Comments on Two-Loop Four-Particle Amplitude in Superstring Theory

It is shown that the four-particle amplitude of superstring theory at two loops obtained in [1,2] is equivalent to the previously obtained results in [3,4,5]. Here the ${\bf Z}_2$ symmetry in hyperelliptic Riemann surface plays an important role in the proof.Comment: 10 pages, Latex fil

Two-Loop Superstrings in Hyperelliptic Language III: the Four-Particle Amplitude

We compute explicitly the four-particle amplitude in superstring theories by using the hyperelliptic language and the newly obtained chiral measure of D'Hoker and Phong. Although the algebra of the intermediate steps is a little bit involved, we obtain a quite simple expression for the four-particle amplitude. As expected, the integrand is independent of all the insertion points. As an application of the obtained result, we show that the perturbative correction to the $R^4$ term in type II superstring theories is vanishing point-wise in (even) moduli space at two loops.Comment: v1, LaTex file, 33 pages; v2, 34 pages, add references and minor correction

Topological Spin Texture in a Quantum Anomalous Hall Insulator

The quantum anomalous Hall (QAH) effect has been recently discovered in experiment using thin-film topological insulator with ferromagnetic ordering and strong spin-orbit coupling. Here we investigate the spin degree of freedom of a QAH insulator and uncover a fundamental phenomenon that the edge states exhibit topologically stable spin texture in the boundary when a chiral-like symmetry is present. This result shows that edge states are chiral in both the orbital and spin degrees of freedom, and the chiral edge spin texture corresponds to the bulk topological states of the QAH insulator. We also study the potential applications of the edge spin texture in designing topological-state-based spin devices which might be applicable to future spintronic technologies.Comment: 5 pages manuscript, 8+ pages supplementary information, 8 figures; published versio

Testing Cosmological Models with Type Ic Super Luminous Supernovae

The use of type Ic Super Luminous Supernovae (SLSN Ic) to examine the cosmological expansion introduces a new standard ruler with which to test theoretical models. The sample suitable for this kind of work now includes 11 SLSNe Ic, which have thus far been used solely in tests involving $\Lambda$CDM. In this paper, we broaden the base of support for this new, important cosmic probe by using these observations to carry out a one-on-one comparison between the $R_{\rm h}=ct$ and $\Lambda$CDM cosmologies. We individually optimize the parameters in each cosmological model by minimizing the $\chi^{2}$ statistic. We also carry out Monte Carlo simulations based on these current SLSN Ic measurements to estimate how large the sample would have to be in order to rule out either model at a $\sim 99.7\%$ confidence level. The currently available sample indicates a likelihood of $\sim$$70-80\%$ that the $R_{\rm h}=ct$ Universe is the correct cosmology versus $\sim$$20-30\%$ for the standard model. These results are suggestive, though not yet compelling, given the current limited number of SLSNe Ic. We find that if the real cosmology is $\Lambda$CDM, a sample of $\sim$$240$ SLSNe Ic would be sufficient to rule out $R_{\rm h}=ct$ at this level of confidence, while $\sim$$480$ SLSNe Ic would be required to rule out $\Lambda$CDM if the real Universe is instead $R_{\rm h}=ct$. This difference in required sample size reflects the greater number of free parameters available to fit the data with $\Lambda$CDM. If such SLSNe Ic are commonly detected in the future, they could be a powerful tool for constraining the dark-energy equation of state in $\Lambda$CDM, and differentiating between this model and the $R_{\rm h}=ct$ Universe.Comment: 33 pages, 9 figures, 1 table. Accepted for publication in AJ. arXiv admin note: text overlap with arXiv:1405.2388, arXiv:1410.0875; text overlap with arXiv:1409.4429 by other author