is constant on these points of intersection, when defined. We investigate the relation between the Gauss map and multisecant planes of the Kummer variety as well.Comment: Minor changes, to appear on the Bulletin of London Mathematical Societ

Conceptual aspects of line tensions

We analyze two representative systems containing a three-phase-contact line: a liquid lens at a fluid--fluid interface and a liquid drop in contact with a gas phase residing on a solid substrate. We discuss to which extent the decomposition of the grand canonical free energy of such systems into volume, surface, and line contributions is unique in spite of the freedom one has in positioning the Gibbs dividing interfaces. In the case of a lens it is found that the line tension is independent of arbitrary choices of the Gibbs dividing interfaces. In the case of a drop, however, one arrives at two different possible definitions of the line tension. One of them corresponds seamlessly to that applicable to the lens. The line tension defined this way turns out to be independent of choices of the Gibbs dividing interfaces. In the case of the second definition,however, the line tension does depend on the choice of the Gibbs dividing interfaces. We provide equations for the equilibrium contact angles which are form-invariant with respect to notional shifts of dividing interfaces which only change the description of the system. Conceptual consistency requires to introduce additional stiffness constants attributed to the line. We show how these constants transform as a function of the relative displacements of the dividing interfaces. The dependences of the contact angles on lens or drop volumes do not render the line tension alone but a combination of the line tension, the Tolman length, and the stiffness constants of the line.Comment: 34 pages, 9 figure

Context Semantics, Linear Logic and Computational Complexity

We show that context semantics can be fruitfully applied to the quantitative analysis of proof normalization in linear logic. In particular, context semantics lets us define the weight of a proof-net as a measure of its inherent complexity: it is both an upper bound to normalization time (modulo a polynomial overhead, independently on the reduction strategy) and a lower bound to the number of steps to normal form (for certain reduction strategies). Weights are then exploited in proving strong soundness theorems for various subsystems of linear logic, namely elementary linear logic, soft linear logic and light linear logic.Comment: 22 page

Singularities of moduli of curves with a universal root

In a series of recent papers, Chiodo, Farkas and Ludwig carry out a deep analysis of the singular locus of the moduli space of stable (twisted) curves with an $\ell$-torsion line bundle. They show that for $\ell\leq 6$ and $\ell\neq 5$ pluricanonical forms extend over any desingularization. This allows to compute the Kodaira dimension without desingularizing, as done by Farkas and Ludwig for $\ell=2$, and by Chiodo, Eisenbud, Farkas and Schreyer for $\ell=3$. Here we treat roots of line bundles on the universal curve systematically: we consider the moduli space of curves $C$ with a line bundle $L$ such that $L^{\otimes\ell}\cong\omega_C^{\otimes k}$. New loci of canonical and non-canonical singularities appear for any $k\not\in\ell\mathbb{Z}$ and $\ell>2$, we provide a set of combinatorial tools allowing us to completely describe the singular locus in terms of dual graph. We characterize the locus of non-canonical singularities, and for small values of $\ell$ we give an explicit description.Comment: 30 pages, to appear in Documenta Mathematic

Bose-Fermi duality and entanglement entropies

Entanglement (Renyi) entropies of spatial regions are a useful tool for characterizing the ground states of quantum field theories. In this paper we investigate the extent to which these are universal quantities for a given theory, and to which they distinguish different theories, by comparing the entanglement spectra of the massless Dirac fermion and the compact free boson in two dimensions. We show that the calculation of Renyi entropies via the replica trick for any orbifold theory includes a sum over orbifold twists on all cycles. In a modular-invariant theory of fermions, this amounts to a sum over spin structures. The result is that the Renyi entropies respect the standard Bose-Fermi duality. Next, we investigate the entanglement spectrum for the Dirac fermion without a sum over spin structures, and for the compact boson at the self-dual radius. These are not equivalent theories; nonetheless, we find that (1) their second Renyi entropies agree for any number of intervals, (2) their full entanglement spectra agree for two intervals, and (3) the spectrum generically disagrees otherwise. These results follow from the equality of the partition functions of the two theories on any Riemann surface with imaginary period matrix. We also exhibit a map between the operators of the theories that preserves scaling dimensions (but not spins), as well as OPEs and correlators of operators placed on the real line. All of these coincidences can be traced to the fact that the momentum lattice for the bosonized fermion is related to that of the self-dual boson by a 45 degree rotation that mixes left- and right-movers.Comment: 40 pages; v3: improvements to presentation, new section discussing entanglement negativit