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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Left-invariant vector fields on super Minkowski space
I'm reading Volume 1 of Quantum Fields and Strings: A Course for Mathematicians.
Let $V$ be a Minkowski vector space of signature $(1,d-1)$, and let $\mathrm{Spin}(V) \curvearrowright S$ be an ...
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Proving that a morphism between supermanifolds as superringed spaces is local
So I've been reading Mathematical Foundations of Supersymmetry by Carmeli, Caston and Fioresi, and I cannot wrap my head around the proof of lemma 4.2.3 from that book. The first part of the lemma (...
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Checking that a form $dx d\psi_1 d\psi_2$ is invariant under a vector field flow on a supermanifold $\mathbb R^{1|2}$
Let's say I have a supermanifold $\mathbb R^{1|2}$ with the standard volume form $dx d \psi_1 d \psi_2$, regarded as a section of $\operatorname{Ber} \Omega^1$, in the sense of Deligne-Morgan-...
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Global triviality of graded fiber bundles with purely odd fibers
I have heard quite a few times that "purely odd directions don't carry topological obstructions", but I can't find/understand the exact statement. The question I want to understand is the ...
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Manifolds as ringed spaces and values of functions
I am currently reading a book on supergeometry which also introduces sheaves and ringed spaces, and in particular it proposes a definition of a differential manifold as a locally ringed space $(M,\...
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$n$th symmetric power of a superspace
Given a vector space $V$, we consider the (trivial) associated even superspace $V\oplus 0$ and odd superspace $0\oplus V$. For any (super) vector space $W$ we define the $n$th symmetric power as
$$
\...
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Superderivative of $G^\infty$ maps $\mathbb{R}^{1,1}_\infty\to\mathbb{R}_\infty$
I am following Rogers's Supermanifolds: Theory and Applications and I might be getting something wrong, because I reach a definition that, as I understand it, doesn't imply what the author states.
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What kind of mathematical objects are, in their tangent, supermodules over superrings?
I am interested in generalizing the notion of Lie groups and Lie algebras to higher categories. One way to do this is to replace the category of vector spaces with a more general category of modules ...
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Definition of Supermanifold
defining a super-domain as a pair $(U\subset\mathbb{R}^n, C^\infty(U)\otimes\bigwedge[\theta^1\cdots\theta^m])$, a supermanifold is usually defined as a topological space $M$ endowed with a sheaf of ...
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Erlangen-Style approach to Homogneoues Fiber Bundles
In connection with a philosophy of physics project I have recently been looking at fiber bundles from an Erlangen-inspired perspective. My inspiration is the following well-known result: Every smooth ...
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Parity operator and tensor products
When we consider usual vector spaces like $\mathbb{C}^n$, it is simple to consider tensor product operations, like $\mathbb{C}^n\otimes \mathbb{C}^m \cong \mathbb{C}^{n\cdot m}$, or $\mathbb{C}^n\...
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What is the space of maps between superplane $\mathbb{R}^{0|2}$ and a smooth manifold $M$?
Within the algebrogeometric approach to supergeometry, a supermanifold of dimension $m|n$ is an ordinary $m$ dimensional smooth manifold $M$ and a sheaf of supercommutative super algebras $\mathbf{C}^...
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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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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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