Questions tagged [supergeometry]

Supergeometry is differential geometry of modules over graded commutative algebras, supermanifolds and graded manifolds. Supergeometry is part and parcel of many classical and quantum field theories involving odd fields, e.g., SUSY field theory, BRST theory, or supergravity.

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Surjective map of super vectorspaces

Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space. The space $V \otimes V$ is also $\mathbb Z_2$-graded for which $(V \otimes V)_0=V_0 \otimes V_0 \oplus V_1 \otimes V_1$ and $(V \otimes V)...
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Questions about $\mathbb{Z}$-graded manifolds (references, concrete approach analogous to supermanifold)

I am trying to learn about $\mathbb{Z}$-graded manifolds. It seems that the theory of $\mathbb{Z}$-graded manifolds has some complications that supermanifolds do not have, and there is fewer ...
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Parity reversed tangent bundle of a supermanifold and the corresponding Q-structure

EDIT: I've copied this question to MathOverflow, due to 9 days waiting for no response. The link to the copy on MathOverflow is: https://mathoverflow.net/questions/428747/parity-reversed-tangent-...
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orthosymplectic super group and super algebra

Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space over $\mathbb C$. Suppose $V$ has an even non-degenerate bilinear form $(-, -)$ which is symmetric on $V_0$, skew symmetric on $V_1$, and ...
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Alternative expression for Riemann curvature tensor

There is the usual expression for the Riemann tensor $$R_{abcd}=\partial_c\Gamma_{adb}-\partial_d\Gamma_{acb}+\Gamma_{ace}{\Gamma^e}_{db}-\Gamma_{ade}{\Gamma^e}_{cb}.$$ However, in the last page of ...
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Formulation of matrix representation of morphisms between free super modules

EDIT: I've copied this question to MathOverflow, due to 9 days waiting for no response. The link to the copy on MathOverflow is: https://mathoverflow.net/questions/417194/formulation-of-matrix-...
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Request for expository articles on supersymmetric geometry.

Various kinds of supersymmetric QFTs are studied in the physics literature. A typical physics talk describes a Lie "superalgebra" by a huge list of operators (with many supercharges, ...
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Tensor product of super vector spaces

The tensor product as well as the product in any category is determined by the corresponding limit diagram. So I should be able to find the product and tensor product in the category of super vector ...
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supersymmetry and quantum groups

I have some background in non-commutative geometry (in particular, I am doing research in quantum groups) and I have of course heard several times about the concept of supergeometry and supergroups. ...
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Supergroup for SO groups vs Spin group

We know that the $\mathbb{Z}/2$ central extension of $ SO(d) $ can give a nontrivial double/universal cover of $SO(d)$ known as the $Spin(d)$ group. They have this relation $$ 1 \to \mathbb{Z}/2 \to ...
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Understanding Moonshine and Heterotic E8xE8

Recently I have become familiar with the conjectured relationship of monstrous moonshine and pure $(2+1)$-dimensional quantum gravity in AdS with maximally negative cosmological constant and, it’s ...
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Defining supermanifolds by equations

I would like to in what sense supermanifolds may be defined by systems of equations in the ordinary flat superspace. I am particularly interested in the approach to supergeometry via ringed spaces. ...
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Intuition for the need of generalizing from mappings to morphisms to functors in supermathematics?

I am currently reading this paper about the categorical formulation of superalgebras and supergeometry, where in definition 2.3 it says that to change the parity of a right supermodule a morphism will ...
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What exactly is the role of the mysterious space underlying the definition of a superspace?

In the intro to chapter 12.3 of this book about the applications of coherent states, it says that classical spaces for bosons are real or complex vector spaces or manifolds, whereas classical spaces ...
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Exterior algebra on a given number of generators?

I just started studying supermanifolds and I need to understand how to construct the exterior algebra on $q$ generators. Can anyone recommend me a a book which deals with it? I know this is an ...
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Geometric meaning of Berezin integration

Berezin integration in a Grassmann algebra is defined such that its algebraic properties are analogous to definite integration of ordinary functions: linearity (taking anticommutativity into account), ...
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