The moment map on symplectic vector space and oscillator representation

The aim of this paper is to show that the canonical quantization of the moment maps on symplectic vector spaces naturally gives rise to the oscillator representations. More precisely, let $(W,\omega)$ denote a real symplectic vector space, on which a Lie group $G$ acts symplectically on the left, where $G$ denotes a real reductive Lie group $\mathrm{Sp}(n,\mathbb R), \mathrm U(p,q)$ or $\mathrm O^*(2n)$ in this paper. Then we quantize the moment map $\mu: W \to \mathfrak g_0^*$, where $\mathfrak g_0^*$ denotes the dual space of the Lie algebra $\mathfrak g_0$ of $G$. Namely, after taking a complex Lagrangian subspace $V$ of the complexification of $W$, we assign an element of the Weyl algebra for $V$ to $$, which we denote by $< \hat{\mu}, X >$, for each $X \in \mathfrak g_0$. It is shown that the map $X \mapsto \mathrm i$ gives a representation of $\mathfrak g_0$ which extends to the one of $\mathfrak g$, the complexification of $\mathfrak g_0$, by linearity. With a suitable choice of the complex Lagrangian subspace $V$ in each case, the representation coincides with the oscillator representation of $\mathfrak g$.Comment: 24pages, no figure; corrected some typos (v2); 27pages, added Section 5 describing a relation between the image of complex Lagrangian subspaces by the moment map and the associated variety of the corresponding irreducible (g,K)-modules (v3); some references are added and replaced (v4

Self-organization on social media: endo-exo bursts and baseline fluctuations

A salient dynamic property of social media is bursting behavior. In this paper, we study bursting behavior in terms of the temporal relation between a preceding baseline fluctuation and the successive burst response using a frequency time series of 3,000 keywords on Twitter. We found that there is a fluctuation threshold up to which the burst size increases as the fluctuation increases and that above the threshold, there appears a variety of burst sizes. We call this threshold the critical threshold. Investigating this threshold in relation to endogenous bursts and exogenous bursts based on peak ratio and burst size reveals that the bursts below this threshold are endogenously caused and above this threshold, exogenous bursts emerge. Analysis of the 3,000 keywords shows that all the nouns have both endogenous and exogenous origins of bursts and that each keyword has a critical threshold in the baseline fluctuation value to distinguish between the two. Having a threshold for an input value for activating the system implies that Twitter is an excitable medium. These findings are useful for characterizing how excitable a keyword is on Twitter and could be used, for example, to predict the response to particular information on social media.Comment: Presented at WebAL-1: Workshop on Artificial Life and the Web 2014 (arXiv:1406.2507

Magnetic instability in AdS/CFT : Schwinger effect and Euler-Heisenberg Lagrangian of Supersymmetric QCD

To reveal the Schwinger effect for quarks, i.e., pair creation process of quarks and antiquarks, we derive the vacuum decay rate at strong coupling using AdS/CFT correspondence. Magnetic fields, in addition to the electric field responsible for the pair creation, causes prominent effects on the rate, and is important also in experiments such as RHIC/LHC heavy ion collisions. In this paper, through the gravity dual we obtain the full Euler-Heisenberg Lagrangian of N=2 supersymmetric QCD and study the Schwinger mechanism with not only a constant electric field but also a constant magnetic field as external fields. We determine the quark mass and temperature dependence of the Lagrangian. In sharp contrast with the zero magnetic field case, we find that the imaginary part, and thus the vacuum decay rate, diverges in the massless zero-temperature limit. This may be related to a strong instability of the QCD vacuum in strong magnetic fields. The real part of the Lagrangian serves as a generating function for non-linear electro-magnetic responses, and is found such that the Cotton-Mouton effect vanishes. Interestingly, our results of the Schwinger / Cotton-Mouton effects coincide precisely with those of N=2 supersymmetric QED.Comment: 21 pages. v2:references added, footnotes adde