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Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions x, y, z, ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann numbers rather than real numbers. The ordinary space dimensions correspond to bosonic degrees of freedom, the anticommuting dimensions to fermionic degrees of freedom. The word "superspace" was first used by John Wheeler in an unrelated sense to describe the configuration space of general relativity; for example, this usage may be seen in his 1973 textbook Gravitation. There are several similar, but not equivalent, definitions of superspace that have been used, and continue to be used in the mathematical and physics literature. One such usage is as a synonym for super Minkowski space. In this case, one takes ordinary Minkowski space, and extends it with anti-commuting fermionic degrees of freedom, taken to be anti-commuting Weyl spinors from the Clifford algebra associated to the Lorentz group. Equivalently, the super Minkowski space can be understood as the quotient of the super Poincaré algebra modulo the algebra of the Lorentz group. A typical notation for the coordinates on such a space is with the overline being the give-away that super Minkowski space is the intended space. Superspace is also commonly used as a synonym for the super vector space. This is taken to be an ordinary vector space, together with additional coordinates taken from the Grassmann algebra, i.e. coordinate directions that are Grassmann numbers. There are several conventions for constructing a super vector space in use; two of these are described by Rogers and DeWitt. A third usage of the term "superspace" is as a synonym for a supermanifold: a supersymmetric generalization of a manifold. Note that both super Minkowski spaces and super vector spaces can be taken as special cases of supermanifolds. A fourth, and completely unrelated meaning saw a brief usage in general relativity; this is discussed in greater detail at the bottom.

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PHYS-817: Supersymmetry

Supersymmetry is the unique quantum extension of the symmetry principles of relativity. This course offers a first but broad introduction covering the role of Supersymmetry in our understanding of bot

Differential operators for superconformal correlation functions

Andrea Manenti

We present a systematic method to expand in components four dimensional superconformal multiplets. The results cover all possible N = 1 multiplets and some cases of interest for N = 2. As an application of the formalism we prove that certain N = 2 spinning chiral operators (also known as \exotic" chiral primaries) do not admit a consistent three-point function with the stress tensor and therefore cannot be present in any local SCFT. This extends a previous proof in the literature which only applies to certain classes of theories. To each superdescendant we associate a superconformally covariant differential operator, which can then be applied to any correlator in superspace. In the case of threepoint functions, we introduce a convenient representation of the differential operators that considerably simplifies their action. As a consequence it is possible to efficiently obtain the linear relations between the OPE coefficients of the operators in the same superconformal multiplet and in turn streamline the computation of superconformal blocks. We also introduce a Mathematica package to work with four dimensional superspace.

2020

The conformal bootstrap is a non-perturbative technique designed to study conformal field theories using only first principles, such as unitarity, crossing symmetry and the existence of an Operator Product Expansion. In this thesis we discuss an application of the bootstrap method in four dimensional conformal field theories. We also consider in detail the special case where the theory is supersymmetric. In particular we focus on the case study of four abelian currents. The non-supersymmetric setup applies to all conformal field theories with a global abelian symmetry group. When we include the assumption of supersymmetry, the current is taken to be the generator of the R-symmetry, which is tied to the stress tensor due to the superconformal algebra. The supersymmetric setup therefore applies to all local superconformal field theories. We start by introducing all the necessary ingredients. In particular, we discuss the formalism of the embedding space and of the conformal frame to study conformal kinematics. We also give a supersymmetric generalization of the conformal frame formula to count three-point tensor structures. Then we address the important problem of expanding superspace correlators in their components. To this aim we introduce a set of differential operators that act in superspace. Using this formalism we are able to compute the linear relations among the operators in the same superconformal multiplet. This is a necessary step in the computation of superconformal blocks, but it will also be useful for other purposes that we discuss before passing to the bootstrap analysis. First we use it to impose the averaged null energy condition on arbitrary superconformal field theories. This will lead to interesting consequences on their local operator spectrum. Next we focus on the case of local superconformal field theories with eight supercharges and we prove that a certain class of operators termed ``exotic primaries'' cannot exist. Finally, after a pedagogical introduction to the notion of the conformal bootstrap, we carry out a detailed study of the correlator of four conserved currents. In particular, we compute the conformal and superconformal blocks and the crossing equations. We conclude by proposing several numerical studies and strategies and by showing some preliminary results for the non-supersymmetric case.

EPFL2020

The concept of supergroup is a generalization of that of group. In other words, every supergroup carries a natural group structure, but there may be more than one way to structure a given group as a supergroup. A supergroup is like a Lie group in that there is a well defined notion of smooth function defined on them. However the functions may have even and odd parts. Moreover, a supergroup has a super Lie algebra which plays a role similar to that of a Lie algebra for Lie groups in that they determine most of the representation theory and which is the starting point for classification.

In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra over the complex numbers. The special case of a 1-dimensional algebra is known as a dual number. Grassmann numbers saw an early use in physics to express a path integral representation for fermionic fields, although they are now widely used as a foundation for superspace, on which supersymmetry is constructed.

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Supersymmetry: Massless Multiplets and Chiral Multiplets PHYS-817: Supersymmetry

Explores massless and chiral multiplets in supersymmetry, quantum field algebra, and superspace structure.

Supersymmetry: Lecture 4 PHYS-817: Supersymmetry

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