Reflection groups in hyperbolic spaces and the denominator formula for Lorentzian Kac--Moody Lie algebras

This is a continuation of our "Lecture on Kac--Moody Lie algebras of the arithmetic type" \cite{25}. We consider hyperbolic (i.e. signature $(n,1)$) integral symmetric bilinear form $S:M\times M \to {\Bbb Z}$ (i.e. hyperbolic lattice), reflection group $W\subset W(S)$, fundamental polyhedron \Cal M of $W$ and an acceptable (corresponding to twisting coefficients) set P({\Cal M})\subset M of vectors orthogonal to faces of \Cal M (simple roots). One can construct the corresponding Lorentzian Kac--Moody Lie algebra {\goth g}={\goth g}^{\prime\prime}(A(S,W,P({\Cal M}))) which is graded by $M$. We show that \goth g has good behavior of imaginary roots, its denominator formula is defined in a natural domain and has good automorphic properties if and only if \goth g has so called {\it restricted arithmetic type}. We show that every finitely generated (i.e. P({\Cal M}) is finite) algebra {\goth g}^{\prime\prime}(A(S,W_1,P({\Cal M}_1))) may be embedded to {\goth g}^{\prime\prime}(A(S,W,P({\Cal M}))) of the restricted arithmetic type. Thus, Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type is a natural class to study. Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type have the best automorphic properties for the denominator function if they have {\it a lattice Weyl vector $\rho$}. Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type with generalized lattice Weyl vector $\rho$ are called {\it elliptic}Comment: Some corrections in Sects. 2.1, 2.2 were done. They don't reflect on results and ideas. 31 pages, no figures. AMSTe

Neurophysiophenomenology – predicting emotional arousal from brain arousal in a virtual reality roller coaster

Arousal is a core affect constituted of both bodily and subjective states that prepares an agent to respond to events of the natural environment. While the peripheral physiological components of arousal have been examined also under naturalistic conditions, its neural correlates were suggested mainly on the basis of simplifed experimental designs.   We used virtual reality (VR) to present a highly immersive and contextually rich scenario of roller coaster rides to evoke naturalistic states of emotional arousal. Simultaneously, we recorded EEG to validate the suggested neural correlates of arousal in alpha frequency oscillations (8-12Hz) over temporo-parietal cortical areas. To fnd the complex link between these alpha components and the participants’ continuous subjective reports of arousal, we employed a set of complementary analytical methods coming from machine learning and deep learning

Surface-enhanced optical third-harmonic generation in Ag island films

Surface-enhanced optical third-harmonic generation (THG) is observed in silver island films. The THG intensity from Ag nanoparticles is enhanced by more than two orders of magnitude with respect to the THG intensity from a smooth and homogeneous silver surface. This enhancement is attributed to local plasmon excitation and resonance of the local field at the third-harmonic wavelength. The diffuse and depolarized component of the enhanced THG is associated with the third-order hyper-Rayleigh scattering in a 2-D random array of silver nanoparticles.Comment: 4 pages, 2 figure

The Geometry and Moduli of K3 Surfaces

These notes will give an introduction to the theory of K3 surfaces. We begin with some general results on K3 surfaces, including the construction of their moduli space and some of its properties. We then move on to focus on the theory of polarized K3 surfaces, studying their moduli, degenerations and the compactification problem. This theory is then further enhanced to a discussion of lattice polarized K3 surfaces, which provide a rich source of explicit examples, including a large class of lattice polarizations coming from elliptic fibrations. Finally, we conclude by discussing the ample and Kahler cones of K3 surfaces, and give some of their applications.Comment: 34 pages, 2 figures. (R. Laza, M. Schutt and N. Yui, eds.

M-Theory on (K3 X S^1)/Z_2

We analyze $M$-theory compactified on $(K3\times S^1)/Z_2$ where the $Z_2$ changes the sign of the three form gauge field, acts on $S^1$ as a parity transformation and on K3 as an involution with eight fixed points preserving SU(2) holonomy. At a generic point in the moduli space the resulting theory has as its low energy limit N=1 supergravity theory in six dimensions with eight vector, nine tensor and twenty hypermultiplets. The gauge symmetry can be enhanced (e.g. to $E_8$) at special points in the moduli space. At other special points in the moduli space tensionless strings appear in the theory.Comment: LaTeX file, 11 page

