The Equivariant Cohomology Theory of Twisted Generalized Complex Manifolds

It has been shown recently by Kapustin and Tomasiello that the mathematical notion of Hamiltonian actions on twisted generalized Kähler manifolds is in perfect agreement with the physical notion of general (2, 2) gauged sigma models with three-form fluxes. In this article, we study the twisted equivariant cohomology theory of Hamiltonian actions on H-twisted generalized complex manifolds. If the manifold satisfies the ${\overline{\partial} \partial}It has been shown recently by Kapustin and Tomasiello that the mathematical notion of Hamiltonian actions on twisted generalized K?hler manifolds is in perfect agreement with the physical notion of general (2, 2) gauged sigma models with three-form fluxes. In this article, we study the twisted equivariant cohomology theory of Hamiltonian actions on H-twisted generalized complex manifolds. If the manifold satisfies the -lemma, we establish the equivariant formality theorem. If in addition, the manifold satisfies the generalized K?hler condition, we prove the Kirwan injectivity in this setting. We then consider the Hamiltonian action of a torus on an H-twisted generalized Calabi-Yau manifold and extend to this case the Duistermaat-Heckman theorem for the push-forward measure. As a side result, we show in this paper that the generalized K?hler quotient of a generalized K?hler vector space can never have a (cohomologically) non-trivial twisting. This gives a negative answer to a question asked by physicists whether one can construct (2, 2) gauged linear sigma models with non-trivial fluxes.     

A Consideration of BRST Cohomology in the First Quantization of RNS String

The difference of Grassmanian even-odd and degree even-odd of (super-) conformal ghosts is stressed. With some modifications in the BRST charges and the commutation relations, it can be shown that the first quantization of RNS strings allows the BRST cohomology analysis.     

Chiral BRST Cohomology of N= 2 Strings at Arbitrary Ghost and Picture Number

We compute the BRST cohomology of the holomorphic part of the N= 1 string at arbitrary ghost and picture number. We confirm the expectation that the relative cohomology at non-zero momentum consists of a single massless state in each picture. The absolute cohomology is obtained by an independent method based on homological algebra. For vanishing momentum, the relative and absolute cohomologies both display a picture dependence – a phenomenon discovered recently also in the relative Ramond sector of N= 1 strings by Berkovits and Zwiebach [1]. Received: 5 January 1998 / Accepted: 16 November 1998     

A New Cohomology Theory of Orbifold

Based on the orbifold string theory model in physics, we construct a new cohomology ring for any almost complex orbifold. The key theorem is the associativity of this new ring. Some examples are computed.Both authors partially supported by the National Science Foundation     

A Cohomology Theory of Grading-Restricted Vertex Algebras

We introduce a cohomology theory of grading-restricted vertex algebras. To construct the correct cohomologies, we consider linear maps from tensor powers of a grading-restricted vertex algebra to “rational functions valued in the algebraic completion of a module for the algebra,” instead of linear maps from tensor powers of the algebra to a module for the algebra. One subtle complication arising from such functions is that we have to carefully address the issue of convergence when we compose these linear maps with vertex operators. In particular, for each ${n \in \mathbb{N}}$ , we have an inverse system ${\{H^{n}_{m}(V, W)\}_{m \in \mathbb{Z}_{+}}}$ of nth cohomologies and an additional nth cohomology ${H_{\infty}^{n}(V, W)}$ of a grading-restricted vertex algebra V with coefficients in a V-module W such that ${H_{\infty}^{n}(V, W)}$ is isomorphic to the inverse limit of the inverse system ${\{H^{n}_{m}(V, W)\}_{m\in \mathbb{Z}_{+}}}$ . In the case of n = 2, there is an additional second cohomology denoted by ${H^{2}_{\frac{1}{2}}(V, W)}$ which will be shown in a sequel to the present paper to correspond to what we call square-zero extensions of V and to first order deformations of V when W = V.     

