In supersymmetry, harmonic superspace is one way of dealing with supersymmetric theories with 8 real SUSY generators in a manifestly covariant manner. It turns out that the 8 real SUSY generators are pseudoreal, and after complexification, correspond to the tensor product of a four-dimensional Dirac spinor with the fundamental representation of SU(2)R. The quotient space , which is a 2-sphere/Riemann sphere. Harmonic superspace describes N=2 D=4, N=1 D=5, and N=(1,0) D=6 SUSY in a manifestly covariant manner. There are many possible coordinate systems over S2, but the one chosen not only involves , but also happen to be a coordinatization of . We only get S2 after a projection over . This is of course the Hopf fibration. Consider the left action of SU(2)R upon itself. We can then extend this to the space of complex valued smooth functions over SU(2)R. In particular, we have the subspace of functions which transform as the fundamental representation under SU(2)R. The fundamental representation (up to isomorphism, of course) is a two-dimensional complex vector space. Let us denote the indices of this representation by i,j,k,...=1,2. The subspace of interest consists of two copies of the fundamental representation. Under the right action by U(1)R -- which commutes with any left action—one copy has a "charge" of +1, and the other of -1. Let us label the basis functions . . The redundancy in the coordinates is given by . Everything can be interpreted in terms of algebraic geometry. The projection is given by the "gauge transformation" where φ is any real number. Think of S3 as a U(1)R-principal bundle over S2 with a nonzero first Chern class. Then, "fields" over S2 are characterized by an integral U(1)R charge given by the right action of U(1)R. For instance, u+ has a charge of +1, and u− of -1. By convention, fields with a charge of +r are denoted by a superscript with r +'s, and ditto for fields with a charge of -r. R-charges are additive under the multiplication of fields. The SUSY charges are , and the corresponding fermionic coordinates are . Harmonic superspace is given by the product of ordinary extended superspace (with 8 real fermionic coordinatates) with S2 with the nontrivial U(1)R bundle over it. The product is somewhat twisted in that the fermionic coordinates are also charged under U(1)R. This charge is given by . We can define the covariant derivatives with the property that they supercommute with the SUSY transformations, and where f is any function of the harmonic variables. Similarly, define and . A chiral superfield q with an R-charge of r satisfies . A is given by a chiral superfield . We have the additional constraint . According to the Atiyah-Singer index theorem, the solution space to the previous constraint is a two-dimensional complex manifold. (Wikipedia).
An example of a harmonic series.
From playlist Advanced Calculus / Multivariable Calculus
B03 Simple harmonic oscillation
Explaining simple (idealised) harmonic oscillation, through a second-order ordinary differential equation.
From playlist Physics ONE
Simple Harmonic Motion (10 of 16): An Explanation
This video provides a step by step explanation of simple harmonic motion. You will learn the key terms that you needed to describe an oscillating mass as well as the important points in its motion where the displacement, force, acceleration and velocity will be the greatest and least. Sim
From playlist Simple Harmonic Motion, Waves and Vibrations
If the Laplacian of a function is zero everywhere, it is called Harmonic. Harmonic functions arise all the time in physics, capturing a certain notion of "stability", whenever one point in space is influenced by its neighbors.
From playlist Fourier
B04 Example problem of simple harmonic oscillation
Solving an example problem of simple harmonic oscillation, which requires calculating the solution to a second order ordinary differential equation.
From playlist Physics ONE
From playlist Fall 2020 Course
Supersymmetry and Superspace, Part 3 - Jon Bagger
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From playlist PiTP 2010
Apollonius and harmonic conjugates | Universal Hyperbolic Geometry 2 | NJ Wildberger
Apollonius introduced the important idea of harmonic conjugates, concerning four points on a line. He showed that the pole polar duality associated with a circle produces a family of such harmonic ranges, one for every line through the pole of a line. Harmonic ranges also occur in the cont
From playlist Universal Hyperbolic Geometry
Schwinger Keldysh in Superspace by R. Loganayagam
Bangalore Area String Meeting URL: http://www.icts.res.in/discussion_meeting/BASM2016/ DATES: Monday 25 Jul, 2016 - Wednesday 27 Jul, 2016 VENUE : Ramanujan Lecture Hall, ICTS, Bangalore DESCRIPTION: This meeting is designed to bring together string theorists working in the Bangalore
From playlist Bangalore Area String Meeting
Exact results in SUSY gauge theories - II by Jaume Gomis
Advanced Strings School 2015 TALKS URL: https://www.icts.res.in/program/all/t... PROGRAM URL: http://www.icts.res.in/program/SS2015 ORGANIZERS: Justin David, Chethan Krishnan and Gautam Mandal DATES: Thursday 11 Jun, 2015 - Thursday 18 Jun, 2015 VENUE: Physics Department, Indian Instit
From playlist Advanced Strings School 2015
Supersymmetry and Superspace, Part 1 - Jon Bagger
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From playlist PiTP 2010
Samson Shatashvili - 1/3 Supersymmetric Vacua and Integrability
"I review the relationship between supersymmetric gauge theories and quantum integrable systems. From the quantum integrability side this relation includes various spin chains, as well as many well-known quantum many body systems like elliptic Calogero-Moser system and generalisations. Fro
From playlist Samson Shatashvili - Supersymmetric Vacua and Integrability
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Channel social media: Instagram: @whatthehectogon https://www.instagram.com/whatthehect... Twitter: @whatthehectogon https://twitter.com/whatthehectogon Any questions? Leave a comment below or email me at the misspelled whatthehectagon@gmail.com Here we take the discrete route, rather
From playlist Analysis
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In this talk, we present the philosophy and the basic concepts of Noncommutative Supergeometry, i.e. Hilbert superspaces, C*-superalgebras and quantum supergroups. Then, we give examples of these structures coming from deformation quantization and we expose an application to renormalizable
From playlist HIM Lectures: Trimester Program "Non-commutative Geometry and its Applications"
The Bouncing Scenario by Patrick Peter
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From playlist Physics of The Early Universe - An Online Precursor
11 January 2017 to 13 January 2017 VENUE: Ramanujan Lecture Hall, ICTS, Bengaluru String theory has come a long way, from its origin in 1970's as a possible model of strong interactions, to the present day where it sheds light not only on the original problem of strong interactions, but
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Along our way to understand the "internal" structure of a vertex algebra we look at the notion of the normally ordered product of two vertex operators and why we need such a definition. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Merch: https://teespring.
From playlist Vertex Operator Algebras
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AWESOME Simple harmonic motion!
In this video show simple harmonic motion on spring and pendulums, used position sensor.
From playlist MECHANICS