# Questions tagged [supersymmetry]

The supersymmetry tag has no usage guidance.

63
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### 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) ...

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### Finding the resolvent of two coupled random matrix system using supersymmetry

I'm trying to follow the result in this work. Let me briefly introduce the problem. I have a $2N\times 2N$ matrix
$$
H=\begin{pmatrix}H_1 & V \\ V^{\dagger} & H_2\end{pmatrix}.
$$
Here both $...

1
vote

0
answers

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### Existence of a minimal ideal with a specific property

Suppose that $R$ is a super-commutative ring (i.e. it is a unital $\mathbb{Z}_2$-graded ring satisfying $xy=(-1)^{|x|\cdot |y|}yx$ where $|x|$ denotes the grading degree of a homogeneous element $x\in ...

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### Link between Extended Supergauge Group and Twistor Theory

In a 1974 paper on supergauge transformations in four dimensions, Wess and Zumino considered an extended supergauge group which contains the conformal group as a subgroup. An interesting thing about ...

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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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1
answer

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### Supersymmetric SYK Model in 3D?

In a 2017 article More on supersymmetric and 2d analogs of the SYK model by Murugan, Stanford and Witten, the authors take a model called the SYK model (named after Sachdev, Ye and Kitaev) and study ...

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### What is a BPS state and why is it the cohomology of a moduli space?

The notion of a BPS state has existed in physics for a long time: I do not understand it completely, but I get the impression they are the supersymmetric analogue of ground states, and then physicists ...

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### Is there Z_n graded supersymmetry?

I have tried searching for something similar to what is described below, but to no avail. It would be great if somebody could show some right references, where this has been done, or explain why such ...

6
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1
answer

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### 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 ...

3
votes

1
answer

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### Supersymmetry charge $Q$ as anti-linear and anti-unitary operator

We know the supersymmetry (SUSY) charge $Q$ satisfies the following relation respect to fermion parity operator $(-1)^F$:
$$
(-1)^F Q + Q (-1)^F :=\{Q, (-1)^F \} =0
$$
which defines the anti-...

2
votes

1
answer

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### Notation on supergeometry — parity

I know that given a manifold $M$ and its corresponding tangent bundle $TM$ we can call $\Pi TM$ the space of forms parametrized (via charts) by $\{x_i\}_{i=1,\dotsc,n}$ and its corresponding cotangent ...

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### Spectral Flow Invariance for Calabi-Yau Sigma Models

I am a mathematician who has become interested in some of the mathematics of string theory, of which I am largely ignorant, so please excuse any idiocies in what follows.
If $X$ is a Calabi-Yau $d$-...

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### 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 ...

2
votes

1
answer

84
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### Sufficient conditions for unitarity of a representation of a Lie Superalgebra

Suppose we have a Lie superalgebra with triangular decomposition:
\begin{equation}
\mathfrak{g} = \mathfrak{g}^{+} \oplus \mathfrak{g}^{0} \oplus \mathfrak{g}^{-}
\end{equation}
I've seen it stated ...

6
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### Variation Formula of APS $\eta$-Invariant and Chern-Simons Theory

In Perturbative Expansion of Chern-Simons Theory with Noncompact Gauge Group, the author proved the variation formula of the APS $\eta$-invariant. I have a few questions about their proof. I will ...

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### The Grassmannian Gr(2,8) and an E7 surprise

Are there any mathematical explanations for the following surprising facts?
$$\int_{Gr(2,8)} c_{\text{top}}(TX(-2)) = 6556 = \frac{1}{2} \deg(E_7/P(\alpha_7)) + 1,$$
and
$$\int_{Gr(2,6)} c_{\text{top}}...

6
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1
answer

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### Branching from $E(6)$ to $SO(10) \times U(1)$

In $E(6)$ inspired models of supersymmetry, the inclusion of Lie subgroups
$$
SO(10) \times U(1) \hookrightarrow E_6
$$
is important object of interest. See here for my motivating example.
In ...

7
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1
answer

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### Implications of gauge symmetry breaking on the spectral side of geometric Langlands?

Let $G$ be a complex reductive algebraic group and $X$ be a smooth compact complex curve. It's easy to see that the space of vacua in B-twisted $N=4$ SUSY Yang--Mills theory is $\mathfrak{h}^*[2]/W$ (...

6
votes

1
answer

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### Question on Witten’s paper “Supersymmetry and Morse theory”

EDIT. I am trying to read the article “Supersymmetry and Morse theory” by E. Witten (JDG 17 (1982)).
This well known article applies some tools developed by physicists (e.g. path integrals) to ...

2
votes

0
answers

135
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### Two definitions of super-Virasoro algebra

Let $A=\mathbb C[x,\epsilon]$ where $x$ is an even variable and $\epsilon$ is an odd variable (thus $A$ is a commutative super-algebra). Let $\mathfrak g$ denote the Lie super-algebra of vector fields ...

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1
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### Are there Type III codes with small but nonzero "index"?

