The Contour Spectrum

We introduce the contour spectrum, a user interface component that improves qualitative user interaction and provides real-time exact quantification in the visualization of isocontours. The contour spectrum is a signature consisting of a variety of scalar data and contour attributes, computed over the range of scalar values w 2!.We explore the use of surface area, volume, and gradientintegral of the contour that are shown to be univariate B-spline functions of the scalar value w for multi-dimensional unstructured triangular grids. These quantitative properties are calculated in real-time and presented to the user as a collection of signature graphs (plots of functions of w) to assist in selecting relevant isovalues w 0 for informative visualization. For timevarying data, these quantitative properties can also be computed over time, and displayed using a 2D interface, giving the user an overview of the time-varying function, and allowing interaction in both isovalue and timestep. The effectiveness of the current system and potential extensions are discussed

Regularization dependence of the OTOC. Which Lyapunov spectrum is the physical one?

We study the contour dependence of the out-of-time-ordered correlation function (OTOC) both in weakly coupled field theory and in the Sachdev-Ye-Kitaev (SYK) model. We show that its value, including its Lyapunov spectrum, depends sensitively on the shape of the complex time contour in generic weakly coupled field theories. For gapless theories with no thermal mass, such as SYK, the Lyapunov spectrum turns out to be an exception; their Lyapunov spectra do not exhibit contour dependence, though the full OTOCs do. Our result puts into question which of the Lyapunov exponents computed from the exponential growth of the OTOC reflects the actual physical dynamics of the system. We argue that, in a weakly coupled $\Phi^4$ theory, a kinetic theory argument indicates that the symmetric configuration of the time contour, namely the one for which the bound on chaos has been proven, has a proper interpretation in terms of dynamical chaos. Finally, we point out that a relation between these OTOCs and a quantity which may be measured experimentally --- the Loschmidt echo --- also suggests a symmetric contour configuration, with the subtlety that the inverse periodicity in Euclidean time is half the physical temperature. In this interpretation the chaos bound reads $\lambda \leq \frac{2\pi}{\beta}= \pi T_{\text{physical}}$.Comment: Comment on regularization dependence in 2d-CFTs added. Published versio

Genus Two Surface and Quarter BPS Dyons: The Contour Prescription

Following the suggestion of hep-th/0506249 and hep-th/0612011, we represent quarter BPS dyons in N=4 supersymmetric string theories as string network configuration and explore the role of genus two surfaces in determining the spectrum of such dyons. Our analysis leads to the correct contour prescription for integrating the partition function to determine the spectrum in different domains of the moduli space separated by the walls of marginal stability.Comment: LaTeX file, 25 page

Closed Contour Fractal Dimension Estimation by the Fourier Transform

This work proposes a novel technique for the numerical calculus of the fractal dimension of fractal objects which can be represented as a closed contour. The proposed method maps the fractal contour onto a complex signal and calculates its fractal dimension using the Fourier transform. The Fourier power spectrum is obtained and an exponential relation is verified between the power and the frequency. From the parameter (exponent) of the relation, it is obtained the fractal dimension. The method is compared to other classical fractal dimension estimation methods in the literature, e. g., Bouligand-Minkowski, box-couting and classical Fourier. The comparison is achieved by the calculus of the fractal dimension of fractal contours whose dimensions are well-known analytically. The results showed the high precision and robustness of the proposed technique

Comments on the Mirror TBA

We discuss various aspects of excited state TBA equations describing the energy spectrum of the AdS_5 \times S^5 strings and, via the AdS/CFT correspondence, the spectrum of scaling dimensions of N = 4 SYM local operators. We observe that auxiliary roots which are used to partially enumerate solutions of the Bethe-Yang equations do not play any role in engineering excited state TBA equations via the contour deformation trick. We further argue that the TBA equations are in fact written not for a particular string state but for the whole superconformal multiplet, and, therefore, the psu(2,2|4) invariance is built in into the TBA construction.Comment: 28 pages, 1 figure, v2: misprints are correcte