# Questions tagged [supermanifolds]

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43
questions

**3**

votes

**1**answer

71 views

### Cohomology of the moduli space of rational curves with $n$ marked points with spin structure

Consider $\mathcal{M}_{0,n}$- the moduli space of rational curves with $n$ marked points. A map
$$
p\colon \pi_1(\mathcal{M}_{0,n})\longrightarrow H_1(\mathcal{M}_{0,n},\mathbb{Z}/2\mathbb{Z})
$$
...

**1**

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**0**answers

73 views

### Understanding the transgression map in AKSZ-BV construction

The transgression map in AKSZ construction is defined as follows:
Let $M$ be a closed oriented $n$-manifold and $\mathcal M = T[1]{M}$.
$\Omega^{0}(\mathcal M) = C^{\infty}(\mathcal M) \cong \Omega^\...

**1**

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**0**answers

29 views

### Poisson reduction in odd/graded Poisson geometry?

I would like to know whether there is any literature on Poisson reduction of $\mathbb Z$- or $\mathbb Z_2$-graded Poisson algebras.
A $\mathbb Z$-graded Poisson algebra with degree $p\in\mathbb Z$ ...

**1**

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**1**answer

133 views

### Question about Berezin line bundle of odd cotangent bundle of supermanifold $\text{Ber}(\Pi T^*\mathcal N)$

The followings are from Mnev's paper about BV formalism.
Example 4.15 (Definition of split supermanifold)
Let $E \to M$ be a rank $m$ vector bundle over $n$-manifold $M$, then there exists a ...

**2**

votes

**1**answer

131 views

### $\mathbb Z$-graded manifold is isomorphic to the structure sheaf of supermanifold locally

In this paper, definition 4.4.1 about supermanifold and definition 4.6.1 about graded manifold:
Definition 4.4.1: An supermanifold $\mathcal{M}$ is a locally ringed space $(M,\mathcal O_M)$ which is ...

**3**

votes

**1**answer

591 views

### Subsupermanifolds defined using ideal, transversal example

I am currently learning about algebraic viewpoint on closed embedded subsupermanifolds. In particular, I am struggling with something which should be ''easy to see''. Namely, I refer to the lemma just ...

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52 views

### Spaces of Bordered (Ordinary, Spin- or Super-) Riemann Surfaces

It is known from the work of Deligne and Mumford that the "space" of punctured/marked Riemann surfaces is a Deligne-Mumford stack. I have few questions regarding similar statement for the spaces of ...

**2**

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84 views

### Is there a definition of divergence on $\mathbb N$-graded supermanifolds?

I mean supermanifolds with an extra $\mathbb Z$ grading in the structure sheaf compatible with supermanifold parity, such that said sheaf is locally generated only in non-negative degrees. Such "N-...

**7**

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**0**answers

91 views

### Infinitesimal description of homogeneous supermanifolds

Lie's third theorem says that if $\mathfrak{g}$ is a real, finite-dimensional Lie algebra, then there is a unique (up to isomorphism) simply-connected Lie group $G$ whose tangent Lie algebra is ...

**3**

votes

**1**answer

75 views

### Stationary phase method on supermanifolds

The classical stationary phase method computes an asymptotic behavior of an integral $\int_M f(x)e^{-\frac{1}{h}S(x)} dx$ as $h\to +0$, where $M$ is (say, compact) manifold in terms of the critical ...

**4**

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113 views

### Will the transgression formula for superconnections give back the transgression formula of connections?

Let $E$ be a vector bundle on a smooth manifold $X$ and $\nabla$ be a connection on $E$, by Chern-Weil theory, the Chern character of $(E,\nabla)$ could be construct as
$$
ch(E,\nabla):=tr(\exp(-\...

**5**

votes

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222 views

### Localization principle in integration over supermanifolds

This post is closely related to the post Localization principle in supersymmetry
and can be considered as a continuation of it, although independent.
In § 9.3 of the book "Mirror symmetry" (K. Hori ...

**2**

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**0**answers

54 views

### Action of orthosymplectic group $SpO(d+1|d)$ on $PGL(d+1|d)$

Suppose $d$ is odd, and consider the super vector space $\mathbb{C}^{d+1|d}$ and its super projectivization $\mathbb{P}^{d|d}$.
In the case $d=1$, a paper of Witten's, concerning the moduli space of ...

**4**

votes

**0**answers

189 views

### What is the good notion of supervariety?

The principal (I think) difference between the notions of manifold in differential (including complex analytic) topology and in algebraic (or especially arithmetic) geometry is that for the former the ...

**6**

votes

**0**answers

201 views

### Physical effects in supersymmetric theories of the underlying supermanifold being split or non-split?

Given an $m$-dimensional $C^{\infty}$ manifold $M$ with an $n$-dimensional vector bundle $E$ over $M$, one can use the transition functions of the manifold and the bundle to construct an ($m,n$)-...

**15**

votes

**6**answers

1k views

### supermanifolds - elementary introduction?

I am looking for an elementary but mathematically precise introductory text on supermanifolds in a modern differential geometric setting.
Elementary in the sense that there is plenty of motivation ...

**2**

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**0**answers

203 views

### What exactly is the role of the mysterious manifold 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 ...

**3**

votes

**1**answer

256 views

### Definition of orthosymplectic supergroups

I found two versions of definitions of orthosymplectic supergroups. It seems that they are not equivalent. I don't know which version of the definition is standard.
The first version of the ...

**3**

votes

**1**answer

119 views

### Reference request: coordinate ring of $OSP(2p|n)$

In the paper, the orthosymplectic supergroup $OSP(2p|n)$ is defined as follows.
Let $A = A_0 \oplus A_1$ be a supercommutative superalgebra, where elements in $A_0$ are even and elements in $A_1$ are ...

**0**

votes

**1**answer

112 views

### How to prove a bracket is super anti-commutative?

On page 12 of the paper, there is a formula about super Poisson bracket on a Lie super group $G$:
\begin{align}
\{\phi, \psi\} = \sum_{\mu, \nu} (-1)^{|\phi||\nu|} r^{\mu \nu} ( R_{\mu} \phi R_{\nu} \...

**1**

vote

**1**answer

120 views

### Construct super Poisson brackets on the coordinate rings of Lie super groups

On line 7 of page 61 of the book a guide to quantum groups, a Poisson bracket is defined on $\mathbb{C}[GL_n]$ for every classical $r$-matrix as follows.
Let $V$ be a vector space with a basis $v_1, \...

**1**

vote

**1**answer

196 views

### 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
\...

**8**

votes

**1**answer

591 views

### 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 ...

**4**

votes

**0**answers

146 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 ...

**6**

votes

**0**answers

283 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:...

**2**

votes

**1**answer

103 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:...

**3**

votes

**0**answers

184 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 ...

**20**

votes

**1**answer

577 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 ...

**5**

votes

**1**answer

512 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 ...

**5**

votes

**0**answers

171 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 ...

**3**

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**0**answers

560 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

**1**answer

441 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

96 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

1k 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 ...

**0**

votes

**1**answer

197 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/...

**2**

votes

**2**answers

374 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 ...

**8**

votes

**2**answers

822 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 \...

**2**

votes

**1**answer

407 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 ...

**2**

votes

**2**answers

443 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 $C(R^{0|1})\to C(R^{0|1})$. ...

**9**

votes

**1**answer

965 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, ...

**13**

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 of ...

**5**

votes

**3**answers

420 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. ...

**8**

votes

**3**answers

831 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 ...