Questions tagged [supermanifolds]

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The supermoduli space of supertori with odd spin structure and metaplectic group actions

I'm trying to understand the description of the supermoduli space of supertori with odd spin structure as a quotient of the super complex upper half plane $\mathbb{H}^{1|1}$. Such a description ...
domenico fiorenza's user avatar
18 votes
3 answers
2k views

Mathematical motivation for supergeometry

Motivated by SUSY, mathematicians began to study $\mathbb{Z}_2$-graded mathematics, or super mathematics. In particular, one can formulate supergeometry just following Grothendieck style (even) ...
Estwald's user avatar
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4 votes
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smooth super scheme which is not smooth

I am following the very nice "Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes" by Bruzzo, Ruiperez and Polishchuk. I am having some problem in order to give ...
User43029's user avatar
  • 596
34 votes
8 answers
6k views

Applications of super-mathematics to non-super mathematics

Supergeometry and more broadly supermathematics has been around for few decades. Since its introduction by physicists, there has been an some mathematical interest in them. Although interesting in its ...
2 votes
1 answer
200 views

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}^...
J.V.Gaiter's user avatar
1 vote
0 answers
91 views

NSR superstring as a map of supermanifolds

On one hand, I know that the NSR superstring is described by a map $\Phi: \Sigma \to X$, where $\Sigma$ is a supermanifold with local coordinates $(\sigma,\theta)=(\sigma^0,\sigma^1 | \bar{\theta},\...
Alec's user avatar
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2 votes
0 answers
52 views

Coordinate free supersymmetric sigma model Lagrangian

I would like to know if there is a coordinate free version of the Lagrangian of the supersymmetric sigma model on a $2$-dimensional spacetime, with target space a Kähler manifold. The action for this ...
Quaere Verum's user avatar
1 vote
0 answers
44 views

The space of ("super")maps from a reversed-parity tangent bundle?

Let $T[1]X$ denote the reversed parity tangent bundle of some smooth manifold $X$, i.e. the super/graded manifold such that the global sections of its structure sheaf are differential forms on $X$. I'...
AlexArvanitakis's user avatar
1 vote
0 answers
74 views

Parity reversed tangent bundle of a supermanifold and the corresponding Q-structure

I asked this question in MathStackExchange 10 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....
Shana's user avatar
  • 237
4 votes
0 answers
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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 ...
user avatar
2 votes
1 answer
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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 ...
Shana's user avatar
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0 answers
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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/...
Shana's user avatar
  • 237
2 votes
1 answer
174 views

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 ...
Shana's user avatar
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7 votes
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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 ...
Markus Zetto's user avatar
6 votes
1 answer
231 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 ...
user's user avatar
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0 answers
356 views

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 $\...
Tim Campion's user avatar
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13 votes
2 answers
807 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 ...
Tim Campion's user avatar
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6 votes
0 answers
209 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 ...
Ramiro Hum-Sah's user avatar
3 votes
1 answer
158 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 $$ p\colon \pi_1(\mathcal{M}_{0,n})\longrightarrow H_1(\mathcal{M}_{0,n},\mathbb{Z}/2\mathbb{Z}) $$ ...
Daniil Rudenko's user avatar
1 vote
0 answers
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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 $p\in\mathbb Z$ ...
AlexArvanitakis's user avatar
2 votes
1 answer
350 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 ...
Andrews's user avatar
  • 79
2 votes
1 answer
213 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 ...
Andrews's user avatar
  • 79
3 votes
1 answer
641 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 ...
Jan Vysoky's user avatar
1 vote
0 answers
68 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 ...
QGravity's user avatar
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7 votes
0 answers
121 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 ...
José Figueroa-O'Farrill's user avatar
3 votes
1 answer
128 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 ...
asv's user avatar
  • 21.1k
4 votes
0 answers
214 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(-\...
Zhaoting Wei's user avatar
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5 votes
0 answers
245 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 ...
asv's user avatar
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2 votes
0 answers
67 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 ...
Yuugi's user avatar
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4 votes
0 answers
269 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 ...
მამუკა ჯიბლაძე's user avatar
6 votes
0 answers
213 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 ($m,n$)-...
Dilaton's user avatar
  • 408
19 votes
7 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 for ...
Arnold Neumaier's user avatar
2 votes
0 answers
232 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 ...
Dilaton's user avatar
  • 408
4 votes
1 answer
807 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 ...
Jianrong Li's user avatar
  • 6,121
3 votes
1 answer
158 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 ...
Jianrong Li's user avatar
  • 6,121
0 votes
1 answer
180 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} \...
Jianrong Li's user avatar
  • 6,121
1 vote
1 answer
181 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, \...
Jianrong Li's user avatar
  • 6,121
1 vote
1 answer
350 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 \...
Jianrong Li's user avatar
  • 6,121
8 votes
1 answer
741 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 ...
Soutrik's user avatar
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4 votes
0 answers
160 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 ...
Jaak van der Smut's user avatar
6 votes
0 answers
317 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:...
Theo Johnson-Freyd's user avatar
3 votes
1 answer
124 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:...
domenico fiorenza's user avatar
4 votes
0 answers
266 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 ...
Ho Man-Ho's user avatar
  • 1,087
20 votes
1 answer
668 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 ...
Nerses Aramian's user avatar
5 votes
1 answer
622 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 ...
asv's user avatar
  • 21.1k
6 votes
0 answers
188 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 ...
domenico fiorenza's user avatar
3 votes
0 answers
618 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 ...
user2133437's user avatar
3 votes
1 answer
571 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 ...
Mozibur Ullah's user avatar
3 votes
0 answers
105 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 ...
Valerie's user avatar
  • 885
4 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 ...
Ma Ming's user avatar
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