K3-fibered Calabi-Yau threefolds I, the twist map

A construction of Calabi-Yaus as quotients of products of lower-dimensional spaces in the context of weighted hypersurfaces is discussed, including desingularisation. The construction leads to Calabi-Yaus which have a fiber structure, in particular one case has K3 surfaces as fibers. These Calabi-Yaus are of some interest in connection with Type II -heterotic string dualities in dimension 4. A section at the end of the paper summarises this for the non-expert mathematician.Comment: 31 pages LaTeX, 11pt, 2 figures. To appear in International Journal of Mathematics. On the web at http://personal-homepages.mis.mpg.de/bhunt/preprints.html , #

Aberrant Long-Range Temporal Correlations in Depression Are Attenuated after Psychological Treatment

The spontaneous oscillatory activity in the human brain shows long-range temporal correlations (LRTC) that extend over time scales of seconds to minutes. Previous research has demonstrated aberrant LRTC in depressed patients; however, it is unknown whether the neuronal dynamics normalize after psychological treatment. In this study, we recorded EEG during eyes-closed rest in depressed patients (N = 71) and healthy controls (N = 25), and investigated the temporal dynamics in depressed patients at baseline, and after attending either a brief mindfulness training or a stress reduction training. Compared to the healthy controls, depressed patients showed stronger LRTC in theta oscillations (4–7 Hz) at baseline. Following the psychological interventions both groups of patients demonstrated reduced LRTC in the theta band. The reduction of theta LRTC differed marginally between the groups, and explorative analyses of separate groups revealed noteworthy topographic differences. A positive relationship between the changes in LRTC, and changes in depressive symptoms was observed in the mindfulness group. In summary, our data show that aberrant temporal dynamics of ongoing oscillations in depressive patients are attenuated after treatment, and thus may help uncover the mechanisms with which psychotherapeutic interventions affect the brain

Magnetism and the phase diagram of MnSb2_2O6_6

Static and dynamic magnetic properties of P\={3}1$m$-phase MnSb$_2$O$_6$ have been studied by means of muon spin relaxation ($\mu$SR), high-frequency electron spin resonance (HF-ESR), specific heat, and magnetisation studies in magnetic fields up to 25\,T. The data imply onset of long-range antiferromagnetic order at $T_N$ =~8~K and a spin-flop-like transition at $B_{SF}\approx 0.7 - 1$~T. Below $T_N$, muon asymmetry exhibits well-defined oscillations indicating a narrow distribution of the local fields. A competing antiferromagnetic phase appearing below $T_2$ =~5.3~K is evidenced by a step in the magnetisation and a slight kink of the relaxation rate. Above $T_N$ , both $\mu$SR and HF-ESR data suggest short-range spin order. HF-ESR data show that local magnetic fields persist up to at least $12\cdot T_{\rm N}\approx 100$\,K. Analysis of the antiferromagnetic resonance modes and the thermodynamic spin-flop field suggest zero-field splitting of $\Delta \approx 18$\,GHz which implies small but finite magnetic anisotropy

Period- and mirror-maps for the quartic K3

We study in detail mirror symmetry for the quartic K3 surface in P3 and the mirror family obtained by the orbifold construction. As explained by Aspinwall and Morrison, mirror symmetry for K3 surfaces can be entirely described in terms of Hodge structures. (1) We give an explicit computation of the Hodge structures and period maps for these families of K3 surfaces. (2) We identify a mirror map, i.e. an isomorphism between the complex and symplectic deformation parameters, and explicit isomorphisms between the Hodge structures at these points. (3) We show compatibility of our mirror map with the one defined by Morrison near the point of maximal unipotent monodromy. Our results rely on earlier work by Narumiyah-Shiga, Dolgachev and Nagura-Sugiyama.Comment: 29 pages, 3 figure

Families of Quintic Calabi-Yau 3-Folds with Discrete Symmetries

At special loci in their moduli spaces, Calabi-Yau manifolds are endowed with discrete symmetries. Over the years, such spaces have been intensely studied and have found a variety of important applications. As string compactifications they are phenomenologically favored, and considerably simplify many important calculations. Mathematically, they provided the framework for the first construction of mirror manifolds, and the resulting rational curve counts. Thus, it is of significant interest to investigate such manifolds further. In this paper, we consider several unexplored loci within familiar families of Calabi-Yau hypersurfaces that have large but unexpected discrete symmetry groups. By deriving, correcting, and generalizing a technique similar to that of Candelas, de la Ossa and Rodriguez-Villegas, we find a calculationally tractable means of finding the Picard-Fuchs equations satisfied by the periods of all 3-forms in these families. To provide a modest point of comparison, we then briefly investigate the relation between the size of the symmetry group along these loci and the number of nonzero Yukawa couplings. We include an introductory exposition of the mathematics involved, intended to be accessible to physicists, in order to make the discussion self-contained.Comment: 54 pages, 3 figure