Recall that a Type III code of rank $r$ is a linear subspace $C \subset \mathbb F_3^r$ which is self-dual for the standard inner product. (These occur only when $r$ is divisible by $4$.) Elements of $...

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### Is the "Ramond sector" invariant of a 3-framed lattice always divisible by 24?

For the purposes of this question, a rank-$r$ (integral) lattice is a full-rank discrete subgroup $L \subset \mathbb R^r$ such that $\langle \ell, \ell' \rangle \in \mathbb Z$ for all $\ell \in L$. It ...

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### 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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1
answer

565
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### supersymmetry and the de Rham complex

In Alvarez-Gaume's paper "Supersymmetry and the index theorem" there is
given a certain supersymmetric Lagrangian whose quantization, apparently, leads to the de Rham Laplacian on the exterior ...

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### Geometric or conceptual way to understand supersymmetry algebra

Is there any geometric or more direct conceptual way to understand a supersymmetry algebra, rather than starting from a Lagrangian including boson and fermion fields, deriving all the expressions ...

6
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### 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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7
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### 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 ...

2
votes

0
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### 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 ...

4
votes

1
answer

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### 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

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### 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 ...

1
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1
answer

166
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### 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
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### Trace of the chiral matrix of a subspace

Let $(V,Q)$ be a pair consisting of a $\mathbb{C}$-vector space $V$ together with a nondegenerate bilinear form $Q$ and let $V_0\subseteq V$ be a linear subspace such that $Q\vert_{V_0}$ is ...

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### How do you get the spectral curve from a Calabi-Yau?

In N=2 Quantum Field Theories and Their BPS Quivers by Alim, Cecotti, Córdova, Espahbodi, Rastogi and Vafa the authors give a recipe which constructs from a pair $(C,\phi)$ consisting of a Riemann ...

2
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### $\mathbb{Z}_2$ graded analog of row operations for supermatrices

I'm working on some research involving supermatrices, and I was wondering if there was a $\mathbb{Z}_2$ graded analog of row operations for supermatrices.
It seems to me that it makes sense to have ...

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### What is the "quaternionic" super Brauer group?

In addition to the two reasonably well-known categories $\mathrm{SuperVect}_{\mathbb R}$ and $\mathrm{SuperVect}_{\mathbb C}$ of real and complex super vector spaces, each of which is monoidally ...

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### Mathematics of Chiral Rings

Let $A$ be a graded vector space, and suppose that two commuting differentials $d_1$ and $d_2$ of degree +1 act on $A$, such that $A$ equipped with either is a chain complex.
We now construct $C(A)$, ...

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1
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### 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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### Is there a symmetric monoidal 2-category "SuperDuperVect"?

Recall that the category $\mathrm{SuperVect}$, as a category, consists of pairs of vector spaces, thought of as formal direct sums $V \oplus W\,\Pi$, where $\Pi$ is the "odd line". (Called &...

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### projective representation of supergroup

In fact, I am not very clear about what I am asking, but I am looking for a concrete example of supergroup which has non-trivial projective representation(some supergroup similar to usual Lie group SO(...

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votes

1
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### 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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2
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### Witten index non-trivial in the context of Quantum Mechanics?

Let $H$ be a self-adjoint Hamiltonian and $H$ admits a decomposition into closed operators $D,D^*$, such that we have $H = D^*D$.
I will now consider the one-dimensional case on a compact set:
So ...

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### 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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### Double vector bundle vs vector bundle over supermanifolds

Double vector bundle is roughly a vector bundle over (horizontal) vector bundle.
Vector bundle is a supermanifold in a nature way (non-nature on the other way around).
My question: is a double ...

3
votes

1
answer

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### 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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### 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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### Classification of (almost) contact structures on $S^3$

Question: Is there a classification of almost contact or contact structures on $S^3$? What is it and references?
The motivation of this question is as follows:
(1) There is one paper showing that a ...

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### wrapping M5-branes on a Riemann surface

AdS/CFT gives us a way to use geometry to study field theory! I am trying to wrap M5-branes on a Riemann surface $\Sigma_{g}$. In my problem, for a Riemann surface in 11d, the normal bundle is max $SO(...

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### Is there a version of supersymmetry for homogeneous spaces?

The notion of "supersymmetry" that I am aware of proceeds as follows. One fixes a spacetime $\mathbb R^n$ and signature; I will write $\mathrm{SO}(n)$ for the corresponding group of orthogonal ...

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### Matrix-tree theorem via supersymmetry (i.e. Grassman algebras)

The matrix-tree theorem states the number of spanning trees of a graph $G$ is equal to a modified determinant of the adjacency matrix or "graph Laplacian", $\Delta_G$:
$$\#\{ \text{spanning ...

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### Is the SUSY Algebra isomorphic for all Kähler Manifolds?

For a Kähler manifold, the graded algebra generated by $\partial,\overline{\partial},\partial^*,\overline{\partial}^\ast$, the Lefschetz operator, and the dual Lefschetz operator, is called the ...