In this work, we propose a testing procedure to distinguish between the different approaches for computing complexity. Our test does not require a direct comparison between the approaches and thus avoids the issue of choice of gates, basis, etc. The proposed testing procedure employs the informationtheoretic measures Loschmidt echo and Fidelity; the idea is to investigate the sensitivity of the complexity (derived from the different approaches) to the evolution of states. We discover that only circuit complexity obtained directly from the wave function is sensitive to time evolution, leaving us to claim that it surpasses the other approaches. We also demonstrate that circuit complexity displays a universal behaviour-the complexity is proportional to the number of distinct Hamiltonian evolutions that act on a reference state. Due to this fact, for a given number of Hamiltonians, we can always find the combination of states that provides the maximum complexity; consequently, other combinations involving a smaller number of evolutions will have less than maximum complexity and, hence, will have resources. Finally, we explore the evolution of complexity in non-local theories; we demonstrate the growth of complexity is sustained over a longer period of time as compared to a local theory.
We propose a new diagnostic for quantum chaos. We show that time evolution of complexity for a particular type of target state can provide equivalent information about the classical Lyapunov exponent and scrambling time as out-of-time-order correlators. Moreover, for systems that can be switched from a regular to unstable (chaotic) regime by a tuning of the coupling constant of the interaction Hamiltonian, we find that the complexity defines a new time scale. We interpret this time scale as recording when the system makes the transition from regular to chaotic behaviour.
We derive expressions for the Ricci curvature tensor and scalar in terms of intrinsic torsion classes of half-flat manifolds by exploiting the relationship between half-flat manifolds and non-compact G 2 holonomy manifolds. Our expressions are tested for Iwasawa and more general nilpotent manifolds. We also derive expressions, in the language of Calabi-Yau moduli spaces, for the torsion classes and the Ricci curvature of the particular half-flat manifolds that arise naturally via mirror symmetry in flux compactifications. Using these expressions we then derive a constraint on the Kähler moduli space of type II string theories on these half-flat manifolds. * Tibra Ali@baylor.edu † Gerald Cleaver@baylor.edu are a special class of SU(3) manifolds which are the minimal ingredient for the N = 2 reduction of ten-dimensional type II string theory [24].Half-flat manifolds had already made their appearance in the work of Hitchin [11], and they play an important role in the construction of a special type of G 2 holonomy manifold. It is interesting to note that Hitchin's work emphasizes the form-aspects of the special geometries in six and seven dimensions, and thus is very close in spirit to topological string/M theories and the related form-theories of gravity.In this paper we were motivated by taking a closer look at the metric sector of the half-flat manifolds M proposed in [9], in particular we wanted to know if the Ricci tensor of M plays any role in the low energy effective theory. At the time of the onset of this work there were no formulae for the Ricci curvature of half-flat manifolds available in the literature, and we decided to derive expressions for it. Recently, as our work of the 'physics' part was nearing completion, which is based on the mathematical results that we derived earlier, we learned in [24] of a paper [30] in the mathematics archives which computes the Ricci curvature of manifolds with SU(3) structure.Thus the first half of our paper will have some overlap with [30] in content but not in technique, language or point of view.The central results of our paper are the following: We have derived expressions, eqs. (4.7) and (4.8), for the Ricci curvature of generic half-flat manifolds in terms of their intrinsic torsion by thinking of them as hypersurfaces that foliate a non-compact G 2 holonomy cylinderà la Hitchin [11]. We then derive the Ricci curvature, eqs. (6.12), (6.13) and (6.14), of the particular class of half-flat manifolds M that arise from mirror symmetry [9]. We then use a consistency argument involving the Ricci curvature of M in L eff to derive a condition, eqs. (6.24) or (6.26), on the Kähler moduli space. This leads, by using one of Hitchin's flow equations, to a formula (6.27) for the flow of the volume of M .At the risk of repeating ourselves, we now give a brief overview of the different sections of our paper. In section two we review properties of six-dimensional manifolds with SU(3) structure, considering in particular those of Calabi-Yau and half-flat manifolds. Intrinsic torsi...
We propose a model for large field inflation in heterotic string theory. The construction applies the near alignment mechanism of Kim, Nilles, and Peloso. By including gaugino condensates and world-sheet instanton non-perturbative effects, we obtain a large effective axion decay constant. § tibra.ali@pitp.ca ‡
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