18
votes
Mathematical motivation for supergeometry
In my view, one motivation for the study of supergeometry (though historically backwards) is Deligne's theorem to the effect that any symmetric tensor (abelian) category "of moderate growth" ...
- 16.2k
15
votes
Mathematical motivation for supergeometry
A computational motivation for the introduction of odd (Grassmann) variables arises in the context of random matrix theory, when one seeks to represent the determinant of a matrix (rather than its ...
- 188k
10
votes
Accepted
supersymmetry and the de Rham complex
I think there is a typo in the references to "Supersymmetry and Morse theory", [21] should be replaced by [22]="Constraints on supersymmetry breaking". The quantization of non-linear sigma models and ...
- 6,920
10
votes
Accepted
Supermanifolds — elementary introduction?
There is a short elementary survey by Hohnhold, Stolz, and Teichner:
Super manifolds: an incomplete survey.
- 37.8k
9
votes
Supermanifolds — elementary introduction?
Some further references, that might be of interest for your purposes:
You can see at this article and the book Supermanifolds
Theory and Applications by Alice Rogers. The article discusses -among ...
- 7,746
8
votes
Supermanifolds — elementary introduction?
Some further (further) references:
Lectures on Supergeometry by G. Sardanashvily;
Lectures on Supergeometry by Tiffany Covolo and Norbert Poncin;
The notes by Witten: 1 and 2;
Supersymmetry for ...
- 11.8k
7
votes
Accepted
How do you get the spectral curve from a Calabi-Yau?
In general there is no way to extract a spectral curve from a Calabi-Yau threefold.
In the study of strings on Calabi-Yaus, one object of interest is the periods, i.e. integrals of the holomorphic ...
- 2,087
7
votes
Accepted
Implications of gauge symmetry breaking on the spectral side of geometric Langlands?
We discussed some conjectural implications in Section 4.2 of the paper. I wouldn't say that the category of sheaves with nilpotent singular support was necessarily the "right" category to consider ...
6
votes
Question on Witten’s paper “Supersymmetry and Morse theory”
You can find much more on the specific family $d_t$ if you search for the key phrase "Witten deformation"; I would try to give some specific references here but I am a little puzzled by the statement ...
- 13.6k
5
votes
Accepted
Supersymmetry charge $Q$ as anti-linear and anti-unitary operator
Suppose you are given a super Hilbert space $\mathcal{H} = \mathcal{H}_0 \oplus \mathcal{H}_1$, with bosonic and fermionic subspaces $\mathcal{H}_0$ and $\mathcal{H}_1$ respectively. Define a new ...
- 54.6k
5
votes
Supermanifolds — elementary introduction?
Supergeometry in mathematics and physics by Kapranov (arXiv, 34 pages, submitted on 22 Dec 2015, last revised 2 Apr 2018). Abstract:
This is a chapter for a planned collective volume entitled "New ...
- 18.4k
4
votes
Supermanifolds — elementary introduction?
I would like to add Riemannian supergeometry by Oliver Goertsches. Abstract:
Motivated by Zirnbauer in J Math Phys 37(10):4986–5018 (1996), we develop a theory of Riemannian supermanifolds up to a ...
- 2,830
4
votes
Geometric or conceptual way to understand supersymmetry algebra
In this paper about a global theory of supermanifolds, Alice Rogers develops the theory of supermanifolds as they underly supersymmetric field theories from a rigorous but physicist-friendly ...
- 418
4
votes
Accepted
Are there Type III codes with small but nonzero "index"?
Index $24$ isn't hard for length $36$.
For example, the Type III code with generator matrix
...
- 79.9k
4
votes
Mathematical motivation for supergeometry
Two examples in hand:
Any flag manifold for a complex reductive Lie group is projective by Bruhat decomposition and $\mathscr{D}$-affine by Beilinson--Bernstein, which is no longer true even for a ...
- 1,391
3
votes
Accepted
Is the "Ramond sector" invariant of a 3-framed lattice always divisible by 24?
The discussion in the comments establishes the conjecture when $r$ is divisible by $24$. When $r$ is merely divisible by $12$, the comments establish that $\Delta^{-r/24} Z_{RR}$ is divisible by $12$. ...
3
votes
Accepted
Chain rule for the superderivative
Direct application of definitions and chain rule gives
$$
D_\theta = (D_\theta \hat{\theta}) \partial_{\hat{\theta}}+(D_\theta \hat{z}) \partial_{\hat{z}}.
$$
Then eliminate $\partial_{\hat{\theta}}$ ...
- 333
3
votes
Geometric or conceptual way to understand supersymmetry algebra
If you are looking for geometric-algebraic interpretations of supersymmetric field theories, then non-commutative geometry -in the sense of A. Connes- seems to be the natural playground. There has ...
- 7,746
3
votes
Supermanifolds — elementary introduction?
Alice Rogers Supermanifolds is a rigorous introduction to supermanifolds in the geometric and algebraic approaches with the emphasis on the geometric. She also discusses applications to physics such ...
- 2,369
2
votes
Definition of orthosymplectic supergroups
They are both standard. It depends on your choice of non-degenerate supersymmetric bilinear form, which doesn't matter. It seems that your definition is for $SPO(2p|n)$. SPO means symplectic-...
- 87
2
votes
Reference request: coordinate ring of $OSP(2p|n)$
Since the orthosymplectic supergroup $G=\mathrm{OSP}(m|n)$ is affine algebraic (indeed, this is a Chevalley supergroup), you can apply results in Masuoka's paper ``Harish-Chandra pairs for algebraic ...
2
votes
Supermanifolds — elementary introduction?
There is also:
Bartocci, Bruzzo, Hernández-Ruipérez, The geometry of supermanifolds (1991)
I don't know if it's "elementary" in your sense.
- 23.3k
2
votes
Geometric or conceptual way to understand supersymmetry algebra
I'm not sure what you mean by "derive".
For a more mathematical and geometric description of the super Poincaré group in general dimension you could check out
Freed, Lectures on field theory and ...
- 1,996
2
votes
Is there Z_n graded supersymmetry?
Another possibility is to consider group elements rather than commutators. If you take the matrices $A=\mathrm{diag}(1,\xi,\ldots,\xi^{n-1})$ and $B=\begin{pmatrix}0&1&0&\cdots&0&0\...
- 16.9k
2
votes
Accepted
Is there Z_n graded supersymmetry?
The realizations (of an algebra through another algebra) you are speaking about are actually homomorphisms. And as such they should map between algebraic structures of the same kind: that is from ...
- 7,746
2
votes
Spectral Flow Invariance for Calabi-Yau Sigma Models
I'll keep this answer here cause it has a couple of comments, but the $\sigma$ I describe here is not what's defined in the question, it is rather the automorphism responsible for the topological ...
- 3,242
2
votes
Notation on supergeometry — parity
Since a Lie algebra is a vector space, $\Pi\mathfrak{g}$ is just the vector space $\mathfrak{g}$ with a $\mathbb{Z}_2$-grading where every element is odd. In particular, if $(t_a)$ is a basis of $\...
- 123
2
votes
Accepted
Supersymmetric SYK Model in 3D?
The disordered SYK model in three dimensions with supersymmetry was studied by Fedor Popov in Supersymmetric tensor model at large $N$ and small $\varepsilon$.
The complications are discussed in A 3d ...
- 188k
1
vote
Sufficient conditions for unitarity of a representation of a Lie Superalgebra
If i have correctly understood your question, i think that the answer can be found at
M. D. Gould, R. B. Zhang, Classification of all star and grade star irreps of gl(n|1), J. of Math. Phys., 31, ...
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