Controlling the path of discretized light in waveguide lattices

A general method for flexible control of the path of discretized light beams in homogeneous waveguide lattices, based on longitudinal modulation of the coupling constant, is theoretically proposed. As compared to beam steering and refraction achievable in graded-index waveguide arrays, the proposed approach enables to synthesize rather arbitrary target paths

Undecidability of Two-dimensional Robot Games

Robot game is a two-player vector addition game played on the integer lattice $\mathbb{Z}^n$. Both players have sets of vectors and in each turn the vector chosen by a player is added to the current configuration vector of the game. One of the players, called Eve, tries to play the game from the initial configuration to the origin while the other player, Adam, tries to avoid the origin. The problem is to decide whether or not Eve has a winning strategy. In this paper we prove undecidability of the robot game in dimension two answering the question formulated by Doyen and Rabinovich in 2011 and closing the gap between undecidable and decidable cases

Blowup formulae in Donaldson-Witten theory and integrable hierarchies

We investigate blowup formulae in Donaldson-Witten theory with gauge group SU(N), using the theory of hyperelliptic Kleinian functions. We find that the blowup function is a hyperelliptic sigma-function and we describe an explicit procedure to expand it in terms of the Casimirs of the gauge group up to arbitrary order. As a corollary, we obtain a new expression for the contact terms and we show that the correlation functions involving the exceptional divisor are governed by the KdV hierarchy. We also show that, for manifolds of simple type, the blowup function becomes a tau-function for a multisoliton solution.Comment: 34 pages, harvmac, typos corrected, final versio

Remarks on twisted theories with matter

We investigate some aspects of N=2 twisted theories with matter hypermultiplets in the fundamental representation of the gauge group. A consistent formulation of these theories on a general four-manifold requires turning on a particular magnetic flux, which we write down explicitly in the case of SU(2k). We obtain the blowup formula and show that the blowup function is given by a hyperelliptic sigma-function with singular characteristic. We compute the contact terms and find, as a corollary, interesting identities between hyperelliptic Theta functions.Comment: 18 pages, harvmac, one figur

Constraints for the existence of flat and stable non-supersymmetric vacua in supergravity

We further develop on the study of the conditions for the existence of locally stable non-supersymmetric vacua with vanishing cosmological constant in supergravity models involving only chiral superfields. Starting from the two necessary conditions for flatness and stability derived in a previous paper (which involve the Kahler metric and its Riemann tensor contracted with the supersymmetry breaking auxiliary fields) we show that the implications of these constraints can be worked out exactly not only for factorizable scalar manifolds, but also for symmetric coset manifolds. In both cases, the conditions imply a strong restriction on the Kahler geometry and constrain the vector of auxiliary fields defining the Goldstino direction to lie in a certain cone. We then apply these results to the various homogeneous coset manifolds spanned by the moduli and untwisted matter fields arising in string compactifications, and discuss their implications. Finally, we also discuss what can be said for completely arbitrary scalar manifolds, and derive in this more general case some explicit but weaker restrictions on the Kahler geometry.Comment: 22 pages, Latex, no figure

Soft masses in superstring models with anomalous U(1) symmetries

We analyze the general structure of soft scalar masses emerging in superstring models involving anomalous U(1) symmetries, with the aim of characterizing more systematically the circumstances under which they can happen to be flavor universal. We consider both heterotic orbifold and intersecting brane models, possibly with several anomalous and non-anomalous spontaneously broken U(1) symmetries. The hidden sector is assumed to consist of the universal dilaton, Kahler class and complex structure moduli, which are supposed to break supersymmetry, and a minimal set of Higgs fields which compensate the Fayet-Iliopoulos terms. We leave the superpotential that is supposed to stabilize the hidden sector fields unspecified, but we carefully take into account the relations implied by gauge invariance and the constraints required for the existence of a metastable vacuum with vanishing cosmological constant. The results are parametrized in terms of a constrained Goldstino direction, suitably defined effective modular weights, and the U(1) charges and shifts. We show that the effect induced by vector multiplets strongly depends on the functional form of the Kahler potential for the Higgs fields. We find in particular that whenever these are charged matter fields, like in heterotic models, the effect is non-trivial, whereas when they are shifting moduli fields, like in certain intersecting brane models, the effect may vanish.Comment: 35 pages, LaTe

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