Subfactors of Index less than 5, Part 1: The Principal Graph Odometer

In this series of papers we show that there are exactly ten subfactors, other than A _∞ subfactors, of index between 4 and 5. Previously this classification was known up to index \({3+\sqrt{3}}\). In the first paper we give an analogue of Haagerup’s initial classification of subfactors of index less than \({3+\sqrt{3}}\), showing that any subfactor of index less than 5 must appear in one of a large list of families. These families will be considered separately in the three subsequent papers in this series.     

$$\Lambda $$CDM: Much More Than We Expected,but Now Less Than What We Want

The \(\Lambda \)CDM cosmological model is remarkable: with just six parameters it describes the evolution of the Universe from a very early time when all structures were quantum fluctuations on subatomic scales to the present, and it is consistent with a wealth of high-precision data, both laboratory measurements and astronomical observations. However, the foundation of \(\Lambda \)CDM involves physics beyond the standard model of particle physics: particle dark matter, dark energy and cosmic inflation. Until this ‘new physics’ is clarified, \(\Lambda \)CDM is at best incomplete and at worst a phenomenological construct that accommodates the data. I discuss the path forward, which involves both discovery and disruption, some grand challenges and finally the limits of scientific cosmology.     

A Remark on CFT Realization of Quantum Doubles of Subfactors: Case Index { < 4}

It is well known that the quantum double \({D(N\subset M)}\) of a finite depth subfactor \({N\subset M}\), or equivalently the Drinfeld center of the even part fusion category, is a unitary modular tensor category. It is big open conjecture that all (unitary) modular tensor categories arise from conformal field theory. We show that for every subfactor \({N\subset M}\) with index \({[M:N] < 4}\) the quantum double \({D(N\subset M)}\) is realized as the representation category of a completely rational conformal net. In particular, the quantum double of \({E_6}\) can be realized as a \({\mathbb{Z}_2}\)-simple current extension of \({{{\rm SU}(2)}_{10}\times {{\rm Spin}(11)}_1}\) and thus is not exotic in any sense. As a byproduct, we obtain a vertex operator algebra for every such subfactor. We obtain the result by showing that if a subfactor \({N\subset M }\) arises from \({\alpha}\)-induction of completely rational nets \({\mathcal{A}\subset \mathcal{B}}\) and there is a net \({\tilde{\mathcal{A}}}\) with the opposite braiding, then the quantum \({D(N\subset M)}\) is realized by completely rational net. We construct completely rational nets with the opposite braiding of \({{{\rm SU}(2)}_k}\) and use the well-known fact that all subfactors with index \({[M:N] < 4}\) arise by \({\alpha}\)-induction from \({{{\rm SU}(2)}_k}\).     

On the Importance of Testing Gravity at Distances Less Than 1cm

If the mechanism responsible for the smallness of the vacuum energy is consistent with local quantum field theory, general arguments suggest the existence of at least one unobserved scalar particle with Compton wavelength bounded from below by one tenth of a millimeter. We show that this bound is saturated if vacuum energy is a substantial component of the energy density of the universe. Therefore, the success of cosmological models with a significant vacuum energy component suggests the existence of new macroscopic forces with range in the sub-millimeter region. There are virtually no experimental constraints on the existence of quanta with this range of interaction.     

What Makes a Theory Testable, or Is Intelligent Design Less Scientific Than String Theory?

I compare the theory of intelligent design to string theory to see on what basis, if any, only the former can be justifiably excluded as being scientific.We shall see that the classic criterion of testability or falsifiability is sometimes not so straightforward, and that there are other criteria to help us make such a distinction.     

Generalized Goodman–Harpe–Jones Construction of Subfactors, I

We construct some exceptional finite depth subfactors and determine their principal graphs from some exceptional integrable lattice models. Some of these subfactors are conjectured to be the same as those coming from certain conformal embeddings (, , and ) for which the principal graphs are previously unknown. Received: 24 November 1995 / Accepted: 14 May 1996     

Generalized Goodman–Harpe–Jones Construction of Subfactors, II

We build some exceptional representations of Birman–Wenzl algebras from the data of certain conformal embeddings. As a result we construct new finite depth subfactors whose principal graphs are completely determined. Received: 16 February 1996 / Accepted: 25 July 1996     

Excited state absorption and stimulated emission of Nd3+ in crystals. Part I: Y3Al5O12, YAlO3, and Y2O3

4 F_3/2 excited state of the Nd³⁺ ion in Y₃Al₅O₁₂, YAlO₃, and Y₂O₃ were measured in a continuous wave pump- and probe experiment in a wide spectral range from 850 nm (780 nm for Y₃Al₅O₁₂) to 1500 nm. The cross sections were determined from a comparison with the emission spectra and the simultaneously measured ground state absorption bleaching. The strongest excited state absorption transitions were found in the 1220–1400 nm spectral region due to transitions to the ²G_9/2 and ⁴G_7/2 levels. The spectral positions of the measured transitions are in good agreement with the theoretically expected transitions calculated from the known Stark-level splittings. Received: 4 December 1997/Revised version: 8 May 1998     

High Refractive Index Ti_3O_5 Films for Dielectric Metasurfaces

Ti_3 O_5 films are deposited with the help of an electron beam evaporator for their applications in metasurfaces.The film of subwavelength(632 nm) thickness is deposited on a silicon substrate and annealed at 400°C. The ellipsometr.y result shows a high refractive index above 2.5 with the minimum absorption coefficient in the visible region, which is necessary for high efficiency of transparent metasurfaces. Atomic force microscopy analysis is employed to measure the roughness of the as-deposited films. It is seen from micrographs that the deposited films are very smooth with the minimum roughness to prevent scattering and absorption losses for metasurface devices. The absence of grains and cracks can be seen by scanning electron microscope analysis, which is favorable for electron beam lithography. Fourier transform infrared spectroscopy reveals the transmission and reflection obtained from the film deposited on glass substrates. The as-deposited film shows high transmission above 60%,whieh is in good agreement with metasurfaces.     

Excited state absorption and stimulated emission of Nd3+ in crystals. Part 2: YVO4, GdVO4, and Sr5(PO4)3F

3+ ion in the crystals YVO₄, GdVO₄, and Sr₅(PO₄)₃F. The measurements were performed in the spectral region of the main laser transitions ⁴F_3/2→⁴I_9/2, ⁴F_3/2→⁴I_11/2, and ⁴F_3/2→⁴I_13/2by a continuous wave pump and probe technique. The calibrated gain and ESA spectra are presented and possible implications of ESA on the laser performance are estimated. It is shown that ESA can be a small loss factor to the laser emission near 1060 nm but does considerably diminish the effective emission cross sections near 1340 nm especially in Nd:YVO₄ and Nd:GdVO₄. Received: 29 January 1998/Revised version: 8 May 1998