# Questions tagged [supermanifolds]

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### Superspace derivation of supersymmetric non-linear sigma model in Supersolutions by Deligne and Freed

I am having a little trouble understanding passage from the linear to the non-linear sigma model in Section 4.1 of Supersolutions by Deligne and Freed. Most of my confusion comes down to the ...
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### One-to-one correspondence between super morphisms $\varphi:S\to TX$ and pairs $(f:C^\infty(X)\to C^\infty(S) ,F\in Der_f(C^\infty(X),C^\infty(S))$

I'm trying to show that for an ordinary manifold $X$ and a supermanifold $S$, supermanifold morphisms $\varphi:S\to TX$ are one-to-one to the pairs $(f,F)$ where $f:C^\infty(X)\to C^\infty(S)$ is a ...
1 vote
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### Formulation of matrix representation of morphisms between free super modules

I asked this question in MathStackExchange 9 days ago but get no response (not a vote nor a comment), so I'm copying it here below. The link to the original question is: https://math.stackexchange.com/...
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### Parity reversed tangent bundle as a supermanifold

I encountered an example in a paper telling that $\underline{SM}(\mathbb{R}^{0|1},X)\cong \pi TX$, where $X$ is some fixed ordinary Riemannian manifold, $\pi TX$ is the supermanifold with base ...
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### Constructions with Superschemes via Kan extensions

Let $\operatorname{CAlg}$ be the category of commutative rings (with unit) and $\operatorname{S-CAlg}$ the category of supercommutative $\mathbb{Z}/2$-graded rings. Then we have an adjoint triple (as ...
170 views

### Chain rule for the superderivative

A one dimensional complex supermanifold $X$ is locally described by an ordinary complex coordinate $z$ and an anticommuting coordinate $\theta$, $\theta^2 = 0$. The superderivative is the square root ...
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1 vote
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### Super version of Poisson brackets of tensor products

Let $A$ be a Poisson super algebra ($A$ is a super algebra and $A$ satisfies super Jacobi identity, super commutativity, super Leibniz rule). Super version of the product of two tensor products is \...
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### 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 ...
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### Mirror Symmetry for Flag Supermanifolds

I recently learned the following two things, and I wish to know how to make them reconciled. (1) As far as I understand, the flag manifolds serve as a tractable class of examples for the very ...
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### Is there any work on "super Fukaya categories"?

There is a well-established notion of "supermanifold", and in the world of supergeometry it makes sense to talk about symplectic structures. Actually, there are various kinds of symplectic structures:...
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### A good reference for learning about super-differentiation & super-integration?

I've looked at a couple of books for basic information for super-differentiation & super-integration - Rogers Supermanifolds, and Khrennikovs Superanalysis. Unfortunately both books lack a clear ...
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### Analytic stuctures on $\mathbb R^n$ and the nilpotent ideal of supermanifolds

I have two questions which are somewhat related: (a) It is a well known result (of Freedman?) in differential topology that $\mathbb R^4$ has exotic smooth structures. Apparently, it is known that ...
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### Double tangent bundle of manifolds, two contradictory arguments

I am considering the double tangent bundle $T(TM)$ of manifolds $M$. Locally, if $M=R^d$ then $T(TM)=R^{4d}=\oplus^3 TM$. My attempt is to see whether $T(TM)\cong \oplus^3 TM$ naturally for any $M$. I ...
255 views

### Invariant definition of graded Poisson bracket

Given a graded manifold with symplectic form $\omega$ of degree $n$, I have seen two expressions for the corresponding Poisson bracket of degree $-n$. Cattaneo-Fiorenza-Longoni, http://www.math.uzh.ch/...
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### Supermanifolds and Grassmann algebras

On the first hand one can define a superdomain $U^{p|q}$ as the super ringed space $(U^p,\mathcal{C}^{\infty p|q})$ where $U^p\subset\mathbb R^p$ is open and $\mathcal{C}^{\infty p|q}$ is the sheaf of ...
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### Is every graded manifold affine, and is this definition of graded manifold the right one?

The following definition is from: Dmitry Roytenberg, "AKSZ-BV formalism and Courant algebroid-induced topological field theories", Letters in Mathematical Physics, 2007 vol. 79 (2) pp. 143-159, ...
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### Two fancy ways of defining differential forms: How does one show that they are equivalent?

Given a smooth manifold M, the following procedures yield the differential graded algebra (Ω*(M),ddR) of differential forms: Procedure 1 (synthetic geometry). For each n, consider the object of ...