Chiral fermions on a compact Riemann surface: A sheaf cohomology analysis

We prove that the correlation functions of a system of chiral fermions on a compact Riemann surface are determined by postulating their behaviour at coincident points and a principle of maximal analyticity. The proof proceeds by a reformulation as a problem of sheaf cohomology. Wick's theorem and the Fay identities are rigorous consequences of our analysis.     

A New Cohomology Theory Associated to Deformations of Lie Algebra Morphisms

We introduce a new cohomology theory related to deformations of Lie algebra morphisms. This notion involves simultaneous deformations of two Lie algebras and a homomorphism between them.This revised version was published online in March 2005 with corrections to the cover date.     

一种新的核磁共振手性溶解剂罗通定

以罗通定为手性溶解剂,用核磁共振方法分析了几种小消旋羧酸。考察了化学计量比和浓度对化学位移不等价（△δ）的影响,并比较了在溶剂ＣＤＣｌ３或Ｃ６Ｄ６中,作为羧酸用手性溶解剂罗通定和（Ｓ）－（－）－α－苯乙胺的效果。     

Chiral nanoemitter array: A launchpad for optical vortices

A chiral arrangement of molecular nanoemitters is shown to support delocalised exciton states whose spontaneous decay can generate optical vortex radiation. In contrast to techniques in which phase modification is imposed upon conventional optical beams, this exciton method enables radiation with a helical wave‐front to be produced directly. To achieve this end, a number of important polarisation and symmetry‐based criteria need to be satisfied. It emerges that the phase structure of the optical field produced by degenerate excitons in a propeller‐shaped array can exhibit precisely the sought character of an optical vortex – one with unit topological charge. Practical considerations for the further development of this technique are discussed, and potential new applications are identified.     

A New Look at Calendar Anomalies: Multifractality and Day-of-the-Week Effect

Stock markets can become inefficient due to calendar anomalies known as the day-of-the-week effect. Calendar anomalies are well known in the financial literature, but the phenomena remain to be explored in econophysics. This paper uses multifractal analysis to evaluate if the temporal dynamics of market returns also exhibit calendar anomalies such as day-of-the-week effects. We apply multifractal detrended fluctuation analysis (MF-DFA) to the daily returns of market indices worldwide for each day of the week. Our results indicate that distinct multifractal properties characterize individual days of the week. Monday returns tend to exhibit more persistent behavior and richer multifractal structures than other day-resolved returns. Shuffling the series reveals that multifractality arises from a broad probability density function and long-term correlations. The time-dependent multifractal analysis shows that the Monday returns’ multifractal spectra are much wider than those of other days. This behavior is especially persistent during financial crises. The presence of day-of-the-week effects in multifractal dynamics of market returns motivates further research on calendar anomalies for distinct market regimes.     

Probability of Local Bifurcation Type from a Fixed Point: A Random Matrix Perspective

Results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems (real, random matrices), time-delay dynamical systems (companion matrices), and a set of mappings known for their properties as universal approximators (neural networks). The eigenvalue spectrum is considered both numerically and analytically using previous work of Edelman et al. Based upon the numerical evidence, various conjectures are presented. The conclusion is that in many circumstances, most bifurcations from fixed points of large dynamical systems will be due to complex eigenvalues. Nevertheless, surprising situations are presented for which the aforementioned conclusion does not hold, e.g., real random matrices with Gaussian elements with a large positive mean and finite variance. PACS numbers: 05.45.−a, 05.45.Tp, 89.75.−k, 89.75.Fb     

A New Look at the Spin Glass Problem from a Deep Learning Perspective

Spin glass is the simplest disordered system that preserves the full range of complex collective behavior of interacting frustrating elements. In the paper, we propose a novel approach for calculating the values of thermodynamic averages of the frustrated spin glass model using custom deep neural networks. The spin glass system was considered as a specific weighted graph whose spatial distribution of the edges values determines the fundamental characteristics of the system. Special neural network architectures that mimic the structure of spin lattices have been proposed, which has increased the speed of learning and the accuracy of the predictions compared to the basic solution of fully connected neural networks. At the same time, the use of trained neural networks can reduce simulation time by orders of magnitude compared to other classical methods. The validity of the results is confirmed by comparison with numerical simulation with the replica-exchange Monte Carlo method.     

