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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2 votes
1 answer
97 views

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 ...
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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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2 votes
1 answer
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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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  • 205
7 votes
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208 views

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 ...
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6 votes
1 answer
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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4 votes
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What is a super-infinity-groupoid, homotopy-theoretically?

There's a sheafy way to write down the $\infty$-category of super-$\infty$-groupoids, as detailed on the nlab. Can this notion be reformulated by saying that a super-$\infty$-groupoid is an ordinary $\...
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13 votes
2 answers
539 views

What is the relationship between spinors and supermanifolds and fermions?

I have the following two impressions about fermions in physics. I'm confused about their accuracy, and their compatibility: To consider the behavior of a fermion, whose intrinsic spin is described by ...
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6 votes
0 answers
158 views

References for superhomology

This question concerns topological string theory. It was known sice its outset, that the BRST-cohomology ("observables") of the weakly coupled topological string B-model on a Calabi-Yau ...
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3 votes
1 answer
122 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 $$ p\colon \pi_1(\mathcal{M}_{0,n})\longrightarrow H_1(\mathcal{M}_{0,n},\mathbb{Z}/2\mathbb{Z}) $$ ...
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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 $p\in\mathbb Z$ ...
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1 answer
225 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 ...
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2 votes
1 answer
164 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 ...
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1 answer
623 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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61 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 ...
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7 votes
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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 ...
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3 votes
1 answer
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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 ...
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4 votes
0 answers
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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(-\...
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5 votes
0 answers
237 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 ...
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2 votes
0 answers
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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 ...
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0 answers
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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 ...
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6 votes
0 answers
208 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 ($m,n$)-...
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16 votes
6 answers
2k 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 ...
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2 votes
0 answers
220 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 ...
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4 votes
1 answer
522 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 ...
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3 votes
1 answer
137 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 ...
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  • 5,875
0 votes
1 answer
133 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} \...
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  • 5,875
1 vote
1 answer
138 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, \...
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1 vote
1 answer
259 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 \...
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8 votes
1 answer
660 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 ...
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4 votes
0 answers
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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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6 votes
0 answers
301 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:...
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3 votes
1 answer
116 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:...
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3 votes
0 answers
203 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 ...
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20 votes
1 answer
630 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 ...
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5 votes
1 answer
570 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 ...
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  • 19k
5 votes
0 answers
179 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 ...
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3 votes
0 answers
582 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 ...
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3 votes
1 answer
531 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 ...
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3 votes
0 answers
100 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 ...
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3 votes
2 answers
2k 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 ...
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  • 1,211
0 votes
1 answer
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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2 votes
2 answers
401 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 ...
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8 votes
2 answers
879 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 \...
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2 votes
1 answer
435 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 ...
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  • 1,211
2 votes
2 answers
469 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 $C(R^{0|1})\to C(R^{0|1})$. ...
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9 votes
1 answer
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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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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 ...
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5 votes
3 answers
493 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. ...
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8 votes
3 answers
942 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 ...
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