The supermanifolds tag has no usage guidance.

**18**

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**1**answer

483 views

### Super-cobordisms

One can construct the $d$-dimensional bordism category by declaring the objects to be the $(d-1)$-dimensional compact manifolds without boundary and the morphisms the $d$-dimensional bordisms between ...

**11**

votes

**2**answers

2k views

### 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 ...

**8**

votes

**1**answer

670 views

### 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, ...

**5**

votes

**3**answers

569 views

### Derivations of C(X)? or Why Must Supermanifolds be Smooth?

What are the derivations of the algebra of continuous functions on a topological manifold?
A supermanifold is a locally ringed space (X,O) whose underlying space is a smooth manifold X, and whose ...

**5**

votes

**3**answers

315 views

### Morphisms of supermanifolds

I am confused regarding supermanifolds. Suppose I consider R^(1,2) (1 "bosonic", 2 "fermionic"), This map (x,a,b) -> (x+ab, a,b) (a,b are fermionic) is supposed to be a morphism of this supermanifold. ...

**5**

votes

**2**answers

645 views

### How can I write down a point in the Berezinian of a super vector space?

A vector space $V$ of dimension $n$ has an associated determinant line $Det(V)$.
An element of $Det(V)$ is represented as a (formal limear combination) of expresstions of the form
$v_1 \wedge v_2 \...

**5**

votes

**0**answers

151 views

### 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:...

**5**

votes

**0**answers

125 views

### Hochschild cohains-type models for smooth functions on shifted cotangent bundles

Let $M$ be a smooth manifold. Then, by the well known Hochschild-Kostant-Rosenberg theorem, the cochain complex $C^*(C^\infty(M))$ of Hochshild cochains on the algebra $C^\infty(M)$ of smooth ...

**4**

votes

**1**answer

315 views

### Localization principle in supersymmetry

In $\S$ 9.3 of the book "Mirror symmetry" (Vafa, Zaslow eds.) the authors formulate the following general localization principle for computation of integrals with respect to both even and odd ...

**3**

votes

**1**answer

322 views

### 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 ...

**3**

votes

**0**answers

104 views

### 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 ...

**3**

votes

**0**answers

122 views

### Cohomology of the classifying space of some Super Lie group

Are there any papers on the cohomology of the classifying space of the general linear supergroup $GL(n, m)$ or unitary supergroup $U(n, m)$?
I know basically nothing about supergeometry. It seems ...

**3**

votes

**0**answers

174 views

### Orthosymplectic group, matrix representations

We have the orthosymplectic $osp(n,m|2k)$. The bosonic part is $so(n,m)\times sp(2k)$. The lie algebra generators are given in eg
http://cds.cern.ch/record/524737/files/0110257.pdf$
where the group ...

**3**

votes

**0**answers

80 views

### 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 ...

**2**

votes

**2**answers

346 views

### Morphisms between supermanifolds R^{0|1}→R^{0|1}

I am confused with morphisms of supermanifolds. Take a simple example $f:R^{0|1}\to R^{0|1}$. By (one of) definition, $f$ is a morphism of superalgebras of functions $C(R^{0|1})\to C(R^{0|1})$. ...

**2**

votes

**1**answer

68 views

### An intrinsic supergeometric description of the Green–Schwarz supersymmetric action

The Green-Schwarz action is a natural supersymmetric extension of the Polyakov action (with a $B$-field which I will omit in what follows since it is not relevant to the question).
For a morphism $X:...

**1**

vote

**1**answer

470 views

### 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 ...

**1**

vote

**1**answer

315 views

### How to caculate the internal hom of supermanifolds?

This is my second question on supermanifolds, the previous one is at
Morphisms between supermanifolds R^{0|1}→R^{0|1}
I've learn the difference between homomorphism and internal-hom of ...

**1**

vote

**2**answers

305 views

### 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 ...

**0**

votes

**1**answer

106 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/...

**0**

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**0**answers

32 views

### Bruhat decompositions for Lie supergroups

Are there some references of Bruhat decompositions for Lie supergroups? I searched on google but did not find any papers about Bruhat decompositions for Lie supergroups. Thank you very much.