1-form symmetries of 4d N=2 class S theories

We determine the 1-form symmetry group for any 4d4d\mathcal{N}=2𝒩=2 class S theory constructed by compactifying a 6d6d\mathcal{N}=(2,0)𝒩=(2,0) SCFT on a Riemann surface with arbitrary regular untwisted and twisted punctures. The 6d6d theory has a group of mutually non-local dimension-2 surface operators, modulo screening. Compactifying these surface operators leads to a group of mutually non-local line operators in 4d4d, modulo screening and flavor charges. Complete specification of a 4d4d theory arising from such a compactification requires a choice of a maximal subgroup of mutually local line operators, and the 1-form symmetry group of the chosen 4d4d theory is identified as the Pontryagin dual of this maximal subgroup. We also comment on how to generalize our results to compactifications involving irregular punctures. Finally, to complement the analysis from 6d, we derive the 1-form symmetry from a Type IIB realization of class S theories.

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4D Pyritohedral Symmetry

Sultan Qaboos University Journal for Science [SQUJS] ◽

10.24200/squjs.vol21iss2pp150-161 ◽

2016 ◽

Vol 21 (2) ◽

pp. 150

Author(s):

Nazife O. Koca ◽

Amal J.H. Al Qanobi ◽

Mehmet Koca

Keyword(s):

Euclidean Space ◽

Maximal Subgroup ◽

Symmetry Group ◽

Coxeter Groups ◽

Dimensional Euclidean Space ◽

Root Lattice

We describe an extension of the pyritohedral symmetry in 3D to 4-dimensional Euclidean space and construct the group elements of the 4D pyritohedral group of order 576 in terms of quaternions. It turns out that it is a maximal subgroup of both the rank-4 Coxeter groups W (F4) and W (H4), implying that it is a group relevant to the crystallographic as well as quasicrystallographic structures in 4-dimensions. We derive the vertices of the 24 pseudoicosahedra, 24 tetrahedra and the 96 triangular pyramids forming the facets of the pseudo snub 24-cell. It turns out that the relevant lattice is the root lattice of W (D4). The vertices of the dual polytope of the pseudo snub 24-cell consists of the union of three sets: 24-cell, another 24-cell and a new pseudo snub 24-cell. We also derive a new representation for the symmetry group of the pseudo snub 24-cell and the corresponding vertices of the polytopes.

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Bounds for the Order of Symmetry Group of Automorphism of Compact Riemann Surface

International Journal of Computer Applications ◽

10.5120/12739-9629 ◽

2013 ◽

Vol 73 (5) ◽

pp. 31-32

Author(s):

Chandra Chutia ◽

Moloya Bhuyan ◽

Rafiqul Islam

Keyword(s):

Riemann Surface ◽

Symmetry Group ◽

Compact Riemann Surface

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Moduli space of branched superminimal immersions of a compact Riemann surface into S4

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700036259 ◽

1999 ◽

Vol 66 (1) ◽

pp. 32-50 ◽

Cited By ~ 2

Author(s):

Bonaventure Loo

Keyword(s):

Riemann Surface ◽

Moduli Space ◽

Moduli Spaces ◽

Compact Riemann Surface ◽

Projective Varieties ◽

Fibre Products ◽

Dimension 2 ◽

Path Connected

AbstractIn this paper we describe the moduli spaces of degree d branched superminimal immersions of a compact Riemann surface of genus g into S4. We prove that when d ≥ max {2g, g + 2}, such spaces have the structure of projectivzed fibre products and are path-connected quasi projective varieties of dimension 2d − g + 4. This generalizes known results for spaces of harmonic 2-spheres in S4.

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Toward High-Resolution Low-Voltage SEM

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100103152 ◽

1988 ◽

Vol 46 ◽

pp. 218-219

Author(s):

Zhifeng Shao

Keyword(s):

Low Voltage ◽

Spherical Aberration ◽

Chromatic Aberration ◽

Limiting Factor ◽

Operating Voltage ◽

Probe Radius ◽

Non Local ◽

Electron Microscopes ◽

Image Position ◽

Scanning Electron Microscopes

Recently, low voltage (≤5kV) scanning electron microscopes have become popular because of their unprecedented advantages, such as minimized charging effects and smaller specimen damage, etc. Perhaps the most important advantage of LVSEM is that they may be able to provide ultrahigh resolution since the interaction volume decreases when electron energy is reduced. It is obvious that no matter how low the operating voltage is, the resolution is always poorer than the probe radius. To achieve 10Å resolution at 5kV (including non-local effects), we would require a probe radius of 5∽6 Å. At low voltages, we can no longer ignore the effects of chromatic aberration because of the increased ratio δV/V. The 3rd order spherical aberration is another major limiting factor. The optimized aperture should be calculated as

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Chromatic aberration and low-voltage SEM

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100127293 ◽

1987 ◽

Vol 45 ◽

pp. 546-547

Author(s):

Zhifeng Shao ◽

A.V. Crewe

Keyword(s):

Electron Energy ◽

Low Voltage ◽

Spherical Aberration ◽

Chromatic Aberration ◽

Primary Electron ◽

Probe Size ◽

Non Local ◽

Optical Treatment ◽

Electron Microscopes ◽

Scanning Electron Microscopes

For scanning electron microscopes, it is plausible that by lowering the primary electron energy, one can decrease the volume of interaction and improve resolution. As shown by Crewe /1/, at V0 =5kV a 10Å resolution (including non-local effects) is possible. To achieve this, we would need a probe size about 5Å. However, at low voltages, the chromatic aberration becomes the major concern even for field emission sources. In this case, δV/V = 0.1 V/5kV = 2x10-5. As a rough estimate, it has been shown that /2/ the chromatic aberration δC should be less than ⅓ of δ0 the probe size determined by diffraction and spherical aberration in order to neglect its effect. But this did not take into account the distribution of electron energy. We will show that by using a wave optical treatment, the tolerance on the chromatic aberration is much larger than we expected.

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