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Symmetry in Maxwell's Equations01:28

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Once the fields have been calculated using Maxwell's four equations, the Lorentz force equation gives the force that the fields exert on a charged particle moving with a certain velocity. The Lorentz force equation combines the force of the electric field and of the magnetic field on the moving charge. Maxwell's equations and the Lorentz force law together encompass all the laws of electricity and magnetism. The symmetry that Maxwell introduced into his mathematical framework may not be...
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In physics, symmetry in a system means that something in the considered system remains unchanged due to a specific operation to which it is subjected. For example, consider a horizontal square. The square looks the same if its right and left sides are interchanged. Hence, it is symmetric under a right-left interchange.
In calculations of electric fields, symmetry is of great use. For example, while calculating electric fields of continuous charge distributions.
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Gauss's Law: Spherical Symmetry01:26

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A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half...
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Gauss's Law: Planar Symmetry01:27

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A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...
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Electric potential can be pictorially represented as a three-dimensional surface. On such a surface, the electric potential is constant everywhere. The equipotential surface is always perpendicular to the electric field lines, and while it is three-dimensional, it can be treated as an equipotential line in a two-dimensional case. These equipotential lines are also always perpendicular to electric field lines. The term equipotential is often used as a noun, referring to an equipotential line or...
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Symmetry TFTs from String Theory.

Fabio Apruzzi1, Federico Bonetti2, Iñaki García Etxebarria3

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Communications in Mathematical Physics
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Summary
This summary is machine-generated.

We introduce the Symmetry Topological Field Theory (SymTFT) to describe higher-form symmetries and

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Area of Science:

  • High Energy Physics
  • String Theory
  • Quantum Field Theory

Background:

  • Topological Field Theories (TFTs) are crucial for understanding symmetries and anomalies in Quantum Field Theories (QFTs).
  • M-theory compactifications provide a powerful framework for constructing diverse QFTs.
  • Higher-form symmetries and their 't Hooft anomalies are key features of modern QFTs.

Purpose of the Study:

  • To determine the d-dimensional topological field theory encoding higher-form symmetries and 't Hooft anomalies.
  • To introduce the Symmetry Topological Field Theory (SymTFT) for QFTs from M-theory compactifications.
  • To utilize differential cohomology for background fields of discrete symmetries.

Main Methods:

  • Reducing the topological sector of 11d supergravity on the boundary of a non-compact space X.
  • Reformulating supergravity using differential cohomology to include torsion.
  • Analyzing M-theory compactifications on Calabi-Yau three-fold cones and 7d super-Yang Mills.
  • Complementing M-theory with a IIB 5-brane web approach.

Main Results:

  • The derivation of the Symmetry Topological Field Theory (SymTFT).
  • Identification of background fields for discrete and higher-form symmetries.
  • Application to specific examples: 7d super-Yang Mills and 5d superconformal field theories from Calabi-Yau geometry.
  • Consistent results obtained from both M-theory and 5-brane web perspectives.

Conclusions:

  • The developed framework successfully captures higher-form symmetries and 't Hooft anomalies.
  • The SymTFT provides a unified approach applicable to both Lagrangian and non-Lagrangian QFTs.
  • The methods allow for broad generalizations in the study of QFTs and their symmetries.