A note on semi-bent functions with multiple trace terms and hyperelliptic curves

Semi-bent functions with even number of variables are a class of important Boolean functions whose Hadamard transform takes three values. In this note we are interested in the property of semi-bentness of Boolean functions defined on the Galois field $F_{2^n}$ (n even) with multiple trace terms obtained via Niho functions and two Dillon-like functions (the first one has been studied by Mesnager and the second one have been studied very recently by Wang, Tang, Qi, Yang and Xu). We subsequently give a connection between the property of semi-bentness and the number of rational points on some associated hyperelliptic curves. We use the hyperelliptic curve formalism to reduce the computational complexity in order to provide a polynomial time and space test leading to an efficient characterization of semi-bentness of such functions (which includes an efficient characterization of the hyperbent functions proposed by Wang et al.). The idea of this approach goes back to the recent work of Lisonek on the hyperbent functions studied by Charpin and Gong

Nonsupersymmetric Brane/Antibrane Configurations in Type IIA and M Theory

We study metastable nonsupersymmetric configurations in type IIA string theory, obtained by suspending D4-branes and anti-D4-branes between holomorphically curved NS5's, which are related to those of hep-th/0610249 by T-duality. When the numbers of branes and antibranes are the same, we are able to obtain an exact M theory lift which can be used to reliably describe the vacuum configuration as a curved NS5 with dissolved RR flux for g_s<<1 and as a curved M5 for g_s>>1. When our weakly coupled description is reliable, it is related by T-duality to the deformed IIB geometry with flux of hep-th/0610249 with moduli exactly minimizing the potential derived therein using special geometry. Moreover, we can use a direct analysis of the action to argue that this agreement must also hold for the more general brane/antibrane configurations of hep-th/0610249. On the other hand, when our strongly coupled description is reliable, the M5 wraps a nonholomorphic minimal area curve that can exhibit quite different properties, suggesting that the residual structure remaining after spontaneous breaking of supersymmetry at tree level can be further broken by the effects of string interactions. Finally, we discuss the boundary condition issues raised in hep-th/0608157 for nonsupersymmetric IIA configurations, their implications for our setup, and their realization on the type IIB side.Comment: 84 pages (57 pages + 4 appendices), 18 figure

Semi-classical quantisation of magnetic solitons in the anisotropic Heisenberg quantum chain

Using the algebro-geometric approach, we study the structure of semi-classical eigenstates in a weakly-anisotropic quantum Heisenberg spin chain. We outline how classical nonlinear spin waves governed by the anisotropic Landau-Lifshitz equation arise as coherent macroscopic low-energy fluctuations of the ferromagnetic ground state. Special emphasis is devoted to the simplest types of solutions, describing precessional motion and elliptic magnetisation waves. The internal magnon structure of classical spin waves is resolved by performing the semi-classical quantisation using the Riemann-Hilbert problem approach. We present an expression for the overlap of two semi-classical eigenstates and discuss how correlation functions at the semi-classical level arise from classical phase-space averaging.Comment: 61 pages, 14 figure

Representations of the quantum Teichmuller space, and invariants of surface diffeomorphisms

We investigate the representation theory of the polynomial core of the quantum Teichmuller space of a punctured surface S. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell decompositions of S. Our main result is that irreducible finite-dimensional representations of this polynomial core are classified, up to finitely many choices, by group homomorphisms from the fundamental group of the surface to the isometry group of the hyperbolic 3--space. We exploit this connection between algebra and hyperbolic geometry to exhibit new invariants of diffeomorphisms of S.Comment: Revised introduction. To appear in Geometry & Topolog

Quantum Deconstruction of 5D SQCD

We deconstruct the fifth dimension of 5D SCQD with general numbers of colors and flavors and general 5D Chern-Simons level; the latter is adjusted by adding extra quarks to the 4D quiver. We use deconstruction as a non-stringy UV completion of the quantum 5D theory; to prove its usefulness, we compute quantum corrections to the SQCD_5 prepotential. We also explore the moduli/parameter space of the deconstructed SQCD_5 and show that for |K_CS| < N_F/2 it continues to negative values of 1/(g_5)^2. In many cases there are flop transitions connecting SQCD_5 to exotic 5D theories such as E0, and we present several examples of such transitions. We compare deconstruction to brane-web engineering of the same SQCD_5 and show that the phase diagram is the same in both cases; indeed, the two UV completions are in the same universality class, although they are not dual to each other. Hence, the phase structure of an SQCD_5 (and presumably any other 5D gauge theory) is inherently five-dimensional and does not depends on a UV completion.Comment: LaTeX+PStricks, 108 pages, 41 colored figures. Please print in colo

Cosmology: macro and micro

A new approach to cosmology and space-time is developed, which emphasizes the description of the matter degrees of freedom of Einstein's theory of gravity by a family of K\"ahler-Einstein Fano manifolds