A supermanifold is a generalization of the concept of manifold. It consists of a manifold whose local charts contain two types of variables: commuting variables (as in any ordinary manifold) as well as anti-commuting variables.
Questions tagged [supermanifolds]
17 questions
21
votes
3 answers
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), scale invariance and independence from the…
pregunton
- 6,358
4
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0 answers
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 literature about $\mathbb{Z}$-graded manifolds than those…
Ji Woong Park
- 438
3
votes
1 answer
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 $\mathbb{C}^n\wedge\mathbb{C}^n \cong \mathbb{C}^{{n \choose 2}}$.…
BVquantization
- 293
3
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0 answers
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 not do the job as morphisms have to preserve…
Dilaton
- 1,217
3
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0 answers
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 for fermions are Grassmann algebras…
Dilaton
- 1,217
3
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0 answers
What is the geometric significance of the definition of supermanifold?
We know that a supermanifold $M$ is a locally ringed space $(M,O_M)$ which is locally isomorphic to $(U, C^\infty(U) \otimes \wedge W^\ast)$, where $U$ is an open subset of $\mathbb{R}^n$, $W$ is a finite dimensional real vector space and the above…
Soutrik
- 41
2
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0 answers
Proving uniqueness of a supermanifold morphism to $\mathbb{R}^{0|0}$
While reading a book about super Lie groups I came upon the claim that there is a unique morphism $\phi: G \to \mathbb{R}^{0|0}$, where $G$ is any supermanifold (I'm using the superspace definition of supermanifolds, i.e. supermanifolds here are…
Konrad Gębik
- 33
2
votes
0 answers
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 being dual to $c=24$ (if I recall) homomorphic…
alex sharma
- 33
1
vote
1 answer
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 super-algebras (call it $C^\infty(\mathcal{M})$)…
BVquantization
- 293
1
vote
0 answers
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 $\mathbf{C}^\infty$ on $M$ (thought of as the sheaf of…
J.V.Gaiter
- 2,659
1
vote
1 answer
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.
Remark: situation is quite clear for me in the case…
Blazej
- 3,200
0
votes
1 answer
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-Bernstein. Let
$$X = \psi_2 \frac{d}{dx} + f(x)…
Mykola Pochekai
- 1,784
0
votes
1 answer
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 following: suppose I am given a graded vector bundle $E…
0
votes
1 answer
Putting an inner product on superspace
I was wondering if there is a natural way of explicitly defining an inner product on superspace.
In this paper, for a point x = $x_1,x_2,...,x_{n+1},\eta_1,\xi_1,\eta_2,\xi_2,...,\eta_m,\xi_m \in \mathbb{R}^{n+1|2m}$ an inner product is defined as…
summersfreezing
- 401
0
votes
1 answer
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.
Letting $\mathbb{R}_{\infty}$ be a real Grassmann…
Albert
- 747