A Quasiparticle Equation of State with a Phenomenological Critical Point

We propose a hybrid parameterization of a quasiparticle equation of state, where a critical point is implemented phenomenologically. In this approach, a quasiparticle model with finite chemical potential is used to describe the quark-gluon plasma phase by fitting to the lattice quantum chromodynamics data at high temperature. On the other hand, the hadronic resonance gas model with excluded volume correction is employed for the hadronic phase. An interpolation scheme is implemented so that the phase transition is a smooth crossover when the chemical potential is smaller than a critical value, or otherwise approximately of the first order according to Ehrenfest’s classification. Also, the thermodynamic consistency is guaranteed for the equation of state related to both the quasiparticle model and the implementation of the critical point.     

On Some Second Hochschild Cohomology Spaces for Algebras of Functions on a Manifold

We show that the second Hochschild cohomology space for the space of smooth functions on a manifold corresponding to cochains defined by continuous operators is the same as the one corresponding to differentiable operators, i.e. is given by the space of skewsymmetric contravariant 2-tensors on the manifold. We do this using a coboundary construction due to Omori, Maeda and Yoshioka.     

The Informational Nature of Quantum Mechanics: A Novel Look at the Interference Experiment

It is argued that the nature of probability is essentially informational rather than physical and that quantum mechanical predictions should be viewed as logical inferences made on the basis of the information content of a given experimental situation. By implementing such a viewpoint, it is possible to maintain a sharp distinction between the physical and statistical aspects of quantum mechanics. The idea is applied to the double-beam interference experiment, reproducing the results of the standard formulation of quantum mechanics in a manner that renders the notion of wave-particle duality superfluous.     

Towards Cohomology of Renormalization: Bigrading the Combinatorial Hopf Algebra of Rooted Trees

The renormalization of quantum field theory twists the antipode of a noncocommutative Hopf algebra of rooted trees, decorated by an infinite set of primitive divergences. The Hopf algebra of undecorated rooted trees, ℋ _R , generated by a single primitive divergence, solves a universal problem in Hochschild cohomology. It has two nontrivial closed Hopf subalgebras: the cocommutative subalgebra ℋ_ladder of pure ladder diagrams and the Connes–Moscovici noncocommutative subalgebra ℋ_CM of noncommutative geometry. These three Hopf algebras admit a bigrading by n, the number of nodes, and an index k that specifies the degree of primitivity. In each case, we use iterations of the relevant coproduct to compute the dimensions of subspaces with modest values of n and k and infer a simple generating procedure for the remainder. The results for ℋ_ladder are familiar from the theory of partitions, while those for ℋ_CM involve novel transforms of partitions. Most beautiful is the bigrading of ℋ _R , the largest of the three. Thanks to Sloane's superseeker, we discovered that it saturates all possible inequalities. We prove this by using the universal Hochschild-closed one-cocycle B ₊, which plugs one set of divergences into another, and by generalizing the concept of natural growth beyond that entailed by the Connes–Moscovici case. We emphasize the yet greater challenge of handling the infinite set of decorations of realistic quantum field theory. Received: 31 January 2000 / Accepted: 7 July 2000     

Gamow Vectors for Resonances: A Lax-Phillips Point of View

Results from the Lax-Phillips Scattering Theory are used to analyze quantum mechanical scattering systems, in particular to obtain spectral properties of their resonances which are defined to be the poles of the scattering matrix. For this approach the interplay between the positive energy projection and the Hardy-space projections is decisive. Among other things it turns out that the spectral properties of these poles can be described by the (discrete) eigenvalue spectrum of a so-called truncated evolution, whose eigenvectors can be considered as the Gamow vectors corresponding to these poles. Further an expansion theorem of the positive Hardy-space part of vectors Sg (S scattering operator) into a series of Gamow vectors is presented.